MICROMECHANICS-BASED PREDICTION OF THERMOELASTIC PROPERTIES OF HIGH ENERGY MATERIALS

Transcription

1 MICROMECHANICS-BASED PREDICTION OF THERMOELASTIC PROPERTIES OF HIGH ENERGY MATERIALS by Biswajit Banerjee A research proposal submitted to the faculty of The University of Utah in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mechanical Engineering The University of Utah January 2002

2 ABSTRACT High energy materials such as polymer bonded explosives are commonly used as propellants. These particulate composites contain explosive crystals suspended in a rubbery binder. However, the explosive nature of these materials limits the determination of their mechanical properties by experimental means. Micromechanics-based alternatives are, therefore, explored in this research. In particular, methods for the determination of the effective thermoelastic properties of polymer bonded explosives are investigated. Polymer bonded explosives are two-component particulate composites with high volume fractions of particles (volume fraction 90%) and high modulus contrast (ratio of Young s modulus of particles to binder of 5,000-10,000). Experimentally determined elastic moduli of one such material, PBX 9501, are used to validate the micromechanics methods examined in this research. The literature on micromechanics is reviewed; rigorous bounds on effective elastic properties and analytical methods for determining effective properties are investigated in the context of PBX Since detailed numerical simulations of PBXs are computationally expensive, simple numerical homogenization techniques have been sought. Two such technqiues explored in this research are the generalized method of cells and the recursive cells method. Effective properties calculated, for PBX-like materials, using these methods have been compared with finite element analyses and experimental data. In addition, some shortcomings of these methods have been identified and improvements suggested.

3 CONTENTS ABSTRACT LIST OF FIGURES LIST OF TABLES iv vii CHAPTERS 1. INTRODUCTION 1 2. HIGH ENERGY COMPOSITES PBX Composition of HMX in PBX Elastic Moduli of -HMX Thermal Expansion Properties of HMX Composition of PBX 9501 Binder Elastic Properties of PBX 9501 Binder Thermal Expansion of PBX 9501 Binder Manufacturing Process for PBX Elastic Properties of PBX Mock Propellants MICROMECHANICS OF COMPOSITES Rigorous Bounds Hashin-Shtrikman Bounds Third Order Bounds Analytical Methods Composite Spheres Assemblage Self-Consistent Schemes Differential Effective Medium Approach Numerical Approximations The Representative Volume Element Finite Difference Approximations Finite Element Approximations Regular Arrays in Two Dimensions Random Distributions in Two Dimensions Approximations using Homogenization Theory Approximations using Stochastic Finite Elements Three Dimensional Approximations Discrete Models Integral Equation Based Approximations Fourier Transform Based Approximations Method of Cells

4 4. THE GENERALIZED METHOD OF CELLS Average Strain Relations Stress-Strain Relations Effective Thermoelastic Properties Shear-Coupled Method of Cells THE RECURSIVE CELL METHOD Subcell Stiffness Matrices Displacement Based Four-Noded Element Displacement Based Nine-Noded Element Mixed Displacement-Pressure Nine Noded Element Modeling a Block of Subcells Boundary Conditions Application of Constraint Equations Application of Specified Displacements Calculating Volume Averaged Stresses and Strains Calculating Effective Properties Calculating Effective Properties of the RVE VALIDATION OF GMC AND RCM Comparisons With Exact Relations Phase Interchange Identity Materials Rigid in Shear The CLM Theorem Symmetric Composites with Equal Bulk Modulus Hill s Equation Comments On Comparisons With Exact Solutions Comparisons With Numerical Results Special Cases : Stress Bridging Corner Bridging : X-Shaped Microstructure Edge Bridging : Five Cases Model A Model B Model C Model D Model E Summary SIMULATION OF PBX MICROSTRUCTURES Manually Generated Microstructures FEM Calculations GMC Calculations Fifty Percent Rule The Two-Step Approach Effective Properties from GMC RCM Calculations Randomly Generated Microstructures Circular Particles - PBX 9501 Dry Blend FEM Calculations ii

5 GMC Calculations RCM Calculations Circular Particles - PBX 9501 Pressed Piece FEM Calculations GMC Calculations RCM Calculations Square Particles - Pressed PBX FEM Calculations GMC Calculations RCM Calculations PROPOSED RESEARCH Current Status of Research Remaining Research Improvements to RCM Further FEM Calculations Calculations for PBX APPENDICES A. PLANE STRAIN STIFFNESS AND COMPLIANCE MATRICES 161 REFERENCES 165 iii

6 LIST OF FIGURES 2.1 HMX particle distribution in the dry blend [8] Monoclinic structure of a -HMX crystal HMX particle sizes in PBX 9501 before and after processing Young s modulus vs. applied strain for PBX 9501 [21] at 22 C and strain rate of 0.001/s Young s modulus vs. strain rate and temperature for glass/estane (21%/70% by volume) mock propellants Young s modulus vs. strain rate and temperature for glass/estane (44%/56% by volume) mock propellants Young s modulus vs. strain rate and temperature for glass/estane (59%/41% by volume) mock propellants Parameters and for the penetrable sphere model (* = Values Computed by Berryman [33].) Comparison of bounds on the bulk and shear modulus of PBX 9501 with experimental values Subcells and notation used in GMC Schematic of the recursive cell method Four noded element Nine noded element A four subcell block modeled with four elements A four subcell block modeled with sixteen elements Schematic of the effect of a uniform displacement applied in the direction Schematic of the effect of a uniform displacement applied in the direction Schematic of the effect displacements, corresponding to a pure shear, applied at the boundary nodes Schematic of the effect displacements, corresponding to a pure shear, applied at the corner nodes The recursive cells method applied to a RVE discretized into blocks of four subcells RVE for a checkerboard

7 6.2 Validation of FEM, RCM and GMC using the phase interchange identity for a checkerboard composite Variation of effective shear moduli with modulus contrast for a checkerboard composite Ratio of effective shear moduli predicted by FEM, RCM and GMC to those predicted by the phase interchange identity for a checkerboard composite with varying modulus contrast Convergence of effective moduli predicted by finite element analyses with increase in mesh refinement for a checkerboard composite with shear modulus contrast of 25, RVE for a square array of disks Error in computation of for a Square Array of Disks Error in computation of for a square array of disks Error in computation of for a Square Array of Disks RVE used for corner stress bridging model Variation of with modulus contrast for X -shaped microstructure Variation of with modulus contrast for X -shaped microstructure Variation of with modulus contrast for X -shaped microstructure Comparison of effective stiffness matrix for corner stress bridging model Progressive stress bridging models A through E Comparison of normalized effective stiffnesses for model A Comparison of effective stiffnesses for model B Stress bridging paths for Model C Why RCM predicts square symmetry for Model C Comparison of effective stiffnesses for Model C Comparison of effective stiffnesses for Model D Comparison of effective stiffnesses for Model E Manually generated microstructures for PBXs Effective stiffnesses for the six model microstructures from from detailed finite element analyses as a a multiple of the binder stiffness Application of fifty percent rule to a model microstructure Schematic of the two-step GMC procedure Ratios of effective stiffnesses calculated using GMC (50% rule) and FEM Ratios of effective stiffnesses calculated using GMC (two-step) and FEM Microstructure used for RCM calculations for model Ratios of effective stiffness calculated using RCM and FEM v

8 7.9 Ratios of effective stiffness calculated using FEM (!#"#$&%'!#"#$ square elements) and FEM (65,000 triangular elements) Microstructure of PBX 9501 [19] Microstructures using circular particles based on the dry blend of PBX Approximate microstructure used for FEM and RCM calculations on the 100 particle model of PBX 9501 based on the dry blend Effective stiffness matrix components for microstructures based on the dry blend of PBX 9501 from FEM calculations Effective stiffness matrix components for microstructures based on the dry blend of PBX 9501 from GMC calculations Effective stiffness matrix components for microstructures based on the dry blend of PBX 9501 from RCM calculations Microstructures using circular particles based on the pressed piece size distribution of PBX Approximate microstructure for the 1000 particle model of PBX Effective stiffness matrix components for microstructures based on pressed PBX 9501 from FEM calculations Effective stiffness matrix components for microstructures based on pressed PBX 9501 from GMC calculations Effective stiffness matrix components for microstructures based on pressed PBX 9501 from RCM calculations Microstructures using square particles based on the pressed piece size distribution of PBX Effective stiffness matrix components from FEM calculations for microstructures containing square particles Microstructure for the 700 particle model of PBX 9501 using square, aligned particles Effective stiffness matrix components from GMC calculations for microstructures containing square particles Effective stiffness matrix components from RCM calculations for microstructures containing square particles vi

9 LIST OF TABLES 2.1 Compositions of common PBX materials Composition of PBX HMX particle size distribution in PBX 9501 [7] Different phases of HMX and transition temperatures Components of the stiffness matrix of ( -HMX (GPa) [13, 14] Elastic properties of ( -HMX Thermal expansion properties of ( -HMX Strain-rate and temperature dependent elastic moduli of PBX 9501 binder Elastic properties of PBX Elastic properties of soda glass Young s modulus of Estane 5703 at various temperatures and strain rates Properties of sugar/binder mock propellant [7] Elastic moduli and CTE of PBX 9501 and its components at room temperature and low strain rate Voigt and Reuss bounds for PBX Hashin-Shtrikman upper and lower bounds for PBX Milton upper and lower bounds for PBX Composite spheres assemblage prediction for PBX Self consistent scheme prediction for PBX Three-phase model prediction for PBX Differential scheme predictions of effective properties Out-of-plane properties for square array of disks Components of effective stiffness and compliance matrices for a square array of disks Components of effective stiffness and compliance matrices for a checkerboard composite Original and translated two-dimensional constituent moduli for checking the CLM condition Comparison of effective moduli for the original and the translated composites

10 6.6 Component properties, exact effective properties and numerically computed effective properties for two-component symmetric composite with equal component bulk moduli Phase properties used for testing Hill s relation and the exact effective moduli of the composite Numerically computed effective properties for a square array of disks with equal component shear moduli Component properties used by Greengard and Helsing [97] Comparison of numerically calculated values of two-dimensional bulk and shear moduli of square arrays of disks The elastic properties of the components of the X shaped microstructure )* ++, )* +-, and ).* // for X-shaped microstructure with highest modulus contrast Materials used to test edge bridging using FEM, GMC and RCM Effective properties of Model A from FEM, GMC and RCM Effective properties of Model B from FEM, GMC and RCM Effective properties of Model C from FEM, GMC and RCM Effective properties of Model D from FEM, GMC and RCM Effective properties of Model E from FEM, GMC and RCM Experimentally determined elastic moduli of PBX 9501 and its constituents [7] Effective stiffness for the six model PBX 9501 microstructures from FEM calculations using 65,000 six-noded triangle elements Effective stiffness for the six model PBX 9501 microstructures from GMC calculations Effective stiffness for the six model PBX 9501 microstructures from RCM calculations Effective stiffness for the six model PBX 9501 microstructures from FEM calculations using 0#1#24350#1#2 square elements Effective stiffness for the four model PBX 9501 microstructures based on the dry blend of PBX Volume fractions of particles and moduli of the dirty binder for the four pressed piece based PBX microstructures Effective stiffness for the four model PBX 9501 microstructures based on the pressed piece of PBX Moduli of the dirty binder for the three PBX microstructures with square particles Effective stiffness for the three pressed PBX 9501 model microstructures containing square particles viii

11 CHAPTER 1 INTRODUCTION High energy (HE) materials are those that decompose rapidly and release large amounts of energy when impacted or ignited. These materials are commonly used as propellants for rockets. In recent years, the issue of safeguarding stockpiles of missiles in the United States has generated renewed interest in the mechanical properties of HE materials. Such material properties are essential for the prediction of the response of containers filled with HE materials under different circumstances. Mechanical properties of HE materials can be determined experimentally. However, the hazards associated with experiments on these materials, as well as the attending costs, make this option unattractive. As computational capabilities have grown and improved numerical techniques developed, numerical determination of the properties of HE materials has become possible. In this research, we explore some numerical, micromechanics-based methods for the determination of the mechanical properties of HE materials. Of the numerous types of HE materials that exist, the ones that are of interest in this research are polymer bonded explosives (PBXs). One reason for this interest is that one such material, PBX 9501, has been extensively tested in various National Laboratories in the United States and thus provides a basis for validating numerical calculations. In addition, PBXs provide unique challenges for micromechanical modeling - these materials are viscoelastic particulate composites, contain high volume fractions of particles, and the modulus contrast between the particles and the binder is extremely high. For example, PBX 9501 contains about 92% by volume of particles and the modulus contrast between particles and the binder, at room temperature and low strain rates, is around 20,000. Some simplifying assumptions are made about PBXs in this research. It is assumed that PBXs are two-component particulate composites with the particles completely surrounded by, and perfectly bonded to, the binder. The components of PBXs are assumed to be isotropic and linear elastic, and only the prediction of elastic moduli and coefficients of thermal expansion (CTEs) of PBXs is addressed. A few PBX materials and their compositions are shown in Chapter 2. Since PBX 9501 is the material that provides experimental validation of our micromechanics models, the composition and

12 2 thermoelastic properties of PBX 9501 and it s components are discussed in detail in Chapter 2. In addition, two mock propellants that do not contain explosive crystals are also discussed. Micromechanics-based methods for the determination of effective properties of composites are reviewed in Chapter 3. These include rigorous bounds on the effective properties, analytical solutions and numerical methods. The upper and lower bounds on the effective elastic moduli of PBX 9501 are found to be too far apart to be of practical use. Bounds on the effective CTE are, however, quite close to each other. Analytical solutions for simplified models are found to underestimate the effective elastic moduli considerably. Hence, numerical methods are the only viable approaches for the determination of effective properties of PBXs. The finite element method (FEM) has been chosen to provide benchmark calculations of effective properties in this research. However, the computational cost involved in detailed FEM calculations has led us to consider simpler numerical approaches to model PBXs. The generalized method of cells (GMC) is a simple approach that has been used to compute the effective properties of composites. A reformulation of this technique is discussed in Chapter 4. It has been discovered that GMC predicts inaccurate shear moduli and does not capture stress bridging effects adequately. An alternative GMC-based approach intended to improve upon GMC is also discussed in Chapter 4. A new technique called the recursive cell method (RCM) has also been developed to remedy the drawbacks of GMC. Chapter 5 discusses the recursive cell method detail. Some improvements to this method are also suggested in this chapter. Effective properties, computed using GMC and RCM, are compared with exact results and numerical simulations in Chapter 6. It is observed that both methods predict relatively accurate elastic moduli directions for low volume fractions of particles and for low modulus contrasts. In addition, GMC and RCM are used to predict the effective properties of some special microstructures. Some shortcomings of the two techniques are elucidated by the results from these validation exercises. Procedures of generating microstructures that model PBXs are discussed in Chapter 7. Microstructures containing circular and square particles are generated and the effective properties are calculated using FEM. The effective properties of these microstructures, calculated using GMC and RCM, are compared with those from FEM calculations. For these microstructures, GMC consistently underestimates the effective properties while the current form of RCM consistently overestimates the effective properties. Some improvements to RCM are suggested in this chapter. The remaining research proposed for the Ph.D. degree is discussed in Chapter 8. The improvements to RCM proposed in this chapter are expected to lead to considerable improvement in the ability to predict the effective properties of PBXs.

13 CHAPTER 2 HIGH ENERGY COMPOSITES High energy materials are usually composites containing two or more components. One of the components is an explosive crystal while the other components act as a binder that provides structural support to the crystals. The polymer bonded explosives (PBXs) considered in this research contain a very high volume fraction of crystals that are considerably stiffer than the binder. Some data on the compositions of such PBXs [1, 2, 3] are shown in Table 2.1. Table 2.1. Compositions of common PBX materials. Binder Type PBX Explosive/Binder Weight (%) Source Fluoropolymer LX-10-1 HMX/Viton 95.5/4.5 [1] (e.g., Viton) PBX 9502 TATB/KEL-F /5 [1] PBX 9010 RDX/KEL-F /10 [2] PBX 9407 RDX/Exon /6 [2] PBX 9207 HMX/Exon /8 [2] Polyeurethene PBX 9011 HMX/Estane 5703F1 90/10 [2] EDC 29 HMX/HTPB 95/5 [3] Polyeurethene PBX 9404 HMX/NC+CEF(1:1) 94/6 [2] (with PBX 9501 HMX/ 95/5 [2] Plasticizers) Estane 5703+BDNPA-F(1:1) EDC 37 HMX/NC+K10(1:8) 91/9 [3] 2.1 PBX 9501 The polymer-bonded explosive of interest in this research is PBX This material is a composite of crystals of HMX (High Melting Explosive) and a binder composed of Estane 5703 and BDNPA/F and a free radical inhibitor such as diphenylamine or Irgonox [4]. A more detailed composition of PBX 9501 is shown in Table Composition of HMX in PBX 9501 PBX 9501 contains a mixture of two different size distributions of HMX particles because the smaller particles fit into the interstitial spaces between the larger particles. The mixture contains

14 Table 2.2. Composition of PBX Component Chemical Weight Volume Composition Fraction Fraction6 HMX 1,3,5,7-tetranitro-1,3,5,7-tetraazacyclooctane Estane 5703 polybutylene adipate and ,4 diphenylmethanediisocyanate-1,4-butanediol BDNPA/F bis-dinitropropylacetal-formal Irgonox Voids a - The volume fraction data have been obtained from Dick et al. [5]. b - McAfee et al. [6] cite volume fractions of and for HMX and binder respectively. Class 1 HMX (coarse) and Class 2 HMX (fine) in a ratio of 3:1 by weight. Class 1 HMX consists of particles primarily between 44 and 300 microns in size. The finer grade Class 2 HMX also has a few coarse particles, but 75% of the particles are less than 44 microns in size [4]. Several HMX particle size distributions for PBX 9501 can be found in the literature that do not necessarily match one another. A good approximation that has been listed by Wetzel [7] is shown in Table 2.3. A plot of the particle distributions of the two grades of HMX in PBX 9501 obtained from data generated by Skidmore et al. [8] is shown in Figure 2.1. The plot illustrates the bimodal distribution of particles in the dry blend. Table 2.3. HMX particle size distribution in PBX 9501 [7]. Particle Class 1 Class 2 Size(micron) HMX HMX % at least 75% % 125 at least 98% % 100% % HMX crystals can exist in three stable phases (9 -HMX,: -HMX, and ; -HMX) depending on temperature and pressure. Data obtained by Leiber [9] on these phases and their ranges of stability are shown in Table 2.4. The : -HMX phase is dominant at or near room temperature when linear elastic behavior is expected. The : -HMX crystal has a monoclinic structure as shown in Figure 2.2. The axis < is the axis of second-order symmetry (or equivalently the plane = -> is the plane of symmetry). At room temperature the lattice parameters =, < and > are approximately in the and the angle : is approximately I#JLK (Bedrov et al. [10]).

15 5 Volume Fraction (%) Coarse HMX (100%) Fine HMX (100%) Particle Diameter (microns) Figure 2.1. HMX particle distribution in the dry blend [8]. Table 2.4. Different phases of HMX and transition temperatures. Phase Stable Region Transitions (M C) N -HMX ( NOQP ) at 116M C P -HMX ( POSR ) at M C R -HMX 162-melt ( NOSR ) at M C c β a b a = b = c o α = γ = 90 = β Figure 2.2. Monoclinic structure of a P -HMX crystal Elastic Moduli of T -HMX Crystals of P -HMX are mildly non-linearly elastic at ambient temperatures. As temperature increases, voids develop in the crystals that may lead to degradation of elastic stiffness prior to

16 ZY Z Z Z Z Z [ ZY Z Z Z Z Z [ b b b b b ZY Z Z Z Z Z [ q d b b b b b ZY Z Z Z Z Z [ d b b b b b d b b b b b 6 melting. However, a linear elastic approximation is adequate for HMX below 40U C. As mentioned in the previous section, crystals of V -HMX are monoclinic in structure. Following Lekhnitskii [11], if the W axis (also referred to as the X axis by Ting [12]) is the axis of second order symmetry, the elastic constitutive relation for a HMX crystal is as shown in equation 2.1 (Voigt notation). \^]] \ \ `` \ _ ` \^] ` \^]-_ acb dfe In compact form, this relation can be written as g ]] g ]-_ g ] ` h g ]-i h g ]-_ g g _ ` h g _i h g ] ` g _ ` g `` h g ` i h h h h gkjj h gkjml g ]-i g _i g ` i h g ii h h h h gkjml h gnll esrut Linear elastic moduli of V -HMX have been estimated using experiments (Zaug [13], Dick et al. [5]) and by molecular dynamics (MD) simulations (Sewell et al. [14]). The data obtained by Zaug [13] and Sewell et al. [14] are the most comprehensive and are shown in Table 2.5. The data obtained by Zaug were calculated from measurements of wave velocities through a single crystal of V -HMX. The values of the 13 elastic coefficients were calculated at a temperature of 107U C using a non-linear least squares simplex fit of the experimental data using the room temperature value of bulk modulus (12.5 GPa) as a benchmark. Molecular dynamics simulations by Sewell et al. [14] show results that are close to those obtained by Zaug and are shown inside round brackets in Table 2.5. acb op]] om om`` om_ ` op] ` op]-_ acb (2.1) rve Table 2.5. Components of the stiffness matrix of V -HMX (GPa) [13, 14] (18.7) 3.9 (4.9) 12.5 (7.7) (-1.7) (17.0) 6.5 (7.3) (3.0) (16.7) (0.2) (8.9) (2.4) Symm. 6.4 (9.3) (9.8) acb (Numbers inside round brackets show values from MD simulations [14].) Leiber [9] has commented that the coupling coefficients ( g ]-i, g _i, and gkjml ) shown in Table 2.5 have a significant effect on the normal and shear stresses and strains and hence isotropy is not a good approximation for V -HMX. However, the assumption of isotropy provides a simple way of

17 7 carrying out mesoscopic simulations of PBX materials and has been utilized in this investigation. Various estimates of isotropic bulk modulus, shear modulus and Young s modulus for HMX are shown in Table 2.6. The values obtained by Zaug [13] were calculated from ultrasonic sound speed measurements and hence at high strain rates. The molecular dynamics simulations of Sewell et al. [14] require the load to be applied instantaneously and therefore high strain rates are involved. The results obtained by Dick et al. [5] are also from high strain rate impact tests. Since HMX is not particularly sensitive to strain rate and we assume that these properties can be used for low strain rate simulations as well. Table 2.6. Elastic properties of w -HMX Bulk Shear Young s Poisson s Source Modulus Modulus Modulus Ratio (GPa) (GPa) (GPa) Zaug [13] Sewell et al. [14] Williams [15] 26.6 Dick et al. [5] Thermal Expansion Properties of HMX The coefficients of thermal expansion of HMX crystals have been obtained using X-ray diffraction by Herrmann [16]. The values obtained show a pronounced anisotropy in the x lattice direction compared to the y and z directions. The angles { (between the y and x lattice directions) and (between the y and z lattice directions) do not change significantly with changing temperature. However, there is a large change in the angle w (between x and z ) of the monoclinic lattice. Molecular dynamics (MD) simulations at room temperature by Bedrov et al. [10] show results similar to those obtained by Herrmann. Table 2.7 shows the coefficients of thermal expansion of HMX obtained experimentally by Herrmann and from molecular dynamics simulations by Bedrov et al Composition of PBX 9501 Binder The binder in PBX 9501 is essentially a combination of Estane 5703 and a plasticizer (BDNPA- F). A free radical inhibitor (Irgonox) is added for further stability of PBX The theoretical maximum volume occupied by the binder in PBX 9501 is about 8% of the total volume. Estane 5703 is amorphous and thermoplastic with a relatively low glass transition temperature (-31} C) and a melting temperature of around 105} C. It contains soft and hard segments that serve to enhance entanglement and lead to low temperature flexibility, high temperature stability and

18 8 Table 2.7. Thermal expansion properties of ~ -HMX. ~ Thermal Expansion Experiments MD ƒ (/K) (Herrmann [16]) (Bedrov et al.[10]) Lattice Parameters a Linear/Angular b c Volume good adhesive properties. Gray et al. [4] state that the plasticizer (BDNPA/F) decreases the binder strength and stiffness. Experimental data produced by Gray et al. [4] show that the mechanical properties of PBX 9501 are affected significantly by the porosity of the mix. The porosity of PBX 9501 is supposedly mostly due to cavitation in the binder as the composite relaxes after it has been isostatically pressed [17]. The voids therefore occupy a significant volume fraction of the binder (20-50%) and affect the mechanical properties of the binder considerably. However, it the experimental data in the literature are ambiguous about what percentage the porosity of PBX 9501 is due to voids in the HMX particles or voids in the binder Elastic Properties of PBX 9501 Binder The elastic properties of the PBX 9501 binder are quite sensitive to strain rate and temperature. This has led to experiments on the binder being carried out at different strain rates and temperatures. Complete binder properties are therefore considerably more difficult to obtain from published experimental data than HMX properties. Few low strain rate tests have been carried out because of the low stiffness of the binder. High strain rate tests using Hopkinson bar type experiments do not yield acceptable accuracy in initial modulus values. Molecular dynamics simulations have not been carried out on the constituents of the binder because Estane 5703 molecules are complex and contain both hard and soft segments. Numerous tests have been carried out on the PBX 9501 binder by Dick et al. [5], Cady et al. [18, 20], Gray et al. [4] and Wetzel [7] at various strain rates and temperatures. Data from these sources on the PBX 9501 binder are shown in Table 2.8. The data show that at high strain rates (keeping temperature constant) the Young s modulus of the binder is many times greater than at low strain rates. This higher stiffness at high strain rates is because the polymer chains have less time to flow. The Poisson s ratio of the binder is close to 0.5, as can be expected of rubbers and elastomers.

19 9 Table 2.8. Strain-rate and temperature dependent elastic moduli of PBX 9501 binder. Temperature Strain Young s Poisson s Source ( C) Rate Modulus Ratio (strains/sec) (MPa) Wetzel [7] Wetzel [7] Wetzel [7] Wetzel [7] Dick et al. [5] Cady et al. [20] Cady et al. [20] Cady et al. [20] Gray et al. [4] Cady et al. [20] Gray et al. [4] Cady et al. [20] Cady et al. [20] Cady et al. [20] Cady et al. [20] Cady et al. [20] Cady et al. [20] Gray et al. [4] Cady et al. [20] Gray et al. [4] Cady et al. [20] Thermal Expansion of PBX 9501 Binder Wetzel [7] cites the coefficient of thermal expansion of Estane 5703 to be between 10 ˆ Š to 20 ˆ Š /K. Since data are not available for the binder, we shall assume the coefficient of thermal expansion of the binder to be the same as that of Estane Manufacturing Process for PBX 9501 The first step in the manufacturing of samples is to mix the constituents and to form molding powder granules or prills of PBX Samples are then isostatically compressed until the porosity is reduced to 1-2%. The theoretical maximum density for the composite (1.860 gm/cc) is used to determine the porosity. The process of isostatic pressing causes less segregation of particles away from the pressing surfaces than unidirectional compression. The preparation of the material is usually carried out at a temperature of 90 C. The size distribution of HMX particles after processing is significantly different from that before processing. Experiments by Skidmore et al. [8] show that the cumulative volume fraction of the finer sized particles is dramatically higher in the pressed PBX

20 compared to that of the dry blend of coarse HMX and fine HMX. Figure 2.3 shows the particle size distributions of the dry blend of coarse and fine HMX particles, that of the molding powder and that of the pressed piece. It is difficult to observe the bimodal distribution of particles in the dry blend because the volume fraction of fines is much smaller than that of the coarse sizes. However, the bimodal distribution of particles is clear in the plot pressed piece size distribution. Pressing considerably increases the volume fraction of smaller sized particles. Volume Fraction (%) Dry Blend Molding Powder Pressed Piece Particle Diameter (microns) Figure 2.3. HMX particle sizes in PBX 9501 before and after processing. Experiments by Skidmore et al. [19] have shown that the consolidation of prills initially involves little damage to the large HMX crystals. As porosity is decreased, there is an increasing incidence of transgranular cracking and twinning in the large HMX crystals. If porosity is decreased to less that 1%, microcracks grow across crystals due to crystal-to-crystal contact and intercrystalline indentation Elastic Properties of PBX 9501 The elastic moduli of polymer bonded explosives are strongly influenced by strain rate and temperature [18] primarily because of the strain rate and temperature dependent behavior of the binder. It has also been observed that these composites (especially during high-rate loading) continue to strain after the maximum stress has been achieved ( [4], [18], [20], [21]). Therefore, some time and history dependent behavior is indicated. In general, the compressive strengths and elastic modulus of polymer-bonded explosives increase with decreasing temperature and increasing strain rate. The

21 11 above observations are also true for PBX In addition, some non-linearity in the stress-strain relationship is indicated for PBX 9501 in the small strain domain. Dick et al. [5] have shown that compressive stiffness increases with increasing volumetric strain for small strains prior to yielding. Temperature and strain rate moduli of PBX 9501 reported by Wetzel [7] and obtained from tests carried out by Wiegand [21], Dick et al. [5] and Gray et al. [4] are shown in Table 2.9. The data show the same trends as the binder but higher stiffness at room temperature. The high strain rate Young s modulus is around 12 times the low strain rate Young s modulus at room temperature. Table 2.9. Elastic properties of PBX Temperature Strain Young s Modulus Poisson s Source ( C) Rate Compressive Tensile Ratio (strains/sec) (GPa) (GPa) Gray et al. [4] Gray et al. [4] Gray et al. [4] Gray et al. [4] Gray et al. [4] Wiegand [21] Dick et al. [5] Wetzel [7] Gray et al. [4] Gray et al. [4] Gray et al. [4] Gray et al. [4] Gray et al. [4] Gray et al. [4] Ultimate compressive strengths of PBX 9501 have been found to be around MPa for low strain rate tests and around MPa for high strain rate tests [5]. Wiegand [21] has shown that after yielding, progressive damage develops in PBX 9501 and the material becomes considerably less stiff as shown in Figure Mock Propellants Various mock propellants have been tested to determine the effects of the binder on material properties. A mock propellant made of monodispersed (650 Œ 50 microns) spherical glass beads with Estane as binder has been tested at the Los Alamos National Laboratory [18] at temperatures ranging from -55 C to 25 C. Glass volume fractions of 21%, 44% and 59% (25%, 50% and 65% by weight) were used in the tests. Low strain rate compression tests at 0.001, 0.1 and 1 s ^Ž as well as high strain rate impact tests at 3500 s ^Ž were conducted on the specimens. The glass beads were

22 Young s Modulus (GPa) Applied Strain Figure 2.4. Young s modulus vs. applied strain for PBX 9501 [21] at 22 C and strain rate of 0.001/s. standard soda lime glass with a density of 2.5 gm/cc. Elastic properties of soda glass are shown in Table Table Elastic properties of soda glass. Young s Poisson s Shear Modulus Ratio Modulus (MPa) (MPa) 50, ,000 The Young s modulus of Estane 5703 has been calculated at various temperatures and strain rates on the basis of the stress-strain curves from experiments carried out by Cady et al. [18]. The PBX 9501 binder is less stiff than Estane 5703, but exhibits qualitatively similar temperature and strain dependence. Table 2.11 shows the Young s modulus of Estane 5703 at different temperatures and strain rates. It can be observed from the soda glass and Estane 5703 moduli that the modulus contrast is around 5-10 times lower than that for PBX 9501 for a glass-estane composite. However, for a test of the micromechanics techniques of interest in this investigation, this contrast is adequate.

23 13 Table Young s modulus of Estane 5703 at various temperatures and strain rates. Temperature Strain Rate Young s Modulus ( C) (/sec) (MPa) Micromechanics techniques can be used to determine the effective elastic properties of glass- Estane composites using the properties shown in Tables 2.10 and These can then be compared with experimentally determined elastic properties of mixtures of the two components (mock propellants). The Young s moduli of the three mock propellants tested by Cady et al. [18] at different temperatures and strain rates are shown in Figures 2.5, 2.6, and 2.7. These data provide an additional check of the accuracy of micromechanics simulations of PBX-like materials. Mock propellants have also been designed using sugar instead of glass beads. A sugar based PBX 9501 mock high energy material has been studied by Wetzel [7]. The cubic sugar crystals take the place of HMX crystals while the binder was chosen to have the same composition as the binder for PBX Elastic properties, densities and volume fractions of the sugar crystals, the binder and the mock are listed in Table The data listed are for a strain rate of /s and room temperature.

24 14 Young s Modulus (MPa) o C 10 o C 0 o C 10 o C 20 o C 30 o C 40 o C 10 0 (/s) Strain Rate Figure 2.5. Young s modulus vs. strain rate and temperature for glass/estane (21%/70% by volume) mock propellants. Table Properties of sugar/binder mock propellant [7]. Property Sugar Binder Mock Volume Fraction Density (gm/cc) Tensile Modulus (GPa) Poisson s Ratio The binder can be modeled as a viscoelastic material. Therefore, the Poisson s ratio of the mock varies with applied stress and time. However, we assume that it is constant over the range of strain rates and temperatures of interest in this research. We review some of the methods of determining effective thermoelastic properties of composites in Chapter 3. Some of these methods are also used to predict effective elastic moduli of PBX 9501 based on the properties of the components at room temperature and low strain rates. The predicted values are compared with experimentally determined values to determine the efficacy of some of these micromechanics methods.

25 15 Young s Modulus (MPa) o C 10 o C 0 o C 10 o C 20 o C 30 o C 40 o C 10 0 (/s) Strain Rate Figure 2.6. Young s modulus vs. strain rate and temperature for glass/estane (44%/56% by volume) mock propellants. Young s Modulus (MPa) o C 10 o C 0 o C 10 o C 20 o C 30 o C 40 o C 10 1 (/s) Strain Rate Figure 2.7. Young s modulus vs. strain rate and temperature for glass/estane (59%/41% by volume) mock propellants.

26 CHAPTER 3 MICROMECHANICS OF COMPOSITES The goal of this research is to develop tools that can predict the effective thermoelastic properties of PBX like materials. The term micromechanics describes a class of methods for determining the effective material properties of composites given the material properties of the constituents. Governing equations based on a continuum approximation are used to solve the problem of effective property determination in micromechanics based methods. The material properties of interest in this research are elastic properties and coefficients of thermal expansion in the domain of infinitesimal strain. We do not discuss methods of determining the effective thermal conductivity or effective specific heats of PBX materials. This is because these properties are relatively close to each other for the components of PBX The high volume fraction of the dispersed component in PBXs as well as the high modulus contrast between the dispersed and the continuous components in the composite provide the main challenges. Accurate yet computationally inexpensive methods that can address these challenges are sought. The data on the properties of PBX 9501 and its components that have been presented in Chapter 2 provide an excellent check of the accuracy of various micromechanics methods in dealing with PBX like materials. In this chapter, we review some micromechanics methods and discuss the effectiveness of these methods in the context of PBX Excellent reviews of micromechanics of composite materials are provided by Hashin [22], Markov [23] and Buryachenko [24]. More detailed expositions on the micromechanics of composites can be found in the monographs by Nemat-Nasser and Hori [25] and Milton [26]. Polymer bonded explosives form part of the class of composites known as particulate composites. The particles of the dispersed component of the composite are distributed in three dimensions. Hence, accurate models of these composites should be three-dimensional. However, for simplicity, we primarily explore two-dimensional models in this research. Micromechanics methods that apply only to aligned fiber composites are, therefore, also reviewed in this chapter. Some bounds on the effective elastic properties of particulate composites based on variational principles of the minimization of strain energy are discussed first. These upper and the lower bounds are found to be quite different from each other. Analytical solutions for the effective elastic

27 ž ž 17 properties are discussed next. Simplified models of the composite are used to obtain these solutions and these are found not to fare much better than the bounds. The final portion of this chapter deals with various numerical techniques that have been used to solve the problem of determining effective properties. Some of the advantages and drawbacks of these methods are also discussed in the context of PBX-like materials. Table 3.1 shows the elastic moduli and the coefficients of thermal expansion (CTE) of HMX, Binder and PBX 9501 at room temperature and low strain rate. These data are used to assess the predictions of some of the techniques discussed in this chapter. Table 3.1. Elastic moduli and CTE of PBX 9501 and its components at room temperature and low strain rate. Material Volume Young s Poisson s Bulk Shear CTE Fraction Modulus Ratio Modulus Modulus (%) (MPa) (MPa) (MPa) (10 /K) HMX Binder PBX Rigorous Bounds The most elementary rigorous bounds on elastic moduli are the Voigt (arithmetic mean) and Reuss (harmonic mean) bounds [27]. In terms of isotropic bulk and shear moduli, these bounds can be expressed as š œp Ÿž œ ž œ Voigt Bounds (3.1) œ (3.2) and, v œp œp œ Reuss Bounds (3.3) œ (3.4)

28 ª ª  ¼ ª Á 18 where p.. L«««ª the volume fraction of the particles, the bulk modulus of the particles, the shear modulus of the particles, the volume fraction of the binder, the bulk modulus of the binder, the shear modulus of the binder, the effective bulk modulus of the composite, and, the effective shear modulus of the composite. The subscript denotes the upper bound while the subscript denotes the lower bound on an effective property. Using the bulk and shear moduli of the components shown in Table 3.1 we can calculate the Voigt and Reuss bounds on the effective moduli of the composite. These values are shown in Table 3.2. The Voigt and Reuss bounds show that the actual elastic moduli are closer to the Reuss bound but considerable difference exists between the lower bounds and the experimental values of composite moduli. Table 3.2. Voigt and Reuss bounds for PBX Elastic Voigt Reuss Experimental Modulus Bound Bound Modulus (MPa) (MPa) (MPa) Bulk Shear Hashin-Shtrikman Bounds Variational principles based on the concept of a polarization field have been used by Hashin and Shtrikman [28] to obtain improved bounds on the effective elastic moduli that have been shown to be optimal for assemblages of coated spheres. For particulate composites these bounds can be written as š ±'² ³ šµ D p # L«¹ ½¼ ³ µ^¾à ª ª ª ± ² ³ ª µ D L«¹ ³ ª µ¾fãlä «mº» Á «mº» Hashin-Shtrikman Upper Bounds (3.5) (3.6)

29 î î î Çï É É ã ã Ê Å ß Æ Þß Æ Þ Ð Û Û Æ Æ Æ Ô Æ Ô Ö Ö ÚÅ ß Ú Å Å Î Õ Ý Õ Î Õ Ý Õ Æ Ô Å Ô Î á Æ ß Æ Ð ß Ð æ Î Å ß Å Æ ß Þ É É Ô Õ Õ ã ã Ú Õ Õ ß Å ß ã î Æ ã Õ î Æ Õ Õ Î ä Î ä ç â â î î Æ Æ Æ Ô Þ Æ Ô Õ Õ è Ý ß ë ß ß Ô ß Ô Ð Õ Õ à à à Ç 19 and, Å ÆÇ ÈfÉËÊ Å ß Ç ÈfÉ ÆÍÌÏÎÑÐÓÒpÔ#ÒLÕ Å ß Ì ÎDÒpÔ#ÒLÕ ÆFÜÀÝ ß Ü where, for any property â, we define and, Ýfá ÙØ ÙØ â ä É â ä É à Hashin-Shtrikman Lower Bounds (3.7) à (3.8) Ô4ç è Ô Ò Ô ÔåÒLÕ Ýfâ Ýfâ Ýfé ÝFê Bounds on the effective coefficient of thermal expansion (î Ýfé ÕmÒLÕ ÕÒpÔ ëíì ) of a two-component isotropic composite can be calculated using the Hashin-Shtrikman bounds [29]. These bounds are where î Ç È Ô.ó Õ ó î äðý î äðý ÐÓÒpÔåÒLÕ ÐÓÒpÔåÒLÕ Õ ñ Ô ñ ÔÓòpñ ÔÓòpñ coefficient of thermal expansion of the particles, and, coefficient of thermal expansion of the binder. ÔÓò ÔÓò Rosen-Hashin Upper Bound (3.9) Rosen-Hashin Lower Bound (3.10) For the components of PBX 9501 listed in Table 3.1 the Hashin-Shtrikman bounds have been calculated and are shown in Table 3.3. The data show that only a very limited improvement is obtained over the Voigt-Reuss bounds for the bulk and shear moduli. The bounds on the coefficient of thermal expansion are within 1% of each other.

30 ÿ ÿ ( ( õ ù õ ÿ õ ÿ ù ( ÿ ù õ 20 Table 3.3. Hashin-Shtrikman upper and lower bounds for PBX Bulk Modulus Shear Modulus Thermal Expansion (MPa) (MPa) (ô õ ö ø /K) Upper Bound Lower Bound Experiments Third Order Bounds The bounds discussed so far have been based only on the volume fractions of the component materials. An improvement over the Hashin-Shtrikman bounds is the application of three point correlation functions to incorporate geometric information into the determination of upper and lower bounds of third order on the effective properties of the composite. A simplification of bounds obtained using statistical methods with three point correlation functions by Beran, Molyneux and McCoy [30, 31] have been provided by Milton [32]. The Milton bounds can be expressed as and, ý úû þ ÿíù ùú û'ü ý ùšþÿ ùšþ ý þ Milton Upper Bounds (3.11) ü ý ÿ ý þ (3.12) õ ù ú ü"! õ ù$# &% ù*) ' õ ) Milton Lower Bounds (3.13) õ ú ü! õ # % õ ) õ +', (3.14) where, for any property -, we define, ý þ - ü - ý þ - ü - ý þ - ü - / / ý - þ10 ü

31 p t r y y n p T 6 6 G L y R[ _ ~ 8 : G y y e j e J J z e t E E J n t p T E 6 n p y T m Y y m 6 z T 21 and, 57698:<;=?>@BA U 6 8:<;1HI>+@V;W=M> D 8 =C<DFEG E*G ;HI>;KJ H ;HI>;KJ H ;W= E*L =M>NA ;KR = E*S =M>;HI> D E*L HI>+@ EL 8 HOC<DFE The Milton bounds depend on two extra geometric parameters, Z+[ ;KJ = ;KJ = HI>+@BA HI>+@X;HI> Q 8B\ Z ] and ^4[ 8 HPC&Q 8B\ ^ ] which incorporate the three-point correlation functions and have been found to lie between 0 and 1. These parameters can be calculated using the following relations (following Berryman [33]): where ƒ u w y terms zx{ Z+[ ^4[ Tz{ elh efh ]O`badc enfig efhfiikj `badc mon j mqp jsr (3.15) elh elh n p Tz{ L _ Z+[ rvuxw y zx{ z}t ]`dadc exf g elhf i `dadc mon j e mqp j r (3.16) n p 8 E Tzx{ } r u w y zx{ is the probability of a triangle (with two sides n and p and an included angle ) having all three vertices lie within particles when placed randomly in the composite. The and } zx{ are Legendre polynomials of order 2 and 4 respectively and are given by zx{ zx{ L y S \ 8 { T \kj : G z For composites with constituents that have a small modulus contrast, the Milton bounds are remarkably close to each other. However, this is not be true for large modulus contrast composites like PBX 9501 [32]. One method of calculating the values of the parameters Z+[ and ^4[ is to convert digital images of PBX 9501 into binary (black and white) images with black representing particles and white representing binder and then following the procedure outlined by Berryman [33, 34, 35, 36]. For PBX 9501, the volume fraction of the particles is close to 92%. It is also observed that the reflectivity of different faces of the HMX crystals on a SEM micrograph varies widely. Hence, it is extremely difficult to obtain a binary image of the PBX 9501 microstructure in order that the two parameters be calculated. Instead, we can make the assumption that the penetrable spheres model (where spheres placed randomly in the RVE may overlap) is representative of high volume fraction particulate composites and use the values of Z [ and ^ [ listed by Berryman [33]. These values are plotted in Figure 3.1 and can be observed to increase linearly with increasing volume fraction. { Y

32 ζ p (Curve Fit) η p (Curve Fit) ζ p (Computed * ) η p (Computed * ) ζ p, η p Volume Fraction Figure 3.1. Parameters +ˆ and 4ˆ for the penetrable sphere model (* = Values Computed by Berryman [33].) Linear extrapolation from the data shown in Figure 3.1 for a volume fraction of 0.92 gives +ˆ = and 4ˆ = Using these values to calculate the Milton bounds gives the values shown in Table 3.4. Table 3.4. Milton upper and lower bounds for PBX Elastic Upper Lower Experimental Modulus Bound Bound Modulus (MPa) (MPa) (MPa) Bulk Shear The Milton bounds are definitely an improvement over the Hashin-Shtrikman bounds. However, the upper and lower bounds are still quite far apart from each other and therefore provide no useful engineering approximation on the effective elastic moduli under consideration. Figure 3.2 shows a comparison of the Voigt-Reuss, Hashin-Shtrikman and Milton bounds on the bulk and shear modulus of PBX 9501 as ratios of the experimentally determined values shown in Table 3.1.

33 23 Bounds/Experimental values Bulk Modulus Voigt/Reuss Bounds Hashin Shtrikman Bounds Milton Bounds Shear Modulus Figure 3.2. Comparison of bounds on the bulk and shear modulus of PBX 9501 with experimental values. 3.2 Analytical Methods Analytical methods for approximating effective elastic moduli of random composites have been developed by researchers since the early 1900s. Early developments were based on dilute dispersions of particles in a continuous matrix assuming that there were no particle-particle interactions. More recent developments have explored concentrated dispersions where particle-particle interactions are allowed. For high volume fraction particulate composites, the composite spheres assemblage, the three-phase model, the self-consistent scheme, and the differential effective medium approach are of interest. Each of these methods makes certain simplifying assumptions about the microstructure of the composites. These methods are discussed briefly and the predicted effective moduli for PBX 9501 are compared with the experimental values shown in Table Composite Spheres Assemblage The composite spheres assemblage (CSA) proposed by Hashin [37] idealizes a particulate composite using spherical particles coated with a layer of binder. The volume of the composite is assumed to be filled completely with various sizes of these coated spheres. The ratio of the radius of a spherical particle to the thickness of its binder coating reflects the volume fraction of particles in

34 ± Ž Š Š Š Š Ž 24 the composite. The solution of the problem involves placing a coated sphere in the effective medium and applying a hydrostatic stress at the boundary of the coated sphere. This approach leads to an expression for the effective bulk modulus that can be written as Š? <Œ ŠOŽ ŠOŽ Š Ž šžœ (3.17) The effective coefficient of thermal expansion for an isotropic composite formed from isotropic components is given by [29] <ŒŸž ŠOŽ Ž Š Ž+ Š & (3.18) This equation requires only the isotropic bulk modulus of the composite to calculate the effective coefficient of thermal expansion. For high concentrations of particles, the shear modulus cannot be found accurately using the CSA model though low expressions that are valid in the dilute limit exist. The CSA prediction for PBX 9501 is shown in Table 3.5. This value matches the lower bound predicted by the Hashin-Shtrikman bounds and is considerably lower than the experimentally determined bulk modulus. Table 3.5. Composite spheres assemblage prediction for PBX CSA Predicted Experimental Predicted Bulk Modulus Bulk Modulus Thermal Expansion q«ª (MPa) (MPa) ( /K) Self-Consistent Schemes The effective stiffness tensor of a dilute distribution of particles in a continuous binder can be expressed as [23] where I Œ ŽF P the volume fraction of particles, the stiffness tensor of the particles, Žœ v ± ² Žœ *³ K the stiffness tensor of the binder, the effective stiffness tensor, and, the tensor that relates the applied strain to the strain in a particle. (3.19)

35 Å Æ Å Æ Æ Å Å Å Æ Å Æ Æ Æ Æ Å Æ ½ Å ½ Æ Å Æ Â Ä 25 When the volume fractions of particles is more than 5%, the binder in the dilute approximation is replaced with a material that possesses the unknown effective elastic properties of the composite. Thus, the expression for the effective stiffness tensor is changed to µn µx¹fº*» ¼ ½Wµ¼P¾MµX¹œ và Á&½WµF¼xÂ.µV o ºÃP½K» ¼o Ä (3.20) The above equation can be solved for the effective stiffness of the composite for various particle shapes. This procedure is called the self-consistent scheme, the effective medium approximation and also the coherent potential approximation. Various types of self-consistent approximations of effective composite properties can be found in the literature. Some of these approximations have been found to generate excellent effective properties at low concentrations of the dispersed component. However, at high concentrations when the modulus contrast between the components is large, these methods do not perform well. An excellent comparison of self-consistent models with the commonly used Mori-Tanaka approximation has been provided by Berryman and Berge [38]. A survey of these methods and a critical review has also been given by Christensen [39]. In general, these methods are unsuitable for materials with both a high volume fraction of particulates as well as a high modulus contrast between the constituents as is seen in composites like PBX For a particulate composite containing a dispersion of elastic spheres, the self-consistent scheme leads to two equations in Å and Æ ¹Çº ¹Çº which have to be solved iteratively. These are» ¼ ½ ¼ ¾ ºÉÈ Ê Å Ê Å ºË Ì» ¼ ½ ¼ ¾ ºÏÎ4Ð ºÉÈkÍ Å Î4Ñ º ÎÒ 4Ì ¹ ¹ ¼I¾ ¼I¾ (3.21) (3.22) For the components of PBX 9501, these equations lead to the effective bulk and shear moduli and the corresponding effective coefficient of thermal expansion are shown in Table 3.6. The predicted values of the moduli are considerably higher than the experimental values (about 10 times for the bulk modulus and about 13 times for the shear modulus). However, these values are lower than any of the upper bounds discussed in the previous sections. On the other hand, the predicted coefficient of thermal expansion is higher than the Hashin-Rosen upper bound. The three-phase model developed by Christensen and Lo [40] is another example of a selfconsistent model that has been used considerably by engineers. In this model, a third outer shell of material possessing the effective properties of the composite is added to the composite sphere

36 â ä æ æ â ä Û Û ì ì 26 Table 3.6. Self consistent scheme prediction for PBX Bulk Modulus Shear Modulus Thermal Expansion Predicted Experimental Predicted Experimental Predicted (MPa) (MPa) (MPa) (MPa) ( Ó<ÔÕqÖ«/K) 11,044 1,111 4, assemblage. This model predicts the same bulk modulus as the CSA model. In addition, the effective shear modulus can be found after solving a quadratic equation of the form ØÚÙlÛIÜ ÛšÝ Þ&ßVà*á â ÙlÛ Ü ÛšÝoÞãàäå Õ (3.23) where Øçæ ØéèKê ëíì Û èkê ëqì èkê ëqì ëíì ìî ëíìî ÛšÝ ëíì ìî ëíìî ÛšÝ ëíì ìî ëíìî ÛšÝ Ý.ï Ýï Ýï ð and, The effective shear modulus of PBX 9501 calculated using the three-phase model is shown in Table 3.7. This model predicts values of effective shear modulus that are lower than the Milton lower bounds shown in Table 3.4. Table 3.7. Three-phase model prediction for PBX Predicted Experimental Shear Modulus Shear Modulus (MPa) (MPa) Differential Effective Medium Approach The differential effective medium approach is another scheme for approximating the effective properties of composites containing a continuous component (binder) and a dispersed component (particles). This scheme has been utilized by various researchers, mostly for low volume fractions of the dispersed component [23, 38, 41]. The idea behind this approach is that an infinitesimal volume fraction of the particle material is added to the binder and the effective properties are calculated using a dilute approximation. Next, the binder is replaced with the effective material and the calculation is carried out again with an infinitesimal volume fraction of particles. This process is repeated until the actual volume fraction

37 ú ñ ø ø ø ù ø ø õ õ 27 of particles is reached. Mathematically, this approach can be represented by a coupled system of linear first order ordinary differential equations of the form where ñ+ò&ó7ô õ ö3 oø?ù ô õûú ñ+ò<ó7ô õ ö3 Iù ô õ ù ú õ ó ù ü ø ö&üýø?ùnþãÿ ù þÿ ù õió ö&ü ùvþ ù ñ ù þ ù ùvþ ù ù þ ù (3.24) Solving the system of equations using a fourth-order Runge-Kutta scheme gives the effective bulk and shear moduli shown in Table 3.8. These results show that the differential scheme underestimates the bulk and shear moduli though the values are inside both the Hashin-Shtrikman and Milton bounds. Table 3.8. Differential scheme predictions of effective properties. Bulk Modulus Shear Modulus (MPa) (MPa) Thermal ò Expansion ( /K) Predicted Experiments Several other approximation schemes exist that generate analytical equations relating the effective elastic moduli to the constituent moduli and volume fractions. Details of the schemes can be found in the monograph by Milton [26] and the paper by Buryachenko [24] and references therein. However, none of these schemes provide sufficiently accurate estimates of the effective moduli of composites with high volume fractions of particles and high modulus contrasts between the particles and the binder. 3.3 Numerical Approximations The effective elastic moduli of a composite can be determined approximately by solving the governing equations using numerical methods. This process involves the determination of a representative volume element (RVE), the approximation of the microstructure of the composite, the choice of appropriate boundary conditions and the solution of the resulting boundary value problem. Numerical solution of such problems requires the RVE to be discretized so that the geometry is adequately approximated. The stresses and strains that solve the problem are then averaged over the

38 '. volume (V) of the RVE. The effective stiffness tensor the relation where are the stresses and $&%(' are the strains. *) +," 28! #" of the composite is then calculated from $-%/. #" (3.25) The earliest numerical approximations of effective elastic moduli were carried out using finite difference schemes by Adams and Doner [42, 43]. These were two-dimensional approximations for regular arrays of fibers in a matrix. Randomly generated microstructures in two dimensions were simulated soon after these preliminary investigations [44]. With improvement in the capabilities of computers many researchers have approached this problem using finite element methods [45, 46], boundary integral methods [47, 48, 49], Fourier transform based methods [50, 51] and so on. Threedimensional simulations are also increasingly being carried out [52, 53]. Recently, researchers have also used the concepts of homogenization theory to solve the problem of determination of effective properties of composites [54, 55]. The method of cells [56] is another numerical technique that has been used with considerable success for fiber-matrix composites. Some recent developments on the determination of an optimum RVE are discussed in this section, followed by a review of the literature on two-dimensional simulations for regular arrays of fibers. Two-dimensional simulations of randomly distributed fibers are discussed next. Finite element methods are the most frequently used solution techniques for these studies. Some boundary integral and Fourier transform based method are also discussed. Three-dimensional simulations have been less frequently carried out because of the high computational costs involved. A few of these simulations are also reviewed in this section. Finally, the method of cells and its application to various micromechanics problems is reviewed The Representative Volume Element Primary to the use of numerical approximations of the effective properties of composites is the concept of the representative volume element (RVE). This concept is similar to the crystallographic unit cell which is the building block in the structure of crystals [57]. Square or cubic RVEs are used for most numerical approximations because of the ease of numerically solving boundary values problems with these geometries. The difficulties involved in generating statistical information about particle distributions and concentrations leads to difficulties in the rigorous determination of RVE sizes. Hence, for most applications, RVE sizes have been rather arbitrary. Sab [58] has shown that if an RVE exists for a random composite material, the homogenized properties of the material can be calculated by the simulation of one single realization of the medium. The ergodic hypothesis, which assumes that the ensemble average is equal to the volume

39 29 average, has been used to arrive at this conclusion. The ensemble average is the mean of a large number of realizations of the microstructure. The volume average, on the other hand, is the average as the RVE volume becomes infinitely large compared to the volume of a particle. However, such a realization may lead to a an extremely large RVE and a more practical approach is to simulate a large number of different realizations on smaller RVEs so that bounds on the effective properties are obtained. Drugan and Willis [59] have shown, using non-local equations for the elastic response, that for a random distribution of identical spheres the RVE size is approximately two sphere diameters. A three-dimensional finite element study on the optimal size of the RVE has been carried out by Gusev [52]. The composite considered was composed of around 26% by volume of identical spheres in a continuous matrix. The modulus contrast between the components was around 20. The simulations by Gusev show that the optimum RVE size is around 3-5 times the sphere diameter. However, it is doubtful if the same conclusions can be drawn for polydisperse composites containing spheres of many different sizes. Two and three dimensional finite element analyses (using triangular and tetrahedral elements) of two phase composites made up of spherical inclusions in a continuous matrix have been carried out by Böhm and Han [53]. The results show that though relatively small RVEs can be used for determining effective elastic moduli, elastic-plastic or other nonlinear behaviors require much larger RVEs to be accurately predicted. The validity of these conclusions for composites with high modulus contrast cannot be ascertained from these numerical studies Finite Difference Approximations Early numerical approximations of effective moduli were carried out on unidirectional fiber composites with two-dimensional approximations of the elastic fields. Regular arrays of circular fibers were modeled using finite difference schemes for transverse normal moduli and longitudinal shear moduli by Adams and Doner [42, 43]. These simulations were carried out for fiber volume fractions up to 78%. Plots were generated for the normalized transverse normal and longitudinal shear stiffnesses at various modulus contrasts between the fibers and the matrix. The results show that beyond a modulus contrast of around 1000, the effective stiffness becomes constant for the Poisson s ratios used. However, we have found that this is not true when one of the components is a rubbery material. Hexagonal random packings of fibers were studied by Adams and Tsai [44] using finite differences and found to generate better approximations of actual fiber composite behavior than regular square array or regular hexagonal array packings. Finite difference approximations were soon replaced by finite element approximations as the primary simulation technique. This was probably due the improved discretization of particle geometries using finite elements. However, the finite difference approximations have recently been

40 30 used by Ostoja-Starzewski et al. [60] to compute the effective properties of a random composite with circular inclusions undergoing damage. The lattice-bond based stochastic model for damage growth is easily parallelized with finite differences. This method has also been parallelized for graded interfaces between particles and matrix and shown to generate close bounds on the effective thermal conductivity for particle volume fractions of about 50% [61] Finite Element Approximations Since actual microstructures of composites are difficult to obtain and simulate, most finite element simulations of the micromechanics of composites have involved square or hexagonal arrays of fibers in two dimensions. With decreasing computational costs, complex two-dimensional and three-dimensional problems in the micromechanics of composites are being investigated with finite elements Regular Arrays in Two Dimensions Finite element approximations of the effective behavior of regular arrays of unidirectional circular fibers were carried out by Adams and Crane [62] using a generalized plane strain assumption. The RVE was chosen to contain one fiber and was discretized using triangles. Each node was assigned four degrees of freedom - two for the in-plane displacements and two for the out-of-plane displacements. This formulation can be used to determine the three-dimensional state of stress from a two-dimensional solution. It is not obvious, however, how the formulation can be used for a random distribution of particles as the boundary conditions become considerably more complex. Standard displacement based finite element formulations in two dimensions have been used to model circular fibers by Zhang and Evans [63] and to model rectangular fibers by Shi et al. [64]. The study by Zhang and Evans modeled a RVE containing a circular fiber coated by a annular matrix ring and validated the concentric cylinders model [65]. Though the circular RVE leads directly to the prediction of isotropic properties (unlike a square or rectangular RVE), this approach is not applicable to high volume fraction PBXs since each fiber is coated uniformly with matrix and there is no fiber-fiber contact. The use of rectangular fibers by Shi et al. to model whisker reinforced composites is a gross simplification of the actual microstructure and fiber geometry. Though this simplification has the advantage of being easily discretized, the sharp corners of the particles lead to high stress concentrations and lead to numerical errors. We explore both circular and square particle distributions in Chapter 7. Displacement based finite element solutions have been found to predict effective properties that overestimate the actual properties. On the other hand, force based solutions provide a lower bound on the actual properties. However, force based finite element methods are not often used

41 31 because of the difficulties involved in their formulation. Recently, Lukkassen et al. [66] have shown how homogenization theory can be used to compute the effective properties of unidirectional fiber composites using a force based finite element approximation of the effective properties of a unit cell. A recent finite element study has been carried out on regular arrays of fibers by Pecullan et al. [67]. This study has found that for a composite with high modulus contrast and a compliant matrix, the force based effective stiffness tensor is more accurate. Force based finite element methods are not utilized in this research because of the difficulties is formulation and implementation of these methods. Interestingly, Pecullan et al. [67] have also observed that replacement of the smallest scale microstructure by the equivalent homogeneous material does not cause large errors in calculation of the effective stiffness tensors. This result is of interest in this research because generation of microstructures occupying more that 86% of the volume is difficult. Instead, we can generate microstructures that occupy about 86% of the volume and replace the remaining volume with a dirty binder (a binder with the effective properties of a mixture of particles and the original binder) without much loss in accuracy. Recent finite element analyses of the micromechanics of two-dimensional composites have focused mostly on determining the effective inelastic response [68, 69, 70]. However, most of these approaches use displacement based finite element methods on regular packings of fibers and do not attempt to solve the problems associated with high volume fractions (0 90%) of fibers Random Distributions in Two Dimensions Models using regular arrays of fibers provide reasonably good approximations of the effective elastic properties of fiber composites. However, for particulate composites, this is not true and the complex microstructure has to be taken into account. This implies that three-dimensional models are required. The high computational cost involved in modeling particulate composite microstructures in three dimensions has led to the development of two-dimensional techniques that perform well for some of these composites. Ramakrishnan et al. [71] have used a generalized plane strain approach to model particulate metal matrix composites. The composites considered in the study had a maximum volume fraction of 40% of particles. Particles of various shapes and sizes were randomly distributed in a twodimensional square RVE. The effective Young s modulus was determined by the application of a uniform unidirectional displacement. The effective bulk modulus was determined by applying equal displacements in the three orthogonal directions. The effective Poisson s ratio was determined from the effective bulk and Young s moduli. The effective coefficient of thermal expansion was also determined using finite elements. It is observed that the shapes of the particles do not have any

42 32 significant effect on the effective elastic properties even though many particles have sharp acute angles and therefore high stress concentrations. Periodic boundary conditions are not used in the approach. The RVE size is also chosen arbitrarily. The use of the bulk modulus to determine the Poisson s ratio assumes that the material is isotropic. However, the two-dimensional approximation automatically implies that each particle extends continuously in the out-of-plane direction and hence makes the material anisotropic. We do not use this approach in this research for these reasons though the authors cite good agreement with experimental data. Random distributions of particles in two dimensions have also been studied by Theocaris et al. [72] in the context of determining the effective Poisson s ratio. Finite element simulations were carried out on a unit cell. Periodic boundary conditions and uniform pre-stresses were applied to the unit cell. The effective elastic properties were determined using a strain energy match between a cell simulating the microstructure and an equivalent homogeneous cell. This study is of interest to us because the procedure of determining the effective moduli is well grounded in theory and uses the Hill condition. It also shows that the Poisson s ratios that are calculated using two-dimensional models are actually two-dimensional Poisson s ratios that have an upper bound of 1.0 instead of the 0.5 for the three-dimensional case. The study by Theocaris et al. [72] also suggests that sharp corners in particles do not have any significant effect on the effective elastic properties - assuming perfect interfacial bonding. Jia and Povirk [73] have used a subgrid scale finite element model based technique to calculate the effective moduli of a two phase composite containing randomly distributed square inclusions. A window of the RVE is chosen in the first stage of the calculations and moved over the RVE. Calculations of the effective properties are carried out at each location of the window. These effective properties are then assigned to a smaller mesh for faster calculations of the overall properties. It is found that the error in the estimation is small for the component properties considered by Jia and Povirk. This approach is similar in some respects to the two-step generalized method of cells technique discussed in Chapter 7. Another technique that has been used to determine the effective properties of two-dimensional composites with complex microstructures is the multiphase finite element method [74]. The approach is to assign different material properties to different Gauss points in a finite element analysis of a complex microstructure. This approximation obviates the need to generate complex meshes to describe the geometry. Mishnaevsky et al. [74] have used the method to determine the elastic fields in metal matrix composites. Microstructures have been obtained for these studies from digital images. This approach can be used to model microstructures where adjacent particles are very close to one another. However, the robustness of the method is still not very well established.

43 33 Kwan et al. [75] have used randomly distributed particles of arbitrary sizes and shapes to study the effective behavior of concrete. Though the approach is similar to most displacement based finite element approaches, it is of interest that the interface particles and the matrix has been modeled using zero-thickness interface elements. These interface elements can be used, in this research, both for the modeling of debonding and cracks as well as for a very thin layer of binder between particles Approximations using Homogenization Theory The mathematical theory of homogenization [76] has recently become an established approach for determining effective properties of periodic composites [77]. The governing differential equations with rapidly varying coefficients are replaced by differential equations with constant or slowly varying coefficients. Asymptotic expansions of the field variables along with the assumption of periodicity lead to this transformed set of equations. The new set of equations are called the Y-periodic homogenization problem because the repeating cell is called 1 in the notation used in the theory. The Y-periodic homogenization problem can be solved using finite elements or other techniques. The assumption of periodicity does not preclude the application of this method to particulate composites. We can always assume that an RVE containing randomly distributed particles is repeated periodically in space. Hassani and Hinton [78] have used finite element analyses along with homogenization theory to solve the effective modulus problem for various ranked laminates and for cellular materials with rectangular holes. Incompatible and hybrid finite elements have also been used to solve the Y- periodic homogenization problem for fiber composites [79]. However, these studies have used microstructures with regular arrays of fibers. Ghosh and co-workers [54, 55] have used homogenization theory along with the Voronoi cell finite element method to model RVEs containing random distributions of particles at volume fractions of up to 50%. In this approach, particle locations are generated within the RVE using a random process. A weighted Voronoi tessellation of these particles is carried out to generate a set of Voronoi cells. Each cell is a multi-sided polygon and contains a single particle. Homogenization theory is used to model the effect of a single particle on the properties of a Voronoi cell. The approach shows good agreement with detailed finite element analyses of the same microstructure. However, for high particle volume fractions (2 80%) the Voronoi tessellation leads to needle shaped cells in two dimensions. If the method is extended to three-dimensions, not only is it difficult to generate particle distributions that fill more than 55% of the volume, generation of weighted Voronoi tessellations becomes considerably more involved. Moreover, the effective properties obtained by finite element analyses depend strongly on the choice of element type and size as can be observed

44 34 from the simulations of randomly oriented short fiber composites by Courage and Schreurs [80] Approximations using Stochastic Finite Elements In continuum descriptions of composites, the constitutive relationship is only a function of spatial position. Stochastic descriptions of the constitutive relation assume that the stiffness tensor is a random field with continuous realizations. In other words, an additional stochastic variable is added the continuum description. Stochastic finite element analyses attempt to solve this modified problem using finite element techniques. These methods are applicable to particulate composites where the particle distributions can vary in a random manner. Ostoja-Starzewski [81] has performed stochastic finite element analyses on two-dimensional composites reinforced by randomly located disks. Numerous simulations have been carried out to obtain bounds on the effective stiffness tensor. These analyses show, for a given RVE size, that the predicted bounds deviate from each other by a 0.5% as the modulus contrast between the components reaches about 20. It is not clear from the data what the difference between the bounds would be for higher modulus contrasts. Huyse and Maes [82] have used stochastic finite element analyses (using a truss network to represent a particulate composite) to determine the autocorrelation and cross-correlation coefficients between various elastic constants. Ostoja-Starzewski [83] has suggested that these correlation coefficients could be easily determined for particulate composites. This information could be then be used to generate bounds on the effective elastic response without resorting to time consuming numerical simulations of different realizations of the microstructure. The analyses of Huyse and Maes [82] show that force based finite element formulations provide better estimates of the effective properties than the displacement based method. Similar conclusions can be drawn form the stochastic finite element analyses carried out by Kamiński and Kleiber [84] Three Dimensional Approximations Most three-dimensional finite element studies of the micromechanics of composites have dealt with periodic microstructures [46]. Tetrahedral elements are the most commonly used in these simulations [85] while some use hexahedral elements [86]. The techniques of computing the effective properties for three-dimensional problem are essentially the same as those discussed for two-dimensional problems. However, three-dimensional analyses provide some insights into the mechanics of composites that are not obvious in two-dimensional studies. Further, techniques of microstructure and mesh generation in three dimensions are instructive in the context of PBX-like particulate composites. In addition, the computational costs of three-dimensional studies lead to the exploration of efficient implementations of numerical techniques that are of interest in this research.

45 35 Various approaches have been used to generate three-dimensional microstructures for simulations of particulate composites. The most commonly used method is random sequential placement of spheres in a RVE, also called the Monte Carlo approach. The three dimensional finite element simulations of Gusev [52] used a Monte Carlo approach to generate realizations of the distribution of identically sized spheres inside a cubic RVE. Only about 30% of the volume of the RVE was filled in the study. Tetrahedral elements were used to discretize the geometry. This approach is not well suited for high volume fraction composites like PBX First, the Monte Carlo approach of placing particles become extremely inefficient beyond volume fractions of 55-60%. Meshing of close-packed particles using tetrahedral elements leads to extremely skewed elements and poor numerical performance. A more practicable approach, for PBX-like materials, is a digital image processing based approach adopted by Garbozci and Day [87]. The method uses digital image processing techniques to generate three-dimensional finite element meshes for complex microstructures. X-ray tomography is used to generate three-dimensional voxelized images. Each voxel is then modeled as an eightnoded linear displacement finite element. Terada et al. [88] have also developed detailed digital image based models of composites that use two-dimensional slices to generate three dimensional microstructures. Hexahedral elements are generated directly from the images for these models also. Though digital techniques appear lucrative, advanced image processing techniques are required to generate microstructures for PBX materials. This is because the high volume fractions and similar densities and reflectivities of particles and binder make it difficult to identify the components of the composite from images. A question that arises for PBXs is whether the binder wets all the particles. In other words, it is of interest to know the amount of strain that leads to interfacial debonding between the particles and the binder. A three-dimensional finite element study of the effect of interfaces on the stresses in composites containing spherical inclusions arranged in a cubic array has been carried out by Dong and Wu [89]. The results indicate that the assumption of perfect bonding used in many micromechanics studies may not be appropriate for high concentrations of particles even when small strains are applied. This is because very high interfacial stresses are developed is the particles are to remain bonded to the binder. Multigrid finite element methods often prove to be considerably more computationally efficient than standard finite element methods. An implementation of a multigrid finite element method based on uniform grids has been used by Zohdi and Wriggers [90] to solve three-dimensional elasticity problems for composites reinforced with spheres. Various errors in approximation have been explored for volume fractions of up to 50% of spheres. Results are compared to various bounds

46 36 and curve fits to the numerical results have been presented. It is doubtful that the resulting equations can be used for composites with other constituents. However, the multigrid method can be used to compute the elastic fields for complex materials because of the high computational efficiency that can be achieved by this methods Discrete Models Discrete models, e.g., spring network models, are receiving renewed attention because some of the discretization issues involved in other numerical models become more tractable. Twodimensional triangular spring network models have been used by Day et al. [91] to determine the effective elastic response of plates containing randomly located holes. The computations show that these models generate quite accurate results. Digital images can easily be resolved into spring networks. Random composites can therefore be easily modeled using these techniques. Toi and Kiyoshe [92] used a three dimensional discrete model consisting of springs and rigid crystals to determine the effective mechanical properties of polycrystals with damage. The microstructure is generated using a three-dimensional Voronoi tessellation of a set of randomly generated points. This method is of interest in this research because particles in PBXs are almost rigid compared to the binder. The high volume fraction of particles makes PBXs appear like polycrystals. However, the large variation in particle sizes in PBXs requires the use of a weighted Voronoi tessellation to generate the microstructures of interest. This process is extremely complex in three dimensions. In addition, certain Voronoi cells have to be filled with binder to account for the 8% of binder in PBX 9501, for example. This assignment of binder to Voronoi cells will necessarily be arbitrary and will lead to pockets of binder as is observed in the square particle distributions discussed in Chapter Integral Equation Based Approximations Boundary integral based methods have been used with some success for determining the effective mechanical and thermal properties of two-dimensional composites (e.g., Rizzo and Shippy [47], Achenbach and Zhu [48], Papathanasiou et al. [93], Helsing [94, 95]). The computations of Rizzo and Shippy [47] for square inclusions avoided calculations of stresses at the corner singularity regions. The calculations of Achenbach and Zhu [48] were carried out on single circular inclusions using standard boundary element techniques. Similar methods have been used to determine the effective elastic moduli of two-dimensional composites with low volume fractions of circular inclusions by Papathanasiou et al. [93]. The interface integral method of Helsing [95, 96] has been used to generate accurate effective elastic properties of periodic composites in two dimensions. An Airy stress function based com-

47 37 plex variable representation of the governing differential equation is converted into the Sherman- Lauricella type integral equation in this technique. The integral equation is solved using a matrix free Nyström algorithm [95]. The use of complex variables leads to the method being applicable only to two-dimensional problems in its current form. If the particles touch each other or have high modulus contrast, convergence is reported to be relatively slow. The implementation of the Nyström algorithm consists of several steps and is quite involved. This has made this method unattractive for this research. The Helsing method has been used to determine accurate effective elastic moduli of RVEs containing large numbers of complex shaped inclusions nearly in contact [97]. This technique promises to be one of the best available for two-dimensional analysis of the low strain rate micromechanics of composites. It is especially suited for problems that involve stress singularities Fourier Transform Based Approximations Complex microstructures have also been studied by Moulinec and Suquet [50] using a Fourier transform based numerical approach to solve the unit cell problem. This approach takes advanced of the assumed periodicity of the elastic fields and by reducing the governing differential equations to the Lippman-Schwinger equation form both in real space and Fourier space. The solution is then obtained using an explicit algorithm that alternates between the real and the Fourier spaces. Discretization of the problem is carried out using a regular grid of pixels or voxels generated from images of microstructures. The advantage of this method is that special consideration is not required for materials that are nearly incompressible (as is needed to avoid element locking in finite element approaches). However, for high modulus contrast between the components, the rate of convergence is slow. This problem has been partially solved using an accelerated convergence method [51, 98] that converges as the square root of the modulus contrast. This method has been applied to twodimensional composites but can easily be extended to three-dimensional problems. The integral equation based method of Helsing and the Fourier transform based method of Moulinec and Suquet appear to be the best for numerically studying the linear elastic micromechanics of polymer bonded explosives. The Fourier transform based method is more lucrative because it can easily be extended to model three-dimensional problems and inelastic material behavior. In general, the numerical simulation of the thermomechanical behavior of particulate composites requires large computational resources. Since such resources may not always be available to an engineer, we next explore some simpler approximations that may be used to generate engineering estimates for the effective thermoelastic properties of composites. In particular, we look at the method of cells [56].

48 3.4 Method of Cells The method of cells (MOC) [56, 99] has been used to model the micromechanical behavior of different types of composites with relative success. The advantage of this method over other numerical techniques is that the full set of effective elastic properties can be calculated in one step instead of solving a number of boundary value problems with different boundary conditions. An averaging technique that satisfies subcell continuity and equilibrium in an average sense using integrals over subcell boundaries is used by the method of cells. The problem of discretization is also minimized because a regular rectangular grid is used. This method has been shown to be more computationally efficient than finite elements for modeling fiber composites [99]. The original method of cells was extended by Paley and Aboudi [100] from using a single subcell to represent the inclusions to a more general version with multiple subcells. This new method has been referred to as the Generalized Method of Cells (GMC) [99]. Comparisons of the results from GMC with finite element analyses for a boron/aluminum composite with a volume fraction of 0.46 of boron fibers showed remarkable agreement [99]. In addition, GMC was found to be more computationally efficient than finite element analyses for square arrays of fibers in two dimensions [101]. Far fewer GMC subcells were found to be necessary than finite elements to arrive at the same degree of accuracy in the solution. However, the computational efficiency of GMC is becomes worse than that of finite elements as the number of subcells increases. This is due to the size of the matrix that is inverted is the square/cube of the number of subcells in two/three dimensions. This leads to large memory requirements and large computational times while modeling complex microstructures. Robertson and Mall [102] attempted an improvement over GMC by extending the MOC approach to three dimensions with the additional restriction that composite normal stresses do not produce any shear stresses in the fiber or matrix. A set of closed form equations were given for the effective elastic constants using the cell model that showed good agreement with experimental data from boron-aluminum composites for fiber volume fractions ranging from 40% to 70%. However, the method does not allow for large grids of subcells. This makes it attractive for modeling complex microstructures. The method of cells has also been extended to solve the three-dimensional problem of short-fiber composites by Aboudi [103]. This formulation of GMC in three dimensions leads to a system of equations of : where N is the number of subcells in each coordinate direction. Orozco [104] has partially solved this problem by identifying the sparsity characteristics of the system of equations and by using the Harwell Boeing suite of sparse solvers. The computational efficiency of GMC has been further improved after reformulation by Pindera and Bednarcyk [105, 106, 107, 108]. The 38

49 39 reformulation has taken advantage of the continuity of tractions across subcells to obtain a system of;5<7=9>@? equations in three dimensions, thereby greatly improving the efficiency of the method. Low et al. [109] have used a two-step homogenization scheme using GMC to determine the effective properties of unidirectional fiber composites with interphase regions. The interphase region is discretized into a number of subcells and the variation of elastic properties along the interphase is modeled by assigning different values to different subcells. This subassembly of subcells is then homogenized using GMC. The interface is then represented as a few homogeneous cells in the next step that generates the final homogeneous effective properties using GMC. A similar concept has been used for modeling PBX microstructures in Chapter 7. GMC has also been extended to model interfacial debonding and the results compared with finite element simulations for square arrays of disks [110]. Displacement jumps across interfaces are modeled with springs in this approach. An alternative approach using a Gaussian distribution based interface debonding model has been developed by Robertson and Mall [111]. Both the Aboudi model and the Robertson and Mall model require some trial and error to determine the appropriate model parameters. These models have been applied to metal matrix composites [111, 112, 113] but comparisons with experimental data have not been provided in most cases. The use of springlike displacement jump factors has also been avoided in the study by Lissenden [114]. Instead, interface debonding is described by a cubic polynomial that relates the interfacial tractions to the interfacial displacements in a smooth fashion. Another approach of modeling interfacial damage within the context of GMC has been to use an uniaxial constitutive law for the interfacial zone and then to increase the size of the zone with progressive damage [115]. A Weibull distribution based probability density function has been used to describe the effective interfacial debonding strain in addition to the interface constitutive law in the progressive damage model to obtain better agreement with experimental data [116]. Difficulties involved in the determination of interfacial properties make these models difficult to assess and validate. There is a lack of coupling between the normal and shear stresses and strains in GMC. Bednarcyk and Arnold [117] claim that this lack of coupling makes for an ultra-efficient micromechanics model. However, our studies have shown that this lack of coupling leads to gross underestimation of shear moduli. Recently, a few attempts have been made to rectify the shear coupling problem. Williams and Aboudi [118] have attempted to solve the problem for periodic arrays of fibers by using a third order expansion for the displacement instead of the first order expansion used in the original method of cells [118]. However, this approach leads to a large system of equations and the efficiencies introduced by Pindera and Bednarcyk are no longer applicable. An alternative approach has been taken by Gan et al. [119] to include normal-shear coupling in the GMC analysis.

50 40 The original GMC assumes that there is traction continuity across all cell and subcell interfaces. The modification made by Gan et al. removes this constraint and instead tries to satisfy subcell equilibrium and compatibility. Results obtained by the new method show a much better prediction of shear moduli than the original GMC without much greater computational requirements. The generalized method of cells has been used for the calculation of effective properties of polymer bonded explosives in this research. The details of the method are provided in Chapter 4 and some results using GMC are provided in Chapters 6 and 7.

51 CHAPTER 4 THE GENERALIZED METHOD OF CELLS A simplified version of the reformulated three-dimensional GMC is described in this chapter. It is assumed that a RVE exists for the composite under consideration. Since we are interested in a random particulate composite, we assume that the RVE is cubic and the subcells are of equal size for simplicity in the following derivation. A schematic of the discretization of the RVE along with the notation used is shown in Figure 4.1. RVE Y,2 β Subcell α X,1 γ Z,3 Figure 4.1. Subcells and notation used in GMC. For simplicity, we forego the derivation of the equations for effective plastic strains and the representation of interfacial debonding. It should be noted, however, that our implementation of GMC includes the capability of variable subcell sizes, plastic strains, and interfacial debonding between subcells. We follow the notation used by Aboudi [56] where possible. A different form of the GMC equations can be found in the report by Bednarcyk and Pindera [107]. The three-dimensional version of GMC can be easily converted into the two-dimensional version by suppressing the equations relating to the third dimension.

52 t I I I I I I ˆ ˆ P ˆ P s I P I ABDC EGFGH JLK IBDC EGFGH MONQPSRTN5U RTN5VXWZY\[ B]C H P_^ BDC EGFGH Ỳ [ Udc BaEbH BDC EGFGH Ỳ [ Vgf BeFGH B]C E@FGH In this derivation, we assume a linear displacement field for each subcell of the form I (4.1) whereh represents the coordinate direction and takes the values 1, 2 or 3, MON P RTN U RTN V W is the global coordinate system of the RVE, MO[ BDC H P RT[ UBaEbH RT[ VBeFGH W is the coordinate system local to a subcellmjilkzmnw+r ABDC EGFGH opm AqB]C E@FGH P R AqB]C E@FGH U R ABDC EGFGH V W are the displacements in a subcellmjilkzmzw+r K BDC EGFGH is the displacement at the center of a subcellmjilkzmnw+r ^ BDC EGFGH is the local variation of displacement in the 1 direction, c BDC EGFGH is the local variation of displacement in the 2 direction, and, f BDC EGFGH is the local variation of displacement in the 3 direction. Williams and Aboudi [118] have used a field containing higher order terms so that the shear and normal displacements can be coupled. However, that formulation leads to a much larger set of equations. We do not explore the approach of Williams and Aboudi in this formulation Average Strain Relations The strain displacement equations for the subcell are given by where r#bdc EGFGH Is Jut v Mjw IA BDC EGFGH Yw s A IBDC EGFGHW (4.2) w&pxj w w[ B]C H P R w UyJ w w[ UBaEbH R w VyJ w w[ V{z BaFGH If each subcellmjinkzmnw has the same dimensionsmv R v R v W then the average strain in the subcell is defined as a volume average of the strain field over the subcell as } r Is BDC EGFGH ~ J x - ƒ t r Is BDC EGFGH (4.3) where is the volume of the subcell, and, - ƒ o t V \ \ \ [ B]C H P [ U BaEbH [ VBaFGH z The average strain in the subcell can then be obtained in terms of the displacement field variables. } r BDC EGFGH PTP ~ J - t ƒ Š w P K BDC EGFGH For example, for the normal strain component in ther PTP direction, equation (4.3) becomes (4.4) z Y ^ BDC EGFGH Yw P [ UŒc BDE H B]C E@FGH Yw P [ V f BeFGH PB]C E@FGHŽ

53 º º º 43 We can obtain similar equations for the other normal strain components. For the shear strains, we get similar equations in terms of the displacement field variables. This is seen from the equation for the shear strain # shown below. # G š œ ž&ÿ G T D G G G n a b d D G G T n a G ª «D G G # D G G # l D Œ G \ G, n e G ª G œ (4.5) Now, G n D ± n e G ª are independent of n ]. After some algebraic manipulation, equation (4.4) can be written as # G T š² D G G (4.6) Similarly, using the fact that G n ] ³ l e G ª are independent of n ] and that G ª n ] ³ l a b are independent of n a G ª, we get # G T šl D G G (4.7) # G ªTª š «ª D G G (4.8) The shear strain equation (4.5) can be reduced to Similarly, D G G š² G \ G (4.9) D G G ª š ª D G G G (4.10) D G G ª š «D G G ª D G G (4.11) For the case where the interfaces between subcells are perfectly bonded, the average strain in the composite RVE is given by µ œ º ¹» ¼ ½ ª ¹ ¾ ¹ ¾ ¹ š œ ¹ where G¾ ½ is the length of a side of the RVE, and, is the number of subcells per side of the RVE. º œ D G G (4.12) In the following derivation, tractions are assumed continuous across subcell interfaces. Displacements and tractions in the RVE are assumed to be periodic. In the shear coupled version of the method of cells presented by Gan et al. [119], the traction continuity assumption is replaced by

54 ÙØ àø âø Â È Â Ä Ã Ã Ä Ã Ã Â È Ä Ã Ã Â Â Â Â Â ÿâ ÿâ þ 44 the satisfaction of equilibrium and compatibility across subcells, thereby making it difficult to apply interfacial jump conditions to account for imperfect interfaces. Assuming displacement continuity across interfaces, if À is the interface between two subcells, then Áq à Áq Á ÈÂÄ Ä ÉÉÉÊË Á È (4.13) Á È ÅXÆGÇ Á Ä ÉÉÉÌÊyÍ Î-Ï ÉÉÉÊË È (4.14) ÈÄ Å Æ Ç Ä ÉÉÉÌÊyÍ Î-Ï ÉÉÉÊË È (4.15) ÉÉÉÌÊyÍ Î-Ð whereñ Í ÒÔÓÓÓ +Ï ÙÝÜ if ÍÛÚ Ò Ï Ò Þ Ù\ß ; if Ò Ï Ù ; Í Ö Ö (4.16) à Ü à if Í Ú Ò Ï ÒáÞ ß ; à if ; Ò Ï Í Ö Ö (4.17) â/ü ÍãÚ Ò Ï ÒÞ âäß ; Ò Ï ifâ. ÍåÖ Ö (4.18) Writing the equations (4.13),(4.14, and (4.15) in terms of the local subcell coordinate systems, we have Áq Á à Á ÈÂÄ Å@ÆGÇ ÉÉÉÌæ çìè@é ề ëíì ÄbÅGÆGÇ Ë Á È (4.19) Ä Å@ÆGÇ ÉÉÉæ ç î êëðïì ÁÂ È è@é Í Î-Ï ÉÉÉæ çìñ#é ò`ëíì ÅXÆGÇ Ë ÁÂ È (4.20) ÉÉÉæ ç ò`ëðïì ÈÄ ÅGÆGÇ ñ,é î Í Î-Ï ÉÉÉæ çó!é ô ëíì Ä Å ÆXÇ Ë È (4.21) ÉÉÉæ ç7î ôó!éëðïì Í Î-Ð Applying these displacement continuity equations on an average basis over the interfaces we get, õ-öø áù Á  Á  à ÈÄ Å@ÆGÇ ÉÉÉæ çìè@é ề ëíì ÄbÅ@ÆGÇ Ë È ÉÉÉæ ç î ềè@éëðïìúüû ý þå Ç û ý ÆGÇ (4.22) õ ö ù Á  Á Â Ä ÅGÆGÇ Í Î-Ï È ÉÉÉæ çñé ò ëíì ÅGÆGÇ Ë È ÉÉÉæ ç òñ,é î ëðïì ú û ý Ä Ç û ý ÆGÇ (4.23) õ ö ù Á  Á Â Ä Å@ÆGÇ Í Î-Ï È ÉÉÉæ çìó!é ô ëíì Ä Å ÆXÇ Ë È ÉÉÉæ çoî ô\ëðïì ó,é ú û ý Ä Ç û ý Å Ç (4.24) Í Î-Ï õö where û ý DÇ û ý DÇ õ ì õ ì ïì ïì û ý aç û ý DÇ Ð

55 P J L L B B P 45 Substituting equation (4.1) into equations (4.22), (4.23), and (4.24) we get, after integration and some algebra, " " $ $ "! #%$ & '! %$ & (*),+ (4.25) '.- / / (*),+ (4.26) #/ '.2 #/3$ '.2 4$ (*),5 (4.27) Let 6 /& be the interface between the two subcells 7 8:9<;:= and 7?> 8@9<;:=. The equations relating adjacent subcells can be expressed in terms of a single coordinate system with its origin at the mid point of the interface between the subcells. The mapping between the subcell based coordinates and the interface based coordinates can be written as A B ( A B ( A C D A J ( J A J $ ( A C D A L ( A C D L ( A & A4CED F G "HI+ %$. F G 'I+ A4CED KEG "H3+ (4.28) KEG '3+ MNG "H3+ AO$ 1 CED MNG '35 Evaluating all quantities in equations (4.25),(4.26) and (4.27) at the interfaces using equations (4.28), we have, # "H "H "[ PRQ QIS B PVQ QIS J Q Q3S L / "0 / TU" #%$ & "H- TY" / $ "Z2 TY" /3$ Equations (4.29),(4.30), and (4.31) can be written as "H "H "H PVQ QIS B PRQ Q3S J Q Q3S L %$? "0 %$? TW(X),+ (4.29) / $ "Z- / 1 $ TW(X),+ (4.30) 4$ "Z2 /3$ TW(X),5 (4.31) \ / (*),+ (4.32) ] / (*),+ (4.33) ^ / (*),+ (4.34)

56 f p n f f f x x p n p n f f f f f f x x f where _a`/bc de `/bc de `bcde gwh f`/bc de gwh f`/bc de gwh f`bcde j f`bcde gskutwvrx xiy{z o f`/bc de gkut v x x3y q f`/bc de gkut v x x3y ƒ ij f`/bc de i[o f`/bc de irq f`bcde h f`bcde h f`bcde h f`/bc de Now, from equations (4.38), (4.39), and (4.40), since h `bcde `/bc de are constant, we have, 46 f m l f l m f m kzh `%l b?c de ij `%l bcde (4.35) kzh `b c1de i0o `b c1de (4.36) kzh `/bc3l d?e iq `/bc3l d?e (4.37) k} f `/bc de ~ m (4.38) kz f `/bc de ~Hm (4.39) k f `/bc de ~[ (4.40) xiy{z x3y x x3y ƒ is linear in y f and f`bcde m f`bcde m and j f`/bc de g*ˆ,m (4.41) o f`/bc de g*ˆ,m (4.42) q f`/bc de g*ˆ, (4.43) Therefore, from equations (4.35), (4.36), (4.37) and equations (4.41),(4.42), and (4.43) we have, x3y{z x3y x x3y ƒ _ f `/bc de g x x3y{z f`/bc de g x x3y f`/bc de g x x3y ƒ h f`bcde k x xiy{z h f`bcde k x xiy h f`bcde k x xiy ƒ h f`%l b.c de g*ˆ,m (4.44) h f`b cde l g*ˆ,m (4.45) h f`bc3l de g*ˆ, (4.46) If we carry out a smoothing operation where the displacement at the center of each subcell is set to be equal to the applied displacement, then all of the above equations involving the displacements at the subcell centers are satisfied. Thus, we can assume a solution of the form f gwh f `bcde (4.47) From equations (4.35), (4.36), (4.37) and (4.47), we have, _ `/bc de `/bc de `/bc de gxj f`/bc de gyo f`/bc de gxq f`bcde f m f m f ij `%l bcde (4.48) i}o `%l b&cde (4.49) iq `%l b?cde (4.50)

57 Š Š Š Š Š Š Š Š Š Š Š Š Š µ µ Š Š Š 47 Using equations ( ) and ( ), and summing over the coordinate directions, we get Œ a Œ Œ I Œ Œ œ % Œ? œ Xž 3 Œ? œ Xž Substituting equations ( ) into equations ( ) we @Ž žwªv««i žwªv««3 žwª ««I ² Œ Ž& Œ Œ Therefore, plugging equations (4.47) into equations ( ), we have, ž Œ Xž «: «3 Ž,µ Œ Xž «± «3 Œ Xž «³u «3 ² Œ (4.51) Œ I (4.52) Œ (4.53) Œ W *Ÿ, } : (4.54) Œ W *Ÿ, Z± (4.55) Œ W *Ÿ, (4.56) Z³u (4.57) (4.58) (4.59) We can show, using the preceding equations, that the average strains in the RVE can be represented as Let us check this for ½ ¾ ¹ º ¼» ž ª ««3 µ š ««3». Using equation (4.12) we have Ž Ž º À» Á Á3ÂÄà Substituting equation (4.6) into the above equation we get Ž Ž º» Á Š Á  Now, if equation (4.57) is multiplied by Æ with ½ Š Á» Á4 Œ Ž ««I µ Œ Ž Ž Å H (4.60) Œ Ž (4.61) :» and summed over Ç and È we get Ž µ Ž (4.62)

58 Ë Ø Ø Ø Ë Ë Ë Ø Ø ç ç ç ä Ë Ê Ê Ê Ê ä ä ä ß ß ß ß Ê Ê Ê ß ß ß Î 48 Comparing equations (4.61) and (4.62), we have, É Ê Ë ËÍÌÏÎ Ð Ð3Ñ Ë!Ò ËÔÓ The rest of the relations in equation (4.60) can be shown to hold in a similar way. In order to relate the subcell strains to the volume averaged strains in the RVE, we apply equation ÎÖ (4.6) to equation (4.57) to get, for Õ, Ùàáâ Substituting for Ý ß Ø Ù&Ú Ùà áâ Ë ÛÜÞÝ<ß Ûã Ð Ð3Ñ and using equation (4.60) we have, Ø Ù&Ú Ë ÛÜYä Ë Ë Ùà áâ å Î Ûã Ë!Ò Ë Ó É ÊEË ËÍÌ Ó (4.63) Using similar methods we can get the full set of relationships between the average RVE strains and the subcell strains. These are Ø Ù&Ú à?ú Ø áú Ë ÛÜ ä Ë ÛÜ ä Ë,ÛÜ Ë Ë Ùà áâ ç ç Ùà áâ è è Ùà áâ å Î å Î å Î Ûã Ûã Ûã É ÊEË ËÍÌ æ É Ê ç ç Ì æ É Ê è è Ì æ (4.64) (4.65) (4.66) and Ù&Ú àú Ø Ù&Ú à?ú áú Ø áú Ë&é Ü Ë é Ü Ë é Ü Ë Ùà áâ ç ç è Ùà áâ Ë Ùà áâ è å Î å Î å Î é ã é ã é ã ç É Ê Ë ç Ìêæ ç É Ê ç è Ìêæ ç É Ê Ë è ÌêÓ (4.67) (4.68) (4.69) The relations between the average strains in the RVE and the average subcell strains can be used to generate relations between the average RVE stresses and the average RVE strains using the traction continuity condition. Gan et al. [119] diverge from the standard GMC formulation at this stage by using subcell equilibrium and compatibility equations to arrive at the stress-strain relations.

59 ( ( ( " Stress-Strain Relations Let the constitutive equation be of the form (in matrix notation) ë,ì.íîïðñóòwôöõ íîïðñ4ëùø í/îï ðñnòûúüíîïðñýþþ (4.70) where ëùì íîïðñ òÿô ë íîïðñ ò ë í/îï ðñ ò ë í/îï ðñ ò uë íîïðñ ò uë í/îï ðñ ò uë í/îï ðñ ò õ íîïðñ ô ë ø íîïðñ ò ôë # íîïðñ í/îï ðñ í/îï ðñ íîïðñ! í/îï ðñ í/îï ðñ íîïðñ í/îï ðñ í/îï ðñ íîïðñ í/îï ðñ í/îï ðñ íîïðñ ò ë# í/îï ðñ ò ë# í/îï ðñ ò ë# í/îï ðñ ò ë# í/îï ðñ ò ë# íîïðñ ò $ ü íîïðñ ô &% íîïðñ % í/îï ðñ % íîïðñ $' Note that we assume that the material is at most orthotropic. Let us express the equations ( ) in terms of the subcell stresses using equation (4.70). Then we have, and ) î+* +, ) ï8*, ) ð*, í/îï ðñ í/îï ðñ í/îï ðñ ë # íîïðñ ë # íîïðñ ë # íîïðñ òûú ò ú òûú í/îï ðñ í/îï ðñ í/îï ðñ ë # íîïðñ ë # íîïðñ ë # íîïðñ òþú ò ú òþú íîïðñ íîïðñ íîïðñ ë # íîïðñ ë # íîïðñ ë # íîïðñ òûú-% íîïðñ ò ú-% íîïðñ òûú-% íîïðñ ýþþ/. ô ýþþ. ô ýþþ/. ô (4.71) (4.72) (4.73)

60 ^ ^ ^ ^ : ; : ; : : ; : ; : L L L V =?> ; =?> D =?> ; D =?> ACB G GH ACB X XH ACB Z ZH IKJ B >L 50 MON1PQ LSRT >L9UWV (4.74) I J B L Y M N PQ L RT L Y9UWV (4.75) IKJ B >Y MON PQ LSRT >Y9UW[ (4.76) From the assumption of traction continuity normal to subcells interfaces, applied in an average sense, we have, I J B > > M N IJ B L L MWN IJ B Y Y MWN I J B]\ > > M N_^ IJ B \ L L MON_^ IJ B D`F Y Y \ MON_^ > > V B < DF L L V (4.77) B Y Y where ^ B < DF L L B Y Y > > are the of normal stresses in the 11 direction, are the of normal stresses in the 22 direction, are the of normal stresses in the 33 direction. Similarly, for the shear stresses, I J B >L IJ B L> IJ B L Y I J B Y L IJ B >Y IJ B Y> M N MWN MWN M N MWN MWN I J B]\ >L IJ B \ L> IJ B \ L Y I J B D`F Y L \ IJ B]\ >Y IJ B D`F Y> \ M N_^ MaN_^ MaN_^ M N_^ MaN_^ MaN_^ >L B < DF L> B < DF L Y (4.78) B Y L >Y B Y> where ^ >L B < DF L Y >Y N_^ N_^ N_^ B < DF L> B Y L B Y> are the of shear stresses in the 12 direction, are the of shear stresses in the 23 direction, are the of shear stresses in the 13 direction. The symmetry of the shear stresses leads to a reduction of one dimension in the subcell stress dependencies for the shear directions as shown in equations ( ).

61 u v u v u u u y y y u u y y y u u u v u u u u u v u v u k k k c dfegh ij c j p d!m`gh c ip dfegh k cld!m`gh ji k c p j dqm8e8h k c pi d!m8ebh 51 k c dngh ij1o (4.79) k c d!m+h j pro (4.80) k c d!e8h ipts (4.81) It is seen from the above assumptions that we can separate the normal components of the equations from the shear components leading to decoupling of the effective normal and shear effects. Let us first look at the normal components of the stress. Substituting equations ( ) into equations (4.77)), we have, m+w ikxzy ebw i x y v gw i xzy d!m8egh i i d!m8egh ji d!m8egh pi cldfegh i i { c dfegh i i { c dfegh i i { dqm8eegh ij dqm8eegh j j dqm8eegh p j c dqm`gh j j { c dqm`gh j j { c dqm`gh j j { dqm8eegh ip dqm8eegh j p dqm8eegh p p We can rewrite equations ( ) in the form u v m+w iy v ebw i y v gw i y d!m8egh i i d!m8egh ji d!m8egh pi Š c dfegh i i { v m+w iky ~ i i7 ˆ c c dfegh i i { ŒŽ c dfegh i i { v ebw i y ~4 ƒ j je ˆ c v gw i y ~ ƒ p pe ˆ c c dqm8e8h p p} k1~ ƒ i i c dqm8e8h p p k ~ ƒ j je c dqm8e8h p p} k ~ ƒ p p d!m8egh ij v m+w i c dqm`gh j j { j j d!m8egh c j j { d!m`gh d!m8egh p j v gw ik ebw i c d!m`gh j j { ŒŽ v m+w ï y d!m8egh ip m+w ik e8w i v gw i cld!m8ebh p p dqm8eegh i i ˆ c o (4.82) d!m8egh j j ˆ c o (4.83) d!m8egh p p ˆ c s (4.84) dqm8eegh i i o (4.85) v e8w i y d!m8egh j p c dqm8e8h p p d!m8egh j j o (4.86) v gw i y dqm8eegh p p c d!m8ebh p p dqm8eegh p p s (4.87) Equation (4.85) can be expressed in matrix form as q i j p š œ k žÿ ƒ i i { l 5 j je { l 5 p pe ˆ c i i o (4.88)

62 Ì Í Ïá Ì Ø Õ Ù á ¾ ¾ Ì Ù Ì Î Á Á À À «Ï Ï Ï Õ Ï Á Ï Ï Ï Ï Ï Ï Á Á Á where $ r! ª «««ª «««W ± ª± «««±³²7 $µ (4.89) ½!À ½ ¾ «««¾ º¹» «««¾». µ (4.90) ¼».... ½. Äa ¾ ¾ «««!À À ÄÅ ¾ «««¾ À ູ» «««¾». µ (4.91) ¼».... ÄÅ. È ¾ ¾ È «««È!À À È È «««È É À ÇÆ ¹» «««]». µ (4.92) ¼».... È È È. ³ Ê «««Ë r ² µ and, (4.93) «««r &Î Î ² (4.94) «««½ +Ï ÄÅ +Ï È +ÏÑÐ The components of the sub-matrices, and in the matrices µ µ and ½ ± are given below. For matrices ÒÔÓ!À À ½ +Ï ÒÖÓ Õ «««Õ À º¹» «««Õ». (4.95) ¼»..... Ï ÒÖÓ Õ Õ «««where ÒÔÓ Ï ÏÞÝ (4.96) Ù+Ú «ÄÅ KÛËÜ For the matrices, à Ó à Ó à Ó!À À à «««É ÄÅ ÇÏ Ó à Ó à Ó À ß¹» «««]». (4.97) ¼».... à. Ó à Ó à Ï Ó à ÏâÝ ³ Ê «««Ë where Ó Ù (4.98) È «ÛCÜ For the matrices, ÏäÐ ÏäÐ ÏäÐ ã8ó ã8ó ã8ó!à ÏäÐ ÏäÐ «««ÏäÐ À É È bïñð ã8ó ã8ó ã8ó À º¹» «««]». (4.99) ¼». ÏÑÐ. ÏÑÐ.. ÏÑÐ. ã8ó ã8ó ã8ó æfá ÏÞÝ å Ê «««C where ã8ó ÏÑÐ á æ ± µ if 1ç èé ëê ; (4.100) ÛCÜ µ otherwise. «52

63 î î í ' ' î î ô ô ô ' ' ô ô í ù ù ' ' ì =< =< ô ô ù ï ô í û ï ï ;: í ï ;: =< î î î G: G G J =< ô ;: =< ;: =< K = K ;: ô =< í ;: =< 2 2 ñ ;: 1 =< 53 The components of the sub-matrices of the ì matrix are The components of the matrices û where û ìlíîëï ð!ñ ò î î íâó ì í ìlíùúï are ð!ñ ò íâó ô ô ð!ñ ò ù ù íâó is a matrix. The components of the sub-matrices ñ ò!ô íâó î î õ õ õ ñ òþö íâó î îø (4.101) ñ ò!ô íâó ô ô õ õ õ ñ òþö íâó ô ô (4.102) ñ ò!ô ù ù íâó õ õ õ ñ òþö ù ù íâó (4.103) ïýü7þ ÿ þ ÿ õ õ õ þ ÿ (4.104) ï ð ö î ò î î î íâó ö î òî î ô íâó õ õ õ ö î òî î ö íâó (4.105) Similarly, equation (4.86) can be expressed in matrix form as, ù ìï î î! #"%$ ô ô are &"(' ) ù ù*,+.-0/ ô ô +.1 ô ô*2 ñ (4.106) For the third normal direction given by equation (4.87), we can get a matrix representation as shown below ù0 ìï56 î î &"(' 7 ô ô &"%8 ù ù,+9- /ù ù +.1 ù ù Combining the three normal direction equations, we get, :; î <=>:; 4 ù 3 ì î ì ì ù ô û ' ' î î0 &" ' $ ' ô ô &" ' ' 8 7 ù ñ (4.107) 1lî î 1 ô ô ù ù (4.108) Since the normal and shear stresses are uncoupled, we can deal with the shear response separately from the normal response. We can write the equations relating the subcell shear stresses to the average shear strains in the RVE in matrix form as :;A B 4C <=>:; ì î ì îù û $ ' ' 7 î D" ' $E8 ' ùf &" ' ' û 8 îùf (4.109) The sub-matrices in the above equation are expanded as shown below. 5A î J õ õ õ J ;IH H ô õ õ õ J J J õ õ õ H ö <K (4.110)

64 R Q R Q R Q R Q R Q R Q O O O U U t t t h h h h Q Q Q Q Q l l h x where LNMPO where f S O where o W O and, STVU S W ^I^ M]\ _ (4.111) W TVUYX[Z jk k k `a Ocbd d ed f U g _?_?_ g g fih _?_?_ g g g _?_?_ f 54 (4.112) S W mim M]\ _ (4.113) W TVU M TVU X[Z o U g _?_?_ g jk k `4n g o h _?_?_ g k Ocbd d. (4.114) ed..... g g _?_?_ o STVU where the vector has S W pip M]\ _ (4.115) M TVU X[Z q U h Osrut U h U h _?_?_ t U hwv!x (4.116) q hiy Osrut hiy hiy _?_?_ t hiywv x (4.117) q U y U r t U y h U y _?_?_ t U y v x (4.118) z { elements. {E} z } O~ h h _?_?_ hƒ (4.119) We thus have a set of equations that relate the subcell stresses to the average strains in the RVE. We now have to develop equations that relate the average stresses in the RVE to average strains in the RVE using equations (4.108) and (4.109). In other words, we need an equation for the composite stress-strain equations of the form, ) 7 Œ Ž. i O5ˆP,Š ti _ (4.120) This equation can be solved for the components of the effective stiffness matrix and the effective coefficients of thermal expansion.

65 Ä Ä ³ ª ³ ³ ª È ³ ³ Ü Õ Ô Õ Ô Õ Ô Õ Ô Õ Ô Õ Ô Õ Ô Ñ š š š Û ³ ª È ³ ³ Î Î 4.3 Effective Thermoelastic Properties For the normal directions, given the global average strains? I 0, I F, 7 Ĩ *, and the temperature change Pš, we can compute the normal stresses. From equation (4.108) we can solve for the šœ variables by inverting the matrix. Let ž be the inverse of the matrix. Then ž is a square matrix of dimension Ÿ % E > %. Expressed in matrix form, ³Ńµ ¹ «««ž cª œ c 4 ± 4²& c 4²± ² ³ À žº«±«žº ž¼² cª ž% &«ž% ± ž½ ±² ž½²#«ž½² ž¾²² 55 (4.121) where, ž½áâ is of the same dimensions as 4Áàthough the elements in each sub-matrix may change. Therefore, the equation (4.108) can be written as ªIÄ žº«±«ž¼«å ž¼² ª ž% &«ž½ ž½ ±² ž½²#«ž¾²ƒ ž¾²² žº«±«ž¼«å ž¼² ž% &«ž½ ž½ ±² ž½²#«ž¾²ƒ ž¾²² ªÇÆÈ žº«±«žº žº«² I 0 # ª ž% &«ž% ± ž% ² ž½²#«ž½² ž½²±² È É Ê I F (4.122) ³ > È žº«±«žº žº«² I Ë È ÊĨ F ŽÌ. PšÍª ž% &«ž% ± ž% ² ª Î I ž½²#«ž½² ž½²±² Ĩ These matrices can be expanded out to get expressions in terms of the components of matrices ž½á Ï ÐÐ µ uâ0. for the stresses š The average composite stress in terms of the subcell stresses is given by Using equation (4.77) we therefore get Ñ uâ Ò Ó Ñ& I cü Ý Ñ± I Ý Ñ±Ĩ cü Ý ÙŒÞ ØÞ ØÞ Ó±ÖŽ Ú]Þ Ú]Þ ÙŒÞ µøwù]ú uâ (4.123) µùfú I ß (4.124) µøàú I ß (4.125) We can then express the average RVE stresses in terms of the average RVE strains as µøwù Ĩ (4.126) â ß â ß â â æ æ â æ Ñ# I áãâ I I 0 & äáåâ ) ) I * # äáåâ ) Pš½Ÿ7áåâ I!æ â I äáãâ )?æ I ¼áãâ ) *æ âĩ (4.127) Ñ I áãâ 0 I 0 & äáåâ I ) I * # äáåâ I Pš½Ÿ7áåâ 0!æ â I äáãâ I?æ I ̽áãâ I *æ âĩ (4.128) Ñ á 0 I & äá âĩ ) I # äá âĩ Pš½Ÿ7á 0 â I äá âĩ I ¼á âĩ âĩ (4.129) where the effective stiffness terms á â are expressed a simple sums of terms of the matrices ž¾á Ï.

66 ì ë í ë û ü û ü û ü ý ÿ ý ÿ ý ë ì ý ÿ ÿ ë í õ õ ë õ õ õ 56 The coefficients of thermal expansion can be found using çè éžêëië îï%ðcçèêñãê ëiëòñãê é ìiì ê ñ ê ñ éžê ìiì ê íií ñãê ñãê íiì ñãê îïnó ñ ìií ê ñãê íií çèiô ë îïàõ ô ì ô í (4.130) where the ô#ö terms are also sums over the components of the matrices ½ø ù. These terms involve complex algebraic expressions and are not presented here. The determination of the effective shear stiffnesses is simpler because of the lack of coupling between the normal and the shear terms. The expressions for the shear terms of the effective stiffness matrix are ñ ê úiú ñ ê ñ ê ð û üþý ð ûü ð ûü ý ÿ ë ë ë (4.131) (4.132) ñ (4.133) Thus the complete set of effective stiffness terms is determined. Equations (4.111), (4.113) and (4.115) show that, and ñ are just volume averages of the shear compliances of the subcell materials. Therefore, the shear stiffnesses predicted by GMC are equal to the Reuss (or harmonic) bounds. Since harmonic bounds do not represent accurate effective shear moduli, as will be shown in Chapters 6 and 7, this feature is a shortcoming of GMC. The shear-coupling approaches developed by Williams and Aboudi [118] and Gan et al. [119] attempt to alleviate this problem. 4.4 Shear-Coupled Method of Cells The generalized method of cells with shear coupling as described by Gan et al. [119] for unidirectional fiber composites in two dimensions is extended to three dimensions in this section. We start with the relations between the average RVE strains and the subcell strains shown in equations ( ). These equations, which are based on the continuity of displacements across subcells, can be written as ëië ð ð ìiì ð íií ë ë ë ÿ ëië ìiì íií (4.134) (4.135) (4.136)

67 : : : : : : ( ( ( ( ( ( : : : 57 "!$# % '& ) * + & * -,.) +0/ (4.137) 5 6 "! %# 7& + * / & *, 6.) +0/ 12 3 (4.138) 6 "! %# '& ) * / & *,.) 6 + / (4.139) In addition to continuity of displacements, GMC assumes that tractions are continuous across subcell boundaries. The shear-coupled method of cells does not require traction continuity across subcells. Instead, subcell equilibrium and compatibility are enforced in an average sense. The equations of equilibrium are <;=?> ; > <;=6@> ;=@> ; > ;D 6@> 6;=6A!CB 3 (4.140) 6 ; 6!CB 3 (4.141) 6;D6 6A!CB 8 (4.142) These equations are approximated using forward differences between average subcell stresses. Thus, ;?.5E ) +0/ 1 2GF, ;?.) +0/ 1 2 >, ;?.) + / 1 E 2GF, ;?.) +0/ 1 2 >, ;?.) 6 + /H1 E 2GF, ;?.) 6 +0/ 1 2!CB 3 (4.143), ;?.5E ) +0/ 1 2GF, ;?.) +0/ 1 2 >, ;?.) + / 1 E 2GF, ;?.) +0/ 1 2 >, ;?.) 6 + /H1 E 2GF, ;?.) 6 +0/ 1 2!CB 3 (4.144), ; 6.5E ) +0/ 1 2GF, ; 6.) +0/ 1 2 >, ; 6.) + / 1 E 2GF, ; 6.) +0/ 1 2 >, ; 6 6.) + /H1 E 2GF, ; 6 6.) +0/ 1 2!CB 8 (4.145) the discretized equilibrium equations are (assuming periodicity of stresses in the RVE) I!KJML 3 If we assume a constitutive equation of the form where the stiffness matrix J is orthotropic, we can express the equilibrium equations in terms of N. E ) + / 1,. E ) + / 1 2GF N.) + / 1,.) + / 1 2 > N.5E ) + / 1,.5E ) +0/ 1 2OF N.) +0/ 1,.) +0/ 1 2 > N 6.5E ) +0/ 1, 6 6.5E ) + / 1 2GF N 6.) + / 1, 6 6.) + / 1 2 > N P P.) + / 1 E,.) +Q/ 1 E 2RF N P P.) + / 1,.) + / 1 2 > N T T.) + /S1 E,.) 6 + /S1 E 2GF N T T.) + / 1,.) 6 + / 12!CB (4.146) N.) + / 1 E,.) +Q/ 1 E 2 F N.) + / 1,.) + / 1 2 > N.) + / 1 E,.) +Q/ 1 E 2 F N.) +0/ 1,.) +0/ 1 2 > N 6.) +0/ 1 E,.) 6 6 +Q/ 1 E 2GF N 6.) + / 1,.) / 1 2 > N P P.5E ) + / 1,.5E ) + / 1 2RF N P P.) + / 1,.) + / 1 2 > N U U.) + /S1 E, 6.) + /S1 E 2GF N U U.) + / 1, 6.) + / 12!CB (4.147) the average subcell strains as

68 { { i [ [ [ [ [ { { i { i i { { o { i { i { i { i { i i [ [ [ { { i { i { i { i { i i [ { { i { i { i { i { i i [ [ [ and VXWZYS[D\ WZYS[D\ VXWZYS[ WZYS[ VMWZYS[h\ WZYS[h\ VMWkYS[ WkYS[ _` ]Q^VacbWkYS[D\ _ _ ]Q^ dge WkYS[D\ _` ] ^VacbWZYS[ _ _ ] ^ dgf WZYS[ i ` ]Q^VajbW5\ YS[ i i ]Q^ doe W5\ YH[ i ` ] ^VacbWZYS[ i i ] ^ dlfwzys[ VMWkY ` ` \]S^ a b ` ` WkY \]S^ d e VXWZYS[ ` ` ] ^ a b ` ` WZYS[ ] ^ d f m m ] ^ a b _` ] ^ d e m m ] ^ a b _` ] ^ d f n n ] ^ acb i ` ] ^ dge n n ] ^ acb i ` ] ^ doocp (4.148) Equations ( ) form a system of qgrts ` equations in the terms of the subcell strains, out of whichqrrvuws ` eyx0z equations are independent. where The compatibility equations are { i ƒ {D {D l _ _ b i ` f i i b _` f ` ` b _i f {D i {D -{DŽ i i o i i b _ _ f ` ` b i i f _ _ b ` ` f i ` b _ _ e _` b i i e _i b ` ` e _ _ b i i e} i i b ` ` e} ` ` b i i e} _i b _` e i ` b _i e _` b i ` e ˆ _ Š ev Š f Œ _ Š ˆ _ Š i ~ e ˆ _ Š _ e Š f Š _ i These equations are discretized using central difference schemes of the form The discretized compatibility equations are 58 _i b _i ocpg~ (4.149) i ` b i ` ocpg~ (4.150) _` b _` ocpg~ (4.151) _` b _i ocpg~ (4.152) _i b i ` ocpg~ (4.153) i ` b _` ocpg~ (4.154) WZY \ WkYS[ WZY W5\ Ÿ [ WZYS[ W YH[ a b _ _ ] ^ doe} ajb _ _ ] ^ dgf a b _ _ ] ^ dgf a b i i ] ^ drev ajb i i ] ^ dgf a b i i ] ^ W5\ YH[ W5\ Y WZYS[ WZY d e a b _i ] ^ doe4a b _i ] ^ dre4acb _i ] ^ dge4ajb _i ] ^ WZY \ W Y \ WZYS[ W YS[ dd Me a b _i ] ^ doe4a b _i ] ^ dre4acb _i ] ^ dge4ajb _i ] ^ dd o p ~ (4.155) WZYS[D\ WkYS[ WZYS[ WZY \ WZYS[ WZY a b i i ]Q^ doe} ajb i i ] ^ dgf a b i i ]Q^ dgf a b ` ` ] ^ drev ajb ` ` ] ^ dgf a b ` ` ] ^ WZY \ WkY [\ WZYS[ WZYS[ d e a b i ` ] ^ d e a b i ` ] ^ d e a b i ` ] ^ d e a b i ` ]Q^ WZYS[D\ WkY [D\ WZYS[ WZY dd e a b i ` ]Q^ doe4a b i ` ] ^ dre4acb i ` ] ^ dge4ajb i ` ] ^ dd o p ~ (4.156)

69 ³ ³ š œ5 žhÿ O } šj œžsÿ0 g š œ žhÿ g š œžsÿ Q R v šj œžsÿ g š œžsÿ Q ª«š œ5 žhÿ O š œ5 žhÿ R šc œžsÿ G šj œžsÿ Q D M ª«š œžsÿ Q O š œ žhÿ R šc œžsÿ G šj œ žsÿ D (4.157) šc œ5 žhÿ G v šc œžsÿ M šc œ žhÿ0 ª±š œž ŸQ R šc œž Ÿ G šc œžsÿ0 G š œžsÿ S ª²š œ5 c R žhÿ š œ5 ž Ÿ š œžsÿ0 š œž ŸQ c ª²šc œžsÿ S R šc œ žsÿ Q G šc œžsÿ0 G š œ žhÿ c t Cḡ (4.158) š œž ŸQ v š œžsÿ š œž Ÿ ª±š œžsÿ S š œ žhÿ š œžsÿ0 š œ žhÿ c ª²šc œž ŸQ R šc œž Ÿ Q G šc œžsÿ0 G š œžsÿ S ª²šc œ5 c R žhÿ R šc œ5 žsÿ Q G šc œžsÿ0 G š œž ŸQ c t Cḡ (4.159) š œžsÿ S v š œžsÿ š œžsÿ Q ª±š œ5 žhÿ š œ5 ž Ÿ š œžsÿ0 š œž ŸQ ª²šc cœžsÿ S R šc œ žsÿ Q G šc œžsÿ0 G š œ žhÿ ª²š c R œž ŸQ š œž Ÿ Q š œžsÿ0 š œžsÿ S c Cḡ (4.160) µ y S if R y¹ ; ¹t if º ;» µ» ¼ S if R» y¹ ; ¹½ if» º ; ¾ µ ¾ y S if X ¾ y¹ ; ¹t if¾ º. where ³ The average RVE strain equations ( ), the equilibrium equations ( ) and the compatibility equations ( ) can be combined into a system of equations relating the subcell strains to the average RVE strains. Unlike the original GMC formulation, the normal and the shear strains are coupled in this formulation through the equilibrium and compatibility conditions. The average subcell strains can be calculated from the applied RVE strains by inverting the system of equations. The subcell stresses can be calculated from the subcell strains using the subcell constitutive equations. As in the original GMC formulation, the average RVE stresses can then be related to the average RVE strain to get the effective stress-strain response. The drawback of the shear-coupled approach is that a much larger system of equations is formed, compared to the reformulated GMC discussed in this chapter. Hence, the method is computationally 59

70 60 expensive. Any computational advantage over finite element analysis based approaches could be lost because of the large size of the matrices that have to be inverted. The requirement that a large matrix be inverted to get the effective properties makes the generalized method of cells very inefficient as the number of subcells increases. When materials such as PBX 9501 are modeled, the number of subcells needed to represent a random distribution of particles necessarily becomes large. In such situations, the method of cells based approaches become inefficient and it may be preferable to perform six different finite element analyses to get the effective properties rather than one method of cells based analysis. This limitation, along with the lack of shear coupling in the original method of cells has led us to develop another micromechanics scheme that we call the Recursive Cell Method (RCM). This method is discussed in the next chapter.

71 CHAPTER 5 THE RECURSIVE CELL METHOD The original GMC technique has been found to provide inadequate shear coupling between adjacent subcells. In addition, the amount of computational time needed to calculate the effective properties using GMC increases dramatically as the number of subcells increases. The recursive cell method (RCM), developed as an alternative to GMC, attempts to resolve these problems without loss in accuracy. A schematic of the recursive cell method is shown in Figure 5.1. The RVE is discretized into subcells as in GMC. However, instead of calculating effective properties of the whole RVE in a single step, the effective properties of small blocks of subcells are determined at a time. The effective properties of the RVE are calculated by combining the effective properties of blocks using a recursive process. The effective properties of each block of subcells may be determined using any accurate numerical technique. We use a finite elements based technique in this research. The RCM recursive scheme has been found to reduce the computational cost and remedy the shear-coupling problem of GMC. Efficient recursion through the subcells requires that the number of subcells per block, in each stage of the recursion, be the same. The first step in the recursive cells method is, therefore, the choice of the number of subcells to be homogenized into a single block for the next homogenization stage. We have chosen blocks of four equal sized subcells for the computations in this research. There is, however, no upper limit to the number of subcells needed to form a block. The only constraint is that if ÁÀG subcells are homogenized into a block, then the RVE has to be discretized so that there are at least =Â subcells on each side whereã is an integer greater than zero. We use a simplified finite element based approach to homogenize each block of subcells. The plane strain assumption is made in the two-dimensional calculations performed in this research. To improve computational efficiency, numerical integration is avoided in our calculations. Instead, explicit forms of the strain-displacement and stress-strain relations are used. These relations and the algebra leading to them are shown in the following sections. After these relations were determined, a large number of validation runs were performed to determine the appropriate boundary conditions to be used in the recursive homogenization process. These boundary conditions are also listed in this

72 62 RVE - Level 0 RVE - Level 1 Final RVE RVE - Level 2 Figure 5.1. Schematic of the recursive cell method. chapter. The RCM technique has been developed for two-dimensional problems so far. However, extension to three dimensions is straightforward. The recursive procedure can be used with techniques other than finite elements such as finite differences or integral equation based methods. However, care should be taken so that the computational efficiency of the recursive procedure is higher than that of explicit calculations using these other methods. 5.1 Subcell Stiffness Matrices Explicit expressions for the stiffness matrix using a displacement formulation have been developed for a four-noded square and a nine-noded square. The nine-noded square element is used in conjunction with a hybrid nine-noded displacement/pressure based element that is used to model nearly incompressible behavior. The explicit form of the stiffness matrix eliminates the need for numerical integrations in the calculations.

73 Î Î Ä Î Ò È Ì Ú Ö ê ì Ö Ó Displacement Based Four-Noded Element A schematic of the four-noded element is shown in Figure 5.2. Since the plane strain assumption is only valid for at most an orthotropic material [11], we assume that the material constituting the element is orthotropic. The nodes 1 through 4 are ordered in a counter clockwise manner. Y 4 3 h 1 2 h X Figure 5.2. Four noded element. where Ä Ä?ÅÇÆQÈ<ÉSÊ@Ë ÍÎ ÏÑÐ=Ò Î Ì ÓÔÅÇÆ È<ÉSÊ@Ë ÍÎ ÏÑÐÕÒ Î Ì The displacement functions for this element, in isoparametric form, are functions Ò Î andó ÎÕÖ (5.1) are the displacements in the Æ and É directions at node, respectively. The shape Ò Ð Ë Å ØAÙÆ«ÊÅ Ø ÙvÉSÊ Ú È (5.2) ÒgÛ Ë Å ØÝÜÞÆ«ÊÅ Ø ÙvÉSÊ Ú È (5.3) Ògß Ë Å ØÝÜÞÆ«ÊÅ ØàÜáÉSÊ Ú È (5.4) Ë Å ØAÙÆ«ÊÅ ØàÜáÉSÊ (5.5) î â4ë ä åôæ5çç ã éæ è<è çè-ê ë ì Ë ä îjïä ííííííã î Ó åíííííí î Ä îdð íííííí (5.6) îjð Ü îdï î Ó ííííííë are given by The strain-displacement relations are

74 OOOOOOOOOO ; These relations can be written in terms of the nodal displacements (ñhòôó õsò ) as öø CùRú ó (5.7) where úû ýüñÿþ õ±þ$ñ õ ñ õ ñ õ ó and, (5.8) ù' The stress-strain relations are!#"%$'&(& $ )) * &() where 5 is the orthotropic stiffness matrix,.7 +#, - /.!#"102&(& 402)3) &() 6 þ þ 6 þ 8 6 þ (5.9) +#, - ó (5.10) (5.11) For the special case of isotropy (9 and : are the Young s modulus and Poisson s ratios, respectively),.º 9 : : : < ; >= : < : : þ2? 2@ (5.12) The element stiffness matrix is given by A CB þ B þ?=þ?=þ ù.gùedgf(hgikj J ó (5.13) wherei ;ML is the Jacobian matrix relating the ó2ng< ; coordinate system to the ó < coordinate system. Performing the integration, we get = P Q 9 R P SQ T = 4 R R U SQ = P SQ T 4 R V R WP Q A = 4 R V Q = P Q 4 X 9 R NZY[Y = 4 = PR SQ U ]\\\ \\\\\\\ (5.14) = 4 P where ^ ; 6 þ þ _6727 <"ó Q ; 6 þ _6727 <"ó 4t ^ ; 6 2 _6727 <"ó R ; 6 þ `6727 <"ó T ^ ; 6 þ þ `=a6727 < ó 9 ^ ; 6727>=a6 þ þb<"ó U ^ ; 6 2 >=a6727 < ó V ^ ; 6727c=a6 2 < It may be noted that there is no dependence of the stiffness matrix on the element size or location.

75 o d o h v o v Displacement Based Nine-Noded Element A schematic of the nine-noded displacement based element is shown in Figure 5.3. This element is used in conjunction with the nine-noded displacement/pressure based hybrid element used to model the nearly incompressible binder of PBXs. The material of the element follows the stressstrain relations shown in equations (5.10), (5.11) and (5.12). In this case, the displacement functions Y h h 2 X Figure 5.3. Nine noded element. for the element are dfemg hi jlk noqpsrut m emg hi jlk noqpsrwt m owx emg (5.15) The shape functions used for this element are (using an isoparametric formulation where the local co-ordinate system [120]) hi j is

76 y{z ~} c ƒ } ` ˆƒ (5.16) y{ } ` ƒ } Š y{z (5.17) y Œ } c ƒ } Ž_ y{z (5.18) y{ } ` ƒ } Ž y z (5.19) y{ } c ƒ } ` y{z (5.20) y W } c a } Ž y Œ y{ y{z (5.21) y{ } ŽE a } Ž y Œ y{ y{z (5.22) y ƒ } ŽE a } > y y{ y{z (5.23) yš } c a } > y y y z (5.24) The stiffness matrix for this element is obtained using equations (5.6) and (5.13) and performing the integrations. The explicit form of the stiffness matrix of the nine-noded element is shown in the following page 66

77 Á ¹ ¹ ¹ ¹ ¹ ¹»»º À 67 (5.25) «ª ª «ª ÂuÂ1ÂuÂuÂ1ÂuÂ1ÂuŸÂuÂ1ÂuÂuÂ1ÂuÂ1ÂuÂ1ÂuŸÂuÂ1ÂuÂuÂ1ÂuÂ1ÂuÂ1ÂuÂuà ª ª ¹ À µ ª ¹ ª ¹ ª ª µ ¹ ¹ ² ª ª «¹ µ ª ¹ ¹ ² ª ««¹ ª ¹ ² «µ «ª ª ª ¹ ² «µ ª ª ³ ²µª ± ½¾ ² ª ± ¼ ± ¼ ª ª µ ² ³ ± ² ª ± ± žuž1žužuž1žuž1žužÿžuž1žužuž1žuž1žuž1žužÿžuž1žužuž1žuž1žuž1žužu Mœ

78 Ô È È ÈÔ È 68 where the following substitutions have been made : Ä Å Æ ÈÉ'Ê±Ë Ç Å2ÅÍÌ ËÎ2ÎˆÏ ÄÑÐÆ ÈÉ Ê±Ë Ç Å2ÅlÒ ËÎ2ÎˆÏ Ó ÐÆÖÕ Ê±Ë Å ÐSÒ ËÎ2ÎˆÏ Ø ÅÆ ÈÉ Ê±Ë Ç Ð2ÐlÌ ËÎ2ÎˆÏ Ù ÅÆ ÈÉ Õ Ê±Ë Å2ÅÍÌ È ËÎ2ÎÚÏ Ù ÐÆÖÕ Û ÈÉ Ê±Ë Å2ÅlÒ È ËÎ2ÎˆÏ ÙÝ ÆÖÕ Û ÈÉ Ê È Ë Å2Å Ò ËÎ2ÎˆÏ Þ ÅÆ ÈÉ Ç Ê È Ë Å2ÅÍÌ_ß ËÎ2ÎˆÏ Þ Ü Æ ÈÉ Ç Ê ß Ë Å2ÅÍÌ È ËÎ2ÎˆÏ Þ Ý Æ Ô à Õ Ê ß Ë Å2Å Ò È ËÎ2ÎˆÏ á ÐÆ ÈÉ Õ Ê Õ Û Ë Å2Å Ò`ß ËÎ2Î Ï â ÅÆ ÈÉ Õ Ê±Ë Ð2ÐlÌ È ËÎ2ÎˆÏ â Ü Æ ÈÉ Õ Ê È Ë Ð2Ð Ì ËÎ2ÎˆÏ â Ý Æ ÈÉ Õ Û Ê È Ë Ð2ÐÒ ËÎ2Î Ï ã ÐÆ Ô à Õ Ê È Ë Ð2ÐÒ`ß ËÎ2ÎˆÏ ã Ü Æ ÈÉ Ê Ç È ËÎ2Î Ì_ß Ë Ð2Ð Ï ä Å Æ ÈÉ'Ê Õ ß Ë Ð2Ð Ò Õ Û Ë Î2Î Ï ä Ð Æ ÈÉ Ê Õ Õ Û Ë Ð2Ð Ò`ß Ë Î2Î Ï å ÐÆ ß Ä Å å Ü Æ Ó Å Õ Û åˆæ Æ Ó Å Õ Û å Î Æ Ó Ð åˆè Æ ß Ø Å Èêé Ó ÅÆ Ê±Ë Å ÐlÌ ËÎ2Î Ï Ø ÐÆ ÈÉ'Ê±Ë Ç Ð2ÐSÒ ËÎ2ÎˆÏ ÙÑÜ Æ ÈÉ Õ Ê È Ë Å2ÅÍÌ ËÎ2Î(Ï Þ ÐÆ Ô à Õ Ê È Ë Å2Å Ò ß ËÎ2ÎˆÏ á Å Æ ÈÉ Õ Ê ß Ë Å2Å Ò Õ Û ËÎ2ÎˆÏ â ÐÆ ÈÉ Õ Û Ê±Ë Ð2ÐSÒ È ËÎ2ÎˆÏ ã ÅÆ ÈÉ Ç Ê ß ËÎ2Î Ì È Ë Ð2Ð Ï ã Ý Æ Ô à Õ Ê ß Ë Ð2ÐÒ È ËÎ2ÎˆÏ å Å Æ Ä Å Õ Û å Ý Æ Ó Å åúç Æ Ø Å Õ Û The stiffness matrix is, like the four noded element, independent of the location and size of the element Mixed Displacement-Pressure Nine Noded Element The binder material used in PBXs is nearly incompressible. This implies that the Poisson s ratio of these materials is close to 0.5 and hence the bulk modulus is large compared to the shear modulus. Hence, the volumetric strain is small and is equal to zero in the limit of incompressibility. The strain is determined from derivatives of displacements. In finite element formulations, the derivatives of displacement are less accurately determined than the nodal displacements. Therefore, any error in the predicted volumetric strain for nearly incompressible materials will lead to large errors in the predicted stresses. Since the external loads are balanced by the stresses, this also implies that the predicted displacements will be inaccurate unless an extremely fine mesh is used. In practice, the displacements predicted by displacement based finite elements for nearly incompressible materials are much smaller than those expected [120]. This behavior is called element locking.

79 ø ü í ñ ø ø þ 69 The problem of element locking can be avoided by using a nonlinear material model such as the Mooney-Rivlin rubber model for the binder. However, a finite element representation of this model requires that the load be applied in multiple steps. For isotropic linear elastic materials undergoing small strains, a displacement and pressure based mixed formulation is adequate [120]. We use a mixed formulation presented by Bathe [120] to model the subcells containing the binder material. The basis of the formulation is the Wu-Hashizu functional form of the principle of virtual work. The Wu-Hashizu functional can be expressed as a sum of volumetric and deviatoric strain ë'ìîíˆïðñóòzôöõ`ëgìî ìùø òzôûúýü8þ ñÿú þ í ð ú í õ ì þ energies and equated to the external virtual work as where í ð ì is the deviatoric strain matrix, is the deviatoric stress matrix, is the volumetric strain, is the hydrostatic pressure, is the external virtual work, is the stress matrix, is the Kronecker delta. is the strain matrix, and, (5.26) (5.27) (5.28) ë ì ì ø ò%ôûú In addition, the volumetric strain and the hydrostatic pressure are related by where ø is the bulk modulus, and, is a weighting function. (5.29) Finite element displacement and pressure interpolation functions for the element are chosen of the form ú ø ú (5.30) (5.31)

80 ?? H I I 5 5 H 70 where is the element displacement vector, are the nodal displacement degrees of freedom, are the element pressure degrees of freedom. are the displacement shape functions, and, are the pressure shape functions. The volumetric strain is given by the sum of the strains in the two coordinate directions and is related to the displacements by! "$#%#'&($)*)+-,/. &1,32 (5.32),/0,/465 where. and 2 are the displacements in the 0 and 4 directions, respectively. The strain-displacement relations for the deviatoric strain components are 798 ;;;;;;;: < $#%#>= B *? Then we can form the following relationships EGF EGFF F?,/. =KJ?,/2 F F F FF,/0,/4 F F F?,/2 =? J,/. F F ;;;;;;;;;;:,3.,/4 &,/2,/0 FFF L (5.33) < =J?,/4,/. =KJ,30?,/2,/0,/4 7M8N 8 (5.34)!NO L (5.35) Let the relationship between the deviatoric stress and the deviatoric strain be PRQ (5.36) where Q 8 is the deviatoric stiffness matrix. For an isotropic, nearly incompressible material with shear moduluss, the deviatoric stiffness matrix is ;;: I S T T T EGF Q 8 < T I F S T T T T S T T T T I H L (5.37) S

81 f ~ z [ f { c z f f p o z j [ { p o 71 Application of the principle of virtual work leads to a system of equations relating the nodal displacements and element pressures to the external applied loads. This system of equations can be written as U WVV X3YOZ\[]'^ ] Z ]`_ba c X3Y e f[ Z Y _ha We can write this equation in more compact form as Statically condensing out the pressure terms, we have, c XdYOZ\[Yegf _ba e f c XhY [ eif _ba>kgl mrn%o l qsrit nvuw ryx (5.38) f@f nvz {}{ z { r n qsrit o n uw ryx (5.39) f@f f ƒ p z {{ z { z o t u x (5.40) Equation (5.40) is an equation that can be solved for the displacements. The pressures do not have to be determined explicitly. The nine noded displacement/pressure element with three pressure degrees of freedom (also called a 9/3 u-p element) has been proven to avoid element locking [120]. We have, therefore, chosen this element for the RCM calculations on subcells containing the binder material. The 9/3 u-p element has the same geometry and node numbering scheme as that of the nine noded element shown in Figure 5.3. The displacement interpolation functions are those shown in equations ( ). The pressure interpolation function is chosen to be ˆ Š$Œd t Ž Therefore, in terms of the isoparametric coordinates Š 9, we have, f@f eif ˆ *Œ x (5.41) t šœ } x (5.42) The three pressure related degrees of freedom are internal to the element. The matrices z{}{, and can be determined by explicit integration. After performing z { z the integrations and inserting the resulting matrices into equation (5.40) we get the explicit form of the stiffness matrix for the mixed nine-noded displacement-pressure element. The form of this stiffness matrix is shown in equation (5.43).

82 72 The following substitutions have been made in equation (5.43). ž Ÿ 9 '! ª/«` (/«Ÿ '! ª ±d² Ÿ }³ }µ ± Ÿ ª ª³ } }³ ± ¹ Ÿ }³ ³ º ±» Ÿ ³ } ³ º ± ¼ Ÿ 9³ º ªµ ± ½ Ÿ }µª³ } }³ ± ¾ Ÿ }³ º µ º ± À Ÿ }³ ª ± Á Ÿ }µª³ }µª³ ± ²Ã ŸÄ ³ }³ ± ²$² Ÿ }³ ± ²Ã Ÿ ³ ³ ±d²ã¹ Ÿ ³ ( i ±h²å» Ÿ µ!µª 6 ±h²ã¼ Ÿ }³ (ḧ 6 ±d²ã½ Ÿ }³> Æ ª i ±h² ¾ Ÿ Æ ª } 6 ±h² À Ÿ ³Ç ( ³ i ±d² Á Ÿ }³ 6 ± $ Ÿ } Æ ª } i ± Ȳ Ÿ (³ i ±É $ Ÿ }³ 6 ± $¹ Ÿ ³ ËÊ The above relations show that the stiffness matrix depends only on the material properties and not on the element location and size. After the stiffness matrices have been calculated, they can be assembled in the usual manner.

83 âã Ø âã âã Ø Ø âã âã Ø âã âã Ø Ø âã ä Ø âã ä Ü ä ä Ü ä Ø âã ä Ü ä æ æå 73 ë Ó Ô ì ìñì ì ìñì ìñì ì ì ìñì ì ìñì ìñì ìñì ì ì ìñì ì ìñì ìñì ìñì ì í Ó Ô Ó Ô Ó Ô Ó Öá Ó Ô Ó Öá Ó Ô Ó Ô Ó ØÝ Ô Ó Ô Ó ÜÔ ä Ó Ô Ó Öá Ó Ôá Ó Ôá Ó Ô Ó Þ Ó ÖÖ Ó ØÔ Ó ÖÖ Ó Ô ä Ó ÔÚ Óà ÜÝ Óà Ü ÓÕß ÓÙÔ Ø ÓÕÔÞ Ó ØÝ Ô Ó Ô ÓÕÔÖ ÓÕÖ ÓÕÔÔ Ó Ôá Óà Ü ÓÕßÝ ÓÕÞ ÓÙØ ÓÙÜÝ ÓÕÛ ÓÕÚ ÓÙØ ÓÕ ÓÕÖ ÓÕÔ Ó ÞÝ Ó ß ÓÃÜ à ÓÙÔáÝ Ó ÔÔ Ó Ö Ó ÔÖ ÓÙÔ ÓÙ ÓÃØ ÓÙÚ ÓÙÛ ÓÃÜ ÓÃØ ÓÙÔÝ ÓÕÖ Ý ÓÕÔÔÝ ÓÕÔ Ó ÔÖÝ ÓÕß ÓÕÞÝ ÓÕÛ ÓÕÚÝ ÓÙØ ÓÙÜ ÓÕÖ ÓÕÔÝ Ó ÔÔ Ó Ö ÓÃÜÝ à ÓÙÔáÝ Ó ÞÝ ÓÙß Ó ÔÖÝ Ó Ô ÓÙÚÝ ÓÙÛ ÓÃØ ÓÙ Ý ÓÙÔÝ Ó ßÝ ÓÕÞ Ó Ô Ó ÔÖ Ó Ö ÓÕÔÔ ÓÙØ ÓÕ ÓÕÖ ÓÕÔ Ó ÔÔÝ ÓÕÖ ÓÕÔÖ ÓÕÔ Ó Þ ÓÕß Óà Ü ÓÕÔáÝ ÓÙÜ ÓÙØ ÓÕÔÝ ÓÕÔÖÝ Ó ß ÓÕÞÝ ÓÕÔá Óà ÜÝ Ó Ö Ý ÓÕÔÔÝ ÓÕÖ ÓÕÔÝ ÓÕÔÖÝ ÓÕÔ Ó ÔÔÝ Ó Ö Óà ÜÝ Ó ÔáÝ ÓÕÔÝ (5.43) ä ÓÙÔÛ ÓÙÖÖ Ó ØÝ Ô ä ÓÙÔÚ âã ÓÕÔß ä Ó Ô Ï Ð ÐÑÐ Ð ÐÑÐ ÐÑÐ Ð Ð ÐÑÐ Ð ÐÑÐ ÐÑÐ ÐÑÐ Ð Ð ÐÑÐ Ð ÐÑÐ ÐÑÐ ÐÑÐ Ð Ò Î Ì Í Ó ÔÞÝ Ó Öá ä ÓÕÔß âã ÓÕÔ Ø ä Ó ÔÞ Ó ÔÛ ä ÓÕÔ Ø ÓÕÖÖ Ó ÔÚ ÓÕÔ ØÝ ä ÓÕÔÚ ÓÙÔÞÝ éê èç ç ä Ó ÖÔ Ó ÖÔ

84 5.2 Modeling a Block of Subcells The present implementation of the recursive cells method performs finite element calculations on blocks of four subcells at a time. A schematic of such a block is shown in Figure 5.4. Y 7 4 îîîîîîîîîîîîîîîî îîîîîîîîîîîîîîîî îîîîîîîîîîîîîîîî îîîîîîîîîîîîîîîî îîîîîîîîîîîîîîîî îîîîîîîîîîîîîîîî îîîîîîîîîîîîîîîî îîîîîîîîîîîîîîîî ïïïïïïïïïïïïïïïï ïïïïïïïïïïïïïïïï ïïïïïïïïïïïïïïïï ïïïïïïïïïïïïïïïï ïïïïïïïïïïïïïïïï ïïïïïïïïïïïïïïïï ïïïïïïïïïïïïïïïï ïïïïïïïïïïïïïïïï ðððððððð ðððððððð ðððððððð ðððððððð ðððððððð ðððððððð ðððððððð ðððððððð ññññññññ ññññññññ ññññññññ ññññññññ ññññññññ ññññññññ ññññññññ ññññññññ 3 4 òòòòòòòò òòòòòòòò òòòòòòòò òòòòòòòò òòòòòòòò òòòòòòòò òòòòòòòò òòòòòòòò óóóóóóóó óóóóóóóó óóóóóóóó óóóóóóóó óóóóóóóó óóóóóóóó óóóóóóóó óóóóóóóó ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ X Figure 5.4. A four subcell block modeled with four elements Each block is periodically repeated in space to form a composite material. If the effective properties of this material are to be determined, periodic boundary conditions have to be applied to the block to set up the required finite element analyses. If four elements are used to model the four subcells in a block as shown in Figure 5.4, forcing displacements on the boundary of the block to be periodic leads to forces that are not periodic on the boundaries. This is because the elements on opposite sides of the boundary can have different stiffness and may require different forces to achieve the same displacement. Three types of displacement boundary conditions are applied for the finite element analyses - two normal displacements in the two coordinate directions and a shear displacement in the plane. Note that along the boundaries where displacements are not applied, the forces sum to zero though they may not be zero at any of the nodes on that boundary. Both the forces and the displacements can be forced to be periodic if sixteen finite elements are used to model the four subcell block as shown in Figure 5.5. In this case, the elements on opposite sides of the boundary have the same stiffness and therefore require the same force to achieve a given displacement. Similar approaches are used with nine-noded elements. It should be noted that the nine-noded displacement-pressure elements are used only for subcells containing the binder. Once new material

85 Y üüüüüü üüüüüü üüüüüü üüüüüü üüüüüü üüüüüü ýýýýýý ýýýýýý ýýýýýý ýýýýýý ýýýýýý ýýýýýý öööööö öööööö öööööö öööööö öööööö öööööö þþþþþþþþþþþ þþþþþþþþþþþ þþþþþþþþþþþ þþþþþþþþþþþ þþþþþþþþþþþ þþþþþþþþþþþ ÿÿÿÿÿÿÿÿÿÿÿ ÿÿÿÿÿÿÿÿÿÿÿ ÿÿÿÿÿÿÿÿÿÿÿ ÿÿÿÿÿÿÿÿÿÿÿ ÿÿÿÿÿÿÿÿÿÿÿ ÿÿÿÿÿÿÿÿÿÿÿ úúúúúú úúúúúú úúúúúú úúúúúú úúúúúú úúúúúú úúúúúú ûûûûûû ûûûûûû ûûûûûû ûûûûûû ûûûûûû ûûûûûû ûûûûûû øøøøøø øøøøøø øøøøøø øøøøøø øøøøøø øøøøøø øøøøøø ùùùùùù ùùùùùù ùùùùùù ùùùùùù ùùùùùù ùùùùùù ùùùùùù X Figure 5.5. A four subcell block modeled with sixteen elements. properties have been generated for a block containing the binder, the displacement based nine-noded element is used for all further recursions. The stiffness matrix for each of the elements can be calculated using the explicit forms shown in the previous section. These matrices can then be assembled by superposition to form the global stiffness matrix for this problem. Explicit forms of the global stiffness matrix for the four-element model and the sixteen-element model have been obtained using Maple 6. These matrices become small enough, after application of boundary conditions, that explicit solutions for the nodal displacements can be obtained. However, such explicit forms are extremely complex for the nine noded elements and it is easier to calculate the element stiffnesses and to assemble them numerically. In the present implementation of the recursive cell method, the global stiffness matrix is assembled numerically. After the global stiffness matrix for a block of subcells has been determined, the appropriate boundary conditions are applied to obtain the displacement solution that, in turn, leads to the effective elastic moduli for the block. 5.3 Boundary Conditions The finite element problem involves the solution of a set of linear simultaneous equations relating the displacements to the applied forces. This system of equations can be written as

86 : :! #"%$'& )( +*-, &.0/ ;: 76 (5.44) The stiffness matrix is singular, and the set of equations can only be solved upon the application of suitable boundary conditions. Three sets of boundary conditions are applied on the finite element representation of the four-subcell block so that the two-dimensional effective properties of the block can be calculated. These are 1. a uniform normal displacement in the < direction ( 1 direction), 2. a uniform normal displacement in the = direction ( 2 direction), and, 3. a shear displacement in the <>= -plane ( 12 plane). A schematic of the four element based model of a block of subcells undergoing a normal displacement in the < direction is shown in Figure 5.6. The figure shows the original shape and the deformed shape and the corresponding locations of the nodes. A uniform displacement? is applied to nodes 3, 6, 9 and node 1 is kept fixed. Nodes 2 and 3 are not allowed to move in the = direction. Similarly, nodes 4 and 7 are not allowed to move in the < direction. Nodes 7,8 and 9 are constrained to move an equal amount in the = direction. The pair of nodes 2 and 8 are constrained so that they move an equal amount in the < direction while nodes 4 and 6 are constrained so that they move an equal amount in the = direction. The applied displacement? and the fixed displacements at nodes 1, 2, 3, 4 and 7 are called the prescribed displacements. The constrained displacements are described by constraint equations. The algebra used to apply constraint equations and the prescribed displacements is discussed in sections and respectively. In equation form, the prescribed displacements for the situation shown in Figure 5.6 are ( "@*-ACB DE"@*FACB DHGI*FAJB (LK *? B DKM*-ACB (LN *FACB (LO *? B (QP *RASB (LT *? The constraint equations for this case are (LU9VW( GM*-ACB D O V D N *-ACB D UXV D P *FACB D T9V D P *FA The effect of a uniform displacement applied in the = direction on the positions of the nodes is shown in Figure 5.7. Boundary conditions similar to those for a displacement in the < direction apply to this case too. Note that the constraint equations are used to satisfy periodicity of the displacements. These constraints lead to stress states that are not purely unidirectional. However,

87 Y X Figure 5.6. Schematic of the effect of a uniform displacement applied in the Y direction Y X Figure 5.7. Schematic of the effect of a uniform displacement applied in the Z direction. for the materials under consideration, the deviations of the stresses from a unidirectional state of stress are small. The application of a pure shear displacement is more problematic. Two schemes have been examined for this process. The first scheme involves prescribing displacements that correspond

88 78 to a pure shear at the boundary nodes. A schematic of this process is shown in Figure 5.8. In this approach, node 1 is fixed and node 9 is assigned displacements of magnitude []\9^_[)` in the a and b directions. Node 3 is assigned a displacement []\ in the a direction and a displacement [)` in the b direction. Similarly, node 7 has prescribed displacements of [)` in the a direction and []\ in the b direction. The nodes on the boundary that are between the corner nodes are assigned displacements such that the boundaries remain straight lines. The values of [ \ and [ ` are chosen so that they correspond to a pure shear displacement. Application of such boundary conditions leads to relatively high stresses in the a and b directions and a relatively stiff response Y X Figure 5.8. Schematic of the effect displacements, corresponding to a pure shear, applied at the boundary nodes. An alternative to this approach of application of shear displacement boundary conditions is shown in Figure 5.9. In this case, the displacements are prescribed only at the corner nodes while the other nodes on the boundary are constrained so that they maintain periodicity. Thus nodes 2 and 8 are constrained to have the same displacements in the a and b directions with node 8 being allowed an additional displacement corresponding to the shear displacement. A similar constraint equation relates the displacements at nodes 4 and 6. This approach is used for the calculations shown in Chapter 6 and 7. The normal stresses generated using this type of shear displacement boundary condition are much smaller than with the previous approach. However, when 9/3 u-p elements are used unrealistic displacements may be obtained at node 5 which do not occur when the first approach for applying shear displacements is used. This issue is currently being explored. The prescribed shear displacements for the approach shown in Figure 5.9 are

89 z Y X Figure 5.9. Schematic of the effect displacements, corresponding to a pure shear, applied at the corner nodes. ced@ffgjh ied@ffgch cqjmf-k]d@h ihjmf-k)lmh cqmif-k)l+h inmif-k]d@h cqomf-k]dqprk)lih ihomf-k]dqp7k)lis The corresponding constraint equations are cqtxuvc>w+f-k]d9h ihtxuwiwmf-k)l+h cqxxuwcllif-k)l+h ihxxuwihlif-k]d9s The application of constraint equations and prescribed displacements to the finite element system of equations shown in equation (5.44) is discussed in the following sections Application of Constraint Equations An equation that relates the displacements of two nodes is called a constraint equation. For example, for the case shown in Figure 5.6 a constraint equation is cll9uwcqxmffg where, c l is the displacement in the y direction at node 2, and, clx is the displacement in the y direction at node 8. In this case, cll is the prime degree of freedom since it has a coefficient of +1. There can be many such constraint equations. In general form, these constraint equations can be written as, {!} d%~ c f ~) (5.45)

90 Š ± ± ± ± ± ± ± ƒ ¼ ± ¼ 80 where 9ƒ when the prime degree of freedom is >. We do not have to divide the ˆ values by to get to this form of the constraint equation. This is because the constraint equations applicable for the three sets of boundary conditions used in the RCM calculations automatically satisfy the requirement that! #ƒ. Using the Lagrange multiplier technique, the original set of equations can then be reduced by one to get a set of equations of the form : Ž ' ] I ˆ!Œ# 0 še E Mƒ_œ ) œž Ž Ÿ ƒr š (5.46) Repeated application of this approach for each of the constraint equations gives us a set of equations with the redundant degrees of freedom removed. If there are constraint equations, the reduced system of equations can be written as Š> >Š ' Ž +ƒ-œ Cª4«5ª7 š (5.47) Œ# Application of Specified Displacements The set of equations remaining after the constraint equations have been applied and the redundant displacements removed from the equation, can be written in matrix form as ± ƒf² (5.48) If we decompose the matrix into parts that are related to the specified (subscript ³ ) and the unspecified (subscript ) displacements, and the force vector into the applied (subscript ) and the reaction (subscript µ ) forces, we have, º¹»½¼ ¾ ²À ² ¾ ²ÂÁ ² Á ¾ (5.49) ± The specified displacements are known. Hence, the matrix equation ¹ Ã Ä ƒ-² r² Á (5.50) is redundant. Therefore, we only need equations à ƒf² ² Á (5.51) to determine the unknown displacements ±. Now, the reactions at the points where no displacements are specified are zero, i.e., Therefore, ² Á ƒ-å (5.52) ƒf² (5.53) After the constraint equations and the prescribed displacements are applied, the unknown nodal displacements can now be obtained from the reduced system of equations. It should be noted that

91 81 there are still some unknown nodal forces in the expressions for the force vector because of the constraint equations. These can be set to zero if we assume that the average forces are zero. For the four element model subjected to a uniform normal displacement in the Æ direction (shown in Figure 5.6), we set Ô Ì ÇžÈÀÉËÊrÇžÈÂÌ9Í-ÎCÏ ÇžÐÒÑ;ÊrÇžÐÒÓXÍFÎCÏ ÇžÈÀÔ@ÍFÎJÏ ÇžÐ Í-ÎCÏ ÇžÐ Õ;ÊrǞРÊrÇžÐÒÖXÍFÎC where Ç È and Ç Ð are the nodal forces in the Æ and Ø directions respectively. The subscripts 2,4,5,6,7,8 and 9 refer to nodes at which the forces are applied. Similar equations are used when a uniform normal displacement is applied in the Ø direction. For the four element model subjected to displacements that correspond to a pure shear (shown in Figure 5.9), we again assume that the constrained nodal forces average to zero, i.e., Ç Ç Ç Ç ÈÀÉ ÊrÇ ÈÂÌ ÍFÎJÏ Ð É ÊrÇ ÐÒÌ Í-ÎCÏ ÈÂÑ ÊrÇ ÈÂÓ ÍFÎJÏ ÐÒÑ ÊrÇ ÐÒÓ ÍFÎJÏ ÇžÈÀÔ9ÍFÎJÏ ÇžÐ Ô@Í-ÎC Once the unknown forces have been removed using the above procedure, the system of equations can be solved for the unknown displacements. We use Gaussian elimination to solve for the displacements. This is in order to eliminate any problems due to ill conditioning of the stiffness matrix which may occur because of the large modulus contrast between the particles and the binder in PBX materials Calculating Volume Averaged Stresses and Strains The effective stiffness matrix ÙSÚ of a block of subcells can be obtained from the relation Û ÜMÝ Í Û ÞHÝ Ù Ú (5.54) where Û ÜMÝ is the volume averaged stress in the block, and, Û ÞHÝ is the volume averaged strain in the block. A block is modeled using either four or sixteen elements as shown in Figures 5.4 and 5.5. Since the elements are all the same size, the volume averaged stress or strain in a block is equal to the ensemble average of the average stresses or strains in each element.

92 ã " ë ë +* " ï ë ë ï 82 The volume averaged strain and stress in an element are given by ß àhá âäã åçæéè ß íiá â ã åçæ è àxê åìë íîê åðï (5.55) (5.56) For the four noded element, using the strain-displacement relations and integrating over the element we can determine the volume averaged strains in an element. These explicit expressions for these are ã ã ß ñòòóòhá â ôhõ ö0 Iøeù#úûøLü ú ølý9 Wø>þ]ÿ ß ñ á â ôhõ ö0 ù Hü ú Hý ú þ]ÿ ß ñˆò žá â ôhõ ö0 Iøeù WøLü ú ølý úûø>þi ùqú ü ú HýX þàÿ (5.57) (5.58) (5.59) The average stresses can be obtained similarly from the stress-strain relations. The expressions for the average element stresses are ß òóò á â ã ôhõ ß á â ã ôhõ ß%ò á â ã ôhõ ö ü ú ý ÿ)ö ý ù ÿãú-ö ù ý ÿ)ö ø ü Wø þ žþ+ ølýx WøÃù ÿ ö úâÿ)ö ým Eù Àÿ)ö ølüx WøLþ]ÿˆú öx Âÿ)ö žþ+ úâÿ)ö ølýx WøÃù ÿ ö úâÿ)ö ým Eù Àÿ)ö ølüx WøLþ]ÿˆú öx Âÿ)ö žþ+ úâÿ)ö ølýx WøÃù ÿ (5.60) (5.61) (5.62) For a RVE composed of many such elements, an arithmetic average of the element average stresses and strains can be taken to calculate the volume average over the RVE. Similar expressions for the average stresses and strains can be obtained for the nine noded elements Calculating Effective Properties The effective properties are related to the volume average stresses and strains by the following relation : ß òóò á ß Há ß)ò žá! â$#% '& ùˆù ù ü & '& ù (& ù ü üˆü & (& ü) '& ù ü) & '& ) ß ñ òóò á ß ñ Há ß-,Eò žá! (5.63) To solve for the six components of the stiffness matrix, we need six independent equations relating the volume averaged stresses to the volume averaged strains.

93 = 2 2 = 3 ]\ ] ] 7 0 ] ] 7 ; ] 7 R ^ 0 R ; R 0 ; ; ; R R ; R ; ; R R ; ; ; R R R ; R 0 ; ; ; R R ; R R ; R 0 c a a a a a 83 For the case where a normal displacement is applied in the. direction, we have /10 2(354'6 0)087 2:94(6 0;<7 =>94(6 0? 7 / = ; ;); 6 7 = 94 ;)? 6 7 / 2 = ? ;)? 6 7 = 94 ;)? 6 7 where / 2 0 3A@B 2 21CED / = 0 3F@B =8=GCED / 2 = 0 3F@H 2 =ICED 7 2 = 0 3L@M 2 = CON When a normal displacement in applied in the P direction, we have / 2; 354 0) ; 6 7 = 94 0? 6 7 / = ; 354 0; ;); 6 7 = 94 ;)? 6 7 / 2 = ; 354 0? ;)? 6 7 = 94 ;)? 6 7 where / 2 ; 3A@B 2 2 CED / = ; 3F@B =8= CED / 2 = ; 3F@H 2 = CED 7 2 = ; 3L@M 2 =GCON For a shear displacement in the.qp -plane, we have, /SR 2(354'6 0)087 2:94(6 0;<7 =>94(6 0? 7 / = R 354 0; ;); 6 7 = 94 ;)? 6 7 / 2 = R 354 0? ;)? 6 7 = 94 ;)? 6 7 where / 2 R 3A@B 2 21CED / = R 3F@B =8=GCED / 2 = R 3F@H 2 =ICED 7 2 =T3L@M R 2 =GCON 2 = (5.64) 2 = (5.65) 0 2 = (5.66) 0 2 3F@J 2 2KCED 7 0 = 3F@J = =IC, and, 2 = (5.67) 2 = (5.68) ; 2 = (5.69) ; 2 3F@J 2 2 CED 7 ; = 3F@J = = C, and, 2 = (5.70) 2 = (5.71) R 2 = (5.72) R 2 3F@J 2 2KCED 7 R = 3F@J = =IC, and, These nine equations may always be independent, especially when a block possesses square symmetry. However, the following combination of these equations always leads to six independent equations in the six unknown effective stiffness matrix terms VVVVVVU W XVVVVVV / 0 / = 0 9 / ; YVVVVVVZ / 2 = 0 9 / R / ; / 2 =9 ; / R VVVVVV / 2 = R [ 2 7 = 7 2 = _ `ba = 7 2 = 7 = 7 2 = _ 2 7 = = _ 7 = 7 2 = _ 7 2 _ 7 = 7 2 = = 7 = = 7 2 = 7 2 _ 7 = 7 2 = VVVVVVU W XVVVVVV 4 0) ; 6 YVVVVVVZ 4 0? 6 4 ;); 6 4 ;)? 6 VVVVVV 4?)? 6 [ This equation can be solved to determine the effective stiffness matrix of a block of subcells. (5.73)

94 fg fg fg fg fg fg fg fg fg fg fg fg fg fg fg fg fg fg fg fg jk jk jk jk jk jk jk jk jk jk jk jk jk jk jk jk jk jk jk jk 5.4 Calculating Effective Properties of the RVE A sample ensemble of particles in a matrix is shown in Figure This ensemble is divided into a grid of subcells in such a way that each subcell of the grid is composed of only one material and the number of divisions per side of the ensemble is an integer factor of Y tu tu tu tu tu tu hi hi hi hi hi hi ƒ ƒ ƒ ƒ hi hi hi hi hi tuhi tuhi tuhi hi hi hi hi hi hi hi ƒ ƒ ƒ ƒ Š Š Š 1 Š Š Š ƒ ƒ ƒ ƒ rs rs rs lm lm lm Š Š Š 2 2 rs rs rs lm lm lm ˆ ˆ ˆ ˆ z{ z{ z{ z{ rs rs rs lm lm lm Œ Œ Œ ˆ ˆ ˆ ˆ 1 z{ z{ z{ z{ 1 1 Œ Œ Œ 1 ˆ ˆ ˆ ˆ z{ Ž z{ Ž z{ Ž z{ Ž vw vw vw vw Œ Œ Œ Ž Ž Ž Ž 3 vw vw vw vw ~ ~ Ž ~ Ž Ž Ž xy xy xy } } vw } vw vw vw 1 ~ ~ ~ 1 xy xy xy } } } 1 1 ~ ~ ~ xy xy xy pq } pq } pq } 2 pq pq pq 2 de de de de de de de de nopq de nopq de nopq de de de de de de de de 1 1 no no no1 1 no no no 1 First Iteration š š š š š š Particles X 2 3 Second Iteration Third Iteration Binder Figure The recursive cells method applied to a RVE discretized into blocks of four subcells. The first iteration is carried out with the four cells around the nodes marked 1. This leads to homogenized cells that are used in the second iteration with the composite cells around the nodes marked 2. The final iteration shown in the figure is for the four composite cells that make up the RVE at this stage. The finite element procedure outlined above is used for each four cell ensemble. This approach has been implemented using a quadtree-based data structure [121]. Finite element analyses are used to calculate the effective moduli at the lowest nodes of the structure. The values obtained at these nodes of the quadtree are assigned to the next higher level and the effective

95 85 properties are calculated at this level. This process is repeated recursively until the final effective properties of the RVE are obtained. The effective properties that have been determined using RCM in Chapters 6 and 7 have been calculated using four-noded elements with sixteen elements being used to represent a block of four subcells. The use of four-noded elements leads to a stiffer response than that observed with ninenoded elements. In addition, since 9/3 u-p elements are not used to model the binder, some element locking may be expected.

96 CHAPTER 6 VALIDATION OF GMC AND RCM In the first section of this chapter, exact relations are used to calculate the effective elastic properties of a few two-component composites. These effective properties are compared with predictions from detailed finite element analyses. The effective properties for these composites are also calculated using GMC and RCM and compared with the exact solutions where possible. The second section of this chapter deals with the prediction of the effective elastic properties of microstructures and component materials for which no exact solutions exist. We use the effective properties predicted by detailed finite element analyses as a benchmark for evaluating GMC and RCM as applied to these microstructures. Microstructures for which both GMC and RCM perform well are discussed followed by some special microstructures. In what follows, the detailed finite element analyses have been carried out with ANSYS 5.6 using four-, six- or eight-noded displacement based finite elements. Periodic displacement boundary conditions have been applied. The GMC results have been calculated using the technique discussed in Chapter 4, without coupling of the normal and the shear behaviors. The RCM calculations have been performed using blocks of four subcells modeled with sixteen four-noded elements. 6.1 Comparisons With Exact Relations Exact relations for the effective elastic properties of two-component composites can be classified into three types. The first type consists of relations that have been determined from the similarity of the two-dimensional stress and strain fields for certain types of materials. These exact relations are called duality relations [122]. The second type of exact relations, called translation-based relations, state that if a constant quantity is added to the elastic moduli of the component materials then the effective elastic moduli are also translated by the same amount. Microstructure independent exact relations, valid for special combinations of the elastic properties of the components, form the third category [123]. Many of these exact relations require either some form of rigidity or incompressibility in the phases of the composite. Since neither GMC nor RCM can deal with purely rigid or purely

97 µ ««ž ž œ œ «² ² «µ µ ««87 incompressible behavior, we have to assume a suitably high value of the modulus or Poisson s ratio to approximate the requirements for the exact relations to hold Phase Interchange Identity A duality-based exact relation is the phase interchange identity [26] for the effective shear modulus of a symmetric two-dimensional two-component isotropic composite. A symmetric composite is invariant with respect to interchange of the components. The phase interchange identity states that the effective shear modulus (œt ) of such a composite is given by œ žfÿ œ œ G (6.1) where œ and œ are the shear moduli of the two components. The linear elastic constitutive relationship for a two-dimensional isotropic material can be written as ) ) ª œ ² ) ± œ ² ³ ² ² œ ) <µ ) (6.2) where ), ) and ª are the stresses, ), ) and are the strains, and, µ and œ are the two-dimensional bulk and shear moduli, respectively. For a material with square symmetry, the shear modulus is not the same all directions and the slightly modified constitutive equation is written as where ) ) ª ± ¹ ² ² ) µ ) is the shear modulus for shear applied along the diagonals of the square, and, is the shear modulus for shear applied along the edges of the square. A checkerboard, as shown in Figure 6.1, is an example of a symmetric composite. However, a checkerboard exhibits square symmetry instead of isotropy, i.e., and are different. Since the phase interchange relation is valid only when the composite is isotropic, we choose the two components with low modulus contrast and compare the shear moduli predicted by finite elements, RCM and GMC. (6.3)

98 88 Figure 6.1. RVE for a checkerboard. The two materials that form the checkerboard composites were assigned the same Young s modulus of 15,300 MPa (the Young s modulus of HMX). The Poisson s ratio of the first component was fixed at 0.32 while that of the second component was varied from 0.1 to The exact effective shear modulus for the checkerboard has been plotted as a solid line in Figure 6.2. The two effective shear moduli, º¼» and º¾½, calculated using finite elements (FEM), RCM and GMC, have been plotted as points in the figure. The results show that all the three methods perform well (the maximum error is 0.1%) in predicting the effective shear modulus when the modulus contrast is small, i.e., when the composite is nearly isotropic. It can also be observed that the values of º» and º ½ are within 1% of each other for the chosen component moduli. Another set of numerical calculations has been performed on the checkerboard microstructure to observe the effect of increasing modulus contrast. In this case, the first component of the checkerboard was assigned a Young s modulus of 15,300 MPa and a Poisson s ratio of For the second component, the Poisson s ratio was fixed at 0.49 and the Young s modulus was varied from 0.7 MPa to 7000 MPa. When the modulus contrast between the components of the checkerboard increases, the material can no longer be considered isotropic and the values of º¼» and º¾½ are considerably different from the effective shear modulus 'À predicted by the phase interchange identity. This can be observed from the plot of the effective º» and º ½ versus the ratio of» and ½ shown in Figure 6.3. The exact effective shear moduli for isotropic, symmetric composites of the two components have been plotted with a solid line in the figure. The corresponding values of º» and º ½ predicted by FEM,

99 89 µ 1 and µ Exact µ 1 (FEM) µ 1 (RCM) µ 1 (GMC) µ 2 (FEM) µ 2 (RCM) Poisson s ratio of the second component Figure 6.2. Validation of FEM, RCM and GMC using the phase interchange identity for a checkerboard composite. RCM and GMC are shown as points on the plot (note that GMC predicts the same values of Á¼Â and Á¾Ã for materials with square symmetry). The FEM-based predictions of Á  (circles) and Á¾Ã (diamonds) show that these are the closest to the exact results. The ratio of Á à to Á¼Â increases as the modulus contrast the two components increases. For relatively low modulus contrast, the effective shear modulus predicted by the phase interchange identity is approximately equal to the mean of Á  and Á¾Ã predicted by FEM. The effective Á  and Á¾Ã predicted by RCM are higher than those predicted by FEM while those predicted by GMC are lower. The ratio of Á  and Á¾Ã to the ÄTÅ predicted using the phase interchange identity is shown in Figure 6.4. It can be observed that the FEM computations produce good approximations to the exact results for shear modulus contrasts of up to 500. For shear modulus contrasts above 500, isotropy is no longer an adequate assumption and the FEM results diverge from those predicted by the phase interchange identity. The plot also shows that the RCM predictions of Á¼Â and Á à are consistently higher than those predicted by FEM while the GMC predictions are consistently lower. The RCM results are closer to the FEM results than are the GMC predictions. The above results show that finite element analyses may provide a benchmark for evaluating the RCM and GMC techniques when exact relations are not available. However, we have to

100 Exact µ 1 (FEM) µ 1 (RCM) µ 1 (GMC) µ 2 (FEM) µ 2 (RCM) µ 1 and µ Ratio of the shear moduli of the two components Figure 6.3. Variation of effective shear moduli with modulus contrast for a checkerboard composite. ascertain whether the finite element results that we compare the RCM and GMC results against have converged to a steady solution. The checkerboard material provides an extreme case to test the convergence of the FEM solution because the corner singularities lead to high stresses. Therefore, very high mesh refinement is required to minimize the effect of high corner stresses on the effective moduli. In this research we perform finite element analyses on ÆÈÇÈÉËÊÌÆÈÇÈÉ square grids. The effective ͼΠand Í¾Ï of a checkerboard with a shear modulus contrast of about 25,000 (corresponding to the highest modulus contrast shown in Figure 6.4) have been calculated using various levels of mesh refinement and plotted in Figure 6.5. These plots show that the effective ͼΠconverges to a steady value when about Ð<ÆÈÑÒÊÓÐ<ÆÈÑ elements are used while Í Ï converges to a steady value when ÆÈÇÈÉÒÊ ÆÈÇÈÉ elements are used to discretize the RVE. The higher number of elements in the plot corresponds to a grid of ÔÈÇGÕÖÊ ÔÈÇGÕ elements. Hence, our choice of ÆÈÇÈÉÖÊ ÆÈÇÈÉ elements is justified and can be expected to generate effective elastic moduli that can be used as benchmarks. The checkerboard microstructure also has another significance with respect to the recursive method of cells. For the RCM procedure that uses blocks of Æ Ê Æ subcells, one of the possible microstructures is a checkerboard. The convergence plot shown in Figure 6.5 suggests that using sixteen four-noded elements to model the block may lead to an overestimation of the effective

101 Ø Þ Ø àß Ø ë 91 Ratio of µ 1 and µ 2 to G * µ 1 (FEM) µ 1 (RCM) µ 1 (GMC) µ 2 (FEM) µ 2 (RCM) Ratio of the shear moduli of the two components Figure 6.4. Ratio of effective shear moduli predicted by FEM, RCM and GMC to those predicted by the phase interchange identity for a checkerboard composite with varying modulus contrast. properties. It may be preferable to model each subcell using more than 16 elements to obtain a better approximation for the effective elastic properties Materials Rigid in Shear The stress-strain response of two-dimensional composites that are rigid with respect to shear can be represented by ßàá Ù<ÚÜÛ)Û Ú Ý)Ý ÛÝ Ù âãû)ûäâãûýäå âãûýæâçý)ýäå å å å ßàíì Ù èéû)û èêý)ý ÛÝ (6.4) where èéû)û, èqý)ý and ë ÛÝ are the stresses,, and Þ ÚÜÛ)Û Ú Ý)Ý ÛÝ âïîñð are the strains, and, are the components of the compliance matrix. Two duality-based relations that are valid for two-components composites composed of such materials are [122] :

102 µ 1 µ µ 1 and µ Number 10 3 of elements Figure 6.5. Convergence of effective moduli predicted by finite element analyses with increase in mesh refinement for a checkerboard composite with shear modulus contrast of 25, If òãó)ó8òçô)ôéõíöòãóô< ô:øúù for each phase (where ù is a constant), then the effective compliance tensor also satisfies the same relationship, i.e., òüû ó)ó òýû ô)ô õöòýû óô ô øúù. This relation is true for all microstructures. 2. If the compliance tensors of the two phases are of the form þ ó øúÿ ó and þ ô øúÿ ô where is a constant matrix, then the effective compliance tensor of a checkerboard of the two phases satisfies the relation òüû ó)ó òýû ô)ô õ öòýû óô ô øúÿ ó ÿ ô ö ó)ó ô)ô õ ö óô ô. The finite element analyses performed in this research are two-dimensional and based on the plane strain assumption. The effective compliance matrix cannot be determined directly from plane strain computations (Appendix A). Therefore, an approximate compliance matrix is calculated for the finite element and RCM validations using the method discussed in Appendix A. The effective compliance matrix can be calculated directly by GMC. Numerical experiments using a square array of disks occupying an area fraction of 70% (as shown in Figure 6.6) have been carried out to check if the first of the above relations can be reproduced by finite element analyses, GMC and RCM. The þ matrices that have been used for

103 & the disks (superscript ) and the matrix (superscript ), and the corresponding values of shown below. These matrices have been chosen so that the value of should be equal to that for the effective compliance matrix. and %$ '& ( )*+( '& )*+( (! " #!, # 93 are is constant and therefore The shear modulus for both materials is - around - times the Young s modulus. Higher values of shear modulus have been tested and found not to affect the effective stiffness matrix terms significantly. Figure 6.6. RVE for a square array of disks. We require the out-of-plane Young s modulus and Poisson s ratio to calculate the effective compliance matrix of the composite (as explained in Appendix A). These properties, calculated using the rule of mixtures (ROM) and from the effective compliance matrix predicted by GMC, are shown in Table 6.1. For the ROM calculations, the values of Young s modulus for the two components are. / +021 and &5. / The corresponding Poisson s ratios are ( and

104 E 7 E 7 94 Table 6.1. Out-of-plane properties for square array of disks. ROM Based GMC Based (9;: <+=4> ) (9;: <+=4> ) The compliance matrices calculated using finite elements (123,000 elements), GMC (4,100 subcells) and RCM (4,100 subcells) are shown in Table 6.2. The corresponding values of? are also shown in the table. The compliance matrices have been calculated using both the ROM based and the GMC based out-of-plane properties. Since the modulus contrast between the two components of the composite is small, the calculated effective properties are expected accurate. However, the results in Table 6.2 show that both the detailed finite element calculations and the RCM calculations lead to around 10% error in the estimation of?. On the other hand, the GMC calculations lead to an error of only about 4.5%. Table 6.2. Components of effective stiffness and compliance matrices for a square array of CDC (9;: CFE ) (9;: <+= FEM GMC RCM G CDC A (9: < ) G CFE A? Error ) (9;: < ) (9: <H ) (%) GMC ROM-Based FEM GMC-Based FEM ROM-Based RCM GMC-Based RCM The above results imply that the finite element analyses and the RCM calculations overestimate the effective properties of the square array of disks. This may be because the rigidity of the material in shear is not well approximated by the finite element calculations. Higher values of the shear modulus of components, e.g., IKJL: <NM, lead to essentially the same effective stiffness matrix components and CFE. Another reason for the error could be that the effective compliance matrix terms, for the 3 direction, are not approximated well by the rule of mixtures or by the GMC calculations. This can be verified by carrying out three-dimensional calculations for this composite.

105 d j b b z 95 The slight difference between the rule of mixtures (ROM) based calculations and the GMC based calculations is probably due to machine precision because the values of OQP RFS and OQP SDS calculated by GMC for two-dimensional problems is essentially a rule of mixtures calculation. The second duality relation for materials that are rigid in shear requires the use of a checkerboard geometry as shown in Figure 6.1. The stress singularities at the corner contacts of the checkerboard leads to relatively large errors in the computation of the effective properties as was observed for the phase interchange identity discussed before. However, it is interesting to observe how well finite elements, GMc and RCM perform in calculating this result. To test the second duality relation, we choose T RUWV XX, TZY UWV XXX and, [ U]\ V X ^`_ ^_ V XZacb Then the compliance matrices for the two components of the composite are d R UWV XX \ V X ^_ ^_ V Xea and d Y UfV XXX \ V X ^_ ^`_ V Xea b The duality relation requires that the effective compliance matrix of the checkerboard composite should be such that gh ikj Pml U O RDR P O YDY P ^ O RP Y l Y Un V Xpo,V Xq _ respec- For the constituent material properties the values of r S and s S are t bvu owv X+x4y and X b tively. The same values are obtained using the rule of mixtures and the O RFS P and O P SDS calculated using GMC. Finite element analyses using around 123,000 elements have been used to calculate the effective stiffness matrix for the checkerboard. The GMC and RCM calculations have used around 4,100 elements to determine the effective stiffness matrix. The effective compliance matrix has been determined using the method described in Appendix A for the finite element and RCM calculations and determined directly using GMC. The results from these three methods are tabulated in Table 6.3. Interestingly, the finite element calculations lead to quite an accurate effective compliance matrix and the deviation from the exact result is only around 7%. The GMC calculations overestimate the compliance matrix and the determinant of the compliance matrix is around 1.5 times higher than the exact result. On the other hand, though the RCM calculations underestimate the compliance matrix, they lead to a value of the determinant that is closer to the exact value than the GMC results. The higher values of effective stiffness, predicted by RCM, are due to each block of z o subcells being modeled using 16 elements. As can be seen from Figure 6.5 the effective stiffness predicted using sixteen elements is considerably higher than that predicted using a more refined mesh. Therefore, a way of improving the performance of RCM would be to use more elements to model a block of subcells.

106 96 Table 6.3. Components of effective stiffness and compliance matrices for a checkerboard composite. {} ~D~ {} ~F ( ; +ƒ4 ) ( ; +ƒ4 ) FEM GMC RCM Q ~D~ ~F ˆ kšœ m Error ( ; Ž ) ( ; Ž ) ( ) (%) FEM GMC RCM The CLM Theorem The Cherkaev, Lurie and Milton (CLM) theorem is a well known translation based exact relation for two-component planar composites (Milton [26] and references therein). For a twodimensional two-component isotropic composite, this theorem can be stated as follows. Let the isotropic bulk moduli of the components be ~ and. Let the shear moduli of the two components be ~ and. The effective bulk and shear modulus of a two-dimensional composite made of these two components are and respectively. Let us now create two new materials that are translated from the original component materials by a constant amount. That is, let the bulk and shear moduli of the translated component materials be given by ~ (translated) (translated) ~ (translated) (translated) ~ ~ The CLM theorem states that the effective bulk and shear moduli of a two-dimensional composite of the two translated materials, having the same microstructure as the original composite, are given by

107 š š š š "œ (translated) œ (translated) 97 š œÿž and, (6.5) œ The requirement of isotropy can be satisfied approximately for numerical experiments by choosing component material properties that are very close to each other. Since our goal is to determine how well GMC and RCM perform for high modulus contrast, choosing materials with small modulus contrast is not adequate. Another alternative is to choose a RVE that represents a hexagonal packing of disks. However, such an RVE is necessarily rectangular and cannot be modeled using RCM in its current form. It should be noted that RCM can easily be modified to deal with elements that are not square and hence to model rectangular regions. Another problem in the application of the CLM theorem is that the value of has to be small if the difference between the original and the translated moduli is large and vice versa. If the value of is small, floating point errors can accumulate and exceed the value of. On the other hand, if is large, the original and the translated moduli are very close to each other and the difference between the two can be lost because of errors in precision. Hence, the numbers have to be chosen carefully keeping in mind the limits on the value of the Poisson s ratio. However, it is interesting to observe the differences between the effective moduli predicted finite elements, GMC and RCM before and after a translation. We, therefore, test the translation idea on a square array of disks occupying a volume fraction of 70% as shown in Figure 6.6. This RVE exhibits square symmetry, i.e., the shear moduli and ª shown in equation (6.3) are not equal. We cannot calculate a unique value of the effective shear modulus for this RVE. Instead, we calculate the value of the effective translated shear modulus from equation (6.5) by first setting «œ equal to and then to ª. We then compare these exact values with the and ª values predicted using finite element analyses, GMC and RCM. The original set of elastic moduli for the RVE is chosen to reflect the elastic moduli of the constituents of PBXs. These moduli are then translated by a constant. The original and the translated constituent two-dimensional moduli are shown in Table 6.4 (phase p represents the particles and phase b represents the binder). The three-dimensional moduli can easily be calculated from the two-dimensional moduli shown in Table 6.4 using the relations given by Jun and Jasiuk [41]. The effective bulk and shear moduli of the original and the translated material have been calculated using finite elements (123,000 elements), GMC (4,100 subcells) and RCM (4,100 subcells). These are shown in Table 6.5. The values of err shown in the table have been calculated using the equations

108 À À Á à 98 Table 6.4. Original and translated two-dimensional constituent moduli for checking the CLM condition.! ± p² ³² } 5 ' µ² ± ³² ¹ ;º»¼k½ ¹ ;º»¼k½ ¹ ;º»¼'½ ¹ ;º»¾k½ Original Translated err À»Á»»ºc ºk½±,º»»Ã º ' "Ä Â º ' "Ä original translated º ǼÅ%ÇÉÈ translated  º ǼÅ%ÇÉÈ original where, or, Table 6.5. Comparison of effective moduli for the original and the translated composites. Orig. Ä ÅÊ È Å ¼ È Trans. err Orig. Trans. err Orig. Trans. Ê Ê Ê Ê Ê Ê ( ;º» ) ( ;º» ) (%) ( ;º» ) ( ;º» ) (%) ( ;º»+Ë ) ( ;º»+Ë ) (%) FEM GMC RCM err Even though the modulus contrast between the two components of the composite is high, the effective properties predicted by FEM, GMC and RCM are close to each other in magnitude. It is also interesting to observe that the calculated values of closely approximate the exact value for an isotropic composite for all the three methods even though the value is small ( À»Á»»º ). The detailed finite element analyses produce the most accurate results for this problem. If we compare the finite element analysis based effective moduli with those calculated using GMC, we observe that the moduli are slightly underestimated by GMC. On the other hand, RCM overestimates the effective properties slightly. If we compare the values produced by the three methods, we observe that GMC produces values of that are closer to than RCM does Symmetric Composites with Equal Bulk Modulus The translation procedure can also be used to generate an exact solution for the effective shear modulus of two-dimensional symmetric two-component composites with both components having the same bulk modulus [26]. This relation is

109 å ç ç Ì å 99 Ì"Í;ÎÏÌ Ô Í;Î ÎÐÌÒÑ,ÎÐÌpÓ Õ}Ö Ú ØÙ Ù Û Ö Ì Ô ÑÝÜ Û ÖŸ Ì Ô Ó Ü (6.6) We test this relation on the checkerboard model shown in Figure 6.1 using the component material properties given in Table 6.6. The exact effective properties for the composite, calculated using equation (6.6), are also given in the table. The values of the effective moduli calculated using finite elements (FEM), GMC and RCM are also shown in Table 6.6. Table 6.6. Component properties, exact effective properties and numerically computed effective properties for two-component symmetric composite with equal component bulk moduli. Ì Í â¹ã Ö äæ Þ ß ÌµÓáà Ô â¹ã ÓÆå Ö ä â¹ã Ö äæ â¹ã Ókå Ö ä Component Component Composite Ñéè â¹ã Ó'å Ö ä ÓÝè â¹ã Ókå Ö ä 0.5( ç Ñéè ç â¹ã Ókå Ö ä Diff. Diff. ) Diff. % % % FEM GMC RCM ÓÝè These results show that the effective two-dimensional bulk modulus is calculated correctly by all the three methods. However, the shear moduli calculated for the checkerboard microstructure are quite different from the exact result. Hence, this exact result does not appear adequate for examining the validity of the approaches. The finite element calculations have been carried out using 123,000 elements and hence are expected to be quite accurate. In addition, if we examine the average of the two shear moduli calculated using FEM, we get a value quite close to the exact value for an isotropic composite. This also suggests that the finite element calculations are accurate. The GMC and RCM calculations, as has been observed before, predict lower and higher values of the shear moduli that the finite element calculations, respectively. The RCM results are higher because only sixteen elements are used to calculate the effective properties. We would get results closer to the finite element results if more elements were used to calculate the effective properties in RCM.

110 î ð ó ð î ô ó ð î ô ë ó í ë ô î Hill s Equation Hill s equation [124] is an exact relation that is independent of microstructure. This equation is valid for composites composed of isotropic components that have the same shear modulus. For a two-dimensional two-component composite, this equation can be written as where, ë ìöõ ë õ ë and binder, respectively, ò¹ó î ì ê ë"ì íïîñð ò¹ó í î òô í î (6.7) are the two-dimensional bulk moduli of the composites, particles is the shear modulus of the composite and its components, and, and òô are the volume fractions of the particles and the binder. We attempt to verify this relationship using the RVE containing an array of disks occupying 70% of the volume as shown in Figure 6.6. Table 6.7 shows the properties of the two components used to compare the predictions of finite elements, GMC and RCM with the exact value of bulk modulus predicted by Hill s equation. Table 6.7. Phase properties used for testing Hill s relation and the exact effective moduli of the composite. Vol. ê ýþæÿ ø û¹ü û¹ü ê ýþæÿ ëµùáú ê ýþkÿ û¹ü Frac. Disks Binder Composite Since the modulus contrast is small, we expect the square array of disks to exhibit nearly isotropic behavior. Therefore, we expect the predictions of finite elements, GMC and RCM to be close to the exact values of the effective properties of the composite. It should be noted that the materials chosen are not quite representative of PBX materials. The numerically calculated values of the effective two-dimensional bulk and shear moduli of the composite are shown in Table 6.8. The finite element calculations have used around 123,000 elements, the GMC calculations have used around 4,100 subcells, and the RCM calculations have used around 65,000 subcells. The effective shear moduli predicted by all the three methods are exact. In case of the effective bulk moduli, the RCM predictions are the most accurate followed by GMC and the finite element based calculations. The finite element based calculations overestimate the effective two-dimensional

111 101 Table 6.8. Numerically computed effective properties for a square array of disks with equal component shear moduli. % diff ( ) ( ) ( ) FEM GMC RCM bulk modulus by around 4.4% while GMC underestimates the bulk modulus by around 4.2%. This test appears to show that the accuracy we should expect from the three methods of calculating the effective properties is around 5% Comments On Comparisons With Exact Solutions The above comparisons with exact solutions show that none of the exact relations provide a definitive test of the accuracy of the finite element calculations of the effective properties for materials such as PBX However, we observe that the finite element calculations are quite accurate for materials with low modulus contrasts, even for extreme geometries like the checkerboard model. In addition, we observe that both GMC and RCM perform well for materials with low modulus contrast. Detailed finite element calculations of the effective properties of composites with high modulus contrasts can be presumed to be quite accurate because of the high level of discretization and from comparisons with some of the exact solutions. If we compare the finite element based solutions with RCM and GMC, the most common finding is that the RCM results are closer to the finite element solutions than the GMC results though RCM overestimates the effective properties while GMC underestimates these properties. Improved estimates are obtained for the effective elastic properties when the amount of discretization of the RVE is increased. This is true for all the three methods - except for the special case of the checkerboard microstructure. For the checkerboard microstructure, for any amount of discretization, the present implementation of RCM predicts the same values as that of a RVE discretized into subcells, for obvious reasons. 6.2 Comparisons With Numerical Results In this section, we compare the predictions of effective elastic properties using finite element analyses, GMC and RCM with accurate numerical results obtained by other researchers. These tests serve the purpose of further confirming the accuracy of finite element procedure used in this research. In addition, we obtain a better estimate of the accuracy of GMC and RCM with respect to the detailed finite element analysis based calculations.

112 The current limit on the amount of discretization possible for the finite element analyses is around elements for a square RVE, that for GMC is around subcells and that for RCM is around!" subcells. These limits apply for a four processor Sun Ultra-80 with 4 Gb of main memory in the absence of other users. In this research, for reasons that have mostly to do with availability of computational resources, we have used #$# elements for finite element analyses, %&% elements for GMC and #'# subcells for RCM for the comparisons of the three methods. Highly accurate estimates of the effective properties of square arrays of disks have been obtained by Greengard and Helsing [97] using an integral equation based method. We utilize these results as a benchmark against which the accuracy of the three methods explored in this research. Square arrays of disks similar to that shown in Figure 6.6 are modeled using finite elements, GMC and RCM. Volume fractions of the disks are varied from 10% to 70%. The two-dimensional effective properties determined using these methods have been compared with those determined by Greengard and Helsing [97]. The material properties of the disks and the binder, used by Greengard and Helsing, are shown in Table 6.9. ( ) / Table 6.9. Component properties used by Greengard and Helsing [97]. +21 *, Disks Binder Contrast The values of the effective two-dimensional bulk and shear moduli have been calculated using finite elements (FEM), GMC and RCM. We have assumed that the results of Greengrad and Helsing (G&H) are the most accurate and have calculated the percentage errors in the estimation of effective properties by FEM, GMC, and RCM with respect to the G&H results. Table 6.10 shows the results of Greengard and Helsing along with those from the FEM, GMC, and RCM calculations and the relative errors in approximation of the two-dimensional bulk and shear moduli. The FEM calculations were performed using nodes and 86. six noded triangular elements. The GMC calculations were carried out using %9% #;<# #76=# subcells. 476: subcells and the RCM models used The data in Table 6.10 show that all the computed effective properties (from FEM, GMC and RCM) are quite close to each other when compared to the modulus contrast between the two

113 103 Table Comparison of numerically calculated values of two-dimensional bulk and shear moduli of square arrays of disks. Vol. >? Frac. G&H FEM % Diff. GMC % Diff. RCM % Diff ACB G&H FEM % Diff. GMC % Diff. RCM % Diff D=B G&H FEM % Diff. GMC % Diff. RCM % Diff components of the composite. Therefore, these methods are performing quite well. Moreover, the FEM results match very well with the results of Greengard and Helsing though the GMC and RCM results differ slightly from the accurate calculations. This shows that the approach we have taken to calculate the values of the effective properties from FEM based results is accurate. It also shows that the GMC and RCM approaches are quite accurate though they could be improved. Figure 6.7 shows plots of the error in the two-dimensional bulk moduli calculated by FEM, GMC and RCM relative to those calculated by Greengard and Helsing. This plot shows that the error in estimation of effective properties is independent of volume fraction. The fluctuations in the error can probably be attributed to errors in discretization. It is observed that GMC performs slightly better than RCM in this case. The relative error in the estimation of the shear A? is shown in Figure 6.8. In this case,

114 J Error in K * (%) GMC RCM FEM Volume Fraction (%) Figure 6.7. Error in computation of E F for a Square Array of Disks. the error in the RCM estimates is larger than that of the GMC estimates. This is because the value of GHF IKJ is usually overestimated by RCM. The overestimation can be reduced if each block in RCM is discretized into more than four elements thus leading to a less stiff response. Figure 6.9 shows the relative error in the estimation of the shear modulus L F. GMC underestimates this shear modulus because it predicts a modulus that represents the Reuss lower bound on the shear modulus. The FEM predictions are about 1% to 5% higher than those predicted by Greengard and Helsing. This is probably because the displacement based finite element calculations produce a response that is stiffer than actual. Interestingly, if the boundaries of the RVE are constrained to remain straight lines in shear, an even higher stiffness is predicted because relatively high normal stresses are generated. The RCM predictions are closer to the FEM predictions and usually higher than the FEM predictions. The reason for this is, once again, the minimal discretization that is used for the RCM calculations. On average, GMC is performs better than RCM for the square arrays of disks for the chosen component material properties. The FEM calculations show excellent agreement with the highly accurate results of Greengard and Helsing. Therefore, we use detailed FEM calculations as benchmarks for estimating the accuracy of predictions from GMC and RCM in future validation checks.

115 Error in µ 1* (%) GMC RCM FEM Volume Fraction (%) Figure 6.8. Error in computation of M NCO for a square array of disks Error in µ 2* (%) GMC RCM FEM Volume Fraction (%) Figure 6.9. Error in computation of M P.Q for a Square Array of Disks.

116 6.3 Special Cases : Stress Bridging Comparisons of effective properties predicted using GMC and RCM with exact relations and other numerically determined results show that these methods perform quite well for low modulus contrast materials even for extreme microstructures like the checkerboard model. From our research, some microstructures have been discovered for which GMC performs less than adequately while there are other microstructures for which RCM does not perform well. Some of these microstructures are discussed in this section. Effective properties are calculated for these microstructures using GMC and RCM. Comparisons are made with the corresponding properties predicted using detailed finite element analyses and, where possible, reasons for the poor performance of GMC or RCM are discussed. The square array of disks represents a situation in which there is no contact between particles. Since the PBXs of interest to this research contain more than 90% particles by volume there are bound to be particles that are either very close to each other or in contact. To check the efficacy of GMC and RCM when particles are in contact, we have simulated a number of bridging models. The bridging is due to contact between particles that leads to preferential stress paths or stress bridging. This problem was first recognized when calculating the effective moduli of a random distribution of particles (with some contact between particles). It was observed that GMC consistently generated low values of the effective stiffness matrix terms. This, in turn, led to the development of the RCM model as an alternative to GMC. We discuss some of these bridging models and the effective elastic properties calculated using GMC, RCM and finite elements for these models Corner Bridging : X-Shaped Microstructure The checkerboard model shown in Figure 6.1 has some stress bridging through corner contacts. However, because of the relatively small modulus contrast between the phases that were used to test the exact relations, the difference between the effective moduli predicted by GMC, RCM and finite elements was not very large. When the modulus contrast is increased to match that of PBX 9501 at room temperature the differences between the results produced by the three methods become pronounced. We can observe this for the RVE shown in Figure In this RVE the particles are square, arranged in the form of an X and occupy a volume fraction of 25%. The particles transfer stress through corner contacts. The material properties used are those for HMX and binder as shown in Table 3.1. The sharp corners in the particles lead to stress singularities. We assume that the high stresses are averaged out during the calculation of effective properties (note that this is also one of the assumptions of the RCM technique). For the X shaped microstructure shown in Figure 6.10,

117 r q q r } q } q } q q q } m } q t { q q } r { 107 Y,2 jk jk jk jk jk jk jk jk jk de de de de de de de de de RS RS RS RS RS RS RS RS RS TU TU TU TU TU TU TU TU TU ^_ ^_ ^_ ^_ ^_ ^_ ^_ ^_ ^_ bc bc bc bc bc bc bc bc bc \] \] \] \] \] \] `a `a `a `a `a `a \] \] \] `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `a `az[ Z[ Z[ Z[ Z[ Z[ Z[ Z[ Z[ XY XY XY XY XY XY XY XY XY VW VW VW VW VW VW VW VW VW hi hi hi hi hi hi hi hi hi fg fg fg fg fg fg fg fg fg X,1 Figure RVE used for corner stress bridging model. we first observe the variation in the effective elastic properties with increase in modulus contrast between the particles and the binder. The five sets of materials that are explored are shown in Table Table The elastic properties of the components of the X shaped microstructure. mrsr mrkt m Model lnm opm q usu vpwxyz{ vpwxyz{ vpwxyz2{ vpwxy 3{ (MPa) (MPa) (MPa) (MPa) All rsr rkt l~} o} q usu vpwxy { vpwxy { vpwxy { vpwxy { (MPa) (MPa) (MPa) (MPa) a b c d e mrsr rsr mrkt3 rkt Model usu usu vpwxy vpwxy a b c d e

118 108 Figure 6.11 shows the variation of the effective ƒ s, normalized with respect to the binder properties, with increasing modulus contrast between the particles and the binder. The effective properties have been calculated using finite elements (FEM), GMC and RCM. It can be observed that the FEM, GMC and RCM estimates are close to each other for modulus contrasts below 200. For higher modulus contrasts, the ratio of the effective modulus predicted by GMC to the modulus of the binder remains relatively constant and much lower than that predicted by FEM and RCM. This shows that GMC does not deal as accurately with corner contacts as FEM or RCM FEM GMC RCM C * 11 /Cb C p 11 /Cb 11 Figure Variation of s with modulus contrast for X -shaped microstructure. Similar plots are shown in Figure 6.12 for the effective property K. For this property, it is again observed that the RCM predictions are closer to the FEM predictions than the GMC predictions. At relatively low modulus contrasts, all the three methods predict approximately the same values of H K. GMC is again found to predict low values of ƒ K at high modulus contrasts, implying that the corner contacts between particles are not accurately taken into account. Figure 6.13 shows the values of ƒ s calculated using FEM, GMC and RCM. Once again, GMC is found to predict values that are quite low compared to those predicted by FEM. ˆ5 The components of PBX 9501 have the material properties that correspond to those of model shown in Table 6.11 and have the highest modulus contrast. The values of ƒ s, H K and H s predicted for the X-shaped microstructure for this case are shown in Table The ratio of the

119 FEM GMC RCM C * 12 /Cb C p 12 /Cb Figure Variation of ŠH ŒK with modulus contrast for X -shaped microstructure FEM GMC RCM 2500 C * 66 /Cb C p 66 /Cb 66 Figure Variation of Š ŽsŽ with modulus contrast for X -shaped microstructure.

120 110 FEM results to the GMC results shows the large error in the GMC predictions compared to the RCM predictions. Therefore, the current version of RCM is a definite improvement over GMC for this microstructure. Table ƒ s, ƒ K and H s for X-shaped microstructure with highest modulus contrast. FEM GMC FEM/GMC RCM RCM/FEM p 3 p 2 p 3 s K ƒ s A comparison of the effective stiffness matrix terms s, K and s calculated using GMC, RCM and FEM is also shown in Figure The figure shows that the effective moduli calculated by GMC are considerably lower than those calculated using finite elements. The moduli calculated using RCM are consistently higher than the finite element results. We conclude that GMC does not see the contact points between particles, especially where shear effects are concerned C * 11 GMC RCM FEM C * 11, C* 12 and C* C * 12 C * 66 0 Figure Comparison of effective stiffness matrix for corner stress bridging model.

121 Edge Bridging : Five Cases Further evaluation of GMC and RCM is performed using models A through E shown in Figure The objective of this study is to explore the behavior of GMC and RCM as stress bridging is increased progressively from corner bridging to partial edge bridging followed by continuous stress bridging. š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š š Model A œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ œ ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ª ž ž ž ž ž ž ž ž ž ž ž ž Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ ž ž ž ž Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Ÿ Model B «««««««««««««««««««««««««««««««««««««««««««««««««««««««««««««««««Y,2 Model C Model D ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ± ± ± ² ² ² ² ± ± ± ± ² ² ² ² X,1 ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² ± ² Model E Figure Progressive stress bridging models A through E. Model A contains a square particle that occupies 25% of the volume and is centered in the RVE and does not have any stress bridging. Model B has three particles that touch at two corners of the central particle. In model C the amount of contact is increased until there is a single line of stress

122 À Å À À Å Å Å Å 112 bridging along the center of the RVE. This microstructure has been chosen such that RCM will predict square symmetry when a ³µ ³ subcell based block is used, even though the microstructure does not have square symmetry. Model D extends the line of bridging to an area of bridging in one direction and model E extends the bridging to both directions. In what follows, the 1 direction corresponds to the axis shown in Figure 6.15 and the 2 direction corresponds to the axis. The material properties of the constituents of PBX 9501 at room temperature, as shown in Table 6.13, are used for these tests. GMC simulations of the RVEs were carried out with ¹¹ & ¹¹ subcells, RCM simulations used ³º»~ µ³º» subcells and finite element calculations were carried out using around ¹½¼.¹¹¹ eight noded quadrilateral elements. Table Materials used to test edge bridging using FEM, GMC and RCM. ¾ ÀÁsÁ ÀÁK ÀÄÃsà (MPa) (MPa) (MPa) (MPa) Particle Binder Model A The effective properties of Model A, calculated using FEM, GMC and RCM, are shown in Table The differences shown in the table are percentages of the values computed using FEM. Since Model A exhibits square symmetry, i.e., ÀHÅ ÁsÁÇÆ ÀƒÅ ÂsÂ, the values of ÀƒÅ Âs are not shown in the table. The values of the effective properties from the three techniques are quite close to each other, especially when we consider the large modulus contrast between the particle and the binder. The percentage differences with respect to the FEM results show that GMC is in better agreement with FEM for the quantities À ÁsÁ and À ÁK while RCM is in better agreement for À ÃsÃ. Table Effective properties of Model A from FEM, GMC and RCM. ÁsÁ ÁK ÀHÅ Ãsà FEM GMC Diff. RCM Diff. (MPa) (MPa) (%) (MPa) (%) Å Ãsà Figure 6.16 shows a comparison of the effective stiffness matrix components ÀHÅ ÁsÁ, ÀHÅ ÂsÂ, ÀƒÅ ÁK for model A, normalized with respect to the binder properties. The results show that, for model A, the binder dominates the effective elastic response. The plots in Figure 6.16 further illustrate and

123 113 that the effective moduli calculated for model A using GMC and RCM are quite accurate. It is also observed that the values of ÈHÉ ÊsÊ and ÈHÉ ËsË are identical. 2 C * 11 C * 22 C * 66 GMC RCM FEM Ratio of C * to C binder C * 12 0 Figure Comparison of normalized effective stiffnesses for model A Model B The effective stiffness matrices for the corner-bridging model (Model B) are shown in Table This model also exhibits square symmetry. The stress bridge along the diagonal leads to much higher stiffness than would occur for a single particle occupying the same volume fraction. This model is similar to he checkerboard model in some respects and exhibits similar trends. The GMC calculations predict values of ÈHÉ ÊsÊ and ÈHÉ ÊKË that are lower by a factor of 18 than the FEM results. The value of È ÌsÌ É calculated using FEM are around 1,400 times that calculated using GMC. Since the FEM calculations are quite accurate, it is clear that the averaging process used in GMC is not adequate to capture the effects of corner stress bridging. On the other hand, the value of È predicted by RCM is around 3 times that predicted by FEM. Therefore, RCM predicts values that are quite close to those predicted by FEM though there is room for improvement. The RCM based values of È ÊKË É and È ÌsÌ É are even closer to the FEM based values - around 1.5 times higher. Thus, RCM does quite well in improving upon the drawback in GMC relating to corner bridging. A plot of the effective stiffness matrix terms, calculated for Model B using FEM, GMC and RCM, is shown in Figure ÊsÊ É

124 114 Table Effective properties of Model B from FEM, GMC and RCM. FEM GMC FEM/GMC RCM RCM/FEM ÍpÎÏÐÑ3Ò ÍpÎÏÐÑ2Ò ÍpÎÏÐÑ3Ò (MPa) (MPa) (MPa) ÓƒÔ ÕsÕ ÓƒÔ Õ Ñ ÓƒÔ ÖsÖ C * 11 C * 22 GMC RCM FEM C * 11, C* 22, C* 12 and C* C * 12 C * Figure Comparison of effective stiffnesses for model B Model C Model C has a continuous path through particles along the -axis (the 1 direction) and another continuous particle path along one diagonal. The stress bridge path along the 1 direction impacts mostly the normal components of stiffness while the bridge along the diagonal impacts the shear stiffness. These paths are shown by dashed lines (for normal stress bridge) and by dotted lines (for shear stress bridge) in Figure The stress bridge in the 1 direction leads to a value of ÓƒÔ ÕsÕ that is considerably higher than Ó ÑsÑ Ô. This model microstructure has been chosen to illustrate why RCM may fail to predict accurate effective properties for certain microstructures. The procedure used by RCM to predict the effective properties of Model C is shown in Figure The schematic shows that in the last recursion performed by RCM, the four subcells appear to belong to a material with square symmetry even though this is not the case. This is why, for this type of microstructure, RCM predicts effective

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126 116 Y,2 ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ø ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú ù ú û ûü ûü ûü ûü ûü û ûü ûü ûü ûü ûü û ûü ûü ûü ûü ûü û ûü ûü ûü ûü ûü û ûü ûü ûü ûü ûü ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ýþ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ ÿ X,1 Square Symmetry Figure Why RCM predicts square symmetry for Model C. Table Effective properties of Model C from FEM, GMC and RCM. FEM GMC FEM/GMC RCM RCM/FEM! (MPa) (MPa) (MPa) "$# %&% " '&' # "$# %(' " )&) # of effective stiffness calculated using FEM, GMC and RCM are shown in Table The value of "*# %&% predicted by GMC is around 1.05 times lower than that predicted by FEM while that predicted by RCM is around 1.05 times that predicted by FEM. The value of " # '&' predicted by GMC is around 1/40 th that predicted by FEM while RCM predicts a value that is around 1.6 times the FEM based prediction. The value of " # %(' predicted by RCM is again quite close to that predicted by FEM - around 1.4 times of the FEM value. On the other hand, GMC predicts a value that is around 1/20 th of the FEM prediction. Finally, the shear stiffness term " # )&) predicted by RCM is around 0.9 times the FEM value while that predicted by GMC is about 1/1900 th of the FEM value. Therefore, RCM

127 C * 11 GMC RCM FEM C * 11, C* 22, C* 12 and C* C * 22 C * 12 C * Figure Comparison of effective stiffnesses for Model C. performs well for this microstructure and the problems associated with Model C are not observed for Model D. GMC still fails to provide adequate predictions of the effective elastic moduli. This test also shows that a continuous area of stress bridging across the RVE may be required for GMC to capture the effects of stress bridging. This issue is explored in Model E. The values of +$, -&-, +*,.&., + -(., and + /&/, for model D are also shown in Figure Table Effective properties of Model D from FEM, GMC and RCM. FEM GMC FEM/GMC RCM RCM/FEM ! (MPa) (MPa) (MPa) + -&-, &., $, -( /&/, Model E For Model E there are two direct stress bridges in the 1 and 2 directions and two diagonal stress bridges. The effective properties for this microstructure, calculated using FEM, GMC and RCM, are shown in Table Both GMC and RCM predict the effective stiffness terms +*, -&-, +*,.&. and + -(., quite accurately. This shows that GMC requires continuous stress paths in the direction

128 C * 11 GMC RCM FEM C * 11, C* 22, C* 12 and C* C * 22 C * 12 C * 66 Figure Comparison of effective stiffnesses for Model D. of the coordinate axes to be able to predict the effect of stress bridging. Once again, the shear stiffness predicted by GMC is just a volume fraction weighted average of the shear moduli of the particles and the binder. Therefore, the extra stiffening of the effective shear response because of the diagonal stress bridges is overlooked by GMC. A plot the effective stiffness components predicted by the three methods is shown in Figure Table Effective properties of Model E from FEM, GMC and RCM. FEM 6789:<; GMC 6789:; FEM/GMC RCM67 89:; RCM/FEM (MPa) (MPa) (MPa) =*>?&? =?(@ > =*> A&A Summary Exact relations for the effective properties of two-component composites have been explored using detailed finite element analyses, GMC and RCM. We observe that detailed finite element analyses predict quite accurate effective properties, even though most of the exact relations can be verified only approximately using our approach. For modulus contrasts of less than 500, GMC and RCM are also quite accurate in predicting effective elastic moduli.

129 C * 11 C * 22 GMC RCM FEM C * 11, C* 22, C* 12 and C* C * 12 C * 66 0 Figure Comparison of effective stiffnesses for Model E. Comparisons with numerical experiments conducted by other researchers also show that all the three techniques are quite accurate for the materials considered. In general, we find that RCM tends to overestimate the effective stiffness slightly. The discretization used for the blocks at the lowest levels of recursion could be increased to obtain a less stiff response. GMC tends to underestimate the effective stiffness, especially in the presence of stress bridging. The shear stiffness is also underestimate by GMC when stress bridging is present. Coupling of the normal-shear stress-strain response could alleviate this problem. For RVEs with stress bridging, RCM is found to predict the effective stiffness better than GMC and the problems associated with lack of normal-shear coupling are dealt with successfully. In the following chapter, we explore methods of modeling microstructures that are representative of PBX materials, specifically PBX We apply GMC and RCM to calculate the effective properties of these microstructures. The effective properties thus obtained are compared to finite element analysis based solutions and to experimental data.

130 CHAPTER 7 SIMULATION OF PBX MICROSTRUCTURES Early simulations of the micromechanical response of PBX 9501 using GMC involved the use of a RVE subdivided into 25 subcells in two dimensions (125 subcells in three dimensions) [15]. The determination of the exact geometry of the RVE and the assignment of constituent phases to subcells in this RVE was based on trial and error until the elastic properties were in agreement with a particular set of experimental data. However, any change in the constituent materials and volume fractions required this process to be repeated until a match was found. This approach defeats part of the purpose of micromechanical modeling as it assumes knowledge of experimentally determined properties of the composite PBX. Most micromechanical calculations for PBXs have been carried out using subgrid models that use highly simplified models of the microstructure (for example, spherical grains coated with binder or spherical voids in an effective PBX material [125]). This is because complicated physical and chemical phenomena are usually involved and complex geometries are not only computationally expensive to model but also are not necessarily accurate representations of the actual microstructure. Closed form solutions from these simple models have been used to provide properties to the macroscopic simulations. More detailed calculations have used microstructures containing ordered arrays of circles or polygons in two or three dimensions to model PBXs [126, 127]. These models do not reflect the microstructure of PBXs and hence have limited use for predicting thermoelastic properties. Better two-dimensional approximations of the microstructure have been constructed from digital images of the material and used by Benson and Conley [128] to study some aspects of the micromechanics of PBXs. However, such microstructures are extremely difficult to generate and require state-of-the-art image processing techniques and excellent images to accurately capture details of the geometry of PBXs. There is also the problem of creating a three dimensional image based on two-dimensional data that requires a large number of cross sections. More recently Baer [129] has used a combination of Monte Carlo and molecular dynamics techniques to generate three-dimensional microstructures that model PBXs. Microstructures containing spheres and oriented cubes have been generated using these techniques and these appear to mimic PBX microstructures well. However, the generation of

131 121 a single realization of these microstructures is very time consuming and often leads to a maximum packing fraction of at the most 70%. Periodicity is also extremely difficult to maintain in the RVEs generated by this method. The method used for simulating most of these microstructures has been finite element analysis. The approach we have taken in this research is to apply our techniques (GMC and RCM) to both manually generated microstructures and randomly generated microstructures. The manually generated microstructures are discussed first and results are shown for a few microstructures. The procedure we use for automatically generated random microstructures is discussed next, followed by a few results for these microstructures using GMC and RCM. The material properties used for the particles and the binder in these simulations are shown in Table 7.1. For the binder, we have used the Young s modulus at 25B C and 0.049/s strain rate determined by Wetzel [7]. The experimentally determined values of the stiffness matrix terms for PBX 9501, at 25B C and 0.05/s strain rate, are also given in Table 7.1. The effective elastic moduli calculated using finite elements (FEM), GMC and RCM have been compared with the values for PBX 9501 shown in Table 7.1. C D Table 7.1. Experimentally determined elastic moduli of PBX 9501 and its constituents [7]. Material E$F G&G = E*F H&H E*F G(H E*F I&I (MPa) (MPa) (MPa) (MPa) Particles Binder PBX Manually Generated Microstructures Six two-dimensional microstructures representing PBXs have been created manually to observe the effect of particle distribution of the effective elastic moduli. Circular particles of various sizes have been used to fill a square RVE. Since PBXs are typically a mixture of coarse and fine grains with the finer grains forming filler between coarser grains, most of the models contain one or a few large particles surrounded by smaller particles. The models are shown in Figure 7.1. The volume fractions of particles in each of these models is around 90J 0.5%. All the models possess square symmetry FEM Calculations Detailed finite element analyses have been performed to determine accurate effective properties of each of the models shown in Figure 7.1. Around 65,000 six noded triangular elements have been

132 122 Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Figure 7.1. Manually generated microstructures for PBXs. used to discretize each RVE and the geometry of the particles has been preserved. To alleviate any element shape problems that could occur because of the close proximity of the circular particles, no particles are allowed to be in contact with each other in the models. In addition, periodic displacement boundary conditions have been applied to determine the stress and strain fields. On the basis of the results from Chapter 6, we assume that the effective properties calculated using FEM are close to the actual values for the microstructures and component materials used. The computed values of the effective stiffness matrix terms K$L M&M, K*L M(N and K*L O&O are shown in Table 7.2. The values of K M&M L for the six models vary from 175 MPa to 240 MPa with a mean of 192 MPa, K*L M(N varies between 75 MPa and 114 MPa with a mean of 91 MPa, and K*L O&O has a range of 8 MPa to 38 MPa with a mean of around 20 MPa. The standard deviation of K M&M L is 17% of the mean,

133 123 that for P*Q R(S is 13% of the mean, and for P$Q T&T it is 59% of the mean. Therefore, the shear stiffness shows the largest variability for the six models. On the whole, the six RVEs predict approximately the same effective properties when compared with the modulus contrast between the particles and the binder. Table 7.2. Effective stiffness for the six model PBX 9501 microstructures from FEM calculations using 65,000 six-noded triangle elements. P*Q R&R FEM/ P*Q R(S FEM/ P$Q T&T UVWX SY UV WX RY UVWX RY FEM/ Expt. Expt. Expt. (MPa) (%) (MPa) (%) (MPa) (%) Model Model Model Model Model Model Mean Std. Dev The effective properties for the six models from FEM calculations are also shown in Figure 7.2. From the Figures 7.2 and 7.1 it can be seen that the ratio P R&R[Z Q P R(S Q increases when, at the edges of the RVE, the amount of binder material decreases. Increase in this ratio is accompanied by an increase in P*Q R&R and a decrease in P$Q R(S. The value of the shear modulus P$Q T&T increases as the amount of binder along the diagonal of the RVE decreases. Models 1 through 3 have a single large particle and many smaller particles and show approximately the same effective behavior. Model 4 has a smaller ratio between the radius of the largest to the smallest particles and generates lower effective P*Q R&R than the average and higher effective P*Q R(S and P T&T Q than the average. Models 5 and 6 which have larger lengths of the boundary containing particles show a stiffer P*Q R&R than the other models even though the volume fractions occupied by particles is slightly lower than in the other cases. The effective stiffness from FEM calculations is also shown as a percentage of the experimentally determined stiffness of PBX 9501 in Table 7.2. It is observed that the FEM calculations underestimate the stiffness by an order of magnitude. The reasons for this could be that more stress bridging is involved in real microstructures leading to increased stiffness. Another reason could be that two-dimensional calculations are not accurate enough. Intuitively, however, it appears that three-dimensional calculations should produce a lower effective stiffness than two-dimensional calculations. It is also possible that the material properties used for the binder are not accurate.

134 C * (FEM) / C (Binder) C * 11 C * 12 C * 66 Figure 7.2. Effective stiffnesses for the six model microstructures from from detailed finite element analyses as a a multiple of the binder stiffness. However, from the data shown in Table 2.8 it can be seen that the modulus of the binder, determined by various researchers, does not differ significantly from the value used in these calculations. Hence, the most probable reason for the low values of the effective stiffness is that stress bridging is not considered in the chosen microstructures GMC Calculations Two approaches have been used to determine the effective properties of the six models using GMC. The first applies a 50% rule to determine initial subcell properties while the second applies of an initial GMC step to determine these. A full GMC calculation is then performed Fifty Percent Rule In this case, subcells are assigned particle material properties if particles occupy more than 50% of the subcell. Binder properties are assigned otherwise. The particles are not resolved well when a square grid is overlaid on the microstructure in this manner. Stress bridging paths are created where there are none in the actual microstructure. This leads to the prediction of higher than actual stiffness values. Figure 7.3 shows the effect of application of the 50% rule on the microstructure of one of the models. The RVE has been discretized into \]]_^`\]] subcells.

135 125 Figure 7.3. Application of fifty percent rule to a model microstructure The Two-Step Approach The large size of the matrix to be inverted in GMC limits the number of subcells that can be used to discretize a RVE. To improve the accuracy of GMC even when the number of subcells is limited, a two-step approach has been devised [130] to determine the effective properties of the RVE. A schematic of the two-step GMC procedure is shown in Figure 7.4. In this approach, the RVE is subdivided into a number of subcells. The cumulative volume fraction of particles that fall in each subcell is calculated. Each subcell is now assumed to be a composite containing a square array of particles, occupying the calculated cumulative subcell volume fraction, in a continuous binder. The effective properties of each subcell are next calculated using the original method of cells procedure [56]. This procedure utilizes one subcell to represent the particles and three subcells to represent the binder in a two-dimensional calculation using four subcells. Most subcells are either completely filled with particle or binder and hence only a limited number of intermediate materials are produced that homogenize the properties of areas that contain both particles and binder. The second step of the calculation involves calculating the effective properties of the full RVE based on the properties calculated for each subcell in the first step. The gives us a better estimate of the effective moduli without the full effect of stress bridging artifacts caused by discretization errors.

136 126 RVE GMC Discretization Homogenized RVE First Homogenization Step Figure 7.4. Schematic of the two-step GMC procedure.

137 Effective Properties from GMC The effective properties of the six microstructures have been calculated using the two approaches discussed above. These are shown in Table 7.3. On the average, the GMC calculations based on the 50% rule predict values of a*b c&c that are around 2.5 times the FEM based values. Surprisingly, the predictions of a c(d b are very close to those predicted by FEM calculations, for all the models. The values of a*b e&e from GMC are, as expected, around 10% of the FEM values. Interestingly, the values of a c&c b from the two-step GMC calculations show some improvement over the 50% rule based GMC results. For models 3 and 4, the values of a*b c&c and a$b c(d are within 5% of the FEM values. For models 5 and 6, these are around 1.5 times the finite element results. However, for models 1 and 2 the value of a*b c&c is still around 2.5 times that of the FEM based values. Though there is some increase in the values of a e&e b calculated by GMC, these are generally around one fifth of the ones calculated using finite elements. Table 7.3. Effective stiffness for the six model PBX 9501 microstructures from GMC calculations. a*b c&c GMC/ a*b c(d GMC/ a$b e&e fghi dj GMC/ FEM fg hi d!j FEM FEM (MPa) (MPa) (MPa) Fifty Percent Rule Approach Model Model Model Model Model Model Mean Std. Dev Two-Step Approach Model Model Model Model Model Model Mean Std. Dev The values of a kmlonqp b a rms b l for the six models, using the 50% rule approach, are shown in Figure 7.5. From the figure, we can observe that the GMC model overestimates a$b c&c in models 1, 2 and 3. This is because errors in discretization in these three models lead to continuous stress

138 128 bridging paths across the RVE. Model 4 predicts t*u v&v quite accurately because the geometry of the particles is better represented than in models 1 through 3. Models 5 and 6 slightly underestimate the value of t*u v&v. However, the reason for this is not obvious C * (GMC 50% Rule) / C * (FEM) C * 11 C * 12 C * 66 Figure 7.5. Ratios of effective stiffnesses calculated using GMC (50% rule) and FEM. The ratio of the effective stiffnesses calculated by the two-step GMC approach to those from FEM calculations, are shown in Figure 7.6. Comparison with the results shown in Figure 7.5 shows that the two-step GMC calculation is an improvement over the 50% rule based GMC. Hence, this approach is used in all further GMC based calculations. It is also observed that the shear moduli are underpredicted by both GMC approaches. One way of rectifying this problem would be to use the shear coupled method of cells discussed in Chapter RCM Calculations Next, the effective elastic moduli were calculated for the six model RVEs using RCM. The RVE was discretized into wxy{z wxy equal sized squares. Each subcell of the RVE was assigned material properties using the 50% rule. An example of the resulting particle distribution for model 4 is shown in Figure 7.7. This figure shows that the microstructure is approximated quite accurately though the edges of the particles are not smooth. The effective stiffness for the six microstructures, calculated using RCM, are shown in Table 7.4. Ratios of RCM to FEM results are also shown in the table. It can be observed that the values of t*u v&v

139 129 C * (GMC Two Step) / C * (FEM) C * 11 C * 12 C * 66 Figure 7.6. Ratios of effective stiffnesses calculated using GMC (two-step) and FEM. Figure 7.7. Microstructure used for RCM calculations for model 4.

140 130 predicted by RCM are about 35 times those predicted by FEM. For }*~ ( this ratio is about 20 and for } & ~ the ratio is around 83. These ratios are also shown in Figure 7.8. Interestingly, it is now model 4 that shows the highest normal stiffness whereas FEM calculations predict that this model should have the lowest normal stiffness of the six. Table 7.4. Effective stiffness for the six model PBX 9501 microstructures from RCM calculations. }$~ & RCM/ }$~ ( RCM/ }*~ & ƒ ˆ RCM/ FEM ƒ FEM ƒ FEM (MPa) (MPa) (MPa) Model Model Model Model Model Model Mean Std. Dev C * (RCM) / C * (FEM) C * 11 C * 12 C * 66 Figure 7.8. Ratios of effective stiffness calculated using RCM and FEM. The question that arises at this point is why does RCM predict effective properties for these microstructures that are 30 to 80 times higher than those predicted by FEM. The best estimates that

141 131 we can expect to obtain with RCM are those from a set of finite element analyses on approximate microstructures such as the one shown in Figure 7.7. These microstructures are based on the 50% rule of assigning element material properties. The effective properties for the six models, determined by finite element analyses on microstructures discretized into Š ŒŽ Š four-noded square elements, are shown in Table 7.5. The ratio of these values with those calculated by RCM are also shown in the table. We can observe that the standard deviation of the values predicted for the six models is quite large compared to those for the FEM calculations using triangular elements as shown in Table 7.2. This implies that discretization errors can have considerable influence on the calculation of effective properties of high volume fraction particulate composites. Table 7.5. Effective stiffness for the six model PBX 9501 microstructures from FEM calculations using Š oœ Š square elements. FEM/ FEM/ RCM/ Œ! Expt. FEM Œ Expt. FEM Œ Expt. FEM * & * RCM/ ( FEM/ RCM/ * & (MPa) (MPa) (MPa) Model Model Model Model Model Model Mean Std. Dev Figure 7.9 shows the ratio of the effective stiffness from FEM calculations using Š ŒŽ Š square elements to those using 65,000 six-noded triangular elements. The figure shows that using square elements to discretize the six models leads to values of & that are, on average, around 12 times those obtained from a FEM model that captures the particle boundaries well. For * ( this ratio is around 10. The variation of the ratio for the six models, for &, is larger than for * & and $ ( and values range from 3 to 10 with a median of around 5. This shows that errors in discretization of particle boundaries can have a disproportionate effect of the effective properties of high modulus contrast composites. A solution to this sproblem would be to compute effective properties of elements at the edges of particles using a method similar to the first step in the two-step GMC approach. We propose to explore this possibility in this research. Table 7.5 shows that the effective stiffnesses predicted by RCM are still many times those predicted by FEM using Š šœž Š square elements. For & these values range from 4 times to 31 times the FEM values, for ( the range is from 3 times to 28 times the FEM values, and

142 C * (FEM 256x256) / C * (FEM) C * 11 C * 12 C * 66 Figure 7.9. Ratios of effective stiffness calculated using FEM ( œ ožÿ œ square elements) and FEM (65,000 triangular elements). for $ & the range is even higher, from 3 times to 181 times the FEM values. These differences be attributed to discretization errors alone. We propose to explore the reasons for the errors and to suggest ways of improving the RCM approximations in this research. Preliminary investigations of the steps in the RCM calculations suggest the following reasons for the errors : 1. At the each level of recursion, a block of 4 subcells is homogenized. The effective stiffnesses of these blocks from RCM calculations are higher than actual because a limited number of elements is used to discretize the blocks. 2. The errors add at each level of recursion because accurate solutions are not obtained. Investigations have showed that at the level of recursion at which sixteen subcells have been homogenized into blocks, the errors in estimation are quite small. However, even at this level, there are blocks which overestimate the effective normal moduli by 4 to 10 times. In addition, the shear moduli can be grossly overestimated or underestimated depending on the geometry. The approximations from RCM can be improved by increasing the number of subcells in a block and the number of elements used to model a block. This would lead to a better estimate of the properties of a block. In addition, the number of recursions should be minimized so that errors accumulate only minimally.

143 133 The stiffness predicted by RCM can be reduced to some extent by using the nine noded formulation discussed in Chapter 5. Another way to deal with the problem would be to use, or ª<«ª<«subcells for each block in the recursive calculation. We propose to explore these approaches in the remainder of this research. 7.2 Randomly Generated Microstructures The microstructure of PBX 9501 is shown in Figure The particles are irregularly shaped and are of a large number of sizes. The volume fraction of particles in PBX 9501 is around 92%. However, if the image is manually assigned materials the highest volume fraction that is obtained is around 70%. Therefore, a digital image is not the best possible source for the generation of microstructures. Instead, we try to simplify the shape of the particles and automatically generate particle distributions that are models of the actual microstructure. Figure Microstructure of PBX 9501 [19]. The preferred method for generating close packed microstructures from a set of particles is to use Monte Carlo based molecular dynamics techniques [131] or Newtonian motion based techniques [132]. In this method, a distribution of particles is allocated to the grid points of a rectangular lattice using a random placement method. Molecular dynamics simulations are then carried out on