ELASTIC PROPERTY PREDICTION OF LONG FIBER COMPOSITES USING A UNIFORM MESH FINITE ELEMENT METHOD

Transcription

1 ELASTIC PROPERTY PREDICTION OF LONG FIBER COMPOSITES USING A UNIFORM MESH FINITE ELEMENT METHOD A Thesis presented to the Faculty of the Graduate School University of Missouri In Partial Fulfillment of the Requirements for the Degree Master of Science by JOSEPH ERVIN MIDDLETON Dr. Douglas E. Smith, Thesis Supervisor AUGUST 2008

2 The undersigned, appointed by the Dean of the Graduate School, have examined the thesis entitled ELASTIC PROPERTY PREDICTION OF LONG FIBER COMPOSITES USING A UNIFORM MESH FINITE ELEMENT METHOD presented by Joseph Ervin Middleton a candidate for the degree of Master of Science and hereby certify that in their opinion it is worthy of acceptance. Advisor: Dr. Douglas E. Smith Readers: Dr. P. Frank Pai Dr. V.S. Gopalaratnam

3 ACKNOWLEDGEMENTS First, I would like that thank my advisor Dr. Douglas E Smith for his guidance throughout my research at the University of Missouri. Also, a big thanks goes to Dr. David Jack for his help with the micromechanical models as well as all of the help in the lab over the last two years. Elijah Caselman, another former lab mate, is also owed a big thanks for developing much of the finite element method used for this research. Next, I thank my committee members Dr. P. Frank Pai and Dr. V.S. Gopalaratnam for their advice and review of my research. Finally, I must acknowledge the National Science Foundation grant DMI for funding this research and Vlastamil Kunc from Oak Ridge National Laboratory for providing the CT scans of a composite part. ii

4 ELASTIC PROPERTY PREDICTION OF LONG FIBER COMPOSITES USING A UNIFORM MESH FINITE ELEMENT METHOD Joseph Ervin Middleton Dr. Douglas E. Smith, Thesis Supervisor ABSTRACT This research numerically evaluates elastic non-isotropic material properties of fiber reinforced composite materials. Calculations are performed with a finite element mesh composed of a uniform array of three dimensional finite elements. To better represent spatial inhomogeneities, an increased number of Gauss points are employed during elemental stiffness calculations. Rather than making a complex finite element mesh with element boundaries at all fiber-matrix interfaces, Gauss points are used to define materials points as in fiber depending on where the point lies within the model. A correction factor is applied to accommodate strain differences within elements that contain both fiber and matrix, allowing for greater accuracy in the uniform mesh elements. In this approach, fewer elements can be used to model a composite system enabling computational time and demands of memory to be reduced. The method also avoids complex meshing routines that are required when fiber-matrix interface geometries are needed. The uniform mesh finite element method developed here is used to predict effective properties for curved long fiber composites and for an actual long fiber composite sample provided by Oak Ridge National Laboratory. The results of the curved fiber uniform finite element models are compared to modified Halpin-Tsai and Tandon- Weng micromechanical models where a good agreement is demonstrated. A convergence analysis is performed to show stability and convergence of results. Image processing techniques are applied to the Oak Ridge National Laboratory sample in order to extract a model from the CT data which was provided. iii

5 TABLE OF CONTENTS Acknowledgements ii Abstract iii List of Tables vi List of Figures viii Chapter 1 Introduction Organization of Thesis Chapter 2 Literature Review Analytical Property Prediction Halpin-Tsai Model Mori-Tanaka Model Numerical Property Prediction Finite Element Method Boundary Element Method Chapter 3 Methods Finite Element Model Finite Element Derivation Strain Correction Factor One Dimensional Three Dimensional Boundary Conditions Composite Micromechanics Models iv

6 3.2.1 Halpin-Tsai Model Tandon-Weng Model Digital Image Processing Spacial Filtering Cropping and Rotation Thresholding Morphological Operations Erosion Dilation Chapter 4 Simple Curved Long Fiber Models Effective Elastic Properties Computed Results Chapter 5 Oak Ridge National Laboratory Composite Sample Split Uniform Mesh Finite Element Model Analysis Procedure Results Complete Uniform Mesh Finite Element Model Analysis Procedure Computed Results Chapter 6 Conclusions and Recommendations Bibliography v

7 LIST OF TABLES 3.1 Halpin-Tsai parameters used for short fiber calculations [1] Elastic properties of matrix and fiber Effective property results Halpin-Tsai and Tandon Weng models with comparison to straight fiber (curvature=0) finite element model Short fiber-based results using Halpin-Tsai and Tandon Weng models with comparison to finite element model with a curvature of Short fiber-based results using Halpin-Tsai and Tandon Weng models with comparison to finite element model with a curvature of Short fiber-based results using Halpin-Tsai and Tandon Weng models with comparison to finite element model with a curvature of Finite element results for 1 st layer in Z-direction of the 3 x 3 x 3 model. (Labels above indicate x, y, and z cube positions respectively, i.e. 111 starts at the origin) Finite element results for 2 nd layer in Z-direction of the 3 x 3 x 3 model. (Labels above indicate x, y, and z cube positions respectively, i.e. 222 is the center cube) Finite element results for 3 rd layer in Z-direction of the 3 x 3 x 3 model. (Labels above indicate x, y, and z cube positions respectively, i.e. 333 is the cube farthest from the origin) Gauss point convergence for model Finite Element results of complete model with 6 G.P. s and 3x3x3 model averaged vi

8 5.6 Finite Element results of complete model with 6 G.P. s and Halpin- Tsai micromechanical model assuming a distribution function of fibers from Equation 5.2 where the parameter n is varied Finite Element results of complete model with 6 G.P. s and Tandon- Weng micromechanical model assuming a distribution function of fibers from Equation 5.2 where the parameter n is varied vii

9 LIST OF FIGURES 3.1 Performing a spacial filter [2] Algorithm for a 3x3 median filter [2] Simple cropping of an image Simple threshold of an image of rice (Source: MATLAB, R2007b, The MathWorks, Natick, MA) Example diagram of morphological operations [3] Panda image example illustrating importance of structural element size [3] Example of noise removal by erosion and dilation [3] Example of erosion and dilation shown at the pixel level. (Source: Model of Long Fiber with curvature of Model of Long Fiber with curvature of Model of Long Fiber with curvature of Model of Long Fiber with curvature of Mesh used for all simple curved long fiber models Elastic modulus property predictions for all curved fiber models Shear modulus property predictions for all curved fiber models Poisson s ratio predictions for all curved fiber models Von Mises stress plot of simple fiber model with curvature of 1 during ɛ xx Von Mises stress plot of simple fiber model with curvature of 1 during ɛ yy viii

10 4.11 Von Mises stress plot of simple fiber model with curvature of 1 during ɛ zz Von Mises stress plot of simple fiber model with curvature of 1 during ɛ xy Von Mises stress plot of simple fiber model with curvature of 1 during ɛ yz Stress plot of simple fiber model with curvature of 1 during ɛ zx Stress plot with deformation of simple fiber model with curvature of 1 during ɛ xx Stress plot with deformation of simple fiber model with curvature of 1 during ɛ yy Stress plot with deformation of simple fiber model with curvature of 1 during ɛ xy Simple model illustrating CT data stacks x3x3 Cube D Model of a region the size of one of the smaller models Slice 1 of 1200 provided by ORNL Showing the entire image processing procedure. a) after cropping b) after thresholding c) after morphological smoothing d) after final rotation and cropping An example showing the progression of morphological operations to remove noise from thresheld image a) original image, b) 1st erosion c) 2nd erosion d) 1st dilation, e) 2nd dilation, f)last dilation and final data image used for analysis Collection of image data into column vector Mesh used for all 3x3x3 split models ix

11 5.9 A representation of the relative distribution of E 11 within the complete model. 3 Layers of results are shown as 3 different bar charts stacked on top of each other. Each chart represents 9 individual model s properties for the Z1, Z2, and Z3 layer Gauss point convergence analysis for model Histogram of E 11 /E m for the 27 split model analyses Histogram of E 22 /E m for the 27 split model analyses Histogram of E 33 /E m for the 27 split model analyses Histogram of G 12 /E m for the 27 split model analyses Histogram of G 23 /E m for the 27 split model analyses Histogram of G 31 /E m for the 27 split model analyses Mesh used for all 3x3x3 split models U2 displacement contour of model-111 s positive and negative X faces during ɛ xx U2 displacement contour of model-111 s positive and negative Y faces during ɛ xx U2 displacement contour of model-111 s positive and negative Y faces during ɛ xx Von Mises stress contours from Z-direction strain as well as corresponding image 696. a)result from complete model F.E. model, b)result from cube 333 F.E. model which corresponds as shown in image, c)image 696 showing fiber slice at level of display for both F.E. results x

12 CHAPTER 1 INTRODUCTION Composite materials are a macroscopic combination of two or more distinct materials separated by a recognizable interface. Composites are widely used for their favorable strength and stiffness to weight ratios and enjoy numerous structural, electrical, tribological, and environmental applications. For this thesis, structural properties will be investigated. A practical definition then, for composites, is a material constructed of a continuous matrix constituent that binds together and gives form to an array of stronger, stiffer reinforcement constituents. The reinforcement constituents investigated in this research are long, cylindrical fibers in a polymer matrix. The resulting composite material has a balance of material properties making it more useful than either material alone. Classification of composites occurs on two levels. The first is with respect to the matrix material. A few examples of these are organic-matrix composites (OCMs), metal-matrix composites (MMCs), and ceramic-matrix composites (CMCs). Organicmatrix composites include two popular types of composites: polymer-matrix composites and carbon-matrix composites. The second level of classification refers to the form of the reinforcement constituent. These include, but are not limited to: particulate reinforcements, whisker reinforcements, continuous fiber laminated composites, and woven composites which include braided or knitted fibers. The long fibers evaluated within this thesis are classified as whisker reinforcements. Manufacturing processes used to form long fiber polymer composites result in 1

13 fibers of varying length and complex disorganized fiber arrangements. As a result, reliable yet cost effective methods for determining homogenized elastic properties of composite materials are important to the composites industry. Determining the effective properties of discontinuous long fiber composites is the focus of this research. No specific materials will be investigated, but rather a method of determining composite material properties which is made of a general matrix material and general fiber material. Materials will be considered to be linear elastic with no investigation into the mechanics of the interface between matrix and fiber. Micromechanical models were found to be less effective for long fibers. More research is needed in long fiber micromechanical models in order to more accurately predict long fiber material properties. The uniform mesh finite element method was successful in predicting material properties for long fiber composites. The uniform mesh finite element method was employed in evaluating a set of regularly arranged curved fiber models as well as a real composite attained from CT scan data. 1.1 Organization of Thesis Chapter One provides an introduction to composite materials and motivates the research. A background of prediction methods is reviewed in Chapter Two with a focus on analytical and numerical property prediction methods. The evaluation methods employed in this research is presented in Chapter Three which includes an overview of the finite element method, analytical property prediction, and image processing. A simple set of curved long fiber models are used in Chapter Four to validate the uniform mesh finite element method and show convergence of results. Chapter Five 2

14 demonstrates the computational method with a material sample which has been provided by Oak Ridge National Laboratory in the form of micro-computed tomography image slices. Because of the large model size, the domain of structural analyisis is split equally by three in all directions leaving twenty-seven individual models to analyze and compare. Also, the entire model is evaluated within the constraints of computation power and memory and compared to the smaller, more refined models. Model periodicity is implemented through a representative volume element approach. Lastly, Chapter Six gives conclusions and recommendations for the research presented. 3

15 CHAPTER 2 LITERATURE REVIEW Over the last half century, composite materials have seen large increases in development in response to their unique and advantageous mechanical properties. The increased application of composites has required a new method for predicting their elastic mechanical properties. Analytical methods [4 7] using simplified methods and assumptions have been widely used due in part to their simplicity and ease of use. Unfortunately, the accuracy of simple analytical methods have been shown to decrease with volume fraction and have yet to be demonstrated on long fiber composites. Underlying assumptions of shape, size, and materials all idealize the composite in order to achieve a relatively accurate solution. The two analytical methods investigated below are the Halpin-Tsai and Mori-Tanaka models. It should be noted that material models in this thesis were developed with a focus on short fiber composites, and will be applied to longer fibers in subsequent sections, where possible. Also, within the last two decades, computational costs in computing have significantly decreased. This has given rise to numeric methods for solving the problem of material property prediction in the fiber composites. By discretizing the domain of the model, these methods can solve the problem without some of the assumptions required in the analytical evaluations. The finite element method and the boundary element method for computing homogenized elastic properties will be discussed. 4

16 2.1 Analytical Property Prediction One main goal in creating analytical property prediction methods is to provide polymer processing communities with a measure for determining elastic properties for injection molded short and long fiber composites. It has been shown that commonly employed analytical methods are based on similar simplifying assumptions which include [8]: 1.) All constituents are linear elastic and isotropic or transversely isotropic materials. 2.) Fibers are cylindrical (and thus axisymmetric), identical in size, and can be characterized by their aspect ratio (l/d). 3.) A perfect bond exists between fiber and matrix. It should be noted that the micromechanical models presented below, Halpin-Tsai and Mori-Tanaka, are representative of ideal materials without defects. For real composites, other factors such as variability in fiber size or in constituent properties must be included for more accuracy when predicting elastic properties in actual products. Despite this, these simplified models have been shown to provide reasonable prediction of material properties of polymer composites. Below, the Halpin-Tsai equations and Mori-Tanaka model will be reviewed Halpin-Tsai Model The first micromechanical model reviewed here is known as the Halpin-Tsai model. The Halpin-Tsai equations are very simple to the useful forms of Hill s [9] generalized self-consistent model with engineering approximations to make them more suitable 5

17 for designing composite materials. Hill assumed the entire composite model to be a cylinder consisting of continuous and perfectly aligned fibers [9]. Statistical homogeneity and traverse isotropy were the only requirements for the cross-section and spacial arrangement of the fibers. By combining the self-consistent approach of Hill with the solutions of Hermans [10] and making a few additional assumptions, Halpin and Tsai provide a simpler analytical form for predicting material properties [5].The Halpin-Tsai equations need only one equation to find all the composite moduli and the longitudinal Poisson s ratio is simply found from the rule of mixtures. It should be noted that the Halpin- Tsai equations are partially empirical. One parameter, ζ, was found by fitting the equations to numerical results. The Halpin-Tsai equations provide reasonable results for E 11 at low aspect ratios, but is less accurate at higher ratios [8]. It also gives better results for E 11 and G 12 at low volume fractions and decreases in accuracy as the volume fraction increases. Hewitt and Malherbe [11] suggested that the parameter ζ should be a function of volume fraction. This was done to improve results for G 12 by fitting a new equation to finite element results of a two dimensional problem. Also, Lewis and Nielson improved upon the Halpin-Tsai equations at high volume fractions [12]. By adding a new function which allowed for the shear modulus predictions to approach infinity as the volume fraction approached its maximum, their modification was shown to improve accuracy of E 11 at high volume fractions [13]. An important flaw in the Halpin-Tsai equations is its inability to accurately predict the transverse Poisson s ratio. It has been shown that predictions can be as much as three times higher than 6

18 experimental and numerical results [8, 14] Mori-Tanaka Model In 1957, Eshelby developed a structural model for a single ellipsoidal inclusion within an infinite matrix [15]. Mori and Tanaka extended Eshelby s single inclusion to one with multiple inclusions which could have stress and strain interactions [4]. This interaction between inclusions has been ignored by Eshelby and therefore lead to poor solutions at high volume fractions. The main assumption made by Mori and Tanaka was that a fiber in a concentrated composite sees the average strain of the matrix [8]. The Mori-Tanaka model gives a much better prediction for ν 23 and slightly better predictions for all other elastic properties when compared to the Halpin-Tsai model [8]. 2.2 Numerical Property Prediction Several numerical methods such as the finite element method and boundary element method have also been widely explored in the literature for predicting the material properties of composites. These methods take on a more brute force technique of accurately predicting the material properties of composites by discretizing the domain of interest. Numerical approaches allow for the representation of very complex geometries while avoiding the simplifying assumptions required by micromechanical models. Despite these advantages, there are problems associated with employing numerical methods when computing elastic properties. First, the domain must be discretized 7

19 through a meshing procedure. This introduces potential issues associated with mesh stability and accuracy and also computer memory constraints. Mesh stability is very important in the convergence of the model s solution. Also, because of the geometries inherent to fiber composites, tetrahedral elements must be used in order to mesh the domain. These elements are constant strain elements requiring a very fine mesh to achieve an accurate solution. This also emphasizes the constraint of computer memory needed. Larger models require much more memory as well as much more time to solve the finite element or boundary element equations Finite Element Method The finite element method has been used in predicting material properties of composite materials which can require significant computational resources, especially for large problems. To reduce the required computational resources, representative volume element (RVE) techniques have been developed. By assuming that the composite is periodic at some size level, the problem can be solved for a smaller RVE of the composite which is often contain properties which are nearly equivalent to the properties of the entire composite. Finite elements are used to fill RVEs in order to attain the solutions. Introducing even more simplicity, RVEs have even been assumed to be as small as one fiber or part of a fiber in order to decrease the problem size [1,8,16,17]. RVE techniques require similar underlying assumptions of regularity and uniformity that the analytical micromechanical models include, however they can give insight that was previously missing. Because the size of the RVE is important for approximating the global elastic 8

20 properties, much research has been performed to investigate it [18 23]. As few as 30 fibers has been found to accurately reflect the global solution within a few percent. Hine, et al. [21] concluded that 100 fibers gave accurate solutions with very little standard deviation. These were very promising studies showing that a supercomputer is not required to utilize this method. Another area of interest investigated by Hine, et al. [21] was effects due to inherent fiber length distributions. In manufactured composites, each fiber is slightly different in length and diameter and the affect of these distributions on overall material properties is of interest. In Hine, et al. [21], over 27,000 fibers of an injected moulded composite part were measured in order to attain length and orientation distribution. These were used to seed a Monte Carlo procedure with 100 fibers. Hine, et al. [21] concluded that using the average length of the sample in the analysis could replace seeding a length distribution for the analysis. Similar to Hine, Gusev, et al. [14] considered the role of distributions of the fiber diameter and spacial orientation with respect to unidirectional short fiber composites. An ultrasonic velocity measurement procedure was developed to obtain elastic constants of the sample. Image analysis was used to measure fiber diameters. Once again, the data seeded a Monte Carlo procedure. This study also investigated three different packing schemes: random, hexagonal array, and square array. Like Hine, et al. [21], the distribution had little effect on the results as long as the mean was known. It was found that the packing scheme of the fibers did have significant effects on the transverse material properties such as Young s modulus, shear modulus, and 9

21 Poisson s ratio. These results were compared to results computed with micromechanical models where it was determined that the micromechanical models give unreliable predictions for transverse material properties. This is because they don t account for the spatial distribution of the fibers within the model [14]. In the late 1990 s, Gusev also developed a multiple inclusion RVE finite element method for predicting properties of composites that are assumed to have some degree of randomness to the arrangement of fibers [19, 20]. Gusev fills a unit cell randomly with a Monte Carlo algorithm to distribute the inclusions throughout the RVE in a manner similar to that found elsewhere [14, 21, 24 28]. Gusev considered spherical inclusions and the process has even been used for packing aligned fibers in RVEs, but becomes incredibly inefficient for misaligned fibers with any relatively high volume fraction. Lusti, et. al. [24] modified Gusev s random packing procedure by increasing the size of the RVE until a certain number of fibers were included and not overlapping. This resulted in a diluted volume fraction of around 0.1%. The size of the RVE was then reduced by systematically moving the already placed fibers without changing their orientation. This allowed them to increase volume fractions up to around 8.0% which was previously impossible. The resulting domain was then meshed with tetrahedral elements to capture the complex geometries of the model. Several methods of simplifying the meshing procedure have been developed. In order to reduce computational expenses, Witcomb, et al. [29, 30] have used a local and global mesh combination to evaluate very fine microstructures. Local analyses are performed within the refined mesh region and are based upon displacements and forces from the global mesh. This translation of force between different meshes of 10

22 different element size becomes difficult since the sizes and number of nodes are so different creating problems in attaining a reliable solution. Zeng, et al. [31] also investigated the simplification of the meshing process. They divided the RVEs into a regular array of subcells and used Gauss quadrature to solve for the stiffness tensor. They defined materials by defining properties at each Gauss point rather than for each element. If the Gauss point were to lie in the fiber, then fiber properties were used and if it were not lying in the fiber, then matrix properties were used. Out of plane modulus was found to be stiffer than that found in literature [31]. Unfortunately, Zeng, et al. [31] ignored the resulting discontinuities in the strain field within each element. This research will incorporate a very similar finite element method with uniform elemental arrays using Gauss quadrature to analyze the long fiber composites. Discontinuities in the strain field within each element is accounted for with a correction factor. Long fibers, which have yet to be studied, will be investigated with the uniform mesh finite element method Boundary Element Method Rather than evaluating the entire three-dimensional solid domain of a model like the finite element method, the boundary element method (BEM) only meshes the surfaces of the model. This is done to solve the partial differential equations of the continuum mechanics through surface integrals or boundary integrals. The boundary element method has a significant computational advantage over finite elements in that the number of degrees is much smaller. This corresponds to fewer equations 11

23 and unknowns in the solution, requiring less computational resources to solve the problem [32]. In the mid 1990 s, Papathanisiou and Ingber [13, 32, 33] modeled short fiber composites successfully. These models included as many as 200 fibers. With the use of the boundary element method and a parallel supercomputer, they were able to study composites which could not be studied by standard micromechanical models or single fiber finite element models. The conclusion was that these simplified micromechanics models suffice at low volume fractions, but lacks accuracy at higher volume fractions. Also, they showed that aligned fibers show an increase in longitudinal modulus compared to randomly distributed alignments [32]. One drawback of this research was the exclusion of the idea of a RVE. This lead to boundary interactions producing artificial fiber alignment in the model. Other assumptions were made with respect to the matrix being incompressible and fibers being rigid in order to simplify the boundary integral calculations. More recently, Nashimura [34] introduced a new method for solving these boundary integrals. The fast multipole method (FMM) decreases the solution time greatly compared to the boundary element method. Solution time was reduced from O(N 2 ) for the BEM to O(N) for the FMM where N is the number of equations in the problem [35]. Lui, et al. [35] has evaluated RVE models with over 5800 fibers and 10 million degrees of freedom. Chen and Liu also developed a BEM with quadratic elements that can represent the matrix and fiber as elastic materials [36]. Their model size was limited to a few thousand elements due to the use of the conventional BEM solution with their quadratic elements [36]. 12

24 CHAPTER 3 METHODS This chapter explains in detail the procedures and methods used to perform the uniform mesh finite element method, including the finite element derivation and boundary conditions applied. Halpin-Tsai and Tandon-Weng micromechanical models will also be explained for comparison to the finite element method. This chapter also explores the method of extracting a model from micro computed-tomography (CT) data scans. These image CT scans were received from Oak Ridge National Laboratory in Tennessee. They were then modified by cropping, thresholding, and morphological operations such as erosion and dilation in order to extract fiber geometry information for uniform mesh procedure. 3.1 Finite Element Model A uniform mesh finite element method is investigated and developed in order to predict material properties for the fiber composites presented. A regular array of finite elements are used, thus relieving the need for a robust meshing algorithm. Instead, an increased number of Gauss points are employed in calculating the stiffness matrix where the spacial distinction between fiber and matrix is decided and incorporated into the Gauss quadrature. The material properties of the composite constituents are assigned to each Gauss point depending on its spacial relation to the fibrous model. The advantage of this procedure is that increasing accuracy requires only to increase the number of Gauss points, not the number of elements, therefore not increasing 13

25 the degrees of freedom of the system and ultimately allowing for very complex fiber models to be able to run on a desktop computer. A special note of acknowledgement is needed for Elijah Caselman s work on the uniform mesh finite element method. Much of the derivations provided below are credited to his thesis [37] Finite Element Derivation Hamilton s principle of representing the differential equations of motion as an equivalent integral equation can be shown to formulate the finite element equations [38]. Hamilton s principle states t2 t 1 δldt = 0 (3.1) where δ is the variational calculus operator and L is the Lagrangian [38] which can be written as L = T Π p = kinetic energy - potential energy (3.2) The potential energy can be shown as a difference between strain energy Π and work done W p as Π p = Π W p (3.3) Strain energy and work done are given, respectively, as 14

26 Π = 1 2 V ε T σdv (3.4) and W p = V b T UdV + t T UdS (3.5) S where ε is the strain vector, σ is the stress vector, b is the body force vector, U is the displacement vector, t is a vector of surface tractions, V is the volume of the body, and S is the surface of the body. A linear relationship between stress and strain may be defined by Hooke s law as σ = [C]ε (3.6) In the case of an isotropic material, the stiffness tensor [C] is symmetric and defined with two unknowns: Young s modulus E and Poisson s ratio ν as seen below [39] [C] = 1 E 1 ν ν ν 1 ν ν ν (1 + ν) (1 + ν) (1 + ν) 1 (3.7) By combining Equations 3.4 and 3.6, applying the variational operator δ, and summing, the equation 15

27 δπ = V δε T [C]εdV (3.8) is obtained. Applying the variational operator δ to W p gives δw p = V b T δudv + t T δuds (3.9) S After combining Equations 3.1, 3.2, 3.3, 3.8, and 3.9, Hamilton s principle can be shown as t2 t 1 V δε T [C]ε + b T δudv + t T δudsdt = 0 (3.10) S In order to integrate over the domain V in Equation 3.10, it is divided into N e subdomains often called finite elements and the displacement vector U over the elemental domain V (e) is interpolated as U = N N 2... N N N N N 8 u 1 v 1 w 1 u 2. = [N]d (e) (3.11) w 8 where d (e) is the elemental displacement vector and [N] is the matrix of shape functions that relates the elemental displacement field to the nodal displacements. There are 8 shape functions for the 8-noded brick element that will be used in the uniform mesh finite element analysis which are written as 16

28 N 1 = 1 (1 ξ)(1 η)(1 ζ) 8 N 2 = 1 (1 + ξ)(1 η)(1 ζ) 8 N 3 = 1 (1 + ξ)(1 + η)(1 ζ) 8 N 4 = 1 (1 ξ)(1 + η)(1 ζ) 8 N 5 = 1 (1 ξ)(1 η)(1 + ζ) 8 N 6 = 1 (1 + ξ)(1 η)(1 + ζ) 8 N 7 = 1 (1 + ξ)(1 + η)(1 + ζ) 8 N 8 = 1 (1 ξ)(1 + η)(1 + ζ) 8 (3.12) The shape functions are shown as functions of the natural element coordinates ξ,η,and ζ. Each natural coordinate ranges from -1 to 1 and is zero at the center of the brick element. For rectangular elements with edges parallel to the global coordinate axes, the relation between global coordinates and natural coordinates is x = x + ξ l x 2 y = ȳ + η l y 2 (3.13) z = z + ζ l z 2 where x,ȳ, and z are the coordinates of the elements centroid and l x, l y, and l z are the respective edge lengths of the element. It can then be shown that the infinitesimal physical coordinates can be related to the infinitesimal natural coordinates as [39] dx = x ξ dξ = l x 2 dξ dy = y η dξ = l y dη (3.14) 2 dz = z ζ dξ = l z 2 dζ 17

29 The infinitesimal elemental volume then becomes dv (e) = dxdydz = l x l y l z dξdηdζ (3.15) The strain vector ε can be written in the form of a displacement gradient and then separated into a matrix of partial differential operators and a displacement vector as ε = ε xx ε yy ε zz γ xy γ yz γ xz = u x v y w z u + v y x v + w z y w + u x z = 0 0 x 0 0 y 0 0 y 0 z z z 0 x y 0 x u v = [B]d (e) (3.16) w where the elemental strain-displacement matrix [B] for the element is written as [39] [B] = [[B 1 ][B 2 ]... [B 8 ]] (3.17) where each [B i ] is defined as [B i ] = 2 N i l x ξ N i l y η N i l y η 0 2 N i l z ζ 2 N i l z ξ 2 N i l z ζ N i l z ζ 0 2 N i l y η 2 N i l x ξ, i = 1, 8 (3.18) 18

30 Combining Equations 3.10, 3.11, 3.16 and summing over all elements, we can now express Hamilton s principle as N e t 2 t 1 V δd (e)t [B][C][B]d (e) δd (e)t b T [N]dV (e) (3.19) (e) S δd (e)t t T [N]dS (e) dt = 0 (3.20) The next step in solving the equation is defining the elemental stiffness matrix and elemental force vector as [K (e) ] = [B] T [C][B]dV (e) (3.21) V (e) P (e) = b T [N]dV (e) + t T [N]dS (e) (3.22) V (e) S respectively. Hamilton s principle can now be written in terms of a forcing vector, displacement vector, and stiffness matrix as N e t 2 t 1 δd (e)t ( [K (e) ]d (e) P (e)) dt = 0 (3.23) where the only non-trivial solution would be when [K (e) ]d (e) P (e) = 0. A simpler representation of this is presented here [K]D = P (3.24) where [K] is the global stiffness matrix, D is the global displacement vector, and P is the global force vector. 19

31 After combining 3.21, 3.15, [K (e) ] can be evaluated using Gauss-Legendre quadrature with N gp Gauss points, ξ i, η i, ζ i, and weights, W i, through [K (e) ] = = [B] T [C][B] l x l y l z dξdηdζ V (e) (3.25) N gp N gp N gp [B(ξ n,η m,ζ l )] T [C(ξ n,η m,ζ l )][B(ξ n,η m,ζ l )] lx l y l z2 W 2 2 l W m W n (3.26) n=1 m=1 l= Strain Correction Factor One Dimensional Conventional finite elements are typically limitted to only one material per element. The elements used in this analysis must be modified to account for the possibility of having 2 separate materials within one element. Multiple materials within a single element will be accomodated by applying a correction factor to the strain-displacement matrix [B] [37]. The strain-displacement factor will first be explained through a one dimensional example of two rods in series, one having matrix properties E m and L m, and the other with fiber properties E f and L f. In this analysis, E is the modulus and L is the rod length. A is the cross sectional area which is the same for both fiber and matrix. It can be shown that the force F in both springs is [37] F = E ma L f F = E ma L m ( x 2 x 1 ) = E fa ( x 3 x 2 ) = E effa ( x 3 x 1 ) (3.27) L f L eff u 1 = E fa u 2 = E effa (u 1 + u 2 ) (3.28) L f L eff by requiring the total length to equal the sums of the individual lengths. This ensures 20

32 continuity in the model and it can be shown that E eff = E m E f E m β + E f (1 β) (3.29) where β is the fiber length fraction defined as β = L f L eff (3.30) (3.31) The strain is equal to the strain-displacement matrix [B] times the elemental displacement vector as shown in Equation Therefore, in order to account for various strain values within a single hybrid element, a correction factor is applied to the strain-displacement matrix [37]: [B] = α[b] (3.32) where if the Gauss point lies within the matrix, the corrections factor is [37] where α α = E f E m β + E f (1 β) (3.33) and if the Gauss point lies within the fiber, the correction factor is [37] α = E m E m β + E f (1 β) (3.34) 21

33 Three Dimensional The one-dimensional procedure may be extended to three dimensions by searching all Gauss points in line with the Gauss points and obtaining a fiber length fraction for each direction as β ξ = N gp W ξ (3.35) N gp Wξ β η = N gp W η (3.36) N gp Wη β ζ = N gp W ζ (3.37) N gp Wζ where W is the Gauss weights at each point lying in a fiber. Correction factors are obtained for each direction and for each Gauss point in each element and the factors enter the strain-displacement matrix calculation in Equation 3.18 as [37] [B i ] = 2 N i α ξ l x ξ N i α η l y η N i α η l y η 0 2 N i α ζ l z ζ 2 N i α ξ l z ξ 2 N i α ζ l z ζ N i α ζ l z ζ 0 2 N i α η l y η 2 N i α ξ l x ξ, i = 1, 8 (3.38) 22

34 3.1.3 Boundary Conditions It has been shown that effective properties attained from a model depend on the boundary conditions imposed [40]. Namat-Nasser [40] found that uniform traction boundary conditions result in a lower bound on properties while uniform displacement boundary conditions result in an upper bound on properties. This is reflective of the bound resulting from the Voigt constant strain assumption and Reuss constant stress assumption in previous models. Appropriate periodic boundary conditions must ensure each RVE has the same deformation mode such that there is no separation or overlap between adjacent RVEs [16]. Essentially, RVEs must be periodically stackable at all times in each of the coordinate directions. These conditions are met by this displacement field u i = ε ik x k + u i (3.39) where u i is the i th component of the displacement vector, ε ik is the average strain tensor, x k is the k th component of the coordinate vector, and u i is the i th component of the periodic displacement vector. It follows that periodic displacement boundary conditions for RVEs are expressed as [16] u j+ i = ε ik x j+ k + u i (3.40) u j i = ε ik x j k + u i (3.41) Where boundary surfaces are referred to as the j + and j - to indicate the positive and negative x j surfaces. The difference between periodic surface nodes must remain 23

35 constant so that subtracting these displacements from positive and negative surfaces must always yield a constant as seen below. u j+ i u j i = ε ik (x j+ k x j+ k ) = cj i (3.42) Note that c j i = 0 yields no relative displacement between the sides of the RVE. Alternatively, axial extensiona dn shear are imposed through non-zero c j i. 3.2 Composite Micromechanics Models Halpin-Tsai Model By combining the self-consistent approach of Hill with the solutions of Hermans and making a few additional assumptions, Halpin and Tsai provide a simpler analytical form for predicting material properties of fiber composites [5, 10]. The Halpin-Tsai equations need only one equation to find all the composite moduli and the longitudinal Poisson s ratio is simply found from the rule of mixtures. It should be noted that the Halpin-Tsai equations are partially empirical where one parameter ζ is found by fitting the equations to numerical results. The Halpin-Tsai equations for the elastic constants are given by [5] P P m = 1 + ζηv f 1 ηv f (3.43) η = (P f/p m ) 1 (P f /P m ) + ζ (3.44) ν 12 = ν f v f + ν m (1 v f ) (3.45) ν 23 = E 22 2G 23 1 (3.46) 24

36 where P represents any moduli and P m and P f are the corresponding moduli of the matrix and fiber. The parameter ζ is dependent upon the moduli being calculated as shown in Table 3.1 where the value of ζ G12 proposed by Hewitt and Malherbe [11] is shown. P P f P m ζ E 11 E f E m 2(l/d) E 22 E f E m 2 G 12 G f G m vf 10 G 23 G f G m K m /G m K m /G m + 2 Table 3.1: Halpin-Tsai parameters used for short fiber calculations [1] The bulk modulus of the matrix K m is found from the simple relationship between material constants as K m = Em. 3(1 2ν m) Tandon-Weng Model Tandon and Weng derive explicit expressions for the elastic constants of a short-fiber composite using the Mori-Tanaka [4] approach. Their formulae for the plane-strain bulk modulus k 23 and the major Poisson ratio ν 12 are coupled, and must be solved iteratively [6, 8]. The Tandon and Weng calculations are given as 25

37 E 11 = E 22 = G 12 = G 23 = E m 1 + v f (A 1 + 2v m A 2 )/A E m (3.47) 1 + v f [ 2v m A 3 + (1 v m )A 4 + (1 + v m )A 5 A]/2A (3.48) G m (1 + v f ) G m G f Gm f)s 1212 (3.49) G m (1 + v f ) G m G f Gm f)s 2323 (3.50) ν 12 = ν ma v f (A 3 ν m A 4 ) A + v f (A 1 + 2ν m A 2 ) (3.51) ν 23 = E 22 2G 23 1 (3.52) The equation for ν 12 shown here was derived by Tucker and Liang [8] and provides a non-iterative formula to the iterative equation of ν 12 presented by Tandon and Weng. The constants are found with Equations 3.53 A 1 = D1(B 4 + B 5 ) 2B 2 A 2 = (1 + D1)B 2 (B 4 + B 5 ) A 3 = B 1 D 1 B 3 A 4 = (1 + D 1 )B 1 2B 3 (3.53) A 5 = (1 D 1 )/(B 4 B 5 ) A = 2B 2 B 3 B 1 (B 4 + B 5 ) The constants B i and D j are found from the following 26

38 B 1 = v f D 1 + D 2 + (1 v f )(D 1 S S 2211 ) (3.54) B 2 = v f + D 3 + (1 v f )(D 1 S S S 2233 ) (3.55) B 3 = v f + D 3 + (1 v f )(S (1 + D 1 )S 2211 ) (3.56) B 4 = v f D 1 + D 2 + (1 v f )(S D 1 S S 2233 ) (3.57) B 5 = v f + D 3 + (1 v f )(S S D 1 S 2233 ) (3.58) D 1 = 1 + 2(µ f µ m )/(λ f λ m ) (3.59) D 2 = (λ m + 2µ m )/(λ f λ m ) (3.60) D 3 = λ m /(λ f λ m ) (3.61) where µ m, µ f, λ m, and λ f are Lame s constants for the matrix and fiber materials. Lame s constants are related to the Young s modulus E and Poisson s ratio ν by λ = Eν (1 + ν)(1 2ν) (3.62) and µ = E 2(1 + ν) (3.63) The S ijkl in Equations 3.54 are the Eshelby tensor components for a spheriodal inclusion defined as 27

39 { ] } 1 S 1111 = 1 2ν m + 3α2 1 [1 2(1 ν m ) α 2 1 2ν m + 3α2 g α 2 1 [ ] 3 α 2 S 2222 = S 3333 = 8(1 ν m ) α ν m g 4(1 ν m ) 4(α 2 1) { [ ] } 1 α 2 S 2233 = S 3322 = 4(1 ν m ) 2(α 2 1) 3 1 2ν m + g 4(α 2 1) { } 1 α 2 S 2211 = S 3322 = 2(1 ν m ) α α 2 4(1 ν m ) α 2 1 (1 2ν m) g [ 1 S 1122 = S 1133 = 1 2ν m + 1 ] [ ν 2(1 ν m ) α 2 m + 1 2(1 ν m ) { [ ] } 1 α 2 S 2323 = S 3232 = 4(1 ν m ) 2(α 2 1) ν m g 4(α 2 1) { 1 S 1212 = S 1313 = 1 2ν m α (1 ν m ) α [ 1 2ν m 3(α2 + 1) 2 α 2 1 where α is the aspect ratio of the fiber, and g is a parameter defined as (3.64) ] 3 g 2(α 2 1) ] } g g = α { α } α (α 2 1) 2 1 cosh 1 (α) 3/2 (3.65) 3.3 Digital Image Processing Image processing can be defined as changing the nature of an image in order to either [2]: 1.) Improve its pictorial information for human interpretation 2.) Render it more suitable for autonomous machine perception This thesis explores digital image processing which analyzes and modifies digital images on a computer. A digital image differs from a photo in that x, y, and f(x, y) are all discrete integers. Usually, these points in the image are called picture elements, or more simply pixels having values that range from 1 to 256 with a brightness value from 0(black) to 255(white). The pixels surrounding any single pixel of interest constitute 28

40 its neighborhood. A neighborhood can be defined as a matrix of elements. For example, a neighborhood could be a 3x3, 5x5, or 3x7 array of pixels. Neighborhoods usually have an odd number of rows and columns so that the pixel of interest is centered. If the neighborhood had an even number of rows or columns, it would be necessary to specify which element would be the current pixel of interest [2, 41]. Understanding the basics of image data structures is important. First, a digital image can be viewed as a matrix of values which correspond to some value of color or intensity. The four basic types of images are binary, greyscale, true color (RGB), and indexed. Binary images are black and white images in which the matrix of values are filled with either an on or off setting. Greyscale images have a scale of shades between 0(black) and 255(white). There are 256 possibilities because the data is stored in 8 bits (8 bits with 2 possibilities per bit = 2 8 = 256 possibilities per pixel). True color images are an extension of greyscale, but with shades of color divided into 3 layers of color: one layer of red, one layer of blue, and one layer of green. These combinations give 16,777,216 color possibilities per pixel(256 3 ) [41]. Lastly, indexed images take advantage of the fact that most images do not have 16+ million colors. For simplification and in order to save memory, an index of colors in the image is used and the images are saved with this smaller index number as its data. This index then refers to a certain color allowing for images to be saved using less space than full 3 matrix layers of RGB color size. In order to conveniently study digital image processing, the different algorithms can be classified based on the nature of the task at hand. These categories are [2] 29

41 Image Enhancement Image Restoration Image Segmentation Image enhancement refers to modifying an image so that the result is more suitable for a particular application [2]. Examples include sharpening an out-of-focus image, highlighting edges, improving contrast or brightness, and removing noise. Image restoration refers to reversing damage done to an image by some known cause such as removing a blur caused by a filter, removing optical distortions, or removing periodic interference. Finally, image segmentation involves subdividing an image into constituent parts, or isolating certain aspects of an image. Examples of image segmentation are finding lines or particular shapes in an image and identifying cars, trees, buildings, or roads in aerial photography. It is important to note that all of these image processing operations share common elements. The same algorithms may be used for more than one category with the goal of the procedure determining its purpose. The rest of this section explains basic image manipulations that can effectively be used to extract fiber/matrix data from the set of slices. Spacial filtering, cropping, rotation, thresholding, and a few morphological operators like erosion and dilation are explained in the following in more detail. 30

42 3.3.1 Spacial Filtering Spacial filters are usually associated with smoothing or edge detection algorithms. They look at a neighborhood mask of pixels and modify the pixel of interest based on a function of these neighborhood pixels. The mask is usually an array with an odd number of rows and columns. As stated before, this allows for the neighborhood to center on the pixel of interest. The combination of neighborhood mask and function defines the filter. Figure 3.1 shows the basic procedure of filtering an image. Figure 3.1: Performing a spacial filter [2]. One widely used filter is known as median filtering where the pixel of interest is an average of its surrounding neighbors. Figure 3.2 shows the function of a pixel for a 3x3 masked median filter. 31

43 Figure 3.2: Algorithm for a 3x3 median filter [2]. In this figure, e is the pixel of interest that is changed from the function and mask. Another important category of filters is known as frequency filters. These frequency filters analyze how quickly and how often the pixels change intensity within an image and modify pixels based on the information [2]. High pass filters leave high frequency components and only affects low frequency parts of an image. Conversely, low pass filters leave low frequency components and only affect high frequency portions of the image. High pass filters provide excellent edge detection within images. A few high pass filters used for this include the Laplacian and Laplacian of Gaussian filters. The Laplacian filter is a 2-D isotropic measure of the 2nd spatial derivative of an image. The Laplacian in 2-D is defined as L(x, y) = 2 I x I y 2 (3.66) where I(x, y) are the intensity values of the image. Since an image is discrete, the mask used is an approximation of this function. The Laplacian of Gaussian is the same filter applied after a Gaussian filter has been performed. This is done to reduce the sensitivity inherent to the Laplacian filter. The Gaussian filter is an averaging technique based on a nonuniformly weighted mask that approximates the Gaussian 32

44 distribution function discretely over the mask. There are many other filters with various advantages and disadvantages. The discussion here is limited to the Fourier Transform and Discrete Fourier Transform since they are widely used in image processing image processing. These transforms allow for tasks which would be impossible to otherwise perform as their efficiencies allow other tasks to be performed more quickly. The Fourier Transform provides, among other things, a powerful alternative to linear spatial filtering. It is also more efficient to use the Fourier transform than a spatial filter for a large filter. The Fourier Transform allows the isolation and processing of particular image frequencies, and can therefore perform high and low filter passes with great precision and speed. There are two items to keep note when performing any type of filtering: edge decisions and using a shadow image to apply a filter. Because masks are not fully defined near edges, deciding what to do with them becomes important. Sometimes the edges are ignored all together performing the filter only on pixels who s masks are fully defined. Another popular way of avoiding these edge effects is assuming the undefined portions of the mask to be zero. Second, the shadow copy is a simple but crucial step that must be mentioned. In performing filtering, a new image must be created based on the old one being filtered. The old image must not be updated as in some other processes or the filtered data would affect the unfiltered data s filtering process. 33

45 3.3.2 Cropping and Rotation These simple image editing techniques may seem trivial but are very important in reducing memory requirements and computational time as well as establishing a coordinate system that is consistent with the scanned data. Cropping removes rows or columns of the image within a specified region and rotation moves pixels so that the region of interest is rotated. An example of a simple crop can be seen below. Figure 3.3: Simple cropping of an image Thresholding Thresholding is a process by which you limit values of an image [3]. For example, a 256 or 8 bit (2 8 = 256) per pixel gray-scale image could have a thresholding procedure applied where all the dark regions or possibly all of the white regions are removed. The procedure for choosing the value at which to threshold is where this procedure can become very complicated. There are many statistical and weighted methods that can be used for determining a threshold value. The procedure is as simple as an if statement in programming language. An example of thresholding a greyscale image 34

46 is shown in Figure 3.4. Figure 3.4: Simple threshold of an image of rice (Source: MATLAB, R2007b, The MathWorks, Natick, MA) Morphological Operations Morphological analysis started in the late 1960 s. It uses Minkowski algebra to manipulate classical two valued black and white images [41]. These set-based operations are used to enhance photos to extract data that may otherwise be overlooked. Structural elements are defined and used as sets for the operations. They can be any shape with round or square being most popular, and are similar to masks described above, but are not normally rectangular in shape. In the case of this research, the focus will be on cleaning up CT data TIFF images in order to achieve a binary representation of each slice that depicts fibers and matrix. Figure 3.5 shows an image and simple morphological operation used to modify its appearance. By using multiple successive iterations of morphological operations, it is possible to remove noise and retain the fiber geometry. It can be seen in Figure 3.5 that the procedure and order of operations, give much 35

47 Figure 3.5: Example diagram of morphological operations [3]. different results. Figure 3.6 is another example of the implications of structural element size. Clearly if the structural element chosen is large, too much data is removed and made impossible to later retrieve. This is why relatively smaller elements are used with multiple iterations in order to achieve larger dilations or erosions. Figure 3.7 is another example of how iterations of erosion and dilation can remove noise but keep inherent pieces of the image. The pepper like noise in Figure 3.7 can be removed without eliminating the rest of the image Erosion Erosion removes pixels from an image. For binary images, erosion is performed by turning a pixel from ON to OFF by some criteria based on the values of neighboring 36

48 Figure 3.6: Panda image example illustrating importance of structural element size [3] Figure 3.7: Example of noise removal by erosion and dilation [3]. pixels. Erosion is defined by the intersection of an image and a structural element as A B = b B A b (3.67) where is Minkowski subtraction, A is the image and B is the structural element of erosion, and b is the movement vector for A [41]. In a sense, if B does not fit somewhere inside A, then that pixel of the image is turned OFF. This process is sometimes referred to as image shrinking [3]. 37

49 Dilation Dilation adds pixels to an image. This is done by turning a pixel from OFF to ON by some criteria based on the values of neighboring pixels. Dilation is defined by the union of an image and a structural element as A B = b B A b (3.68) where is Minkowski addition, A is the image and B is the structural element of erosion, and b is the movement vector for A [41]. This process is sometimes referred to as image growing [3]. Dilation and erosion are essentially inverse operations of one another. Their relationship can be shown as A B = Ā ˆB (3.69) or A B = Ā ˆB (3.70) where the complement of an erosion is equal to the dilation of the complement [2]. A simple example of erosion between A and B is shown below in Figure 3.8. The pixels are either added by dilation or removed by erosion. 38

50 Figure 3.8: Example of erosion and dilation shown at the pixel level. (Source: 39

51 CHAPTER 4 SIMPLE CURVED LONG FIBER MODELS In this chapter, the uniform mesh method is applied to predict the elastic properties of simple long fiber models. The models are all regularly packed arrays of long fibers that are 1mm long with different curvatures. The fibers are laid in a planer orientation in order to simplify the procedure and to provide for a better comparison among models. The results of the uniform mesh method is compared with results in the literature where possible. In addition, FEA results will be compared with the Halpin-Tsai and Tandon-Weng micromechanical models that are used for short fiber composites. Each model is defined as a planar array of regularly stacked and positioned fibers. The models are different only in fiber curvature where a simple quadratic function is used to define fibers having a constant curvature. An engineering definition of curvature was used here as the second derivative of the function that defines the fiber s geometry. The models used here appear in Figures and have a fiber volume fraction of with the same RVE, fiber length, and number of fibers. The x direction is the axial direction of the fibers, y direction is normal to this in the plane of fibers, and z direction is normal to the fiber axis and plane. 40

52 Figure 4.1: Model of Long Fiber with curvature of 0. Figure 4.2: Model of Long Fiber with curvature of Effective Elastic Properties The elastic properties are predicted by the uniform mesh finite element method, Halpin-Tsai equations, and a modified Tandon-Weng equation. The process to predict material properties for the curved fibers with the uniform mesh finite element method is discussed in detail below. ABAQUS (Simulia, Providence, RI) solves a system based on an input file that contains the nodal coordinates, connectivity matrix for elements, materials, boundary 41

53 Figure 4.3: Model of Long Fiber with curvature of 2. Figure 4.4: Model of Long Fiber with curvature of 4. conditions, and requested output. Fortran was used to write the input file for the regular array of parallelpiped elements used in the analysis. The mesh used for the regular array of simple curved long fiber is shown below where there are 120 elements in the x-direction, 40 elements in the y-direction, and 10 elements in the z-direction. These models were analyzed with 48,000 elements using 6 Gauss Points for integration in each direction. This problem had 163,722 degrees of freedom and was solved with ABAQUS Because the uniform mesh did not define the fibers, an algorithm was needed to 42

54 Figure 4.5: Mesh used for all simple curved long fiber models. distinguish between fiber and matrix within the elements. The Gauss point properties were determined by a simple algorithm based on the definitions of the fibers y = f(x) = c 2 (x x f) 2 + y f ; x (x min, x max ) (4.1) where c is the curvature of the fibers, x f is the x location of the middle of the fiber, and y f is the y location of the center of the bottom most portion of the fiber. x min and max are predetermined based on the fiber length of 1mm. The equation for solving for these two value is given as s = xmax x min 1 + [f (x)] 2 dx (4.2) For the straight fiber case, x f and y f its centroid. The Z direction is not needed since the fiber lies on the xy-plane, therefore the z-location is known. 43

55 Since the length of the fiber is known, its centroid is defined, and the function for the fiber is specified, testing a simple geometric calculation based on an ellipsoidal cross section of a round fiber cut at an angle could give enough information to determine if the Gauss point is in the fiber. The model is sliced parallel to the yz-plane to determine a fiber cross section ellipsoid can be determined create a condition for the fiber to lie inside or out of the fiber. The conditions for being in a fiber are (z gp z f ) 2 a 2 + (y gp y f ) 2 b 2 1 (4.3) x x min (4.4) x x max (4.5) where z gp and y gp are the Gauss point locations and a and b are the minor and major axes of the cross sectional ellipse, respectively. After the algorithm for defining fibers inside the uniform mesh was developed, an ABAQUS UEL user subroutine developed previously by Caselman [37] was used to implement the unique element type described in Chapter 3. The routine is called at the beginning and end of each solution iteration. Input to the subroutine is the current nodal displacement vector, nodal coordinates, and material properties and the subroutine returns the elemental stiffness matrix and forcing vector. The periodic boundary conditions are definted through the ABAQUS *EQUATION command. ABAQUS then assembles the global matrices and vectors and solves the system of equations for the unknown nodal displacements. In order for the UEL subroutine to return the correct elemental matrices and 44

56 vectors, it must include the functions definted above in Equations The correction factors in Equations 3.33 and 3.34 are based on these Gauss point material properties and are used to calculate the elemental stiffness matrices in Equation The elemental force vector is found simply by multiplying the calculated stiffness matrix with the given elemental displacement vector from ABAQUS. Since the uniform mesh method uses a modified linear 8-node brick element, standard parameters like stress and are not easily output through the solution. In order to easily gain these plots, standard ABAQUS C3D8 8-node, 3 degrees of freedom per node, brick elements are overlayed with the same nodal coordinates and connectivity matrix as in Caselman [37]. So that these would have a negligible affect on the results, they were given modulus values on the order of 10 8 times smaller than the fiber or matrix. Unfortunately, the strains or stresses are then only calculated at the C3D8 Gauss points so all contour plots to follow do not show as much detail as is in the analysis. Once the finite element solution is performed, the resulting reaction forces are used to compute the effective material properties for the RVE. The average strain is defined as ε ij = 1 V V ε ij dv (4.6) where V is the volume of the RVE. The strain tensor ε ij can be written in terms of displacement as 45

57 ε ij = 1 2 (u i,j + u j,i ) (4.7) The average strain in Equation 4.6 is transformed to a the boundary integral using Equation 4.7 and the Gauss Theorem as [1] ε ij = 1 2V S (u i n j + u j n i )ds (4.8) where n j is the j th component of the unit normal vector to the boundary surface S of the RVE. For a parallelepiped element in which the boundary faces are normal to the coordinate axes, the normal will have only one non-zero component and Equation 4.8 reduces to [16] ε ij = 1 2V [ (u j+ i S j = 1 2V (cj i S j + c i js i ) = cj i l xl k + c i jl j l k 2l i l j l k = 1 2 ] u j i )ds + (u i+ j u i j )ds S i c j i l i + c i jl j l i l j (4.9) where l i refers to the length of the RVE in the x i direction, and c j i defining the periodic boundary condition given in Equation is the constant Therefore, the average strain is found from the size of the RVE and the periodic boundary conditions above. In a similar manner, average stress is defined in terms of the stress tensor σ ij as 46

58 σ ij = 1 V V σ ij dv (4.10) The equations of equilibrium can be written as σ ij,j = 0 (4.11) given that there are no body forces. Using the equilibrium equation it can be shown that [42] (σ ik x j ),k = σ ik,k x j + σ ij x j,k = σ ij δ jk = σ ij (4.12) Combining Equations 4.10, 4.12 and again using Gauss s theorem, it can be shown that σ ij = 1 V S σ ik x j n k ds (4.13) By the definition of periodic boundaries, the stress at two corresponding points on opposite surfaces must be equal. Similar to the derivation of Equation 4.9, the average stress can be shown as [16] σ ij = 1 V = 1 V [ ] (σ + im x+ j )ds (σ im x j )ds S m + Sm σ + im (x+ j x j )ds (4.14) S + m 47

59 when m j, x + j = x j, and for m = j, then x+ j x j = l j and therefore [16] σ ij = l j V S j σ ij ds = R ij S j (4.15) where R ij is the sum of the reactions forces on the boundary face S j where j does not indicate summation. Once the average stress and strain have been obtained through Equations 4.9and 4.18, the effective elastic properties are evaluated by σ = [ C] ε (4.16) where [ C] is the average or effective stiffness matrix of the composite, σ is the averaged stress vector obtained from 4.18, and ε is the averaged strain vector from The averaged stress vector is related to the stress tensor by σ = [σ 11, σ 22, σ 33, σ 23, σ 31, σ 12 ] T. The average strain vector and tensor are related similarly. For an orthotropic composite, the average stiffness tensor is related to the average material properties through the following [ C] = 1/Ē11 ν 21 /Ē22 ν 31 /Ē ν 12 /Ē11 1/Ē22 ν 32 /Ē ν 13 /Ē11 ν 23 /Ē22 1/Ē /Ḡ /Ḡ (4.17) /Ḡ13 From Equation 4.17 it is shown that for an orthotropic material there are nine independent material properties (E 11, E 22, E 33, ν 12, ν 23, ν 13, G 12, G 23, G 13 ). Therefore, nine independent equations are needed to solve for the nine independent material properties. The nine equations are obtained from six independent strain conditions defined 48

60 by six sets of deformation constants, c j i : set 1 : c 1 1 = 0.05l x, all other c j i = 0 (4.18) set 2 : c 2 2 = 0.05l y, all other c j i = 0 (4.19) set 3 : c 3 3 = 0.05l z, all other c j i = 0 (4.20) set 4 : c 2 1 = 0.025l y, c 1 2 = 0.025l x, all other c j i = 0 (4.21) set 5 : c 3 2 = 0.025l z, c 2 3 = 0.025l y, all other c j i = 0 (4.22) set 6 : c 3 1 = 0.025l z, c 1 3 = 0.025l x, all other c j i = 0 (4.23) The six strain sets represent three uniaxial extension conditions and three pure shear conditions. From the three uniaxial extension conditions nine nontrivial equations will be obtained, only six of which are independent, and from the three pure shear conditions the three remaining independent equations will be obtained. To ensure the effective stiffness matrix is of the form in equation 3.7, Lagrange multipliers are used as in Caselman [37] when solving equation 3.6 to impose the necessary symmetry constraints. This is done by first forming average stress and strain matrices from the average stress and strain vectors obtained from the six finite element runs so that Equation 3.6 becomes [σ] = [C][ε] (4.24) By multiplying both sides of Equation 4.24 by the transpose of the average strain matrix [ε] T the following unconstrained least squares equation is obtained [ε] T [σ] = [C][ε] T [ε] (4.25) 49

61 Equation 4.25 can then be expanded as follows: ε T ik σ k1 ε T ik σ k2. ε T ik σ k6 = 36 1 ε T ik ε kj 0 ij 0 ij 0 ij 0 ij 0 ij 0 ij ε T ik ε kj 0 ij 0 ij 0 ij 0 ij 0 ij 0 ij ε T ik ε kj 0 ij 0 ij 0 ij 0 ij 0 ij 0 ij ε T ik ε kj 0 ij 0 ij 0 ij 0 ij 0 ij 0 ij ε T ik ε kj 0 ij 0 ij 0 ij 0 ij 0 ij 0 ij ε T ik ε kj C 11 C 12 C 13. C 66 (4.26) 36 1 or b = [A]C (4.27) where i and j are the components of the stress and strain matrices and i, j 1, 2,.., 6. The 6 6 orthotropic stiffness matrix contains 12 non-zero constants. Due to symmetry, the number of constants reduces to 9 as described above. Therefore, 27 constraints must be applied to the solution process. These constraint equations are combined with Equation 4.27 as follows (see e.g. [43]) b 0 = [A] [X] [X] [0] C λ 63 1 (4.28) (4.29) where λ is a vector of Lagrange multipliers, and [X] is a matrix of symmetry and zero constraints. Therefore, the modified C vector is computed by solving C λ = [A] [X] [X] [0] b (4.30) 50

62 4.2 Computed Results The ability of the uniform mesh finite element model to predict properties for long fibers will now be investigated. The results for the long fiber models are compared to the Halpin-Tsai and modified Tandon-Weng models. The material properties used will be an idealized set following Tucker and Liang [8] where E f /E m = 30 as given in Table 4.1. P roperty F iber Matrix E(GP a) 30 1 ν Table 4.1: Elastic properties of matrix and fiber The results for the Halpin-Tsai and Tandon-Weng models for straight fibers as well as the results for the set of long fiber models is shown below in Table 4.2 and??. Elastic Halpin T andon Curvature constants T sai W eng of 0 E 11 /E m E 22 /E m ν ν G 12 /E m G 23 /E m Table 4.2: Effective property results Halpin-Tsai and Tandon Weng models with comparison to straight fiber (curvature=0) finite element model. Table 4.2 shows good agreement for E 11 in comparison to the Halpin-Tsai results, but the Tandon-Weng result are 9% higher. This could be due to the fact that the aspect ratio is much higher than allowed for these micromechanical models. The 51

63 shear terms of the uniform mesh method are within 15% of both micromechanical models. The Poisson s ratio ν 23 underpredicts the Halpin-Tsai results by nearly 45%, however, similar trends related to the Halpin-Tsai equations appear elsewhere [8, 21]. The Tandon-Weng model is also higher for ν 23 as compared to the uniform mesh. Note, however, that ν 12 is in good agreement between the uniform mesh and Halpin- Tsai model, but the Tandon-Weng results are nearly 40% higher than either of these. Next, related calculations based on the short fiber suspension mechanics models of Advani and Tucker [7] and also Jack and Smith [44] are applied to the curved fiber models presented. These short fiber models depend on unidirectional fiber composite properties which can be evaluated with any micromechaniics model such as the Halpin-Tsai or Tandon-Weng models. The orientations tensors a ij and a ijkl (see Advani and Tucker [7]) are employed to account for the curvature of the fibers through a length averaged orientation tensor evaluated through a ij = N f 1 L I I=1 L I p i (s)p j (s)ds (4.31) and a ijkl = N f 1 L I I=1 L I p i (s)p j (s)p k (s)p l (s)ds (4.32) where N f is the number of fibers, L I is the fiber length, and the p i are components of the unit vector which define the fiber direction [7, 44]. Results for the Halpin-Tsai and Tandon-Weng models for the curved fiber models are compared with the uniform mesh finite element analysis in Tables 4.3, 4.4, and

64 Elastic Halpin T andon F EA constants T sai W eng E 11 /E m E 22 /E m E 33 /E m ν ν ν ν ν ν G 12 /E m G 23 /E m G 31 /E m Table 4.3: Short fiber-based results using Halpin-Tsai and Tandon Weng models with comparison to finite element model with a curvature of 1. Similar trends shown in the results of Table 4.2 are also seen for all curved fiber models. The Halpin-Tsai and Tandon-Weng models both predict higher moduli in all directions for curvatures of 1, 2, and 4 except for the E 22 predicted for a curvature of 4. Again, this could be due to the fact that the aspect ratio is so high for the long fibers. The shear moduli G 23 and G 31 of the uniform mesh method are within 15% of both micromechanical models, but G12 is 30 50% lower than the micromechanical models. All Poisson s ratios are computed with high errors indicating that traditional micromechanical models are not applicable to the long fiber composites studied here. 53

65 Elastic Halpin T andon F EA constants T sai W eng E 11 /E m E 22 /E m E 33 /E m ν ν ν ν ν ν G 12 /E m G 23 /E m G 31 /E m Table 4.4: Short fiber-based results using Halpin-Tsai and Tandon Weng models with comparison to finite element model with a curvature of 2. Elastic Halpin T andon F EA constants T sai W eng E 11 /E m E 22 /E m E 33 /E m ν ν ν ν ν ν G 12 /E m G 23 /E m G 31 /E m Table 4.5: Short fiber-based results using Halpin-Tsai and Tandon Weng models with comparison to finite element model with a curvature of 4. 54

66 From a comparison of the curved fiber analyses, it is seen that E 11, ν 13, and ν 23 all decrease as the fibers curvature increases as shown in Figures 4.6 and 4.8. Even a curvature of 1 decreases E 11 by over 27.9% compared to straight fibers. Properties which increase as curvature increases includes ν 12, ν 21, ν 31, G 12, and G 23. The largest increases between a curvature of 0 and 4 was ν 21 and ν 31 with a 184% and 100% increase respectively E 11/E m E 22/E m E 33/E m 4 Effective Property C urvature Figure 4.6: Elastic modulus property predictions for all curved fiber models. Contour plots of Von Mises stress fields for each of the 6 loading cases for the curvature of 1 model appears in Figures 4.9, 4.10, 4.11, 4.12, 4.13, and These are indicative of the other curved fiber model results and reveal the fiber locations within the uniform mesh as well as the general stress state of the RVE. 55

67 Property Prediction G 12/E m G 23/E m G 31/E m C urvature Figure 4.7: Shear modulus property predictions for all curved fiber models nu21 nu [m /m ] C urvature Figure 4.8: Poisson s ratio predictions for all curved fiber models. 56

68 Figure 4.9: Von Mises stress plot of simple fiber model with curvature of 1 during ²xx. Figure 4.10: Von Mises stress plot of simple fiber model with curvature of 1 during ²yy. Figure 4.11: Von Mises stress plot of simple fiber model with curvature of 1 during ²zz. 57

69 Figure 4.12: Von Mises stress plot of simple fiber model with curvature of 1 during ²xy. Figure 4.13: Von Mises stress plot of simple fiber model with curvature of 1 during ²yz. Figure 4.14: Stress plot of simple fiber model with curvature of 1 during ²zx. 58

70 Stress contour plots on the deformed mesh verify the periodicity in the solution as seen in Figure Arched deformations are seen in all curved models. The straightening of the fibers during axial loading causes the model change shape and begins arching the RVE. Figure 4.17 presents the model s periodicity well. Figure 4.15: Stress plot with deformation of simple fiber model with curvature of 1 during ɛ xx. Figure 4.16: Stress plot with deformation of simple fiber model with curvature of 1 during ɛ yy. 59

71 Figure 4.17: Stress plot with deformation of simple fiber model with curvature of 1 during ɛ xy. 60

72 CHAPTER 5 OAK RIDGE NATIONAL LABORATORY COMPOSITE SAMPLE For this research, Oak Ridge National Laboratory (ORNL) has provided the CT image data in TIFF format of a fiber composite specimen. These CT images will provide a means of distinguishing between fiber and matrix in the uniform mesh finite element method in order to predict material properties for the ORNL composite. CT imaging is used to obtain the geometric definition of fibers within a moded composite part. Extracting the relevant data out of the scanned images required additional evaluation in this research as the CT images can be grainy, distorted, and sometimes incomplete. The fibers are relatively small compared to the resolution of what the scanners were developed for - medical examinations of the human anatomy. Since fibers can be as small as micrometers in diameter, specialized micro-ct scanners must be used. Even at these very high resolutions, the structural data is imperfect and needs to be digitally enhanced and then extracted. A simple representation of the scanned stack of data is shown below in Figure 5.1 where each slice represents a CT image. Each layer of the stack is referred to as a slice and is regularly spaced apart from the other layers. Each layer is a matrix of data scanned as a TIFF, DICOM, or BMP. This information can be manipulated in order to retrieve the data stored within the images. Some research is performed where the lengths or diameters are measured from the scans, and these data points are used to seed Monte Carlo algorithms [21]. Because of the shear size of the model, this uniform mesh finite element model was 61

73 Z Y X Figure 5.1: Simple model illustrating CT data stacks. evaluated with a poor mesh size for the entire composite, and also split up by 3 in each direction giving 27 smaller pieces of the whole. This chapter analyzes and discusses both models with the 3x3x3 split model first. The figures below give a simple representation of the models. The first drawing, Figure 5.2 illlustrates the size and relation of the two different approaches and the second, Figure 5.3 shows the fibers segmented and modeled in 3D by using Amira (Visage Imaging, Carlsbad, CA). 62

74 Model 133 Model 233 Model 333 Model 123 Model 223 Model 323 Model 113 Model 213 Model 313 z = 696 slices ( mm) Model 112 Model 212 Model 312 Z Y Model 111 Model 211 Model 311 y = 816 px. ( mm) X ORIGIN x = 852 px. ( mm) Figure 5.2: 3x3x3 Cube 5.1 Split Uniform Mesh Finite Element Model Analysis Procedure This section explains the image manipulations used to extract fiber/matrix data from the set of CT slices obtained from ORNL. The CT data will then be used to determine if the finite element Gauss points are within a fiber or not in order to implement the UEL subroutine with ABAQUS as described in Chapter 4. Cropping, rotation, thresholding, and the erosion and dilation morphological operators were used. The first slice in the set of CT data from ORNL appears in Figure 5.4 in its original form. The first step in the image processing for the TIFF CT data files was to crop the images. This was done in order to save memory since each of the 696 slices analyzed 63

75 3 2 1 Figure 5.3: 3D Model of a region the size of one of the smaller models. was over 11 MB in size. Also, the fact that most of the boarder data outside the actual sample as seen in Figure 5.4 was not needed to define the finite element model. The cropping operation was performed in MATLAB which provided a convenient computational tool for manipulating pixel values. Following the cropping operation, each grayscale TIFF image was thresheld in order to further distinguish between the fibers and the matrix. Fibers were found to have an intensity value of around 252, so a thresholding procedure was done with the cutoff being 245. This produced a clearer binary image with some noise still present. To further improve the quality of each image, morphological filtering processes were employed which include both erosion and dilation. The morphological processes were 64

76 Figure 5.4: Slice 1 of 1200 provided by ORNL. found to be very sensitive to the size of the structural element used since the image resolution resulted in approximately 7 pixels across a fiber diameter. This relationship of fiber diameter to image resolution limited the useful size of the smoothing mask. An octagonal shape, diamond shape, square shape, and circular shape mask were all examined, but only the smallest diamond shape consisting of 4 pixels gave reasonable results. All other shapes were so large that the entire structures of the fibers were lost in one iteration of erosion. Even though a very smooth definition of fibers could be extracted, a few iterations of erosion followed by iterations of dilation resulted in a very reasonable looking binary image of fiber and mesh compared to the original image. The process of cropping, thresholding, eroding, dilating, and finally rotating 65

77 were all automated through the stack of data with MATLAB. Below, Figure 5.5 shows the progression of these steps. Figure 5.5: Showing the entire image processing procedure. a) after cropping b) after thresholding c) after morphological smoothing d) after final rotation and cropping Figure 5.6 shows the iterations of the morphological steps in smoothing the image of the first slice of data. It is clear that the noise left behind from the thresholding is significantly reduced giving clearer representation of fiber and matrix geometry. 66

78 a b c d e f Figure 5.6: An example showing the progression of morphological operations to remove noise from thresheld image a) original image, b) 1st erosion c) 2nd erosion d) 1st dilation, e) 2nd dilation, f)last dilation and final data image used for analysis. 67

79 Once all images had been processed, the binary pixel data of the entire model composed of 696 slices which were each 852x816 pixels was arranged into a binary column vector of data. The vector was arranged in a very systematic way as shown in Figure 5.7 so that the Gauss point location algorithm could easily access component data. This type of arrangement for the data allowed for x,y, and z coordinates to relate to an index number for the vector of data. Start DATA(1) DATA(852) 1 2 DATA = DATA(694381) Finish DATA(695232) Figure 5.7: Collection of image data into column vector. The index for pixel location is determined from INDEX = (K 1) (I 1) J (5.1) where 68

80 I = 816 F LOOR[(y gp + y o )/l p ] J = F LOOR[(x gp + x o )/l p ] + 1 K = F LOOR[(z gp + z o )/l p ] + 1 l p = where l p is the size of one side of the square pixels, and x o,y o, and z o are vectors from the global coordinate origin to the coordinate origin of each smaller 3x3x3 geometry. The FLOOR function is used in FORTRAN (The Fortran Co., Tucson, AZ) to round down any real number to its closest integer value. Figure 5.8 shows the finite element mesh used for each of the 3x3x3 split model analyses. The mesh is 36 elements in the x direction by 34 elements in y, and 29 elements in z giving 35,496 total elements and 116,559 total degrees of freedom. Figure 5.8: Mesh used for all 3x3x3 split models. 69

81 5.1.2 Results The results for the 3x3x3 models will be presented in this section. The material properties used in the analysis are the same as in Table 4.1 where E f /E m = 30. It should be noted that each of the 27 small models were analyzed as RVEs where appropriate periodic boundary conditions were used for all models. The results are presented in groups of 9 which share the same z-locations where labels are given in Figure 5.2. M aterial P roperties v f E 11 /E m E 22 /E m E 33 /E m ν ν ν ν ν ν G 12 /E m G 23 /E m G 31 /E m Table 5.1: Finite element results for 1 st layer in Z-direction of the 3 x 3 x 3 model. (Labels above indicate x, y, and z cube positions respectively, i.e. 111 starts at the origin) Table 5.1 contains all models having relative z location of 1 (i.e. 221). First note that the volume fraction of the center bottom model is significantly less than the others on this level. All other models were above 40% volume fraction and model-221 was only composed of about 32% fibers indicating a resin-rich core within the model. 70

82 This essentially turns the composite into a complicated box beam. The second item of interest from the model data is the fact that the composite sample is significantly stiffer in the y-direction. This is not surprising given that the slice data shows that a majority of fibers are aligned in this direction. M aterial P roperties v f E 11 /E m E 22 /E m E 33 /E m ν ν ν ν ν ν G 12 /E m G 23 /E m G 31 /E m Table 5.2: Finite element results for 2 nd layer in Z-direction of the 3 x 3 x 3 model. (Labels above indicate x, y, and z cube positions respectively, i.e. 222 is the center cube) In a similar manner, Table 5.2 contains results from all models having a relative z location of 2. As in z location 1, the center model for this layer also has a comparatively low volume fraction, around 34%. Also, the y-direction E 22 is significantly higher than that in the other two directions. The x-direction seems to be slightly less compliant than the z-direction, but not in all cases. Finally, Table 5.3 contains results for all models with a relative z location of 3. Again, the center model at location 223 has the lowest fiber volume fraction by far 71

83 M aterial P roperties v f E 11 /E m E 22 /E m E 33 /E m ν ν ν ν ν ν G 12 /E m G 23 /E m G 31 /E m Table 5.3: Finite element results for 3 rd layer in Z-direction of the 3 x 3 x 3 model. (Labels above indicate x, y, and z cube positions respectively, i.e. 333 is the cube farthest from the origin) with just 34%. Also, E 22 is similarly higher than the other two directions in all models. Figure 5.9 below gives a visual representation of the distribution of the E 11 modulus for all 27 models. Similar trends are present for both E 22 and E

84 E11/Em Z Y Y2 X1 X2 Y1 X Z E11/Em Y Y2 X1 X2 Y1 X Z1 E11/Em Y Y2 X1 X2 Y1 X3 Figure 5.9: A representation of the relative distribution of E 11 within the complete model. 3 Layers of results are shown as 3 different bar charts stacked on top of each other. Each chart represents 9 individual model s properties for the Z1, Z2, and Z3 layer. 73

85 To show convergence in results, an analysis was performed using the 111 model looking at 2,4,6, and 8 Gauss points. It can be seen below in Table 5.10 that there is a clear and steady convergence in all computed results. As a result, 6 Gauss points were used as the standard number for all of the analyses of the 27 smaller models given above since its error is less than 2% compared to 10 Gauss points E 11/E m E 22/E m E 33/E m 6 Predicted Property Number of Gauss Points Figure 5.10: Gauss point convergence analysis for model

86 Elastic constants GP s GP s GP s GP s GP s E 11 /E m E 22 /E m E 33 /E m ν ν ν ν ν ν G 12 /E m G 23 /E m G 31 /E m Table 5.4: Gauss point convergence for model

87 Histograms for the 27 sectioned models are shown below in Figures 5.11 through 5.16 where a spread of the data over the 27 models is seen for each property predicted. It can be seen that E 11, E 33, G 23, and G 31 all had the largest variation compared to the respective means. Overall, E 33 had the largest spread with one standard deviation being over 18% of the mean value Frequency E11/Em Figure 5.11: Histogram of E 11 /E m for the 27 split model analyses. 76

88 Frequency E22/Em Figure 5.12: Histogram of E 22 /E m for the 27 split model analyses Frequency E33/Em Figure 5.13: Histogram of E 33 /E m for the 27 split model analyses. 77

89 8 7 6 Frequency G12/Em Figure 5.14: Histogram of G 12 /E m for the 27 split model analyses Frequency G23/Gm Figure 5.15: Histogram of G 23 /E m for the 27 split model analyses. 78

90 Frequency G31/Em Figure 5.16: Histogram of G 31 /E m for the 27 split model analyses. 79

91 5.2 Complete Uniform Mesh Finite Element Model This section of the chapter will consider the analysis of the entire Oak Ridge National Laboratory long fiber composite sample as a whole and compare the results to the 3x3x3 model averaged as well as the modified Halpin-Tsai and Tandon-Weng equations Analysis Procedure The complete sample model evaluated here was prepared in the same manner as the smaller models in Section 5.1 except that a larger element size is used to remain within the available computational resources. The complete model mesh is 54 elements (xdirection) by 52 elements (y-direction) by 44 elements (z-direction) giving 123,552 elements and 393,534 degrees of freedom total. The same data set from the CT images is described above to assign properties within the larger model. Figure 5.17 shows the uniform mesh for the complete model. 80

92 Figure 5.17: Mesh used for all 3x3x3 split models. 81

93 5.2.2 Computed Results Table 5.5 shows the results for the complete model along with the average values obtained from the 27 smaller models in Section 5.1. Note that there is very good agreement between the complete model results and theose from the averaged smaller RVEs. The complete model is stiffest in the E 22 direction and has a fiber volume fraction of 43.4% which is similar to the average from the smaller models, as expected. M aterial Complete 3x3x3 M odel 3x3x3 Standard P roperties M odel M ean Deviation v f E 11 /E m E 22 /E m E 33 /E m ν ν ν ν ν ν G 12 /E m G 23 /E m G 31 /E m Table 5.5: Finite Element results of complete model with 6 G.P. s and 3x3x3 model averaged. Similar to Chapter 4, the Halpin-Tsai and Tandon-Weng models were used with the random fiber computations of Advani and Tucker [7] and Jack and Smith [44] was employed for comparison to the uniform mesh finite element analysis. A simple fiber distribution function ψ(θ, φ) = c sin 2n θ cos 2n φ (5.2) 82

94 was used for comparison where n is increased to achieve a high degree of alighnment(n = 0 represents a uniformly distributed random fiber orientation with isotropic material properties) [44]. The results comparing these models are presented in Tables 5.6 and 5.7. Tables 5.6 and 5.7 both indicate that the distributions are most near the finite element results when n = 2. Results are marginal in comparison, especially for the Poisson s ratios and shear moduli. Possibly a more complex distribution function could have better approximated the composite sample. Material Complete n = 0 n = 1 n = 2 n = 3 n = 4 P roperties Model E 11 /E m E 22 /E m E 33 /E m ν ν ν ν ν ν G 12 /E m G 23 /E m G 31 /E m Table 5.6: Finite Element results of complete model with 6 G.P. s and Halpin-Tsai micromechanical model assuming a distribution function of fibers from Equation 5.2 where the parameter n is varied. 83

95 Material Complete n = 0 n = 1 n = 2 n = 3 n = 4 P roperties Model E 11 /E m E 22 /E m E 33 /E m ν ν ν ν ν ν G 12 /E m G 23 /E m G 31 /E m Table 5.7: Finite Element results of complete model with 6 G.P. s and Tandon-Weng micromechanical model assuming a distribution function of fibers from Equation 5.2 where the parameter n is varied. 84

96 The periodic boundary condition were checked by looking at displacements of opposite faces in ABAQUS. As seen in Figures 5.18, 5.19, and 5.20 below, the RVE maintained periodicity in all directions. The images appear mirrored views since each face is a mirror of its negative face. Positive X Face Negative X Face Figure 5.18: U2 displacement contour of model-111 s positive and negative X faces during ɛ xx.. Positive Y Face Negative Y Face Figure 5.19: U2 displacement contour of model-111 s positive and negative Y faces during ɛ xx. Finally, a strain contour plot of the complete model as well as a corresponding 85

97 Positive Z Face Negative Z Face Figure 5.20: U2 displacement contour of model-111 s positive and negative Y faces during ɛ xx. smaller model appears in Figure 5.5 and the related CT image. It can be seen that the bright colored regions of high strain correspond to regions lacking fibers and teh darker areas correspond to fiber-rich regions. 86

98 a c b *Note: Image C is not a magnification of image B, but rather a refined F.E. analysis. Figure 5.21: Von Mises stress contours from Z-direction strain as well as corresponding image 696. a)result from complete model F.E. model, b)result from cube 333 F.E. model which corresponds as shown in image, c)image 696 showing fiber slice at level of display for both F.E. results 87