CHARACTERIZATION OF RATE EFFECTS ON MODE-I DOUBLE-CANTILEVERED BEAM TESTS THROUGH FINITE ELEMENT MODELING. A Thesis by. Christopher Todd Pepper

Transcription

1 CHARACTERIZATION OF RATE EFFECTS ON MODE-I DOUBLE-CANTILEVERED BEAM TESTS THROUGH FINITE ELEMENT MODELING A Thesis by Christopher Todd Pepper Bachelor of Science, University of Oklahoma, 001 Submitted to the Department of Aerospace Engineering and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Master of Science May 011

2 Copyright 011 by Christopher Todd Pepper All Rights Reserved

3 CHARACTERIZATION OF RATE EFFECTS ON MODE-I DOUBLE- CANTILEVERED BEAM TESTS THROUGH FINITE ELEMENT MODELING The following faculty members have examined the final copy of this thesis for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Master of Science with a major in Aerospace Engineering. Suresh Raju Keshavanarayana, Committee Chair Walter Horn, Committee Member Hamid M. Lankarani, Committee Member iii

4 ABSTRACT Tests of Mode-I delamination in Double Cantilever Beam (DCB) specimens were modeled with the finite element solver, LS-DYNA, to simulate the behavior of composite materials at high speed loading rates (between in/s and 100 in/s). Cohesive elements were used to model the crack growth behavior. The investigation examines the dynamic effects of the loading block or loading hinge on the simulations. Recommendations are made for which material models, boundary conditions, and other modeling techniques to use for best results. A factor for strain energy release rate is also introduced to account for reductions in G Ic at high loading rates. The adjustment factor would apply to a test-derived value of G Ic and would result in a reduction ranging from 1.6% at in/s to 87% at 100 in/s. The implications of the finding are that testing would need to indicate a power-law increase in the apparent G Ic value in order to maintain a constant actual G Ic. iv

5 TABLE OF CONTENTS Chapter Page Abstract... iv Table of Contents... v List of Tables... vii List of Figures... viii List of Abbreviations... xi List of Symbols... xiii CHAPTER 1 Introduction Background... CHAPTER Objectives... 7 CHAPTER Test Specimen Geometry... 8 CHAPTER 4 Model Overview Material Properties Boundary Conditions Cohesive Element Instability CHAPTER 5 Theory and Equations Beam Flexural Stiffness Compliance Method Energy Methods Loading Block Kinetic Energy Specimen Kinetic Energy... 9 CHAPTER 6 LS-DYNA Results Elastic Cohesive Element Block Mass Specimen, Mixed-Mode Cohesive Element Mixed-Mode Cohesive Element Spring and Damper Loading Results Summary Block Mass Specimen, Tvergaard-Hutchinson Element in/s Crosshead Speed... 8 v

6 TABLE OF CONTENTS (CONTINUED) Chapter Page in/s Crosshead Speed in/s Crosshead Speed in/s Crosshead Speed CHAPTER 7 Conclusions References Appendices Appendix A Units Appendix B Software Versions Appendix C Elastic Cohesive Results Block Mass Specimen, Elastic Cohesive Element Initial Loading Loading during Crack Extension Block Rotation Force Smoothing Mass-less Block Specimen, Elastic Cohesive Element Initial Loading Loading during Crack Extension Block Rotation Summary Hinge-Tab Specimen, Elastic Cohesive Element Initial Loading Adjustment to G c for High Velocity in Brittle Material Conclusions on Elastic Cohesive Elements Appendix D LS-DYNA Input Files vi

7 LIST OF TABLES Table Page Table 1 Material Properties of Aluminum Block [14] Table Material Properties of APC- [5] Table G Ic Results for 100 in/s Table 4 G Ic Results for 0 in/s Table 5 G Ic Results for 8 in/s Table 6 G Ic Results for in/s Table 7 Units Used for Modeling Table 8 Forcing Function Parameters with a Block Mass... 7 Table 9 Failure Times and Deflections with a Block Mass... 7 Table 10 Forcing Function Parameters with a Mass-less Block Table 11 Failure Times and Deflections with a Mass-less Block Table 1 Forcing Function Parameters with a Hinge-Tab Table 1 Failure Times and Deflections with a Mass-less Block vii

8 LIST OF FIGURES Figure Page Figure 1. Double Cantilever Beam Specimen with Loading Blocks... 1 Figure. Displacement of Track Rod with/without Damping []... Figure. Effects of Low Strain Rate on Strain Energy Release Rate []... 5 Figure 4. Summary of G Ic Values with Respect to Crosshead Speed []... 6 Figure 5. Optional Hinge Tab Configurations... 8 Figure 6. Loading Block Mass and Dimensions... 9 Figure 7. Beam Specimen Dimensions... 9 Figure 8. Test Specimen Assembly Figure 9. Assumed Planes of Symmetry in DCB Test... 1 Figure 10. Initial Mesh (0.5 Wide) Overlay with Specimen Geometry... 1 Figure Wide Mesh for Double-Cantilever Strip Model with Block... 1 Figure Wide Mesh for Double-Cantilever Strip Model with Hinge Tab Figure 1. Block End of Model Figure 14. Transition of Mesh to Fine Grid Figure 15. Cohesive Zone of Model Figure 16. Cohesive Element Instability Figure 17. Free-Body Diagram of DCB Specimen Figure 18. Free-Body Diagram of Cantilever Beam to Crack Tip... 0 Figure 19. Force-Displacement Curve [4]... Figure 0. Kinetic Energy versus Crosshead Speed Predicted by Derived Equations... 0 Figure 1. Traction-Deflection Model [18]... Figure. Spring and Damper Connections in Model... 5 Figure. True-Scale Deflection of Beam before and after Crack Propagation... 8 Figure 4. Displacement of Pin vs. Time, 100 in/s Crosshead Speed... 9 Figure 5. Crosshead Speed vs. Time, 100 in/s... 9 viii

9 LIST OF FIGURES (CONTINUED) Figure Page Figure 6. Pin Element (Spring and Damper) Z-Forces vs. Time, 100 in/s Figure 7. Pin Force vs. Deflection, 100 in/s Figure 8. Pin Force vs. Deflection to First Failure, 100 in/s... 4 Figure 9. Crack Tip Nodal Velocity, 100 in/s... 4 Figure 0. Crack Tip Opening Displacement and Rate... 4 Figure 1. Specimen Energy Distribution vs. Crosshead Displacement, 100 in/s Figure. Influence of Crosshead Speed on Energy at Peak Load Figure. Crosshead Speed vs. Time, 0 in/s Figure 4. Pin Force vs. Deflection, 0 in/s Figure 5. Pin Force vs. Deflection to First Failure Figure 6. Beam Deflection (measured by pin) vs. Time, 8 in/s Figure 7. Crosshead Speed vs. Time, 8 in/s Figure 8. Pin Force vs. Deflection, 8 in/s... 5 Figure 9. Pin Force vs. Deflection to First Failure, 8 in/s... 5 Figure 40. Beam Deflection (measured by pin) vs. Time, in/s Figure 41. Crosshead Speed vs. Time, in/s Figure 4. Pin Force vs. Deflection, in/s Figure 4. Pin Force vs. Deflection to First Failure, 8 in/s Figure 44. Traction-Deflection Model Figure 45: Apparent G Ic vs. Loading Rate for APC- Block Specimen... 6 Figure 46: Dynamic Adjustment Factor for APC- Block Model... 6 Figure 47. Forcing Function for APC- (w/ Free Block) at 0 in/s Figure 48. Forcing Function Amplitude vs. Loading Velocity for APC- (w/ Free Block)... 7 Figure 49. Loading for APC- (w/ Free Block) at in/s Figure 50. Loading for APC- (w/ Free Block) at 4 in/s Figure 51. Loading for APC- (w/ Free Block) at 8 in/s ix

10 LIST OF FIGURES (CONTINUED) Figure Page Figure 5. Loading for APC- (w/ Free Block) at 0 in/s Figure 5. Counter-Rotation of the Loading Block Figure 54. Beam Deflection beneath Loading Pin at in/s (with Free Block) Figure 55. Beam Deflection beneath Loading Pin at 4 in/s (with Free Block) Figure 56. Beam Deflection beneath Loading Pin at 8 in/s (with Free Block) Figure 57. Beam Deflection beneath Loading Pin at 0 in/s (with Free Block) Figure 58. Measured Crosshead Speed with Smoothed Force Function... 8 Figure 59. Initial Forcing Function for APC- (w/ Mass-less Block) at in/s Figure 60. Loading for APC- (w/ Mass-less Block) at in/s Figure 61. Loading for APC- (w/ Mass-less Block) at 4 in/s Figure 6. Loading for APC- (w/ Mass-less Block) at 8 in/s Figure 6. Loading for APC- (w/ Mass-less Block) at 0 in/s Figure 64. Comparison of Pin Force with and without Block Mass at 8 in/s Figure 65. Beam Deflection beneath Loading Pin at in/s (with Mass-less Block) Figure 66. Beam Deflection beneath Loading Pin at 4 in/s (with Mass-less Block)... 9 Figure 67. Beam Deflection beneath Loading Pin at 8 in/s (with Mass-less Block)... 9 Figure 68. Beam Deflection beneath Loading Pin at in/s (with Mass-less Block) Figure 69. Hinge-Tab Configuration Figure 70. Initial Forcing Function for APC- (w/ Hinge-Tab) at in/s Figure 71. Initial Forcing Function for APC- (w/ Hinge-Tab) at 8 in/s Figure 7. Initial Forcing Function for APC- (w/ Hinge-Tab) at 0 in/s Figure 7. Amplitude of Nodal Loading vs. Crosshead Speed Figure 74. Damping Ratio of Nodal Forces vs. Mass of Loading Block x

11 LIST OF ABBREVIATIONS a Crack Length a o Initial Crack Length A 1 A 1 E I A A P c a APC- b C DCB ft F G Carbon Fiber/PEEK Composite Material Width of Test Specimen Compliance Double Cantilever Beam Feet Work applied to Test Specimen Strain Energy Release Rate G I Strain Energy Release Rate for Mode-I Delamination (opening) G Ic Fracture Toughness (Critical Value of Strain Energy Release Rate) h in lb f lb m L m P Total Thickness of DCB Specimen Inch Pound-force Pound-mass Length of Test Specimen mass Load Applied to each Block P c Critical Load Applied at Onset of Crack Extension t T Time Kinetic Energy t ply Thickness per Ply U Elastic Strain Energy xi

12 LIST OF ABBREVIATIONS (CONTINUED) y Crack Tip Opening Rate at Two Ply Thicknesses in Front of Crack Tip ct X Amplitude of damped harmonic loading function xii

13 LIST OF SYMBOLS Total Deflection of Test Specimen at Loading Point δ Velocity at Loading Point (also crosshead speed) Frequency of loading function d d Damped Frequency of loading function Damped period of loading function Damping Ratio of loading function xiii

14 CHAPTER 1 INTRODUCTION A key measure to a composite material s resistance to delamination is its Mode-I fracture toughness, also referred to as the strain energy release rate. This material property, G Ic, can be determined by testing of a double-cantilever beam (DCB) specimen shown in Figure 1. The specimen is typically a unitape beam that is pre-cracked by inserting a Teflon film of predefined width in the middle layer, which prevents bonding of the upper and lower half of the beam during the cure cycle. The specimen is then loaded in tension normal to the crack (Mode-I opening) until the delamination extends to a point of instability. The point at which delamination begins has been shown to correlate to the fracture toughness, which is a function of crack length and critical load value. P Initial Delamination Free End P Figure 1. Double Cantilever Beam Specimen with Loading Blocks Traditional testing using this specimen geometry is limited to low loading rates to avoid dynamic effects [1] and material brittleness that occurs when loaded quickly. Testing by Barbezat et al. [] showed that if the loading excited the natural frequency of the specimen, then the test data generated was not smooth and needed filtering. A high rate test machine uses a pickup unit, which is a key for dynamic testing. The pickup unit is a device that separates the loading piston from the specimen. When the test is started, the piston takes an instant to begin moving. The pickup unit doesn t transfer any load to the specimen until the desired load rate of the piston has been reached. As seen in the Figure, the damped track rod provides a much smoother load. The undamped unit produces harmonic oscillations that can excite the natural frequencies 1

15 of the test specimen and cloud the results. The assumption herein is that a pickup unit and a damped track rod would result in smooth and constant load application to the loading block. Ideal Damped Displacement Undamped Time Figure. Displacement of Track Rod with/without Damping [] The property G Ic is calculated using the equations 1 through 4, which are quoted from ASTM D 558 [] for completeness. In general terms, G is the energy released per infinitesimal change in crack length, per unit width. The derivation of these equations is summarized in Chapter 4. G 1 du G I 1) b da P δ G I (Modified Beam Theory) ) b a n P δ G I (Compliance Calibration) ) b a P A 1 C b / I (Modified Compliance Calibration) 4) h The effects of loading rate on this property have been investigated, and it is generally understood that there is a decrease in fracture toughness as the loading rate increases [4]. In relation to airframe fracture of composite parts, it is important to understand the fracture

16 mechanisms so that premature failure will not occur if the airframe is subjected to dynamic, rather than static, load conditions Background As will be summarized below, much of the testing for strain rate effects on fracture toughness of composites began between 1980 and The fracture toughness as a function of strain rate was investigated by several researchers during this span. At the time, toughened epoxies were still being refined. It should be noted that the behaviors of the resin system may not correlate completely to modern toughened epoxy resins. Nevertheless, the data found offers a starting point for understanding the material behavior. Gillespie et al. [5] investigated the effect of loading rate on thermo-plastic graphite/apc- (graphite/poly-ether-ether-ketone) and thermoset Cycom 98 (graphite/epoxy) composite specimens. The loading rate investigated included cross-head speeds of up to 0.17 in/s. The results showed that for graphite/apc- material, the G Ic value increased 1% above the nominal value with increased loading rate, before dipping back down below the nominal value at the highest loading rate. One of the key conclusions drawn from the paper is that the APC- material exhibited sub-critical crack extension. In other words, the load-displacement curve became nonlinear prior to reaching the critical load. The same behavior was not observed in the graphite/epoxy testing, which remained linear up to the critical load. The approach of how to express G Ic as a function of loading rate is also developed by Gillespie et. al [5]. As the crack extends through the length of the specimen, the velocity of the crack tip opening may change. Therefore it is insufficient to use cross-head speed alone as the variable. Instead, the crack tip opening rate (CTOR), y ct, at a distance of ply thicknesses ahead of the crack tip, is adopted as the parameter to measure changes in G Ic. The expression for CTOR is given by Equation 5 [5]. t ply y ct δ 5) a

17 Smiley and Pipes [4,6] continued the study on APC- and graphite/epoxy (AS4/501-6) at even higher speeds. Crosshead speeds of up to 6.4 in/s were tested. Smiley noted a constant G Ic for APC- up to a CTOR of approximately in/s. For a= inches, this corresponds to a crosshead speed of 0.15 in/s. From there G Ic begins to slope downward and drops off as much as 75% at higher rates. Smiley also noted that at higher velocities the material becomes brittle and that less plastic deformation occurs. Kistner [7] investigated AS4/APC- (fiberglass/peek) and AS4/50 (fiberglass/epoxy) DCB specimens and the combined effects of strain rate and temperature on G Ic. Crosshead speeds of up to 0. in/s were employed in the testing. At room temperature, the plot of G Ic versus CTOR appears to be relatively flat above 0.0 in/s loading rate. There is a reduction in G Ic of 1% and 14% in APC- and 50, respectively, for the tests at higher speeds compared to lower speeds. Kistner concluded that at a given temperature, the strain rate effects are unclear because the scatter of data points is so large. Mall et al. [8] tested a plain weave graphite/peek material at crosshead speeds up to 0.66 in/s. Mall presented the G Ic as a function of strain rate at the crack tip, rather than crack tip opening rate. The CTOR and strain rate at the crack tip are proportional, so the trend of G Ic versus strain rate is comparable to other studies. Mall s data shows a linear reduction in G Ic with increased strain rate of up to 60%. The curves generated do not agree with the testing by Smiley for the similar APC- material, which indicated a nonlinear reduction. Mall s graphite/peek material also had approximately ½ times the fracture toughness of Smiley s graphite/apc- [4]. Clearly, either advance in the resin toughness had occurred between the two sets of tests or the testing parameters were fundamentally different. Daniel et al. [9] tested rate effects on DCB specimens made with T00 Thronel fibers and F-185 elastomer-modified epoxy. Testing was conducted on straight and tapered specimens up to a crosshead speed of 0. in/s. Daniel s testing demonstrates a linear reduction in G Ic starting at as low as a 0.0 in/s crosshead speed. At 0. in/s, the reduction in G Ic for the uniform width specimens is 17%. 4

18 As summarized by the aforementioned research, the effect of loading rate on G I is unclear. Figure, illustrating test results from several sources, indicates the uncertainty in the relationship between G Ic and strain rates. Mall et al. [8] showed a steady decline in G Ic, which becomes significant at higher strain rates. Other testing showed an increase in fracture toughness with increasing load rate. Strain Energy Release Rate, G Ic (in-lb f /in ) Graphite/PEEK Fabric [8] T00/F185 [9] Graphite/APC- [6] AS4/501-6 [14] AS4/501-6 [6] E E E E E E E E-0. Crack Tip Opening Rate, y ct (in/s) Figure. Effects of Low Strain Rate on Strain Energy Release Rate [] The unclear trends in the data suggest a few possible explanations. One is that no clear trend can be determined under dynamic conditions. However, the rates of loading tested have barely been in the dynamic region. In fact most of the tests were quasi-static. The fact that each study, independently, found the experimental result was repeatable doesn t completely support this hypothesis either. For example, Smiley [4] displays hundreds of experimental data points all showing on the same curve. A second possibility owes to the different material systems investigated. The resin properties of the newer toughened epoxy laminates may behave much differently than some of the older 5

19 materials investigated. Some of the materials may have a much more brittle nature than others, influencing the crack growth mechanisms and thus the results. Examination of Figure shows that even for the same material (AS4/501-6) there are differences in one study vs. another. Therefore this cannot be the entire explanation. However, if material differences are the primary explanation, then the rate effects would need to be studied by physical testing of each material. Still a third option is that some difference in experimental procedure or test setup led to the differences between the trends in the data. One example would be the loading blocks vs. loading hinges on the test specimen. At speeds greater than ~40 in/s, the test specimen experiences dynamic behavior []. It is the purpose of this paper to investigate the dynamic effects of the specimen and loading blocks on the test results. As illustrated by Figure 4, many papers have been written investigating the effects of strain rate on G Ic. However the dynamic region of loading is still largely unexplored. By simulating fracture at high speeds, it is the intent of this paper to provide some expectations of the test specimen behavior under ideal conditions. This study is aimed at identifying the role of specimen vibrations on the measured responses for quantifying the material toughness. 0 Graphite/PEEK Fabric [8] T00/F185 [9] Strain Energy Release Rate, G Ic (in-lb f /in ) Graphite/APC- [6] AS4/501-6 [14] AS4/501-6 [6] Region of Interest for Dynamic Loading Crosshead Speed, (in/s) Figure 4. Summary of G Ic Values with Respect to Crosshead Speed [] 6

20 CHAPTER OBJECTIVES The objective of this investigation is to determine the effects of the mass of the loading blocks and the beam flexural stiffness on the DCB experiment at high loading rates. A secondary objective is to prove or disprove whether it is possible to model a Mode-I DCB test at high speeds using LS-DYNA software and cohesive elements for crack behavior. Three cohesive material models will be used and the results for each type will be recorded. Using the best results obtained from the modeling, G Ic values will be calculated for four cross-head speeds. The results at different speeds will be used to plot trendlines for predicted behavior of fracture toughness with respect to loading rate. Additionally, kinetic energy terms versus loading rate will also be investigated, as will internal energy and work. 7

21 CHAPTER TEST SPECIMEN GEOMETRY Investigation of the rate effects on the DCB test required a theoretical specimen to be used. Figure 5 through Figure 7 illustrate the theoretical test specimen used for modeling. The dimensions follow recommended guidelines from ASTM specifications [] and are similar to dimensions used in previous research. The test parameter with the greatest variability is the loading block. Smiley et al. [4,6] used a hinge tab installed with fasteners instead of a loading block. Daniel et al. [8] used a bonded reverse-hinge tab with the hinge at the tip of the beam. Mall [] also used the reversehinge tab but with fasteners to attach to the specimen. The hinge tab attachment configurations are illustrated in Figure 5. Kistner [7] used a loading block with fasteners through the laminate. There are multiple ways to properly use the loading block or hinge tabs, but the most effective should allow free rotation at the loading axis and should not induce any heel-toe effects, which could be introduced by having fasteners that are offset from the loading line. The loading block in Figure 6 meets these requirements and is used as the baseline model in this study. The hinge tab is also modeled as a second configuration to be included in the study. Hinge Tab w/ Fasteners Reverse Hinge Tab w/ Fasteners Figure 5. Optional Hinge Tab Configurations 8

22 Density = 0.10 lb m /in =.64 E-04 slinch/in Mass =.56 E-0 lb m = 6.6 E-05 slinch I oy = 1.5 E-0 lb m *in =.9 E-05 slinch*in Figure 6. Loading Block Mass and Dimensions Based on a survey of previous testing, a typical specimen dimension of 10.0 x 1.0 x 0.1 was chosen. The half beam thickness, h/, is 0.06 inches or 0.10 inches. Both thicknesses were investigated. The thinner beam is representative of 4 plies of uni-tape and the thicker beam is representative of 40 plies of uni-tape..10 Figure 7. Beam Specimen Dimensions 9

23 The initial crack length is also created in conformance with ASTM specifications [] as shown in Figure 8. A.00-inch crack is defined to be measured from the load line to the initial crack tip. Figure 8. Test Specimen Assembly 10

24 CHAPTER 4 MODEL OVERVIEW An LS-DYNA model of the theoretical beam specimen was developed in MSC.Patran, and included a loading block/pin and beam specimen. LS-DYNA [16] is a finite element solver with applications to large deflection static problems and dynamic problems, including fluids. LS- DYNA was chosen for modeling because of its ability to model high speed dynamic responses and for its comprehensive material library, which includes cohesive elements that can be used to model crack growth. The model is constructed in space such that the X-axis is parallel with the long direction of the beam, the Y-axis is parallel to the width dimension of the beam, and the Z-axis is the direction of loading of the block. The origin of the global coordinate system is at the mid-ply of the half-beam, in line with the loading pin. The number of elements considered in the modeling needs to be carefully considered to balance file size and run time with accuracy of model. Planes of symmetry are first identified in the real structure to reduce the number of elements and nodes. The model is assumed to have two planes of symmetry shown in Figure 9. The loading applied to the upper block is assumed to be the same magnitude as the loading on the lower block. The XY plane of the DCB specimen becomes a plane of symmetry with this assumption. It is also assumed that the beam specimen does not twist. Therefore the XZ plane is also treated as a plane of symmetry. The nodes positioned on the XZ plane are restrained from translation in the Y-axis. With these two planes of symmetry, the number of elements is reduced by 75% compared to a full model of the entire test specimen. The elements chosen for the model are hexahedral elements with 8-nodes. They are invoked in the LS-DYNA key file using the keyword *ELEMENT_SOLID_ORTHO [16]. All sections of like elements were created as orthotropic, and later changed to isotropic material properties as needed for the loading block and pin sections. The *SECTION_SOLID card [16] is also used to control the manner in which the Hex elements are formulated. The cohesive elements are called 11

25 out specifically as such and all other element sections are categorized under a default constant stress solid. Z X Y Z xz plane xy plane Y Figure 9. Assumed Planes of Symmetry in DCB Test The first model generated is overlayed with the full-width beam specimen in Figure 10. The solve time proved to be too long, even with this reduced mesh. In addition there were hourglassing modes present with the elements (even with the full integration), believed to be caused by an excessive aspect ratio. In order to reduce the solve time and hourglassing errors, the mesh was refined to a single strip of elements with a 0.0-inch width in the Y-axis. With the single strip of elements, the model represents a plane-stress condition. It is believed that a planestress model will not significantly affect the results for two reasons. The first is that the material modeled has a low Poisson s ratio and does not create significant transverse stresses because of its unitape construction. The second is that the true specimen is only 1 inch wide, which is typically regarded as somewhere between plane stress and plane strain, even with an isotropic material. 1

26 Figure 10. Initial Mesh (0.5 Wide) Overlay with Specimen Geometry Elements Representing Pin Hex Elements w/ Isotropic Aluminum Properties Hex Elements w/ Orthotropic Composite Properties Thick Hex Elements w/ Cohesive Properties Figure Wide Mesh for Double-Cantilever Strip Model with Block 1

27 Loaded Node Hex Elements w/ Isotropic Aluminum Properties Hex Elements w/ Orthotropic Composite Properties Thick Hex Elements w/ Cohesive Properties Figure Wide Mesh for Double-Cantilever Strip Model with Hinge Tab The mesh is created to minimize the number of elements in the loading block and minimize element size near the crack tip. The transition from a coarser mesh to an extremely fine mesh is illustrated in Figure 1 through Figure 15. δ Block Mesh Size 0.05 x 0.0 x 0.05 X x Y x Z Z X Beam Mesh Size 0.05 x 0.0 x 0.01 Beam Mesh Transition Figure 1. Block End of Model 14

28 Beam Mesh Transition Initial Crack Tip Figure 14. Transition of Mesh to Fine Grid Top Surface of Beam Beam Mesh Size 0.01 x 0.0 x 0.01 Free End of Beam Cohesive Mesh Size 0.01 x 0.0 x Or 0.01 x 0.0 x Bond Surface to Lower Beam NOTE: The cohesive elements illustrated have been stretched so they will be distinctly visible in the figure. Figure 15. Cohesive Zone of Model 15

29 4.1. Material Properties There are four sections of the model for which the material properties are defined: pin, block, composite beam, and resin (cohesive element). The pin and block are assumed to be aluminum alloys. The properties for both are the same. TABLE 1 MATERIAL PROPERTIES OF ALUMINUM BLOCK [14] Modulus of Elasticity Poisson s Ratio Density Alloy E (Msi) (slinch/in ) 7050-T *10-4 TABLE MATERIAL PROPERTIES OF APC- [5] Density (slinch/in ) 1.45*10-4 [17] Modulus of Elasticity (Msi) Poisson s Ratio E a E b E c Shear Modulus (Msi) Damping G 1 G G 1 G Sig-F Msi 1000 psi G is the shear modulus for frequency independent damping. With lack of sufficient data for this parameter, the initial value used is the same as the static shear modulus. Sig-F is the limit stress for frequency independent, frictional, damping. G should be 50 to 1000 times greater than Sig-F [16], and the values used comply with this guideline by using a ratio of Boundary Conditions The bottom surface of the cohesive elements is translation-constrained along the global X and Y axes, and rotational-constrained about the global Y-axis. One side of the strip falling on the X-Z plane is translational-constrained in the global Y axis. An optional constraint that was later deleted was a translational constraint on the pin in the global X-axis. The laboratory testing would include this restraint. However, since the cohesive elements are also constrained in the X- 16

30 axis this doesn t allow the beam to shorten in that direction as it is bent vertically. For small deflections this didn t affect results significantly, and so it was removed. 4.. Cohesive Element Instability Without an X-constraint on the cohesive nodes, one error encountered in modeling emerged in the form of cohesive element instability. At approximately seconds in the simulation of the block model, at 0 in/s crosshead speed, the cohesive elements began to fail close to the free end of the beam. In order to eliminate this instability, the additional translational constraint along the beam length was imposed on the bottom edge of the cohesive elements. 17

31 Bottom edge of cohesive elements Instability developing over time leads to premature failure of elements away from crack tip. Figure 16. Cohesive Element Instability 18

32 CHAPTER 5 THEORY AND EQUATIONS The beam bending and energy theories are outlined in this section to document the underlying theory in the DCB test. The equations are a summary of those developed by Berry and refined by subsequent testing in composites [1,4] Beam Flexural Stiffness Consider the free-body diagram of the double-cantilever-beam test in Figure 17 and Figure 18. An end-load, P, is applied normal to the crack. It is assumed that each half of the beam from the end up to the crack tip can be treated as a cantilevered beam. The Euler-Bernoulli beam equations can be used to show the theoretical deflection of the beam as a function of the applied load and crack length. P a L P Figure 17. Free-Body Diagram of DCB Specimen 19

33 z / (-)M (+)M P P x a Figure 18. Free-Body Diagram of Cantilever Beam to Crack Tip The moment as a function of position is related to the curvature of the beam as follows. This leads to a solution for the deflected shape of the beam through integration. d w M(x) P x E I 6) dx d w dx P x E I 7) dw dx P x E I dx P x E I c 1 8) P x P x w(x) c1 dx c1 x c 9) E I 6 E I At the free end of the beam, the bending moment, and therefore d w/dx are equal to zero. At the crack tip, the beam is cantilevered and the slope of the deflected beam shape is zero. Therefore dy/dx is zero. 0

34 1 Boundary Conditions: 0 w(a) 10) 0 dx dw a x 11) Substitution of the boundary conditions leads to a solution for the constants c 1 and c. I E a P c 0 c I E a P dx dw c I E x P dx dw 1 1 a x 1 1) I E a P - c 0 c a I E a P I E 6 a P c a c I E 6 a P z(a) 1 1) The resulting equation for the deflected shape of the lower beam is therefore I E a P I E x a P I E 6 x P w(x) 14) By symmetry, the upper half of the beam would deflect to the equation I E a P I E x a P I E 6 x P w(x) 15) The deflection of the upper half of the specimen, /, is then δ I E a P w(0) 16) Equation 16 can be simplified to show the ratio P/, which will be shown to be an important parameter in the compliance method [1]. a I E δ P 17)

35 5.. Compliance Method Testing of a DCB specimen will produce data points for applied load vs. deflection. The plot should appear linear up to the point of crack instability [4]. As noted previously, some materials exhibit a nonlinearity prior to peaking, which is known as subcritical crack extension. Assuming a linear response, however, the reciprocal of the slope of load vs. deflection prior to crack growth is equal to the compliance, C. Load ΔP Δδ 1 C Displacement, Figure 19. Force-Displacement Curve [4] C Δδ ΔP 18) Combining equations 17 and 18, the compliance can be expressed, at least in theory, as relating to the beam properties [4]. C a E I 19) C is a variable that changes as a function of crack length. Once the applied strain energy exceeds the fracture toughness of the material, the crack grows rapidly and the load is decreased until the crack is stabilized at a new length [4]. When load is increased after this point, the slope of the Load-Displacement curve (1/C) is reduced because of the increased crack length. The

36 deflection measured per unit increase in load is therefore greater during the second load cycle. Using this procedure, it is possible to measure several P c values during the test. From the beam theory, and from the compliance method, the relation between load and deflection is (combining (17) and (19)) E I a P δ δ C 0) Introducing a constant A 1, where A 1 E, a more generalized equation for load vs. I deflection is shown as [4] P 1 A a 1 n δ 1) The coefficient n is equal to from beam theory. This assumption holds up well when fitted to data points [,4]. 5.. Energy Methods For a specimen containing a crack, the condition for crack growth occurs when the energy available for crack growth, s, exceeds the work required to create a unit of new crack area. A portion of the work, F, applied to the specimen by force is stored as elastic strain energy, U. This allows the specimen to deform without crack growth until F>U, absorbing some of the applied energy through bending [11]. d(f - U) b da ds b da ) Typically, the strain energy release rate is expressed as [4] 1 d(f - U) G ) b da

37 F is the work applied to the specimen by external force. U is the elastic strain energy, or the energy released by deformation of the DCB specimen without crack extension. G may alternatively be expressed with a kinetic energy term, denoted as T [4]. 1 d(f - U - T) G 4) b da In other words, G is the rate difference (per crack growth rate) of energy between the applied work and the energy absorbed through the specimen by elastic deformation, per unit width. Energy applied to the specimen that is not absorbed through strain is released by material fracture. The kinetic energy term, T will be discussed later, but for now it is neglected. The work and elastic strain energy are related to the applied force and deflection as follows. F U δ P(z) dz 5) 0 4

38 Substituting equation (0), F U δ 0 δ C dδ δ C 6) Substituting equation (0) for, F U P C 7) Substituting equation 19 for C, the equation for the term F-U is simplified to a function of applied load and crack length. F U P a E I P a E I 8) The quantity G can now be solved by substitution of equation (8) into equation () [4]. G 1 b d(f - U) da 1 b d da P a E I 9) crack length [4]. Differentiation of the equation yields an expression for G as a function of applied load and G P a b E I 0) An additional parameter, A, is now introduced such that A P a c, where P c is the critical loading at crack extension [4]. G Ic A b E I 1) Using the definition A 1, the equation shows that 1. Therefore E I ( E I) A 1 the above equation for G Ic [4]. G b A E I ( A 1 Ic ) E I) 5

39 A A b 1 G Ic ) Equation () is a general expression for G Ic whereby A 1 and A are both determined from testing and are independent of beam theory. At low speed testing, T would be negligible since v is small. A mass moving at a velocity v has a kinetic energy of m. v /. As the loading rate increases, the kinetic energy term should increase parabolically. If the term is neglected, then one would expect that results for G would be artificially inflated for higher velocities. T b ρ u w dx dz 4) ~ mass density Since all the elements are modeled in the LSDYNA model, the kinetic energy as a function of time is then n 1 T mi u i w i 5) i 1 Because there are two distinct parts in the test, the loading block and the specimen, the kinetic energy calculation is broken down as the sum of the two components. T T b T s 6) 6

40 5..1. Loading Block Kinetic Energy The test apparatus applies a vertical movement only to the loading block. The term u represents the velocity in the plane normal to the load application, and is therefore neglected in the equation as a simplifying assumption. The translational portion of the block kinetic energy is simply calculated as T t 1 m δ 7) This represents the energy associated with the linear translation of the block at the crosshead speed of v δ /. An additional rotational velocity is imparted to the loading block. At a given time during the experiment, the assumed small angle of rotation of the block is expressed as dw dx 1 θ tan 8) dw dx x 0 x 0 θ for small angles 9) From Equation (1), is θ P a E I 40) Employing equation 17, substitution for P reveals that for small angles a E I E I a θ δ 4 δ a 41) The exact expression for the angle is θ tan 1 P a E I tan 1 4 δ a 4) d θ(δ, a) ω where δ(t) and a(t) 4) dt 7

41 Using the quotient rule, and assuming that is small such that equation 9 is accurate, then δ a - δ a ω 44) 4 a The angular velocity of the specimen, ω θ, is then calculated assuming that a is a constant. When a is not a constant, the crack is growing at an unstable rate and the rotational velocity of the block becomes a function of the crack growth rate as shown above. The rotational kinetic energy of the block mass is T r 1 I 45) shows that where I is the mass moment of inertia of the block mass. Substitution of Equation 6 9 δ a Tr I 46) The total kinetic energy of the block is derived by summing equation 46) and 7) (times since there are two blocks) δ 9 I T m 47) 4 4 a 8

42 Specimen Kinetic Energy The term w represents the vertical velocity of the specimen and loading block, which is assumed to be a function of x only. The loading block bond to the specimen should apply a uniform deformation over the width of the specimen. Employing Euler-Bernoulli beam theory, from equation (15), the kinetic energy of the specimen itself can be derived. I E a P I E x a P I E 6 x P dt d dt dw w 48) δ a 4 x δ a 4 x δ dt d dt dw 49) δ a 4 δ a 4 x δ dt dw x 50) Substitution leads to specimens dy dx δ a 4 x δ a 4 x δ h ρ 1 T b 0 a -l s 51) l ~ length of specimen on opposite side of loading block from crack tip a l a l s dx 4 1 a 4 x a 16 x 9 a 4 x a 8 x a 16 x δ h b ρ dx δ a 4 x δ a 4 x δ h b ρ T 5) a -l s 4 x a 8 x a 16 x a 16 x a 40 x a 11 x δ h b ρ T 5) 4 l a a 8 ) (a a 16 ) (a a 16 a a 40 ) (a a 11 a δ h b ρ T s l l l l l 54)

43 Figure 0 plots the predicted kinetic energy for the block specimen of APC- material using the derived equations above. Predicted Kinetic Energy (lbf-in) Block Mass Block Inertia Specimen Total Crosshead Speed (in/s) Figure 0. Kinetic Energy versus Crosshead Speed Predicted by Derived Equations 0

44 CHAPTER 6 LS-DYNA RESULTS The results are discussed in three following sections: elastic cohesive elements, mixedmode cohesive elements, and Tvergaard-Hutchinson elements. The modeling was performed in this order, and each section then discusses the effects of different variables, namely velocity, on the mesh Elastic Cohesive Element This section of results included an error in the cohesive element property card that caused the crack growth to behave as a very brittle material. The cohesive element material was input with the COHESIVE_ELASTIC card [16]. The behavior after crack growth is representative of a brittle material rather than a toughened epoxy composite, even though the defined strain energy release rate based on the traction-separation curve was an order of magnitude greater than what was successfully used in subsequent modeling. Crack propagation and failure occurs swiftly. The results for this section are included in Appendix C because they are unrealistically brittle. However, the results prior to crack growth should be unaffected. The results after crack growth are also useful for characterizing the behavior of a brittle material. They are also important for understanding the method that led to the successful runs in the following sections. 1

45 6.. Block Mass Specimen, Mixed-Mode Cohesive Element As noted in the previous section, the elastic cohesive element did not produce the desired results and was much too brittle. It was therefore hypothesized that another material model that has more plastic deformation will yield more realistic results. An alternate cohesive element is now incorporated into the model using the COHESIVE_MIXED_MODE material card [16]. Another lesson learned from the previous models is that the nodal forces at the loading block do not provide an accurate representation of the pin forces applied to the model. The forces cannot be directly measured as would be the case with a static model. To overcome this impasse, a spring is added to the model between the prescribed deflection and the loading block. This spring is discussed in detail in Section 6... Other changes made to the model include using a fully integrated solid element. The previous models documented have used a constant stress solid for simplicity. The fully integrated solid is expected to eliminate hourglassing concerns.

46 6..1. Mixed-Mode Cohesive Element The material behavior is described in terms of a general elastic-plastic traction-separation theory, similar to that described by Tvergaard and Hutchinson [18]. Consider first the Mode-I fracture toughness of the cohesive elements is represented by the area under the tractionseparation curve, as shown in Figure 1 [16]. G Ic max Traction, t( ) E n Nodal Deflection, f Figure 1. Traction-Deflection Model [18] G Ic δf 0 t(δ( dδ σ max δ f 55) The quantities G Ic and max are taken to be 9.99 in-lb f /in and 11 ksi, respectively, as found by Gillespie [5]. However 1, f, and k are unknown. Therefore an additional equation is needed as are some assumptions relating 1 and f. δ f δ 1 56) The elastic stiffness of the element is represented by the portion of the traction curve below the yield point, 1. σ max E n 57) δ1

47 The solution of the above equations for f and E n (lbs/in ) yield the following equations [16]. G Ic δ f 58) σ max σmax E 59) n G Ic (9.995 in lb f /in ) δ in 60) f (11000 lb /in ) (11000 lb f /in ) f 7 E n lb f /in 61) (9.995 in lb f /in ) The mixed-mode element utilizes a combined damage model whereby the simulation will keep track of loading past the yield point and account for a reduced stiffness that occurs on unloading and reloading after the damage has been incurred. 4

48 6... Spring and Damper Loading In order to measure the force applied to the pin, two springs are added to the model between independent nodes, 1 and, and the pin nodes, 785 and The independent nodes motion is defined by a prescribed velocity in the Z-direction, with no movement allowed in the X or Y directions. The spring and damper connect to the pin center and effectively carry the block along the prescribed velocity. Both spring and damper are limited to applying forces in the Z-direction. The pin center is therefore unconstrained in the X and Y directions. Mass elements of slinch ( lb m ) were added to nodes 1 and to avoid errors in the execution of LSDYNA. δ Node 1, Node 785, 7866 Z X Figure. Spring and Damper Connections in Model 5

49 The spring constant is determined relative to the natural frequency of the beam specimen. A Nastran solution for the natural frequency of a.06 thick half-beam shows the first mode natural frequency is 144 Hz. The spring stiffness, k, is set to a value much higher than that of the specimen so that its presence will not affect the results. A factor of 10 is multiplied by the critical spring stiffness. The critical spring stiffness is the value which a simple mass-spring system would match the natural frequency of the beam above. m (0.0 in)(0.5 in)(0.6 4 k 10 ω mblock 6) in)( slinch/in ) slinch 4 6 block 6) k 10 (144/s) ( slinch). 10 lb f /in 64) 6

50 A damper is also added to the spring after initial runs showed that the block continued to oscillate about the desired velocity. The critical value of damping is calculated in the following equations based on a simple mass-spring system. The damping constant is rounded up to 0.15 lb f /(in/s) to ensure over-damping. c k m 65) c (. 10 slinch/s ) ( slinch) 0.14 lb f in/s 66) 6... Results Summary The simulation was attempted at several velocities and more than a dozen of combinations of variables. A time-step of seconds was used based on element attributes. None of the simulations were successful with the MIXED_MODE (MAT_18) material model. The failure in the elements occurred in less than 10-4 seconds and did not propagate from the crack tip. The reason for such an early failure could not be found, however it is noted that neither the deflection nor the stress in the elements was near the critical fracture point. 7

51 6.. Block Mass Specimen, Tvergaard-Hutchinson Element The simulation was repeated with the same parameters as described in Section 6., but with a MAT_COHESIVE_TH material card instead of the mixed mode material. Using this model did provide some more realistic results, which are discussed in this section. This element uses the same traction curve as the mixed-mode element. It does not include sustained damage modeling that could occur on unloading and reloading. It does, however, include a penetration stiffness multiplier that I discovered was required to prevent negative volume errors in the elements in/s Crosshead Speed The model was run at 50 in/s (symmetric condition results in 100 in/s crosshead speed) for a duration of 0.1 seconds with a time-step of seconds. By cohesive element definition, the G Ic value is known to be 9.99 in-lb f /in, but an apparent G Ic value can be calculated from the data points. Before we get into the fracture toughness, the model needs to be verified. First, the displacement of the pin should be linear. Figure 4 verifies that the pin deflection is linear and translates at a speed of 50 in/s, which equates to a 100 in/s crosshead speed if both blocks are moving apart at the same speed. Figure. True-Scale Deflection of Beam before and after Crack Propagation 8

52 Node Displacement, / (in) Time (s.) A785 B7866 Figure 4. Displacement of Pin vs. Time, 100 in/s Crosshead Speed Another verification of the loading rate is performed by measuring the velocity at the pin. The speed starts at 0 but quickly ramps up to 100 in/s. The speed fluctuates, but the impulses seen during the test are not significant enough to be seen on the linear displacement vs. time. 140 Chrosshead Speed, d /dt (in/s) Pin LH Pin RH Time (s.) Figure 5. Crosshead Speed vs. Time, 100 in/s 9

53 The pin forces are measured using two springs and two dampers. The spring forces are the majority of the overall pin force acting on the block. As seen in Figure 6, the damper forces fluctuate wildly, but the average damper force over any length of time will be close to zero. It is therefore assumed that the pin force can be measured directly by the springs only. By removing the damper forces, the data is much more clearly defined, and essentially filtered from most of the beam vibrations Element Shear Forces (lbf) Spring1 Spring Damper1 Damper Time (s.) Figure 6. Pin Element (Spring and Damper) Z-Forces vs. Time, 100 in/s 40

54 Force, P (lbf) Before Crack Growth After Crack Growth Critical Load Displacement, (in.) Figure 7. Pin Force vs. Deflection, 100 in/s Figure 7 illustrates the force-displacement data acquired from the simulation. Examination of the cracking data yielded three points where delamination had stalled before reaching a critical load value, at which time the delamination continued to grow again. These three points are identified on the figure. At this high-speed, it is very difficult to identify the second and third points. The first critical load value is reached after the delamination had spread for a small length. This is termed sub-critical crack extension and was noted by Gillespie in tests of APC- [5]. After the first critical load point, there is a vertical drop-off in load. Smiley characterized this as brittle-unstable crack growth, and noted that APC- exhibited such behavior [4]. Of particular interest is the first G Ic value. The force and displacement are plotted in Figure 8 and a trendline is fitted to the data points. The force is nearly linear with a slope of.8 lbs/in. 41

55 y =.8x 10 Force, P (lbf) Displacement, (in.) Figure 8. Pin Force vs. Deflection to First Failure, 100 in/s Also of interest is the crack tip opening rate. Figure 9 shows the nodal z-velocity at.01 in front of the crack tip. Even before crack growth, the fluctuation in velocity is large. It would be difficult to use this parameter to measure G Ic effects without taking an average velocity over a period of time Nodal Z-Velocity (in/s) E78076 F Time (s.) Figure 9. Crack Tip Nodal Velocity, 100 in/s 4

56 The best way determined to find the CTOR is to measure the difference in nodal Z- displacement between a node 0.01 ahead of the crack tip and at the top of the cohesive element directly at the initial crack tip. The results are shown in Figure 0. A linear trend-line fitted to the data points from the dplot file indicates a crack tip opening rate of in/s up until the delamination growth begins around seconds. Comparatively, Smiley and Pipes [4] present an equation based on crosshead speed that predicts a CTOR of in/s. Note that in this study, it was discovered that because the cohesive element is not significantly rigid, the node at the crack tip actually deflects to a certain small z-coordinate, thereby making it necessary to include both the nodes at the crack tip and ahead of the crack tip to determine the opening rate. Crack Tip Opening Displacement (in.) Nodal Z Displacement Linear Trend y = x Time (s.) Figure 0. Crack Tip Opening Displacement and Rate Figure 1 plots the energy in the specimen as manifested by kinetic energy, internal energy and work applied to the block. Internal energy and kinetic energy are measured directly by LS-DYNA and adjusted for a 1-inch wide symmetric specimen. The work is calculated using trapezoidal numerical integration of the force-deflection curve, or U+T, whichever is greater. At 4

57 the critical load, there is a tremendous loss in energy as shown by the separation between the work and internal energy lines. Figure shows the internal energy and kinetic energy, measured at the first critical load occurrence, for all crosshead speeds. As indicated, the internal energy at crack propagation decreases greatly with respect to loading rate. Energy (lbf-in) Kinetic Energy Internal Energy Work Displacement, (in.) Figure 1. Specimen Energy Distribution vs. Crosshead Displacement, 100 in/s Energy at Peak Load (lbf-in) Kinetic Energy Internal Energy Crosshead Speed, d /dt (in/s) Figure. Influence of Crosshead Speed on Energy at Peak Load 44

58 The apparent G Ic values are computed using different methods and documented in Table. The G Ic is known to be 9.99 in-lb/in, but the simulation results are indicating a much higher fracture toughness than that. The lowest estimate is Smiley s equation with the KE adjustment. TABLE G IC RESULTS FOR 100 IN/S G ic-app Method (in-lb f /in ) Modified Beam Theory 7.8 Compliance Calibration Method 69.5 Modified Compliance Calibration Method 67. Smiley/Pipes 61.4 Smiley/Pipes with KE adjustment

59 in/s Crosshead Speed The crosshead speed of this run is verified in Figure. The result is a nearly linear deflection response at 0 in/s. 0 Chrosshead Speed, d /dt (in/s) Pin LH Pin RH Time (s.) Figure. Crosshead Speed vs. Time, 0 in/s The pin forces are measured using two springs and two dampers. The spring forces are the majority of the overall pin force acting on the block. It is assumed again that the pin force can be measured directly by the springs only, as shown in Figure 4. 46

60 Before Crack Growth After Crack Growth Critical Loads Force, P (lbf) Displacement, (in.) Figure 4. Pin Force vs. Deflection, 0 in/s Figure 4 illustrates the force-displacement data acquired from the spring elements in the simulation. Examination of the cracking data yielded three points where delamination had stalled before reaching a critical load value, at which time the delamination continued to grow again. These three points are identified on the figure. The second and third points are more clearly visible at 0 in/s than at 100 in/s, but still questionable given the vibrations in the model. The first critical load value is reached after the delamination had spread for a small length. The curve also shows sub-critical crack extension and brittle-unstable fracture behavior. 47

61 The force and displacement are plotted in Figure 5 and a trendline is fitted to the data points. The force is nearly linear with a slope of lbs/in. 10 y = 08.55x 100 Force, P (lbf) Displacement, (in.) Figure 5. Pin Force vs. Deflection to First Failure 48

62 The apparent G Ic values are computed using several methods and documented in Table 4. The G Ic is known to be 9.99 in-lb/in, but the simulation results are indicating a much higher fracture toughness. The lowest estimate is Smiley s equation with the KE adjustment. TABLE 4 G IC RESULTS FOR 0 IN/S G ic Method (in-lb f /in ) Modified Beam Theory 44.7 Compliance Calibration Method.0 Modified Compliance Calibration Method 6.0 Smiley/Pipes 4.5 Smiley/Pipes with KE adjustment

63 in/s Crosshead Speed The crosshead speed of this run is verified in Figure 7. The deflection response measured by the pin nodes is nearly linear at a rate of 8 in/s. However, as shown in Figure 7, the velocity of the pin node fluctuates significantly. While this is expected with the spring and dampers used to load the pin, it is noted that there is potential for refinement of the spring and damper constants that might produce a more constant loading rate..5 Displacement, (in) A785 B Time (s.) Figure 6. Beam Deflection (measured by pin) vs. Time, 8 in/s 50

64 Chrosshead Speed, d /dt (in/s) Pin LH Pin RH Time (s.) Figure 7. Crosshead Speed vs. Time, 8 in/s The pin forces are measured using two springs and two dampers. The spring forces are the majority of the overall pin force acting on the block. It is assumed again that the pin force can be measured directly by the springs only, as shown in Figure 8. 51

65 Force, P (lbf) Before Crack Growth After Crack Growth Critical Loads Displacement, (in.) Figure 8. Pin Force vs. Deflection, 8 in/s Figure 8 illustrates the force-displacement data acquired from the simulation. The Compliance methods all require more than one data point for calculation of G Ic. With the data above, the nd and rd data points are extremely difficult to determine. Judgment was used to pick the last two points. However, because of the lack of a clear crack arrest with this model, only the first critical load value is a considered a reliable data point. The curve shows brittle-unstable fracture behavior after the first critical loading point. 5

66 The force and displacement are plotted in Figure 9 and a trendline is fitted to the data points. The force is nearly linear with a slope of lbs/in. The reciprocal of the slope is the Compliance, C y = x 50 Force, P (lbf) Displacement, (in.) Figure 9. Pin Force vs. Deflection to First Failure, 8 in/s 5

67 The apparent G Ic values are computed using several methods and documented in Table 4. The G Ic is known to be 9.99 in-lb/in, but the simulation results are indicating a much higher fracture toughness. The closest estimate is the Compliance Calibration Method. However, all calculations except for the Modified Beam Theory require the nd and rd data points, which have been called into question. These values are listed in red to indicate the uncertainty surrounding their derivation. TABLE 5 G IC RESULTS FOR 8 IN/S G ic Method (in-lb f /in ) Modified Beam Theory 19.4 Compliance Calibration Method 8.8 Modified Compliance Calibration Method 6. Smiley/Pipes 1.7 Smiley/Pipes with KE adjustment

68 6..4. in/s Crosshead Speed The crosshead speed of this run is verified in Figure 40. The deflection response measured by the pin nodes is nearly linear at a rate of in/s. However, as shown in Figure 41, the velocity of the pin node fluctuates significantly. While this is expected with the spring and dampers used to load the pin, it is noted that there is potential for refinement of the spring and damper constants that might produce a more constant loading rate Displacement, (in) Time (s.) A785 B7866 Figure 40. Beam Deflection (measured by pin) vs. Time, in/s 55

69 Chrosshead Speed, d /dt (in/s) Time (s.) Pin LH Pin RH Figure 41. Crosshead Speed vs. Time, in/s 56

70 Force, P (lbf) Before Crack Growth After Crack Growth Critical Loads Displacement, (in.) Figure 4. Pin Force vs. Deflection, in/s Figure 4 illustrates the force-displacement data acquired from the simulation. The Compliance methods all require more than one data point for calculation of G Ic. With the data above, the nd and rd data points unable to be determined. Only the first critical load value is a considered a reliable data point. The curve shape is anomalous and doesn t conform to expected results for brittle-stable or brittle-unstable fracture. 57

71 The force and displacement are plotted in Figure 4 and a trendline is fitted to the data points. The force is nearly linear with a slope of lbs/in. The model is believed to be adversely affected at lower speeds by spring and damper coefficients that are more suited toward higher speed simulations. 5 0 y = x Force, P (lbf) Displacement, (in.) Figure 4. Pin Force vs. Deflection to First Failure, 8 in/s 58

72 The apparent G Ic values are computed using only the Modified Beam Theory. Other compliance methods could not be used without more than one reliable critical load point at different crack lengths. The G Ic is known to be 9.99 in-lb/in, and the simulation generates an apparent strain energy release rate of 10. in-lb/in. TABLE 6 G IC RESULTS FOR IN/S G ic Method (in-lb f /in ) Modified Beam Theory

73 CHAPTER 7 CONCLUSIONS The composite specimen was loaded with three different element types: Elastic Cohesive, Mixed-Mode Cohesive, and Tvergaard-Hutchinson. The best results were obtained using the Tvergaard-Hutchinson elements defined by MAT_185 (see Appendix D for material card). Many simulations were run other than are documented herein. These simulations failed for various reasons, leading to the following recommendations for this type of modeling: Nodal forces extracted from the NODFORC keyword do not provide accurate measurement of the force applied to the block. A spring and damper are required to apply the prescribed velocity. The cohesive elements have a tendency to be unstable because they are not true solid elements (and cannot react tensor stresses). Be cautious of boundary conditions or complex loading that can cause instability and premature failure. Elastic cohesive elements tend to be too brittle. Use of a bilinear or trilinear traction curve yields better results. MAT_185 is recommended for pure Mode-I loading. Dynamic results are sensitive to element aspect ratios of solids. Keep the length, width, and thickness of hex elements under a ratio of to prevent hourglassing. Sufficient nodal mass (element density) is required to increase the time-step and reduce run time. Over-stiffened springs and dampers will reduce the time-step as well. The damper used in the block simulations is believed to be too strong. It is recommended to reduce the damping factor. Using the cohesive model shown in Figure 44, the G Ic for the APC- specimen modeled is assumed to be a constant 9.99 in-lb f /in [5], independent of the loading rate. The cohesive element G Ic is the actual G Ic of the material. 60

74 G Ic =9.995 in-lb f /in max = 11 ksi Traction, t( ) E n =1.11E+7 psi Nodal Deflection, f =1.817E- in Figure 44. Traction-Deflection Model The apparent G Ic value, on the other hand, is that value measured using the forcedeflection data only. Recall that G Ic using the Modified Beam Theory [] is proportional to the area under the force-deflection curve, where the parameters P. / represent the area up to initial failure. P δ G I 67) b a Using Equation 67 to calculate the apparent G Ic for the specimen, it is shown that there is a power-law increase in G Ic-app with loading rate, as illustrated in Figure 45. The area under the force-deflection curve becomes larger with each increase in loading rate. One might assume that the energy lost due to kinetic energy becomes a significant factor, however the indications are that the kinetic energy is still a fairly small portion of the total energy of the system (see Figure ). The more likely scenario is that the plastic zone size, as defined with the cohesive material model, is larger for the higher velocities. The load is not as localized on the crack tip, thereby allowing greater absorption of load before failure. 61

75 100.0 Apparent Actual GIc-apparent (in-lbf/in ) 10.0 y = 7.46x Cross-Head Speed (in/s) Figure 45: Apparent G Ic vs. Loading Rate for APC- Block Specimen In order to adjust G Ic-apparent to the actual G Ic, an adjustment factor,, is introduced such that G G Ic actual η 68) Ic-apparent P δ G I η 69) b a The adjustment factor, plotted in Figure 46, shows a decreasing adjustment factor with respect to crosshead speed. The significance of the factor is that it shows an additional knockdown that would otherwise be unaccounted for in the test results. If the experiment data indicates a constant G Ic versus loading rate, then the adjustment factor would modify the data to show a corresponding decrease. If the experimental data showed a decreasing G Ic versus loading rate, then the adjustment would indicate an even more severe reduction to the curve. 6

76 1.000 Dynamic Adjustment Factor, y = 1.79x Cross-Head Speed (in/s) Figure 46: Dynamic Adjustment Factor for APC- Block Model In conclusion, the APC- specimen showed an apparent increase in G Ic of 67% over the range of in/s to 100 in/s loading rate. Based on the apparent increase in G Ic, the adjustment factor,, is developed to use in conjunction with the Modified Beam Theory, to return G Ic to its predicted value based on the cohesive element strength. One last recommendation for determining G Ic at high speeds is to test specimens with varying initial crack lengths. The only method that does not require multiple failure points within the same specimen is the Modified Beam Theory, which was the only sound method that could be used for some of the results summarized. Using different initial crack lengths would allow for practical use of the more accurate Compliance Calibration Methods. 6

77 REFERENCES 64

78 REFERENCES [1] Berry, J. P. Determination of Fracture Surface Energies by the Cleavage Technique. Journal of Applied Physics. 1 (1967): 6. [] Barbezat, M. and H. H. Kausch. Mechanical characterization of polymers and composites with a servohydraulic high-speed tensile tester. Journal de Physique III, Dec [] ASTM D 558, Standard Test Method for Mode-I Interlaminar Fracture Toughness of Unidirectional Fibre-reinforced Polymer Matrix Composites [4] Smiley, A. J., and R. B. Pipes. "Rate Effects on Mode-I Interlaminar Fracture Toughness in Composite Materials." Journal of Composite Materials. 1 (1987): 670 [5] Gillespie, J. W. Jr., Carlsson, L., Pipes, R. B., Rothschilds, R., Trethewey, B, Smiley, A. Delamination Growth in Composite Materials. National Aeronautics and Space Administration. NASA CR [6] Smiley, A. J. Rate Sensitivity of Interlaminar Fracture Toughness in Composite Materials, M. S. Thesis, Center for Composite Materials, University of Delaware, Newark, Delaware (December 1985). [7] Kistner, Mark D., Strain Rate Sensitivity of Polymer-Matrix Composites under Mode-I Delamination. Wright Research and Development Center. WRDC-TR [8] S. Mall, G.E. Law and M. Katouzian, Loading Rate Effect on Interlaminar Fracture Toughness of a Thermoplastic Composite, Journal of Composite Materials 1987; 1; 569. [9] Daniel, I. M., Shareef, I., and Aliyu, A. A., Rate Effects on Delamination Fracture Toughness of a Toughened Graphite/Epoxy, Toughened Composites, ASTM STP 97, Norman J. Johnston, Ed., American Society for Testing and Materials, Philadelphia, 1987, pp [10] Raju, I.S. Simple Formulae for Strain-energy Release Rates with Higher Order and Singular Finite Elements. National Aeronautics and Space Administration. NASA CR [11] Mallick, P. K. Composite Engineering Handbook. Marcell Dekker, Inc. New York, NY [1] Wilkins, K. J., Eisenmann, J. R., Camin, R. A., Margolis, W. S., and Benson, R. A., Characterizing Delamination Growth in Graphite-Epoxy, Damage in Composite Materials, ASTM STP 775, K. L. Reifsnider, Ed., American Society for Testing and Materials, 198, pp [1] Whitney, J. M., Browning, C. E., and Hoogsteden, W., A Double Cantilever Beam Test for Characterizing Mode-I Delamination of Composite Materials, Journal of Reinforced Plastics and Composites 198; 1;

79 REFERENCES [14] Aliyu, A. A. and I. M. Daniel, Effects of Strain Rate on Delamination Fracture Toughness of Graphite/Epoxy, Delamination and Debonding of Materials, ASTM STP 876, W. S. Johnson, ed. American Society for Testing and Materials, Philadelphia, pp (1985). [15] (008; 009). Metallic Materials Properties Development and Standardization (MMPDS- 04) - Includes Change Notice 1. (pp: -77). Battelle Memorial Institute. Online version available on May 9, 011 at: 86&VerticalID=0 [16] LS-DYNA Keyword User's Manual. Version 971. Vol. 1. Livermore, CA: Livermore Software Technology Corporation (LSTC), 007. Print. [17] Military Handbook - MIL-HDBK-17-F: Composite Materials Handbook, Volume - Polymer Matrix Composites Materials Properties. (pp: 4-04). U.S. Department of Defense. Online version available on May 9, 011 at: 0&VerticalID=0 [18] Tvergaard, V., and Hutchinson, J.W., Crack Growth Resistance and Fracture Process Parameters in Elastic-Plastic Solids, Journal of the Mechanics and Physics of Solids 199; 40;

80 APPENDICES 67

81 APPENDIX A UNITS The English System of units are used consistently throughout the modeling and analysis conducted in this document. All measurable quantities are listed in Table 7 along with the base unit used to describe each quantity. For some dimensions, an equivalent unit that may be more commonplace is listed along with the conversion factor. For example, the LS-DYNA model energy is converted into foot-pounds by multiplying the base units by 1. The base unit for mass used in the analysis is termed the slinch, which is equivalent to 1 slugs. This mass unit was chosen to ensure the units of force, stress, and fracture toughness are automatically expressed as lb f, psi, and in-lb f /in, respectively, with no necessary factoring of the results. TABLE 7 UNITS USED FOR MODELING Dimension Definition Base Unit Equivalent Unit 86.1 lb Mass slinch m Time Distance Area Distance Volume Distance in Moment of Inertia Distance 4 4 Second (s) Inch (in.) 1 ft 1 in 1 ft 144 in slinch Density Mass/Volume Velocity Distance/Time in s Acceleration Distance/Time in s Force Mass*Acceleration Stress Force/Area Energy Force*Distance lb f in Strain Energy Release Rate (G Ic ) lb in 86.1 lb m in 1 ft 1 s 1 ft 144 s lb f slinch in s f in Energy/Distance in lb f in 1 1 ft lb f 68

82 APPENDIX B SOFTWARE VERSIONS The following software and versions were used to complete the research documented herein: CATIA v LS-DYNA v. 971 LS-PREPROST v..1 Microsoft Excel v. 007 Microsoft Word v. 007 MSC Patran v. 008r 69

83 APPENDIX C ELASTIC COHESIVE RESULTS The results are discussed in three following sections: Block Mass Specimen, Zero-Mass Block Specimen, and Hinge Tab Specimen. The modeling was performed in this order, and each section then discusses the effects of different variables, namely velocity, on the mesh Block Mass Specimen, Elastic Cohesive Element The block mass model was the first model investigated. Velocities of, 4, 8, and 0 inches/s displacement rate were applied at the pin center. This section of results included an error in the cohesive element property card that caused the crack growth to behave as a very brittle material. The cohesive element material was input with the COHESIVE_ELASTIC card [16]. The behavior after crack growth is representative of a brittle material rather than a toughened epoxy composite. Crack propagation and failure occurs swiftly. However, the results prior to crack growth should be unaffected. The results after crack growth are also useful for characterizing the behavior of a brittle material. 70

84 APPENDIX C Initial Loading It was observed through post-processing that the constant velocity applied to the loading block was converted within LS-DYNA by means of a damped oscillatory forcing function, F(t). The force function versus time is characterized by the equation P(t) X e sin( t) ζω n t d 54) A plot of the forcing function for a crosshead speed of 0 in/s is shown in Figure 47. The data points are taken from two loaded nodes at the pin center, and are summed to find the force on a.0 wide strip. The sum is then multiplied by 50 to factor up the loads for a full 1 inch wide specimen. The values for X,, n, and d are derived from the LS-DYNA data points measured at 1 s increments. The interval is critical for developing the full curve, but 1 s was found to be sufficient for any loading velocity. As shown in the figure, the assumed formula for P(t) agrees extremely well with the data points Equation 54 LS-DYNA 100 Force (lbf) E+00.0E E E E-05 Time (s.) Figure 47. Forcing Function for APC- (w/ Free Block) at 0 in/s Initially, the shape of the forcing function was a surprising observation because it does not match the low rate test data. A rapid increase in force, and thus acceleration, is needed to start the motion of the block in the vertical direction. If this force is maintained, the velocity of the 71

85 APPENDIX C block will continue to increase versus time. However, a constant velocity is desired and therefore a reduction in force is required to slow the acceleration of the block after the initial impulse. A compressive force applied alternately with a tension force will provide a smooth deflection response, much like any designed control system. It is believed that this is a phenomenon that is driven by the software and may not be realistic. The amplitude of the forcing function varies linearly proportional to the crosshead speed, as shown in Figure 48. The n and d values are constant for any loading velocity. For the configuration tested, the values are.900e+05 Hz and.90e+05 Hz, respectively. These values depend on the material properties and specimen geometry. 50 Force Amplitude, X (lbf) y = x Crosshead Speed (in/s) Figure 48. Forcing Function Amplitude vs. Loading Velocity for APC- (w/ Free Block) In the preceding section, the strain energy release rate was developed assuming a linear loading function. Using a damped oscillatory loading function increases the complexity of the solution and identifies the need for a new set of equations for high speed testing. 7

86 APPENDIX C TABLE 8 FORCING FUNCTION PARAMETERS WITH A BLOCK MASS Cross-Head Speed Pin Velocity Force Amplitude Logarithmic Decrement Damping Ratio Damped Period Damped Frequency Undamped Frequency (in./s) (in./s) X (lb f ) d (s) d (Hz) n (Hz) E E E E TABLE 9 FAILURE TIMES AND DEFLECTIONS WITH A BLOCK MASS Cross-Head Speed Pin Velocity Initial Failure Time Total Failure Time Pin Z- Onset Final Pin Z- Deflection (in./s) (in./s) (s.) (s.) (in.) (in.) E E E E E E E-0.89E E E-0.0E E E-04 1.E-0 4.E-0 1.E-0 7

87 APPENDIX C Loading during Crack Extension Prior to crack extension, the nodal force function is fairly predictable. However, as shown in the following figures, once crack growth begins the force fluctuates erratically. The conclusion being that under dynamic conditions, only the initial G Ic value would be practical to measure. 0 0 Prior to Crack Growth After Crack Growth 10 Force (lb f ) E E E-0 1.5E-0.0E-0 Time (s.) Figure 49. Loading for APC- (w/ Free Block) at in/s 74

88 APPENDIX C Prior to Crack Growth After Crack Growth Force (lb f ) E+00.0E E E E E-0 1.E-0 1.4E-0 1.6E-0 Time (s.) Figure 50. Loading for APC- (w/ Free Block) at 4 in/s Prior to Crack Growth After Crack Growth 40 0 Force (lb f ) E+00.0E E E E E-0 1.E-0 1.4E-0 Time (s.) Figure 51. Loading for APC- (w/ Free Block) at 8 in/s 75

89 APPENDIX C Prior to Crack Growth After Crack Growth Force (lb f ) E+00.0E E E E E-0 1.E-0 1.4E-0 Time (s.) Figure 5. Loading for APC- (w/ Free Block) at 0 in/s 76

90 APPENDIX C Block Rotation Counter-rotation of the block under dynamic conditions was also observed in the first three LS-DYNA models. Figure 5 illustrates this unexpected reaction. Under static conditions, the block should rotate about the pin in the opposite direction as shown. This phenomenon is also evident in the measurement of beam deflection. If the deflection is measured at the pin center (node 785), then the deflection vs. time is linear with a slope equal to the velocity applied to the pin node. If the deflection is measured at the mid-ply of the half-beam (node 816) as shown in Figure 5, the response will be linear until the counter-rotation of the loading block becomes significant. At the onset of the rotation there is a characteristic decrease (or increase) in deflection even as the pin continues on with a constant vertical motion. Figure 5. Counter-Rotation of the Loading Block 77

91 APPENDIX C The deflection of the half-beam mid-ply is measured at nodes 816 and 816. These nodes start at the origin of the X and Z coordinate plane. Hence deflection at any time can conveniently be measured as the Z coordinate of the nodes at any time increment. This provides more precision when the deflections are small and when using the NODOUT keyword in the input file. The average deflection of the two nodes at any given point is then taken. Figure 54 illustrates deflection vs. time for a crosshead speed of in/s. Up to 1 ms the trend is linear and note the slope is equal to 1 in/s (speed of half of beam specimen). There is some slight nonlinearity after 1 ms, but doesn t become significant until about 1.5 ms. The negative slope of the curve is an indicator of counter-rotation centered between the pin and the crack. The increased slope (velocity) is an indicator of counter-rotation centered on the opposite side of the pin. Note that the delamination begins to propagate at about 0.77 ms. At this velocity, the specimen response remains linear even after the initial crack growth Prior to Crack Growth After Crack Growth Deflection (in.) Counter-Rotation Evident 0 0.0E E E-0 1.5E-0.0E-0 Time (s.) Figure 54. Beam Deflection beneath Loading Pin at in/s (with Free Block) 78

92 APPENDIX C Prior to Crack Growth After Crack Growth Deflection (in.) Counter-Rotation Evident E E E-0 1.5E-0 Time (s.) Figure 55. Beam Deflection beneath Loading Pin at 4 in/s (with Free Block) Prior to Crack Growth After Crack Growth Deflection (in.) Counter-Rotation Evident E+00.0E E E E E-0 1.E-0 1.4E-0 Time (s.) Figure 56. Beam Deflection beneath Loading Pin at 8 in/s (with Free Block) 79

93 APPENDIX C Prior to Crack Growth After Crack Growth Deflection (in.) Counter-Rotation Evident 0.0E+00.0E E E E E-0 1.E-0 1.4E-0 Time (s.) Figure 57. Beam Deflection beneath Loading Pin at 0 in/s (with Free Block) 80

94 APPENDIX C Force Smoothing After observing the oscillatory force function, I hypothesized that some smoothing of the force curve would provide an almost identical result. For example, the average force applied to the block over the first period of loading could be applied without the wild fluctuations associated with the sine component of the equation. Then the average force for the second period could be applied and so on. This would first provide a stair-step loading function which could be smoothed by curve fitting. Using this approach, the general form of the smoothed force function was derived to be of the form ζωnt P(t) c1 e 70) The constant c 1 was found by setting the value of P( d /) equal to the average of the sinusoidal force function over the first period. This provides a solution for c 1 equal to c 1 X(1 ζ )(e ζω τ / n d e -ζ τ / n d ) 71) All of the parameters for c 1 are derived from the original oscillatory function. This approach was tested on the APC- specimen in an attempt to approximate a 0 in/s crosshead speed. The results were not as desired. Figure 58 illustrates the pin crosshead speed 81