AN ABSTRACT OF A THESIS HYDROSTATIC STRESS EFFECTS IN METAL PLASTICITY

AN ABSTRACT OF A THESIS HYDROSTATIC STRESS EFFECTS IN METAL PLASTICITY Phillip A. Allen Master of Science in Mechanical Engineering Classical metal plasticity theory assumes that hydrostatic pressure has no influence on the yield and postyield behavior of metals. Plasticity textbooks from the earliest to the most modern infer that there is negligible hydrostatic effect on the yielding of metals. Even modern finite element programs direct the user to assume the same. Therefore, the object of this study is to use finite element analysis to examine the effect of internal hydrostatic stress on the yield behavior of metals using the von Mises (hydrostatic independent) and Drucker-Prager (hydrostatic dependent) constitutive models. Several nonlinear finite element analyses were used in order to assess how well the von Mises and Drucker-Prager failure criteria could estimate the yield and postyield behavior of metals. A variety of finite element models (FEM s) were created to demonstrate the response of the failure theories to a wide range of internal hydrostatic stress. The geometries modeled include: smooth tensile, notched round bar, equal-arm bend, double-edge notch tension, and modified double-edge notch tension specimens. The equal-arm bend specimen is used by engineers at NASA to simulate the loading condition of a problem area in a new fuel turbopump housing for the Space Shuttle. Accurate load-displacement or load-microstrain test data or both was available for each of the geometries tested and was used as a measure of the FEM s performance. The two materials tested in this study were 2024-T351 and Inconel 100 (IN100). The elastic and plastic properties for each material are presented. Also, various methods of determining the material constants for the constitutive models are discussed. Internal hydrostatic stress effects cannot always be considered negligible. For every geometry tested, the von Mises constitutive model overestimated the load for a given value of strain or displacement. For given displacements at failure, the Drucker- Prager FEM s predicted loads that were 3% to 10% lower than the von Mises values. For given failure loads, the Drucker Prager FEM s predicted strains that were 20% to 65% greater than the von Mises values, and, therefore, did a much better job of matching the overall specimen response. Therefore, the Drucker-Prager constitutive model is preferable for modeling geometries with a significant hydrostatic stress influence.

HYDROSTATIC STRESS EFFECTS IN METAL PLASTICITY A Thesis Presented to the Faculty of the Graduate School Tennessee Technological University by Phillip A. Allen In Partial Fulfillment of the Requirements of the Degree MASTER OF SCIENCE Mechanical Engineering August 2000

CERTIFICATE OF APPROVAL OF THESIS HYDROSTATIC STRESS EFFECTS IN METAL PLASTICITY by Phillip A. Allen Graduate Advisory Committee: Chairperson Member Member date date date Approved for the Faculty Dean of Graduate Studies Date ii

STATEMENT OF PERMISSION TO USE In presenting this thesis in partial fulfillment of the requirements for a Master of Science degree at Tennessee Technological University, I agree that the University Library shall make it available to borrowers under rules of the Library. Brief quotations from this thesis are allowable without special permission, provided that accurate acknowledgement of the source is made. Permission for extensive quotation from or reproduction of this thesis may be granted by my major professor when the proposed use of the material is for scholarly purposes. Any copying or use of the material in this thesis for financial gain shall not be allowed without my written permission. Signature Date iii

DEDICATION This thesis is dedicated to my wife Shannon. Without her unselfish sacrifices, love, and support this work could not have been accomplished. iv

ACKNOWLEDGEMENTS I would like to thank my major professor, Dr. Chris Wilson, for his guidance, his insistence on excellence, and his Christian example. I would also like to express thanks to the other committee members, Dr. George Buchanan and Dr. Dale Wilson, for their comments and assistance. David Loy also deserves special thanks for all the help he provided. I am grateful to several people at Marshall Space Flight Center for their assistance, guidance, and advice. These people include Dr. Greg Swanson, Rob Wingate, Jim Hawkins, Jeff Rayburn, and Dr. Preston McGill. Funding for this research was provided by the National Aeronautics and Space Administration. v

TABLE OF CONTENTS vi Page LIST OF TABLES...viii LIST OF FIGURES... ix LIST OF SYMBOLS, ACRONYMS, AND ABBREVIATIONS...xiii CHAPTER 1. INTRODUCTION... 1 2. TECHNICAL BACKGROUND... 4 A Classical View of Metal Plasticity... 4 Yield Functions... 4 Hardening Rules... 9 Flow Rules... 13 Hydrostatic Stress Deviations From Classical Theory... 15 3. RESEARCH PROCEDURE... 31 Finite Element Modeling... 31 Smooth (Unnotched) Tensile Bar Specimen... 32 Notched Round Bar Specimens... 36 Equal-Arm Bend Specimen... 42 Double-Edge Notch Tension Specimen... 47 Modified Double-Edge Notch Tension Specimen... 54 Required Material Properties... 57 4. RESULTS... 64 Material Property Calculations... 64 2024-T351 Properties... 64 IN100 Properties... 66 Finite Element Model Results... 68 Smooth Tensile Bar Results... 68 Notched Round Bar Results... 72 Equal-Arm Bend Results... 81 Double-Edge Notch Tension Results... 87 Modified Double-Edge Notch Tension Results... 99 5. CONCLUSIONS AND RECOMMENDATIONS... 105

Page BIBLIOGRAPHY... 108 APPENDICES APPENDIX A - SCRIPT FILE... 113 APPENDIX B MATERIAL PROPERTY TABLES... 115 APPENDIX C TABLES OF TEST DATA... 118 VITA... 137 vii

LIST OF TABLES Page Table 2.1. Summary of Experimental Results for Constants in Equation (2.27) [20] 22 Table 3.1. Alloy Composition of 2024- T351 and IN100 [28] 59 viii

LIST OF FIGURES Page Figure 1.1. Schematic of Space Shuttle Main Rocket Engine Fuel Pump Turbine Housing Illustrating Problem Area [7]... 2 Figure 2.1. von Mises Yield Surface in Principal Stress Space [10]... 9 Figure 2.2. Isotropic Hardening for the von Mises Yield Function... 10 Figure 2.3. Illustration of the Bauschinger Effect... 11 Figure 2.4. Kinematic Hardening for the von Mises Yield Function... 12 Figure 2.5. True Stress Versus True Strain Curves in Tension and Compression for 4310 Steel [12]... 12 Figure 2.6. True Stress Versus True Strain Curves in Tension and Compression for 4330 Steel [12]... 13 Figure 2.7. Flow Stress (Effective Stress) as a Function of Strain for Tempered Pearlite Tested at Various Pressures [14]... 16 Figure 2.8. Flow Stress (Effective Stress) as a Function of Strain for Tempered Martensite Tested at Various Pressures [14]... 17 Figure 2.9. Plastic Stress-Strain Relations in Tension for Nittany No.2 Brass Under Hydrostatic Pressure [18]... 19 Figure 2.10. Effect of Hydrostatic Pressure on the Stress-Strain Curves in Compression for 4330 Steel [20]... 20 Figure 2.11. Effect of Hydrostatic Pressure on the Stress-Strain Curves in Compression for Aged Maraging Steel [20]... 20 Figure 2.12. Dependence of Yielding on Mean Stress in 4330 Steel [20]... 21 Figure 2.13. Dependence of Yielding on Mean Tension for Aged Maraging Steel [20] 22 Figure 2.14. Plastic Volume Increase as a Function of True Plastic Strain for 4310 and 4330 Steels [20]... 24 Figure 2.15. Schematic of σ eff versus I 1 [10]... 25 Figure 2.16. Cohesive Force as a Function of the Separation Between Atoms (Adapted from [21])... 26 Figure 2.17. Comparison of Drucker-Prager and von Mises Yield Surfaces in Principal Stress Space [10]... 29 Figure 3.1. Engineering Drawing of the Smooth Tensile Bar Specimen (All Dimensions Are in Inches)... 33 Figure 3.2. Axisymmetric Q4 Element with Node Numbers and Degrees of Freedom.. 33 Figure 3.3. Schematic of Axisymmetric Model of a Smooth Tensile Bar Specimen Utilizing Two Planes of Symmetry... 34 Figure 3.4. Finite Element Model of the Smooth Tensile Bar... 35 Figure 3.5. Engineering Drawing of the Notched Round Bar Specimen (All Dimensions Are in mm) [10]... 37 ix

x Page Figure 3.6. Schematic of Axisymetric Model of a NRB Utilizing Two Planes of Symmetry [10]... 38 Figure 3.7. Fine Mesh FEM of NRB (ρ = 0.005 in.)... 39 Figure 3.8. Coarse Mesh FEM in the Notch Region of the NRB (ρ = 0.005 in.)... 40 Figure 3.9. Medium Mesh FEM in the Notch Region of the NRB (ρ = 0.005 in.)... 41 Figure 3.10. Fine Mesh FEM in the Notch Region of the NRB (ρ = 0.005 in.)... 42 Figure 3.11. Engineering Drawing of the Equal Arm Bend Specimen (Dimensions in inches, Nominal Dimensions Used in FEM)... 43 Figure 3.12. Q4 Element with Node Numbers and Degrees of Freedom... 43 Figure 3.13. Schematic of the Equal-Arm Bend FEM Utilizing One Symmetry Plane.. 45 Figure 3.14. Equal Arm Bend Finite Element Model Utilizing One Plane of Symmetry46 Figure 3.15. Mesh in the Fillet Region of the Equal Arm Bend Specimen... 47 Figure 3.16. Engineering Drawing of the DENT Specimen (Dimensions in inches)... 48 Figure 3.17. Schematic of the 2-D DENT FEM Utilizing Two Symmetry Planes... 48 Figure 3.18. Illustration of the 2-D DENT FEM... 50 Figure 3.19. Coarse Mesh in the Notch Region for the 2-D DENT FEM... 51 Figure 3.20. Medium Mesh in the Notch Region for the 2-D DENT FEM... 51 Figure 3.21. Fine Mesh in the Notch Region for the 2-D DENT FEM... 52 Figure 3.22. H8 3-D Element With Node Numbers and Degrees of Freedom... 52 Figure 3.23. Schematic of the Three Symmetry Planes of the 3-D DENT Model... 53 Figure 3.24. Illustration of the 3-D DENT FEM... 53 Figure 3.25. Illustration of One Side of the modified DENT Specimen Showing the New Notch Cut into the Existing Notch on the DENT Specimen... 54 Figure 3.26. Illustration of the Modified DENT Finite Element Model... 55 Figure 3.27. True Stress versus True Strain Curve for 2024-T351 at Room Temperature [6]... 58 Figure 3.28. True Stress versus True Strain Curve for IN100 at Room Temperature [27]... 58 Figure 3.29. Linear Drucker-Prager Model: Yield Surface and Flow Direction in the p-t Plane (Adapted from [5])... 60 Figure 3.30. Schematic of Calculation of a from Uniaxial Tension and Uniaxial Compression Data... 62 Figure 4.1. 2024-T351 Isotropic Hardening Parameters for the von Mises and Drucker- Prager Constitutive Models... 66 Figure 4.2. IN100 Isotropic Hardening Parameters for the von Mises and Drucker-Prager Constitutive Models... 67 Figure 4.3. Load-Displacement Results for 2024-T351 Smooth Tensile Bar... 69 Figure 4.4. Broken Tensile Specimen with Angled Fracture Surface... 70 Figure 4.5. Effective Stress (psi) in the Smooth Tensile Bar at 0.15 in. Gage Displacement... 70 Figure 4.6. Hydrostatic Pressure (psi) in the Smooth Tensile Bar at 0.15 in. Gage Displacement... 71

xi Page Figure 4.7. Stress in the z-direction (psi) in the Smooth Tensile Bar at 0.15 in. Gage Displacement... 71 Figure 4.8. Equivalent Plastic Strain in the Smooth Tensile Bar at 0.15 in. Gage Displacement... 72 Figure 4.9. Load-Displacement Plot for Coarse, Medium, and Fine Mesh NRB's with ρ = 0.005 in... 73 Figure 4.10. Effective Stress as a Function of Distance Across the Neck of the Coarse, Medium, and Fine Mesh NRB's with ρ = 0.005 in. at Failure Load... 74 Figure 4.11. Mean Stress as a Function of Distance Across the Neck of the Coarse, Medium, and Fine Mesh NRB's with ρ = 0.005 in. at Failure Load... 74 Figure 4.12. Radial Stress as a Function of Distance Across the Neck of the Coarse, Medium, and Fine Mesh NRB's with ρ = 0.005 in. at Failure Load... 75 Figure 4.13. Equivalent Plastic Strain as a Function of Distance Across the Neck of the Coarse, Medium, and Fine Mesh NRB's with ρ = 0.005 in. at Failure Load... 75 Figure 4.14. Effective Stress, Mean Stress, and Radial Stress as a Function of Distance Across the Neck of the Fine Mesh NRB with ρ = 0.005 in. at Failure Load... 76 Figure 4.15. Load-Displacement Results for the NRB with ρ = 0.005 in.... 78 Figure 4.16. Load-Displacement Results for the NRB with ρ = 0.010 in.... 78 Figure 4.17. Load-Displacement Results for NRB with ρ = 0.020 in.... 79 Figure 4.18. Effective Stress (psi) in the NRB Specimen (ρ = 0.010 in.) at Failure Load... 79 Figure 4.19. Hydrostatic Pressure (psi) in the NRB Specimen (ρ = 0.010 in.) at Failure Load... 80 Figure 4.20. Stress in the z-direction (psi) in the NRB Specimen (ρ = 0.010 in.) at Failure Load... 80 Figure 4.21. Equivalent Plastic Strain in the NRB Specimen (ρ = 0.010 in.) at Failure Load... 81 Figure 4.22. Load-Microstrain Results for the Equal-Arm Bend Specimen... 83 Figure 4.23. Effective Stress, Mean Stress, and Stress in the x-direction as a Function of Distance Across the Fillet Section of the Equal-Arm Bend FEM at Point A... 84 Figure 4.24. Equivalent Plastic Strain as a Function of Distance Across the Fillet Section of the Equal-Arm Bend FEM at Point A... 84 Figure 4.25. Effective Stress (psi) in the Equal-Arm Bend Specimen at Point A... 85 Figure 4.26. Hydrostatic Pressure (psi) in the Equal-Arm Bend Specimen at Point A 86 Figure 4.27. Stress in the y-direction (psi) in the Equal-Arm Bend Specimen at Point A... 86 Figure 4.28. Equivalent Plastic Strain in the Equal-Arm Bend Specimen at Point A.. 87 Figure 4.29. Load-Displacement Plot for Coarse, Medium, and Fine Mesh 2-D DENT Finite Element Models... 88

Page Figure 4.30. Effective Stress as a Function of Distance Across the Notch Region of the Coarse, Medium, and Fine Mesh 2-D DENT FEM's at Failure Load... 89 Figure 4.31. Mean Stress as a Function of Distance Across the Notch Region of the Coarse, Medium, and Fine Mesh 2-D DENT FEM's at Failure Load... 90 Figure 4.32. Stress in the x-direction as a Function of Distance Across the Notch Region of the Coarse, Medium, and Fine Mesh 2-D DENT FEM's at Failure Load... 90 Figure 4.33. Equivalent Plastic Strain as a Function of Distance Across the Notch Region of the Coarse, Medium, and Fine Mesh 2-D DENT FEM's at Failure Load... 91 Figure 4.34. Effective Stress, Mean Stress, Stress in the x-direction, and Stress in the y- Direction as a Function of Distance Across the Notch Region of the Fine Mesh 2-D DENT FEM at Failure Load... 92 Figure 4.35. Load-Displacement Results for the 2-D DENT Specimen... 94 Figure 4.36. Load-Displacement Results for the 3-D DENT Specimen... 94 Figure 4.37. Linear Portion of the Load-Displacement Results for the 2-D DENT Specimen... 95 Figure 4.38. Linear Portion of the Load-Displacement Results for the 3-D DENT Specimen... 95 Figure 4.39. Effective Stress (psi) for the 2-D DENT FEM at Failure Load... 96 Figure 4.40. Stress in the y-direction (psi) for the 2-D DENT FEM at Failure Load... 96 Figure 4.41. Equivalent Plastic Strain for the 2-D DENT FEM at Failure Load... 97 Figure 4.42. Effective Stress (psi) for the 3-D DENT FEM at Failure Load... 97 Figure 4.43. Hydrostatic Pressure (psi) for the 3-D DENT FEM at Failure Load... 98 Figure 4.44. Stress in the y-direction (psi) for the 3-D DENT FEM at Failure Load... 98 Figure 4.45. Equivalent Plastic Strain for the 3-D DENT FEM at Failure Load... 99 Figure 4.46. Load-Displacement Results for the Modified DENT Specimen... 101 Figure 4.47. Linear Portion of the Load-Displacement Results for the Modified DENT Specimen... 101 Figure 4.48. Load-Microstrain Results for the Modified DENT Specimen... 102 Figure 4.49. Effective Stress (psi) for the Modified DENT FEM at Failure Load... 103 Figure 4.50. Stress in the y-direction (psi) for the Modified DENT FEM at Failure Load... 103 Figure 4.51. Equivalent Plastic Strain for the Modified DENT FEM at Failure Load.. 104 xii

LIST OF SYMBOLS, ACRONYMS, AND ABBREVIATIONS Symbol a a o d f g i j k p q 1, q 2 t x A B C E F I 1, 2, 3 J 1, 2, 3 K P R R o S 1, 2, 3 F U 1, 2, 3 Description Slope of Effective Stress Versus the First Stress Invariant Equilibrium Atomic Spacing Modified Yield Strength Yield Function Plastic Potential Function Tensor Notation Subscript Tensor Notation Subscript Yield Strength in Pure Shear Hydrostatic Pressure Adjustable Parameters in Tvergaard s Modified Yield Function Variable in Drucker-Prager Yield Function in ABAQUS Separation Distance Between Atoms Material Constant in Hu s Yield Function Material Constant in Hu s Yield Function Material Constant in Hu s Yield Function Young s Modulus Void Volume Fraction Stress Invariants Deviatoric Stress Invariants Ratio of Yield Stress in Triaxial Tension to Yield Stress in Triaxial Compression Load Average of the Radial Displacements of an Ellipsoidal Void Initial Radius of Spherical Void Deviatoric Principal Stresses Void Volume Fraction Displacements in the x Direction xiii

V 1, 2, 3 W 1, 2, 3 W f Displacements in the y Direction Displacements in the z Direction Fracture Work β Angle of the Slope of the Yield Surface in the p-t stress plane dφ Positive Constant in General Flow Rule ε True Strain ε 1, 2, 3 Principal Strains ε pl Plastic Strain ε Equivalent Plastic Strain pl eq γ Surface Energy λ Lattice Wavelength µε Microstrain ν Poisson s Ratio ρ Notch Root Radius σ (a) True Stress (b) Cohesive Stress σ 1, 2, 3 Principal Stresses σ c Theoretical Cohesive Strength σ eff Effective Stress ~ σ Modified Effective Stress eff σ m σ max σ xx, yy, zz σ ys σ ysc τ xy, xz, yz Mean Stress Maximum Applied Stress Normal Stresses Yield Strength Compressive Yield Strength Shear Stresses Dilation Angle Unix Stream Editor Double-Edge Notch Tension Finite Element Analysis Finite Element Model National Aeronautics and Space Administration Notched Round Bar ψ sed DENT FEA FEM NASA NRB 2-D Two Dimensional 3-D Three Dimensional xiv

CHAPTER 1 INTRODUCTION Classical metal plasticity theory assumes that hydrostatic pressure has no influence on the yield and postyield behavior of metals. This assumption is based on the early work of Bridgman [1]. In the 1940 s, Bridgman tested smooth tensile bars made from different materials under the conditions of varying external hydrostatic pressures up to 450 ksi (3100 MPa). His tests showed no significant influence on yield until the highest pressures were reached. The primary effects of hydrostatic pressure were larger strains before fracture and, therefore, increased ductility. Because of the influence of Bridgman s work, plasticity textbooks from the earliest (e.g. Hill [2]) to the most modern (e.g. Lubliner [3]) infer that there is negligible hydrostatic effect on the yielding of metals. Even modern finite element programs such as ANSYS [4] and ABAQUS [5] direct the user to assume the same. Engineers often make calculations based on the assumption that the effect of hydrostatic stress is negligible, when in reality the effects of hydrostatic stress can have a significant influence on material yield behavior. It is well documented that large tensile hydrostatic stresses develop in sharply notched or cracked geometries. Wilson [6] has demonstrated that for these cases, a yield criterion that has a hydrostatic dependence parameter provides a solution that is approximately ten percent more accurate than those given by the von Mises yield function. This increase in accuracy might not be significant 1

2 to an engineer using large factors of safety, but it is very meaningful to a high performance applications designer using minimal factors of safety. Additionally, any increase in understanding which leads to more accurate solutions should be considered beneficial to the field of engineering. Therefore, the objective of this study is to examine the effect of hydrostatic stress on the yield behavior of metals using the von Mises and Drucker-Prager constitutive models in the ABAQUS finite element program. The structural engineers at the National Aeronautics and Space Administration s (NASA) Marshall Space Flight Center are especially interested in this research because of a current design problem with a new fuel pump turbine housing for the main rocket engines of the Space Shuttle (Figure 1.1). This housing is designed to last 60 cycles. Region of Cracking Turbine Housing (forged IN100 material) Figure 1.1. Schematic of Space Shuttle Main Rocket Engine Fuel Pump Turbine Housing Illustrating Problem Area [7]

3 Instead, several housings have developed cracks in a fillet section after as few as 10 cycles. The housing undergoes plastic deformation upon each firing of the engine, and the fillet area develops a significant internal hydrostatic pressure. Therefore, it is probable that a yield function that more accurately models plastic behavior through the inclusion of hydrostatic stress effects will be able to help disclose the source of the premature failures. In the next chapter, a classical view of metal plasticity is presented. Following the discussion of classical plasticity theory, results of previous research that deviate from the tenets of classical metal plasticity are discussed. The finite element modeling procedures for several different geometries are presented, and the methods for obtaining the required material properties for a yield function that depends on hydrostatic stress are given. The material property calculations and finite element modeling results are presented. Finally, conclusions and recommendations are presented.

CHAPTER 2 TECHNICAL BACKGROUND In this chapter, a classical view of metal plasticity is presented including a discussion of yield functions, hardening rules, and flow rules. Next, the results of several researchers that studied the effects of hydrostatic stress are presented. These results deviate from classical plasticity theory and illustrate the effects of hydrostatic stress on yielding and ductile fracture. A Classical View of Metal Plasticity Plastic material behavior is a more complex phenomenon than elastic material behavior. In the elastic range, the strains are linearly related to the stresses by Hooke s law, and the strains are uniquely determined by the stresses. In general, plastic strains are not uniquely determined by the stresses. Plastic strains depend on the whole loading history or how the stress state was reached [8]. Therefore, to completely describe material behavior in the plastic range, one must determine the appropriate yield function, hardening rule, and flow rule. Yield Functions A yield function is a mathematical relationship that predicts the onset of yielding in a material. Some background information defining various stress quantities must be 4

5 presented before a mathematical description of yield functions is given. The principal stresses are given by σ 1, σ 2, and σ 3. The cubic equation 3 2 σ I σ I σ I = 0 (2.1) 1 2 3 is solved to give the principal stresses, where the three roots of σ are principal stresses and I 1, I 2, and I 3 are the stress invariants. The stress invariants are expressed in the cartesian coordinate system as I I 1 = σ xx + σ yy + σ zz, (2.2) ( σ σ + σ σ σ σ ) = τ + τ + τ + 2 2 2 2 xy yz zx xx yy yy zz zz xx (2.3) and I 2 2 2 ( σ τ + σ τ σ τ ) = σ σ σ + τ τ τ +, (2.4) 3 xx yy zz 2 xy yz zx xx yz yy zx zz xy where σ xx, σ yy, and σ zz are the normal stresses and τ xy, τ xz, and τ yz are the shear stresses in the cartesian coordinate system [9]. The hydrostatic or mean stress is defined as σ = 1 m I 1 3, (2.5) and the hydrostatic pressure is given as p = σ m = 1 I 1. (2.6) 3 Bridgman [1] conducted experiments studying the effect of an externally applied pressure on the yield and postyield behavior of metals. He found that there was no

6 significant effect on the yield point until high external pressures (450 ksi) were reached. (Bridgman s work is discussed in more detail in the Hydrostatic Stress Deviations From Classical Theory section.) Early developers of plasticity theory interpreted Bridgman s results to mean that hydrostatic stress, whether externally applied or internally generated, has a negligible effect on the yield behavior of metals. These early plasticity researchers, therefore, developed the first tenet of classical metal plasticity the hydrostatic stress has no effect on yielding. This rationale led to the development of a plasticity theory that subtracts the hydrostatic stress from the principal stresses resulting in the deviatoric stresses S 1, S 2, and S 3. The deviatoric stresses are 1 S 1 = σ 1 I 1, (2.7) 3 1 S 2 = σ 2 I 1, (2.8) 3 and 1 S 3 = σ 3 I 1. (2.9) 3 The deviatoric stresses are the roots of the cubic equation 3 2 S J S J S J = 0. (2.10) 1 2 3 J 1, J 2, and J 3 are the deviatoric stress invariants: J 1 = 0, (2.11)

7 2 2 2 [( σ σ ) + ( σ σ ) + ( σ σ ) ] 1 J 2 = 1 2 2 3 3 1, (2.12) 6 and J ( σ σ )( σ σ )( σ σ ) 3 = 1 m 2 m 3 m. (2.13) In terms of the deviatoric stresses, the deviatoric stress invariants are and 2 2 2 ( S + S S ) 1 J 2 = 1 2 + 3 (2.14) 2 J =. (2.15) 3 S1S 2S3 In classical metal plasticity theory, a yield function, f, is a function of the principal stresses written in the form of ( σ, σ σ ) f = f. (2.16) 1 2, 3 By definition, when f < 0 the material behaves elastically, and when f = 0 yielding occurs and the material behavior is plastic. Assuming that yield is independent of hydrostatic stress leads to a yield function ( ) f = f J, J. (2.17) 2 3 The von Mises yield function is often used for classical metal plasticity calculations. This function states that yield is independent of hydrostatic stress and only depends on J 2 in the form of

8 f 2 ( J ) J =, (2.18) 2 2 k where k is the yield strength in pure shear and is a function of plastic strain for hardening materials. The von Mises or effective stress is defined as σ 2 2 2 [( σ σ ) + ( σ σ ) + ( σ ) ] 1 = 3J = σ 2 eff 2 1 2 2 3 3 1. (2.19) Setting f ( J 2 ) equal to zero in Equation (2.18) leads to 2 J 2 = k, (2.20) which can be interpreted as the von Mises yield surface in Haigh-Westergaard (principal stress) space. The yield surface for the von Mises yield function is a circular cylinder of radius, k, whose axis is defined in the direction of the hydrostatic pressure (Figure 2.1). A yield locus is a curve made by intersecting the yield surface with a plane perpendicular to the cylinder axis. For the von Mises yield function, a yield locus taken anywhere along the hydrostatic pressure axis is a circle of radius k, thus demonstrating the function s hydrostatic independence. The hydrostatic stress is zero on the plane passing through principal stress space origin. This plane is defined as the π plane and is given by the equation σ 1 + σ 2 + σ 3 = 0.

9 σ m = 1/3I 1 σ 3 k σ 2 σ 1 Figure 2.1. von Mises Yield Surface in Principal Stress Space [10] Hardening Rules If a material exhibits strain hardening, the yield surface may change shape or location or both as the material deforms plastically. This effect can be approximated for many materials by using isotropic hardening wherein the yield surface expands equally in all directions. Considering the von Mises yield function, the radius of the yield surface increases from k 1 to k 2 as the material hardens. A graphical representation of isotropic hardening for the von Mises yield function is given in Figure 2.2. Points a and b represent arbitrary points on the yield surfaces, and the line from a to b is an arbitrary path through principal stress space connecting the two points.

10 σ3 b k 2 a k 1 σ1 σ2 Figure 2.2. Isotropic Hardening for the von Mises Yield Function As implied by the name, a material that obeys isotropic hardening has the same yield behavior in both tension and compression. This is approximately true for some materials, but it is not an accurate description of material behavior in general. Many materials exhibit a behavior referred to as the Bauschinger effect, wherein the yield strength for tension is different than that for compression. The Bauschinger effect is illustrated graphically in Figure 2.3. Upon initial loading of a specimen, stress and strain are linearly related until the tensile yield strength, σ ys, is reached at point 1. The load is then increased on the specimen causing plastic deformation and bringing the stress in the specimen up to a maximum stress, σ max, at point 2. When the load is reversed, plastic strains develop at point 3 before σ ys is reached. This effect is very important when a reversal of loading (locally or globally) is to be considered. A kinematic hardening model can describe the behavior of materials with a significant Bauschinger effect. This is accomplished by shifting the axis of the yield surface in principal stress space while maintaining the same radius as the initial yield

11 σ σmax 2 σys 1 ε 3 σys Figure 2.3. Illustration of the Bauschinger Effect surface. Kinematic hardening for the von Mises yield function is graphically represented in Figure 2.4. Since almost no material hardens in a pure isotropic or kinematic fashion, a linear combination of both models is often used to describe real materials. A material behavior related to the Bauschinger effect is the strength-differential phenomenon. The term strength-differential refers to the difference between tensile and compressive yield strengths for many materials. Many plasticity researchers including Drucker [11] and Spitzig [12] have studied the causes of this phenomenon and the resultant effect on material behavior. Spitzig conducted compression and tension tests on 4310 and 4330 steel to investigate the strength-differential effect in high strength steels, and the results of his tests are given in Figure 2.5 and Figure 2.6. The yield strength in

12 σ3 b a σ1 σ2 Figure 2.4. Kinematic Hardening for the von Mises Yield Function Figure 2.5. True Stress Versus True Strain Curves in Tension and Compression for 4310 Steel [12]

13 Figure 2.6. True Stress Versus True Strain Curves in Tension and Compression for 4330 Steel [12] compression for the 4310 steel increases by approximately 4.5% from 151 ksi to 158 ksi, and the yield strength in compression for the 4330 steel increases by approximately 4.3% from 210 ksi to 219 ksi. Flow Rules Flow rules for plastic behavior are analogous to Hooke s law for elastic behavior. Hooke s law defines the relationship between stress and elastic strains, while flow rules define the relationship between stresses and plastic strain increments. A general form of a flow rule relating stresses to plastic strain increments is given by dε pl ij g = dφ, (2.21) σ ij

14 pl where dε ij are the plastic strain increments, g is the plastic potential function, and dφ is a positive constant [8]. The pair of indices i and j range from 1 to 3 or x to z. Associated flow occurs when g = f, where f is the yield function. Bridgman [1] made the observation that volume change during plastic deformation is nearly elastic, and, therefore, he assumed metals were incompressible. The influence of Bridgman s observation lead to the second basic tenet of classical metal plasticity metal incompressibility. For an incompressible material, the sum of the plastic strain increments (or plastic dilatation rate) must be zero. This can be written in terms of the principal strain increments dε 1, dε 2, and dε 3 as pl pl pl pl dε = dε + dε + dε 0. (2.22) ii 1 2 3 = where ε is the plastic portion of dε 1. pl d 1 Equation (2.21) is written in associated form as dε pl ij f = dφ. (2.23) σ ij pl Drucker and Prager [13] showed that dε ii can be summed from Equation (2.23) to obtain f ε ii = 3dφ. (2.24) I d pl 1 Since the hydrostatic stress is 1 I 3 1, pl dε ii must equal zero if the yield function does not depend on hydrostatic stress.

15 Hydrostatic Stress Deviations From Classical Theory Since the 1940 s, many have considered Bridgman s experiments on the effects of hydrostatic pressure on metals the definitive study. In his study, Bridgman tested smooth (unnotched) tensile bars made from a variety of common metals. He conducted tensile tests under the conditions of hydrostatic pressures up to 450 ksi and found that there was no significant effect on the yield point until the higher pressures were reached. His studies revealed that the primary effect of hydrostatic pressure was increased ductility. Bridgman also measured the volume of the material in the gage section and found that this volume did not change, even for very large changes in plastic strain. Since the volume in the gage section did not change, he concluded that metals have incompressible plastic strains. His two observations about metal behavior no effect of hydrostatic pressure on yielding and incompressibility for plastic strain changes have become the standard tenets for studies of metal plasticity [1]. Bridgman continued to study the effects of external hydrostatic pressure for many years, and, in 1952, he wrote a comprehensive summary of his work in his book Studies in Large Plastic Flow and Fracture with Special Emphasis on the Effects of Hydrostatic Pressure [14]. In this book, he reexamined his earlier results and made observations that many plasticity books failed to notice. On p. 64 of his book, Bridgman writes: The fact that a curve is obtained with haphazard pressures indicates that the effect of pressure as such on the strain hardening is unimportant, the role of pressure being merely to permit the large strains without fracture which determine the strain hardening. This is indeed the case to a first approximation. In nearly all the work tabulated above, no consistent correlation was apparent between pressure and the stress-strain points, in

16 view of the sometimes large scatter arising from other factors. By the time the last series of measurements was being made under the arsenal contract, however, skill in making the measurements had so increased, and probably also the homogeneity of the material of the specimens had also increased because of care in preparation, that it was possible to establish a definite effect of pressure on the strain hardening curve [14]. Representative results of Bridgman s later tests are given in Figure 2.7 and Figure 2.8. These data clearly demonstrate a strong dependence of flow stress on hydrostatic pressure for both tempered pearlite and tempered martensite. For example, the flow stress for tempered pearlite at a strain of 2.75 increased from 255 ksi at atmospheric pressure to 315 ksi when pressurized to approximately 360 ksi (Figure 2.7). Therefore, Bridgman clearly demonstrated in his later work a definite external hydrostatic pressure effect on yielding. Unfortunately, he failed to consider the effect of internally generated hydrostatic stresses. Figure 2.7. Flow Stress (Effective Stress) as a Function of Strain for Tempered Pearlite Tested at Various Pressures [14]

17 Figure 2.8. Flow Stress (Effective Stress) as a Function of Strain for Tempered Martensite Tested at Various Pressures [14] In the 1950 s and 60 s, Hu conducted several experiments to test the validity of a hydrostatic independent yield condition. He felt that the biaxial tension-tension and biaxial tension-torsion experiments used by early plasticity researchers to check the validity of Bridgman s work were not sensitive enough to detect the effect of hydrostatic stresses on the yielding of metals. Hu stated that the results of these experiments have led to the false conclusion that the effect of hydrostatic stresses on plastic behavior of metals is insignificant as assumed, even though the influence of hydrostatic pressure on simple tension and compression has long been known [15]. In the 1950 s Hu conducted biaxial-stress tests on aluminum alloys to check the validity of assumptions made in theories of plastic flow for metals. A tubular specimen

18 was stressed by applying an internal pressure and axial tension. His test results did not agree with the stress-strain relations formulated using the von Mises yield criterion and they did not support the theory of metal incompressibility by the classical flow theories [16,17]. He later performed pressurized tension tests on Nittany No. 2 brass and found the effect of hydrostatic pressure on plastic stress-strain relations to be quite significant as shown in Figure 2.9 [18]. For example, Hu found the yield strength of the brass to be approximately 45 ksi with no externally applied pressure (No. 1, Figure 2.9), but the yield strength increased to about 55 ksi when the specimen was pressurized to 53.2 ksi (No. 7, Figure 2.9). His tests also demonstrated a threefold increase in ductility with increased external pressure. From his findings, Hu suggested that a yield criterion for metals should include the influence of hydrostatic stress and could be written in simple form as 2 2 2 A + BI1 CI1 J = +, (2.25) where A, B, and C are experimentally determined material constants. Hu [15] also suggested that for many metals, C is negligible and therefore the yield function could be written as J = +. (2.26) 2 2 A BI1

19 Figure 2.9. Plastic Stress-Strain Relations in Tension for Nittany No.2 Brass Under Hydrostatic Pressure [18] In the 1970 s Richmond, Spitzig, and Sober [12,19,20] also conducted experiments that challenged the basic tenets of classic metal plasticity. They studied the effects of hydrostatic pressure on yield strength for four steels: 4310, 4330, maraging steel, and HY80. They conducted compression and tension tests in a Harwood hydrostatic-pressure unit at pressures up to 160 ksi (1100 MPa). Richmond found that hydrostatic pressure had a significant effect on the stressstrain response of the steels as shown in Figure 2.10 and Figure 2.11. For 4330 steel, the compressive yield strength increased from 1520 MPa to 1610 MPa as pressure was increased to 1100 MPa, and for the aged maraging steel, the compressive yield strength increased from 1810 MPa to 1890 MPa as pressure was increased to 1100 MPa

20 Figure 2.10. Effect of Hydrostatic Pressure on the Stress-Strain Curves in Compression for 4330 Steel [20] Figure 2.11. Effect of Hydrostatic Pressure on the Stress-Strain Curves in Compression for Aged Maraging Steel [20]

21. Richmond also found that the yield strength was a linear function of hydrostatic pressure. This is shown graphically in Figure 2.12 and Figure 2.13. Richmond proposed that for high-strength steels the yielding process is described by the yield function f ( I, J ) J + ai d, (2.27) 1 2 = 3 2 1 where a and d are material constants proportional to those suggested by Drucker and Prager [13] for soils. The experimental values of a and d obtained by Richmond are listed in Table 2.1. Figure 2.12. Dependence of Yielding on Mean Stress in 4330 Steel [20]

22 Figure 2.13. Dependence of Yielding on Mean Tension for Aged Maraging Steel [20] Table 2.1. Summary of Experimental Results for Constants in Equation (2.27) [20] Material Name a d, MPa a/d, MPa -1 HY80 Steel 0.008 606 13x10-6 Unaged Maraging Steel 0.017 1005 17x10-6 4310 Steel 0.025 1066 23x10-6 4330 Steel 0.025 1480 17x10-6 Aged Maraging Steel 0.037 1833 20x10-6

23 Another interesting result that emerged from Richmond s tests was a strong correlation between the coefficients a and d. He found that the ratio of a/d was nearly constant for all of the steels as listed in Table 2.1. Richmond also suggested that the ratio a/d is a property of the bulk iron lattice similar to the elastic constants E and ν. It was previously shown that Equation (2.24) will equal zero if the yield function is independent of hydrostatic stress. In contrast, if Equation (2.27) is true, then Equation (2.24) will not be equal to zero. Instead, taking the partial derivative of f with respect to I 1 from Equation (2.27) results in f I 1 = a. (2.28) Combining Equations (2.24) and (2.28) produces the expression ε ii = 3 adφ > 0. (2.29) d pl Therefore, the plastic volume must increase for increasing plastic strain. Richmond measured the plastic volume increase for varying levels of plastic strain and compared this data to the plastic volume increase predicted by Equation (2.29). His measured values of plastic volume increase were only about one-fifteenth of that predicted by the normality flow rule as illustrated in Figure 2.14. The true or equivalent plastic strain, ε, plotted in Figure 2.14 is defined as the sum of the plastic strain increments and can be pl eq written as pl pl 2 pl pl 2 pl 2 [( dε 1 dε 2 ) + ( dε 2 dε 3 ) + ( dε 3 dε 1 ) ] 2 pl pl ε eq =. (2.30) 3 0.5

24 Figure 2.14. Plastic Volume Increase as a Function of True Plastic Strain for 4310 and 4330 Steels [20] Richmond also conducted pressurized compression and tension tests on two polymers crystalline polyethylene and amorphous polycarbonate. These tests were done to see if the plasticity theories developed for metals were compatible with other materials. He found that hydrostatic pressure had a significant effect on the stress-strain response of the polymers and that the effective stress was a linear function of hydrostatic pressure. In other words, Richmond established that the polymers plastic response could be described by the same plasticity theories that he developed for metals. Therefore, the fact that soils, metals, and polymers are all affected in a similar manner by hydrostatic pressure is a unifying concept. The yield function developed by Richmond for the steels and polymers was identical to a yield function formulated in the 1950 s by Drucker and Prager [13]. The Drucker-Prager yield criterion is a modification of the von Mises criterion that includes

25 the influence of hydrostatic pressure on fracture and was originally formulated to solve soil mechanics problems. The Drucker-Prager yield function is defined as f ( I J ) ai + 3J, 1, 2 1 2 d = (2.31) where d is the modified yield strength in absence of mean stress and a is a material constant related to the theoretical cohesive strength of the material, σ c. The material constant a is determined graphically as the slope of the graph of σ eff versus I 1, as illustrated in Figure 2.15. The value of I 1 for σ eff = 0 is equal to the theoretical cohesive strength of the material, and the value of I 1 = σ uniaxial /3 corresponds to the yield strength for a tensile test. Also, the value of I 1 = 0 leads to σ eff = d. The theoretical cohesive strength can be defined as the stress required to overcome the cohesive force between the neighboring atoms. The cohesive stress, σ, between two atoms as a function of atomic separation is illustrated in Figure 2.16, where σeff d a 1 0 σc I 1 Figure 2.15. Schematic of σ eff versus I 1 [10]

26 σ a o σc λ/2 x Figure 2.16. Cohesive Force as a Function of the Separation Between Atoms (Adapted from [21]) a o is the equilibrium spacing, λ is the lattice wavelength, and x is the separation between atoms [21]. A good approximation of the cohesive stress is given by 2πx σ = σ c sin. (2.32) λ For small displacements, sin(x) x, and therefore Equation (2.32) can be written as 2πx σ = σ c. (2.33) λ Assuming elastic behavior leads to Ex σ =. (2.34) a o Combining Equations (2.33) and (2.34) and solving for σ c leads to

27 σ c λe =. (2.35) 2πa o Assuming a o λ/2 allows Equation (2.35) to be simplified to σ = E c π. (2.36) The value for cohesive strength given by Equation (2.36) generally overpredicts the actual value of σ c since it does not account for the dislocations and lattice imperfections inherent in most engineering materials or that slip occurs plane by plane. Dieter [21] lists a range of σ c from E/15 to E/4 with a typical value for σ c of E/5.5. An expression for cohesive strength can also be derived by considering the energetics of the fracture process. The fracture work done per unit area is given by λ πx λ W f = 2 σ c sin dx = σ c. (2.37) λ π 0 2 If the expression for fracture work is equated to the energy required to form two new fracture surfaces 2γ, one can solve for λ to give 2πγ λ =. (2.38) σ c Substituting Equation (2.38) into Equation (2.37) and solving for σ c gives Eγ σ c =. (2.39) a o

28 In practice, the theoretical cohesive strength is difficult to determine. Therefore, an alternate method for determining the material constant a is needed. When a can be properly determined, the Drucker-Prager yield surface is a rightcircular cone in principal stress space as shown in Figure 2.17. The axis of the cone is the hydrostatic pressure axis, and the apex of the cone is located at a hydrostatic stress equal to the cohesive strength. The yield surface for an actual material probably does not come to a sharp apex as the linear Drucker-Prager model predicts. The sharp point of the cone could cause numerical difficulty for flow calculations, and, therefore, ABAQUS provides hyperbolic and exponential Drucker-Prager constitutive models that round off the end of the cone [5]. For small amounts of hydrostatic stress, the cylinder of the von Mises yield criterion can approximate the cone. As the hydrostatic stress increases, the deviation from the cylinder can be considerable, and the Drucker-Prager yield surface is preferable (Figure 2.17). Therefore, because of its hydrostatic dependency, the Drucker- Prager yield criterion should result in more accurate modeling of geometries that have a high hydrostatic stress influence such as cracks and notches. A completely different approach to dealing with hydrostatic pressure has been used in fracture mechanics. Void growth analysis during ductile failure is an area of study in which hydrostatic pressure has long been recognized as a significant factor. In the 1960 s Rice and Tracey [22] developed a semi-empirical relationship to approximate the growth of a single void that may form during ductile failure. They assumed that the

29 Drucker-Prager σ 3 k σ m = 1/3I 1 σ 2 von Mises σ 1 Figure 2.17. Comparison of Drucker-Prager and von Mises Yield Surfaces in Principal Stress Space [10] initial void was spherical but became ellipsoidal as it deformed. This equation is dependent on mean stress (σ m = I 1 /3) and can be written as pl R ε eq I pl dε eq R σ 1 ln = 0.283 exp, (2.40) 0 0 2 ys where R 0 is the radius of the initial spherical void and R is the average of the radial displacements of the ellipsoidal void. Expanding on this work, Gurson [23] developed a failure criterion that assumes that the material behaves as a continuum and, therefore, the effects of each void have an averaged effect on the global behavior. The Gurson yield function is also a function of mean stress and is written as f 3J = σ 2 2 ys I1 + 2F cosh 2σ ys 2 ( 1+ F ), (2.41)

30 where F is the void volume fraction and S ij is the deviatoric stress written in tensor notation. When F = 0, Equation (2.41) reduces to the von Mises yield failure theory with isotropic hardening. Tvergaard [24] modified the Gurson model by adding two adjustable parameters q 1 and q 2 that are used to calibrate the equation with experimental data. Tvergaard s modified equation is f 3J = σ 2 2 ys q2 I1 + 2q1F cosh 2 σ ys 2 [ 1+ ( q F ) ] 1. (2.42) After calibrating Equation (2.42) with experimental data, Tvergaard found that failure could be reasonable predicted when q 1 = 2 and q 2 = 1. This chapter presented a classical view of metal plasticity. Also, the work of various researchers that challenged the tenets of classical metal plasticity was presented. These researchers demonstrated that hydrostatic stress can have a significant effect on the yield and postyield behavior of materials. The next chapter presents the research procedure used to quantitatively investigate the effects of hydrostatic stress, in particular the effects of hydrostatic tension, in metal plasticity.

CHAPTER 3 RESEARCH PROCEDURE In this chapter, the procedure used to conduct the research for this thesis is presented. The chapter begins with a general discussion of finite element modeling. Next, several different test geometries are presented, and specific details are given concerning the creation of the finite element models for each geometry. The chapter ends with a discussion of the required material property inputs for each of the constitutive models used in the finite element analysis. Finite Element Modeling Several nonlinear finite element analyses (FEA) were used in order to assess how well the von Mises and Drucker-Prager failure criteria could estimate yield and postyield behavior. A wide range of finite element models (FEM s) were created, from smooth tensile to notched specimens with high initial stress concentrations, in order to demonstrate the response of the failure theories to a wide range of internal hydrostatic stress. Accurate load-displacement or load-microstrain test data (or both) was available for each of the geometries tested and was used as a measure of the FEM s performance. The FEM s meshes were created using the modeling programs FASTQ [25] or Patran [26]. The FASTQ and Patran meshes were translated into an ABAQUS [5] input file, and the job control commands and material properties were added using a text editor. 31

32 All of the finite elements models were loaded in displacement control with full integration and large strain effects activated. The FEM s were then analyzed and postprocessed in the finite element program ABAQUS. Electronic hard copies (*.jpg files) of the desired contour plots were recorded using the screen capture program Snapshot while postprocessing each model. The ABAQUS database files were gleaned of the required information by using a Unix stream-editor (sed) script and the data filter function in Microsoft Excel (Appendix A). Finally, the load-displacement or loadmicrostrain data was plotted alongside the mechanical test data using Excel. Smooth (Unnotched) Tensile Bar Specimen A smooth (unnotched) tensile bar made of 2024-T351 was modeled as a baseline measure of the constitutive model s performance under low hydrostatic influence (Figure 3.1). Since the smooth tensile bar has one dominant state of stress until the onset of necking, it should provide a comparison of the performance of the constitutive models when hydrostatic influence is minimal. After necking begins though, a triaxial state of stress will exist, and the presence of the resulting hydrostatic stress will be significant. Wilson [6] created the axisymmetric finite element model of the tensile bar used in this research. The FEM has 552 Q4 elements (type CAX4 in ABAQUS, Figure 3.2). As illustrated in Figure 3.3, only one-quarter of the tensile bar gage section was modeled by utilizing two symmetry planes. An illustration of the finite element model is given in Figure 3.4.

33 Figure 3.1. Engineering Drawing of the Smooth Tensile Bar Specimen (All Dimensions Are in Inches) Figure 3.2. Axisymmetric Q4 Element with Node Numbers and Degrees of Freedom

Figure 3.3. Schematic of Axisymmetric Model of a Smooth Tensile Bar Specimen Utilizing Two Planes of Symmetry 34

35 B C A Figure 3.4. Finite Element Model of the Smooth Tensile Bar The tensile bar FEM has three distinct areas as illustrated in Figure 3.4. The first area, A, is the tapered region where the specimen will neck down. This area is 20 elements wide by 12 elements high and is slightly tapered on the right side to account for the taper from 0.250 in. to 0.248 in. in Figure 3.1. The initial length to height ratio of the elements in the neck region is approximately 3 to 1. This initial aspect ratio is used with

36 the expectation that the element aspect ratio will be close to one after element deformation at the maximum load. The length bias for the elements in the neck region is approximately uniform since all of the elements in the neck level will be highly stressed after yielding. The second area, B, is the region where a uniform displacement is applied and is 6 elements wide by 5 elements high. The third area, C, has a transition mesh between the previously described areas. Notched Round Bar Specimens Wilson [6] created the axisymmetric models of 2024-T351 notched round bar (NRB) tensile specimens used in this research. The NRB finite element models were composed of axisymmetric Q4 elements (Figure 3.2). Two planes of symmetry were utilized as illustrated in Figure 3.6. Three variations of the NRB finite element models were created by making the notch root radius, ρ, equal to 0.020 in., 0.010 in., and 0.005 in., respectively. These dimensions are equivalent to those used in NRB tests by Wilson [6]. Each variation of the NRB model was meshed with three levels of mesh refinement. The coarse, medium, and fine meshes for each variation typically had 250, 500, and 1000 elements in the notch region, respectively. An illustration of the fine mesh NRB FEM with ρ =0.005 in. is given in Figure 3.7, and representative meshes in the notch region for each level of mesh refinement are illustrated in Figures 3.8 through 3.10. All of the NRB finite element models have four distinct areas as shown in Figure 3.7, and the details of the areas of the fine mesh FEM s will be described. The first area,

37 A, is the notch region. This area is 34 elements wide by 28 elements high, and the initial length to height ratio of the elements in the neck region is approximately 3 to 1. This initial aspect ratio is used with the expectation that the element aspect ratio will be close to one after element deformation at the maximum displacement. The element widths are biased toward the notch root to ensure the smallest elements will be in the area of highest stress. The second area, B, is the region where a uniform displacement is applied and was 6 elements wide by 10 elements high. The other two areas, C and D, are transition meshes between the previously described areas. 12.7 12.7 57.15 114.3 12.7 45 19.0 DIA. (TYP.) 6.35 0.508 RADIUS Figure 3.5. Engineering Drawing of the Notched Round Bar Specimen (All Dimensions Are in mm) [10]

Figure 3.6. Schematic of Axisymetric Model of a NRB Utilizing Two Planes of Symmetry [10] 38

39 B D C A Figure 3.7. Fine Mesh FEM of NRB (ρ = 0.005 in.)

Figure 3.8. Coarse Mesh FEM in the Notch Region of the NRB (ρ = 0.005 in.) 40

Figure 3.9. Medium Mesh FEM in the Notch Region of the NRB (ρ = 0.005 in.) 41

42 Figure 3.10. Fine Mesh FEM in the Notch Region of the NRB (ρ = 0.005 in.) Equal-Arm Bend Specimen The next specimen modeled was an IN100 equal-arm bend specimen tested by Pratt & Whitney (Figure 3.11). This specimen is used in low cycle fatigue testing and was designed to simulate the geometry and loading condition of a problem area in the Space Shuttle main rocket engine housing. The equal-arm bend FEM was created using Q4 plane strain elements (type CPE4 in ABAQUS, Figure 3.12) and one symmetry plane

43 Loading Axis Figure 3.11. Engineering Drawing of the Equal Arm Bend Specimen (Dimensions in inches, Nominal Dimensions Used in FEM) Figure 3.12. Q4 Element with Node Numbers and Degrees of Freedom

44 resulting in a total of 1339 elements. The symmetry plane and boundary conditions are illustrated in Figure 3.13. Plane strain elements were used because the thickness to width ratio in the fillet region was approximately 5 to 1. The loading was applied to the finite element model by filling the hole for the loading-pin with elements and applying a displacement to the node in the center of the loading-pin hole. An illustration of the finite element model is given in Figure 3.14. The mesh in the fillet region in Figure 3.14 is difficult to see and, therefore, a closer view of the mesh in the fillet region is shown in Figure 3.15. The equal-arm bend FEM has five distinct areas as illustrated in Figure 3.14. The first area, A, is the fillet region, which is 25 elements wide by 15 elements high. The initial length to height ratio of the elements in the neck region is approximately 2 to 1. This initial aspect ratio is used with the expectation that the element aspect ratio will be close to unity after element deformation at the maximum strain. Because this geometry does not have a sharp notch, all of the elements in the neck level will be highly stressed after yielding. Therefore, the length bias for the elements in the neck region is approximately uniform. The second area, B, has a transition mesh between the fillet region and the main arm of the model, and the third area, C, is the region of main arm of the model. The fourth area, D, is the region surrounding the loading-pin hole, and the final area, E, is the circular region of the loading pin hole.

Figure 3.13. Schematic of the Equal-Arm Bend FEM Utilizing One Symmetry Plane 45

46 E D C B A Figure 3.14. Equal Arm Bend Finite Element Model Utilizing One Plane of Symmetry

47 Figure 3.15. Mesh in the Fillet Region of the Equal Arm Bend Specimen Double-Edge Notch Tension Specimen The next geometry analyzed was an IN100 double-edge notch tension (DENT) specimen tested by NASA (Figure 3.16). Both two-dimensional (2-D) and threedimensional (3-D) FEM s were created since the thickness to width ratio was approximately unity in the notch region. Three 2-D DENT models were created by utilizing a coarse, medium, and fine mesh in the notch region. The 2-D models were created using Q4 plane stress elements (type CPS4 in ABAQUS). Only the gage section of the specimen was modeled by utilizing two planes of symmetry as illustrated in Figure 3.17. The coarse, medium and fine meshes had approximately 150, 300, and 630

48 Figure 3.16. Engineering Drawing of the DENT Specimen (Dimensions in inches) Figure 3.17. Schematic of the 2-D DENT FEM Utilizing Two Symmetry Planes

49 elements in the notch region, respectively. An illustration of the medium mesh 2-D FEM is given in Figure 3.18, and the meshes in the notch region for each level of mesh refinement are illustrated in Figures 3.19 through 3.21. The 2-D DENT finite element models have three distinct areas as illustrated in Figure 3.18, and the specifics of the medium mesh FEM will be described. The first area, A, is the notch region. This area is 25 elements wide by 12 elements high. The initial length to height ratio of the elements in the notch region is approximately 1.5 to 1 with the expectation that the element aspect ratio will be close to one after element deformation at the maximum displacement. The width of the elements in the notch region is only slightly biased toward the notch root since all of the elements in the notch region will be highly stressed after yielding. The second area, B, is the region where a uniform displacement is applied and is 9 elements wide by 4 elements high. The third area, C, has a transition mesh between the previously described areas. The 3-D DENT finite element model was created using 4740 H8 elements (type C3D8 in ABAQUS, Figure 3.22). Three symmetry planes were utilized in creating this FEM (Figure 3.23) by dividing the specimen along the longitudinal axis, along the transverse axis, and through one-half of the thickness. An illustration of the 3-D finite element model is given in Figure 3.24. The 3-D DENT finite element model has three volumes that are 10 elements deep, and these three volumes correspond identically to the three areas of the medium mesh 2-D model.

50 B C A Figure 3.18. Illustration of the 2-D DENT FEM

51 Figure 3.19. Coarse Mesh in the Notch Region for the 2-D DENT FEM Figure 3.20. Medium Mesh in the Notch Region for the 2-D DENT FEM

52 Figure 3.21. Fine Mesh in the Notch Region for the 2-D DENT FEM Figure 3.22. H8 3-D Element With Node Numbers and Degrees of Freedom

53 Figure 3.23. Schematic of the Three Symmetry Planes of the 3-D DENT Model Figure 3.24. Illustration of the 3-D DENT FEM

54 Modified Double-Edge Notch Tension Specimen A notch (ρ = 0.020 in.) was cut into the existing two notches on the IN100 DENT specimen geometry to create the modified Double-Edge Notch Tension specimen (Figure 3.25). This was done to create a geometry that would provide a more severe stress concentration and higher hydrostatic stress than that provided by the DENT specimen. The 2-D modified DENT finite element model was created by modifying the geometry and mesh of the medium mesh 2-D DENT FEM in order to reflect the presence of the new notch. The modified DENT model had a total of 434 Q4 plane stress elements (type CPS4 in ABAQUS), and all other modeling parameters were left unchanged from the 2-D DENT model. An illustration of the modified DENT finite element model is given in Figure 3.26. A 3-D modified DENT finite element model was not created because of the similarity in geometry and specimen behavior to the DENT specimen. 0.050 in. 0.020 in. Figure 3.25. Illustration of One Side of the modified DENT Specimen Showing the New Notch Cut into the Existing Notch on the DENT Specimen

55 C D B A Figure 3.26. Illustration of the Modified DENT Finite Element Model

56 The modified DENT finite element model has four distinct areas as illustrated in Figure 3.26. The first area, A, is the new notch region. This area is 22 elements wide by 6 elements high. The initial length to height ratio of the elements in the notch region is approximately unity. The width of the elements in the notch region is only slightly biased toward the notch root since all of the elements in the notch region will be highly stressed after yielding. The second area, B, is the region surrounding the old notch and is approximately 8 elements high by 25 elements wide. Area B has several elements that are badly skewed and have poor aspect ratios. The stress in these elements is low compared to the elements surrounding the notch, so it is assumed that the poor shape of these elements has a negligible effect on the overall behavior of the model. The third area, C, is the region where a uniform displacement is applied to simulate the test and is 9 elements wide by 4 elements high. The final area, D, has a transition mesh between the second and third areas.

57 Required Material Properties Several material properties are required as input for the von Mises and Drucker- Prager constitutive models in ABAQUS. Both constitutive models require the elastic material properties of Young s modulus, E, and Poisson s ratio, ν. The von Mises model with hardening also necessitates a table of true stress, σ, versus plastic strain, ε pl. The elastic material properties and the table of σ versus ε pl for 2024-T351 were obtained from Wilson s uniaxial tensile tests [6], and the same data types for Inconel 100 (IN100) were gathered from Pratt and Whitney tensile test data [27]. True stress versus true strain plots for 2024-T351 and IN100 at room temperature are given in Figure 3.27 and Figure 3.28. It is notable that IN100 has an upper and lower yield point. The alloy compositions of 2024-T351 and IN100 are given in Table 3.1.

58 80000 70000 60000 True Stress (psi) 50000 40000 30000 20000 10000 0 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 True Strain Figure 3.27. True Stress versus True Strain Curve for 2024-T351 at Room Temperature [6] 200000 180000 160000 140000 True Stress (psi) 120000 100000 80000 60000 40000 20000 0 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 True Strain Figure 3.28. True Stress versus True Strain Curve for IN100 at Room Temperature [27]

59 Table 3.1. Alloy Composition of 2024- T351 and IN100 [28] Nominal Composition % Alloy Designation Al Cr Co Cu Mg Mn Mo Ni Ti V 2024-T351 93.50 4.40 1.50 0.60 IN100 5.50 10.00 15.00 3.00 60.85 4.70 0.95 The plastic material properties for the Drucker-Prager constitutive model require careful explanation and, therefore, will be discussed in some detail. The linear Drucker- Prager yield function is written in ABAQUS as f = t p tan β d, (3.1) where 3 1 1 1 r t = 3J 2 1 + 1. (3.2) 2 K K 3J 2 β is the slope of the linear yield surface in the p-t stress plane, p is the hydrostatic pressure, d is the effective cohesion of the material, and K is the ratio of the yield stress in triaxial tension to the yield stress in triaxial compression. The variables used by the linear Drucker-Prager yield function are shown graphically in Figure 3.29.

60 t dε pl β ψ hardening β d 0 p Figure 3.29. Linear Drucker-Prager Model: Yield Surface and Flow Direction in the p-t Plane (Adapted from [5]) The flow potential, g, for the linear Drucker-Prager model is defined as g = t p tanψ, (3.3) where ψ is the dilation angle in the p-t plane (Figure 3.29). The dilation angle controls the movement of an arbitrary point on the yield surface during the hardening process. Setting ψ = β results in associated flow. Therefore, the original Drucker-Prager model is available by setting ψ = β and K = 1 [5]. Associated flow was used in this research as a first-order approximation. In order to conveniently compare ABAQUS Drucker-Prager material property variables with those used by Richmond, et al. in their material testing, one must correlate

61 the variables in Equations (2.31) and (3.1). Recalling that p = 1 I 3 1 and t = 3J 2 for K=1, Equation (3.1) can be written as f 1 = 3J 2 + I1 tan β d. (3.4) 3 Comparing Equations (2.31) and (3.4) leads to the conclusion that 1 ai 1 = I 1 tan β. (3.5) 3 Solving for β yields ( 3a) β = tan 1. (3.6) In this way, the material constants of the original Drucker-Prager theory and the material constants used in the ABAQUS linear Drucker-Prager constitutive model can be related. The linear Drucker-Prager constitutive model in ABAQUS requires, in addition to the elastic constants, values for β, ψ, K, and a table of modified effective stress, ~ σ eff, versus ε pl. The modified effective stress is defined as ~ σ = σ (1 a). (3.7) eff eff The challenge comes in calculating a value for a. It was reported in Chapter 2 that the ratio of a/d in Richmond s tests [20] was nearly constant for the high strength steels, and, therefore, a/d is possibly a material constant similar to Young s modulus or Poisson s ratio. Since Young s modulus for IN100 is similar to E for the high strength steels, it was

62 assumed that a for IN100 is approximately equal to the values of a reported by Richmond (Table 2.1). Estimating a value for a for 2024-T351 is more difficult though because no extensive test data of σ eff versus I 1 is readily available. The value for a can be estimated for any material though by conducting uniaxial tension and uniaxial compression tests and plotting the yield results in σ eff versus I 1 space. The slope a of the line connecting the compressive yield, σ ysc, and the tensile yield, σ ys, in Figure 3.30 is σ ysc σ ys a =. (3.8) σ + σ ysc ys As mentioned in Chapter 2, the material cohesive strength, σ c, can be estimated as E/15 to E/4 with a typical value of E/5.5 [21]. Therefore, a can also be estimated as initial yield strength divided by an approximate cohesive strength. σeff Uniaxial Compression Test a σysc Uniaxial Tensile Test 1 σys σysc 0 σys σc I 1 Figure 3.30. Schematic of Calculation of a from Uniaxial Tension and Uniaxial Compression Data

63 In summary, this chapter presented a procedure to investigate the differences between the von Mises and Drucker-Prager yield functions. The first step was to create finite element models of several different geometries. The next step was to analyze the geometries by using the von Mises and Drucker-Prager constitutive models in ABAQUS. To complete this step, several material properties must be obtained to be used in the constitutive models, and the final part of this chapter described the procedure for obtaining these properties. The next chapter presents the results of the finite element analyses.

CHAPTER 4 RESULTS This chapter begins with a discussion of material property calculations. Next, the finite element results for each geometry are presented and compared to specimen test results. For some geometries, various stress quantities at failure are given as a function of the distance across the specimen neck. Also, contour plots of various stress and strain quantities are presented and discussed. Material Property Calculations Two metals were used to conduct this research. The first metal is 2024-T351, which is one of the materials that Wilson [6] used for smooth tensile and NRB tests. The second material used in this research is Inconel 100 (IN100), which is a nickel-base superalloy developed for high strength at elevated temperatures. This metal is of special interest to this research because the Space Shuttle fuel turbopump housing is made from an IN100 forging. 2024-T351 Properties The 2024-T531 material properties for Young s modulus, E, and Poisson s ratio, ν, used were 10.4x10 6 psi and 0.33, respectively. A table of true stress, σ, versus plastic strain, ε pl, values used in the isotropic hardening von Mises constitutive model is given in Appendix B and is shown graphically in Figure 4.1. To use the original form of the linear Drucker-Prager constitutive model, the ratio of the yield stress in triaxial tension to 64

65 the yield stress in triaxial compression, K, was set equal to one, and the friction angle, β, and the dilation angle, ψ, were equated. The following method was used to estimate β and create the table of modified effective stress, ~ σ eff, versus ε pl. Following Dieter s recommendation [21], the theoretical cohesive strength, σ c, was estimated as E/5.5. Then taking a as approximately equal to the initial yield strength over σ c leads to a value for a of 0.029. Substituting this number for a into Equation (3.6) gives a β of approximately 5. Once a (or β) is known, ~ σ eff is calculated by using Equation (3.7). The values of ~ σ eff calculated for each increment of ε pl are given in Appendix B and are illustrated in Figure 4.1. Six compression tests were performed in an attempt to validate the estimate of a for 2024-T351. The compression specimens were one-inch long cylinders and were cut from the broken halves of Wilson s [6] NRB specimens. The compressive yield could not be measured with great accuracy because of the crude nature of the tests, but σ ysc was clearly in the range of 56 ksi to 60 ksi. The value for the compressive yield calculated using Equation (3.8) with a = 0.029 and σ ys = 55.4 ksi is 58.7 ksi which is consistent with the compression test results.

66 80000 75000 70000 Stress (Psi) 65000 60000 True Stress Modified Effective Stress 55000 50000 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 0.160 0.180 0.200 Plastic Strain Figure 4.1. 2024-T351 Isotropic Hardening Parameters for the von Mises and Drucker- Prager Constitutive Models IN100 Properties Poisson s ratio for IN100 was assumed to be 0.30, and the lower yield stress was used for σ ys. The true stress versus plastic strain curve used in the isotropic hardening von Mises constitutive model is given in Figure 4.2. A table of σ versus ε pl for IN100 is included in Appendix B. The Lüder effect from the upper and lower yield strength was neglected in constructing this curve. As was done for 2024-T351, K was set equal to one, and β and ψ were equated in order to use the original form of the linear Drucker-Prager constitutive model. Richmond s test results (Table 2.1) for high strength steels was used to estimate a value for a of 0.022. Using Equation (3.6), β was calculated to be

67 approximately 3.8. The values of ~ σ eff were calculated for increments of ε pl using Equation (3.7) and are illustrated in Figure 4.2. A table of ~ σ eff versus ε pl for IN100 is included in Appendix B. Young s modulus was assumed to be 29.9x10 6 psi for all of the models except for the equal-arm bend model. For this model, E had to be reduced to 26.0x10 6 psi in order to match the test data in the linear range. The reason for the change in E is a matter of debate, but it is possible that the measured value of E changes with loading type. This is supported by the fact that in addition to normal stresses, the equal-arm bend specimen was under the influence of large bending stresses. Also, Pratt and Whitney [27] tensile test data for IN100 has widely scattered values for E ranging from 25x10 6 to 35x10 6 psi. 185000 180000 175000 Stress (psi) 170000 165000 True Stress Modified Effective Stress 160000 155000 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 Plastic Strain Figure 4.2. IN100 Isotropic Hardening Parameters for the von Mises and Drucker-Prager Constitutive Models

68 Finite Element Model Results The finite element analysis results for the geometries modeled in this research are presented in this section. The FEA results are compared to load-displacement or loadmicrostrain test data or both. Contour plots of various stress and strain quantities are presented for each geometry. Also, for some geometries, various stress quantities at the point of failure or load-reversal are given as a function of the distance across the neck of the specimen. Smooth Tensile Bar Results Load-displacement curves from the smooth tensile bar finite element models are plotted along with Wilson s [6] tensile test data in Figure 4.3. Since the tensile bar does not have a large hydrostatic pressure influence until necking, it was expected that the von Mises and Drucker-Prager constitutive models would give similar results. The von Mises curve slightly overestimates the load for a given value of displacement in the plastic region, while the Drucker-Prager curve slightly underestimates the data. After a gage displacement of 0.15 in., the curves for both FEM s fall sharply below the test data. This is partially due to the fact that the test specimen failed with an angled fracture along a line of maximum shear (Figure 4.4). The axisymmetric finite element models cannot accurately model failure along slip planes. Instead, the FEM s assume that the neck region continues to decrease in diameter uniformly (uniform necking) until final failure. Contour plots of the effective stress σ eff, hydrostatic pressure p, stress in the z-direction

69 pl σ zz, and equivalent plastic strain ε eq for the Drucker-Prager FEM at a gage displacement of 0.15 in. are shown in Figures 4.5 through 4.8. 3500.00 3000.00 2500.00 Load P (lbs) 2000.00 1500.00 1000.00 Test Specimen 500.00 von Mises FEA Drucker-Prager FEA Gage Length = 1.0 in. 0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Gage Displacement v (in.) Figure 4.3. Load-Displacement Results for 2024-T351 Smooth Tensile Bar

70 Figure 4.4. Broken Tensile Specimen with Angled Fracture Surface Figure 4.5. Effective Stress (psi) in the Smooth Tensile Bar at 0.15 in. Gage Displacement

71 Figure 4.6. Hydrostatic Pressure (psi) in the Smooth Tensile Bar at 0.15 in. Gage Displacement Figure 4.7. Stress in the z-direction (psi) in the Smooth Tensile Bar at 0.15 in. Gage Displacement

72 Figure 4.8. Equivalent Plastic Strain in the Smooth Tensile Bar at 0.15 in. Gage Displacement Notched Round Bar Results Load-displacement results for the coarse, medium, and fine mesh NRB finite element models with ρ = 0.005 in. are shown in Figure 4.9. One can see that the FEM s response at the gage point is practically identical. The identical response is expected since the gage point is far enough removed from the notch root to prevent mesh refinement from having any significant effect on the load-displacement results. To truly see the difference in the data given by the different mesh densities, one must examine the pl stress and strain components in the neck region. The values of σ eff, σ m, σ rr, and ε eq as a function of distance across the neck (from the centerline of the NRB out to the surface of

73 7000 6000 5000 Load P (lb.) 4000 3000 2000 1000 Coarse Mesh Medium Mesh Fine Mesh Gage Length = 0.4 in. 0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 Gage Displacement v (in.) Figure 4.9. Load-Displacement Plot for Coarse, Medium, and Fine Mesh NRB's with ρ = 0.005 in. the notch root) are shown in Figures 4.10 through 4.13. All quantities plotted are values averaged at the nodes. The three mesh densities do not produce much variation in results until the outer surface of the notch is reached. The increased level of accuracy on at the outer surface of the notch may or may not be of consequence depending on the level of accuracy required. As expected, the fine mesh gave the most accurate results at the notch tip. For example, σ rr should equal zero on the free surface of the notch tip. As the mesh is refined, σ rr approaches zero at the free surface in Figure 4.12. The correlation between σ eff, σ m, and σ rr across the neck is shown in Figure 4.14. It is notable that σ m and σ rr peak at approximately the same distance across the neck, and σ eff has a plateau region and then climbs swiftly to its maximum after it crosses the area of maximum σ m. After completing this convergence study, the fine mesh FEM s were chosen.

74 8.00E+04 7.50E+04 7.00E+04 6.50E+04 Effective Stress (psi) 6.00E+04 5.50E+04 5.00E+04 4.50E+04 4.00E+04 Coarse Mesh Medium Mesh Fine Mesh 3.50E+04 0.00E+00 2.50E-02 5.00E-02 7.50E-02 1.00E-01 1.25E-01 Distance Across Neck (in.) Figure 4.10. Effective Stress as a Function of Distance Across the Neck of the Coarse, Medium, and Fine Mesh NRB's with ρ = 0.005 in. at Failure Load 95000 90000 85000 80000 Mean Stress (psi) 75000 70000 65000 60000 55000 50000 Coarse Mesh Medium Mesh Fine Mesh 45000 0.00E+00 2.50E-02 5.00E-02 7.50E-02 1.00E-01 1.25E-01 Distance Across Neck (in.) Figure 4.11. Mean Stress as a Function of Distance Across the Neck of the Coarse, Medium, and Fine Mesh NRB's with ρ = 0.005 in. at Failure Load

75 7.00E+04 6.00E+04 5.00E+04 Radial Stress (psi) 4.00E+04 3.00E+04 2.00E+04 Coarse Mesh 1.00E+04 Medium Mesh Fine Mesh 0.00E+00 0.00E+00 2.50E-02 5.00E-02 7.50E-02 1.00E-01 1.25E-01 Distance Across Neck (in.) Figure 4.12. Radial Stress as a Function of Distance Across the Neck of the Coarse, Medium, and Fine Mesh NRB's with ρ = 0.005 in. at Failure Load 0.45 0.4 0.35 Equivalent Plastic Strain 0.3 0.25 0.2 0.15 0.1 Coarse Mesh Medium Mesh Fine Mesh 0.05 0 0.00E+00 2.50E-02 5.00E-02 7.50E-02 1.00E-01 1.25E-01 Distance Across Neck (in.) Figure 4.13. Equivalent Plastic Strain as a Function of Distance Across the Neck of the Coarse, Medium, and Fine Mesh NRB's with ρ = 0.005 in. at Failure Load

76 1.00E+05 9.00E+04 8.00E+04 7.00E+04 6.00E+04 Stress (psi) 5.00E+04 4.00E+04 3.00E+04 2.00E+04 1.00E+04 Effective Stress Mean Stress Radial Stress 0.00E+00 0.00E+00 2.50E-02 5.00E-02 7.50E-02 1.00E-01 1.25E-01 Distance Across Neck (in.) Figure 4.14. Effective Stress, Mean Stress, and Radial Stress as a Function of Distance Across the Neck of the Fine Mesh NRB with ρ = 0.005 in. at Failure Load Load-displacement curves from NRB finite element models are plotted along with Wilson s [6] NRB test data in Figures 4.15, 4.16, and 4.17. The FEA curves extend beyond the specimen test data because the finite element models do not have a built in fracture mechanism to indicate when fracture has occurred. Results for the three NRB geometries are very similar with the hydrostatic effects becoming slightly more apparent as the notch radius is decreased. For a given displacement, the von Mises FEM s always overestimates the load in the plastic region. Considering the failure displacement, the Drucker-Prager FEM s predict loads that are about 10% lower than the von Mises values. For the failure load, the Drucker Prager FEM s predict strains that are from 20% to 65% greater than the von Mises values, and, therefore, do a much better job of matching the overall specimen response. The shift of the Drucker-Prager curves from above the test

77 specimen data (Figure 4.15) to slightly below the test data (Figure 4.17) could possibly be prevented with a more judicious choice of a. Also, the mismatch in the linear response in Figures 4.16 and 4.17 indicates a variation in Young s modulus for the test specimens. Contour plots of various stress, strain, and displacement components were analyzed for all three variations of the NRB geometries (ρ = 0.005, 0.010, and 0.020 in.). The distributions throughout the NRB s for the different stress components were similar, and, therefore, only the contour results for the NRB with ρ = 0.010 in. will be presented. pl Contour plots of the σ eff, p, stress in the z-direction, σ zz, and ε eq for the Drucker-Prager FEM at a point immediately before test specimen fracture are shown in Figures 4.18 pl through 4.21. As expected, σ eff and ε eq are greatest in the vicinity of the notch tip. It is also notable that at failure σ m (σ m = - p) is greatest in the interior of the NRB at a point approximately 0.012 in. from the notch tip (Figure 4.19).

78 6000.00 5000.00 4000.00 Load P (lbs) 3000.00 2000.00 Test Specimen 1000.00 von Mises FEA Drucker-Prager FEA Gage Length = 0.4 in. 0.00 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 Gage Displacement v (in.) Figure 4.15. Load-Displacement Results for the NRB with ρ = 0.005 in. 6000.00 5000.00 4000.00 Load P (lbs) 3000.00 2000.00 Test Specimen 1000.00 von Mises FEA Drucker-Prager FEA Gage Length = 0.4 in. 0.00 0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.010 Gage Displacement v (in.) Figure 4.16. Load-Displacement Results for the NRB with ρ = 0.010 in.

79 6000.00 5000.00 4000.00 Load P (lbs) 3000.00 2000.00 Test Data 1000.00 von Mises FEA Drucker-Prager FEA Gage Length = 0.4 in. 0.00 0.000 0.002 0.004 0.006 0.008 0.010 0.012 Gage Displacement v (in.) Figure 4.17. Load-Displacement Results for NRB with ρ = 0.020 in. Figure 4.18. Effective Stress (psi) in the NRB Specimen (ρ = 0.010 in.) at Failure Load

80 Figure 4.19. Hydrostatic Pressure (psi) in the NRB Specimen (ρ = 0.010 in.) at Failure Load Figure 4.20. Stress in the z-direction (psi) in the NRB Specimen (ρ = 0.010 in.) at Failure Load

81 Figure 4.21. Equivalent Plastic Strain in the NRB Specimen (ρ = 0.010 in.) at Failure Load Equal-Arm Bend Results Pratt & Whitney test data [27] was provided for a five-cycle proof test, but only the initial loading portion of the first cycle was considered in this research. Load versus microstrain FEM curves for the equal-arm bend specimen are plotted alongside the Pratt & Whitney test data in Figure 4.22. The FEM strain data is the maximum strain at the integration points of the element at the fillet root. Once again, the von Mises model overestimates the load for a given value of strain in the postyield region. Considering the strain at load reversal, the Drucker-Prager FEM predicts loads that are about 5% lower than the von Mises values. For the maximum load, the Drucker Prager FEM predicts

82 strains that are approximately 20% greater than the von Mises values, and, therefore, does a much better job of matching the overall specimen response. The correlation between σ eff, σ m, and σ xx across the fillet section at the end of the initial loading cycle (Point A on Figure 4.22) is shown in Figure 4.23. As expected, σ xx approaches zero on both free surfaces of the fillet section. The minimum value of σ eff is in the middle of the fillet section where the bending stress is the lowest, and large values of σ eff are on both sides of the fillet section due to the combination of bending and tensile stresses. Also, the effective stress reaches a plateau and then begins to rise as at the point of maximum mean stress, which is similar to the stress behavior in the NRB. The mean stress is negative on the left side of the fillet section due to the compressive loading. Equivalent plastic strain across the fillet section at point A is given in Figure 4.24. One can see that there is no plastic deformation across the middle of the fillet section, and the peak of the equivalent plastic strain is on the outer surface of the fillet.

83 700.00 600.00 500.00 A Load, P (lbs.) 400.00 300.00 200.00 100.00 P&W Test Data von Mises FEA Drucker-Prager FEA 0.00 0.00 5000.00 10000.00 15000.00 20000.00 25000.00 30000.00 35000.00 40000.00 Microstrain µε (in./in.) Figure 4.22. Load-Microstrain Results for the Equal-Arm Bend Specimen

84 2.00E+05 1.50E+05 1.00E+05 Stress (psi) 5.00E+04 0.00E+00 0.00E+00 2.00E-02 4.00E-02 6.00E-02 8.00E-02 1.00E-01 1.20E-01 1.40E-01 1.60E-01-5.00E+04 Effective Stress Mean Stress Stress in x-dir. -1.00E+05 Distance Across Fillet Section (in.) Figure 4.23. Effective Stress, Mean Stress, and Stress in the x-direction as a Function of Distance Across the Fillet Section of the Equal-Arm Bend FEM at Point A 3.00E-02 2.50E-02 Equivalent Plastic Strain 2.00E-02 1.50E-02 1.00E-02 5.00E-03 0.00E+00 0.00E+00 2.00E-02 4.00E-02 6.00E-02 8.00E-02 1.00E-01 1.20E-01 1.40E-01 1.60E-01 Distance Across Fillet Section (in.) Figure 4.24. Equivalent Plastic Strain as a Function of Distance Across the Fillet Section of the Equal-Arm Bend FEM at Point A

85 pl Contour plots of σ eff, p, stress in the y-direction, σ yy, and ε eq for the Drucker- Prager FEM at the end of the initial loading cycle (Point A on Figure 4.22) are shown in Figures 4.25 through 4.28. (Only partial views of the FEM s are presented in Figures pl 4.25 through 4.28.) As expected, σ yy and ε eq are greatest in the vicinity of the fillet. The effective stress is almost equal on both sides of the fillet section but is actually greatest on the surface of the side opposite the fillet due to the interaction of the bending and tensile stresses. Once again, σ m (σ m = -p) is greatest in the interior of the specimen at failure at a distance of approximately 0.035 in. from the surface of the fillet (Figure 4.26). Figure 4.25. Effective Stress (psi) in the Equal-Arm Bend Specimen at Point A

86 Figure 4.26. Hydrostatic Pressure (psi) in the Equal-Arm Bend Specimen at Point A Figure 4.27. Stress in the y-direction (psi) in the Equal-Arm Bend Specimen at Point A

87 Figure 4.28. Equivalent Plastic Strain in the Equal-Arm Bend Specimen at Point A Double-Edge Notch Tension Results Load-displacement results for the coarse, medium, and fine mesh DENT finite element models are shown in Figure 4.29. The three finite element meshes predict essentially identical load-displacement curves. The identical response is expected since the gage point is far enough away from the notch to prevent mesh density from having any significant effect on the load-displacement results. To truly see the difference in the data given by the different mesh densities, one must examine the stress and strain pl components in the notch region. The values of σ eff, σ m, σ xx, and ε eq as a function of

88 distance across the neck (from the centerline of the DENT out to the surface of the notch root) are shown 16000 14000 12000 10000 Load P (lb.) 8000 6000 4000 2000 Coarse Mesh Medium Mesh Fine Mesh Gage Length = 0.5 in. 0 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 Gage Displacement v (in.) Figure 4.29. Load-Displacement Plot for Coarse, Medium, and Fine Mesh 2-D DENT Finite Element Models in Figures 4.30 through 4.33. All quantities plotted are values averaged at the nodes. The three mesh densities do not produce much variation in results until the outer surface of the notch is reached. As expected, σ eff andε steadily increase from the centerline to the surface of the notch. The mean stress and the stress in the x-direction should decrease as the surface of the notch is approached, but, instead, both of these quantities increased. pl eq

89 In fact, as the mesh was refined, the values for σ m and σ xx seemed to become increasingly unstable. This behavior is believed to be due to the inability of plane stress elements to mathematically model the material behavior around the notch due to the increased constraint from the notch. Plane strain elements do not develop this peculiar behavior in areas of high constraint. Therefore, a possible solution would be to replace a certain number of plane stress elements around the notch with plane strain elements. After completing this convergence study, the medium mesh FEM was chosen as a good balance between computation speed and accuracy. 3.00E+05 2.80E+05 2.60E+05 Effective Stress (psi) 2.40E+05 2.20E+05 2.00E+05 1.80E+05 Coarse Mesh Medium Mesh Fine Mesh 1.60E+05 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 Distance Across Neck (in.) Figure 4.30. Effective Stress as a Function of Distance Across the Notch Region of the Coarse, Medium, and Fine Mesh 2-D DENT FEM's at Failure Load

90 1.10E+05 1.05E+05 1.00E+05 Mean Stress (psi) 9.50E+04 9.00E+04 8.50E+04 Coarse Mesh Medium Mesh Fine Mesh 8.00E+04 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 Distance Across Neck (in.) Figure 4.31. Mean Stress as a Function of Distance Across the Notch Region of the Coarse, Medium, and Fine Mesh 2-D DENT FEM's at Failure Load 8.00E+04 7.00E+04 6.00E+04 Stress in x-direction (psi) 5.00E+04 4.00E+04 3.00E+04 2.00E+04 1.00E+04 Coarse Mesh Medium Mesh Fine Mesh 0.00E+00 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140-1.00E+04 Distance Across Neck (in.) Figure 4.32. Stress in the x-direction as a Function of Distance Across the Notch Region of the Coarse, Medium, and Fine Mesh 2-D DENT FEM's at Failure Load

91 2.00E-01 1.80E-01 1.60E-01 1.40E-01 Equivalent Plastic Strain 1.20E-01 1.00E-01 8.00E-02 6.00E-02 4.00E-02 Coarse Mesh Medium Mesh Fine Mesh 2.00E-02 0.00E+00 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140 Distance Across Neck (in.) Figure 4.33. Equivalent Plastic Strain as a Function of Distance Across the Notch Region of the Coarse, Medium, and Fine Mesh 2-D DENT FEM's at Failure Load The correlation between σ eff, σ m, σ xx, and σ yy across the notch region for the fine mesh 2-D DENT model is shown in Figure 4.34. The stress in the y-direction exhibits the same unstable behavior around the notch tip as observed in σ m and σ xx. The mean stress is almost constant across the neck section of the DENT specimen as opposed to the NRB, which had a sharp peak in mean stress. The difference in mean stress response of these two specimens is due to the difference in notch acuity (0.05 in. versus 0.005 in.) and clearly demonstrates the effect that constraint has on internal hydrostatic pressure.

92 3.50E+05 3.00E+05 2.50E+05 Stress (psi) 2.00E+05 1.50E+05 1.00E+05 Effective Stress Mean Stress Stress in x-direction Stress in y-direction 5.00E+04 0.00E+00 0.000 0.020 0.040 0.060 0.080 0.100 0.120 0.140-5.00E+04 Distance Across Neck (in.) Figure 4.34. Effective Stress, Mean Stress, Stress in the x-direction, and Stress in the y- Direction as a Function of Distance Across the Notch Region of the Fine Mesh 2-D DENT FEM at Failure Load Load-displacement data for the 2-D and 3-D DENT finite element models are plotted alongside NASA test data [29] in Figure 4.35 and Figure 4.36, and the linear portions of the load-displacement data are shown in Figure 4.37 and Figure 4.38. The slope of the linear portion of the finite element results is slightly less than that of the test specimen data indicating that the test specimen had a modulus of elasticity greater than 29.9x10 6 psi. Also, the test data crosses the FEM curves at a strain of approximately 0.0003. For all of the FEM s, the von Mises yield criterion overestimates the load for a given value of displacement in the postyield region. Considering the failure displacement, the Drucker-Prager FEM s predict loads that are about 3% lower than the von Mises

93 values. For the failure load, the Drucker Prager FEM s predict strains that are about 35% greater than the von Mises values, and, therefore, do a much better job of matching the overall specimen response. The 2-D plane stress model gives load values that are about 2% lower than those given by the 3-D model and, therefore, has a lower stiffness than the 3-D model. Contour plots of the effective stress σ eff, stress in the y-direction σ yy and pl equivalent plastic strain ε eq for the 2-D Drucker-Prager FEM at a point immediately before test specimen failure are shown in Figures 4.39 through 4.41. A contour plot of hydrostatic pressure is not shown because this variable cannot be plotted when using plane stress elements in ABAQUS. These same variables along with hydrostatic pressure p are plotted for the 3-D DENT FEM in Figures 4.42 through 4.45. As expected, σ eff, pl σ yy, and ε eq are greatest in the vicinity of the notch tip. Once again, the mean stress (σ m = -p) is greatest in the interior of the specimen at failure at a distance of approximately 0.022 in. from the surface of the notch tip (Figure 4.43).

94 14000.00 12000.00 10000.00 Load P (lbs) 8000.00 6000.00 4000.00 Test Data 2000.00 von Mises FEA Drucker-Prager FEA Gage Length = 0.5 in. 0.00 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 Gage Displacement v (in.) Figure 4.35. Load-Displacement Results for the 2-D DENT Specimen 14000.00 12000.00 10000.00 Load P (lbs) 8000.00 6000.00 4000.00 Test Data 3-D von Mises FEA 2000.00 3-D Drucker-Prager FEA Gage Length = 0.5 in. 0.00 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 Gage Displacement v (in.) Figure 4.36. Load-Displacement Results for the 3-D DENT Specimen

95 14000.00 12000.00 10000.00 Load P (lbs) 8000.00 6000.00 4000.00 Test Data 2000.00 von Mises FEA Drucker-Prager FEA Gage Length = 0.5 in. 0.00 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 Gage Displacement v (in.) Figure 4.37. Linear Portion of the Load-Displacement Results for the 2-D DENT Specimen 14000.00 12000.00 10000.00 Load P (lbs) 8000.00 6000.00 4000.00 Test Data 3-D von Mises FEA 2000.00 3-D Drucker-Prager FEA Gage Length = 0.5 in. 0.00 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 Gage Displacement v (in.) Figure 4.38. Linear Portion of the Load-Displacement Results for the 3-D DENT Specimen

96 Figure 4.39. Effective Stress (psi) for the 2-D DENT FEM at Failure Load Figure 4.40. Stress in the y-direction (psi) for the 2-D DENT FEM at Failure Load

97 Figure 4.41. Equivalent Plastic Strain for the 2-D DENT FEM at Failure Load Figure 4.42. Effective Stress (psi) for the 3-D DENT FEM at Failure Load

98 Figure 4.43. Hydrostatic Pressure (psi) for the 3-D DENT FEM at Failure Load Figure 4.44. Stress in the y-direction (psi) for the 3-D DENT FEM at Failure Load

99 Figure 4.45. Equivalent Plastic Strain for the 3-D DENT FEM at Failure Load Modified Double-Edge Notch Tension Results Load-displacement data for the modified DENT finite element models are plotted alongside NASA test data [30] in Figure 4.46, and the linear portion of the loaddisplacement data is shown in Figure 4.47. Similar to the DENT results, the slope of the linear portion of the finite element results is less than that of the test specimen data indicating that the test specimen had a modulus of elasticity greater than 29.9x10 6 psi. Once again, the von Mises FEM overestimates the load for a given value of displacement in the postyield region. Considering the failure displacement, the Drucker-Prager FEM predicts loads that are about 3% lower than the von Mises values. For the failure load,

100 the Drucker Prager FEM predicts strains that are about 35% greater than the von Mises values, and, therefore, does a much better job of matching the overall specimen response. NASA strain gage data [30] for the modified DENT specimen are plotted alongside load versus microstrain FEM data in Figure 4.48. The microstrain data for the finite element models were taken as maximum strain at the integration points averaged over two or three elements at the notch tip. The strain results were averaged over these elements in an attempt to match dimensions of the region that the strain gage was attached on the test specimen. The test data is in between the two and three element average curves, but three-element average is a closer match. For the three-element average, both the von Mises and Drucker-Prager constitutive models slightly underestimate the strain for a given load, and the response of the two models is very similar until very large strains (µε>30,000) are reached. In the large strain region, the Drucker-Prager data curve is about 3% lower that the von Mises curve and is a closer match to the test data. No attempt was made to determine the linear and nonlinear error range of the strain gage data. pl Contour plots of σ eff, σ yy and ε eq for the modified DENT model at a point immediately before test specimen failure are shown in Figures 4.49 through 4.51. As pl expected, σ eff, σ yy, and ε eq are greatest in the vicinity of the notch tip.

101 14000.00 12000.00 10000.00 Load P (lbs) 8000.00 6000.00 4000.00 Test Data 2000.00 von Mises FEA Drucker-Prager FEA Gage Length = 0.5 in. 0.00 0.000 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.016 0.018 Gage Displacement v (in.) Figure 4.46. Load-Displacement Results for the Modified DENT Specimen 14000.00 12000.00 10000.00 Load P (lbs) 8000.00 6000.00 4000.00 Test Data 2000.00 von Mises FEA Drucker-Prager FEA Gage Length = 0.5 in. 0.00 0.0000 0.0005 0.0010 0.0015 0.0020 0.0025 Gage Displacement v (in.) Figure 4.47. Linear Portion of the Load-Displacement Results for the Modified DENT Specimen

102 12000 10000 8000 Load, P (lbs.) 6000 4000 2000 Test Specimen von Mises FEA - 2 Element Average Drucker-Prager FEA - 2 Element Average von Mises FEA - 3 Element Average Drucker-Prager FEA - 3 Element Average 0 0 10000 20000 30000 40000 50000 60000 Microstrain µε (in./in.) Figure 4.48. Load-Microstrain Results for the Modified DENT Specimen

103 Figure 4.49. Effective Stress (psi) for the Modified DENT FEM at Failure Load Figure 4.50. Stress in the y-direction (psi) for the Modified DENT FEM at Failure Load

104 Figure 4.51. Equivalent Plastic Strain for the Modified DENT FEM at Failure Load In summary, this chapter described the results of the finite element analyses. First, the material properties were presented for the two metals tested. Next, loaddisplacement or load-microstrain plots or both were presented along with contour plots of several stress and strain quantities. For every geometry tested, the von Mises constitutive model overestimated the load for a given value of strain or displacement. For given displacements at failure, the Drucker-Prager FEM s predicted loads that were from 3% to 10% lower than the von Mises values. For given failure loads, the Drucker Prager FEM s predicted strains that were from 20% to 65% greater than the von Mises values, and, therefore, did a much better job of matching the overall specimen response. Also, the contour plots revealed that for every FEM, the maximum hydrostatic stress occurred at a small distance inside the notch or fillet.