1. MSU DMG Plasticity-Damage Theory 1.0 Bammann, D. J., Chiesa, M. L., Horstemeyer, M. F., Weingarten, L. I., "Failure in Ductile Materials Using Finite Element Methods," Structural Crashworthiness and Failure, eds. T. Wierzbicki and N. Jones, Elsevier Applied Science, The Universities Press (Belfast) Ltd, 1993. Main References Horstemeyer, M.F., Lathrop, J., Gokhale, A.M., and Dighe, M., “Modeling Stress State Dependent Damage Evolution in a Cast Al-Si-Mg Aluminum Alloy,” Theoretical and Applied Fracture Mechanics, Vol. 33, pp. 31-47, 2000. Bammann, D. J., "Modeling Temperature and Strain Rate Dependent Large of Metals," Applied Mechanics Reviews, Vol. 43, No. 5, Part 2, May, 1990.

2. What is a constitutive law? A mathematical description of material behavior to satisfy continuum theory relating stress and strain Conservation of mass1 Balance of linear momentum3 Balance of angular momentum3 Balance of energy1 Number of equations Number of equations = 8 Number of unknowns=17 Constitutive Law equations =9 Background

3. Restrictions on Constitutive Laws 1. Physical admissibility 2. Material memory 3. Frame indifference 4. Equipresence 5. Local action

4. Physical Admissibility of ISVs ISVs are useful to model collective effects of changing material structure involving multiple mechanisms at multiple length scales e.g. dislocation interactions phase transformations distributed voids/cracks etc. PREMISE: all scales/treatments beyond quantum mechanics arephenomenological to some extent. Degree of “rigor” is related to the degree of resolution selected in solving the problem.

5. strain L  P  stress-strain relation Dislocation Internal State Variable

6. dislocation density

7. strain L  P  stress-strain relation Internal State Variable (damage) Observable State Variable (strain)

8. Observable State Variables (strain, strain rate, temperature) Internal State Variable (damage)

9. Observable State Variables (strain, strain rate, temperature) Internal State Variables (dislocations, damage)

10. ACTUAL EFFECTIVE CONTINUUM Porous Creep-Plastic Material Domain is occupied by dense material and voids/cracks Domain is occupied by voids/cracks Damage is defined by volume fraction or area fraction

11. Porous Creep-Plastic Material ACTUAL EFFECTIVE CONTINUUM Kachanov (1959)Rabotnov (1960)

12. ACTUAL EFFECTIVE CONTINUUM For example, the MSU DMG model,

13. Thermodynamical Framework of MDU DMG 1.0 Internal State Variable Model Elastic and Inelastic parts of Helmholtz Free Energy Stress-strain are thermodynamic conjugates Temperature-entropy are thermodynamic conjugates Backstress-kinematic (anisotropic) hardening are thermodynamic conjugates Global stress-isotropic hardening are thermodynamic conjugates Energy release rate-void volume fraction are thermodynamic conjugates

14. Frost-Ashby deformation mechanism map

15. Motivate evolution equations from Kocks-Mecking where dislocation density evolves as a dislocation storage minus recovery event. In an increment of strain dislocations are stored inversely proportional to the mean free path, which in a Taylor lattice is inversely proportional to the square root of dislocation Density. Dislocations are annihilated or “recover” due to cross slip or climb in a manner proportional to the dislocation density Evolution of dislocation density A scalar measure of the stored elastic strain in such a lattice is

16. Temperature Dependent Yield Rather than introducing several flow rules, we propose a temperature dependence for the initial value of the internal strength that emulates all of the mechanisms at a very low strain rate

17. Introduce a flow rule of the form Linear Elasticity From dislocation mechanics, (statistically stored dislocations)

18. Recovery included for the same compression curve. In this case the model accurately captures both the hardening and recovery through the isotropic hardening variable. data   STRAIN STRESS MPa Recovery with the Plasticity Constants (Now we have 3 constants: Yield, Isotropic Hardening with recovery) Y=C 3 H=C 15 R d =C 13

19. The small strain fit can be improved by including the short transient a which saturates at small strains as a function of its hardening and recovery parameters Recovery with the Isotropic and Anisotropic Plasticity Constants (Now we have 5 constants: Yield, Isotropic and Kinematic Hardening with recovery) Y=C 3 h=C 9 r d =C 7 H=C 15 R d =C 13

20. Stress State Dependence (Now we have 7 constants: Yield, Isotropic and Kinematic Hardening with recovery and stress state dependence) Y=C 3 h=C 9 r d =C 7 H=C 15 Rd=C 13 C a torsion C b tension/compression

21. High Strain Rate Effects Strain rate stress 10 3 /sec (Now we have 8 constants: Yield, Isotropic and Kinematic Hardening with recovery+ Strain rate dependence on Yield) V=C 1 Y=C 3 h=C 9 r d =C 7 H=C 15 R d =C 13 C a torsion C b tension/compression C 1 is the additional stress related to the added strain rate

22.  10 -1 [1/s]  10 [1/s] model10 [1/s] model10 -1 [1/s] Six parameter fit of 304L SS compression data with only the long transient k but including the effects of rate dependence of yield through the parameters V and f. The strain dependent rate effect is captured by the static recovery parameter R sk in the isotropic hardening. Strain Rate Effects on Static Recovery Creep Effects in Isotropic Hardening (9 constants) V=C 1 Y=C 3 h=C 9 r d =C 7 H=C 15 R d =C 13 R s =C 17 C a torsion C b tension/compression

23. Seven parameter fit of 304l SS compression curve including the short transient a. This fit will more accurately capture material response during changes in load path direction Strain Rate Effects on Static Recovery Creep Effects in Isotropic and Kinematic Hardening (10 constants) V=C 1 Y=C 3 h=C 9 r d =C 7 r s =C 17 H=C 15 R d =C 13 R s =C 17 C a torsion C b tension/compression

24. Temperature Effects (add even numbers to yield and hardening)

25. Model prediction for 304L stainless steel tension tests or 304L stainless steel is depicted in Figure 1. Strain rate dependent model correlation

26. History is important to predict the future!! 304L SS 25% wrong answer if history is not considered!!

27. Rate and temperature history change Load at 25C, 6000 1/s Reload at 269C at.0004 1/s Load at 269C, 5200 1/s Reload at 25C.0004 1/s

28. Rate change - decreasing at 25C - increasing at 269C

29. Number of ISV Model Constants 2: Bilinear Hardening 3: Yield+ Nonlinear Hardening 5: Yield+Nonlinear Isotropic and Kinematic Hardening 7: Yield+Nonlinear Isotropic and Kinematic Hardening to distinguish between tension, compression, and torsion 8: Yield+Nonlinear Isotropic and Kinematic Hardening to distinguish between tension, compression, and torsion to examine high strain rates 10: Yield+Nonlinear Isotropic and Kinematic Hardening to distinguish between tension, compression, and torsion to examine high strain rates and low rate creep using static recovery 12: Yield+Nonlinear Isotropic and Kinematic Hardening to distinguish between tension, compression, and torsion to examine high strain rates and low rate creep using static recovery and damage/failure 23: Yield+Nonlinear Isotropic and Kinematic Hardening to distinguish between tension, compression, and torsion to examine high strain rates and low rate creep using static recovery and damage/failure for temperature dependence 54: Yield+Nonlinear Isotropic and Kinematic Hardening to distinguish between tension, compression, and torsion to examine high strain rates and low rate creep using static recovery and damage/failure for temperature depedence including all of the nucleation, growth and coalescence for damage/failure

30. Kinematic vs. Isotropic Hardening 1 2 3 If all hardening occurs uniformly by statistically stored dislocations, (and the texture is random), the yield surface would grow isotropically “the same in every direction, independent of the direction of loading”. The radius of the yield surface, is given by , the internal strength of the material. This type of loading is illustrated in the figures. The material deforms elastically and the stress increases linearly until the initial yield surface is reached and the material hardens and the yield surface grows until unloading begins at point 1. Upon reversal of load the material deforms elastically until point 3 is reached. If geometrically necessary dislocations form pileups at grain boundaries (small effect) or at particles (larger effect), the material exhibits an apparent softening upon load reversal To model this, the yield surface is allowed to translate to the same stress point 1 (red surface). Now upon load reversal, plastic flow begins at point 2. Real material would begin a combination of these two exaggerated figures. This is a short transient and  represents the center of the yield surface. In some cases, we used to use  as long transient to model texture effects. But now we introduce a structure tensor for this effect. 1 2 3  

31. The evolution of damage is based upon the analytic solution of Cocks and Ashby Growth of spherical void in a power law creeping material under a three dimensional state of stress Cocks and Ashby using a bound theorem calculated the approximate growth rate of the void We utilize the functional form in the evolution of our damage state variable Failure occurs when a critical level of damage has accumulated and the material becomes unstable

32. Brittle vs. Ductile Fracture A. Very ductile, soft metals (e.g. Pb, Au) at room temperature, other metals, polymers, glasses at high temperature. B. Moderately ductile fracture, typical for ductile metals C. Brittle fracture, cold metals, ceramics.

33. Ductile Fracture (Dislocation Mediated) a)Necking b)Void Nucleation c)Void Coalescence d)Void Growth e)Fracture (cup and cone)

34. Ductile Fracture Macroscopic Prominent necking “Cup & Cone” fracture face “Dull” fracture surface +/-45 shear bands visible on side of neck

35. Ductile Fracture Microscopic Dimples with inclusions Inclusions suggest processing problem? Direction of dimples can indicate load history or misaligned photo TEM or SEM?

38. Ductile vs Brittile

39. Monotonic Microstructure-Property Model (Macroscale) Plasticity/Damage Inputs –21 constants for plasticity determined from different strain rate and temperature tests under compression –6 constants for void nucleation, void growth, and void coalescence equation determined from torsion, tension, and compression tests Microstructure-Defect Inputs –Silicon (volume fraction, size distribution, nearest neighbor distance) –Porosity (volume fraction, size distribution, nearest neighbor distance) –Dendrite Cell Size distribution –Other inclusion features (eg, oxides, etc) Outputs –Time and location of failure on complex geometrical component using FEA –Bauschinger effect –Various strain rate and temperature histories –Various loading path sequences (eg, fatigue followed by tensile loading, etc) –Implemented in ABAQUS finite element code Future –Addition of chemical corrosion effects D CB A E Stress (from highest to lowest) D A C E B Inclusion (from most severe to less severe) B E A D C Damage (from most severe to less severe) A D E C B

40. Microstructure-Property Model Equations (Macroscale) stress-strain relations Dislocation-plasticity internal state variables Damage internal state variables

41. Kinematics of Damage Framework Multiplicative Decomposition of the deformation gradient Damage definition

42. Damage Descriptions D 1  v a, ˙ D 1  ˙  v a  ˙ v a Barbee et al. (1972), Davison et al. (1977), etc. D 2   v a 1  v a, ˙ D 2  ˙  v a  ˙v a 1  v a  2 Davison et al. (1977) D 3  1  exp(  v a ), ˙ D 3  ˙  v a  ˙ v a  exp  v a  Kolmogorov (1937), Avarami (1939), Johnson (1949) D 4  v a, ˙ D 4  ˙  ˙ v a Gurson, Needleman, Tvergaard, LeBlond, McDowell, etc. D  dV v 2  v dN dV 2  v a  N- total number of nucleation sites V- void volume v

43. Description of Damage # voids/unit volume measured in intermediate configuration average void volume total volume of voids damage definition damage in terms of nucleation density and void volume and coalescence

44. Philosophy Of Modeling IVS Model Void Growth Void/Void Coalescence Void/Particle Coalescence Fem Analysis Idealized Geometry Realistic RVE Geometry Monotonic/Cyclic Loads Crystal Plasticity Experiment Fracture of Silicon Growth of Holes Experiment Uniaxial Monotonic Torsional Monotonic Notch Tensile Fatigue Crack Growth Cyclic Plasticity FEM Analysis Torsion Compression Tension Monotonic/Cyclic Loads Continuum Model Cyclic Plasticity Damage Structural Analysis Steering Knuckle Upper Control Arm Experiments FEM Model Cohesive Energy Critical Stress Analysis Fracture Interface Debonding Atomistics Experiment Fracture Interface Debonding ISV Model Void Nucleation FEM Analysis Idealized Geometry Realistic Geometry Micromechanics Mesomechanics Macromechanics ISV Model Void Growth Void/Crack Nucleation

45. Evolution of Damage

46. Figure 2.3. The damage model encompasses the limiting cases shown by (a) a single void growing in and (b) just void nucleation.

47. Scales of Importance for A356 Al Control Arm Electronic Principles (void-crack nucleation) Nm Gave bi-material elastic interfacial energy and moduli Atomistic (void-crack nucleaction) Nm Gave critical stresses for interface debonding Microscale (void-crack nucleation) 1-20 mm Gave temperature dependence on void-crack nucleation and microstructural morphological effects such as particle size, shape, and spacing Mesoscale I (silicon-porosity interactions) 1-200 mm Gave coalescence affects of second phase particles with casting porosity Mesoscale II (pore-pore interactions) 200-500 mm Gave coalescence effects of casting porosity interactions that considered void volume fraction, size, shape, and distribution at different temperatures and strain rates Macroscale (constitutive model formulation) mm-cm Material model was developed that included lower scale effects of microstructures that allows analysis of history effects Structural scale (control arm, etc.) cm-m Predicted and experimentally verified simulations at structural scale to validate multiscale methodology

48. Al-Si Atomistic Simulations were used to determine Damage nucleation mechanism

49. Atomistic simulations: An elastic fracture criterion approximates solution fairly accurately

50. Particle Parameters 1. Size 2. Shape 3. Number density 4. Temperature 5. Prestrain anisotropy 6. Microporosity 7. Loading direction Void Nucleation Parametric Study

51. Temperature was the main influence parameter

52. Temperature Dependence of Damage Nucleation Quantified

53. FE calculations were matched to the void nucleation experimental data

54.  =  vc  determined from simulation  is known from experiments v is a single void growth model (known) c is coalescence that needs to be determined from these simulations Mesoscale I: silicon-pore coalescence quantified

55. Mesoscale II (pore-pore interactions) Porosity parameters 1. size 2. shape 3. number density 4. temperature 5. prestrain anisotropy 6. microporosity 7. loading direction volume fraction is same in all calculations

56. Mesoscale I (pore-pore interactions): Parametric Results temperature, size, and microporosity are first order influences on void coalescence

57. (a) (b) (c) Figure 4.16. One void and two void configurations.

58. Figure 4.17. Void volume fraction normalized by the initial void volume fraction versus von Mises strain illustrating the two void aluminum material will experience greater void growth than the one void case given the same initial void volume fractions. The boundary condition in this case is biaxial (stretch forming) displacement boundary conditions at room temperature. (  r/  a=0.5)

59. Figure 4.19. Void volume fraction normalized by the initial void volume fraction versus von Mises strain illustrating that greater void growth occurs as the temperature increases. The boundary condition is plane strain in the axisymmetric geometry with one void aluminum material. (  r/  a=0.0)

60. Figure 4.18. Void volume fraction normalized by the initial void volume fraction versus von Mises strain illustrating that greater void growth occurs as the temperature increases. The boundary condition is plane strain in the axisymmetric geometry with two void aluminum material. (  r/  a=0.0)

61. 294K600K Figure 4.20. Effective plastic strain contours at the same snapshot in time illustrate the increase in plastic deformation as the temperature increases. This increase in plastic deformation enhances void growth. The boundary condition is plane strain, axisymmetric geometry with the two void aluminum material. (  r/  a=0.0)

62. von Mises Strain Normalized Void Volume Fraction 1 void 2 void, 4 diameters 2 void, 5 diameters 2 void, 6 diameters Figure 4.22. The critical intervoid distances were determined by comparing responses of multiple voids growing to the single void case. The six void diameter case was identical to the single void case below a strain of 0.0005. This calculation was a biaxial, axisymmetric calculation at room temperature.

63. Boundary condition 1Boundary condition 2Boundary condition 3 Figure 4.23. The critical intervoid distances determined to enhance void growth. The critical length, L, is determined for different loading conditions, temperatures, and void configurations.

64. Figure 4.24. Void volume fraction normalized by the initial void volume fraction versus von Mises strain illustrating the increase in void growth as the stress triaxiality increasess with two voids over one void. These calculations were performed under constant triaxiality conditions, at room temperature, and for the axisymmetric geometry.

65. ISV Microstructure-Property Model Equations Dislocation-plasticity internal state variables Damage internal state variables Particle size Particle Volume fraction Nearest neighbor distance Dimensionless grain size Grain size Dimensionless grain size Damage Damage rate

66. Physical Observations of Damage Evolution damage tensioncompression

67. Void Nucleation Rate Nucleation rate differs as a function of tension, compression, and torsion Damage rate is directly tied to stress state

68. model strain rate/ temperature dependence on yield stress state dependence strain rate/temperature dependence on hardening Model/Experiment Comparisons

69. model dendrite cell size on hardening Load reversal Bauschinger effect

70. Contour plots of total void volume fraction comparing the finite element simulations at first element failure, total damage (  ) is SDV14, assuming initial random and homogeneous distributions. peak value

71. Pictorial illustration of porosity distribution for the 98% of failure load specimen from (a) x-ray tomography, (b) image analysis, and (c) finite element simulation with an initially random distribution of porosity at a level of 0.001. (a) (b) (c)

72. Total void volume fraction along (a) radial distance and (b) axial distance determined from 98% of failure load specimen from x-ray tomography, image analysis, and finite element analysis with averaged regions similar to those taken for the tomography and image analysis measurements. (a) (b)

73. Comparison of (a) experiment and (b) microstructure-property model failure prediction (damage=SDV14) for weapons carrier analysis. failure sites (a) (b) first damage initiation site

74. initial failure site (a) (b) Comparison of microstructure-property model failure prediction (damage=SDV14) with experiment for Control Arm 2. The worst case microstructure/inclusion content was assumed in the calculation.

75. Under USCAR/USAMP Lightweight Metals CRADA, Control Arm was optimized Sandia model was used to optimize a redesign: 25% weight saved 50% increase in load-bearing capacity 100% increase in fatigue life $2 less per part.

76. D CB A E Stress (from highest to lowest) D A C E B Inclusion (from most severe to less severe) B E A D C Damage (from most severe to less severe) A D E C B initial failure site (a) (b) Region 3 Region 1 mode l experiment Result: To optimize a redesign such that 25% weight saved 50% increase in load-bearing capacity 100% increase in fatigue life $2 less per part GM CADILLAC CONTROL ARM LIGHTWEIGHT DESIGN Objective: To employ multiscale material modeling to reduce the weight of components

77. Powder metal compaction/sintering Porosity Pore coalescence Casting Particles Porosity Pore Pore growth nucleation +coalescence +growth Rolling/Extrusion/Forging/Stamping Particles Pore nucleation Defect volume fraction Defect size (m) 10 -3 10 -6 10 -4 10 -5 10 -7 10 -5 10 -4 10 -3 10 -2 10 -1 10 -0 Ductile fracture Manufacturing process Defect type Dominant damage mechanism under monotonic loads Relationship of Manufacturing Process, Defect, and Ductile Fracture Mechanisms