2. MODELING PLASTICITY/DAMAGE USING INTERNAL STATE VARIABLES
DOUGLAS J. BAMMANN
CENTER FOR MATERIALS AND ENGINEERING SCIENCES
SANDIA NATIONAL LABS/CA

3. Phenomenological Modeling Using State Variables
Kinematics
Thermodynamics
Physical basis (micromechanics)
Parameter determination
Microstructural (physical) constants
fixed state experiments
evolving state experiments
analytical solutions - simple tests - nonlinear least squares
numerical integration - optimization
Numerical implementation
Validation

4. Coleman-Gurtin applied to plasticity (Kratochvil-Dillon, Werne-Kelly, Teodosiu-Sidoroff, Perzyna,Anand ...
Assume multiplicative decomposition of deformation gradient into elastic and plastic parts (unloading elastically defines stress free or intermediate configuration - this assumption couples kinematics with constitutive model)
Assume that the free energy, defined with respect to the intermediate configuration, depends upon the elastic strain and deformation type (or defects) state variables
Determine restrictions of symmetry and invariance
Develop evolution equations for the state variables (generally motivated from the microstructural effects they represent)
Conjugate thermodynamic driving forces are defined as the derivative of the the free energy w.r.t. the state variables and are the stress like variables in the flow rule
Chosen as a result of the appearance of the product of these two terms in the dissipation inequality
Determine material parameters
Microstructural constants
Fixed state experiments
Evolving state experiments

5. The plasticity variables (those related to deviatoric plastic flow from dislocations) are motivated as follows
Evolution of variables cast in hardening minus recovery format
hardening is based upon a dislocation storage mechanism and has temperature dependence of shear modulus
dynamic recovery is motivated by cross slip and operates on same time scale as plastic flow and therefore its rate dependence is determined by the kinetics of slip
thermal recovery is motivated by diffusional climb and operates on a completely different time scale and introduces a strong rate dependence
Scalar variable represents statistically stored dislocations and gives rise to most of the hardening
Dislocations are stored inversely proportional to mean free path
Tensor variable describes geometrically necessary dislocations
Tensor variable hardens in direction of plastic flow and recovers in the direction of the current value of the tensor variable
Under constant temperature, strain rate and loading direction both variables reach steady state when hardening is balanced by recovery

6. Configurations
Kinematics of continuum mechanics begins by introducing a map from material space (manifold) to physical space (R3). Thus the map is from a reference configuration in material space to the current configuration in physical space. The reference configuration is not to be confused with the Lagrangian configuration which is simply a previously occupied configuration in physical space.
To simplify matters we will consider the reference configuration to be the initial or Lagrangian configuration as we develop our 1D continuum x Bt X x0 BR
Bt0

7. KINEMATICS
Following Bilby (1956), Kroner (1960) decompose the deformation gradient into elastic and plastic parts

8. Strain can be defined with respect to any configuration based upon the Lagrange change in square length per unit square length
Define a Lagrange total and plastic strain w.r.t. reference configuration
Then the elastic strain in this configuration is defined as
These can be mapped forward to the intermediate configuration

9. The velocity gradient is defined as usual and is naturally composed of elastic, plastic, damage and thermal parts. These can be mapped to any other configuration.
Velocity gradient
Notice that the plastic part of the velocity gradient is naturally defined with respect to the
natural configuration as
And the velocity gradient w.r.t any configuration can be split into elastic and plastic parts
And then from algebra

10. The decomposition of the deformation gradient into elastic and plastic parts
Results in additional degrees of freedom
Requires specifying expressions for plastic stretching (strain rate) and plastic spin
These are easy to specify in crystal plasticity, but more difficult to motivate in phenomenological models
Most models ignore plastic spin
This can be shown to be critical in attempting to predict evolving anisotropy (from texture)

11. INTERNAL STATE VARIABLE THEORY
What are state variables?
Variable whose current value represents some observable state of the material
Can be initialized (can be measured without knowing anything about the past)
Field theory
How are kinematics related to the state variables?
Define thermodynamics with respect to specifically defined kinematic configuration
How does kinematics or the geometric structure of the state variables affect the degrees of motion of the continuum

12. Y = Y0 s = s0 ep= 0 Y = Y0+ DY s = s0 + Ds ep= b/2 Y = Y0 s = s0 ep= b
The free energy of the crystal depends upon the dislocations present while the plastic deformation is governed by the transport of dislocations
Y = Y0
s = s0
ep= 0
Y = Y0+ DY
s = s0 + Ds
ep= b/2
Y = Y0
s = s0
ep= b

13. Thermodynamics - 1st Law
Thermodynamics - Coleman and Gurtin (1967) (Formulation will be small strain, 1D to simplify concepts)
Kinematics:
Thermodynamics - 1st Law
Rate of change of internal energy is the sum of the rate at which work is being done on the body, , balanced by rate at which heat is supplied to the body, h
2nd Law

14. It follows from the 2nd law
Assume that the free energy depends upon the elastic strain, the temperature, and two state variables, the elastic strain resulting from the distribution of geometrically necessary dislocations leading to kinematic hardening, gnand the elastic strain associated with the density of statistically stored dislocations giving rise to isotropic hardening, ss
It follows from the 2nd law
Where, and  are conjugate thermodynamic stresses

15. And the dissipation inequality reduces to
Then neglecting elastic heating effects and neglecting
conduction for high strain rate applications, the energy
balance reduces to
And the dissipation inequality reduces to
The kinematics of plasticity introduced degrees of freedom requiring more”constitutive” equations. Similarly, the extra degrees of freedom also requires temporal evolution equations to complete the system.
This is where the physics of the smaller length scales enters!

16. To complete the system we need
Analogous to elasticity, assume a free energy of the form
Then,
To complete the system we need
An expression for the plastic strain rate
Evolution equations for the state variables  
Now let’s take a very brief look at dislocation models of plasticity to determine the necessary forms

17. Statistically Stored Dislocationsb
Zero net burgers vector -b

18. Geometrically Necessary Dislocationsb
Net burgers vector is 2 b b

19. Geometrically or Kinematically Necessary Dislocations and/or Boundaries
void
overlap
Polycrystal deforms, grains rotate, lose continuity. Geometrically necessary dislocations permit reassembly of polycrystal

20. Some Important Deformation Mechanisms - Klahn, Mukherjee, Dorn
Drag
Mechanisms
Thermally
Activated
SHEAR STRESS
Athermal
Diffusion
Controlled
200
400
600
800
TEMPERATURE K

21. Frost-Ashby deformation mechanism map

22. 
Example - Thermally Activated Motion (reaction-rate theory, Eyring (1936))
Assume the number of times per second that a dislocation segment overcomes an energy barrier under the action of an applied stress can be written as a thermally activated process.

Q0 = height of energy barrier
bA = work done by applied stress in overcoming energy barrier or effectively lowering barrier
k = Boltzman constant
 = temperature
A = area swept in glide plane when dislocation segment moves to top of barrier
b = Burgers vector

23. Frequency of jumps backward over the barrier
Net forward reaction rate
If l is the length of the dislocation freed after each successful jump, and V is the volume of the crystal, the strain after each jump is
And the strain rate is then
Where N is the number of dislocation segments (activation sites) per unit volume

24. Force - Distance Diagram (constant rate and temperature)
Distance along
glide path
lb
lb
lb
lb
A/l

lb
lb - force of applied stress
Lb - force of average value of spatially fluctuating long range
internal stress from other dislocations
lb - short range obstacle strength

25. Ono (1968) and Kocks et al. (1975) showed that a large range of obstacle shapes could be described by the following
Where F is the activation energy characterizing the strength of a single obstacle. It also determines the rate sensitivity of the internal strength, .

26. Deformation Mechanisms - Frost and Ashby
Elastic collapse - stress exceeds ideal shear strength
Thermally activated motion
Peierls force - lattice resistance
Power Law Creep

27. Harper-Dorn Creep
Power Law Breakdown
Diffusional Flow
Net plastic Flow

28. Viscous Drag
Phonon and Electron

29. Evolution of dislocation density
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 l, 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 
A scalar measure of the stored elastic strain in such a lattice is

30. Model Development
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

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

32. Heat conduction reduces to
The tensor variable is motivated by the continuum theory of dislocations (recall the nonlocal workshop where we introduced the elastic curvature as a state variable resulting in a natural gradient and natural bacck stress). Here we simplify to a local form and choose
Heat conduction reduces to
As dislocations are stored, the dissipation is reduced until recovery becomes dominant. Then the heat dissipated approaches the assumption that 90-95% of plastic work is dissipated as heat.

33. The tensor variable is motivated by the continuum theory of dislocations (recall the nonlocal workshop where we introduced the elastic curvature as a state variable resulting in a natural gradient and natural back stress). Here we simplify to a local form and choose
For a wide range of temperatures and strain rates, plastic flow is a thermally activated strain rate. By choosing an appropriate form for the activation energy we get a flow rule of the form

34. Two parameter fit of 304L SS data at 800C for both large and small strain offset definitions of yield. For two parameters, the model reduces to rate independent bilinear hardening.
Data
Large offset
Small offset
STRESS MPa
STRAIN

35. Recovery included for the same compression curve
Recovery included for the same compression curve. In this case the model accurately captures both the hardening and recovery through the isotropic hardening variable .
STRESS MPa
data s 
STRAIN

36. 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

37. 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 Rsk .
model10 [1/s]
model10-1[1/s]
k 10 [1/s]
k 10-1[1/s]

38. Five 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

39. Model prediction for 304L stainless steel tension tests or 304L stainless steel is depicted in Figure 1.

40. Model prediction of compression tests and compression reload demonstrating temperature and history effects. The 800C test was quenched after being strained to 23% and reloaded at room temperature. The temperature history effect is demonstrated by the reload curve being much softer than the 20C curve.

41. Kinematic vs. Isotropic Hardening1 2 3 a
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 k, 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 a represents the center of the yield surface.
In some cases, we used to use a as long transient to model texture effects. But now we introduce a structure tensor for this effect. k 1 2 3

42. The proportion of kinematic and isotropic hardening can be determined from reverse loading tests
To satisfy yield at both unloading and reverse yielding (assuming state variables don’t change during elastic unloading) 1 1 2 2
Then the kinematic and isotropic hardening proportions can be determined as

43. The offset used to to determine yield has a large effect upon the proportion of kinematic to isotropic hardening
A small offset definition of yield stress (such as a strain of %) results in a greater proportion of kinematic hardening
A larger offset (the standard 0.2 % strain) will generally result in a prediction of a domination isotropic hardening
The truth (experimental data) is most closely approximated by the smallest offset and is most important in problems involving unloading at small strains

44. J. Hodowany’s Ph.D. dissertation (Caltech, 1997) showed quantitative measurements of the conversion of plastic work into heat as a function of plastic strain
Hodowany’s measurements indicate that in the early stages of deformation, the amount of plastic work converted to heat may be quite small
As deformation progresses, the heat conversion more nearly equals the plastic work rate

45. Results of J. Hodowany
(Ph.D. dissertation, Caltech, 1997)
Kolsky (split Hopkinson) bar arrangement
High speed measurements of
Temperature (IR detector)
Plastic work (strain gages on input and output bars)

46. Primary results involve a parameter b which is the fraction of plastic work converted to heat

47. Fraction of plastic work converted to heat

48. Fraction of plastic work converted to heat for 2024 Al

49. Simulation for a “fully annealed” material

50. Simulation for an initially hardened material

51. The effect of damage on the elastic moduli can be determined from considering a voided cylinder loaded in tension
The effect of the porosity can be viewed as a reduction of the load bearing area
The stress is then
This derivation treats voids as an area distribution. The effective elastic moduli for a volumetric distribution has been determined using self consistent techniques by Budiansky and OConnel

52. In the same manner, the voids tend to concentrate the stress in the matrix material, thereby enhancing the plastic flow
For metals, the plastic flow will generally have a strong dependence upon the stress, for example
In the presence of damage, the stress will be concentrated and the resulting flow rule takes the form

53. 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

54. It follows from the 2nd law
Assume that the free energy depends upon the elastic strain, the temperature, and two state variables, the elastic strain resulting from the distribution of geometrically necessary dislocations leading to kinematic hardening, gnand the elastic strain associated with the density of statistically stored dislocations giving rise to isotropic hardening, ss
It follows from the 2nd law
Where, ,  and  are conjugate thermodynamic stresses

55. And the dissipation inequality reduces to
Then neglecting elastic heating effects and neglecting
conduction for high strain rate applications, the energy
balance reduces to
And the dissipation inequality reduces to
The kinematics of plasticity introduced degrees of freedom requiring more”constitutive” equations. Similarly, the extra degrees of freedom also requires temporal evolution equations to complete the system.
This is where the physics of the smaller length scales enters!

56. natural configuration
The deformation gradient is decomposed multiplicatively into thermal, elastic, deviatoric plastic and damage parts and make simplifying assumptions about the directionality of thermal and damage gradients
natural configuration
Define strain with respect to natural configuration

57. natural configuration as
The velocity gradient is defined as usual and is naturally composed of elastic, plastic, damage and thermal parts. These can be mapped to any other configuration.
Velocity gradient
Notice that the plastic part of the velocity gradient is naturally defined with respect to the
natural configuration as
And the velocity gradient w.r.t any configuration can be split into elastic and plastic parts
And then from algebra

58. Since natural configuration is most closely related to lattice, we must relate velocity gradients in current configuration (where we solve momentum balance) to strain rates in the lattice (where we model material response)
The current configuration velocity gradient is defined as
where
Now
And assuming, that elastic strains are small
Then, the elastic strain rate in the intermediate configuration is approximately equal to
the current configuration elastic velocity gradient

59. The thermodynamics are formulated with respect to the natural configuration following Coleman and Gurtin. The free energy is assumed to depend upon the elastic strain, temperature, temperature gradient, and internal variables related to dislocations in cells and cell walls, and voids.
Assume the free energy with respect to the natural configuration
Recall the Clausius-Duhem inequality
Substituting, the dissipation becomes

60. The balance of energy becomes
Neglecting thermal-elastic effects and for high rate processes where conduction is negligible

61. As a result of thermodynamic restrictions we define “thermodynamic stresses” as the derivative of free energy with respect to state variables. The stresses appear in our flow rule and the evolution of the state variables is motivated from micromechanics. These are related through the free energy.
To satisfy thermodynamics:
Define conjugate stresses to the deformation like state variables:

62. The elastic strain associated with the dislocations is motivated by considering the energy of a Frank network
The energy of a Frank network of dislocations of density d is
Since the stress associated with the dislocations is given by
We can identify the strain associated with the dislocation network as,

63. The dislocation velocity is assumed to depend upon the magnitude of the “net stress” (effective stress) which is the sum of the applied stress and any stresses due to the state variables which were introduced in the thermodynamics
The net stress is the Cauchy stress minus the two dislocation stresses plus the void stress
To model isotropic hardening assume dislocation cell stress is in the direction of Cauchy minus
back stress due to dislocations in cell interiors and that the void stress concentrates this stress ;
Then the effective stress is
The dislocation velocity would then depend upon
And for small porosity

64. We now have reduced the problem for determining the free energy as a function of two scalar “strains” related to a scalar dislocation density and a scalar porosity as well as a tensor strain related to dislocation distribution

65. Now assume linear elasticity with respect to the natural configuration, mapping forward to the current configuration, and take the material derivative of the Cauchy stress. The Budiansky solution is utilized to describe the dependence of the elastic moduli on the damage.
Linear elasticity
Elastic moduli degraded by voids
Velocity gradient decomposition
The thermal and damage parts of the velocity gradient were discussed in the kinematics,
hence we need to address the plastic part (dislocation related) of the velocity gradient

66. Motivation for evolution comes from Kocks-Mecking
Motivate evolution equations from Kocks-Mecking where dislocation density evolves
as a dislocation storage minus recovery event. Hardening is proportional to
dislocation cell size which is inversely proportional to square root of dislocation
density while recovery is proportional to dislocation density
Then,

67. For small concentrations of damage the flow rule reduces to the more familiar one. The evolution of the state variable comes form dislocation micromechanics and ductile void growth of Cocks and Ashby. The plastic spin is assumed zero.
Flow rule
Evolution of hardening state variables
Evolution of void growth

68. For constant true strain rate, isothermal, viscoplastic uniaxial stress states (and in the absence of damage effects - compression), an analytical solution is obtained
Solution for hardening variables
Resulting flow stress
Saturation stress and strain

69. Then the dissipation inequality takes the form
Analogous to elasticity, assume a free energy of the form
Then,
Then the dissipation inequality takes the form

70. SUMMARY - CURRENT CONFIGURATION
Linear Elasticity
State Variables

71. Flow Rule -(Cocks, 1989) (Marin, McDowell-1996)
where

72. Elongation at Failure Radius Test Analysis 0.390 0.043 0.044 0.156
0.021
0.023
0.078
0.014
0.015
0.039
0.011
0.013

73. Aluminum disks were impacted by hardened tool steel rods
Aluminum disks were impacted by hardened tool steel rods. At 3600 in/s this loading resulted in partial failure of the back surface.
Experiment
Simulation

74. Aluminum disks were impacted by hardened tool steel rods
Aluminum disks were impacted by hardened tool steel rods. At 4200 in/s this loading resulted in complete penetration of the disk..
Experiment
Simulation

75. Application of the BCJ State Variable Model to Submarine Vulnerability Experiments (Steel Plates Loaded by Focused Blasts)
Analysis
Experiment
Technical Approach:
Model blast pressure time history with Eulerian code.
Apply the BCJ constitutive model, utilizing the strain rate and temperature dependence and ductile failure capabilities
Problem:
Predict response of steel (HY100) plate with initial 1 inch diameter hole loaded by blast wave produced by 38 grams of explosive.

76. The model has been used to predict response of steel (HY100) plate with initial 1 inch diameter hole loaded by blast wave produced by 38 grams of explosive. The blast pressure time history is modeledwith Eulerian code
Analysis
Experiment .