1. MSU DMG Plasticity-Damage Theory 1.0
Main References
Bammann, D. J., "Modeling Temperature and Strain Rate Dependent Large of Metals," Applied Mechanics Reviews, Vol. 43, No. 5, Part 2, May, 1990.
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.
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 , 2000.

2. Background
What is a constitutive law? A mathematical
description of material behavior to satisfy
continuum theory relating stress and strain
Number of equations
Conservation of mass 1
Balance of linear momentum 3
Balance of angular momentum 3
Balance of energy 1
Number of equations = 8
Number of unknowns=17
Constitutive Law equations =9

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

6. dislocation density

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

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

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

10. Porous Creep-Plastic Material
ACTUAL
EFFECTIVE CONTINUUM
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. EFFECTIVE CONTINUUM
ACTUAL
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. 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

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

18. Recovery with the Plasticity Constants
(Now we have 3 constants: Yield, Isotropic
Hardening with recovery)
Recovery included for the same compression curve. In this case the model accurately captures both the hardening and recovery through the isotropic hardening variable .
Y=C3
H=C15
Rd=C13
STRESS MPa
data s 
STRAIN

19. Recovery with the Isotropic and Anisotropic Plasticity Constants
(Now we have 5 constants: Yield, Isotropic and Kinematic
Hardening with recovery)
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
Y=C3
h=C9
rd=C7
H=C15
Rd=C13

20. Stress State Dependence (Now we have 7 constants: Yield, Isotropic and Kinematic Hardening with recovery and stress state dependence)
Y=C3
h=C9
rd=C7
H=C15
Rd=C13
Ca torsion
Cb tension/compression

21. High Strain Rate Effects
(Now we have 8 constants: Yield, Isotropic and Kinematic Hardening with recovery+ Strain rate dependence on Yield)
stress
V=C1
Y=C3
h=C9
rd=C7
H=C15
Rd=C13
Ca torsion
Cb tension/compression
Strain rate
103/sec
C1 is the additional stress related to
the added strain rate

22. Strain Rate Effects on Static Recovery
Creep Effects in Isotropic Hardening (9 constants)
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 in the isotropic hardening.
model10 [1/s]
V=C1
Y=C3
h=C9
rd=C7
H=C15
Rd=C13
Rs=C17
Ca torsion
Cb tension/compression
model10-1[1/s]
k 10 [1/s]
k 10-1[1/s]

23. Strain Rate Effects on Static Recovery
Creep Effects in Isotropic and Kinematic Hardening (10 constants)
Seven parameter fit of 304LS compression curve including the short transient a . This fit will more accurately capture material response during changes in load path direction
V=C1
Y=C3
h=C9
rd=C7
rs=C17
H=C15
Rd=C13
Rs=C17
Ca torsion
Cb tension/compression

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

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

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

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

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)
Necking
Void Nucleation
Void Coalescence
Void Growth
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
Ductile Fracture

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 C B A E
Stress
(from highest
to lowest)
Inclusion
(from most severe
to less severe)
Damage

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

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

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) mm
Gave coalescence affects of second phase particles with casting porosity
Mesoscale II (pore-pore interactions) 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. ISV Microstructure-Property
Model Equations
Damage
Dislocation-plasticity internal state variables
Dimensionless grain size
Damage internal state variables
Grain size
Particle size
Damage rate
Particle Volume fraction
Nearest neighbor distance
Dimensionless grain size

49. Physical Observations of Damage Evolution
compression
tension
damage

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

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

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

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

54. (b)
(c)
(a)
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

55. (b)
(a)
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.

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

57. (a)
initial failure site
(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.

58. 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.

59. ARM LIGHTWEIGHT DESIGN
GM CADILLAC CONTROL
ARM LIGHTWEIGHT DESIGN D C B A E
Stress
(from highest
to lowest)
Inclusion
(from most severe
to less severe)
Damage
Objective: To employ multiscale material modeling
to reduce the weight of components
initial failure site
(a)
(b)
Region 3
Region 1
model
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

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