Large Deformation Plasticity of
Amorphous Solids, with Application
and Implementation into Abaqus
Kristin M. Myers
January 11, 2007
Plasticity ES 246 - Harvard
References:
[1] Anand, L., Gurtin, M.E., 2003. “A theory of amorphous solids undergoing large deformations, with
application to polymeric glasses.” International Journal of Solids and Structures 40, 1465-1487.
[2] Boyce, M. C., Arruda, E. M., 1990. “An experimental and analytical investigation of the large strain
compressive and tensile response of glassy polymers.” Polymer Engineering and Science 30 (20), 12881298.
[3] Lubliner, J. Plasticity Theory. 1990. Macmillan Publishing Company. (Chapter 8)
[4] Abaqus 6.5-4 Documentation “Getting Started with ABAQUS/EXPLICIT.” Hibbitt, Karlsson &
Sorensen,INC.
Motivation – Examples of Materials
•
Amorphous Solids –
– polymeric and metallic glasses (i.e. Polycarbonate)
– Rubber degradation
•
Biomaterials
– Soft Collageneous Biological Tissue
(i.e. cartilage, cervical tissue, skin, tendon, etc.)
– Engineering Collagen Scaffolds
(i.e. skin, nerve, tendon etc.)
Material Characteristics:
1. Large stretches – elastic & inelastic
2. Highly non-linear relationships between stress/strain
3. Time-Dependent; viscoplasticity
4. Strain hardening or softening after initial yield
5. Non-linearity of tension & compression behavior (Bauschinger effect)
Experimental Results – Polycarbonate
From Boyce and Arruda
COMPRESSION
TENSION
• Large deformation regime
• Strain-softening after initial yield
• Back stress evolution after yield drop to create strain-hardening
Kinematics – Multiplicative Decomposition of the
Deformation Gradient
p
e
d x  F  X d l Segment of the “current
F X 
configuration”
e
F X 
dl  F
p
 X d X
Segment of the “relaxed
configuration”
“Relaxed Configuration”: Intermediate
configuration created by elastically
unloading the current configuration and
relieving the part of all stresses.
F   y( X , t )
Deformation Gradient
FF F
Decomposition of deformation gradient into its elastic
and plastic components (Kroner-Lee)
e
p
v  y ( X , t )
Velocity tensor
1

L  grad v  F F
Velocity Gradient
Kinematics – Multiplicative Decomposition
of the Deformation Gradient II
LL F L F
e
e
p
e 1
1
e
e
e
e

L  F F  D W
e
D  sym L
e
1
p
p
p
p

L  F F  D W
p
e
D  sym L
p
p
W  skwL
e
e
W  skwL
p
p
• Conditions of Plastic Flow
– Incompressible
det F  1
p
J  det F  det F
tr L  0
p
W 0
p
p
p

F D F
p
– Irrotational
L D
p
p
e
Principle of Objectivity
Principle of Material Frame Indifference
Smooth time-dependent rigid transformations of the Eulerian Space:
y( X , t ) 
 y ( X , t )  Q(t ) y( X , t )  q(t )
*
Principle of Relativity: relation of the two motions is equivalent
Q(t )  Relative motion of two observers Eulerian bases
g  Qg
*
To be objective (in general):
F
 Q F
F 
 Q F
e
e
e
T
D 
 Q D Q
e
e
*
T
The relaxed and reference configurations are invariant to
the transformations of the Eulerian Space
T

L 
 Q L Q  QQ
e
G  QGQ
T
F
P
e
e T
T
W 
 QW Q  Q Q
D
P
Principal of Virtual Power
• External expenditure of power = internal energy


~e
1 P ~ P
~
~
Wext   t (n)  v dA   f b  v dVt  Wint   T  L  J T  D dVt
dP
P
 Macroscopic Force Balance
divT  f b  0
P
• Internal energy Wint is invariant under all changes
in frame
*
Wint  Wint
–  Microforce Balance


Wint   T  D  J 1T P  D dVt
P
e
P
Dissipation Inequality and
Constitutive Framework
• 2nd Law of Thermodynamics: The temporal increase
in free energy ψ of any part P be less than or equal to
the power expended on P
  J T  D  T  D  0
e
P
P
• Constitutive framework:   ˆ ( F , F )
Free energy, stress, and
e
P
ˆ
T  T (F , F )
internal variables are a
function of deformation. T P  Tˆ P ( F e , F P , D P ,  )
e
P
e
P
P
i

  h (F , F , D , )
Constitutive Theory – Framework
• Frame Indifference
– Euclidean Space
– Amorphous Solids: material are invariant under all rotations
of the Relaxed and Reference Configuration
  ˆ ( F , F )
   (C , B )
e
P
ˆ
T  T (F , F )
T  F T (C , B ) F
P
e
P
P
ˆ
T  T (F , F , D , )
e
P
P
i

  h (F , F , D , )
T  T (C , B , D ,  )
e
P
P
i

  h (C , B , D ,  )
e
P
P
e
e
P
P
e
P
e
P
P
eT
P
Constitutive Theory – Thermodynamic
Restrictions and Flow Rule


e
P
 
C 
B
e
P
C
B
  (C e , B P )  e
F
T  2 J F 
e


C


e
P



(
C
,
B
)
P
P


T  2symo 

Y
P


B


1
S back
Energy dissipated per unit volume (in the
relaxed configuration) must be purely
dissipatative. Dissipative FLOW STRESS:
e
FLOW RULE:
Define:
Plug into dissipation inequality
  (C e , B P ) 

 2 symo 
P


B


Y (C , B , D ,  )  D  0
P
e
P
P
P
Y (C , B , D ,  )  S o  S back
P
e
P

S 0  2 symo C T
e
P
e

 e  (C e , B P ) 

 2 symo  C
e


C


Constitutive Equations
material parameters
• Free Energy
   
e
 GE
e
e 2
o
 1 / 2K tr E
e 2
P
Constitutive prescription
 P   (P )
• Equations for Stress
T  2G E o  K tr E 1
e
e
eT
T  R TR
e
e
e
Te Stress conjugate to Ee
T R T R
e
e
eT
= Cauchy Stress
Constitutive Equations
material parameters
• FLOW RULE for Plastic
STRETCHING
P
P
P

F D F
F
T  B
P
D   P 
2

e
o
P
o




P
Constitutive prescription
 X ,0  1
Effective Stress:
1/ m
 

 s  p 

 P   o 
 
1
2
T  B
e
o
P
o
DP=(magnitude)(DIRECTION)
• Evolution of Internal Variables

s 
s  ho 1  ~  P
 s ( ) 
 s

 1 P
 scv 
  g o 
= evolution of shear resistance
(captures strain softening)
= change in free-volume from
initial state
Saturation value:
~
s ( )  scv 1  b(cv  ) 
Micrograph by Roeder et al, 2001
Evolution of the Back Stress: Langevin Statistics
undeformed
Amorphous polymeric materials:
• Wavy kinked fibrous network structure
• Resistance of the network in tension
• Have finite distensibility (maximum stretch L )
• Once material overcomes the resistance to intermolecular chain motion
chains will align w/principle plastic stretch (Bp,λp)
•Alignment decreases the configurational entropy
 creates an internal network back stress Sback
Force-stretch relationship:
- Initially compliant behavior followed
by increase in stiffness as the limiting
stretch is approached
deformed
L
1  (P )
 P
3
P
 P
   R  
 L
P
Force
L0
2
L
Stretch


 x  ....


 
L(  )  coth(   )  (  ) 1
L
Limiting extensibility
P
 L  1   
   R  P  L  
 3   L 
Parameters:
•Rubbery Modulus
•Limiting stretch
L
L0
State Variables in Summary: In VUMAT
C**********************************************************************
C STATE VARIABLES - Variables that need to be evolved with TIME
C
STATEV(1) = Fp(1,1) -- PLASTIC DEFORMATION GRADIENT, (1,1) COMP.
C
STATEV(2) = Fp(1,2) -- PLASTIC DEFORMATION GRADIENT, (1,2) COMP.
C
STATEV(3) = Fp(1,3) -- PLASTIC DEFORMATION GRADIENT, (1,3) COMP.
C
STATEV(4) = Fp(2,1) -- PLASTIC DEFORMATION GRADIENT, (2,1) COMP.
C
STATEV(5) = Fp(2,2) -- PLASTIC DEFORMATION GRADIENT, (2,2) COMP.
C
STATEV(6) = Fp(2,3) -- PLASTIC DEFORMATION GRADIENT, (2,3) COMP.
C
STATEV(7) = Fp(3,1) -- PLASTIC DEFORMATION GRADIENT, (3,1) COMP.
C
STATEV(8) = Fp(3,2) -- PLASTIC DEFORMATION GRADIENT, (3,2) COMP.
C
STATEV(9) = Fp(3,3) -- PLASTIC DEFORMATION GRADIENT, (3,3) COMP.
C
C
STATEV(10)= Internal variable S - shear resistance
C
C
STATEV(11)= dFp(1,1) -- incre in PLASTIC DEFORMATION GRADIENT, (1,1) COMP.
C
STATEV(12)= dFp(1,2) -- incre in PLASTIC DEFORMATION GRADIENT, (1,2) COMP.
C
STATEV(13)= dFp(1,3) -- incre in PLASTIC DEFORMATION GRADIENT, (1,3) COMP.
C
STATEV(14)= dFp(2,1) -- incre in PLASTIC DEFORMATION GRADIENT, (2,1) COMP.
C
STATEV(15)= dFp(2,2) -- incre in PLASTIC DEFORMATION GRADIENT, (2,2) COMP.
C
STATEV(16)= dFp(2,3) -- incre in PLASTIC DEFORMATION GRADIENT, (2,3) COMP.
C
STATEV(17)= dFp(3,1) -- incre in PLASTIC DEFORMATION GRADIENT, (3,1) COMP.
C
STATEV(18)= dFp(3,2) -- incre in PLASTIC DEFORMATION GRADIENT, (3,2) COMP.
C
STATEV(19)= dFp(3,3) -- incre in PLASTIC DEFORMATION GRADIENT, (3,3) COMP.
C
C
STATEV(20)= Internal variable eta: eta=0 at virgin state of the material,
C
and change in free volume with time evolution
C
C**********************************************************************
Material Parameters in Summary: In
VUMAT
C---------------------------------------------------------------------C
MATERIAL PARAMETERS
C
C
Elastic Properties
C
EG = elastic shear modulus
C
EK = elastic bulk modulus
C
Langevin Properties (Statistical Mechanics)
C
MU_R = rubbery modulus
C
LAMBDA_L = network locking stretch
C
C
D0 = reference (initial) plastic shear-strain rate
C
m = plastic strain rate dependency (m=0; rate independent)
C
ALPHA = coefficent of pressure dependency
C
Internal Variable S coefficients (s monitors the isotropic resistance to deformation
C
H0 = initial hardening rate
C
SCV = equilibrium hardening strength
C
SO = initial resistance to flow (yield point)
C
Coefficients for ETA - free volume
C
G0 = coefficent of plastic dilantancy
C
b = coefficient for evolving eta
C
NCV = equilibrium value for free volume
C----------------------------------------------------------------------
•
•
•
•
VUMAT Program
F_t = F at start of step
F_tau = F at end of step
U_tau = U at end of step
For the first time step
–
–
–
–
–
–
–
–
•
Initialize state variables
Fp_tau = 1
Fe_tau=F_tau
Calculate Ce_tau
Calculate Ee_tau
Calculate Te_tau
Calculate T_tau
Rotate Cauchy stress to Abaqus
Stress and update Abaqus stress
variables
For other time steps
–
–
–
–
–
–
–
–
–
Get state variables from last step
Calculate Fp_tau
Normalize Fp
Calculate Fp_tau_inv
Calculate Fe_tau
Calculate Ce_tau
Calculate Ee_tau
Calculate Te_tau
Calculate pressure
– Calculate Tmendel; Mendel stress
– Calculate μ Bp_tau_dev; Back
Stress (USE LANGEVIN)
– Calculate Tflow; Flow Stress
– Calculate tau: Equivalent Shear
Stress
– IF tau is not ZERO THEN
•
•
•
•
EVOLVE DP; calculate ANUp;
EVOLVE dFp
EVOLVE S
EVOLVE eta
– IF tau is ZERO
• Do not evolve state variables
– Update Fp, F, C, U, T
– Update state variables
– Update Abaqus stresses