On Classical and a Gradient Model of Finite
Viscoplasticity
Amit Acharya
Outline
ƒ Phenomenological models with plastic flow co-axial with
elastic stretch
• General Theory
• Finite Element Numerics
ƒ Crystal Plasticity (with ‘Simple’ Gradient Hardening)
• General Theory
• Finite Element Numerics
• Results
¾
¾
¾
Polycrystal size effects
Cleavage of ductile materials
Orientation dependence of Fracture
Finite Phenomenological Viscoplasticity
ƒ Constitutive Description
• Elasticity – Isotropic Hyperelasticity
• Rate Dependent
• Strain Hardening
ƒ Finite Element Implementation
•
•
•
•
Multiplicative Decomposition of Deformation Gradient
Plastic flow coaxial with elastic stretch
Rate-independent case is a trivial (one line) modification
Recovers fully-nonlinear hyperelastic response in the absence of
inelasticity
• Nonlinear Viscoelasticity is a straightforward modification
Constitutive Model
Multiplicative Decomposition
Free Energy/vol
Cauchy Stress
Flow-rule
(No plastic spin)
Rate-Dependence
F = F eF p
ψ = ψ (C e )
T = Fe
∂ψ eT
F
e
∂C
F F F
e
p
p −1
F
e −1
T dev
= γ dev
T
γ = p ( γ ,σ s ) ; σ s = T dev
2
Material Update
Kinematics
Isotropy
for any F( )
Discrete Update
Formula
Weber & Anand, 1990
A
2
; n(A ) = F( ) N (A) λ(A)
T = ∑ σ A neA ⊗ neA
A
F e F p F p −1 F e−1 =
Flow Rule
( )
C( ) = ∑ λ( ) N (A) ⊗ N (A)
A
γ
A A
A
S
n
⊗
n
∑
e
e
2σ s A
γ 
A
A
A 
p
A
th
dev
N
N
F
T
⊗
;
:
=
e-value
of
S
S
A


∑
e
e
2σ s  A

 ∆γ
A
A
A 
p
⊗
F p = exp 
N
N
F
; ∆γ = γ∆t
S
e
e  n
s ∑
 2σ A

⇒ F p =
Note: F0p = I
⇒ det ( F p ) = 1
since det [ exp ( ⋅)] = exp ( tr ( ⋅) )
Material Update
Elastic predictor
Coaxiality and
Uniqueness of
Polar Decomposition
 −∆γ
A
A
A 
p −1
′
⊗
F e = FF p −1 = F ′ exp 
S
N
N
;
F
=
FF
∑ e
e 
n
 2σ s A

 ∆γ
A
A
A 
⊗
R eU e exp 
S
N
N
∑
 = R′U ′
e
e
s
 2σ A

 ∆γ
A
A
A 
⊗
= U′
R e = R′ ; U e exp 
S
N
N
e
e 
s ∑
 2σ A

λ ′ A = λeA exp  S A

N ′ A = N eA
∆γ  

2σ s  


A = 1, 2,3
Update Equations
ε eA = ε ′ A − ∆γ
S A (ε e )
2σ
s
(S )
ε = ln ( λ )
∆γ = p ( γ n + ∆γ ,σ s ) ∆t
 ∆γ
A
A
A 
p
′
′
F p = exp 
N
N
F
S
⊗
∑

n
 2σ s A

U e = ∑ λeA N ′ A ⊗ N ′ A
A
Linearization of Virtual Work
R = ∫ T : δ L dv = ∫ τ : δ L dv0
V
V0
; δ L :=
∂δ x
∂x
dR = ∫ dτ : δ L dv0 + ∫ T : d δ L dv
V0
V
dτ = ∑ dτ A neA ⊗ neA + ∑τ A d ( neA ⊗ neA )
A
A

 N ′A ⊗ N ′A  T
A
A 2
′
′
′
= ∑ dτ n′ ⊗ n′ + F ′ ∑ d 
F
+
d
L
τ
+
τ
d
L
;
µ
=
λ
( )

A
′
µ
A

 A 
A
A
A
Linearization: Geometric Stiffness
(3 distinct e-values of Predictor Stretch)
Serrin, Cayley-Hamilton, Characteristic Equation:
(Simo, 1991)
I 3′


′
′
II
B
B
I
I
II
−
⊗
+
⊗
−
(
)
 B′

µ ′A


A
A


2τ A  + µ ′A ( B′ ⊗ m ′ + m ′ ⊗ B′ )
 N ′A ⊗ N ′A  T
F ′ ∑ d 
 : dD
F′ = ∑ A 

A
2
−1
A
A
µ′
A D′  + ( I ′µ ′ − 4 µ ′ + I ′µ ′
m′ ⊗ m′ 

 A 
1 A
3 A )
A
 ′

I
 − ( I ⊗ m′ A + m′ A ⊗ I )

 µ ′A

m ′ A := n′ A ⊗ n′ A
D′ A :=  µ ′ A − µ ′ A+1   µ ′ A − µ ′ A+ 2 
symmetric
Linearization: Material Stiffness
(3 disinct e-values of Predictor Stretch)

A
A
A
′
′
δ
D
:
n
n
d
τ
δ
D
:
⊗
=
∑
∑

A
A
∑ ( n′
B
A
⊗ n′ A ) Η BA ( n′B ⊗ n′ B )  : dD

−1
 B ∆γ  B 1 A B 1 
κ  d A d B  κσ sτ d B
C
Η :=  ΑA +
ΑA 
δ − i i − 1 +
τ τ +
s 
s  A
s ∑
3
2  ∆γ Jτ 
2τ 
2τ C



∆t ∂σ p
recall γ = p ( γ ,σ s )
κ :=
Asymmetry due
(1 − ∆t∂γ p )
s
 ∂τ A 
Α :=  B 
 ∂ε 
i := (1,1,1)
p
to Cauchy stress
in flow rule
−1
Nonlinear elasticity
(symmetric)
;
τ s = τ dev
2
; τ d A = Ath e-value of τ dev
If F in flow-rule (e.g. viscoelasticity) then additional asymmetry
Linearization:
Coalescence of e-values of Predictor Stretch
ƒ Geometric Stiffness is ‘highly’ singular in case of 1 or 2
repeated eigenvalues of the Predictor Stretch tensor
ƒ These are not exceptional deformation states (e.g. the
undeformed state)
ƒ Perturbation of distinct e-value results (as suggested in
Simo, 1990) does not work
ƒ Need separate limit expressions
Linearization:
Three repeated e-values of Predictor Stretch
ƒ Finite incremental response is completely elastic
e
• i.e. F = F ′
ƒ ‘Approximation’ (perhaps exact!)
dτ = {Β diag
 ∂τ A 
− Β off − diag } dD + Β off − diag tr ( dD ) I ; Β =  B 
 ∂ε e 
Carlson & Hoger, 1986
Linearization:
2 distinct e-values of Predictor Stretch
ƒ Material Stiffness from expression for 3 distinct evalues
case
ƒ Use results from Hill (1978) for Lagrangian spin to derive
singularity-free “limit” of geometric stiffness
• Quadratic convergence up to 10−9 (nondimensional) for residual
• Residual can be taken down to zero up to round-off
• Expression involves non-unique e-vectors and principal dyads