A Introduction of Finite Deformation Crystal Plasticity Theories

A Introduction of Finite Deformation Crystal Plasticity Theories
Haiming ZHANG (章海明), Associate Prof.

hm.zhang@sjtu.edu.cn
153-1725-6001
cunfuhaiming
Institute of Forming Technology & Equipment,

School of Materials Science and Engineering
April 22, 2019
ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 1 / 27

§ Outline
1 Background
The lenth scale of crystal plasticity models
The methodologies of crystal plasticity models
2 Finite deformation theories

Kinematics
Stress tensors
Elasto-plasticity decomposition
Marcpscopic stress-strain law
3 Single crystal plasticity theory

Transition from the macroscopic stress to the microscopic driving stress
Microscopic mechanism of the plastic flow
General constitutive equations of crystal plasticity models
4 Examples of applications
Yield surfaces and anisotropy
Stress and strain partition at the grain scale

§ Background
I The VON M ISES and T RESCA yield surfaces

plane Von Mises
(Deviatoric Plane) Yield Surface
Hydrostatic
Tresca Axis
Yield Curve
Von Mises
Yield Curve Tresca
Yield Surface
The VON M ISES and T RESCA yield surfaces in principal stress coordinates
§ Background
I The VON M ISES and T RESCA yield surfaces

plane Von Mises
Hydrostatic
Tresca Axis
Yield Curve
Von Mises
Yield Curve Tresca
Yield Surface
σ2 von Mises
Re
Tresca
Re
2
Re
√3 σ1
O Re
45°
§ Background
I Crystal plasticity models: a multiscale simulation strategy
ms
macro-scale
Time scale
meso/micro-scale
µs
micro/nano-scale
µm mm
Length scale
The length scale and time scale of crystal plasticity models.

§ Background
I Crystal plasticity models: a multiscale simulation strategy
ms
macro-scale
Time scale
meso/micro-scale
Crystal plasticiy models
µs
micro/nano-scale
µm mm
Length scale
The length scale and time scale of crystal plasticity models.

§ Background
I The methodologies of crystal plasticity models

Forming force

§ Background

Forming force
Crystal plasticity simulation is

always a multi-scale problem
Microstructure
RVE, PDEs solver
Continuous dislocation dynamics

§ Background

Forming force
Crystal plasticity simulation is

always a multi-scale problem
Microstructure
RVE, PDEs solver
Continuous dislocation dynamics
> Anisotropy, strain localization, damage initiation, recrystallization nucleation, · · ·

> Earing prediction, microstructure design, tool design, component properties, · · ·
§ Finite deformation
I Deformation gradient
> An arbitrary element dX in a deformable

body.

I Deformation gradient
> An arbitrary element dX in a deformable

body. After a certain deformation, dX
becomes dx.
q
> A displacement field χ (X, t) which maps
dX to dx, i.e.,
dx := χ (X + dX, t) − χ (X, t)
p > Deformation gradient

∂x ∂xi
F = ∇χ (X, t) = ; Fij =
∂X ∂Xj

I Deformation measurements
> The G REEN strain tensor E := 12 FT F − I , measures

q
the change from dX2 to dx2 , with respect to dX2 , that is:
|dx|2 = dxT dx = (FdX)T · (FdX) = dXT · FT F · dX.
p dXT · 2Ė · dX = dxT dx − dXT dX.


q
The G REEN strain tensor is just one of myriad deformation
measurements.


q
measurements.
∂v
dXT · 2Ė · dX = dẋT dx + dxT dẋ; with dẋ = dx.
∂x

∂v ∂v
dxT FT · 2Ė · F-1 dx = dxT · ( )T + · dx.
∂x ∂x


q
measurements.
∂v
∂x

∂v ∂v
∂x ∂x
∂v
> The velocity gradient tensor L = , and the
∂x
1 T
deformation rate tensor D = 2 (L + L).


q
measurements.
∂v
∂v ∂X ∂x
> L= = ḞF-1
∂X ∂x ∂v ∂v
∂x ∂x
D = FT · Ė · F-1
∂v
> The velocity gradient tensor L = , and the
∂x
1 T
deformation rate tensor D = 2 (L + L).

I Stress measurements
> Following the N EWTON’s dynamics, a body’s motion is produced by the action of two
kinds of externally applied forces, i.e., surface forces f and body forces ρb.

t
f
> Following E ULER’s equations of motion, internal contact forces (traction t) and
moments are transmitted from point to point in the body.

t
f
f
t
> Following E ULER’s equations of motion, internal contact forces (traction t) and
moments are transmitted from point to point in the body.
> Following N EWTON’s law of motion, the net force of the internal forces (t) and the
body force ρb of any points (infinitesimal volume V ) is zero, that is:
I Z
tdS + ρbdV = 0
∂V V


> Defining a stress tensor σ(σij , i, j = 1, 2, 3), which yields:

lim σji dSj = ti dS, with dSj = nj dS, that is, σji nj = ti , or σ T n = t
dS→0

σ 23
σ 22
σ 23 σ 21
σ 22 t
σ 21
σ 23
σ 22
σ 21

dS→0
I
t
> σ is the C AUCHY stress tensor. Its divergence is divσ = lim dS
v →0 ∂v v

σ 23
σ 22
σ 23 σ 21
σ 22 t
σ 21
σ 23
σ 22
σ 21

dS→0
I
t
v →0 ∂v v
I Z
> According to the principle of divergence, tdS + ρbdV = 0 is equivalent with:
∂v v
Z Z
divσ dv + ρb dv = 0
v v

σ 23
σ 22
σ 23 σ 21
σ 22 t
σ 21
σ 23
σ 22
σ 21

dS→0
I
t
v →0 ∂v v
I Z
> According to the principle of divergence, tdS + ρbdV = 0 is equivalent with:
∂v v
Z Z
divσ dv + ρb dv = 0 ⇒ divσ + ρb = 0
v v

> It is not difficult to derive that the C AUCHY stress tensor σ is work conjugate to the
velocity gradient tensor L or the deformation rate tensor D.

> It is not difficult to derive that the C AUCHY stress tensor σ is work conjugate to the
velocity gradient tensor L or the deformation rate tensor D.
> Other frequently used stress tensors:

• The first P IOLA -K ICHHOFF stress tensor: P = det(F)σF-T , which is work
conjugate to the deformation gradient F.
• The second P IOLA -K ICHHOFF stress tensor: S = det(F)F-1 σF-T , which is work
conjugate to the G REEN strain tensor E.

I Elasto-plasticity decomposition
n
ts ∂x
F = ∂X
C
C0

n
ts ∂x
F = ∂X
C
C0
Fp Fe
> Perform the multiplicative decomposition of the deformation gradient F as: F = Fe Fp

> It yields a additive decomposition of the velocity gradient as:
L = ḞF-1 = Ḟe Fe-1 + Fe · Ḟp Fp-1 · Fe-1 = Le + Lp
> We can also get the additive decomposition of the deformation rate tensor D as:
D = De + Dp

n
ts nα0 ∂x nα
F = ∂X
sα0 sα
C
C0
Fp Fe
nα0
sα0
C
> Perform the multiplicative decomposition of the deformation gradient F as: F = Fe Fp

> It yields a additive decomposition of the velocity gradient as:
L = ḞF-1 = Ḟe Fe-1 + Fe · Ḟp Fp-1 · Fe-1 = Le + Lp
> We can also get the additive decomposition of the deformation rate tensor D as:
D = De + Dp

§ Finite deformation theory
I General constitutive equations for plastic flow

> Kinematic equations: L = ḞF-1 ; Le = Ḟe Fe-1 ; Lp = Fe · Ḟp Fp-1 · Fe-1 .
> A yield function: f (σ, β) − σy 6 0.
O O O
> H OOKE’s law (for hypo-elastic materials): σ = C : Le = C : De .
> A specific plastic flow law: Dp = λ̇Np .

§ Finite deformation theory
I General constitutive equations for plastic flow

> Kinematic equations: L = ḞF-1 ; Le = Ḟe Fe-1 ; Lp = Fe · Ḟp Fp-1 · Fe-1 .
> A yield function: f (σ, β) − σy 6 0.
• VON M ISES yield function for instance: f (σ, β) =
1
q
(σ11 − σ22 )2 − (σ22 − σ33 )2 + (σ33 − σ11 )2 + 6 σ12
2 2 2

√ + σ23 + σ31 60
2
O O O
> H OOKE’s law (for hypo-elastic materials): σ = C : Le = C : De .
O
• σ is a specific objective rate of C AUCHY stress tensor, for example, the J AUMANN
OJ
objective rate of C AUCHY stress is: σ = σ̇ + σ · skew(L) − skew(L) · σ
O
• C is the elastic modulus.
• Np represents the plastic flow direction. Following the D RUCKER’ postulate, the
plastic flow direction is the unit direction along the gradient of a plastic potential
p ∂G(σ), h ∂G(σ), h
G(σ) in the stress space, that is: N := / . For the VON
∂σ ∂σ
0
σ
M ISES flow law, for instance, G(σ) = σ̄ vM , and Np = vM .
σ̄
ZHANG, √ (SJTU)
3Haiming Multiscale Constitutive Theories 12 / 27
§ Single crystal plasticity theory
I Transition from the macroscopic stress to the microscopic stress


sα
nα
> Assuming the crystal is embedded

in a stress field σ(x, t), which
applies a traction t = σ T · nα on the
slip plane.
sα
τα t
nα
σ(x, t)


slip plane.
sα
τα t > The driving force of the plastic flow,
nα i.e., the resolved shear stress (RSS)
τ α on the slip systems, then is:
τ α = tT · sα = nαT · σsα
σ(x, t)


slip plane.
sα
τα t > The driving force of the plastic flow,
nα i.e., the resolved shear stress (RSS)
τ α on the slip systems, then is:
τ α = tT · sα = nαT · σsα
> Now we get the S CHMID equation,
which relates the macroscopic
stress tensor σ with the microscopic
stress τ α , that is:
σ(x, t) τ α = σ : (sα ⊗ nα ) = σ : Sα

I Microscopic plastic Flow
driving force τ
velocity v
h
s
w

I Microscopic shear rate to the macroscopic plastic deformation
> Supposing an arbitrary infinite-

γh driving force τ
simal element dx at the point x,
the velocity’s differential of dx due
x+dx the movement of dislocations is:
dx · n dx dx · n γh
dv = lim s = γ̇s ⊗ n · dx
x velocity v t→0 h t
h
γ
n
s
w

x+dx v+dv the movement of dislocations is:
dx · n dx
v
dx · n γh
h
> Superposing the contribution of all
the active slip systems, dv is
γ re-written
n as:
P α α
dv = γ̇ s ⊗ nα · dx
α
s
w


x+dx v+dv the movement of dislocations is:
dx · n dx
v
dx · n γh
h
> Superposing the contribution of all
the active slip systems, dv is
γ re-written
n as:
P α α
dv = γ̇ s ⊗ nα · dx
α
s
w > We get what we want, the plastic
velocity gradient tensor:
∂v
Lp =
P α α
= γ̇ s ⊗ nα .
∂x α

I The microscopic plastic flow law
x+dx v+dv
dx · n dx
v
x velocity v
h
γ
n
s
w
> The plastic velocity gradient tensor:

∂v
Lp = = γ̇ α sα ⊗ nα
∂x

γh =b driving force τ
x+dx v+dv
dx · n dx
v
x velocity v
h
γ
n
s
w

∂v
∂x
v ∆t b
> ∆γ = N
w h
x+dx v+dv
dx · n dx
v
x velocity v
h
γ
n
s
w

∂v
∂x
v ∆t b N
> ∆γ = N = bv ∆t = ρbv ∆t
w h wh
x+dx v+dv
dx · n dx
v
x velocity v
h
γ
n
s
w

∂v
∂x
v ∆t b N
> ∆γ = N = bv ∆t = ρbv ∆t⇒ γ̇ α = bρα
mv
α
w h wh

> The flow law: γ̇ α = bρα
mv
α


mv
α
> The movement of dislocations is a thermal-acticated process, the velocity is

described by an
( A RRHENIUS equation as:
q )
|τ − τbα | − τµα p
α
α α ∆F
v = l vD exp − 1− sign (τ α )
kB T τ0


mv
α

described by an
q )
α
α α ∆F
kB T τ0
• τ0 represents the short-range resistances, e.g., the P EIRES stress and the solid
solution strengthening.
• τb represents the back-stress, e.g., the resistance due to the polar dislocations in
the dislocation sub-structures.
• τµα represents the long-range resistance, e.g., the resistance due to the elastic
stresspfiled of forest dislocations. τµ can be either written physically as:
τµα ∝ Gαβρβ , or written as the function of the accumulated shear rate γ β , i.e.,
τµα = H γ β


mv
α

described by an
q )
α
α α ∆F
kB T τ0
• τ0 represents the short-range resistances, e.g., the P EIRES stress and the solid
solution strengthening.
• τb represents the back-stress, e.g., the resistance due to the polar dislocations in
the dislocation sub-structures.
• τµα represents the long-range resistance, e.g., the resistance due to the elastic
stresspfiled of forest dislocations. τµ can be either written physically as:
τµα ∝ Gαβρβ , or written as the function of the accumulated shear rate γ β , i.e.,
τµα = H γ β
> With the assumption of p = q = 1 and γ̇0 = bρα α
m l vD , for most metals deformed
under the temperature T < 0.3Tm , the shear strain rate can be simplified as:
α n
τ − τbα
γ̇ α = γ̇0 sign (τ α )
τµα + τ0
I The resistance of the movement of dislocations

> The long-range (athermal) resistance τµα
b
b
D
Forest
Dislocation dislocations Grain boundary
> The short-range (thermal activated) resistance τ0
Dislocation
Atoms
Solid solution
Peierls potention for movement atom

I General constitutive equations of crystal plasticity models

> The kinematic equations: L = ḞF-1 ; Le = Ḟe Fe-1 ; Lp = Fe · Ḟp Fp-1 · Fe-1
O O O
> The H OOKEN law: σ = C : Le == C : De .
R
> A specific hardening law: σy = H λ̇dt, T , ξi


O O O

α n
p P α α α τ − τbα
sign (τ α ) sα ⊗ nα
P
D = γ̇ s ⊗ n = γ̇0
τµα + τ0
R


O O O

α n
P
D = γ̇ s ⊗ n = γ̇0
τµα + τ0
R
τµα = Hµ
R β
γ̇ dt, T , ξi , τbα = Hb
R β
γ̇ dt, T , ξi


O O O

α n
P
D = γ̇ s ⊗ n = γ̇0
τµα + τ0
R
τµα = Hµ
R β
γ̇ dt, T , ξi , τbα = Hb
R β
γ̇ dt, T , ξi
> The resolved shear stress: τ α = σ : (sα ⊗ nα )

§ Yield surfaces and anisotropy of metals
I Two extreme of metals’ yield surfaces

The VON M ISES and T RESCA yield surfaces
plane Von Mises
Hydrostatic
Tresca Axis
Yield Curve
Von Mises
Yield Curve Tresca
Yield Surface
σ2 von Mises
Re
Tresca
Re
2
Re
√3 σ1
O Re
45°
I The S CHMID yield surfaces of single crystals

According to the S CHMID’s law, the percolative plastic yielding of crystalline
materials starts as the RSS on a potential slip system reaches its critical value
1/n
P σ : Sα n

α
τcrit , that is: |σ : Sα | = τcrit
α
, which implies: lim τα =1
n→∞ α crit
α
τ
α
τcrit /s1α n1α , αcritα
s2 n2
fcc, cube orienta- bcc, cube orienta-
tion, 12 systems tion, 48 systems
The multiplane S CHMID yield surfaces of Cube orientation of fcc and bcc materials.
I The S CHMID yield surfaces of single crystals

According to the S CHMID’s law, the percolative plastic yielding of crystalline
materials starts as the RSS on a potential slip system reaches its critical value
1/n
P σ : Sα n

α
τcrit , that is: |σ : Sα | = τcrit
α
, which implies: lim τα =1
n→∞ α crit
α
τ
α
τcrit /s1α n1α , αcritα
s2 n2
fcc, cube orienta- bcc, cube orienta-
tion, 12 systems tion, 48 systems
n=50,16,8,2 n=50,16,8,2
The multiplane S CHMID yield surfaces of Cube orientation of fcc and bcc materials.
I CP based virtual laboratory: predict the anisotropy of sheet metals and

identify the advanced yield functions
Y‘ Y (TD)
Sheet metal
Transverse direction Loading direction
X‘
z (001) x (100)
Crystal X (RD)
y (010)
Rolling direction

I CP based virtual laboratory: predict the anisotropy of sheet metals and

identify the advanced yield functions
Y‘ Y (TD)
Sheet metal
Transverse direction Loading direction
X‘
z (001) x (100)
Crystal X (RD)
y (010)
Rolling direction

I CPFEM: the prediction of earings in the deep drawing process
49 Exp. Yld2004-18p CP sim.
48
47
Earing profiles/mm
46
45
44
43
42
41
0 50 100 150 200 250 300 350
Angle from rolling direction
Earing prediction of deep drawing processes by the crystal plasticity simulation.

§ Examples of applications
I CPFEM: simulates the micro-deep drawing process
Fine grain Coarse grain
CPFEM simulations of micro-deep drawing processes in consideration of the grain size effect.

I High resolution crystal plasticity simulation: stress and strain partition at the
grain scale

I High resolution crystal plasticity simulation: stress and strain partition at the
grain scale

§ The end
What can you do?
You are encouraged to do the CPFEM simulations!

§ The end
Thank you for your attention!
Haiming Zhang
hm.zhang@sjtu.edu.cn
Institute of Forming Technology & Equipment,

School of Materials Science and Engineering,
Shanghai Jiao Tong University,
Shanghai, China

A Introduction of Finite Deformation Crystal Plasticity Theories

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A Introduction of Finite Deformation Crystal Plasticity Theories

Haiming ZHANG (章海明), Associate Prof.

Institute of Forming Technology & Equipment,

April 22, 2019

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 1 / 27

2 Finite deformation theories

3 Single crystal plasticity theory

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 2 / 27

I The VON M ISES and T RESCA yield surfaces

I The VON M ISES and T RESCA yield surfaces

I Crystal plasticity models: a multiscale simulation strategy

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 4 / 27

I Crystal plasticity models: a multiscale simulation strategy

Crystal plasticiy models

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 4 / 27

I The methodologies of crystal plasticity models

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 5 / 27

I The methodologies of crystal plasticity models

Crystal plasticity simulation is

Continuous dislocation dynamics

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 5 / 27

I The methodologies of crystal plasticity models

Crystal plasticity simulation is

Continuous dislocation dynamics

> Anisotropy, strain localization, damage initiation, recrystallization nucleation, · · ·

> An arbitrary element dX in a deformable

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 6 / 27

> An arbitrary element dX in a deformable

p > Deformation gradient

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 6 / 27

> The G REEN strain tensor E := 12 FT F − I , measures

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 7 / 27

> The G REEN strain tensor E := 12 FT F − I , measures

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 7 / 27

> The G REEN strain tensor E := 12 FT F − I , measures

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 7 / 27

> The G REEN strain tensor E := 12 FT F − I , measures

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 7 / 27

> The G REEN strain tensor E := 12 FT F − I , measures

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 7 / 27

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 8 / 27

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 8 / 27

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 8 / 27

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 9 / 27

> Defining a stress tensor σ(σij , i, j = 1, 2, 3), which yields:

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 9 / 27

> Defining a stress tensor σ(σij , i, j = 1, 2, 3), which yields:

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 9 / 27

> Defining a stress tensor σ(σij , i, j = 1, 2, 3), which yields:

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 9 / 27

> Defining a stress tensor σ(σij , i, j = 1, 2, 3), which yields:

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 9 / 27

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 10 / 27

> Other frequently used stress tensors:

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 10 / 27

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 11 / 27

> Perform the multiplicative decomposition of the deformation gradient F as: F = Fe Fp

ZHANG, Haiming (SJTU) Multiscale Constitutive Theories 11 / 27

> Perform the multiplicative decomposition of the deformation gradient F as: F = Fe Fp