0 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Chair for Continuum Mechanics – Institute of Engineering Mechanics
S. Wulfinghoff, T. Böhlke, E. Bayerschen
Single Crystal Gradient Plasticity – Part I
Chair for Continuum Mechanics
Institute of Engineering Mechanics (Prof. Böhlke)
Department of Mechanical Engineering
KIT – University of the State of Baden-Wuerttemberg and
National Laboratory of the Helmholtz Association www.kit.edu
Motivation
1 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Materials exhibit strong size effects, when the length scale associated
with non-uniform plastic deformation is in the order of microns.
Examples:
Fine-grained steelsPrecipitation hardening
Torsion of thin wires Fleck et al.
(1994)
Norm
aliz
ed
torq
ue
Deformation
grain size
mechanicalstrength
B. Eghbali (2007)
Motivation
2 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
iam.kit.eduiam.kit.edu
golem.de
keramverband.de
wbk.kit.edu
wbk.kit.edu nanotechnologyinvesting.us palminfocenter.com
Understanding micro-plasticity Design of materials
Dimensioning of micro-components and micro-systems
How to identify physically based continuum mechanical
micro-plasticity models?
Motivation
3 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Atomistic and DDD-simulations are limited to very small systems
Classical continuum mechanical plasticity fails at the micro-scale
Need for a Continuum Dislocation Theory to bridge the scales
DFG Research Group 1650 "Dislocation based Plasticity"
Dislocation Microstructure
4 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Cell structures at different
deformation stages
Spiral source Single dislocations
Dash (1957) Göttler (1973)Göttler (1973)
Mughrabi (1971) Mughrabi (1971)
weltderphysik.de
Important features of the microstructure
Total line length/density
Dislocation sources
Dislocation motion/transport
Lattice distorsion
Literature
5 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Fundamental dislocation theorye.g. Taylor (1934); Orowan (1935); Schmid and Boas (1935); Hall (1951); Petch (1953)
Kinematics and crystallographic aspects of GNDsNye (1953); Bilby, Bullough and Smith (1955); Kröner (1958); Mura (1963); Arsenlis and Parks (1999)
Thermodynamic gradient theoriese.g. Fleck et al. (1994); Steinmann (1996); Menzel and Steinmann (2001); Liebe and Steinmann (2001);
Reese and Svendsen (2003); Berdichevski (2006); Ekh et al. (2007); Gurtin, Anand and Lele (2007);
Fleck and Willis (2009); Bargmann et al. (2010); Miehe (2011)
Slip resistance dependent on GNDs and SSDse.g. Becker and Miehe (2004); Evers, Brekelmans and Geers (2004); Cheong, Busso and
Arsenlis (2005)
Micromorphic approache.g. Forest (2009); Cordero et al. (2010); Aslan et al. (2011)
Continuum Dislocation Dynamicse.g. Hochrainer (2006); Hochrainer, Zaiser and Gumbsch (2007); Hochrainer, Zaiser and
Gumbsch (2010); Sandfeld, Hochrainer and Zaiser (2010); Sandfeld (2010)
Outline
6 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Single crystal kinematics at small strains
Free-energy ansatz
Field and boundary equations
Constitutive modeling
Numerical examples
Kinematics of Single Crystals at Small Strains
7 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Displacement gradient
H = He + Hp
Plastic part of the displacement
gradient
Hp =∑
α
γαdα ⊗ nα
Example:
Homogeneous plastic shear
γα
nα
dα ∂dαγα = 0
lα
Additive decomposition of the strain tensor
ε = sym(H) = εe + εp
Flow rule
εp =∑
α
γαMα Mα =1
2(dα ⊗ nα + nα ⊗ dα)
Kinematics of Single Crystals at Small Strains
8 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Introduction of a pure edge dislocation (with negative sign)
nα
dα
lα
local lattice deformation
Real crystal ⇒ discontinuous slip
Local lattice deformation not captured precisely by classical theory
Kinematics of Single Crystals at Small Strains
9 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Graphical representation of the averaging process of γα in continuum
mechanics
γα
nα
dα
lαcontinuum average process
γα + ∂dαγα ddα
Kinematics of Single Crystals at Small Strains
9 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Graphical representation of the averaging process of γα in continuum
mechanics
γα
nα
dα
lαcontinuum average process
γα + ∂dαγα ddα
⇒ Dislocations of equal sign lead to non-homogeneous plastic shear
∂dαγα 6= 0
⇒ Geometrically neccessary dislocations
Kinematics of Single Crystals at Small Strains
10 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Edge and screw dislocation
densities
ρα⊢ := −∂dα
γα = −∇γα ·dαγα
nα
dα
lαb
γα−ρα⊢ ddα
Dislocation density tensor
α =∑
α
[(−∇γα ·dα)lα ⊗ dα + (∇γα · lα)dα ⊗ dα]
Kinematics of Single Crystals at Small Strains
10 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Edge and screw dislocation
densities
ρα⊢ := −∂dα
γα = −∇γα ·dα
ρα⊙ := ∂lαγα = ∇γα · lα
γα
nα
dα
lαb
γα−ρα⊢ ddα
Dislocation density tensor
α =∑
α
[(−∇γα ·dα)lα ⊗ dα + (∇γα · lα)dα ⊗ dα]
Free Energy Ansatz
11 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
nα
dαlα
elastic lattice deformation
Free Energy Ansatz
11 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
nα
dαlα
elastic lattice deformation
Illustrated elastic lattice deformation is averaged by classical theory
and thereby not accounted for correctly
Free Energy Ansatz
11 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
nα
dαlα
elastic lattice deformation
Illustrated elastic lattice deformation is averaged by classical theory
and thereby not accounted for correctly⇒ Free energy W ≥ We := 1
2εe ·C[εe]
Free Energy Ansatz
11 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
nα
dαlα
elastic lattice deformation
Illustrated elastic lattice deformation is averaged by classical theory
and thereby not accounted for correctly⇒ Free energy W ≥ We := 1
2εe ·C[εe]
Ansatz
Free energy W = We(εe) + Wd(ρα⊢, ρα
⊙)
Free Energy Ansatz
11 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
nα
dαlα
elastic lattice deformation
Illustrated elastic lattice deformation is averaged by classical theory
and thereby not accounted for correctly⇒ Free energy W ≥ We := 1
2εe ·C[εe]
Ansatz
Free energy W = We(εe) + Wd(ρα⊢, ρα
⊙)
Dissipation Dtot =∑
α
D(γα) =∑
α
τDα γα
Field and Boundary Equations
12 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Virtual work of external forces δWext and internal forces δWint
δWext!= δWint ∀δu, δγα
Field and Boundary Equations
12 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Virtual work of external forces δWext and internal forces δWint
δWext!= δWint ∀δu, δγα
δWint = δ
∫
V
WdV +
∫
V
∑
α
τDα δγαdV
=
∫
V
[
∂εeWe · δεe +
∑
α
(
∂ρα
⊢Wdδρ
α⊢ + ∂ρα
⊙Wdδρ
α⊙
)
]
dV
+
∫
V
∑
α
τDα δγαdV.
Field and Boundary Equations
12 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Virtual work of external forces δWext and internal forces δWint
δWext!= δWint ∀δu, δγα
δWint = δ
∫
V
WdV +
∫
V
∑
α
τDα δγαdV
=
∫
V
[
∂εeWe · δεe +
∑
α
(
∂ρα
⊢Wdδρ
α⊢ + ∂ρα
⊙Wdδρ
α⊙
)
]
dV
+
∫
V
∑
α
τDα δγαdV.
δWe = σ · δεe σ := ∂εeWe stress
δWd =∑
α
ξα · ∇δγα ξα := −∂ρα
⊢Wddα + ∂ρα
⊙Wdlα defect stress
Field and Boundary Equations
13 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Principle of virtual displacements yields
Field equations
div (σ) + f = 0, τDα = τα − (−1)div (ξα) ∀x ∈ V
τα := σ ·Mα resolved shear stress
(−1)div (ξα) backstress due to GNDs
Field and Boundary Equations
13 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Principle of virtual displacements yields
Field equations
div (σ) + f = 0, τDα = τα − (−1)div (ξα) ∀x ∈ V
τα := σ ·Mα resolved shear stress
(−1)div (ξα) backstress due to GNDs
Boundary equations
σn = t, ξα ·n = Ξα ∀x ∈ ∂Vσ
u = u, γα = γα ∀x ∈ ∂Vu
t traction
Ξα microtraction, results from evaluation of the PovD
Constitutive Modeling
14 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Strain energy (linearized theory)
We = 1
2εe ·C[εe]
Constitutive Modeling
14 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Strain energy (linearized theory)
We = 1
2εe ·C[εe]
Constitutive Modeling
14 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Strain energy (linearized theory)
We = 1
2εe ·C[εe]
Dissipation
D(γα) = γ0τ0
∣
∣
∣
γα
γ0
∣
∣
∣
m+1
⇒ τDα = sgn(γα)τ0
∣
∣
∣
γα
γ0
∣
∣
∣
m
Constitutive Modeling
15 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Defect energy (quadratic)
Wd = 1
2~ρ ·E ~ρ, ~ρ = (ρ1
⊢, ρ2
⊢, ..., ρN
⊢, ρ1
⊙, ρ2⊙, ..., ρN
⊙ )
Constitutive Modeling
15 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Defect energy (quadratic)
Wd = 1
2~ρ ·E ~ρ, ~ρ = (ρ1
⊢, ρ2
⊢, ..., ρN
⊢, ρ1
⊙, ρ2⊙, ..., ρN
⊙ )
Leads to linear dependence of the defect stress on the dislocation
densities
Constitutive Modeling
15 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Defect energy (quadratic)
Wd = 1
2~ρ ·E ~ρ, ~ρ = (ρ1
⊢, ρ2
⊢, ..., ρN
⊢, ρ1
⊙, ρ2⊙, ..., ρN
⊙ )
Leads to linear dependence of the defect stress on the dislocation
densities
Example – uncoupled quadratic ansatz:
Wd =1
2S0L
2∑
α
[
|ρα⊢|
2 + |ρα⊙|
2]
; S0, L = const.
Gurtin et al. (2007)
Constitutive Modeling
15 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Defect energy (quadratic)
Wd = 1
2~ρ ·E ~ρ, ~ρ = (ρ1
⊢, ρ2
⊢, ..., ρN
⊢, ρ1
⊙, ρ2⊙, ..., ρN
⊙ )
Leads to linear dependence of the defect stress on the dislocation
densities
Example – uncoupled quadratic ansatz:
Wd =1
2S0L
2∑
α
[
|ρα⊢|
2 + |ρα⊙|
2]
; S0, L = const.
Gurtin et al. (2007)
Genereralized nonlinear ansatz
Wd = cn
∑
α
[
|ρα⊢L|n + |ρα
⊙L|n]
; cn, n, L = const.
Numerical Results
16 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Infinite 2d-layer, two slip systems:
symmetric
slip-
directions
τ
|γ1|t = 0.1 s
t = 1 s
without gradients
Numerical Results
16 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Infinite 2d-layer, two slip systems:
symmetric
slip-
directions
τ
|γ1|t = 0.1 s
t = 1 s
without gradients
Quadratic defect energy
leads to smooth strain curves
Numerical Results
16 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
Infinite 2d-layer, two slip systems:
symmetric
slip-
directions
τ
|γ1|t = 0.1 s
t = 1 s
without gradients
Quadratic defect energy
leads to smooth strain curves
long-range dislocation interactions
Numerical Results
17 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
τ
|ρ1⊢|
dislocation density
linear theory
long-range
influence of
dislocation pile-ups
Numerical Results
17 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
τ
|ρ1⊢|
dislocation density
linear theory
long-range
influence of
dislocation pile-ups
Quadratic energy
leads to a long-range influence of the dislocation pile-ups at the
boundaries on the bulk-dislocations
18 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT
End of Part I
The support of the German Research Foundation (DFG) in the project
"Dislocation based Gradient Plasticity Theory" of the DFG Research
Group 1650 "Dislocation based Plasticity" under Grant BO 1466/5-1 isgratefully acknowledged.