eserved@d = *@let@token Single Crystal Gradient Plasticity -- Part I · 2012-08-31 · Kinematics of Single Crystals at Small Strains 9 12.07.2012 S. Wulﬁnghoff, T. Böhlke, E.

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


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


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


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


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


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


Single crystal kinematics at small strains

Free-energy ansatz

Field and boundary equations

Constitutive modeling

Numerical examples

Kinematics of Single Crystals at Small Strains


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)



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



Graphical representation of the averaging process of γα in continuum

mechanics

γα

nα

dα

lαcontinuum average process

γα + ∂dαγα ddα



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



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]



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


nα

dαlα

elastic lattice deformation

Free Energy Ansatz


nα

dαlα


Illustrated elastic lattice deformation is averaged by classical theory

and thereby not accounted for correctly

Free Energy Ansatz


nα

dαlα



and thereby not accounted for correctly⇒ Free energy W ≥ We := 1

2εe ·C[εe]

Free Energy Ansatz


nα

dαlα




2εe ·C[εe]

Ansatz

Free energy W = We(εe) + Wd(ρα⊢, ρα

⊙)

Free Energy Ansatz


nα

dαlα




2εe ·C[εe]

Ansatz

Free energy W = We(εe) + Wd(ρα⊢, ρα

⊙)

Dissipation Dtot =∑

α

D(γα) =∑

α

τDα γα

Field and Boundary Equations


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.





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



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



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


Strain energy (linearized theory)

We = 1

2εe ·C[εe]




We = 1

2εe ·C[εe]




We = 1

2εe ·C[εe]

Dissipation

D(γα) = γ0τ0

∣

∣

∣

γα

γ0

∣

∣

∣

m+1

⇒ τDα = sgn(γα)τ0

∣

∣

∣

γα

γ0

∣

∣

∣

m



Defect energy (quadratic)

Wd = 1

2~ρ ·E ~ρ, ~ρ = (ρ1

⊢, ρ2

⊢, ..., ρN

⊢, ρ1

⊙, ρ2⊙, ..., ρN

⊙ )




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




Wd = 1

2~ρ ·E ~ρ, ~ρ = (ρ1

⊢, ρ2

⊢, ..., ρN

⊢, ρ1

⊙, ρ2⊙, ..., ρN

⊙ )


densities

Example – uncoupled quadratic ansatz:

Wd =1

2S0L

2∑

α

[

|ρα⊢|

2 + |ρα⊙|

2]

; S0, L = const.

Gurtin et al. (2007)




Wd = 1

2~ρ ·E ~ρ, ~ρ = (ρ1

⊢, ρ2

⊢, ..., ρN

⊢, ρ1

⊙, ρ2⊙, ..., ρN

⊙ )


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


Infinite 2d-layer, two slip systems:

symmetric

slip-

directions

τ

|γ1|t = 0.1 s

t = 1 s

without gradients

Numerical Results



symmetric

slip-

directions

τ

|γ1|t = 0.1 s

t = 1 s

without gradients

Quadratic defect energy

leads to smooth strain curves

Numerical Results



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


τ

|ρ1⊢|

dislocation density

linear theory

long-range

influence of

dislocation pile-ups

Numerical Results


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


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.

eserved@d = *@let@token Single Crystal Gradient Plasticity -- Part I · 2012-08-31 · Kinematics of Single Crystals at Small Strains 9 12.07.2012 S. Wulﬁnghoff, T. Böhlke, E.

Documents