Single Crystal Gradient Plasticity Part I

Chair for Continuum Mechanics Institute of Engineering Mechanics (Prof. Böhlke) Department of Mechanical Engineering S. Wulfinghoff, T. Böhlke, E. Bayerschen Single Crystal Gradient Plasticity Part I Chair for Continuum Mechanics Institute of Engineering Mechanics 0 KIT 12.07.2012 University of the S. State Wulfinghoff, Baden-Wuerttembergand T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT 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 steels Precipitation hardening Torsion of thin wires Fleck et al. (1994) mechanical strength grain size Normalized torque Deformation B. Eghbali (2007) 1 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Motivation iam.kit.edu iam.kit.edu keramverband.de golem.de Understanding micro-plasticity Design of materials wbk.kit.edu wbk.kit.edu nanotechnologyinvesting.us palminfocenter.com Dimensioning of micro-components and micro-systems How to identify physically based continuum mechanical micro-plasticity models? 2 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

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" 3 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Dislocation Microstructure Mughrabi (1971) Mughrabi (1971) Dash (1957) Göttler (1973) Göttler (1973) weltderphysik.de Spiral source Cell structures at different deformation stages Single dislocations Important features of the microstructure Total line length/density Dislocation sources Dislocation motion/transport Lattice distorsion 4 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Literature Fundamental dislocation theory e.g. Taylor (1934); Orowan (1935); Schmid and Boas (1935); Hall (1951); Petch (1953) Kinematics and crystallographic aspects of GNDs Nye (1953); Bilby, Bullough and Smith (1955); Kröner (1958); Mura (1963); Arsenlis and Parks (1999) Thermodynamic gradient theories e.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 SSDs e.g. Becker and Miehe (2004); Evers, Brekelmans and Geers (2004); Cheong, Busso and Arsenlis (2005) Micromorphic approach e.g. Forest (2009); Cordero et al. (2010); Aslan et al. (2011) Continuum Dislocation Dynamics e.g. Hochrainer (2006); Hochrainer, Zaiser and Gumbsch (2007); Hochrainer, Zaiser and Gumbsch (2010); Sandfeld, Hochrainer and Zaiser (2010); Sandfeld (2010) 5 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Outline Single crystal kinematics at small strains Free-energy ansatz Field and boundary equations Constitutive modeling Numerical examples 6 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Kinematics of Single Crystals at Small Strains Displacement gradient Example: Homogeneous plastic shear H = H e + H p Plastic part of the displacement gradient H p = γ α d α n α l α α Additive decomposition of the strain tensor n α γ α d α dα γ α = 0 ε = sym(h) = ε e + ε p Flow rule ε p = α γ α M α M α = 1 2 (d α n α + n α d α ) 7 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Kinematics of Single Crystals at Small Strains Introduction of a pure edge dislocation (with negative sign) n α l α d α local lattice deformation Real crystal discontinuous slip Local lattice deformation not captured precisely by classical theory 8 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Kinematics of Single Crystals at Small Strains Graphical representation of the averaging process of γ α in continuum mechanics γ α + dα γ α dd α γ α n α l α d α continuum average process 9 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Kinematics of Single Crystals at Small Strains Graphical representation of the averaging process of γ α in continuum mechanics γ α + dα γ α dd α γ α n α l α d α continuum average process Dislocations of equal sign lead to non-homogeneous plastic shear dα γ α 0 Geometrically neccessary dislocations 9 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Kinematics of Single Crystals at Small Strains γ α ρ α dd α Edge and screw dislocation densities ρ α := d α γ α = γ α d α Dislocation density tensor l α γ α n α d α b α = α [( γ α d α )l α d α + ( γ α l α )d α d α ] 10 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Kinematics of Single Crystals at Small Strains γ α ρ α dd α Edge and screw dislocation densities ρ α := d α γ α = γ α d α ρ α := lα γ α = γ α l α Dislocation density tensor l α γ α n α d α b α = α [( γ α d α )l α d α + ( γ α l α )d α d α ] 10 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Free Energy Ansatz l α n α dα elastic lattice deformation 11 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Free Energy Ansatz l α n α dα elastic lattice deformation Illustrated elastic lattice deformation is averaged by classical theory and thereby not accounted for correctly 11 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Free Energy Ansatz l α n α dα elastic lattice deformation Illustrated elastic lattice deformation is averaged by classical theory and thereby not accounted for correctly Free energy W W e := 1 2 ε e C[ε e ] 11 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Free Energy Ansatz l α n α dα elastic lattice deformation Illustrated elastic lattice deformation is averaged by classical theory and thereby not accounted for correctly Free energy W W e := 1 2 ε e C[ε e ] Ansatz Free energy W = W e (ε e ) + W d (ρ α, ρα ) 11 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Free Energy Ansatz l α n α dα elastic lattice deformation Illustrated elastic lattice deformation is averaged by classical theory and thereby not accounted for correctly Free energy W W e := 1 2 ε e C[ε e ] Ansatz Free energy W = W e (ε e ) + W d (ρ α, ρα ) Dissipation D tot = α D( γ α ) = τα D γ α α 11 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Field and Boundary Equations Virtual work of external forces δw ext and internal forces δw int δw ext! = δw int δu, δγ α 12 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Field and Boundary Equations Virtual work of external forces δw ext and internal forces δw int δw ext! = δw int δu, δγ α δw int = δ = V V + [ V WdV + V τα D δγ α dv α ε e W e δε e + α τα D δγ α dv. α ( ρ α W d δρ α + ρ α W d δρ α ) ] dv 12 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Field and Boundary Equations Virtual work of external forces δw ext and internal forces δw int δw ext! = δw int δu, δγ α δw int = δ = V V + [ V WdV + V τα D δγ α dv α ε e W e δε e + α τα D δγ α dv. α ( ρ α W d δρ α + ρ α W d δρ α ) ] dv δw e = σ δε e σ := ε e W e stress δw d = ξ α δγ α ξ α := ρ α W d d α + ρ α W d l α defect stress α 12 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Field and Boundary Equations Principle of virtual displacements yields Field equations div (σ) + f = 0, τ D α = τ α ( 1)div (ξ α ) x V τ α := σ M α ( 1)div (ξ α ) resolved shear stress backstress due to GNDs 13 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Field and Boundary Equations Principle of virtual displacements yields Field equations div (σ) + f = 0, τ D α = τ α ( 1)div (ξ α ) x V τ α := σ M α ( 1)div (ξ α ) resolved shear stress backstress due to GNDs Boundary equations σn = t, ξ α n = Ξ α x V σ u = ū, γ α = γ α x V u t traction Ξ α microtraction, results from evaluation of the PovD 13 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Constitutive Modeling Strain energy (linearized theory) W e = 1 2 ε e C[ε e ] 14 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Constitutive Modeling Strain energy (linearized theory) W e = 1 2 ε e C[ε e ] 14 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Constitutive Modeling Strain energy (linearized theory) W e = 1 2 ε e C[ε e ] Dissipation D( γ α ) = γ 0 τ 0 γ α γ0 m+1 τ D α = sgn( γ α )τ 0 γ α γ0 m 14 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Constitutive Modeling Defect energy (quadratic) W d = 1 2 ρ E ρ, ρ = (ρ1, ρ2,..., ρn, ρ1, ρ2,..., ρn ) 15 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Constitutive Modeling Defect energy (quadratic) W d = 1 2 ρ E ρ, ρ = (ρ1, ρ2,..., ρn, ρ1, ρ2,..., ρn ) Leads to linear dependence of the defect stress on the dislocation densities 15 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Constitutive Modeling Defect energy (quadratic) W d = 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: W d = 1 2 S 0L 2 α [ ρ α 2 + ρ α 2] ; S 0, L = const. Gurtin et al. (2007) 15 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Constitutive Modeling Defect energy (quadratic) W d = 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: W d = 1 2 S 0L 2 α [ ρ α 2 + ρ α 2] ; S 0, L = const. Gurtin et al. (2007) Genereralized nonlinear ansatz [ W d = c n ρ α L n + ρ α L n] ; c n, n, L = const. α 15 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Numerical Results Infinite 2d-layer, two slip systems: τ symmetric slipdirections t = 1 s without gradients t = 0.1 s γ 1 16 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Numerical Results Infinite 2d-layer, two slip systems: τ symmetric slipdirections t = 1 s without gradients t = 0.1 s γ 1 Quadratic defect energy leads to smooth strain curves 16 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Numerical Results Infinite 2d-layer, two slip systems: τ symmetric slipdirections t = 1 s without gradients t = 0.1 s γ 1 Quadratic defect energy leads to smooth strain curves long-range dislocation interactions 16 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Numerical Results τ dislocation density linear theory long-range influence of dislocation pile-ups ρ 1 17 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT

Numerical Results τ dislocation density linear theory long-range influence of dislocation pile-ups ρ 1 Quadratic energy leads to a long-range influence of the dislocation pile-ups at the boundaries on the bulk-dislocations 17 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 is gratefully acknowledged. 18 12.07.2012 S. Wulfinghoff, T. Böhlke, E. Bayerschen: Continuum Dislocation Microplasticity Chair for Continuum Mechanics, KIT