Micromechanically-Based Modelling in Nonlocal Crystal Plasticty

Martin Becker ∗ andChristian Miehe

Institut fur Mechanik im Bauwesen (LS I), Universitat Stuttgart, Pfaffenwaldring 7, D-70550 Stuttgart

Within purely local descriptions inhomogeneous plastic deformations induce an incompatibility of the intermediate config-uration introduced in connection with the multiplicative split of large strain crystal plasticity. Extending the local model byinclusion of higher order gradients of the inelastic deformation restores on the one hand the compatibility of the intermedi-ate configuration and leads on the other hand to size effects in the material behavior in analogy to experimentally observedphenomena. The underlying physical interpretation is the storage of geometrically necessary dislocations (GNDs) for plasti-cally inhomogeneous deformations. We briefly address the relation between deformation incompatibilities and the storage ofGNDs. Furthermore we consider the corresponding numerical implementation, validate our approach by means of a bench-mark problem and give a short comparison to other dislocation density based as well as rather phenomenological models.

1 Introduction

A size dependence of the plastic material response is a well-known effect observed in experiments on metal plasticity. Theseinclude especially microtorsion as well as microbending experiments on thin wires, micro- or nanoindentation tests and asa further example the observation of the Hall-Petch effect. Under these circumstances developing strain gradients play adominant role. Affiliated with the resulting inhomogeneous deformation state is an incompatibilty of the intermediate config-uration in multiplicative plasticity. The incompatibility of the local inelastic deformation can be identified with a continuummeasure for so-called geometrically necessary dislocations. This viewpoint, which traces back toNYE [1953], motivatedthe formulation of various extensions to purely local theories in crystal plasticity by many authors. A direct consequence ofthe aforementioned incompatibility is to include dependencies on higher order gradients of the inelastic deformation in theconstitutive functions such as the free energy or the hardening relations.

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Fig. 1 For locally homogeneous plastic deformations (left figure) a purely local description in terms of the elastic (Fe) andplastic (Fp) deformation gradient defines a compatible intermediate configuration. For plastically inhomogeneous deforma-tions (right figure)Fe andFp alone define an incompatible intermediate configuration. By inclusion of the incompatibilitymeasures CurlT Fp and CurlT Fe into the theoretical description a compatible intermediate configuration is restored.

2 Incompatibility, storage of geometrically necessary dislocations

The incompatibility of the intermediate configuration can be measured by the closure failure of a line integral over the in-finitesimal placement incrementsdx along a so-calledBurgers circuitC in the intermediate configurationB. This closurefailure can be identified with the cumulative macrosopicBurgers vectorβ of all dislocations piercing the planeAC enclosedby the circuitC. Furthermore, with the plastic deformation mapFp, the Burgers vector can be evaluated for a referentialcircuit C and finally be recast into a surface integral representation by application of Stokes’ theorem

β =∮

Cdx =

∮

CFp · dX =

∫

ACCurlT Fp · dN dA (1)

With this final representation we can identifyAT :=CurlT Fp as a tensorial incompatibility measure which, as originallyintroduced byNYE [1953], is denoted asdislocation density tensorand gives a direct measure for the density of GNDs storedin a plastically inhomogeneously distorted crystal. Following the arguments ofASHBY [1970] andFLECK ET AL . [1994] wederive, with the assumption of seperate single slip characterized byFp

α = 1+ γαsα⊗Mα, the GND densityραG on systemα

∗ Corresponding author: e-mail:becker@mechbau.uni-stuttgart.de, Phone: +0049 (0)711 685 6326, Fax: +0049 (0)711 685 6347

PAMM · Proc. Appl. Math. Mech. 4, 213–214 (2004) / DOI 10.1002/pamm.200410088

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ραG :=

1b|AT

α | =1b

√(∇γα · sα)2 + (∇γα ·Tα)2 =

1b

√ρα

G,edge2 + ρα

G,screw2 (2)

where, from micromechanical observations, we can identifyραG,edge=1/b∇γα ·sα andρα

G,screw=1/b∇γα ·Tα as the densitiesof geometrically necessary edge/screw dislocations which have to be stored on systemα in order to accomodate a slip gradient∇γα in the slip- (sα)/tangential- (Tα = sα ×Mα) direction on that system. Hereb denotes the Burgers vector length. Notethat in this representation we do not have any GND storage for gradients∇γα ·Mα, perpendicular to the slip plane.

3 Constitutive model problem for viscoplastic flow response

For the constitutive model problem of single crystal strain gradient plasticity we assume a multiplicative split of the totaldeformation gradientF=FeFp where the evolution law forFp is Fp−1=Fp−1(−∑m

α=1 γαSα ⊗Mα). Here we determinethe rate of plastic shearingγα from the viscoplastic slip lawγα = γ0

{|τα|/ταr

}1/m sign(τα) with the resolved slip system

shear stressτα and the slip resistanceταr = cµb(

∑mβ=1 Gαβ [ρβ

SSD + ρβGND])1/2 on systemα. ρSSD denotes the density of

statistically stored andρGND the density of geometrically necessary dislocations.Gαβ characterizes an interaction matrix (e.g.FRANCIOSI [1980]) which accounts for the dependence of the obstacle strength on the relative orientations of the interactingsystems. For theρSSD-evolution we employ a saturation type evolution law (GILLIS & G ILLMAN [1965], ESSMANN &RAPP [1973]). Within the numerical treatment of strain gradient crystal plasticity we constitute the strong form of the coupledproblem by the standard stress equilibrium and the viscoplastic slip law. Then in the finite element discretization we introduceas nodal degrees of freedom the displacement fieldu and additionaly the plastic slipsγα with respect to which we linearizethe nonlinear system of coupled equations in view of an iterative solution procedure.

4 Benchmark problem: Constraint simple shear test

In order to validate our approach and to compare the results with solutions from the literature we consider the benchmarkproblem of constraint simple shear of an infinitely wide strip under plane strain conditions. Therefore a strip of varying heightH with a symmetric double slip arrangement for the constitutive model of section 3 and a discretization of 1× 50 four-nodedelements is subjected to simple shear with prescribed displacements at the top. The slip variablesγ1 andγ2 are fixed to zeroduring the whole process at the top and bottom nodes. Furthermore along the left and right strip boundary all nodal degrees offreedom are linked. Due to constraint plastic slip plastic strain gradients develop, giving rise to inhomogeneous deformations

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Fig. 2 Normalized shear profile at constant shearΓ = 0.01 (left), macroscopic Kirchhoff stress vs. applied shearΓ (right).

resulting in boundary layers with reduced slip. This leads to a strong size dependence in the material behavior with boundarylayers developing over the full strip height forH < 0.125mm and an increase in the baoundary layer-thickness with ongoingdeformation (up to a saturation thickness). The statistically stored dislocation (SSD) densities accumulate towards the middleof the strip whereas the GND densities accumulate at the walls due to the inhomogeneous deformation. The size dependenceis also displayed by the load-deflection curve which, in accordance with the experimentally observed Hall Petch relation,indicates the relationship:the smaller the stronger, though in contrast to the linear Hall Petch relation we observe a slightlynon-linear proportionality. Comparing these results furthermore with the likewise dislocation density based model ofEVERS

ET AL . [2003], the rather phenomenological approach bySVENDSEN & REESE[2003] or the discrete dislocation simulationsby SHU ET AL . [2001] reveals a good agreement in the observed phenomena.

Acknowledgements Support for this research was provided by the Deutsche Forschungsgemeinschaft (DFG) under grant SFB 404/A8.

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[2] U. Essmann and M. Rapp, Acta. Metall.21, 1305-1317 (1973).

[3] Evers et al., J. Mech. Phys. Sol., submitted (2003).

[4] N. A. Fleck et al., Acta Metall. Mater.42, 475-487 (1994).

[5] P. Franciosi et al., Acta Metall.28, 273-283 (1980).

[6] P. P. Gillis and J. J. Gilman, J. Appl. Phys.36, 3380-3386 (1965).

[7] J. F. Nye, Acta Metall.1, 153-162 (1953).

[8] S. Reese and B. Svendsen, Kluwer Series on Solid Mechanicsand Its Application108, C. Miehe, editor, 141-150, (2003).

[9] Shu et al., J. Mech. Phys. Sol.,49, 1361-1395 (2001).

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