Exponential Map Algorithm for Stress Updates in Anisotropic Multiplicative Elastoplasticity for Single Crystals

Abstract

SUMMARYThis paper presents a new stress update algorithm for large-strain rate-independent single-crystal plasticity.The theoretical frame is the well-established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. A distinct feature of the present formulation is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts, resulting in a very convenient representation of the Schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of Eulerian vectors. The key contribution of this paper is an algorithmic formulation of the exponential map exp : sI(3) -+ SL(3) for updating the special linear group SL(3) of unimodular plastic deformation maps. This update preserves exactly the plastic incompressibility condition of the anisotropic plasticity model under consideration. The resulting fully implicit stress update algorithm treats the possibly redundant constraints of singlecrystal plasticity by means of an active set search. It exploits intrinsically the simple representation of the Schmid stresses by formulating the retum algorithm and the associated consistent elastoplastic moduli in terms of Eulerian vectors updates. The performance of the proposed algorithm is demonstrated by means of a representative numerical example.

Simulation of the deformation of polycrystalline nanostructured Ti by computational homogenization

“…effect of gradients, modeling of twinning or the behavior of grain boundaries) remains still open. The basic ideas of crystal plasticity homogenization were presented in the pioneer work of Taylor [2] and its precise mathematical implementation was developed in the 1960s and 1970s by Hill [3], Rice [4] and Hill and Rice [5], Since then, the goal was to establish a general theory including a precise formulation for finite deformations [6,7] and an accurate description of the single crystal hardening evolution. Regarding the latter, the first models were phenomenological [8,9] and described the evolution of the critical resolved shear stress (CRSS) of the different slip systems as a function of the accumulated shear strain.…”

3d vs. 2d Modelling of Cracking and Plasticity in Polycrystalline Materials

“…A crystal plasticity formulation [18][19][20] is adopted to describe the constitutive behaviour of polycrystalline grains in 3D. The deformation gradient F = ∂x ∂X with Jacobian J = det F > 0 maps tangent vectors of material lines in the reference configuration B ∈ R 3 onto tangent vectors of deformed lines in the current configuration B t ∈ R 3 and is decomposed into an elastic and a plastic part.…”

Finite strain viscoplasticity with nonlinear kinematic hardening: Phenomenological modeling and time integration

“…• Exponential scheme, also referred as Euler scheme with exponential map (see, for example, [39], [29], [30], [5]). …”