Numerical homogenisation of an elasto-plastic model-material with large elastic strains: macroscopic yield surfaces and the Eulerian normality rule

Abstract

This article presents the details of a numerical technique for computing the macroscopic response of a material with a given micro-structure to arbitrary prescribed loading histories. The method uses classical concepts of homogenisation theory in combination with the finite element method and focusses on the computation of macroscopic yield surfaces and inelastic strain rates. It places no restrictions on the magnitude of deformation and allows arbitrary combinations of stress-or strain control including the prescription of histories of the Cauchy stress. The method is illustrated by analysing a model material, consisting of a non-linearly elastic matrix with stiff elasto-plastic inclusions, which exhibits macroscopically associative elastoplastic material behaviour with finite elastic strains. Yield surfaces and the directions of plastic flow after a prior finite simple shear deformation are computed for this material and are shown to be consistent with an additive decomposition of the Eulerian strain rate into elastic and plastic parts and a suitable formulation of the normality rule in Cauchy stress space; a novel version of the latter is derived which is valid for an arbitrary reference configuration.

Idealized vs. Realistic Microstructures: An Atomistic Simulation Case Study on γ/γ′ Microstructures

“…This is accomplished by using fictitious constraints through which the displacement of two equivalent points a and b located on opposite surfaces of the cube-shaped simulation box are coupled with the macroscopic deformation gradient (${\overline{F}}_{ij}$) as follows [29]: $u_i^a-u_i^b=a_j{\stackrel{false¯}{F}}_{ij},i,j=1\dots 3$ The components of the macroscopic deformation gradient can be imposed through the displacement of so-called control nodes, which are additional/auxiliary nodes introduced in the three principal Cartesian directions: $u_i^p={\stackrel{false¯}{F}}_{ip},i,p=1\dots 3,$ where the superscript p refers to the index of the auxiliary node under consideration. It can be shown, through the discretized weak form of the boundary value problem, that the reaction forces on the auxiliary nodes weighted by the volume of the unit cell correspond to the components of the macroscopic first Piola–Kirchhoff (PK) stress (${\overline{P}}_{ij}$) tensor [30]. Consequently, we can prescribe, component-wise, either the deformation gradient or the first PK stress tensor.…”

Determination of Mechanical Properties of Polycrystals by Using Crystal Plasticity and Numerical Homogenization Schemes

“…A first order homogenization scheme is applied in the presented study. To calculate the homogenized quantities like stress and strain tensor components, a weighted average of the considered quantity over all integration points of the unit cell for given time increments is calculated on the basis of Schmidt . Alternatively, it is also possible to calculate the homogenized quantities from the displacements and forces of the control nodes.…”

“…The displacement of these nodes is coupled with all nodes of the corresponding face. [29] It is known from literature [30,31] that linear displacement boundary conditions tend to overestimate the macroscopic quantities while traction boundary conditions lead to an underestimation if the size of the RVE is not sufficient. In this work, periodic boundary conditions were used because the convergence rate with increasing RVE size towards the effective macroscopic value is more efficient www.steel-research.de compared to linear displacement or traction boundary conditions.…”

“…These nodes are additional nodes on each positive face of the unit cell. The displacement of these nodes is coupled with all nodes of the corresponding face . It is known from literature that linear displacement boundary conditions tend to overestimate the macroscopic quantities while traction boundary conditions lead to an underestimation if the size of the RVE is not sufficient.…”

Investigation of the effect of microstructural parameters on the initial yield surface of non-isothermal composites

“…Lissenden [10] investigated initial yielding and characterized the hardening behavior of a highly anisotropic material by construct yield surfaces. By combining the homogenization theory with the finite element method, Schmidt [11] presented the details of a numerical technique for computing the macroscopic response of a material with a given microstructure to arbitrarily prescribed loading Brought to you by | New York University Bobst Library Technical Services Authenticated Download Date | 6/22/15 10:07 AM histories. In light of the aforementioned investigations, it was learned that few studies concerning the influence of different fiber arrays and fiber cross-section shapes on yield surfaces have been reported.…”