On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles

Abstract

A new method for determining the overall behavior of composite materials comprised of nonlinear inelastic constituents is presented. Upon use of an implicit timediscretization scheme, the evolution equations describing the constitutive behavior of the phases can be reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a transformation strain which is a nonuniform field (not even uniform within the phases). In this first part of the study the variational technique of Ponte Castañeda is used to approximate the nonuniform eigenstrains by piecewise uniform eigenstrains and to linearize the nonlinear thermoelastic problem. The resulting problem is amenable to simpler calculations and analytical results for appropriate microstructures can be obtained. The accuracy of the proposed scheme is assessed by comparison of the method with exact results.

On convergence properties of variational constitutive updates for elasto‐visco‐plasticity

“…We consider uniaxial tension/compression tests along direction 1 under applied strain rate |ε 11 Two sources of errors must be distinguished:…”

“…The particular mathematical structure of these incremental updates, inherited from its variational nature, has been exploited in various directions such as mesh adaption [9,10] or numerical and semi-analytical homogenization [7,[11][12][13]. Variational constitutive updates are not limited to simple J 2 elasto-visco-plasticity and extensions have been proposed to more complex plasticity models [14,15], non-linear viscoelasticity [16] or coupled thermo-mechanical boundary-value problems [17], to list only a few examples.…”

A homogenization model for porous ductile solids under cyclic loads comprising a matrix with isotropic and linear kinematic hardening

“…The effective (macroscopic) elastic and plastic behavior of the porous material will be treated independently at this level (see similar work in Aravas and Ponte Castañeda (2004) and Danas and Aravas (2012)). The independent treatment is of course an assumption, as discussed in detail by Lahellec and Suquet (2007), but as we will show later in the results section, it is a sufficiently accurate one for low to moderate porosities considered here. By decoupling elastic and plastic regions (similar to numerous earlier studies, e.g.…”

“…In other cases, however, that these errors could be large, the decoupling strategy should probably be used with caution. To amend this, recently, Lahellec and Suquet (2007) proposed an incremental variational formulation for materials with a hereditary behavior described by two potentials: a free energy and a dissipation function. This method has been introduced mainly to deal with the coupled elasto-plastic response of composites in an attempt to resolve the cyclic response of these materials (see also recent work by Brassart et al (2011)).…”

An incremental-secant mean-field homogenization method with second statistical moments for elasto-plastic composite materials

“…The variational forms pioneered by [11] is a second-order secant formulation, also called modified-secant MFH, as demonstrated in [38]. Finally, the incremental variational formulations recently proposed for visco-elastic and elasto-(visco-)plastic composite materials by [42][43][44][45][46][47] also account for the second statistical moment. This paper is organised as follows.…”