Homogenization of elasto-(visco) plastic composites based on an incremental variational principle

Abstract

International audienceAn original approach is proposed in order to compute the homogenized response of composite materials with elasto-(visco) plastic constituents. The formulation is based on an incremental variational principle according to which the local stress-strain relation derives from a single incremental potential constructed from a free energy and a dissipation function. Both rate-dependent and rate-independent plasticity are handled within the same framework through the choice of the dissipation function. The key feature of the model is the explicit use of the elastic trial strain in order to define a Linear Comparison Composite whose mechanical response coincides with the response of the actual composite at a given time step. The hereditary character of the behavior is accounted for through internal variables. The method was successfully applied to several two-phase elasto-plastic and elasto-viscoplastic composites made of a continuous matrix reinforced by ellipsoidal inclusions. General loading conditions, including cyclic ones, were considered. The proposed method provides accurate predictions of the macroscopic response in many cases, and competes with previously proposed schemes in elasto-(visco) plasticity

An affine formulation for the self-consistent modeling of elasto-viscoplastic heterogeneous materials based on the translated field method

“…It was shown that predictions are very close to the ones obtained with the incremental variational approach developed by Lahellec and Suquet (2007b) for two-phase fibrereinforced composites. It is also thought that such translated field approaches are simpler to formulate than the recent incremental variational approaches for non-linear elastic-viscoplastic materials (Brassard et al, 2012;Suquet, 2007b, 2013) whose development within a self-consistent framework for polycrystalline materials has not been achieved yet.…”

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.…”

A combined incremental-secant mean-field homogenization scheme with per-phase residual strains for elasto-plastic composites

“…Suquet (1995) actually showed that the variational forms pioneered by Ponte Castañeda (1992) correspond to a second-order secant formulation, which was called modified secant. Recently, incremental variational formulations, which also correspond to a second-moment estimation, were proposed for visco-elastic composites by Lahellec and Suquet (2007a,b), for thermoelastic composites by Lahellec et al (2011), for elasto-(visco-)plastic composites by Brassart et al (2011Brassart et al ( , 2012, and for elastovisco-plastic composites with isotropic and kinematic hardening by Lahellec and Suquet (2013). The method proposed by Brassart et al (2011Brassart et al ( , 2012 can also be seen as a secant method based on the elastic trial strain, instead of the total strain as in the original secant formulation, which allows the case of non-monotonic and non-radial loading conditions to be simulated.…”

“…As a result, when compared to the direct numerical simulations of a representative volume element (RVE), the method remains accurate for low fiber ratios or low damage values only. In the variational approach proposed by Brassart et al (2012), the plastic strain field at the beginning of each time interval is supposed to be uniform within the RVE, threatening the accuracy of the method in some cases, as when UD composites made of elastic fibers are loaded in the longitudinal direction.…”

“…Contrarily to the incrementaltangent approach, there is no need to made isotropic the tensor used in the Eshelby term, as it is intrinsically so. In this formulation, the residual strains in the different phases, and thus the plastic residual strains, are piece-wise continuous, removing a major assumption in the variational approach proposed by Brassart et al (2012). Another advantage of the method is expected when damage induced strain softening will be considered in a future work.…”