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Self-consistent estimates for the rate-dependent elastoplastic behaviour of polycrystalline materials. (English) Zbl 0976.74010
The paper deals with a new Hill-type formulation for rate-dependent elasto-plasticity. Recalling Kröner’s vs. Hill’s versions of nonlinear self-consistent model, the authors conclude that for the rate-dependent elasto-plasticity Kröner’s approach is inadequate and that Hill’s concept would be more appropriate in the rate-dependent case. Essential features of Hill’s approach to the self-consistent modelling of nonlinear heterogeneous materials are the reduction of nonlinear homogenization procedure to a quasi-linear one through an adequate linearization process, and the derivation of self-consistent concentration equations for the solution of Eshelby-type problems. An alternative linearization method at the local level is adopted as a consistent affine formulation, in the specific context of rate-dependent elasto-plasticity. At each point of the representative volume element of a multiphase material and at any time instant, the authors derive the inelastic strain rate from a potential depending on internal parameters which are described by evolution equations. According to the proposed affine approach, for sufficiently large time, the authors derive the linearized constitutive equations for the total strain rate and for the rate of internal parameters. Using the solution of these differential equations for internal variables, simpler linearized constitutive equations are derived within the framework of rate-dependent elasto-plasticity. At any time a linear viscoplastic homogenization problem, in the presence of eigenstrains, has to be solved. By use of the correspondence principle, i.e. the Laplace-Carson transform, the problem is reduced to an elastic one. The authors put into evidence the following difficulties: a) due to the affine approach, the whole nonlinear problem has an implicit nature; b) due to the viscoelastic coupling, local and global responses need to be known not only at a fixed time, but also at any previous time. The authors undertake the following solution steps: first, they discretize the whole time interval; second, they elaborate an iterative scheme to simultaneously determine the associated unknown variables (stresses and internal parameters derived from the affine model) defined at intermediate times, in such a way that the variables satisfy the concentration and constitutive equations, for some given macroscopic loading path, at any intermediate time. The predictions of the proposed theory for typical mechanical response of rate-dependent elasto-plastic face-centred cubic polycrystals are analyzed and compared with the corresponding predictions from Taylor’s and Kröner-Weng’s model.

MSC:
74C10 Small-strain, rate-dependent theories of plasticity (including theories of viscoplasticity)
74E15 Crystalline structure
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