Constitutive relations of solid materials are usually formulated at the engineering, or macroscopic, scale. However, as the loadings become more complex, an accurate description of their response requires the introduction of more internal variables, whose physical meaning is not always clear and for which calibration of more material parameters is needed. Micromechanical approaches provide an alternative to this phenomenological formulation of macroscopic constitutive relations, based on the observation that all solid materials are heterogeneous at a small enough scale. Micromechanical models for composite materials and polycrystals fall into one of three categories:

1/ Analytical models.

2/ Full-field simulations.

3/ Reduced-order models.

First, the present study presents a theoretical comparison between two micromechanical models for composites with constituents having linear viscoelastic behavior:

1/ Analytical model [1]: a common practice in multiscale problems for heterogeneous materials with well separated scales, is to look for homogenized, or effective, constitutive relations. To derive the effective behavior of linear viscoelastic heterogeneous media, the Laplace–Carson transform is classically used: the so called correspondence principle, Mandel [2]. This functional transform allows to define a symbolic linear elastic composite in the Laplace domain. Linear homogenization models are then applied to this fictitious elastic body to derive its effective properties. The viscoelastic effective properties (for instance the effective relaxation function) are then deduced by the inversion of the Laplace–Carson transform.

2/ Reduced-order model [3]: the Nonuniform Transformation Field Analysis (NTFA) is a compromise between analytical models and full-field simulations. Analytical models, which are available only for specific microstructures, provide effective constitutive relations which can be used in macroscopic structural computations, but often fail to deliver sufficiently detailed information at small scale. At the other extreme, full-field simulations provide detailed local fields, in addition to the composite effective response, but come at a high cost when used in nested Finite Element Methods. The NTFA method is a technique of model reduction which achieves a compromise between both approaches. It is based on the observation that the transformation strains (viscous strains, eigenstrains) often exhibit specific patterns called NTFA modes. It delivers both effective constitutive relations and localization rules which allow for the reconstruction of local fields upon post-processing of macroscopic quantities.

The theoretical comparison realized in this paper underlines a huge analogy between the two approaches: the evolution laws enabling to compute the internal variables are similar (a system of differential equation of order one). Nevertheless, in the NTFA approach the internal variables solve a system of coupled differential equations and the numerical coupled term isn't negligible. Second, the present study presents the accuracy of the two models for three-phase particulate composites with constituents having linear viscoelastic behavior with or without swelling: the accuracies are assessed by comparison with full-field simulations on different tests. The predictions (results for the overall response of the composite as well as for the average response of the phases) of both models are in excellent agreement with full-field simulations for various loading conditions, monotonic as well as non proportional loading, creep and relaxation.

References:

1 J.M. Ricaud, R.Masson, Int. J. Solids Structures 46, 1599-1606 (2009)

2 J. Mandel, Un Principe de Correspondance pour les Corps viscoélastiques Linéaires Vieillissants. In: Hult, J. (Ed.), Mechanics of Visco-Elastic Media and Bodies. Springer, Berlin, 44–55, (1974)

3 J.C. Michel, P. Suquet, Int. J. Solids Structures 40, 6937-6955 (2003)