Homogenization of processes in nonlinear visco-elastic composites

Visintin, Augusto

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 10 (2011) no. 3, pp. 611-644.

The constitutive behaviour of a multiaxial visco-elastic material is here represented by the nonlinear relation

$$ϵ-A\left(x\right):{\int }_0^t\sigma (x,\tau )d\tau \in \alpha (\sigma ,x),$$

which generalizes the classical Maxwell model of visco-elasticity of fluid type. Here $\alpha (·,x)$ is a (possibly multivalued) maximal monotone mapping, $\sigma$ is the stress tensor, $ϵ$ is the linearized strain tensor, and $A\left(x\right)$ is a positive-definite fourth-order tensor. The above inclusion is here coupled with the quasi-static force-balance law, $-÷\sigma =\overrightarrow{f}$. Existence and uniqueness of the weak solution are proved for a boundary-value problem.

Convergence to a two-scale problem is then derived for a composite material, in which the functions $\alpha$ and $A$ periodically oscillate in space on a short length-scale. It is proved that the coarse-scale averages of stress and strain solve a single-scale homogenized problem, and that conversely any solution of this problem can be represented in that way. The homogenized constitutive relation is represented by the minimization of a time-integrated functional, and is rather different from the above constitutive law. These results are also retrieved via De Giorgi’s notion of $\Gamma$-convergence. These conclusions are at variance with the outcome of so-called analogical models, that rest on an (apparently unjustified) mean-field-type hypothesis.

Classification : 35B27, 49J40, 73E50, 74QXX

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