Homogenization of processes in nonlinear visco-elastic composites
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 3, p. 611-644

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

ϵ-A(x):0tσ(x,τ)dτα(σ,x),

which generalizes the classical Maxwell model of visco-elasticity of fluid type. Here α(·,x) is a (possibly multivalued) maximal monotone mapping, σ is the stress tensor, ϵ is the linearized strain tensor, and A(x) is a positive-definite fourth-order tensor. The above inclusion is here coupled with the quasi-static force-balance law, -÷σ=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 α 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 Γ-convergence. These conclusions are at variance with the outcome of so-called analogical models, that rest on an (apparently unjustified) mean-field-type hypothesis.

Published online : 2018-06-21
Classification:  35B27,  49J40,  73E50,  74QXX
@article{ASNSP_2011_5_10_3_611_0,
     author = {Visintin, Augusto},
     title = {Homogenization of processes in nonlinear visco-elastic composites},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 10},
     number = {3},
     year = {2011},
     pages = {611-644},
     zbl = {1242.35033},
     mrnumber = {2905380},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2011_5_10_3_611_0}
}
Visintin, Augusto. Homogenization of processes in nonlinear visco-elastic composites. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 10 (2011) no. 3, pp. 611-644. http://www.numdam.org/item/ASNSP_2011_5_10_3_611_0/

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