The constitutive behaviour of a multiaxial visco-elastic material is here represented by the nonlinear relation
which generalizes the classical Maxwell model of visco-elasticity of fluid type. Here is a (possibly multivalued) maximal monotone mapping, is the stress tensor, is the linearized strain tensor, and is a positive-definite fourth-order tensor. The above inclusion is here coupled with the quasi-static force-balance law, . 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 and 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 -convergence. These conclusions are at variance with the outcome of so-called analogical models, that rest on an (apparently unjustified) mean-field-type hypothesis.
@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}, pages = {611--644}, publisher = {Scuola Normale Superiore, Pisa}, volume = {Ser. 5, 10}, number = {3}, year = {2011}, mrnumber = {2905380}, zbl = {1242.35033}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2011_5_10_3_611_0/} }
TY - JOUR AU - Visintin, Augusto TI - Homogenization of processes in nonlinear visco-elastic composites JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze PY - 2011 SP - 611 EP - 644 VL - 10 IS - 3 PB - Scuola Normale Superiore, Pisa UR - http://www.numdam.org/item/ASNSP_2011_5_10_3_611_0/ LA - en ID - ASNSP_2011_5_10_3_611_0 ER -
%0 Journal Article %A Visintin, Augusto %T Homogenization of processes in nonlinear visco-elastic composites %J Annali della Scuola Normale Superiore di Pisa - Classe di Scienze %D 2011 %P 611-644 %V 10 %N 3 %I Scuola Normale Superiore, Pisa %U http://www.numdam.org/item/ASNSP_2011_5_10_3_611_0/ %G en %F 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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