Keywords: structured continua, mixed finite elements, dynamics.

A wide class of theoretical and technological problems are related to the mechanical description and the practical use of bodies endowed with a numerous population of microcracks scattered throughout the volume. When microcracks are dilute in the sense that the interactions between them are not prominent, and also the microcrack distribution is (approximately) periodic, standard homogenization procedures can be profitably applied to describe the influence of the microcracks on the gross behavior. In contrast, when microcracks are dense in a way that the interactions between them are prominent, and also their distribution is not homogeneous, involving non-negligible gradients, standard homogenization procedures cannot be used in a standard way; precisely the microcracked body has to be considered strictly as a complex body and its description falls naturally within the setting of multi-field theories representing complex bodies. Here we pay attention to bodies with a dense population of microcracks scattered throughout a "soft" matrix of material.

The specific goals that are claimed to be reached in the paper are listed here along with relevant physical and numerical motivations:

• the proposal of novel variational formulations of mixed type for generalized continua with microstructure. Existing formulations from the recent literature [2] are limited to displacement-based variational principles that may not be accurate enough as to the a-posteriori computation of the macro and micro stress fields. The issue is however crucial in that micro-stresses (and self-forces) are essential for the capability of the physical model to reproduce the actual behavior of micro-cracked materials. Two continuous variational formulations of Hellinger-Reissner type are proposed that differ as to the regularity of displacement and stress fields. The "natural" finite-element discretizations are then proposed and the pros and cons of the two highlighted.
• Though the constitutive law is linear-elastic, the capability of the generalized continuum investigated in the paper to exhibit localized deformations has been shown under the hypothesis of a static regime [1]. The second goal of the paper is the extension of the model to the dynamic regime. The mixed-nature of the proposed approach ensures that the evolution of the macro and micro stress and displacement fields is determined and relevant localizations assessed by snapshots of the solutions at fixed time instants.

Among the goals for further research, is an analysis of the generalized acoustic tensor which is currently at an advanced stage so as to analytically confirm the predictions of the numerical approach.

1

P.M. Mariano, "Multifield Theories in mechanics of solids", Adv. Appl. Mech., 38, 1-93, 2001. doi:10.1016/S0065-2156(02)80102-8

2

P.M. Mariano, F.L. Stazi, "Strain localization in elastic microcracked bodies", Comp. Methods Appl. Mech. Engrg., 190, 5657-5677, 2001. doi:10.1016/S0045-7825(01)00200-6

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