Reggiani, Dario (2024) Variational Problems with Non-Standard Growth. [Tesi di dottorato]

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Abstract

In this thesis diverse applications of the calculus of variations to continuum mechanics are analysed. Three relevant case studied are present and a common feature of the problems is the nonstandard character of the bulk energy, encoded in the growth condition or in the presence of multiple wells. Chapters two and three are devoted to the derivation, by means of Gamma convergence, of a reduced model for brittle membranes in the presence of interpenetration and incompressibility constraint, respectively. In chapter four we investigate lower semicontinuity and relaxation properties of free discontinuity functionals when the bulk part of the energy presents a non-standard growth. Such a non-standard character of the functional is typically expressed in terms of a point dependent integrability of the deformation gradient, which may be captured, in a functional setting, in terms of Orlicz type spaces. Finally, in the last chapter we derive a quantitative rigidity estimate for incompatible fields in the presence of a multi-well structure. We then use it to derive, by means of Gamma convergence, a strain gradient plasticity model in two dimensions assuming that the nonlinear elastic energy density presents multiple ground states.

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