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Linear \(\Gamma \)-limits of multiwell energies in nonlinear elasticity theory. (English) Zbl 1160.74321
Summary: We derive linearized theories from nonlinear elasticity theory for multiwell energies. Under natural assumptions on the nonlinear stored energy densities, the properly rescaled nonlinear energy functionals are shown to \(\Gamma \)-converge to the relaxation of a corresponding linearized model. Minimizing sequences of problems with displacement boundary conditions and body forces are investigated and found to correspond to minimizing sequences of the linearized problems. As applications of our results we discuss the validity and failure of a formula that is widely used to model multiwell energies in the regime of linear elasticity. Applying our convergence results to the special case of single well densities, we also obtain a new strong convergence result for the sequence of minimizers of the nonlinear problem.

MSC:
74B20 Nonlinear elasticity
74G65 Energy minimization in equilibrium problems in solid mechanics
74B15 Equations linearized about a deformed state (small deformations superposed on large)
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