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Analysis and approximation of a strain-limiting nonlinear elastic model. (English) Zbl 1327.74032

Summary: Elastic solids with strain-limiting response to external loading represent an interesting class of material models, capable of describing stress concentration at strains with small magnitude. A theoretical justification of this class of models comes naturally from implicit constitutive theory. We investigate mathematical properties of static deformations for such strain-limiting nonlinear models. Focusing on the spatially periodic setting, we obtain results concerning existence, uniqueness and regularity of weak solutions, and existence of renormalized solutions for the full range of the positive scalar parameter featuring in the model. These solutions are constructed via a Fourier spectral method. We formulate a sufficient condition for ensuring that a renormalized solution is in fact a weak solution, and we comment on the extension of the analysis to nonperiodic boundary-value problems.

MSC:

74B20 Nonlinear elasticity
74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010)
74G30 Uniqueness of solutions of equilibrium problems in solid mechanics
35Q74 PDEs in connection with mechanics of deformable solids
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