Singular perturbation of an elastic energy with a singular weight. (English) Zbl 1485.49003
Summary: We study the singular perturbation of an elastic energy with a singular weight. The minimization of this energy results in a multi-scale pattern formation. We derive an energy scaling law in terms of the perturbation parameter and prove that, although one cannot expect periodicity of minimizers, the energy of a minimizer is uniformly distributed across the sample. Finally, following the approach developed by G. Alberti and S. Müller [Commun. Pure Appl. Math. 54, No. 7, 761–825 (2001; Zbl 1021.49012)] we prove that a sequence of minimizers of the perturbed energies converges to a Young measure supported on piecewise-linear periodic functions of slope \(\pm 1\) whose period depends on the location in the domain and the weights in the energy.
MSC:
| 49J05 | Existence theories for free problems in one independent variable |
| 49J45 | Methods involving semicontinuity and convergence; relaxation |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 74G65 | Energy minimization in equilibrium problems in solid mechanics |
| 35B25 | Singular perturbations in context of PDEs |
Keywords:
solid-to-solid phase transitions; elastic energy; singular perturbation; microstructure; Young measure; scaling lawCitations:
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