Exact constructions in the (non-linear) planar theory of elasticity: From elastic crystals to nematic elastomers

Abstract

In this article we deduce necessary and sufficient conditions for the presence of `Conti-type', highly symmetric, exactly-stress free constructions in the geometrically non-linear, planar $n$-well problem, generalising results of [CKZ17]. Passing to the limit $n\to \infty$, this allows us to treat solid crystals and nematic elastomer differential inclusions simultaneously. In particular, we recover and generalise (non-linear) planar tripole star type deformations which were experimentally observed in [MA80,MA80a,KK91]. Further we discuss the corresponding geometrically linearised problem.

arXiv comment49 pages, 18 figures

References in corpus (4)

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• Time evolution of tetragonal-orthorhombic ferroelastics
• Effective behavior of nematic elastomer membranes
• Convex Integration Arising in the Modelling of Shape-Memory Alloys: Some Remarks on Rigidity, Flexibility and Some Numerical Implementations

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• On a probabilistic model for martensitic avalanches incorporating mechanical compatibility
• On the Scaling of the Cubic-to-Tetragonal Phase Transformation with Displacement Boundary Conditions