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MiS Preprint
68/2006

A $\Gamma$-convergence result for thin martensitic films in linearized elasticity

Peter Hornung

Abstract

The linearly elastic energy of a thin film $\Omega_h$ of thickness $h$ with displacement $u$ is given by the functional $E^h(u) = \int_{\Omega_h} W(\nabla u)\ dx$. We consider materials whose energy density $W$ is linearly frame indifferent and vanishes on two linearized wells which are compatible in the plane but incompatible in the thickness direction. We prove compactness of displacement sequences satisfying $E^h(u^{(h)})\leq Ch^2$ and show that the limiting two-dimensional displacements can only have two different interface directions. Our main result is the derivation of the $\Gamma$-limit of the functionals $\frac{1}{h^2} E^h$ as $h\to 0$. It is given by a weighted sum over the lengths of the interfaces.

Received:
Jul 28, 2006
Published:
Jul 28, 2006
MSC Codes:
74B05, 74M25
Keywords:
linearized elasticity, martensitic materials

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A -convergence result for thin martensitic films in linearized elasticity