Mathematics Colloquium: The repulsion property in nonlinear elasticity and a numerical scheme to circumvent it

Speaker:  Pablo Negron, University of Puerto Rico

Title:  The repulsion property in nonlinear elasticity and a numerical scheme to circumvent it

Abstract :  For problems in the calculus of variations that exhibit the Lavrentiev phenomenon, it is known that the \textit{repulsion property} holds, that is, if one approximates the global minimizer in these problems by smooth functions, then the approximate energies will blow up. Thus, standard numerical schemes, like the finite element method, may fail when applied directly to these types of problems. In this paper we prove that the repulsion property holds for variational problems in three-dimensional elasticity that exhibit cavitation. In addition, we propose a numerical scheme that circumvents the repulsion property, which is an adaptation of the Modica and Mortola functional for phase transitions in liquids, in which the phase function is coupled to the mechanical part of the stored energy functional, via the determinant of the deformation gradient. We show that the corresponding approximations by this method satisfy the lower bound $\Gamma$--convergence property in the multi-dimensional non--radial case. The convergence to the actual cavitating minimizer is established for a spherical body, in the case of radial deformations.

Time:  Friday, November 17, 3:30pm – 4:20pm

Place:  Exploratory Hall, room 4106

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