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Analysis of microstructures and phase transition phenomena in one-dimensional, non-linear elasticity by convex optimization

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Abstract

We propose a general method for determining the theoretical microstructure in one-dimensional elastic bars whose internal deformation energy is given by nonconvex polynomials. We use nonconvex variational principles and Young measure theory to describe the optimal energetic configuration of the body. By using convex analysis and classical characterizations of algebraic moments, we can formulate the problem as a convex optimal control problem. Therefore, we can estimate the microstructure of several models by using nonlinear programming techniques. This method can determine the minimizers or the minimizing sequences of nonconvex, variational problems used in one-dimensional, nonlinear elasticity.

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  1. Departamento de Matemáticas, Universidad de los Andes, Cra 1 No. 18 A-70, Bogotá, Colombia

    René J. Meziat & Jorge Villalobos

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  1. René J. Meziat
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  2. Jorge Villalobos
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Correspondence to René J. Meziat.

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Meziat, R.J., Villalobos, J. Analysis of microstructures and phase transition phenomena in one-dimensional, non-linear elasticity by convex optimization. Struct Multidisc Optim 32, 507–519 (2006). https://doi.org/10.1007/s00158-006-0029-7

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