Analysis of Microstructures and Phase Transition Phenomena in One-Dimensional, Non-Linear Elasticity by Convex Optimization

We propose a general method to determine the theoretical microstructure in one dimensional elastic bars whose internal deformation energy is given by non-convex polynomials. We use non-convex 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 non-linear programming techniques. This method can determine the minimizers or the minimizing sequences of non-convex, variational problems used in one-dimensional, non-linear elasticity.