Introduction to Structured Deformations: Foundations and Applications


Thursday, March 18th-29th 10:00, 2019
Turing Room at DeustoTech

José Matias & Marco Morandotti
Universidade de Lisboa(Lisboa, Portugal), Politecnico di Torino(Torino, Italy)

Abstract: The scope of these lectures is to provide the audience with an overview of the theory of Structured Deformations, including the essential theoretical tools to address problems that are relevant in mechanics and the description of some specific problems. Alongside classical theories, this new and promising theory sits at the interface between Continuum Mechanics and Calculus of Variations. It provides a multiscale geometry that captures the contributions at the macroscopic level of both smooth and non-smooth geometrical changes (disarrangements) at submacroscopic levels. In the framework of an energetic formulation, and therefore suitable to be studied with variational methods, structured deformations are a powerful tool to bridge mechanical responses at different length scales.

The lectures will include some preliminaries on the functional analytic tools needed for the variational framework. Typical energies for studying the deformations of continuum bodies are of integral type, with (possibly) non-convex energy densities. One can then relax these energies to a target structured deformation and seek for an integral representation in terms of suitable bulk and surface energy densities.

Lectures: The plan of the lectures is the following:

  • Lecture 1: Structured Deformations: definition à la Del Piero-Owen [4]; Definition à la Choksi-Fonseca [3]. Relaxation and integral presentation.
  • Lecture 2: Applications to:
    • Finding explicit formulae for relaxed energies coming from purely interfacial initial energies[1].
    • A problem of optimal design in the context of Structured Deformations MMZ[5].
    • Dimension reduction in the context of Structured Deformations CMMO[2].

    To conclude, the connection with possible numerical applications will be discussed, as well as some open problems.

References

[1] A. C. Barroso, J. Matias, M. Morandotti, and D. R. Owen: Explicit Formulas for Relaxed Energy Densities Arising from Structured Deformations. Math. Mech. Complex Syst., 5(2) (2017), 163-189.
[2] G. Carita, J. Matias, M. Morandotti, and D. R. Owen: Dimensio Reduction in the Context of Structured Deformations. J. Elas., 133 (2018), 1-35.
[3] R. Choksi and I. Fonseca: Bulk and Interfacial Energies for Structured Deformations of Continua. Arch. Rational Mech. Anal., 138 (1997), 37-103.
[4] G. Del Piero and D. R. Owen: Structured deformations of continua, Arch. Rational Mech. Anal. 124 (1993), 99-155.
[5] J. Matias, M. Morandotti, and E. Zappale: Optimal Design of Fractured Media with Prescribed Macroscopic Strain. J. Math. Anal. Appl., 449 (2017), 1094-1132.