March 29, 10:00-12:00, room 4, RM0018; April 13, 20, 27, 16:00-18:00, Aula Seminari, RM004; May 4, 11, 16:00-18:00, Aula Sseminari, RM004
This proposed course would cover the material on the book [11]. Broadly speaking, first order structured deformations introduced in [8] provide a mathematical framework to capture the effects at the macroscopic level of geometrical changes at submacroscopic levels. This theory was broadened by [16] in order to allow for geometrical changes at the level of second order derivatives (second order structured deformations). The theory of structured deformations was further enriched in [10] in order to consider different levels of microstructure, that is, hierarchical structural deformations.
Starting from this mechanical formulation of the theory, and upon describing the needed mathematical framework, namely recalling some basic properties of spaces of bounded variation and spaces of bounded hessian, the variational formulation for first order structured deformations in [6] is presented as well as two different variational formulations for second order structured deformations, in [2] and in [9]. A variational formulation for hierarchical structured deformations in [4] is also presented. Different applications in this context will be discussed, namely:
(1) Dimension reduction in the context of structured deformations [13] and [7];
(2) Derivation of explicit formulae for the relaxed energy densities [3] and [15];
(3) Optimal design in the context of structured deformations [14];
(4) Homogenization in the context of structured deformations [1];
(5) Upscaling and spatial localization of non-local energies [12].
Finally, some open problems and possible generalizations of the theory will be discussed.
References:
[1] M. Amar, J. Matias, M. Morandotti, and E. Zappale: Periodic homogenization in the context of structured deformations. ZAMP, 73, 173 (2022)
[2] A. C. Barroso, J. Matias, M. Morandotti and D. R. Owen: Second-order structured deformations: relaxation, integral representation and examples. Arch. Rational Mech. Anal., 225 (2017), 1025–1072.
[3] 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),
[4] A. C. Barroso, J. Matias, M. Morandotti, D. R. Owen and E. Zappale The variational modeling of hierarchical structured deformations. Submitted to J. Elasticity (2022).
[5] R. Choksi, G. Del Piero, I. Fonseca, and D. R. Owen: Structured deformations as energy minimizers in models of fracture and hysteresis. Mathematics and Mechanics of Solids 4 (1999), 321–356.
[6] R. Choksi and I. Fonseca: Bulk and interfacial energy densities for structured deformations of continua. Arch. Rational Mech. Anal., 138 (1997), 37–103.
[7] G. Carita, J. Matias, M. Morandotti, and D. R. Owen: Dimension reduction in the context of structured deformations. J. Elast. 133 Issue 1 (2018), 1–35.
[8] G. Del Piero and D. R. Owen: Structured deformations of continua. Arch. Rational Mech. Anal., 124 (1993), 99–155.
[9] I. Fonseca, A. Hagerty, and R. Paroni: Second-order structured deformations in the space of functions of bounded hessian. J. Nonlinear Sci., 29(6) (2019), 2699–2734.
[10] L. Deseri and D. R. Owen: Elasticity with hierarchical disarrangements: a field theory that admits slips and separations at multiple submacroscopic levels. J. Elasticity, 135 (2019), 149–182.
[11] J. Matias, M. Morandotti and D. R. Owen Energetic relaxation of structured deformations. A Multiscale Geometrical Basis for Variational Problems in Continuum Mechanics. Book to be published by SpringerBriefs on PDEs and Data Science.
[12] J. Matias, M. Morandotti, D. R. Owen, and E. Zappale: Upscaling and spatial localization of non-local energies with applications to crystal plasticity, Math. Mech. Solids, 26 (2021), 963–997.
[13] J. Matias and P. M. Santos: A dimension reduction result in the framework of structured deformations. Appl. Math. Optim. 69 (2014), 459–485.
[14] J. Matias, M. Morandotti, and E. Zappale: Optimal design of fractured media with prescribed macroscopic strain. Journal of Mathematical Analysis and Applications 449 (2017), 1094–1132.
[15] M. Šilhavý: The general form of the relaxation of a purely interfacial energy for structured deformations. Math. Mech. Com- plex Syst., 5(2) (2017), 191–215.
[16] D. R. Owen and R. Paroni: Second-order structured deformations. Arch. Rational Mech. Anal. 155 (2000), 215–235.