Effective theories for metric gradient flows in solid mechanics
Final Report Abstract
Nonlinear dissipative evolution equations present a variety of challenging mathematical problems ranging from existence theory, to approximation of solutions, to the effective behavior of evolutionary systems depending on a small parameter. In this context, the variational approach to gradient flows in Hilbert or general metric spaces is of overriding importance by providing efficient tools for modeling, analysis, and simulations. In this project, we have focused on gradient flows and other evolutionary PDEs to describe models in solid mechanics featuring elastic energies and viscous dissipations. In the first part, we derived existence results for nonlinear problems in thermoviscoelasticity and we have studied their relation to geometrically linear counterparts. The second part of the project was devoted to the derivation of effective theories for thin viscoelastic plates, rods, and ribbons by means of dimension reduction. The methodology was mainly based on tools from the calculus of variations including the modern theory of metric gradient flows, evolutionary Γ-convergence, and quantitative geometric rigidity estimates. Besides its applications to Materials Science, the project has contributed to the understanding of evolution equations with strain-rate dependent dissipation potentials, in particular coupled to other effects such a heat transfer.
Publications
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Derivation of a one-dimensional von Kármán theory for viscoelastic ribbons. Nonlinear Differential Equations and Applications NoDEA, 29(2).
Friedrich, Manuel & Machill, Lennart
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Nonlinear and Linearized Models in Thermoviscoelasticity. Archive for Rational Mechanics and Analysis, 247(1).
Badal, Rufat; Friedrich, Manuel & Kružík, Martin
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One-dimensional viscoelastic von Kármán theories derived from nonlinear thin-walled beams. Calculus of Variations and Partial Differential Equations, 62(7).
Friedrich, Manuel & Machill, Lennart
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Derivation of a von Kármán plate theory for thermoviscoelastic solids. Mathematical Models and Methods in Applied Sciences, 34(14), 2749-2824.
Badal, Rufat; Friedrich, Manuel & Machill, Lennart
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Derivation of effective theories for thin 3D nonlinearly elastic rods with voids. Mathematical Models and Methods in Applied Sciences, 34(04), 723-777.
Friedrich, Manuel; Kreutz, Leonard & Zemas, Konstantinos
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Geometric rigidity on Sobolev spaces with variable exponent and applications. Nonlinear Differential Equations and Applications NoDEA, 32(1).
Almi, Stefano; Caponi, Maicol; Friedrich, Manuel & Solombrino, Francesco
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Variational Models with Eulerian–Lagrangian Formulation Allowing for Material Failure. Archive for Rational Mechanics and Analysis, 249(1).
Bresciani, Marco; Friedrich, Manuel & Mora-Corral, Carlos
