Abstract : Many models in mechanics, physics and biology invoke thin structures and physical processes therein. With this minisymposium we intend to bring together mathematicians working on the modeling, mathematical analysis and numerics of such models. Topics of particular interest include variational models for mechanical thin films and rods, e.g., featuring wrinkling, prestrain, microstructure, disclocations and their numerical treatment.
Eliot Fried (Okinawa Institute of Science and Technology)
Alessandra Pluda (University of Pisa)
Paolo Piovano (Politecnico di Milano)
Diane Guignard (University of Ottawa)
David Padilla-Garza (TU Dresden)
Talks in Minisymposium :
[01343] Mesoscale modeling of systems of planar wedge disclinations and edge dislocations
Author(s) :
pierluigi cesana (kyushu university)
Abstract : Planar wedge disclinations are rotational mismatches at the level of the crystal lattice entailing a violation of rotational symmetry. Alongside dislocations, disclinations are observed in classes of Shape-Memory Alloys undergoing the austenite-to-martensite transformation and in crystal plasticity. In this talk, I will describe some recent results on the modeling of planar wedge disclinations and edge dislocations via an energy minimization principle. We model disclinations and dislocations as the solutions to minimum problems for isotropic elastic energies under the constraint of kinematic incompatibility. Our main result is the analysis of the energetic equivalence of systems of disclination dipoles and edge dislocations in the asymptotics of their singular limit regimes. The material of this talk is mainly based on a collaboration with Prof M. Morandotti & L. De Luca https://arxiv.org/abs/2207.02511
[01792] A reduced model for plates arising as low energy Gamma-limit in nonlinear magnetoelasticity
Author(s) :
Marco Bresciani (University of Erlangen)
Martin Kruzik (Czech Academy of Sciences)
Abstract : We investigate the problem of dimension reduction for plates in nonlinear magnetoelasticity. The model features a mixed Eulerian-Lagrangian formulation, as magnetizations are defined on the deformed set in the actual space.
We consider low-energy configurations by rescaling the elastic energy according to the linearized von Kármán
regime.
First, we identify a reduced model by computing the $\Gamma$-limit of the magnetoelastic energy, as the thickness of the plate goes to zero. Then, we introduce applied loads given by mechanical forces and external magnetic fields and we prove that, under clamped boundary conditions, sequences of almost minimizers of the total energy converge to minimizers of the corresponding energy in the reduced model.
Subsequently, we study quasistatic evolutions driven by time-dependent applied loads and a rate-independent dissipation. We prove that energetic solutions for the bulk model converge to energetic solutions for the reduced model and we establish a similar result for solutions of the approximate incremental minimization problem. Both these results provide a further justification of the reduced model in the spirit of the evolutionary $\Gamma$-convergence.
[01825] Variational Modeling of Stress-Driven Rearrangement Instabilities
Author(s) :
Paolo Piovano (Politecnico di Milano)
Abstract : Variational models in the context of the theory of stress-driven rearrangement instabilities are considered to describe the morphology of crystalline materials under stress due to the interaction with other adjacent materials. The existence and regularity of energy minimizers is discussed in various settings, from two to higher dimensions, and in the framework of a two-phase free-boundary problem by letting free also the contact interface with the other materials, both in its coherent and incoherent portions.
[04118] A novel dimensional reduction for the equilibrium study of inextensional material surfaces Author links open overlay panel
Author(s) :
Eliot Fried (Okinawa Institute of Science and Technology)
Yi-Caho Chen (University of Houston)
Roger Fosdick (University of Minnesota)
Abstract : A general framework is developed for finding the equations describing the equilibrium of an inextensional material surface with arbitrary flat reference shape that is deformed by applying tractions or moments to its edge. Euler--Lagrange equations are derived, leading to a complete and definitive set of equilibrium equations, which are a system of ordinary-differential equations for the spatial directrix.
[04148] Vector Fields on Flexible Curves and Surfaces
Author(s) :
Alessandra Pluda (University of Pisa)
Abstract : We consider a variant of the Helfrich functional defined on couples of flexible surfaces and vector fields defined on the surface. The energy includes a Willmore type contribution, a Frank type contribution and a coupling term between the orientation of the surfactants and the curvature of the interface.
I will discuss short term existence and stability for the associated gradient flow.
This talk is based on joint project with Georg Dolzmann and Christopher Brand.
[04480] Numerical approximation of the deformation of thin plates
Author(s) :
Andrea Bonito (Texas A&M University)
Diane Guignard (University of Ottawa)
Angelique Morvant (Texas A&M University)
Ricardo H Nochetto (University of Maryland)
Shuo Yang (Yanqi Lake Beijing Institute of Mathematical Sciences and Applications)
Abstract : We study the elastic behavior of prestrained and bilayer plates which can undergo large deformations and achieve nontrivial equilibrium shapes. The mathematical model consists of a fourth order minimization problem subject to a nonlinear and nonconvex metric constraint. We introduce a numerical method based on discontinuous Galerkin finite elements for the space discretization and a discrete gradient flow for the energy minimization. We discuss the properties of the method and present several insightful numerical experiments.
[04525] A homogenized bending theory for prestrained plates
Author(s) :
Klaus Böhnlein (TU Dresden)
Stefan Neukamm (TU Dresden)
David Padilla-Garza (TU Dresden)
Oliver Sander (TU Dresden)
Abstract : Nonlinear plate theory described the energy of an incompressible and inextensible thin elastic
sheet. In this work, we show a general rigorous derivation of a generalization of such a model for
non-euclidean plates with microheterogeneous structures. We also analyze the limiting energy in
some examples and discover interesting and counter-intuitive phenomena.
[04690] A simple numerical approach for elastic rods
Author(s) :
Patrick Dondl (Albert-Ludwig-University Freiburg)
Abstract : We derive a discrete version of the Kirchhoff elastic energy for rods undergoing bending and torsion and prove Gamma-convergence to the continuous model. This discrete energy is given by the bending and torsion energy of an interpolating conforming polynomial curve and provides a simple formula for the bending energy depending in each discrete segment only on angle and adjacent edge lengths. For the liminf-inequality, we need to introduce penalty terms to ensure arc length parametrization in the limit. For the recovery sequence a discretization with equal euclidean distance between consecutive points is constructed. Particular care is taken to treat the interaction between bending and torsion by employing a discrete version of the Bishop frame. To obtain a local description of the energy without any restrictions on a reference configuration we employ a variant of parallel transport where the singularity for antiparallel directions of current and reference configuration is removed.
