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.
[04525] A homogenized bending theory for prestrained plates
Format : Talk at Waseda University
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.
[04480] Numerical approximation of the deformation of thin plates
Format : Talk at Waseda University
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.
[04690] Rod shaped structures in plants
Format : Talk at Waseda University
Author(s) :
Patrick Dondl (Albert-Ludwig-University Freiburg)
Abstract : During biological evolution, plants have developed a wide variety of body plans and concepts that enable them to adapt to changing environmental conditions. The trade-off between flexural and torsional rigidity is an important example of sometimes conflicting mechanical requirements, the adaptation to which can be quantified by the dimensionless twist-to-bend ratio. In this work, we derive a model for the optimization of the bending and torsional rigidities of non-homogeneous elastic rods. Using a phase field approximation of the optimization problem we compute optimal structures and relate the resulting shapes to the morphology of plant stems.
[01825] Variational Modeling of Stress-Driven Rearrangement Instabilities
Format : Talk at Waseda University
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.
[01792] A reduced model for plates arising as low energy Gamma-limit in nonlinear magnetoelasticity
Format : Talk at Waseda University
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.
[04118] A novel dimensional reduction for the equilibrium study of inextensional material surfaces Author links open overlay panel
Format : Talk at Waseda University
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.
[01343] Mesoscale modeling of systems of planar wedge disclinations and edge dislocations
Format : Talk at Waseda University
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
