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In the context of finite elasticity, we propose plate models describing the spontaneous bending of nematic elastomer thin films due to variations along the thickness of the nematic order parameters. Reduced energy functionals are deduced…

Analysis of PDEs · Mathematics 2017-02-03 Virginia Agostiniani , Antonio DeSimone

We investigate finite-strain elastoplastic evolution in the nonassociative setting. The constitutive material model is formulated in variational terms and coupled with the quasistatic equilibrium system. We introduce measure-valued…

Analysis of PDEs · Mathematics 2025-05-08 Ulisse Stefanelli , Andreas Vikelis

This paper investigates the homogenization, dimension reduction, and linearization of a composite plate subjected to external loading within the framework of non-linear elasticity problem. The total elastic energy of the problem is of order…

Analysis of PDEs · Mathematics 2025-10-24 Amartya Chakrabortty , Georges Griso , Julia Orlik

In this study, we use discrete dislocation dynamics (DDD) simulation to investigate the effect of heterogeneous dislocation density on the transition between quasi-elastic deformation and plastic flow in face-centered cubic single crystals.…

Materials Science · Physics 2019-07-24 Xu Zhang , Jian Xiong , Haidong Fan , Michael Zaiser

This paper focuses on the simultaneous homogenization and dimension reduction of periodic composite plates within the framework of non-linear elasticity. The composite plate in its reference (undeformed) configuration consists of a periodic…

Analysis of PDEs · Mathematics 2026-02-03 Amartya Chakrabortty , Georges Griso , Julia Orlik

Quasistatic evolutions of critical points of time-dependent energies exhibit piecewise smooth behavior, making them useful for modeling continuum mechanics phenomena like elastic-plasticity and fracture. Traditionally, such evolutions have…

Optimization and Control · Mathematics 2026-01-09 Stefano Almi , Massimo Fornasier , Jona Klemenc , Alessandro Scagliotti

We prove a linearization result for quasistatic fracture evolution in nonlinear elasticity. As the stiffness of the material tends to infinity, we show that rescaled displacement fields and their associated crack sets converge to a solution…

Analysis of PDEs · Mathematics 2024-11-21 Manuel Friedrich , Pascal Steinke , Kerrek Stinson

We study the evolution of solidification microstructures using a phase-field model computed on an adaptive, finite element grid. We discuss the details of our algorithm and show that it greatly reduces the computational cost of solving the…

Materials Science · Physics 2009-10-31 Nikolas Provatas , Nigel Goldenfeld , Jonathan Dantzig

In this paper, the starting point of our analysis is a coupled system of linear elasticity and Stokes equation. We consider two small parameters: the thickness $h$ of the thin plate and the pore scale $\varepsilon(h)$ which depends on $h$.…

Analysis of PDEs · Mathematics 2025-01-16 Marin Bužančić , Pedro Hernandez-Llanos , Igor Velčić , Josip Žubrinić

We show that nonlinearly elastic plates of thickness $h\to 0$ with an $\varepsilon$-periodic structure such that $\varepsilon^{-2}h\to 0$ exhibit non-standard behaviour in the asymptotic two-dimensional reduction from three-dimensional…

Analysis of PDEs · Mathematics 2015-11-19 Mikhail Cherdantsev , Kirill Cherednichenko

We provide a rigorous justification of the classical linearization approach in plasticity. By taking the small-deformations limit, we prove via \Gamma-convergence for rate-independent processes that energetic solutions of the quasi-static…

Analysis of PDEs · Mathematics 2011-11-07 Alexander Mielke , Ulisse Stefanelli

We propose a model for quasistatic growth of cavities and cracks in two-dimensional nonlinear elasticity. Cavities and cracks are modeled as discrete and compact subsets of a planar domain, respectively, and deformations are defined only…

Analysis of PDEs · Mathematics 2025-07-22 Marco Bresciani , Manuel Friedrich

Elasto-plastic models are among the most successful ways to study the critical properties of the plastic yielding transition of amorphous solids. Typically these models are studied under a condition of constant transition rates from one…

Statistical Mechanics · Physics 2017-12-05 E. A. Jagla

We consider the cylindrical bending problem for an infinite plate as modelled with a family of generalized continuum models, including the micromorphic approach. The models allow to describe length scale effects in the sense that thinner…

Classical Analysis and ODEs · Mathematics 2021-03-31 Gianluca Rizzi , Geralf Hütter , Angela Madeo , Patrizio Neff

In this article, we discuss the stability of soft quasicrystalline phases in a coupled-mode Swift-Hohenberg model for three-component systems, where the characteristic length scales are governed by the positive-definite gradient terms.…

Soft Condensed Matter · Physics 2016-04-20 Kai Jiang , Jiajun Tong , Pingwen Zhang

We investigate a linear, fully coupled thermoelasticity problem for a highly heterogeneous, two-phase medium. The medium in question consists of a connected matrix with disconnected, initially periodically distributed inclusions separated…

Analysis of PDEs · Mathematics 2017-02-13 Michael Eden , Adrian Muntean

A standard elasto-plasto-dynamic model at finite strains based on the Lie-Liu-Kr\"oner multiplicative decomposition, formulated in rates, is here enhanced to cope with spatially inhomogeneous materials by using the reference (called also…

Analysis of PDEs · Mathematics 2023-04-13 Tomáš Roubíček , Giuseppe Tomassetti

We derive, via simultaneous homogenization and dimension reduction, the $\Gamma$-limit for thin elastic plates of thickness $h$ whose energy density oscillates on a scale $\eh$ such that $ \eh^2 \ll h\ll \eh$. We consider the energy scaling…

Analysis of PDEs · Mathematics 2014-10-09 Igor Velcic

We use the notion of stochastic two-scale convergence to solve the problem of stochastic homogenization of the elastic plate in the bending regime.

Analysis of PDEs · Mathematics 2016-12-13 Peter Hornung , Matthaus Pawelczyk , Igor Velcic

The quasistatic behavior of a simple 2D model of a cohesive powder under isotropic loads is investigated by Discrete Element simulations. The loose packing states, as studied in a previous paper, undergo important structural changes under…

Materials Science · Physics 2008-10-27 Francisco Gilabert , Jean-Noël Roux , Antonio Castellanos