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Injective weak solutions in second-gradient nonlinear elasticity

Timothy J. Healey, Stefan Krömer (2008)

ESAIM: Control, Optimisation and Calculus of Variations

We consider a class of second-gradient elasticity models for which the internal potential energy is taken as the sum of a convex function of the second gradient of the deformation and a general function of the gradient. However, in consonance with classical nonlinear elasticity, the latter is assumed to grow unboundedly as the determinant of the gradient approaches zero. While the existence of a minimizer is routine, the existence of weak solutions is not, and we focus our efforts on that question...

Junction of elastic plates and beams

Antonio Gaudiello, Régis Monneau, Jacqueline Mossino, François Murat, Ali Sili (2007)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the linearized elasticity system in a multidomain of 𝐑 3 . This multidomain is the union of a horizontal plate with fixed cross section and small thickness ε, and of a vertical beam with fixed height and small cross section of radius r ε . The lateral boundary of the plate and the top of the beam are assumed to be clamped. When ε and r ε tend to zero simultaneously, with r ε ≫ ε 2 , we identify the limit problem. This limit problem involves six junction conditions.

L p -approximation of Jacobians

Jan Malý (1991)

Commentationes Mathematicae Universitatis Carolinae

The paper investigates the nonlinear function spaces introduced by Giaquinta, Modica and Souček. It is shown that a function from Cart p ( Ω , 𝐑 m ) is approximated by 𝒞 1 functions strongly in 𝒜 q ( Ω , 𝐑 m ) whenever q < p . An example is shown of a function which is in cart p ( Ω , 𝐑 2 ) but not in cart p ( Ω , 𝐑 2 ) .

L p - L q -Time decay estimate for solution of the Cauchy problem for hyperbolic partial differential equations of linear thermoelasticity

Jerzy Gawinecki (1991)

Annales Polonici Mathematici

We prove the L p - L q -time decay estimates for the solution of the Cauchy problem for the hyperbolic system of partial differential equations of linear thermoelasticity. In our proof based on the matrix of fundamental solutions to the system we use Strauss-Klainerman’s approach [12], [5] to the L p - L q -time decay estimates.

Linear approximation in Continuum Mechanics

Giuseppe Grioli (1994)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Some critical remarks are made about the theory of Linear Elasticity, questioning on its validity in general. An alternative linear approximation of the exact theory is proposed.

Local Parameterization and the Asymptotic Numerical Method

H. Mottaqui, B. Braikat, N. Damil (2010)

Mathematical Modelling of Natural Phenomena

The Asymptotic Numerical Method (ANM) is a family of algorithms, based on computation of truncated vectorial series, for path following problems [2]. In this paper, we present and discuss some techniques to define local parameterization [4, 6, 7] in the ANM. We give some numerical comparisons of pseudo arc-length parameterization and local parameterization on non-linear elastic shells problems

Localization effects for eigenfunctions near to the edge of a thin domain

Serguei A. Nazarov (2002)

Mathematica Bohemica

It is proved that the first eigenfunction of the mixed boundary-value problem for the Laplacian in a thin domain Ω h is localized either at the whole lateral surface Γ h of the domain, or at a point of Γ h , while the eigenfunction decays exponentially inside Ω h . Other effects, attributed to the high-frequency range of the spectrum, are discussed for eigenfunctions of the mixed boundary-value and Neumann problems, too.

Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity

Z. Belhachmi, J.-M. Sac-Epée, S. Tahir (2009)

Mathematical Modelling of Natural Phenomena

We consider mixed and hybrid variational formulations to the linearized elasticity system in domains with cracks. Inequality type conditions are prescribed at the crack faces which results in unilateral contact problems. The variational formulations are extended to the whole domain including the cracks which yields, for each problem, a smooth domain formulation. Mixed finite element methods such as PEERS or BDM methods are designed to avoid locking for nearly incompressible materials in plane elasticity....

Material constraints in continuum mechanics

Stuart S. Antman (1981)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

Si dimostra che ci sono valide ragioni per considerare la teoria standard dei vincoli interni, nella meccanica dei continui, insufficientemente generale. In particolare, con l’unica eccezione dell’iperelasticità, l’extra-stress dovrebbe dipendere anche dai moltiplicatori di Lagrange, cioè, dallo stress che non effettua lavoro (virtuale).

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