On the role of the strong ellipticity condition in nonlinear elasticity
type: Item
instance of: scholarly article
title: On the role of the strong ellipticity condition in nonlinear elasticity
zbMATH Open document ID: 0531.73009
DOI: 10.1016/0020-7225(83)90133-7
published in: International Journal of Engineering Science
publication date: 1983-01-01T00:00:00Z
review text: A class of nonlinear elastic materials is defined by an inequality (I), suggested by Gårding's inequality for elliptic systems. Some integral inequalities (J) involving the strain energy function W are derived from (I), assuming strong ellipticity of the material. Under additional restrictions on W, bounds on weak solutions of the boundary value problem of place (BVP) are obtained from (J). For special non-elliptic materials, inequalities are derived which would imply Lyapunov instability of solutions of the (BVP), according to results of \textit{R. J. Knops} and \textit{B. Straughan} [Trends Appl. pure Math. Mech., Collect. pap. present. Conf. Univ. Lecce 1975, 187-206 (1976; Zbl 0352.73044)]. The question whether these results apply to any reasonably realistic class of constitutive equations is not discussed.
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74S30
zbMATH DE Number: 3843229
zbMATH Keywords: growth conditions
zbMATH Keywords: Gårding's inequality for elliptic systems
zbMATH Keywords: integral inequalities
zbMATH Keywords: strong ellipticity
zbMATH Keywords: bounds on weak solutions
zbMATH Keywords: boundary value problem of place
zbMATH Keywords: non-elliptic materials
zbMATH Keywords: Lyapunov instability of solutions
author: M. Aron
MaRDI profile type: Publication
full work available at URL: 0020-7225(83)90133-7
OpenAlex ID: W2051708453
cites work: On the thermostatics of continuous media
cites work: Q5510245
cites work: Q5518064
cites work: Static implications of the strong-ellipticity condition
cites work: Constitutive inequalities for isotropic elastic solids under finite strain
cites work: Q5630603
cites work: Q4066685
cites work: Ordinary differential equations of non-linear elasticity. II: Existence and regularity theory for conservative boundary value problems
cites work: The second law of thermodynamics and stability
cites work: Strict convexity, strong ellipticity, and regularity in the calculus of variations
cites work: Energy criteria for finite hyperelasticity
cites work: On the ellipticity of the equations of nonlinear elastostatics for a special material
cites work: On the failure of ellipticity of the equations for finite elastostatic plane strain
cites work: On the nation of hyperbolicity
cites work: On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics
cites work: The elastica and the problem of the pure bending for a non-convex stored energy function
cites work: Co-existent phases in the one-dimensional static theory of elastic bars
cites work: The equilibrium and post-buckling behavior of an elastic curve governed by a non-convex energy
cites work: Integral estimates for the displacement and strain energy in nonlinear elasticity in terms of the body force
cites work: Sufficient conditions for continuous dependence in nonlinear hyperelasticity
cites work: A continuous dependence result for a mixed boundary value problem in finite elastostatics
cites work: Q3927181
cites work: Linear elliptic differential systems and eigenvalue problems
cites work: Further study of spatial decay estimates for semilinear parabolic equations
cites work: On stability and uniqueness in finite elasticity
cites work: Energy decay in a problem of linear thermoelasticity
cites work: Nonexistence of global solutions in nonlinear Cauchy elastodynamics
Recommended article: Q4936465
Recommended article: A criterion for the strong ellipticity of the equilibrium equations of an isotropic nonlinearly elastic material
Recommended article: On strong ellipticity for implicit and strain-limiting theories of elasticity
Recommended article: Q5427566
Recommended article: Strong ellipticity of comparison solids in elastoplasticity with volumetric non-associativity
Recommended article: On the equivalence of strong ellipticity in the material and spatial settings of finite elasticity
Recommended article: On the existence of solutions of a strongly nonlinear problem of elasticity
Recommended article: On the strong ellipticity of the anisotropic linearly elastic materials
Recommended article: Nonlinear elastodynamics of materials with strong ellipticity condition: Carroll-type solutions
Recommended article: A reformulation of the strong ellipticity conditions for unconstrained hyperelastic media
description: scientific article; zbMATH DE number 3843229