Results 1  10
of
22
On Dual Configurational Forces
, 2006
"... The dual conservation laws of elasticity are systematically reexamined by using both Noether’s variational approach and Coleman–Noll–Gurtin’s thermodynamics approach. These dual conservation laws can be interpreted as the dual configurational force, and therefore they provide the dual energy–momen ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
The dual conservation laws of elasticity are systematically reexamined by using both Noether’s variational approach and Coleman–Noll–Gurtin’s thermodynamics approach. These dual conservation laws can be interpreted as the dual configurational force, and therefore they provide the dual energy–momentum tensor. Some previously unknown and yet interesting results in elasticity theory have been discovered. As an example, we note the following duality condition between the configuration force (energy–momentum tensor) P and the dual configuration force (dual energy–momentum tensor) L, P LðP: FÞ1 rðP xÞ: This and other results derived in this paper may lead to a better understanding of configurational mechanics and therefore of mechanics of defects.
Canonical elastic moduli
 JOURNAL OF ELASTICITY 19:189212 (1988)
, 1988
"... Every linear planar anisotropic elastic material is equivalent, under a linear change of coordinates, to an orthotropic material. Consequently, up to linear changes of variables, there are just two "canonical" planar elastic moduli which determine the properties of any linearly elastic mat ..."
Abstract

Cited by 6 (3 self)
 Add to MetaCart
Every linear planar anisotropic elastic material is equivalent, under a linear change of coordinates, to an orthotropic material. Consequently, up to linear changes of variables, there are just two "canonical" planar elastic moduli which determine the properties of any linearly elastic material. Extensions to threedimensional elasticity and applications are indicated.
On dual conservation laws in linear elasticity: Stress function formalism. Nonlinear Dyn
"... Abstract. Dual conservation laws of linear planar elasticity theory have been systematically studied based on stress function formalism. By employing generalized symmetry transformation or the Lie–Bäcklund transformation, a class of new dual conservation laws in planar elasticity have been discove ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
(Show Context)
Abstract. Dual conservation laws of linear planar elasticity theory have been systematically studied based on stress function formalism. By employing generalized symmetry transformation or the Lie–Bäcklund transformation, a class of new dual conservation laws in planar elasticity have been discovered based on the Noether theorem and its Bessel–Hagen generalization. The physical implications of these dual conservation laws are discussed briefly. Key words: conservation laws, elasticity, Jintegral, Lie group, Lie–Bäcklund transformation 1.
On Noether’s Theorem for the Euler–Poincaré equation on the diffeomorphism group with advected quantities
 Foundations of Computational Mathematics
, 2012
"... ..."
(Show Context)
Application of Lie transformation group methods to classical theories of plates and rods
 Int. J. Solids Structures (submitted
"... Abstract—In the present paper, a class of partial differential equations related to various plate and rod problems is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived. A general statement of the ass ..."
Abstract

Cited by 3 (1 self)
 Add to MetaCart
(Show Context)
Abstract—In the present paper, a class of partial differential equations related to various plate and rod problems is studied by Lie transformation group methods. A system of equations determining the generators of the admitted point Lie groups (symmetries) is derived. A general statement of the associated groupclassification problem is given. A simple intrinsic relation is deduced allowing to recognize easily the variational symmetries among the ”ordinary ” symmetries of a selfadjoint equation of the class examined. Explicit formulae for the conserved currents of the corresponding (via Noether’s theorem) conservation laws are suggested. Solutions of groupclassification problems are presented for subclasses of equations of the foregoing type governing stability and vibration of plates, rods and fluid conveying pipes resting on variable elastic foundations and compressed by axial forces. The obtained groupclassification results are used to derive conservation laws and groupinvariant solutions readily applicable in plate statics or rod dynamics. 1.
Solutions of Navier Equations and Their Representation Structure 1
, 810
"... Navier equations are used to describe the deformation of a homogeneous, isotropic and linear elastic medium in the absence of body forces. Mathematically, the system is a natural vector (field) O(n, R)invariant generalization of the classical Laplace equation, which physically describes the vibrati ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
Navier equations are used to describe the deformation of a homogeneous, isotropic and linear elastic medium in the absence of body forces. Mathematically, the system is a natural vector (field) O(n, R)invariant generalization of the classical Laplace equation, which physically describes the vibration of a string. In this paper, we decompose the space of polynomial solutions of Navier equations into a direct sum of irreducible O(n, R)submodules and construct an explicit basis for each irreducible summand. Moreover, we explicitly solve the initial value problems for Navier equations and their wavetype extension—Lamé equations by Fourier expansion and Xu’s method of solving flag partial differential equations. 1
Conservation laws in anisotropic elasticity I. Basic framework
 Proceedings of the Royal Society of London A
, 1993
"... A complete class of first order conservation laws for two dimensional deformations in general anisotropic elastic materials is derived. The derivations are based on Stroh's formalism for anisotropic elasticity. The general procedure proposed by P. J. Olver for the construction of conservation ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
A complete class of first order conservation laws for two dimensional deformations in general anisotropic elastic materials is derived. The derivations are based on Stroh's formalism for anisotropic elasticity. The general procedure proposed by P. J. Olver for the construction of conservation integrals is followed. It is shown that the conservation laws are intimately connected with Cauchy's theorem for complex analytic functions. Realform conservation laws that are valid for degenerate or nondegenerate materials are given. 1.
Invariant theory, equivalence problems and the calculus of variations
 In Invariant Theory and Tableaux
, 1988
"... Abstract: This paper surveys some recent connections between classical invariant theory and the calculus of variations, stemming from the mathematical theory of elasticity. Particular problems to be treated include the equivalence problem for binary forms, covariants of biforms, canonical forms for ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract: This paper surveys some recent connections between classical invariant theory and the calculus of variations, stemming from the mathematical theory of elasticity. Particular problems to be treated include the equivalence problem for binary forms, covariants of biforms, canonical forms for quadratic variational problems, and the equivalence problem for particle Lagrangians. It is shown how these problems are interrelated, and results in one have direct applications to the other. 1.
Canonical Forms and Conservation Laws in Linear Elastostatics
, 1997
"... In this paper, we shall review earlier work on canonical forms in linear elasticity, and applications to the classification of conservation laws (pathindependent integrals). ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
In this paper, we shall review earlier work on canonical forms in linear elasticity, and applications to the classification of conservation laws (pathindependent integrals).