Constitutive laws and existence questions in incompressible nonlinear elasticity

In this paper, we first give a justification of the specific form of the constitutive law of incompressible materials. The existence and uniqueness of the indeterminate pressure is established in general. In the framework of nonlinear incompressible elasticity, we then prove an existence and uniqueness result for the pure displacement boundary value problem with sufficiently small body forces.

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Resumé Dans cet article, on donne d'abord une justification de la forme que prend la loi de comportement d'un matériau incompressible. On établit l'existence et l'unicité de la pression indéterminée en général. On montre ensuite, dans le cadre de l'élasticité non linéaire incompressible, un résultat d'existence et d'unicité pour le problème aux limites en déplacement pur avec des forces volumiques suffisamment petites.

     

Cartesian currents,weak diffeomorphisms and existence theorems in nonlinear elasticity

Summary In this paper we introduce some new classes of functions, among these a class of weak diffeomorphisms. In these classes we prove by direct methods the existence of minimizers for several kinds of variational integrals. In particular, we prove the existence of one-to-one orientation-preserving maps that minimize suitable energies associated with hyperelastic materials. The minimizers are also proved to satisfy equilibrium equations. Finally radial deformations are discussed in connection with cavitation.     

Injectivity and self-contact in nonlinear elasticity

Let be a bounded open connected subset of ³ with a sufficiently smooth boundary. The additional condition det dx vol () is imposed on the admissible deformations : ¯ of a hyperelastic body whose reference configuration is ¯. We show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate. We also extend J. Ball's theorems on existence to this case by establishing the existence of a minimizer of the energy in the space W ^1,p (;³), p > 3, that is injective almost everywhere.     

A Riemannian approach to strain measures in nonlinear elasticity

The isotropic Hencky strain energy appears naturally as a distance measure of the deformation gradient to the set SO(n)$\mathrm{SO}(n)$ of rigid rotations in the canonical left-invariant Riemannian metric on the general linear group GL(n)$\mathrm{GL}(n)$. Objectivity requires the Riemannian metric to be left-GL(n)$\mathrm{GL}(n)$-invariant, isotropy requires the Riemannian metric to be right-O(n)$\mathrm{O}(n)$-invariant. The latter two conditions are only satisfied for a three-parameter family of Riemannian metrics on the tangent space of GL(n)$\mathrm{GL}(n)$. Surprisingly, the final result is basically independent of the chosen parameters.     

An optimization algorithm applied to the Morrey conjecture in nonlinear elasticity

For a long time it has been studied whether rank-one convexity and quasiconvexity give rise to different families of constitutive relations in planar nonlinear elasticity. Stated in 1952 the Morrey conjecture says that these families are different, but no example has come forward to prove it. Now we attack this problem by deriving a specialized optimization algorithm based on two ingredients: first, a recently found necessary condition for the quasiconvexity of fourth-degree polynomials that distinguishes between both classes in the three dimensional case, and secondly, upon a characterization of rank-one convex fourth-degree polynomials in terms of infinitely many constraints.After extensive computational experiments with the algorithm, we believe that in the planar case, the necessary condition mentioned above is also necessary for the rank-one convexity of fourth-degree polynomials. Hence the question remains open.     

Complex-potential method in the nonlinear theory of elasticity

For materials characterized by a linear relation between Almansi strains and Cauchy stresses, relations between stresses and complex potentials are obtained and the plane static problem of the theory of elasticity is thus reduced to a boundary-value problem for the potentials. The resulting relations are nonlinear in the potentials; they generalize well-known Kolosov's formulas of linear elasticity. A condition under which the results of the linear theory of elasticity follow from the nonlinear theory considered is established. An approximate solution of the nonlinear problem for the potentials is obtained by the small-parameter method, which reduces the problem to a sequence of linear problems of the same type, in which the zeroth approximation corresponds to the problem of linear elasticity. The method is used to obtain both exact and approximate solutions for the problem of the extension of a plate with an elliptic hole. In these solutions, the behavior of stresses on the hole contour is illustrated by graphs. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 133–143, January–February, 2000.     

Shear and necking instabilities in nonlinear elasticity

This paper examines the behavior of a class of nonlinearly elastic rods under tension. These rods can suffer longitudinal extension, transverse contraction, shear, and flexure. The deformation is governed by an eighth order, quasilinear system of ordinary differential equations. A convergent perturbation process shows that there is a very rich collection of solutions that bifurcate from the trivial affine configuration. Global results are obtained for hyperelastic rods. These solutions describe instabilities of the shear and necking type and can exhibit hysteresis.

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Résumé On considère le comportement d'une classe de poutres nonlinéairement élastiques en état de tension. Ces poutres peuvent souffrir une extension longitudinale, une contraction transversale, un cisaillement et une fiexion. La déformation est gouvernée par un système quasi-linéaire d'équations différentielles ordinaires du huitième ordre. Un procès convergeant de perturbations montre qu'il y a une collection très riche de solutions qui bifurquent de la configuration banale affine. Dans le cas hypérélastique on obtient des résultats globaux. On montre que ces solutions décrivent des instabilités du type de cisaillement et d'étranglement et peuvent exhiber une hystérésis.

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The research of the first author was supported by National Science Foundation Grant MPS 73-08587 A02. Some of the results reported here were obtained by the second author in his master's thesis.     

Stability of homogeneous deformations in nonlinear elasticity

The stability of homogeneous deformations of a compressible, elastic body of cylindrical shape is studied. Two types of boundary conditions are considered: one in which the displacements on the bases are controlled and one in which the displacements on the lateral surface are controlled. As particular examples, the stability conditions for uniaxial tension and simple shear are examined.     

Some properties of the set of fourth-order tensors,with application to elasticity

In this paper the algebraic structure of the set of fourth-order tensors is examined. Special emphasis is given to the two different tensor products which can be defined over this set, when it is regarded as a vector space and as the set of all linear transformations over a vector space, respectively.The new formalism introduced here makes it possible to write in a simple form the restrictions imposed on the elasticity tensors, in finite elasticity, by balance of angular momentum and by the principle of material frame-indifference.A reassessment of the various elasticity tensors used in the literature is presented, and some classical results are re-stated in a simple and natural way.

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Riassunto Questo articolo tratta della struttura algebrica dell' insieme dei tensori del quarto ordine. Particolare rilievo è dato alla definizione di prodotto tensoriale tra tensori del secondo ordine, che è diversa a seconda che essi siano considerati transformazioni lineari su di uno spazio vettoriale, oppure essi stessi elementi di uno spazio vettoriale.Il nuovo formalismo introdotto rende semplice la determinazione delle restrizioni imposte ai tensori elastici, in Elasticità finita, dal bilancio della quantità di moto e dal principio di indifferenza materiale.Diventa anche possibile un riordinamento dei tensori elastici piò frequentemente usati nella letteratura. Ciò permette di ritrovare, in modo semplice e naturale, alcuni risultati classici.

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Dedicated to Guido Stampacchia,in mortem     

Saint-Venant's problem and semi-inverse solutions in nonlinear elasticity

Saint-Venant's problem consists in finding elastic deformations of an infinite prismatic body taking given values for the cross-sectional resultants of force and moment. Using the center manifold approach we show that all deformations having sufficiently small bounded strains lie on a finite-dimensional manifold. In particular, the flow on this manifold is described by a set of equations having exactly the form of the classical rod equations. Moreover, the set of semi-inverse solutions can be analyzed locally.     

The existence of solutions in nonlinear elastodynamics

I.Intr0ductionIntheclassicalelasticity,thelinearconstitutivelawisassumed,wherethehyper-potentialisapositiveorsemi-positivedefinitequadraticform.Forsuchsystems,theexistence,uniquenessandrelativepr0pertiesofsolutionstostaticanddynamicproblemscanbefound[l'2'…     

Preface: Frontiers and current applications in elasticity

<正>The elasticity,viz.,the deformation recoverability without energy dissipation,is known as a basic aspect in various deformation behaviors of solid materials.Typical examples are small and large elastic deformations characteristic of hard solids(e.g.,metals)and soft solids(e.g.,elastomers),respectively.The theory of elasticity for both small and     

Hypoelastic form of equations in nonlinear elasticity theory

It is shown that the complete system of equations of elasticity theory for an isotropic medium admits a unique representation in the hypoelastic form (the tensor of the rate of change of stresses is a linear function of the tensor of strain rates with coefficients depending on the invariants of the stress tensor). It is necessary to this end that the hypothesis be satisfied on the determination of strains by stresses which are unknown. Any arbitrariness in the choice of the coefficients of the hypoelastic relation may result in the thermodynamic identity being infringed.     

On the static and dynamic equilibrium of concentrated loads in linear acoustics and linear elasticity

The displacement field in an unbounded linear elastic fluid subjected to a time-dependent point force is obtained by using integral transform techniques. Differentiation of the displacement field yields the pressure field. It is shown that the pressure on the surface of a spherical ball B _r of radius r centered at the point where the force is applied is statically equivalent in the limit as r0 to only one-third of the force. The remaining two-thirds are carried by the inertia terms. It is also shown, by an independent reasoning, that a point force cannot be carried in static equilibrium by a linear elastic fluid.The displacement field corresponding to an unbounded isotropic linear-elastic solid subjected to a time-dependent point force (the Stokes solution) is also obtained by using integral transform techniques. As is well-known, the tractions of the Stokes solution on the surface of a spherical ball B _r are statically equivalent in the limit as r0 to the force itself; consequently, the inertia terms do not contribute to the dynamic equilibrium of B _r. The contrast between the response of a fluid and that of an isotropic solid under the action of a point force is discussed.     

Characteristic cones of the equations in the nonlinear theory of elasticity

The roots of the equation for the characteristic normals for two systems of differential equations in the nonlinear theory of elasticity are investigated. The first model is constructed using a thermodynamic identity. The second is a very simple hypoelastic model (the deviator of the stress-rate tensor is proportional to the deviator of the strain-rate tensor). It is shown that the roots of the equations for the normals to the characteristics for the second model are the same as the first-order terms in the expansion of the roots of the first model with respect to the strain-tensor deviator.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 126–132, May–June, 1974.The author is grateful to S. K. Godunov for discussions.     

Extended variational principle in nonlinear theory of elasticity

From the extended energy functional, a unified variational principle and an extended variational principle are given and proved in this paper. It is about the non-conservative, abrupt and divided domains static (or kinetic) system and by using the undetermined energy functional, we can immediately deduce various variational principles.