On the dissipation associated with equilibrium shocks in finite elasticity

Equilibrium fields with discontinuous displacement gradients can occur in finite elasticity for certain materials. The presence of such equilibrium shocks affects the energy balance in the elastostatic field, and the present paper is concerned with a notion of dissipation associated with this energy balance. A dissipation inequality is proposed for three-dimensional equilibrium shocks for both compressible and incompressible materials. The consequences of this inequality are studied for weak shocks in plane strain for compressible materials and for shocks of arbitrary strength in anti-plane strain for a class of incompressible materials. A thermodynamic argument for the dissipation inequality is also given.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 between the California Institute of Technology and the Office of Naval Research.     

On the physical meaning of an uniqueness condition in finite elasticity

With the help of the concept of work done by the deformation of an elastic body an uniqueness condition in nonlinear elasticity is interpreted. A work theorem is reformulated.

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Resume A l'aide de concept de travail donné par la deformation d'un materiaux élastique on discute la signification physique d'une condition d'unicité en élasticité non linéaire. Une théorème de travail est reformulé.

     

An explicit method to determine response coefficients in finite elasticity

Some classical solutions to problems involving controllable deformations of isotropic incompressible elastic solids are recapitulated. It is shown that, when only one deformation mechanism is inhomogeneous, a parametric differentiation of the load-deformation relations yields a possibility to determine explicitly the material response coefficients from data conveniently measurable at tests. The method is illustrated for two cases; extension, inflation and torsion of a tube and inflation of a spherical shell.

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Résumé Quelques solutions classiques de problèmes en déformation pour des solides élastiques isotropes incompressibles sont rappelies. On montre que, quand un seul mécanisme est nonhomogène, une dérivation parametrique des relations force-déformation rend possible la détermination explicite des coefficients characteristiques du solide à partir de données expérimentales facilement accessibles.Deux examples illustrent la méthode: cas d'un tube soumis à une extension, une expansion et une torsion et cas d'une coque sphérique en expansion.

     

On the dissipation inequality in equilibrium shocks

This paper presents expressions for the dissipation inequality corresponding to an equilibrium shock. These expressions are independent of the orientation of the shock. A sufficient condition on the constitutive equations for positive dissipation is given in the case of anti-plane deformations.     

Energetic bounds in finite elasticity

Summary We study the conditions under which the internal work of deformation in an elastic isotropic body in finite deformations may be bounded by results obtained from a suitably defined linear infinitesimal problem. The values of the constants appearing in the principal inequalities are calculated and discussed for a certain class of extensional deformations.     

Inequalities sufficient for material stability in finite elasticity

Summary Some restrictions on the response function of isotropic elastic solids under finite strain are discussed. These restrictions follow when a strong localization of the classical Hadamard condition of infinitesimal stability is adopted as an a priori inequality under certain side conditions which are not fullfilled by all of the possible strained states.

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Sommario Vengono discusse alcune restrizioni sulla funzione di risposta dei solidi elastici isotropi in deformazioni finite. Queste restrizioni sono determinate da una versione locale della classica definizione di stabilità infinitesima di Hadamard, qui considerata come una diseguaglianzaa priori sotto certe condizioni che non sono verificate da tutte le possibili configurazioni deformate.

     

Neo-Hookean fiber-reinforced composites in finite elasticity

The response of a transversely isotropic fiber-reinforced composite made out of two incompressible neo-Hookean phases undergoing finite deformations is considered. An expression for the effective energy-density function of the composite in terms of the properties of the phases and their spatial distribution is developed. For the out-of-plane shear and extension modes this expression is based on an exact solution for the class of composite cylinder assemblages. To account for the in-plane shear mode we incorporate an exact result that was recently obtained for a special class of transversely isotropic composites. In the limit of small deformation elasticity the expression for the effective behavior agrees with the well-known Hashin-Shtrikman bounds. The predictions of the proposed constitutive model are compared with corresponding numerical simulation of a composite with a hexagonal unit cell. It is demonstrated that the proposed model accurately captures the overall response of the periodic composite under any general loading modes.     

Dual reciprocal states in finite elasticity

For certain equilibrium states of an elastic body dual states (in the sense of Hill [1] and Ogden [2]) can be constructed, roughly speaking by interchanging the roles of stress and deformation. Furthermore, for each of the original and dual states, a reciprocal state (in the sense of Adkins [3] and Shield [4]) can be found by interchanging initial and final coordinates. Although the resulting reciprocal states are not dually related, a closed chain of eight dual and reciprocal states can be constructed. In the case of plane strain simple formulae relating the strain-energy functions for all eight states can be written down.     

Uniquencess for the penny-shaped crack equilibrium problem in linear elasticity

Two uniqueness theorems for the equilibrium problem of an elastic body containing a circular crack (penny-shaped crack) are proved.

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Sommario Si dimostrano due teoremi di unicità per il problema al contorno associato all'equilibrio di un corpo elastico tridimensionale contenente una fessura circolare.

     

Inhomogeneous inverse problem in finite elasticity

For a class of materials for which the principal strain directions always coincide with the principal stress directions one can determine the stress field in an inhomogeneously deformed body from given boundary conditions and a known strain field without knowing the constitutive equations. Each point of the inhomogeneously deformed body contains information such as that derived from an individual homogeneous identification test. The practically important two-dimensional case leads to a problem of solving a linear hyperbolic system where two differential equations describe the principal stress components. The problem can be reduced to that of integration along characteristics.Under a certain globality condition the existence, uniqueness, and correctness of the solution are guaranteed in the whole test piece. It is shown that the globality condition is closely related to whether or not the test piece is isotropic and elastic. The influence of experimental error on the correctness of problem formulation is discussed.

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Abstrakte Für die Klasse der Materialen, für welche die Grundspannungsrichtungen und die Grundverformungsrichtungen gleich sind, kann man das Spannungsfeld in einem inhomogen deformierten Körper für gegebene Randbedingungen und einem bekannten Verformungsfeld ohne Kenntnis der detailierten Form der Dehnungs-Spannungsgleichung bestimmen. Jeder Punkt des inhomogen deformierten Körpers liefert dieselbe Information wie ein individueller homogener Test. Der praktisch wichtige zweidimensionale Fall führt zu einem Problem der Lösung eines linear hyperbolischen Systems von zwei Differentialgleichungen für die Grundspannungskomponenten. Das Problem kann auf die Integration entlang der Charakteristik des Systems reduziert werden.Unter einer Bedingung der Globalität das die Existent, Eindeutigkeit und Richtigkeit der Lösung in der ganzen Testprobe gewährleistet sind. Es wird gezeigt, dass diese Globalitätsbedingung damit zusammenhängt eng, ob die Testprobe isotrop und elastisch ist. Der Einfluss der experimentellen Fehler auf die Richtigkeit der Formulation des Problems wird auch analysiert.

     

Bui's path-independent integral in finite elasticity

A path-independent integral has been stated by Bui in the presence of a straight crack in a two-dimensional deformation field. Such an integral isdual to the Rice integral in the sense that it is based on the complementary stress energy density. Here we establish a boundary-independent integral in finite elasticity from which Bui's result follows as a particular case.

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Sommario Un integrale indipendente dal cammino intorno al vertice di una frattura in un campo di deformazione bi-dimensionale è stato stabilito da Bui. Tale integrale èduale all'integrale di Rice, nel senso che si basa sulla densità di energia complementare o degli sforzi. Qui si propone un integrale invariante in un continuo tridimensionale soggetto a deformazioni finite. Si mostra che il risultato di Bui segue come caseo particolare.

     

On the equilibrium and admissibility coupling in NT vaults of general shape

In the paper a approach for dealing with masonry vault analysis is outlined, based on the selection of membrane stress surfaces obeying equilibrium and masonry material admissibility at the meanwhile.Thereafter the problem is expressed in function of a suitably defined stress function allowing some simplification and the search of the solution is, then, set up by an energetic approach.In general, assuming the stress function as a generic function of the membrane surface, it is demonstrated that, under gravitational loads, the equilibrium of the vault implies its admissibility. This result is very significant since coupling of equilibrium and strength explains why, anciently, it was so easy to build up masonry vaults, by simply hypothesizing a resistant shape under the assigned loads and, at the same time, resulting also in the satisfaction of the material requirements as regards to stress admissibility.A “direct stress” approach is outlined, as well, for obtaining analytical solutions for each vault shape, and, as an example, its application to the case of the barrel vaults is reported.     

Necessary conditions at the boundary for minimizers in finite elasticity

Dedicated to Bernard Coleman on the occasion of his 60^th birthday     

Some research directions in finite elasticity theory

Summary A number of topics in finite elasticity theory which appear to lend themselves to further development were briefly discussed. These include (i) the effect of kinematic constraints which are exactly, or approximately, satisfied; (ii) the mechanics of elastic membranes; (iii) the applicability of results in finite elasticity theory to problems involving stress relaxing materials; (iv) the development of necessary and sufficient conditions for material stability of isotropic elastic materials; (v) the conditions for bifurcation solutions to exist in deformed elastic bodies.

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Zusammenfassung Es werden einige Themen der nichtlinearen Elastizitätstheorie, die sich für weitere Entwicklungen anbieten, kurz besprochen. Darunter sind (i) die Wirkung kinematischer Zwangsbedingungen, die exakt oder näherungsweise erfüllt werden; (ii) die Mechanik elastischer Membranen; (iii) die Anwendbarkeit von Ergebnissen in der nichtlinearen Elastizitätstheorie auf Probleme, die Materialien mit Spannungsrelaxation einschließen; (iv) die Entwicklung notwendiger und hinreichender Bedingungen für materielle Stabilität isotroper elastischer Materialien; (v) die Bedingungen für die Existenz von Verzweigungslösungen in deformierten elastischen Körpern.

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Constitutive equations in finite elasticity of rubbers

A constitutive model is derived for the elastic behavior of rubbers at arbitrary three-dimensional deformations with finite strains. An elastomer is thought of as an incompressible network of flexible chains bridged by permanent junctions that move affinely with the bulk material. With reference to the concept of constrained junctions, the chain ends are assumed to be located at some distances from appropriate junctions. These distances are not fixed, but are altered under deformation. An explicit expression is developed for the distribution function of vectors between junctions (an analog of the end-to-end distribution function for a flexible chain with fixed ends). An analytical formula is obtained for the strain energy density of a polymer network, when the ratio of the mean-square distance between the ends of a chain and appropriate junctions is small compared with the mean-square end-to-end distance of chains. Stress–strain relations are derived by using the laws of thermodynamics. The governing equations involve three adjustable parameters with transparent physical meaning. These parameters are found by fitting experimental data on plain and particle-reinforced elastomers. The model ensures good agreement between the observations at uniaxial tension and the results of numerical simulation, as well as an acceptable prediction of stresses at uniaxial compression, simple shear and pure shear, when its parameters are found by matching observations at uniaxial tensile tests.