Virtual displacements principle,existence and uniqueness for elastic no tension bodies

Aim of this paper is the formulation of the virtual displacement principle, and some its applications, for bodies made of masonry like–material, i.e. the elastic no tension material (ENT). In place of the common adoption of continuously diffused smeared cracks, used to include both cracks and fracture strains, the paper considers the direct presence of discontinuities of the displacement fields, i.e. the cracks occurring in masonry bodies. The paper makes use of the notion of the new boundary of the body which includes the fractures, following the approach of Volpert and Hudjaev (Analysis in classes of discontinuous functions and equations of mathematical physics, Nijhoff, Dordrecht, 1985). In this framework the paper extends to the ENT bodies a previous formulation of the principle of virtual displacements, given by the Author, for the simpler case of the rigid no tension bodies. The consequent study of the existence and uniqueness of the admissible equilibrium concludes the paper.     

An existence and uniqueness theorem in linear elastodynamics

Summary Laplace transform with respect to time is applied to the boundary-initial-value problem of linear elastodynamics in order to produce a simpler elliptic boundary-value problem. Existence and uniqueness of the weak solution of the latter problem are proved.

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Sommario Si studiano le questioni di esistenza e unicità per il problema al contorno di tipo ellittico che risulta dalla trasformazione secondo Laplace rispetto al tempo del problema nei valori iniziali e al contorno della elastodinamica lineare.

     

On divergence-free stress fields and zero-energy stress functions in elasticity

Applying two identities for divergence-free non-symmetric and symmetric second-order tensors, novel type of first- and second-order stress functions are proposed for three-dimensional elasticity problems. It is shown that self-equilibrated but non-symmetric 3D stress fields can be generated by one first-order stress function vector, whereas a self-equilibrated and symmetric 3D stress field can be generated by one Airy-type second-order stress function. Assuming linearly elastic materials, the zero-energy modes of the stress functions introduced are derived and investigated. It is pointed out that the structure of the zero-energy modes of the proposed first-order stress function vector is the same as that of the rigid-body displacements in the linear theory of elasticity.     

On uniqueness, existence and objectivity of S-R decomposition theorem

I.IntroductionThedecompositionofadeformationgradientfunctionFisafundamentalproblemforfinitedeformationtheory.Greenstraintheoryandpolardecompositiontheorywereemployedbymostscientists,However,thesetheorieshaveallsomeconsiderabledrawbacks.Inl979,ChenZhidadev…     

On uniqueness for the traction problem in finite elasticity

In many problems of interest the (Cauchy) surface traction is given as a function of position on the deformed surface. A class of loadings sufficiently general to include these problems is considered, and within the context of the fraction problem in finite elasticity, a number of uniqueness results are established. This work extends results obtained for the mixed problem by Gurtin and Spector.

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Resumé Dans plusieurs proble\`mes interessants la traction surfacique de Cauchy est donnée comme une fonction de la position dans la surface deformée. Une telle classe des charges, suffisamment géerale, est considerée et un nombre des résultats d'unicité est établit dans le cadre du probleme de traction de l'elasticité non linéaire. Cet ouure prolonge des résultats pour le proble\`me mixte obtenus par Gurtin et Spector.

     

On uniqueness in finite elasticity with general loading

In many problems of interest the (Cauchy) surface traction is given as a function of position on the deformed surface. A class of loadings sufficiently general to include these problems is considered and within the context of finite elasticity a number of uniqueness results are established. A key ingredient is the result of Gurtin and Spector that uniqueness holds in any convex, stable set of deformations.

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Résumé Dans plusieurs problémes interessants la traction surfacique de Cauchy est donnée comme une fonction de la position dans la surface deformée. Une telle classe des charges, suffisamment générale, est considerée et un nombre des résultats d'unicité et établit dans le cadre d'élasticité non linéaire. Le résultat de Gurtin et Spector (unicité est valable sur un ensemble stable et convexe queleonque des déformations) est utilisé.

     

On the completeness of solutions of Boussinesq,Timpe, Love,and Michell in axisymmetric elasticity

The completeness of Love's solution was proved by Noll, Gurtin, and Carlson if the meridional half-section is z-convex. In this paper it is pointed out that the condition of z-convexity is unnecessary for the completeness of Love's solution.     

The completeness and a new derivation of the Stroh formalism of anisotropic linear elasticity

In this paper we present a new, simpler and unified derivation of the Stroh formalism of anisotropic linear elasticity, for both nondegenerate and degenerate cases. It is based on the potential representation and Jordan canonical representation theorems. The completeness of the Stroh formalism is proved in the derivation process itself. This new approach is also extended to piezoelastic problems. Besides, we show that the eigenvalues of the fundamental elastic matrix in planar anisotropic elasticity are always distinct, except for the case of isotropy. The project supported by the National Natural Science Foundation of China (19525207 and 19891180).     

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.

     

Coaxiality of strain and stress in anisotropic linear elasticity

For a linearly elastic anisotropic body there are at least two rotations of the principal axes of strain such that the stress and strain tensors become coaxial. These rotations correspond to critical points for the stored energy, viewed as a function of the relative orientation between the body and the strain tensor.Supported by Gruppo Nazionale per la Fisica Matematica of C.N.R. (Italy).     

The existence and uniqueness of solutions of generalized variational inequalities arising from elasticity with friction

ＩｎｔｒｏｄｕｃｔｉｏｎＡｔｐｒｅｓｅｎｔ,ｔｈｅｒｅｓｅａｒｃｈｏｎｖａｒｉａｔｉｏｎａｌｉｎｅｑｕａｌｉｔｉｅｓｃａｕｓｅｄｂｙｅｌａｓｔｉｃｉｔｙｗｉｔｈｆｒｉｃｔｉｏｎａｒｅｓｔｉｌｌｒｅｓｔｒｉｃｔｅｄｔｏｔｈｅｅｑｕｉｖａｌｅｎｔｍａｘｉｍｕｍｏｒｍｉｎｉｍｕｍｅｎｅｒｇｙｐｒｉｎｃｉｐｌｅｓ .Ｓｏｍｅｆａｍｏｕｓｅｘｐｅｒｔｓ,ＧＵＯＹｏｕ＿ｚｈｏｎｇ[1],ＺＨＯＵＳｈｕ＿ｚｈｉ[2 ],Ｇ .Ｆ .ＣａｒｅｙａｎｄＪ.Ｔ .Ｏｄｅｎ[3]ａｎｄＪ.Ｔ .ＯｄｅｎａｎｄＬ .Ｃａｍｐｏｓ[4 ],ａｌｌｈａｖ…     

On the existence and uniqueness of solutions generated by equivalent linearization

The existence and uniqueness of approximate solutions generated by the generalized method of equivalent linearization is considered. For the stationary analysis of systems with harmonic or Gaussian random excitation, it is shown that even though the equivalent linear system may not be unique, a simple element-by-element substitute system exists. Furthermore, this system is at least as good as any other similarly defined substitute system.     

On uniqueness for the displacement problem in the dynamical theory of elasticity

By means of a weighted form of Garding's inequality, we prove a uniqueness theorem for the initial boundary value problem of finite elastodynamics in unbounded domains.

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Sommario Mediante un'opportuna forma pesata della disuguaglianza di Garding, che ricaviamo preliminarmente, dimostriamo un teorema di unicità per il problema degli spostamenti in elastodinamica non lineare relativo a corpi occupanti regioni non limitate dello spazio.

     

On the completeness and uniqueness of the Papkovich-Neuber and the non-axisymmetric Boussinesq,Love, and Burgatti solutions in general cylindrical coordinates

The exact conditions under which the Papkovich-Neuber, non-axisymmetric Boussinesq, non-axisymmetric Love and non-axisymmetric Burgatti solutions (to be defined below) are complete and unique for a general (not necessarily axi-symmetric) problem in general (not necessarily circular) cylindrical coordinates are investigated. They form four problems. The first one is the conjecture that any component of the curvilinear form of the Papkovich-Neuber solution can be omitted and also the general uniqueness of this solution in term of three harmonic functions. The remaining three concern the completeness and uniqueness of the non-axisymmetric Bousinesq, non-axisymmetric Love, non-axisymmetric Burgatti and a non-axisymmetric augmented Love solutions. Love's solution is shown to be incomplete for the case of a hollow sphere with the internal void pressurised.