On discontinuous strain fields in finite elastostatics

A general method for the study of piece-wise homogeneous strain fields in finite elasticity is proposed. Critical homogeneous deformations, supporting strain jumping, are defined for any anisotropic elastic material under constant Piola–Kirchhoff stress field in three-dimensional elasticity. Since Maxwell’s sets appear in the neighborhood of singularities higher than the fold, the existence of a cusp singularity is a sufficient condition for the emergence of piece-wise constant strain fields. General formulae are derived for the study of any problem without restrictions or fictitious stress–strain laws. The theory is implemented in a simple shearing plane strain problem. Nevertheless, the procedure is valid for any anisotropic material and three-dimensional problems.     

On discontinuous strain fields in incompressible finite elastostatics

Piece-wise homogeneous three-dimensional deformations in incompressible materials in finite elasticity are considered. The emergence of discontinuous strain fields in incompressible materials is studied via singularity theory. Since the simplest singularities, including Maxwell’s sets, are the cusp singularities, cusp conditions for the total energy function of homogeneous deformations for incompressible materials in finite elasticity will be derived, compatible with strain jumping. The proposed method yields simple criteria for the study of discontinuous deformations in three-dimensional problems and for any homogeneous incompressible material. Furthermore the homogeneous stress tensor is also not restricted. Neither fictitious nor simplified constitutive relations are invoked. The theory is implemented in a simple shearing problem.     

Crack mouth widening energy-release rate and its application

Unlike the classical crack extension energy release rate, the crack mouth widening energy release rate is related to the Mode I stress intensity factor by application of the principle of virtual work. Solutions are derived for cracked beams and pipes while the method applies in a straightforward manner to other crack configurations. The results are shown to agree well with those available in the open literature.     

On some conservation laws of conservative and non-conservative dynamic systems

This study is concerned with the derivation of conservation laws of both conservative and non-coaservative dynaraical systems with finite numbers of degrees of freedom. First, the derivation of generators of the infinitesimal transformations of the generalized coordinates and time from Noether's basic identity is discussed. In the second part, a special class of conservation laws of conservative dynamical systems which are called action integral conservation laws is developed.     

Conservation laws in elasticity. III. Planar linear anisotropic elastostatics

The first order conservation laws for an arbitrary homogeneous linear planar elastic material are completely classified. In all cases, both isotropic and anisotropic, besides the standard Betti reciprocity laws, there are two infinite-dimensional families of conservation laws, each depending on an arbitrary analytic function of two complex variables.     

On the failure of ellipticity and the emergence of discontinuous deformation gradients in plane finite elastostatics

Summary This investigation concerns equilibrium fields with discontinuous displacement gradients, but continuous displacements, in the theory of finite plane deformations of possibly anisotropic, compressible elastic solids. Elastostatic shocks of this kind, which resemble in many respects gas-dynamical shocks associated with steady flows, are shown to exist only if and when the governing field equations of equilibrium suffer a loss of ellipticity. The local structure of such shocks, near a point on the shockline, is studied with particular attention to weak shocks, and an example pertaining to a shock of finite strength is explored in detail. Also, necessary and sufficient conditions for the dissipativity of time-dependent equilibrium shocks are established. Finally, the relevance of the analysis carried out here to localized shear failures-such as those involved in the formation of Lüders bands-is discussed.Since this paper was submitted for publication, Professor James R. Rice has pointed out to us that the dissipation inequality (6.28), which was in essence postulated in the present work, could have been deduced from the thermodynamic requirement of a positive rate of entropy production together with the energy identity (6.23). It is presumed in such a derivation that the underlying quasi-static time dependent equilibrium shock constitutes an isothermal process.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington, D.C.     

An improved third-order finite difference weighted essentially nonoscillatory scheme for hyperbolic conservation laws

In this article, we present an improved third-order finite difference weighted essentially nonoscillatory (WENO) scheme to promote the order of convergence at critical points for the hyperbolic conservation laws. The improved WENO scheme is an extension of WENO-ZQ scheme. However, the global smoothness indicator has a little different from WENO-ZQ scheme. In this follow-up article, a convex combination of a second-degree polynomial with two linear polynomials in a traditional WENO fashion is used to compute the numerical flux at cell boundary. Although the same three-point information is adopted by the improved third-order WENO scheme, the truncation errors are smaller than some other third-order WENO schemes in L_∞ and L₂ norms. Especially, the convergence order is not declined at critical points, where the first and second derivatives vanish but not the third derivative. At last, the behavior of improved scheme is proved on a variety of one- and two-dimensional standard numerical examples. Numerical results demonstrate that the proposed scheme gives better performance in comparison with other third-order WENO schemes.     

On group properties and conservation laws for second-order quasi-linear differential equations

A sufficient condition for the absence of tangent transformations admitted by second-order quasi-linear differential equations and a sufficient condition for linear autonomy of operators of the Lie group of transformations admitted by second-order weakly nonlinear differential equations are found. A theorem on the structure of the first-order conservation laws for second-order weakly nonlinear differential equations is proved. A classification of second-order linear differential equations with two independent variables in terms of first-order conservation laws is proposed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 3, pp. 64–70, May–June, 2009.     

Equivalence conditions between linear Lagrangian finite element and node‐centred finite volume schemes for conservation laws in cylindrical coordinates

A complete set of equivalence conditions, relating the mass‐lumped Bubnov–Galerkin finite element (FE) scheme for Lagrangian linear elements to node‐centred finite volume (FV) schemes, is derived for the first time for conservation laws in a three‐dimensional cylindrical reference. Equivalence conditions are used to devise a class of FV schemes, in which all grid‐dependent quantities are defined in terms of FE integrals. Moreover, all relevant differential operators in the FV framework are consistent with their FE counterparts, thus opening the way to the development of consistent hybrid FV/FE schemes for conservation laws in a three‐dimensional cylindrical coordinate system. The two‐dimensional schemes for the polar and the axisymmetrical cases are also explicitly derived. Numerical experiments for compressible unsteady flows, including expanding and converging shock problems and the interaction of a converging shock waves with obstacles, are carried out using the new approach. The results agree fairly well with one‐dimensional simulations and simplified models. Copyright © 2013 John Wiley & Sons, Ltd.     

Discontinuous deformation gradients in plane finite elastostatics of incompressible materials

Summary This investigation is concerned with the possibility of the change of type of the differential equations governing finite plane elastostatics for incompressible elastic materials, and the related issue of the existence of equilibrium fields with discontinuous deformation gradients. Explicit necessary and sufficient conditions on the deformation invariants and the material for the ellipticity of the plane displacement equations of equilibrium are established. The issue of the existence, locally, of elastostatic shocks—elastostatic fields with continuous displacements and discontinuous deformation gradients—is then investigated. It is shown that an elastostatic shock exists only if the governing field equations suffer a loss of ellipticity at some deformation. Conversely, if the governing field equations have lost ellipicity at a given deformation at some point, an elastostatic shock can exist, locally, at that point. The results obtained are valid for an arbitrary homogeneous, isotropic, incompressible, elastic material.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 with the Office of Naval Research in Washington D.C.     

Uniformity of stresses inside an elliptic inclusion in finite plane elastostatics

We consider an elastic inclusion embedded in a particular class of harmonic materials subjected to uniform remote stress. Using complex variable techniques, we show that if the Piola stress within the inclusion is uniform, the inclusion is necessarily an ellipse except in the special case when the (uniform) remote stress assumes a particular form. In addition, we obtain the complete solution for an elliptic inclusion with uniform interior stress for any uniform remote stress distribution.     

On symmetries, conservation laws and invariant solutions of the foam-drainage equation

This study deals with symmetry group properties and conservation laws of the foam-drainage equation. Firstly, we study the classical Lie symmetries, optimal systems, similarity reductions and similarity solutions of the foam-drainage equation which are obtained through the Lie group method of infinitesimal transformations. Secondly, using the new general theorem on non-local conservation laws and partial Lagrangian approach, local and non-local conservation laws are also studied and, finally, non-classical symmetries are derived.     

On the traction problem in linear elastostatics

A sharp uniqueness class is determined for the traction problem of linear elastostatics in exterior domains and in the half space. In particular, it is shown that this problem has a most one solution in the class of all vector fields u such that either u=o(r) or % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafy4bIeTbaK% aaaaa!3782!\[\hat \nabla \]u=o(1), as r+, with w and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafy4bIeTbaK% aaaaa!3782!\[\hat \nabla \]u respectively rigid displacement and symmetric part of u.