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.     

Anti-plane shear fields with discontinuous deformation gradients near the tip of a crack in finite elastostatics

Summary This paper reconsiders the problem of determining the elastostatic field near the tip of a crack in an all-round infinite body deformed by a Mode III loading at infinity to a state of anti-plane shear. The problem is treated for a class of incompressible, homogeneous, isotropic elastic materials whose constitutive laws permit a loss of ellipticity in the governing displacement equation of equilibrium at sufficiently severe shearing strains. The analysis represents a generalization of that reported in an earlier study and, as before, is carried out for the small-scale nonlinear crack problem, in which a crack of finite length is replaced by a semi-infinite one, and the nonlinear field far from the crack-tip is matched to the near field predicted by the linearized theory. The methods employed in the present paper are necessarily largely qualitative, since they apply to all materials in the class considered. The principal feature of the resulting elastic field is the presence of two symmetrically located curves issuing from the crack-tip and bearing discontinuities in displacement gradient and stress.The results communicated in this paper were obtained in the course of an investigation supported in part by Contract N00014-75-C-0196 with the Office of Naval Research in Washington, D.C.     

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.     

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.     

The screw dislocation problem in incompressible finite elastostatics: a discussion of nonlinear effects

The jump conditions arising in the formulation of dislocation problems in finite elastostatics are discussed and a full field solution of the anti-plane shear type is given for the screw dislocation problem. The solution is valid for the most general homogeneous isotropic incompressible nonlinear elastic solid. The level of nonlinearity is defined for this solution and compared to "dislocation core" estimates in materials science.     

Uniqueness theorems for displacement fields with locally finite energy in linear elastostatics

Uniquencess theorems are proved for the fundamental boundary value problems of linear elastostatics in bodies of arbitrary shape. The displacement fields are required to have finite strain energy in bounded portions of the bodies and satisfy the principle of virtual work. For bounded bodies, the total strain energy is finite and uniquencess is proved without additional hypotheses. In particular, no restrictions other than the energy condition are placed on the field singularities that may occur at sharp edges and corners. For unbounded bodies, uniqueness can be proved as in the bounded case if the total strain energy is finite. Sufficient conditions for this are shown to be the finiteness of the strain energy in bounded portions of the body together with the growth restriction % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% GabaqaaiGacaGaamqadaabaeaafiaakeaadaWdraqaaiaabwhadaWg% aaWcbaGaaeyAaaqabaGccaGGOaGaaeiEaiaacMcacaqG1bWaaSbaaS% qaaiaabMgaaeqaaOGaaiikaiaabIhacaGGPaGaaeizaiaabIhacaqG% 9aGaaGimaiaacIcacaqGYbGaaiykaiaacYcacaqGYbGaeyOKH4Qaey% OhIukaleaacqGHPoWvdaWgaaadbaGaaeOCaiaacYcacqaH0oazaeqa% aaWcbeqdcqGHRiI8aaaa!5E73!\[\int_{\Omega _{{\text{r}},\delta } } {{\text{u}}_{\text{i}} ({\text{x}}){\text{u}}_{\text{i}} ({\text{x}}){\text{dx = }}0({\text{r}}),{\text{r}} \to \infty } \] on the displacement fieldu _i , where _r, is the portion of the body that lies between concentric spheres with radiir andr+ and >0.This research was supported by the Air Force Office of Scientific Research. Reproduction in whole or part is permitted for any purpose of the United States Government.Prepared under Contract No. F 49620-77-C-0053 for Air Force Office of Scientific Research.     

A coupled continuous and discontinuous finite element method for the incompressible flows

In this paper, we develop a coupled continuous Galerkin and discontinuous Galerkin finite element method based on a split scheme to solve the incompressible Navier–Stokes equations. In order to use the equal order interpolation functions for velocity and pressure, we decouple the original Navier–Stokes equations and obtain three distinct equations through the split method, which are nonlinear hyperbolic, elliptic, and Helmholtz equations, respectively. The hybrid method combines the merits of discontinuous Galerkin (DG) and finite element method (FEM). Therefore, DG is concerned to accomplish the spatial discretization of the nonlinear hyperbolic equation to avoid using the stabilization approaches that appeared in FEM. Moreover, FEM is utilized to deal with the Poisson and Helmholtz equations to reduce the computational cost compared with DG. As for the temporal discretization, a second‐order stiffly stable approach is employed. Several typical benchmarks, namely, the Poiseuille flow, the backward‐facing step flow, and the flow around the cylinder with a wide range of Reynolds numbers, are considered to demonstrate and validate the feasibility, accuracy, and efficiency of this coupled method. Copyright © 2016 John Wiley & Sons, Ltd.     

On Sanders' energy-release rate integral and conservation laws in finite elastostatics

Sanders showed in 1960, within the framework of two-dimensional elasticity, that in any body a certain integral I around a closed curve containing a crack is path-independent. I is equal to the rate of release of potential energy of the body with respect to crack length. Here we first derive, in a simple way, Sanders' integral I for a loaded elastic body undergoing finite deformations and containing an arbitrary void. The strain energy density need not be homogeneous nor isotropic and there may be body forces. In the absence of body forces, for flat continua, and for special forms of the strain energy density, it is shown that I reduces to the well-known vector and scalar path-independent integrals often denoted by J, L, and M.     

Numerical comparison of hybridized discontinuous Galerkin and finite volume methods for incompressible flow

In this study, a hybridizable discontinuous Galerkin method is presented for solving the incompressible Navier–Stokes equation. In our formulation, the convective part is linearized using a Picard iteration, for which there exists a necessary criterion for convergence. We show that our novel hybridized implementation can be used as an alternative method for solving a range of problems in the field of incompressible fluid dynamics. We demonstrate this by comparing the performance of our method with standard finite volume solvers, specifically the well‐established finite volume method of second order in space, such as the icoFoam and simpleFoam of the OpenFOAM package for three typical fluid problems. These are the Taylor–Green vortex, the 180‐degree fence case and the DFG benchmark. Our careful comparison yields convincing evidence for the use of hybridizable discontinuous Galerkin method as a competitive alternative because of their high accuracy and better stability properties. Copyright © 2014 John Wiley & Sons, Ltd.     

Deformations with discontinuous gradients in plane elastostatics of compressible solids

We investigate certain issues pertaining to plane deformations with discontinuous gradients sustained by compressible, isotropic, hyperelastic materials. Conditions on the elastic potential which are necessary and sufficient for the existence of such deformations are derived. An alternative, explicit set of criteria is deduced from these, which involves jump conditions restricting the deformation invariants on either side of the discontinuity. This result, which is expressed in terms of the local amounts of shear and dilatation, characterizes all possible two-phase states sustained by a given elastic potential. Some implications of ellipticity loss on the existence of such states are considered.     

Finite strain analysis of crack tip fields in incompressible hyperelastic solids loaded in plane stress

A new method is developed to determine the dominant asymptotic stress and deformation fields near the tip of a Mode-I traction free plane stress crack. The analysis is based on the fully nonlinear equilibrium theory of incompressible hyperelastic solids. We show that the dominant singularity of the near tip stress field is governed by the asymptotic solution of a linear second order ordinary differential equation. Our method is applicable to any hyperelastic material with a smooth work function that depends only on the trace of the Cauchy-Green tensor and is particularly useful for materials that exhibit severe strain hardening. We apply this method to study two types of soft materials: generalized neo-Hookean solids and a solid that hardens exponentially. For the generalized neo-Hookean solids, our method is able to resolve a difficulty in the previous work by Geubelle and Knauss (1994a). Our theoretical results are compared with finite element simulations.     

Analysis of local singular fields near the corner of a quarter-plane with mixed boundary conditions in finite plane elastostatics

We consider a quarter-plane of compressible hyperelastic material of harmonic-type undergoing finite plane deformations. The plane is subjected to mixed (free–fixed) boundary conditions. In contrast to the analogous case from classical linear elasticity, we find that the deformation field is smooth in the vicinity of the vertex and is actually bounded at the vertex itself. In particular, the normal displacement remains positive eliminating the possibility of material interpenetration. Finally, explicit expressions for Cauchy and Piola stress distributions are obtained in the vicinity of the vertex.     

FE analysis of stress and strain fields in finite random composite bodies

In this paper the mechanical behaviour of finite random heterogeneous bodies is considered. The analysis of non-local interactions between heterogeneities in microscopically heterogeneous materials is necessary when the spatial variation of the load or the dimensions of the body, relative to the scale of the microstructure, cannot be ignored. Microstructures can be periodic but generically they are random. In the first case, an exact calculation can be performed but in the second case recourse has to be made either to simulation or to some scheme of approximation. One such scheme is based on a stochastic variational principle. The novelty of the present work is that a stochastic variational principle is projected directly onto a finite-element basis so that all subsequent analysis is performed within a finite-element framework. The proposed formulation provides expressions for the local stress and strain fields in any realization of the medium, from which expressions for statistically-averaged quantities can be derived. Then an approximation of Hashin-Shtrikman type is developed, which generates a FE-based numerical procedure able to take account of interactions between random inclusions and boundary layer effects in finite composite structures. Finally, two examples are presented, namely a cylinder with square cross-section subjected to mixed boundary conditions of different types on different faces and a rectangular body containing a centre crack. The results show that in the vicinity of the boundary or close to the crack tip, the strain and the stress in the matrix and in the inclusions differ considerably from those obtained by the formal application of conventional homogenization.     

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 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.