Traction Boundary Value Problem Research Articles

Stress Independence of Poisson's Ratio and Divergence-Free Body Forces

In simply connected bodies, the compatibility conditions in 2D elasticity are independent of Poisson's ratio (and hence there is a reduction in the number of constants) for divergence-free body forces for the boundary-value problem of traction. It is shown that that for divergence-free body forces in two fields with different compliances and the same stress (hence, inducing a reduction in constants and independence of Poisson's ratio) the displacements differ by a field that is a solution for λ + 2μ = 0, where λ and μ are the Lame constants for the elastic material, which is a constant rotation. Seen from another aspect, in a Cosserat spectral orthogonal and complete decomposition of the stress [12], the terms associated with Poisson's ratio in the stress representation have as coefficient the inner product of the body force with a displacement which is an eigenfunction at λ + 2μ = 0. This displacement is orthogonal to a divergence- free body force vector, which results in the vanishing of the coefficient of the Poisson's ratio dependence term in the orthogonal decomposition. Due to the completeness and orthogonality of the eigenfunctions, divergence-free body force is a necessary and sufficient condition for stress independence of Poisson's ratio in simply connected 2D bodies, while for multiply connected ones, the Michell generalized conditions also have to be satisfied, as shown here as well. Thus, both approaches, the CLM [1] theorem, which deals with reduction of constants, and the Cosserat spectral decomposition, which shows explicitly the material dependence of the solution, lead to the same conditions for stress independence of Poisson's ratio both for simply connected and multiply connected 2D bodies.

Signorini’s method for live loads and second-order effects

In this paper we deal with Signorini’s method for live loads to derive approximate solutions of some pure traction boundary value problems in finite elastostatics. Then, the obtained solutions are used to determine the second-order constitutive constants of a homogeneous and isotropic elastic material.

Saint-Venant end effects for plane deformations of linear piezoelectric solids

The recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity. In this paper, we investigate the decay of Saint-Venant end effects for plane deformations of a piezoelectric semi-infinite strip. First of all, we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid. Just as in the mechanical case, not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non-trivial state of plane deformation. The governing system of four second-order partial differential equations for the two in-plane displacements and electric potential are overdetermined in general. Sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation. The resulting traction boundary-value problem with prescribed surface charge is an oblique derivative boundary-value problem for a coupled elliptic system of three second-order partial differential equations. The special case of a piezoelectric material transversely isotropic about the poling axis is then considered. Thus the results are valid for the hexagonal crystal class 6 mm. The geometry is then specialized to be a two-dimensional semi-infinite strip and the poling axis is the axis transverse to the longitudinal direction. We consider such a strip with sides traction-free, subject to zero surface charge and self-equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity. A formulation of the problem in terms of an Airy-type stress function and an induction function is adopted. The governing partial differential equations are a coupled system of a fourth and third-order equation for these two functions. On seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second-order ordinary differential equations. This problem is the piezoelectric analog of the well-known eigenvalue problem arising in the case of an anisotropic elastic strip. It is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth-order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti-symmetric deformations and electric fields. Assuming completeness of the eigenfunctions, the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part. Numerical results are given for PZT-5H, PZT-5, PZT-4 and Ceramic-B. It is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials.

Stress formulation in 3D elasticity and application to spherically uniform anisotropic solids

We formulate the boundary value problem of traction for inhomogeneous anisotropic elastic materials in terms of stresses following the method introduced by Pobedria and apply it to spherically anisotropic materials. An example of spherically symmetric deformation of spherically uniform anisotropic materials is presented.

Exponential decay of end effects in anti-plane shear for functionally graded piezoelectric materials

This paper is concerned with the investigation of the effect of material inhomogeneity on the decay of Saint–Venant end effects in functionally graded linear piezoelectric materials. Saint–Venant's principle and related results for elasticity theory have received considerable attention in the literature but it is only recently that analogous issues in piezoelectricity have been investigated. The current rapidly developing smart–structures technology provides motivation for the investigation of such problems. We examine the decay of Saint–Venant end effects in the context of anti–plane shear deformations for linear inhomogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations (PDEs) of equilibrium are shown to be a coupled system of second–order PDEs with variable coefficients for the mechanical displacement u and electric potential φ . The traction boundary–value problem with prescribed surface charge is formulated as an oblique derivative boundary–value problem for this elliptic system. The axial decay of solutions on a semi–infinite strip subjected to non–zero boundary conditions only at the near end is investigated. This analysis is carried out for a subclass of special graded materials, namely, laterally inhomogeneous solids. Two specific piezoceramics that have received widespread application are considered. For the first (corresponding to the hexagonal 6 mm crystal–symmetry class), it is shown that the mechanical and electrical problems essentially decouple, and that the decay rates for mechanical and electrical Saint–Venant end effects coincide and are equal to that of linear inhomogeneous isotropic anti–plane elasticity, for which results on the decay rate have been obtained previously. For the second class of materials (corresponding to the cubic 4∼3m symmetry) the situation is quite different. The boundary–value problem involves a full coupling of mechanical and electrical effects. Energy decay estimates using differential–inequality methods are used to obtain an explicit estimated decay rate (a lower bound for the actual decay rate) in terms of a single dimensionless piezoelectric coupling constant d and the decay rate for an associated linear inhomogeneous isotropic anti–plane shear elasticity problem. For fixed material inhomogeneity, the decay rate is shown to be monotone decreasing with increasing values of d In the limit as d→0 we recover the purely mechanical case. Thus, for this class of functionally graded materials, piezoelectric end effects are predicted to penetrate further into the strip than their functionally graded isotropic elastic counterparts , confirming recent results obtained in other contexts in linear piezoelectricity. For both classes of materials, the influence of material inhomogeneity on the decay rate is reflected solely by the shear modulus of an associated inhomogeneous isotropic anti–plane shear elasticity problem. It is shown that, for both material classes, material inhomogeneity has a significant influence on the decay of end effects.

The effect of an elastic triangular inclusion on a crack

The interaction between an elastic triangular inclusion and a crack is investigated. The problem is formulated using the boundary integral equations for traction boundary value problems derived by Chau and Wang as basic equations. By using the continuity condition of traction and displacement on interface as supplement equations, a set of equations for solving the interaction problem between an inclusion and a crack are obtained, which are solved by using a new boundary element method. The results in terms of stress intensity factors (SIFs) are calculated for a variety of crack-inclusion arrangements and the elastic constants of the matrix and the inclusion. The results are valuable for studying new composite materials.

On complex variable method in finite elasticity

We highlight the alternative presentation of the Cauchy-Riemann conditions for the analyticity of a complex variable function and consider plane equilibrium problem for an elastic transversely isotropic layer, in finite deformation. We state the fundamental problems and consider traction boundary value problem, as an example of fundamental problem-one. A simple solution of "Lame's problem" for an infinite layer is obtained. The profile of the deformed contour is given; and this depends on the order of the term used in the power series specification for the complex potential and on the material constants of the medium.

Uniqueness and Complementary Energy in Nonlinear Elastostatics

Global uniqueness of the smooth stress and deformation to within the usual rigid-body translation and rotation is established in the null traction boundary value problem of nonlinear homogeneous elasticity on a n-dimensional star-shaped region. A complementary energy is postulated to be a function of the Biot stress and to be para-convex and rank-(n-1) convex, conditions analogous to quasi-convexity and rank-(n-2) of the stored energy function. Uniqueness follows immediately from an identity involving the complementary energy and the Piola-Kirchhoff stress. The interrelationship is discussed between the two conditions imposed on the complementary energy, and between these conditions and those known for uniqueness in the linear elastic traction boundary value problem.

Saint-Venant's Principle for Antiplane Shear Deformations of Linear Piezoelectric Materials

This paper is concerned with investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venant's principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. We examine the decay of Saint-Venant end effects in the context of antiplane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are shown to be a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential $\varphi$. The traction boundary-value problem with prescribed surface charge is formulated as an oblique derivative boundary-value problem...

Saint-Venant End Effects in Anti-plane Shear for Classes of Linear Piezoelectric Materials

This paper is concerned with further investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venant's principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. The decay of Saint-Venant end effects is investigated in the context of anti-plane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential ϕ. The traction boundary-value problem with prescribed surface charge can be formulated as an oblique derivative boundary-value problem for this elliptic system. Energy-decay estimates using differential inequality methods are used to study the axial decay of solutions on a semi-infinite strip subjected to non-zero boundary conditions only at the near end. This analysis is carried out for a rather general class of materials (the tetragonal \({\bar 4}\) crystal class). The boundary-value problem involves a full coupling of mechanical and electrical effects. There are four independent material constants appearing in the problem. An explicit estimated decay rate (a lower bound for the actual decay rate) is obtained in terms of two dimensionless piezoelectric parameters d0,r, the first of which provides a measure of the degree of piezoelectric coupling. The estimated decay rate is shown to be monotone decreasing with increasing values of the coupling parameter d0. In the limit as d0→0, we recover the exact decay rate for the purely mechanical case. Thus, for the tetragonal \({\bar 4}\) class of materials, piezoelectric end effects are predicted to penetrate further into the strip than their elastic counterparts, confirming recent results obtained in other contexts in linear piezoelectricity.

A Monte Carlo solution method for linear elasticity

A probabilistic method is proposed and implemented to solve linear elasticity problems. The method, called walk on the boundary method (WBM), uses the same governing equations as the boundary element method. Unlike in finite element and boundary element methods, WBM does not require any meshing. Also, error estimates for WBM are easier to obtain than in finite element and boundary element methods. Furthermore, WBM obtains a point solution at a specific point of interest instead of a full field solution as in finite element and boundary element methods. WBM is developed for general traction boundary value problems in antiplane shear, plane strain, and 3D elasticity. Numerical implementations are performed for three example problems: (1) antiplane shear problem with a centrally located circular hole being loaded by uniformly applied traction, (2) plane strain problem with centrally located circular hole being loaded by uniform tension, and (3) 3D elasticity problem with a centrally located spherical cavity being loaded by uniform tension. Results from the three example problems compared favorably with results from the analytical and finite element solution. Three critical issues associated with the WBM for linear elasticity are pointed out for its further improvement.

Exact solution for finite electrode layers embedded at the interface of two piezoelectric half-planes

The exact solution is obtained for two bonded dissimilar piezoelectric half-planes with discontinuous thin electrode layers embedded at the interface. In contrast with existing models on interfacial insulating cracks or rigid electrode layers, the present model is based on the assumption that tractions and displacements are continuous across the thin electrode layers. As a result, the present model leads to a mixed boundary value problem, rather than a conventional generalized traction or displacement boundary value problem to which the existing models have led. Detailed discussion is given for the case of practical significance in which two piezoelectric half-planes are poled in opposite directions perpendicular to the interface. In this case, the electroelastic field exhibits the inverse square-root singularity and an oscillatory singularity does not appear. It is found that the interfacial tractions and the normal electrical displacement vanish along all electrode-free parts of the interface. On the other hand, along the interfaces between piezoelectric half-planes and the embedded electrode layer, the normal traction and the normal electrical displacement exhibit electrode-tip singularity and the sign of the normal traction is determined by the sign of the electric charge of the electrode layer. The implications of these results to reliability of layered piezoelectric devices are discussed.

Some Remarks Concerning a Boundary-Value Problem in Nonlinear Elastostatics

The paper is concerned with solvability of a mixed boundary value problem of place and traction that describes the straightening of annular cylindrical sectors composed of unconstrained isotropic hyperelastic solids. The existence and uniqueness of solutions is discussed and several examples are exhibited.

Boundary integral equations of unique solutions in elasticity

The properties of the fundamental solution are derived in linear elastostatics. These properties are used to show that the conventional displacement and traction boundary integral equations yield non-unique displacement solutions in a traction boundary value, problem. The condition for the existence of unique displacement solutions is proposed for the traction boundary value problem. The degrees of freedom of the displacement solution are removed by the condition to obtain the boundary integral equations of unique solutions for the traction boundary value problems. Numerical example is presented to demonstrate the accuracy and efficiency of the present equations.

More on stress invariance conditions for the traction boundary value problem of plane linear elasticity

Cherchaev (1992) gave a sufficient condition for a change in the elastic tensor field to preserve the stress state of a plane linearly elastic solid subject to prescribed traction forces on its boundary. This condition turned out to have a number of fruitful applications in the mechanics of composite materials and was later extended by Dundurs and Markenstoff (1993). The present work addresses and answers the question whether there is a yet more general condition. We show that: (i) if the change in the elastic tensor field is required to be hyperelastic, the extended condition of Dundurs and Markenscoff is not only sufficient but also necessary; (ii) if the change in the elastic tensor field is relaxed so as to be elastic, a more general necessary and sufficient condition exists. In proving these two conclusions, several orthogonal decompositions are constructed for third- and fourth-order tensors presenting index permutation symmetries. Such decompositions are probably also useful for solving other problems in mechanics.

A conforming mixed finite element method for the pure traction problem of linear elasticity

A conforming mixed finite element method using the method of reduced integration is developed for the two-dimensional pure traction boundary value problem of linear elasticity. The convergence is uniform as the material becomes nearly incompressible.

Multigrid Methods for the Pure Traction Problem of Linear Elasticity: Mixed Formulation

A multigrid method using conforming P - 1 finite elements is developed for the two-dimensional pure traction boundary value problem of linear elasticity. The convergence is uniform even as the material becomes nearly incompressible. A heuristic argument for acceleration of the multigrid method is discussed as well. Numerical results with and without this acceleration as well as performance estimates on a parallel computer are included.

A Remarkable Tensor in Plane Linear Elasticity

It is shown that any two-dimensional elastic tensor can be orthogonally and uniquely decomposed into a symmetric tensor and an antisymmetric tensor. To within a scalar multiplier, the latter turns out to be equal to the right-angle rotation on the space of two-dimensional second-order symmetric tensors. On the basis of these facts, several useful results are derived for the traction boundary value problem of plane linear elasticity.

Saint-Venant's principle in linear isotropic elasticity for incompressible or nearly incompressible materials

One of the unresolved issues on Saint-Venant's principle concerns the energy decay estimates established in the literature for the traction boundary-value problem of three-dimensional linear isotropic elastostatics for a cylinder. For the semi-infinite cylinder with traction-free lateral surface and self-equilibrated loads at the near end, it has been shown that the stresses decay exponentially from the end and results were obtained for the estimated decay rate, which is a lower bound for the exact decay rate. These results are, however, generally conservative in that they underestimate the exact decay rate. Another shortcoming, which motivated the present investigation, is that the estimated decay rates tend to zero as the Poisson's ratio ν tends to the value 1/2. Thus for the limiting case of an incompressible material, these methods fail to establish exponential decay. The purpose of the present paper is to remedy this defect. In particular, an exponential decay estimate is established with estimated decay rate independent of Poisson's ratio. Thus, in particular, the results here hold in the incompressible limit as ν → 1/2. An alternative treatment directly for the incompressible case has been given recently. It should be noted that the stresses in the three-dimensional traction boundary-value problem do depend on Poisson's ratio ν and that stress decay estimates for the cylinder problem with estimated decay rates dependent on ν are, in fact, to be expected. However, in the absence of such results that do not deteriorate as ν → 1/2, we obtain here an estimated decay rate that is independent of ν.

On the Cavities and Rigid Inclusions Correspondence and the Cosserat Spectrum

The correspondence in two-dimensional elasticity between the stress fields of cavities and rigid inclusions has been obtained by Dundurs [1] and Markenscoff [3]. It was shown that if the limit of the stress of the inclusion boundary-value problem, which depends on the elastic constants, exists when the Poisson's ratio v tends to 1, then this solves the traction boundary-value problem for the cavity problem since it satisfies equilibrium and boundary conditions, and, by the uniqueness theorem, exists and is unique. In three dimensions the solution of the traction boundary-value problem of elasticity does depend on Poisson's ratio since the Beltrami-Mitchell compatability conditions for the stress depend on Poisson's ratio. So the similar argument for the correspondence between cavities and rigid inclusions cannot in principle be made. However, the Beltrami-Mitchell compatability conditions are independent of v if the dilatation is a constant or a linear function of the position. In this case we can show that the same result goes through for the correspondence. In order to investigate the behavior of the solutions in the vicinity of v = 1, we use some results obtained for the Cosserat spectrum by Mikhlin [4], Maz'ya and Mikhlin [3], see also [6]. The existence of the limit for 2D and 3D when v tends to 1 is proved on the basis of the fact that the eigenvalue ω = — 1 of the Cosserat spectrum is isolated.