A circular inclusion with imperfect interface in finite plane elastostatics

We consider a circular elastic inclusion embedded in a particular class of harmonic materials subjected to remote uniform stresses. The imperfect interface can be rate dependent as well as rate independent. First, we study the situation in which both rate-depending slip and diffusional relaxation are present on the sharp inclusion-matrix imperfect interface. It is found that in general, the internal Piola stresses within the inclusion are spatially non-uniform and decay with two relaxation times. Interestingly, the average mean Piola stress within the circular inclusion is time independent. Some extreme cases for the imperfect interface are discussed in detail. Particularly, we find a simple condition leading to internal uniform Piola stresses that decay only with a single relaxation time. Second, we investigate a rate-independent spring-type imperfect interface on which normal and shear tractions are proportional to the corresponding displacement jumps. It is found that in general, the internal Piola stresses are intrinsically non-uniform. A special kind of the spring-type interface leading to internal uniform Piola stresses is also found.     

The sector problem in plane elastostatics

Using a biorthogonality relation, it is shown how solutions for the generalized plane stress problem for a sector whose radial edges are ‘stress-free’ or ‘built-in’ can be obtained for physically meaningful boundary functions prescribed on the curved edge. Numerical results are presented for the reproduction of the edge stresses when a self-equilibrating load is prescribed on the curved edge of the sector having its radial edges stress-free.     

On Mitchell conditions for plane problems in elastostatics

Summary The main objective of the present work is the investigation of the conditions of single-valuedness for plane problems of elastostatics. It has been shown that in addition to the usual conditions further conditions, referred to as supplementary conditions of single-valuedness or generalized Mitchell's, conditions in the case of isotropic materials, are to be, satisfied by the strains in order that the corresponding displacements and rigid body rotations should be single-valued along those bounding curves on which displacements and tractions are alternatively prescribed.It has also been proved that both the supplementary conditions of single-valuedness and the so called kinematic boundary conditions are natural boundary conditions of the principle of minimum complementary energy as a variational principle.     

On complex-variable formulation for finite plane elastostatics of harmonic materials

Summary Although complex-variable formulation provides a powerful method for linear plane elastostatics, finite plane elastostatics does not admit an efficient complex-variable method. Here, to explore the potential of complex-variable methods to finite elastostatics, a new derivation is presented for a complex-variable formulation for plane-strain deformation of compressible hyperelastic harmonic materials, developed first by Varley and Cumberbatch [8]. The present derivation is remarkable in its mathematical conciseness, and has the potential to stimulate further interest in the complex-variable methods for finite plane elastostatics. To demonstrate the powerfulness of the complex-variable method, a complete solution is given for an interface crack in a special class of compressible harmonic materials, which defines a real parameter similar to the Dundurs's parameter in linear elasticity. In particular, this parameter becomes unity and all oscillatory singularities disappear when the asymptotic behavior of the harmonic materials obeys a constitutive restriction proposed by Knowles and Sternberg [10]. Finally, main features of finite deformation of the interface crack are discussed with a comparison to the well-known results of linear elasticity.     

Neutral spherical inhomogeneities in a twisted circular cylindrical bar

We consider the torsion problem of a circular cylindrical bar which is filled up with composite spherical inclusions. The composite inclusions consist of a core and coating both of which are spherically orthotropic with the volume fractions of the core being the same in every composite inclusion. The center points of the spherical inhomogeneities are on the axis of revolution of the circular cylinder. The neutral inhomogeneity in the considered problem of elastic equilibrium is defined as a foreign body (inclusion) which can be introduced in a host body without disturbing the elastic field (displacements, stresses) in it. The conditions of the neutral inhomogeneity for the twisted circular cylindrical bar are derived, and some special cases of inhomogeneity are analyzed. The present paper gives a new example for neutral inhomogeneity in the field of elasticity.     

Three-phase elliptical inclusions with internal uniform hydrostatic stresses in finite plane elastostatics

We consider the internal stress field of a three-phase elliptical inclusion bonded to an infinite matrix through an interphase layer when the matrix is subjected to remote uniform stresses. The elastic materials comprising all the three phases belong to a particular class of harmonic materials, and the formed interfaces are two confocal ellipses. A condition leading to internal uniform hydrostatic stresses is derived. This condition relates the two remote principal stresses with the geometric parameters (the thickness of the interphase layer and the aspect ratio of the elliptical inclusion) of the three-phase elliptical inclusion. When this condition is met, the hoop stress in the interphase layer along the entire interphase/inclusion interface is also uniform. Five special situations of practical importance are discussed in considerable detail to demonstrate the unique phenomena inherent in harmonic materials. Our discussions indicate that when this condition is met, it is permissible for the two remote principal stresses to have opposite signs and that for given geometric and material parameters, the remote loading ratio is no longer constant and multiple external loading states exist leading to internal uniform hydrostatic stresses. It is found that this condition can be written into a hyperbola for the two remote principal stresses when the interphase layer is extremely compliant or relatively stiff or when the inclusion is almost rigid. When the magnitudes of the remote stresses are sufficiently large, this condition becomes a very simple one relating the remote loading ratio with the geometric parameters of the composite. Interestingly, it is clearly observed from the simple condition that for given geometric parameters of the three-phase elliptical inclusion, there exist two different values of the remote loading ratio, both of which lead to an internal uniform hydrostatic stress state.     

Volume integral equation method for multiple circular and elliptical inclusion problems in antiplane elastostatics

A Volume Integral Equation Method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing interacting multiple isotropic and anisotropic circular/elliptical inclusions subject to remote antiplane shear. This method is applied to two-dimensional problems involving long parallel cylindrical inclusions. A detailed analysis of the stress field at the interface between the matrix and the central inclusion is carried out for square and hexagonal packing of isotropic and anisotropic inclusions. The effects of the number of isotropic and anisotropic inclusions and various fiber volume fractions on the stress field at the interface between the matrix and the central circular/elliptical inclusion are also investigated in detail. The accuracy of the method is validated by solving single isotropic and orthotropic circular/elliptical inclusion problems and multiple isotropic circular and elliptical inclusion problems for which solutions are available in the literature.     

Transient heat conduction in a medium with multiple circular cavities and inhomogeneities

A two‐dimensional transient heat conduction problem of multiple interacting circular inhomogeneities, cavities and point sources is considered. In general, a non‐perfect contact at the matrix/inhomogeneity interfaces is assumed, with the heat flux through the interface proportional to the temperature jump. The approach is based on the use of the general solutions to the problems of a single cavity and an inhomogeneity and superposition. Application of the Laplace transform and the so‐called addition theorem results in an analytical transformed solution. The solution in the time domain is obtained by performing a numerical inversion of the Laplace transform. Several numerical examples are given to demonstrate the accuracy and the efficiency of the method. The approximation error decreases exponentially with the number of the degrees of freedom in the problem. A comparison of the companion two‐ and three‐dimensional problems demonstrates the effect of the dimensionality. Copyright © 2009 John Wiley & Sons, Ltd.     

Laminate elastostatics

A general formulation is presented for a class of plane elastostatic, laminated body problems which have the characteristic that the internal material interfaces and the external boundary are level surfaces of the same curvilinear coordinate system. The laminated body is regarded as a material continuum characterized by material property functions which possess finite jumps across material interfaces. The stress field within the body is considered as being induced by external boundary tractions and by temperature, the body being stress-free in the reference state. It is shown that the problem involves the determination of a single piecewise continuous stress function which is in turn expressed in terms of the material property functions and three continuous stress functions which have discontinuous normal derivatives at material interfaces. The theory reduces to that of ordinary elasticity when the material property functions become uniformly constant. In the case in which the number of material interfaces increases without limit, a correspondence principle is derived which relates the stress field in the laminated body to the stress field in a ‘corresponding’ homogeneous, anisotropic body.     

Design of neutral elastic inhomogeneities in plane elasticity in the case of non-uniform loading

We use the concept of imperfect interface to design a (stress) neutral elastic inhomogeneity in the case when the inhomogeneity-matrix system is subjected to plane deformations. Of particular interest is the fact that the prescribed stress field inside the matrix is assumed to be non-uniform. Using complex variable techniques, we derive conditions on the functions describing the (imperfect) interface to ensure neutrality in several cases of practical interest.     

Uncoupled solutions in elastostatics

The paper presents two new forms of uncoupled solutions of elastostatic equations. These solutions are valid for compressible and incompressible bodies and are presented by harmonic functions. The derivation procedure is based on the author's previous results concerning integration of differential equations in elastostatics.     

The plane circular crack problem in the linearized couple-stress theory

The problem of the plane circular crack in a homogeneous and isotropic elastic body under uniform uniaxial tension normal to the plane of the crack is considered within the linearized couple-stress theory. The stress intensity factors, of value in fracture mechanics, are calculated.     

Elastic circular inclusion in an infinite plane containing two cracks

The elastic fields in an elastic circular inclusion and surrounding infinite matrix containing two cracks symmetrically situated, are determined when the matrix is subjected to loads at infinity. In this problem, the elastic properties of inclusion could differ from those of the matrix. The Muskhelishvili's technique is used. The solution depends upon two sets of suitable complex potentials Φ_m(z), Ψ_m(z), Φ_i(z), Ψ_i(z) for matrix and inclusion respectively, which solves the problem.     

On the uniqueness of solutions for the Dirichlet boundary value problem of linear elastostatics in the circular domain

Summary The uniqueness and mathematical stability of the Dirichlet boundary value problem of linear elastostatics is studied. The problem is posed as a set of partial differential equations in terms of displacements and Dirichlet-type of boundary conditions (displacements) for arbitrary bounded domains. Then for the circular interior domain the closed form analytical solution is obtained, using an extended version of the method of separation of variables. This method with corresponding complete solution allows for the derivation of a necessary and sufficient condition for uniqueness. The results are compared with existing energy and uniqueness criteria. A parametric study of the elastic characteristics is performed to investigate the behaviour of the displacement field and the strain energy distribution, and to examine the mathematical stability of the solution. It is found that the solution for the circular element with hourglass-like boundary conditions will be unique for all v ≠ 0.5, 0.75, 1.0 and will be mathematically stable for all v ≠ 0.75. Locking of the circular element occurs for v = 0.75 as the energy tends to infinity.     

Error bounds in nonlinear elastostatics

The boundary value problem of place and traction in nonlinear hyper-elastostatics is considered. As a consequence of convexity of the strain energy function in some neighborhood of a nondegenerate critical point in a quotient space the constitutive equations are invertible. Complementary functionals and a generalized interaction energy lead to estimates for the error energy and error norm without use of the orthogonality conditions. Introduction of extended singular Green states leads formally to pointwise estimates for some field quantities. Numerical results for the von Kármán plate are presented.     

A semi-analytical solution for multiple circular inhomogeneities in one of two joined isotropic elastic half-planes

The paper presents a semi-analytical method for solving the problem of two joined, dissimilar isotropic elastic half-planes, one of which contains a large number of arbitrary located, non-overlapping, perfectly bonded circular elastic inhomogeneities. In general, the inhomogeneities may have different elastic properties and sizes. The analysis is based on a solution of a complex singular integral equation with the unknown tractions at each circular boundary approximated by a truncated complex Fourier series. A system of linear algebraic equations is obtained by using a Taylor series expansion. Apart from round-off, the only errors introduced into the solution are due to truncation of the Fourier series. The resulting semi-analytical method allows one to calculate the elastic fields everywhere in the half-planes and inside the inhomogeneities. Numerical examples are included to demonstrate the effectiveness of the approach.     

Dual conservation laws in elastostatics

Several new dual conservation laws are given in this study under the assumption of the existence of the complementary strain energy density function. They are valid for both finite deformation and infinitesimal deformation of elastic solid. Applications of these laws to fracture mechanics are discussed.     

Existence theorems in micropolar elastostatics

The present paper is concerned with some fundamental three-dimensional boundary value problems of micropolar elasticity. Existence theorems are derived by the method of potentials.