On displacement methods in planar anisotropic elasticity

Two displacement formulation methods are presented for problems of planar anisotropic elasticity. The first displacement method is based on solving the two governing partial differential equations simultaneously/ This method is a recapitulation of the orignal work of Eshelby, Read and Shockley [7] on generalized plane deformations of anisotropic elastic materials in the context of planar anisotropic elasticity.The second displacement method is based on solving the two governing equations separately. This formulation introduces a displacement function, which satisfies a fourth-order partial differential equation that is identical in the form to the one given by Lekhnitskii [6] for monoclinic materials using a stress function. Moreover, this method parallels the traditional Airy stress function method and thus the Lekhnitskii method for pure plane problems. Both the new approach and the Airy stress function method start with the equilibrium equations and use the same extended version of Green's theorem (Chou and Pagano [13], p. 114; Gao [11]) to derive the expressions for stress or displacement components in terms of a potential (stress or displacement) function (see also Gao [10, 11]). It is therefore anticipated that the displacement function involved in this new method could also be evaluated from measured data, as was done by Lin and Rowlands [17] to determine the Airy stress function experimentally.The two different displacement methods lead to two general solutions for problems of planar anisotropic elasticity. Although the two solutions differ in expressions, both of the depend on the complex roots of the same characteristic equation. Furthermore, this characteristic equation is identical to that obtained by Lekhnitskii [6] using a stress formulation. It is therefore concluded that the two displacement methods and Lekhnitskii's stress method are all equivalent for problems of planar anisotropic elasticity (see Gao and Rowlands [8] for detailed discussions).     

Matrix inversions and singular solutions in the linear theories of elasticity

Elementary matrix multiplications and inversions lead to Galerkin type representations for the steady-state vibration equations of linear classical elasticity, thermoclasticity, Mindlin's couple stress and Eringen's micropolar theories of elasticity. The use of representations and the equations satisfied by potentials is further illustrated by obtaining singular solutions. Some of the results agree with known solutions.     

Existence theorems in the theory of mixtures

This paper is concerned with boundary-value problems of the linear theory for binary mixtures of elastic bodies. First, a counterpart of the Boussinesq-Somigliana-Galerkin solution in classical elastostatics is established and the fundamental solutions in the equilibrium theory of homogeneous and isotropic mixtures are derived. Then, representations of Somigliana type for the displacement fields are presented. The potentials of single layer and double layer are used to reduce the boundary-value problems to singular integral equations. Existence and uniqueness results are established.     

On the representation of three-dimensional elasticity solutions with the aid of complex valued functions

In this paper the representation of three-dimensional displacement fields in linear elasticity in terms of six complex valued functions is considered. The representation includes the complex Muskhelishvili formulation for plane strain as a special case. The completeness of the complex representation for regular solutions is shown and a relationship to the Neuber/Papkovich solutions is given.     

Microstructural effects in non linear stratified composites

In this paper, we analyze the microstructural effects on non linear elastic and periodic composites within the framework of asymptotic homogenization. We assume that the constitutive laws of the individual constituents derive from strain potentials. The microstructural effects are incorporated by considering the higher order terms, which come from the asymptotic series expansion. The complete solution at any order requires the resolution of a chain of cell problems in which the source terms depend on the solution at the lower order. The influence of these terms on the macroscopic response of the non linear composite is evaluated in the particular case of a stratified microstructure. The analytic solutions of the cell problems at the first and second order are provided for arbitrary local strain–stress laws which derive from potentials. As classically, the non-linear dependence on the applied macroscopic strain is retrieved for the solution at the first order. It is proved that the second order term in the expansion series also exhibits a non linear dependence with the macroscopic strain but linearly depends on the gradient of macroscopic strain. As a consequence, the macroscopic potential obtained by homogenization is a quadratic function of the macroscopic strain gradient when the expansion series is truncated at the second order. This model generalizes the well known first strain gradient elasticity theory to the case of non linear elastic material. The influence of the non local correctors on the macroscopic potential is investigated in the case of power law elasticity under macroscopic plane strain or antiplane conditions.     

Integral formulas in symmetric and asymmetric elasticity

Several new integral representations of the solutions to some problems of the moment and nonmoment theories of elasticity for heterogeneous bodies are proposed in terms of the solutions to the same problems for homogeneous bodies. In particular, these integral representations can be used to substantiate the homogenization procedure for composite mechanics problems.     

Dimension Reduction in the Context of Structured Deformations

In the present paper the basic boundary value problems (BVPs) of the full coupled linear theory of elasticity for triple porosity materials are investigated by means of the potential method (boundary integral equation method) and some basic results of the classical theory of elasticity are generalized. In particular, the Green’s identities and the formula of Somigliana type integral representation of regular vector and regular (classical) solutions are presented. The representation of Galerkin type solution is obtained and the completeness of this solution is established. The uniqueness theorems for classical solutions of the internal and external BVPs are proved. The surface (single-layer and double-layer) and volume potentials are constructed and their basic properties are established. Finally, the existence theorems for classical solutions of the BVPs are proved by means of the potential method and the theory of singular integral equations.     

Couple stress theory for solids

By relying on the definition of admissible boundary conditions, the principle of virtual work and some kinematical considerations, we establish the skew-symmetric character of the couple-stress tensor in size-dependent continuum representations of matter. This fundamental result, which is independent of the material behavior, resolves all difficulties in developing a consistent couple stress theory. We then develop the corresponding size-dependent theory of small deformations in elastic bodies, including the energy and constitutive relations, displacement formulations, the uniqueness theorem for the corresponding boundary value problem and the reciprocal theorem for linear elasticity theory. Next, we consider the more restrictive case of isotropic materials and present general solutions for two-dimensional problems based on stress functions and for problems of anti-plane deformation. Finally, we examine several boundary value problems within this consistent size-dependent theory of elasticity.     

On the steady vibrations of elastic materials with voids

This paper is concerned with the linear elastodynamics of homogeneous and isotropic materials with voids. First, the singular solutions corresponding to concentrated forces in the case of steady vibrations are established. Then, representations of Somigliana type for the displacement field and the change in the volume fraction field are presented. Radiation conditions of Sommerfeld type are derived. The potentials of single layer and double layer are used to reduce the boundary value problems to singular integral equations for which Fredholm's basic theorems are valid. Existence and uniqueness results for exterior problems are established.     

Specific features of the nonlinearly elastic behavior of cylindrical compressible bodies in torsion

The torsion problem of a cylinder with a circular transverse cross section twisted by end moments that are equal in magnitude and opposite in direction is considered for various models of nonlinearly elastic compressible media. The problem is solved by the semi-inverse method of elasticity theory. The Poynting effect, which consists of variation in the length of a shaft in torsion, is treated qualitatively and quantitatively. The results of the numerical and asymptotic (only terms that are quadratic relative to the displacement gradient are conserved) solutions for various models of the nonlinearly elastic behavior of materials are compared. An analysis of the results shows that in some cases, the quasilinear model is not applicable for studying the behavior of nonlinearly elastic compressible media. Rostov State Construction University, Rostov-on-Don 344022. Rostov State University, Rostov-on-Don 344090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 2, pp. 188–193, March–April, 2000.     

On the approximate solution of elliptic,self adjoint boundary value problems

Summary Integral representations for the solutions to linear elliptic self adjoint boundary value problems are derived in terms of two functions which are generalisations of the single and double layer potentials used in the theory of harmonic functions. The generalised potentials are constructed in terms of a fundamental solution which is an approximation to the exact kernel of the boundary value problem in question. The representations so obtained are shown to provide a basis from which strict approximations to the solutions of boundary value problems can be developed. In particular the structure of the integral equation representing the given boundary value problem is precisely determined.     

Diagonalization of a three-dimensional system of equations in terms of displacements of the linear theory of elasticity of transversely isotropic media

A dynamic three-dimensional system of linear equations in terms of displacements of the theory of elasticity of transversely isotropic media is given explicit expressions for phase velocities and polarization vectors of plane waves. All the longitudinal normals are found. For some values of the elasticity moduli, the system of equations is reduced to a diagonal shape. For static equations, all the conditions of the system ellipticity are determined. Two new representations of displacements through potential functions that satisfy three independent quasi-harmonic equations are given. Constraints on elasticity moludi, at which the corresponding coefficients in these representations are real, different, equal, or complex, are determined. It is shown that these representations are general and complete. Each representation corresponds to a recursion (symmetry) operator, i.e., a formula of production of new solutions.     

On the dissipation associated with equilibrium shocks in finite elasticity

Equilibrium fields with discontinuous displacement gradients can occur in finite elasticity for certain materials. The presence of such equilibrium shocks affects the energy balance in the elastostatic field, and the present paper is concerned with a notion of dissipation associated with this energy balance. A dissipation inequality is proposed for three-dimensional equilibrium shocks for both compressible and incompressible materials. The consequences of this inequality are studied for weak shocks in plane strain for compressible materials and for shocks of arbitrary strength in anti-plane strain for a class of incompressible materials. A thermodynamic argument for the dissipation inequality is also given.The results communicated in this paper were obtained in the course of an investigation supported by Contract N00014-75-C-0196 between the California Institute of Technology and the Office of Naval Research.     

A finite element implementation of the stress gradient theory

In this contribution, a finite element implementation of the stress gradient theory is proposed. The implementation relies on a reformulation of the governing set of partial differential equations in terms of one primary tensor-valued field variable of third order, the so-called generalised displacement field. Whereas the volumetric part of the generalised displacement field is closely related to the classic displacement field, the deviatoric part can be interpreted in terms of micro-displacements. The associated weak formulation moreover stipulates boundary conditions in terms of the normal projection of the generalised displacement field or of the (complete) stress tensor. A detailed study of representative boundary value problems of stress gradient elasticity shows the applicability of the proposed formulation. In particular, the finite element implementation is validated based on the analytical solutions for a cylindrical bar under tension and torsion derived by means of Bessel functions. In both tension and torsion cases, a smaller is softer size effect is evidenced in striking contrast to the corresponding strain gradient elasticity solutions.

     

Singular integral equations in elastic dielectrics

Galerkin representations for the displacement vector, polarization vector and the potential field are obtained by elementary matrix inversions of the equations of equilibrium. Matrices of fundamental solutions of an infinite elastic dielectric continuum subjected to a concentrated body force, an electric force, and a charge density, are constructed. Theorems are proved on the discontinuity of double layer potentials and R, M, M operators of single layer potentials. By means of these theorems, the solution of the two basic boundary value problems has been reduced to the solution of a system of seven singular integral equations.     

Mixed Plane Problems in Linearized Solid Mechanics: Exact Solutions

The paper analyzes the exact solutions to mixed plane problems of linearized solid mechanics in cases of statics, dynamics, stability, and fracture. The exact solutions have a universal form for compressible and incompressible, elastic and plastic bodies and account for stresses and displacements expressed in terms of analytical functions of complex variables. To obtain these solutions, the use is made of complex variable theory, in particular, the Riemann–Hilbert methods and Keldysh–Sedov formula. When the initial (residual) stresses tend to zero, the exact solutions go over into the corresponding exact solutions of classical linear solid mechanics, which are based on the complex representations due to Muskhelishvili, Lekhnitskii, and Galin     

Displacement function and simplifying of plane elasticity problems of two-dimensional quasicrystals with noncrystal rotational symmetry

The paper systematically investigates the plane elasticity problems of two-dimensional quasicrystals with noncrystal rotational symmetry. First, applying their independent elastic constants, the equilibrium differential equations of the problems in terms of displacements are derived and it is found that the plane elasticity of pentagonal quasicrystals is the same as that of decagonal. Then by introducing displacement functions, huge numbers of complicated partial differential equations of the problems are simplified to a single or a pair of partial differential equations of higher order, which is called governing equations, such that the problems can be easily solved. Finally, some solving methods of these governing equations obtained are introduced briefly.     

Stress singularities in an anisotropic body of revolution

To fill the gap in the literature on the application of three-dimensional elasticity theory to geometrically induced stress singularities, this work develops asymptotic solutions for Williams-type stress singularities in bodies of revolution that are made of rectilinearly anisotropic materials. The Cartesian coordinate system used to describe the material properties differs from the coordinate system used to describe the geometry of a body of revolution, so the problems under consideration are very complicated. The eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions by directly solving the three-dimensional equilibrium equations in terms of the displacement components. The correctness of the proposed solution is verified by convergence studies and by comparisons with results obtained using closed-form characteristic equations for an isotropic body of revolution and using the commercial finite element program ABAQUS for orthotropic bodies of revolution. Thereafter, the solution is employed to comprehensively examine the singularities of bodies of revolution with different geometries, made of a single material or bi-materials, under different boundary conditions.     

A Theory of General Solutions of Plane Problems in Two-Dimensional Octagonal Quasicrystals

A theory of general solutions of plane problems is developed for the coupled equations in plane elasticity of two-dimensional octagonal quasicrystals. In virtue of the operator method, the general solutions of the antiplane and inplane problems are given constructively with two displacement functions. The introduced displacement functions have to satisfy higher order partial differential equations, and therefore it is difficult to obtain rigorous analytic solutions directly and is not applicable in most cases. In this case, a decomposition and superposition procedure is employed to replace the higher order displacement functions with some lower order displacement functions, and accordingly the general solutions are further simplified in terms of these functions. In consideration of different cases of characteristic roots, the general solution of the antiplane problem involves two cases, and the general solution of the inplane problem takes three cases, but all are in simple forms that are convenient to be applied. Furthermore, it is noted that the general solutions obtained here are complete in x ₃-convex domains.      

Equivalent and separable strain-energy functions in compressible,finite elasticity

A material non-uniqueness is identified for isotropic, homogeneous, hyperelastic, compressible materials. This non-uniqueness is a special case of the hyperelastic null Lagrangian. Equivalence classes of strain-energy functions (SEFs) are then obtained. This equivalence can be used to extend in a natural way some of the common SEF used in incompressible elasticity to compressible elasticity. The familiar Valanis–Landel separable form of the SEF for incompressible materials is extended in this way. Some necessary restrictions are imposed on this new form and its behaviour in uniaxial tension is discussed. Previous problems in using the Valanis–Landel form of the SEF in compressible elasticity have been overcome.