Complex Variable Formalism Research Articles

Adhesive contact between two-dimensional anisotropic elastic bodies

Adhesion plays a vital role in the design of smart and intelligent high-tech devices such as modern optical, microelectromechanical, and biomedical systems. However, in the literature, adhesive contact is mostly considered for contact of rigid substrates and transversely isotropic and isotropic elastic materials. The composite materials are increasingly used in the mart and intelligent high-tech devices. Since the composite materials are generally anisotropic and contact bodies are all deformable, it is more practical to consider the adhesive contact of two anisotropic elastic materials. In this paper, an adhesive contact model of anisotropic elastic bodies is established, and the closed-form solutions for two-dimensional adhesive contact of two anisotropic elastic bodies are derived. The full-field solutions and the relation for the contact region and applied force are developed using the Stroh complex variable formalism, the analytical continuation method, and concepts of the JKR adhesive model. We will show that the frictionless contact of two anisotropic elastic materials is just a special case of the present contact problem, and its solutions can be obtained by setting the work of adhesion equal to zero. In addition, we also show that our present solutions are valid for the problems of indentation by a rigid punch on an elastic half-space through a proper placement of the contact radius and the corresponding material constant. Numerical results are provided to demonstrate the accuracy, applicability, and versatility of the developed solutions.

Discussion of the linearized version of the Steigmann-Ogden surface model in plane deformation and its application to inclusion problems

We re-examine two linearized versions of the Steigmann-Ogden model of a general elastic cylindrical surface under plane deformation. The first version appears quite commonly in the literature and is distinguished by the fact that the bending moment of the surface is defined by the nominal curvature of the deformed surface (which is independent of the change in the length scale of the surface during deformation). The second version is a modified version in which the bending moment is defined by the actual curvature of the deformed surface. In both versions, the tangential force of the surface is determined by its stretch during deformation. We demonstrate from an energetic point of view that the first version is self-consistent while the second is not in that it does not yield a unique positive-definite surface strain energy density. Accordingly, we propose a refined and self-consistent linearized version of the Steigmann-Ogden model in which the bending moment of a material surface remains dependent on the actual curvature of the deformed surface while the corresponding tangential force relies on the stretch of the surface, the initial curvature of the surface and the actual curvature of the deformed surface. We formulate, in the context of the complex variable formalism of elasticity, the boundary condition corresponding to this refined version for an elastic composite system with an arbitrary curved interface (including the case of an arbitrarily-shaped inclusion embedded within an elastic matrix) under plane deformation. Some additional formulae are also given for the analysis of the boundary value problem corresponding to an arbitrarily-shaped inclusion subjected to the proposed refined boundary condition.

Coupled stretching-bending boundary element analysis for unsymmetric magneto-electro-elastic laminates with multiple holes, cracks and inclusions

Based upon the corresponding boundary element method (BEM) for the elastic composite laminates under coupled stretching-bending (CSB) deformation, the BEM for magneto-electro-elastic (MEE) laminates is developed in this study. By using the Stroh-like complex variable formalism, recently we derived Green's functions for the MEE laminates with or without holes/cracks/inclusions. Its associated fundamental solutions employed in BEM for the MEE laminated plates under CSB deformation are then derived accordingly, with which no meshes are required for the boundaries of holes/cracks/inclusions. At each corner of the plate, three nodes are usually suggested to cover all the nodal tractions (including corner force) at, behind and ahead of the corner. If the triple nodes are placed at the same position, the auxiliary equations would be necessary to replace the trivial equations for displacement-prescribed corner and are therefore constructed in this study. Since the proposed MEE-BEM is purposely organized in the same form as its corresponding elastic case, the associated techniques in dealing with singular integrals, complete solutions for boundary and internal nodes, dual coordinates for inclined holes/cracks/inclusions, and evaluation of stress intensity factors are now successfully extended from elastic to MEE laminates. Moreover, the subregion technique is employed to cover the problems with multiple holes, cracks, and inclusions.

Discussion on the conformal mapping of a half-plane onto a unit disk in anisotropic elasticity and related applications

The conformal mapping, which transforms a half-plane into a unit disk, has been used widely in studies involving an isotropic elastic half-plane under anti-plane shear or plane deformation. However, very little attention has been paid to the possibility of utilizing this mapping in the study of an anisotropic elastic half-plane under the same deformation. In this paper, we discuss a general case of an arbitrarily located anisotropic elastic half-plane that corresponds to several affine counterparts (resulting from corresponding complex variable formalism). We show that this mapping is indeed applicable to each of the affine half-planes if and only if the key parameters in the mapping satisfy simple geometrical conditions. In addition, we introduce the application of this mapping with the corresponding geometrical conditions to the related study of anisotropic thin films under two-dimensional deformation.

Equi-stress boundaries in two- and three-dimensional elastostatics: The single-layer potential approach

The well-known developments in elastostatics concerning the equi-stressness criterion of optimality for two-dimensional multi-connected unbounded solids under the bulk-dominating load are generalized toward the transient three-dimensional case with rotational symmetry. This paper advances our previous work by focusing specifically on explicitly identifying the optimal equi-stress surfaces through a simple regular integral equation which involves the single-layer potential kernel associated with the axially symmetric Laplacian. Its two-dimensional analogue is also obtained as a competitive counterpart to the commonly used complex-variable formalism. In both cases, the equations are reformulated as a minimization problem, solved numerically with a standard genetic algorithm over a wide variety of governing parameters thus permitting comparison of the shape optimization results in spatial and plane elasticity for multi-connected domains.

A Unified Definition of Stress Intensity Factors for Cracks/Corners/Interface Cracks/Interface Corners in Anisotropic/Piezoelectric/Viscoelastic Materials

To have a definition covering all the possible situations of singular orders (real or complex, distinct or repeated), the matrix form near tip solutions of interface corners written by the Stroh's complex variable formalism of anisotropic elasticity were employed in this paper. Starting from the matrix form near tip solutions for anisotropic elastic materials, identical mathematical expressions were applied to the cases of piezoelectric materials and viscoelastic materials by the ways such as the expansion of the elastic properties to include the piezoelectric effects, and the use of the Laplace transform to make the viscoelastic stress-strain relations look like linear elastic materials. The stress intensity factors defined through the matrix form expression are therefore unified for several different situations. With this unified definition, some special cases are reduced and compared with the definitions proposed in the literature.

Boundary element analysis for viscoelastic solids containing interfaces/holes/cracks/inclusions

With the aid of the elastic–viscoelastic correspondence principle, the boundary element developed for the linear anisotropic elastic solids can be applied directly to the linear anisotropic viscoelastic solids in the Laplace domain. Green's functions for the problems of two-dimensional linear anisotropic elastic solids containing holes, cracks, inclusions, or interfaces have been obtained analytically using Stroh's complex variable formalism. Through the use of these Green's functions and the correspondence principle, special boundary elements in the Laplace domain for viscoelastic solids containing holes, cracks, inclusions, or interfaces are developed in this paper. Subregion technique is employed when multiple holes, cracks, inclusions, and interfaces exist simultaneously. After obtaining the physical responses in Laplace domain, their associated values in time domain are calculated by the numerical inversion of Laplace transform. The main feature of this proposed boundary element is that no meshes are needed along the boundary of holes, cracks, inclusions and interfaces whose boundary conditions are satisfied exactly. To show this special feature by comparison with the other numerical methods, several examples are solved for the linear isotropic viscoelastic materials under plane strain condition. The results show that the present BEM is really more efficient and accurate for the problems of viscoelastic solids containing interfaces, holes, cracks, and/or inclusions.

Green's Functions for Anisotropic/Piezoelectric Bimaterials and Their Applications to Boundary Element Analysis

The Green’s function for anisotropic bimaterials has been investigated around three decades ago. Since the mathematical formulation of piezoelectric elasticity can be organized into the same form as that of anisotropic elasticity by just expanding the dimension of the corresponding matrix to include the piezoelectric effects, the extension of the Green’s function to piezoelectric bimaterials can be obtained immediately through the associated anisotropic bimaterials. In this paper, the Green’s function for the bimaterials bonded together with one anisotropic material and one piezoelectric material is derived by applying Stroh’s complex variable formalism with the aid of analytical continuation method. For this problem, the interfacial condition of electric field depends on the electric conductivity of anisotropic elastic materials. Employing these Green’s functions, a special boundary element satisfying the interfacial continuity conditions of anisotropic/piezoelectric bimaterials is developed. With the embedded Green’s functions, this special boundary element preserves two special features: (1) the interface continuity conditions are satisfied exactly and no meshes are needed along the interface; (2) the materials below and above the interface can be any kinds of piezoelectric or anisotropic elastic materials. To show the advantages of the present special boundary element, several numerical examples such as orthotropic/isotropic bimaterials, PZT-7A/PZT-5H bimaterials and anisotropic/piezoelectric bimaterials are illustrated and compared with the solutions calculated by other numerical methods. The numerical results show that the present special boundary element is not only accurate but also efficient.

The first-order orbital equation

We derive the first-order orbital equation employing a complex variable formalism. We then examine Newton’s theorem on precessing orbits and apply it to the perihelion shift of an elliptic orbit in general relativity. It is found that corrections to the inverse-square gravitational force law formally similar to that required by general relativity were suggested by Clairaut in the 18th century.

Stroh-Like Complex Variable Formalism for the Bending Theory of Anisotropic Plates

Based upon the knowledge of the Stroh formalism and the Lekhnitskii formalism for two-dimensional anisotropic elasticity as well as the complex variable formalism developed by Lekhnitskii for plate bending problems, in this paper a Stroh-like formalism for the bending theory of anisotropic plates is established. The key feature that makes the Stroh formalism more attractive than the Lekhnitskii formalism is that the former possesses the eigenrelation that relates the eigenmodes of stress functions and displacements to the material properties. To retain this special feature, the associated eigenrelation and orthogonality relation have also been obtained for the present formalism. By intentional rearrangement, this new formalism and its associated relations look almost the same as those for the two-dimensional problems. Therefore, almost all the techniques developed for the two-dimensional problems can now be applied to the plate bending problems. Thus, many unsolved plate bending problems can now be solved if their corresponding two-dimensional problems have been solved successfully. To illustrate this benefit, two simple examples are shown in this paper. They are anisotropic plates containing elliptic holes or inclusions subjected to out-of-plane bending moments. The results are simple, exact and general. Note that the anisotropic plates treated in this paper consider only the homogeneous anisotropic plates. If a composite laminate is considered, it should be a symmetric laminate to avoid the coupling between stretching and bending behaviors.

Explicit solutions for the coupled stretching–bending problems of holes in composite laminates

Although the classical lamination theory was developed long time ago, it is still not easy to apply this theory to find the analytical solutions for the curvilinear boundary value problems especially when the stretching and bending are coupled each other. To overcome the difficulties, recently we developed a Stroh-like formalism for the general composite laminates. By using this formalism, most of the relations for the coupled stretching–bending problems can be organized into the forms of Stroh formalism for two-dimensional anisotropic elasticity problems. With this newly developed Stroh-like formalism, it becomes easier to obtain an analytical solution for the coupled stretching–bending problems of holes in composite laminates. Because the Stroh-like formalism is a complex variable formalism, the analytical solutions for the whole field are expressed in complex form. Through the use of some identities derived in this paper, the resultant forces and moments around the hole boundary are obtained explicitly in real form. Due to the lack of analytical solutions for the general cases, the comparison is made with the existing analytical solutions for some special cases. In addition, to show the generality of our analytical solutions, several numerical examples are presented to discuss the coupling effect of the laminates and the shape effect of the holes.

Stroh-like formalism for the coupled stretching–bending analysis of composite laminates

Due to the two-dimensional nature of thin plates, the lamination theory considering the composite laminates with in-plane and plate bending problems coupling each other is treated in this paper by using complex variable formulation. By following the steps of Stroh formalism for two-dimensional linear anisotropic elasticity, a displacement complex variable formalism developed by the other researchers was introduced and re-derived in a different but more Stroh-like way. In addition, a brand-new mixed formalism (mixed use of displacements and stresses as basic functions) is established to compensate the displacement formalism. In order to transfer all the related formulae and mathematical techniques of the Stroh formalism to these two formalisms, the general solutions for the basic equations of lamination theory and their associated eigenrelations have been purposely arranged in the form of Stroh formalism. Moreover, by using the presently developed mixed formalism, the explicit expressions for the fundamental matrix and eigenvectors are obtained first time for the most general composite laminates. Furthermore, letting the coupling stiffness vanish, the formalism has been reduced to the case of symmetric laminates and checked by a recently developed Stroh-like formalism for the plate bending problems. The comparison between Stroh formalism for two-dimensional problem, Stroh-like formalism for plate bending problem, displacement formalism and mixed formalism is then made at the end of this paper, and through their connection some useful relations are obtained.

Development of the boundary element method for 2D piezoelectricity

Abstract The analytical and numerical foundations are laid out for the formulation of the boundary element method (BEM) for plane piezoelectric solids. We extend a physical interpretation of Somigliana's identity to piezoelectricity and give a direct formulation of the BEM in terms of the continuous distributions of point forces/charges and displacement/electric potential discontinuities in the infinite piezoelectric domain. We adopt Stroh's complex variable formalism for piezoelectricity to derive the point force/charge and the displacement/electric potential discontinuity, their dipoles and continuous distributions systematically. The duality relations between the force/charge and the displacement/electric potential solutions, embedded in the Stroh formalism, are exploited as the foundations for the analytic and the numerical approaches to the piezoelectric boundary value problems in two dimensions. These approaches enable us to solve important problems of piezoelectricity with arbitrary geometry and composition.

A dislocation and point force approach to the boundary element method for mixed mode crack analysis of plane anisotropic solids

In this paper we formulate a direct boundary element method (BEM) for plane anisotropic elasticity (i.e., the in‐plane deformation decoupled from the out‐of‐plane deformation) based on distributions of point forces and dislocation dipoles. According to a physical interpretation of Somigliana's identity the displacement field in a finite body R is represented by the continuous distributions of point forces and dislocation dipoles along the imaginary boundary ?R of the finite domain R embedded in an infinite body. We adopt Stroh's complex variable formalism for anisotropic elasticity and represent the point force and the dislocation, their dipoles, and continuous distributions systematically exploiting the duality relations between the point force and the dislocation solutions. Explicit formulas for the displacement and the traction formulations, obtained by analytical integration of the boundary integrals, are given. We apply these formulas to mixed mode crack problems for multiply cracked anisotropic bodies by extending the physical interpretation of Somigliana's identity to cracked bodies and representing the crack by the continuous distribution of dislocation dipoles. With the help of the conservation integrals of anisotropic elasticity, we will demonstrate the capability of the method to determine the mixed mode stress intensity factors (KI and KII ) accurately.

Dislocation and point-force-based approach to the special Green's Function BEM for elliptic hole and crack problems in two dimensions

In this paper we give the theoretical foundation for a dislocation and point-force-based approach to the special Green's function boundary element method and formulate, as an example, the special Green's function boundary element method for elliptic hole and crack problems. The crack is treated as a particular case of the elliptic hole. We adopt a physical interpretation of Somigliana's identity and formulate the boundary element method in terms of distributions of point forces and dislocation dipoles in the infinite domain with an elliptic hole. There is no need to model the hole by the boundary elements since the traction free boundary condition there for the point force and the dislocation dipole is automatically satisfied. The Green's functions are derived following the Muskhelishvili complex variable formalism and the boundary element method is formulated using complex variables. All the boundary integrals, including the formula for the stress intensity factor for the crack, are evaluated analytically to give a simple yet accurate special Green's function boundary element method. The numerical results obtained for the stress concentration and intensity factors are extremely accurate. © 1997 John Wiley & Sons, Ltd.

Computational modeling of elastic and plastic multiple cracks by the fundamental solutions

The fundamental solutions of elasticity are used to establish a numerical method for elastic and plastic multiple crack problems in two dimensions. The continuous distributions of the point forces, dislocations, and the plastic sources are used systematically to model the crack, non-crack boundary, and the plastic deformation. Use of these singularities are guided strictly by the physical interpretation of the problem. We adopt Muskhelishvili's complex variable formalism that facilitate the analytical evaluation of the integrals representing the continuous distributions of the singularities. The resulting numerical method is concise and accurate enough to be used for elastic and plastic multiple crack problems.

Magneto-elastic stresses of an infinite anisotropic medium with cylindrical hole

The paper is concerned with the derivation of a two dimensional theory of magneto-elasticity of anisotropic solids with cylindrical hole. The theory has been developed in complex variable formalism and illustrated by considering a particular type of magnetic field.

Two-dimensional elastic inclusion problems

ABSTRACTAn account is given of Eshelby's point-force method for solving elastic inclusion problems, and of his equations relating an in homogeneity to its equivalent inclusion. The introduction of complex variable formalism enables explicit solutions to be found in various two-dimensional cases. Strain energies are calculated. The equilibrium shape of an elliptic inclusion exhibits an interesting feature not previously expected. A fresh analysis of stress magnification effects is developed.