Stroh Complex Research Articles

Integral Equations of Plane Magnetoelectroelasticity for a Cracked Bimaterial With Thin Inclusions

Based on the combined application of the Stroh formalism and the theory of functions of complex variable, we deduce dual integral equations for a magnetoelectroelastic bimaterial. For the first time, we construct the integral representations of the Stroh complex potentials and the explicit expressions for all kernels in terms of the parameters and matrices of the applied formalism. This noticeably reduces the amount of computations required to get the governing equations of the boundary-element methods. The explicit formulas are obtained for the principal parts of the complex potentials. These formulas enable us to consider homogeneous magnetoelectromechanical loads applied at infinity. The obtained equations, together with previously developed models of thin deformable inclusions, are introduced in the computational algorithm of the boundary-element method with jump functions. The solution of test problems reveals high accuracy and efficiency of the proposed approach. Some solutions are presented for new problems posed for a magnetoelectroelastic bimaterial with thin inclusion.

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.

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.

The anisotropic elastic solid with an elliptic hole or rigid inclusion

The two-dimensional problem of an elliptic hole in a solid of general anisotropy subject to an arbitrarily prescribed traction on the hole surface is studied. Stroh's complex formalism is adopted here but real-form solutions are obtained for the displacement and the hoop stress around the hole. For an arbitrarily prescribed traction, the solutions are in the form of an infinite series. However, through the use of a conjugate function they can be expressed in closed form directly in terms of the applied traction. We also consider an elliptic rigid inclusion subject to a force and a torque. Again, real-form solutions are obtained for the interface stress, the hoop stress around the rigid inclusion and the rotation of the rigid inclusion. When there is no torque applied at the inclusion, the traction vector at the surface of the rigid inclusion is in the direction of the applied force and is a constant when the ellipse is a circle. This is an unexpected result since the material surrounding the rigid inclusion is of general anisotropy.

Solutions for the anisotropic elastic wedge at critical wedge angles

The classical solution for an isotropic elastic wedge loaded by uniform tractions on the sides of the wedge becomes infinite everywhere in the wedge when the wedge angle 2α equals π, 2π or 2α* where tan 2α* = 2α*. When the wedge is loaded by a concentrated couple at the wedge apex the solution also becomes infinite at 2α = 2α*. A similar situation occurs when the wedge is anisotropic except that 2α* is governed by a different equation and depends on material properties. Solutions which do not become infinite everywhere in the wedge are available for isotropic elastic wedges. In this paper we present solutions for the anisotropic elastic wedge at critical wedge angles. The main feature of the solutions obtained here is that they are in a real form even though Stroh's complex formalism is employed.

TWO-DIMENSIONAL PROBLEMS OF THE ANISOTROPIC ELASTIC SOLID WITH AN ELLIPTIC INCLUSION

The two-dimensional elliptic inclusion problems have been widely studied and many important results have been obtained. Solutions for which the inclusion as well as the matrix are of general anisotropic elastic materials are presented in this paper using the Stroh formalism. New identities which relate Stroh's complex eigenvalues and eigenvectors to real quantities obtainable directly from the elastic constants are derived, through which several solutions are expressed in real form. The special cases in which inclusion is rigid or a void are also studied