Real Form Solutions Research Articles

An Anisotropic Elastic Solid With an Elliptical Inhomogeneity Under a Non-Uniform In-Plane And Anti-Plane Remote Loading

Summary We consider the case of an anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic matrix subjected to a non-uniform remote loading described by remote stresses and strains which are linear functions of the two in-plane coordinates. The internal stresses and strains within the elliptical inhomogeneity are found to be linear functions of the two in-plane coordinates. In addition, we obtain explicit real-form solutions describing the elastic field inside the inhomogeneity as well as hoop stress vectors and hoop stresses on the matrix side and on the inhomogeneity side. We also obtain the corresponding solutions for the two limiting cases in which the elliptical inhomogeneity takes the form of a hole or a rigid inhomogeneity. The solution method presented here can be extended to accommodate the more general scenario in which the remote applied stresses and strains are arbitrary-order polynomials of the two in-plane coordinates.

Real-form solution for an anisotropic elastic elliptical inhomogeneity under uniform heat flux

We use the extended Stroh sextic formalism for thermo-anisotropic elasticity to present an elegant solution of the problem of an anisotropic elastic elliptical inhomogeneity embedded in an infinite anisotropic elastic matrix subjected to uniform remote heat flux. We prove rigorously that the remote thermal stresses cannot be set to zero in a generally anisotropic elastic matrix. In particular, a real-form solution is derived for the internal thermoelastic field describing stresses, strains, and displacements within the elliptical inhomogeneity.

TU20. APPLICATION OF REGULARISED STRUCTURAL EQUATION MODELLING TO A BIOPSYCHOSOCIAL MEDIATION MODEL

This article presents three-dimensional (3D) analysis of generally anisotropic elastic, piezoelectric, piezomagnetic, and magneto-electro-elastic solids by boundary element method (BEM). The associated Green's functions of displacements and tractions in both complex and real forms together with their first derivatives are derived completely in the first time by using Radon–Stroh formalism. Like the Stroh formalism for two-dimensional problems, the solutions for the solids with these different material types all bear exactly the same mathematical forms distinguished by the contents and dimensions of the related matrices and vectors. This feature provides a big advantage for computer programming. To emphasize the development of 3D-BEM, the system of algebraic equations to calculate the nodal displacements and nodal tractions along the boundary of 3D solids is formulated. The relations for calculating strains and stresses at the boundary nodes as well as at the interior points are also derived. After the employment of Green's functions to the newly formulated 3D-BEM, we found that the 3D-BEM with complex form solution is computationally more efficient than that with real form solution, although the opposite performance was observed for a single point calculation of Green's function. To verify the correctness of our solutions, numerical examples of a cube with or without through hole are presented as illustrations of our successful implementation in the BEM.

Three-dimensional boundary element analysis for anisotropic elastic solids and its extension to piezoelectric and magnetoelectroelastic solids

This article presents three-dimensional (3D) analysis of generally anisotropic elastic, piezoelectric, piezomagnetic, and magneto-electro-elastic solids by boundary element method (BEM). The associated Green's functions of displacements and tractions in both complex and real forms together with their first derivatives are derived completely in the first time by using Radon–Stroh formalism. Like the Stroh formalism for two-dimensional problems, the solutions for the solids with these different material types all bear exactly the same mathematical forms distinguished by the contents and dimensions of the related matrices and vectors. This feature provides a big advantage for computer programming. To emphasize the development of 3D-BEM, the system of algebraic equations to calculate the nodal displacements and nodal tractions along the boundary of 3D solids is formulated. The relations for calculating strains and stresses at the boundary nodes as well as at the interior points are also derived. After the employment of Green's functions to the newly formulated 3D-BEM, we found that the 3D-BEM with complex form solution is computationally more efficient than that with real form solution, although the opposite performance was observed for a single point calculation of Green's function. To verify the correctness of our solutions, numerical examples of a cube with or without through hole are presented as illustrations of our successful implementation in the BEM.

Green’s function for the composite laminates with bending extension coupling

By employing the recently developed Stroh-like formalism for the composite laminates with bending extension coupling, the explicit closed form solutions for an infinite composite laminate subjected to concentrated forces and moments at any arbitrary points have been found analytically. Although Stroh-like formalism is a complex variable formulation, through the use of some identities developed in the literature real form solutions of the present problem are also obtained in this paper. Because the concentrated forces and moments will influence the mechanical behavior of the composite in different ways when they are applied in different directions, three different loading cases are discussed. The results show that the solutions corresponding to the inplane concentrated forces f ̂ 1 , f ̂ 2 and out-of-plane concentrated moments m ̂ 1 , m ̂ 2 have exactly the same form as that of the corresponding two-dimensional problems. While for the cases under the concentrated force f ̂ 3 and pure torsion m ̂ 3 , new types of solutions are obtained explicitly.

Explicit Expressions of the Fundamental Elasticity Matrices of Stroh-Like Formalism for Symmetric/Unsymmetric Laminates

AbstractBased upon our recent development of Stroh-like forma lism for symmetric/unsymmetric laminates, most of the relations for bending problems can be organized into the forms of Stroh formalism for two-dimensional problems. Through the use of Stroh-like formalism, the fundamental elasticity matrices Ni, S, H and L appear frequently in the real form solutions of plate bending problems. Therefore, the determination of these matrices becomes important in the analysis of plate bending problems. In this paper, by following the approach for two-dimensional problems, we obtain the explicit expressions of the fundamental elasticity matrices for symmetric and unsymmetric laminates, which are all expressed in terms of the extensional, bending and coupling stiffnesses of the composite laminates.

Exact solution for mixed boundary value problems at anisotropic piezoelectric bimaterial interface and unification of various interface defects

The special mixed boundary value problem in which a debonded conducting rigid line inclusion is embedded at the interface of two piezoelectric half planes is solved analytically by employing the 8-D Stroh formalism. Different from existing interface insulating crack model and interface conducting rigid line inclusion model, the presently analyzed model is based on the assumption that all of the physical quantities, i.e., tractions, displacements, normal component of electric displacements and electric potential, are discontinuous across the interface defect. Explicit solutions for stress singularities at the tips of debonded conducting rigid line inclusion are obtained. Closed form solutions for the distribution of tractions on the interface, surface opening displacements and jump in electric potential on the debonded inclusion are also obtained, in addition real form solutions for these physical quantities are derived. Various forms of interface defect problems encountered in practice are solved within a unified framework and the stress singularities induced by those interface defects are discussed in detail. Particularly, we find that the analysis of interface cracks between the embedded electrode layer and piezoelectric ceramics can also be carried out within the unified framework.

Solving the punch problems by analogy with the interface crack problems

By cutting through the contact/crack surface, the contact/crack problems can be treated as a half-plane problem with the displacements/tractions prescribed along a specified region. Through the conversion to half-plane problems, the analogy between the contact problems and crack problems has been noticed before. However, to the authors' knowledge, no detailed analogous relations have been provided. Moreover, no contact problems have been solved directly by just using the corresponding solutions of the crack problems, or vice versa. In this paper, special consideration will be focused upon the connections between the punch problems and the interface crack problems with one of the materials to be rigid. Similar to the analogy between line forces and line dislocations, cracks and rigid fine inclusions, or holes and rigid inclusions, we may now solve the punch problems by analogy with the interface crack problems. In addition, very simple explicit solutions for the contact pressure and the surface deformation are derived in this paper. Moreover, a general procedure to get the real form solution is also described. Finally, three representative punch problems are solved completely. They are the indentation by a flat-ended punch, a flat-ended punch tilted by a couple, and the indentation by a parabolic punch. The explicit full field solutions, and the real form solutions for the contact pressure and surface deformation of these three problems are all provided. Based upon these closed-form solutions, several numerical examples were done and their related stresses contours, surface deformations and contact pressures were also plotted to help us see more clearly the physical behaviors of the punch problems.

Piezoelectric solid with an elliptic inclusion or hole

The two-dimensional problem of an elliptic hole in a solid of anisotropic piezoelectric material is studied. The Stroh formalism is adopted here. Real form solutions are obtained along the hole boundary in the case of an arbitrarily prescribed vector field on the hole surface. For an elliptic rigid inclusion of electric conductor subjected to a line force, a torque, and a line charge, a real form solution at the interface is obtained. Finally, general solutions for an elliptic piezoelectric inclusion with uniform loading at infinity are investigated.

Line Force, Charge, and Dislocation in Anisotropic Piezoelectric Composite Wedges and Spaces

Two-dimensional problems of anisotropic piezoelectric composite wedges and spaces are studied. The Stroh formalism is employed to obtain the basic real-form solution in terms of two arbitrary constant vectors for a particular wedge. Explicit real-form solutions are then obtained for (i) a composite wedge subjected to a line force and a line charge at the apex of the wedge and (ii) a composite space subjected to a line force, line charge, line dislocation, and an electric dipole at the center of the composite space. For the composite wedge the surface traction on any radial plane θ = constant and the electric displacement Dθ normal to the radial plane θ = constant vanish everywhere. For the composite space these quantities may not vanish but they are invariant with the choice of the radial plane.

Explicit expressions of the Barnett-Lothe tensors for anisotropic materials

The three Barnett-Lothe tensorsS, H, andL appear frequently in the real form solutions of two-dimensional anisotropic elasticity problems. Explicit expressions for the components of these tensors are presented for general anisotropic materials. The special cases of monoclinic materials with the plane of material symmetry at x3=0, x2=0, and x1=0 are then deduced. For monoclinic materials with the symmetry plane at x2=0 or x1=0, the locations of image singularities for the Green's functions for a half-space have a special geometry.

Barnett-Lothe tensors and their associated tensors for monoclinic materials with the symmetry plane at x 3=0

The three Barnett-Lothe tensors S, H, L and the three associated tensors S(θ), H(θ), L(θ) appear frequently in the real form solutions to two-dimensional anisotropic elasticity problems. Explicit expressions of the components of these tensors are derived and presented for monoclinic materials whose plane of material symmetry is at x 3=0. We use the algebraic formalism for these tensors but the results are derived not by the straight-forward substitution of the complex matrices A and B into the formulae. Instead, we find the product −AB -1, whose real and imaginary parts are SL -1 and L -1, respectively. The tensors S, H, L are then determined from SL -1 and L -1. For S(θ), H(θ), L(θ) we again avoid the direct substitution by employing an alternate approach. The new approaches require minimal algebra and, at the same time, provide simple and concise expressions for the components of these tensors. Although the new approaches can be extended, in principle, to monoclinic materials whose plane of symmetry is not at x 3=0 and to materials of general anisotropy, the explicit expressions for these materials are too complicated. More studies are needed for these materials.

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.

Line Inclusions in Anisotropic Elastic Solids

A line inclusion located at x2 = 0, |x1| < 1 in the anisotropic elastic medium of infinite extent under uniform loading at infinity is considered. Stroh’s formalism is used to find the displacement and stress fields. The inclusion can be rigid or elastic. Conditions on the loading under which the line inclusion does not disturb the homogeneous field are derived. For the rigid inclusion, a real form solution is obtained for the stress and displacement along x2 = 0. When the inclusion is elastic (and anisotropic), a pair of singular Fredholm integral equations of the second kind is derived for the difference in the stress on both surfaces of the inclusion. The pair can be decoupled and asymptotic solutions of the integral equation are obtained when λ, which represents the relative rigidity of the matrix to the inclusion, is small. For the general cases, the integral equation is solved by a numerical discretization. Excellent agreements between the asymptotic and numerical solutions are observed for small λ.