Rigid Line Inclusion Research Articles

Quadruple Trigonometrical Series Equations and Their Application to an Inclusion Problem in the Theory of Elasticity

Closed form solution of quadruple series equations involving cosine kernels has been obtained by reducing the series equations into triple Abel's type integral equations which in turn are reduced to a single integral equation. Making use of finite Hilbert transforms the solution of the single integral equation is obtained in closed form. This solution is used to solve an electrostatic problem. The results of this paper have also been used in a two‐dimensional elastostatic problem under anti‐plane shear and the effect of rigid line inclusions with thickness on the Griffith cracks has been examined. The expressions for shear stress and stress intensity factor at the tip of the crack are obtained. Finally, some numerical results for the stress intensity factor and shear stress distribution are obtained.

Green’s functions for an anisotropic thin plate with a crack or an anticrack

Based on the octet formalism established by the authors, Green’s functions for an infinite anisotropic elastic thin plate with a Griffith crack or an anticrack (rigid line inclusion) are studied. The plate can be laminated or consist of a continuously inhomogeneous material in the thickness direction. Two systems of problems of the plate under non-self-equilibrated loads are solved. The first is associated with discontinuous in-plane displacements and slopes, in-plane concentrated forces and out-of-plane concentrated moments. The second is associated with a transverse concentrated force. The Green functions for an infinite plate with a crack or an anticrack and the surface Green functions for an infinite plate with a crack are obtained in an exact closed form. The surface Green functions and associated stress resultants at any point throughout the line where the crack is located are presented in a real form. The intensity factor of membrane stress resultants and bending moments are obtained in a real form for the Griffith crack subjected to concentrated loads.

Interaction between cracks and rigid-line inclusions by an integral equation approach

An integral equation approach is presented to investigate the interaction between cracks and rigid-line inclusions embedded in an infinite isotropic elastic matrix subject to remote loading. The relevant fundamental solutions in the integral formulation are presented. Special tip elements are used to simulate the \(\) variation of the discontinuous displacements over the crack surfaces, and the axial and shear forces along the rigid-line inclusions. The stress intensity factors at the crack tips and at the ends of the rigid-line inclusions are computed and compared with available solutions.

Explicit Solutions For the Magnetoelastic Fields with a Rigid Line Inclusion

ABSTRACTA general solution to the magnetoelastic problem with a rigid line inclusion is presented. Based upon the complex variable theory, the proposed analysis dealing with sectionally holomorphic functions can be reduced to find the solution of the Hilbert problem. It is indicated that the magnetoelastic stress fields near the inclusion tip possess a square root singularity just like that of the corresponding crack problem. The stress singularity coefficients which are defined in this study to characterize the near tip fields are similar to the stress intensity factors for crack problem. Numerical results of the stress distribution in the vicinity of inclusion tip are also displayed in graphic form to elucidate the effect of various parameters.

Determination of stress intensity factors for bimaterial interface stationary rigid line inclusions by boundary element method

The stress intensity factors for a rigid line inclusion lying along a bimaterial interface are calculated by the boundary element method with the multiregion and the discontinuous traction singular elements. The relationships between the stress intensity factors and the inclusion surface stresses are derived. The numerically computed stress intensity factors for the bimaterial interface rigid line inclusion in the infinite body are proved to be in good agreement within 3% when compared with the previous exact solutions. In the finite bimaterial models, the stress intensity factors for the center and edge rigid line inclusions at the interface are computed with the variation of the rigid line inclusion length and the shear modulus ratio under the uniaxial and biaxial loading conditions.

Numerical Method of Singular Integral Equations for Anticrack Analysis by Body Force Method.

It is known that each stress component is intensified at the tips of an anticrack (rigid line inclusion) as well as a crack, proportionally to γ-1/2. This paper deals with the numerical method of the singular integral equations of the Cauchy type for the analysis of stress-singularities at anticrack tips by using the body force method. In the singular integral equations, the densities of the body forces which should be distributed along the anticrack surfaces are to be obtained so that the boundary conditions on the anticrack surfaces can be satisfied. The numerical method of the singular integral equations is based on the Lobatto-Chebyshev integration formula. In order to demonstrate the effectiveness of the proposed method, some numerical examples are given. The calculated stresssingularities at anticrack tips converge very fast with the increase of the number of the collocation points and some of them are compared with the exact solutions in order to verify the accuracy.

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.

Antiplane problems of periodical rigid line inclusions between dissimilar anisotropic materials

Antiplane problems of periodical rigid line inclusions between dissimilar anisotropic materials

Analysis of the Singularity for the Vertical Contact Problem of Line Crack-Inclusion

Taking the short-fiber composite materials as engineering back-ground, utilizing the existing basic solutions of single inclusion and single crack, the plane problem of vertical contact interactions between line crack and rigid line inclusion in infinite plane (matrix) from the viewpoint of crack fracture mechanics is studied. According to boundary conditions, a set of standard Cauchy-type singular integral equations of the problem is obtainable. Besides, singular indexes, stresses and stress intensity factors around the contact point are expressed. Numerical examples are given to provide references to engineering.

An interface inclusion between two dissimilar piezoelectric materials

The generalized two-dimensional problem of a dielectric rigid line inclusion, at the interface between two dissimilar piezoelectric media subjected to piecewise uniform loads at infinity, is studied by means of the Stroh theory. The problem was reduced to a Hilbert problem, and then closed-form expressions were obtained, respectively, for the complex potentials in piezoelectric media, the electric field inside the inclusion and the tip fields near the inclusion. It is shown that in the media, all field variables near the inclusion-tip show square root singularity and oscillatory singularity, the intensity of which is dependent on the material constants and the strains at infinity. In addition, it is found that the electric field inside the inclusion is singular and oscillatory too, when approaching the inclusion-tips from inside the inclusion.

ANTIPLANE PROBLEMS OF PERIODICAL RIGID LINE INCLUSIONS BETWEEN DISSIMILAR ANISOTROPIC MATERIALS

The antiplane shear problems of periodical rigid line inclusions between dissimilar anisotropic materials are dealt with. By using the complex variable method, the closed form solutions are obtained. The stress distribution in the immediate vicinity of the rigid line is examined. The corresponding formulation between dissimilar isotropic materials and in homogeneous anisotropic medium can be derived from the special cases of those in the present paper, and the limit conditions are in agreement with the previously known results.

Dynamic Response of a Piezoelectric Material with a Conducting Rigid Inclusion

The problem of a conducting rigid inclusion embedded in an infinite piezoelectric matrix is considered under the action of combined electromechanical impact loads. By using integral transform techniques, the mixed initial-boundary value problem for the case of anti-plane shear load and in-plane electric field is transformed into two systems of dual integral equations, the solutions of which give the singularity coefficients of electroelastic field near the inclusion tips in closed-form in the Laplace transform domain. Numerical results for the stress singularity coefficient in the physical space are presented graphically by numerically solving the resulting Fredholm integral equation and carrying out the numerical inversion of Laplace transform for a PZT-5H material with a conducting rigid line inclusion.

Green's Function for a Point Heat Source in an Anisotropic Body Containing an Elliptic Hole or a Rigid Inclusion

AbstractThe thermoelastic problem associated with a point heat source embedded in an anisotropic body containing an elliptic hole or a rigid inclusion is considered in this paper. By using the formalism of Stroh [1], the approach of analytic function continuation and the technique of conformal mapping, the expression for the temperature, displacements and stress functions is expressed in explicit matrix form. The present derived solutions are also valid for some special problems such as a crack or a rigid line inclusion if one lets the minor axis of the ellipse approach to zero. The stress intensity factors induced by a point heat source are also obtained.

Antiplane Shear of a Strip Containing a Periodic Array of Rigid Line Inclusions

Motivated by the increased use of fiber-reinforced materials, the authors illustrate how the effective elastic modulus of an isotropic material can be increased by the insertion of rigid inclusions. Specifically, the authors consider the two-dimensional antiplane shear problem for a strip of material. The strip is reinforced by introducing a periodic array of ribbon-like, rigid inclusions perpendicular to the faces of the strip. The strip is then subjected to a prescribed uniform displacement difference between its faces. The problem is reduced in standard fashion to a mixed-boundary value problem in a rectangular domain, whose closed-form solution, given in terms of integrals of Jacobian elliptic functions, is obtained via triple sine series techniques. The effective shear modulus of the reinforced strip can now be calculated and compared with the shear modulus of a strip without inclusions. Also obtained are the stress singularity factors at the end tips of the inclusions. Numerical results are presented for several different reinforcement geometries.

Analysis of Conducting Rigid Inclusion at the Interface of Two Dissimilar Piezoelectric Materials

This paper is concerned with the electro-elastic analysis of a conducting rigid line inclusion at the interface of two bonded piezoelectric materials. By combining the analytic function theory and the Stroh formalism, we were able to obtain closed-form expressions for the field variables. Both the mechanical stresses and the electric displacement are shown to have at least one of the following behaviors: (i) traditional square root singularity; (ii) nonsquare root singularity; and (iii) oscillatory singularity, which depend upon the electro-elastic mismatch at the interface. By using the static equilibrium conditions, the rigid rotation vector of the inclusion is determined and the extended stress singularity factors (ESSF) are evaluated.

Antiplane shear of a strip containing a staggered array of rigid line inclusions

Motivated by the increased use of fibre-reinforced materials, we illustrate how the effective elastic modulus of an Isotropic and homogeneous material can be increased by the insertion of rigid inclusions. Specifically, we consider the two-dimensional antiplane shear problem for a strip of material. The strip is reinforced by introducing two sets of ribbon-like, rigid inclusions perpendicular to the faces of the strip. The strip is then subjected to a prescribed uniform displacement difference between its faces, see Figure 1. It should be noted that the problem posed is equivalent to that of the uniform antiplane shear problem for an infinite two-dimensional material containing a staggered array of rigid inclusions (see [1] for a review of antiplane problems in the literature). The problem is reduced in standard fashion [2–6] to a mixed boundary value problem in a rectangular domain, whose closed form solution given in terms of integrals of Weierstrassian Elliptic functions, is obtained via triple sine series techniques. The effective shear modulus of the reinforced strip can now be calculated and compared with the shear modulus of a strip without inclusions. Also obtained are the stress singularity factors at the end tips of the inclusions. Numerical results are presented for several different reinforcement geometries.

Interaction between a rigid line inclusion and an elastic circular inclusion

In this paper, the interaction problem of a rigid line inclusion and an elastic circular inclusion has been reduced to solve a normal Cauchy-type singular integral equation. The stress intensity factors at the ends of the rigid line inclusion and the interface stresses of the inclusions are obtained.

Rigid line inclusions under anti-plane deformation and in-plane electric field in piezoelectric materials

The problems of the rigid line inclusions under anti-plane deformation and in-plane electric field in piezoelectric materials are studied by using the complex variable method of Muskhelishvili. The rigid line inclusions are considered, respectively, as a dielectric and conductor. A square-root singularity of the field variables is identified. The singularity exists only if the shear loading is acting perpendicular to the rigid lines and the electric loading is parallel to them. For the rigid dielectric line inclusion, the singularity coefficients of both stress and electric field depend on the material properties, and the mechanical and electrical loads applied at infinity, whereas the electric displacement is a constant on the entire plane. The values of the J-integral are always negative. For the rigid conductor line inclusion, the field singularity coefficients corresponding to the field variables applied at infinity become independent of the material properties and other respective loading parameters. The values of the J-integral can be either positive or negative.

Anisotropic Elasticity: Theory and Applications

1. Matrix Algebra 2. Linear Anisotropic Elastic Materials 3. Antiplane Deformations 4. The Lekhnitskii Formalism 5. The Stroh Formalism 6. The Structures and Identities of the Elasticity Matrices 7. Transformation of the Elasticity Matrices and Dual Coordinate Systems 8. Green's Functions for Infinite Space, Half-space, and Composite Space 9. Particular Solutions, Stress Singularities, and Stress Decay 10. Anisotropic Matrials with an Elliptic Boundary 11. Anisotropic Media with a Crack or a Rigid Line Inclusion 12. Steady State Motion and Surface Waves 13. Degenerate and Near Degenerate Materials 14. Generalization of the Stroh Formulism 15. Three-Dimensionsal Deformations

A general duality principle in elasticity

Abstract Anisotropic elasticity problems are considered, where the field variables depend on only two spatial coordinate variables, say, the rectangular Cartesian coordinates, x1 and x1. The equilibrium equations are then identically satisfied (no body forces), if the stress components, σij, are such that σj1 = ∂Φj/∂x2 and σj2 = −∂Φj/∂x1, j = 1, 2, 3, where Φ with the component Φj is the stress vector potential. In terms of the six-dimensional vector field η = [u, Φ], the basic elasticity equations reduce to {∂/∂x2}[u, Φ]T = N{∂/∂x1}[u, Φ]T, where u is the displacement field and N is called the fundamental elasticity matrix, given by the elastic moduli of the solid. This formulation and the corresponding duality remove the distinction between the displacement vector field u, and the stress vector-potential field Φ. Indeed, these two vector fields are, in many respects, each other's dual, in the sense that, for a given solid with given geometry and elasticity, the solution to a class of prescribed mixed displacement-traction boundary-value problems also provides the solution of the corresponding dual mixed traction-displacement boundary-value problems, without a change in geometry or elasticity of the solid. This duality principle is first proved rigorously. Then the results are applied to a general mixed boundary-value problem of a multiply connected heterogeneous finite solid with arbitrary piecewise constant anisotropic elasticity. A number of interesting illustrations involving dislocations and concentrated forces, cracks and anticracks (rigid line inclusions), interface problems, and the half-space Green functions, are also presented as special cases of the general duality principle. In addition to the general duality principle which provides a two-way relation between u and Φ, a number of other one-way relations between the displacement field u and the stress vector potential Φ of the same problem, as well as u of one problem and u of its dual, and Φ of one problem and Φ of s dual, are also presented.