Elliptical inhomogeneity with polynomial eigenstrains embedded in orthotropic materials

A closed-form solution for elastic field of an elliptical inhomogeneity with polynomial eigenstrains in orthotropic media having complex roots is presented. The distribution of eigenstrains is assumed to be in the form of quadratic functions in Cartesian coordinates of the points of the inhomogeneity. Elastic energy of inhomogeneity–matrix system is expressed in terms of 18 real unknown coefficients that are analytically evaluated by means of the principle of minimum potential energy and the corresponding elastic field in the inhomogeneity is obtained. Results indicate that quadratic terms in the eigenstrains induce zeroth-order elastic strain components, which reflect the coupling effect of the zeroth- and second-order terms in the polynomial expressions on the elastic field. In contrast, the first-order terms in the eigenstrains only produce corresponding elastic fields in the form of the first-order terms. Numerical examples are given to demonstrate the normal and shear stresses at the interface between the inhomogeneity and the matrix. Furthermore, the solution reduces to known results for the special cases.     

A rigid line in a confocal elliptic inhomogeneity embedded in an infinite medium under antiplane shear

An effective method is developed and used to investigate the antiplane problem of a rigid line in a confocal elliptic inhomogeneity embedded in an infinite medium. The analytical solution is obtained. The proposed method is based upon the use of conformal mapping and the theorem of analytic continuation. Special solutions which are verified by comparison with existing ones are provided. Finally, the characteristics of stress singularity at the tip of the rigid line inhomogeneity are analyzed and the extension forces for the crack and the rigid line inhomogeneity are derived. The project supported by the National Natural Science Foundation of China, the State Education Commission Foundation and the Failure Mechanics Lab of SEdC.     

The interaction of an edge dislocation with an inhomogeneity of arbitrary shape in an applied stress field

A general, approximate solution is presented for an edge dislocation interacting with an inhomogeneity of arbitrary shape under combined dislocation and applied stress fields. The solution shows that the contributions of the dislocation stress field and the applied stress field to the interaction follow a simple superposition principle. The dislocation stress field has a short range effect, while the applied stress field has a long range effect. As special cases, explicit solutions for some common inhomogeneity shapes are obtained for the interaction induced by the applied stress field.     

Stress and electric fields in an infinite electrostrictive plate with an elliptic inhomogeneity

The stress and electric fields in electrostrictive materials under general electric loading at infinity are obtained in this paper. It is shown that the pseudo total stresses are continuous in the whole body. The elliptic inhomogeneity problem is first discussed in this paper and its solution is also given. The results show that the stress in the inhomogeneity is not uniform which is different from the solution of Eshelby theory for elastic materials. When the inhomogeneity and matrix have the same dielectric permittivity or the matrix is a non-electrostrictive material, the stress field is uniform in the inhomogeneity. The form of stress function is simple when the inhomogeneity degenerates to a circle.     

Asymptotic computation of the inhomogeneity energy in a body in an external stress field

We consider a problem on an ellipsoidal inhomogeneity in an infinitely extended homogeneous isotropic elastic medium. The inhomogeneity differs from the ambient body in the elastic moduli (Poisson’s ratio ν and shear modulus μ) and in that it has intrinsic strains. We use the equivalent inclusion method to write out expressions for the Helmholtz and Gibbs free energy of the inhomogeneity as quadratic forms in the intrinsic strains and strains at infinity. The general expressions for the coefficients of these quadratic forms are written out as three rank four tensors characterizing the contribution to the energy by the plastic strain (ɛ _p ²), by the strain at infinity (ɛ ₀²), and (only for the Gibbs energy) by the cross term ɛ ⁰ ɛ _p .     

Stress state of an orthotropic material with an elliptic crack under linearly varying pressure

The static equilibrium of an elastic orthotropic medium with an elliptic crack subject, on its surface, to linearly varying pressure is studied. The stress state of the elastic medium is represented as a superposition of the principal and perturbed states. Use is made of Willis’ approach based on the triple Fourier transform in spatial variables, the Fourier-transformed Green’s function for an anisotropic material, and Cauchy’s residue theorem. The contour integrals are evaluated using Gaussian quadratures. The results for particular cases are compared with those obtained by other authors. The influence of orthotropy on the stress intensity factors is studied __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 73–81, July 2006.     

Thickness effect of a thin film on the stress field due to the eigenstrain of an ellipsoidal inclusion

Solutions of the stress field due to the eigenstrain of an ellipsoidal inclusion in the film/substrate half-space are obtained via the Fourier transforms and Stroh eigenrelation equations. Based on the acquired solutions, the effect of a thin film’s thickness on the stress field is investigated with two types of ellipsoidal inclusions considered. The results in this paper show that if the thickness of the thin film increases, its effect on the stress field will become weaker, and can even be neglected. In the end, a guide rule is introduced to simplify the calculation of similar problems in engineering.     

Refined analysis of the stress state of orthotropic elliptic cylindrical shells with variable geometrical parameters

An approach developed earlier to solve boundary-value problems is used to analyze the behavior of the stress-strain state of orthotropic elliptic cylindrical shells with variation in the geometric parameters of their cross section at constant volume (weight) Translated from Prikladnaya Mekhanika, Vol. 44, No. 9, pp. 53–62, September 2008.     

DYNAMIC STRESS FIELD AROUND THE MODE Ⅲ CRACK TIP IN AN ORTHOTROPIC FUNCTIONALLY GRADED MATERIAL

The problem of a Griffith crack in an unbounded orthotropic functionally graded material subjected to antipole shear impact was studied. The shear moduli in two directions of the functionally graded material were assumed to vary proportionately as definite gradient. By using integral transforms and dual integral equations, the local dynamic stress field was obtained. The results of dynamic stress intensity factor show that increasing shear moduli’s gradient of FGM or increasing the shear modulus in direction perpendicular to crack surface can restrain the magnitude of dynamic stress intensity factor.     

Buckling of an orthotropic graded coating with an embedded crack bonded to a homogeneous substrate

To simulate buckling of nonuniform coatings, we consider the problem of an embedded crack in a graded orthotropic coating bonded to a homogeneous substrate subjected to a compressive loading. The coating is graded in the thickness direction and the material gradient is orthogonal to the crack direction which is parallel with the free surface. The elastic properties of the material are assumed to vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the crack orientation. The loading consists of a uniform compressive strain applied away from the crack region. The graded coating is modeled as a nonhomogeneous medium with an orthotropic stress–strain law. Using a nonlinear continuum theory and a suitable perturbation technique, the plane strain problem is reduced to an eigenvalue problem describing the onset of buckling. Using integral transforms, the resulting plane elasticity equations are converted analytically into singular integral equations which are solved numerically to yield the critical buckling strain. The Finite Element Method was additionally used to model the crack problem. The main objective of the paper is to study the influence of material nonhomogeneity on the buckling resistance of the graded layer for various crack positions, coating thicknesses and different orthotropic FGMs.     

Time-harmonic dynamical stress field in a system comprising a pre-stressed orthotropic layer and pre-stressed orthotropic half-plane

The time-harmonic dynamical stress field in a system comprising a pre-stressed orthotropic layer and orthotropic half-plane is studied within the scope of the piecewise homogeneous body model utilizing the three-dimensional linearized theory of elastic waves in an initially stressed body. The main focus is on the influence of the mechanical properties of the constituent materials and the initial stresses present on the “resonance” values of the normal stress acting on the interface plane and on the “resonance” values of the frequency of the external point-located force. The numerical results are presented and discussed. In particular, it is shown that the values of the normal stress decrease with a decrease in the modulus of elasticity of the materials along the thickness of the covering layer.     

Elastic field of an isotropic matrix with a nanoscale elliptical inhomogeneity

In traditional continuum mechanics, the effect of surface energy is ignored as it is small compared to the bulk energy. For nanoscale materials and structures, however, the surface effects become significant due to the high surface/volume ratio. In this paper, two-dimensional elastic field of a nanoscale elliptical inhomogeneity embedded in an infinite matrix under arbitrary remote loading and a uniform eigenstrain in the inhomogeneity is investigated. The Gurtin–Murdoch surface/interface elasticity model is applied to take into account the surface/interface stress effects. By using the complex variable technique of Muskhelishvili, the analytic potential functions are obtained in the form of an infinite series. Selected numerical results are presented to study the size-dependency of the elastic field and the effects of surface elastic moduli and residual surface stress. It is found that the elastic field of an elliptic inhomogeneity under uniform eigenstrain is no longer uniform when the interfacial stress effects are taken into account.     

Singular stress field near interface edge in orthotropic/isotropic bi-materials

Based on the asymptotic fields near the singular points in two-dimensional isotropic and orthotropic elastic materials, the eigenequation as well as the displacement and singular stress fields near the interface edge, with an arbitrary wedge angle for the orthotropic material, in orthotropic/isotropic bi-materials are formulated explicitly. The advantages of the developed approach are that not only the eigenequation is directly given in a simple form, but also the eigenvector is no longer needed for the determination of the asymptotic fields near the interface edge. This approach differs from the known methods where the eigenequation is constantly expressed in terms of a determinant of matrix, and the eigenvector is required for the determination of the asymptotic fields. Therefore, the solution proposed in this paper is more convenient and effective for the analysis of the singular stresses near the interface edge in the orthotropic/isotropic bi-material. To demonstrate the validity of the presented formulae, an example is selected for the comparison of analytical and FEM results. According to the theoretical analyses, the influences of the wedge angle and material constant of the orthotropic material on the singular stresses near the interface edge are discovered clearly. The results obtained may give some references to certain engineering designs such as the structural repair or strengthening.     

The stress state of an elastic orthotropic medium with a penny-shaped crack

The static-equilibrium problem for an elastic orthotropic space with a circular (penny-shaped) crack is solved. The stress state of an elastic medium is represented as a superposition of the principal and perturbed states. To solve the problem, Willis approach is used, which is based on the triple Fourier transform in spatial variables, the Fourier-transformed Greens function for an anisotropic material, and Cauchys residue theorem. The contour integrals obtained are evaluated using Gauss quadrature formulas. The results for particular cases are compared with those obtained by other authors. The influence of anisotropy on the stress intensity factors is studied.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 76–83, December 2004.     

Boundary element analysis for elastic and elastoplastic problems of 2D orthotropic media with stress concentration

Both the orthotropy and the stress concentration are common issues in modern structural engineering. This paper introduces the boundary element method (BEM) into the elastic and elastoplastic analyses for 2D orthotropic media with stress concentration. The discretized boundary element formulations are established, and the stress formulae as well as the fundamental solutions are derived in matrix notations. The numerical procedures are proposed to analyze both elastic and elastoplastic problems of 2D orthotropic media with stress concentration. To obtain more precise stress values with fewer elements, the quadratic isoparametric element formulation is adopted in the boundary discretization and numerical procedures. Numerical examples show that there are significant stress concentrations and different elastoplastic behaviors in some orthotropic media, and some of the computational results are compared with other solutions. Good agreements are also observed, which demonstrates the efficiency and reliability of the present BEM in the stress concentration analysis for orthotropic media. The project supported by the Basic Research Foundation of Tsinghua University, the National Foundation for Excellent Doctoral Thesis (200025) and the National Natural Science Foundation of China (19902007). The English text was polished by Keren Wang.     

Plane problems in piezoelectric media with elliptic inclusion

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Inclusion of an arbitrary polygon with graded eigenstrain in an anisotropic piezoelectric half plane

This paper presents an exact closed-form solution for the Eshelby problem of a polygonal inclusion with graded eigenstrains in an anisotropic piezoelectric half plane with traction-free on its surface. Using the line-source Green’s function, the line integral is carried out analytically for the linear eigenstrain case, with the final expression involving only elementary functions. The solutions are applied to the semiconductor quantum wire (QWR) of square, triangular, and rectangular shapes, with results clearly illustrating various influencing factors on the induced fields. The exact closed-form solution should be useful to the analysis of nanoscale QWR structures where large strain and electric fields could be induced by the non-uniform misfit strain.