Exact Analytical Solutions of Three-Dimensional Static Thermoelastic Problems for a Transversally Isotropic Body in Curvilinear Coordinate Systems

An approach to the solution of three-dimensional static thermoelastic problems for a transversally isotropic (the case of rectangular anisotropy) body is proposed. The results of construction of the general analytic solutions to thermoelastic problems for canonical bodies are systematized. The exact analytic solutions of three-dimensional problems are obtained. It is assumed that the bodies under consideration are thermoelastic and their boundary surface corresponds to the coordinate surfaces in coordinate systems that allow separating the variables in the three-dimensional Laplace equation. The stress concentration near cavities and inclusions is studied. The stress intensity factors near elliptic and hyperbolic cracks are determined. Formulas are presented for the stress intensity factors on the surface of a rigid elliptic inclusion and inside the body near a homogeneity under various thermal effects     

Exact analytic solutions of three-dimensional boundary-value problems of the statics of a transversely isotropic body of canonical form (Survey)

Conclusion Thus, we have described some of the approaches used to solve three-dimensional problems of the statics of isotropic and transversely isotropic bodies, and we have systematized results from the construction of analytic general solutions to problems of the theory of elasticity of transversely isotropic bodies of canonical form. These solutions have in turn been used to solve numerous specific problems on stress concentration near notches, cavities, inclusions, and cracks, as well as problems concerning the contact interaction of an elastic half-space with elliptical and parabolic dies. Translated from Prikladnaya Mekhanika, Vol. 33, No. 10, pp. 3–30, October, 1997.     

Stress singularities in an anisotropic body of revolution

To fill the gap in the literature on the application of three-dimensional elasticity theory to geometrically induced stress singularities, this work develops asymptotic solutions for Williams-type stress singularities in bodies of revolution that are made of rectilinearly anisotropic materials. The Cartesian coordinate system used to describe the material properties differs from the coordinate system used to describe the geometry of a body of revolution, so the problems under consideration are very complicated. The eigenfunction expansion approach is combined with a power series solution technique to find the asymptotic solutions by directly solving the three-dimensional equilibrium equations in terms of the displacement components. The correctness of the proposed solution is verified by convergence studies and by comparisons with results obtained using closed-form characteristic equations for an isotropic body of revolution and using the commercial finite element program ABAQUS for orthotropic bodies of revolution. Thereafter, the solution is employed to comprehensively examine the singularities of bodies of revolution with different geometries, made of a single material or bi-materials, under different boundary conditions.     

Curve cracks lying along a parabolic curve in anisotropic body

ＣＵＲＶＥＣＲＡＣＫＳＬＹＩＮＧＡＬＯＮＧＡＰＡＲＡＢＯＬＩＣＣＵＲＶＥＩＮＡＮＩＳＯＴＲＯＰＩＣＢＯＤＹＨｕＹｕａｎ－ｔａｉ（胡元太）ＺｈａｏＸｉｎｇ－ｈｕａ（赵兴华）（ＳｈａｎｇｈａｉＵｎｉｖｅｒｓｉｔｙ;ＳｈａｎｇｈａｉＩｎｓｔｉｔｕｔｅｏｆＡ...     

Analysis of rotating solids with cracks by the boundary element method

The use of the boundary element method for the solution of linear elastic fracture mechanics problems, without body forces, is quite extensive since the method is intrinsically well suited to the analysis of high stress gradients associated with crack problems. The crack-tip stresses for rotating bodies are similar to the stresses for stationary bodies and therefore all the advantages of the boundary element procedure can be encompassed in the extension of the method to the solution of rotating bodies with cracks. In the present analysis, the additional volume integral that arises from the treatment of inertial body forces is eliminated by using the well-known particular integral procedure. The matrix ill-conditioning that results from the need to model co-planar crack surfaces of non-symmetrical cracks is avoided by using the multi-region approach. The accuracy of the numerical solutions is improved by utilizing quarter-point elements with traction singular enhancement at the crack-tip. The procedure is then applied to the solution of arbitrary cracks in two- and three-dimensional rotating bodies.     

Thermostressed state of a piezoelectric bodywith a plane crack under symmetric thermal load

The paper addresses a thermoelectroelastic problem for a piezoelectric body with an arbitrarily shaped plane crack in a plane perpendicular to the polarization axis under a symmetric thermal load. A relationship between the intensity factors for stress (SIF) and electric displacement (EDIF) in an infinite piezoceramic body with a crack under a thermal load and the SIF for a purely elastic body with a crack of the same shape under a mechanical load is established. This makes it possible to find the SIF and EDIF for an electroelastic material from the elastic solution without the need to solve specific problems of thermoelasticity. The SIF and EDIF for a piezoceramic body with an elliptic crack and linear distribution of temperature over the crack surface are found as an example __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 3, pp. 96–108, March 2008.     

Stress state of a piezoelectric ceramic body with a plane crack under antisymmetric loads

The static equilibrium of an electroelastic transversely isotropic space with a plane crack under antisymmetric mechanical loads is studied. The crack is located in the plane of isotropy. Relationships are established between the stress intensity factors (SIFs) for an infinite piezoceramic body and the SIFs for a purely elastic body with a crack of the same form under the same loads. This makes it possible to find the SIFs for an electroelastic body without the need to solve specific electroelasitc problems. As an example, the SIFs are determined for a piezoelastic body with penny-shaped and elliptic cracks under shear __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 32–42, February 2006.     

Stress state of a transversely isotropic piezoceramic body with a spheroidal cavity

The exact solution is found to the three-dimensional electroelastic problem for a transversely isotropic piezoceramic body with a spheroidal cavity. The solutions of static electroelastic problems are represented in terms of harmonic functions. The case of stretching the piezoceramic medium at a right angle to the spheroid axis of symmetry is analyzed numerically. The dependence of the stress concentration factor on the geometry of the spheroid and the electromechanical characteristics of the material is studied.Translated from Prikladnaya Mekhanika, Vol. 40, No. 11, pp. 92–105, November 2004.This revised version was published online in April 2005 with a corrected cover date.     

Stress analysis in two dimensional electrostrictive material with an elliptic rigid conductor

This paper presents the governing equations of electrostrictive materials. The stress and electric field solutions for an infinite plate with a rigid elliptic conductor under applied load at infinity are given. The asymptotic expansions of the solution for a narrow elliptic conductor show that the stresses and the electric fields near the end of a narrow elliptic conductor possess r⁻¹ and r^−1/2 forms respectively in a local coordinate system with the origin at its focus.     

压电材料中三维I型断裂力学分析

讨论了不可导通情况下三维横观各向刚性压电材料中受拉伸和电载荷作用的平片裂纹Ⅰ型断裂力学问题．使用自限部分概念，从二维线性压电理论出发，严格得到了一组以裂纹面位移间断和电势间断为未知变量的超奇异积分方程组；应用二维超奇异积分的主部分析法，从理论上分析得到了裂纹前沿应力和电势奇性指数以及应力和电位移奇性场，从而找到了以裂纹面位移间断和电势间断表示的应力和电位移强度因子、能量释放率表达式；为所得到的超奇异积分方程组建立了数值法，并用此计算了若干典型的平片裂纹问题，数值结果令人满意．     

Free nonaxisymmetric vibrations of radially polarized hollow piezoceramic cylinders of finite length

The three-dimensional problem of free nonaxisymmetric vibrations of hollow piezoceramic cylinders with axial polarization is considered. An efficient numerical analytic method to solve boundary-value problems is proposed. The original three-dimensional problem of electroelasticity is reduced to a two-dimensional problem by representing the displacement components as standing circumferential waves. Spline collocation with respect to the axial coordinate is used to reduce this two-dimensional problem to an eigenvalue boundary-value problem with respect to the radial coordinate. This problem is solved by the stable discrete-orthogonalization and incremental-search methods. Numerical results are presented and the natural frequencies of the cylinders are analyzed in a wide range of their geometric characteristics     

A computational study of local stress intensity factor solutions for kinked cracks near spot welds in lap-shear specimens

In this paper, the local stress intensity factor solutions for kinked cracks near spot welds in lap-shear specimens are investigated by finite element analyses. Based on the experimental observations of kinked crack growth mechanisms in lap-shear specimens under cyclic loading conditions, three-dimensional and two-dimensional plane-strain finite element models are established to investigate the local stress intensity factor solutions for kinked cracks emanating from the main crack. Semi-elliptical cracks with various kink depths are assumed in the three-dimensional finite element analysis. The local stress intensity factor solutions at the critical locations or at the maximum depths of the kinked cracks are obtained. The computational local stress intensity factor solutions at the critical locations of the kinked cracks of finite depths are expressed in terms of those for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. The three-dimensional finite element computational results show that the critical local mode I stress intensity factor solution increases and then decreases as the kink depth increases. When the kink depth approaches to 0, the critical local mode I stress intensity factor solution appears to approach to that for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. The two-dimensional plane-strain computational results indicate that the critical local mode I stress intensity factor solution increases monotonically and increases substantially more than that based on the three-dimensional computational results as the kink depth increases. The local stress intensity factor solutions of the kinked cracks of finite depths are also presented in terms of those for vanishing kink depth based on the global stress intensity factor solutions and the analytical kinked crack solutions for vanishing kink depth. Finally, the implications of the local stress intensity factor solutions for kinked cracks on fatigue life prediction are discussed.     

Nonaxisymmetric electroelastic vibrations of a hollow sphere made of functionally gradient piezoelectric material

The nonaxisymmetric problem of natural vibrations of a hollow sphere made of functionally gradient piezoelectric material is solved based on 3D electroelasticity. The properties of the material change continuously along a radial coordinate according to an exponential law. The external surface of the sphere is free of tractions and either insulated or short-circuited by electrodes. After separation of variables and representation of the components of the displacements and of the stress tensor in terms of spherical functions, the initially three-dimensional problem is reduced to a boundary-value problem for the eigenvalues expressed by ordinary differential equations. This problem is solved by a stable discrete-orthogonalization technique in combination with a step-by-step search method with respect to the radial coordinate. Moreover, a numerical investigation is performed based on the algorithm used for solving the problem. In particular, we investigate the influence of the geometric and electric parameters on the frequency spectrum at the nonaxisymmetry of natural vibrations of an inhomogeneous piezoceramic thick-walled sphere.     

The equilibrium of a piezoceramic cylindrical body with a parabolic crack

A conjugate electroelastic field in a piezoceramic cylinder with a parabolic crack under static loading is investigated. Uniformly tensile stresses and an electric potential are applied to the end faces of the cylinder. The following two types of electric conditions are considered at the crack boundary: the electric potential is continuous across the crack and the normal component of the electric-displacement vector on the crack surface is equal to zero. For each of these cases, expressions for some quantities characterizing the disturbed field in the crack plane and formulas to calculate the stress intensity factors are presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 72–80, March, 2000.     

Stress intensity factors for three-dimensional cracks in functionally graded materials using enriched finite elements

A three-dimensional enriched finite-element methodology is presented to compute stress intensity factors for three-dimensional cracks contained in functionally graded materials (FGMs). A general-purpose 3D finite-element based fracture analysis program, FRAC3D, is enhanced to include this capability. First, using available solutions from the literature, comparisons have been made in terms of stresses under different loading conditions, such as uniform tensile, bending and thermal loads. Mesh refinement studies are also performed. The fracture solutions are obtained for edge cracks in an FGM strip and surface cracks in a finite-thickness FGM plate and compared with existing solutions in the literature. Further analyses are performed to study the behavior of stress intensity factor near the free surface where crack front terminates. It is shown that three-dimensional enriched finite elements provide accurate and efficient fracture solutions for three-dimensional cracks contained in functionally graded materials.     

The Stress State of a Piezoceramic Medium with an Elliptic Inclusion under Pure Shear and Pure Bending

Explicit analytical solutions to electroelastic problems for an infinite transversely isotropic medium with a tunnel elliptic inclusion are constructed. At a sufficient distance from the inclusion, the medium is subjected to pure shear or pure bending. It is assumed that the medium and inclusion are dissimilar piezoceramic materials whose axes of symmetry coincide with each other and with the minor axis of the ellipse. The stresses and the projections of the electromagnetic induction vector acting in the medium beyond the inclusion are determined for each case of loading at infinity     

Exact solutions for the plane problem in piezoelectric materials with an elliptic or a crack

Based on the complex potential approach, the two-dimensional problems in a piezoelectric material containing an elliptic hole subjected to uniform remote loads are studied. The explicit, closed-form solutions satisfying the exact electric boundary condition on the hole surface are given both inside and outside the hole. When the elliptic hole degenerates into a crack, the field intensity factors are obtained. It is shown that the stress intensity factors are the same as that of isotropic material, while the electric displacement intensity factor depends on both the material properties and the mechanical loads, but not on the electric loads. In other words, the uniform electric loads have no influence on the field singularities. It is also shown that the impermeable crack assumption used previously to simply the electric condition is not valid to crack problems in piezoelectric materials.     

Stress and electric displacement analyses in piezoelectric media with an elliptic hole and a small crack

In this paper, the interactions between an elliptic hole and an arbitrary distributed small crack in plane piezoelectric medium, which are often happened in engineering problems, are discussed. The Green’s functions in a piezoelectric plate with an elliptic hole for a generalized line dislocation and a generalized line force are presented. The small crack is represented by unknown continuous distributed dislocations. By considering traction free conditions on the surface of the small crack, the problem is then reduced to a group of singular integral equations which are solved by using a special numerical technique. Accuracy of the present method is confirmed by comparing the numerical results with those in literatures for PZT-4 when the elliptic hole is degenerated into a crack. The generalized stress intensity factors of cracks and the generalized stress on the edge of the elliptic hole are shown graphically. It is shown that the small crack may have shielding or amplifying effects on the main elliptic hole or crack, which depends on the location and orientation of the small crack. The hole near a crack can significantly reduce the stress intensity factor of the crack. The direction of the electric field is important to shielding effect.     

The Stress State of a Transversally Isotropic Piezoceramic Body with a Parabolic Crack in a Uniform Heat Flow

The explicit solution is constructed for a static thermoelastic problem for an infinite transversally isotropic piezoceramic body containing a heat-insulated parabolic crack in the isotropy plane. The crack surface is assumed free of forces. The body is under a uniform heat flow, which is perpendicular to the crack surface and is far from the crack itself. The problem is solved for two cases of electric conditions on the crack surface. In the first case, an electric potential is absent on the crack surface and, in the second case, the normal component of the electric-displacement vector is equal to zero. The intensity factors, which depend on the heat flow, crack geometry, and the thermoelectroelastic properties of the piezoceramic body, are determined for the force field and electric displacement near the crack tip     

Exact solutions for piezoelectric screw dislocations near two asymmetrical edge cracks emanating from an elliptical hole

The interaction between piezoelectric screw dislocations and two asymmetrical interfacial cracks emanating from an elliptic hole under combined mechanical and electric load at infinity is dealt with. The closed-form solutions are derived for complex potentials and generalized stress fields. In the limiting cases, some well-known results can be obtained from the present solutions. Moreover, some new exact solutions are shown. The stress intensity factor and the energy release rate at the right tip due to a screw dislocation near the right interfacial crack are also calculated. The results show that the shielding effect of dislocation on crack expanding decreases with the increase in dislocation azimuth angle and the distance between the dislocation and the crack tip, and the repulsion acting on the dislocation from the other half plane demotes crack propagation. The increasing of the length of the other crack promotes crack growth, but the increasing of the minor semi-axis demotes it.