Interaction of an Ellipsoidal Inclusion with an Elliptic Crack in an Elastic Material under Triaxial Tension

The interaction of an elastic ellipsoidal inclusion with an elliptic crack in an infinite elastic medium under triaxial loading is analyzed. The stress state in the elastic space is represented as a superposition of the principal state and perturbed states, which are due to the presence and interaction of the inclusion and the crack. The analytical solution of the problem is found using the method of equivalent inclusion, the potential of an inhomogeneous ellipsoid, and a system of harmonic functions for an elliptic crack. The effect of triaxial loading on the stress intensity factors is analyzed     

Stress Concentration Near an Elliptic Crack in the Interface Between Elastic Bodies under Steady-State Vibrations

The paper addresses the three-dimensional problem on steady-state vibrations of an elastic body consisting of two perfectly joined dissimilar half-spaces with an elliptic mode I crack located in one of the half-spaces normally to the interface. The problem is reduced to a boundary integral equation for the crack opening function. The integration domain of the equation is bounded by the crack domain, and the interaction between the crack and the interface is described by a regular kernel. The equation is solved using the mapping method. Numerical results are obtained for the case where the surfaces of the elliptic crack are subjected to harmonic loading with constant amplitude. The dependences of the stress intensity factors on the wave number are presented for various relationships among the mechanical constants that ensure the absence of near-surface waves     

Interaction of an Ellipsoidal Inclusion with an Elliptic Crack in an Elastic Medium

An approach is considered to how to allow for the interaction between an ellipsoidal heterogeneity (inclusion) and an elliptic crack in an elastic medium. Using the superposition of perturbed stress states, the boundary conditions are satisfied on the ellipsoidal surface by the method of equivalent inclusion and on the crack surface by the least-squares method. A numerical analysis is carried out. Typical mechanical effects are revealed. In the calculations, the stress state near the ellipsoidal heterogeneity is approximated by a polynomial of the second degree in Cartesian coordinates, whereas the load on the crack surface is simulated by a polynomial of the fourth degree in Cartesian coordinates. In particular cases, the results are in good agreement with the data obtained by other authors     

Mixed-mode I/II fields around a crack with a cohesive zone ahead of the crack tip

Considering a cohesive zone ahead of the crack tip, the linear elastic crack problem under Mode I, Mode II or mixed-mode conditions is formulated in an elliptic coordinate system, so that the cohesive surfaces are conveniently represented by straight line segments. It is shown that the displacement and stress fields around the crack tip and the cohesive zone, expressed in terms of elliptic coordinates, have a simple mathematical form, which does not contain a stress singularity at the crack tip due to the existence of the cohesive zone.     

The behavior of a flat elliptical crack in an anisotropic elastic body

A method devolving upon the computation of certain influence coefficients is herein presented for determining the material displacements and stress in the vicinity of the edge of an elliptic crack within an arbitrarily anisotropic elastic body. In particular, compact line integral expressions for the stress intensity factors about the circumference of the crack and for the magnitude of the crack face displacement are derived. In all cases, the elastic body is assumed subject to uniform stress states far from the crack. Numerical results for a special example are also shown.     

On the anti-plane shear of an elliptic nano inhomogeneity

The elastic field of an elliptic nano inhomogeneity embedded in an infinite matrix under anti-plane shear is studied with the complex variable method. The interface stress effects of the nano inhomogeneity are accounted for with the Gurtin–Murdoch model. The conformal mapping method is then applied to solve the formulated boundary value problem. The obtained numerical results are compared with the existing closed form solutions for a circular nano inhomogeneity and a traditional elliptic inhomogeneity under anti-plane. It shows that the proposed semi-analytic method is effective and accurate. The stress fields inside the inhomogeneity and matrix are then systematically studied for different interfacial and geometrical parameters. It is found that the stress field inside the elliptic nano inhomogeneity is no longer uniform due to the interface effects. The shear stress distributions inside the inhomogeneity and matrix are size dependent when the size of the inhomogeneity is on the order of nanometers. The numerical results also show that the interface effects are highly influenced by the local curvature of the interface. The elastic field around an elliptic nano hole is also investigated in this paper. It is found that the traction free boundary condition breaks down at the elliptic nano hole surface. As the aspect ratio of the elliptic hole increases, it can be seen as a Mode-III blunt crack. Even for long blunt cracks, the surface effects can still be significant around the blunt crack tip. Finally, the equivalence between the uniform eigenstrain inside the inhomogeneity and the remote loading is discussed.     

Elastoplastic near field solution for mode II crack loaded by remote shear stress in an infinite plate

A suitable elastic stress field near the crack line which satisfied the far field boundary conditions and the boundary conditions of the crack surfaces has been obtained and successful analysis has been made of a near crack line field for an infinite elastic-perfectly plastic medium containing a quasi-statically propagating plane stress crack subjected to far field shear stress. It is shown that the solutions of the problem of mode II crack loaded by remote shear stress from the Westergaard method in some previous papers is used as the elastic stress field near the crack line, are inappropriate.     

Equilibrium of a Transversally Isotropic Body with an Elliptic Crack under Thermal Action

A nonuniform thermal effect on the surface of an elliptic crack in a transversally isotropic medium is studied. Use is made of the previous result on the analogy between elastic and thermoelastic problems for an arbitrary smooth flat crack in isotropic and transversally isotropic materials. A numerical analysis is performed to study how the stress intensity factors depend on the parameters and orientation of the crack     

Elastic state of a transversely isotropic piezoelectric body with an arbitrarily oriented elliptic crack

The elastic stress state in a piezoelectric body with an arbitrarily oriented elliptic crack under mechanical and electric loads is analyzed. The solution is obtained using triple Fourier transform and the Fourier-transformed Green’s function for an unbounded piezoelastic body. Solving the problem for the case of a crack lying in the isotropy plane, for which there is an exact solution, demonstrates that the approach is highly efficient. The distribution of the stress intensity factors along the front of a crack in a piezoelectric body under uniform mechanical loading is analyzed numerically for different orientations of the crack __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 39–48, February 2008.     

Elliptic cavity in a three-dimensional anisotropic body with inclined strata deformed by line loads and dislocations

Summary Analytical solutions are proposed for the stress and displacement fields in a quasi three-dimensional elastic anisotropic body containing an elliptic cavity or rigid inclusion. The directions of the principal elastic axes are allowed to be inclined arbitrarily with respect to the axes of the elliptic cavity. As an application, expressions for the stress intensity factors are formulated when the cavity reduces to a colinear crack.     

深埋椭圆形片状裂纹的偏折扩展

基于无限大弹性基体深埋椭圆形片状裂纹的变形场,推导了椭圆形片状裂纹的能量释放率,采用能量平衡方法建立了椭圆形片状裂纹承受拉应力和剪应力时的复合断裂准则. 考虑裂纹在拉-剪应力作用下的偏折扩展,分析了裂纹的偏折方向,提出了椭圆形片状裂纹发生偏折扩展时的初始偏折位置的确定方法.      

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.     

Opening-Function Simulation of the Three-Dimensional Nonstationary Interaction of Cracks in an Elastic Body

Integral relations between three-dimensional dynamic displacements (stresses) in an infinite elastic body with arbitrarily located plane cracks and discontinuities in the displacements of the opposite crack faces are presented. The influence of opening cracks on each other is considered in the problem on crack faces loaded by pulse forces. This problem is reduced to a system of boundary integral equations of the wave-potential type in a time domain. The dynamic mode I stress intensity factors are determined for two coplanar elliptic cracks under forces in the form of the Heaviside function     

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.     

Exact Analytical Solutions of Static Electroelastic and Thermoelectroelastic Problems for a Transversely Isotropic Body in Curvilinear Coordinate Systems

An approach to the solution of three-dimensional static problems for a transversely isotropic (rectilinear anisotropy) body is expounded and the solutions for piezoceramic canonical bodies are systematized. The result of the study is explicit analytical solutions of three-dimensional problems. Bodies are examined whose boundary surface is the coordinate surfaces in coordinate systems that permit the separation of the variables in the three-dimensional Laplace equation. The stress concentration in bodies near necks, cavities, inclusions, and cracks is investigated. The stress intensity factors of the force field and electric induction near elliptic and parabolic cracks are determined. The contact interaction of a piezoceramic half-space with elliptic and parabolic dies is studied. The bodies are under various mechanical, thermal, and electric loads     

Numerical studies of loss of ellipticity near singularities in an elastic material

Numerical solutions of anti-plane shear crack problems and screw dislocation problems are presented for materials in which the equilibrium equation varies in type locally from elliptic to hyperbolic as a result of deformation. These results show the emergence of surfaces of discontinuity in the displacement field in some materials. In other materials they show a chaotic mixture of elliptic phases at intermediate distances from the singularity. A statistical analysis applied to the numerical solutions demonstrates the role of elastic stability in the mechanics of these deformations.     

Static Equilibrium of an Elastic Orthotropic Medium with an Elliptic Crack under Bending

The static-equilibrium problem for an elastic orthotropic space with an elliptic crack is solved. The stress state of the space is represented as a superposition of the principal and perturbed states. To solve the problem, Willis' approach is used. It is based on the Fourier transform in spatial variables, the Fourier-transformed Green function for anisotropic material, and Cauchy's residue theorem. The contour integrals appearing during solution are evaluated using Gaussian quadratures. 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. 41, No. 8, pp. 72–81, August 2005.     

弹性半平面中边界裂纹研究的新方法

讨论了载荷作用在裂纹面上的弹性半平面边界裂纹问题.研究以线弹性断裂力学为基础,采用复变函数方法以及Riemann-Hilbert(R-H)边值问题的一般理论,将问题分拆为含有限裂纹的全平面问题与无裂纹的半平面问题的叠加,计算得到裂纹尖端的应力强度因子.与文献结果比较,该方法具有精度高的优点.     

The effect of an elastic triangular inclusion on a crack

IntroductionWiththedevelopmentofparticleandfiberreinforcedcomposites,theinclusion_crackinteractionproblemisbecominganimportantfieldbeingstudied .Andasamodel,itisalsousedtostudytheeffectsofmaterialdefectsonthestrengthandfractureofengineeringstructure.TheinterationbetweencircularinclusionandcrackwasstudiedinRefs.[1 -6 ] ;InRefs.[7-1 2 ] ,theinterationbetweenlineinclusionandcrackswasdiscussed ;TheinterationbetweenellipticalinclusionandcrackwasstudiedinRefs.[1 3,1 4] .However,withthedevelopmento…     

带裂纹的椭圆孔口问题的应力分析

断裂现象与材料和结构中的孔洞、缺口或裂纹等缺陷密切相关,这是因为缺陷附近的应力集中明显.该文利用复变方法,通过保角映射研究了带裂纹的椭圆孔洞的平面弹性问题,给出了应力强度因子的解析解.并由此计算了两互相垂直的裂纹问题.