Nonlinear mode II crack-tip fields for some Hookean materials

This paper presents a modified nonlinear Mode II crack model which is shown to satisfy the nonpenetrating crack surface boundary condition for homogeneous isotropic Hookean materials taking into account finite deformations. A recent investigation of the problem by Knowles [1] reveals apparent interpenetration of the crack surfaces which is considered nonphysical and therefore invalid. This observation is confirmed when a general solution based on Knowles's perturbation boundary layer method to characterize the finite deformation effects on Mode II crack-tip fields for the materials is derived. By deducing complete 2nd order solutions of the problem, the Poynting nonlinear effect becomes self-evident at the crack tip and for Hookean materials with there always exists the penetrating phenomenon between upper and lower crack surfaces.     

A special crack tip displacement discontinuity element

Based on the analytical solution to the problem of a constant discontinuity in displacement over a finite line segment in the x, y plane of an infinite elastic solid and the note of the crack tip element by Crouch, in the present paper, the special crack tip displacement discontinuity element is developed. Further the analytical formulas for the stress intensity factors of crack problems in general plane elasticity are given. In the boundary element implementation the special crack tip displacement discontinuity element is placed locally at each crack tip on top of the non-singular constant displacement discontinuity elements that cover the entire crack surface. Numerical results show that the displacement discontinuity modeling technique of a crack presented in this paper is very effective.     

含圆孤裂纹系的压电材料反平面应变问题

应用复变函数解析延展原理，并通过求解Ｒｉｅｍａｎｎ－Ｈｉｌｂｅｒｔ问题，得到了含圆弧裂纹压电材料反平面应变问题的一般解，对单个圆弧裂纹的情形，给出了封闭形式的复函数解和场强度因子，结果表明，当无限远处或裂纹表面同时受机械载荷（应力τ＾∞或Ｔｚ）和电载荷（电位移Ｄ＾∞或电荷ｑ）联合作用时，应力强度因子仅与机械载荷有关，而电位移动强度因子仅与电载荷有关。     

Diffraction of a plane stress wave by a semi-infinite crack in a general anisotropic elastic material

The problem of a semi-infinite crack subjected to an incident stress wave in a general anisotropic elastic solid is considered. The plane wave impinges the crack at a general oblique angle and is of any of the three types propagating in that direction. A related problem of a semi-infinite crack loaded by a pair of concentrated forces moving along the crack surfaces is also considered. In contrast to the conventional approach by Laplace transforms, a Stroh-like formalism is employed to construct the solution directly in the time domain. The solution is shown to depend on a Wiener–Hopf factorization of a symmetric matrix. Closed-form solution of the stress intensity factors is derived. A remarkably simple expression for the energy release rate is obtained for normal incidence.     

An antisymmetric problem of a penny-shaped crack in a piezoelectric medium

Summary  An exact, three-dimensional analysis is developed for a penny-shaped crack in an infinite transversely isotropic piezoelectric medium. The crack is assumed to be parallel to the plane of isotropy, with its faces subjected to a couple of concentrated normal forces and a couple of point electric charges that are antisymmetric with respect to the crack plane. The fundamental solution of a concentrated force and a point charge acting on the surface of a piezoelectric half-space is employed to derive the integral equations for the general boundary value problem. For the above antisymmetric crack problem, complete expressions for the elastoelectric field are obtained. A numerical calculation is finally performed to show the piezoelectric effect in piezoelectric materials. It is noted here that the present analysis is an extension of Fabrikant's theory for elasticity. Received 30 August 1999; accepted for publication 1 March 2000     

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     

二维半圆孔附近的动态热应力分布

在稳态温度场中,由于弹性平面内半圆孔或半圆坑等缺陷在介质裂纹扩展和疲劳损伤机理中所起的作用,因此深入了解半圆孔或半圆坑的表面动态热应力分布显得很有必要．分析在半无限弹性平面内一半圆孔附近施加稳态调谐温度场条件下的半圆孔孔边动态热应力分布问题,利用共形映照等复变函数的方法,给出了汉克函数表示的此问题的解析解,并给出了相应动态热应力分布系数的数值结果．     

Three-dimensional fracture analysis using tetrahedral enriched elements and fully unstructured mesh

In the absence of automated and customized methods and tools, some of today’s existing methods for solving three-dimensional fracture problems require comprehensive finite element meshing, labor-intensive analysis and post-processing efforts. In this study, a tetrahedral enriched element method and related applications are presented that demonstrate employment of fully unstructured tetrahedral meshes for general mixed-mode three-dimensional fracture problems. As in the case of hexahedral enriched elements, the tetrahedral enriched elements also alleviate the needs of pre- and post-processing the finite element model, allowing direct computation of stress intensity factors in the solution phase. In addition, when tetrahedral enriched elements are used, the crack front region can also be meshed using unstructured elements allowing direct use of automatic free-meshing programs. The applications presented are plane-strain central crack problem, mode-I surface crack in a plate, inclined penny-shaped crack, edge-cracked bar under constant heat flux and lens-shaped crack embedded in a large elastic body. The results obtained are in good comparative agreement with those available in the literature. Thus, it is concluded that the enriched tetrahedral elements can be applied efficiently and accurately on a general three-dimensional fracture problem allowing usage of fully unstructured finite element meshes.     

Dynamic crack growth under Rayleigh wave

A nonuniform crack growth problem is considered for a homogeneous isotropic elastic medium subjected to the action of remote oscillatory and static loads. In the case of a plane problem, the former results in Rayleigh waves propagating toward the crack tip. For the antiplane problem the shear waves play a similar role. Under the considered conditions the crack cannot move uniformly, and if the static prestress is not sufficiently high, the crack moves interruptedly. For fracture modes I and II the established, crack speed periodic regimes are examined. For mode III a complete transient solution is derived with the periodic regime as an asymptote. Examples of the crack motion are presented. The crack speed time-period and the time-averaged crack speeds are found. The ratio of the fracture energy to the energy carried by the Rayleigh wave is derived. An issue concerning two equivalent forms of the general solution is discussed.     

Crack propagation in the power-law nonlinear viscoelastic material

I.IntroductionCrazingdamageisacommonphenon1enonoffractureofpolymericmaterials.Theformationofcrazezoneisamid-stateinthefractureprocessofthematerialsfromperfectstatetofaiIurc.Microscopically,inthisregionthereexistssomefibrilslinkingthetwocracksurfacesandres…     

Axially symmetric thermal stress of an external circular crack under general thermal conditions

Summary The paper presents a solution for the linear thermoelastic problem of determining axisymmetric stress and displacement fields in an isotropic elastic solid of infinite extent weakened by an external circular crack under general mechanical loadings and general thermal conditions. The mechanical loadings and thermal conditions applied on the crack faces are axisymmetric, being non-symmetric about the crack plane. In similar lines of [7], equations of equilibrium of an elastic solid conducting heat have been solved using Hankel transforms and Abel operators of the first kind. Expressions for stress, displacement, temperature and heat flux functions are obtained in terms of Abel transforms of the first kind of the jumps of stress, displacement, temperature and heat flux at the crack plane. Two types of thermal conditions, that is, general surface temperatures and general heat flux on faces of the crack are considered. In both the cases, closed form solutions have been obtained for the unknown functions solving Abel type of integral equations. Explicit expressions for stresses, displacements, temperature fields, stress intensity factors have been obtained. Two special cases of thermal conditions in which: (i) crack faces are subjected to constant non-symmetric temperatures over a circular ring area, (ii) crack faces are subjected to constant non-symmetric heat flux over a circular ring area, have been considered. In some special cases, results have been compared with those from the literature.     

Intersonic crack growth under time-dependent loading

The problem investigated in this paper is a mode II crack extending at a uniform intersonic speed in an otherwise unbounded elastic solid subjected to time dependent crack-face tractions. The fundamental solution for this problem is obtained analytically, which is then used to construct the general solution for an intersonic crack subjected to arbitrary time-dependent loading. For time-independent loading, this solution reduces to Huang and Gao’s [J. Appl. Mech 68 (2001) 169] fundamental solution. We have also studied two crack-face loadings that are of interest for engineering applications.     

An Interface Crack for a Functionally Graded Strip Sandwiched Between Two Homogeneous Layers of Finite Thickness

In this paper, the behavior of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness subjected to an uniform tension is resolved using a somewhat different approach, named the Schmidt method. The Fourier transform technique is applied and a mixed boundary value problem is reduced to two pairs of dual integral equations in which the unknown variables are the jumps of the displacements across the crack surface. To solve the dual integral equations, the jumps of the displacements across the crack surfaces are expanded in a series of Jacobi polynomials. This process is quite different from those adopted in previous works. Numerical examples are provided to show the effects of the crack length, the thickness of the material layer and the materials constants upon the stress intensity factor of the cracks. It can be obtained that the results of the present paper are the same as ones of the same problem that was solved by the singular integral equation method. As a special case, when the material properties are not continuous through the crack line, an approximate solution of the interface crack problem is also given under the assumption that the effect of the crack surface interference very near the crack tips is negligible. Contrary to the previous solution of the interface crack, it is found that the stress singularities of the present interface crack solution are the same as ones of the ordinary crack in homogenous materials.     

A general solution and the application of space axisymmetric problem in piezoelectric material

ＡＧＥＮＥＲＡＬＳＯＬＵＴＩＯＮＡＮＤＴＨＥＡＰＰＬＩＣＡＴＩＯＮＯＦＳＰＡＣＥＡＸＩＳＹＭＭＥＴＲＩＣＰＲＯＢＬＥＭＩＮＰＩＥＺＯＥＬＥＣＴＲＩＣＭＡＴＥＲＩＡＬＷａｎｇＺｉ－ｋｕｎｇ（王子昆）ＣｈｅｎＧｅｎｇ－ｃｈａｏ（陈庚超）（ＸｉａｎＪｉａｏ...     

Dynamic penny-shaped cracks in multilayer sandwich composites

A point force method is proposed for obtaining the dynamic elastic response of a multilayer sandwich composite in the presence of a penny-shaped crack under a harmonic loading. The sandwich composite is a multilayered solid whose lower half is the mirror image of the upper half with the center plane as the mirror. The crack is lying on the mirror plane of the composite. The solution of the mode I dynamic crack problem is formulated by integrating the Green’s function of a time-harmonic surface normal point force over the crack surface with an unknown point force distribution. The dual integral equations of the unknown point force distribution are established by considering the boundary conditions, which can be reduced to a Fredholm integral equation of the second kind. A complete solution of the crack problem under consideration can be obtained by solving this Fredholm integral equation. It will be shown that the results obtained by this approach are the same as some existing solutions.     

承受弯曲板断裂问题的特征根

把Rekner型平板裂纹尖端位移场展开式推广到任意切口、任意边界的多材料问题．证明了Reissner型平板断裂问题的特征根等价于相类似的反平面切口问题和平面切口问题两部分的特征根迭加．     

On the asymptotics of the stress and strain fields near a crack tip

The problem is solved under the plane strain conditions for a crack of general form, which in general is neither a mode I nor a mode II crack. We assume that the strains are small and the material is nonlinearly elastic. The mathematical statement of the problem is reduced to the eigenvalue problem for a system of ordinary nonlinear differential equations. Its solution is obtained numerically. We show that, for an incompressible material with power-law relations between the stress and strain deviators, the solution (the well-known HRR-asymptotics [1, 2]) exists only for mode I and II cracks. In the general case, we can only speak of approximate solutions. A similar conclusion can be made for different-modulus materials. We analyze the results of the preceding papers [1–7], where specific cases of the problem were considered.     

同震断层三维裂纹扩展波动时域超奇异积分法

应用波动时域超奇异积分法将P波、S波和磁电热弹多场耦合作用下同震断层任意形状三维裂纹扩展问题转化为求解以广义位移间断率为未知函数的超奇异积分方程组问题;定义了广义应力强度因子,得到裂纹前沿广义奇异应力增量解析表达式;应用波动时域有限部积分概念及体积力法,为超奇异积分方程组建立了数值求解方法,编制了FORTRAN程序,以三维矩形裂纹扩展问题为例,通过典型算例,研究了广义应力强度因子随裂纹位置变化规律;分析了同震断层裂纹扩展中力、磁、电场辐射规律.      

Harmonic vibrations of a half-space with a surface-breaking crack under conditions of out-of-plane deformation

The paper presents a solution of the problem of determining the stress state in an elastic isotropic half-space with a crack intersecting its boundary under harmonic longitudinal shear vibrations. The vibrations are excited by a regular action of a harmonic shear load on the crack shores. The solution method is based on the use of the discontinuous solution of the Helmholtz equation, which allows one to reduce the original problem to a singular integro-differential equation for the unknown jump of the displacement on the crack surface. The solution of this equation is complicated by the existence of a fixed singularity of its kernel. Therefore, one of the main results is the development of an efficient approximate method for solving such equations, which takes into account the true asymptotics of the unknown function. The latter allows one to obtain a high-precision approximate formula for calculating the stress intensity factor.     

A unified treatment of axisymmetric adhesive contact problems using the harmonic potential function method

A unified treatment of axisymmetric adhesive contact problems is provided using the harmonic potential function method for axisymmetric elasticity problems advanced by Green, Keer, Barber and others. The harmonic function adopted in the current analysis is the one that was introduced by Jin et al. (2008) to solve an external crack problem. It is demonstrated that the harmonic potential function method offers a simpler and more consistent way to treat non-adhesive and adhesive contact problems. By using this method and the principle of superposition, a general solution is derived for the adhesive contact problem involving an axisymmetric rigid punch of arbitrary shape and an adhesive interaction force distribution of any profile. This solution provides analytical expressions for all non-zero displacement and stress components on the contact surface, unlike existing ones. In addition, the newly derived solution is able to link existing solutions/models for axisymmetric non-adhesive and adhesive contact problems and to reveal the connections and differences among these solutions/models individually obtained using different methods at various times. Specifically, it is shown that Sneddon’s solution for the axisymmetric punch problem, Boussinesq’s solution for the flat-ended cylindrical punch problem, the Hertz solution for the spherical punch problem, the JKR model, the DMT model, the M-D model, and the M-D-n model can all be explicitly recovered by the current general solution.