含圆形嵌体弹性平面中径向裂纹问题的超奇异积分方程方法

根据含圆形嵌体平面问题在极坐标下的弹性力学基本解,使用Betti互换定理,在有限部积分意义下将问题归结为两个以裂纹岸位移间断为基本未知量、对于Ⅰ型和Ⅱ型问题相互独立的超奇异积分方程,对含圆形嵌体弹性平面中的径向裂纹问题进行了研究.根据有限部积分原理,建立了问题的数值算法.计算结果表明,嵌体半径、裂纹位置及材料剪切弹性模量等都对裂纹应力强度因子具有较为明显的影响.     

Interaction between a rigid inclusion and a line crack under uniform heat flux

The thermoelastic displacement boundary value problem for a rigid inclusion interacting with a line crack in an infinite plane subjected to a uniform heat flux is studied, in which the rigid body rotation of the inclusion is considered. To solve the prescribed problem, we use the principle of superposition to decompose it into two groups of problems, which are further reduced to several basic subproblems including Green’s functions of edge dislocation and heat source couple, as well as the problem of a plane containing the inclusion under uniform heat flux and the problem of the inclusion subjected to a small rotation. The problems are solved using the complex variable method along with the rational mapping function technique. The variations of the stress intensity factors at the crack tips and the rigid body rotation angles with various crack lengths and heat flux angles are shown. The effects of the inclusion shape and size are also investigated.     

Interaction between curved crack and elastic inclusion in an infinite plate

Summary In this paper, the curved-crack problem for an infinite plate containing an elastic inclusion is considered. A fundamental solution is proposed, which corresponds to the stress field caused by a point dislocation in an infinite plate containing an elastic inclusion. By placing the distributed dislocation along the prospective site of the crack, and by using the resultant force function as the right-hand term in the equation, a weaker singular integral equation is obtainable. The equation is solved numerically, and the stress intensity factors at the crack tips are evaluated. Interaction between the curved crack and the elastic inclusion is analyzed. Received 8 October 1996; accepted for publication 27 March 1997     

Effect of a flat inclusion on stress intensity factor of a semi-infinite crack

The elastic plane interaction between an arbitrarily located and oriented flat inclusion and a semi-infinite crack subjected to a remote Mode I loading is considered. The method uses distributions of edge dislocations to formulate integral expressions of flat inclusion (including crack) tractions and is shown to be very accurate by a test problem. The stress intensity factors of the main crack tip are presented for a variety of crack inclusion geometries. It is seen that the flat inclusion could either yield a stress enhancement or stress shielding effect to the main crack tip depending upon the location, orientation and thickness of the flat inclusion, and depending upon the modulus ratios of the flat inclusion to matrix.     

On the asymptotic crack-tip stress fields in nonlocal orthotropic elasticity

The crack-tip stress fields in orthotropic bodies are derived within the framework of Eringen’s nonlocal elasticity via the Green’s function method. The modified Bessel function of second kind and order zero is considered as the nonlocal kernel. We demonstrate that if the localisation residuals are neglected, as originally proposed by Eringen, the asymptotic stress tensor and its normal derivative are continuous across the crack. We prove that the stresses attained at the crack tip are finite in nonlocal orthotropic continua for all the three fracture modes (I, II and III). The relative magnitudes of the stress components depend on the material orthotropy. Moreover, non-zero self-balanced tractions exist on the crack edges for both isotropic and orthotropic continua. The special case of a mode I Griffith crack in a nonlocal and orthotropic material is studied, with the inclusion of the T-stress term.     

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…     

Small edge crack in a semi-infinite solid subjected to thermal stratification

The mode I stress intensity factor for a small edge crack in an elastic half-space is found when the space is in contact with two stratified fluids of different temperatures, the boundary between the fluids oscillating sinusoidally over the solid surface. The variation in the stress intensity factor, which may lead to thermal fatigue crack growth, is examined as a function of time, crack depth, amplitude and temporal frequency of oscillation, surface heat transfer coefficient and material properties of the half-space. It is shown how this ‘boundary layer’ solution may be applied to problems involving finite geometries.     

A penny-shaped crack in a filament-reinforced matrix—II. The crack problem

Using the filament model developed in the previous paper, the elastostatic interaction problem between a penny-shaped crack and a slender inclusion or filament in an elastic matrix is formulated. For a single filament as well as multiple identical filaments located symmetrically around the crack the problem is shown to reduce to a singular integral equation. The solution of the problem is obtained for various geometries and filament to-matrix stiffness ratios, and the results relating to the angular variation of the stress intensity factor and the maximum filament stress are presented.     

Interaction between annular crack and rigid disc inclusion in a transversely isotropic solid

An analytical approach has been presented to analyze the asymmetric and axisymmetric interactions between an annular crack and a rigid disc inclusion embedded in a transversely isotropic full-space. With the aid of a method of potential functions, Hankel and Abel transforms, the solution of the problems is reduced to a system of Fredholm integral equations, which are solved by using a numerical method. In each case, the stiffness of the disc inclusion and the stress intensity factor at the tips of the annular crack for different degrees of material anisotropy and different ratios of the inner and outer radius of the crack are illustrated graphically. Several limiting cases such as penny-shaped crack and external crack along with some exact solutions are presented to demonstrate the efficiency of the method.     

Boundary element analysis of interaction between an elastic rectangular inclusion and a crack

The interaction between an elastic rectangular inclusion and a kinked crack inan infinite elastic body was considered by using boundary element method. The new complexboundary integral equations were derived. By introducing a complex unknown function H(t)related to the interface displacement density and traction and applying integration by parts,the traction continuous condition was satisfied automatically. Only one complex boundaryintegral equation was obtained on interface and involves only singularity of order l/ r. Toverify the validity and effectiveness of the present boundary element method, some typicalexamples were calculated. The obtained results show that the crack stress intensity factorsdecrease as the shear modulus of inclusion increases. Thus, the crack propagation is easiernear a softer inclusion and the harder inclusion is helpful for crack arrest.     

Thermal stresses in a two-dimensional infinite medium containing a rigid inclusion embedded in a line crack

This paper examines the problem of finding thermal stresses, caused by a symmetric indentation of a line crack by an inclusion in an infinite isotropic elastic heat conducting solid. The thermal and elastic problems are reduced to a system of triple integral equations. In each case the solution of the triple integral equations is obtained in a closed form. The expressions for the stress intensity factor at the edge of the line crack, the strain energy density function and the resultant pressure applied to the inclusion are obtained. The expression for the displacement component is also obtained. Finally the results for the physical quantities are displayed graphically.     

Heat conduction and thermal stress induced by an electric current in an infinite thin plate containing an elliptical hole with an edge crack

A uniform electric current at infinity was applied to a thin infinite conductor containing an elliptical hole with an edge crack. The electric current gives rise to two states, i.e., uniform and uneven Joule heat. These two states must be considered to analyze the heat conduction problem. The uneven Joule heat gives rise to uneven temperature and thus to heat flux, and to thermal stress.Using a rational mapping function, problems of the electric current, the Joule heat, the temperature, the heat flux, the thermal stress are analyzed, and each of their solutions is obtained as a closed form. The distributions of the electric current, the Joule heat, the temperature, the heat flux and the stress are shown in figures.The heat conduction problem is solved as a temperature boundary value problem. Solving the thermal stress problem, dislocation and rotation terms appear, which complicates this problem. The solutions of the Joule heat, the temperature, the heat flux and the thermal stress are nonlinear in the direction of the electric current. The crack problems are also analyzed, and the singular intensities at the crack tip of each problem are obtained. Mode II (sliding mode) stress intensity factor (SIF) is produced as well as Mode I (opening mode) SIF, for any direction of the electric current. The relations between the electric current density and the melting temperature and between the electric current density and SIF are investigated for some crack lengths in an aluminum plate.     

A MIXED MODE FRACTURE CRITERION BASED ON THE MAXIMUM TANGENTIAL STRESS IN BRITTLE INCLUSION

A closed-form solution for predicting the tangential stress of an inclusion located in mixed mode Ⅰ and Ⅱ crack tip field was developed based on the Eshelby equivalent inclusion theory. Then a mixed mode fracture criterion, including the fracture direction and the critical load, was established based on the maximum tangential stress in the inclusion for brittle inclusioninduced fracture materials. The proposed fracture criterion is a function of the inclusion fracture stress, its size and volume fraction, as well as the elastic constants of the inclusion and the matrix material. The present criterion will reduce to the conventional one as the inclusion having the same elastic behavior as the matrix material. The proposed solutions are in good agreement with detailed finite element analysis and measurement.     

Problem of a crack on the boundary of a rigid inclusion in an elastic plate

The problem of equilibrium of a thin elastic plate containing a rigid inclusion is considered. On part of the interface between the elastic plate and the rigid inclusion, there is a vertical crack. It is assumed that, on both crack edges, the boundary conditions are given as inequalities describing the mutual impenetrability of the edges. The solvability of the problem is proven and the character of satisfaction of the boundary conditions is described. It is also shown that the problem is the limit problem for a family of other problems posed for a wider region and describing equilibrium of elastic plates with a vertical crack as the rigidity parameter tends to infinity.     

Interaction of general plane P wave and cylindrical inclusion partially debonded from its viscoelastic matrix

The interaction of a general plane P wave and an elastic cylindrical inclusion of infinite length partially debonded from its surrounding viscoelastic matrix of infinite extension is investigated. The debonded region is modeled as an arc-shaped interface crack between inclusion and matrix with non-contacting faces. With wave functions expansion and singular integral equation technique, the interaction problem is reduced to a set of simultaneous singular integral equations of crack dislocation density function. By analysis of the fundamental solution of the singular integral equation, it is found that dynamic stress field at the crack tip is oscillatory singular, which is related to the frequency of incident wave. The singular integral equations are solved numerically, and the crack open displacement and dynamic stress intensity factor are evaluated for various incident angles and frequencies. The project supported by the National Natural Science Foundation of China (19872002) and Climbing Foundation of Northern Jiaotong University     

黏弹性功能梯度材料裂纹问题的有限元方法

针对组分材料体积含量任意分布的黏弹性功能梯度材料裂纹问题建立有限元分析途径. 通过Laplace变换,将黏弹性问题转化到象空间中求解,基于反映材料非均匀的梯度单元和裂纹尖端奇异特性的奇异单元计算象空间中的位移、应力和应变场,应用虚拟裂纹闭合方法得到应变能释放率,分别由应力和应变能释放率确定应力强度因子. 给出这些断裂参量在物理空间和象空间之间的对应关系,由数值逆变换求出其在物理空间的相应值. 文中分析两端均匀受拉的黏弹性边裂纹板条,首先针对松弛模量表示为空间函数和时间函数乘积的特殊梯度材料进行计算,结合对应原理验证方法的有效性. 然后分析组分材料体积含量具有任意梯度分布的情形,由Mori-Tanaka方法预测象空间中的等效松弛模量. 计算结果表明,蠕变加载条件下,应变能释放率随时间增加,其增大程度与黏弹性组分材料体积含量相关. 由于梯度材料的非均匀黏弹性性质,产生应力重新分布,导致应力强度因子随时间变化,其变化范围与组分材料的体积含量分布方式有关.     

The Crack-Inclusion Interaction and the Analysis of Singularity for the Horizontal Contact

ＩｎｔｒｏｄｕｃｔｉｏｎＷｈｉｌｅｗｅｓｔｕｄｙｔｈｅｓｔｒｅｎｇｔｈａｎｄｔｈｅｃｒａｃｋｏｆｐｒａｃｔｉｓｉｎｇｃｏｍｐｏｎｅｎｔｓ,ｔｈｅｍａｔｅｒｉａｌｄｅｆｅｃｔｉｏｎｓｈｏｕｌｄｂｅｃｏｎｓｉｄｅｒｅｄ .ＩｎｔｈｅｏｐｉｎｉｏｎｓｏｆＣｒａｃｋＭｅｃｈａｎｉｃｓ,ｔｈｅｍａｔｅｒｉａｌｄｅｆｅｃｔｉｏｎｃａｎｂｅｒｅｄｕｃｅｄｔｏｐｌａｎａｒｃｒａｃｋｓａｎｄｉｎｃｌｕｓｉｏｎｓ.Ｂｅｓｉｄｅｓ,ｔｈｅｐｒｏｂｌｅｍｏｆｓｈｏｒｔ＿ｆｉｂｅｒｃｏｍｐｏｓｉｔｅｍａｔｅｒｉａｌｓｕｃｈａｓ…     

A crack along the interface of a circular inclusion embedded in an infinite solid

The two-dimensional problem of an arc shaped crack lying along the interface of a circular elastic inclusion embedded in an infinite matrix with different elastic constants is considered. Based on the complex variable method of Muskhelishvili, closed-form solutions for the stresses and the displacements around the crack are obtained when general biaxial loads are applied at infinity. These solutions are then combined with A.A. Griffith's virtual work argument to give a criterion of crack extension, namely the de-bonding of the interface. The critical applied loads are expressed explicitly in terms of a function of the inclusion radius and the central angle subtended by the crack arc. In the case of simple tension the critical load is inversely proportional to the square-root of the inclusion radius. By analyzing the variation of the cleavage stress near the crack tip, the deviation of the crack into the matrix is discussed. The case of uniaxial tension is worked out in detail.     

The effect of couple stresses on the tip of a crack

The effect of couple stresses at a crack tip is investigated by considering two particular problems. A formally exact solution is obtained (for couple-stress and micropolar elasticity) for the case of a semi-infinite crack with a prescribed internal stress. Secondly, the problem of a finite crack in an infinite medium (with couple stresses) under uniform tension at infinity, is solved by matched expansions when the couple stress parameter is small compared with the crack length. In each case it is shown that the energy release rate from a crack tip tends to the classical elastic value as the couple stress (or micropolar) parameter tends to zero.     

A new integral equation formulation of two-dimensional inclusion–crack problems

A new integral equation formulation of two-dimensional infinite isotropic medium (matrix) with various inclusions and cracks is presented in this paper. The proposed integral formulation only contains the unknown displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks. In order to solve the inclusion–crack problems, the displacement integral equation is used when the source points are acting on the inclusion–matrix interfaces, whilst the stress integral equation is adopted when the source points are being on the crack surfaces. Thus, the resulting system of equations can be formulated so that the displacements on the inclusion–matrix interfaces and the discontinuous displacements over the cracks can be obtained. Based on one point formulation, the stress intensity factors at the crack tips can be achieved. Numerical results from the present method are in excellent agreement with those from the conventional boundary element method.