Investigation of the Scattering of Harmonic Elastic Waves by Two Collinear Symmetric Cracks Using the Non-Local Theory

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Investigation of a griffith crack subject to uniform tension using the non-local theory by a new method

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The scattering of harmonic elastic anti-plane shear waves by two collinear cracks in the piezoelectric plate by using the non-local theory

In this paper, the dynamic interaction between two collinear cracks in a piezoelectric material plate under anti-plane shear waves is investigated by using the non-local theory for impermeable crack surface conditions. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. These equations are solved using the Schmidt method. This method is more reasonable and more appropriate. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The project supported by the Natural Science Foundation of Heilongjiang Province and the National Natural Science Foundation of China(10172030, 50232030)     

Investigation of the Dynamic Behavior of a Griffith Crack in a Piezoelectric Material Strip Subjected to the Harmonic Elastic Anti-plane Shear Waves by Use of the Non-local Theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material strip subjected to the harmonic anti-plane shear waves is investigated by use of the non-local theory for impermeable crack surface conditions. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near at the crack tip. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the thickness of the strip, the circular frequency of incident wave and the lattice parameter.     

The scattering of harmonic elastic anti-plane shear waves by a finite crack in infinitely long strip using the non-local theory

In this paper, the scattering of harmonic anti-plane shear waves by a finite crack in infinitely long strip is studied using the non-local theory. The Fourier transform is applied and a mixed boundary value problem is formulated. Then a set of dual integral equations is solved using the Schmidt method instead of the first or the second integral equation method. A one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress occurring at the crack tips. Contraty to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the width of the strip and the lattice parameter. Supported by the Post Doctoral Science Foundation of Heilongjiang Province, the Natural Science Foundation of Heilongjiang Province and the National Foundation for Excellent Young Investigators.     

各向异性介质中SH波引起的裂纹扩展

本文利用Green函数法,求解各向异性介质中半无限长裂纹在SH波作用下,以任意速度扩展的问题。首先,利用Laplace变换和Cagniard-de Hoop反演法求解各向异性介质中反平面问题的Green函数,并利用它建立了求解裂纹扩展问题的积分方程。因为方程为Abel型的,所以可得到在SH波作用下,半无限长裂纹扩展问题的解析解。还可求得裂纹端点附近的应力和裂纹表面上位移的表达式。并对裂纹端点附近的奇异性进行讨论。最后讨论了裂纹尖端附近任一点的能量关系。并应用Griffith的能量准则,对裂纹扩展规律进行了讨论。     

A crack perpendicular to the bimaterial interface in finite solid

The dislocation simulation method is used in this paper to derive the basic equations for a crack perpendicular to the bimaterial interface in a finite solid. The complete solutions to the problem, including the T stress and the stress intensity factors are obtained. The stress field characteristics are investigated in detail. It is found that when the crack is within a weaker material, the stress intensity factor is smaller than that in a homogeneous material and it decreases when the distance between the crack tip and interface decreases. When the crack is within a stiffer material, the stress intensity factor is larger than that in a homogeneous material and it increases when the distance between the crack tip and interface decreases. In both cases, the stress intensity factor will increase when the ratio of the size of a sample to the crack length decreases. A comparison of stress intensity factors between a finite problem and an infinite problem has been given also. The stress distribution ahead of the crack tip, which is near the interface, is shown in details and the T stress effect is considered.     

THE EVALUATION OF STRESS INTENSITY FACTORS OF PLANE CRACK FOR ORTHOTROPIC PLATE WITH EQUAL PARAMETER BY F2LFEM

In this paper, the evaluation of stress intensity factor of plane crack problems for orthotropic plate of equal-parameter is investigated using a fractal two-level finite element method (F2LFEM). The general solution of an orthotropic crack problem is obtained by assimilating the problem with isotropic crack problem, and is employed as the global interpolation function in F2LFEM. In the neighborhood of crack tip of the crack plate, the fractal geometry concept is introduced to achieve the similar meshes having similarity ratio less than one and generate an infinitesimal mesh so that the relationship between the stiffness matrices of two adjacent layers is equal. A large number of degrees of freedom around the crack tip are transformed to a small set of generalized coordinates. Numerical examples show that this method is efficient and accurate in evaluating the stress intensity factor (SIF).     

弹性T项对裂尖参数和裂纹扩展路径稳定性的作用

研究了弹性T项在主裂纹与近裂尖微空洞干涉问题中对裂尖参数和裂纹扩展路径稳定性的影响作用.利用“伪力”法,并通过解决主裂纹与近裂尖微空洞干涉问题,对远场纯1型载荷和远场弹性T项载荷下,该干涉问题中弹性T项的影响作用进行了分析从数值结果可以看出：由于空洞的存在;释放了弹性T项所引起的应力,弹性T项对裂尖参数;应力强度因子和J积分都有直接显著的影响,因而,它对该载荷下的裂纹扩展路径的稳定性有控制作用。     

Investigation of the dynamic behavior of two parallel symmetric cracks in piezoelectric materials use of non-local theory

In this paper, the dynamic behavior of two parallel symmetric cracks in piezoelectric materials under harmonic anti-plane shear waves is investigated by use of the non-local theory for permeable crack surface conditions. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the problem to obtain the stress occurs near the crack tips. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations that the unknown variables are the jumps of the displacement along the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided. Contrary to the previous results, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the frequency of the incident wave, the distance between two cracks and the lattice parameter of the materials, respectively. Contrary to the impermeable crack surface condition solution, it is found that the dynamic electric displacement for the permeable crack surface conditions is much smaller than the results for the impermeable crack surface conditions. The results show that the dynamic field will impede or enhance crack propagation in the piezoelectric materials at different stages of the dynamic load.     

Finite element analysis of a bimaterial interface crack

In this investigation, the enriched element method developed by Benzley was extended to treat the stress analysis problem involving a bimaterial interface crack. Unlike crack problems in isotropic elasticity, where the stress singularity at the crack tip is of the inverse square root type, the interface crack contains an additional oscillatory singularity. Although the effect of this oscillatory characteristic is confined to a region very close to the crak tip, it nevertheless requires proper treatment in order to obtain accurate predictions on the stress intensity factors. Using appropriate crack tip stress and displacement expressions, the enriched element method can model the stress singularity for an interface crack exactly. The finite element implementation of this method has been made on the code APES. Stress intensity factor results predicted by the modified APES program compare favorably with those available in the literature. This indicates tha the enriched element technique provides an accurate and efficient numerical tool for the analysis of bimaterial interface crack problems.     

On the uses of special crack tip elements in numerical rock fracture mechanics

Numerical methods such as boundary element methods are widely used for the stress analysis in solid mechanics. These methods are also used for crack analysis in rock fracture mechanics. There are singularities for the stresses and displacements at the crack tips in fracture mechanics problem, which decrease the accuracy of the numerical results in areas very close to the crack ends. To overcome this, higher order elements and isoperimetric higher order elements have been used. Recently, special crack tip elements have been proposed and used in most of the numerical fracture mechanics models. These elements can drastically increase the accuracy of the results near the crack tips, but in most of the models only one special crack tip element has been used for each crack end. In this study the uses of higher order crack tip elements are discussed and a higher order displacement discontinuity method is used to investigate the effect of these elements on the accuracy of the results in some crack problems. The useful shape functions for two special crack tip elements, are derived and given in the text and appendix for both infinite and semi-infinite plane problems. In this analysis both Mode I and Mode II stress intensity factors are computed . Some example problems are solved and the computed results are compared with the results given in the literature. The numerical results obtained here are in good agreement with those cited in the literature. For the curved crack problem, the strain energy release rate, G can be calculated accurately in the vicinity of the crack tips by using the higher order displacement discontinuity method with a quadratic variation of displacement discontinuity elements and with two special crack tip elements at each crack end.     

On anti-plane shear behavior of a Griffith permeable crack in piezoelectric materials by use of the non-local theory

In this paper, the non-local theory of elasticity is applied to obtain the behavior of a Griffith crack in the piezoelectric materials under anti-plane shear loading for permeable crack surface conditions. By means of the Fourier transform the problem can be solved with the help of a pair of dual integral equations with the unknown variable being the jump of the displacement across the crack surfaces. These equations are solved by the Schmidt method. Numerical examples are provided. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularity is present at the crack tip. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length and the lattice parameter of the materials, respectively. The project supported by the National Natural Science Foundation of China (50232030 and 10172030)     

THE PROBLEM OF I-II COMBINED PLANE CRACK SOLVED WITH WESTERGAARD STRESS FUNCTION

直接获得I-II复合型平面裂纹问题裂纹尖端区域的应力场是一个比较复杂的问题,在此应用Westergaard应力函数求解I-II复合型平面裂纹问题,导出了裂纹尖端区域应力分量的表达式。该方法推导过程简单,物理概念清晰,其结果与一般断裂力学教材和文献中的结果一致。同时,应用叠加原理将裂纹面上的作用力转化为裂纹外边界的受力,给出了解决裂纹面上有作用力的I-II复合型平面裂纹问题的解题方法。     

双周期裂纹压电材料反平面剪切问题断裂力学分析

研究压电材料双周期裂纹反平面剪切与平面电场作用的问题．运用复变函数方法，获得了该问题严格的闭合解，并由此给出了裂纹尖端应力强度因子和电位移强度因子的精确公式．数值算例显示了裂纹分布特征对材料断裂行为的重要影响．叠间小裂纹能够对主裂纹的应力和电位移场起着屏蔽作用，相反行间小裂纹却起着放大作用，至于钻石形分布裂纹的影响规律则更为复杂．对于某些特殊情形给予了解答并导出一系列有意义的结果。     

Retardation of a crack with connections between the faces using an induced thermoelastic stress field

This paper considers local temperature variations near the tip of a crack in the presence of regions in which the crack faces interact. It is assumed that these regions are adjacent to the crack tip and are comparable in size to the crack size. The problem of local temperature variations consists of delay or retardation of crack growth. For a crack with connections between the crack faces subjected to external tensile loads, an induced thermoelastic stress field, and the stresses at the connections preventing crack opening, the boundary-value problem of the equilibrium of the crack reduces to a system of nonlinear singular integrodifferential equations with a Cauchy kernel. The normal and tangential stresses at the connections are found by solving this system of equations. The stress intensity factors are calculated. The energy characteristics of cracks with tip regions are considered. The limiting equilibrium condition for cracks with tip regions is formulated using the criterion of limiting stretching of the connections.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 133–143, January–February, 2005     

Strain energy density prediction of crack initiation and direction in cracked T-beams and pipes

In this paper, the S-theory is applied to determine crack initiation and direction for cracked T-beams and circumferentially cracked pipes. It makes use of a parameter called strain energy density factor, S, which is a function of the stress intensity factors. The strain energy density theory provides a more general treatment of fracture mechanics problems by virtue of its ability in describing the multiscale feature of material damage and in dealing with mixed mode crack propagation problem. A simple method for obtaining approximate stress intensity factors is also applied. It takes into account the elastic crack tip stress singularity while using the elementary beam theory. Some basic loading conditions in beams and pipes are studied.     

粘弹性双材料界面裂缝的缝端位移场及动态断裂分析

本文研究的是经常在实际工程中遇到的粘弹性双材料界面裂缝的动断裂问题.由于粘弹性自身的复杂性,使得粘弹性双材料界面裂缝缝端应力的奇异性较弹性呈现出更为复杂的形式,从而使动断裂问题的分析变得更为困难.根据此情况,本文采用复阻尼理论反映粘弹性体的运动规律,用复势理论和平面问题复变函数解答的科洛索夫公式推导了粘弹性双材料界面裂缝缝端位移场及动态应力强度因子的求解公式,利用特解边界元进行了粘弹性双域耦合动力响应计算,按求得的公式用位移外推法计算了单边裂纹板在动荷载作用下的动态应力强度因子.分析了粘性,弹模比和缝长对动态应力强度因子的影响,得出了一些有益的结论.     

各向异性复合材料中界面屈服/脱粘对裂纹扩展的影响

本研究针对层状复合材料中正交于层合面的裂纹,研究裂纹前方层状界面发生屈服或脱粘现象对该裂纹前沿应力场的扰动。通过利用叠加原理,借用滑移型位错密度表征界面的屈服或脱粘。利用Chebyshev数值积分法求解相应的位错密度的奇异积分方程,得到沿界面屈服/脱粘区域的位错密度分布及裂端区应力场。结果表明,若层状复合材料界面为发生屈服或脱粘,将减弱独立层裂尖的应力奇异性,进而抑制独立层中裂纹的扩展。     

变分法在含边裂纹正交各向异性板中的应用

文中给出了含边裂纹正交各向异性板问题的特征函数展开式，然后利用变分法及边值条件确定展开式中的未知系数，从而可求得裂端应力强度因子。该方法即可应用于应力边界值和位移边界值问题，又适应混合边界值问题。数值计算结果证实了该方法的有效性。