The analysis of crack problems with non-local elasticity

IntroductionTheclassicalconhnuummechanicshasbeenusedtosolvemanyproblemsinmacrofracturemechanics,butencountersdifficulheswhentheeffectofITilcrocharacteristicdimensionshouldbetakenintoaccount.Thestressfieldverynearthecracktipisstillnotclear.Somephenomenaofshortcrackscannotbeexplained["']andsomemechanismoffracturehasnotbeensolvedyet.Thenon-localelashcitytheoryseemsattractivetotheseproblems.Thetheoryofnon-localelasticity,establishedanddevelopedbyEringenetal[3),connectstheclassicalcontinuummechan…     

三维压电弹性体断裂分析的基本解

求解了在材料的特征方程有重根时，三维压电弹性体的单位集中不连续位移和不连续电势基本解。讨论了重根对断裂力学问题解的影响。     

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

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｌａｓｔｆｏｕｒｄｅｃａｄｅｓｈａｖｅｗｉｔｎｅｓｓｅｄｔｈｅｉｎａｕｇｕｒａｔｉｏｎｏｆａｎｏｖｅｌｔｈｅｏｒｙｏｆｍａｔｅｒｉａｌｂｏｄｉｅｓ,ｎａｍｅｄｔｈｅｎｏｎ＿ｌｏｃａｌｍｅｃｈａｎｉｃｓ.ＴｈｉｓｗａｓｄｏｎｅｐｒｉｍａｒｉｌｙｄｕｅｔｏｔｈｅｅｆｆｏｒｔｓｏｆＥｄｅｌｅｎ[1],Ｅｒｉｎｇｅｎ[2 ],ＧｒｅｅｎａｎｄＲｉｖｌｉｎ[3].Ａｃｃｏｒｄｉｎｇｔｏｔｈｅｎｏｎ＿ｌｏｃａｌｔｈｅｏｒｙ ,ｔｈｅｓｔｒｅｓｓａｔａｐｏｉｎｔＸｉｎａｂｏｄｙｄｅｐｅｎｄｓｎｏ…     

双轴载荷作用下源于椭圆孔的分支裂纹的一种边界元分析

利用一种边界元方法来研究双轴载荷作用下无限大板中源于椭圆孔的分支裂纹．该边界元方法由Crouch与Starfied建立的常位移不连续单元和笔者提出的裂尖位移不连续单元构成．在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界,文中算例说明本数值方法对计算平面弹性裂纹的应力强度因子是非常有效的。该文对双轴载荷作用下无限大板中源于椭圆孔的分支裂纹的数值结果进一步证实本数值方法对计算复杂裂纹的应力强度因子的有效性,同时该数值结果可以揭示双轴载荷及裂纹体几何对应力强度因子的影响。     

Stress intensity factors for asymmetric branched cracks in plane extension by using crack-tip displacement discontinuity elements

Stress intensity factors are important in the analysis of cracked materials. They are directly related to the fracture propagation and fatigue crack growth criteria. Based on the analytical solution (Crouch, S.L., 1976. Solution of plane elasticity problems by displacement discontinuity method, Int. J. Numer. Methods Eng. 10, pp. 301–343; Crouch, S.L., Starfield, A.M., 1983. Boundary Element Method in Solid Mechanics, with Application in Rock Mechanics and Geological Mechanics, London, Geore Allon and Unwin, Bonton, Sydney) to the problem of a constant discontinuity in displacement over a finite line segment in the x, y plane of an infinite elastic solid, recently, the crack-tip displacement discontinuity element which can be classified as the left and right crack-tip displacement discontinuity elements are developed by the author Yan, X., (in press. A special crack-tip displacement discontinuity element, Mechanics Research Communications) to model the crack-tip fields to more accurately compute the stress intensity factors of cracks in general plane elasticity. In the boundary element implementation the left or the right crack-tip displacement discontinuity element is placed locally at the corresponding left or right crack tip on top of the ordinary non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. To prove further the efficiency of the suggested approach and provide more results of the stress intensity factors, in this study, analysis of an asymmetric branched crack bifurcated from a main crack in plane extension is carried out.     

THE MIXED BOUNDARY-VALUE PROBLEM FOR NON-LOCAL ASYMMETRIC ELASTICITY

ＩｎｔｒｏｄｕｃｔｉｏｎＩｔｉｓｗｅｌｌ＿ｋｎｏｗｎｔｈａｔｔｈｅｒｅｅｘｉｓｔｓｔｈｅａｒｇｕｍｅｎｔｂｅｔｗｅｅｎＡｔｋｉｎｓｏｎ（ｓｅｅ［１～４］）ａｎｄＥｒｉｎｇｅｎａｎｄｃｏ＿ｗｏｒｋｅｒｓ（ｓｅｅ［５～７］）ｏｖｅｒｔｈｅｎｏｎ＿ｌｏｃａ...     

The influence of elastic waves on dynamic stress intensity factors (three-dimensional problems)

Summary The fundamental solutions of the displacement discontinuity for three-dimensional problems in Laplace space are deduced in thsi paper. The displacement discontinuity method and the equivalent stress method were combined and used to determine dynamic stress intensity factors for three-dimensional time-dependent crack problems. The stress intensity factors were calcualted for dynamically loaded cracks with rectangular, circular, and elliptical crack fronts. The influence of elasticity waves (in particular surface waves) on the magnitude of the stress intensity factor and on the displacement of the crack surfaces was analysed.On leave from the Central-South University of Technology, Changsha, Hunan Province, P. R. China.     

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

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｓｅｖｅｒａｌｐｒｅｖｉｏｕｓｐａｐｅｒｓ［１,２,３］,Ｅｒｉｎｇｅｎｄｉｓｃｕｓｓｅｄｔｈｅｓｔａｔｅｏｆｓｔｒｅｓｓｎｅａｒｔｈｅｔｉｐｏｆａｓｈａｒｐｌｉｎｅｃｒａｃｋｉｎａｎｅｌａｓｔｉｃｐｌａｔｅｓｕｂｊｅｃｔｔｏｕｎｉｆｏｒｍｔｅｎｓｉｏｎ,ｓｈｅａｒａｎｄａｎｔｉ＿ｐｌａｎｅｓｈｅａｒ．Ｔｈｅｆｉｅｌｄｅｑｕａｔｉｏｎｓｅｍｐｌｏｙｅｄｉｎｔｈｅｓｏｌｕｔｉｏｎｏｆｔｈｅｓｅｐｒｏｂｌｅｍｓａｒｅｔｈｏｓｅｏｆｔｈｅｔｈｅｏｒｙｏｆｎｏｎ＿ｌｏｃａｌｅ…     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

The G2 constant displacement discontinuity method – Part I: Solution of plane crack problems

A new constant displacement discontinuity element was presented in a previous paper applied initially for the numerical solution of either isolated straight cracks or for co-linear cracks of the three fundamental deformation modes I, II and III due to the special form of the solution. It was based on the strain-gradient elasticity theory in its simplest possible Grade-2 variant. The assumption of the G2 expression for the stresses has resulted to a better average stress value at the mid-point of the straight displacement discontinuity compared to the classical elasticity solution. This new element gave considerably better predictions of the stress intensity factors compared to the constant displacement discontinuity element and the linear displacement discontinuity element. Moreover, it preserved the simplicity and hence the high speed of computations. In this Part I, the solution for this element is extended for the analysis of cracks of arbitrary shape in an infinite plane isotropic elastic body and it is validated against three known analytical solutions.     

Three-dimensional interfacial fracture analysis of a one-dimensional hexagonal quasicrystal coating

In this paper, the three-dimensional (3D) interfacial fracture is analyzed in a one-dimensional (1D) hexagonal quasicrystal (QC) coating structure under mechanical loading. A planar interface crack with arbitrary shape is studied by a displacement discontinuity method. Fundamental solutions of interfacial concentrated displacement discontinuities are obtained by the Hankel transform technique, and the corresponding boundary integral-differential equations are constructed with the superposition principle. Green’s functions of constant interfacial displacement discontinuities within a rectangular element are derived, and a boundary element method is proposed for numerical simulation. The singularity of stresses near the crack front is investigated, and the stress intensity factors (SIFs) as well as energy release rates (ERRs) are determined. Finally, relevant influencing factors on the fracture behavior are discussed.     

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.     

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.      

On the tangential stress anomaly in the displacement discontinuity method

It is shown that the anomaly associated with the incorrect evaluation of tangential stresses in the displacement discontinuity (DD) method, commonly used to solve crack problems, is related to the order of singularity of the fundamental solutions of linear elasticity. It is established here that a minimum of linear variation of the shear DD distribution is needed to obtain the correct tangential stress jump across a crack. Alternatively, a correction term (‘patch’) that improves tangential stresses is derived. It is also shown that need for higher functionality is a fundamental requirement rather than a convenient artifice for obtaining better accuracy.     

具有自由边的复合材料层合板的解析解

从三维弹性力学基本方程出发,通过假设自由边的边界位移函数,建立了正交异性层合板的状态方程,给出了对边自由,对边简支矩形板的解析解.此解满足层合板的基本方程和层间连续条件.用本文的方法比较容易处理层合板的自由边.算例表明,数值结果具有较高的精度.     

Foundation of Polar Linear Elasticity for Fibre-Reinforced Materials

This study is motivated by evidence suggesting that the equations of polar elasticity of fibre-reinforced materials are non-elliptic even within the regime of infinitesimal deformations. In its endeavour to resolve this issue, which in symmetric-stress elasticity emerges in the regime of finite deformations only, it lays the foundation for development of a second-gradient theory of linear elasticity. Complete formulation of this new theory is achieved for locally transverse isotropic materials; namely, materials having embedded a single unidirectional family of arbitrarily shaped fibres which are resistant in bending, stretching and twist. The associated analysis shows that, indeed, the obtained Navier-type displacement equations are not elliptic. They accordingly predict that there exist in the material weak discontinuity surfaces, which may indeed be activated within the infinitesimal deformation regime. Surfaces containing the fibres are certainly such surfaces of weak discontinuity; this result may be not irrelevant to numerous practical situations where straight metallic fibres in fibre-reinforced concrete structures emerge partially de-bonded and exposed from their concrete matrix. Nevertheless, the analysis reveals further that additional surfaces of weak discontinuity may well exist in the locally transverse isotropic material of interest. An extension framework is also outlined towards cases of fibrous composites containing two or more families of non-perfectly flexible fibres.     

A numerical analysis of perpendicular cracks under general in-plane loading with a hybrid displacement discontinuity method

In this paper, a numerical analysis of perpendicular cracks under general in-plane loading is performed by using a hybrid displacement discontinuity method which consists of the non-singular displacement discontinuity element presented by Crouch and Starfied and the crack tip displacement discontinuity elements by the author. In the boundary element implementation the left or the right crack tip displacement discontinuity element is placed locally at corresponding left or right crack tip on top of the ordinary non-singular displacement discontinuity elements that cover the entire crack surface and the other boundary. The present numerical results show that the numerical approach is simple, yet very accurate for calculating numerically stress intensity factors for perpendicular cracks under general in-plane loading.     

FATIGUE GROWTH MODELING OF MIXED-MODE CRACK IN PLANE ELASTIC MEDIA

This paper presents an extension of a displacement discontinuity method with cracktip elements （a boundary element method） proposed by the author for fatigue crack growth analysis in plane elastic media under mixed-mode conditions. The boundary element method consists of the non-singular displacement discontinuity elements presented by Crouch and Starfield and the crack-tip displacement discontinuity elements due to the author. In the boundary element implementation the left or right crack-tip element is placed locally at the corresponding left or right crack tip on top of the non-singular displacement discontinuity elements that cover the entire crack surface and the other boundaries. Crack growth is simulated with an incremental crack extension analysis based on the maximum circumferential stress criterion. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not required because of an intrinsic feature of the numerical approach. Crack growth is modeled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. At the same time, the element characteristics of some related elements are adjusted according to the manner in which the boundary element method is implemented. As an example, the fatigue growth process of cracks emanating from a circular hole in a plane elastic plate is simulated using the numerical simulation approach.     

含两类附加函数的扩展等参有限元法

基于扩展有限元的基本思想,提出一类指数型间断函数来模拟。由于裂纹或节理等非连续结构所导致的位移不连续现象,该附加函数是以到间断处的垂直距离为自变量,且随距离的增大而呈指数衰减,同时,在非连续结构末端引入能反映其奇异场特性的三角基函数。本文用弱解形式推导了扩展有限元格式,并论证了两类附加函数在单元公共边上能够保持位移连续性这一要求。最后,编制了二维4节点和三维8节点的扩展等参有限元程序,并分别给出了算例,结果表明在模拟裂纹追踪时,扩展有限元法可行且有效。     

激波作用下三维涡的形成及其界面捕捉计算

针对三维多介质可压缩流体,给出了可压缩多介质流体三维高精度数值计算方法,以及界面捕捉方程和带重新初始化的三维LevelSet方法,对初始压力间断和密度间断条件形成激波、接触间断以及稀疏波的三维复杂流场相互作用情况进行数值计算,给出流场中涡的形成过程和界面位置。并对计算方法进行理论验证。