Anti-plane analysis of an orthotropic strip weakened by several moving cracks

In this paper several finite cracks with constant length (Yoffe-type crack) propagating in an orthotropic strip were studied. The distributed dislocation technique is used to carry out stress analysis in an orthotropic strip containing moving cracks under anti-plane loading. The solution of a moving screw dislocation is obtained in an orthotropic strip by means of Fourier transform method. The stress components reveal the familiar Cauchy singularity at the location of dislocation. The solution is employed to derive integral equations for a strip weakened by moving cracks. Finally several examples are solved and the numerical results for the stress intensity factor are obtained. The influences of the geometric parameters, the thickness of the orthotropic strip, the crack size and speed have significant effects on the stress intensity factors of crack tips which are displayed graphically.     

Cracked orthotropic strip with clamped boundaries

The stress intensity factor at the tip of a semi-infinite crack in an orthotropic infinite strip was determined. Clamped strip boundaries were considered.     

周期裂纹削弱的无限长板条的应力分析

作出了周期裂纹削弱的无限长板条的应力分析．假设这些裂纹均在水平位置,又板条承受 y方向的拉伸力p．此时边值问题归结为一个复杂混合边值问题．发现,对此问题言,特征展开变分原理方法 ( eigenfunction expansion variational method,简称为EEVM)是非常有效的．研究了裂纹端的应力强度因子和T-应力．从拉伸力作用下的弹性变形考虑,开裂板条可等价于一不开裂的正交异性板条．还分析了等价正交异性板条的弹性性质．最后给出了算例和数值结果．     

The Mathematical Study of an Edge Crack in Two Different Specified Models under Time-Harmonic Wave Disturbance

Mechanics of Composite Materials - This paper devotes to determining a stress intensity factor (SIF) at the tip of an edge crack in two models considered. Problem-1 is an orthotropic strip of a...     

条状功能梯度材料中偏心裂纹对反平面简谐波的散射问题

利用Schmidt方法研究了条状功能梯度材料中偏心裂纹对反平面简谐波的散射问题,裂纹垂直于条状功能梯度材料的边界．通过Fourier变换,问题可以转换为对一对未知变量是裂纹表面位移差的对偶积分方程求解．为了求解对偶积分方程,把裂纹表面的位移差展开为Jacobi多项式级数形式,进而得到了功能梯度参数、裂纹位置以及入射波频率对应力强度因子影响的规律．     

Two problems with mixed boundary conditions for an elastic orthotropic strip

Two problems are considered for an elastic orthotropic strip: the contact problem and the crack problem. Both problems are reduced to integral equations of the first kind with different kernels, containing a singularity: logarithmic for the first problem and singular for the second problem. Regular and singular asymptotic methods are employed to construct approximate solutions of these integral equations. Numerical results are presented.     

正交异性双材料的Ⅱ型界面裂纹尖端场

通过引入含16个待定实系数和两个实应力奇异指数的应力函数,再借助边界条件,得到了两个八元非齐次线性方程组.求解该方程组,在双材料工程参数满足适当条件下,确定了两个实应力奇异指数.根据极限唯一性定理,求出了全部系数,得到了应力函数的表示式.代入相应的力学公式,推出了当特征方程组两个判别式都小于0时,每种材料的裂纹尖端应力强度因子、应力场和位移场的理论解.裂纹尖端附近的应力和位移有混合型断裂特征,但没有振荡奇异性和裂纹面相互嵌入现象作为特例,当两种正交异性材料相同时,可以推出正交异性单材料Ⅱ型断裂的应力奇异指数、应力强度因子公式、应力场、位移场表示式.     

Mathematical modelling and the study of the influence of initial stresses on the SIF and ERR at the crack tips in a plate-strip of orthotropic material

A plate-strip fabricated from the orthotropic material and containing a crack whose edges are parallel to the face planes of the plate is considered. It is assumed that the strip is stretched (or compressed) initially along the crack edges by uniformly distributed external normal forces acting on the simply supported ends of the plate-strip. After this initial stretching (or compression) the crack edges are loaded by additional uniformly distributed normal (opening) forces. As a result of the action of these additional forces the stress concentration characterized by the stress intensity factor (SIF) of mode I or by the energy release rate (ERR) of mixed mode arises at the crack tips. In this paper, the influence of the initial stresses on the SIF or ERR is modelled mathematically by the use of the three-dimensional linearized theory of elasticity. The aim of the present investigations is to study the effect of the mechanical–orthotropic properties of the plate-strip material on this influence by the use of the finite element method (FEM) modelling of the corresponding boundary-value problem.     

位于两不同正交各向异性半平面间张开型界面裂纹的性能分析

利用Schmidt方法分析了位于正交各向异性材料中的张开型界面裂纹问题．经富立叶变换使问题的求解转换为求解两对对偶积分方程,其中对偶积分方程的变量为裂纹面张开位移．最终获得了应力强度因子的数值解．与以前有关界面裂纹问题的解相比,没遇到数学上难以处理的应力振荡奇异性,裂纹尖端应力场的奇异性与均匀材料中裂纹尖端应力场的奇异性相同．同时当上下半平面材料相同时,可以得到其精确解．     

Stress analysis of orthotropic planes weakened by cracks

The stress fields in an orthotropic infinite plane containing Volterra type climb and glide edge dislocations are derived. The dislocation solutions are utilized to formulate integral equations for dislocation density functions on the surfaces of smooth cracks. The integral equations are of Cauchy singular type and are solved for several different cases of crack configurations and arrangements. The results are used to evaluate modes I and II stress intensity factors for multiple smooth cracks.     

Progressive thermal stress distribution around a crack under Joule heating in orthotropic materials

Stress intensity factor and stress distribution at crack tips are classical problems in solids, which are closely related to the failure and reliability of materials. A crack in a nonlinearly coupled anisotropic medium, on the other hand, is much more difficult to analyze. Using the generalized complex variable method, the thermal stress problem of a crack embedded in an orthotropic medium has been analyzed, and the progressive thermal stress distributions have been obtained in closed-forms. The analysis shows that the thermal stress intensity factors are linear functions of remote thermal flux while are nonlinear functions of remote current; the thermal stress distributions under produced by thermal flux and Joule heating are similar, but not identical; the thermal stress intensity factors are linear functions with respect to the thermal expansion coefficients; with the increase of crack length, the thermal stress intensity factor caused by Joule heat increases rapidly; the thermal stress intensity factors are directly proportional to the temperature difference between the upper and lower crack surfaces and the left and right half crack surfaces divided by the square root of the crack length, and the ratios are only determined by the material parameters. These results provide a powerful tool for the failure and reliability analysis of conductive materials, and suggested that thermal stress analysis may be localized.     

On the peculiarities of a crack growth path in anisotropic solid

One of the possibilities to increase the resistance of a structure to catastrophic fracture is to force a main line crack to deviate from its path. In this connection the influence of the elastic moduli of an anisotropic material on the possibilities of crack rotation are studied. In particular a linear elastic problem for a straight Mode I crack, located on a symmetry axis of an orthotropic plane is considered. The strength properties of the material are supposed to be isotropic. For studying a direction of a crack growth path several crack models are considered. It is shown that a thin elongated elliptical hole as a crack model leads to more plausible results concerning crack rotation conditions than an ideal cut model. The maximal tensile stresses are taken as a crack growth criterion. It is shown that for some class of orthotropic materials a crack deviates from the straight path just after it starts to grow even in the conditions of uniaxial normal tension. The problem of the stability of a straight crack path under Mode I loading is also considered. This problem is reduced to the problem of the fracture direction determination for thin elongated elliptical cavity slightly inclined to the initial direction. In the frame of the proposed approach the conditions of instability are obtained. It is shown that for some class of orthotropic materials a straight crack path is unstable in the conditions of uniaxial normal tension. This class of materials is wider than one for which a crack deviates from the straight crack path just after its start. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A strip with a boundary crack under the action of a heat source

We study the influence of instantaneous heat sources on the stress-strain state and the stress intensity coefficients of a strip with a boundary crack. It is assumed that mixed homogeneous boundary conditions are given on the boundaries of the strip while the lateral surfaces are thermally insulated.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 36, 1992, pp. 97–101.     

复合材料桥纤维拔出问题的动态裂纹模型

在一无限的正交各向异性体的弹性平面上,对具有桥纤维平行自由表面的一个内部中央裂纹,进行了弹性分析．提出了复合材料桥纤维拔出的一个动态模型．由于纤维破坏是由最大拉应力支配,纤维断裂并且裂纹扩展将以自相似的方式出现．通过复变函数的方法将所讨论的问题转化为Reimann-Hilbert混合边界值问题的动态模型,呈现一简单的和容易的解．求得了正交异性体中扩展裂纹受运动的阶梯载荷、瞬时脉冲载荷作用下问题的解析解,并利用这一解,通过迭加最终求得该模型的解．     

功能梯度材料有限宽板的反平面断裂问题研究

研究了功能梯度材料有限宽板中与板边平行的III型裂纹问题．假设材料的剪切模量沿板宽度方向呈指数规律变化,利用Fourier变换将问题描述为奇异积分方程,并进一步将未知的位错密度函数表示为Chebyshev多项式的级数式,从而将奇异积分方程化为线性代数方程组进行配点数值求解．基于数值结果,讨论了材料非均匀性参数、板和裂纹的几何参数等对应力强度因子（SIF）的影响．研究表明,SIF随裂纹长度的增大而增大,随裂纹所在区域材料刚度的增大而减小;板越窄,SIF对非均匀性参数的变化越敏感,且变化规律也越复杂．随着非均匀性参数的增大,SIF既可能增大也可能减小还可能基本保持不变,这主要取决于板的相对宽度和裂纹的相对位置．当裂纹位于板的中央或当板较宽时,SIF对非均匀性参数的变化都不太敏感．     

A strip yield model solution for an internally cracked piezoelectric strip

An analysis of the crack closure and fatigue crack growth rate have been carried out for an infinitely long poled piezoelectric ceramic strip weakened by a straight hair line internal crack. The ceramic under consideration is assumed to be mechanically more brittle. The crack faces are perpendicular to the poled direction of the strip. The crack faces open in Mode-I deformation on account of in-plane tension applied to the edges of the strip together with either an in-plane electric displacement prescribed on edges of the strip or a uniform constant electric field prescribed on its edges. As a result, a yield zone is formed ahead of each tip of the crack. The yield zones developed are then arrested by applying a normal, cohesive, linearly varying yield point-stress to their rims. For each case, the Fourier transform method is used to find a solution. The resulting integral equations are solved numerically. Expressions are derived for the crack opening displacement and the crack growth rate. The variations in these quantities are plotted in relation to the affecting parameters, viz., the strip thickness, the yield zone length, the electric displacement, and material constants. A case study is presented graphically for PZT-4, PZT-5H, and BaTiO₃ ceramics. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 5, pp. 647–664, September–October, 2008.     

On time-dependent behavior of cross-ply laminated strips with viscoelastic interfaces

The two-dimensional problem of a simply supported laminated orthotropic strip with viscoelastic interfaces under static loading is studied. State-space formulations are developed based on the exact elasticity equations governing orthotropic media and the Kelvin–Voigt constitutive relation of interfaces. Since the response of the strip is time-dependent, the power series expansion technique is adopted to model the variations of elastic fields with time. Results show that the response of the laminated strip with viscoelastic interfaces changes remarkably with time, which is also significantly different from that of a plate with perfect interfaces or with viscous interfaces. Note that from the present analysis, the response for a laminated plate with spring-like interfaces or with viscous interfaces can be easily obtained because they are just two particular cases of the present Kelvin–Voigt model.     

Effect of Thermomechanical Loading on an Edge Crack of Finite Length in an Infinite Orthotropic Strip

Mechanics of Composite Materials - The purpose of this article is to determine the effect of thermal loadings on the stress intensity factor of an edge crack of finite length in an orthotropic...     

有限高狭长压电体中半无限反平面裂纹分析

利用保角变换和复变函数方法,研究了裂纹面上受反平面剪应力和面内电载荷共同作用下的有限高狭长压电体中半无限裂纹的断裂问题,给出了电不可通边界条件下裂纹尖端场强度因子和机械应变能释放率的解析解．当狭长体高度趋于无限大时,可得到无限大压电体中半无限裂纹的解析解．若不考虑电场作用,所得解可退化为纯弹性材料的已知结果．此外,通过数值算例,分析了裂纹面上受载长度、狭长体高度以及机电载荷对机械应变能释放率的影响规律．     

Buckling Delamination of a Rectangular Orthotropic Thick Plate with an Embedded Rectangular Crack Subjected to an Axial Compressive Force

Mechanics of Composite Materials - A 3D buckling delamination problem for a simply supported rectangular orthotropic composite thick plate containing a rectangular crack and subjected to an axial...