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.     

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.     

Forced vibration of an initially stressed thick rectangular plate made of an orthotropic material with a cylindrical hole

The forced vibration of an initially statically stressed rectangular plate made of an orthotropic material is studied. The plate is simply supported along all its edges and contains an internal across-the-width cylindrical hole of rectangular cross section with rounded corners. The initial stresses are created by uniformly distributed normal forces applied to opposite end faces of the plate. Because of the hole, these stresses are not uniform in the plate and significantly affect the stress field caused by additional time-harmonic dynamical forces acting on the upper face of the plate. Hence, for solving the boundary-value problem considered, the superposition principle is unsuitable. Therefore, our investigations are carried out within the framework of the three-dimensional linearized theory of elastic waves in initially stressed bodies. The corresponding boundary- value problems on determining the initial and additional, dynamical stress states are solved by using the three-dimensional finite-element method. Numerical results on stress concentrations around the cylindrical hole and the fundamental frequencies, and on the influence of the initial stresses on the frequencies are presented.     

Toughening effect of ferroelectric ceramics induced by domain switching and dislocations

The multi-scale analysis of fracture toughness of ferroelectric ceramics under complicate mechanical–electrical coupling effect is carried out in this paper. The generalized stress intensity factor (SIF) arising from spontaneous strains and polarization transformation in switching domain zones is accurately obtained by using an extended Eshelby theory. Taking BaTiO₃ ferroelectric ceramic for example, it is discovered that the crack propagation can be induced by domain switching arising from negative electrical field when the crack surface is parallel to the isotropic plane, and the obtained critical electric displacement intensity factor (EDIF) approximates closely to that obtained by the Green’s function method. Additionally, as pinning dislocations and slip dislocations can strongly influence properties of ferroelectric devices and induce the property degradation, it is necessary to investigate the dislocation toughening effects on fatigue and fracture mechanisms. The results show that the dislocation shielding and anti-shielding effects on mode II SIF, mode I SIF and EDIF are obviously different when a dislocation locates at a position near the crack tip. Through the calculation of the critical applied EDIF for crack propagation by using mechanical energy release rate (MERR) theory, it is discovered that the slip angles obviously influence fracture toughness, and the mode II SIF arising from dislocation has little influence on fracture toughness, however, the mode I SIF and EDIF arising from dislocation have great influences on fracture toughness.     

Stress concentration in an anisotropic half-plane with holes and cracks

On the basis of general representations of the generalized complex potentials for a multiconnected half-plane, which the authors have obtained, we solve problems for a multiconnected half-plane with holes and cracks when external forces or dies act on the boundary of the half-plane. Using conformal mapping for an ellipse and the method of least squares, we reduce these problems to solving a system of linear algebraic equations. For different anisotropic materials we give the results of studies of the stress distributions and the variation of the stress intensity factors for a half-plane with a crack in the case of tension at infinity, internal pressure on the edges of the crack, and the action of normal forces on the rectilinear boundary. Two figures, 2 tables. Bibliography: 2 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 63–72.     

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．     

无限体中受均匀拉伸的椭圆片状裂纹周界上各点应力强度因子的讨论

本文从Green-Sneddon解[1]的裂纹表面位移场结果出发,应用坐标变换推出了无限体中受均匀拉伸的椭圆片状裂纹周界上任意点、任意方位上的应力强度因子K₁(x₁,z₁,α)表达式.从而补充了Irwin的工作[3],证明了对椭圆周界上某一确定点而言,沿法线平面上所得的应力强度因子为最大值.并指出了一些著作中,对[3]中有关内容所作的错误解释.还推荐了一个更为直观的以极角来表示的椭圆周界上任意点处的应力强度因子表达式.     

Determination of the stress state of a circular disk with a crack

With the use of complex potentials from the solution of a contact problem for slits in a multiply-connected region, a solution is found for a problem of the theory of elasticity for an isotropic circular disk with an arbitrary radial crack. The case of an edge crack is among the cases for which a solution is found. The types of loading examined are uniform tension on an outside edge, internal pressure on the edges of cracks, and concentrated forces at arbitrary points of a disk. The unknown coefficients in the complex potentials are found from the boundary conditions on the outside edge of the disk by the series method, the colocation method, or the least squares method. Detailed numerical studies are conducted to determine the effect of the geometric characteristics and the points of application of concentrated forces on the character of the stress distribution and the stress intensity factor. A comparative characteristic of the methods used to find the coefficients is presented.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19 pp. 50–61, 1988.     

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...     

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)     

中心裂纹夹紧矩形板的拉伸*

本文利用单裂纹基本解及无限板条的Fourier变换解,将含有中心裂纹的夹紧矩形板的拉伸问题,化归为解一组奇异积分方程,进而使用Gauss-Jacobi求积公式,计算了中心裂纹的应力强度因子及夹紧边的法向应力,在应力强度因子表中还作了数值结果比较.     

Deformation of sandwich panels cylindrically bent by point forces. 2. Development of procedure

Our proposed method [1] is extended for the cylindrical bending of an asymmetrical composite panel upon piecewise constant loading and point forces, with and without allowance for transversal stiffness (perfect compliance) in shear or tension/compression along the normal. A set of fundamental functions was obtained for all the cases examined. The properties of functions were studied taking account of the discontinuous nature of the surface loading. The set of functions was normalized for initial values of the variable coordinate. Integral relationships required for analysis were derived and an identical expression of the unit function was represented in terms of the fundamental function set. The boundary problem of a panel supported along the surface of its lower face layer with free ends is reduced to the Cauchy problem. The solution is greatly simplified for a panel symmetrical relative to its mean plane. Asymptotic formulas were obtained for the case of infinite panel length. Relationships are give for the stresses and layer deflections, which permit consideration of all the features of the stress state in addition to simplified calculations for actual panel design.Communication 1, see [1].Institute of Polymer Mechanics, Latvian Academy of Sciences, Riga LV-1006, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 1, pp. 34–65, January–February, 1997.     

Delamination buckling of a rectangular orthotropic composite plate containing a band crack

The delamination buckling problem for a rectangular plate made of an orthotropic composite material is studied. The plate contains a band crack whose faces have an initial infinitesimal imperfection. The subsequent development of this imperfection due to an external compressive load acting along the crack is studied through the use of the three-dimensional geometrically nonlinear field equations of elasticity theory for anisotropic bodies. A criterion of initial imperfection is used in determining the critical forces. The corresponding boundary-value problems are solved by employing the boundary-form perturbation technique and the FEM. Numerical results for the critical force are presented.     

一维六方准晶双材料中圆孔边共线界面裂纹的反平面问题

研究了一维六方准晶双材料中圆孔边不对称共线界面裂纹的反平面问题。利用Stroh公式和复变函数方法得到了声子场和相位子场耦合作用下的复势函数,给出了裂纹尖端应力强度因子和能量释放率的解析表达式。通过数值算例,讨论了圆孔半径和裂纹长度对应力强度因子的影响,以及耦合系数、声子场应力和相位子场应力对能量释放率的影响。结果表明:当圆孔半径不变时,应力强度因子随右裂纹长度的增大趋向稳定值。当相位子场应力取一定值时,能量释放率达到最小值,说明特定的相位子场应力可以抑制裂纹的扩展。     

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.     

Cohesive zone model for a transverse breathing crack in a rotor

The breathing mechanism of a transversely cracked rotor and its influence on a rotor system that appears due to the shaft weight is studied. This breathing mechanism is based on experimental and simulation result for the crack shape reported in the literature. If the crack depth is small, the crack closure line is a straight line while for larger crack depths the crack closure becomes more curved. For both cases, a method is proposed for the evaluation of the stiffness losses in the cross section that contains the crack. This method is based on a cohesive zone model (CZM) instead of linear elastic fracture mechanics (LEFM) approach, because LEFM is valid only for the fully open crack and cannot be extended to other intermediate situations. As the crack is closed, the stress intensity factor (SIF) will not appear at the boundary between the closed cracked areas and the open cracked areas. The CZM is developed for mode-I plane strain conditions and accounts explicitly for triaxiality of the stress state by using constitutive relations. The proposed model gives more realistic results than models based on LEFM for the stiffness losses of the crack rotor system for a wide range of the crack depth. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamic response of planar interface crack between viscoelastic bodies

IntroductionThe dynamic stress intensity factor (SIF) plays an important role in dynamic fracture underboth harmonic and transient loads. It predicts whether or not the fracture toughness of thematerial will be exceeded and catastrophic crack propagation will follow. The dynamic SIFof interfaCe cracks between two dissimilar elastic materials has been studied widely. Kundull]studied the dynamic SIF of interface crack under transient loading with the method based onBetti's reciprocal theore…     

Energy release rate in a prestressed sandwich plate-strip containing interface cracks

Within the framework of a piecewise homogeneous body model, by utilizing the three-dimensional linearized theory of elasticity, the problem of opening of interface cracks in an initially stressed sandwich plate-strip is studied. To solve the corresponding boundary-contact problem, the FEM is employed. Numerical results for the influence of the initial stresses on the energy release rate (ERR) at various values of problem parameters are presented and discussed. In particular, it is established that the ERR in creases (de creases) with growing initial tension (compression) of the plate-strip.     

变刚度矩形薄板的弯曲与稳定计算

本文研究两对边简支,其它两边任意支承,板的刚度沿简支边方向按任意规律变化的矩形板.采用文[1]提出的有限板条元素法求解,其特点与习用的有限元法或有限单条法不同.不是先建立单元或单条的刚度矩阵,然后拼装总刚阵再求解,而是确立各个板条元素的变位和内力的传播关系.实例计算表明,该法是一个简便有效的方法.