An explicit elastic solution for a brittle film with periodic cracks

A two-dimensional explicit elastic solution is derived for a brittle film bonded to a ductile substrate through either a frictional interface or a fully bonded interface, in which periodically distributed discontinuities are formed within the film due to the applied tensile stress in the substrate and consideration of a “weak form stress boundary condition” at the crack surface. This solution is applied to calculate the energy release rate of three-dimensional channeling cracks. Fracture toughness and nominal tensile strength of the film are obtained through the relation between crack spacing and tensile strain in the substrate. Comparisons of this solution with finite element simulations show that the proposed model provides an accurate solution for the film/substrate system with a frictional interface; whereas for a fully bonded interface it produces a good prediction only when the substrate is not overly compliant or when the crack spacing is large compared with the thickness of the film. If the section is idealized as infinitely long, this solution in terms of the energy release rate recovers Beuth’s exact solution for a fully cracked film bonded to a semi-infinite substrate. Interfacial shear stress and the edge effect on the energy release rate of an asymmetric crack are analyzed. Fracture toughness and crack spacing are calculated and are in good agreement with available experiments.     

含周期性裂纹正交各向异性板平面问题的应力场分析

通过引入适当的Westergaard应力函数,采用复变函数方法和待定系数法对含周期性裂纹正交各向异性纤维增强复合材料板的Ⅰ 型、Ⅱ型问题中裂纹尖端附近的应力场进行了力学分析。在远处对称载荷与斜对称载荷作用下,先给出Ⅰ型、Ⅱ型问题在裂纹尖端处的应力强度因子,然后导出用应力强度因子表示的Ⅰ型、Ⅱ型裂纹问题应力场的解析表达式。此外,应力场大小与材料常数有关,这是正交各向异性材料不同于各向同性材料的特征。由于裂纹的周期分布,应力强度因子的大小取决于形状因子。结果表明,形状因子随着裂纹长度的增加而增大,随着裂纹间距的增大而逐渐下降,当裂纹间距趋于无穷大时,退化为含单个中心裂纹正交各向异性纤维增强复合材料板的结果。      

On some numerical methods for the solution of the plane elasticity problem for bodies with cracks by means of singular integral equations

Comparison is given of the accuracy of calculation of stress intensity factors at the crack tips by various methods when solving plane elasticity problems for bodies with cruciform and edge cracks. It is shown that, within the range of quadrature formulas for singular integrals discussed, the type of the formula chosen for the solution of an equation, if used correctly, affects negligibly the accuracy of the stress intensity factor evaluation at the crack tip, and in view of this a method is proposed based on simple relationships.

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Résumé On compare l'exactitude des calculs des facteurs d'intensité de contrainte aux extrémités d'une fissure à l'aide de diverses méthodes à l'occasion de la solution de problèmes d'élasticité plane dans des corps présentant des fissures cruciformes et des fissures de bord. On montre que dans les limites des formules de quadrature des intégrales singulières qui sont discutées, le type de formule choisie pour la solution d'une équation n'a qu'une influence négligeable si elle est utilisée correctement sur l'exactitude du facteur d'intensité de contrainte évalué à l'extrémité de la fissure. Dans cette optique, on propose une méthode basée sur des relations simples.

     

压电复合材料的共线周期性裂纹平面问题的电弹性场分析

利用复变函数知识、半逆解法及待定系数法, 研究了压电复合材料的共线周期性裂纹问题, 给出了在电不可渗透边界条件下的应力、电位移、应力强度因子、电位移强度因子和机械应变能释放率的解析解。当裂纹间距趋于无穷时, 共线周期性裂纹退化为一条单裂纹, 得到了压电复合材料一条单裂纹的结果。通过数值算例讨论了共线周期性裂纹的裂纹长度、裂纹间距和机电载荷对机械应变能释放率的影响规律。结果表明, 机械应变能释放率随着共线周期性裂纹的裂纹长度、共线周期性裂纹的裂纹间距、机械载荷和正电场的增大而增大, 随着负电场的增大而减小。     

In-plane problem for wave propagation through elastic solids with a periodic array of cracks

Summary In the context of wave propagation in damaged (elastic) solids, an analytical method previously introduced for scalar problems, is now applied to study the (vector) problem for normal penetration of a longitudinal plane wave into a periodic array of collinear cracks. Reduced the problem to an integral equation holding over the openings, an approximation of one-mode type leads to analytical solutions and then to explicit representations for the wave fields and the scattering parameters. Some graphs will finally compare our results with the numerical ones by other authors.     

The orthogonal projection method for the numerical solution of a field problem

A numerical method, based on the orthogonal projection method of Hilbert space, is given. Two approximate solutions, which bound the exact solution from above and below, are described. The error of the approximate solution is calculated directly. An example of a calculation of temperature field in a nuclear reactor shell is presented.     

An efficient numerical method for the solution of the second boundary value problem of elasticity for 3D-bodies with cracks

The second boundary value problem of elasticity for 3D-bodies containing cracks is considered. Presentation of the solution in the form of the double layer potential reduces the problem to a system of 2D-integral equations which kernels are similar for the body boundary and crack surfaces. For discretization of these equations, Caussian approximation functions centered at a set of nodes homogeneously distributed on the body and crack surfaces are used. For such functions, calculation of the elements of the matrix of the discretized problem is reduced to five standard 1D-integrals that can be tabulated. For planar cracks, these integrals are calculated in closed analytical forms. The method is mesh free, and for its performing, only node coordinates and surface orientations at the nodes should be defined. Calculation of stress intensity factors at the crack edges in the framework of the method is discussed. Examples of an elliptical crack, a lens-shaped crack, and a spherical body subjected to concentrated and distributed surface forces are considered. Numerical results are compared with the solutions of other authors presented in the literature. Convergence of the method with respect to the node grid steps is analyzed. An efficient algorithm of the node grid generation is proposed.     

On the elastic plastic initial-boundary value problem and its numerical integration

An interative approach is proposed for the numerical analysis of elastic–plastic continua. This approach gives after convergence an implicit scheme of integration of the evolution problem, and is concerned with elastic-perfectly plastic materials and with hardening standard materials. Under a generalized assumption of positive hardening, the proof of convergence of the iterative solutions is given. Some numerical examples by the finite element method are also discussed.     

Solutions for dilating shear cracks in elastic plane

A general two-dimensional solution has been obtained analytically for the crack with interating rough faces. The interaction has been modelled at a large scale as (1) Mohr-Coulomb friction and (2) the crack opening being a known function of sliding. The detailed consideration of a crack with saw-like faces has shown that the energy release rate is less than for a conventional shear crack. Tensile stress concentration caused by the crack opening reduces the angle of kinking though not sufficiently to make the crack propagate in its own plane. At a certain magnitude of shear loading the opening reaches its maximum value determined by the height of asperities. Starting from this point the zone where the maximum opening is reached rapidly increases and the dependencies of crack face displacements and the area of crack opening on the load become nonlinear. This however does not affect values of the stress intensity factors. Key words: Dilation, friction, opening, saw-like faces, shear crack, stress intensity factors.     

Vibrations of elastic infinite and semi-infinite media containing cracks

The problems of stress distribution in an infinite medium and in an elastic half-plane containing line cracks, when the pressure which opens the crack is periodic in time, are considered. These are (1) a cruciform crack in an elastic infinite medium, (2) an edge crack perpendicular to the surface of an elastic half-plane, and (3) their corresponding “exterior” problems. The integral equations corresponding to these problems are obtained. Expressions for the stress intensity factor and the crack energy are derived and numerical results are presented. The equivalence of the stress intensity factor and the crack energy for “exterior” and “interior” problems as established by Stallybrass for the static case is obtained from the dynamic results by letting the frequency tend to zero.     

The elastic contact problem for dissimilar orthotropic semi-infinite and infinite strips

This article examines the stress state which develops when the end of a symmetrically loaded orthotropic semi-infinite strip is supported on an orthotropic infinite strip. Both smooth and bonded contact between the strips are considered. By using Fourier transforms the problem is reduced to a system of singular integral equations. The dependence of the order of the stress singularity on the orthotropic properties of the materials is investigated. Numerical solutions are presented for the contact stresses corresponding to different loadings and material properties.     

The numerical solution of boundary-value problems for an elastic body with an elliptic hole and linear cracks

Using the boundary-element method which is a combination of a fictitious load and a displacement discontinuity, numerical solutions are obtained for two-dimensional (plane deformation) boundary-value problems for the elastic equilibrium of infinite and finite homogeneous isotropic bodies having elliptic holes with cracks and cuts of finite length. Using the method of separation of variables, the boundary-value problem is solved in the case of an infinite domain containing an elliptic hole with a linear cut on whose contour the symmetry conditions are fulfilled.     

A boundary integral solution for the problem of multiple interacting cracks in an elastic material

A boundary integral equation method for the solution of a class of two-dimensional elasticity problems involving multiple interacting cracks in an elastic material is presented. The method is used to obtain a procedure for the numerical evaluation of the crack tip stress intensity factors for this class of problems. The stress intensity factors for some specific problems are computed using this procedure.

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Résumé On présente une méthode par équations intégrales de contour pour solutionner une classe de problèmes d'élasticité bidimensionnelle comportant de multiples fissures interactives dans un matériau élastique. On utilise la méthode pour obtenir une procédure d'évaluation numérique des facteurs d'intensité de contraintes à l'extrémité de la fissure, pour cette classe de problèmes. En utilisant cette procédure, on peut calculer les facteurs d'intensité de contraintes correspondant à divers problèmes spécifiques.