On interface crack models with contact zones situated in an anisotropic bimaterial

Summary A boundary value problem for two semi-infinite anisotropic spaces with mixed boundary conditions at the interface is considered. Assuming that the displacements are independent of the coordinate x ₃, stresses and derivatives of displacement jumps are expressed via a sectionally holomorphic vector function. By means of these relations the problem for an interface crack with an artificial contact zone in an orthotropic bimaterial is reduced to a combined Dirichlet-Riemann problem which is solved analytically. As a particular case of this solution, the contact zone model (in Comninou's sense) is derived. A simple transcendental equation and an asymptotic formula for the determination of the real contact zone length are obtained. The classical interface crack model with oscillating singularities at the crack tips is derived from the obtained solution as well. Analytical relations between fracture mechanical parameters of different models are found, and recommendations concerning their implementation are given. The dependencies of the contact zone lengths on material properties and external load coefficients are illustrated in graphical form. The practical applicability of the obtained results is demonstrated by means of a FEM analysis of a finite-sized orthotropic bimaterial with an interface crack. Received 19 October 1998; accepted for publication 13 November 1998     

Buckling of an orthotropic graded coating with an embedded crack bonded to a homogeneous substrate

To simulate buckling of nonuniform coatings, we consider the problem of an embedded crack in a graded orthotropic coating bonded to a homogeneous substrate subjected to a compressive loading. The coating is graded in the thickness direction and the material gradient is orthogonal to the crack direction which is parallel with the free surface. The elastic properties of the material are assumed to vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the crack orientation. The loading consists of a uniform compressive strain applied away from the crack region. The graded coating is modeled as a nonhomogeneous medium with an orthotropic stress–strain law. Using a nonlinear continuum theory and a suitable perturbation technique, the plane strain problem is reduced to an eigenvalue problem describing the onset of buckling. Using integral transforms, the resulting plane elasticity equations are converted analytically into singular integral equations which are solved numerically to yield the critical buckling strain. The Finite Element Method was additionally used to model the crack problem. The main objective of the paper is to study the influence of material nonhomogeneity on the buckling resistance of the graded layer for various crack positions, coating thicknesses and different orthotropic FGMs.     

理想正交各向异性弹塑性材料平面应力条件下Ⅰ型静止裂纹尖端场

在本文中,以 Hill 的塑性理论为基础,详细地讨论了理想正交各向异性弹塑性材料,平面应力条件下Ⅰ型静止裂纹尖端场解。裂纹尖端应力场不包含应力间断线,但包含弹性区。分析的结果表明(i)对于平面应力静止裂纹问题,应力场解不是唯一的,场解中的自由参数必须由远场条件来确定。(ii)裂纹尖端的应力、应变的奇异性,无论是各向异性材料还是各向同性材料,都是相同的。但在各向异性材料中,各向异性参数影响着应力、应变的幅度和分布。     

The effect of material and geometry on the non-linear vibrations of orthotropic circular cylindrical shells

The extensive use of circular cylindrical shells in modern industrial applications has made their analysis an important research area in applied mechanics. In spite of a large number of papers on cylindrical shells, just a small number of these works is related to the analysis of orthotropic shells. However several modern and natural materials display orthotropic properties and also densely stiffened cylindrical shells can be treated as equivalent uniform orthotropic shells. In this work, the influence of both material properties and geometry on the non-linear vibrations and dynamic instability of an empty simply supported orthotropic circular cylindrical shell subjected to lateral time-dependent load is studied. Donnell׳s non-linear shallow shell theory is used to model the shell and a modal solution with six degrees of freedom is used to describe the lateral displacements of the shell. The Galerkin method is applied to derive the set of coupled non-linear ordinary differential equations of motion which are, in turn, solved by the Runge–Kutta method. The obtained results show that the material properties and geometric relations have a significant influence on the instability loads and resonance curves of the orthotropic shell.     

Two Nonlinear Models of Brittle Fracture for Solids

This paper considers isotropic and orthotropic nonlinear constitutive relations for brittle materials in the case of plane stresses. Numerical solution algorithms based on the finite-element method are developed. The resulting material models are incorporated in the PIONER software. The correctness of crack path determination is examined by solving a test problem of crack propagation. The isotropic model gives mesh-dependent results, whereas the orthotropic model provides an adequate solution. It is shown that solutions obtained for the isotropic model are close to those obtained by eliminating failed elements.     

利用焦耳效应提高含裂纹金属构件抗裂性能问题的研究

设一无限大金属薄板中含有一个线裂纹,对金属板施加恒定的电流场,在两个裂尖处产生的热量远远大于其余地方产生的热量,可简化成两个点热源．经求解得到了问题的解析解,包括裂纹尖端附近区域温度、应力、应变、应变能密度因子的解析表达式．计算结果表明,裂纹尖端处的材料发生熔化而形成一个焊点,裂纹尖端明显纯化,可抑制裂纹的进一步扩展,提高含裂纹金属构件的抗裂性能．     

Analytical solution of cracked shell resting on elastic foundation

The elastostatic problem for cracked shallow spherical shell resting on linear elastic foundation is considered. The problem is formulated for a homogeneous isotropic material within the confines of a linearized shallow shell theory. By making use of integral transforms and asymptotic analysis, the problem is reduced to the solution of a pair of singular integral equations. The stress distribution obtained, around the crack tip, is similar to that of the elasticity solutions. The numerical results obtained agree well with those of previous work, where the elastic supports were neglected. The influences of the shell curvature and the modulus of subgrade reaction on the stress intensity factor are given.     

Stress intensity factor for a kinked interfacial crack in dissimilar anisotropic materials under antiplane deformation

The asymptotic problem of a kinked interfacial crack in dissimilar anisotropic materials under antiplane deformation is investigated. The linear transformation method for the problem of the anisotropic bimaterial with a straight interface is proposed. The stress intensity factor for the kinked interfacial crack in the anisotropic composite is obtained from the solution of the transformed problem of the kinked interfacial crack in the isotropic bimaterial based on the linear transformation method. The effects of the material parameters as well as the kink angle on the stress intensity factor are discussed from numerical results of the stress intensity factor. The finite element analysis is carried out to verify the stress intensity factor obtained by using the linear transformation. The influence of the material orientations on the stress intensity factor is investigated for the kinked crack in the bimaterial consisting of dissimilar inclined orthotropic materials.     

Wedging an Orthotropic Body by a Rectangular Wedge of Finite Length

Galin's method is used to derive equations that relate the basic parameters of the problem on wedging an orthotropic space by a rigid rectangular wedge. To eliminate the stress singularity at the wedging crack tips, a Leonov–Panasyuk–Dugdale prefracture zone is assumed to exist at the crack front. The equations are derived using the COD criterion     

理想正交各向异性弹塑性材料平面应变条件下静止裂纹尖端场

以Hill唯象理论为基础,建立正交各向异性弹塑性材料的本构关系,给出理想正交各向异性弹塑性材料在平面应变条件下混合型静止裂纹尖端的弹塑性场.与J.Pan的解不同,采用自相似假定,可以用解析方法求得不存在应力间断的应力场.对满塑性区条件和应变的奇异性加以讨论,这些为建立断裂准则提供了理论的依据.     

Scattering of harmonic antiplane shear waves by an arc-shaped interfacial crack in functionally graded annular bi-material guide rail

Abstract

The dynamic behavior of an arc-shaped interfacial crack in an orthotropic functionally graded annular bi-material structure is investigated. In order for the analysis to be executable, the material properties are assumed to vary with the power function of the radial coordinates. By applying the separation variable method, the boundary value problem of the partial differential equation describing the fracture problem of this article can be transformed into a Cauchy kernel singular integral equation with the unknown jump of displacements across the crack surfaces. The obtained integral equation is solved numerically by Lobatto–Chebyshev collocation method to show the effects of the geometric and physical parameters upon the dynamic stress field near the crack tips.

Communicated by Kuang-Hua Chang.     

Vibration analysis of nonhomogeneous orthotropic cylindrical shells including combined effect of shear deformation and rotary inertia

This paper is the result of an investigation on the vibration of non-homogeneous orthotropic cylindrical shells, based on the shear deformation theory. Assume that the Young’s moduli, shear moduli and density of the orthotropic material are continuous functions of the coordinate in the thickness direction. The basic equations of non-homogeneous orthotropic cylindrical shells with the shear deformation and rotary inertia are derived in the framework of Donnell-type shell theory. The ends of a non-homogeneous orthotropic cylindrical shell are considered as simply supported. The basic equations are reduced to the sixth-order algebraic equation for the frequency using the Galerkin method. Solving this algebraic equation, the lowest values of non-dimensional frequency parameters for non-homogeneous orthotropic cylindrical shells with and without shear deformation and rotary inertia are obtained. Calculations, effects of shear stresses and rotary inertia, orthotropy, non-homogeneity and shell geometry parameters on the lowest values of non-dimensional frequency parameter are described. The results are verified by comparing the obtained values with those in the existing literature.     

Effect of elastic constants on crack channeling in a thin film bonded to an orthotropic substrate

A channeling crack confined in an orthotropic film bonded to an orthotropic substrate under a steady-state condition is investigated. The problem is formulated based on a modified Stroh formalism and an orthotropy rescaling technique, in order to determine the necessary material parameters required to describe the steady-state energy release rate. A closed form of the energy release rate is obtained with the exception of the normalized energy release rate for the transformed bimaterial structure that consists of the orthotropic film and isotropic substrate. The normalized energy release rates for the transformed problem are shown to depend on only four material parameters and are numerically calculated using finite element analyses. The periodic channels in the film layer of the bimaterial structure are also considered. The steady-state energy release rates for the periodic channeling cracks are obtained as a function of the ratio of the film thickness to the crack spacing for various combinations of the material parameters.     

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

In this paper, the dynamic behavior of two collinear cracks in the anisotropic elasticity material plane subjected to the harmonic anti-plane shear waves is investigated by use of the nonlocal theory. To overcome the mathematical difficulties, a one-dimensional nonlocal kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress field near the crack tips. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near crack tips. The nonlocal elasticity solutions yield a finite hoop stress at the crack tips, thus allowing us to using the maximum stress as a fracture criterion. The magnitude of the finite stress field not only depends on the crack length but also on the frequency of the incident waves and the lattice parameter of the materials.     

Specification of crack opening and surface sliding in orthotropic viscoelastic material

Modelling of crack opening and surface sliding in an orthotropic viscoelastic material is made by introducing two coefficients: one for the surface displacement and surface friction. The material possesses orthotropy in two dimensions and viscoelastic property consisting of a Kelvin element in series with a spring. The method of Laplace transform is applied to obtain a closed form solution to the problem. Explicit expressions of Mode I and II stress intensity factors are computed together with crack surface opening. Trade-off between the Mode I and II stress intensity factors depends on the nature of material orthotropy.     

Non-local theory for cracks in laminated media

Non-local theory for composite is applied to solve the Mode III crack problem under impact loading. The composite is a laminate made up of two different orthotropic layers. Based on the classical boundary conditions, all quantities are expressed in terms of a single unknown function. Integral transform is used to reduce the problem to a system of dual integral equations. The solution is obtained by using the Copson's method [1], which can be generalized to include arbitrary loading applied to the crack.     

复合材料的裂纹动力学问题的模型

复合材料产生裂纹后，其纤维处形成“桥连”，这是一个不可避免的现象。由于桥连问题很复杂，在数学方法的处理上有很大困难，至今人们研究的大多是桥连的静力学问题，而对其动力学问题研究得很少。只有建立复合材料的桥连动力学模型，才能更好地研究复合材料的断裂动力学问题。为了便于分析复合材料的问题，将桥连处用载荷代替，当裂纹高速扩展时，其纤维也连续地断裂。通过复变函数论的方法，将所讨论的问题转化为Riemann-Hilbet问题。利用建立的动态模型和自相似方法，得到了正交异性体中扩展裂纹受运动的集中力Px/t及均布载荷作用下位移、应力和动态应力强度因子的解析解，并通过迭加原理，最终求得了该模型的解。     

The stress intensity variation of an orthotropic perforated plate containing an internal crack

Summary The elastic problem of an infinite orthotropic plate with different roots of its characteristic equation, when the fibers are oriented perpendicularly to an internal crack, and is weakened by an elliptic hole, is solved using Lekhnitskii's theory. The plate is subjected to prescribed stresses at infinity, while the boundary conditions are given at the flanks of the crack, at the rim of the perforation and at infinity.Using the complex-variable method, the solution of the problem is reduced to the evaluation of Cauchy-type integrals concerning the analytic functions of the problem. The numerical solution of the problem revealed an intense variation of mode-I stress intensity factors (SIF) at the crack tips due to the increase of either the crack length, or the distance of the near-by rack tip from the center of the hole. Furthermore, it was found that orthotropy strongly influences the intensity of stresses at the crack tips. These findings are in complete agreement with results given in a previous work by the authors, concerning a similar problem for an orthotropic plate, which, however, constitutes a special case, where the material presents equal roots for its characteristic equation [1].

---

Spannungsintensitätsvarianten einer orthotropen Platte mit einem Loch und innerem Riß

Zusammenfassung Das elastische Problem einer unendlichen, orthotropen Platte mit verschiedenen Wurzeln ihrer charakteristischen Gleichung wird im Falle, daß die Fasern senkrecht zu einem inneren Riß verlaufen und daß sie durch ein elliptisches Loch geschwächt sind, im Rahmen der Lekhnitskii-Theorie gelöst. Die Platte wird im Unendlichen einer vorgeschriebenen Spannung unterworfen, während die Randbedingungen an den Rißflanken, am Lochrand und im Unendlichen gegeben sind.Mit Hilfe der Methode der komplexen Variable wird die Lösung des Problems zurückgeführt auf die Berechnung von Cauchy-Integralen der analytischen Funktionen des Problems. Die numerische Lösung des Problems zeigt eine starke Abhängigkeit von Mode-I-Spannungsintensitätsfaktoren (SIF) an den Rißspitzen von der Rißlänge oder des Ligaments zwischen der Rißplatte und dem Loch. Weiterhin wurde gezeigt, daß Orthotropie starken Einfluß auf die Spannungsintensität hat. Diese Beobachtungen stehen in voller Übereinstimmung mit Resultaten aus der Arbeit [1] über ein ähnliches Problem für eine orthotrope Platte, wo allerdings die Wurzel der charakteristischen Gleichung identisch sind.

     

EXACT SOLUTION FOR ORTHOTROPIC MATERIALS WEAKENED BY DOUBLY PERIODIC CRACKS OF UNEQUAL SIZE UNDER ANTIPLANE SHEAR

Orthotropic materials weakened by a doubly periodic array of cracks under far-field antiplane shear are investigated, where the fundamental cell contains four cracks of unequal size. By applying the mapping technique, the elliptical function theory and the theory of analytical function boundary value problems, a closed form solution of the whole-field stress is obtained. The exact formulae for the stress intensity factor at the crack tip and the effective antiplane shear modulus of the cracked orthotropic material are derived. A comparison with the finite element method shows the efficiency and accuracy of the present method. Several illustrative examples are provided, and an interesting phenomenon is observed, that is, the stress intensity factor and the dimensionless effective modulus are independent of the material property for a doubly periodic cracked isotropic material, but depend strongly on the material property for the doubly periodic cracked orthotropic material. Such a phenomenon for antiplane problems is similar to that for in-plane problems. The present solution can provide benchmark results for other numerical and approximate methods.     

界面下主微裂纹间干涉的研究

涉及两相正交各向异性体界面干涉问题的研究,多裂纹问题被分解为只含单裂纹的子问题,利用位错理论和裂面应力自由条件,列出一组可数值求解位错密度函数的奇异积分方程,从耐 注得应力强度因子。