Transparent boundary conditions for the harmonic diffraction problem in an elastic waveguide

This work concerns the numerical finite element computation, in the frequency domain, of the diffracted wave produced by a defect (crack, inclusion, perturbation of the boundaries, etc.) located in a 3D infinite elastic waveguide. The objective is to use modal representations to build transparent conditions on some artificial boundaries of the computational domain. This cannot be achieved in a classical way, due to non-standard properties of elastic modes. However, a biorthogonality relation allows us to build an operator, relating hybrid displacement/stress vectors. An original mixed formulation is then derived and implemented, whose unknowns are the displacement field in the bounded domain and the normal component of the normal stresses on the artificial boundaries. Numerical validations are presented in the 2D case.     

Multiple crack fatigue growth modeling by displacement discontinuity method with crack-tip elements

This paper presents a numerical approach for modeling multiple crack fatigue growth in a plane elastic infinite plate. It involves a generation of Bueckner’s principle, a displacement discontinuity method with crack-tip elements (a boundary element method) proposed recently by the author and an extension of Paris’ law to a multiple crack problem under mixed-mode loading. Because of an intrinsic feature of the boundary element method, a general multiple crack growth problem can be solved in a single-region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Crack extension is conveniently modeled by adding new boundary elements on the incremental crack extension to the previous crack boundaries. Fatigue growth modeling of an inclined crack in an infinite plate under biaxial cyclic loads is taken into account to illustrate the effectiveness of the present numerical approach. As an example, the present numerical approach is used to study the fatigue growth of three parallel cracks with same length under uniaxial cyclic load. Many numerical results are given.     

Growth and intersection of cracks using boundary-element analysis

Existing procedures which are available for the determinationof the path traversed by a crack growing in a two-dimensionallinear elastic body are limited to simple geometric configurationswhere the growth of all the crack tips is the same. In thispaper, an important advance has been made; a crack may intersectother cracks, while a crack growing from a boundary may intersectother boundaries. Simulations of crack growth and intersectionsare performed for a number of configurations and computationalprocedures are given which control the rate of crack growth.In order to assess the robustness of these new extensions, cracksat or near circular holes and multiple cracks growing into anedge crack are considered, and detailed results given.     

Simulation of short crack propagation using a hybrid boundary element technique

Service life of cyclically loaded components is often determined by the propagation of short fatigue cracks, which is highly influenced by microstructural features such as grain boundaries. A two-dimensional model to simulate the growth of such stage I-cracks is presented. The crack is discretised by dislocation discontinuity boundary elements and the direct boundary element method is used to mesh the grain boundaries. A superposition procedure couples these different boundary element methods to employ them in one model. Varying elastic properties of the grains are considered and their influence on short crack propagation is studied. A change in crack tip slide displacement determining short crack propagation is observed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On stresses in a strip with a crack

The solution of the problem of a loaded crack in an infinite strip is given using the method of superposition of three problems (a loaded crack in the infinite plane; an infinite homogeneous strip with normal and tangent stresses that are given on nonhomogeneous boundaries; an infinite strip with longitudinal generators which are free from load and an arbitrary load at the end), which makes it possible to satisfy the boundary conditions exactly.Translated from Dinamicheskie Sistemy, No. 9, pp. 65–71, 1990.     

Global approaches for accurate loading analyses and crack path predictions at single and two-cracks systems

Based on the work by Eshelby, the path-independent J_k-, M-, L- and interaction- or I_k-integrals were introduced and applied to cracks for the accurate calculation of crack tip loading quantities. Applying the FE-method to solve boundary value problems with cracks, numerically inaccurate values are observed within the crack tip region affecting the accuracy of local approaches. Simulating crack paths, local approaches face further problems as cracks are running towards interfaces, internal boundaries or other crack faces. Within global approaches, path-independent integrals are calculated along remote contours far from the crack tip, essentially exploiting numerically reliable data requiring special treatment only for the near-tip crack faces. To provide path-independence, additional integrals along interfaces, internal boundaries and crack faces are necessary. In this paper, new global approaches of path-independent integrals are presented and applied to the calculation of crack paths at two-cracks systems. A second focus is directed to the accurate loading analysis and crack path prediction considering anisotropic properties and material interfaces. The numerical model provides crack paths which are in good agreement with those obtained from crack growth experiments. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Crack growth in elastic materials with internal boundaries and interfaces

This work presents numerical methods used for predicting crack paths in technical structures based on the theory of linear elastic fracture mechanics. The FE-method is used in combination with an efficient remeshing algorithm and a post processor to calculate crack tip loading. The interaction of cracks and internal boundaries and interfaces is investigated. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

平面弹性裂纹分析的一种有效边界元方法

提出了一种简单而有效的平面弹性裂纹应力强度因子的边界元计算方法．该方法由Crouch与Starfield建立的常位移不连续单元和闫相桥最近提出的裂尖位移不连续单元构成A·D2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．算例(如单向拉伸无限大板中心裂纹、单向拉伸无限大板中圆孔与裂纹的作用)说明平面弹性裂纹应力强度因子的边界元计算方法是非常有效的．此外,还对双轴载荷作用下有限大板中方孔分支裂纹进行了分析．这一数值结果说明平面弹性裂纹应力强度因子的边界元计算方法对有限体中复杂裂纹的有效性,可以揭示双轴载荷及裂纹体几何对应力强度因子的影响．     

反平面裂纹在裂纹自由表面附近的弹塑性分析

裂纹自由面附近的弹塑性场和弹塑性边界是裂纹弹塑性分析的重要内容,但现有的方法难以对其进行有效描述．该文发展了裂纹线场分析方法的研究思路,将裂纹面视为裂纹线的拓展部分,对理想弹塑性Ⅲ型裂纹进行了裂纹面附近弹塑性场的分析,得出了裂纹面附近弹塑性应力场、塑性区长度和弹塑性边界的单位法向量．分析结果表明,可放弃传统的小范围屈服条件．     

Stretching of an Elastoplastic Plate with an Arbitrarily Oriented Crack

The problem of determination of the crack resistance of an elastoplastic plate, weakened by a rectilinear slit in the form of a cut, under the conditions of uniaxial stretching is considered. The material of the body is assumed to be incompressible, reinforcing, and obeying the Mises condition of plasticity. The straining theory of plasticity is used. The solutions are obtained in the elastic and plastic regions in the form of asymptotic expansions in the neighborhood of the end of the crack. Based on the conditions that the crack borders are unloaded and the elastic and plastic solutions are conjugate on the contour of the plastic region, unknown constants of integration are found. The two-leafed contours of the plastic region are obtained. The critical load is determined.     

Numerical crack path prediction considering orthotropic effects and experimental verification

For a reliable prediction of crack paths, on the one hand the accurate calculation of crack tip loading quantities is inevitable, on the other hand orthotropic features of the fracture toughness need to be taken into account. The interplay of crack tip loading and material response due to fracture is still unclear and seems to have a crucial effect on crack path predictions. Numerical tools for the accurate calculation of crack tip loading quantities using path-invariant J-integrals and interaction integrals (I-integral) are presented. Here, global approaches are beneficial when considering crack tips approaching other crack faces or internal boundaries. Curved crack faces have to be taken into account and special treatment regarding crack face integrals is necessary. Experimental investigations are carried out at standard CT-specimens of rolled aluminum alloy Al-7075 exhibiting a directional orthotropy of the fracture toughness. Considering that property, the numerically predicted crack paths based on FE calculations show very good agreement with subcritically grown paths obtained from experiments. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Spherulite boundaries and certain aspects of interspherulitic fracture

The morphology and properties of the interspherulitic boundaries in polyethylene specimens with ring-type spherulitic microstructure have been the subject of a microscopic and electron-microscopic investigation. It is shown that the structure and the strength of the boundaries depend on the orientation of the structural elements, of which the spherulites are composed. The strongest boundaries are those where the c axes of the crystals are oriented perpendicular to the interface. The parts of the boundaries where the crystals are in lateral contact serve as centers of brittle crack nucleation. The morphology of the interspherulitic boundaries is discussed with reference to the mechanism of spherulite formation. The possibility of controlling the boundary structure by regulating the supermolecular microstructure of the polymer is examined.Paton Institute of Electrowelding, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 3, pp. 391–395, May–June, 1969.     

A Boundary Integral Equation Formulation for Problems Involving Non-linear Power-law Materials

A boundary integral equation method is derived in the strainplane for problems involving power-law elastic materials anda reciprocal theorem, whereby the stress intensity factor ofa crack is related to integrals around finite boundaries inthe strain plane, is developed. In order to check these methodsa comparison with known exact results is effected. In addition,an integral equation formulation is obtained for a crack ina square box, and the paper concludes with an extension to notchproblems.     

Finite Element Modelling of Cracks Based on the Partition of Unity Method

In this paper a finite element model for the analysis of brittle materials in the post cracking regime is presented. The model allows the representation of failure zones several times smaller than the structure itself using relatively coarse finite element meshes. The formulation is based on the partition of unity method. Discontinuous shape functions are used to enrich the continuous approximation of the displacement field where a crack has opened [2]. The magnitude of the displacement jump is determined by extra degrees of freedom at existing nodes. The crack path is completely independent of the structure of the mesh and is continuous across element boundaries. To model inelastic deformations around the crack tip a cohesive crack model is used. A representative numerical example illustrates the performance of the proposed model.     

On corner singularities in Reissner-Mindlin's theory of elastic plates

In the framework of linear elasticity, singularities occur in domains with non-smooth boundaries. Particularly in Fracture Mechanics, the local stress field near stress concentrations is of interest. In this work, singularities at re-entrant corners or sharp notches in Reissner-Mindlin plates are studied. Therefore, an asymptotic solution of the governing system of partial differential equations is obtained by using a complex potential approach which allows for an efficient calculation of the singularity exponent λ. The effect of the notch opening angle and the boundary conditions on the singularity exponent is discussed. The results show, that it can be distinguished between singularities for symmetric and antisymmetric loading and between singularities of the bending moments and the transverse shear forces. Also, stronger singularities than the classical crack tip singularity with free crack faces are observed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

Scattering of a SH-wave by an elastic fiber of nonclassical cross section with an interface crack

The problem of interaction of a plane time-harmonic SH-wave with an elastic fiber of quasi-square or quasi-triangular cross section, when an interface crack is present between an infinite elastic matrix and the fiber, is considered. The modified null-field method taking into account the asymptotic behavior of the solution at crack tips is exploited for obtaining numerical results. The effects of fiber shape, fiber/matrix material combination, debonding (crack size), and direction of wave incidence on the scattering amplitude in the far zone are analyzed. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 2, pp. 245–254, March–April, 2008.     

An axisymmetric steady-state thermoelastic problem of an external circular crack in an isotropic thick plate

A steady state thermoelastic mixed boundary value problem for an isotropic thick plate is considered in this paper. The faces of an external circular crack situated in the mid-plane of the plate are opened up by the application of temperature while the bounding surface of the plate are maintained at a constant zero temperature. Solution valid for large values of the ratio of the plate thickness to the diameter of the crack has been obtained. Expressions for various quantities of physical interest are derived by finding iterative solutions of the equations and the results are shown graphically.