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.     

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．     

Interaction of multiple cracks in a rectangular plate

This paper is concerned with interaction of multiple cracks in a finite plate by using the hybrid displacement discontinuity method (a boundary element method). Detail solutions of the stress intensity factors (SIFs) of the multiple-crack problems in a rectangular plate are given, which can reveal the effect of geometric parameters of the cracked body on the SIFs. The numerical results reported here illustrate that the boundary element method is simple, yet accurate for calculating the SIFs of multiple crack problems in a finite plate.     

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part I: Theoretical solutions

An extended displacement discontinuity (EDD) boundary integral equation method is proposed for analysis of arbitrarily shaped planar cracks in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack surface. Green's functions for unit point EDDs in an infinite three-dimensional medium of 2D hexagonal QC are derived using the Hankel transform method. Based on the Green's functions and the superposition theorem, the EDD boundary integral equations for an arbitrarily shaped planar crack in an infinite 2D hexagonal QC body are established. Using the EDD boundary integral equation method, the asymptotic behavior along the crack front is studied and the classical singular index of 1/2 is obtained at the crack edge. The extended stress intensity factors are expressed in terms of the EDDs across crack surfaces. Finally, the energy release rate is obtained using the definitions of the stress intensity factors.     

Crack opening displacement for two unequal straight cracks with coalesced plastic zones – A modified Dugdale model

The problem investigated is of an infinite plate weakened by two collinear unequal hairline straight quasi-static cracks. Uniform constant tension is applied at infinity in a direction perpendicular to the rims of the cracks. Consequently the rims of the cracks open in Mode I type deformation. The tension at infinity is increased to the limit such that the plastic zones developed at the two adjacent interior tips of cracks get coalesced. To arrest the crack from further opening normal cohesive variable stress distribution is applied on the rims of the plastic zones. Closed form analytic expressions are obtained for load bearing capacity and crack opening displacement (COD). An illustrative case is discussed to study the behavior of load bearing capacity and crack opening displacement with respect to affecting parameters viz. crack length, plastic zone length and inter crack distance between the two cracks. Results obtained are reported graphically and analyzed.     

Mathematical model for crack arrest of an infinite plate weakened by a finite and two semi-infinite cracks

The problem of an unbounded plate weakened by three quasi-static coplanar and collinear straight cracks: two semi-infinite cracks and a finite crack situated symmetrically between two semi-infinite cracks, is investigated. The plate is subjected to uniform unidirectional in-plane tension at infinite boundary. Developed plastic zones are arrested by distributing cohesive yield point stress over their rims. The solution is obtained using complex variable technique. Closed form analytic expressions are derived for load bearing capacity and crack-tip-opening displacement (CTOD). A case study is presented for CTOD and load bearing capacity versus crack length, plastic zone length and inter-crack distance etc. Results are presented graphically and analyzed.     

Crack-tip-opening displacement for four symmetrically situated cracks with coalesced interior yield zones

Crack-tip opening displacements are obtained for four collinear straight cracks, weakening an unbounded homogeneous and isotropic elastic-perfectly plastic plate. The cracks are so configured that two symmetrically situated and interiorly lying cracks are of equal-lengths. Other two exteriorly lying, collinear straight cracks (surrounding the interiorly lying straight cracks) are of mutually equal-lengths. Thus an exterior and an interior crack-set are symmetrically oriented with respect to the other interior–exterior collinear cracks-set configuration. Uniform constant load prescribed at remote boundary of the plate, opens the crack in self-similar fashion developing a strip-yield zone ahead each tip of the cracks. It is assumed that the strip-yield zone developed at each of interior tips of an exteriorly and interiorly lying crack-set configuration gets coalesced. The developed yield zones are subjected to normal cohesive yield stress to arrest the crack from further opening. The solution of the problem is obtained by superposing the solutions of the two auxiliary problems, appropriately derived from the given problem. Each of the auxiliary problems, in turn, is solved using complex variable technique. Expressions are derived for quantities of interest viz. crack-tip opening displacement (CTOD), length of each developed yield zone. The effect of applied load and closing load on the parameters CTOD and strip yield zone affecting the crack arrest is presented graphically and concluded.     

Analysis of arbitrarily shaped planar cracks in two-dimensional hexagonal quasicrystals with thermal effects. Part II: Numerical solutions

The extended displacement discontinuity (EDD) boundary element method is developed to analyze an arbitrarily shaped planar crack in two-dimensional (2D) hexagonal quasicrystals (QCs) with thermal effects. The EDDs include the phonon and phason displacement discontinuities and the temperature discontinuity on the crack face. Green's functions for uniformly distributed EDDs over triangular and rectangular elements for 2D hexagonal QCs are derived. Employing the proposed EDD boundary element method, a rectangular crack is analyzed to verify the Green's functions by discretizing the crack with rectangular and triangular elements. Furthermore, the elliptical crack problem for 2D hexagonal QCs is investigated. Normal, tangential, and thermal loads are applied on the crack face, and the numerical results are presented graphically.     

Mixed mode axisymmetric cracks in transversely isotropic infinite solid cylinders

The present study examined mixed mode cracking in a transversely isotropic infinite cylinder. The solutions to axisymmetric Volterra climb and glide dislocations in an infinite circular cylinder of the transversely isotropic material are first obtained. The solutions are represented in terms of the biharmonic stress function. Next, the problem of a transversely isotropic infinite cylinder with a set of concentric axisymmetric penny-shaped, annular, and circumferential cracks is formulated using the distributed dislocation technique. Two types of loadings are considered: (i) the lateral cylinder is loaded by two self-equilibrating distributed shear stresses; (ii) the curved surface of the cylinder is under the action of a distributed normal stress. The resulting integral equations are solved by using a numerical scheme to compute the dislocation density on the borders of the cracks. The dislocation densities are employed to determine stress intensity factors for axisymmetric interacting cracks. Finally, a good amount of examples are solved to depict the effect of crack type and location on the stress intensity factors at crack tips and interaction between cracks. Numerical solutions for practical materials are presented and the effect of transverse isotropy on stress intensity factors is discussed.     

The free-discontinuity problem of inverse crack identification from boundary measurements

The inverse crack identification of planar cracks from elastostatics boundary measurements is regarded as free-discontinuity problem in respect to the unknown displacement field and the discontinuity region of the cracked body. The proposed solution strategy is based on the variational approximation of the sharp interface problem by elliptic functionals developed by Ambrosio and Tortorelli. The numerical calculation is realized by the finite element method. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analysis of anti-plane interface cracks in one-dimensional hexagonal quasicrystal coating

The displacement discontinuity method is extended to study the fracture behavior of interface cracks in one-dimensional hexagonal quasicrystal coating subjected to anti-plane loading. The Fredholm integral equation of the first kind is established in terms of displacement discontinuities. The fundamental solution for anti-plane displacement discontinuity is derived by the Fourier transform method. The singularity of stress near the crack front is analyzed, and Chebyshev polynomials of the second kind are numerically adopted to solve the integral equations. The displacement discontinuities across crack faces, the stress intensity factors, and the energy release rate are calculated from the coefficients of Chebyshev polynomials. In combination with numerical simulations, a comprehensive study of influencing factors on the fracture behavior is conducted.     

A strong discontinuity based adaptive refinement approach for the modeling of crack branching

In the current work, the physical phenomena of dynamic fracture of brittle materials involving crack growth, acceleration and consequent branching is simulated. The numerical modeling is based on the approach where the failure in the form of cracks or shear bands is modeled by a jump in the displacement field, the so called ‘strong discontinuity’. The finite element method is employed with this strong discontinuity approach where each finite element is capable of developing a strong discontinuity locally embedded into it. The focus in this work is on branching phenomena which is modeled by an adaptive refinement method by solving a new sub-boundary value problem represented by a finite element at the growing crack tip. The sub-boundary value problem is subjected to a certain kinematic constraint on the boundary in the form of a linear deformation constraint. An accurate resolution of the state of material at the branching crack tip is achieved which results in realistic dynamic fracture simulations. A comparison of resulting numerical simulations is provided with the experiment of dynamic fracture from the literature. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

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)     

Modeling quasi-static crack growth with the embedded finite element method on multiple levels

The current work presents the multilevel approach of the embedded finite element method which is obtained by combining features of the method of domain decomposition with those of the standard embedded finite element method. The conventional requirement of fine mesh in a possible failure zone is rendered unnecessary with the new approach thereby reducing the computational expense. In addition, it is also possible to stop a propagating crack-tip in the middle of a finite element. In this approach, the finite elements at the failure-prone zone where cracks or shear bands, referred to as strong discontinuities which represent jumps in the displacement field, can form and propagate based on some failure criterion are treated as separate sub-boundary value problems which are adaptively discretized during the run time into a number of sub-elements and subjected to a kinematic constraint on their boundary. Each sub-element becomes equally capable of developing a strong discontinuity depending upon its state of stress. A linear displacement based constraint is applied initially which is modified accordingly as soon as a strong discontinuity propagates through the boundary of the main finite element. At the local equilibrium, the coupling between the quantities at two different levels of discretization is obtained by matching the virtual energies due to admissible variations of the main finite element and its constituent sub-elements. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stress intensity factor for multiple cracks in bonded dissimilar materials using hypersingular integral equations

This paper deals with the multiple inclined or circular arc cracks in the upper half of bonded dissimilar materials subjected to shear stress. Using the complex variable function method, and with the help of the continuity conditions of the traction and displacement, the problem is formulated into the hypersingular integral equation (HSIE) with the crack opening displacement function as the unknown and the tractions along the crack as the right term. The obtained HSIE are solved numerically by utilising the appropriate quadrature formulas. Numerical results for multiple inclined or circular arc cracks problems in the upper half of bonded dissimilar materials are presented. It is found that the nondimensional stress intensity factors at the crack tips strongly depends on the elastic constants ratio, crack geometries, the distance between each crack and the distance between the crack and boundary.     

夹层压电材料中垂直于界面的共线双裂纹动力学问题分析

采用Schmidt方法分析了在简谐反平面剪切波作用下,两个半空间夹层压电材料中的共线裂纹的动力学行为．压电材料层内裂纹垂直于界面,电边界条件假设为可导通．通过Fourier变换,使问题的求解转换为两对三重积分对偶方程．通过数值计算,给出了裂纹的几何尺寸、压电材料常数、入射波频率等对于应力强度因子的影响．结果表明,在不同的入射波频率范围,动力场将阻碍或促使压电材料内裂纹的扩展．与不可导通电边界条件相比,导通裂纹表面的电位移强度因子比不可导通裂纹的电位移强度因子要小许多．     

Strip electro-mechanical yielding model for piezoelectric plate cut along two equal collinear cracks

The multiple-crack problems for piezoelectric ceramics till now have not yet address the crack opening arrest problem. The present work addresses this paucity. A 2-D strip-electro-mechanical yielding model is proposed for a transversely isotropic piezoelectric media weakened by two internal equal collinear straight cracks. The infinite boundary is prescribed with combined uniform constant in-plane mechanical and electrical loads. Developed mechanical and electric strip zones are arrested by prescribing over their rims uniform, normal, cohesive yield point stress and saturation limit electric displacement. Two cases are considered when saturation zone is bigger than developed yield zone and vice versa. Stroh formulation together with complex variable technique is employed to obtain the solution. Closed form expressions are derived for saturation zone length, yield zone length, crack opening displacement (COD), crack opening potential jump (COP) and energy release rate (ERR). An illustrative numerical study is prescribed to determine the effect of various parameters on the crack growth arrest and presented graphically. The results reveal that the model is capable of crack arrest under small-scale mechanical and electric yielding.     

A new boundary element technique for solving plane problems of linear elasticity: improved theory and an application to fracture mechanics

An improved boundary element method for solving plane problems of linear elasticity theory is described. The method is based on the Muskhelishvili complex variable representation for the displacement and stress fields. The paper shows how to take account of symmetry about the x and/or y axes.The potential accuracy of the method is illustrated by its application to the calculation of stress intensity factors associated with cracks in both a square and a circular plate. The crack problem is solved using a Gauss-Chebyshev representation of a singular integral equation by a set of linear algebraic equations. The integral equation involves an analytic function which takes account of the presence of the external boundary. This function is determined directly using the boundary element method.Numerical results are believed to be more accurate than the existing published values which are quoted to four significant figures.     

Contact interaction of the borders of edge cracks in a flexible half plane

A singular integral equation method is used to study the closure of edge cracks during bending of an isotropic half space. The cases of a single crack and of a periodic system of cracks perpendicular to the edge of the half space are considered. The effects of the free edge and the spatial period of the cracks on the distribution of the contact reaction and limiting load are investigated. The numerical results are compared with published data obtained without taking crack closure into account, as well as with results on the contact problem for bending of an infinite plate. Ivano-Frankovsk Sector, Institute of Applied Mechanical and Mathematical Problems, National Academy of Sciences of Ukraine. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 29, pp. 83–89, 1999.     

Crack Arrest Model for a Plate Weakened by Internal and External Cracks - a Modified Dugdale Approach

A modified Dugdale model solution is obtained for an elastic-perfectly-plastic plate weakened by one internal and two external straight collinear hairline cracks. The tension applied to the infinite boundary of the plate opens the rims of cracks with forming a plastic zone ahead of each tip of the internal crack and also at each finitely distant tip of the two external cracks. The developed plastic zones are closed by normal cohesive linearly varying yield-point stress distributions applied to their rims. The problem is solved using the complex-variable technique. A case study is carried out to find the load required to prevent the cracks from further growing with respect to affecting parameters. The results obtained are reported graphically and analyzed.