Strip yield analysis of two collinear unequal cracks in an infinite sheet

Plastic zone sizes and crack tip opening displacements (CTODs) are obtained for two collinear cracks in an infinite sheet subjected to known remote stress. Analysis is conducted by assuming the crack accompanying plastic zones as a fictitious crack and formulating integral equations based upon traction free and no stress singularity conditions. In addition, critical remote stress, plastic zone sizes, and CTODs when the adjacent plastic zones touched are obtained by assuming the coalesced fictitious cracks as a single fictitious crack and formulating integral equations based upon no stress singularity and zero coalesced point displacement conditions. Extensive numerical results are presented.     

Exact solutions of two semi-infinite collinear cracks in a strip

The stress intensity factors (SIFs) are calculated for an infinitely long strip of finite height containing two straight semi-infinite collinear cracks, which is a very useful model in simulating the interaction of faults in the study of tectonic earthquake. The new solutions are obtained by complex function method. It is shown that two well-known exact solutions for the crack problems are the limiting cases of the present results.     

Strip yield crack analysis for multiple site damage in infinite and finite panels – A weight function approach

An analytical approach to the Dugdale strip yield model for multiple site damage is presented by using the weight function method. Two example problems, an array of periodic collinear cracks in an infinite sheet and a coalesced center crack in a finite width panel, are analyzed by the closed-form weigh function; the effect of finite boundary is considered. Results are extensively verified against available analytical and numerical solutions. The capability of the closed-form weight function for the strip yield model analysis of multiple site damage is demonstrated.     

Weight functions for interface cracks in dissimilar anisotropic piezoelectric materials

Weight functions proposed for interface cracks in dissimilar isotropic materials (Gao, 1991; Chen and Hasebe, 1994) are extended to treat those in piezoelectric materials. The difficulties in separating the eight distinct complex arguments are overcome. The pseudo-orthogonal properties of the eigenfunction expansion form found in isotropic dissimilar cases(Chen and Hasebe, 1994) are proved to be valid in the present cases although the mathematical manipulations performed here seem much more complicated than those in isotropic dissimilar materials. Several path-independent integrals are obtained and all the coefficients in the eigenfunction expansion form, including the K _I, K _II, K _III and K _e, could be calculated by the weight functions introduced in this paper. It is concluded that the weight functions presented here provide a powerful tool to calculate the dominant parameters at the interface crack tip without any special treatment to the singular stress field of the near-tip region.     

Approximate Green's functions for singular and higher order terms of an edge crack in a finite plate

An edge crack in a finite plate (FSECP) subjected to wedge forces is solved by the superposition of the analytical solution of a semi-infinite crack, and the numerical solution of a FSECP with free crack faces, which is solved by the Williams expansion. The unknown coefficients in the expansion are determined by a continuous least squares method after comparing it with the direct boundary collocation and the point or discrete least squares methods. The results are then used to validate the stress intensity factor (SIF) formula provided by Tada et al. that interpolates the numerical results of Kaya and Erdogan, and an approximate crack face opening displacement formula obtained in this paper by Castigliano's theorem and the SIF formula of Tada et al. These approximate formulae are accurate except for point forces very close to the outer edge, and can be used as Green's functions in the crack-closure based crack growth analysis, as well as in interpreting the size effect of quasi-brittle materials. Green's functions for coefficients relevant to the second to the fifth terms in the crack tip asymptotic field are also provided. Finally, a FSECP with a uniform pressure over a part of the crack faces is solved to illustrate the application of the obtained Green's functions and to further assess their accuracy by comparing with a finite element analysis.     

Closed-form solution for the size of plastic zone in an edge-cracked strip

This paper is concerned with the problem of plastic zone at the tip of an edge crack in an isotropic elastoplastic strip under anti-plane deformations. By means of complex potential and Dugdale model, the stress intensity factor and the size of plastic zone are obtained in closed-form. Furthermore, the analytic solutions for an edge crack at the free boundary of a half-space and a semi-infinite crack heading towards a free surface are determined as the limiting cases of the strip geometries.     

Weight function,stress intensity factor and crack opening displacement solutions to periodic collinear edge hole cracks

Periodic collinear edge hole cracks and arbitrary small cracks emanating from collinear holes, which are two typical multiple site damages occurred in the aircraft structures, are studied by using the weigh function method. An explicit closed form weight function for periodic edge hole cracks in an infinite sheet is obtained and further used to calculate the stress intensity factor and crack opening displacement for various loading cases. Compared to finite element method, the present weight function is accurate and highly efficient. The interactions of the holes and cracks on the stress intensity factor and crack opening displacement are quantitatively determined by using the present weight function. An approximate weight function method is also proposed for arbitrary small cracks emanating from multiple collinear holes. This method is very useful for calculating the stress intensity factor for arbitrary small cracks.     

A fast and accurate analysis of the interacting cracks in linear elastic solids

This paper presents a fast and accurate solution for crack interaction problems in infinite- and half- plane solids. The new solution is based on the method of complex potentials developed by Muskhelishvili for the analysis of plane linear elasticity, and it is formulated through three steps. First, the problem is decomposed into a set of basic problems, and for each sub-problem, there is only one crack in the solid. Next, after a crack-dependent conformal mapping, the modified complex potentials associated with the sub-problems are expanded into Laurent’s series with unknown coefficients, which in turn provides a mechanism to exactly implement in the form of Fourier series the boundary condition in each sub-problem. Finally, taking into account the crack interaction via a perturbation approach, an iterative algorithm based on fast Fourier transforms (FFT) is developed to solve the unknown Fourier coefficients, and the solution of the whole problem is readily obtained with the superposition of the complex potentials in each sub-problem. The performance of the proposed method is fully investigated by comparing with benchmark results in the literatures, and superb accuracy and efficiency is observed in all situations including patterns where cracks are closely spaced. Also, the new method is able to cope with interactions among a large number of cracks, and this capability is demonstrated by a calculation of effective moduli of an elastic solid with thousands of randomly-spaced cracks.     

Weight Function for a Crack in a Two-dimensional Orthotropic Medium Under Impact Shear Loading

The paper examines the elastodynamic response of an infinite two-dimensional orthotr- opic medium containing a central crack under impact shear loading. Laplace and Fourier integral transforms are employed to reduce the problem to a pair of dual integral equations in the Laplace transformed plane. These equations are reduced to integral differential equations, which have been solved in the low frequency domain by iterations. To determine time dependence, these equations are inverted to yield the dynamic stress intensity factor (SIF) for shear point force loading that corresponds to the weight function for the crack under shear loading. It is used to derive SIF for polynomial loading.     

Determination of approximate point load weight functions for embedded elliptical cracks

This paper presents the application of weight function method for the calculation of stress intensity factors in embedded elliptical cracks under complex two-dimensional loading conditions. A new general mathematical form of point load weight function is proposed based on the properties of weight functions and the available weight functions for two-dimensional cracks. The existence of this general weight function form has simplified the determination of point load weight functions significantly. For an embedded elliptical crack of any aspect ratio, the unknown parameters in the general form can be determined from one reference stress intensity factor solution. This method was used to derive the weight functions for embedded elliptical cracks in an infinite body and in a semi-infinite body. The derived weight functions are then validated against available stress intensity factor solutions for several linear and non-linear stress distributions. The derived weight functions are particularly useful for the fatigue crack growth analysis of planer embedded cracks subjected to fluctuating non-linear stress fields resulting from surface treatment (shot peening), stress concentration or welding (residual stress).     

Stress intensity factor for center-cracked plate with crack surface interference

This paper presents a simple and physically acceptable analysis of stress intensity factor (SIF) for the center-cracked infinite and finite-width plates. The analysis includes the effect of crack surface interference (i.e., the upper and lower crack surfaces are not allowed to overlap) that influences both the SIF at the tension-side crack tip and the crack opening displacement (COD) profile. For an infinite plate, exact solutions are obtained by superimposing the classical (overlapping) solutions. For a finite-width plate, where the SIF solutions cannot be found in closed form, the solutions are carried out numerically. The overlapping SIF solutions from the weight function method are used. An example is given for the case of a finite-width plate under bending. It was found that the overlapping solutions underestimate the stress intensity factor at the tension-side crack tip up to 15%. The analysis results are also compared with the finite element solutions for verification purpose.     

A weight function method for mixed modes hole‐edge cracks

A weight function approach is proposed to calculate the stress intensity factor and crack opening displacement for cracks emanating from a circular hole in an infinite sheet subjected to mixed modes load. The weight function for a pure mode II hole‐edge crack is given in this paper. The stress intensity factors for a mixed modes hole‐edge crack are obtained by using the present mode II weight function and existing mode I Green (weight) function for a hole‐edge crack. Without complex derivation, the weight functions for a single hole‐edge crack and a centre crack in infinite sheets are used to study 2 unequal‐length hole‐edge cracks. The stress intensity factor and crack opening displacement obtained from the present weight function method are compared well with available results from literature and finite element analysis. Compared with the alternative methods, the present weight function approach is simple, accurate, efficient, and versatile in calculating the stress intensity factor and crack opening displacement.     

Weight functions and stress intensity factors for quarter-elliptical corner cracks in fastener holes

Approximate weight functions for a quarter‐elliptical crack in a fastener hole were derived from a general weight function form and two reference stress intensity factors. Closed‐form expressions were obtained for the coefficients of the weight functions. The derived weight functions were validated against numerical data by comparison of stress intensity factors calculated for several nonlinear stress fields. Good agreements were achieved. These derived weight functions are valid for the geometric range of 0.5 ≤a/c≤ 1.5 and 0 ≤a/t≤ 0.8 and R/t= 0.5; and are given in forms suitable for computer numerical integration. The weight functions appear to be particularly suitable for fatigue crack growth prediction of corner cracks in fastener holes and fracture analysis of such cracks in complex stress fields.     

A novel weight function for RMS stress intensity factor determination in surface cracks

This paper discusses the problem of stress intensity factor determination in surface cracks. In particular, the concept of root mean square stress intensity factors (RMS SIF) is discussed for the general class of semi-elliptical surface cracks. The weight function SIF derivation method is considered, problems with the existing techniques are highlighted, and a novel technique for the derivation of the RMS SIF weight functions for surface cracks is presented and results are compared with numerical solutions for a variety of loadings and geometries.     

有限高磁电弹性体中双半动态与静态裂纹分析

狭长体中的裂纹是断裂力学中经常采用的研究模型。含有共线无限长裂纹的条形磁电弹性体,当面内的力电磁和反平面的剪应力作用在左边裂纹尖端附近的一段裂纹面上时,往往会产生动态断裂。利用复变函数法中的拱形变换公式,导出了磁电全非渗透型边界条件下左裂纹尖端动态的应力强度因子以及机械应变能释放率的解析解。当运动速度趋于零时退化为静止状态下的解。通过数值算例分析了断裂机理,讨论了静止状态下狭长体和裂纹的几何尺寸、外力、电场和磁场分别对能量释放率的影响,为相关器件的设计与制造提供了帮助。     

Stress intensity factor for an embedded elliptical crack under arbitrary normal loading

In this paper we introduce the boundary value problem of three-dimensional classical elasticity for an infinite body containing an elliptical crack. Using the method of simultaneous dual integral equations, the problem is transformed to the system of linear algebraic equations. Stress intensity factor is obtained in the form of the Fourier series expansion. Several solutions for specific cases of applied polynomial stress fields are derived and compared with existing results. Eligibility of the method for more complicated stress fields is demonstrated on the example of partially loaded elliptical crack.     

Doubly periodic arrays of cracks and thin inhomogeneities in an infinite magnetoelectroelastic medium

A two-dimensional boundary element method for the analysis of a magnetoelectroelastic medium containing doubly periodic sets of cracks or thin inclusions is developed in this paper. The integral equations and closed-form expressions for corresponding kernels are obtained. Based on the quasi-periodicity of extended displacement and stress function, the integral representations for average stress, strain, electric displacement, magnetic induction etc. are developed. The algorithm of effective properties determination is given. The numerical examples prove the efficiency and high accuracy of the proposed approach in determination of stress, electric displacement and magnetic induction intensity factors and effective properties of the material containing doubly periodic arrays of cracks or thin inclusions.     

Approximate weight function method for elliptical arc through cracks in mechanical joints

Mechanical joints such as bolted, riveted or pinned joints are widely used to join the constituent parts of structural components. Reliable stress intensity factor analysis of arbitrary cracks in mechanical joints is required for the safety evaluation or fracture mechanics design. It has been reported that cracks in mechanical joints usually nucleate as the corner crack and grow as the elliptical arc through crack. The weight function method is a useful technique to calculate the stress intensity factor using the appropriate weight function for a cracked body and the stress field of an uncracked body. In this paper, the weight function method for the two surface points of elliptical arc through cracks in mechanical joints is developed to analyze the mixed-mode stress intensity factors. Unknown coefficients included in the weight function are determined using the reference stress intensity factors obtained from finite element analysis.     

Transient thermal stress intensity factors for an internal longitudinal semi-elliptical crack in a thick-walled cylinder

In this paper, the stress intensity factors are derived for an internal semi-elliptical crack in a thick-walled cylinder subjected to transient thermal stresses. First, the problem of transient thermal stresses in a thick-walled cylinder is solved analytically. Thermal and mechanical boundary conditions are assumed to act on the inner and outer surfaces of the cylinder. The quasi-static solution of the thermoelasticity problem is derived analytically using the finite Hankel transform and then, the stress intensity factors are extracted for the deepest point and the surface points of the semi-elliptical crack using the weight function method. The results show to be in accordance with those cited in the literature in the special case of steady-state problem. Using the closed-form relations extracted for the transient thermal stress intensity factors, some conclusive results are drawn.     

Application of the crack compliance method to long axial cracks in pipes with allowance for geometrical nonlinearity and shape imperfections (dents)

Application of the crack compliance method to the analysis of thin-walled rings with a radial crack has two features: a crack is considered as a concentrated angular compliance and the deformation of all other sections of the rings is calculated as for a curvilinear beam. The latter can be most conveniently found by the method of initial parameters where the values of generalized forces and displacements at the end of some zone are determined as a linear combination of their values at the beginning of the zone. The goal of the study is to derive and apply the method of initial parameters equations taking into account the influence of circumferential stresses on the ring curvature. As far as the authors know, this is the first time that the stress intensity factor has been derived for an elastic thin-walled pipe with a radial crack in a geometrically nonlinear formulation. Here, an increase in pressure leads to a somewhat slowed increase in the stress intensity factor. In addition, a number of problems for dents are considered. The effect of the dent shape on the stress-strain state is analyzed. An expression for the stress intensity factor for a complex defect, a crack emanating from the dent apex, is presented.