Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The conventional multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.     

The impact response of a half plane crack in a transversely isotropic solid due to 3-D mixed mode loading

Three-dimensional analysis of a half plane crack in a transversely isotropic solid is performed. The crack is subjected to two opposed pairs of shear line loads on its faces. Transform methods are used to reduce the boundary value problem to a set of coupled integral equations that can be solved by the Wiener-Hopf technique. The Cagniard-de Hoop method is employed to invert the transforms. Exact expressions are derived for the mode II and III stress intensity factors as functions of time and position along the crack edge. Some features of the solutions are discussed through numerical results.     

Boundary integral equation solution of three‐dimensional elastostatic problems in transversely isotropic solids using closed‐form displacement fundamental solutions

The boundary integral equation method is used for the solution of three‐dimensional elastostatic problems in transversely isotropic solids using closed‐form fundamental solutions. The previously published point force solutions for such solids were modified and are presented in a convenient form, especially suitable for use in the boundary integral equation method. The new presentations are used as a basis for accurate numerical computations of all Green's functions necessary in the BEM process without inaccuracy and redundant computations. The validity of the new presentation is shown through three numerical examples. Copyright © 2000 John Wiley & Sons, Ltd.     

Neural approach to estimate the stress intensity factor of semi‐elliptical cracks in rotating cracked shafts in bending

In the last decades, neural network approach has often been used to study various and complex engineering problems, such as optimization or prediction. In this paper, a methodology founded on artificial neural networks (ANNs) was used to calculate the stress intensity factor (SIF) in different points of the front of a semi‐elliptical crack present in a rotating shaft, taking into account the shape and depth of the crack, the angle of rotation, and the location of the point in the front. In the event of rotating machines, such as shafts, it is crucial to know the SIF along the crack front because this parameter, according to the Paris Law, is related to the performance of the crack during its propagation. Previously, it was necessary to achieve the data for the ANN training, for this a quasi‐static numerical model was made, which simulates a rotating cracked shaft with a semi‐elliptical crack. The numerical solutions cover a wide range of crack depths and shapes, and rotation angles. The values of the SIF estimated by the ANNs were contrasted with other solutions available in the literature finding a good agreement between them. The proposed neural network methodology is an alternative that offers a very good option for the SIF estimation, because it is efficient and easy to use, does not require high computational costs, and can be used to analyse the propagation of cracks contained in rotating shafts by means of the Paris Law taking into account the nonlinear behaviour of the shaft.     

Interfaces Between two Dissimilar Elastic Materials

In this paper the near tip solutions for interface corners written in terms of the stress intensity factors are presented in a unified expression. This single expression is applicable for any kinds of interface corners including corners and cracks in homogeneous materials as well as interface corners and interface cracks lying between two dissimilar materials, in which the materials can be any kinds of linear elastic anisotropic materials or piezoelectric materials. Through this unified expression of near tip solutions, the singular orders of stresses and their associated stress/electric intensity factors for different kinds of interface problems can be determined through the same formulae and solution techniques. This unified feature of solving interface problems is then implemented numerically through several different interface problems. Moreover, in order to improve the accuracy and efficiency of numerical computation, a special boundary element based upon the Green's function of bimaterials is introduced in this paper.     

BEM analysis of laterally loaded piles in multi-layered transversely isotropic soils

This paper presents a theory for the static analysis of laterally loaded piles embedded in multi-layered transversely isotropic soils. Boundary element method (BEM) is applied to the pile–soil model where the floating pile is modeled as a Bernoulli–Euler beam using the finite difference method and the layered soil is represented utilizing a decoupled analytical layer-element solution as a kernel function for its high accuracy and efficiency. Several numerical examples presented reveal that the pile behavior is affected synthetically by both transverse isotropy and stratified character of soil and the pile's size and physical properties.     

Boundary element analysis of three‐dimensional cracks in anisotropic solids

This paper presents a boundary element analysis of linear elastic fracture mechanics in three‐dimensional cracks of anisotropic solids. The method is a single‐domain based, thus it can model the solids with multiple interacting cracks or damage. In addition, the method can apply the fracture analysis in both bounded and unbounded anisotropic media and the stress intensity factors (SIFs) can be deduced directly from the boundary element solutions. The present boundary element formulation is based on a pair of boundary integral equations, namely, the displacement and traction boundary integral equations. While the former is collocated exclusively on the uncracked boundary, the latter is discretized only on one side of the crack surface. The displacement and/or traction are used as unknown variables on the uncracked boundary and the relative crack opening displacement (COD) (i.e. displacement discontinuity, or dislocation) is treated as a unknown quantity on the crack surface. This formulation possesses the advantages of both the traditional displacement boundary element method (BEM) and the displacement discontinuity (or dislocation) method, and thus eliminates the deficiency associated with the BEMs in modelling fracture behaviour of the solids. Special crack‐front elements are introduced to capture the crack‐tip behaviour. Numerical examples of stress intensity factors (SIFs) calculation are given for transversely isotropic orthotropic and anisotropic solids. For a penny‐shaped or a square‐shaped crack located in the plane of isotropy, the SIFs obtained with the present formulation are in very good agreement with existing closed‐form solutions and numerical results. For the crack not aligned with the plane of isotropy or in an anisotropic solid under remote pure tension, mixed mode fracture behavior occurs due to the material anisotropy and SIFs strongly depend on material anisotropy. Copyright © 2000 John Wiley & Sons, Ltd.     

Stress intensity factors for penny-shaped cracks perpendicular to graded interfacial zone of bonded bi-materials

This paper examines the stress intensity factors that are associated with a penny-shaped crack perpendicular to the interface of a bi-material bonded with a graded interfacial zone. Elastic modulus of the graded interfacial zone is assumed to be an exponential function of the depth. The stress intensity factors are calculated numerically using a so-called generalized Kelvin solution based boundary element method. Three cases of normal or shear tractions acting on the crack surfaces are examined. Values of the stress intensity factors are examined by taking into account the effects of the following four parameters: (a) the crack front position; (b) the non-homogeneity parameter of the graded interfacial zone; (c) the crack distance to the graded interfacial zone; and (d) the graded interfacial zone thickness. The numerical results are compared well with existing solutions under some degenerated conditions. These results are useful to furthering our knowledge on fracture behavior of bi-material systems with or without a graded interfacial zone.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

Analysis of three-dimensional interface cracks using enriched finite elements

Many important interface crack problems are inherently three-dimensional in nature, e.g., debonding of laminated structures at corners and holes. In an effort to accurately analyze three-dimensional interface fracture problems, an efficient computational technique was developed that utilizes enriched crack tip elements containing the correct interface crack tip asymptotic behavior. In the enriched element formulation, the stress intensity factors K _I, K _II, and K _III are treated as additional degrees of freedom and are obtained directly during the finite element solution phase. In this study, the results that should be of greatest interest are obtained for semi-circular surface and quarter-circular corner cracks. Solutions are generated for uniform remote tension and uniform thermal loading, over a wide range of bimaterial combinations. Of particular interest are the free surface effects, and the influence of Dundurs’ material parameters on the strain energy release rate magnitudes and corresponding phase angles.     

Dynamic stress intensity factors around two parallel cracks in a functionally graded layer bonded to dissimilar half-planes subjected to anti-plane incident harmonic stress waves

The time-harmonic problem of determining the stress field around two parallel cracks in functionally graded materials (FGMs) is studied. The Fourier transform technique is used to reduce the boundary conditions to four simultaneous integral equations which are then solved by expanding the differences of crack surface displacements in a series. The unknown coefficients in the series are obtained by the Schmidt method. Numerical calculations are carried out for dynamic stress intensity factors (DSIF) in FGMs.     

Analyzing interaction between coplanar square cracks using an efficient boundary element‐free method

This paper examines the interaction between coplanar square cracks by combining the moving least‐squares (MLS) approximation and the derived boundary integral equation (BIE). A new traction BIE involving only the Cauchy singular kernels is derived by applying integration by parts to the traditional boundary integral formulation. The new traction BIE can be directly applied to a crack surface and no displacement BIE is necessary because all crack boundary conditions (both upper and lower ones) are incorporated. A boundary element‐free method is then developed by combining the derived BIE and MLS approximation, in which the crack opening displacement is first expressed as the product of weight functions and the characteristic terms, and the unknown weight is approximated with the MLS approximation. The efficiency of the developed method is tested for isotropic and transversely isotropic media. The interaction between two and three coplanar square cracks in isotropic elastic body is numerically studied and the case of any number of coplanar square cracks is deduced and discussed. Copyright © 2012 John Wiley & Sons, Ltd.     

含垂直极化方向穿透的共线半无限裂纹的狭长压电体的平面问题

通过构造新的广义保角映射, 利用广义复变函数方法研究了在电不可渗透边界条件下含有沿垂直于极化方向穿透的共线半无限裂纹的狭长压电体的平面问题, 给出了裂纹尖端处的各场强度因子的解析解。此外, 当狭长体高度和裂纹尺寸按一定的趋势变化时, 还可以得到压电复合材料另外几种新构型的平面问题的解析解。通过数值算例分析得到了两裂纹之间的距离、 狭长体的高度及裂纹面上的受载长度对场强度因子的影响规律。     

Dynamic stress intensity factors of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material under a time‐harmonic elastic P‐wave

The dynamic behavior of two 3D rectangular permeable cracks in a transversely isotropic piezoelectric material is investigated under an incident harmonic stress wave by using the generalized Almansi's theorem and the Schmidt method. The problem is formulated through double Fourier transform into three pairs of dual integral equations with the displacement jumps across the crack surfaces as the unknown variables. To solve the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Finally, the relations among the dynamic stress field and the dynamic electric displacement filed near the crack edges are obtained, and the effects of the shape of the rectangular crack, the characteristics of the harmonic wave, and the distance between two rectangular cracks on the stress and the electric intensity factors in a piezoelectric composite material are analyzed. Copyright © 2015 John Wiley & Sons, Ltd.     

Unique real‐variable expressions of displacement and traction fundamental solutions covering all transversely isotropic elastic materials for 3D BEM

A general, efficient and robust boundary element method (BEM) formulation for the numerical solution of three‐dimensional linear elastic problems in transversely isotropic solids is developed in the present work. The BEM formulation is based on the closed‐form real‐variable expressions of the fundamental solution in displacements U_ik and in tractions T_ik, originated by a unit point force, valid for any combination of material properties and for any orientation of the radius vector between the source and field points. A compact expression of this kind for U_ik was introduced by Ting and Lee (Q. J. Mech. Appl. Math. 1997; 50 :407–426) in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector. Working from this expression of U_ik, and after a revision of their final formula, a new approach (based on the application of the rotational symmetry of the material) for deducing the derivative kernel U_{ik, j} and the corresponding stress kernel Σ_ijk and traction kernel T_ik has been developed in the present work. These expressions of U_ik, U_{ik, j}, Σ_ijk and T_ik do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex‐valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational‐symmetry axis. The expressions of U_ik, U_{ik, j}, Σ_ijk and T_ik have been presented in a form suitable for an efficient computational implementation. The correctness of these expressions and of their implementation in a three‐dimensional collocational BEM code has been tested numerically by solving problems with known analytical solutions for different classes of transversely isotropic materials. Copyright © 2007 John Wiley & Sons, Ltd.     

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。     

Three dimensional boundary element analysis of internal cracks under sliding contact load with a spherical indenter

Using boundary element based three dimensional modelling for linear fracture mechanics, we present an analysis of cracking in a homogeneous medium subject to contact load. The proposed iterative solution procedure allows a simultaneous treatment of a reasonable number of partially closed cracks. It is shown that the most probable direction of propagation of a vertical internal crack is strongly dependent on its size compared to the contact radius and its location with respect to the axis of maximum normal load.     

Approximate weight function technique using single-layer potential for a cracked circular disc

An efficient weight function technique using the indirect boundary integral method was presented for cracked circular discs. The crack opening displacement field was presented by a single layer whose kernel was a modified form of the fundamental solution in elastostatics. The application of a single-layer potential to the weight function method leads to a unique closed-form SIF (stress intensity factor) solution. The solution can be applied to a cracked circular discs with or without an internal hole or opening. For these crack geometries over a wide range of crack ratios, the SIF solution can be applied without any modification.

The calculation procedure of SIFs for the various cracked circular discs using only one analytical solution is very simple and straightforward. The information necessary in the analysis includes only two or three reference load cases. In most cases the SIF solution using two reference SIFs gives reasonably accurate results while the SIF solution with three reference load cases may be used to improve the solution accuracy of the crack configurations, with an internal opening or hole, compared with the solutions of the available literature.     

Sharp evaluation of the Oore–Burns integral for cracks subjected to arbitrary normal stress field

This paper presents a new approximation formula for the Oore–Burns integral related to three‐dimensional weight functions. The approach drastically reduces the computational time of the Oore–Burns integral with respect to previous formulations because the mesh over the integration domain can be very coarse without loss of accuracy. This is made possible by analytic evaluation of the coefficient of δ^1/2 of the deviation between the integral and its Riemann sum (δ is the size of the mesh over the crack). In the case of a penny‐shaped crack, the new equation can be considered as an explicit formulation of the exact weight function of Galin. In order to confirm the accuracy of our new formulation, we consider the case of penny‐shaped cracks under different types of mode I loading. Predictions of the stress intensity factor are compared with analytical predictions along the crack, and the new equation appears to be stable with respect to the refinement of the mesh. Furthermore, it is accurate even when the stress field is represented with high‐order polynomial terms. Finally, we apply our approximation of the Oore–Burns integral to an elliptical crack with small eccentricity under uniform pressure. Agreement with the Irwin solution is excellent.