Collinear Zener-Stroh crack problem in plane elasticity

This paper investigates the collinear Zener-Stroh crack problem in plane elasticity. Two cracks in series are chosen as an example in analysis. Different to the Griffith crack problem, an initial displacement discontinuity exists in the Zener-Stroh crack problem, which in turn is the increment of displacement when a moving point goes around the crack in a closed loop. From the traction free condition along the cracks, the dislocation distribution function as well as the complex potential with undetermined coefficients is suggested. The involved undetermined coefficients can be evaluated from the condition of the assumed initial displacement discontinuity. Finally, a closed form solution for the problem is obtained and the calculated stress intensity factors at crack tips are presented. A problem for two collinear Zener-Stroh cracks with different lengths is also studied.     

Numerical solution for the T-stress in branch crack problem with infinitesimal branch length

This paper investigates the T-stress in a branch crack problem with infinitesimal branch length. The branch crack is composed of a main crack and many branches. The ratio of the lengths for branch to main crack is very small. A singular integral equation method is suggested to solve the problem numerically and the stress intensity factor and T-stress can be evaluated immediately. Many computed results for T-stress under different conditions for branches are presented. It is found from the computed results that the interaction for T-stress among branches is very complicated.     

Evaluation of the stress intensity factors and the T-stress in periodic crack problem

This paper investigates the T-stress at crack tips in the periodic crack problem. Remote tension in the y-direction is applied to cracks with an arbitrary inclined angle. The original stress field can be considered a superposition of a uniform stress field and a perturbation stress field. The problem of evaluating the stresses in the perturbation field can be considered a superposition of many single crack problems. A Fredholm integral equation is suggested for the solution of the perturbation stress field. In the equation, the loading on the crack face is chosen as unknown quantity. Once the integral equation is solved, the stress intensity factors and the T-stress at the crack tip can be evaluated immediately. For solving the integral equation and evaluating stresses in the perturbation field, the remainder estimation technique is suggested for evaluating the influences on the central crack from infinite cracks. The technique can considerably improve convergence in computation. Many results for the stress intensity factors and the T-stresses in periodic cracks are presented. It is shown that the interaction is significant for the closer cracks.     

A rigorous derivation for T-stress in line crack problem

In this paper, a rigorous derivation for T-stress in line crack problem is presented. Similar to the edge crack case, this paper provides the T-stress dependence on loading with the Dirac delta function property.     

Axisymmetric crack problem of thick-walled cylinder with loadings on crack surfaces

This study is concerned with the fracture of an infinite thick-walled cylinder. The inner surface of the cylinder is stress free and the outer is rigidly fixed. The cylinder having a ring-shaped crack located at the symmetry plane is subjected to distributed compressive load on its surfaces. The Hankel and Fourier transform techniques are used for the solution of the field equations. By applying the boundary conditions, the singular integral equation in terms of crack surface displacement derivative is derived. By using an appropriate quadrature formula, the integral equation is reduced to a system of linear algebraic equations. Numerical results are obtained for the stress intensity factors at the edges of the crack, surfaces of which are subjected to uniform, linear and parabolic load distributions.     

The effect of T-stress on plane strain dynamic crack growth in elastic–plastic materials

In this paper, the effects of T‐stress on steady, dynamic crack growth in an elastic–plastic material are examined using a modified boundary layer formulation. The analyses are carried out under mode I, plane strain conditions by employing a special finite element procedure based on moving crack tip coordinates. The material is assumed to obey the J₂ flow theory of plasticity with isotropic power law hardening. The results show that the crack opening profile as well as the opening stress at a finite distance from the tip are strongly affected by the magnitude and sign of the T‐stress at any given crack speed. Further, it is found that the fracture toughness predicted by the analyses enhances significantly with negative T‐stress for both ductile and cleavage mode of crack growth.     

T-stress solutions for two-dimensional crack problems in anisotropic elasticity using the boundary element method

The importance of a two‐parameter approach in the fracture mechanics analysis of many cracked components is increasingly being recognized in engineering industry. In addition to the stress intensity factor, the T stress is the second parameter considered in fracture assessments. In this paper, the path‐independent mutual M‐integral method to evaluate the T stress is extended to treat plane, generally anisotropic cracked bodies. It is implemented into the boundary element method for two‐dimensional elasticity. Examples are presented to demonstrate the veracity of the formulations developed and its applicability. The numerical solutions obtained show that material anisotropy can have a significant effect on the T stress for a given cracked geometry.     

Numerical solution for multiple crack problem in an infinite plate under compression

This paper investigates a numerical solution for multiple crack problem in an infinite plate under remote compression. The influence of friction is taken into account. In the first step of the solution, we make a full contact assumption on the crack faces. The full contact assumption means that one component of the dislocation distribution vanishes, and the first mode stress intensity factors (K ₁) at the crack tips become zero. On the above-mentioned assumption, the problem can be solved by using integral equation method, and the second mode stress intensity factors (K ₂) at the crack tips can be evaluated. Meantime, after solving the integral equation the normal contact stress on the crack faces can be evaluated. The next step is to examine the full contact assumption. If the contact stresses on the crack faces are definitely negative, the solution is true. Otherwise, the obtained solution is not true. It is found from present study that in most cases the full contact condition is satisfied, and only in a few cases the full contact condition is violated. Numerical examples are given. It is found that the friction can lower the stress intensity factors at crack tips in general.     

A generalized crack problem in plane elasticity

The plane isotropic elasticity problem of a simple curvilinear crack with non-coincident edges (contrary to the idealization usually made) is considered. The maximum opening between the edges of the crack may be as great as 0.2 of the crack length. For the solution of this problem, the model of replacing the real crack by a continuous distribution of poles (concentrated forces and edge dislocations) along a single are lying between the real crack edges is introduced. The problem is reduced to an almost singular integral equation and an approximate method for its numerical solution is proposed. An application to the case of a symmetric crack in an infinite plane medium under uniform loading at infinity is also made.     

T-stress in orthotropic functionally graded materials: Lekhnitskii and Stroh formalisms

A new interaction integral formulation is developed for evaluating the elastic T-stress for mixed-mode crack problems with arbitrarily oriented straight or curved cracks in orthotropic nonhomogeneous materials. The development includes both the Lekhnitskii and Stroh formalisms. The former is physical and relatively simple, and the latter is mathematically elegant. The gradation of orthotropic material properties is integrated into the element stiffness matrix using a “generalized isoparametric formulation” and (special) graded elements. The specific types of material gradation considered include exponential and hyperbolic-tangent functions, but micromechanics models can also be considered within the scope of the present formulation. This paper investigates several fracture problems to validate the proposed method and also provides numerical solutions, which can be used as benchmark results (e.g. investigation of fracture specimens). The accuracy of results is verified by comparison with analytical solutions.     

A moving interface crack between two dissimilar functionally graded strips under plane deformation with integral equation methods

In this paper a finite interface crack with constant length (Yoffe-type crack) propagating along the interface between two dissimilar functionally graded strips with spatially varying elastic properties under in-plane loading is studied. By utilizing the Fourier transformation technique, the mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. The influences of the geometric parameters, the graded parameter, the strip thickness and crack speed on the stress intensity factors and the probable kink angle are investigated. The numerical results show that the graded parameters, the thicknesses of the functionally graded strips and the crack speed have significant effects on the dynamic fracture behavior of functionally graded material structures.     

含裂纹损伤结构强度的多源数据融合方法研究

工程结构在制造工艺过程中或使用期间会产生裂纹,对结构断裂路径的预测和研究是防治工程安全问题发生的重要手段。在考虑裂纹尖端应力场常数项T应力的基础上对传统的最大周向应力准则(Maximum tangential stress criterion, MTS)和最小应变能密度因子准则(Minimum strain energy density criterion, SED)进行修正,采用Python语言对ABAQUS的前、后处理和有限元计算模块进行二次开发,通过计算最优解的粒子群算法(Particle swarm optimization, PSO)将修正后的准则编入裂纹自动扩展程序脚本中。利用上述二次开发程序对初始纯I型裂纹的扩展路径进行模拟,结果表明：采用ABAQUS脚本程序模拟结果与相关文献实验结果吻合,表明了程序的有效性,进而实现考虑T应力的多种断裂准则对裂纹扩展路径的预测;当T应力值处于一定范围内时,修正的MTS准则无法预测裂纹发生的偏转现象,扩展路径呈直线,此时可采用修正的SED准则进行预测。

     

T-stress Solutions for Cracks Emanating from a Circular Hole in a Finite Plate

The boundary element method is employed to obtain T-stress solutions for cracks emanating from a circular hole in finite rectangular plates. Numerical values of the T-stress are obtained using the M-contour integral approach. A range of crack lengths are analyzed for two hole sizes, and the cases of a single crack and double-cracks emanating from the hole in the plate under both uniform remote tension and simple bending are considered. For completeness, stress intensity factor solutions are also presented. These results will be useful for failure assessments using two-parameter linear elastic fracture mechanics.     

Analysis of a mode III crack problem in a functionally graded coating-substrate system with finite thickness

This paper presents an analysis of the static problem of model III crack of a functionally graded coating-substrate system with an internal crack perpendicular to the interface under antiplane shear loading when the coating layer and substrate have finite thickness. After the Fourier transform method is employed, the expressions of the displacement components can be obtained. Integral transforms are employed to reduce the problem to a singular integral equation that can be solved numerically. The influences of the nonhomogeneity constant, relative crack length and thickness ratio are quantitatively studied.     

Three-dimensional crack problem analysis using boundary element method with finite-part integrals

A set of hypersingular integral equations of a three-dimensional finite elastic solid with an embedded planar crack subjected to arbitrary loads is derived. Then a new numerical method for these equations is proposed by using the boundary element method combined with the finite-part integral method. According to the analytical theory of the hypersingular integral equations of planar crack problems, the square root models of the displacement discontinuities in elements near the crack front are applied, and thus the stress intensity factors can be directly calculated from these. Finally, the stress intensity factor solutions to several typical planar crack problems in a finite body are evaluated. This revised version was published online in August 2006 with corrections to the Cover Date.     

Evaluation of the T-stress in branch crack problem

This paper investigates the T-stress in the branch crack problem. The problem is modeled by a continuous distribution of dislocation along branches, and the relevant singular integral equation is obtained accordingly. After discretization of the singular integral equation, the balance for the number of equations and unknowns is well designed. After the singular integral equation is solved, the equation for evaluating the T-stress is derived. The merit of present study is to provide necessary equation for evaluating T-stress, rather than to provide the integral equation. Many computed results for T-stress under different conditions for branch crack are presented. It is found from the computed results that the interaction for T-stress among branches is complicated.     

Eigenvalue and eigenfunction analysis arising from degenerate scale problem of BIE in plane elasticity

A new boundary integral equation (BIE) of plane elasticity is suggested with the use of a novel kernel. The relevant homogenous equation is also suggested. The equation is studied in a discrete form, or it is reduced to an algebraic equation. From the condition that the value of a determinant vanishes, the degenerate scale (or the eigenvalue) and the non-trivial solution (or the eigenfunction) are obtained approximately. Except for the notch with symmetric configuration for two axes, computed results prove that there are two degenerate scales in general. The dependence of the eigenvalue and eigenfunction with respect to the translation or the rotation of notch is investigated. It is found that the eigenvalues are invariant with respect to the translation and the rotation of notch. However, the eigenfunctions are changed when the notch has a rotation. Several numerical examples that include a rectangular notch, a half-ring-shaped notch and a complicated notch configuration are presented with the computed eigenvalues and eigenfunctions.     

An interaction energy integral method for T-stress evaluation in nonhomogeneous materials under thermal loading

An interaction energy integral method is developed for the finite element evaluation of the T-stress in nonhomogeneous materials under thermal loading. A domain-independent integral expression for extracting the T-stress is proposed for nonhomogeneous materials even when the integral domain intersects the interface. Then it is set in the extended finite element method (XFEM) so that the T-stress can be solved with high accuracy and efficiency. Several representative examples are solved to show the validity and the domain-independence of the method. A crack problem in a functionally graded thermal barrier coating (TBC) is also analyzed. Finally, the influences of material continuity on the T-stress are investigated. It can be found that the discontinuity of both thermal expansion coefficient and Young’s modulus affects the T-stress dramatically.     

New Fredholm integral equation for multiple crack problem in plane elasticity and antiplane elasticity

A singular integral equation for the multiple crack problem of plane elasticity is formulated in this paper. In the formulation we choose the crack opening displacement (COD) as unknown function and the resultant force as the right hand term of the equation. After using Vekua's regularization procedure or making inversion of the Cauchy singular integral in the equation, a new Fredholm integral equation is obtainable. The obtained Fredholm integral equation is compact in form and easy for computation. After solving the equation, the CODs of the cracks and the stress intensity factors (SIFs) at the crack tips can be derived immediately. Similar formulation for the multiple crack problem of antiplane elasticity is also presented. Finally, numerical examples are given to demonstrate the use of the proposed integral equation approach.