Stress intensity factor analysis of a three-dimensional interface crack between dissimilar anisotropic materials

A new numerical method to calculate the stress intensity factors (SIFs) of a three-dimensional interface crack between dissimilar anisotropic materials was developed. In this study, the M-integral method was employed for mode separation of the SIFs. The moving least-square method was utilized to calculate the M-integral. Using the M-integral with the moving least-square method, SIFs can be automatically calculated with only the nodal displacements from the finite element method (FEM). Here, SIFs analyses of some typical three-dimensional problems are demonstrated. Excellent agreement was achieved between the numerical results obtained by the present method and the corresponding results proposed by other researchers. In addition, the SIFs of a single-edge crack, a through crack, and a semi-circular crack between two anisotropic solids in three-dimensional structures were analyzed.     

Stress intensity factor analyses of three-dimensional interface cracks using tetrahedral finite elements

A numerical method is proposed for evaluating the stress intensity factors of a three-dimensional bimaterial interfacial crack using tetrahedral finite elements. This technique is based on the M₁-integral method and employs the moving least-squares approximation. Stress or strain in the M₁-integral equation is automatically approximated from the nodal displacements obtained by the finite element analysis using the moving least-squares method. Therefore, the presented method needs no elemental information from the finite element analysis. In this study, stress intensity factor analyses of some typical three-dimensional interface crack problems using the tetrahedral finite elements are demonstrated.     

A semi-theoretical and experimental approach for the determination of the stress intensity factors

The stress intensity factors for plexiglass plates containing edge cracks and subjected to either pure bending or tension are determined herein. The method of investigation was based on a semi-theoretical and experimental approach, where the stress intensity factors are expressed in terms of the measured diameter of the caustic, the crack length, and the width of the specimen. First, two basic crack arrangements (single and double edge cracks) were studied and then the method was utilized for the investigation of more complicated crack arrangements which are difficult or maybe impossible to be investigated otherwise. In particular, the stress intensity factor for plates having a sharp V-notch of various angles θ, and semi-infinite plates containing equal parallel edge cracks subjected to pure bending and tension respectively, were investigated in order to verify the validity of this method.     

Near-tip fields and intensity factors for interfacial cracks in dissimilar anisotropic piezoelectric media

A complete form of stress and electric displacement fields in the vicinity of the tip of an interfacial crack, between two dissimilar anisotropic piezoelectric media, is derived by using the complex function theory. New definitions of real-valued stress and electric displacement intensity factors for the interfacial crack are proposed. These definitions are extensions of those for cracks in homogeneous piezoelectric media. Closed form solutions of the stress and electric displacement intensity factors for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media are also obtained by using the mutual integral.     

Stress intensity factor analysis at an interfacial corner between anisotropic bimaterials under thermal stress

A numerical method using a path-independent H-integral based on the Betti reciprocal principle was developed to analyze the stress intensity factors of an interfacial corner between anisotropic bimaterials under thermal stress. According to the theory of linear elasticity, asymptotic stress near the tip of a sharp interfacial corner is generally singular as a result of a mismatch of the materials’ elastic constants. The eigenvalues and the eigenfunctions are obtained using the Williams eigenfunction method, which depends on the materials’ properties and the geometry of an interfacial corner. The order of the singularity related to the eigenvalue is real, complex or power-logarithmic. The amplitudes of the singular stress terms can be calculated using the H-integral. The stress and displacement fields around an interfacial corner for the H-integral are obtained using finite element analysis. A proposed definition of the stress intensity factors of an interfacial corner involves a smooth expansion of the stress intensity factors of an interfacial crack between dissimilar materials. The asymptotic solutions of stress and displacement around an interfacial corner are uniquely obtained using these stress intensity factors.     

Fracture analysis of magnetoelectroelastic solids by using path independent integrals

A solution scheme based on the fundamental solution for a generalized edge dislocation in an infinite magnetoelectroelastic solid is presented to analyze problems involving single, multiple and slowly growing impermeable cracks. The fundamental solution for a generalized dislocation is obtained by extending the complex potential function formulation used for anisotropic elasticity. The solution for a continuously distributed dislocation is derived by integrating the solution for an edge dislocation. The problem of a system of cracks subjected to remote mechanical, electric and magnetic loading is formulated in terms of set of singular integral equations by applying the principle of superposition and the solution for a continuously distributed dislocation. The singular integral equation system is solved by using a numerical integration technique based on Chebyshev polynomials. The J_i and M-integrals for single crack and multi-cracks problems are derived and their dependence on the coordinate system is investigated. Selected numerical results for the M-integral, total energy release rate and mechanical energy release rate are presented for single, double and multiple crack problems. The case of a slowly growing crack interacting with a stationary crack is also considered. It is found that M-integral presents a reliable and physically acceptable measure for assessment of fracture behaviour and damage of magnetoelectroelastic materials.     

Dislocation tri-material solution in the analysis of bridged crack in anisotropic bimaterial half-space

The problem of an edge-bridged crack terminating perpendicular to a bimaterial interface in a half-space is analyzed for a general case of elastic anisotropic bimaterials and specialized for the case of orthotropic bimaterials. The edge crack lies in the surface layer of thickness h bonded to semi-infinite substrate. It is assumed that long fibres bridge the crack. Bridging model follows from the assumption of “large” slip lengths adjacent to the crack faces and neglect of initial stresses. The crack is modelled by means of continuous distribution of dislocations, which is assumed to be singular at the crack tip. With respect to the bridged crack problems in finite dissimilar bodies, the reciprocal theorem (Ψ-integral) is demonstrated as to compute, in the present context, the generalized stress intensity factor through the remote stress and displacement field for a particular specimen geometry and boundary conditions using FEM. Also the application of the configurational force mechanics is discussed within the context of the investigated problem.     

Conducting cracks in dissimilar piezoelectric media

Complete stress and electric fields near the tip of a conducting crack between two dissimilar anisotropic piezoelectric media, are obtained in terms of two generalized bimaterial matrices proposed in this paper. It is shown that the general interfacial crack-tip field consists of two pairs of oscillatory singularities. New definitions of real-valued stress and electric field intensity factors are proposed. Exact solutions of the stress and electric fields for basic interface crack problems are obtained. An alternate form of the J integral is derived, and the mutual integral associated with the J integral is proposed. Closed form solutions of the stress and electric field intensity factors due to electromechanical loading and the singularities for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media, are also obtained by using the mutual integral.     

Stress intensity factor analyses of interface cracks between dissimilar anisotropic materials using the finite element method

Delamination along an interface between dissimilar materials is the primary cause of failure in microstructures like electronic packages, micro-electro-mechanical systems (MEMS), and so on. Fracture mechanics is a powerful tool for the evaluation of delamination. However, many materials used in microstructures such as composite materials and single crystals are anisotropic materials. Stress intensity factors of an interface crack between dissimilar anisotropic materials, which were proposed by Hwu, are useful for evaluating the reliability of microstructures. However, numerical methods that can analyze the stress intensity factors of an interface crack between anisotropic materials have not been developed. We propose herein a new numerical method for the analysis of an interface crack between dissimilar anisotropic materials. The stress intensity factors of an interface crack are based on the generalized plane strain condition. The energy release rate is obtained by the virtual crack extension method in conjunction with the finite element method for the generalized plane strain condition. The energy release rate is separated into individual modes of the stress intensity factors K_I, K_II, and K_III, using the principal of superposition. The target problem to be solved is superposed on the asymptotic solution of displacement in the vicinity of an interface crack tip, which is described using the Stroh formalism. Analyses of the stress intensity factors of center interface cracks between semi-infinite dissimilar anisotropic media subjected to concentrated self-balanced loads on the center of crack surfaces and to uniform loads are demonstrated. The present method accurately provides mode-separated stress intensity factors using relatively coarse meshes for the finite element method.     

Stress intensity factors for cracks in thin plates

This paper presents stress intensity factor solutions for several crack configurations in plates. The loadings considered include internal pressure, and also combined bending and tension. The dual boundary element method is used to model the plate and mixed mode stress intensity factors are evaluated by a crack surface displacement extrapolation technique and the J-integral technique. Several cases including centre crack, edge crack and cracks emanating from a hole in finite width plates are presented.     

Analysis of multiple interfacial cracks in an orthotropic bi-material subjected to anti-plane shear loading

The present work concerns with the elasto-static problem of double interfacial cracks located between two dissimilar orthotropic plates. The dimensions of the bi-material composite, are assumed to be finite. The crack faces are subjected to anti-plane shear traction. Finite Fourier transforms are applied to reduce the problem to a triple series equations, and then to a system of singular integral equations with Cauchy type singularity. That are solved numerically using Gauss-Chebyshev integration formulae. The stress intensity factors, are determined in a closed form expressions. The obtained results agreed with the previous analytical ones. Further, a parametric study is introduced to investigate the effects of the geometric and elastic characteristics of the composite on the values of the stress intensity factors.     

The Time-Averaged Elastodynamic J-Integral

The time-averaged path independent J-integral for a stationary crack subjected to time-harmonic elastic waves is introduced. It can be determined from remote fields providing an alternative approach to compute the stress intensity factors. The J-integral is evaluated for a semi-infinite crack impinged by a plane sheer wave at an oblique angle.     

An investigation of stress intensity factors for plates with equal and unequal parallel edge cracks

A semi-theoretical and experimental method is used to investigate thin plates with parallel edge cracks of equal and unequal length loaded in tension. Plates with equal and unequal periodic parallel edge cracks are treated under two separate cases:

1. varying the length of the crack, and
2. varying the crack spacing.

Experimental observations show that in plates where the width of the specimen is much larger than the length of the cracks the stress intensity factor is in agreement with that of equal parallel edge cracks of semi-infinite plates. However, when the ratio of the length of the crack to the width of the specimen is no longer small, the stress intensity factor deviates considerably from that of the semi-infinite plates. Furthermore, in the case of unequal periodic parallel edge cracks, the size of the caustic of the shorter crack becomes very small (it cannot be observed and measured) when the difference in the depth of the larger and short crack becomes equal or greater than one half of the crack spacing.     

J-integral analysis of the interaction between an interface crack and parallel subinterface cracks in dissimilar anisotropic materials

The J-integral analysis is presented for the interaction problem between a macro-interface crack and subinterface microcracks parallel to the former in the near-tip process zone in dissimilar anisotropic composite materials. Elementary solutions respectively considering an interface crack and a subinterface crack subjected to different loads are given from which the interaction problem is deduced to a system of integral equations with the aid of superimposing technique (i.e., the so called `Pseudo-Traction Method' abbreviated as PTM). After the integral equations are solved numerically, a consistent relation among three kinds of the J-integral values is obtained. They are induced from the macro-interface crack tip, the microcracks, and the remote field, respectively. This consistent relation of the J-integral can be used to confirm the numerical results derived by using whatever kind of technique. With the aid of J-integral analysis, the interaction behaviors between an interface crack and parallel subinterface cracks are investigated in detail, and some special physical phenomena are obtained.     

Irreversible growth of a spherical cavity in rubber-like material: A fracture mechanics description

A procedure is presented for determining stress intensity factors for single and double edge cracks in simply supported undamped Bernoulli–Euler beams under a moving load. The approach is based on using modal analysis to determine the equivalent load on the beam, then linear elastic fracture mechanics is used to calculate stress intensity factors (SIF). The results show that SIF is a function of time, speed of the moving load and crack size and location.     

Determination of the stress intensity factor for parallel and symmetric edge cracks under pure bending

An experimental and semi-theoretical investigation concerning thin plates having “Parallel and Symmetric” (P.A.S.)² edge cracks and subjected to pure bending will be discussed in this paper. Experiments were performed extensively on plates with P.A.S. equal edge cracks in order to study the effect of the variation of the length of the cracks and crack spacing seperately.Observations show that when crack spacing becomes larger than crack length, the diameter of the caustic for the equal P.A.S. edge cracks approaches the diameter of the caustic of the double edge cracks. Based on this fact, the equation that expresses the semi-theoretical and experimental stress intensity factor for the double edge cracks was first modified and then utilized in order to express the strese intensity factor of P.A.S. edge cracks. Moreover, an alternative formula for the stress intensity factor was established which is based on a cubic interpolating polynomial. For large crack spacing the cubic interpolating polynomial converges to unity and as a result, the diameter of the caustic and the stress intensity factor approach those of a double edge crack.For the case of unequal P.A.S. edge cracks, a crack closure phenonmenon under the pure bending condition was investigated. Experimental measurements showed that the relative position of the shorter crack with respect to the longer cracks for which the crack closure phenomenon happens is related not only to the crack spacing of the parallel edge cracks in the tension region, but also strongly related to the variation of the existing symmetric edge crack in the compression region.     

各向异性板应力强度因子的变分解法

本文在文献的基础上给出了含边缘裂纹各向异性板的应力与位移展开式, 并应用广义变分原理求解含对称中心裂纹、对称边缘裂纹与对称孔边裂纹正交各向异性板的应力强度因子。所得结果的收敛性是令人满意的。同时在各向同性的情况下, 所得结果与文献之结果非常符合。      

A theory for the fracture of thin plates subjected to bending and twisting moments

Stress fields near the tip of a through crack in an elastic plate under bending and twisting moments are reviewed assuming both Kirchhoff and Reissner plate theories. The crack tip displacement and rotation fields based on the Reissner theory are calculated here for the first time. These results are used to calculate the J-integral (energy release rate) for both Kirchhoff and Reissner plate theories. Invoking Simmonds and Duva's [16] result that the value of the J-integral based on either theory is the same for thin plates, a universal relationship between the Kirchhoff theory stress intensity factors and the Reissner theory stress intensity factors is obtained for thin plates. Calculation of Kirchhoff theory stress intensity factors from finite elements based on energy release rate is illustrated. A small scale yielding like model of the crack tip fields is discussed, where the Kirchhoff theory fields are considered to be the far field conditions for the Reissner theory fields. It is proposed that, for thin plates, fracture toughness and crack growth rates be correlated with the Kirchhoff theory stress intensity factors.     

Determination of modes I and II stress intensity factors for oblique single edge cracks subjected to tension

A semi-theoretical and experimental approach is employed herein to investigate plates with 90, 75, 60, 45 and 30° oblique single edge cracks subjected to tensile loading. A series of tension tests were performed and the effect of the variation of the length of the cracks on the stress intensity factors were studied. Experimental measurements show that the mode one and two stress intensity factors are related mainly to the size of the caustic (optical singularity measured on a screen at a distance Z₀ from the screen), the length and the angle of the crack, and the width of the specimen. The calculated values of the semi-theoretical stress intensity factors were corroborated with the theoretical results; and then the method was further utilized to investigate more complicated oblique edge cracks subjected to tension. In particular, the stress intensity factors of plates with oblique 45° equal parallel edge cracks were studied by this method and the results were checked with the stress intensity factors of single oblique edge cracks.     

The effect of material combinations and relative crack size to the stress intensity factors at the crack tip of a bi-material bonded strip

Several types of singular stress fields may appear at the corner where an interface between two bonded materials intersects a traction-free edge depending on the material combinations. Since the failure of the multi-layer systems often originates at the free-edge corner, the analysis of the edge interface crack is the most fundamental to simulate crack extension. In this study, the stress intensity factors for an edge interfacial crack in a bi-material bonded strip subjected to longitudinal tensile stress are evaluated for various combinations of materials using the finite element method. Then, the stress intensity factors are calculated systematically with varying the relative crack sizes from shallow to very deep cracks. Finally, the variations of stress intensity factors of a bi-material bonded strip are discussed with varying the relative crack size and material combinations. This investigation may contribute to a better understanding of the initiation and propagation of the interfacial cracks.