Asymmetrical Cracks Parallel to an Interface between Dissimilar Materials

This paper solves a plane strain problem for two bonded dissimilar planes containing a crack parallel to the interface in each layer. The bimaterial system is loaded by tractions distributed along the crack surfaces. Based on the Fourier transform, the problem is reduced to a system of Cauchy type singular integral equations which contain exact and explicit kernel functions. The solution of these equations is obtained easily by utilizing Gauss–Chebyshev integral formulae for various material combinations and geometrical parameters. Several numerical results of stress intensity factors, energy release rate and stress distribution along the interface are presented to exhibit the interaction among cracks and interface. This revised version was published online in July 2006 with corrections to the Cover Date.     

Weight functions for bimaterial interface cracks

An efficient approach using the analytically decoupled near-tip displacement solution for bimaterial interface cracks presented in this paper involves: (1) the calculation of the decoupled strain energy release rates G _I and G _II associated respectively with the decoupled stress intensity factors K _I and K _II and (2) the extension of Rice's displacement derivative representation of Bueckner's weight function vectors beyond the homogeneous media. It is shown that the stress intensity factors for a bimaterial interface crack predicted by the present approach agree very well with those solutions available in the literature. The computational efficiency is enhanced through the use of singular elements in the crack-tip neighborhood.As reported in the homogeneous case, the calculated weight function for a bimaterial interface crack is load-independent but depends strongly on geometry and constraint conditions. Due to the coupling nature of the stress intensity factors of a bimaterial interface crack, the invariant characteristics of the dimensionless weight function vectors are different from those of a crack in homogeneous material. In addition, the elastic constants of two constituents can significantly alter the weight function behavior for a cracked bimaterial medium.Due to the load-independent characteristic of the weight functions, the stress intensity factors for a bimaterial interface crack can be obtained accurately and inexpensively by performing the sum of worklike products between the applied loads and the weight functions for the cracked bimaterial body under any loading conditions once the weight functions are explicitly predetermined. The same calculation can also be applied for the identical cracked bimaterial medium with different constraint conditions by including the self-equilibrium forces that contain both the external loads and the reaction forces induced at the constraint locations. Moreover, the physical interpretation of the weight functions can provide a guidance for damage tolerant design application.     

Modeling bimaterial interface cracks using the numerical manifold method

The numerical manifold method (NMM) is explored for simulations of bimaterial interface cracks. Two special types of physical covers with customized cover functions are introduced to describe the weak discontinuity across the material interface, the partially cracked elements as well as the interface crack tip singularity. Three typical bimaterial crack problems are simulated. The mixed-mode stress intensity factors are evaluated by the virtue of the domain form of the interaction integral and compared with the available reference solutions. Good agreements have demonstrated the validity and accuracy of the developed program.     

Dual boundary element analysis of cracked plates: singularity subtraction technique

The present paper is concerned with the formulation of the singularity subtraction technique in the dual boundary element analysis of the mixed-mode deformation of general homogeneous cracked plates.The equations of the dual boundary element method are the displacement and the traction boundary integral equations. When the displacement equation is applied on one of the crack surfaces and the traction equation is applied on the other, general mixed-mode crack problems can be solved in a single region boundary element formulation, with both crack surfaces discretized with discontinuous quadratic boundary elements.The singularity subtraction technique is a regularization procedure that uses a singular particular solution of the crack problem to introduce the stress intensity factors as additional problem unknowns. The single-region boundary element analysis of a general crack problem restricts the availability of singular particular solutions, valid in the global domain of the problem. A modelling strategy, that considers an automatic partition of the problem domain in near-tip and far-tip field regions, is proposed to overcome this difficulty. After the application of the singularity subtraction technique in the near-tip field regions, regularized locally with the singular term of the Williams' eigenexpansion, continuity is restored with equilibrium and compatibility conditions imposed along the interface boundaries. The accuracy and efficiency of the singularity subtraction technique make this formulation ideal for the study of crack growth problems under mixed-mode conditions.     

Boundary integral equation formulation for Interface cracks in anisotropic materials

We present a boundary integral formulation for anisotropic interface crack problems based on an exact Green's function. The fundamental displacement and traction solutions needed for the boundary integral equations are obtained from the Green's function. The traction-free boundary conditions on the crack faces are satisfied exactly with the Green's function so no discretization of the crack surfaces is necessary. The analytic forms of the interface crack displacement and stress fields are contained in the exact Green's function thereby offering advantage over modeling strategies for the crack. The Green's function contains both the inverse square root and oscillatory singularities associated with the elastic, anisotropic interface crack problem. The integral equations for a boundary element analysis are presented and an example problem given for interface cracking in a copper-nickel bimaterial.     

An interaction integral method for computing mixed mode stress intensity factors for curved bimaterial interface cracks in non-uniform temperature fields

In finite element analysis the interaction integral has been a useful tool for computing the stress intensity factors for fracture analysis. This work extends the interaction integral to account for non-uniform temperatures in the calculation of stress intensity factors for three dimensional curvilinear cracks either in a homogeneous body or on a bimaterial interface. First, the derivation of the computational algorithm, which includes the additional terms developed by the non-zero gradient of the temperature field, is presented in detail. The algorithm is then implemented in conjunction with commercial finite element software to calculate the stress intensity factors of a crack undergoing non-uniform temperatures on both a homogeneous and a bimaterial interface. The numerical results displayed path independence and showed excellent agreement with available analytical solutions.     

Evaluations of the T-stress for interface cracks by the boundary element method

The path independent M-integral is applied to computation of the T-stress for interface cracks between dissimilar materials. The unique relation between the M-integral and T-stress is found for a properly selected auxiliary solution. The problem of a semi-infinite interface crack between dissimilar materials loaded by a point force applied to the crack tip in a direction parallel to the interface suits such an auxiliary solution. A new subregion boundary element method is applied to solve the given bimaterial interface crack problem. Numerical results for centre-cracked plate, single-edged notch and double-edged notch specimens are included.     

Delamination analysis of composites by new orthotropic bimaterial extended finite element method

The extended finite element method (XFEM) is further improved for fracture analysis of composite laminates containing interlaminar delaminations. New set of bimaterial orthotropic enrichment functions are developed and utilized in XFEM analysis of linear‐elastic fracture mechanics of layered composites. Interlaminar crack‐tip enrichment functions are derived from analytical asymptotic displacement fields around a traction‐free interfacial crack. Also, heaviside and weak discontinuity enrichment functions are utilized in modeling discontinuous fields across interface cracks and bimaterial weak discontinuities, respectively. In this procedure, elements containing a crack‐tip or strong/weak discontinuities are not required to conform to those geometries. In addition, the same mesh can be used to analyze different interlaminar cracks or delamination propagation. The domain interaction integral approach is also adopted in order to numerically evaluate the mixed‐mode stress intensity factors. A number of benchmark tests are simulated to assess the performance of the proposed approach and the results are compared with available reference results. Copyright © 2011 John Wiley & Sons, Ltd.     

Stress intensity factors and energy release rates of delaminations in composite laminates

Due to the oscillatory characteristics of stresses near interface crack tips, the stress intensity factor K_i, i = I, II, III, should be modified and the energy release rate G_i, i = 1, 2, 3, of each fracture mode calculated by the virtual crack closure method may not exist. Based upon a near-tip solution for interface cracks between dissimilar anisotropic media, a proper definition for the stress intensity factors and energy release rates for general anisotropic bimaterial interface cracks is provided in this paper, which is applicable for the delaminated composites. Moreover, this definition can be reduced to the classical definition for a crack tip in homogeneous media when the two materials become the same. A simple quadratic relation between K_i and _Gi is derived, which is further reduced explicitly for orthotropic bimaterials. The influence of fiber orientation and the coupling among opening, shearing and tearing mode fracture are studied numerically. The results show that the classical stress intensity factors and energy release rates are still the dominant stress intensity and energy release rate of the mixed mode condition induced by the interface.     

Comparison of several BEM-based approaches in evaluating crack-tip field intensity factors in piezoelectric materials

From the viewpoint of fracture mechanics, of importance is the near-tip field which can be characterized as field intensity factors. In this paper, the crack-tip field intensity factors of the stress and electric displacement in two dimensional piezoelectric solids are evaluated by using four approaches including the displacement extrapolation, the stress method, the J-integral and the modified crack closure integral method (MCCI) based on a boundary element method (BEM). The strongly singular displacement boundary integral equations (BIEs) are applied on the external boundary of the cracked solid, while the hypersingular traction BIEs are used on the crack faces. Three numerical examples are presented to show the path independence and the high accuracy of the J-integral in piezoelectric materials and to analyze the pros and cons of these approaches in evaluating the field intensity factors.     

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.     

Performance of variable order singular elements in analyzing interfacial fractures

This work concerns the development of singular boundary elements and the investigation of their numerical performance in analyzing interfacial cracks. In the vicinity of such cracks arise singular stress fields with variable order of singularity depending on the material characterizing parameters. The development of these elements which approximate displacement and traction functions is accomplished through controlled relocation of the mid-side node determined by compatibility and continuity requirements which must obey shape functions. These elements were applied to simulate the elastic behavior of cracks which are perpendicular and terminate on the interface of a bimaterial structure. Their efficiency in conjunction to the boundary only element method, are demonstrated in crack opening displacement diagrams and crack tip stress tabulated results.     

Partition of unity enrichment for bimaterial interface cracks

Partition of unity enrichment techniques are developed for bimaterial interface cracks. A discontinuous function and the two‐dimensional near‐tip asymptotic displacement functions are added to the finite element approximation using the framework of partition of unity. This enables the domain to be modelled by finite elements without explicitly meshing the crack surfaces. The crack‐tip enrichment functions are chosen as those that span the asymptotic displacement fields for an interfacial crack. The concept of partition of unity facilitates the incorporation of the oscillatory nature of the singularity within a conforming finite element approximation. The mixed‐mode (complex) stress intensity factors for bimaterial interfacial cracks are numerically evaluated using the domain form of the interaction integral. Good agreement between the numerical results and the reference solutions for benchmark interfacial crack problems is realized. Copyright © 2004 John Wiley & Sons, Ltd.     

Axisymmetric boundary integral equation analysis of interface cracks between dissimilar materials

The numerical boundary integral equation (BIE) method with quadratic quarter-point crack-tip singular elements is used to analyse interface cracks between dissimilar material in axisymmetry. Such crack problems present modelling difficulties using conventional procedures for obtaining the stress intensity factors. This is because of the oscillatorily singular nature of the stresses in the vicinity of the bimaterial interface crack-tip. Analytical expressions for the direct evaluation of the fracture characterising parameters from the BIE numerical results of displacements or tractions are derived. Three different crack problems are investigated, two of which have known solutions in the literature. Excellent agreement between the BIE results and these other established solutions are obtained even with relatively coarse mesh discretisations. The present study illustrates the ease with which the BIE method may be used in the fracture analysis of both straight and curved binaterial interface cracks.     

A complex variable boundary element method for solving interface crack problems

The problem of finite bimaterial plates with an edge crack along the interface is studied. A complex variable boundary element method is presented and applied to determine the stress intensity factor for finite bimaterial plates. Using the pseudo-orthogonal characteristic of the eigenfunction expansion forms and the well-known Bueckner work conjugate integral and taking the different complex potentials as auxiliary fields, the interfacial stress intensity factors associated with the physical stress-displacement fields are evaluated. The effects of material properties and crack geometry on stress intensity factors are investigated. The numerical examples for three typical specimens with six different combinations of the bimaterial are given. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analyzing the interaction between collinear interfacial cracks by an efficient boundary element-free method

A new traction boundary integral equation is presented for analyzing the interaction effect of any number of collinear interface cracks in a two-dimensional bimaterial. The dislocation densities on every crack surface are expressed in the products of the characteristic terms and the weight functions, and the unknown weight functions are approximated using the moving least-squares technique based on the constructed orthogonal basis functions. An efficient numerical integral method is employed to evaluate the Cauchy principal integrals that appear in the meshless method. The boundary element-free method is established, and a series of numerical results is presented. The interaction between the collinear interfacial cracks is analyzed.     

Virtual crack closure technique for an interface crack between two transversely isotropic materials

The virtual crack closure technique makes use of the forces ahead of the crack tip and the displacement jumps on the crack faces directly behind the crack tip to obtain the energy release rates \({{\mathcal {G}}}_I\) and \({\mathcal {G}}_{II}\). The method was initially developed for cracks in linear elastic, homogeneous and isotropic material and for four noded elements. The method was extended to eight noded and quarter-point elements, as well as bimaterial cracks. For bimaterial cracks, it was shown that \({\mathcal {G}}_I\) and \({\mathcal {G}}_{II}\) depend upon the virtual crack extension \(\varDelta a\). Recently, equations were redeveloped for a crack along an interface between two dissimilar linear elastic, homogeneous and isotropic materials. The stress intensity factors were shown to be independent of \(\varDelta a\). For a better approximation of the Irwin crack closure integral, use of many small elements as part of the virtual crack extension was suggested. In this investigation, the equations for an interface crack between two dissimilar linear elastic, homogeneous and transversely isotropic materials are derived. Auxiliary parameters are used to prescribe an optimal number of elements to be included in the virtual crack extension. In addition, in previous papers, use of elements smaller than the interpenetration zone were rejected. In this study, it is shown that these elements may, indeed, be used.     

Fracture mechanics analysis of cracked 2-D anisotropic media with a new formulation of the boundary element method

A new formulation of the boundary element method (BEM) is proposed in this paper to calculate stress intensity factors for cracked 2-D anisotropic materials. The most outstanding feature of this new approach is that the displacement and traction integral equations are collocated on the outside boundary of the problem (no-crack boundary) only and on one side of the crack surfaces only, respectively. Since the new BEM formulation uses displacements or tractions as unknowns on the outside boundary and displacement differences as unknowns on the crack surfaces, the formulation combines the best attributes of the traditional displacement BEM as well as the displacement discontinuity method (DDM). Compared with the recently proposed dual BEM, the present approach doesn't require dua elements and nodes on the crack surfaces, and further, it can be used for anisotropic media with cracks of any geometric shapes. Numerical examples of calculation of stress intensity factors were conducted, and excellent agreement with previously published results was obtained. The authors believe that the new BEM formulation presented in this paper will provide an alternative and yet efficient numerical technique for the study of cracked 2-D anisotropic media, and for the simulation of quasi-static crack propagation.     

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.     

Driving traction on phase boundary interfaces in solids with boundary elements

A method of computing driving tractions for phase transformations is presented. The driving traction calculation requires evaluating the jump in the displacement gradient and the average stress acting across the interface. A bimaterial Green's function is used in a boundary element formulation for the calculation. The Green's function satisfies all interface boundary conditions for the bimaterial providing an efficient formulation for evaluating the necessary jump and average interface terms. We derive a boundary integral equation for calculating the driving traction and apply the developed numerical tool to driving traction calculations in copper-aluminum-nickel.