Finite element study of a penny-shaped crack along the interface in a bi-material cylinder

A contemporary approach to the analysis of interface cracks in bi-material cylinders using finite elements is presented. From results obtained with a commercial finite element code using regular and singular isoparametric elements, three fracture mechanics techniques are considered to study the interface crack problem and are presented in a fundamental manner. These are the stress intensity factor evaluation by the crack opening displacement method, the strain energy release rate evaluation using the modified crack closure integral method, and the J-integral evaluation using the virtual crack extension technique. Only the finite element results in the vicinity of the crack are then needed. The accuracy of the proposed approach is assessed by solving standard test problems with known solutions. In particular, the mode I problem of a penny-shaped crack in a homogeneous isotropic cylinder under remote tension loading is used as a standard test case. Finally, the mixed-mode (I and II) problem of a penny-shaped crack along the interface in a bi-material cylinder under three loading conditions is studied in detail. Numerical results are presented to quantify the combined effects of geometry and material discontinuities on the strain energy release rate.     

Fracture characterization of Carbon fiber-reinforced polymer-concrete bonded interfaces under four-point bending

A combined analytical and experimental approach is presented to characterize both mode-II and mixed mode fracture of Carbon fiber-reinforced polymer-concrete bonded interfaces under four-point bending load, and closed-form solutions of compliance and energy release rate of the mode-II (four-point symmetric end-notched flexure) and mixed (four-point asymmetric end-notched flexure) mode fracture specimens are provided. The transverse shear deformation in each sub-layer of bi-material bonded beams is included by modeling each sub-layer as an individual first order shear deformable beam, and the effect of interface crack tip deformation on the compliance and energy release rate are taken into account by applying the interface deformable bi-layer beam theory (i.e., the flexible joint model). The improved accuracy of the present analytical solutions for both the compliance and energy release rate is illustrated by comparing with the solutions predicted by the conventional rigid joint model and finite element analysis. The fracture of Carbon fiber-reinforced polymer-concrete bonded interface is experimentally evaluated using both the four-point symmetric and asymmetric end-notched flexure specimens, and the corresponding values of critical energy release rates are obtained. Comparisons of the compliance rate-changes and resulting critical energy release rates based on the rigid joint model, the present theoretical model, and numerical finite element analysis demonstrate that the crack tip deformation plays an important role in accurately characterizing the mixed mode fracture toughness of hybrid material bonded interfaces under four-point bending load. The improved solution of energy release rates for the four-point symmetric and asymmetric end-notched flexure specimens by the flexible joint model can be used to effectively characterize hybrid material interface, and the fracture toughness values obtained for the Carbon fiber-reinforced polymer-concrete interface under mode-II and mixed mode loading can be employed to predict the interface fracture load of concrete structures strengthened with composites.     

Parametric quadratic programming method for elastic contact fracture analysis

A solution procedure for elastic contact fracture mechanics has been proposed in this paper. The procedure is based on the quadratic programming and finite element method (FEM). In this paper, parametric quadratic programming method for two-dimensional contact mechanics analysis is applied to the crack problems involving the crack surfaces in frictional contact. Based on a linear complementary contact condition, the parametric variational principle and FEM, a linear complementary method is extended to analyze contact fracture mechanics. The near-tip fields are properly modeled in the analysis using special crack tip elements with quarter-point nodes. Stress intensity factor solutions are presented for some frictional contact fracture problems and are compared with known results where available.     

On the three-dimensional problem of an interface crack under uniform heat flow in a bimaterial periodically-layered space

This paper deals with the study of thermal stresses due to a plane crack lying on an interface in a microperiodic two-layered composite under a uniform perpendicular heat flow. An approximate analysis is carried out within the framework of linear stationary thermoelasticity with microlocal parameters. Taking this approach, a general method of solving the resulting boundary-value problem is presented. Effective results have been achieved through the use of suitable potential solutions and reducing the thermal crack problem to its skew-symmetric isothermal mechanical counterpart. An illustrative example is given in the paper by considering an insulated stress-free penny-shaped crack. For this problem the solution is obtained in terms of elementary functions and discussed from the point of view of fracture theory.     

Analysis of stress intensity factors for three-dimensional interface crack problems in electronic packages using the virtual crack closure technique

In this study the fracture mechanics parameters, including the strain energy release rate, the stress intensity factors and phase angles, along the curvilinear front of a three-dimensional bimaterial interface crack in electronic packages are considered by using finite element method with the virtual crack closure technique (VCCT). In the numerical procedure normalized complex stress intensity factors and the corresponding phase angles (Rice, J Appl Mech 55:98–103, 1988) are calculated from the crack closure integrals for an opening interface crack tip. Alternative procedures are also described for the cases of crack under inner pressure and crack faces under large-scale contact. Validation for the procedure is performed by comparing numerical results to analytical solutions for the problems of interface crack subjected to either remote tension or mixed loading. The numerical approach is then applied to study interface crack problems in electronic packages. Solutions for semi-circular surface crack and quarter-circular corner crack on the interface of epoxy molding compound and silicon die under uniform temperature excursion are presented. In addition, embedded corner delaminations on the interface of silicon die and underfill in flip-chip package under thermomechanical load are investigated. Based on the distribution of the fracture mechanics parameters along the interface crack front, qualitative predictions on the propensity of interface crack propagation under thermomechanical loads are given.     

Three-dimensional fracture analysis in transversely isotropic solids

In this paper a boundary element formulation for three-dimensional crack problems in transversely isotropic bodies is presented. Quarter-point and singular quarter-point elements are implemented in a quadratic isoparametric element context. The point load fundamental solution for transversely isotropic media is implemented. Numerical solutions to several three-dimensional crack problems are obtained. The accuracy and robustness of the present approach for the analysis of fracture mechanics problems in transversely isotropic bodies are shown by comparison of some of the results obtained with existing analytical solutions. The approach is shown to be a simple and useful tool for the evaluation of stress intensity factors in transversely isotropic media.     

A mixed-mode crack analysis of rectilinear anisotropic solids

A simple numerical method is presented for analysing the mixed mode of rectilinear anisotropic solids. The method is formulated on the basis of a finite element and the crack closure integral approach in conjunction with fundamental relationships in fracture mechanics. A simple and efficient solution procedure is developed involving only the known auxiliary solution for evaluating the strain energy release rate. The finite element solution converges to an accurate solution for small crack extensions. Numerical examples are presented to demonstrate the accuracy of the proposed approach.     

Three-dimensional numerical investigation of brittle bond fracture

A fracture mechanics based analysis of interface bond failure is presented. The bond edge is regarded as an interface crack front loaded under combined mode 1, 2 and 3 loading, and results are obtained for the critical stress for initiation of bond failure and the location along the bond edge where failure is initiated. A numerical procedure is formulated to study the propagation of the interface crack following initiation. Assuming that the crack propagates at the interface, a criterion for propagation is formulated, and it is shown that the crack front shape predicted is consistent with the basic interface fracture mechanics assuming quasi-static crack propagation. Results for the bond strength are presented for different fracture criteria and different bond shapes.     

Analysis of beam-type fracture specimens with crack-tip deformation

A novel bi-layer beam model is developed to account for local effects at the crack tip of a bimaterial interface by modeling a bi-layer composite beam as two separate shear deformable beams. The effect of interface stresses on the deformations of sub-layers, which is referred to as the elastic foundation effect in the literature, is considered in this model by introducing two interface compliance coefficients; thus a flexible joint condition at the crack tip is considered in contrast to the rigid joint condition used in the conventional bi-layer model. An elastic crack tip deformable model is presented, and the closed-form solutions of local deformation at the crack tip are then obtained. By applying this novel crack tip deformation model, the new terms due to the local deformations at the crack tip, which are missing in the conventional composite beam solutions of compliance and energy release rate (ERR) of beam-type fracture specimens, are recovered. Several commonly used beam-type fracture specimens are examined under the new light of the present model, and the improved solutions for ERR and mode mixity are thus obtained. A remarkable agreement achieved between the present and available solutions illustrates the validity of the present study. The significance of local deformation at the crack tip is demonstrated, and the improved solutions developed in this study provide highly accurate predictions of fracture properties which can actually substitute the full continuum elasticity analysis such as the finite element analysis. The new and improved formulas derived for several specimens provide better prediction of ERR and mode mixity of beam-type fracture experiments.*Author for correspondence (E-mail address: qiao@uakron.edu)     

MODELLING OF 3-D MIXED-MODE FRACTURES NEAR PLANAR BIMATERIAL INTERFACES USING SURFACE INTEGRALS

Mixed-mode fractures of arbitrary orientation with respect to a planar bimaterial interface have been effectively modelled using a surface integral approach. By requiring only that the surface of the fracture be discretized, the surface integral method circumvents the practical difficulties associated with having to mesh the interacting dual singularities in stress along the three-dimensional (3-D) crack front and at the interface. The key elements of this numerical capability are discussed in detail. These include: the derivation of the fundamental solutions for a generalized fracture event near a planar bimaterial interface, formulation of the governing integral equation including its decomposition into singular and non-singular terms, development of analytical and numerical techniques for performing the singular integrations, and efficient numerical integration of the non-singular terms using non-dimensionalized surface approximations of the dipole solutions. The problem of a pressurized planar crack near a bimaterial interface was used to assess convergence. The effect of material contrast and crack shape on tendencies for crack growth were also examined.     

Film cracking and debonding in a coated fiber

A fracture mechanics based methodology for the determination of interface fracture toughness from crack spacing in a thin coated fiber is presented. The coating (film) may be regarded as the matrix material in typical experiments employing this configuration. Matrix crack spacing is considered to be the result of a competitive process between matrix segmentation and interface debonding which are assumed to be governed by critical energy release rate criteria. Matrix cracks are assumed to form by the process of channeling in the circumferential direction and steady state conditions are assumed at the matrix crack front in the channeling direction. Energy release rates are determined using domain integral procedures in conjunction with the finite element method. The minimum crack spacing is obtained as a function of applied stress for different values of interface fracture toughness. A methodology to relate the saturated crack spacing to interface fracture toughness is developed. Interfaces are classified into three categories: weak, intermediate and strong. It is shown that in experiments of this type, quantitative information about the interface fracture toughness can be obtained for intermediate interfaces while qualitative information may be obtained for weak and strong interfaces.     

The use of interlayers in modeling interface crack propagation

Delaminations are a common mode of failure at interfaces between two material layers which have dissimilar elastic constants. There is a well-known oscillatory nature to the singularity in the stress fields at the crack tips in these bimaterial delaminations, which creates a lack of convergence in the modewise energy release rates. This makes constructing fracture criteria somewhat difficult. An approach used to overcome this is to artificially insert a thin, homogeneous, isotropic layer (the interlayer) at the interface. The crack is positioned in the middle of this homogeneous interlayer, thus modifying the original ‘bare’ interface crack problem into a companion ‘interlayer’ crack problem. Individual modes I and II energy release rates are convergent and calculable for the companion problem and can be used in the construction of a fracture criterion or locus. However, the choices of interlayer elastic and geometric properties are not obvious. Moreover, a sound, consistent, and comprehensive methodology does not exist for utilizing interlayers in the construction and application of mixed-mode fracture criteria in interface fracture mechanics. These issues are addressed here. The role of interlayer elastic modulus and thickness is examined in the context of a standard interface fracture test specimen. With the help of a previously published analytical relation that relates the bare interface crack stress intensity factor to the corresponding interlayer crack stress intensity factor, a suitable thickness and elastic modulus are identified for the interlayer in a bimaterial four-point bend test specimen geometry. Interlayer properties are chosen to make the interlayer fracture problem equivalent to the bare interface fracture problem. A suitable mixed-mode phase angle and a form for the fracture criterion for interlayer-based interface fracture are defined. A scheme is outlined for the use of interlayers for predicting interface fracture in bimaterial systems such as laminated composites. Finally, a simple procedure is presented for converting existing bare interface crack fracture loci/criteria into corresponding interlayer crack fracture loci.     

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.     

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.     

The stress intensity factors of slightly undulating interface cracks of anisotropic bimaterials

Summary An interface crack in anisotropic dissimilar materials is treated here as a slightly undulating one. Based on Lekhnitskii-Eshelby-Stroh (LES) representation and perturbation analysis, analytic solutions are given for displacement and stress fields of both anisotropic half-planes. Many failures, e.g., occurring in the interfaces between thin-film and substrate or fiber and matrix, are always slightly undulating. For these reasons, the unevenness effect of crack faces must be taken into account to determine the practical stress intensity factors for predicting the interface fracture behaviours. And the solutions to the first order of unevenness are determined by considering only the lateral stresses, because other far-field stresses have dominated the zeroth order solution.     

COMPARISON OF ANALYTICAL, NUMERICAL AND EXPERIMENTAL SOLUTIONS TO PROBLEMS OF DEEPLY CRACKED WELDED JOINTS IN BENDING

This paper presents analytical and numerical solutions to the analysis of welded specimens when loaded in three-point bending, and compares the results with those obtained experimentally. In each case the crack is located within the weld material, and runs parallel to the weld. Two analytical models are presented for deriving limit loads using slip-line field theory. Due to the welding process, the material behaviour in the weld-base interface is complex, and this is described in the analytical solutions using a number of material zones. The analytical solutions also provide the η and d_∞ coefficients which are used to determine the J and crack tip opening displacement (CTOD) parameters. These solutions are then compared with numerical results obtained using the finite element method. Good agreement is obtained between the numerical and analytical results, and it is shown that for overmatched specimens a better analytical solution can be obtained by using a slip-line field geometry which passes predominantly through the base material. When the analytical η solutions are applied to the experimental results it is shown that, in the case of undermatched weldments, J can be used as a fracture characterising parameter, but it cannot be used in the case of an overmatched specimen.     

Dynamic crack growth along an elastoplastic bimaterial interface

Summary The near-tip stress and deformation rate fields of a crack dynamically propagating along an interface between dissimilar elastic-plastic bimaterials are presented in this paper. The elastic-plastic materials are characterised by theJ ₂-flow theory with linear plastic hardening. The solutions are assumed to be of variable-separable form with a power-law singularity in the radial direction. Two distinct solutions corresponding to the tensile and shear solutions exist with slightly different singularity strengths and very different mixities at the crack tip. The phenomenon of discrete and determinate mixities at the interfacial crack tip is confirmed in dynamic crack growth. This is not an artifact of the variable-separable solution assumption, arising from the linear-hardening material model. The dynamic crack analysis shows that the mixity of the near-tip field is mainly determined by the given material parameters and affected slightly by the crack propagation velocity. A significant variation of the mixity is observed near to the coalescing point of the tensile and shear solutions. The strength of the singularity is almost determined by the smaller strain-hardening alone, and dynamic inertia decreases the stress intensity. The asymptotic solutions reveal that the crack propagation velocity changes only the stress field of the tensile mode significantly. With increasing the crack propagation velocity, the stress singularity of the tensile solutions decreases obviously and the stress triaxiality at the tip (=0) falls considerably at the unity effective stress. These observations imply that the fracture toughness of the interface crack under tensile mode may be significantly higher than that under quasi-static conditions.     

Stress intensity factors as the fracture parameters for delamination crack growth in composite laminates

A “mutual integral” approach is used to calculate the mixed-mode stress intensity factors for a free-edge delamination crack in a laminate under tensile loading conditions. This “mutual integral” approach, for generalized plane strain conditions, is based on the application of the path-independent J integral to a linear combination of three solutions: one, the problem of the laminate to be solved using the quasi 3-D finite element method, the second, an “auxiliary” solution with a known asymptotic singular solution, and the third, the particular solution due to the out-of-plane loading. A comparison with the exact solutions is made to determine the accuracy and efficiency of this numerical method. With this “mutual integral” approach, it was found that the calculated mixed-mode stress intensity factors of the free-edge delamination crack remain relatively constant as the crack propagates into the laminate. It was also found that the fracture criterion based on the mixed-mode stress intensity factors is more consistent with the experimental observations than the criterion based on the total energy release rate, and hence demonstrates the importance of the ability to calculate each individual component of the stress intensity factors. Furthermore, it was found that the fracture toughness measurements from double cantilever beam specimens can be used directly to predict the onset of delamination crack growth between two dissimilar laminae. Using these fracture toughness measurements from the double cantilever beam specimens, some examples are given to show that the fracture criterion based on the mixed-mode stress intensity factors can accurately predict the failure load for various laminates under tensile loading conditions.     

A new method for evaluation of stress intensities for interface cracks

A new method is presented for calculating the values of K_I and K_II in the elasticity solution at the tip of an interface crack. The method is based on an evaluation of the J-integral by the virtual crack extension method. Expressions for calculating K_I and K_II by using the displacements and the stiffness derivative of the finite element solution and asymptotic crack tip displacements are derived. The method is shown to produce very accurate solutions even with coarse element mesh.     

Asymptotic stress intensity factor density profiles for smeared-tip method for cohesive fracture

The paper presents a computational approach and numerical data which facilitate the use of the smeared-tip method for cohesive fracture in large enough structures. In the recently developed K-version of the smeared tip method, the large-size asymptotic profile of the stress intensity factor density along a cohesive crack is considered as a material characteristic, which is uniquely related to the softening stress-displacement law of the cohesive crack. After reviewing the K-version, an accurate and efficient numerical algorithm for the computation of this asymptotic profile is presented. The algorithm is based on solving a singular Abel's integral equation. The profiles corresponding to various typical softening stress-displacement laws of the cohesive crack model are computed, tabulated and plotted. The profiles for a certain range of other typical softening laws can be approximately obtained by interpolation from the tables. Knowing the profile, one can obtain with the smeared-tip method an analytical expression for the large-size solution to fracture problems, including the first two asymptotic terms of the size effect law. Consequently, numerical solutions of the integral equations of the cohesive crack model as well as finite element simulations of the cohesive crack are made superfluous. However, when the fracture process zone is attached to a notch or to the body surface and the cohesive zone ends with a stress jump, the solution is expected to be accurate only for large-enough structures.