Interface crack between two interface deformable piezoelectric layers

Based on an interface deformable piezoelectric bi-layer beam model, a bonded piezoelectric bi-material beam with an interface crack perpendicular to the poling axis is analyzed within the framework of the theory of linear piezoelectricity. The layer-wise approximations of both the elastic displacements and electric potential are employed, and each sub-layer is modeled as a single linearly elastic Timoshenko beam perfectly bonded together through a deformable interface. Using the impermeable crack assumption, the closed form solutions for the energy release rate (ERR) and crack energy density (CED) are derived for the layered piezoelectric beam subjected to combined uniformly distributed electromechanical loading. Based on superposition principle, both the ERR and CED and their components are all reduced to the functions of the crack tip loading parameters. Loading dependence of the total CED with respect to the applied electric field is manifested with the analytical results, showing that there is a transformation from an even dependence to an odd dependence for the normalized CED when the applied mechanical loading increases. Compared with the commonly used equivalent single layer model, the proposed analysis augments the crack driving force by alleviating the stress concentration along the interface and thus increases the loading parameters at the crack tip. The proposed model provides improved solutions for fracture analysis of piezoelectric layered structures and sheds light on the loading dependence of the fracture parameters (i.e., the ERR and CED) with respect to the applied electromechanical loadings.     

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.     

Measurement of mixed-mode fracture parameters near cracks in homogeneous and bimaterial beams

An investigation of deformation fields and evaluation of fracture parameters near mixed-mode cracks in homogeneous and bimaterial specimens under elastostatic conditions is undertaken. A modified edge notched flexural geometry is proposed for testing bimaterial interface fracture toughness. The ability of the specimen in providing a fairly wide range of mode mixities is demonstrated through direct optical measurements and a simple flexural analysis. A full field optical shearing interferometry called Coherent Gradient Sensing (CGS) is used to map crack tip deformations in real time. Experimental measurements and predictions based on beam theory are found to be in good agreement. Also, for a large stiffness mismatch bimaterial system, the interface crack initiation toughness is evaluated as a function of the crack tip mode mixity.     

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.     

Numerical evaluation of elastodynamic energy fracture parameters in 2-D heterogeneous media

The elastodynamic energy fracture parameters for a stationary crack in 2-D heterogeneous media are evaluated with a presented generalized Domain Integral Method (DIM). The method, incorporated with the finite element solutions, is demonstrated to be patch-independent in a generalized sense. In the context of dynamic response, the near-tip region is always involved in the calculation. The method is used for determination of the associated Energy Release Rate (ERR) for the cases when the crack tip is away from the material interface, with the formulation valid for both small and large elastic deformations. Numerical results for such problems appear to be very insensitive to the crack-tip finite element models. As to the instances when the tip terminates normally at the material interface, the ERR is not feasible for use as a fracture criterion. The generalized DIM is then applied for calculation of the alternative elastodynamic energy parameter J/R^λ₀. The exponential order λ, with regard to the strength of stress singularity, is also properly evaluated in the calculation. No particular singular finite element is required throughout the study. © 1998 John Wiley & Sons, Ltd.     

Fracture parameters for interfacial cracks: an experimental-finite element study of crack tip fields and crack initiation toughness

Crack tip measurements and analysis of interfacial parameters for PMMA-aluminum bimaterial system are presented. A variety of crack tip mode-mixities are obtained by subjecting asymmetric four-point-bend specimens to different boundary loads. The crack tip fields are mapped using the optical method of Coherent Gradient Sensing (CGS). The complex stress intensity factors and the associated crack tip mixities () are measured from CGS fringe patterns. The asymptotic expansion field for interface cracks is used for extracting fracture parameters by accounting for higher order contributions to the experimental data. The measurements are compared with complementary finite element computations. A linear relationship between crack tip mixity and the applied load mixity is experimentally demonstrated in this large elastic mismatch system. The fracture load and hence the energy release rate G _cr () at crack initiation is measured as applied load mixities are varied. Limited discussion on the influence of surface roughness prior to bonding on the fracture toughness is included. Positive and negative shear on the crack plane produce different failure responses in this bimaterial system and the observed asymmetry is akin to the one predicted by the T&H model that includes crack tip nonlinearty.     

A single leg bending test for interfacial fracture toughness determination

A single leg bending test is described and its suitability for interfacial fracture toughness testing is evaluated. The test specimen consists of a beam-type geometry comprised of two materials, one top and one bottom, with a split at one end along the bimaterial interface. A portion of the bottom material in the cracked section of the beam is removed and the geometry is loaded in three-point bending. Thus, the reaction force of the support at the cracked end is transmitted only into the material comprising the top portion of the beam. The test is analyzed by a crack tip element analysis and the resulting expressions for energy release rate and mode mixity are verified by comparison with finite element results. It is shown that, by varying the thicknesses of the two materials, the single leg bending test can be used to determine the fracture toughness of most bimaterial interfaces over a reasonably wide range of mode mixities.     

On the analysis of a mixed mode bending sandwich specimen for debond fracture characterization

The mixed mode bending specimen originally developed for mixed mode delamination fracture characterization of unidirectional composites has been extended to the study of debond propagation in foam cored sandwich specimens. The compliance and strain energy release rate expressions for the mixed mode bending sandwich specimen are derived based on a superposition analysis of solutions for the double cantilever beam and cracked sandwich beam specimens by applying a proper kinematic relationship for the specimen deformation combined with the loading provided by the test rig. This analysis provides also expressions for the global mode mixities. An extensive parametric analysis to improve the understanding of the influence of loading conditions, specimen geometry and mechanical properties of the face and core materials has been performed using the derived expressions and finite element analysis. The mixed mode bending compliance and energy release rate predictions were in good agreement with finite element results. Furthermore, the numerical crack surface displacement extrapolation method implemented in finite element analysis was applied to determine the local mode mixity at the tip of the debond.     

Tapered beam on elastic foundation model for compliance rate change of TDCB specimen

A modified beam theory is developed to predict compliance rate change of tapered double cantilever beam (TDCB) specimens for mode-I fracture of hybrid interface bonds, such as polymer composites bonded to wood. The analytical model treats the uncracked region of the specimen as a tapered beam on generalized elastic foundation (TBEF), and the effect of crack tip deformation is incorporated in the formulation. A closed-form solution is obtained to compute the compliance and compliance vs. crack length rate change. The present TBEF model is verified with finite element analyses and experimental calibration data of compliance for wood-wood and wood-composite bonded interfaces. The compliance rate change can be used with experimental critical fracture loads to determine the respective critical strain energy release rates or fracture toughness of interface bonds. The present analytical model, which accounts for the crack tip deformation, can be efficiently and accurately used for compliance and compliance rate-change predictions of TDCB specimens and reduce the experimental calibration effort that is often necessary in fracture studies. Moreover, the constant compliance rate change obtained for linear-slope TDCB specimens can be applied with confidence in mode-I fracture tests of hybrid material interface bonds.     

Face/core interface fracture characterization of mixed mode bending sandwich specimens

Debonding of the core from the face sheets is a critical failure mode in sandwich structures. This paper presents an experimental study on face/core debond fracture of foam core sandwich specimens under a wide range of mixed mode loading conditions. Sandwich beams with E‐glass fibre face sheets and PVC H45, H100 and H250 foam core materials were evaluated. A methodology to perform precracking on fracture specimens in order to achieve a sharp and representative crack front is outlined. The mixed mode loading was controlled in the mixed mode bending (MMB) test rig by changing the loading application point (lever arm distance). Finite element analysis was performed to determine the mode‐mixity at the crack tip. The results showed that the face/core interface fracture toughness increased with increased mode II loading. Post failure analysis of the fractured specimens revealed that the crack path depends on the mode‐mixity at the crack tip, face sheet properties and core density.     

An Experimental Study on the Elastic–Plastic Fracture in a Ductile Material under Mixed-Mode Loading

Abstract:  Stereo vision was used to measure the crack-tip parameters, such as J integral, plastic mixity and elastic mixity of mixed-mode fracture specimens, and to study the applicability of the Shih's plane strain solution to the mixed-mode crack-tip fields. The fracture specimen used in this study was a compact tension shear (CTS) specimen made of 2024-O aluminum. The in-plane strain and stress fields near the mixed-mode crack tip of the CTS specimen were determined using the deformation field measured by the stereo vision. It is observed that the J integral values computed along rectangular contours surrounding the mixed-mode crack-tip approach constant values after r / h  > 0.5. The in-plane strains determined experimentally at several points near the crack tip and at several radial lines emerging from the crack tip are compared with the values calculated using Shih's plane-strain solution and the HRR slope, named after the investigations of Hutchinson, Rice and Rosengren respectively. It is found that the measured values follow the trends of the Shih's plane-strain solution. The elastic mixity evaluated using the measured crack-tip stress fields is close to that obtained from analytical solution. However, the evaluated plastic mixity deviates from the analytical solution.     

Interface crack initiation at V-notches along adhesive bonding in weakly bonded polymers subjected to mixed-mode loading

In this paper, interface crack initiation at V-notches along adhesive in bonded Polycarbonate (PC) and Poly Methyl Methacrylate (PMMA) subjected to mixed-mode loading conditions was investigated based on a combined experimental, finite element and matched asymptotic analysis. The V-notch specimens with an adhesive interface starting from its tip made at different notch angles were tested under three-point bending conditions. The experimental observations show that the specimens mainly fail by cracks along the interface. Also, the load at the crack initiation increases when the notch angle increases. The computational results are then used to explain and to correlate with the experimental data. A two-fold criterion developed by Leguillon (Eur J Mech A/Solids 21:61?C72 2002) that requires a simultaneous satisfaction of both Griffith energy and stress conditions for the crack initiation at a notch in the specimen made of a homogeneous brittle material is first extended for V-notch specimens under mixed-mode loading conditions and then used to estimate the crack initiation load. The estimated loads appear to agree well with the experimental data. Finally, an inverse method is proposed to estimate the values of fracture toughness at different mode mixity ratios.     

A fracture mechanics analysis of cutting and machining

The process of cutting and machining is analysed using concepts developed in the fracture analysis of beam specimens. Increasing cutting forces and decreasing tool rake angles lead to a sequence of deformation processes from elastic bending to elastic-plastic bending and finally to shear yielding in the chip. The conditions for each mode of deformation are identified. Fracture toughness is included in the analysis as is, in addition, the notion of root rotation at the crack tip. Under some circumstances this gives rise to the condition of the tool tip touching the crack tip during which energy is transferred directly to the fracture process. The tool-chip interface is characterised by Coulomb friction and by the inclusion of an adhesion toughness to model the effects of hot polymeric chips sticking to the rake face of the tool. The combined effects of bending and shearing leading to chip curling and coiling are also analysed.     

Arbitrarily oriented microcracks near the tip of an interface macrocrack in bonded dissimilar anisotropic materials

It is well known that microcracking in brittle materials results in a reduction of the stress intensity factor (SIF) and energy release rate (ERR). The reduced SIF or ERR represents crack tip shielding which is of significant interest to micromechanics and material science researchers. However, the effect of microcracking on the SIF and ERR is a complicated subject even for isotropic homogeneous materials, and becomes much more formidable in case of interface cracks in bonded dissimilar solids. To unravel the micromechanics of interface crack tip shielding in bonded dissimilar anisotropic solids, an interface crack interacting with arbitrarily oriented subinterface microcracks in bonded dissimilar anisotropic materials is studied. After deducing the fundamental solutions for a subinterface crack under concentrated normal and tangential tractions, the present interaction problem is reduced to a system of integral equations which is then solved numerically. A J‐integral analysis is then performed with special attention focused on the J₂‐integral in a local coordinate system attached to the microcracks. Theoretical and numerical results reassert the conservation law of the J‐integral derived for isotropic materials 1 , 2 also to be valid for bonded dissimilar anisotropic materials. It is further concluded that there is a wastage when the remote J‐integral transmits across the microcracking zone from infinity to the interface macrocrack tip. In order to highlight the influence of microstructure on the interfacial crack tip stress field, the crack tip SIF and ERR in several typical cases are presented. It is interesting to note that the Mode I SIF at the interface crack tip is quite different from the ERR in bonded dissimilar anisotropic materials.     

The effects of constraint on crack tip fields and fracture toughness

The detailed stresses, deformations and porosities in the vicinity of a blunting crack in different fracture specimens, Single Edge Notch (SEN), Three Point Bending (TPB) specimens, and Small Scale Yielding (SSY) model were studied by the larger deformation finite element method. The presence and subsequent growth of smaller scale voids were taken into account by using a modified Gurson's model to describe the constitutive behavior of the material. The dependences of the stresses, deformations and porosities on specimen configurations were associated with the crack tip constraint. The porosity in the tip region, along with the void coalescence criterion, were used to predict the macroscopic fracture toughness as a function of the constraint, and a comparison with experimental data was performed in this paper.     

Energy release rate and mode-mixity of adhesive joint specimens

Fracture behaviour of adhesive joints under mixed mode loading is analysed by using the beam/adhesive-layer (b/a) model, in which, the adherends are beamlike and the adhesive is constrained to a thin flexible layer between the adherends. The adhesive layer deforms in peel (mode I), in shear (mode II) or in a combination of peel and shear (mixed mode). Macroscopically, the ends of the bonded part of the joints can be considered as crack tips. The energy release rate of a single-layer adhesive joint is then formulated as a function of the crack tip deformation and the mode-mixity is defined by the shear portion of the total energy release rate. The effects of transversal forces and the flexibility of the adhesive layer are included in the b/a-model, which can be applied to joints with short crack length as well as short bonding length. The commonly used end-loaded unsymmetric semi-infinite joints are examined and closed-form solutions are given. In comparison to the singular-field model in the context of linear elastic fracture mechanics, the b/a-model replaces the singularity at the crack tip with a stress concentration zone. It is shown that the b/a-model and the singular-field model yield fundamentally different mode-mixities for unsymmetric systems. The presented closed-form b/a-model solutions facilitates parametric studies of the influence of unbalance in loading, unsymmetry of the adherends, as well as the flexibility of the adhesive layer, on the mode mixity of an adhesive joint.     

Delamination in elastic shear deformable circular layers

This work extends the analytical solution of an interface crack in straight layered structures to circular layered structures. A small segment at the vicinity of an interface crack tip in a circular laminated beam is analyzed by a novel shear deformable bi-layered circular beam theory. Two concentrated forces are found existing at the crack tip due to the requirement of the equilibrium condition. Closed-form solution of the total energy release rate of the interface crack is obtained as the half of the product of the concentrated forces and the corresponding displacement gradient discontinuities at the crack tip. Closed-form expressions of the mode I and II components of the energy release rate are also obtained by global and local methods. Numerical verifications are conducted by analyzing the interlaminar delamination of a circular beam with an edged crack and comparing with the baseline results obtained through finite element analysis. Excellent agreements between the present method and finite element analysis on the predictions of total energy release rate and mode partition verify the accuracy and efficiency of the present solution.     

Continuum aspects of directionally dependent cracking of an interface between copper and alumina crystals

Directionally dependent cracking along the interface of a Cu/Al₂O₃ bicrystal has been analyzed using continuum mechanics. This work extends previous analyses by considering the elastic anisotropy and plastic deformation of copper. The goals of the analysis are: (1) To provide a possible continuum explanation of the experimentally observed directionally dependent cracking, and (2) to understand the effect of continuum deformation on the competition between dislocation nucleation at a crack tip (i.e., tip blunting and alleviation of the stress concentration) and cleavage. First, the mode mixity of the elastic fields at the crack tip is calculated by considering the traction vector on the interface as derived from anisotropic elasticity. This is compared to the results from isotropic elasticity. The effects of anisotropy, as pertaining to dislocation nucleation and cleavage, are discussed. Elastic-plastic FEM analyses within both the ‘small strain' and finite deformation formulations have been performed for the two crack directions in. Furthermore, a variety of hardening laws, including ideal plasticity, were used to investigate the robustness of the solutions. Comparisons between the FEM analyses show that the general sectorial nature of the crack tip stress fields isthe same for all hardening laws chosen. The effects of geometrical hardening and softening are pronounced in a finite deformation, ideal plasticity analysis. However, even moderate hardening greatly diminishes these effects. If the hardening law saturates, the resulting near tip fields are similar to those of an ideally plastic material. The two crack tip orientations affect the nature of the localization of the plastic deformation. These effects are discussed in the context of different hardening laws. In addition, the interface traction phase angle is changed significantly by the plastic deformation with consequences regarding dislocation nucleation and cleavage. Overall, no clear continuum-based explanation of directionally dependent fracture was found, but it is clear that elastic anisotropy and plastic deformation should, in general, be taken into account. Finally, critical problems encountered in this type of analysis are identified and some directions for future research are suggested.     

Characterization of isochromatic fringe patterns for a dynamically propagating interface crack

The isochromatic fringes surrounding a crack propagating along a bimaterial interface have been developed and characterized. A parametric investigation has also been conducted to study the influence of various fracture parameters on this isochromatic fringe pattern. The relevant fracture parameters of interest were the crack-tip velocity, the mode mixity of loading and the non-singular stress field component. In all the cases the fringe pattern was compared with the more familiar patterns that are generated for the case of crack propagation in homogeneous media. It was found that both the crack tip velocity and the mode mixity of loading have a significant effect on the size and shape of the isochromatic fringe pattern surrounding a crack tip propagating along a bimaterial interface. However, the non-singular stress field component was found not to have a substantial effect on the fringe pattern. This is in contrast with the case of crack propagation in homogeneous media, where the non-singular stress field component determines the tilt of the fringe contours. The paper also presents an appropriate scheme to analyze experimental fringe contours to extract the various fracture parameters of interest. Finally, this scheme is employed to analyze actual experimental data from a typical bimaterial interface fracture experiment.     

Effect of mode mixity on K-dominance and three dimensionality in cracked plates

The method of Coherent Gradient Sensing (CGS) in transmission, in conjunction with two and three dimensional finite element methods, is used to study the effect of mode mixity on crack tip stress fields. Using a two dimensional finite element analysis the outer bounds of the region of K-dominance were determined. A three dimensional finite analysis was utilized to study the effect of mode mixity on the three dimensional nature of the stress field in the immediate vicinity of the crack tip and to obtain an inner bound of the region of K-dominance. It was noted that increasing mode mixity leads to an increased rotation of the three dimensional zone, keeping its shape and size unchanged. In contrast, the region of K-dominance is seen to dramatically depend on mode mixity, both in shape and size. In addition, an analysis of the CGS interferograms was conducted to obtain an estimate of the regions of K-dominance experimentally. A least squares fit data analysis technique was used to extract fracture parameters, namely the stress intensity factors K _I, K _II and subsequently the crack tip phase angle, . The data points used for the least square fitting were obtained from the determined regions of K-dominance. The same fracture parameters were also evaluated from the finite element analysis, and good agreement was found between experimental measurements and finite element predictions.