An elastic-plastic finite element analysis of a blunting interface crack with microvoid damage

A finite strain elastic-plastic finite element analysis is performed on a crack which lies on an interface between two dissimilar materials. The materials above and below the interface are assumed to be different from each other in yield stress or in strain-hardening exponent. Gurson's constitutive equation for porous plastic materials is used in order to take into account the effect of the microvoid nucleation and growth on the fields near the tip of a crack.It is found that the microvoids have larger effects on the crack tip blunting and stress fields for a bimaterial than for a homogeneous material. It is also found that the plastic strain and the microvoid volume fraction localize in a few narrow bands which grow into the softer material from the intersection of the interface and the blunted crack tip at inclinations of about 15° 45°.     

Three-dimensional analysis of interface cracks in rubber materials

Three-dimensional and plane stress finite element analyses were carried out to investigate the stress fields and fracture parameters of interface cracks in rubber materials. The tearing energy computations for any arbitrarily shaped 3D crack front in dissimilar materials were obtained using the virtual crack extension method. The finite element results obtained are validated against existing alternate solutions and experimental data for cracks in homogeneous as well as bimaterial cases. The effects of different rubber material models, tearing energy distributions, crack extension angles, and three-dimensional regions at the crack tip for interface cracks are presented and discussed. It is shown that nonlinear materials have larger three-dimensional effects near the interface crack front, and that this effect increases as the material mismatch increases.     

Plane stress near-tip field analysis of steady-state crack growth along a linear-hardening elastic-plastic interface

Summary In this work the asymptotic near-tip stress and velocity fields of a crack propagating steadily and quasi-statically along a ductile interface are presented for plane stress cases. The elastic-plastic materials are characterized by the J₂-flow theory with linear plastic hardening. The solutions are assumed to be of variable-separable form with a power singularity in the radial distance to the crack tip. It is found that two distinct solutions exist with slightly different singularity strengths and very different mixities on the interface ahead of the crack tip. One of the solutions corresponds to a tensile-like mode and the other corresponds to a shear-like mode. An interface will change the near-tip fild of the tensile solution obviously, whereas the shear-like solution maintains its original structure as in homogeneous materials. In cases the elastic bimaterial parameter differs from zero, the two solutions can coalesce at some high strain-hardening. An interface between two high strain-hardening materials only slightly affects the stress and velocity distribution around the tip, whereas the singularity strength deviates from the homogeneous solutions. The strength of the singularity is predominantly determined by the smaller strain-hardening material. Poisson's ratio affects variation of the singularity as a function of strain-hardening slightly if the coalescing point of the variable-separable solution is not approached. Only for the very distinct elastic moduli the near-tip field approaches the rigid interface solution.     

Kinking of an interface crack in an orthotropic bimaterial

We investigated the asymptotic problem of a kinked interface crack in an orthotropic bimaterial under in‐plane loading conditions. The stress intensity factors at the tip of the kinked interface crack are described in terms of the stress intensity factors of the interface crack prior to the kink combined with a dimensionless matrix function. Using a modified Stroh formalism and an orthotropy rescaling technique, the matrix function was obtained from the solutions of the corresponding problem in transformed bimaterial. The effects of orthotropic and bimaterial parameters on the matrix function were examined. A reduction in the number of dependent material parameters on the matrix function was made using the modified Stroh formalism. Moreover, the explicit dependence of one orthotropic parameter on the matrix function was determined using an orthotropic rescaling technique. The effects of the other material parameters on the matrix function were numerically examined. The energy release rate was obtained for a kinked interface crack in an orthotropic bimaterial.     

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.     

Fracture mechanics parameters for cracks on a slightly undulating interface

Typical bimaterial interfaces are non-planar due to surface facets or roughness. Crack-tip stress fields of an interface crack must be influenced by non-planarity of the interface. Consequently, interface toughness is affected. In this paper, the crack-tip fields of a finite crack on an elastic/rigid interface with periodic undulation are studied. Particular emphasis is given to the fracture mechanics parameters, such as the stress intensity factors, crack-tip energy release rate, and crack-tip mode mixity. When the amplitude of interface undulation is very small relative to the crack length (which is the case for rough interfaces), asymptotic analysis is used to convert the non-planarity effects into distributed dislocations located on the planar interface. Then, the resulting stress fields near the crack tip are obtained by using the Fourier integral transform method. It is found that the stress fields at the crack tip are strongly influenced by non-planarity of the interface. Generally speaking, non-planarity of the interface tends to shield the crack tip by reducing the crack-tip stress concentration.     

Small-scale crack blunting at a bimaterial interface with Coulomb friction

The deformation field near the tip of a tensional crack impinging upon a normally loaded bimaterial frictional interface is studied. By allowing the two materials to slide with respect to each other along the interface the high stresses around the crack tip are relaxed. The region of slip as well as the stress distribution inside the slip zone is obtained as a function of the far field loading parameter.     

The extended finite element method for fracture in composite materials

Methods for treating fracture in composite material by the extended finite element method with meshes that are independent of matrix/fiber interfaces and crack morphology are described. All discontinuities and near‐tip enrichments are modeled using the framework of local partition of unity. Level sets are used to describe the geometry of the interfaces and cracks so that no explicit representation of either the cracks or the material interfaces are needed. Both full 12 function enrichments and approximate enrichments for bimaterial crack tips are employed. A technique to correct the approximation in blending elements is used to improve the accuracy. Several numerical results for both two‐dimensional and three‐dimensional examples illustrate the versatility of the technique. The results clearly demonstrate that interface enrichment is sufficient to model the correct mechanics of an interface crack. Copyright © 2008 John Wiley & Sons, Ltd.     

The J-integral at Dugdale cracks perpendicular to interfaces of materials with dissimilar yield stresses

The J-integral is applied to a Dugdale crack perpendicular to an interface of materials with equal elastic properties but different yield stresses. It is shown that the integral is path independend with certain limitations to the integration path. Three essentially different paths can be distinguished. The first integration path is totally within the first material, it provides the local crack driving force. Performing the integral around the plastic zone in both materials gives the global crack driving force. An interface force can be defined by evaluating the integral along both sides of the plastically deformed region of the interface. A comparison of these three integrals reveals that the global crack driving force is equal to the sum of the local crack driving force and of the interface force. The derived expression for the J-integral are compared with the crack tip opening displacement published recently. This reveals that the local J describes the plastic deformation at the crack tip. Therefore it represents the crack driving force in bimaterials as it does the conventional J-integral in case of homogeneous materials. The analyses are also extended to cyclic plasticity, where an out-of-phase effect is observed. Finally it is discussed how these results can be used to explain fatigue tests at bimaterial specimens.     

Dynamic response of bimaterial and graded interface cracks under impact loading

The interfacial fracture in bimaterial and functionally graded material (FGM) under impact loading conditions is investigated using experimental and numerical techniques that are valid for both type of interfaces. Experiments are conducted on epoxy based specimens in three point bend configuration and the complex SIF is measured using an electrical strain gage mounted close to the crack-tip. A complementary two-dimensional finite element simulation is performed using tup force and support reactions as input tractions, and the SIF-time history is determined using a displacement extrapolation technique. The experimentally determined SIF-histories match closely with numerical simulation up to the time of fracture initiation. The test results show that the mode-mixity remains nearly constant through out the test in both the materials, and the mixity values correspond to their respective static counterparts. The general dynamic response of the bimaterial and FGM specimens in terms of impact load, support reaction and the magnitude of complex SIF are comparable, and the mode-mixity is the parameter that distinguishes the graded interface from the bimaterial case.     

Material‐dependent crack‐tip enrichment functions in XFEM for modeling interfacial cracks in bimaterials

A novel set of enrichment functions within the framework of the extended finite element method is proposed for linear elastic fracture analysis of interface cracks in bimaterials. The motivation for the new enrichment set stems from the revelation that the accuracy and conditioning of the widely accepted 12‐fold bimaterial enrichment functions significantly deteriorates with the increase in material mismatch. To this end, we propose an 8‐fold material‐dependent enrichment set, derived from the analytical asymptotic displacement field, that well captures the near‐tip oscillating singular fields of interface cracks, including the transition to weak discontinuities of bimaterials. The performance of the proposed material‐dependent enrichment functions is studied on 2 benchmark examples. Comparisons are made with the 12‐fold bimaterial enrichment as well as the classical 4‐fold homogeneous branch functions, which have also been used for bimaterials. The numerical studies clearly demonstrate the superiority of the new enrichment functions, which yield the most accurate results but with less number of degrees of freedom and significantly improved conditioning than the 12‐fold functions.     

A Nonlinear Finite Element Eigenanalysis of Singular Plane Stress Fields in Bimaterial Wedges Including Complex Eigenvalues

A finite element formulation is developed for the analysis of variable-separable singular stress fields in power law hardening materials under conditions of plane stress. The displacement field within a sectorial element is assumed to be quadratic in the angular coordinate and of the power type in the radial direction as measured from the singular point. An iteration scheme that combines the Newton method and matrix singular value decomposition is used to solve the nonlinear homogeneous eigenvalue problem, where the eigenvalues and eigenfunctions are obtained simultaneously. The formulation and iteration scheme apply when the eigenvalue is complex. The examples considered include the single material crack and wedge to demonstrate convergence, and the bimaterial interface crack and the bimaterial wedge to demonstrate geometric versatility and the ability to handle complex eigenvalues. It is found that the real part of the complex eigenvalue for the interface crack agrees with the HRR value. In this case the associated complex eigenfunction is converted into an approximate real-valued eigenfunction that is valid for any mode-mix. In addition, the behavior of separable solutions near certain 'wedge paradox' geometries where non-separable solutions occur is investigated. This revised version was published online in July 2006 with corrections to the Cover Date.     

A constraint based parameter for quantifying the crack tip stress fields in welded joints

By means of finite element analyses of plane strain crack tip stress fields from homogeneous and heterogeneous modified boundary layer formulations, as well as homogeneous and mismatched full field solutions, a new constraint parameter β_m has been established for overmatched welded joints, allowing the material mismatching effect on the crack tip stress fields to be quantified. In the case of complete specimens, both geometry and material mismatching affect the crack tip stress fields, and a total constraint parameter β_T can be defined. This approach allows to quantify the stress fields directly from the values of the remote applied load.     

Two-parameter characterization of the near-tip stress fields for a bi-material elastic-plastic interface crack

A particular case of interface cracks is considered. The materials at each side of the interface are assumed to have different yield strength and plastic strain hardening exponent, while elastic properties are identical. The problem is considered to be a relevant idealization of a crack at the fusion line in a weldment. A systematic investigation of the mismatch effect in this bi-material plane strain mode I dominating interface crack has been performed by finite strain finite element analyses. Results for loading causing small scale yielding at the crack tip are described. It is concluded that the near-tip stress field in the forward sector can be separated, at least approximately, into two parts. The first part is characterized by the homogeneous small scale yielding field controlled by J for one of the interface materials, the reference material. The second part which influences the absolute value of stresses at the crack tip and measures the deviation of the fields from the first part can be characterized by a mismatch constraint parameter M. Results have indicated that the second part is a very weak function of distance from the crack tip in the forward sector, and the angular distribution of the second part is only a function of the plastic hardening property of the reference material.     

Crack-tip fields for matrix cracks between dissimilar elastic and creeping materials

The stress fields near the tip of a matrix crack terminating at and perpendicular to a planar interface under symmetric in-plane loading in plane strain are investigated. The bimaterial interface is formed by a linearly elastic material and an elastic power-law creeping material in which the crack is located. Using generalized expansions at the crack tip in each region and matching the stresses and displacements across the interface in an asymptotic sense, a series asymptotic solution is constructed for the stresses and strain rates near the crack tip. It is found that the stress singularities, to the leading order, are the same in each material; the stress exponent is real. The oscillatory higher-order terms exist in both regions and stress higher-order term with the order of O(r°) appears in the elastic material. The stress exponents and the angular distributions for singular terms and higher order terms are obtained for different creep exponents and material properties in each region. A full agreement between asymptotic solutions and the full-field finite element results for a set of test examples with different times has been obtained.     

Effect of a graded interlayer on the stress intensity factor of cracks in a joint under thermal loading

In a bimaterial joint with and without a graded interlayer, the stress intensity factor of cracks perpendicular to the interface was calculated for a thermal loading by a homogeneous change in the temperature. In joints without an interlayer, the stress intensity factor increases to infinity as the crack approaches the interface for the case of the Young’s moduli E₁/E₂>1 (crack in material 1). Introducing a graded interlayer with a continuous transition in the material properties between the two joined materials leads to a continuous change in the stress intensity factor if the crack propagates from material 1 into material 2. Results are presented for different transition functions of the material properties and for different thickness ratios of the layers. The possible beneficial effect of a graded interlayer is discussed.     

Crack Tip Shielding or Anti-shielding due to Smooth and Discontinuous Material Inhomogeneities

This paper describes a theoretical model and related computational methods for examining the influence of inhomogeneous material properties on the crack driving force in elastic and elastic-plastic materials. Following the configurational forces approach, the crack tip shielding or anti-shielding due to smooth (e.g. graded layer) and discontinuous (e.g. bimaterial interface) distributions in material properties are derived. Computational post-processing methods are described to evaluate these inhomogeneity effects. The utility of the theoretical model and computational methods is demonstrated by examining a bimaterial interface perpendicular to a crack in elastic and elastic-plastic compact tension specimens.     

BIMATERIAL INTERFACE CRACK TIP STRESS ANALYSIS: ANISOTROPIC THEORY

Abstract— Near crack tip stress and displacement fields are obtained for anisotropic bimaterial interface cracks. A contact zone model is used in order to get rid of the unphysical oscillatory interpenetration between the edges of the crack. Semi-infinite and the finite crack problems have been studied. Using the near crack tip results of this model crack branching angles can be predicted. These results are illustrated by numerical results for various materials.     

Crack deflection at an interface of alumina/metal joint: A numerical analysis

Deflection of a crack at the bimaterial interface is the initial mechanism required for obtaining enhanced toughness in bimaterial system. In this paper, a criterion is presented to predict the competition between crack deflection and penetration at the interface, using an energy release rate criterion. The finite element methods are used to calculate the strain energy release rates at the crack tip of alumina–metal bimaterial that either deflect or penetrate at the interface as a function of elastic mismatch and length of the deflected or penetrated crack. The effects of the elastic properties of two bonded materials were highlighted in order to evaluate the conditions for the crack deflection by the interface as well as the distance between the crack tip and the interface.