Asymptotic solution for mode III crack growth in J 2-elastoplasticity with mixed isotropic-kinematic strain hardening

Mode III fracture propagation is analyzed in a J ₂-flow theory elastoplastic material characterized by a mixed isotropic/kinematic law of hardening. The asymptotic stress, back stress and velocity fields are determined under small-strain, steady-state, fracture propagation conditions. The increase in the hardening anisotropy is shown to be connected with a decrease in the thickness of the elastic sector in the crack wake and with an increase of the strength of the singularity. A second order analytical solution for the crack fields is finally proposed for the limiting case of pure kinematic hardening. It is shown that the singular terms of this solution correspond to fully plastic fields (without any elastic unloading sector), which formally are identical to the leading order terms of a crack steadily propagating in an elastic medium with shear modulus equal to the plastic tangent modulus in shear.     

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

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

Scale effects in the initiation of cracking of a scarf joint

The singular stress field at the interface-corner of a bi-material scarf joint is analysed for a strip of finite width, w, under remote tension and bending. The two substrates are taken as linear elastic and isotopic. The intensity of the singular stress field is calculated using a domain integral method, and is plotted as a function of joint geometry and material mismatch parameters. It is envisaged that the intensity of singularity can serve as a valid fracture criterion provided the zone of nonlinearity is fully embedded within the singular elastic field. It is assumed that fracture initiates when the magnitude of the corner singularity attains a critical value; consequently, the fracture strength of the joint depends upon the size of the structure. In addition, the interfacial stress intensity factor and the associated T-stress are determined for an edge interfacial crack. When the crack is short with respect to the width of the strip, the stress intensity factor is dominated by the presence of the corner singularity; a boundary layer formulation is used to determine the coupling between the crack tip field and the interface-corner field. The solution suggests that an interfacial crack grows unstably with a rapidly increasing energy release rate, but with only a small change in mode mix. This revised version was published online in July 2006 with corrections to the Cover Date.     

Elastic-plastic analysis of an arbitrarily-oriented mode III crack touching an interface

In this paper we investigate a semi-infinite crack terminating at an arbitrarily oriented interface between two elastic-plastic materials under an anti-plane shear loading. An analytical solution is first developed for general power-law hardening materials under a mode III loading. If both materials have the same hardening exponent, the formulation results in a nonlinear eigenequation which can be solved numerically. The stress singularities are determined as a function of two material constants: the hardening exponent n and parameter G which represents the relative resistance of the two materials. In addition to the power of the singularity, the stress, strain and displacement asymptotic fields are also determined. If the hardening exponents are not the same, the leading order terms of an expansion model ensure the stress continuity across the interface. The results show that the stress singularity mainly depends upon the material having the larger hardening exponent, with the highest stresses in the material having the smaller hardening exponent. By taking the hardening exponent n , the perfectly plastic bimaterial problem is studied. It has been found that if the crack lies in the less stiff material, the entirely plastic asymptotic fields around the crack tip can be determined. On the other hand, if the crack lies in the stiffer material, the crack-tip fields are partially elastic and partially plastic. For both cases, unique asymptotic fields can be determined explicitly. For those cases when the materials present a strain hardening property, different mathematical models are established.     

Singularity of dynamic stress fields around an interface crack between viscoelastic bodies

The singular nature of the dynamic stress fields around an interface crack located between two dissimilar isotropic linearly viscoelastic bodies is studied. A harmonic load is imposed on the surfaces of the interface crack. The dynamic stress fields around the crack are obtained by solving a set of simultaneous singular integral equations in terms of the normal and tangent crack dislocation densities. The singularity of the dynamic stress fields near the crack tips is embodied in the fundamental solutions of the singular integral equations. The investigation of the fundamental solutions indicates that the singularity and oscillation indices of the stress fields are both dependent upon the material constants and the frequency of the harmonic load. This observation is different from the well-known −1/2 oscillating singularity for elastic bi-materials. The explanation for the differences between viscoelastic and elastic bi-materials can be given by the additional viscosity mismatch in the case of viscoelastic bi-materials. As an example, the standard linear solid model of a viscoelastic material is used. The effects of the frequency and the material constants (short-term modulus, long-term modulus and relaxation time) on the singularity and the oscillation indices are studied numerically.     

Higher order asymptotic solutions of V-notch tip fields for damaged nonlinear materials under antiplane shear loading

The higher order solutions of stress and deformation fields near the tip of a sharp V-notch in a power-law hardening material with continuous damage formation are analytically investigated under antiplane shear loading condition. The interaction between a macroscopic sharp notch and distributed microscopic damage is considered by describing the effect of damage in terms of a damage variable in the framework of damage mechanics. A deformation plasticity theory coupled with damage and a damage evolution law are formulated. A hodograph transformation is employed to determine the solution of damaged nonlinear notch problem in the stress plane. Then, inversion of the stress plane solution to the physical plane is performed. Consequently, higher order terms in the asymptotic solutions of the notch tip fields are obtained. Analytical expressions of the dominant and second order singularity exponents and associated angular distribution functions of notch tip stress and strain are presented. Effects of damage and strain hardening exponents and notch angle on the singular behavior of the notch tip quantities are discussed detailly. It is found that damage can lead to a weaker singularity of the dominant term of stress on one hand, but to stronger singularities of the second order term of stress and the dominant and second order terms of strain compared to that for undamaged case on the other. Also, both hardening exponent and notch angle have important effects on the notch tip quantities. Moreover, reduction of the notch tip solutions to a damaged nonlinear crack problem is carried out, and higher order solutions of the crack tip fields are obtained. Effects of damage and hardening exponents on the dominant and second order terms in the crack tip solutions are detailly discussed. Discussions on some other special cases are also presented, which shows that if damage exponent equals to zero, then the present solutions can be easily reduced to the solutions for undamaged cases. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

Dynamic asymptotic mode III crack tip field in rate dependent materials

The asymptotic field at a dynamically growing crack tip in strain-rate sensitive elastic-plastic materials is investigated under anti-plane shear loading conditions. In the conventional viscoplasticity theory, the rate sensitivity is included only in the flow stress. However, it is often found that the yield strength is also affected by previous strain rates. The strain rate history effects in metallic solids are observed in strain rate change tests in which the flow stress decreases gradually after a rapid drop in strain rate. This material behavior may be explained by introducing the rate sensitivity in the hardening rule in addition to the flow rule. The strain-rate history effect is pronounced near the propagating crack where the change of strain rates take place. Effects of the rate dependency in the flow rule and the hardening rule on the crack propagation are analyzed. The order of the stress singularity in the asymptotic field is determined in terms of material parameters which characterize the rate sensitivity of the material. The results show that an elastic sector is present in the wake zone when the rate-dependency is considered only in the hardening rule. Terminal crack propagation speed is determined by applying the critical stress fracture criterion and the critical strain criterion to the asymptotic fields under the small scale yielding condition.     

Asymptotic and finite element analyses of mode III dynamic crack growth at a ductile-brittle interface

In this work, dynamic crack growth along a ductile-brittle interface under anti-plane strain conditions is studied. The ductile solid is taken to obey the J ₂ flow theory of plasticity with linear isotropic strain hardening, while the substrate is assumed to exhibit linear elastic behavior. Firstly, the asymptotic near-tip stress and velocity fields are derived. These fields are assumed to be variable-separable with a power singularity in the radial coordinate centered at the crack tip. The effects of crack speed, strain hardening of the ductile phase and mismatch in elastic moduli of the two phases on the singularity exponent and the angular functions are studied. Secondly, full-field finite element analyses of the problem under small-scale yielding conditions are performed. The validity of the asymptotic fields and their range of dominance are determined by comparing them with the results of the full-field finite element analyses. Finally, theoretical predictions are made of the variations of the dynamic fracture toughness with crack velocity. The influence of the bi-material parameters on the above variation is investigated.     

Interfacial Micromechanics of Hybrid Metal Matrix Composites

Hybrid metal matrix composites (MMCs) in which adjoining matrices have different plastic responses have potential to offer more attractive damage tolerance properties than conventional MMCs \[1]. At the interface of two dissimilar metal matrices, both capable of deforming plastically, the accentuated free-edge stresses may cause intermatrix decohesion to form an interface crack. In this article, the solution of a local-global analysis to determine the mechanics environment governing onset of such intermatrix debonding is presented. First an effective method of asymptotic local analysis for the singular interfacial stress in plastically deforming hybrid MMCs \[1-5] is summarized and new results are presented. Then, the generalized stress intensity factor that scales the free-edge interfacial stresses is solved by a global analysis for several illustrative cases. The global analysis is performed via elastic-plastic finite-element method. The solutions presented completely characterize the dominant mechanical environment that governs the onset of intermatrix decohesion. The solutions, with proper interpretation, are also applicable to hybrid MMCs at elevated temperatures with matrices deforming by power-law creep. \[6]     

Elastoplastic crack analysis for pressure-sensitive dilatant materials-Part II: Interface cracks

The problem of a plane strain crack lying along an interface between a rigid substrate and an elastic-plastic material has been studied. The elastic-plastic material exhibits pressure-sensitive yielding and plastic volumetric deformation. Two-term expansions of the asymptotic solutions for both closed frictionless and open crack-tip models have been obtained. The Mises effective stress in the interfacial crack-tip fields is a decreasing function of the pressure-sensitivity in both open and closed-crack tip models. The variable-separable solution exists for most pressure-sensitive materials and the limit values for existence of the variable-separable solution vary with the strain-hardening exponents. The governing equations become singular as the pressure-sensitivity limit is approached. Strength of the leading stress singularity is equal 1/(n+1) for both crack-tip models, regardless of the pressure-sensitivity. The second-order fields have been solved as an additional eigenvalue problem and the elasticity terms do not enter the second-order solutions as n2. The second exponents for the closed crack model are negative for the weak strain hardening, whereas the negative second-order eigenvalue in the open crack model slightly grows with the pressure-sensitivity. The second-order solutions are of significance in characterising the crack-tip fields. The leading-order solution contains the dominant mode I components for both open and closed crack-tip models when the materials do not have substantial strain hardening. The second-order solutions are generally mode-mixed and depend significantly on the pressure-sensitivity.     

Analysis of elastoplastic sharp notches

In this paper the elastoplastic solutions with higher-order terms for apex V-notches in power-law hardening materials have been discussed. Two-term expansions of the plane strain and the plane stress solutions have been obtained. It has been shown that the leading-order singularity approaches the value for a crack when the notch angle is not too large. In plane strain cases the elasticity does not enter the second-order solutions when the notch opening angle is too small. For a large notch angle, the two-term expansions of the plane strain near-tip fields are described by a single amplitude parameter. The plane stress solutions generally contain the elasticity terms. The boundary layer formulations based on the small-strain plasticity theory confirm that a dominance zone exists ahead of the notch tip. Finite element results give good agreement to the asymptotic solutions under both plane strain and plane stress conditions. The second-order terms cannot improve the predictions significantly. The near-tip fields are dominated by a single parameter. Finite element calculations under the finite strain J ₂-flow plasticity theory revealed that the finite strains can only affect local characterization of the asymptotic solution. The asymptotic solution has a large dominance zone around the notch tip. For an apex notch bounded to a rigid substrate the leading-order singularity falls with the notch angle significantly more slowly than in the homogeneous material. It vanishes at the notch angle about 135° for all power-hardening exponents. The elasticity effects enter the second-order solutions when the notch angle becomes large enough. The tip fields are characterized by the hydrostatic stress and the shear stress ahead of the notch.     

Interface crack and nonideal interface concept (Mode III)

Asymptotic behaviour of displacements and stresses in a vicinity of the interface crack tip situated on a nonideal interface between two different elastic materials is investigated. The nonideal interface is described by special transmission conditions along the material bonding. The corresponding modelling boundary value problem is reduced to a singular integral equation with fixed point singularities. It is shown from the solution to the problem that asymptotic behaviour of displacement and stresses near the crack tip essentially depends on the model parameters. Some numerical examples are presented and discussed with respect to the stress singularity exponent and the generalized stress intensity factors.     

On the dynamic crack propagation in an interphase with spatially varying elastic properties under inplane loading

This article provides a comprehensive theoretical investigation on a finite crack with constant length (Yoffe type crack) propagating in an interfacial layer with spatially varying elastic properties under inplane loading. The analytical formulations are developed using Fourier transforms and solving the resulting singular integral equations in terms of the opening and sliding displacements of the crack. The dynamic stress intensity factors and energy release rate are analyzed to study the dynamic fracture property of this inherent mixed mode crack problem. Numerical examples are provided to show the effects of the material properties, the thickness of the interfacial layer, the crack position and speed upon the dynamic fracture behaviour, and the singularity transition between the current crack and the corresponding interfacial crack for thin interphase.     

Stress singularities for crack normal to and terminating at bimaterial interface on orthotropic half-plates

The problem of a crack normal to and terminating at an interface in two joined orthotropic plates is considered and the eigenequation for the asymptotic behavior of stresses at the crack tip on the interface is given in an explicit form. It is found that the singular stress field around the crack tip can be separated into two independent fields, respectively of the mode I and II. Also it is found that for both the mode I and II deformations the effects of elastic constants on the stress singularity order can be respectively expressed by three material parameters, two of which are the same for both the mode I and mode II deformations.     

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.     

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

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

Determination of high‐order fields for multianisotropic material antiplane V‐notches and inclusions by the asymptotic expansion technique and an overdeterministic method

In this paper, the singular behavior for anisotropic multimaterial V‐notched plates is investigated under antiplane shear loading condition. Firstly, the elastic governing equations are transformed into eigen ordinary differential equations through introducing the asymptotic expansions of displacements near the notch tip. The stress singularity exponents, including the higher‐order terms, and the corresponding eigen angular functions are then obtained by solving the established equations by using the interpolating matrix method. Thus, using the combination of the results from finite element analyses and the derived asymptotic expansion, an overdeterministic method is employed to calculate the amplitudes of the coefficients in the asymptotic expansions. Finally, the stress and displacement fields in the vicinity of the notch tip, consisting of both singular terms and higher‐order terms, are determined. The effects of material properties and geometry characteristic on the singular behaviour of the notch tip are discussed in detail.