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.     

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 perturbation analysis of combined mode I and III dynamic crack propagation

Summary In the present paper the asymptotic stress and deformation fields of dynamic crack extension in materials with linear plastic hardening under combined mode I (plane strain and plane stress) and anti-plane shear loading conditions (mode III) are investigated. The governing equations of the asymptotic crack-tip fields are formulated from two groups of angular functions, one for the in-plane mode and the other for the anti-plane shear mode. It was assumed that all stresses and deformations are of separable functional forms ofr and , which represent the polar coordinates centered at the actual crack tip. Perturbation solutions of the governing equations were obtained. The singularity behavior and the angular functions of the crack-tip in-plane and the anti-plane stresses obtained from the perturbation analysis show that, regardless of the mixity of the crack-tip field and the strain-hardening, the in-plane stresses under the combined mode I and mode III conditions have stronger singularity in the whole mixed mode steady-state crack growth than that of the anti-plane shear stresses. The anti-plane shear stresses perturbed from the plane strain mode I solutions lose their singularity for small strain hardening, whereas the angular stress functions perturbed from the plane stress mode I have a nearly analogous uniform distribution feature compared to pure mode III cases. An obvious deviation from the unperturbed solution is generally to be observed under combined plane strain mode I and anti-plane mode III conditions, especially for a large Mach number in a material with small strain-hardening; but not under plane stress and mode III conditions. The crack propagation velocity decreases the singularities of both pure mode and perturbed crack-tip fields.     

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.     

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.     

Crack tip fields in ductile crystals

Results on the asymptotic analysis of crack tip fields in elastic-plastic single crystals are presented and some preliminary results of finite element solutions for cracked solids of this type are summarized. In the cases studied, involving plane strain tensile and anti-plane shear cracks in ideally plastic f c c and b c c crystals, analyzed within conventional small displacement gradient assumptions, the asymptotic analyses reveal striking discontinuous fields at the crack tip.For the stationary crack the stress state is found to be locally uniform in each of a family of angular sectors at the crack tip, but to jump discontinuously at sector boundaries, which are also the surfaces of shear discontinuities in the displacement field. For the quasi-statically growing crack the stress state is fully continuous from one near-tip angular sector to the next, but now some of the sectors involve elastic unloading from, and reloading to, a yielded state, and shear discontinuities of the velocity field develop at sector boundaries. In an anti-plane case studied, inclusion of inertial terms for (dynamically) growing cracks restores a discontinuous stress field at the tip which moves through the material as an elastic-plastic shock wave. For high symmetry crack orientations relative to the crystal, the discontinuity surfaces are sometimes coincident with the active crystal slip planes, but as often lie perpendicular to the family of active slip planes so that the discontinuities correspond to a kinking mode of shear.The finite element studies so far attempted, simulating the ideally plastic material model in a small displacement gradient type program, appear to be consistent with the asymptotic analyses. Small scale yielding solutions confirm the expected discontinuities, within limits of mesh resolution, of displacement for a stationary crack and of velocity for quasi-static growth. Further, the discontinuities apparently extend well into the near-tip plastic zone. A finite element formulation suitable for arbitrary deformation has been used to solve for the plane strain tension of a Taylor-hardening crystal panel containing, a center crack with an initially rounded tip. This shows effects due to lattice rotation, which distinguishes the regular versus kinking shear modes of crack tip relaxation. and holds promise for exploring the mechanics of crack opening at the tip.     

Continuous near-tip fields for a dynamic crack propagating in a power-law elastic-plastic material

By use of the J ₂ flow theory and the rectangular components of field quantities, the near-tip asymptotic fields are studied for a dynamic mode-I crack propagating in an incompressible power-law elastic-plastic material under the plan strain conditions. Through assuming that the stress and strain components near a dynamic crack tip are of the same singularity, the present paper constructs with success the fully continuous dominant stress and strain fields. The angular variations of these fields are identical with those corresponding to the dynamic crack propagation in an elastic-perfectly plastic material (Leighton et al., 1987). The dynamic asymptotic field does not reduce to the quasi-static asymptotic field in the limit as the crack speed goes to zero. This revised version was published online in July 2006 with corrections to the Cover Date.     

Plastic intensity factors for cracked plates

An elastic-plastic analysis is performed for two problems relevant to fracture mechanics: a semiinfinite body with an edge crack in a far out-of-plane shearing field and an infinite plate under plane stress conditions containing a finite line crack in a remote tensile field. Amplitudes of the dominant singularity in the plastic region at the crack tip, the plastic stress and strain intensity factors, are calculated for applied stress levels approaching the yield stress. A technique is developed for using the dominant singular solution in conjunction with the finite element method to make accurate calculations for the near-tip fields. Additionally, a comparative study of deformation theory with flow theory is performed for cracks in an anti-plane shear field. Elastic fracture mechanics is extended to high levels of applied stress for which the plastic zone is no longer small compared to the crack length by relating the critical stress for fracture initiation to the plastic intensity factors.     

Tensile crack tip fields in elastic-ideally plastic crystals

Crack tip stress and deformation fields are analyzed for tensile-loaded ideally plastic crystals. The specific cases of (0 1 0) cracks growing in the [1 0 1] direction, and (1 0 1) cracks in the [0, 1, 0] direction, are considered for both fcc and bcc crystals which flow according to the critical resolved shear stress criterion. Stationary and quasistatically growing crack fields are considered. The analysis is asymptotic in character; complete elastic-plastic solutions have not been determined. The near-tip stress state is shown to be locally constant within angular sectors that are stressed to yield levels at a stationary crack tip, and to change discontinuously from sector to sector. Near tip deformations are not uniquely determined but fields involving shear displacement discontinuities at sector boundaries are required by the derived stress state. For the growing crack both stress and displacement must be fully continuous near the tip. An asymptotic solution is given that involves angular sectors at the tip that elastically unload from, and then reload to, a plastic state. The associated near-tip velocity field then has discontinuities of slip type at borders of the elastic sectors. The rays, emanating from the crack tip, on which discontinuities occur in the two types of solutions are found to lie either parallel or perpendicular to the family of slip plane traces that are stressed to yield levels by the local stresses. In the latter case the mode of concentrated shear along a ray of discontinuity is of kink type. Some consequences of this are discussed in terms of the dislocation generation and motion necessary to allow the flow predicted macroscopically.     

Elastoplastic crack analysis for pressure-sensitive dilatant materials — Part I: Higher-order solutions and two-parameter characterization

An elastoplastic solution with higher-order terms for cracks in materials exhibiting pressure-sensitive yielding and plastic volumetric deformation is presented in this paper. Two-term expansions of the plane strain and plane stress solutions for a crack in a homogeneous material are obtained. It is shown that a variable-separable solution form under plane strain conditions exists only for weakly pressure-sensitive materials and the limit values of the pressure-sensitivity factor depend on the strain-hardening exponent. The second-order plane strain terms have to be solved as an eigenvalue problem and the elastic terms enter the second-order solutions only when the material has substantial strain-hardening. It follows that the second stress amplitude factor must be determined by the applied load. The values of the second exponents in the stress expansion are slightly larger than zero for most hardening materials and behave as an increasing function of the pressure-sensitivity factor. The finite element computations confirm that the second-order terms under plane strain conditions will increase dominance of the asymptotic solution remarkably. The plane stress analysis shows that the amplitudes of both leading-order and second-order solution are determined by the J-integral for most pressure-sensitive dilatant materials. The variable-separable asymptotic solution exists for all available values of the pressure-sensitivity factor. Because of rapid changes in leading-order terms of the stress component 295-1 at 160° the second-order solution will not significantly improve the prediction of the asymptotic solution in the whole tip field. Numerical results based on the incremental theory of plasticity show that the asymptotic solution characterizes the near-tip fields. Finite strains dominate in the region 295-2 under plane strain conditions. The two-parameter boundary layer formulation with different T-stresses predicts that the higher-order terms are only weakly dependent on the distance to the crack tip and vary significantly with in the forward sector.     

Evaluation of the elastic-plastic mixity parameters on the base of different crack propagation criteria. Part 1. Preliminary analysis

In order to evaluate the mechanical behavior around small-scale yielding crack tip for both plane strain and plane stress, the asymptotic governing equations and their boundary conditions by considering fracture mechanisms are formulated. A total deformation theory of plasticity with a power-law hardening is used. The analysis of the near-tip fields is carried out for both the maximum tensile and shear stress crack growth direction criteria, as well as for the complete range of mixity parameters and various strain-hardening levels. The new scheme of mixed-mode problem solution is proposed. Realationships between elastic and plastic mixity parameters are given as functions of the crack growth direction criterion and the strain-hardening exponent.Translated from Problemy Prochnosti, No. 1, pp. 60–75, January–February, 2005     

Two parameter approach for elastic-plastic fracture of short cracked specimens under mixed mode loading

For mode-I loading, in order to describe the near-tip stress field in a specimen under large scaled yielding, two parameter approaches such as J-T, J-Q and J-A₂ theories have been developed and proved well for their validity and limit. In this work elastic-plastic finite element analysis were performed to investigate the effects of mode mixity and T-stress upon near-tip stress distribution for a small-scale-yield model with the modified boundary layer and CTS (Compact Tension-Shear) configuration under large-scale-yield state. As the results, some peculiar characteristics were found as follows; As the mode mixity increases, normal stresses _rr and near the crack tip in the small-scale-yield model get significantly affected by the positive T-stress as well as the negative T-stress, while the shear stress _r is little affected by T-stress. Also, the near-tip stress distribution of short cracked CTS specimens under the large-scale-yield state agree fairly well with that of the small-scale-yield model with an appropriate positive T-stress. The two parameters approach with J-integral and T-stress seems to be a good tool for describing the near-tip stress field under a mixed mode loading and large-scale-yield state.     

Non-singular stress and velocity fields for crack-tip plastic zones and conditions for uniqueness

Non-singular plastic stress and velocity fields, for the tip of a crack of finite thickness and root radius, are developed as an elastic-plastic crack model that is likely to be more physically realistic than the classical infinitesimal crack with a plastic crack-tip singularity. With a non-singular plastic zone the velocity-field equations are not uniquely determined by the boundary conditions, under large geometrical changes, and they must therefore have the form of a wide set of kinematically-admissible velocity fields. These virtual velocity fields are used to establish the critical work-hardening rate to give a sufficient condition for uniqueness of the crack-tip velocity field in elastic-plastic fracture; it is shown that proof of uniqueness of the velocity field is likely to be an essential requirement for the valid application of elastic-plastic fracture mechanics.The elastic infinitesimal-crack model is shown to give an inadequate representation of the circumferential T-stress distribution at the surface of a crack of finite root radius, and this requires the adoption of a finite-thickness elliptical crack model to give approximate consistency between the elastic stress field and the non-singular plastic stress field at the crack tip.     

Dynamic growth of anti-plane shear cracks in ideally plastic crystals

A near-tip asymptotic analysis is given for the stress and deformation field near the tip of crack propagating dynamically under anti-plane shear in an ideally plastic single crystal. A paricular class of orientations of the crack relative to the crystals is considered so that the yield is so simple diamond shape (relative to directions along perpendicular to the crack) in the plane of the anti-plane shear stresses. The near-tip solution is shown to consists of sectors which carry constant stresses, at yield levels, corresponding to adjacent vertices on the diamond-shaped yield locus, and which are joined along an elastic-plastic shock discontinuity. All plastic flow in the near-tip region occurs in the shock. Plastic strains and particle velocity are finite at the crack tip. The plastic strain is proportional to the elastic strain at onset of yielding and is inversely proportional to the elastic Mach number associated with the speed of crack growth.     

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.     

Creep crack growth in brittle materials

During steady state crack growth by diffusive cavitation at grain boundaries, crack tip fields are relaxed due to the presence of a cavitation zone. In the present analysis, analytic solutions for the actual crack tip stress fields and the crack velocity in the presence of cavitation zone consisting of continuously distributed cavities ahead of the crack tip are derived using the smeared volume concept. Results indicate that the r ^–1/2 singularity is now attenuated to r ^{–1/2 +} (0<<1/2) singularity. The singularity attenuation parameter is a function of the crack velocity and material parameters. The crack growth rate is related to the mode I stress intensity factor K by K ² at relatively high load, K ⁿ at intermediate load, and approaches zero at small load near K _th. Meanwhile, the cavitation zone extends further into the material due to the stress relaxation at the crack tip and the subsequent stress redistribution. Such relaxation effects become very distinct at low crack velocity and low applied load. Key words: Creep crack growth, brittle material, diffusive cavity growth, sintering stress, crack tip stress field.     

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)     

Study of Fatigue Behavior of Epoxy‐Carbon Composites under Mixed Mode I/II Loading

This paper presents an experimental assessment of the initiation and propagation of interlaminar cracks under mixed mode I/II dynamic fracture loading of a composite material with an MTM45‐1 epoxy matrix and unidirectional IM7 carbon‐fiber reinforcement. The aims of the experimental program developed for this purpose are to determine, on the one hand, the initiation curves of the fatigue delamination process, understood as the number of load cycles needed to generate a fatigue crack, and on the other, the crack growth rate (delamination rate) for different percentages of static Gc, in both cases for two mode mixities (0.2 and 0.4) and for a tensile ratio R = 0.1. All this with the goal of quantifying the influence of the degree of mode mixity on the overall behavior of the laminate under fatigue loading. The results show that the energy release rate increases with increasing loading levels for both degrees of mode mixity and that the fatigue limit is located around the same percentages. However, crack growth rate behavior differs from one degree of mode mixity to the other. This difference in the behavior of the material may be due to the varying influence of mode I loading on the delamination process.

     

Simulation of hydride-induced steady-state crack growth in metals – Part II: general near tip field

Hydride-induced steady-state crack propagation in metals is investigated under conditions of constant temperature, plane strain, small-scale yielding and small-scale hydride precipitation, by taking into account the coupling of the operating physical processes. It is shown that the near-tip field depends on a normalized stress intensity factor, which incorporates both effects of the applied stress intensity factor and the crack velocity. According to Part I of the present study, when the normalized stress intensity factor tends to zero, the crack-tip field near the threshold stress intensity factor is produced, which is characterized by a constant hydrostatic stress in the hydride precipitation zone. As the value of the normalized stress intensity factor increases, the evolution of the near-tip field for crack propagation from stage-I to stage-II regime is produced: the actual size of the hydride precipitation zone decreases, the hydrostatic stress increases, deviating from the level of the plateau, and the near-tip field tends to that of a hydrogen-free metal. The near-tip field depends strongly on hydrogen concentration, far from the crack tip. The stage-II crack growth velocity is predicted and the experimentally observed effect of metal yield stress and temperature on crack velocity is confirmed.The present study is part of the project on Hydrogen Embrittlement and Fracture in Hydride Forming Metals, financed by the Foundation for Knowledge and Competence Development and performed in cooperation with Studsvik Nuclear AB and Barsebäck Kraft AB.     

A finite element investigation of quasi-static and dynamic asymptotic crack-tip fields in hardening elastic-plastic solids under plane stress

The asymptotic structures of crack-tip stress and deformation fields are investigated numerically for quasi-static and dynamic crack growth in isotropic linear hardening elastic-plastic solids under mode I, plane stress, and small-scale yielding conditions. An Eulerian type finite element scheme is employed. The materials are assumed to obey the von Mises yield criterion and the associated flow rule. The ratio of the crack-tip plastic zone size to that of the element nearest to the crack tip is of the order of 1.6 × 10⁴. The results of this study strongly suggest the existence of crack-tip stress and strain singularities of the type r ^s (s < 0) at r=0, where r is the distance to the crack tip, which confirms the asymptotic solutions of variable-separable type by Amazigo and Hutchinson [1] and Ponte Castañeda [2] for quasi-static crack growth, and by Achenbach, Kanninen and Popelar [3] for dynamic crack propagation. Both the values of the parameter s and the angular stress and velocity field variations from the present full-field finite element analysis agree very well with those from the analytical solutions. It is found that the dominance zone of the r ^s-singularity is quite large compared to the size of the crack-tip active plastic zone. The effects of hardening and inertia on the crack-tip fields as well as on the shape and size of the crack-tip active plastic zone are also studied in detail. It is discovered that as the level of hardening decreases and the crack propagation speed increases, a secondary yield zone emerges along the crack flank, and kinks in stress and velocity angular variations begin to develop. This dynamic phenomenon observable only for rapid crack growth and for low hardening materials may explain the numerical difficulties, in obtaining solutions for such cases, encountered by Achenbach et al. who, in their asymptotic analysis, neglected the existence of reverse yielding zones along the crack surfaces.