Constraint effects on the elastic–plastic fracture behaviour in strain gradient solids

ABSTRACT Crack‐tip constraint effects (or T‐stress effects) on the elastic–plastic fracture behaviour in strain gradient materials are analysed in the present study. The T‐stress effects on the stress distributions along the plane ahead of the stationary and growing crack tip, respectively, are analysed by using the Fleck and Hutchinson strain gradient plasticity formation. For a steadily growing crack, the T‐stress effects on the steady‐state fracture toughness are analysed by adopting the embedded fracture process zone model. In addition, the analysis for the growing crack is applied to an interfacial cracking experiment for a metal/ceramic system, and the material length‐scale parameter appearing in the strain gradient plasticity theory is predicted. In the present analyses, a new finite element method specially designed for strain gradient problems by Wei and Hutchinson is adopted.     

The flow theory of mechanism-based strain gradient plasticity

The flow theory of mechanism-based strain gradient (MSG) plasticity is established in this paper following the same multiscale, hierarchical framework for the deformation theory of MSG plasticity in order to connect with the Taylor model in dislocation mechanics. We have used the flow theory of MSG plasticity to study micro-indentation hardness experiments. The difference between deformation and flow theories is vanishingly small, and both agree well with experimental hardness data. We have also used the flow theory of MSG plasticity to investigate stress fields around a stationary mode-I crack tip as well as around a steady state, quasi-statically growing crack tip. At a distance to crack tip much larger than dislocation spacings such that continuum plasticity still applies, the stress level around a stationary crack tip in MSG plasticity is significantly higher than that in classical plasticity. The same conclusion is also established for a steady state, quasi-statically growing crack tip, though only the flow theory can be used because of unloading during crack propagation. This significant stress increase due to strain gradient effect provides a means to explain the experimentally observed cleavage fracture in ductile materials [J. Mater. Res. 9 (1994) 1734; Scripta Metall. Mater. 31 (1994) 1037; Interface Sci. 3 (1996) 169].     

Interface crack problems with strain gradient effects

In this paper, the strain gradient theory proposed by Chen and Wang (2001a, 2002b) is used to analyze an interface crack tip field at micron scales. Numerical results show that at a distance much larger than the dislocation spacing the classical continuum plasticity is applicable; but the stress level with the strain gradient effect is significantly higher than that in classical plasticity immediately ahead of the crack tip. The singularity of stresses in the strain gradient theory is higher than that in HRR field and it slightly exceeds or equals to the square root singularity and has no relation with the material hardening exponents. Several kinds of interface crack fields are calculated and compared. The interface crack tip field between an elastic-plastic material and a rigid substrate is different from that between two elastic-plastic solids. This study provides explanations for the crack growth in materials by decohesion at the atomic scale.     

J-M based criteria for bifurcation assessment of a crack in elastic-plastic materials under mixed mode I-II loading

In this work, a set of J-M^p based criteria is proposed to assess the bifurcation angle of an elastic-plastic crack in plane strain under mixed mode loading. We first establish a criterion in order to study the competition between the tensile fracture (T-type fracture) and the shear fracture (S-type fracture) in the frame of RKR conception [J. Mech. Phys. Solids 21 (1973) 395]. Based on the generalized HRR [ASTM STP 560 (1974) 187] solution and on the knowledge of the mixity parameter M^p, the RKR based criterion is transformed into a J-M^p based criterion. When the crack grows in tensile manner, the maximum circumferential stress criterion is adopted to determine the crack bifurcation angle. When the crack grows in shear manner, the traditional theory of plasticity, according to which the plastic flow develops essentially along one of the slip bands immediately ahead of the crack tip, is used to determine the crack growth direction. All these proposed criteria are given as function of the mixity parameter M^p and the hardening exponent n of the material. Finally, we carry out some numerical computations by applying the proposed criteria and then compare the obtained results with experimental data gathered in literature. It is shown that these criteria are physically reasonable and highly accurate. Another advantage is that the parameters used are easy to identify numerically and experimentally, therefore, these criteria are simple to apply in engineering structures.     

Crack tip fields in a viscoplastic solid: monotonic and cyclic loading

The asymptotic stress and strain field near the tip of a plane strain Mode I stationary crack in a viscoplastic material are investigated in this work, using a unified viscoplastic model based on Chaboche (Int J Plast 5(3):247–302, 1989). Asymptotic analysis shows that the near tip stress field is governed by the Hutchinson–Rice–Rosengren (HRR) field (Hutchinson in J Mech Phys Solids 16(1):13–31, 1968; Rice and Rosengren in J Mech Phys Solids 16(1):1–12, 1968) with a time dependent amplitude that depends on the loading history. Finite element analysis is carried out for a single edge crack specimen subjected to a constant applied load and a simple class of cyclic loading history. The focus is on small scale creep where the region of inelasticity is small in comparison with typical specimen dimensions. For the case of constant load, the amplitude of the HRR field is found to vanish at long times and the elastic K field dominates. For the case of cyclic loading, we study the effect of stress ratio on inelastic strain and find that the strain accumulated per cycle decreases with stress ratio.     

RESIDUAL STRESS FIELDS DURING FATIGUE CRACK GROWTH

This study outlines the distinction between (1) residual stresses at an ideal crack tip, undergoing reversed deformation in the absence of crack closure, and (2) additional residual stresses generated due to plasticity induced closure upon fatigue crack growth. Residual stresses resulting from reversed deformation in plane strain were higher compared to the plane stress case, while residual stresses generated behind the crack tip were more significant in plane stress compared to plane strain. The origin of these residual stresses was studied for two specimen geometries over a wide range of loading conditions. We define a new crack tip parameter, S_tt as the applied stress level that corresponds to the development of tensile stresses immediately ahead of crack tips. The S_tt levels were significantly higher for a fatigue crack than for an ideal crack. We attribute the difference in S_tt levels between these two cases to plasticity induced closure. The results demonstrate the importance of the S_tt parameter, since the stresses ahead of crack tips could remain compressive even when the crack surfaces are open. Moreover, the study emphasizes the need, when describing fatigue crack growth, to incorporate both the closure concept and residual stress field ahead of a crack tip.     

The effects of shear and near tip deformations on energy release rate and mode mixity of edge-cracked orthotropic layers

In this paper semi-analytical expressions are derived for the energy release rate and the stress intensity factors of edge-cracked homogeneous and orthotropic layers subject to arbitrary generalized end forces. The expressions are accurate for long and short cracks. Following the work of Li et al. [Li S, Wang J, Thouless MD. The effects of shear on delamination in layered materials. J Mech Phys Solids 2004;52(1):193-214] for isotropic bi-material layers, the derivation extends the method proposed by Suo [Suo ZG. Delamination specimens for orthotropic materials. J Appl Mech 1990;57(3):627-34] for axial forces and bending moments in order to include the contribution of the shear forces. The shear contribution to the fracture parameters depends on the shear deformations along the layer and the elastic near tip deformation of the material. Li et al. [Li S, Wang J, Thouless MD. The effects of shear on delamination in layered materials. J Mech Phys Solids 2004;52(1):193-214] derived semi-analytical expressions for the fracture parameters that depend on the crack tip stress resultants, the elastic constants and five numerically-determined constants globally describing the effect of shear. In this paper analogous constants are derived for orthotropic layers and defined by semi-analytical expressions that highlight their physical significance and allow separation of the different contributions. The derivation is based on the assumption that the near tip deformation can be described by means of relative rotations between the cross sections of the different sub-layers at the crack tip (root rotations). The root rotations depend linearly on the crack tip stress resultants through compliance coefficients that are derived numerically in the paper for a wide range of orthotropic materials. Applications to different mixed mode delamination and peeling problems, for which accurate two-dimensional finite element solutions can be found in the literature, highlight the accuracy of the proposed expressions.     

Models of Interface Separation Accompanied by Plastic Dissipation at Multiple Scales

Two continuum mechanical models of interface fracture for interfaces joining materials where at least one undergoes plastic deformation are reviewed and examined critically. The embedded process zone model (EPZ model) has an adhesive zone, characterized by a work of separation and an interface strength, embedded within a continuum model of the adjoining materials. The SSV model imposes an elastic, plasticity-free layer of prescribed thickness between the interface and the surrounding elastic-plastic continuum. Crack advance requires the work of separation to be supplied by the local elastic crack tip field. The objective of each model is to provide the relation between the macroscopic interface toughness (the total work of fracture) and the work of separation. Under steady-state crack growth, the total work of fracture is the work of separation plus the work of plastic dissipation, the latter often far exceeding the former. It will be argued that each model has its own domain of validity, subject to the accuracy of conventional continuum plasticity at small scales, but neither is able to capture the dramatic trends which have been observed in macroscopic toughness measurements stemming from alterations in interface bonding conditions. A unified model is proposed which coincides with the two models in their respective domains of validity and provides a transition between them. Interface separation energy and interface strength (the peak separation stress) each play a central role in the unified model. Strain gradient plasticity is used to illustrate the effect of plastic deformation at the micron scale on the link between interface and macroscopic properties. This revised version was published online in July 2006 with corrections to the Cover Date.     

Influence of isotropic strain hardening on plane strain dynamic growth in elastic-plastic materials

In this paper, dynamic crack growth in an elastic-plastic material is analysed under mode I, plane strain, small-scale yielding conditions using a finite element procedure. The material is assumed to obey J₂ incremental theory of plasticity with isotropic strain hardening which is of the power-law type under uniaxial tension. The influence of material inertia and strain hardening on the stress and deformation fields near the crack tip is investigated. The results demonstrate that strain hardening tends to oppose the role of inertia in decreasing plastic strains and stresses near the crack tip. The length scale near the crack tip over which inertia effects are dominant also diminishes with increase in strain hardening. A ductile crack growth criterion based on the attainment of a critical crack tip opening displacement is used to obtain the dependence of the theoretical dynamic fracture toughness on crack speed. It is found that the resistance offered by the elastic-plastic material to high speed crack propagation may be considerably reduced when it possesses some strain hardening.     

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.     

Computations of fatigue crack growth with strain gradient plasticity and an irreversible cohesive zone model

Computations of fatigue crack growth with a first-order strain gradient plasticity (SGP) model and an irreversible cohesive zone model are reported. SGP plays a significant role in the model predictions and leads to increased fatigue crack growth rates relative to predictions with classical plasticity. Increased magnitudes of tractions and material separation at the crack tip together with reduced crack closure appear as the cause for accelerated crack growth in SGP. Under plane strain conditions SGP appears as an essential feature of the development of the crack closure zone. Size effects are explored relative to changes in internal material length scale as well as to structural length scales.     

Analysis of in-plane transonically propagating interface crack with a finite contact zone

The report of Lambros and Rosakis [(1995) J Mech Phys Solids 43(2): 169–188] has focused attention on steady-state transonic interfacial crack growth accounting for the phenomenon of crack face contact in elastic/rigid bimaterial but could not handle issues relating to energy transmission across the interface. The present paper attempts to provide a complete explicit expression of the asymptotic fields induced by transonically propagating interfacial crack in elastic/elastic bimaterial for in-plane case. The energy distribution on the contact area, crack tip and two singular characteristic lines is analysed thoroughly and compared with the dynamic separated J-integrals. The length of the contact zone is also discussed briefly by establishing energy fracture criterion that satisfies contact condition. The two-dimensional in-plane asymptotic deformation field surrounding the contact area of a crack propagating transonically along an elastic/elastic bimaterial interface is observed and discussed thoroughly.     

The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K _IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K ^H _IC for a single hydride platelet at the crack tip has been also estimated.     

Finite element analysis of a multilayer piezoelectric actuator taking into account the ferroelectric and ferroelastic behaviors

Ferroelasticity and ferroelectricity are the non-linear behaviors exhibited by piezoceramics, especially in the case of high electric field or stress. Many studies have focused on the role of ferroelastic and ferroelectric switching in fracture of actuators. However, engineering reliability analyzes are carried out with tools like finite element software that do not take into account these non-linear phenomena. To overcome such a problem, a simplified phenomenological constitutive law describing the non-linear behavior of piezoceramics has been developed and implemented in the commercial software ABAQUS. This finite element tool is used to study the effects of applied voltage on the electroelastic field concentrations ahead of electrodes in a multilayer piezoelectric actuator. The study lies on the experimental observations made by Shindo et al. [Y. Shindo, M. Yoshida, F. Narita, K. Horiguchi, Electroelastic field concentrations ahead of electrodes in multilayer piezoelectric actuators: experiment and finite element simulation, J. Mech. Phys. Solids 52 (2004) 1109–1124]. Electroelastic analysis on piezoceramics with surface electrode showed that high values of stress and electric displacement arose in the neighborhood of the electrode tip. Thus, the strain, stress and electric displacement concentrations were calculated and the numerical results showed that ferroelectric switching arose in the area of the electrode tip, causing a change in remnant polarization and remnant strain.     

Numerical investigation on stable crack growth in plane stress

Large deformation finite element analysis has been carried out to investigate the stress-strain fields ahead of a growing crack for compact tension (a/W=0.5) and three-point bend (a/W=0.1 and 0.5) specimens under plane stress condition. The crack growth is controlled by the experimental J-integral resistance curves measured by Sun et al. The results indicate that the distributions of opening stress, equivalent stress and equivalent strain ahead of a growing crack are not sensitive to specimen geometry. For both stationary and growing cracks, similar distributions of opening stress and triaxiality can be found along the ligament. During stable crack growth, the crack- tip opening displacement (CTOD) resistance curve and the cohesive fracture energy in the fracture process zone are independent of specimen geometry and may be suitable criteria for characterizing stable crack growth in plane stress. This revised version was published online in August 2006 with corrections to the Cover Date.     

Families of asymptotic tensile crack tip stress fields in elastic-perfectly plastic single crystals

In this work, two families of asymptotic near-tip stress fields are constructed in an elastic-ideally plastic FCC single crystal under mode I plane strain conditions. A crack is taken to lie on the (010) plane and its front is aligned along the direction. Finite element analysis is first used to systematically examine the stress distributions corresponding to different constraint levels. The general framework developed by Rice (Mech Mater 6:317–335, 1987) and Drugan (J Mech Phys Solids 49:2155–2176, 2001) is then adopted to generate low triaxiality solutions by introducing an elastic sector near the crack tip. The two families of stress fields are parameterized by the normalized opening stress prevailing in the plastic sector in front of the tip and by the coordinates of a point where elastic unloading commences in stress space. It is found that the angular stress variations obtained from the analytical solutions show good agreement with finite element analysis.     

The cohesive zone model in a problem of delayed hydride cracking of zirconium alloys

The crack tip model with the cohesive zone ahead of a finite crack tip has been presented. The estimation of the length of the cohesive zone and the crack tip opening displacement is based on the comparison of the local stress concentration, according to Westergaard's theory, with the cohesive stress. To calculate the cohesive stress, von Mises yield condition at the boundary of the cohesive zone is employed for plane strain and plane stress. The model of the stress distribution with the maximum stress within the cohesive zone is discussed. Local criterion of brittle fracture and modelling of the fracture process zone by cohesive zone were used to describe fracture initiation at the hydride platelet in the process zone ahead of the crack tip. It was shown that the theoretical K _IH-estimation applied to the case of mixed plane condition within the process zone is qualitatively consistent with experimental data for unirradiated Zr-2.5Nb alloy. In the framework of the proposed model, the theoretical value of K ^H _IC for a single hydride platelet at the crack tip has been also estimated.     

A finite element analysis of plane strain dynamic crack growth in materials displaying the Bauschinger effect

In this paper dynamic crack growth in an elastic-plastic material is analyzed under mode I plane strain small-scale yielding conditions using a finite element procedure. The main objective of this paper is to investigate the influence of anisotropic strain hardening on the material resistance to rapid crack growth. To this end, materials that obey an incremental plasticity theory with linear isotropic or kinematic hardening are considered. A detailed study of the near-tip stress and deformation fields is conducted for various crack speeds. The results demonstrate that kinematic hardening does not oppose the role of inertia in decreasing the plastic strains and stresses near the crack tip with increase in crack speed to the same extent as isotropic strain hardening. A ductile crack growth criterion based on the attainment of a critical crack opening displacement at a small micro-structural distance behind the tip is used to obtain the dependence of the theoretical dynamic fracture toughness with crack speed. It is found that for any given level of strain hardening, the dynamic fracture toughness displays a much more steep increase with crack speed over the quasi-static toughness for the kinematic hardening material as compared to the isotropic hardening case.     

A simple damage-accumulation model for constraint effects on ductile fracture

A simple model is presented to account for the effects of void-type damage on crack initiation and propagation in ductile steels under plane strain conditions by virtue of elementary fracture mechanics solutions. Multiple primary voids from large inclusions are uniformly distributed ahead of the crack tip. The growth of these primary voids is followed by nucleation of a large population of secondary voids from second-phase particles. A critical accumulative damage based on the length ratio of the damage zone to the spacing of primary voids, is employed as a failure criterion, including contributions from two populations of voids. Damage accumulation depends much on the strain and stress states such as stress triaxilities, which are extracted from existing results instead of detailed computation. Results show the dependence of fracture toughness on the size of damage zones associated with constraints. Initiation of crack growth is insensitive to the constraints since nucleation of fine voids is determined by local deformation. The model captures the transition in mechanisms from void-by-void growth to multiple void interactions in terms of a decreasing trend in the slopes of fracture resistance curves. At high constraints and large damage zone, a steady-state crack advance is identified with constant toughness. Damage accumulation from the growth of primary voids determines subsequent crack growth resistance and the study demonstrates its dependences on the crack-tip constraints.     

Temperature dependence of material length scale for strain gradient plasticity and its effect on near‐tip opening displacement

This paper investigates the temperature dependence of the material length scale in the conventional mechanism‐based strain gradient (CMSG) plasticity theory. The work reported here also examines the plastic strain gradient effect on the opening displacement near a sharp crack tip. The study examines the mechanical properties of two typical structural steels (S355 and S690) in onshore and offshore structures at two different temperatures (20 and 300 °C) through both the uniaxial tension test and the indentation test. The CMSG‐based finite element analysis then confirms a constant material length scale for these two steels at the two tested temperatures, despite the apparent temperature dependence of the macroscopic material parameters measured from the tension test. Using the calibrated material length scale, the subsequent numerical study demonstrates that the magnitude of the near‐tip crack opening displacement computed by the CMSG theory remains significantly lower than that computed from the classical plasticity.