Interfacial crack-tip constraints and <Emphasis Type="Italic">J</Emphasis>-integral for bi-materials with plastic hardening mismatch

This paper investigates interfacial crack tip stress fields and the J-integral for bi-materials with plastic hardening mismatch via detailed elastic-plastic finite element analyses. For small scale yielding, the modified boundary layer formulation with the elastic T-stress is employed. For fully plastic yielding, plane strain single-edge- cracked specimens under pure bending are considered. Interfacial crack tip stress fields are explained by modified Prandtl slip-line fields. It is found that, for bi-materials consisting of two elastic-plastic materials, increasing plastic hardening mismatch increases both crack-tip stress constraint in the lower hardening material and the J-contribution there. The implication of asymmetric J-integral in bi-materials is also discussed.     

Constraint effect on the near tip stress fields due to difference in plastic work hardening for bi-material interface cracks in small scale yielding

The change in near-tip stress field in Small Scale Yielding (SSY) for cracks located at an interface between two materials with different plastic work hardening is investigated. The difference in hardening is termed hardening mismatch, and is quantified through the parameter n, which is the difference in hardening exponent between the two materials. For cracks in elastic-ideally plastic materials the stress level in front of the crack tip is mainly controlled by the angular extent of the part where the slip lines are curved, often referred to as a centered fan like slip line sector. It is shown that for an elastic-ideally plastic material coupled to a material with non-zero hardening, an increase in stress is observed due to an extension of this centered fan like slip line sector. The angular extension of the centered fan like sector is dependent on the radial distance from the crack tip. Further, the change in stress depends strongly on hardening mismatch, increasing as n increases. For the situation with coupling between two non-zero hardening materials it is shown that the full field stress solution develops in a self-similar manner, but differs from the homogeneous case due to a coupling between the radial and angular stress field dependence. The amplitude of the change in stress field is to a rather good approximation only controlled by the hardening mismatch, n, and is more or less independent of the absolute values of hardening exponent of the two materials.     

Plastic Deformation at the Tip of a Tensile Crack in a Non-linear Kinematic Hardening Material

The plastic deformation at the tip of a tensile crack in a non-linear kinematic hardening material under small-scale yielding conditions is investigated, with a view to quantifying the functional dependence of crack-tip plastic blunting size on material's strain hardening parameters. It is shown by dimensional analysis that, for materials being characterised by the Armstrong-Frederick non-linear kinematic hardening rule, the crack-tip blunting parameter depends parametrically on only two non-dimensional parameters; the functional dependence is determined using a parametric finite element analysis.     

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.     

EFFECT OF ELASTIC MISMATCH ON THE GROWTH OF A CRACK INITIALLY TERMINATED AT AN INTERFACE IN ELASTIC PLASTIC BIMATERIALS

Abstract A crack perpendicular to, and initially with the tip on, a bimaterial interface is studied. An asymptotic analysis is performed and crack growth proceeds straight ahead at constant remote load. Mode I conditions and plane strain are assumed. The materials on both sides of the interface are elastic perfectly-plastic with different elastic properties and the same yield stress. A finite element analysis is made and crack growth is simulated by an element relaxation technique. Because of the interface, the crack-tip driving force is not constant, which is reflected in the near-tip state. The development of the plastic zone and the crack opening displacements is presented for different elastic mismatches. Small scale yielding like results are obtained after a crack extension of about the plastic zone size from the interface, i.e. long before a square-root singular stress field may be expected to embed the plastic zone. An important observation is that the development of the crack opening displacement at the initial stage of growth is reversed when plasticity is introduced, as compared to the prediction by an elastic model. A region of stable crack growth is identified at the initial phase of growth into a stiffer material, solely due to elastic mismatch.     

Finite deformation cohesive polygonal finite elements for modeling pervasive fracture

In the present study, mode I crack subjected to cyclic loading has been investigated for plastically compressible hardening and hardening–softening–hardening solids using the crack tip blunting model where we assume that the crack tip blunts during the maximum load and re-sharpening of the crack tip takes place under minimum load. Plane strain and small scale yielding conditions have been assumed for analysis. The influence of cyclic stress intensity factor range (\(\Delta \hbox {K})\), load ratio (R), number of cycles (N), plastic compressibility (\({\upalpha })\) and material softening on near tip deformation, stress–strain fields were studied. The present numerical calculations show that the crack tip opening displacement (CTOD), convergence of the cyclic trajectories of CTOD to stable self-similar loops, plastic crack growth, plastic zone shape and size, contours of accumulated plastic strain and hydrostatic stress distribution near the crack tip depend significantly on \(\Delta \hbox {K}\), R, N, \({\upalpha }\) and material softening. For both hardening and hardening–softening–hardening materials, yielding occurs during both loading and unloading phases, and resharpening of the crack tip during the unloading phase of the loading cycle is very significant. The similarities are revealed between computed near tip stress–strain variables and the experimental trends of the fatigue crack growth rate. There was no crack closure during unloading for any of the load cycles considered in the present study.     

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.     

Effects of non-singular stresses on crack-tip fields for pressure-sensitive materials,Part 1: Plane strain case

Mode I near-tip stress fields for elastic perfectly plastic pressure-sensitive materials under plane strain and small-scale yielding conditions are presented. A Coulomb-type yield criterion described by a linear combination of the effective stress and the hydrostatic stress is adopted in the analysis. The finite element computational results sampled at the distance of a few crack opening displacements from the tip show that, as the pressure sensitivity increases, the magnitudes of the normalized radial and hoop stress ahead of the tip decrease, the total angular span of the singular plastic sectors decreases, and the angular span of the elastic sectors bordering the crack surfaces increases. When non-singular T stresses are considered along the boundary layer of the small-scale yielding model, the near-tip stresses decrease as the T stress decreases. The plastic zone shifts toward the crack surfaces as the T stress increases. When the discontinuities of the radial stress and the out-of-plane normal stress along the border between the plastic sector and the elastic sector are allowed, the angular variations of the asymptotic crack-tip fields agree well with those of the finite element computations. Variation of the Q stresses for pressure-sensitive materials can be found from the asymptotic solutions when the plastic zone size ahead of the tip is relatively larger than the crack opening displacement. In addition the T stress is shown to have strong effects on the plastic zone sizes and shapes which could affect the toughening of pressure-sensitive materials.     

Studies on the fracture mechanics parameters of weldment with mechanical heterogeneity

The fully plastic solutions of welded centre-cracked strip for plane stress problem were carefully investigated with the fully plastic finite element method. It was introduced for assessing the fracture mechanics parameters of weldment with mechanical heterogeneity that there existed an equivalent yielding stress and equivalent strain hardening exponent in the vicinity of crack tip keeping the assessment of fracture mechanics parameters of weldment in the same way as the homogeneous material. The equivalent yielding stress and equivalent strain hardening exponent of various matched weldment were computed and the effect of weld metal width were calculated and discussed on equivalent yielding stress and equivalent strain hardening exponent near crack tip. The engineering approach was given for estimating the fracture mechanics parameters of weldment with mechanical heterogeneity in elastic-plastic range.     

Plane strain crack closure and cyclic hardening

This study is focused on the understanding of the mechanical effects of cyclic hardening on crack tip plasticity and on plasticity-induced crack closure. Various finite element analyses were conducted using abaqus. Cyclic hardening is found to affect both crack closure and the shape of the plastic zone at the crack tip. Crack growth modelling in plane strain conditions in a cyclically hardening material is discussed. An empirical formula is provided which allows the calculation of the crack tip plastic zone size under plane strain conditions in a cyclically hardening material. The effects of overloads are also examined.     

Orowan mechanism for abrupt localized flow and fracture initiation in metals ahead of an yielding notch

Plane strain elastic-plastic stresses are determined in Mises yielding solid at the root of an yielding crack like notch. This external edge notch is infinitely deep, and has a small finite (fixed) flank angle with a small tip root blunting radius. A boundary value type approach has been followed throughout, to solve this famous Orowan-lrwin problem. Firstly, a fictitious elastic stress field is calculated, considering a misfit in the bulk volume loading; these elastic stress expressions are valid when the notch is fully loaded. Secondly, the plastic stresses are determined in the compressibility gradient, maintaining the continuity of stresses and their derivatives at the yielded-unyielded interface. Our calculations reveal that: Orowan mechanism is fairly dominant below the notch root, as well as on ± 45° planes. It is concluded that the flow-localization in the Mises solid is due to a reverse slip, caused by the sudden release of a favourable critical mismatch stress concentration. Some elastic strain energy density is seen to be getting released from the bulk volume, while unloading the misfit load. The mismatch has been created entirely due to the compressibility-incompressibility difference, as suggested by Orowan.

Following Orowan, it is shown here that, before the onset of a stable crack extension, the increase in stress concentration at the notch tip root, is directly proportional to the strength of mismatch strain-localization below the notch, and inversely proportional to the plane strain plastic zone size on the crack extension plane. For a large scale yielding situation, compressive stresses and pure distortion regions are seen to occur at a far field within the plastic enclave.     

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.     

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.     

Failure of a Porous Solid from a Deep Notch

A finite strain finite element method is used to examine the stress state near the tip of a deep notch in an elastic-plastic porous solid. The notch is loaded in mode I plane strain tension and small scale yielding is assumed. Two rate independent strain hardening material models are used: a version of the Gurson model (1977) and the more recent FKM model developed by Fleck, Kuhn and McMeeking (1992). Under increasing K _I, void growth is initially stable and independent of mesh dimension. Localization of plastic flow sets in at a finite value K _i, and the deformation field is mesh-size dependent thereafter. The initiation of crack growth at the notch root is assumed to occur when a critical level of porosity is attained. The results show that the shape of the plastic zone for both the Gurson and the FKM material is highly dependent on the initial porosity. In the case of low initial porosity, the plastic zone shape is similar to that of a fully dense material; at higher initial porosities the plastic zone is concentrated ahead of the notch tip. The effect of the initial void volume fraction on the porosity field and the critical stress intensity factor is studied, and the mesh-size dependence of the results is discussed. The analysis is useful for prediction of the notched strength of porous metals. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

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°.     

Plane-strain mixed-mode near-tip fields in elastic perfectly plastic solids under small-scale yielding conditions

Within the context of the small-strain approach, plane-strain mixed-mode near-tip fields of a stationary crack in an elastic perfectly plastic Mises solid under small-scale yielding conditions are examined by finite element methods. Steady-state stress fields in the immediate vicinity of the crack tip are obtained as the remote loading of the elastic K-field increases. Asymptotic crack-tip solutions consisting of constant stress sectors, centered fan sectors, and an elastic sector are then constructed accordingly. The asymptotic crack-tip stress solutions agree well with the numerical results for a whole spectrum of mixed-mode loadings. Our mixed-mode near-tip solution with an elastic sector differs from that of Saka et al. by one (plastic) constant stress sector situated between the elastic sector and the neighbouring fan sector. The effect of the existence of the elastic sector on the near-tip fields is discussed in the light of the computational results. The plastic mixity factor of the near-tip field is given as a function of the elastic mixity factor of the prescribed K-field. This function is well bounded by that of the perfectly plastic limit of the corresponding solutions for power-law hardening materials given by Shih. Some issues pertaining to the numerical procedures such as the implementation of the small-scale yielding assumption are also addressed.     

Plastic yielding at the tip of a blunt notch during static and fatigue loading

Rice's analytical Mode III solution for the relationship between anti-plane stress and anti-plane strain was used to determine the small scale plastic yielding at the tip of a two-dimensional blunt notch. The results were applied to fatigue loading. The plastic zone size and crack opening displacement derived in the present analysis were determined as functions of applied stress, geometric factors (notch radius and length) and material properties (yield stress and the work hardening rate). The minimum stress intensity required for plastic yielding at a blunt notch tip was postulated to be the experimentally observed threshold stress intensity for fatigue crack initiation. The threshold stress intensity so determined depends not only on the notch geometry but also on material properties. There is good agreement with calculated and measured values of the threshold stress intensity for fatigue crack initiation.     

Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading

An analytical study is carried out on the elastic–plastic stress and strain distributions and on the shape of the plastic zone ahead of parabolic notches under antiplane shear loading and small scale yielding. The material is thought of as obeying an elastic-perfectly-plastic or a strain hardening law. When the notch root radius becomes zero, the analytical frame matches the solutions for the crack case due to Hult–McClintock (elastic-perfectly-plastic material) and Rice (strain hardening material). The analytical frame provides an explicit link between the plastic stress and the elastic stress at the notch tip. Neuber’solution for blunt notches under antiplane shear is also obtained and the conditions under which such a solution is valid are discussed in detail by using elastic and plastic notch stress intensity factors. Finally, revisiting Glinka and Molski’s equivalent strain energy density (ESED), these factors are used also to give, under antiplane shear loading, the increment of the strain energy at the notch tip with respect to the linear elastic case.