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.     

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.     

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.     

The mode III full-field solution in elastic materials with strain gradient effects

The near-tip asymptotic field and full-field solution are obtained for a mode III crack in an elastic material with strain gradient effects. The asymptotic analysis shows that, even though the near-tip field is governed by a single parameter B (similar to the mode III stress intensity factor), the near-tip field is very different from the classical K_III field; stresses have r ^-3/2 singularity near the crack tip, and are significantly larger than the classical K _III field within a zone of size l to the crack tip, where l is an intrinsic material length, depending on microstructures in the material. This high-order stress singularity, however, does not violate the boundness of strain energy around a crack tip. The parameter B of the near-tip asymptotic field has been determined for two anti-plane shear loadings: the remotely imposed classical K _III field, and the arbitrary shear stress tractions on crack faces. The mode III full-field solution is obtained analytically for an elastic material with strain gradient effects subjected to remotely imposed classical K _III field. It shows that the near-tip asymptotic field dominates within a zone of size 0.5 l to the crack tip, while strain gradient effects are clearly observed within 5l. It is also shown that the conventional way to evaluate the crack tip energy release rate would lead to an incorrect, infinite value. A new evaluation gives a finite crack tip energy release rate, and is identical to the J-integral. This revised version was published online in August 2006 with corrections to the Cover Date.     

Potential theory approach to a dislocation problem

The BCS dislocation model of a crack under anti-plane shear is interpreted in terms of classical plasticity and a yield criterion of BCS type. The model leads to complete and consistent elastic-plastic and limit load solutions which, however, exhibit unbounded stresses at the crack tip. Using a potential theoretic formulation a generalization of the BCS yield criterion is dealt with and a slight modification of the yield criterion results in more physically acceptable bounded stresses. However, it is necessary that the yield strength vanish at the crack tip. It is felt that the BCS model is not valid in obtaining stress and strain fields in a real material. The model is extremely useful for measuring crack tip displacements.     

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

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

Local crack plasticity and its influences on the global elastic stress field

This paper presents the background and development of a novel ‘plastic inclusion’ approach for dealing with the local plasticity which occurs at the tip of a growing fatigue crack. Localised plasticity arises from crack growth mechanisms and essentially blunts the crack, creates a reversed cyclic plastic zone, and induces shear along the crack flanks, along with the possible generation of wake contact stresses which act on the applied elastic stress field at the boundary of the elastic–plastic enclave surrounding the crack. The paper outlines the development of a meso-scale model of the elastic stress field around a growing crack that explicitly incorporates these interaction effects. The outcome is a modified crack tip stress intensity factor that includes some aspects of the magnitude of plastic wake-induced crack tip shielding and which the authors propose has the potential to help resolve some long-standing controversies associated with plasticity-induced closure. A full-field approach is developed for stress using photoelasticity and also for displacement using digital image correlation.     

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.     

Temperature fields near a running crack tip

Near a running crack tip, the plastic work rate is high. According to the theory of irreversible thermodynamics, the plastic work will be almost completely converted into heat which may lead to high temperature rise at the running crack tip. The plastic zone is regarded as the zone of the heat source, and the plastic work rate as the strength of the heat source. In this paper, the plastic work rate is derived from the solution of stress and strain fields obtained by Chitaley and McClintock[1] for a steady state crack growth under anti-plane shear in an elastic perfectly-plastic material. The dependence of the thermal conductivity on temperature has been considered and a non-linear model for temperature fields has been proposed. The numerical results for glass have been given and compared with other papers.     

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.     

Near-tip field of an anti-plane crack in a micro-cracked power-law solid

Asymptotic near-tip field is investigated for an anti-plane (mode III) crack in a power-law solid permeated by a distribution of micro-cracks. The micro-crack location is assumed to be random, while the micro-crack orientation is taken to be non-random. The anisotropic nature of this kind of damage gives rise to anisotropic constitutive equations for the overall macroscopic strains and stresses. The structure of the asymptotic field at a macro-crack tip is analyzed by solving a nonlinear eigenvalue problem. It is shown that under the assumptions made in this analysis the asymptotic crack tip field of the damaged solid has the same structure as the mode III HRR-field of the undamaged solid. Numerical results are presented for the angular functions, the contours of constant effective shear stress, the normalization constant arising in the near-tip field, and the crack opening displacement. By means of these results, the effects of the micro-crack density and orientation on the crack-tip field will be explored.     

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.     

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.     

Experimental and numerical investigations on the influence of the loading direction on the fatigue crack growth

During a service loading fatigue cracks can be subjected to a mixed mode loading if, due to the alteration of the loading direction, the basic crack modes (Modes I, II and III) are combined. An alteration of the loading direction, e.g. can occur either occasionally paired with an overload (mixed mode overload) or permanently in terms of a mixed mode block loading as a combination of normal and shear stresses.Within the scope of this paper, experimental investigations on both mixed mode overloads, which are interspersed into a Mode I baseline level loading, and mixed mode block loadings are presented. The experimental investigations show that the retardation effect decreases with an increasing amount of Mode II of the overload. Due to the block loading, the fatigue crack growth rate is retarded as well, and the crack is also deflected. The kinking angle depends on the fraction of shear stresses. Furthermore, a detailed elastic–plastic finite element analysis of the fatigue crack growth after mixed mode overloads is presented in order to understand the mechanism of the load interaction effects. By such numerical simulations, it can be shown that, due to mixed mode overloads, plastic deformations occur, which on the one hand reduce the near-tip closure and on the other hand cause a far-field closure. Also the stress distribution before and after the crack tip changes. A mixed mode overload causes lower closure and the crack tip deformations become asymmetrical, which is a reason for the smaller retardation effect of a mixed mode overload.     

Elastic plastic mode III crack under internal shear

A complete solution is presented for the problem of a mode III crack in an infinite elastic perfectly-plastic solid under internal shear stress. This problem is the anti-plane strain equivalent of a mode I crack with internal pressure. The problem is transformed into a boundary value problem for a potential function. The particular case when the applied stress σA is equal to the yield stress σ0 is solved analytically, and the distance to the elastic-plastic boundary is obtained in closed form. The general case when σA σ0 is solved numerically by using the Boundary Element Method for potential problems. Numerical results are given for the distance to the elastic-plastic boundary and the crack tip opening displacement. The extent of the plastic zone ahead of the crack tip is shown to vary linearly with the ratio σA/σ0) when 0.5 ≤ (σA/σ0) ≤ 1. This revised version was published online in August 2006 with corrections to the Cover Date.     

Debonding of circular second phase particles

The plastic relaxation of a shear crack situated normal to the interface of a second phase particle of circular cross section is quantitatively analyzed. The ratio of applied stress to yield stress and the relative displacement of crack faces at the tips in the matrix and at the interface of the second phase particle are related to the crack parameters—namely the length of the crack, the width of the plastic zone along the interface and the width of the plastic zone in the matrix. The effect of the shear modulus and size of the second phase particle on the behavior of the plastic zones is determined. A critical value of the relative displacement of the crack faces at the tip is used as the criterion to determine the tendency to brittle extension of the crack into the matrix or along the interface. Conclusions are made on the debonding of the second phase particle from the matrix.     

Asymptotic analysis of growing crack stress/deformation fields in porous ductile metals and implications for stable crack growth

Asymptotic stress and deformation fields near a quasi-statically growing plane strain tensile crack tip in porous elastic-ideally plastic material, characterized by the Gurson-Tvergaard yield condition and associated flow rule, are derived for small uniform porosity levels throughout the range 0 to 4.54 percent. The solution configuration resembles that for crack growth in fully dense, elastically compressible, elastic-ideally plastic Huber-Mises material for this porosity range, except that the angular extents and border locations of near-tip solution sectors vary with porosity level, as do the stress and deformation fields within sectors. Increasing porosity is found to result in a dramatic reduction in maximum hydrostatic stress level, greater than that for a stationary crack; it also causes a significant angular redistribution of stresses, particularly for a range of angles ahead of the crack and adjacent to the crack flank. The near-tip deformation fields derived are employed to generalize a previously-developed, successful ductile crack growth criterion. Our model predicts that for materials having the same initial slopes of their crack growth resistance curves, but different levels of uniform porosity, higher porosity results in a substantially greater propensity for stable crack growth.     

Strain localization below the root of an yielding notch

Prandtl-Orowan mechanism for plastic crack extension is discussed. A method of calculating strain localization below an yield notch in Prager's elastic-plastic material is described. Firstly, a boundary value problem is solved for an elastic notch under the combined loadings of normal and shear stresses acting on its flank faces. Secondly, the stresses and strains in the compressibility gradient are calculated on the crack extension plane assuming a forward slip. Thirdly, taking a sticking plastic friction on the flank faces, it is found that a large transverse strain localizes at the plastic incompressibility-compressibility gradient ahead of an yielding notch. Finally, it is suggested that Rayleigh-Lin type fluid mechanics instability criterion may help to understand such plastic flow localization problem.