Analytical solutions for crack-tip sectors in perfectly plastic Mises materials under mixed in-plane and out-of-plane shear loading conditions

In this paper, analytical solutions for asymptotic crack-tip plastic sectors in perfectly plastic Mises materials are derived under mixed in-plane and out-of-plane shear loading conditions. Plastic strains in crack-tip plastic sectors are considered to be singular and non-singular. Sectors with singular plastic strains have the solution of centered fan type, and sectors with non-singular plastic strains have the solution of either centered fan or constant stress type. The requirement of stress continuity along the border between a constant stress and a centered fan sectors is then discussed. Discontinuities of the normal and out-of-plane shear stresses in the radial direction between two constant stress sectors are assumed in assembling the crack-tip fields under mixed mode II/III and I/III conditions. Crack-tip fields under mixed mode II/III and I/III conditions with small contributions of mode III are then presented to show the existence of asymptotic crack-tip fields for perfectly plastic materials under mixed in-plane and out-of-plane shear loading conditions. The trends of the angular variations of the mode III stresses under the mixed mode II/III and I/III conditions are generally in agreement with those of the available asymptotic and finite element analyses for low strain hardening materials.     

Mixed-mode fracture of ductile thin-sheet materials under combined in-plane and out-of-plane loading

Cracks in thin structures often are subjected to combined in-plane and out-of-plane loading conditions leading to complex mixed mode conditions in the crack tip region. When applied to ductile materials, large out-of-plane displacements make both experimentation and modeling difficult. In this work, the mixed-mode behavior of thin, ductile materials containing cracks undergoing combined in-plane tension (mode I) and out-of-plane shear (mode III) deformation is investigated experimentally. Mixed-mode fracture experiments are performed and full, three-dimensional (3D) surface deformations of thin-sheet specimens from aluminum alloy and steel are acquired using 3D digital image correlation. General characteristics of the fracture process are described and quantitative results are presented, including (a) the fracture surface, (b) crack path, (c) load-displacement response, (d) 3D full-field surface displacement and strain fields prior to crack growth, (e) radial and angular distributions of the crack-tip strain fields prior to crack growth and (f) singularity analysis of the crack-tip strains prior to crack growth. Results indicate that the introduction of a mode III component to the loading process (a) alters the crack tip fields relative to those measured during nominally mode I loading and (b) significantly increases the initial and stable critical crack-opening-displacement. The data on strain fields in both AL6061-T6 aluminum and GM6208 steel consistently show that for a given strain component, the normalized angular and radial strains at all load levels can be reasonably represented by a single functional form over the range of loading considered, confirming that the strain fields in highly ductile, thin-sheet material undergoing combined in-plane tension and out-of-plane shear loading can be expressed in terms of separable angular and radial functions. For both materials, the displacement and strain fields are (a) similar for both mixed-mode loading angles Φ = 30° and Φ = 60° and (b) different from the fields measured for Mode I loading angle Φ = 0°. Relative to the radial distribution, results indicate that the in-plane strain components do not uniformly exhibit the singularity trends implicit in the HRR theory.     

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.     

Analytical solutions of fully plastic crack-tip higher order fields under antiplane shear

Analytical solutions of higher order fields in a fully plastic power-law hardening material are presented. By the use of hodograph transformation and asymptotic analysis the stress and strain exponents, angular distributions of shear stresses and strains are analytically determined. Special cases, such as linearly elastic, perfectly plastic materials are discussed. Similar characteristics between mode III and mode I plane strain, and mode II plane stress are examined. Comparison of four-term asymptotic solutions with exact and leading term solutions in an infinite strip with a semi-infinite crack under constant displacements along its edges is provided.     

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.     

Singular stress fields at V-notch tips in elastoplastic pressure-sensitive materials

Summary Elastoplastic solutions with the higher-order terms for V-notches in materials exhibiting pressure-sensitive yielding and plastic volumetric deformations are presented. It is shown that under plane strain conditions the variable-separable solution exists within some limited pressure-sensitivities. The limit values grow significantly with increasing notch angle. The leading singularity is a decreasing function of notch angle. The small notch angle can hardly affect the singularity. The plane stress fields are generally more singular than the plane strain ones under the same conditions. The pressure-sensitivity does not affect the plane strain field singularity, but the angular stress distributions. The plane stress singularity is slightly increased by the high pressure-sensitivities at the large notch angles. The second-order exponent grows significantly with increasing notch angle. At a notch angle greater than 60°, the elasticity enters the second-order terms in all materials under plane strain conditions, while the plane stress second-order solutions contain the elasticity effects for all notches. It implies that the second-order terms in the notch analysis may not give a significant improvement in characterising the full stress fields. For an apex notch bounded to a rigid substrate, the leading-order singularity falls with increasing notch angle more slowly than that in the homogeneous pressure-sensitive materials. It vanishes at a notch angle of about 125° for all strain-hardening exponents. The elasticity affects the second-order solutions when notch angle becomes large. Whereas the stress fields are dominated by the hoop stress under assumptions of the traction-free crack surfaces, the shear stress is significant for large angle notches.     

Stress, deformation and damage fields near the tip of a crack in a damaged nonlinear material

The stress, strain, displacement and damage fields near the tip of a crack in a power-law hardening material with continuous damage formation under antiplane longitudinal shear loading are investigated analytically. The interaction between a major crack and distributed microscopic damage is considered by describing the effect of damage in terms of a damage variable D. A deformation plasticity theory coupled with damage and a damage evolution law are formulated. A hodograph transformation is employed to determine the singularity and angular distribution of the crack-tip quantities. Consequently, analytical solutions for the antiplane shear crack-tip fields are obtained. Effects of the hardening exponent n and the damage exponent m on the crack-tip fields are discussed. It is found that the present crack-tip stress and strain solutions for damaged nonlinear material are similar to the well-known HRR fields for virgin materials. However, damage leads to a weaker singularity of stress, and to a stronger singularity of strain compared to that for virgin materials, respectively. The stress associated with damage always falls below the HRR field for virgin material; but the distribution of strain associated with damage lies slightly above the HRR field for r/(J/0) > 1.5 while the difference becomes negligible when r/(J/0) > 2. The limiting distributions of stress and strain may indeed be given by the HRR field.     

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.     

Crack-tip fields in elastic-plastic material under plane stress mode I loading

New results on the crack-tip fields in an elastic power-law hardening material under plane stress mode I loading are presented. Using a generalized asymptotic expansion of the stress function, higher-order terms are found which have newly-discovered characteristics. A series solution is obtained for the elastic-plastic crack-tip fields. The expansion of stress fields contains both the and terms where t_i is real and t_k is complex; the terms σ⁽ⁱ⁾ _pq(θt_i) and σ^(k) _rsθt_k) are real and complex functions of θ respectively. Comparing the results with that for the plane strain mode I loading shows that: (1) the effect of higher-order solutions on the crack-tip fields is much smaller; and (2) the path-independent integral J also controls the second-order or third-order term in the asymptotic solutions of the crack-tip fields for most of the engineering materials (1 < n < 11) in plane stress, while the J-integral does not control the second and the third-order terms for the plane strain mode I case for n > 3. These theoretical results imply that the crack-tip fields can be well characterized by the J-integral, and can be used as a criterion for fracture initiation under plane stress mode I loading. This is in agreement with existing full-field solutions and experimental data that J at crack growth initiation is essentially independent of in-plane specimen geometry. The comparison confirms the theoretical asymptotic solutions developed in this study. This revised version was published online in August 2006 with corrections to the Cover Date.     

Mode I plane stress crack-tip constraint for elastic-plastic materials with different strain hardening

Finite element analyses and simulations have been undertaken to investigate the triaxial constraint in the crack-tip regions of a stationary crack and a steady-state growing crack under mode I plane stress for elastic-plastic materials with different strain hardening. The results show that the triaxial constraint in the crack-tip region is independent of specimen geometry, and material strain hardening, both for a stationary and an extending crack quasi-statically. The triaxial constraints for the various configurations examined are in better accordance with those required by the HRR solution for a stationary crack, which defines the low and similar constraints in crack-tip regions for different material strain hardening in the plane stress case. Along the entire ligament ahead of a crack tip, the constraint level transites gradually from that defined by the HRR solution within the near tip zone to that characterized by the stress intensity factor K _I in the far field.     

On fast fracture in an elastic-(plastic)-viscoplastic solid – 1. Stress and velocity fields with loading and unloading processes

An asymptotic analysis of the near-tip field is given for fast crack propagation in an elastic-plastic-viscoplastic solid. The plasticity of the material is characterised by power law hardening, and the visco-plasticity covers primary, secondary and tertiary creep depending on a parameter q being smaller, equal to and larger than zero, respectively. The yield condition used is Von Mises criterion. Explicit results are given for the order of the crack-tip singularity, the angular position at which unloading occurs, and the angular variations of stresses and velocities in the near crack-tip fields. In particular, it is shown that the eigenvalue, which determines the order of stress singularity, relates only to the viscoplastic parameters but is independent of the crack-tip speed, boundary and loading conditions. Also, it is found that the plasticity effect cannot explicitly enter the asymptotic stress field. Otherwise, additional assumptions would be required.     

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.     

Quasi-statically growing crack-tip fields in elastic perfectly plastic pressure-sensitive materials under plane strain conditions

Quasi-statically growing crack-tip fields in elastic perfectly plastic pressure-sensitive materials under plane strain conditions are investigated in this paper. The materials are assumed to follow the Drucker-Prager yield criterion and the normality flow rule. The asymptotic mode I crack-tip fields are assumed to follow the five-sector assembly of Drugan et al. (1982) for Mises materials. The crack-tip sectors, in turns, from the front of the crack tip are a constant stress sector, a centered fan sector, a non-singular plastic sector, an elastic sector and finally a trailing non-singular plastic sector bordering the crack face. The results of the asymptotic analysis show that as the pressure sensitivity increases, the plastic deformation shifts to the front of the tip, the angular span of the elastic unloading sector increases, and the angular span of the trailing non-singular plastic sector bordering the crack surface decreases. As the pressure sensitivity increases to about 0.6, the angular span of the trailing non-singular plastic sector almost vanishes. The effects of the border conditions between the centered fan sector and the first non-singular plastic sector on the solutions of the crack-tip fields for both Mises and pressure-sensitive materials are investigated in details. This revised version was published online in August 2006 with corrections to the Cover Date.     

Effects of plastic anisotropy on crack-tip behaviour

For a crack in a homogeneous material the effect of plastic anisotropy on crack-tip blunting and on the near-tip stress and strain fields is analyzed numerically. The full finite strain analyses are carried out for plane strain under small scale yielding conditions, with purely symmetric mode I loading remote from the crack-tip. In cases where the principal axes of the anisotropy are inclined to the plane of the crack it is found that the plastic zones as well as the stress and strain fields just around the blunted tip of the crack become non-symmetric. In these cases the peak strain on the blunted tip occurs off the center line of the crack, thus indicating that the crack may want to grow in a different direction. When the anisotropic axes are parallel to the crack symmetry is retained, but the plastic zones and the near-tip fields still differ from those predicted by standard isotropic plasticity.     

The shielding effects of the crack-tip plastic zone

In this paper we demonstrate that a plastically deformed zone around a stressed crack tip can be, mechanically, identified with an inclusion of transformation strain by means of Eshelby equivalent inclusion method. Thus, the shielding effect of the plastic zone can be quantitatively evaluated by the present transformation toughening theory. A closed-form solution to determine the change in the stress intensity factor induced by the plastic zone is given both for plane stress and plane strain mode I cracks under small-scale yielding conditions. By using the present solution, the effects of the strain-hardening behavior of the material, the plane stress and plane strain states and the T-stress on the crack-tip shielding effects are identified.     

Constraint effects on plastic crack-tip fields for plane strain mode-I, II and III cracks in non-hardening materials

To explore constraint effects on fully plastic crakc-tip fields, analytical solutions are examined for mode-I, II and III loading in non-hardening materials under plane strain conditions. The results reveal that under mode-II and III loading the crack-tip stress fields are unique, and thus can be characterized by a `single parameter'. Under mode-I loading, however, the crack-tip stress field is non-unique but can be characterized by two sets of solutions or `two parameters'. One set of the solutions is the well-known Prandtl field and the other is a plastic T-stress field. This conclusion corroborates the observation of McClintock (1971) that the slip-line field is non-unique for plane strain tensile cracks. A two-term plastic solution which combines the Prandtl field and the plastic T-stress field with two parameters B ₁ and B ₂ can then characterize the crack-tip stress field of plane strain mode-I crack over the plastic region and quantify the magnitude of crack-tip constraints. These characters are similar to those for hardening materials. Analyses and examples show that the two-term plastic solution can match well with the slip-line field or finite element results over plastic region. Thus the parameters B ₁ and B ₂ can be used to characterize the constraint level for mode-I finite-sized crack specimens in non-hardening materials under plane strain conditions.     

Dynamic asymptotic mode III crack tip field in rate dependent materials

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

The strain-hardening effect in HRR plane fields according to T-criterion

The strain-hardening effect on fracture is investigated with the aid of the T-criterion using HRR stress fields [1–3] around a crack tip in a power hardening material. Using the appropriate components of strain energy density for the elastic-plastic as well as a nearly elastic expression of the T-criterion, we find the fracture angles, as well as fracture stresses in materials possessing an elastic extended up to a perfectly plastic behavior, by considering plane mixed-mode deformation at the crack tip.Significant influence of the strain hardening coefficient, n on the fracture stress, as well as the hardening parameter mainly appeared in plane strain conditions. This phenomenon was observed almost independently of the solution applied, which provides a nearly elastic or an elastic-plastic expression of the T-criterion describing the fracture conditions.     

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.