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.     

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.     

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.     

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

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

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

This is the second half of a two-part finite element investigation of quasi-static and dynamic crack growth in hardening elastic-plastic solids under mode I plane stress, steady state, and small-scale yielding conditions. The hardening materials are assumed to obey the von Mises yield criterion and the associated flow rule, and are characterized by a Ramberg-Osgood type power-law effective stress-strain curve. The asymptotic feature of the crack-tip stress and deformation fields, and the influence of hardening and crack propagation speed on these fields as well as on the size and shape of the crack-tip active plastic zone, are addressed in detail. The results of this study strongly suggest the existence of stress and strain singularities of the type [ln(R _o/r )]^s (s>0) at r=0, where r is the distance to the crack tip and R ₀ is a length scaling parameter, which is consistent with the predictions of asymptotic analyses of variable-separable type by Gao et al. [1–4]. Difficulties in estimating the values of R ₀ and s by fitting the results of the present full-field study to the type of singularities shown above are analyzed, and quantititive differences between the results of this study and those of the asymptotic analyses are discussed. As expected, findings presented here share many similarities with those reported in the first part of this study [5] for crack growth in linear hardening solids. A prominent common feature of crack growth in these two types of hardening materials is that as the level of hardening decreases and the crack propagation speed increases, a secondary yield zone emerges along the crack surface, and kinks in the angular variations of the stress and velocity fields begin to develop near where elastic unloading is taking place.     

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.     

Strain and stress fields near the blunted tip of a crack under mixed mode loading and the implications for fracture

The strain distribution in the vacinity of a blunted crack-tip is analysed by slip line theory under the conditions of plane-strain, small-scale yielding, and mixed-mode loading of Modes I and II. A generalized crack-tip opening displacement is introduced by which the strain and stress fields near the blunted crack-tip are determined uniquely over a wide range of Mode I and II combinations. Also, coupled experimental and finite-element analyses under the condition of large-scale yielding reveal that the initiation of stable crack growth occurs when the generalized crack-tip opening displacement attains a critical value which is constant for the material tested. The finite-element analysis is based on the finite deformation theory of elastic-plastic materials. The generalized crack-tip opening displacement criterion is found to be superior to the J-integral and the usual COD for the characterization of the initiation of stable crack growth. The plastic work in a small circular region at the crack-tip is found to be equivalent to the generalized crack-tip opening displacement, as a fracture criterion.     

Steady crack growth in a thin, ductile plate under small-scale yielding conditions: Three-dimensional modeling

This work employs high resolution, finite element computations to investigate key features of the elastic–plastic fields near a steadily advancing crack at quasi-static rates under three-dimensional, small-scale yielding conditions. The model represents a structurally thin component constructed of a material (e.g., Al and Ti alloys) with flow stress and fracture toughness properties that together limit the size of the in-plane plastic zone during steady-growth to no more than several multiples of the plate thickness. The computational approach generalizes the streamline integration procedure used previously for two-dimensional studies into three dimensions to represent steady-state growth on a fixed mesh in a boundary-layer framework. The plate thickness provides the only geometrical length scale. Crack extension occurs at the remotely applied, fixed loading without the need for a local growth criterion. In the first computations of this type, the present work considers a straight crack front advancing under local and global mode I loading with zero T-stress in a moderately hardening material. Applied remote loads at steady growth generate plastic zone sizes ahead of the advancing crack front ranging from 0.25 to 6.4 times the thickness. Key results include: (1) the crack-front fields exhibit a self-similar scaling characterized by a non-dimensional loading parameter; (2) three-dimensional effects extend to distances of approximately 1.5–2.5 times the thickness ahead of the advancing crack front for key values of this loading parameter, beyond which the fields (elastic–plastic then linear-elastic at greater distances) become uniform over the thickness; and (3) crack opening profiles on the outside surface reveal a “wedge-like”, opening shape which simplifies the definition of a crack-tip opening angle.     

Solutions of the second elastic-plastic fracture mechanics parameter in test specimens

Extensive finite element analyses have been conducted to obtain solutions of the A-term, which is the second parameter in three-term elastic-plastic asymptotic expansion, for test specimens. Three mode I crack plane-strain test specimens, i.e. single edge cracked plate (SECP), center cracked plate (CCP) and double edge cracked plate (DECP) were studied. The crack geometries analyzed included shallow to deep cracks. Solutions of A-term were obtained for material following the Ramberg-Osgood power law with hardening exponent of n = 3, 4, 5, 7 and 10. Remote tension loading was applied which covers from small-scale to large-scale yielding. Based on the finite element results, empirical equations to predict the A-terms under small-scale yielding (SSY) to large-scale yielding conditions were developed. In addition, by using the relationships between A and other commonly used second fracture parameters such as Q factor and A₂-term, the present solutions can be used to calculate parameters A₂ and Q as well. The results presented in the paper are suitable to calculate the second elastic-plastic fracture parameters for test specimens for a wide range of crack geometries, material strain hardening behaviors and loading conditions.     

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.     

Plane Stress Crack Tip Field Around a Rapidly Growing Ductile/Rigid Interface Crack

Plane stress dynamic crack growth along a ductile/rigid interface is investigated. The ductile material is taken to be ideally plastic and obey the J₂ flow theory of plasticity. Under steady-state conditions, the asymptotic structure of the crack-tip stress, velocity and strain fields has been obtained. The study reveals that two types of crack-tip sectors exist, namely uniform and nonuniform plastic sectors and that the stress, strain and velocity fields are bounded (nonsingular) in all sectors. In a uniform sector, the rectangular Cartesian components of the stress, strain and velocity fields are constant, and there is no plastic strain accumulation. In a nonuniform sector, the stress, strain and velocity components at a point depend on the angular position of the point in the crack-tip polar coordinate system and are governed by a system of simultaneous ordinary differential equations. This is a sector plastic strains can accumulate. A general crack-tip sector assembly is obtained for a practical range of crack growth speeds. Several nontrivial families of admissible solutions of the crack-tip fields based on this general assembly of uniform and nonuniform crack-tip sectors are presented and discussed. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Mode III crack growth in linear hardening materials with strain gradient effects

The flow-theory version of couple stress strain gradient plasticity is adopted for investigating the asymptotic fields near a steadily propagating crack-tip, under Mode III loading conditions. By incorporating a material characteristic length, typically of the order of few microns for ductile metals, the adopted constitutive model accounts for the microstructure of the material and can capture the strong size effects arising at small scales. The effects of microstructure result in a substantial increase in the singularities of the skew-symmetric stress and couple stress fields, which occurs also for a small hardening coefficient. The symmetric stress field turns out to be non-singular according to the asymptotic solution for the stationary crack problem in linear elastic couple stress materials. The performed asymptotic analysis can provide useful predictions about the increase of the traction level ahead of the crack-tip due to the sole contribution of the rotation gradient, which has been found relevant and non-negligible at the micron scale.     

Finite-element solutions of crack growth in incompressible elastic-plastic solids with various yielding extents and loadings: Detailed comparisons with analytical solutions

We have conducted numerical finite element studies of plane strain quasistatic crack growth in elastic-plastic material for a wide range of applied loading conditions and yielding extents, especially general yielding. To facilitate precise comparisons with previousanalytical results, we have employed a fully incompressible, nonhardening material model. A reduced/selected integration procedure is successfully used to enforce material incompressibility. For crack growth under bending-dominant conditions, we employ an experimentally-measured applied load versus crack length history for a compact tension specimen that experiences crack growth from small-scale yielding through general yielding conditions. A constant crack tip opening angle crack growth criterion is employed to investigate crack growth under tension-dominant loadings in the same geometry. We have also conducted a small-scale yielding crack growth simulation employing a highly refined mesh, and several additional general yielding stationary crack solutions to further explore the effects of different far-field loading combinations. Detailed comparisons of the finite element results with Drugan and Chen's [1] ‘m-family’ of asymptotic analytical solutions are made in an effort to assess the latter's accuracy and range of applicability, and to identify their asymptotically indeterminate parametersm andR as functions of crack growth history. Among several interesting results, we find that Drugan and Chen's near-tip characterizing parameter has a nearly constant value ofm ≈ 1.23 for the entire crack growth process from small-scale yielding through general yielding conditions underbending-dominant loading when specimens have traction-free sides. However, we findm to vary significantly from that value as general yielding conditions are approached intension-dominant loading situations, and whenever specimen sides are subjected to uniform applied loading. The numerical solutions confirm that Chen and Drugan's [2] global approximate analytical solutions for general yielding crack growth are remarkably accurate to substantial distances from the crack tip under a wide variety of loading conditions. The fully incompressible material model employed also facilitates great physical insight into the global stress and deformation fields accompanying general yielding crack growth: numerous figures display the slip lines (which are characteristics for both the stress and velocity fields) throughout the plastically deforming regions.     

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.     

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.     

Strain energy release rates for an interface crack in orthotropic media--a finite element investigation

Strain energy release rate (SERR) components for an interface crack in two-dimensional orthotropic media were obtained using finite element (FE) analysis. The elastic analysis of interface cracks results in oscillatory singularity. This is prevalent over a very small zone near the crack-tip, where the traction free crack faces undergo unacceptable deformations resulting in the interpenetration of crack faces. The individual and total strain energy release rates are calculated using modified crack closure integral (MCCI) method. Although the total SERR converges, it is observed that the individual SERR components are dependent on the values of the smallest element size (Δa) at the crack-tip. It is observed that both the crack opening and sliding displacements are oscillatory when the interpenetration is allowed in the contact zone. The contact zone length (r_c) calculated using Suo's analytical expression [Singularities, interfaces and cracks in dissimilar anisotropic media. Proc. Royal Soc. London, Ser A427 (1990) 331] is in good agreement with the results from FE analysis and MCCI calculations. However, for the chosen material properties, the estimated contact zone length based on the analytical expression proposed by Ni and Nemat-Nasser [J. Mech. Phys. Solids 39 (1991) 113] exhibits a large deviation from the present FE results. It is seen that the mode-II behavior dominates the crack growth, even under mode-I loading.     

Finite-element solutions of crack growth in incompressible elastic-plastic solids with various yielding extents and loadings: Detailed comparisons with analytical solutions

We have conducted numerical finite element studies of plane strain quasistatic crack growth in elastic-plastic material for a wide range of applied loading conditions and yielding extents, especially general yielding. To facilitate precise comparisons with previous analytical results, we have employed a fully incompressible, nonhardening material model. A reduced/selected integration procedure is successfully used to enforce material incompressibility. For crack growth under bending-dominant conditions, we employ an experimentally-measured applied load versus crack length history for a compact tension specimen that experiences crack growth from small-scale yielding through general yielding conditions. A constant crack tip opening angle crack growth criterion is employed to investigate crack growth under tension-dominant loadings in the same geometry. We have also conducted a small-scale yielding crack growth simulation employing a highly refined mesh, and several additional general yielding stationary crack solutions to further explore the effects of different far-field loading combinations. Detailed comparisons of the finite element results with Drugan and Chen's [1] m-family of asymptotic analytical solutions are made in an effort to assess the latter's accuracy and range of applicability, and to identify their asymptotically indeterminate parameters m and R as functions of crack growth history. Among several interesting results, we find that Drugan and Chen's near-tip characterizing parameter has a nearly constant value of m 1.23 for the entire crack growth process from small-scale yielding through general yielding conditions under bending-dominant loading when specimens have traction-free sides. However, we find m to vary significantly from that value as general yielding conditions are approached in tension-dominant loading situations, and whenever specimen sides are subjected to uniform applied loading. The numerical solutions confirm that Chen and Drugan's [2] global approximate analytical solutions for general yielding crack growth are remarkably accurate to substantial distances from the crack tip under a wide variety of loading conditions. The fully incompressible material model employed also facilitates great physical insight into the global stress and deformation fields accompanying general yielding crack growth: numerous figures display the slip lines (which are characteristics for both the stress and velocity fields) throughout the plastically deforming regions.     

Effect of loading rate on dynamic fracture initiation toughness of brittle materials

Numerical simulation is carried out to investigate the effect of loading rate on dynamic fracture initiation toughness including the crack-tip constraint. Finite element analyses are performed for a single edge cracked plate whose crack surface is subjected to uniform pressure with various loading rate. The first three terms in the Williams’ asymptotic series solution is utilized to characterize the crack-tip stress field under dynamic loads. The coefficient of the third term in Williams’ solution, A ₃, was utilized as a crack tip constraint parameter. Numerical results demonstrate that (a) the dynamic crack tip opening stress field is well represented by the three term solution at various loading rate, (b) the loading rate can be reflected by the constraint, and (c) the constraint A ₃ decreases with increasing loading rate. To predict the dynamic fracture initiation toughness, a failure criterion based on the attainment of a critical opening stress at a critical distance ahead of the crack tip is assumed. Using this failure criterion with the constraint parameter, A ₃, fracture initiation toughness is determined and in agreement with available experimental data for Homalite-100 material at various loading rate.     

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.