Dynamic crack growth along an elastoplastic bimaterial interface

Summary The near-tip stress and deformation rate fields of a crack dynamically propagating along an interface between dissimilar elastic-plastic bimaterials are presented in this paper. The elastic-plastic materials are characterised by theJ ₂-flow theory with linear plastic hardening. The solutions are assumed to be of variable-separable form with a power-law singularity in the radial direction. Two distinct solutions corresponding to the tensile and shear solutions exist with slightly different singularity strengths and very different mixities at the crack tip. The phenomenon of discrete and determinate mixities at the interfacial crack tip is confirmed in dynamic crack growth. This is not an artifact of the variable-separable solution assumption, arising from the linear-hardening material model. The dynamic crack analysis shows that the mixity of the near-tip field is mainly determined by the given material parameters and affected slightly by the crack propagation velocity. A significant variation of the mixity is observed near to the coalescing point of the tensile and shear solutions. The strength of the singularity is almost determined by the smaller strain-hardening alone, and dynamic inertia decreases the stress intensity. The asymptotic solutions reveal that the crack propagation velocity changes only the stress field of the tensile mode significantly. With increasing the crack propagation velocity, the stress singularity of the tensile solutions decreases obviously and the stress triaxiality at the tip (=0) falls considerably at the unity effective stress. These observations imply that the fracture toughness of the interface crack under tensile mode may be significantly higher than that under quasi-static conditions.     

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.     

Determination of the asymptotic crack tip fields for a crack perpendicular to an interface between elastic-plastic materials

Summary The singular behavior near a crack tip at the interface between two power-law hardening materials with the crack perpendicular to the interface is studied for both Mode I and Mode II loading under either plane strain or plane stress conditions. The mathematical model developed can be expressed as a fourth order ordinary differential equation with homogeneous boundary condition. A shooting method is applied to obtain the eigenvalues and to solve the differential equation with homogeneous boundary conditions. When both materials have the same hardening exponent,N, another material parameter, , representing the relative resistance of two materials to plastic deformation, is introduced to reflect the joint effect of the two materials on the singularity. Results indicate that if both materials have the sameN, the singularity at the crack tip is reduced as increases; however, when becomes large there appears to be little change in the singularity for a fixedN. When the hardening exponents are not the same, the mathematical model assumes stress continuity across the interface. The results show that the order of the singularity depends largely on the softer material, with the largest stresses in the harder material.     

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.     

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.     

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.     

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.     

Asymptotic solution for mode III crack growth in J 2-elastoplasticity with mixed isotropic-kinematic strain hardening

Mode III fracture propagation is analyzed in a J ₂-flow theory elastoplastic material characterized by a mixed isotropic/kinematic law of hardening. The asymptotic stress, back stress and velocity fields are determined under small-strain, steady-state, fracture propagation conditions. The increase in the hardening anisotropy is shown to be connected with a decrease in the thickness of the elastic sector in the crack wake and with an increase of the strength of the singularity. A second order analytical solution for the crack fields is finally proposed for the limiting case of pure kinematic hardening. It is shown that the singular terms of this solution correspond to fully plastic fields (without any elastic unloading sector), which formally are identical to the leading order terms of a crack steadily propagating in an elastic medium with shear modulus equal to the plastic tangent modulus in shear.     

Interface corner stress states: plasticity effects

When a butt joint fails, failure often initiates in the region where the interface intersects the stress-free edge. Asymptotic solutions for the stress field found at this type of interface corner are presented for an idealized butt joint with rigid adherends and a thin, essentially semi-infinite, adhesive bond. Linear elastic, power law hardening, and perfectly plastic adhesive models are considered. A stress singularity of type Kr(<0) exists when the adhesive is either linear elastic or power law hardening. The impact of material properties on the order of the stress singularity and the effect of load level and bond thickness on the value of the interface corner stress intensity factor K are detailed. Slip theory is used to determined the asymptotic, interface corner stress field for a perfectly plastic adhesive. This solution indicates that there is a high level of hydrostatic tension, equal to 1.5 _y , in the yielded material along the interface. The three asymptotic solutions are used to construct interface normal stress distributions that closely approximate full, finite element results for an idealization butt joint when small scale yielding conditions apply.     

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.     

An approximate, analytical approach to the `HRR'-solution for sharp V-notches

The well-known so-called `HRR-solution' (Hutchinson, 1968 and Rice and Rosengren, 1968) considers the elasto-plastic stress field in a power-law strain hardening material near a sharp crack. It provides a closed form explicit expression for the stress singularity as a function of the power-law exponent `n' of the material, but the stress angular variation functions are not found in closed form. More recently, similar formulations have appeared in the literature for sharp V-notches under mode I and II loading conditions. In such cases not only is the angular variation of the stress fields obtained numerically, but so is the singularity exponent of the stress field. In the present paper, approximate but accurate closed form solutions are first reported for sharp V-notches with an included angle greater than /6 radians. Such solutions, limited here to Mode I loading conditions, allow a very satisfactory estimate of the angular stress components in the neighbourhood of the notch tip, in the entire range of notch angles and for the most significant values of n (i.e. from 1 to 15). When the notch opening angle tends towards zero, and the notch approaches the crack case, the solution becomes much more complex and a precise evaluation of the parameters involved requires a best-fitting procedure which, however, can be carried out in an automatic way. This solution is also reported in the paper and its degree of accuracy is discussed in detail.     

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°.     

HRR fields for damaged materials

The paper presents an investigation of the interaction between a macroscopic crack and distributed damage in an elastic-plastic material based on the HRR field model for virgin materials. This is achieved by describing the mechanical effects of the distributed micro-cracks in terms of the damage variable D on the HRR fields. Damage evolution equation and the constitutive equations coupled with damage are formulated and the resulting boundary value problems are solved numerically. Material constants , n and m ₀are varied to examine their effects on the resulting stress distributions. It is found that the HRR fields for damaged and virgin materials are surprisingly similar although the severity of damage equivalent stress is of several orders of magnitude higher than the conventional plastic equivalent stress without damage consideration. Furthermore, it is shown theoretically and justified numerically that the J-integral loses its path independency for damaged materials, causing the amplitude of the singularity K _D to remain an unknown variable in the asymptotic analysis.     

Singular behaviour near the tip of a sharp V-notch in a power law hardening material

This paper presents an asymptotic analysis of the near-tip stress and strain fields of a sharp V-notch in a power law hardening material. First, the asymptotic solutions of the HRR type are obtained for the plane stress problem under symmetric loading. It is found that the angular distribution function of the radial stress presents rapid variation with the polar angle if the notch angle is smaller than a critical notch angle; otherwise, there is no such phenomena. Secondly, the asymptotic solutions are developed for antisymmetric loading in the cases of plane strain and plane stress. The accurate calculation results and the detailed comparisons are given as well. All results show that the singular exponent s is changeable for various combinations of loading condition and plane problem.     

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.     

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.     

The asymptotic structure of small-scale yielding interfacial free-edge joint and crack-tip fields

Summary The problem of the small-scale yielding (SSY) plane-strain asymptotic fields for the interfacial free-edge joint singularity is examined in detail, and comparisons are made with the interfacial crack tip. The geometries are idealized as isotropic elasto-plastic materials with Ramberg-Osgood power-law hardening properties bonded to a rigid elastic substrate. The resulting fields are shown to be singular and are presented in terms of radial and angular distributions of stress and displacement, and as idealized plastic slip-line sectors. A fourth-order Runge-Kutta numerical method provides solutions to fundamental equations of equilibrium and compatibility that are verified with those of a highly focused finite element (FE) analysis. It is shown that, as in the case of the crack, the asymptotic singular fields are only dependent on the hardening parameter and only a small range of interfacial mode-mix ratios are permitted. The order for the stress singularity may be formulated in terms of the hardening parameter and the elastic solution for incompressible material. The rigid-slip-line field for the interfacial free-edge joint is presented, and it is shown that there is some significant similarity between the asymptotic fields of the deviatoric polar stresses for the joint and the crack-tip having an elastic wedge sector.     

Creep crack growth in brittle materials

During steady state crack growth by diffusive cavitation at grain boundaries, crack tip fields are relaxed due to the presence of a cavitation zone. In the present analysis, analytic solutions for the actual crack tip stress fields and the crack velocity in the presence of cavitation zone consisting of continuously distributed cavities ahead of the crack tip are derived using the smeared volume concept. Results indicate that the r ^–1/2 singularity is now attenuated to r ^{–1/2 +} (0<<1/2) singularity. The singularity attenuation parameter is a function of the crack velocity and material parameters. The crack growth rate is related to the mode I stress intensity factor K by K ² at relatively high load, K ⁿ at intermediate load, and approaches zero at small load near K _th. Meanwhile, the cavitation zone extends further into the material due to the stress relaxation at the crack tip and the subsequent stress redistribution. Such relaxation effects become very distinct at low crack velocity and low applied load. Key words: Creep crack growth, brittle material, diffusive cavity growth, sintering stress, crack tip stress field.     

On fast fracture in an elastic-(plastic)-viscoplastic solid Part II – The motion of crack

The motion of a crack in an elastic-(plastic )-viscoplastic medium is studied in terms of an energetic analysis. Combined with the stress and velocity fields obtained in Part 1, Kishimoto's energy integral, , is used as a crack driving force to determine its motion. The major results obtained are: (1) dependence of crack speed on a modified near-field parameter, K _I tip, (or equivalently, a modified dynamic energy release, G _I tip), which is different from the usual stress intensity factor K _I of an elastic crack-tip field but is related to it; (2) influence of inelastic effect, such as the viscoplastic exponent n, on the motion of the crack; and (3) stability condition of crack motion. In particular, for the last point, it has been found that, for a given loading and material coefficients, there exist two possible motions of the crack: one is stable crack growth and the other is unstable fracture. The lower and upper bounds of crack motion are also discussed. It is finally shown that the maximum crack velocity is lower than the Rayleigh wave speed, and is dependent on the viscoplastic exponent of the material.     

Reinvestigation of Elastic-Plastic Growth of Mode III Cracks in Power Law Hardening Materials

This work deals with asymptotic analysis of near tip fields for mode III cracks growing quasi-statically in steady in power law hardening materials. In the analysis the stress expansion originally suggested by Gao and Hwang was modified. Attention was focused on the behavior of the asymptotic solutions as hardening exponent n. Analytical expressions of the dominant terms of deformation velocity and flow factor in the centered fan sector have been obtained by expanding them in series of the small parameter s=2/(n-1). Results showed that the asymptotic solution of near tip fields for power hardening materials goes, as , to its corresponding solution for elastic-perfectly plastic materials.