A finite element analysis of plane strain dynamic crack growth at a ductile-brittle interface

In this work, steady, dynamic crack growth under plane strain, small-scale yielding conditions along a ductile-brittle interface is analysed using a finite element procedure. 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 behaviour. The objectives of this work are to establish the validity of an asymptotic solution for this problem which has been derived recently [12], and to examine the effect of changing the remote (elastic) mode-mixity on the near-tip fields. Also, the influence of crack speed on the stress fields and crack opening profiles near the propagating interface crack tip is assessed for various bi-material combinations. Finally, theoretical predictions are made for the variation of the dynamic fracture toughness with crack speed for crack growth under a predominantly tensile mode along ductile-brittle interfaces. Attention is focused on the effect of mismatch in stiffness and density of the constituent phases on the above aspects.     

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.     

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.     

Plane stress near-tip field analysis of steady-state crack growth along a linear-hardening elastic-plastic interface

Summary In this work the asymptotic near-tip stress and velocity fields of a crack propagating steadily and quasi-statically along a ductile interface are presented for plane stress cases. The elastic-plastic materials are characterized by the J₂-flow theory with linear plastic hardening. The solutions are assumed to be of variable-separable form with a power singularity in the radial distance to the crack tip. It is found that two distinct solutions exist with slightly different singularity strengths and very different mixities on the interface ahead of the crack tip. One of the solutions corresponds to a tensile-like mode and the other corresponds to a shear-like mode. An interface will change the near-tip fild of the tensile solution obviously, whereas the shear-like solution maintains its original structure as in homogeneous materials. In cases the elastic bimaterial parameter differs from zero, the two solutions can coalesce at some high strain-hardening. An interface between two high strain-hardening materials only slightly affects the stress and velocity distribution around the tip, whereas the singularity strength deviates from the homogeneous solutions. The strength of the singularity is predominantly determined by the smaller strain-hardening material. Poisson's ratio affects variation of the singularity as a function of strain-hardening slightly if the coalescing point of the variable-separable solution is not approached. Only for the very distinct elastic moduli the near-tip field approaches the rigid interface solution.     

The effect of T-stress on plane strain dynamic crack growth in elastic–plastic materials

In this paper, the effects of T‐stress on steady, dynamic crack growth in an elastic–plastic material are examined using a modified boundary layer formulation. The analyses are carried out under mode I, plane strain conditions by employing a special finite element procedure based on moving crack tip coordinates. The material is assumed to obey the J₂ flow theory of plasticity with isotropic power law hardening. The results show that the crack opening profile as well as the opening stress at a finite distance from the tip are strongly affected by the magnitude and sign of the T‐stress at any given crack speed. Further, it is found that the fracture toughness predicted by the analyses enhances significantly with negative T‐stress for both ductile and cleavage mode of crack growth.     

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.     

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.     

Dynamic fields near a crack tip growing in an elastic-perfectly-plastic solid

A complete asymptotic solution is given for the fields in the neighborhood of the tip of a steadily advancing crack in an incompressible elastic-perfectly-plastic solid.For Mode I crack growth in the plane strain condition, the following noteworthy results are revealed: (1) The entire crack tip in steady crack growth is surrounded by a plastic region, and no elastic unloading is predicted by the complete dynamic asymptotic solution. Thus, the elastic unloading region predicted by the result of neglecting the important influence of the inertia terms in the equations of motion. (2) Unlike the quasi-static solution, the dynamic solution yields strain fields with a logarithmic singularity everywhere near the crack tip. (3) The stress field varies throughout the entire crack tip neighborhood, but does display behavior which can be approximated by a constant field followed by an essentially centered-fan field and then by another constant field, especially for small crack growth speeds. Indeed, the stress field reduces to that for the stationary crack, as the crack tip velocity - measured by the Mach number, M - reduces to zero; the strain field, however, does not reduce to that for the static solution, as M vanishes. (4) There are two shock fronts emanating from the crack tip across which certain stress and strain components undergo jump discontinuities. The location of the shock fronts and the magnitude of the jumps depend on the crack growth speed. The stress jump vanishes while the strain jump becomes unbounded, as the crack tip speed goes to zero.Finally, the Mode III steady-state crack growth is reviewed and, on the basis of Mode I and Mode III results, it is concluded that ductile fracture criteria for nonstationary cracks must be based on solutions which include the inertia effects, and that for this purpose, quasi-static solutions may be inadequate. Then, a possible ductile fracture criterion is suggested and discussed.One interesting feature of the complete dynamic asymptotic solution is that, unlike the quasi-static solution, it yields the same strain singularity for all three fracture modes.     

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.     

Analytic and numerical studies on mode I and mode II fracture in elastic-plastic materials with strain gradient effects

This paper presents a study on fracture of materials at microscale (∼1 μm) by the strain gradient theory (Fleck and Hutchinson, 1993; Fleck et al., 1994). For remotely imposed classical K fields, the full-field solutions are obtained analytically or numerically for elastic and elastic-plastic materials with strain gradient effects. The analytical elastic full-field solution shows that stresses ahead of a crack tip are significantly higher than their counterparts in the classical K fields. The sizes of dominance zones for mode I and mode II near-tip asymptotic fields are 0.3l and 0.5l,while strain gradient effects are observed within land 2l to the crack tip, respectively, where l is the intrinsic material length in strain gradient theory and is on the order of microns in strain gradient plasticity (Fleck et al., 1994; Nix and Gao, 1998; Stolken and Evans, 1997). The Dugdale–Barenblatt type plasticity model is obtained to provide an estimation of plastic zone size for mode II fracture in materials with strain grain effects. The finite element method is used to investigate the small-scale-yielding solution for an elastic-power law hardening solid. It is found that the size of the dominance zone for the near-tip asymptotic field is the intrinsic material lengthl. For mode II fracture under the small-scale-yielding condition, transition from the remote classical K _IIfield to the near-tip asymptotic field in strain gradient plasticity goes through the HRR field only when K _IIis relatively large such that the plastic zone size is much larger than the intrinsic material length l. For mode I fracture under small-scale-yielding condition, however, transition from the remote classical K _I field to the near-tip asymptotic field in strain gradient plasticity does not go through the HRR field, but via a plastic zone. This revised version was published online in July 2006 with corrections to the Cover Date.     

A finite element analysis of plane strain dynamic crack growth in materials displaying the Bauschinger effect

In this paper dynamic crack growth in an elastic-plastic material is analyzed under mode I plane strain small-scale yielding conditions using a finite element procedure. The main objective of this paper is to investigate the influence of anisotropic strain hardening on the material resistance to rapid crack growth. To this end, materials that obey an incremental plasticity theory with linear isotropic or kinematic hardening are considered. A detailed study of the near-tip stress and deformation fields is conducted for various crack speeds. The results demonstrate that kinematic hardening does not oppose the role of inertia in decreasing the plastic strains and stresses near the crack tip with increase in crack speed to the same extent as isotropic strain hardening. A ductile crack growth criterion based on the attainment of a critical crack opening displacement at a small micro-structural distance behind the tip is used to obtain the dependence of the theoretical dynamic fracture toughness with crack speed. It is found that for any given level of strain hardening, the dynamic fracture toughness displays a much more steep increase with crack speed over the quasi-static toughness for the kinematic hardening material as compared to the isotropic hardening case.     

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.     

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.     

Crack-tip fields for matrix cracks between dissimilar elastic and creeping materials

The stress fields near the tip of a matrix crack terminating at and perpendicular to a planar interface under symmetric in-plane loading in plane strain are investigated. The bimaterial interface is formed by a linearly elastic material and an elastic power-law creeping material in which the crack is located. Using generalized expansions at the crack tip in each region and matching the stresses and displacements across the interface in an asymptotic sense, a series asymptotic solution is constructed for the stresses and strain rates near the crack tip. It is found that the stress singularities, to the leading order, are the same in each material; the stress exponent is real. The oscillatory higher-order terms exist in both regions and stress higher-order term with the order of O(r°) appears in the elastic material. The stress exponents and the angular distributions for singular terms and higher order terms are obtained for different creep exponents and material properties in each region. A full agreement between asymptotic solutions and the full-field finite element results for a set of test examples with different times has been obtained.     

Analytical forms of higher-order asymptotic elastic-plastic crack-tip fields in a linear hardening material under antiplane shear

An analytical study of the higher-order asymptotic solutions of the stress and strain fields near the traction-free crack tip under antiplane shear in a linear hardening material is investigated. The results show that every term of the asymptotic fields is controlled by both elasticity and plasticity and all the higher-order asymptotic fields are governed by linear nonhomogeneous equations. The first four term solutions are presented analytically and the first four terms are described by two independent parameters J and K ₂. The amplitude of the second order term solution is only dependent on the material properties, but independent of loading and geometry. This paper focuses on the case with traction-free crack surface boundary conditions. The effects of different crack surface boundary conditions, such as clamped and mixed surfaces, on the crack-tip fields are also presented. Comparison of multi-term solution with leading term solution, and finite element solution in an infinite strip with semi-infinite crack under constant displacements along the edges is provided.     

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.     

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.     

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.     

An asymptotic analysis of dynamic crack growth in rubber

Asymptotic analyses of the mechanical fields in front of stationary and propagating cracks are important for several reasons. For example, they facilitate the understanding of the mechanical and physical state in front of crack tips, and they enable prediction of crack growth. Furthermore, efficient modelling of arbitrary crack growth by use of XFEM (extended finite element method) requires accurate knowledge of the asymptotic crack tip fields. The present study focuses on the asymptotic fields in front of a crack that propagates dynamically in rubber. Static analyses of this type of problem have been made in previous studies. In order to be able to compare the present results with these earlier studies, the constitutive model from Knowles and Sternberg (J. Elast. 3:67–107, 1973) was adopted. It is assumed that viscoelastic stresses become negligible compared with the singular elastic stresses close to the crack tip. The present analysis shows that in materials with a significant hardening, the inertia term in the equations of motion becomes negligible in the asymptotic analysis. However, for a neoHookean type of model, inertia comes into play and causes a maximum theoretical crack speed that equals the shear wave speed.     

Analysis of elastoplastic sharp notches

In this paper the elastoplastic solutions with higher-order terms for apex V-notches in power-law hardening materials have been discussed. Two-term expansions of the plane strain and the plane stress solutions have been obtained. It has been shown that the leading-order singularity approaches the value for a crack when the notch angle is not too large. In plane strain cases the elasticity does not enter the second-order solutions when the notch opening angle is too small. For a large notch angle, the two-term expansions of the plane strain near-tip fields are described by a single amplitude parameter. The plane stress solutions generally contain the elasticity terms. The boundary layer formulations based on the small-strain plasticity theory confirm that a dominance zone exists ahead of the notch tip. Finite element results give good agreement to the asymptotic solutions under both plane strain and plane stress conditions. The second-order terms cannot improve the predictions significantly. The near-tip fields are dominated by a single parameter. Finite element calculations under the finite strain J ₂-flow plasticity theory revealed that the finite strains can only affect local characterization of the asymptotic solution. The asymptotic solution has a large dominance zone around the notch tip. For an apex notch bounded to a rigid substrate the leading-order singularity falls with the notch angle significantly more slowly than in the homogeneous material. It vanishes at the notch angle about 135° for all power-hardening exponents. The elasticity effects enter the second-order solutions when the notch angle becomes large enough. The tip fields are characterized by the hydrostatic stress and the shear stress ahead of the notch.