Steady-state crack growth and fracture work based on the theory of mechanism-based strain gradient plasticity

Mode I steady-state crack growth is analyzed under plane strain conditions in small scale yielding. The elastic-plastic solid is characterized by the mechanism-based strain gradient (MSG) plasticity theory [J. Mech. Phys. Solids 47 (1999) 1239, J. Mech. Phys. Solids 48 (2000) 99]. The distributions of the normal separation stress and the effective stress along the plane ahead of the crack tip are computed using a special finite element method based on the steady-state fundamental relations and the MSG flow theory. The results show that during the steady-state crack growth, the normal separation stress on the plane ahead of the crack tip can achieve considerably high value within the MSG strain gradient sensitive zone. The results also show that the crack tip fields are insensitive to the cell size parameter in the MSG theory. Moreover, in the present research, the steady-state fracture toughness is computed by adopting the embedded process zone (EPZ) model. The results display that the steady-state fracture toughness strongly depends on the separation strength parameter of the EPZ model and the length scale parameter in the MSG theory. Furthermore, in order for the results of steady crack growth to be comparable, an approximate relation between the length scale parameters in the MSG theory and in the Fleck-Hutchinson strain gradient plasticity theory is obtained.     

Constraint effects on the elastic–plastic fracture behaviour in strain gradient solids

ABSTRACT Crack‐tip constraint effects (or T‐stress effects) on the elastic–plastic fracture behaviour in strain gradient materials are analysed in the present study. The T‐stress effects on the stress distributions along the plane ahead of the stationary and growing crack tip, respectively, are analysed by using the Fleck and Hutchinson strain gradient plasticity formation. For a steadily growing crack, the T‐stress effects on the steady‐state fracture toughness are analysed by adopting the embedded fracture process zone model. In addition, the analysis for the growing crack is applied to an interfacial cracking experiment for a metal/ceramic system, and the material length‐scale parameter appearing in the strain gradient plasticity theory is predicted. In the present analyses, a new finite element method specially designed for strain gradient problems by Wei and Hutchinson is adopted.     

Finite deformation finite element analyses of tensile growing crack fields in elastic-plastic material

Finite deformation finite element analyses of plane strain stationary and quasi-statically growing crack fields in fully incompressible elastic-ideally plastic material are reported for small-scale yielding conditions. A principal goal is to determine the differences between solutions of rigorous finite deformation formulation and those of the usual small-displacement-gradient formulation, and thereby assess the validity of the (nearly all) extant studies of ductile crack growth that are based on a small-displacement-gradient formulation. The stationary crack case with a significantly blunted tip is studied first; excellent agreement in stress characteristics at all angles about the crack tip and up to a radius of about three times the crack tip opening displacement is shown between Rice and Johnson's [1] approximate analytical solution and our numerical solution. Outside this radius, the numerical results agree very well with Drugan and Chen's [2] small-displacement-gradient analytical characteristics solution in the region of principal plastic deformation. Thus we identify accurate analytical representations for the stress field throughout the plastic zone of a blunted stationary crack. For the growing crack case, the macroscopic difference in crack tip opening profiles between previous small-displacement-gradient solutions and the present results is shown to be negligible, as is the difference in the stress fields in plastic regions. The stress characteristics again agree very well with analytical results of [2]. The numerical results suggest—in agreement with a recent analytical finite deformation study by Reid and Drugan [3]—that it is the finite geometry changes rather than the additional spin terms in the objective constitutive equation that cause any differences between the small-displacement-gradient and the finite deformation solutions, and that such differences are nearly indistinguishable for growing cracks.     

Interface crack problems with strain gradient effects

In this paper, the strain gradient theory proposed by Chen and Wang (2001a, 2002b) is used to analyze an interface crack tip field at micron scales. Numerical results show that at a distance much larger than the dislocation spacing the classical continuum plasticity is applicable; but the stress level with the strain gradient effect is significantly higher than that in classical plasticity immediately ahead of the crack tip. The singularity of stresses in the strain gradient theory is higher than that in HRR field and it slightly exceeds or equals to the square root singularity and has no relation with the material hardening exponents. Several kinds of interface crack fields are calculated and compared. The interface crack tip field between an elastic-plastic material and a rigid substrate is different from that between two elastic-plastic solids. This study provides explanations for the crack growth in materials by decohesion at the atomic scale.     

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

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

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.     

Asymptotic fields near the crack tip in elastic-perfectly plastic crystals

The basic equations of plane strain problem for the elastic-perfectly plastic crystals with double slip systems have been presented in the basis of three dimensional flow theory of crystal plasticity. Using these equations the stationary crack tip stress and deformation fields are analysed for tensile load. The fields involve an elastic angular sector and are fully continuous. An asymptotic solution is also obtained for the steadily growing crack that consists of five angular sectors: two plastic angular sectors in the front of the crack tip connected with the boundary on which the associated velocity field has discontinuities; a secondary plastic angular sector near the crack face; two elastically unload angular sectors connected with the boundary on which the discontinuity of the associated velocity field occurs. The asymptotoic solution is not unique. A family of solutions is obtained. Finally, the application of these solutions on both FCC and BCC crystals is discussed.     

Analysis of size effects based on a symmetric lower-order gradient plasticity model

Size effects become significant as soon as strain gradients are high. For instance the stress state around the crack tip cannot be described using conventional plasticity theory due to the extreme strain gradients and micro-structural metallurgical changes. To give an accurate prediction of the structure integrity and to quantify the material failure process, it is necessary to combine the strain gradients into constitutive equations. The present paper deals with developing a symmetric lower-order gradient-dependent constitutive model, which contains only the first-order gradient of equivalent plastic strain as regulator and introduces an intrinsic material length scale to take into account the micro-structure characteristics of materials. The flow stress is assumed linearly depending on the square root of first gradient of the equivalent plastic strain. Our analytical solutions for bending, torsion, void growth and interface stress fields show that the present model is computationally efficient to implement and may catch size effects in different geometries, in comparing with the known experiments. Furthermore, we are briefly discussing the micro-indentation based on the gradient plasticity. The quadratic dependence of the hardness on the inverse of the indentation depth is confirmed. The introduced intrinsic material length can be identified from the micro-indentation test.     

Crack tip fields in ductile crystals

Results on the asymptotic analysis of crack tip fields in elastic-plastic single crystals are presented and some preliminary results of finite element solutions for cracked solids of this type are summarized. In the cases studied, involving plane strain tensile and anti-plane shear cracks in ideally plastic f c c and b c c crystals, analyzed within conventional small displacement gradient assumptions, the asymptotic analyses reveal striking discontinuous fields at the crack tip.For the stationary crack the stress state is found to be locally uniform in each of a family of angular sectors at the crack tip, but to jump discontinuously at sector boundaries, which are also the surfaces of shear discontinuities in the displacement field. For the quasi-statically growing crack the stress state is fully continuous from one near-tip angular sector to the next, but now some of the sectors involve elastic unloading from, and reloading to, a yielded state, and shear discontinuities of the velocity field develop at sector boundaries. In an anti-plane case studied, inclusion of inertial terms for (dynamically) growing cracks restores a discontinuous stress field at the tip which moves through the material as an elastic-plastic shock wave. For high symmetry crack orientations relative to the crystal, the discontinuity surfaces are sometimes coincident with the active crystal slip planes, but as often lie perpendicular to the family of active slip planes so that the discontinuities correspond to a kinking mode of shear.The finite element studies so far attempted, simulating the ideally plastic material model in a small displacement gradient type program, appear to be consistent with the asymptotic analyses. Small scale yielding solutions confirm the expected discontinuities, within limits of mesh resolution, of displacement for a stationary crack and of velocity for quasi-static growth. Further, the discontinuities apparently extend well into the near-tip plastic zone. A finite element formulation suitable for arbitrary deformation has been used to solve for the plane strain tension of a Taylor-hardening crystal panel containing, a center crack with an initially rounded tip. This shows effects due to lattice rotation, which distinguishes the regular versus kinking shear modes of crack tip relaxation. and holds promise for exploring the mechanics of crack opening at the tip.     

Applications of the element-free Galerkin method for singular stress analysis under strain gradient plasticity theories

It is known that the plasticity models affect characterization of the crack tip fields. To predict failure one has to understand the crack tip stress field and control the crack. In the present work the element-free Galerkin methods for gradient plasticity theories have been developed and implemented into the commercial finite element code ABAQUS and used to analyze crack tip fields. Based on the modified boundary layer formulation it is confirmed that the stress singularity in the gradient plasticity theories is significantly higher than the known HRR solution and seems numerically to equal to 0.78, independently of the strain-hardening exponent. The strain singularity is much lower than the known HRR one. The crack field in gradient plasticity under small-scale yielding condition consists of three zones: The elastic K-field, the plastic HRR-field dominated by the J-integral and the hyper-singular stress field. Even under gradient plasticity there exists an HRR-zone described by the known J-integral, whereas the hyper-singular zone cannot be characterized by J. The hyper-singular zone is very small (r ? J/σ₀) and contained by the HRR zone in the infinitesimal deformation framework. The finite strains under the gradient plasticity will not eliminate the stress singularity as r → 0, in contrast to the known finite strain results under the Mises plasticity. Numerically no significant changes in characterization of the stress field were found in comparison with the infinitesimal deformation theory. Since the hyper-singular stress field is much smaller than the HRR zone and in the same size as the fracture process zone, one may still use the known J concept to control the crack in the gradient plasticities. In this sense the gradient plasticity will not change characterization of the crack.     

Crack tip fields at a ductile single crystal-rigid material interface

Small-scale yielding around a stationary crack along a ductile single crystal–rigid material interface is analyzed. Plane strain conditions are assumed to prevail and geometry changes are neglected. The analyses are carried out using both continuum slip and discrete dislocation plasticity theory for model fcc and bcc crystal geometries having either two or three slip systems. Numerical and analytical asymptotic solutions are presented for continuum slip plasticity theory. Solutions exhibiting both slip bands and kink bands are obtained. The addition of a third slip system to ductile single crystals having two slip systems is found to have a significant effect on the interface crack-tip fields. The results illustrate the role that each of the formulations considered can play in elucidating crack tip fields in single crystals.     

A model on twinning-induced crack kinking out of a metal-ceramics interface

A continuum model is proposed to study the effects of deformation twinning on interface crack kinking in metal/ceramics layered materials. At the final stage of material failure, plastic work hardening exhausts and lattice rotation becomes main mechanism after competing with dislocation gliding. The crack-tip plasticity is established in terms of the second gradient of microrotation due to the coupling effect of the twins. The formed twinning structures not only shield the crack tip, but inhibit further dislocation emission by increasing the near-tip stress levels. A Dislocation-Free Zone (DFZ) can exist in the immediate vicinity of the tip. The model is based on the equivalence of the stresses derived from twin-based crack-tip plasticity, macroscopic plasticity and elasticity on the boundary. The two-parameter characterization of near-tip stress fields is used for the outer plastic zone to account for constraint effects. Crack kinking out of the interface follows the direction of the maximum flow stress from the crack-tip plasticity. The DFZ size and the crack-tip shielding ratio, as well as the kink angle, are obtained for various values of low hardening exponents and crack-tip constraints.     

Analysis of crack tip plasticity for microstructurally small cracks using crystal plasticity theory

Crack tip plastic zone sizes and crack tip opening displacements (CTOD) for stationary microstructurally small cracks are calculated using the finite element method. To simulate the plastic deformation occurring at the crack tip, a two-dimensional small strain constitutive relationship from single crystal plasticity theory is implemented in the finite element code ANSYS as a user-defined plasticity subroutine. Small cracks are modeled in both single grains and multiple grains, and different crystallographic conditions are considered. The computed plastic zone sizes and CTOD are compared with those found using conventional isotropic plasticity theory, and significant differences are observed.     

Fracture analysis in the conventional theory of mechanism-based strain gradient (CMSG) plasticity

In a remarkable series of experiments, Elssner et al. (1994) and Korn et al. (2002) observed cleavage cracking along a bimaterial interface between Nb and sapphire. The stress required for cleavage cracking is around the theoretical strength of the material. Classical plasticity models fall short to reach such a high stress level. We use the conventional theory of mechanism-based strain gradient plasticity (Huang et al., 2004) to investigate the stress field around the tip of an interface crack between Nb and sapphire. The tensile stress at a distance of 0.1 m to the interface crack tip reaches 13.3_Y, where _Y is the yield stress of Nb. This stress is nearly 4 times of that predicted by classical plasticity theory (3.6_Y) at the same distance to the crack tip, and is high enough to trigger cleavage cracking in materials and interfaces. This is consistent with Elssner et al.'s (1994) and Korn et al.'s (2002) experimental observations.     

Representation of crack-tip plasticity in pressure sensitive geomaterials

This paper presents a model based on dislocation theory for representation of crack tip plasticity in a pressure sensitive material. The basic equations are derived from a simple model of the crack tip plasticity which represents the plasticity by superdislocations placed at the effective centres of the complete slip process distributed around the crack tip. The positions and strengths of the superdislocations are determined by imposing the following conditions:(i) the total stress-intensity factor at the crack tip is zero(ii) the total stress minus the self stress of the superdislocation acting at the superdislocation position is just equal to the friction stress due to the Mohr-Coulomb yield criterion and(iii) the total crack opening displacement produced by the model is maximized.Condition (iii) enables the angle of the slip band on which the superdislocation lies to be determined. The results are presented in a dimensionless form allowing the discussion of particular cases and the recognition of the dominant parameters. A series of parametric studies are carried out to demonstrate that the model can capture the essentials of the crack tip plasticity.     

Tensile crack tip fields in elastic-ideally plastic crystals

Crack tip stress and deformation fields are analyzed for tensile-loaded ideally plastic crystals. The specific cases of (0 1 0) cracks growing in the [1 0 1] direction, and (1 0 1) cracks in the [0, 1, 0] direction, are considered for both fcc and bcc crystals which flow according to the critical resolved shear stress criterion. Stationary and quasistatically growing crack fields are considered. The analysis is asymptotic in character; complete elastic-plastic solutions have not been determined. The near-tip stress state is shown to be locally constant within angular sectors that are stressed to yield levels at a stationary crack tip, and to change discontinuously from sector to sector. Near tip deformations are not uniquely determined but fields involving shear displacement discontinuities at sector boundaries are required by the derived stress state. For the growing crack both stress and displacement must be fully continuous near the tip. An asymptotic solution is given that involves angular sectors at the tip that elastically unload from, and then reload to, a plastic state. The associated near-tip velocity field then has discontinuities of slip type at borders of the elastic sectors. The rays, emanating from the crack tip, on which discontinuities occur in the two types of solutions are found to lie either parallel or perpendicular to the family of slip plane traces that are stressed to yield levels by the local stresses. In the latter case the mode of concentrated shear along a ray of discontinuity is of kink type. Some consequences of this are discussed in terms of the dislocation generation and motion necessary to allow the flow predicted macroscopically.     

A coherent gradient sensor for crack tip deformation measurements: analysis and experimental results

A first order diffraction analysis of an optical interferometer, Coherent Gradient Sensor (CGS), for measuring surface gradients is presented. Its applicability in the field of fracture mechanics is demonstrated by quantitatively measuring the gradients of out-of-plane displacements around a crack tip in a three point bent fracture specimen under static loading. This method has potential for the study of deformation fields near a quasi-statically or dynamically growing crack.     

Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials

The near-tip field of a dynamically propagating crack in an incompressible power-law hardening material is studied using the asymptotic method as an extension of Leighton et al. (Journal of the Mechanics and Physics of Solids, 1987, Vol.35, pp.541–563). The crack is subjected to tensile loads and propagates steadily under plane strain conditions. The material deformation is described by the J ₂ flow theory of plasticity with infinitesimal displacement gradients. Results show that, in the present crack-tip field, (a) the angular variations of stresses, strains and particle velocities around the crack tip are fully continuous; (b) the stresses and strains at the crack tip are bounded; (c) there is a free parameter _eq0 which cannot be determined in the asymptotic analysis. Comparisons indicate that the present asymptotic solution matches well with full-field numerical results, and the parameter _eq0 can characterize the effects of the far field on the crack-tip field. Furthermore, the present solution approaches that of Leighton et al. (1987) in the limit as the material hardening exponent goes to infinity, but does not reduce to the accepted solution for quasi-statically growing cracks in the limit as the crack speed goes to zero.     

Numerical analysis of plane cracks in strain-gradient elastic materials

The classical linear elastic fracture mechanics is not valid near the crack tip because of the unrealistic singular stress at the tip. The study of the physical nature of the deformation around the crack tip reveals the dominance of long-range atomic interactive forces. Unlike the classical theory which incorporates only short range forces, a higher-order continuum theory which could predict the effect of long range interactions at a macro scale would be appropriate to understand the deformation around the crack tip. A simplified theory of gradient elasticity proposed by Aifantis is one such grade-2 theory. This theory is used in the present work to numerically analyze plane cracks in strain-gradient elastic materials. Towards this end, a 36 DOF C¹ finite element is used to discretize the displacement field. The results show that the crack tip singularity still persists but with a different nature which is physically more reasonable. A smooth closure of the structure of the crack tip is also achieved.