Crack tip singular fields in ductile crystals with taylor power-law hardening. I: Anti-plane shear

Asymptotic singular solutions of the HRR type are presented for anti-plane shear cracks in ductile crystals. These are assumed to undergo Taylor hardening with a power-law relation between stress and strain at sufficiently large strain. Results are given for several crack orientations in fcc and bcc crystals. The neartip region divides into angular sectors which are the maps of successive flat segments and vertices on the yield locus. Analysis is simplified by use of new general integrals of crack tip singular fields of the HRR type. It is conjectured that the single crystal HRR fields are dominant only over part of the plastic region immediately adjacent to the crack tip, even at small scale yielding, and that their domain of validity vanishes as the perfectly plastic limit is approached. This follows from the fact that while in the perfectly plastic limit the HRR stress states approach the correct discontinuous distributions of the complete elasticideally plastic solutions for crystals (Rice and Nikolic, J. Mech. Phys. Solids33, 595 (1985)), the HRR displacement fields in that limit remain continuous. Instead, the complete elastic-ideally plastic solutions have discontinuous displacements along planar plastic regions emanating from the tip in otherwise elastically stressed material. The approach of the HRR stress fields to their discontinuous limiting distributions is illustrated in graphical plots of results. A case examined here of a fcc crystal with a crack along a slip plane is shown to lead to a discontinuous near-tip stress state even in the hardening regime.Through another limiting process, the asymptotic solution for the near-tip field for an isotropic material is also derived from the present single crystal framework.     

Effects of the strain-hardening exponent on two-parameter characterizations of surface-cracks under large-scale yielding

The influence of strain hardening exponent on two-parameter J-Q near tip opening stress field characterization with modified boundary layer formulation and the corresponding validity limits are explored in detail. Finite element simulations of surface cracked plates under uniaxial tension are implemented for loads exceeding net-section yield. The results from this study provide numerical methodology for limit analysis and demonstrate the strong material dependencies of fracture parameterization under large scale yielding. Sufficient strain hardening is shown to be necessary to maintain J-Q predicted fields when plastic flow progresses through the remaining ligament. Lower strain hardening amplifies constraint loss due to stress redistribution in the plastic zone and increases the ratio of tip deformation to J. The onset of plastic collapse is marked by shape change and/or rapid relaxation of tip fields compared to those predicted by MBL solutions and thus defining the limits of J-Q dominance. A radially independent Q-parameter cannot be evaluated for the low strain hardening material at larger deformations within a range where both cleavage and ductile fracture mechanisms are present. The geometric deformation limit of near tip stress field characterization is shown to be directly proportional to the level of stress the material is capable of carrying within the plastic zone. Accounting for the strain hardening of a material provides a more adjusted and less conservative limit methodology compared to those generalized by the yield strength alone. Results from this study are of relevance to establishing testing standards for surface cracked tensile geometries.     

Damage accumulation for growth of anti-plane crack in work-hardening material

Elastic–plastic solutions of an anti-plane crack in an infinite body are used in conjunction with a continuum damage model to describe the conditions necessary for the onset of crack instability, fatigue crack propagation due to cyclic loading, and rates of crack growth due to time dependent events. A power law relates the stress to the strain of the material. The damage, which invokes nucleation, growth and coalescence of microvoids due to elevated strain, is confined to the plastic zone surrounding the crack tip. For applied loading below the yield stress, the small-scale and large-scale yielding solutions are used to determine the influence of strain hardening on crack instability and failure. Crack growth due to cyclic loading and time-dependent deformations are studied using the small-scale yielding solution of the deformation theory of plasticity.     

Asymptotic crack-tip fields for dynamic crack growth in compressive power-law hardening materials

The effect of material compressibility on the stress and strain fields for a mode-I crack propagating steadily in a power-law hardening material is investigated under plane strain conditions. The plastic deformation of materials is characterized by the J₂ flow theory within the framework of isotropic hardening and infinitesimal displacement gradient. The asymptotic solutions developed by the present authors [Zhu, X.K., Hwang K.C., 2002. Dynamic crack-tip field for tensile cracks propagating in power-law hardening materials. International Journal of Fracture 115, 323–342] for incompressible hardening materials are extended in this work to the compressible hardening materials. The results show that all stresses, strains, and particle velocities in the asymptotic fields are fully continuous and bounded without elastic unloading near the dynamic crack tip. The stress field contains two free parameters σ_eq0 and s₃₃₀ that cannot be determined in the asymptotic analysis, and can be determined from the full-field solutions. For the given values of σ_eq0 and s₃₃₀, all field quantities around the crack tip are determined through numerical integration, and then the effects of the hardening exponent n, the Poisson ratio ν, and the crack growth speed M on the asymptotic fields are studied. The approximate behaviors of the proposed solutions are discussed in the limit of ν → 0.5 or n → ∞.     

Effect of crack-tip shape on the near-tip field in glassy polymer

The physical nature of a crack tip is not absolutely sharp but blunt with finite curvature. In this paper, the effects of crack-tip shape on the stress and deformation fields ahead of blunted cracks in glassy polymers are numerically investigated under Mode I loading and small scale yielding conditions. An elastic–viscoplastic constitutive model accounting for the strain softening upon yield and then the subsequently strain hardening is adopted and two typical glassy polymers, one with strain hardening and the other with strain softening–rehardening are considered in analysis. It is shown that the profile of crack tip has obvious effect on the near-tip plastic field. The size of near-tip plastic zone reduces with the increase of curvature radius of crack tip, while the plastic strain rate and the stresses near crack tip enhance obviously for two typical polymers. Also, the plastic energy dissipation behavior near cracks with different curvatures is discussed for both materials.     

Anti-plane shear cracks in ideally plastic crystals

Cracks in ductile single crystals are analyzed here for geometries and orientations such that two-dimensional states of anti-plane shear constitute possible deformation fields. The crystals are modelled as ideally plastic and yield according to critical resolved shear stresses on their slip systems. Restrictions on the asymptotic forms of stress and deformation fields at crack tips are established for anti-plane loading of stationary and quasistatically growing cracks, and solutions are presented for several specific orientations in f.c.c. and b.c.c. crystals. The asymptotic solutions are complemented by complete elastic-plastic solutions for stationary and growing cracks under small scale yielding, based on previous work by Rice (1967, 1984) and Freund (1979). Remarkably, the plastic zone at a stationary crack tip collapses into discrete planes of displacement and stress discontinuity emanating from the tip; plastic flow consists of concentrated shear on the displacement discontinuities. For the growing crack these same planes, if not coincident with the crack plane, constitute collapsed plastic zones in which velocity and plastic strain discontinuities occur, but across which the stresses and anti-plane displacement are fully continuous. The planes of discontinuity are in several cases coincident with crystal slip planes but it is shown that this need not be the case, e.g., for orientations in which anti-plane yielding occurs by multi-slip, or for special orientations in which the crack tip and the discontinuity planes are perpendicular to the activated slip plane.     

Pseudo plane stress plastic field for mode I crack growth

The pseudo plane stress field for a mode I crack growth is analyzed for both perfectly plastic and power law hardening plastic materials. When finite strain is taken into account, it is found that for perfectly plastic materials, the plastic domain is a narrow strip ahead of the crack tip. For power law hardening plastic material, the plastic domain contains a strip and a region ahead of the strip. The fracture criterion is discussed. The energy dissipated in the plastic strip is found to be proportional to the square of the thickness. Singular solutions to the field are ruled out by analysis.     

Asymptotic fields in steady crack growth with linear strain-hardening

The asymptotic stress and velocity fields of a crack propagating steadily and quasi-statically into an elastic-plastic material are presented. The material is characterized by J₂-flow theory with linear strain- hardening. The possibility of reloading on the crack flanks is taken into account. The cases of anti-plane strain (mode III), plane strain (modes I and II), and plane stress (modes I and II) are considered. Numerical results are given for the strength of the singularity and for the distribution of the stress and velocity fields in the plastic loading, elastic unloading and plastic reloading regions, as functions of the strain-hardening parameter. An attempt is made to make a connection with the perfectly-plastic solutions in the limit of vanishing strain-hardening.     

VISCOPLASTIC SOLUTION TO FIELD AT STEADILY PROPAGATING CRACK TIP IN LINEAR- HARDENING MATERIALS

An elastic-viscoplastic constitutive model was adopted to analyze asymptotically the tip-field of moving crack in linear-hardening materials under plane strain condition. Under the assumption that the artificial viscosity coefficient was in inverse proportion to power law of the rate of effective plastic strain, it is obtained that stress and strain both possess power law singularity and the singularity exponent is uniquely determined by the power law exponent of the rate of effective plastic strain. Variations of zoning structure according to each material parameter were discussed by means of numerical computation for the tip-field of mode Ⅱ dynamic propagating crack, which show that the structure of crack tip field is dominated by hardening coefficient rather than viscosity coefficient. The secondary plastic zone can be ignored for weak hardening materials while the secondary plastic zone and the secondary elastic zone both have important influence on crack tip field for strong hardening materials. The dynamic solution approaches to the corresponding quasi-static solution when the crack moving speed goes to zero, and further approaches to the HR (Hui-Riedel) solution when the hardening coefficient is equal to zero.     

I型定常扩展裂纹尖端的弹黏塑性场

考虑材料在扩展裂纹尖端的黏性效应，假设黏性系数与塑性应变率的幂次成反比，对幂硬化材料中平面应变扩展裂纹尖端场进行了弹黏塑性渐近分析，得到了不含间断的连续解，并讨论了I型裂纹数值解的性质随各参数的变化规律. 分析表明应力和应变均具有幂奇异性，并且只有在线性硬化时，尖端场的弹、黏、塑性才可以合理匹配. 对于I型裂纹，裂尖场不含弹性卸载区. 当裂纹扩展速度趋于零时，动态解趋于准静态解，表明准静态解是动态解的特殊形式；如果进一步考虑硬化系数为零的极限情况，便可退化为Hui和Riedel的非线性黏弹性解.     

Effect of large elastic strains on cavitation instability predictions for elastic–plastic solids

For an infinite solid containing a void, the cavitation instability limit is defined as the remote stress–and strain state, at which the void grows without bound, driven by the elastic energy stored in the surrounding material. Such cavitation limits have been analysed by a number of authors for metal plasticity as well as for nonlinear elastic solids. The analyses for elastic–plastic solids are here extended to consider the effect of a large initial yield strain, and it is shown how the critical stress value decays for increasing value of the yield strain. Analyses are carried out for remote hydrostatic tension as well as for more general axisymmetric remote stress field, with an initially spherical void. Different levels of strain hardening are considered.     

Field structure at mode III dynamically propagating crack tip in elastic-viscoplastic materials

An elastic-viscoplastic mechanics model is used to investigate asymptotically the mode Ⅲ dynamically propagating crack tip field in elastic-viscoplastic materials. The stress and strain fields at the crack tip possess the same power-law singularity under a linear-hardening condition. The singularity exponent is uniquely determined by the viscosity coefficient of the material. Numerical results indicate that the motion parameter of the crack propagating speed has little effect on the zone structure at the crack tip. The hardening coefficient dominates the structure of the crack-tip field. However, the secondary plastic zone has little influence on the field. The viscosity of the material dominates the strength of stress and strain fields at the crack tip while it does have certain influence on the crack-tip field structure. The dynamic crack-tip field degenerates into the relevant quasi-static solution when the crack moving speed is zero. The corresponding perfectly-plastic solution is recovered from the linear-hardening solution when the hardening coefficient becomes zero.     

The near tip stress and strain fields in cracked single crystals

The numerical analyses of stationary mathematically sharp Mode I crack in FCC and BCC crystals with elastic-ideally plastic (EIP) and fast hardening saturation (FHS) law are carried out in the present paper. From the calculated results, it is shown that: for the cases of small strain, EIP crystal cracks, the features of concentrated deformation patterns and the stress state in near-crack tip deformation fields are identical to the earlier analytical solutions, but along the angular sector boundaries, there exist narrow complex stress zones. The overall characteristics of deformation patterns for the cases of EIP and FHS are similar. The behaviours of crack tip opening can be characterized by crack-tip-opening-displacement (CTOD). For the case of FHS, finite deformation BCC crystal crack, our calculations are qualitatively in agreement with recent experimental observations. The project supported by National Natural Science Foundation of China     

Field structure at mode Ⅲ dynamically propagating crack tip in elastic-viscoplastic materials

An elastic-viscoplastic mechanics model is used to investigate asymptotically the mode Ⅲ dynamically propagating crack tip field in elastic-viscoplastic materials. The stress and strain fields at the crack tip possess the same power-law singularity under a linear-hardening condition. The singularity exponent is uniquely determined by the viscosity coefficient of the material. Numerical results indicate that the motion parameter of the crack propagating speed has little effect on the zone structure at the crack tip. The hardening coefficient dominates the structure of the crack-tip field. However, the secondary plastic zone has little influence on the field. The viscosity of the material dominates the strength of stress and strain fields at the crack tip while it does have certain influence on the crack-tip field structure. The dynamic crack-tip field degenerates into the relevant quasi-static solution when the crack moving speed is zero. The corresponding perfectly-plastic solution is recovered from the linear-hardening solution when the hardening coefficient becomes zero.     

Tangential continuity of elastic/plastic curvature and strain at interfaces

The continuity vs discontinuity of the elastic/plastic curvature & curvature rate, and strain & strain rate tensors is examined at non-moving surfaces of discontinuity, in the context of a field theory of crystal defects (dislocations and disclinations). Tangential continuity of these tensors derives from the conservation of the Burgers and Frank vectors over patches bridging the interface, in the limit where such patches contract onto the interface. However, normal discontinuity of these tensors remains allowed, and Kirchhoff-like compatibility conditions on their normal discontinuities across the concurring interfaces are derived at multiple junctions. In a simple plane case and in the absence of surface-disclinations, the compatibility of the normal discontinuities in the elastic curvatures assumes the form of a Young’s law between the grain-to-grain disorientations and the sines of the dihedral angles. Complete continuity of the plastic strain rate tensor at triple junctions also derives from the compatibility of the normal discontinuities in the plastic strain rates in such conditions.     

Mode I crack tip field with strain gradient effects

A plane strain mode I crack tip field with strain gradient effects is investigated. A new strain gradient theory is used. An elastic-power law hardening strain gradient material is considered and two hardening laws, i. e. a separation law and an integration law are used respectively. As for the material with the separation law hardening, the angular distributions of stresses are consistent with the HRR field, which differs from the stress results^[19]; the angular distributions of couple stresses are the same as the couple stress results^[19]. For the material with the integration law hardening, the stress field and the couple stress field can not exist simultaneously, which is the same as the conclusion^[19], but for the stress dominated field, the angular distributions of stresses are consistent with the HRR field; for the couple stress dominated field, the angular distributions of couple stresses are consistent with those in Ref. [19]. However, the increase in stresses is not observed in strain gradient plasticity because the present theory is based on the rotation gradient of the deformation only, while the crack tip field of mode I is dominated by the tension gradient, which will be shown in another paper. Supported by the National Science Foundation of China (No. 19704100), Science Foundation of Chinese Academy of Sciences (Project KJ951-1-20), CAS K. C. Wong Post-doctoral Research Award Fund and the Post Doctoral Science Fund of China.     

Application of corotational rates of the logarithmic strain in constitutive modeling of hardening materials at finite deformations

In this paper a finite deformation constitutive model for rigid plastic hardening materials based on the logarithmic strain tensor is introduced. The flow rule of this constitutive model relates the corotational rate of the logarithmic strain tensor to the difference of the deviatoric Cauchy stress and the back stress tensors. The evolution equation for the kinematic hardening of this model relates the corotational rate of the back stress tensor to the corotational rate of the logarithmic strain tensor. Using Jaumann, Green–Naghdi, Eulerian and logarithmic corotational rates in the proposed constitutive model, stress–strain responses and subsequent yield surfaces are determined for rigid plastic kinematic and isotropic hardening materials in the simple shear problem at finite deformations.     

Logarithm strain singularity at tip of mode I growing crack in elasticand perfectly-plastic material

Reanalyzed in detail is the stress and strain distribution near the tip of a Mode I steadily growing crack in an elastic and perfectly-plastic material. The crack tip region is divided into five angular sectors, one of which is singular in character and represents a rapid transition zone that becomes a line of strain discontinuity in the limit as crack tip is approached. It is shown for an incompressible material (ν=0.5) under plane strain that the local strain in all the angular sectors possesses the same logarithm singularity, i.e., In r where r is the radial distance measured from the crack tip. This result also prevails for the compressible material ( v < 0.5) and resolves a long standing controversy concerning the strain singularity in the sector just ahead of the crack tip.     

Steady crack growth in elastic–plastic fluid-saturated porous media

An asymptotic solution is obtained for stress and pore pressure fields near the tip of a crack steadily propagating in an elastic–plastic fluid-saturated porous material displaying linear isotropic hardening. Quasi-static crack growth is considered under plane strain and Mode I loading conditions. In particular, the effective stress is assumed to obey the Drucker–Prager yield condition with associative or non-associative flow-rule and linear isotropic hardening is adopted. Both permeable and impermeable crack faces are considered. As for the problem of crack propagation in poroelastic media, the behavior is asymptotically drained at the crack-tip. Plastic dilatancy is observed to have a strong effect on the distribution and intensity of pore water pressure and to increase its flux towards the crack-tip.     

Plane-strain crack-tip stress field for anisotropic perfectly-plastic materials

Plane-strain crack-tip stress solutions for anisotropic perfectly-plastic materials are presented. These solutions are obtained using the plane-strain slip-line theory developed by Rice (1973). The plastic anisosotropy is described by the Hill quadratic yield condition. The crack-tip stress solutions under symmetric (Mode I) and anti-symmetric (Mode II) conditions agree well with the low-hardening solutions for the corresponding power-law hardening materials. The crack-tip stress solutions under mixed Mode I and II conditions are also presented. All the solutions indicate that the general features of the slip-line field near a crack tip in orthotropic plastic materials with the elliptical yield contours in the Mohr plane are the same as those associated with isotropic plastic materials. However, the angular variations of the crack-tip stress fields for the materials with large plastic orthotropy differ substantially from those for isotropic plastic materials. Modifications due to polygonal yield contours are outlined and implications of solutions to the fracture analysis of ductile composite materials containing macroscopic flaws are discussed.