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.     

可压缩理想塑性体中逐步扩展裂纹顶端的弹塑性场

本文利用理想塑性固体平面应变问题的基本方程,分析了可压缩理想塑性体中逐步扩展裂纹顶端的弹塑性场,得到了关于应力的渐近场,分析了弹性卸载区的演变过程和修正的中心扇形区的发展过程,预示了出现二次塑性区的可能性,弹性可压缩性的影响明显表现在经典的中心扇形区必需加以修正,垂直于板面方向的应力偏量不再为零,而且随着新裂纹面的形成,裂纹前方的均匀应力场和紧连着的修正的中心扇形区的应力偏量将发生变化,这种变化是由于垂直于板面方向的应力偏量发生变化造成的。     

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.     

Analytic solutions to crack tip plastic zone under various loading conditions

Based on stress field equations and Hill yield criterion, the crack tip plastic zone is determined for orthotropic materials and isotropic materials under small-scale yielding condition. An analytical solution to calculating the crack tip plastic zone in plane stress states is presented. The shape and size of the plastic zone are analyzed under different loading conditions. The obtained results show that the crack tip plastic zones present “butterfly-like” shapes, and the elastic–plastic boundary is smooth. The size of the plastic zone for orthotropic composites is less at the crack tip for various loading conditions, compared with the case of isotropic materials. Crack inclination angle and loading conditions affect greatly the size and shape of crack tip plastic zone. The mode I crack has a crucial effect on the plastic zone for mixed mode case in plane stress state. The plastic zone for pure mode I crack and pure mode II crack have a symmetrical distribution to the initial crack plane.     

Effects of pre-stress on crack-tip fields in elastic,incompressible solids

A closed-form asymptotic solution is provided for velocity fields and the nominal stress rates near the tip of a stationary crack in a homogeneously pre-stressed configuration of a nonlinear elastic, incompressible material. In particular, a biaxial pre-stress is assumed with stress axes parallel and orthogonal to the crack faces. Two boundary conditions are considered on the crack faces, namely a constant pressure or a constant dead loading, both preserving an homogeneous ground state. Starting from this configuration, small superimposed Mode I or Mode II deformations are solved, in the framework of Biot's incremental theory of elasticity. In this way a definition of an incremental stress intensity factor is introduced, slightly different for pressure or dead loading conditions on crack faces. Specific examples are finally developed for various hyperelastic materials, including the J₂-deformation theory of plasticity. The presence of pre-stress is shown to strongly influence the angular variation of the asymptotic crack-tip fields, even if the nominal stress rate displays a square root singularity as in the infinitesimal theory. Relationships between the solution with shear band formation at the crack tip and instability of the crack surfaces are given in evidence.     

Elastic-plastic fields near crack tip growing step-by-step in a perfectly plastic solid

In this paper, based on the three-dimensional flow theory of plasticity, the fundametal equations for plane strain problem of elastic-perfectly plastic solids are presented. By using these equations the elastic-plastic fields near the crack tip growing step-by-step in an elastic incompressible-perfectly plastic solid are analysed.The first order asymptotic solutions for the stress field and velocity fields near the crack tip are obtained. The solutions show the evolution process of elastic unloading domain and the development process of central fan domain and reveal the possibility of the presence of the secondary plastic domain. The second order asymptotic solution for stress field is also presented.     

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.     

Analyses of steady quasistatic crack growth in plane strain tension in elastic-plastic materials with non-isotropic hardening

Steady state crack propagation problems of elastic-plastic materials in Mode I, plane strain under small scale yielding conditions were investigated with the aid of the finite element method. The elastic-perfectly plastic solution shows that elastic unloading wedges subtended by the crack tip in the plastic wake region do exist and that the stress state around the crack tip is similar to the modified Prandtl fan solution. To demonstrate the effects of a vertex on the yield surface, the small strain version of a phenomenological J₂, corner theory of plasticity (Christoffersen, J. and Hutchinson, J. W. J. Mech. Phys. Solids,27, 465 C 1979) with a power law stress strain relation was used to govern the strain hardening of the material. The results are compared with the conventional J₂ incremental plasticity solution. To take account of Bauschinger like effects caused by the stress history near the crack tip, a simple kinematic hardening rule with a bilinear stress strain relation was also studied. The results are again compared with the smooth yield surface isotropic hardening solution for the same stress strain curve. There appears to be more potential for steady state crack growth in the conventional J₂ incremental plasticity material than in the other two plasticity laws considered here if a crack opening displacement fracture criterion is used. However, a fracture criterion dependent on both stress and strain could lead to a contrary prediction.     

Plasticity aspects of interfacial fracture for polymer/glass specimens

Experimental results suggest that the interfacial fracture resistance is minimal for approximate near tip Mode I accompanied by positive and negative near tip Mode II. Finite-strain FE analysis is made for an elastic–plastic medium bonded to an ideally elastic medium with an interface crack. Small-scale plasticity conditions are invoked and examined in relation to the elastic–plastic stress distribution along the bond line. Plasticity engenders a tendency to turn near tip biaxiality towards pure Mode I regardless of the mixed-mode loading. High levels of hydrostatic stress are attained. For different mode mixities of the applied load, the dependence of the elastic–plastic normal bond stress on load level is examined. It is found that under positive Mode II loading, the normal bond stress σ_yy tends to saturate as the load level rises. This does not occur for Mode I and negative Mode II loading. In addition, deformation patterns inside the plastic zone are examined for mixed-mode situations. A displacement criterion based on the normal bond crack opening suggests a dependence of the critical load level on the extent of mixed mode. Under positive mode II fracture, traces of the ductile material are found at the top of the elastic substrate. Some of these conclusions appear to be consistent with the fracture patterns observed for LD-polyethylene/glass interfacial mixed-mode fracture.     

Asymptotic fields around an interfacial crack with a cohesive zone ahead of the crack tip

This paper considers an interfacial crack with a cohesive zone ahead of the crack tip in a linearly elastic isotropic bi-material and derives the mixed-mode asymptotic stress and displacement fields around the crack and cohesive zone under plane deformation conditions (plane stress or plane strain). The field solution is obtained using elliptic coordinates and complex functions and can be represented in terms of a complete set of complex eigenfunction terms. The imaginary portion of the eigenvalues is characterized by a bi-material mismatch parameter ε = arctanh(β)/π, where β is a Dundurs parameter, and the resulting fields do not contain stress singularity. The behaviors of “Mode I” type and “Mode II” type fields based on dominant eigenfunction terms are discussed in detail. For completeness, the counterpart for the Mode III solution is included in an appendix.     

Effect of poisson ratio on the perfectly plastic field at a rapidly propagating plane-strain crack-tip

All the stress components at a rapidly propagating crack-tip in elastic perfectly-plasticmaterial are the functions ofθonly.Making use of this condition and the equations ofsteady-state motion,plastic stress-strain relations,and Mises yield condition with Poissonratio,in this paper,we derive the general expression of perfectly plastic field at a rapidlypropagating plane-strain crack-tip.Applying this general expression with Poisson ratio toModeⅠcrack,the perfectly plastic field at the rapidly propagating tip of ModeⅠplane-strain crack is obtained.This perfectly plastic field contains a Poisson ratio,and thus,wecan obtain the effect of Poisson ratio on the perfectly plastic field at the rapidly propagatingtip of ModeⅠplane-strain crack.     

Perfectly plastic stress field at a stationary crack tip

Under the hypothesis that all the perfectly plastic stress components at a orach tip are the functions of θ only, making use of yield conditions and equilibrium equations. we derive the generally analytical expressions of the perfectly plastic stress field at a crack tip. Applying these generally analytical expressions to the concrete cracks, the analytical expressions of perfectly plastic stress fields at the tips of Mode Ⅰ Mode Ⅱ, Mode Ⅲ and Mixed Mode Ⅰ-Ⅱ cracks are obtained.     

Limitations to the small scale yielding approximation for crack tip plasticity

Recent finite-element results by S.G. Larsson and A.J. Carlsson suggest a limited range of validity to the ‘small scale yielding approximation’, whereby small crack tip plastic zones are correlated in terms of the elastic stress intensity factor. It is shown with the help of a model for plane strain yielding that their results may be explained by considering the non-singular stress, acting parallel to the crack at its tip, which accompanies the inverse square-root elastic singularity. Further implications of the non-singular stress term for crack tip deformations and fracturing are examined. It is suggested that its effect on crack tip parameters, such as the opening displacement and J-integral, is less pronounced than its effect on the yield zone size.     

THE EXACT SOLUTIONS OF ELASTIC－PLASTIC CRACK LINE FIELD FOR MODE ⅡPLANE STRESS CRACK

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Crack tip behavior of flexible bio-materials

An elastic constitutive model is proposed to describe the mechanical property of bio-materials that possesses strain limits. Analytical solution for the Mode I crack tip behavior is obtained. The tensile strain limit can be reached by approaching the crack tip in any direction while the compression strain limit can only be reached in two sectors of the crack tip domain.     

狭长体中非对称快速传播裂纹的分析解

用复变函数方法得到了弹性狭长体中含有一非对称半元限裂纹的动力学问题的分析解，当裂纹速度Ｖ→０时，此动力学的解可还原静力学的解，该问题Ｉ型与ＩＩ型静态与动态应力强度因子ＫＩ，ＫＩＩ得以确定，并且具有解析的形式。     

A finite element study of the near crack tip deformation of a ductile material under mixed mode loading

In ductile fracture, voids near a crack tip play an important role. From this point of view, a large deformation finite element analysis has been made to study the deformation, stress and strain, and void ratio near the crack tip under mixed mode plane strain loading conditions, employing Gurson's constitutive equation which has taken into account the effects of void nucleation and growth. The results show that: (i) one corner of the crack tip sharpens while the other corner blunts, (ii) the stress and strain distributions except for the near crack tip region, can be superimposed by normalizing distance from the crack tip by a crack tip deformation length, i.e., a steady-state solution under a mixed mode condition has been obtained, (iii) the field near a crack tip can be divided into four characteristic fields (K field, HRR field, blunted crack tip field, and damaged region), and (iv) the strain and void volume fraction become concentrated in the sharpened part of a crack tip with increasing Mode II component.     

静止裂纹顶端塑性区内应力场

本文对不可压缩的理想塑性材料裂纹顶端塑性区内的应力场进行了数学分析,证明了当塑性区包围着裂纹顶端而应力函数可用分离变匱型的级数展开且该级数展开的首项与第一类渐近解相同时,第一类渐近解即是塑性区内应力场的精确解。本文又提出了第二类渐近解,说明应力场的渐近解不是唯一的。     

An inclined surface crack subject to biaxial loading

The elastic–plastic stress fields and mode mixity parameters for semi-elliptical surface cracks on biaxial loaded plates have been investigated using detailed three-dimensional finite element calculations. Different degrees of mode mixity are given by combinations of the far-field stress level, biaxial stress ratio and inclined crack angle. These analyses were performed for different surface flaw geometries to study the combined load biaxiality and mode mixity effects on the crack-front stress fields and the size and shape of the plastic zones. It is clear from considering the local stress distributions along the crack front that the elastic crack tip singularities have been derived for several particular cases of mixed mode biaxial loading. By theoretical analysis, the new formulae have been introduced for both the elastic and plastic mode-mixity parameters, accounting for ratios between the I/II, II/III and III/I modes. Particular attention was paid to the strong variations of the mode-mixity parameters along the semi-elliptical surface crack front. The mixed-mode behavior of the crack growth direction angle along the semi-elliptical crack front for different combinations of biaxial loading and inclination crack angles was also determined. It was done using methods based on the maximum tangential stress and the strain energy density criteria.     

Stress and strain field near tip of mode III growing crack in materials with creep behavior

Using a proposed constitutive relation for materials with creep behavior, the stress and strain distribution near the tip of a Mode III growing crack is examined. Asymptotic equations of the crack tip field are derived and solved numerically. The stresses remain finite at the crack tip. Obtained qualitatively is the crack tip velocity and the local autonomy of the near tip field solution is discussed.