ELASTIC PERFECTLY-PLASTIC FIELDS AT A RAPIDLY PROPAGATING CRACK-TIP

All the stress components at a rapidly propagating crack-tipin an elastic perfectly-plastic material are the functions of θ only.Making use of this condition and theequations of steady-state motion.stress-strain relations and yield conditions,we obtain thegeneral solutions in both the cases of anti-plane and in-plane strain.Applying these twogeneral solutions to propagating Mode III and Mode I cracks.respectively,the elasticperfectly-plastic and the perfectly-plastic fields at the rapidly propagating tips of Mode IIIand Mode I crocks are derived.     

Anisotropic plastic stress fields at a slowly propagating crack tip

Under the condition that any perfectly plastic stress components at a crack tip are nothing but the functions of 0 only making use of equilibrium equations. Hill anisotropic yield condition and unloading stress-strain relations, in this paper, we derive the general analytical expressions of anisotropic plastic stress fields at the slowly steady propagating tips of plane and anti-plane strain. Applying these general analytical expressions to the concrete cracks, the analytical expressions of anisotropic plastic stress fields at the-slowly steady propagating tips of Mode I and Mode III cracks are obtained. For the isotropic plastic material, the anisotropic plastic stress fields at a slowly propagating crack tip become the perfectly plastic stress fields.     

Anisotropic plastic stress field near a singular point

On condition that any perfectly plastic stress component near a singular point is nothing but the function of θ only, making use of equilibrium equations and Hill anisotropic yield condition, we derive the general analytical expressions of the anisotropic plastic stress field near a singular point in both the cases of anti-plane and in-plane strains. Applying these general analytical expressions to the concrete cracks and the plane-strain bodies with a singular point, the anisotropic plastic stress fields at the tips of Mode Ⅰ, Mode Ⅱ, Mode Ⅲ and mixed mode Ⅰ-Ⅱ cracks, and the limit loads of anisotropic plastic plane-strain bodies with a singular point are obtained.     

Anisotropic plastic fields at a rapidly propagating crack-tip

Under the condition that all the stress components at a crack-tip are the functionsofθonly,making use of the equations of steady-state motion,stress-strain relationsand Hill anisotropic yield conditions,we obtain the general solutions at a crack-tip inboth the cases of anti-plane and in-plane strains.Applying these general solutions tothe concrete cracks,the anisotropic plastic fields at the rapidly propagating tips ofmodeⅢand modeⅠcracks are derived.     

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 rapidly propagating plane-stress crack tip

Under the condition that all the perfectly plastic stress components at a crack tip are the functions of only, making use of the Treasca yield condition, steady-state moving equations and elastic perfectly-plastic constitutive equations, we derive the generally analytical expressions of perfectly palstic stress field at a rapidly propagating plane-stress crack tip. Applying these generally analytical expressions to the concrete crack, we obtain the analytical expressions of perfectly plastic stress field at the rapidly propagating tips of models I and II plane-stress cracks.     

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.     

Anisotropic plastic fields at a rapidly propagating plane-stress crack-tip

Under the condition that all the stress components at a crack-tip are the functions ofθonly,making use of the equations of steady-state motion.Hill anisotropic yield condition and stress-strain relations,we obtain the general solution of anisotropic plastic field at a rapidly propagating plane-stress crack-tip.Applying this general solution to four particular cases of anisctropy,the general solutions of these four particular cases are derived.Finally,we give the anisotropic plastic field at the rapidly propagating plane-stress modeⅠcrack-tip in the case of X=Y=Z     

Mode III Stress-intensity factors in cracked orthotropic plates—An analogy with propagating cracks in isotropic media

In this paper, the method of reflected caustics—which was used to evaluate Mode III SIF's in stationary cracks in isotropic plates—was extended to deal with stationary cracks in orthotropic plates. Furthermore, a correspondence between the anisotropic stationary case and the case of a Mode III dynamic crack, traversing an isotropic plate, is developed by analyzing appropriately the governing equations of the two problems. For this purpose the singulardisplacement field for rectilinearly orthotropic cracked bodies was combined with either Yoffé's model for steady-state, or Broberg's model for transient-crack propagation. Graphs are given where the equivalence between these cases can be established. In this way, the dynamic problem of the propagating crack in an isotropic medium can be readily simulated by considering the experimentally easier anisotropic stationary case.     

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.     

PERFECTLY PLASTIC FIELDS AT A RAPIDLY PROPAGATING PLANE-STRESS CRACK TIP

Under the condition that all the perfectly plastic stress components at a crack tiP arethe functions ofθonly,making use of the Mises yield condition,steady-state movingequations and elastic perfectly-plastic constitutive equations,we derive the generallyanalytical expressions of perfectly plastic fields at a rapidly propagating plane-stress cracktip.Applying these generally analytical expressions to the concrete crack,we obtain theanalytical expressions of perfectly plastic fields at the rapidly propagating tips of,modesⅠandⅡplane-stress cracks.     

SIF-based fracture criterion for interface cracks

The complex stress intensity factor K governing the stress field of an interface crack tip may be split into two parts, i.e.,■ and s~(-iε), so that K = ■ s~(-iε), s is a characteristic length and ε is the oscillatory index. ■ has the same dimension as the classical stress intensity factor and characterizes the interface crack tip field. That means a criterion for interface cracks may be formulated directly with■, as Irwin(ASME J. Appl. Mech. 24:361–364, 1957) did in 1957 for the classical fracture mechanics. Then, for an interface crack,it is demonstrated that the quasi Mode I and Mode II tip fields can be defined and distinguished from the coupled mode tip fields. Built upon SIF-based fracture criteria for quasi Mode I and Mode II, the stress intensity factor(SIF)-based fracture criterion for mixed mode interface cracks is proposed and validated against existing experimental results.     

Kinking of a crack under dynamic loading conditions

Summary A method is presented to analyze elastodynamic stress intensity factors at the tip of a branch which emanates at velocity v and under an angle from the tip of a semi-infinite crack, when the faces of the semi-infinite crack are subjected to impulsive normal pressures. By taking advantage of self-similarity, the system of governing equations is reduced to a set of two Laplace's equations in half-plane regions. The solutions to these equations, which are coupled along the real axes of the half-planes, are obtained by using complex function theory together with summations over Chebychev polynomials. For small values of the Mode I and Mode II stress intensity factors and the corresponding flux of energy into the crack tip have been computed.     

Mixed Mode Dynamic Fracture in Particulate Reinforced Functionally Graded Materials

A detailed analytical and experimental investigation is presented to understand the dynamic fracture behavior of functionally graded materials (FGMs) under mode I and mixed mode loading conditions. Crack-tip stress, strain and displacement fields for a mixed mode crack propagating at an angle from the direction of property gradation were obtained through an asymptotic analysis coupled with a displacement potential approach. This was followed by a comprehensive series of experiments to gain further insight into the behavior of propagating cracks in FGMs. Dynamic photoelasticity coupled with high-speed photography was used to obtain crack tip velocities and dynamic stress fields around the propagating cracks. Birefringent coatings were used to conduct the photoelastic study due to the opaqueness of the FGMs. Dynamic fracture experiments were performed using different specimen geometries to develop a dynamic constitutive fracture relationship between the mode I dynamic stress intensity factor (K _ID ) and crack-tip velocity ( ) for FGMs with the crack moving in the direction of increasing fracture toughness. A similar -K _ID relation was also obtained for matrix material (polyester) for comparison purposes. The results obtained show that crack propagation velocities in FGMs were about 80% higher than the polyester matrix. Crack arrest toughness was found to be about 10% lower than the value of local fracture toughness in FGMs.     

Interaction of two off-axis cracks exhibiting orthotropic and general anisotropic behavior under arbitrary extension

Obtained in this work are the stress intensity factors for two interacting cracks, one of which coincides with the axis of material orthotropy and the other is oriented in a general direction. The direction of the applied stress can be varied with reference to the cracks and/or the axes of material anisotropy. Mode I and II stress intensity factors are displayed to show their variations with the geometric, material and load parameters.     

THE PRECISE AND NEW ANALYSIS FOR A MODE Ⅲ GROWING CRACK IN AN ELASTIC-PERFECTLY PLASTIC SOLID

The near crack line field analysis method has been used to investigate into ModeⅢ quasistatically propagating crack in an elastic-perfectly plastic material.Thesignificance of this paper is that the usual small scale yielding theory has been brokenthrough.By obtaining the general solutions of the stresses and the displacement rate ofthe near crack line plastic region,and by matching the general solutions with theprecise elastic fields(not the usual elastic K-dominant fields)at the elastic-plasticboundary,the precise and new solutions of the stress and deformation fields,the sizeof the plastic region and the unit normal vector of the elastic-plastic boundary havebeen obtained near the crack line.The solutions of this paper are sufficiently precisenear the crack line region because the roughly qualitative assumptions of the smallscale yielding theory have not been used and no other roughly qualitative assumptionshave been taken,either.The analysis of this paper shows that the assumingly“steady-state cas     

Perturbation of a dynamic planar crack moving in a model viscoelastic solid

Weight functions, which give stress intensity factors in terms of applied loading, are constructed, for three-dimensional time-dependent loading of a semi-infinite crack, propagating at uniform speed. Both a model problem, governed by a scalar wave equation, and the full vectorial problem for Mode I loading, are considered. The medium through which the crack propagates is viscoelastic; the approach is general but explicit formulae are given when the medium is a Maxwell fluid. The weight functions are exploited to develop formulae for the first-order perturbations of stress intensity factors when the crack edge is no longer straight but becomes slightly wavy. Implications for stability, and for “crack front waves” in the case of the Mode I problem, are discussed.     

两种各向异性材料界面共线裂纹的反平面问题

本文研究两种各向异性材料界面共线裂纹的反平面剪切问题。利用复变函数方法，提出了一般问题公式和某些实际重要问题的封闭形式解。考察了裂纹尖端附近的应力分布并给出了应力强度因子公式。从本文解签的特殊情形，可以直接导出两种各向同性材料界面裂纹，均匀各向异性材料共线裂纹以及均匀各向同性材料共线裂纹的相应问题公式，其中包括已有的经典结果。     

An exact transient analysis of plane wave diffraction by a crack in an orthotropic or transversely isotropic solid

A transient plane strain analysis of diffraction of plane waves by a semi-infinite crack in an unbounded orthotropic or transversely isotropic solid is performed. The waves approach the crack at a general oblique angle, and are of two types, a normal stress pulse and a shear stress pulse, i.e. a P- and an SV-wave, respectively, in the isotropic limit. A class of materials that includes this limit and beryl, cobalt, ice, magnesium and titanium is chosen for illustration, and exact solutions are obtained for the initial/mixed boundary value problems.In contrast to related work, a factorization in the Laplace transform space is used to simplify the solution forms and the Wiener-Hopf component of the solution process, and to yield a more compact expression for the Rayleigh wave speed. Calculations for this speed, the two allowable, direction-dependent, plane wave speeds, and quantities related to the Mode I and Mode II dynamic stress intensity factors are given for the five anisotropic materials mentioned.