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.     

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.     

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

Under the condition that all the perfectly plastic stress components at a crack tip are the functions of θ only, making use of equilibrium equations and Von-Mises yield condition containing Poisson ratio, in this paper, we derive the generally analytical expressions of perfectly plastic stress field at a stationary plane-strain crack tip. Applying these generally analytical expressions to the concrete cracks, the analytical expressions of perfectly plastic stress fields at the stationary tips of Mode Ⅰ, Mode Ⅱ and Mixed-Mode Ⅰ-Ⅱ plane-strain cracks are obtained. These analytical expressions contain Poisson ratio.     

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 MIXED-MODE CRACK TIP UNDER PLANE AND ANTI-PLANE STRAIN

Under the condition that any Perfectly plastic stress component at a crack tip isnothing but the function of θ only. making use of equilibrium equations, stress-strain-rate relations, compatibility equations and yield condition. in this paper. we derive thegeneral analytical expressions of the perfectly plastic stress field at a Mixed-Mode cracktip under plane and anti-plane strain. Applying this general analytical expressions to theMixed-Mode cracks. we can obtain the analytical expressions of perfectly plastic stressfields at the tips of Mixed-ModeⅠ-Ⅲ.Ⅱ-Ⅲ andⅠ-Ⅱ-Ⅲ 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.     

On perfectly stress field at a mixed-mode crack tip under plane and anti-plane strain

In [1], under the condition that all the perfectly plastic stress components at a crack tip are functions of ϕ only, making use of equilibrium equations, stress-strain rate relations, compatibility equations and yield condition. Lin derived the general analytical expressions of the perfectly plastic stress field at a mixed-mode crack tip under plane and anti-plane strain. But in [1] there were several restrictions on the proportionality factor γ in the stress-strain rate relations, such as supposing that γ is independent of ϕ and supposing that γ=c or cr⁻¹. In this paper, we abolish these restrictions. The cases in [1], γ=cr^d (n=0 or-1) are the special cases of this paper.     

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.     

Contact zone assessment for a fast growing interface crack in an anisotropic bimaterial

Summary A plane strain problem for a crack with a frictionless contact zone at the leading crack tip expanding stationary along the interface of two anisotropic half-spaces with a subsonic speed under the action of various loadings is considered. The cases of finite and infinite-length interface cracks under the action of a moving concentrated loading at its faces are considered. A finite-length crack for a uniform mixed-mode loading at infinity is considered as well. The associated combined Dirichlet-Riemann boundary value problems are formulated and solved exactly for all above-mentioned cases. The expressions for stresses and the derivatives of the displacement jumps at the interface are presented in a closed analytical form for an arbitrary contact zone length. Transcendental equations are obtained for the determination of the real contact zone length, and the associated closed form asymptotic formulas are found for small values of this parameter. It is found that independently of the types of the crack and loading, an increase of the crack tip speed leads to an increase of the real contact zone length and the correspondent stress intensity factor. The latter increase significantly for an interface crack tip speed approaching the Ragleigh wave speed.     

Hydrostatic stress-dependent perfectly-plastic stress fields at a stationary plane-stress crack-tip

Under the condition that all the stress components at a crack-tip are the functions of θ only, making use of equilibrium equations and hydrostatic stress-dependent yield condition, in this paper, we derive the generally analytical expressions of the hydrostatic stress-dependent perfectly-plastic stress fields at a stationary plane-stress crack-tip. Applying these generally analytical expressions to the concrete cracks, the analytical expressions of hydrostatic stress-dependent perfectly-plastic stress fields at the tips of mode Ⅰ and mode Ⅱ cracks are obtained.     

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.     

Supplementary study on anisotropic plastic stress fields at a stationary plane-stress crack-tip

The results in Ref.[1]are not suitable for the cases of a≥2 .For this reason,we use the method in Ref.[1]to derive the general expressions of the anisotropic plastic stress fields at a stationary plane-stress crack-tip for both of the cases of a=2 and a>2 .As an example,we give the analytical expressions of the anisotropic plastic stress fields at the stationary tips of modeⅠand modeⅡplane-stress cracks for the case of a=2.     

垂直磁压电材料界面三维裂纹的超奇异积分法

应用有限部积分概念和广义位移基本解,垂直于磁压电双材料界面三维复合型裂纹问题被转 化为求解一组以裂纹表面广义位移间断为未知函数的超奇异积分方程问题. 进而,通过主部 分析法精确地求得裂纹尖端光滑点附近的奇性应力场解析表达式. 然后,通过将裂纹表面 位移间断未知函数表达为位移间断基本密度函数与多项式之积,使用有限部积分法对超奇异 积分方程组建立了数值方法. 最后,通过典型算例计算,讨论了广义应力强度因子的变化规 律.     

Interaction of three interfacial Griffith cracks between bonded dissimilar orthotropic half planes

The paper deals with the interaction of a pair of outer cracks on a central crack situated at the interface of two dissimilar orthotropic half-planes. The mixed boundary value problem is reduced to solving a pair of simultaneous singular integral equations which have finally been solved numerically by using Jacobi polynomials. The analytical expressions for stress intensity factors at the central crack tip and the expression of the strain energy release rate have been derived for general loading. Numerical values of the interaction effects of the outer cracks on the central crack have been calculated through stress magnification factors. It is seen that the interaction effects are either shielding or amplification depending on the size of the outer cracks and their spacing from the central crack.     

Analysis of crack moving and curving in anisotropic solids

The general equations for a dynamically curved crack in an anisotropic solid are derived, and the asymptotic fields of a moving crack under arbitrary distributed loading on the crack surface are calculated from them. For a moving crack under mixed-mode loading conditions a general Muskhelishvili type approach is proposed to calculate intensity factors due to crack surface loading in anisotropic materials. The kinking and curving caused by dynamic loading in anisotropic materials are calculated using the maximum normal stress ratio criterion. The results show that cracks in anisotropic solids may deviate from the straight path and approach a direction parallel to the stiff axis even under symmetric loading and that a crack will tend to deviate more from the crack path to the direction of the stiff axis as the crack speed becomes higher.     

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.     

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.     

各向异性介质中SH波引起的裂纹扩展

本文利用Green函数法,求解各向异性介质中半无限长裂纹在SH波作用下,以任意速度扩展的问题。首先,利用Laplace变换和Cagniard-de Hoop反演法求解各向异性介质中反平面问题的Green函数,并利用它建立了求解裂纹扩展问题的积分方程。因为方程为Abel型的,所以可得到在SH波作用下,半无限长裂纹扩展问题的解析解。还可求得裂纹端点附近的应力和裂纹表面上位移的表达式。并对裂纹端点附近的奇异性进行讨论。最后讨论了裂纹尖端附近任一点的能量关系。并应用Griffith的能量准则,对裂纹扩展规律进行了讨论。     

SINGULAR SOLUTIONS OF ANISOTROPIC PLATE WITH AN ELLIPTICAL HOLE OR A CRACK

In the present paper, closed form singular solutions for an infinite anisotropic plate with an elliptic hole or crack are derived based on the Stroh-type formalism for the general anisotropic plate. With the solutions, the hoop stresses and hoop moments around the elliptic hole as well as the stress intensity factors at the crack tip under concentrated in-plane stresses and bending moments are obtained. The singular solutions can be used for approximate analysis of an anisotropic plate weakened by a hole or a crack under concentrated forces and moments.They can also be used as fundamental solutions of boundary integral equations in BEM analysis for anisotropic plates with holes or cracks under general force and boundary conditions.     

Non-overlapping conditions for crack face displacement of anisotropic bodies

Crack tip stress and displacement fields are useful for studying the fracture behavior of cracks in both isotropic and anisotropic materials. Under certain boundary conditions, crack surfaces could overlap, a condition that could be more prevalent for the anisotropic case as compared with isotropic materials. Conditions can be derived for different loading conditions and material properties such that overlap of the crack faces would not occur.