关于平面应变和反平面应变复合型裂纹尖端的理想塑性应力场

文献[1]在裂纹尖端的理想塑性应力分量都只是θ的函数的条件下,利用平衡方程、应力应变率关系、相容方程和屈服条件导出了平面应变和反平面应变复合型裂纹尖端的理想塑性应力场的一般解析表达式。但文献[1]对应力应变率关系式中的比例因子λ(r,θ)作了很多限制,即假定λ与θ无关,并假定λ=c或cr-1。本文取消了对λ的这些限制。而文献[1]所研究的λ=crn(n=0或-1)的情形,只是本文的一个特殊情况。     

Dynamic crack propagation in a 2D elastic body: The out-of-plane case

Already in 1920 Griffith has formulated an energy balance criterion for quasistatic crack propagation in brittle elastic materials. Nowadays, a generalized energy balance law is used in mechanics [F. Erdogan, Crack propagation theories, in: H. Liebowitz (Ed.), Fracture, vol. 2, Academic Press, New York, 1968, pp. 498-586; L.B. Freund, Dynamic Fracture Mechanics, Cambridge Univ. Press, Cambridge, 1990; D. Gross, Bruchmechanik, Springer-Verlag, Berlin, 1996] in order to predict how a running crack will grow. We discuss this situation in a rigorous mathematical way for the out-of-plane state. This model is described by two coupled equations in the reference configuration: a two-dimensional scalar wave equation for the displacement fields in a cracked bounded domain and an ordinary differential equation for the crack position derived from the energy balance law. We handle both equations separately, assuming at first that the crack position is known. Then the weak and strong solvability of the wave equation will be studied and the crack tip singularities will be derived under the assumption that the crack is straight and moves tangentially. Using the energy balance law and the crack tip behavior of the displacement fields we finally arrive at an ordinary differential equation for the motion of the crack tip.     

Material force method within the framework of the X-FEM-distribution of nodal material forces

The Material Force Method (MFM) and the Extended Finite Element Method (X-FEM), both have been major subjects of computational fracture mechanics in recent time. Thus combining the advantages of both concepts [1, 2] seems a promising approach to describe the behaviour of discontinuities such as cracks in otherwise continuous bodies. As the X-FEM models a crack independently of the mesh, the crack tip is in general not located at a node which is why a central question is, which nodal material forces do contribute to the resulting material force vector at the crack tip. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Problems of a cut in a three-dimensional elastic wedge

The Ritz variational method is applied to problems of a crack (a cut) in the middle half-plane of a three-dimensional elastic wedge. The faces of the elastic wedge are either stress-free (Problem A) or are under conditions of sliding or rigid clamping (Problems B and C respectively). The crack is open and is under a specified normal load. Each of the problems reduces to an operator integrodifferential equation in relation to the jump in normal displacement in the crack area. The method selected makes it possible to calculate the stress intensity factor at a relatively small distance from the edge of the wedge to the cut area. Numerical and asymptotic solutions [Pozharskii DA. An elliptical crack in an elastic three-dimensional wedge. Izv. Ross Akad. Nauk. MTT 1993;(6):105–12] for an elliptical crack are compared. In the second part of the paper the case of a cut reaching the edge of the wedge at one point is considered. This models a V-shaped crack whose apex has reached the edge of the wedge, giving a new singular point in the solution of boundary-value problems for equations of elastic equilibrium. The asymptotic form of the normal displacements and stress in the vicinity of the crack tip is investigated. Here, the method employed in [Babeshko VA, Glushkov YeV, Zinchenko ZhF. The dynamics of Inhomogeneous Linearly Elastic Media. Moscow: Nauka; 1989] and [Glushkov YeV, Glushkova NV. Singularities of the elastic stress field in the vicinity of the tip of a V-shaped three-dimensional crack. Izv. Ross Akad. Nauk. MTT 1992;(4):82–6] to find the operator spectrum is refined. The new basis function system selected enables the elements of an infinite-dimensional matrix to be expressed as converging series. The asymptotic form of the normal stress outside a V-shaped cut, which is identical with the asymptotic form of the contact pressure in the contact problem for an elastic wedge of half the aperture angle, is determined, when the contact area supplements the cut area up to the face of the wedge.     

Computation of crack propagation in 3-dimensional anisotropic solids

This contribution presents ideas, how crack propagation in three-dimensional solids composed of anisotropic materials can be predicted using the Griffith energy principle. Since the work of Irwin the change of potential energy caused by a straight elongation of a crack in an isotropic two-dimensional homogeneous structure can be expressed in quadratic terms of the stress intensities at the crack tip. This result was generalized in the last decades using methods of asymptotic analysis by many authors [1] to more complicated geometries, to anisotropic and inhomogeneous materials. With the energy release rate at hand, quasi-static scenarios of crack propagation can be simulated for plane problems [2], but this is still a complicated task for three-dimensional problems [3]. We show an idea how the change of energy caused by propagation of a crack surface in a fully three-dimensional solid of nearly arbitrary shape can be computed in anisotropic materials. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermo-mechanical modeling of turbine blades taking into account structural defects

In turbine blades of aero-engines typical defects are cracks due to high mechanical and thermal loads. The extended finite element method (XFEM) is used for simulations of fracture mechanics problems with cracks. Discontinuities in the displacement and temperature field are allowed and the crack opening displacement and crack tip stress field are reproduced accurately. Since crack closure and non-physical penetration of the crack surfaces may occur under certain load conditions, it becomes necessary to enforce the non-penetration condition for crack surfaces. This contact formulation is assumed to be frictionless. The node-to-segment approach proposed in [3] is extended to ten-node tetrahedral elements with quadratic shape functions. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

纯弯曲矩形截面梁Ⅰ型单边裂纹端部的应力应变场及裂纹失稳扩展临界应力的计算

本文应用文[1]的分析方法,研究了纯弯曲矩形载面梁Ⅰ型单边裂纹端部的应力应变场,给出了裂纹尖端的应力应变分量和计算裂纹端部弹性变形区和变形强化区宽度的公式以及计算裂纹失稳扩展临界应力的方程组。最后用计算实例对裂纹失稳扩展临界应力方程组进行了验证,最大误差不超过0.18%.     

裂纹间作用机制探讨及微裂纹区对主裂纹的作用效应研究

采用迭加原理和Kachanov提出的简化方法,研究了裂纹间的相互作用机理,分析了不同裂纹布置形式所产生的增强或屏蔽效应,发现当微裂纹沿着或垂直于最大拉应力方向布置时都不产生最大的作用效应,这有别于Ortiz的结论．还探讨了混凝土类材料的微裂纹的产生机制及微裂纹区对主裂纹尖端产生的作用效应,得出微裂纹区对主裂纹是起增强的作用,增强程度随微裂纹密度和微裂纹长度的增大而增大的结论．     

Investigation of self-healing of cracks in polymers

The effect of the molecular weight of a polymer and the environment on self-healing of cracks in polyvinyl acetate was investigated. It was confirmed that any change of the relaxation spectrum of the polymer affects the self-healing process occurring at the crack tip.See [1] for report 1.Deceased.L. Ya. Karpov Scientific-Research Physicochemical Institute, Moscow. Translated from Mekhanika Polimerov, No. 3, pp. 445–448, May–June, 1970.     

Bases of singular solutions in problems of mechanics of cracks

The mechanics of cracks serves as a remarkable example of a consistent application of the general results of the theory of elliptic boundary value problems to domains with piecewise smooth boundaries (see key papers [1–3] and also monographs [4, 5]). In the first place, this applies to the analysis of the behavior of stress-strain state in the vicinity of the crack tip and to operations involving the most important characteristics of this state, because explicit formulas that enable a reasonable prediction of the fracture process are available only for isotropic elastic materials. Meanwhile, the modern engineering practice involves the study of damages in anisotropic and composite bodies.     

Numerical aspects of the XFEM — The XFEM p–Version

The XFEM is known to approximate the displacements and stresses around a crack tip in a very efficient way. But as we will present in this paper we have to deal with a phenomenon coming along with this method that compels us to use higher order shape functions for those elements that are enriched by the crack tip functions. For the computation of the stress–intensity–factors we are using a J–integral over a circular domain Ω. The accuracy of the results depend on • the radius of Ω • the number of elements used in the XFEM computation • the number of nodes which were enriched by the crack tip functions (number of layers) and • the shape functions which were used for the standard FE term For more information about the XFEM we refer to [1]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

轴对称金属模具电磁热裂纹止裂中热应力场的分析

为求解金属模具脉冲放电止裂瞬间裂纹尖端附近的热应力场,选择具有半埋藏环形裂纹的金属凹模为研究对象,采用复变函数方法求解了凹模内外环面均匀通入强脉冲电流放电止裂时的热应力场．理论分析结果证实:由于放电瞬间脉冲电流的绕流集中效应,使金属凹模内部环形裂纹尖端附近金属迅速升温,金属熔化形成堆焊,并由于瞬间温升产生热压应力场．研究结果表明:应用电磁热效应止裂技术可以减小裂纹尖端的应力集中,形成的热压应力场有效地阻止金属模具中干线裂纹源的开裂趋势,达到了裂纹止裂目的．     

On acoustic and electric waves diffraction in piezoelectric medium by a permeable half-plane crack

The problem of electric and acoustic waves diffraction by a half-plane crack in a transversal isotropic piezoelectric medium is investigated. The crack is assumed to be electric permeable and free of tractions. The so-called “quasi-hyperbolic approximation” [15] is adopted. Applying Laplace transformations and Wiener–Hopf technique a closed form solution is obtained. By the means of Cagniard–de Hoop method a detailed dynamic full electroacoustic wavefield’s investigation is conducted. Mode conversion between electric and acoustic waves, effect of electroacoustic head wave, Bleustein–Gulyaev surface wave and the wavefield structure depending on the type of the incident wave (acoustic or electric) and its angle of incidence are analyzed in details. The dynamic field intensity factors at the crack tip depending on the angle of incidence and on time are derived explicitly. Numerical analysis is presented.     

On acoustic and electric waves diffraction in piezoelectric medium by a permeable half-plane crack

The problem of electric and acoustic waves diffraction by a half-plane crack in a transversal isotropic piezoelectric medium is investigated. The crack is assumed to be electric permeable and free of tractions. The so-called “quasi-hyperbolic approximation” [15] is adopted. Applying Laplace transformations and Wiener–Hopf technique a closed form solution is obtained. By the means of Cagniard–de Hoop method a detailed dynamic full electroacoustic wavefield’s investigation is conducted. Mode conversion between electric and acoustic waves, effect of electroacoustic head wave, Bleustein–Gulyaev surface wave and the wavefield structure depending on the type of the incident wave (acoustic or electric) and its angle of incidence are analyzed in details. The dynamic field intensity factors at the crack tip depending on the angle of incidence and on time are derived explicitly. Numerical analysis is presented.     

基于裂纹尖端二阶弹性解的断裂过程区尺度

基于Westergaard应力函数裂纹尖端二阶弹性解,推导了裂纹尖端微裂区的轮廓线和特征尺寸的解析表达式;采用幂函数模型描述的拉应变软化模型,确定了在最大拉应力强度理论和最大拉应变强度理论下断裂过程区（FPZ）临界值的解析表达式;将基于Westergaard应力函数一阶弹性解及二阶弹性解、Muskhelishvili应力函数和Duan-Nakagawa模型确定的FPZ临界值进行了比较．结果表明裂纹尖端微裂区和FPZ临界值随着Poisson比的减小而增加并逐渐趋近于应用最大拉应力强度理论确定的结果;二阶弹性解确定的裂纹尖端微裂区和FPZ临界值大于一阶弹性解的值;FPZ临界值随着拉应变软化指数的增加而增加;二阶弹性解确定的FPZ临界值的精度远高于一阶弹性解确定的值.     

On the computation of material forces in continuum‐atomistic modelling

Material forces govern the behavior and evolution of defects in solids. In elastic materials these forces which are associated with the Eshelby stress tensor are used to describe fracture sensitivities and can be employed to compute the J‐integral [2]. Since crack propagation begins with a variety of fundamental processes which occur within highly localized ultra–fine volume of material that constitute the fracture process zone surrounding a crack tip [3], the question of appropriate growth criteria, i.e. how far and in which direction a crack will glide under a certain loading condition is implied by the material force. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical crack path prediction considering orthotropic effects and experimental verification

For a reliable prediction of crack paths, on the one hand the accurate calculation of crack tip loading quantities is inevitable, on the other hand orthotropic features of the fracture toughness need to be taken into account. The interplay of crack tip loading and material response due to fracture is still unclear and seems to have a crucial effect on crack path predictions. Numerical tools for the accurate calculation of crack tip loading quantities using path-invariant J-integrals and interaction integrals (I-integral) are presented. Here, global approaches are beneficial when considering crack tips approaching other crack faces or internal boundaries. Curved crack faces have to be taken into account and special treatment regarding crack face integrals is necessary. Experimental investigations are carried out at standard CT-specimens of rolled aluminum alloy Al-7075 exhibiting a directional orthotropy of the fracture toughness. Considering that property, the numerically predicted crack paths based on FE calculations show very good agreement with subcritically grown paths obtained from experiments. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

各向异性材料动态裂纹扩展特性和动态应力强度因子

基于各向异性材料力学,研究了无限大各向异性材料中Ⅲ型裂纹的动态扩展问题．裂纹尖端的应力和位移被表示为解析函数的形式,解析函数可以表达为幂级数的形式,幂级数的系数由边界条件确定．确定了Ⅲ型裂纹的动态应力强度因子的表达式,得到了裂纹尖端的应力分量、应变分量和位移分量．裂纹扩展特性由裂纹扩展速度M和参数alpha反映,裂纹扩展越快,裂纹尖端的应力分量和位移分量越大;参数alpha对裂纹尖端的应力分量和位移分量有重要影响．     

A criterion for the growth of cracks with bonds in the end zone

A fracture criterion which takes account of the work done in the deformation of bonds in the end zone of a crack is proposed for analysing the quasistatic growth of a crack with bonds in the end zone. The energy condition that the deformation energy release rate at the crack tip is equal to the rate of deformation energy consumption by the bonds in the end zone of the crack (the first fracture condition) corresponds to the state of limit equilibrium of the crack tip. The rupture of bonds at the trailing edge of the end zone is determined by the condition for their limiting traction (the second fracture condition). Starting from these two conditions, the processes of subcritical and quasistatic crack growth are considered for the case of a rectilinear crack at interface of materials and the two basic fracture parameters, the critical external load and the size of the end zone of the crack in the state of limit equilibrium, are determined. Analytical expressions are obtained for the deformation energy release rate at the crack tip and the rate of deformation energy consumption by the bonds and, also, the dependences of the critical external load and size of the end zone of the crack on the crack length in the case of a rectilinear crack in a homogeneous body with bond tractions which are constant and independent of the external load. The limit cases of a crack which is filled with bonds and a crack with a short end zone are considered.