Shape sensitivity of a plane crack front

The 3D‐elasticity model of a solid with a plane crack under the stress‐free boundary conditions at the crack is considered. We investigate variations of a solution and of energy functionals with respect to perturbations of the crack front in the plane. The corresponding expansions at least up to the second‐order terms are obtained. The strong derivatives of the solution are constructed as an iterative solution of the same elasticity problem with specified right‐hand sides. Using the expansion of the potential and surface energy, we consider an approximate quadratic form for local shape optimization of the crack front defined by the Griffith criterion. To specify its properties, a procedure of discrete optimization is proposed, which reduces to a matrix variational inequality. At least for a small load we prove its solvability and find a quasi‐static model of the crack growth depending on the loading parameter. Copyright © 2003 John Wiley & Sons, Ltd.     

Fractographic study of macrocrack propagation in oriented Kapron

The fracture surface of an oriented Kapron monofilament has been studied at electron-microscope and optical magnifications with the object of establishing the details of the micromechanism of crack propagation associated with the fracture of polymeric materials. Microparabolic figures can be observed in the specular zone of the fracture surface. This relief may be assumed to originate in the interaction of the main crack and the submicroscopic cracks present in loaded polymers. In the region of high main-crack velocities it is possible to observe a self-oscillatory motion of the tip of the main crack leading to the formation on the fracture surface of a system of bands parallel to the main crack front.Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 4, pp. 645–648, July–August, 1971.     

论复合型脆断的周向应变因子准则

本文讨论了线弹性断裂力学的复合型断裂准则.推荐了周向应变因子准则.     

Design of a new fracture specimen by 3D singularity analyses

A new design of a fracture mechanics specimen is presented. It ensures a valid square‐root singularity along the whole threedimensional crack front under mode‐I conditions. Within the proposed concept it is possible to adjust the angle between the crack front and the free surface in advance. Therefore, the shape of the whole crack front can be influenced in a desired manner. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Non-local theory solution of two collinear mode-I cracks in piezoelectric materials

In this paper, the basic solution of two collinear cracks in a piezoelectric material plane subjected to a uniform tension loading is investigated by means of the non-local theory. Through the Fourier transform, the problem is solved with the help of two pairs of integral equations, in which the unknown variables are the jumps of displacements across the crack surfaces. To solve the integral equations, the jumps of displacements across the crack surfaces are directly expanded in a series of Jacobi polynomials. Numerical examples are provided to show the effects of the interaction of two cracks, the materials constants and the lattice parameter on the stress field and the electric displacement field near crack tips. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularities are present at crack tips. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to using the maximum stress as a fracture criterion in piezoelectric materials.     

幂律非线性粘弹性材料中的裂纹扩展*

对蠕变不可压幂律非线性粘弹性材料中裂纹的蠕变扩展进行了分析,为描述银纹带中的力学行为,假设在裂纹尖端邻域中断裂过程区中分布着阻抗裂纹张开的粘聚应力б_f,.通过对均匀应力参考状态平凡解的摄动,将非线性粘弹性问题化成线性问题处理.对于幂指数.n≌1的弱非线性情况得到了应力与位移表达式.提出断裂过程区局域能量判据,导出了裂纹孕育时间t*与蠕变扩展率a的预测公式.     

各向异性体的断裂条件及其应力空间几何

本文根据固体断裂裂纹扩展的能量条件,探索各向异性体的断裂条件及其应力空间几何,结论是:在恒温条件下,断裂的应力空间曲面为二次曲面.当流体静应力大于零时,为椭球面;当流体静应力小于零时,为单叶双曲面.本文所得的结论具有一定的普遍性,前人的一些结论可作为本文的特殊形式.     

双材料界面裂纹平面问题的半权函数法

应用半权函数法求解双材料界面裂纹的平面问题．由平衡方程、应力应变关系、界面的连续条件以及裂纹面零应力条件推导出裂尖的位移和应力场,其特征值为lambda及其共轭．设置特征值为lambda的虚拟位移和应力场,即界面裂纹的半权函数A·D2由功的互等定理得到应力强度因子KⅠ和KⅡ以半权函数与绕裂尖围道上参考位移和应力积分关系的表达式．数值算例体现了半权函数法精度可靠、计算简便的特点．     

An asymptotic form of the energy functional for an elastic body with a crack and a rigid inclusion. The plane problem

The plane problem in the linear theory of elasticity for a body with a rigid inclusion located within it is investigated. It is assumed that there is a crack on part of the boundary joining the inclusion and the matrix and complete bonding on the remaining part of the boundary. Zero displacements are specified on the outer boundary of the body. The crack surface is free from forces and the stress state in the body is determined by the bulk forces acting on it. The variation in the energy functional in the case of a variation in the rigid inclusion and the crack is investigated. The deviation of the solution of the perturbed problem from the solution of the initial problem is estimated. An expression is obtained for the derivative of the energy functional with respect to a zone perturbation parameter that depends on the solution of the initial problem and the form of the vector function defining the perturbation. Examples of the application of the results obtained are studied.     

A singular perturbation approach to integral equations occurring in poroelasticity

Singular perturbation theory is used to solve the integral equationswhich occur when treating finite-length crack problems in porouselastic materials. The method provides the stress intensityfactors which characterize the near crack tip stress and displacementfields for small times. The method also gives the stress andpore pressure fields on the fracture plane for small times relativeto the diffusive time scale. In this paper, the authors treatcrack problems which are unmixed in the pore pressure boundarycondition on the fracture plane. The Abelian result that smalltimes correspond, in Laplace transform space, to large valuesof the transform variable is used to formulate the problemsin terms of a small parameter. Rescaling on this small parameterleads to inner problems which are eigensolutions of the semi-infiniteproblems treated earlier by the authors. The outer solutionsare given by elastic eigensolutions together with appropriatefluid dipole responses. These outer solutions give the completestress and pore pressure fields except in the neighbourhoodof the crack tips; in this region the outer solutions are asymptoticallymatched with inner solutions. The full outer solutions are givenhere as an asymptotic expansion for small times and enable thedevelopment of the outer fields to be followed in real time.A reciprocal theorem in Laplace transform space is used to checkthe small-time solutions. The inner problem is rescaled to asemi-infinite crack problem, so eigensolutions of this semi-infiniteproblem are used together with the known asymptotic behaviourof the real solution to identify the stress intensity factor.The stress intensity factor is then related to an integral involvingthe inner limit of the outer solution together with the eigensolutionof the semi-infinite problem. Using this integral, we recoverthe result for the stress intensity factor found using singularperturbation theory. A ‘nearly’ invariant integralanalogous to the invariant M integral used in elastostaticsis derived. Unfortunately, the poroelastic analogue is not invariant,although it is used to verify the small-time results.     

Cohesive zone model for a transverse breathing crack in a rotor

The breathing mechanism of a transversely cracked rotor and its influence on a rotor system that appears due to the shaft weight is studied. This breathing mechanism is based on experimental and simulation result for the crack shape reported in the literature. If the crack depth is small, the crack closure line is a straight line while for larger crack depths the crack closure becomes more curved. For both cases, a method is proposed for the evaluation of the stiffness losses in the cross section that contains the crack. This method is based on a cohesive zone model (CZM) instead of linear elastic fracture mechanics (LEFM) approach, because LEFM is valid only for the fully open crack and cannot be extended to other intermediate situations. As the crack is closed, the stress intensity factor (SIF) will not appear at the boundary between the closed cracked areas and the open cracked areas. The CZM is developed for mode-I plane strain conditions and accounts explicitly for triaxiality of the stress state by using constitutive relations. The proposed model gives more realistic results than models based on LEFM for the stiffness losses of the crack rotor system for a wide range of the crack depth. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Role of the supermolecular structure of PMM in the morphology of fracture and features introduced by the type of loading

The spherulitic supermolecular structure of PMM, expressed in the fracture surface, determines the fracture pattern and kinetics. The effect of the type of loading on the morphology of the fracture surface is described. Banding of the fracture surface is attributed to periodic energy pulses leading to quasi-brittle fracture at the moving crack front and selective local crack development at the band edges.Riga Lenin Komsomol Institute of Civil Aviation Engineers. Translated from Mekhanika Polimerov, Vol. 4, No. 5, pp. 776–782, September–October, 1968.     

裂纹与线夹杂垂直接触问题的奇性分析

以短纤维复合等作为工程背景,采用现有文献中单裂纹和单夹杂的基本解,对于限平项（基体）上,裂纹和线夹杂的垂直接触问题从断理解力学的角度作了研究,得到了问题的积分方程,推出了接触点的性线指数,奇性应力及以此表示的接触点附近三个区域内的应力强度因子表达式,并给出一些数值结果,可供工程实际参考。     

Analysis of the Stress State of the Surface Layer of a Polymeric Mass upon Filling of Bulky Compression Molds

The stress state of the surface layer of a polymeric mass during filling of bulky compression molds is analyzed. It is shown that, at particular rheological characteristics of the mass, temperature, and filling rates, cracking of the surface layer occurs, which leads to defects in the finished products. A physical analysis of this process makes it possible to conclude that the cracks arise due to the normal stresses operating in the front region of the moving polymeric mass. It is found that, under certain flow conditions, areas with a pressure lower than the atmospheric one appear on the surface of the polymer. If the tensile stresses arising in these local regions are higher than the tensile strength of the mass, the continuity of the composition is broken in the direction determined by the greatest rate of the normal deformation. To confirm the reliability of the crack-formation mechanism proposed, the distribution of the pressure and normal stresses over the free surface is calculated based on a numerical method. These calculations show that, by comparing the stress level achieved in the front region with the tensile-strength characteristics of the polymeric composition, it is possible to predict, with a sufficient accuracy, the possibility of crack formation in the surface layer of such a mass under given flow conditions and thus to solve the question on flawless manufacturing of products.     

Influence of Multiple Micro Cracks on the Damage Behavior of a Macro-Crack Tip北大核心CSCD

The solution of an infinite plane containing a macro crack and a cluster of micro cracks under uniaxial tensile load was presented based on Muskhelishvili’s complex function method and the stepwise recursive method. The stress field and stress intensity factor K were obtained. Combined with the damage mechanics, damage parameter D of the macro-crack tip and the micro-crack tip under uniaxial tension was redefined, and the influence of different damage zone forms on the damage of the crack tip was analyzed. The results show that, both the chain-distribution and the reverse-chain-distribution micro cracks have an amplifying effect on the macro crack growth, and the damage parameter increases with the decrease of the inclination angle of the micro crack and the reduction of the distance between the macro crack and the micro cracks. For a relatively small inclination angle of the micro crack, the damage parameters of the macro crack and the micro crack heightens, and the damage parameter of the macro crack increases with the micro-crack length. For evenly distributed micro cracks in the continuous damage zone, the micro cracks have an amplifying effect on the macro-crack growth, and the damage parameter of the macro crack increases with the micro-crack number. © 2022 Editorial Office of Applied Mathematics and Mechanics. All rights reserved.     

含有直线状裂纹正交异性板的平面问题的应力函数

本文的解析对象为含有一与主轴呈任意角度直线状裂纹的无限大正交异性板的平面问题.采用加权积分法导出了能够表现裂纹尖端附近有限应力集中特征的应力函数.这样的计算模型消除了裂纹尖端的奇异性,可以比较真实地反映非金属材料微裂区的力学行为.     

平面应力状态断裂强度的椭圆准则分析

简要分析了近年来提出的一个断裂准则——椭圆准则的基本特征,导出了它在主应力坐标系下的基本方程.根据所导出的基本方程,获得了平面应力条件下椭圆准则理论断裂强度曲线的完整描述关系,并分析讨论了破坏发生的方位及断裂形式与材料本征力学性质参数之间的联系.与既有理论结果及实验现象的对比解释了椭圆准则在材料相关参数确定方面的局限性.当应力状态相关材料特征参数在拉伸区和压缩区均作为常数时,获得了铸铁和混凝土平面应力状态下的断裂强度曲线.与相关实验数据的对比表明,它们在拉伸区能较好地吻合,但压缩区的差异十分显著,进一步证实了材料特征参数随应力状态变化规律对椭圆准则发展的必要性.     

A study of the influence of the orientation of a planar surface crack in the shape of a limaçon of pascal on the stress concentration in a half-space

We study the stress state in the vicinity of a planar surface crack whose boundary is described by the limaçon of pascal. The problem is solved by a conformal mapping of the region occupied by the crack onto part of a disk in the plane. This makes it possible to apply a numerical-analytic method for solving systems of double singular integral equations of the mathematical theory of cracks. We present the graphs of the dependence of the stress intensity factor on the angular coordinate for cracks that are part of a limaçon of Pascal.     

Rupture in soft biological tissues modeled by a phase-field method

We present an application of the phase-field method of fracture to the simulation of artery rupture at large strains. To achieve this, the crack driving force function associated with the evolution of the crack phase-field is modified to account for the inherent anisotropy of the soft biological tissues. The phase-field methods present a promising and innovative approach to the thermodynamically consistent modeling of fracture. A key advantage lies in the prediction of the complex crack topologies where the cohesive zone approaches to fracture are known to suffer. A regularized crack surface functional is introduced that Γ-converges to a sharp crack topology for vanishing length scale parameter. The evaluation of the phase-field follows the minimization of this crack surface functional. The phase-field variable can be treated as a geometric quantity whose evolution is coupled to the anisotropic bulk response in a modular format in terms of a crack driving state function. A stress-based anisotropic failure criterion is introduced whose maximum value from the deformation history drives the irreversible crack phase-field. The formulation is verified by the finite element based simulation of a real arterial cross-section undergoing rupture in a two-dimensional setting when subjected to inflation pressure. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

双I—型裂纹断裂动力学问题的非局部理论解

研究了非局部理论双中I－型裂纹弹性波散射的力学问题,并利用富里叶变换使本问题的求解转换为三重积分方程的求解,进而采用新方法和利用一维非局部积分核代替二维非局部积分核来确定裂纹尖端的应力状态,这种方法就是Schmidt方法,所得结是比艾林根研究断裂静力学问题的结果准确和更加合理,克服了艾林根研究断裂静力学问题时遇到的数学困难,与经典弹性解相比,裂纹尖端不再出现物理意义下不合理的应力奇异性,并能够解释宏观裂纹与微观裂纹的力学问题。