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。     

Derivatives of the energy functional for 2D‐problems with a crack under Signorini and friction conditions

We consider the two‐dimensional elasticity problem for an elastic body with a crack under unilateral constraints imposed at the crack. We assume that both the Signorini condition for non‐penetration of the crack faces and the condition of given friction between them are fulfilled. The problem is non‐linear and can be described by a variational inequality. Varying the shape of the crack by a local coordinate transformation of the domain, the first derivative of the energy functional to the problem with respect to the crack length is obtained, which gives the criterion for the crack growing. The regularity of the solution is discussed and the singular solution is performed. Copyright © 2000 John Wiley & Sons, Ltd.     

Analysis of a mode-III crack in the presence of surface elasticity and a prescribed non-uniform surface traction

We consider an elastic solid incorporating a mode-III crack in which the crack faces incorporate the effects of surface elasticity and are further subjected to prescribed non-uniform surface tractions. The surface elasticity is modelled using the continuum-based model of Gurtin and Murdoch. Using complex variable techniques, the corresponding problem is reduced to the solution of a first order Cauchy singular integro-differential equation which, in turn, leads to the complete solution of the aforementioned crack problem valid everywhere in the domain of interest (including at the crack tip). Finally, we note that, as a particular case of our analysis, the classical decomposition of a mode-III crack problem in linear elasticity continues to hold even in the presence of surface elasticity.     

夹杂和裂纹的相互作用及端点相交的奇性性态分析

利用单根裂纹和单根夹杂的基本解,通过弹性力学的线性叠加原理,将平面裂纹和夹杂相互作用的问题归结为解一组带有柯西型奇异积分的积分方程组,计算了裂纹和夹杂端点的应力强度因子,给出了一些数值例子,并对夹杂和裂纹水平接触时的情形作了奇性分析,结果可作为研究夹杂尖端引起的裂纹及其扩展的工程分析的计算模型。     

一维正方准晶沿准周期方向穿透的抛物线裂纹问题

利用广义复变函数方法研究了一维正方准晶材料中周期平面的抛物线裂纹问题,通过建立广义保角映射,将物理平面的抛物线裂纹外映射到数学平面里的单位圆内.得出了声子场和相位子场的应力分量在像平面下的复表示,并且得到了抛物线裂纹尖端的应力强度因子.并在特殊情况下,所得结果与Griffith裂纹的结果一致.     

一维六方准晶压电材料中幂函数型曲线裂纹的反平面问题

依据一维六方准晶压电材料反平面问题的基本方程,利用复变函数方法,通过引入适当的保角映射,研究了一维六方准晶压电材料中幂函数型曲线裂纹的反平面问题,并利用Cauchy积分理论,得到电不可通和电可通边界条件下的应力场和位移场的复表示以及裂纹尖端场强度因子的解析表达式.     

Saint–Venant torsion of a circular bar with radial cracks incorporating surface elasticity

The Saint–Venant torsion problem of a circular cylinder containing a radial crack with surface elasticity is studied. The surface elasticity is incorporated into the crack faces by using the continuum-based surface/interface model of Gurtin and Murdoch. Both an internal crack and an edge crack are considered. By using the Green’s function method, the boundary value problem is reduced to a Cauchy singular integro-differential equation of the first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials and the collocation method. Due to the incorporation of surface elasticity, the stresses exhibit the logarithmic singularity at the crack tips. The torsion problem of a circular cylinder containing two symmetric collinear radial cracks of equal length with surface elasticity is also solved by using a similar method. The strengths of the logarithmic singularity and the size-dependent torsional rigidity are calculated.     

Nonsmooth domain optimization for elliptic equations with unilateral conditions

In the paper we consider elliptic boundary problems in domains having cuts (cracks). The non-penetration condition of inequality type is prescribed at the crack faces. A dependence of the derivative of the energy functional with respect to variations of crack shape is investigated. This shape derivative can be associated with the crack propagation criterion in the elasticity theory. We analyze an optimization problem of finding the crack shape which provides a minimum of the energy functional derivative with respect to a perturbation parameter and prove a solution existence to this problem.     

On elastic bodies with thin rigid inclusions and cracks

This paper is concerned with the analysis of equilibrium problems for two‐dimensional elastic bodies with thin rigid inclusions and cracks. Inequality‐type boundary conditions are imposed at the crack faces providing a mutual non‐penetration between the crack faces. A rigid inclusion may have a delamination, thus forming a crack with non‐penetration between the opposite faces. We analyze variational and differential problem formulations. Different geometrical situations are considered, in particular, a crack may be parallel to the inclusion as well as the crack may cross the inclusion, and also a deviation of the crack from the rigid inclusion is considered. We obtain a formula for the derivative of the energy functional with respect to the crack length for considering this derivative as a cost functional. An optimal control problem is analyzed to control the crack growth. Copyright © 2010 John Wiley & Sons, Ltd.     

Similarity and dimensional analysis in the plane problem of the propagation of a vertical hydraulic fracture crack in an elastic medium

The problem of the growth of a vertical hydraulic fracture crack in an unbounded elastic medium under the pressure produced by a viscous incompressible fluid is studied qualitatively and by numerical methods. The fluid motion is described in the approximation of lubrication theory. Near the crack tip a fluid-free domain may exist. To find the crack length, Irwin’s fracture criterion is used. The symmetry groups of the equations describing the hydraulic fracture process are studied for all physically meaningful cases of the degeneration of the problem with respect to the control parameters. The condition of symmetry of the system of equations under the group of scaling and time-shift transformations enables the self-similar variables and the form of the time dependence of the quantities involved in the problem to be found. It is established that at non-zero rock pressure the well-known solution of Spence and Sharp is an asymptotic form of the initial-value problem, whereas the solution of Zheltov and Khristianovich is a limiting self-similar solution of the problem. The problem of the formation of a hydraulic fracture crack taking into account initial data is solved using numerical methods, and the problem of arriving at asymptotic mode is investigated. It is shown that the solution has a self-similar asymptotic form for any initial conditions, and the convergence of the exact solutions to the asymptotic forms is non-uniform in space and time.     

Closed-form solutions for a rectangular layer with central crack under anti-plane shear stress

We consider the problem of determining the stress distributionin a finite rectangular elastic layer containing a Griffithcrack which is opened by internal shear stress acting alongthe length of the crack. The mode III crack is assumed to belocated in the middle plane of the rectangular layer. The followingtwo problems are considered: (A) the central crack is perpendicularto the two fixed lateral surfaces and parallel to the othertwo stress-free surfaces; (B) all the lateral surfaces of therectangular layer are clamped and the central crack is parallelto the two lateral surfaces. By using Fourier transformations,we reduce the solution of each problem to the solution of dualintegral equations with sine kernels and a weight function whichare solved exactly. Finally, we derive closed-form expressionsfor the stress intensity factor at the tip of the crack andthe numerical values for the stress intensity factor at theedges of the cracks are presented in the form of tables.     

A contact problem for a smooth rigid disc inclusion in a penny-shaped crack

The present paper examines the problem of the complete indentation of the surface of a penny-shaped crack by a smooth rigid disc inclusion. The integral equation governing the problem is solved numerically to evaluate the axial stiffness of the rigid inclusion and the stress intensity factors at the tip of the penny-shaped crack.     

Problem of an elastic half-plane with a crack emerging orthogonally at the boundary

The problem of the half-plane, in which a finite crack emerges orthogonally at the boundary, is studied. On the edges of the crack a self-balancing load is applied. A detailed investigation is carried out for an integral equation with respect to the unknown derivative of the displacement jump, to which the problem can be reduced. The exact solution of the integral equation is constructed with the aid of the Mellin transform and the Riemann boundary value problem for the halfplane. The asymptotic behavior of the solution at both ends of the crack is elucidated. First the asymptotic behavior of the solution at the point of emergence of the crack is obtained and the dependence of this asymptotic behavior on the type of the load is established. For a special form of the load one obtains a simple expression of the stress intensity coefficient. In the case of a general load, the asymptotic behavior is used for the construction of an effective approximate solution on the basis of the method of orthogonal polynomials. As a result, the problem reduces to an infinite algebraic system, solvable by the reduction method.Translated from Dinamicheskie Sistemy, No. 4, pp. 45–51, 1985.     

Unsteady propagation of longitudinal shear cracks

In all problems of unsteady crack propagation which have been solved to date [1 to 3], it has been assumed that the crack propagates at a constant speed. This assumption was not prompted by physical considerations of the problem, but by the methods of solution, therefore, the applicability of the results is limited. It would be more realistic to consider the speed of crack propagation as a function of time based on explicit physical hypotheses. Unfortunately, the general case of the resultant problem cannot be solved by existing methods. However, the problem of longitudinal shear cracks i.e. the plane problem in which the displacement is parallel to the crack boundary, may be solved for an arbitrary given variation in crack propagation speed, utilizing the method developed in connection with the theory of supersonic flows [4 and 5].

Note that equilibrium problems of longitudinal shear cracks have been studied in [6 and 7].     

裂纹与弹性夹杂的相互影响*

本文利用无限域上单根弹性夹杂和单根裂纹产生的位移和应力,将裂纹与弹性夹杂的相互影响问题归为解一组柯西型奇异积分方程,然后用此对夹杂分枝裂纹解答的奇性性态作了理论分析,并求得了振荡奇性界面应力场,对于不相交的夹杂裂纹问题,具体计算了端点的应力强度因子及夹杂上的界面应力,结果令人满意。     

Nonlinear breathing crack identification from time-domain sensitivity analysis

Breathing cracks are often encountered in engineering structures and detection of such cracks at the earliest stage is important for safety and serviceability of the structures. In this paper, a new breathing crack identification approach based on the time-domain sensitivity analysis is proposed. For the forward problem, the crack is simply treated as a nonlinear oscillator and how to model a structure with breathing cracks is specified. As regards the inverse crack identification, it is formulated as a nonlinear least-squares optimization problem and a new sensitivity-based approach is developed to get the solution. In doing so, the sensitivity analysis of the possibly non-differentiable breathing crack models is necessarily proceeded through a smoothing strategy. Moreover, to enhance the convergence, the trust-region constraint is introduced and the Tikhonov regularization is naturally called to tackle the constraint. Numerical examples are studied to verify the feasibility and efficiency of the proposed approach in breathing crack identification.     

A conductive crack and a remote electrode at the interface between two piezoelectric materials

An interaction of a tunnel conductive crack and a distant strip electrode situated at the interface between two piezoelectric semi-infinite spaces is studied. The bimaterial is subject by an in-plane electrical field parallel to the interface and by an anti-plane mechanical loading. Using the presentations of electromechanical quantities at the interface via sectionally-analytic functions the problem is reduced to a combined Dirichlet-Riemann boundary value problem. Solution of this problem is found in an analytical form excepting some one-dimensional integrals calculations. Closed form expressions for the stress, the electric field and their intensity factors, as well as for the crack faces displacement jump are derived. On the base of these presentations the energy release rate is also found. The obtained solution is compared with simple particular case of a single crack without electrode and the excellent agreement is found out. An auxiliary plane problem for open and closed cracks between two isotropic materials is also considered. The mathematical model of this problem is identical to the above one, therefore, the obtained solution is used for this model. It is compared with finite element solution of a similar problem and good agreement is found out.     

有一条裂纹的圆形焊接问题

本文讨论了在带圆孔的无限平面中焊接一个不同材料的带裂纹的近似圆板的问题.该问题化为求解解析函数边值问题然后又转化为求解沿裂纹的奇异积分方程.后者的数字解法也已给出.文末并对Ⅰ型、Ⅱ型情况得出了应力强度因子的公式以及数字结果.     

采用新方法研究非局部理论中Ⅰ-型裂纹的断裂问题

采用新的方法研究非局部理论中Ⅰ_型裂纹的断裂问题,进而确定裂纹尖端的应力状态,这种方法就是Schmidt方法·　所得结果比艾林根研究同样问题的结果准确和更加合理,克服了艾林根研究同样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上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·     

On a finite crack partially penetrating two circular inhomogeneities and some related problems

We investigate a Mode-III finite slit crack partially penetrating two circular inhomogeneities embedded in an unbounded matrix. In order to obtain analytical solutions, it is assumed that the two circular inhomogeneity-matrix interfaces are Apollonius circles with respect to the two crack tips (or equivalently the two crack tips are just mutually image points with respect to each one of the two circular interfaces). Particularly closed-form expressions of the stress intensity factors at the two crack tips are obtained even though only series form solutions to the original boundary value problem can be derived. The loadings considered in this research include: (i) remote uniform anti-plane shearing; (ii) a straight screw dislocation at any position of the three-phase composite system; (iii) a Zener-Stroh crack. The results are verified by comparison with existing solutions. The related problem of a circular hole partially merged in two circular inhomogeneities is also addressed, with closed-form expressions of the stress concentration factors derived.