Hydraulic fracturing with a simplified fluid model and XFEM

In hydraulic fracturing, the pressure exerted by the fracking fluid onto the surrounding solid, is typically obtained from solving the Reynolds equation. This is not always robust and leads to a complex behaviour at the crack front (toughness or viscous regimes). In the presented work, the Reynolds equation is replaced by a simplified fluid model, where pre-defined pressure distributions are assumed which lead to a simpler and robust coupled problem. The surface integration of the fluid pressure onto the surrounding solid is outlined without any limitations on the distributions. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

含椭圆孔或裂纹压电介质平面问题的基本解

应用复变函数的方法，并基于精确的电边界条件，导出了含一椭圆孔或裂纹的横观各向同性压电体在任意集中力和集中电荷作用下的复变函数解，即Ｃｒｅｎ函数解·叠加该解，得到了裂纹表面作用任意集中载荷或分布载荷时的一般解·这些解不但澄清了从前文献中一些不合理的结果，同时也为应用边界元法求解更复杂的压电介质断裂力学问题提供了基本解·     

Invariant Integrals for the Equilibrium Problem for a Plate with a Crack

We consider the equilibrium problem for a plate with a crack. The equilibrium of a plate is described by the biharmonic equation. Stress free boundary conditions are given on the crack faces. We introduce a perturbation of the domain in order to obtain an invariant Cherepanov–Rice-type integral which gives the energy release rate upon the quasistatic growth of a crack. We obtain a formula for the derivative of the energy functional with respect to the perturbation parameter which is useful in forecasting the development of a crack (for example, in study of local stability of a crack). The derivative of the energy functional is representable as an invariant integral along a sufficiently smooth closed contour. We construct some invariant integrals for the particular perturbations of a domain: translation of the whole cut and local translation along the cut.     

The Method of Smooth Domains in the Equilibrium Problem for a Plate with a Crack

A free boundary problem is considered of the equilibrium of an elastic plate with a crack. We suppose that some boundary mutual nonpenetration conditions are given on the crack faces in the form of simultaneous equalities and inequalities. We suggest a new approach to posing the problem in a smooth domain although it was stated in a domain with cuts originally. We treat the constraints on the components of the displacement vector and stress tensor on the crack faces as interior constraints, i.e., constraints given on subsets of the smooth domain of a solution.     

非对称载荷作用的外部圆形裂纹问题

使用边界积分方程方法,研究了三维无限弹性体中受非对称载荷作用的外部圆形裂纹问题。通过使用Fourier级数和超几何函数,将问题的二维边界奇异积分方程简化为Abel型方程,获得了一般非对称载荷作用的外部圆形裂纹问题的应力强度因子精确解,比用Hankel变换法得到的结果更为一般。结果表明:边界积分方程法在解析分析方面还有很大的潜力。     

泊松比对静止平面应变裂纹尖端的理想塑性应力场的影响

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,利用平衡方程和含有泊松比的Mises屈服条件,本文导出了静止平面应变裂纹尖端的理想塑性应力场的一般解析表达式.将这些一般解析表达式用于具体裂纹,我们就可以得到静止平面应变Ⅰ型、Ⅱ型及Ⅰ-Ⅱ复合型裂纹尖端的理想塑性应力场的解析表达式,这些表达式含有泊松比.     

两非均匀半平面粘结的非均匀平面的裂纹问题*

本文对两个非均匀半平面粘结的非均匀平面的裂纹问题作了分析,文中假定两种材料的泊松比v相同,但杨氏模量随坐标x按不同形式的指数函数变化.本文使用非均匀平面问题的单裂纹解及富氏变换方法, 使问题归为一个柯西型奇异积分方程,最后对应力强度因子的计算给出了若干数值例子.     

The Cruciform Crack Problem and Sigmoidal Transformations

A sigmoidal transformation is an S-shaped one-to-one mapping of a compact interval onto itself. Various sigmoidal transformations are discussed, some of which are used to find approximate solutions of an integral equation, with a fixed singularity, arising from a problem in elasticity. In particular it is shown that, in this context, a sigmoidal transformation due to Sidi gives the best results. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.     

Elastodynamic Problem of an Interface Linear Crack under Harmonic Loading

The present work is devoted to application of boundary integral equations to the 2D problem for a linear crack located on the bimaterial interface under harmonic loading. The system of linear algebraic equations is derived to solve the problem numerically. The distribution of the displacements and tractions at the bonding interface and the surface of the crack are obtained for the case of the tension–compression wave which propagates normally to the interface. The results are compared with those obtained for the static case. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

含圆形孔口和水平直裂纹的平面问题的应力分析

利用复变函数方法和积分方程理论研究了既含有圆形孔口又含有水平裂纹的无限大平面的平面弹性问题,将复杂的解析函数的边值问题化成了求解只在裂纹上的奇异积分方程的问题.此外,还给出了裂纹尖端附近的应力场和应力强度因子的公式.     

The Half-plane with an Edge Crack in Plane Elastostatics

The Wiener-Hopf technique is applied to the problem of a half-planewith a crack perpendicular to its edge. A closed solution isobtained giving the stress and displacement fields in the formof eigenfunction expansions, valid over the whole region, exceptthe neighbourhood of the tip of the crack. The behaviour inthis region is obtained by an asymptotic technique. The coefficientsof the expansions involve a generalized factorial function,and a method for the computation of this function is given,thus enabling numerical values of the stresses and displacementsto be obtained for the given cases of stretching and bendingthe half-plane.     

The Stress Boundary Value Problem for an Ideal Fibre-reinforced Rectangular Plate with a Hole

The title problem is examined for plane deformations of an idealfibre-reinforced rectangular plate with a hole. The solutionis found to depend on simultaneous ordinary differential equationswhich may be integrated without difficulty. This is in markedcontrast with the corresponding problems for an anisotropicelastic body and enables approximate solutions for stronglyanisotropic bodies to be derived without difficulty. The particularcases of a line crack in the plate at an angle to the fibresand an arbitrary symmetrical hole are examined in detail.     

Bounds for an Optimization Problem with a Linear Programming Structure

The problem of estimating the value of a quadratic cost indexminimized with respect to a linear system constraint arisesin many design studies in control theory and regression analysis.In particular, the assessment of several choices of system inputscould lead to inefficient procedures based on the executionof an optimization routine for each input subset. To avoid suchpitfalls, a set of bounds on the optimal cost have been constructedusing information resident in a set of system moments. The momentsare easily calculated for systems in weighting function formand the bounds obtained from explicit formulae. The derivationof these bounds is presented in some detail using Hilbert spaceanalysis to obtain a canonical form, and randomized solutionsto give the subsequent bound formulae. A simple example is presentedto illustrate the results.     

复杂水力计算中的高次方程问题

将实际工程中的复杂水力计算问题归纳为对高次方程的求解,并提出快速、正确解决这类问题的数值方法——中点迭代法,对解决实际工程问题具有指导意义.     

A quasi-optimal convergence result for fracture mechanics with XFEM

The aim of this Note is to give a convergence result for a variant of the eXtended Finite Element Method (XFEM) on cracked domains using a cut-off function to localize the singular enrichment area. The difficulty is caused by the discontinuity of the displacement field across the crack, but we prove that a quasi-optimal convergence rate holds in spite of the presence of elements cut by the crack. The global linear convergence rate is obtained by using an enriched linear finite element method. To cite this article: E. Chahine et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

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.     

非均匀双周期裂纹场的反平面问题

本文从虎克定律与平衡方程出发,利用复变函数映象的理论.将裂纹发生的矩形区域保角变换到ζ平面的上半平面去.再根据H.И.穆斯黑里什维利[1]的理论,对非均匀双周期裂纹场的反平面问题,我们求得闭合解,并推出了应力强度因子.     

三角形弹性夹杂对裂纹的影响

研究了三角形弹性夹杂和裂纹之间的相互影响问题。应用Chau和Wang导出的面力边值问题的边界积分方程为基本方程,用夹杂和基体交界面上的面力和位移的连续性条件为补充方程,从而得到了一组能够解决夹杂和裂纹相互影响问题的方程,最后的方程组用一种新的边界单元法求解。计算了各种不同的夹杂和基体的材料常数以及夹杂和基体之间不同距离情况下裂纹尖端的应力强度因子。文中结果对研究新型复合材料有一定的应用价值。