应力强度因子分析的三角形Williams单元

弹性断裂分析的Williams广义参数单元计算模型中忽略了紧邻裂尖的微区域,为了进一步完善该计算模型,本文提出并建立了三角形Williams单元。首先围绕裂尖将奇异区均匀分割为有限个三角形单元,利用改进的Williams级数建立该单元的整体位移场计算模型;其次沿径向将该三角形单元进一步离散为多个相似四边形微单元和裂尖三角形微单元,并利用经典有限元理论建立微单元的局部位移场计算模型;然后利用整体位移场控制各微单元结点位移,并在此基础上研究建立裂尖奇异区三角形Williams单元及其控制方程。该单元模型中含有与裂尖应力强度因子相关的参数,能够直接计算裂尖处的应力强度因子。最后结合算例详细分析了三角形Williams单元计算模型中径向离散因子、离散数、Williams级数项对计算结果的影响。算例分析表明,三角形Williams单元所得的应力强度因子具有对奇异区尺寸不敏感的优点,且收敛快,计算精度高。     

冲击载荷作用下圆孔缺陷对裂纹动态扩展行为的影响规律

空腔和裂纹缺陷通常共存于深部地下岩体中,它们共同影响着岩体的结构安全性与稳定性。为了探究动力扰动载荷下圆形空腔对裂隙岩体内裂纹扩展行为的影响规律,提出了不同圆孔倾角的直裂纹空腔圆弧开口试件（circular opening specimen with straight crack cavity, COSSCC）,利用自制大型落锤冲击实验装置进行动态加载实验,同时采用裂纹扩展计系统测试了裂纹的动态起裂时刻与裂纹扩展速度等各种断裂力学参数,随后采用有限差分软件Autodyn进行裂纹扩展路径与圆孔周围应力场的数值分析,并采用有限元软件Abaqus计算裂纹的动态起裂韧度与裂纹扩展过程中的动态扩展韧度。结果表明:（1）当圆孔倾角θ小于10°时,裂纹扩展路径会偏折并穿过圆孔表面;当圆孔倾角θ为20°与30°时,裂纹扩展路径向圆孔方向发生偏折但不会穿过圆孔,圆孔具有明显的裂纹扩展引导作用; 当圆孔倾角θ为40°与50°时,裂纹扩展路径不会发生偏折,圆孔引导作用明显减弱。（2）当裂纹扩展路径达到圆孔空腔附近时,裂纹尖端的拉伸应力区与圆孔边缘的拉伸应力区发生重合,此时裂纹扩展速度显著增大,裂纹动态断裂韧度显著减小。（3）裂纹的偏折方向与裂纹尖端最大周向应力的方向基本一致。（4）裂纹动态断裂韧度始终小于裂纹起裂韧度,且裂纹动态断裂韧度与裂纹动态扩展速度呈负相关关系。裂纹动态扩展速度越大,裂纹动态断裂韧度越小。     

直接计算应力强度因子的扩展有限元法

系统地给出了直接计算应力强度因子的扩展有限元法。该方法以常规有限元法为基础,利用单位分解法思想,通过在近似位移表达式中增加能够反映裂纹面的不连续函数及反映裂尖局部特性的裂尖渐进位移场函数,间接体现裂纹面的存在,从而无需使裂纹面与有限元网格一致,无需在裂尖布置高密度网格,也不需要后处理就可以直接计算出应力强度因子,并且大大简化了前后处理工作。最后通过两个简单算例验证了该方法的精度,分析了影响计算结果的因素,并与采用J积分计算的应力强度因子作了对比,得出了两种方法计算精度相当的结论。     

基于有限元方法的航空发动机叶片应力强度因子计算

为研究叶片裂纹尖端的应力奇异性,以某型航空发动机压气机叶片为例,利用有限元方法研究了叶片裂纹尖端应力强度因子的计算方法,并研究了旋转叶片振动状态下裂尖应力强度因子随裂纹长度的变化规律。建立计算模型时,在裂纹尖端划分了三维奇异单元,在裂尖外围划分了过渡单元。计算结果表明：研究旋转叶片振动状态下的裂尖应力奇异性,仅利用I型应力强度因子就具有足够的精度;对于同一裂纹,绝大多数情况下叶盆面应力强度因子大于叶背面应力强度因子,故研究叶片应力强度因子时只需研究叶盆应力强度因子即可;随着裂纹扩展,叶盆面I型应力强度因子不断增大。本文的研究方法及结论为进一步研究叶片的裂纹扩展规律及损伤容限奠定了基础。     

各向异性双材料界面裂纹扩展裂尖场

应用界面断裂力学理论和Stroh方法,研究了广义平面变形下动态裂纹沿着各向异性双材料界面扩展时的裂尖奇异应力及动态应力强度因子.双材料界面的动态裂尖区域特性主要由两个实矩阵W和D确定,且裂尖奇异应力和动态应力强度因子可以由包含这两个矩阵的柯西奇异积分方程确定,同时给出了动态应力强度因子和能量释放率的显示表达式.算例得出当裂纹以小速度扩展时,裂尖振荡因子ε与静态时几乎相同,当界面裂纹扩展速度接近瑞利波速时,ε趋于无穷大;同时得出应力强度因子及能量释放率随裂纹扩展速度的变化关系.     

断裂问题分析的Williams广义参数单元

结合Ⅱ型断裂问题.研究建立了裂尖区应力强度因子计算的Williams广义参数单元和过渡单元.结合Williams级数解和广义参数有限元法,研究建立了弹性断裂问题的Williams广义参数单元计算格式;同时为了方便连接奇异区的Williams单元和常规区域的普通等参单元,建立了过渡单元模型.结合算例详细分析了计算模型中径向高散因子、离散数以及Williams级数项对计算结果的影响,并给出了建议值,同时研究了矩形板尺寸对Ⅱ型应力强度因子的影响.证实了解析解的局限性.计算结果表明,由于Williams单元位移模型中含有与应力强度因子直接相关的参数,所以可以避免传统有限元法需通过其他物理量间接计算应力强度因子的缺陷,且Williams单元具有较高的精度,构造使用方便.     

改进型扩展比例边界有限元法

结合了扩展有限元法(extended finite elementmethods,XFEM)和比例边界有限元法(scaled boundary finite elementmethods,SBFEM)的主要优点,提出了一种改进型扩展比例边界有限元法(improvedextended scaled boundary finite elementmethods,$i$XSBFEM),为断裂问题模拟提供了一条新的途径.类似XFEM,采用两个正交的水平集函数表征材料内部裂纹面,并基于水平集函数判断单元切割类型;将被裂纹切割的单元作为SBFE的子域处理,采用SBFEM求解单元刚度矩阵,从而避免了XFEM中求解不连续单元刚度矩阵需要进一步进行单元子划分的缺陷;同时,借助XFEM的主要思想,将裂纹与单元边界交点的真实位移作为单元结点的附加自由度考虑,赋予了单元结点附加自由度明确的物理意义,可以直接根据位移求解结果得出裂纹与单元边界交点的位移;对于含有裂尖的单元,选取围绕裂尖单元一圈的若干层单元作为超级单元,并将此超级单元作为SBFE的一个子域求解刚度矩阵,超级单元内部的结点位移可通过SBFE的位移模式求解得到,应力强度因子可基于裂尖处的奇异位移(应力)直接获得,无需借助其他的数值方法.最后,通过若干数值算例验证了建议的$i$XSBFEM的有效性,相比于常规XFEM,$i$XSBFEM的基于位移范数的相对误差收敛性较好;采用$i$XSBFEM通过应力法和位移法直接计算得到的裂尖应力强度因子均与解析解吻合\较好.      

模拟裂纹扩展的一种有限元局部动态子划分方法

提出了一种有限元子划分结合子结构的方法来模拟裂纹扩展问题。提出的方法中,将单元分为三类:被裂纹贯穿的单元,包含裂尖的单元和常规单元。对前两类单元进行子划分,每个单元的归类随裂纹的扩展而动态变化。覆盖一条裂纹的前两类单元子划分后构成一个子结构,子结构也是动态的,跟随裂纹的扩展而逐步扩大。本文的方法可以使裂纹沿任意路径扩展而不受初始网格的限制,裂纹扩展后无需对结构整体的网格重划分,结构整体分析的总自由度也不变。用该方法计算无限大平面中心裂纹的应力强度因子,模拟三点弯梁跨中裂纹的扩展,验证了计算精度,并进一步用该方法模拟了非均质材料中裂纹的扩展,考核了对复杂裂纹扩展问题的适用性。     

扩展有限元裂尖场精度研究

论述了扩展有限元方法和基本原理,研究了单元类型(四边形单元和三角形单元、线性单元和二次单元)、网格密度、J积分区域半径等因素对裂尖局部应力场(应力强度因子)计算精度的影响。研究发现,上述因素对裂尖应力强度因子计算的收敛速度与稳定性影响不大,证实了XFEM可以用较少的节点获得较高的裂尖场精度,并提出了通过固定裂尖附加区半径可以进一步改善XFEM的收敛速度。     

基于扩展有限元法的裂尖场精度研究

扩展有限元方法基于单元分解的基本思想,通过引入位移加强函数来表征裂纹的不连续性和裂尖的奇异性。在裂尖加强单元与常规单元之间有一层混合单元,当对裂尖特定区域进行加强时,混合单元个数相应增加,混合单元个数与计算精度存在一定联系。本文提出一种正方形裂尖加强区域的选择方式,可得到较单个加强和圆形加强精度更高、更稳定的计算结果。对于不同长度的裂纹,表征裂尖场奇异性所需的裂尖加强范围存在较大差异,以正方形裂尖加强方式进行计算,得到了不同裂纹长度下最优的加强尺寸。     

A numerical analysis of cracks emanating from an elliptical hole in a 2-D elasticity plate

This paper is concerned with the stress intensity factors (SIFs) of cracks emanating from an elliptical hole in an infinite or a finite plate under biaxial loads by using a boundary element method, which consists of the non-singular displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements due to the author. In the boundary element implementation the left or the right crack-tip element is placed locally at the corresponding left or right crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. A few numerical examples are included to show that the present approach is very efficient and accurate for the calculating the SIFs of crack problems in an infinite or a finite plate. The present numerical results of cracks emanating from an elliptical hole under biaxial loads can reveal the effect of the elliptical aspect ratio and the transverse load on the SIFs.     

An effective boundary element method for analysis of crack problems in a plane elastic plate

A simple and effective boundary element method for stress intensity factor calculation for crack problems in a plane elastic plate is presented. The boundary element method consists of the constant displacement discontinuity element presented by Crouch and Starfield and the crack-tip displacement discontinuity elements proposed by YAN Xiangqiao. In the boundary element implementation the left or the right crack-tip displacement discontinuity element was placed locally at the corresponding left or right each crack tip on top of the constant displacement discontinuity elements that cover the entire crack surface and the other boundaries. Test examples (i. e. , a center crack in an infinite plate under tension, a circular hole and a crack in an infinite plate under tension) are included to illustrate that the numerical approach is very simple and accurate for stress intensity factor calculation of plane elasticity crack problems. In addition, specifically, the stress intensity factors of branching cracks emanating from a square hole in a rectangular plate under biaxial loads were analysed. These numerical results indicate the present numerical approach is very effective for calculating stress intensity factors of complex cracks in a 2-D finite body, and are used to reveal the effect of the biaxial loads and the cracked body geometry on stress intensity factors.     

Hypersingular integral equation method for a three-dimensional crack in anisotropic electro-magneto-elastic bimaterials

Using Green’s functions, the extended general displacement solutions of a three-dimensional crack problem in anisotropic electro-magneto-elastic (EME) bimaterials under extended loads are analyzed by the boundary element method. Then, the crack problem is reduced to solving a set of hypersingular integral equations (HIE) coupled with boundary integral equations. The singularity of the extended displacement discontinuities around the crack front terminating at the interface is analyzed by the main-part analysis method of HIE, and the exact analytical solutions of the extended singular stresses and extended stress intensity factors (SIFs) near the crack front in anisotropic EME bimaterials are given. Also, the numerical method of the HIE for a rectangular crack subjected to extended loads is put forward with the extended crack opening dislocation approximated by the product of basic density functions and polynomials. At last, numerical solutions of the extended SIFs of some examples are obtained.     

拉伸载荷作用下单边缺陷-边裂纹应力强度因子研究

利用杂交位移不连续法研究拉伸载荷作用下矩形板中单边缺陷-边裂纹(半圆孔裂纹和半方孔裂纹)问题,给出了这三种平面弹性裂纹问题的应力强度因子的详细数值解。通过半圆孔裂纹问题和半方孔裂纹问题与单边裂纹问题的应力强度因子的比较,发现半圆孔和半方孔对单边裂纹有屏蔽影响。此外,本文的研究结果表明,杂交位移不连续法用于分析平面弹性有限体中复杂裂纹问题的应力强度因子简单且又准确。     

一种新的计算各向异性材料裂纹尖端应力强度因子杂交元法

利用有限元特征分析法研究了平面各向异性材料裂纹端部的奇性应力指数以及应力场和位移场的角分布函数,以此构造了一个新的裂纹尖端单元。文中利用该单元建立了研究裂纹尖端奇性场的杂交应力模型,并结合Hellinger-Reissner变分原理导出应力杂交元方程,建立了求解平面各向异性材料裂纹尖端问题的杂交元计算模型。与四节点单元相结合,由此提出了一种新的求解应力强度因子的杂交元法。最后给出了在平面应力和平面应变下求解裂纹尖端奇性场的算例。算例表明,本文所述方法不仅精度高,而且适应性强。     

A new boundary element approach of modeling singular stress fields of plane V-notch problems

In this paper, a new boundary element (BE) approach is proposed to determine the singular stress field in plane V-notch structures. The method is based on an asymptotic expansion of the stresses in a small region around a notch tip and application of the conventional BE in the remaining region of the structure. The evaluation of stress singularities at a notch tip is transformed into an eigenvalue problem of ordinary differential equations that is solved by the interpolating matrix method in order to obtain singularity orders (degrees) and associated eigen-functions of the V-notch. The combination of the eigen-analysis for the small region and the conventional BE analysis for the remaining part of the structure results in both the singular stress field near the notch tip and the notch stress intensity factors (SIFs).Examples are given for V-notch plates made of isotropic materials. Comparisons and parametric studies on stresses and notch SIFs are carried out for various V-notch plates. The studies show that the new approach is accurate and effective in simulating singular stress fields in V-notch/crack structures.     

Stress state around cracks on the boundary of a hole in a photoelastic orthotropic plate under creep

The stress–strain state near cracks on the boundary of a circular hole in a linear elastic orthotropic composite plate under tension is analyzed. The distribution of stress intensity factors (SIFs) at the crack tip is found from photoelectric measurements. The dependence of the SIFs on the ratio of crack length to hole radius and on the mechanical properties of the material is established     

ANALYSIS OF 3-D NOTCHED/CRACKED STRUCTURES BY USING EXTENDED BOUNDARY ELEMENT METHOD 1)

在线弹性理论中,三维 V 形切口/裂纹结构尖端区域存在多重应力奇异性,常规数值方法不易求解. 本文提出和建立了三维扩展边界元法 (XBEM),用于分析三维线弹性 V 形切口/裂纹结构完整的位移和应力场. 先将三维线弹性 V 形切口/裂纹结构分为尖端小扇形柱和挖去小扇形柱后的外围结构. 尖端小扇形柱内的位移函数采用自尖端径向距离 $r$ 的渐近级数展开式表达,其中尖端区域的应力奇异指数、位移和应力特征角函数通过插值矩阵法获得. 而级数展开式各项的幅值系数作为基本未知量. 挖去扇形域后的外围结构采用常规边界元法分析. 两者方程联立求解可获得三维 V 形切口/裂纹结构完整的位移和应力场,包括切口/裂纹尖端区域精细的应力场. 扩展边界元法具有半解析法特征,适用于一般三维 V 形切口/裂纹结构完整位移场和应力场的分析,其解可精细描述从尖端区域到整体结构区域的完整应力场. 作者研制了三维扩展边界元法程序,文中给出了两个算例,通过计算结果分析,表明了扩展边界元法求解三维 V 形切口/裂纹结构完整应力场的准确性和有效性.     

An analytical solution for the stress field and stress intensity factor in an infinite plane containing an elliptical hole with two unequal aligned cracks

The existing analytical solutions are extended to obtain the stress fields and the stress intensity factors (SIFs) of two unequal aligned cracks emanating from an elliptical hole in an infinite isotropic plane. A conformal mapping is proposed and combined with the complex variable method. Due to some difficulties in the calculation of the stress function, the mapping function is approximated and simplified via the applications of the series expansion. To validate the obtained solution, several examples are analyzed with the proposed method, the finite element method, etc. In addition, the effects of the lengths of the cracks and the ratio of the semi-axes of the elliptical hole (a/b) on the SIFs are studied. The results show that the present analytical solution is applicable to the SIFs for small cracks.