直接施加本质边界条件的FE／EFG耦合算法及其数值实现

提出一种可以直接施加本质边界条件的有限元与无网格Galerkin（FE／EFG）耦合算法。将问题域分成FE和EFG两种类型的子域,采用转换矩阵耦舍两子域的交界面;通过另一转换矩阵将无网格区域本质边界上的名义位移转换成真实位移,从而可在其上直接施加本质边界条件;采用二次转换实现两种转换矩阵之间的协调。提出全域统一采用单元...     

裂纹扩展独立于网格的一种有限元单元子划分方法

提出了一种有限元模拟裂纹扩展的单元子划分结合子结构的方法。本方法中,裂纹可以进入或穿过一个单元,或沿单元的边界扩展,因此裂纹可以沿任意路径扩展而不受初始网格的限制。对上述几类包含裂纹的单元按照裂纹的路径进行子划分,覆盖一条裂纹的所有子划分单元就组成了一个子结构,子结构规模随裂纹的扩展而增大。子结构中因单元子划分而新增的结点自由度,通过自由度的凝聚用初始网格结点的自由度表示,因此结构整体分析的总自由度不变。以上述方法为基础建立了裂纹萌生和扩展的准则。用本文的方法分析了单（双）材料无限大平面中心（界面）裂纹的裂尖场,验证了本文方法的精度,并模拟了颗粒复合材料中微裂纹在颗粒、基体和界面中逐步扩展的过程,考核了本文方法对复杂裂纹扩展问题模拟的适用性。     

动态断裂力学的无限相似边界元法

对弹性动力学的相似边界元法进行了进一步研究,推导了相应的计算公式,并在此基础上提出了动态断裂力学的无限相似边界元法.与传统的边界元法相比,相似边界元法由于只需在少数单元上进行数值积分,大大减少了计算量.对动态断裂力学问题,无限相似边界元法由于在裂纹尖端的边界上设置了逼近于裂纹尖端的无限个相似边界单元,可直接得到裂纹尖端具有奇异性的应力,而不需要设置奇异单元,从而突破了奇异单元对应力奇异性阶次的局限.另外,还讨论了无限相似边界元法得到的无限阶的线性代数方程组的求解方法.     

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。     

Analysis of the electromechanical behavior of ferroelectric ceramics based on a nonlinear finite element model

A nonlinear finite element (FE) model based on domain switching was proposed to study the electromechanical behavior of ferroelectric ceramics. The incremental FE formulation was improved to avoid any calculation instability. The problems of mesh sensitivity and convergence, and the efficiency of the proposed nonlinear FE technique have been assessed to illustrate the versatility and potential accuracy of the said technique. The nonlinear electromechanical behavior, such as the hysteresis loops and butterfly curves, of ferroelectric ceramics subjected to both a uniform electric field and a point electric potential has been studied numerically. The results obtained are in good agreement with those of the corresponding theoretical and experimental analyses. Furthermore, the electromechanical coupling fields near (a) the boundary of a circular hole, (b) the boundary of an elliptic hole and (c) the tip of a crack, have been analyzed using the proposed nonlinear finite element method (FEM). The proposed nonlinear electromechanically coupled FEM is useful for the analysis of domain switching, deformation and fracture of ferroelectric ceramics.The project supported by the National Natural Science Foundation of China (10025209, 10132010 and 90208002), the Research Grants of the Council of the Hong Kong Special Administrative Region, China (HKU7086/02E) and the Key Grant Project of the Chinese Ministry of Education (0306)     

An Application of the Kane and Mindlin Theory to Crack Problems in Plates of Arbitrary Thickness

Classical plane solutions of the theory of elasticity, which are sometimes more than 100 years old, are still used today and provide a framework for the analysis of many practical problems. But, strictly speaking, these analytical solutions are only applicable to plates with vanishing thickness or infinite thickness, where the stress state could be classified as plane stress or plane strain, respectively. However, the through-the-thickness stresses that exist in a plate of given thickness have a significant impact in a number of practical applications; and these stresses are often inevitably ignored due to the lack of analytical tools. This paper presents new analytical results for crack tip opening displacement (CTOD) for the through-the-thickness crack in infinite plates with various thicknesses. These results are based on the solution for an edge dislocation in infinite plate of arbitrary thickness and an application of the distributed dislocation technique. The analytical predictions of the CTOD and the constraint factor are compared with the three-dimensional elasto-plastic finite element (FE) results. It is shown that both analytical and numerical results are in good agreement when the numerical calculations are not affected by the size of the FE mesh and by the boundaries of the FE model.     

动载下裂纹应力强度因子计算的数值流形元法

相较于传统有限元,数值流形方法（numerical manifold method, NMM） 的一个显著优点是在处理裂纹问题时网格无需与裂纹重合,这就方便了岩体破坏过程的模拟. 基于包含裂尖增强函数的NMM,采用Newmark 隐式动力学算法进行时间积分,重点研究了动力载荷条件下裂纹动态应力强度因子（dynamic stress intensity factor,DSIF） 的求解方法. 针对典型的线弹性动力裂纹问题,给出了NMM 的数值算例. 结果表明NMM 能够准确计算动载荷条件下裂纹的DSIF,并且具有较好的收敛性.      

EVALUATION ON STRESS INTENSITY FACTOR OF CRACK UNDER DYNAMIC LOAD USING NUMERICAL MANIFOLD METHOD

相较于传统有限元,数值流形方法（numerical manifold method, NMM） 的一个显著优点是在处理裂纹问题时网格无需与裂纹重合,这就方便了岩体破坏过程的模拟. 基于包含裂尖增强函数的NMM,采用Newmark 隐式动力学算法进行时间积分,重点研究了动力载荷条件下裂纹动态应力强度因子（dynamic stress intensity factor,DSIF） 的求解方法. 针对典型的线弹性动力裂纹问题,给出了NMM 的数值算例. 结果表明NMM 能够准确计算动载荷条件下裂纹的DSIF,并且具有较好的收敛性.     

THE EVALUATION OF STRESS INTENSITY FACTORS OF PLANE CRACK FOR ORTHOTROPIC PLATE WITH EQUAL PARAMETER BY F2LFEM

In this paper, the evaluation of stress intensity factor of plane crack problems for orthotropic plate of equal-parameter is investigated using a fractal two-level finite element method (F2LFEM). The general solution of an orthotropic crack problem is obtained by assimilating the problem with isotropic crack problem, and is employed as the global interpolation function in F2LFEM. In the neighborhood of crack tip of the crack plate, the fractal geometry concept is introduced to achieve the similar meshes having similarity ratio less than one and generate an infinitesimal mesh so that the relationship between the stiffness matrices of two adjacent layers is equal. A large number of degrees of freedom around the crack tip are transformed to a small set of generalized coordinates. Numerical examples show that this method is efficient and accurate in evaluating the stress intensity factor (SIF).     

分区界面元-有限元-无限元混合模型

利用界面元良好的相容性,引入过渡界面元的概念．实现了界面元与有限元二种数值计算方法的结合,并提出了一种界面元-有限元-无限元混合模型。这种混合模型既可以发挥界面元计算精度高、适用于不连续变形等优点．又能够充分利用有限元的计算效率和无限元方便处理无限域介质的特点,较为和谐地解决了计算精度和计算效率的矛盾。数值算例表明,本文所建立的混合模型的有效性,揭示此类混合模型具有广阔的工程应用前景。     

近场动力学与有限元方法耦合求解热传导问题

求解含裂纹等不连续问题一直是计算力学的重点研究课题之一,以偏微分方程为基础的连续介质力学方法处理不连续问题时面临很大的困难. 近场动力学方法是一种基于积分方程的非局部理论,在处理不连续问题时有很大的优越性. 本文提出了求解含裂纹热传导问题的一种新的近场动力学与有限元法的耦合方法. 结合近场动力学方法处理不连续问题的优势以及有限元方法计算效率高的优势,将求解区域划分为两个区域,近场动力学区域和有限元区域. 包含裂纹的区域采用近场动力学方法建模,其他区域采用有限元方法建模. 本文提出的耦合方案实施简单方便,近场动力学区域与有限元区域之间不需要设置重叠区域. 耦合方法通过近场动力学粒子与其域内所有粒子(包括近场动力学粒子和有限元节点)以非局部方式连接,有限元节点与其周围的所有粒子以有限元方式相互作用. 将有限元热传导矩阵和近场动力学粒子相互作用矩阵写入同一整体热传导矩阵中,并采用Guyan缩聚法进一步减小计算量. 分别采用连续介质力学方法和近场动力学方法对一维以及二维温度场算例进行模拟,结果表明,本文的耦合方法具有较高的计算精度和计算效率. 该耦合方案可以进一步拓展到热力耦合条件下含裂纹材料和结构的裂纹扩展问题.      

裂缝分析的比例边界有限元与有限元耦合的虚拟结构面模型

比例边界有限元法SBFEM(Scaled Boundary Finite Element Method)是一种半解析数值方法,在裂缝分析特别是强度因子计算上具有相当高的精度。本文提出了一种用于裂缝分析的基于虚拟结构面的SBFEM与常规FEM的耦合分析方法。首先选取裂缝周边一定范围的计算域,并将结构分成不含裂缝区域和含裂缝区域两部分。然后,对不含裂缝区域,采用FEM进行网格离散;对含裂缝区域,采用SBFEM进行网格离散;两者相互独立,在这两个域内,分别采用各自相应的位移模式。最后通过在SBFEM网格的外边界设置虚拟耦合结构面的模式,实现有限元网格和比例边界有限元网格的耦合。通过两个经典的含裂缝平板的算例研究,探讨了本文方法在I型开裂和混合型开裂分析中,影响应力强度因子精度的因素。算例表明,SBFEM具有的降维和半解析性质,使本文方法在裂缝分析中的前处理简单易行,且计算结果具有相当高的计算精度。     

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.     

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.      

任意斜裂纹转子的耦合振动研究

对包含不同类型裂纹(横裂纹、横-斜裂纹以及任意斜裂纹)的转子的耦合振动进行研究,以揭示裂纹转子在不同方向上刚度参数的变化规律及其交叉耦合机理,特别是由此引发的振动特征. 对于包含不同类型裂纹的转子轴段,采用六自由度Timoshenko梁单元模型对其进行单元建模,并基于应变能理论推导计算柔度参数和刚度矩阵. 在此基础上, 采用纽马克-$\beta$数值算法求解裂纹转子的运动方程,获得裂纹转子在单故障或多故障激励(不平衡激励、扭转激励或不平衡激励加扭转激励)作用下的耦合振动响应,进而分析耦合振动谱特征. 与横裂纹和横-斜裂纹相比,任意斜裂纹使转子刚度矩阵的交叉耦合效应更显著,导致转子发生更强烈的弯-扭耦合甚至是纵-弯-扭耦合振动.无论是在不平衡激励还是扭转激励作用下, 弯曲振动与扭转振动幅度都更大. 而且,包含不同类型裂纹的转子的耦合振动特征频率,例如旋转基频与二倍频、扭转激励频率及其边带成分的幅值,对裂纹面方向角具有不同的敏感性. 所得的这些研究结果,可以为转子裂纹的特征参数辨识与诊断提供理论依据.      

无网格与有限元的耦合在动态断裂研究中的应用

提出将无网格Galerkin法与有限元耦合的方法用于分析动态裂纹扩展问题,只在裂尖附近区域沿裂纹扩展方向布置无网格结点,而在其他区域采用一般的有限元,区域交界处的结点采用MLS方法插值,然后将求得的结点值再分配到有限单元的相关结点上,保证了无网格区域和有限元区域的交界处位移的连续。避免了网格的再生成,同时也克服了单纯使用无网格Galerkin法所带来的边界条件难处理及计算效率较低的缺点。数值算例显示这种方法是有效的。     

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.     

双轴载荷作用下源于椭圆孔的分支裂纹的一种边界元分析

利用一种边界元方法来研究双轴载荷作用下无限大板中源于椭圆孔的分支裂纹．该边界元方法由Crouch与Starfied建立的常位移不连续单元和笔者提出的裂尖位移不连续单元构成．在该边界元方法的实施过程中,左、右裂尖位移不连续单元分别置于裂纹的左、右裂尖处,而常位移不连续单元则分布于除了裂尖位移不连续单元占据的位置之外的整个裂纹面及其它边界,文中算例说明本数值方法对计算平面弹性裂纹的应力强度因子是非常有效的。该文对双轴载荷作用下无限大板中源于椭圆孔的分支裂纹的数值结果进一步证实本数值方法对计算复杂裂纹的应力强度因子的有效性,同时该数值结果可以揭示双轴载荷及裂纹体几何对应力强度因子的影响。     

Evolving crack patterns in thin films with the extended finite element method

This paper develops the extended finite element method (XFEM) to evolve patterns of multiple cracks, in a brittle thin film bonded to an elastic substrate, with a relatively coarse mesh, and without remeshing during evolution. A shear lag model describes the deformation in three dimensions with approximate field equations in two-dimensions. The film is susceptible to subcritical cracking, obeying a kinetic law that relates the velocity of each crack to its energy release rate. At a given time, the XFEM solves the field equations and calculates the energy release rate of every crack. For a small time step, each crack is extended in the direction of maximal hoop stress, and by a length set by the kinetic law. To confirm the accuracy of the XFEM, we compare our simulation to the exiting solutions for several simple crack patterns, such as a single crack and a set of parallel cracks. We then simulate the evolution of multiple cracks, initially in a small region of the film but of different lengths, showing curved crack propagation and crack tip shielding. Starting with multiple small cracks throughout the film, the XFEM can generate the well-known mud crack pattern.     

Multiple axial cracks in a coated hollow cylinder due to thermal shock

The transient thermal stress problem of an inner-surface-coated hollow cylinder with multiple pre-existing surface cracks contained in the coating is considered. The transient temperature, induced thermal stress, and the crack tip stress intensity factor (SIF) are calculated for the cylinder via finite element method (FEM), which is exposed to convective cooling from the inner surface. As an example, the material pair of a chromium coating and an underlying steel substrate 30CrNi2MoVA is particularly evaluated. Numerical results are obtained for the stress intensity factors as a function of normalized quantities such as time, crack length, convection severity, material constants and crack spacing.