Two kinds of contact problems in three-dimensional icosahedral quasicrystals

Two kinds of contact problems, i.e., the frictional contact problem and the adhesive contact problem, in three-dimensional(3D) icosahedral quasicrystals are discussed by a complex variable function method. For the frictional contact problem, the contact stress exhibits power singularities at the edge of the contact zone. For the adhesive contact problem, the contact stress exhibits oscillatory singularities at the edge of the contact zone. The numerical examples show that for the two kinds of contact problems, the contact stress exhibits singularities, and reaches the maximum value at the edge of the contact zone. The phonon-phason coupling constant has a significant effect on the contact stress intensity, while has little impact on the contact stress distribution regulation. The results are consistent with those of the classical elastic materials when the phonon-phason coupling constant is 0. For the adhesive contact problem, the indentation force has positive correlation with the contact displacement, but the phonon-phason coupling constant impact is barely perceptible. The validity of the conclusions is verified.     

正交各向异性弹塑性摩擦接触问题的数值求解

采用正交各向异性摩擦定律对三维弹塑性摩擦接触问题进行分析,基于参变量变分原理,经过有限元离散,将问题化为线性互补问题模型,之后给出一个求解互补问题的非内点光滑化算法．对三维接触问题,滑动方向的确定一直是个难点,为此,该文采用作者提出的组合规划法和迭代法对各向异性摩擦本构模型进行分析,数值结果说明了模型与算法的正确性。     

解三维摩擦接触问题的一个二阶锥线性互补法

针对三维摩擦接触问题的求解,给出了一种基于参变量变分原理的二阶锥线性互补法. 首先,基于三维Coulomb摩擦锥在数学表述上属于二阶锥的事实,利用二阶锥规划对偶理论,建立了三维Coulomb摩擦接触条件的参变量二阶锥线性互补模型,它是二维Coulomb摩擦接触条件参变量线性互补模型在三维情形下的自然推广;随后,利用参变量变分原理与有限元方法,建立了求解三维摩擦接触问题的二阶锥线性互补法. 较之于将三维Coulomb摩擦锥进行显式线性化的线性互补法,该方法无需对三维Coulomb摩擦锥进行线性化,因而在保证精度的前提下所解问题的规模要小很多. 最后通过算例展示了该方法的特点.      

Development of a solution method for frictional contact problems with complex loading

In this work, solution methods for frictional contact problems are extended to the case of moving punches and to the external loading history-dependent system states. To solve the frictional contact problems in the contact area, an iterative method is developed and implemented. Solutions of two-dimensional problems are constructed using the boundary element method. Numerical analysis is aimed at the quantitative study of effects such as the interaction of contact pressure and friction forces, estimates of the friction force differences due to the differences in the choice of local basis for the calculation of normal pressure and friction forces, and evaluation of the effects of complex loading (rotation of the rigid punch after its preliminary penetration into the solid). We find that, for the same definition of the friction force, different initial approximations lead to the same solution. At the same time, the friction forces defined either as projections onto the common tangent plane or as projections onto the plane tangent to the punch can differ quite substantially. Similar conclusions are derived for the solutions corresponding to single or multiple loading steps. The work relies on the variational principle for the solution of contact problems and numerical algorithms developed for the problems with one-sided constraints. The variational principle was first applied by Signorini [1] to the determination of the stress-strain state in a linearly deformed body in a rigid smooth shell. The modern view of the problem and its generalizations to the frictional problems and some other problems involving unilateral constraints in given in the monograph [2]. Finite difference and finite element methods in application to the problems with unilateral constraints are described in [3]. Analytical solution methods are developed in the monographs [4–6].     

点群10mm十次对称二维准晶中的两类接触问题

采用复变函数法探讨了在一个刚性压头作用下十次对称二维准晶材料的两类接触问题,即具有有限摩擦的接触问题以及粘结接触问题.特别地对于平底压头,获得了表征声子场和相位子场的全纯函数的显式表达式,以及在压头上的接触应力分布.结果显示,对于具有有限摩擦的接触问题,接触应力在接触区边缘具有实指数奇异性-1／2±β,其中β由准晶体的材料常数及静摩擦系数确定;而对于粘结接触问题,接触应力在接触区边缘具有振荡型奇异性-1／2±iε,其中ε由准晶体的材料常数确定.     

摩擦接触问题的比例边界等几何B可微方程组方法

摩擦接触问题是计算力学领域最具挑战性的问题之一,接触系统的泛函具有非线性、非光滑的特点,导致接触算法的收敛性与精确性难以保证.因此将比例边界等几何分析(scaled boundary isogeometric analysis,SBIGA)与B可微方程组(B dierential equation,BDE)相结合,提出了求解二维摩擦接触问题的比例边界等几何B可微方程组方法.在比例边界等几何坐标变换的基础上,通过虚功原理推导了关于边界控制点变量的接触平衡方程,表示成B可微方程组形式的接触条件可被严格满足,求解B可微方程组的算法的收敛性有理论保证.此比例边界等几何B可微方程组方法(SBIGA-BDE)只需在接触体边界进行等几何离散,使问题降低一维,能精确描述接触边界,并可通过节点插入算法进行真实接触区域的识别.此外,由于几何建模和数值分析使用相同的基函数,节约了划分网格的时间.以赫兹接触问题和悬臂梁摩擦接触问题为例,通过与解析解及数值计算软件ANSYS计算结果进行对比,验证了该方法求解二维摩擦接触问题的有效性及高精度等特点.      

Block pivot methods for solving frictional contact problems

Based on elementary group theory, the block pivot methods for solving two-dimensional elastic frictional contact problems are presented in this paper. It is proved that the algorithms converge within a finite number of steps when the friction coefficient is “relative small”. Unlike most mathematical programming methods for contact problems, the block pivot methods permit multiple exchanges of basic and nonbasic variables. The project supported by the National Natural Science Foundation of China     

接触分析的光滑模型及迭代算法

利用变分不等式和基于信息熵的凝聚函数把有摩擦接触问题模型化为一个标准的凸二次规划问题,极大地简化了这一复杂的问题,同时引入摩擦方向约束并构造了以无摩擦解为初值的迭代算法,在较摩擦系数时计算也能保证收敛,算例表明算法高效可靠。     

Two Kinds of Contact Problems in Dodecagonal Quasicrystals of Point Group 12 mm

In this paper,two kinds of contact problems in 2-D dodecagonal quasicrystals were discussed using the complex variable function method:one is the finite frictional contact problem,the other is the adhesive contact problem.The analytic expressions of contact stresses in the phonon and phason fields were obtained for a flat rigid punch,which showed that:(1) for the finite frictional contact problem,the contact stress exhibited power-type singularities at the edge of the contact zone;(2) for the adhesive contact problem,the contact stress exhibited oscillatory singularities at the edge of the contact zone.The distribution regulation of contact stress under punch was illustrated;and the low friction property of quasicrystals was verified graphically.     

Effective numerical methods for elasto-plastic contact problems with friction

Several effective numerical methods for solving the elasto-plastic contact problems with friction are presented. First, a direct substitution method is employed to impose the contact constraint conditions on condensed finite element equations, thus resulting in a reduction by half in the dimension of final governing equations. Second, an algorithm composed of contact condition probes and elasto-plastic iterations is utilized to solve the governing equation, which distinguishes two kinds of nonlinearities, and makes the solution unique. In addition, Positive-Negative Sequence Modification Method is used to condense the finite element equations of each substructure and an analytical integration is introduced to determine the elasto-plastic status after each time step or each iteration, hence the computational efficiency is enhanced to a great extent. Finally, several test and practical examples are presented showing the validity and versatility of these methods and algorithms. The Project Supported by National Natural Science Foundation of China.     

传热与接触两类问题耦合作用的有限元分析

考虑传热接触耦合作用的热力学分析问题大量存在于工程中,分析的难点是必须考虑热与可移动接触边界间的耦合作用。针对这类问题的求解,该文给出了接触边界上热交换与温度边界条件,并在此基础上建立了两类变分方程,一类是热力学变分泛函,其考虑了接触区域对结构热传导的影响;另一类是二维热弹性接触分析的参数变分原理,可以方便地对接触问题进行求解。文中给出有限元分析的离散公式,并进一步给出两类问题耦合分析的迭代算法,其中接触分析的惩罚因子是可以消除的,数值结果验证了该文的理论与算法。     

弹塑性接触问题的非光滑非线性方程组方法

将求解三维弹性摩擦接触问题的非光滑非线性方程组方法推广到弹塑性(Mises材料)情形，提出了两种应用方法：一种是将非光滑非线性方程组方法和求解弹塑性问题常用的Newton—Raphson迭代方法结合起来；另一种是将问题写成统一的非光滑非线性方程组，直接求解。数值算例验证了两种方法的有效性，并进行了结果比较。     

Experimental and numerical analysis of fretting crack formation based on 3D X-FEM frictional contact fatigue crack model

Nowadays, numerical simulation of 3D fatigue crack growth is easily handled using the eXtended Finite Element Method coupled with level set techniques. The finite element mesh does not need to conform to the crack geometry. Most difficulties associated to complex mesh generation around the crack and the re-meshing steps during the possible propagation are hence avoided. A 3D two-scale frictional contact fatigue crack model developed within the X-FEM framework is presented in this article. It allows the use of a refined discretization of the crack interface independent from the underlying finite element mesh and adapted to the frictional contact crack scale. A stabilized three-field weak formulation is also proposed to avoid possible oscillations in the local solution linked to the LBB condition when tangential slip is occurring. Two basic three-dimensional numerical examples are presented. They aim at illustrating the capacities and the high level of accuracy of the proposed X-FEM model. Stress intensity factors are computed along the crack front. Finally an experimental 3D ball/plate fretting fatigue test with running conditions inducing crack nucleation and propagation is modeled. 3D crack shapes defined from actual experimental ones and fretting loading cycle are considered. This latter numerical simulation demonstrates the model ability to deal with challenging actual complex problems and the possibility to achieve tribological fatigue prediction at a design stage based on the fatigue crack modeling.     

三维h自适应有限元分析的资料转移

研究了两类用于三维h自适应网络管理的资料转移方法。文中将降低解的精度以节省计算时间的简单方法（简法）与在资料转移中确保解的精度的严格方法（严法）进行了比较分析，并通过数值算例进行论证，结果表明：对于简单问题，两类方法都能满足精度，完成资料转移。对于复杂应力状态问题，严法更可靠，特别是在网络稀疏时，严法明显优越。     

Implicit frictional-contact model for soft particle systems

We introduce a novel numerical approach for the simulation of soft particles interacting via frictional contacts. This approach is based on an implicit formulation of the Material Point Method, allowing for large particle deformations, combined with the Contact Dynamics method for the treatment of unilateral frictional contacts between particles. This approach is both precise due to the treatment of contacts with no regularization and artificial damping parameters, and robust due to implicit time integration of both bulk degrees of freedom and relative contact velocities at the nodes representing the contact points. By construction, our algorithm is capable of handling arbitrary particle shapes and deformations. We illustrate this approach by two simple 2D examples: a Hertz contact and a rolling particle on an inclined plane. We also investigate the compaction of a packing of circular particles up to a solid fraction well above the jamming limit of hard particles. We find that, for the same level of deformation, the solid fraction in a packing of frictional particles is above that of a packing of frictionless particles as a result of larger particle shape change.     

非光滑方程组方法在求解非匹配网格接触问题中的应用

将非光滑方程组方法与Mortar StS接触模型(Mortar Segment-to-Segment)相结合,来求解接触面网格非匹配时的弹性接触问题.其中,非光滑方程组方法是求解弹性摩擦接触问题的有效方法,具有精确满足接触条件、迭代算法收敛性有理论保证的优点,但目前仅用于求解网格匹配的接触问题.Mortar StS接触模型可以较为方便地处理网格非匹配接触问题,其特点是不引入过多约束,满足接触分片检验条件,但目前大都采用“试验-误差”迭代方法求解控制方程,对于复杂接触问题,其收敛性不易保证.因此,将二者结合来处理网格非匹配接触问题,既可以提高求解精度,又能使得算法的收敛性得到理论保证.数值算例对接触分片检验和算法的计算精度进行了验证.     

Quadratic programming with parametric vectors in three-dimensional contact problems

The present study is concerned with 3-D contact problems by parametric quadratic programming(PQP). The stiffness matrix of a 3-D contact element is introduced by means of penalty function expression of the contact pressure and frictional force, and two constitutive relations of 3-D contact problems can be obtained, similar to elasto—plastic problems. The penalty factors can be eliminated by using a definite numerical technique, so that a parametric linear complementary problem is obtained. As an application of the theory presented, some numerical examples are illustrated.     

空间接触问题的参变量方法

本文借助于参变量方法给出了空间接触问题的解法.空间接触单元刚度矩阵是通过接触压力和摩擦力的惩罚函数表示式而引入的.两种空间接触本构关系(圆锥型和棱锥型)可以类似于弹塑性问题的模型得到,一定的数学技巧可使惩罚因子最后得以消去,而获得参变量线性互补问题.两个实例证明了这个方法的有效性.     

水工圆形隧洞围岩衬砌摩擦滑动接触的新解法

在实际工程中,围岩和衬砌接触时,它们之间并非完全光滑,也并非可以承受任意大的摩擦力.如果围岩与衬砌之间的剪应力大于所能承受的最大静摩擦力,接触面间将发生切向滑动,定义接触面上产生最小滑动量的状态为衬砌的真实工作状态,这种接触即为摩擦滑动接触.以库仑摩擦模型模拟围岩和衬砌之间的摩擦滑动接触,在考虑支护滞后效应的前提下,利用平面弹性复变函数方法列出了应力边界条件、应力连续条件以及位移连续条件的方程, 再结合最优化理论,建立了具有一般性的摩擦滑动接触解法.在利用混合罚函数法求解最优化问题的过程中,减少了设计变量的个数,极大地简化了优化模型,提升了优化过程的迭代速度以及优化结果的精度.以此为基础,获得了围岩和衬砌相互作用下圆形水工隧洞的应力解析解.该方法可以求解光滑接触和完全接触两种极限情况,具有一般性.同时,利用一种精确的计算方法得到了不同情况下满足完全接触条件摩擦系数的阈值,还分析了衬砌和围岩边界上切向应力的变化规律.      

Interfacial slip and creep in rolling contact incorporating a cylinder with an elastic layer

A systematic approach for investigating the interfacial behaviour of tyred systems is proposed. A two-dimensional contact model of an elastic strip, shrink-fitted onto a wheel, and subjected to different rolling contact conditions, has been adopted to illustrate the method. The model combines existing techniques to explore individual elastic contact problems and it enables us to characterise the behaviour at the strip/substrate interface caused by loads induced by a quasi-static application of stationary and moving loads on the surface of the layer. The solution is compared to the stationary load case and regimes of local slip, full stick, separation and frictional creep are identified and collated for a variety of loading conditions, materials and geometries. Further, this article presents an investigation of frictional shakedown for layered systems subjected to periodic contact loading. The term shakedown is here referred to as the possibility of developing interfacial residual stresses at the layer/substrate interface such that frictional slip, originally activated by the applied external contact load, ceases after a few loading cycles. The possible applicability of the Melan’s theorem for elastic frictional system is investigated and preliminary results presented.