三维摩擦接触问题的增量线性互补方法

借助罚因子建立了三维摩擦接触本构模型,接触条件被表示为类似于非关联流动的弹塑性本构关系的形式。采用增量描述,对Coulomb摩擦定律采用Taylor展开作线性近似,导出了接触问题的互补虚功方程,然后基于有限元离散建立了三维摩擦接触问题的增量线性互补方法。数值算例表明了本文方法的有效性。     

干气密封滑动摩擦界面切向接触刚度分形模型

为揭示干气密封滑动摩擦界面高频微幅自激摩擦振动规律,用分形参数表征摩擦界面形貌特性,根据重新建立的微凸体接触变形方式,以及对非协调弹性体在切向力作用下初始滑动问题的研究,建立了干气密封滑动摩擦界面切向接触刚度分形模型. 通过数值对切向接触刚度的影响因素进行了分析,研究结果表明:切向接触刚度随分形维数、真实接触面积和材料特性系数的增大而增大;切向接触刚度随特征尺度、摩擦系数的增大逐渐减小. 相比于分形维数、特征尺度和材料特性系数对切向接触刚度的影响,摩擦系数的影响相对较小. 这些研究结果为进一步研究干气密封高频微幅自激摩擦振动奠定了基础.      

大变形中摩擦接触问题的数值模拟及应用

大变形的摩擦接触是复杂的非线性问题 ,本文介绍了一种处理摩擦接触问题的数值方法。采用接触单元技术模拟接触界面 ,基于弹塑性理论形式的非经典Coulomb摩擦定律及罚函数方法建立了摩擦接触的增量本构关系。结合大变形的增量分析格式给出了积分摩擦接触本构方程的回映方法。这种处理摩擦接触问题的方法计算简单、使用方便。给出的计算实例及应用实例说明了方法的精度与稳定性     

从规划法求解看有摩擦接触解的不唯一性

以单点接触问题为例研究了有摩擦接触问题的几个特点，给出了这类问题出现不唯一解的条件。进而从二次规划求解角度分析了解的不稳定性与不唯一性的数学原理，文中给出了规划法求解有摩擦接触问题应注意的问题     

The Method of Integrodifferential Relations for Linear Elasticity Problems

Some possible modifications of the governing equations of the linear theory of elasticity are considered. The stress–strain relation is specified by an integral equality instead of the local Hooke’s law. The modified integrodifferential boundary value problem is reduced to the minimization of a nonnegative functional under differential constraints. A numerical algorithm based on polynomial approximations of unknown functions (stresses and displacements) is developed and applied to linear elasticity problems. The bilateral estimation criteria of solution errors are proposed in order to analyze the algorithm convergence rate. The numerical results obtained by applying the integrodifferential relation method and the conventional variational method are compared and discussed.     

On the Boundary Value Problems of Linear Elasticity with Constraints

In this paper classic boundary value problems of linear elastostatics are studied. Displacement, mixed and traction type boundary conditions are considered for an internally constrained, non-homogeneous, anisotropic material. Existence of solutions and constraint stability results are presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

A New Method for Solution of 3D Elastic-Plastic Frictional Contact Problems

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Frictional Contact Problems for Thin Elastic Structures and Weak Solutions of Sweeping Processes

The linearized equilibrium equations for straight elastic strings, beams, membranes or plates do not couple tangential and normal components. In the quasi-static evolution occurring above a fixed rigid obstacle with Coulomb dry friction, the normal displacement is governed by a variational inequality, whereas the tangential displacement is seen to obey a sweeping process, the theory of which was extensively developed by Moreau in the 1970s. In some cases, the underlying moving convex set has bounded retraction and, in these cases, the sweeping process can be solved by directly applying Moreau’s results. However, in many other cases, the bounded retraction condition is not fulfilled and this is seen to be connected to the possible event of moving velocity discontinuities. In such a case, there are no strong solutions and we have to cope with weak solutions of the underlying sweeping process.     

Linear Response in Finite Elasticity

This paper presents objective finite elastic equations that reproduce the linear stress-strain behaviour observed in uniaxial tension tests with some materials. More generally, the new equations reduce to linear elastic equations when the deformation is a pure stretch. Further, the response of the proposed equations for the case of isotropic materials in uniaxial tension-compression and simple shear is examined. It is also shown how the new equations can be considered as linear approximations of general equations of finite elasticity.     

Characterizing Frictional Contact Loading via Isochromatics

Isochromatic patterns in the vicinity of frictional contacts furnish vital clues for characterizing friction. Though friction effects are evident in a diametrally loaded circular disk, three-point loading provides better results towards highlighting friction. In this paper, a new method of characterizing friction at loading contacts using photoelastic isochromatics patterns is presented. Location of isotropic points (IPs) formed in three-point and four-point loadings of circular disk is used as a main tool to quantify the friction component using theoretical analysis. Bifurcation of isochromatic fringe loops near the distributed loads is explained by the presence of anti-symmetric Hertzian shear traction in addition to Hertzian normal traction. The classical solution by Flamant for point load at the edge of half plane is used to derive stresses in circular disk for all required loading configurations. A semicircualr ring under three-point loading is examined using photoelasticity to understand the isochromatics pattern theoretically by considering normal and shear traction components at loaded regions.     

Formulating Dynamic Multi-Rigid-Body Contact Problems with Friction as Solvable Linear Complementarity Problems

A linear complementarity formulation for dynamic multi-rigid-body contact problems with Coulomb friction is presented. The formulation, based on explicit Euler integration and polygonal approximation of the friction cone, is guaranteed to have a solution for any number of contacts and contact configuration. A model with the same property, based on the Poisson hypothesis, is formulated for impact problems with friction and nonzero restitution coefficients. An explicit Euler scheme based on these formulations is presented and is proved to have uniformly bounded velocities as the stepsize tends to zero for the Newton–Euler formulation in body co-ordinates.     

About Clapeyron's Theorem in Linear Elasticity

We examine some elementary interpretations of the classical theorem of Clapeyron in linear elasticity theory. As we show, a straightforward application of this theorem in the purely mechanical setting leads to an apparent paradox which can be resolved by referring either to dynamics or to thermodynamics. These richer theories play an essential part in understanding the physical significance of this theorem.     

The Compatibility Constraint in Linear Elasticity

It is shown that the St.Venant–Beltrami equation of compatibility can be regarded as an internal constraint on the admissible strains, a constraint maintained by reactive stress fields that are collectively characterized in a global manner. It is also shown that such reactive stresses have an important role in a formal integration of the mixed boundary-value problem of linear elasticity. The main tools – Beltrami's map, Donati's theorem, and Cesàro's formula – have been available for a century or so; they are here used in a form as close to original as possible.     

Linear Constitutive Relations in Isotropic Finite Elasticity

For simple shearing and simple extension deformations of a homogeneous and isotropic elastic body, it is shown that a linear relation between the second Piola-Kirchhoff stress tensor and the Green-St. Venant strain tensor does not predict a physically reasonable response of the body. This constitutive relation implies that the slope of the curve between an appropriate component of the first Piola-Kirchhoff stress tensor and a deformation measure is an increasing functions of the deformation measure. This revised version was published online in August 2006 with corrections to the Cover Date.     

Linear Equations of Motion in Finite Elasticity

In this note we show that it may be possible and useful to construct valid strain-energy functions that lead directly to linear equilibrium equations for problems in isotropic homogeneous unconstrained nonlinear elasticity. While it is possible to make some general progress the final outcome will depend on the geometry and kinematics of the problem under consideration. Specific examples are given to show how exact solutions, via the linear equations of motion, can be found to non-trivial problems for physically meaningful constitutive models.      

The Cauchy Relations in Linear Elasticity Theory

In linear elasticity, we decompose the elasticity tensor into two irreducible pieces with 15 and 6 independent components, respectively. The vanishing of the piece with 6 independent components corresponds to the Cauchy relations. Thus, for the first time, we recognize the group-theoretical underpinning of the Cauchy relations.     

Periodic Homogenization and Material Symmetry in Linear Elasticity

Here homogenization theory is used to establish a connection between the symmetries of a periodic elastic structure associated with the microscopic properties of an elastic material and the material symmetries of the effective, macroscopic elasticity tensor. Previous results of this type exist but here more general symmetries on the microscale are considered. Using an explicit example, we show that it is possible for a material to be fully anisotropic on the microscale and yet the symmetry group on the macroscale can contain elements other than plus or minus the identity. Another example demonstrates that not all material symmetries of the macroscopic elastic tensor are generated by symmetries of the periodic elastic structure.     

三维摩擦接触问题不动点算法

三维摩擦接触问题具有多重非线性性质,使求解变得比较困难,为了解决此问题,建立了三维摩擦接触问题的模型,介绍了一种非光滑混合不动点算法.该算法克服了在接触面上由于可能的滑动状态有无穷多个而难以确定的难点,算法未引入人工变量,计算量较小,计算结果精确满足接触状态条件,收敛性得到保证.根据此算法编制程序,将叠合悬臂梁算例数值计算结果与商用有限元软件进行比较,也表明了不动点算法的有效性.     

Analytical Solution for Sliding Rounded-Edge Contact

Reliable analysis of the local stress fields in the vicinity of sliding frictional contacts of engineering components is a pre-requisite for the reliable assessment of structural integrity. In this paper we present the use of Muskhelishvili potentials to derive an analytical solution for a semi-infinite punch with a rounded edge pressed against a half-pane substrate, and a general numerical solution for a kind of Cauchy integral involved in the analytical contact solution. Using these solutions, the effect of the friction coefficient on the normal traction distribution is investigated. Numerical results show that the peak normal traction value is altered in comparison with the frictionless case solution, but this variation is mild (less than 5%), provided the friction coefficient does not exceed about 0.6. Finally, the adaptation of the analytical result to the solution of practical contact problems is addressed.     

New Solution of Axisymmetric Contact Problem of Elasticity

We consider two problems of elasticity, namely, the Boussinesq problem about the action of a lumped force on a half-space and the related problem about the interaction of the half-space with a cylindrical rigid punch with plane base. In the classical statement, these problems have singular solutions. In the Boussinesq problem, the displacement under the action of the force is infinitely large, and in the punch problem, the infinitely large variable is the pressure on the punch boundary. In the present paper, these problems are solved with the use of relations of generalized elasticity derived regarding a medium element of small but finite dimensions rather than a traditional infinitesimal element. The structure parameter of the medium contained in the solutions can be determined experimentally. The obtained generalized solutions of the problems under study are regular.