The Method of Fictitious Domains in the Signorini Problem

We justify the method of fictitious domains for an elliptic equation with nonlinear Signorini boundary conditions. The method makes it possible to construct a family of auxiliary problems defined in a wider domain and possessing the property that their solutions converge in an appropriate sense to a solution of the original problem.     

Signorini接触问题的边界变分不等式

本以Signorini接触问题为背景，讨论了变分不等式与边值问题的等价性，利用Green公式，基本解和基本解法向导数的性质，将区域型变分不等式化成等价的边界型变分不等式，并证明了边界变分不等式解的存在唯一性，为使用边界元方法数值求解提供理论依据。     

A Boundary Element Approximation of a Signorini Problem with Friction Obeying Coulomb Law

In this work, a Signorini problem with Coulomb friction in two dimensional elasticity is considered. Based on a new representation of the derivative of the double-layer potential, the original problem is reduced to a system of variational inequalities on the boundary of the given domain. The existence and uniqueess of this system are established for a small frictional coefficient. The boundary element approximation of this system is presented and an error estimate is given.     

The method of fundamental solutions for Signorini problems

We investigate the use of method of fundamental solutions (MFS)for the numerical solution of Signorini boundary value problems.The MFS is an ideal candidate for solving such problems becauseinequality conditions alternating at unknown points of the boundarycan be incorporated naturally into the least-squares minimizationscheme associated with the MFS. To demonstrate its efficiency,we apply the method to two Signorini problems. The first isa groundwater flow problem related to percolation in gentlysloping beaches, and the second is an electropainting application.For both problems, the results are in close agreement with previouslyreported numerical solutions.     

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.     

细长杆的Cosserat动力学模型和变分原理

利用Cosserat理论建立了细长杆的三维非线性动力学模型,借助伪刚体法和变分原理得到了Cosserat杆的包括各种形变的三维空间运动方程.     

The Signorini problem with Coulomb friction for a hyperelastic body under finite deformations

The quasi-static three-dimensional problem of elasticity theory for a hyperelastic body under finite deformations, loading by bulk and surface forces, partial fastening and unilateral contact with a rigid punch and in the presence of time-dependent anisotropic Coulomb friction is considered. The equivalent variational formulation contains a quasi-variational inequality. After time discretization and application of the iteration method, the problem arising with “specified” friction is reduced to a non-convex miniumum functional problem, which is studied by Ball's scheme. The operator in contact stress space is determined. It is shown that a threshold level of the coefficient of friction corresponds to each level of loading, below which there is at least one fixed point of the operator. If the solution at a certain instant of time is known, the iteration process converges to the solution of the problem at the next, fairly close instant of time.     

A quasi-static Signorini contact problem for a thermoviscoelastic beam

This paper is concerned with the existence, uniqueness and numerical solution of a system of equations modelling the evolution of a quasi-static thermoviscoelastic beam that may be in contact with two rigid obstacles. A finite element approximation is proposed and analysed and some numerical results are given. Work partially supported by the Brazilian institution CNPq.     

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.     

Existence of a solution for a Signorini contact problem for Maxwell-Norton materials

The aim of this article is to study the quasistatic evolutionof a Maxwell–Norton three-dimensional viscoelastic solidwith contact constraints. After introducing the appropiate functionalframework, we will discretize the problem in time using an implicitscheme whose resultant variational inequality is well posed.By using monotonicity arguments together with compensated compactnesstechniques, we will prove that the corresponding discrete solutionconverges to a solution of the continuous problem.     

梯形板弯曲问题的康托洛维奇解

在康托洛维奇对矩形板弯曲问题的有效近似解的基础上,本文进一步探讨了在不同边界条件下的梯形板弯曲问题的康氏解法.将板的位移用一级近似位移函数ω(x,y)=u(x,y)v(y)表示,式中, 在x方向的位移采用广义梁函数,用最小势能原理建立起对应于不同边界条件下的关于y方向位移函数v(y)的变系数常微分方程,求解微分方程,并利用边界条件,求出v(y)的精确解,从而可得到近似程度较高的梯形板弯曲问题的解.     

New Approach to Solving a Two-Dimensional Problem for an Orthotropic Plate

A new approach to solving a two-dimensional problem for an orthotropic multiply connected plate is proposed in place of the standard approach which reduces the problem to integrating a fourth-order differential equation. The new method reduces the problem to integrating second-order differential equations that can be solved successively using a perturbation technique.     

Optimal regularity for the Signorini problem

We prove under general assumptions that solutions of the thin obstacle or Signorini problem in any space dimension achieve the optimal regularity C ^1,1/2. This improves the known optimal regularity results by allowing the thin obstacle to be defined in an arbitrary C ^1,β hypersurface, β > 1/2, additionally, our proof covers any linear elliptic operator in divergence form with smooth coefficients. The main ingredients of the proof are a version of Almgren’s monotonicity formula and the optimal regularity of global solutions.     

无限板圆孔边四不等长裂纹问题

利用有限截项法研究了无限远处受任意角度单向均拉、无限大板圆孔边四不等长裂纹的应力强度因子和裂纹面张开位移问题,结果表明,当裂纹长与半径的比值大于2时,有限截项法和复变函数法所得应力强度因子的结果逼近程度较好.将圆孔边四条裂纹退化为两条裂纹时,应力强度因子的结果在裂纹长与半径的比值大于1.5时与复变函数解吻合较好;裂纹面张开位移的值仅在裂纹起始处与文献已有结果有差异,与已有的有限截项法结果均一致,即还原了两条裂纹的结果.