Numerical simulation of the non‐linear crack problem with non‐penetration

Here the numerical simulation of some plane Lamé problem with a rectilinear crack under non‐penetration condition is presented. The corresponding solids are assumed to be isotropic and homogeneous as well as bonded. The non‐linear crack problem is formulated as a variational inequality. We use penalty iteration and the finite‐element method to calculate numerically its approximate solution. Applying analytic formulas obtained from shape sensitivity analysis, we calculate then energetic and stress characteristics of the solution, and describe the quasistatic propagation of the crack under linear loading. The results are presented in comparison with the classical, linear crack problem, when interpenetration between the crack faces may occur. Copyright © 2004 John Wiley & Sons, Ltd.     

Two‐grid variational multiscale algorithms for the stationary incompressible Navier‐Stokes equations with friction boundary conditions

Two‐grid variational multiscale (VMS) algorithms for the incompressible Navier‐Stokes equations with friction boundary conditions are presented in this article. First, one‐grid VMS algorithm is used to solve this problem and some error estimates are derived. Then, two‐grid VMS algorithms are proposed and analyzed. The algorithms consist of nonlinear problem on coarse grid and linearized problem (Stokes problem or Oseen problem) on fine grid. Moreover, the stability and convergence of the present algorithms are established. Finally, Numerical results are shown to confirm the theoretical analysis. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 546–569, 2017     

Contact problems with friction for hemitropic solids: boundary variational inequality approach

We study the interior and exterior contact problems for hemitropic elastic solids. We treat the cases when the friction effects, described by Tresca friction (given friction model), are taken into consideration either on some part of the boundary of the body or on the whole boundary. We equivalently reduce these problems to a boundary variational inequality with the help of the Steklov–Poincaré type operator. Based on our boundary variational inequality approach we prove existence and uniqueness theorems for weak solutions. We prove that the solutions continuously depend on the data of the original problem and on the friction coefficient. For the interior problem, necessary and sufficient conditions of solvability are established when friction is taken into consideration on the whole boundary.     

The primal-dual active set method for a crack problem with non-penetration

The Lamé problem in a 2D domain with a crack under anon-penetration condition is considered as a variational inequality.A primal–dual active set method is proposed as an efficientnumerical solution technique and compared to a previously employediterative method for a penalized formulation. Sufficient conditionsfor monotonic convergence of a discretized version of the proposedalgorithm are given and numerical experiments are presented.     

Asymptotic modeling of Signorini problem with Coulomb friction for a linearly elastostatic shallow shell

In 2002–2003, Paumier studied the Signorini problem with friction in the linear Kirchhoff–Love theory of plates using the convergence method. In 2008, Léger and Miara generalized this study to the case of linearized shallow shell but without friction. The purpose of this paper is to extend those results to the case of linearized shallow shell with a Coulomb friction law. Copyright © 2015 John Wiley & Sons, Ltd.     

Asymptotics of the energy functional for a fourth-order mixed boundary value problem in a domain with a cut

We study a fourth-order elliptic equation in a domain with a curvilinear cut. There are unilateral constraints for the solution on the cut. Considering rather general sufficiently smooth perturbations of the domain, we study the asymptotics of the energy functional. We deduce a formula for the derivative of the energy functional with respect to the perturbation parameter of the domain.     

Lyapunov‐type inequalities for a class of fractional boundary value problems with integral boundary conditions

In this paper, Lyapunov‐type inequalities are derived for a class of fractional boundary value problems with integral boundary conditions. As an application, we obtain a lower bound for the eigenvalues of corresponding equations.     

A quasistatic contact problem with slip‐dependent coefficient of friction

We consider a mathematical model which describes the bilateral quasistatic contact of a viscoelastic body with a rigid obstacle. The contact is modelled with a modified version of Coulomb's law of dry friction and, moreover, the coefficient of friction is assumed to depend either on the total slip or on the current slip. In the first case, the problem depends upon contact history. We present the classical formulations of the problems, the variational formulations and establish the existence and uniqueness of a weak solution to each of them, when the coefficient of friction is sufficiently small. The proofs are based on classical results for elliptic variational inequalities and fixed point arguments. We also study the dependence of the solutions on the perturbations of the friction coefficient and obtain a uniform convergence result. Copyright © 1999 John Wiley & Sons, Ltd.     

Homogenization of the Signorini boundary‐value problem in a thick junction and boundary integral equations for the homogenized problem

We consider a mixed boundary‐value problem for the Poisson equation in a thick junction Ω_ε which is the union of a domain Ω₀ and a large number of ε—periodically situated thin cylinders. The non‐uniform Signorini conditions are given on the lateral surfaces of the cylinders. The asymptotic analysis of this problem is done as ε→0, i.e. when the number of the thin cylinders infinitely increases and their thickness tends to zero. We prove a convergence theorem and show that the non‐uniform Signorini boundary conditions are transformed in the limiting variational inequalities in the region that is filled up by the thin cylinders as ε→0. The convergence of the energy integrals is proved as well. The existence and uniqueness of the solution to this non‐standard limit problem is established. This solution can be constructed by using a penalty formulation and successive iteration. For some subclass, these problems can be reduced to an obstacle problem in Ω₀ and an appropriate postprocessing. The equations in Ω₀ finally are also treated with boundary integral equations. Copyright © 2010 John Wiley & Sons, Ltd.     

Stability of the essential spectrum for 2D‐transport models with Maxwell boundary conditions

We discuss the spectral properties of collisional semigroups associated to various models from transport theory by exploiting the links between the so‐called resolvent approach and the semigroup approach. Precisely, we show that the essential spectrum of the full transport semigroup coincides with that of the collisionless transport semigroup in any L^p‐spaces (1 <p < ∞) for three 2D‐transport models with Maxwell‐boundary conditions. Copyright © 2005 John Wiley & Sons, Ltd.     

Multiple solutions for Kirchhoff‐type problems with variable exponent and nonhomogeneous Neumann conditions

The existence of at least three weak solutions for a class of differential equations with ‐Kirchhoff‐type and subject to small perturbations of nonhomogeneous Neumann conditions is established under suitable assumptions. Our technical approach is based on variational methods. In addition, an example to illustrate our results is given.     

Variational and numerical analysis of a quasistatic viscoelastic contact problem with normal compliance and friction

In this paper, a class of generalized evolution variational inequalities arising in quasistatic friction contact problem for viscoelastic materials is introduced and studied. Under some suitable assumptions, we obtain an existence and uniqueness theorem of the solution for the generalized evolution variational inequalities by using Banach’s fixed point theorem. Moreover, we study two numerical approximation schemes of the problem: semidiscrete scheme and fully discrete scheme. For both schemes, we prove the existence of the solution and derive the error estimations.     

Signorini problem with a solution dependent coefficient of friction (model with given friction): approximation and numerical realization

Contact problems with given friction and the coefficient of friction depending on their solutions are studied. We prove the existence of at least one solution; uniqueness is obtained under additional assumptions on the coefficient of friction. The method of successive approximations combined with the dual formulation of each iterative step is used for numerical realization. Numerical results of model examples are shown.This research was supported under the grant No. 101/01/0538 of the Grant Agency of the Czech Republic, by the projects CEZ:J17/98:272401 and MSM 113200007 of the Ministry of Education od the Czech Republic.This revised version was published online in April 2005 with a corrected missing date string.     

Long-term failure of a layered viscoelastic composite material with a crack under a time-dependent load

The long-term failure of a layered viscoelastic composite caused by precritical propagation of a coin-shaped crack is studied. It is assumed that the crack is located inside a viscoelastic layer (the layer of binder) parallel to the layer orientation. The crack development due to stretching of the composite massive by uniformly distributed external forces increasing with time is described. It is assumed that these forces act perpendicularly to the plane of crack propagation. The investigation is carried out within the framework of Boltzmann-Volterra linear theory for resolving integral operators with difference kernels describing the deformation of a material with time-dependent rheological properties. An irrational function of the viscoelastic integral operator is presented in the form of a proper continued fraction and transformed using the method of operator continued fractions. Numerical solutions are obtained for resolving integral operators with the kernel in the form of Rabotnov exponential-fractional function. The kinetics of crack growth with a prefailure zone commensurable with the crack length is described. A comparison with the results obtained in terms of the concept of thin structure of the crak tip is given.Timoshenko Institute of Mechanics, Ukrainian National Academy of Sciences, Kiev, Ukraine. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 545–558, July–August, 2000.     

均衡及约束凸优化问题公共解的一般迭代算法

梯度投影法在解决约束凸极小化问题中起到了重要的作用. 基于Tian的一般迭代算法, 本文将梯度投影法和平均算子方法相结合, 首次提出隐式和显式的复合迭代算法, 寻求均衡问题和约束凸极小化问题的公共解. 在适当条件下, 获得了强收敛定理.     

Steady solutions of the Navier–Stokes equations with threshold slip boundary conditions

We establish the wellposedness of the time‐independent Navier–Stokes equations with threshold slip boundary conditions in bounded domains. The boundary condition is a generalization of Navier's slip condition and a restricted Coulomb‐type friction condition: for wall slip to occur the magnitude of the tangential traction must exceed a prescribed threshold, independent of the normal stress, and where slip occurs the tangential traction is equal to a prescribed, possibly nonlinear, function of the slip velocity. In addition, a Dirichlet condition is imposed on a component of the boundary if the domain is rotationally symmetric. We formulate the boundary‐value problem as a variational inequality and then use the Galerkin method and fixed point arguments to prove the existence of a weak solution under suitable regularity assumptions and restrictions on the size of the data. We also prove the uniqueness of the solution and its continuous dependence on the data. Copyright © 2006 John Wiley & Sons, Ltd.     

Mixed antiplane boundary‐value problem for a piecewise‐homogeneous elastic body with a semi‐infinite interfacial crack

Antiplane stress state of a piecewise‐homogeneous elastic body with a semi‐infinite crack along the interface is considered. The longitudinal displacements along one of the crack edges on a finite interval, adjacent to the crack tip, are known. Shear stresses are applied to the body along the crack edges and at infinity. The problem reduces to a Riemann–Hilbert boundary‐value matrix problem with a piecewise‐constant coefficient for a complex potential in the class of symmetric functions. The complex potential is found explicitly using a Gaussian hypergeometric function. The stress state of the body close to the singular points is investigated. The stress intensity factors are determined. Copyright © 2016 John Wiley & Sons, Ltd.     

Analysis of a time discretization for an implicit variational inequality modelling dynamic contact problems with friction

Some dynamic contact problems with friction can be formulated as an implicit variational inequality. A time discretization of such an inequality is given here, thus giving rise to a so‐called incremental solution. The convergence of the incremental solution is established, and then the limit is shown to be the unique solution of the variational inequality. This paper contains therefore not only some new results concerning the numerical aspect of some models of contact and friction but also a constructive existence result. Copyright © 2001 John Wiley & Sons, Ltd.     

On the differential variational inequalities of parabolic‐elliptic type

We consider a model of infinite dimensional differential variational inequalities formulated by a parabolic differential inclusion and an elliptic variational inequality. The existence of global solution and global attractor for the semiflow governed by our system is proved by using measure of noncompactness. Copyright © 2017 John Wiley & Sons, Ltd.     

一类型变分不等式解的存在性唯一性问题及其对力学中的Signorini问题的应用

在本文中,我们研究了一类变分不等式解的存在性和唯一性问题,作为应用,讨论了力学中的著名的Signorini问题,改进了这一问题有解的条件.