Explicit solution of the frictional contact problem of anisotropic materials indented by a moving stamp with a triangular or parabolic profile

The frictional contact problem of anisotropic materials under a moving rigid stamp is solved exactly. Inside the contact region, the Coulomb friction law is applied. Both Galilean transformation and Fourier transform are employed to get the appropriate fundamental solutions, which can lead to real solutions of physical quantities no matter whether the eigenvalues are real or complex. The complicated mixed boundary value problem is converted to singular integral equations of the second kind, which are solved analytically in terms of elementary functions for either a triangular or a parabolic stamp. Explicit formulae of surface stresses are obtained. Numerical analyses are performed in detail to reveal the surface damage mechanism. It is also found that in the frictionally moving contact problem, the friction coefficient has a more important role than the moving velocity.     

β−type fractional Sturm‐Liouville Coulomb operator and applied results

In this article, β‐type fractional Sturm‐Liouville Coulomb operator is considered by Hilfer fractional derivative. Fundamental spectral theory is investigated for the aforementioned problem. In this context, it is shown that the operator is self‐adjoint, eigenfunctions correspond to the distinct eigenfunctions are orthogonal, and eigenvalues are real. Furthermore, applications of this problem are given by the Adomian decomposition method and the results are shown with visual graphs.     

Asymptotic and spectral analysis of non‐selfadjoint operators generated by a filament model with a critical value of a boundary parameter

We consider a class of non‐selfadjoint operators generated by the equation and the boundary conditions, which govern small vibrations of an ideal filament with non‐conservative boundary conditions at one end and a heavy load at the other end. The filament has a non‐constant density and is subject to a viscous damping with a non‐constant damping coefficient. The boundary conditions contain two arbitrary complex parameters. In our previous paper (Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169), we have derived the asymptotic approximations for the eigenvalues and eigenfunctions of the aforementioned non‐selfadjoint operators when the boundary parameters were arbitrary complex numbers except for one specific value of one of the parameters. We call this value the critical value of the boundary parameter. It has been shown (in Mathematical Methods in the Applied Sciences 2001; 24 (15) : 1139–1169) that the entire set of the eigenvalues is located in a strip parallel to the real axis. The latter property is crucial for the proof of the fact that the set of the root vectors of the operator forms a Riesz basis in the state space of the system. In the present paper, we derive the asymptotics of the spectrum exactly in the case of the critical value of the boundary parameter. We show that in this case, the asymptotics of the eigenvalues is totally different, i.e. both the imaginary and real parts of eigenvalues tend to ∞as the number of an eigenvalue increases. We will show in our next paper, that as an indirect consequence of such a behaviour of the eigenvalues, the set of the root vectors of the corresponding operator is not uniformly minimal (let alone the Riesz basis property). Copyright © 2003 John Wiley & Sons, Ltd.     

A one‐dimensional computational model for hyperelastic string structures with Coulomb friction

In this paper, we consider a new model for the simulation of textiles with frictional contact between fibers and no bending resistance. In the model, one‐dimensional hyperelasticity and the Capstan equation are combined, and its connection with conventional hyperelasticity and Coulomb friction models is shown. Then, the model is formulated as a problem with the rate‐independent dissipation, and we prove that the problem possesses proper convexity and continuity properties. The article concludes with a numerical algorithm and provides numerical experiments along with a comparison of the results with a real measurement. Copyright © 2016 John Wiley & Sons, Ltd.     

Extremal eigenvalue intervals of symmetric tridiagonal interval matrices

Computing the extremal eigenvalue bounds of interval matrices is non‐deterministic polynomial‐time (NP)‐hard. We investigate bounds on real eigenvalues of real symmetric tridiagonal interval matrices and prove that for a given real symmetric tridiagonal interval matrices, we can achieve its exact range of the smallest and largest eigenvalues just by computing extremal eigenvalues of four symmetric tridiagonal matrices.     

Existence for an energetic problem in contact mechanics with dry friction

A simplified contact problem with dry friction is considered for a non–linear elastic system with two degrees of freedom. The simplification consits in neglecting kinetic energies or equivalently inertial forces. By proving existence it is shown under which conditions such simplifications are justifiable. The main focus is on the influence of a curved obstacle surface on the question of existence. The friction is modelled according to the Coulomb law and the coefficient of friction may vary along the obstacle surface. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On measure‐valued solutions to a two‐dimensional gravity‐driven avalanche flow model

This paper concerns measure‐valued solutions for the two‐dimensional granular avalanche flow model introduced by Savage and Hutter. The system is similar to the isentropic compressible Euler equations, except for a Coulomb–Mohr friction law in the source term. We will partially follow the study of measure‐valued solutions given by DiPerna and Majda. However, due to the multi‐valued nature of the friction law, new more sensitive measures must be introduced. The main idea is to consider the class of x‐dependent maximal monotone graphs of non‐single‐valued operators and their relation with 1‐Lipschitz, Carathéodory functions. This relation allows to introduce generalized Young measures for x‐dependent maximal monotone graph. Copyright © 2005 John Wiley & Sons, Ltd.     

Homogenization of contact problem with Coulomb's friction on periodic cracks

We consider the elasticity problem in a domain with contact on multiple periodic open cracks. The contact is described by the Signorini and Coulomb‐friction conditions. The problem is nonlinear, the dissipative functional depends on the unknown solution, and the existence of the solution for fixed period of the structure is usually proven by the fix‐point argument in the Sobolev spaces with a little higher regularity, H^1+α. We rescaled norms, trace, jump, and Korn inequalities in fractional Sobolev spaces with positive and negative exponents, using the unfolding technique, introduced by Griso, Cioranescu, and Damlamian. Then we proved the existence and uniqueness of the solution for friction and period fixed. Then we proved the continuous dependency of the solution to the problem with Coulomb's friction on the given friction and then estimated the solution using fixed‐point theorem. However, we were not able to pass to the strong limit in the frictional dissipative term. For this reason, we regularized the problem by adding a fourth‐order term, which increased the regularity of the solution and allowed the passing to the limit. This can be interpreted as micro‐polar elasticity.     

One‐dimensional approximations of the eigenvalue problem of curved rods

In this work we analyse the asymptotic behaviour of eigenvalues and eigenfunctions of the linearized elasticity eigenvalue problem of curved rod‐like bodies with respect to the small thickness ? of the rod. We show that the eigenfunctions and scaled eigenvalues converge, as ? tends to zero, toward eigenpairs of the eigenvalue problem associated to the one‐dimensional curved rod model which is posed on the middle curve of the rod. Because of the auxiliary function appearing in the model, describing the rotation angle of the cross‐sections, the limit eigenvalue problem is non‐classical. This problem is transformed into a classical eigenvalue problem with eigenfunctions being inextensible displacements, but the corresponding linear operator is not a differential operator. Copyright © 2001 John Wiley & Sons, Ltd.     

Motion analysis of digitally controlled machines

The non‐linear dynamics of digital position control is investigated in the presence of Coulomb friction and round‐off errors. Periodic, transient chaotic, chaotic motions are found even in linear stable cases. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Unified analysis of preconditioning methods for saddle point matrices

Short and unified proofs of spectral properties of major preconditioners for saddle point problems are presented. The need to sufficiently accurately construct approximations of the pivot block and Schur complement matrices to obtain real eigenvalues or eigenvalues with positive real parts and non‐dominating imaginary parts are pointed out. The use of augmented Lagrangian methods for more ill‐conditioned problems are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of a unilateral contact problem taking into account adhesion and friction

In this paper, we investigate a contact problem between a viscoelastic body and a rigid foundation, when both the effects of the (irreversible) adhesion and of the friction are taken into account. We describe the adhesion phenomenon in terms of a damage surface parameter according to Frémond?s theory, and we model unilateral contact by Signorini conditions, and friction by a nonlocal Coulomb law. All the constraints on the internal variables as well as the contact and the friction conditions are rendered by means of subdifferential operators, whence the highly nonlinear character of the resulting PDE system. Our main result states the existence of a global-in-time solution (to a suitable variational formulation) of the related Cauchy problem. It is proved by an approximation procedure combined with time discretization.     

A Domain Decomposition Method for Frictional Contact Problem

The bilateral or unilateral contact problem with Coulomb friction between two elastic bodies is considered [1]. An algorithm is introduced to solve the resulting finite element system by a non‐overlapping domain decomposition method. The global problem is transformed to a smaller problem on the contact surface. The solution is obtained by using a successive approximation method, in each step of this algorithm we solve two intermediate problems the first with prescribed tangential pressure and the second with prescribed normal pressure.     

DD‐type Domain Decomposition Method for Frictional Contact Problem

The bilateral or unilateral contact problem with Coulomb friction between two elastic bodies is considered [1]. An algorithm is introduced to solve the resulting finite element system by a non‐overlapping domain decomposition method. The global problem is transformed to a smaller problem on the contact surface. The solution is obtained by using a successive approximation method, in each step of this algorithm we solve two intermediate problems the first with prescribed tangential pressure and the second with prescribed normal pressure.     

Maxwell and Lamé eigenvalues on polyhedra

In a convex polyhedron, a part of the Lamé eigenvalues with hard simple support boundary conditions does not depend on the Lamé coefficients and coincides with the Maxwell eigenvalues. The other eigenvalues depend linearly on a parameter s linked to the Lamé coefficients and the associated eigenmodes are the gradients of the Laplace–Dirichlet eigenfunctions. In a non‐convex polyhedron, such a splitting of the spectrum disappears partly or completely, in relation with the non‐H² singularities of the Laplace–Dirichlet eigenfunctions. From the Maxwell equations point of view, this means that in a non‐convex polyhedron, the spectrum cannot be approximated by finite element methods using H¹ elements. Similar properties hold in polygons. We give numerical results for two L‐shaped domains. Copyright © 1999 John Wiley & Sons, Ltd.     

Inertia‐based spectrum slicing for symmetric quadratic eigenvalue problems

In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos‐based method for computing all real eigenvalues contained in a given interval of large‐scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory‐efficient by representing the computed pseudo‐Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.     

The equilibrium of a rigid body on a plane with anisotropic dry friction

The problem of the conditions for the static equilibrium of a body, resting on a rough plane at one, two or three points, is considered. It is assumed that an arbitrary system of active forces is applied to the body, while the friction on the rough supporting plane is anisotropic. This model generalizes the well-known isotropic model of Coulomb dry friction. Explicit analytic formulae, which express the necessary and sufficient conditions for static equilibrium, are obtained. The investigation procedure uses the idea of an anisotropic force of static friction, which enables analytical results for the equilibrium conditions to be obtained more easily.     

A subspace shift technique for nonsymmetric algebraic Riccati equations associated with an M‐matrix

The worst situation in computing the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation associated with an M‐matrix occurs when the corresponding linearizing matrix has two very small eigenvalues, one with positive and one with negative real part. When both eigenvalues are exactly zero, the problem is called critical or null recurrent. Although in this case the problem is ill‐conditioned and the convergence of the algorithms based on matrix iterations is slow, there exist some techniques to remove the singularity and transform the problem to a well‐behaved one. Ill‐conditioning and slow convergence appear also in close‐to‐critical problems, but when none of the eigenvalues is exactly zero, the techniques used for the critical case cannot be applied. In this paper, we introduce a new method to accelerate the convergence properties of the iterations also in close‐to‐critical cases, by working on the invariant subspace associated with the problematic eigenvalues as a whole. We present numerical experiments that confirm the efficiency of the new method.Copyright © 2012 John Wiley & Sons, Ltd.     

An H‐matrix Type Preconditioner For Frictional Contact Problems

The bilateral or unilateral contact problem with Coulomb friction between two elastic bodies is considered [1]. An algorithm is introduced to solve the resulting finite element system by a non‐overlapping domain decomposition method [2, 3]. The global problem is transformed to a independant local problems posed in each bodie and a problem posed on the contact surface (the interface problem). The solution is obtained by using a successive approximation method, in each step of this algorithm we solve two intermediate problems the first with prescribed tangential pressure and the second with prescribed normal pressure [8]. Our preconditioner construction is based on the application of the H‐matrix technique [6, 7] together with the representation of the H^1/2 seminorm by a sum of partial seminorms [4]. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Collapse of Keldysh chains and stability boundaries of non‐conservative systems

Eigenvalue problems for non‐selfadjoint linear differential operators smoothly dependent on a vector of real parameters are considered. Bifurcation of eigenvalues along smooth curves in the parameter space is studied. The case of a multiple eigenvalue with the Keldysh chain of arbitrary length is investigated. Explicit expressions describing bifurcation of eigenvalues are found. The obtained formulae use eigenfunctions and associated functions of the adjoint eigenvalue problems as well as the derivatives of the differential operator taken at the initial point of the parameter space. These results are important for the stability theory and sensitivity analysis of non‐conservative systems. Mechanical examples are considered and discussed in detail.