 • Applications in Mechanics and Engineering.
• Hemivariational Inequalities: Applications in Mechanics and Engineering.
• School Works - Other Essays!
• Publications on Analysis and Numerical Solution of Variational Inequalities.
• Nanocrystalline Materials.
• Records: Mathematical Theory (Translations of Mathematical Monographs).

Numerous discontinuous Galerkin schemes for the Kirchhoff plate bending problem are extended to the variational inequality. Numerical results are presented to illustrate convergence orders of the different methods. This chapter constructs and analyzes a model for the dynamic behavior of nonlinear viscoelastic beam, which is acted upon by a horizontal traction, that may come in contact with a rigid or reactive foundation underneath it.

We use a model, first developed and studied by D. Gao, that allows for the buckling of the beam when the horizontal traction is sufficiently large. In contrast with the behavior of the standard Euler—Bernoulli linear beam, it can have three steady states, two of which are buckled.

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Moreover, the Gao beam can vibrate about such buckled states, which makes it important in engineering applications. We describe the contact process with either the normal compliance condition when the foundation is reactive, or with the Signorini condition when the foundation is perfectly rigid. We use various tools from the theory of pseudomonotone operators and variational inequalities to establish the existence and uniqueness of the weak or variational solution to the dynamic problem with the normal compliance contact condition.

The main step is in the truncation of the nonlinear term and then establishing the necessary a priori estimates. Then, we show that when the viscosity of the material approaches zero and the stiffness of the foundation approaches infinity, making it perfectly rigid, the associated solutions of the problem with normal compliance converge to a solution of the elastic problem with the Signorini condition. In this chapter we present an energy-consistent numerical model for the dynamic frictional contact between a hyperlastic body and a foundation.

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5. Our contribution has two traits of novelty. The first one arises from the specific frictional contact model we consider, which provides intrinsic energy-consistent properties. The second trait of novelty consists in the construction and the analysis of an energy-consistent scheme, based on recent energy-controlling time integration methods for nonlinear elastodynamics. Some numerical results for representative impact problems are provided.

They illustrate both the specific properties of the contact model and the energy-consistent properties of the numerical scheme. In this chapter we study a dynamic frictional contact problem with normal compliance and non-clamped contact conditions, for thermo-viscoelastic materials. The weak formulation of the problem leads to a general system defined by a second order quasivariational evolution inequality coupled with a first order evolution equation.

## Hemivariational Inequalities: Applications in Mechanics and Engineering

We state and prove an existence and uniqueness result, by using arguments on parabolic variational inequalities, monotone operators and fixed point. Then, we provide a numerical scheme of approximations and various numerical computations. We consider two classes of evolution contact problems on two dimensional domains governed by first and second order evolution equations, respectively. The contact is represented by multivalued and nonmonotone boundary conditions that are expressed by means of Clarke subdifferentials of certain locally Lipschitz and semiconvex potentials.

### 1st Edition

For both problems we study the existence and uniqueness of solutions as well as their asymptotic behavior in time. For the first order problem, that is governed by the Navier—Stokes equations with generalized Tresca law, we show the existence of global attractor of finite fractal dimension and existence of exponential attractor. For the second order problem, representing the frictional contact in antiplane viscoelasticity, we show that the global attractor exists, but both the global attractor and the set of stationary states are shown to have infinite fractal dimension.

Such problems lead to a new type of variational forms, the hemivariational inequalities, which also lead to multivalued differential or integral equations.

## articlesrevues1

Innovative numerical methods are presented for the treament of realistic engineering problems. This book is the first to deal with variational theory of engineering problems involving nonmonotone multivalue realations, their mechanical foundation, their mathematical study existence and certain approximation results and the corresponding eigenvalue and optimal control problems.

Applications in Engineering Mechanics with Wayne Whiteman and

All the numerical applications give innovative answers to as yet unsolved or partially solved engineering problems, e. The book closes with the consideration of hemivariational inequalities for fractal type geometries and with the neural network approach to the numerical treatment of hemivariational inequalities.

JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser. Engineering Computational Intelligence and Complexity. This book is the first to deal with variational theory of engineering problems involving nonmonotone multivalue realations, their mechanical foundation, their mathematical study existence and certain approximation results and the corresponding eigenvalue and optimal control problems.

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6. All the numerical applications give innovative answers to as yet unsolved or partially solved engineering problems, e. The book closes with the consideration of hemivariational inequalities for fractal type geometries and with the neural network approach to the numerical treatment of hemivariational inequalities.

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