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The balayage method: boundary control of a thermo-elastic plate

Walter Littman, Stephen Taylor (2008)

Applicationes Mathematicae

We discuss the null boundary controllability of a linear thermo-elastic plate. The method employs a smoothing property of the system of PDEs which allows the boundary controls to be calculated directly by solving two Cauchy problems.

The dynamical Lame system : regularity of solutions, boundary controllability and boundary data continuation

M. I. Belishev, I. Lasiecka (2002)

ESAIM: Control, Optimisation and Calculus of Variations

The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input → state” map in L 2 -norms is established. A structure of the reachable sets for arbitrary T > 0 is studied. In general case, only the first component u ( · , T ) of the complete state { u ( · , T ) , u t ( · , T ) } may be controlled, an approximate controllability occurring in the subdomain filled with the shear (slow) waves. The controllability results are applied to the problem of the boundary data continuation....

The dynamical Lame system: regularity of solutions, boundary controllability and boundary data continuation

M. I. Belishev, I. Lasiecka (2010)

ESAIM: Control, Optimisation and Calculus of Variations

The boundary control problem for the dynamical Lame system (isotropic elasticity model) is considered. The continuity of the “input → state" map in L2-norms is established. A structure of the reachable sets for arbitrary T>0 is studied. In general case, only the first component u ( · , T ) of the complete state { u ( · , T ) , u t ( · , T ) } may be controlled, an approximate controllability occurring in the subdomain filled with the shear (slow) waves. The controllability results are applied to the problem of the boundary data continuation....

The interface crack with Coulomb friction between two bonded dissimilar elastic media

Hiromichi Itou, Victor A. Kovtunenko, Atusi Tani (2011)

Applications of Mathematics

We study a model of interfacial crack between two bonded dissimilar linearized elastic media. The Coulomb friction law and non-penetration condition are assumed to hold on the whole crack surface. We define a weak formulation of the problem in the primal form and get the equivalent primal-dual formulation. Then we state the existence theorem of the solution. Further, by means of Goursat-Kolosov-Muskhelishvili stress functions we derive convergent expansions of the solution near the crack tip.

The relaxation of the Signorini problem for polyconvex functionals with linear growth at infinity

Jarosław L. Bojarski (2005)

Applicationes Mathematicae

The aim of this paper is to study the unilateral contact condition (Signorini problem) for polyconvex functionals with linear growth at infinity. We find the lower semicontinuous relaxation of the original functional (defined over a subset of the space of bounded variations BV(Ω)) and we prove the existence theorem. Moreover, we discuss the Winkler unilateral contact condition. As an application, we show a few examples of elastic-plastic potentials for finite displacements.

The unbonded contact problem of a rectangular plate resting on an elastic foundation

Luigi Ascione, Renato S. Olivito, Giuseppe Spadea (1983)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

In questo lavoro viene analizzato il problema di equilibrio statico di una piastra rettangolare in contatto unilaterale e senza attrito con un mezzo elastico. Si esaminano i due modelli di fondazione alla Winkler e di semispazio elastico. Il problema viene risolto mediante discretizzazione agli elementi finiti utilizzando un approccio di tipo «penalty». La rapida convergenza del metodo e la sua efficienza sono dimostrate dagli esempi studiati, che riguardano sia piastre quadrate che rettangolari...

The weak solution of an antiplane contact problem for electro-viscoelastic materials with long-term memory

Ammar Derbazi, Mohamed Dalah, Amar Megrous (2016)

Applications of Mathematics

We study a mathematical model which describes the antiplane shear deformation of a cylinder in frictionless contact with a rigid foundation. The material is assumed to be electro-viscoelastic with long-term memory, and the friction is modeled with Tresca's law and the foundation is assumed to be electrically conductive. First we derive the classical variational formulation of the model which is given by a system coupling an evolutionary variational equality for the displacement field with a time-dependent...

Theoretical analysis of discrete contact problems with Coulomb friction

Tomáš Ligurský (2012)

Applications of Mathematics

A discrete model of the two-dimensional Signorini problem with Coulomb friction and a coefficient of friction ℱ depending on the spatial variable is analysed. It is shown that a solution exists for any ℱ and is globally unique if ℱ is sufficiently small. The Lipschitz continuity of this unique solution as a function of ℱ as well as a function of the load vector f is obtained. Furthermore, local uniqueness of solutions for arbitrary ℱ > 0 is studied. The question of existence of locally Lipschitz-continuous...

Thick obstacle problems with dynamic adhesive contact

Jeongho Ahn (2008)

ESAIM: Mathematical Modelling and Numerical Analysis

In this work, we consider dynamic frictionless contact with adhesion between a viscoelastic body of the Kelvin-Voigt type and a stationary rigid obstacle, based on the Signorini's contact conditions. Including the adhesion processes modeled by the bonding field, a new version of energy function is defined. We use the energy function to derive a new form of energy balance which is supported by numerical results. Employing the time-discretization, we establish a numerical formulation and investigate...

Topology optimization of quasistatic contact problems

Andrzej Myśliński (2012)

International Journal of Applied Mathematics and Computer Science

This paper deals with the formulation of a necessary optimality condition for a topology optimization problem for an elastic contact problem with Tresca friction. In the paper a quasistatic contact model is considered, rather than a stationary one used in the literature. The functional approximating the normal contact stress is chosen as the shape functional. The aim of the topology optimization problem considered is to find the optimal material distribution inside a design domain occupied by the...

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