Modeling of Contact Problems
In this chapter we deal with the mathematical modeling of the processes of contact between a deformable body and a foundation. We present the physical setting, the variables which determine the state of the system, the balance equations, the material’s be
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Modeling of Contact Problems
In this chapter we deal with the mathematical modeling of the processes of contact between a deformable body and a foundation. We present the physical setting, the variables which determine the state of the system, the balance equations, the material’s behavior which is reflected in the constitutive law, and the boundary conditions for the system variables. In particular, we provide a description of the frictional contact conditions, including versions of the Coulomb law of dry friction and its regularizations. Then we extend our description to the case of piezoelectric materials, i.e., materials which present a coupling between mechanical and electrical properties. In this chapter, all the variables are assumed to have sufficient degree of smoothness consistent with developments they are involved in. Moreover, as usual in the literature devoted to Contact Mechanics, everywhere in the rest of the book we denote vectors and tensors by bold-face letters.
6.1 Physical Setting We consider the general physical setting shown in Fig. 6.1 that we describe in what follows. A deformable body occupies, in the reference configuration, an open bounded connected set ˝ Rd (d D 1; 2; 3) with boundary @˝ D , assumed to be Lipschitz. We denote by D .i / the unit outward normal vector and by x D .xi / 2 ˝ D ˝ [ the position vector. Here and below, the indices i; j; k; l run from 1 to d ; an index that follows a comma indicates a derivative with respect to the corresponding component of the spatial variable x and the summation convention over repeated indices is adopted. We denote by Sd the space of secondorder symmetric tensors on Rd or, equivalently, the space of symmetric matrices of order d . We recall that the canonical inner products and the corresponding norms on Rd and Sd are given by
S. Mig´orski et al., Nonlinear Inclusions and Hemivariational Inequalities, Advances in Mechanics and Mathematics 26, DOI 10.1007/978-1-4614-4232-5 6, © Springer Science+Business Media New York 2013
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6 Modeling of Contact Problems
Fig. 6.1 The physical setting; C is the contact surface
fN GN
GD W - body
f0
GC g0 - gap foundation
u v D ui vi ; W D ij ij ;
kvkRd D .v v/1=2 kkSd D . W /1=2
for all u D .ui /; v D .vi / 2 Rd ; for all D .ij /; D .ij / 2 Sd ;
respectively. We also assume that the boundary is composed of three sets D , N , and C , with mutually disjoint relatively open sets D , N , and C , such that meas .D / > 0. The body is clamped on D and time-dependent surface tractions of density fN act on N . The body is, or can arrive, in contact on C with an obstacle, the so-called foundation. At each time instant C is divided into two parts: one part where the body and the foundation are in contact and the other part where they are separated. The boundary of the contact part is a free boundary, determined by the solution of the problem. We assume that in the reference configuration there exists a gap, denoted by g0 , between C and the foundation,
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