Contact Problems for Nonlinearly Elastic Materials: Weak Solvability Involving Dual Lagrange Multipliers

Abstract

A class of problems modelling the contact between nonlinearly elastic materials and rigid foundations is analysed for static processes under the small deformation hypothesis. In the present paper, the contact between the body and the foundation can be frictional bilateral or frictionless unilateral. For every mechanical problem in the class considered, we derive a weak formulation consisting of a nonlinear variational equation and a variational inequality involving dual Lagrange multipliers. The weak solvability of the models is established by using saddle-point theory and a fixed-point technique. This approach is useful for the development of efficient algorithms for approximating weak solutions.2000 Mathematics subject classification: primary 74M10; secondary 74M15, 35J50, 35J66.

Existence and uniqueness result for a class of mixed variational problems

“…In the present work, the behavior of the materials is described by using the subdifferential of a proper, convex, lower semicontinuous functional and the contact is modeled with Signorini’s condition with zero gap. The results extend and improve the results obtained in [ 5 , 6 ], where a unilateral frictionless contact model for nonlinearly elastic materials is analyzed.…”

“…The proof is analogous to Theorem 6.1 in [ 5 ]. From ( 24 ) and ( 25 ), it is easy to verify that ( 3 ) and ( 5 ) hold.…”

“…Next, we introduce the space of admissible displacement fields U defined by Then is a Hilbert space, where From Proposition 2.1 and Remark 2.2 in [ 5 ] we know that can be organized as a Hilbert space. Let be the dual space of .…”

“…Furthermore, we define by Let us introduce the following subset of : Obviously, . Moreover, from ( 23 ) it follows that and Then, combining ( 18 )–( 23 ), from Problem 4.1 in [ 5 ], Problem 13 can be written as the following variational formulation:…”

A Class of Generalized Mixed Variational-Hemivariational Inequalities I: Existence and Uniqueness Results

“…The most representative recent results in this area are the following: Cojocaru-Matei [5] who have discussed the unique solvability for a class of frictional contact problems governed by the p-Laplace operator, which can be formulated as a mixed variational inequality; Matei et al [19] have employed the Lagrange multipliers method to consider a deformable body in frictionless unilateral contact with a moving rigid obstacle, and explored an efficient algorithm approximating the weak solution for a more general case of a two-body contact problem including friction; Han-Reddy [10] who have analyzed the finite element method for a class of mixed variational inequalities of the second kind which arises in elastoplastic problems; Sofonea-Matei [32] who have considered a new class of mixed variational problems, and proved existence, uniqueness as well as continuous dependence results by applying generalized saddle point formulations and various estimates, combined with a fixed point argument. We refer the reader to [13,14,15,16,17,18,19,20,30,31] and the references therein for a more detailed discussion of this topic.…”