Nonlinear Inclusions and Hemivariational Inequalities

In this chapter we present preliminary material from functional analysis which will be used subsequently. The results are stated without proofs, since they are standard and can be found in many references. For the convenience of the reader we summarize definitions and results on normed spaces, Banach spaces, duality, and weak topologies which are mostly assumed to be known as a basic material from functional analysis. We then recall some standard results on measure theory that will be applied repeatedly in this book. We assume that the reader has some familiarity with the notions of linear algebra and general topology.

In this chapter we introduce function spaces that will be relevant to the subsequent developments in this monograph. The function spaces to be discussed include spaces of continuous and continuously differentiable functions, smooth functions, Lebesgue and Sobolev spaces, associated with an open bounded domain in \({\mathbb{R}}^{d}\). In order to treat time-dependent problems, we also introduce spaces of vector-valued functions, i.e., spaces of mappings defined on a time interval [0, T] with values in a Banach or a Hilbert space.

In this chapter we present basic material on the set-valued mappings, nonsmooth analysis, subdifferential calculus, and operators of monotone type. For set-valued mappings we concentrate on measurability and continuity issues which we need in subsequent chapters. The section on nonsmooth analysis is devoted to results on the generalized differentiation for locally Lipschitz superpotentials. Next, we provide a result on the subdifferential of the integral superpotentials which is an essential tool in Chaps. 4 and 5 of the book. Finally, we recall the results on single and multivalued operators of monotone type in Banach spaces. The surjectivity results for such operators play a crucial role in our existence results for stationary and evolutionary inclusions. Most of the results presented in this chapter are stated without proofs.

In this chapter we study stationary operator inclusions, i.e., inclusions in which the derivatives of the unknown with respect to the time variable are not involved. We start with a basic existence result for abstract operator inclusions. Then we use it in order to prove the existence of solutions for various operator inclusions of subdifferential type. We also prove that, under additional assumptions, the solution of the corresponding inclusions is unique. Finally, we specialize our existence and uniqueness results in the study of stationary hemivariational inequalities. The theorems presented in this chapter will be applied in the study of static frictional contact problems in Chap. 7.

In this chapter we study evolutionary inclusions of second order. These are multivalued relations which involve the second-order time derivative of the unknown. We start with a basic existence result for such inclusions. Then we provide results on existence and uniqueness of solutions to evolutionary inclusions of the subdifferential type, i.e., inclusions involving the Clarke subdifferential operator of locally Lipschitz functionals. We also prove an existence and uniqueness result for integro-differential evolutionary inclusions. Next, we consider a class of hyperbolic hemivariational inequalities for which we provide a theorem on existence of solutions and, under stronger hypotheses, their uniqueness. We conclude this chapter with a result on existence and uniqueness of solutions to the evolutionary integro-differential hemivariational inequality with the Volterra integral term. The results provided below represent the dynamic counterparts of theorems presented in Chap. 4 and will be used in the study of the dynamic frictional contact problems in Chap. 8.

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.

In this chapter we illustrate the use of the abstract results obtained in Chap. 4, in the study of three representative static frictional contact problems for deformable bodies. In the first two problems we model the material’s behavior with a nonlinear elastic constitutive law and with a viscoelastic constitutive law with long memory, respectively, and we describe the frictional contact with subdifferential boundary conditions. In the third problem the deformable body is assumed to be piezoelectric and, therefore, we model its behavior with an electro-elastic constitutive law. And, again, the contact conditions, including the electrical conditions on the contact surface, are of subdifferential type. For each problem we provide a variational formulation. For the first two problems it is in a form of a hemivariational inequality for the displacement field and, for the third one, it is in a form of a system of hemivariational inequalities in which the unknowns are the displacement and electric potential fields. Then, we use the abstract existence and uniqueness results presented in Chap. 4 to prove the weak solvability of the corresponding contact problems and, under additional assumptions, their unique weak solvability. Finally, we present concrete examples of constitutive laws and frictional contact conditions for which our results work and provide the related mechanical interpretation. Everywhere in this chapter we use the notation introduced in Chap. 6.

In this chapter we apply the abstract results of Chap. 5 in the study of three dynamic frictional contact problems. In the first two problems we model the material’s behavior with a nonlinear viscoelastic constitutive laws with short and long memory, respectively. In the third problem the body is supposed to be piezoelectric and, therefore, the process is mechanically dynamic and electrically static. In all problems under investigation we describe frictional contact with subdifferential boundary conditions. For each problem we deliver a variational formulation. For the first two problems it is in a form of a hemivariational inequality for the displacement field and, for the third one, it is in a form of a system of hemivariational inequalities in which the unknowns are the displacement and electric potential fields. Next, we use the abstract existence and uniqueness results presented in Chap. 5 to prove the weak solvability of the corresponding contact problems and, under additional assumptions, their unique weak solvability. Everywhere in this chapter we use the notation introduced in Chap. 6.