Currently various engineering applications require scientific analysis of contact phenomena. In order to study such problems a framework in the form of variational and hemivariational inequalities complemented by theory of Finite Element Method emerged. Literature in the field of Contact Mechanics grows rapidly. Main ideas and mathematical tools were introduced in monographs [6, 14, 17, 18, 20]. Other comprehensive studies include [10, 19, 21, 22]. Theory of nonsmooth optimization used in our approach was presented in  and applied to a sample static contact problem in . Numerical analysis of contact problems can be found for instance in papers [3, 8, 9, 12, 13, 16] and for a recent study .
This paper is a continuation of our previous work , where we reformulated time-independent hemivariational inequality as optimization problem and used it to solve a static mechanical contact problem. Here, we consider time-dependent hemivariational inequality, and use similar idea to solve it numerically. In this case we prove existence and uniqueness of a solution using fixed point argument and apply this abstract framework to solve a quasistatic mechanical contact problem. This approach allows us to substitute Coulomb’s law of dry friction by more general, nonmonotone friction law with dependence on tangential component of velocity. Additionally, the friction bound is determined by normal component of displacement, describing penetration of the foundation.
The rest of this paper is organized as follows. In Section 2 we consider an abstract problem in the form of hemivariational inequality and show that under appropriate assumptions it has a unique solution. Section 3 contains a discrete scheme which approximates solution to introduced abstract problem. We also prove the theorem concerning numerical error estimate. An application of presented theory to a mechanical contact model is introduced in Section 4. Finally, in Section 5, we describe computational algorithm used to solve mechanical contact problem. Then we present results of simulations and empirical error estimation for a set of sample data.
2 Hemivariational inequality
We start with basic notation used in this paper. For a normed space , we denote by its norm, by its dual space and by the duality pairing of and . By we denote a generic constant (value of may differ in different equations).
Let us now assume that is locally Lipschitz continuous. The generalized directional derivative of at in the direction is defined by
The generalized subdifferential of at is a subset of the dual space given by
If is a locally Lipschitz function of variables, then we denote by and the Clarke subdifferential and generalized directional derivative with respect to -th variable of , respectively.
Let now be a reflexive Banach space and be a Banach space. Let be a linear and continuous operator from to , and . We denote by the adjoint operator to . Let be a time interval with . Let , , and let . The functional is assumed to be locally Lipschitz continuous with respect to its second argument. We formulate the considered hemivariational inequality as follows.
Problem : Find such that for all ,
We introduce the following assumptions.
is linear and bounded,
is symmetric, i.e., for all ,
there exists such that for all .
Operator is Lipschitz continuous, i.e., there exists such that
is locally Lipschitz continuous for all ,
there exist such that
for all ,
there exist such that
for all .
We remark that assumption (c) is equivalent to the following condition
for all . In the special case when does not depend on its first variable, we obtain a relaxed monotonicity condition, i.e. for all
So, condition (c) is a generalization of .
Define an operator by the formula
We can reformulate Problem as the following one.
Problem : Find such that for all ,
Let us now present the following theorem.
Under assumptions , , , , and , Problem has a unique solution .
Proof. We use a fixed point argument. Given , define
Then . Consider the auxiliary problem of finding a function such that
for all . Applying Theorem 4.2 in  with , we know that there exists a unique element which solves this inequality for each . Let us show that the function . For simplicity, with , denote , , for . We get that
Using strong monotonicity of the operator guaranteed by (c), assumptions , (c) and we obtain
By the smallness assumption , , and from continuity of and , we deduce that is a continuous function. This allows us to define an operator via the relation
we derive from the previous inequality that
This shows that is a history dependent operator. Applying Theorem 3.20 in , we know that the operator has a unique fixed point . Moreover, by the definition of , is a solution to Problem . Uniqueness of a solution to Problem is a consequence of uniqueness of fixed point.
3 Numerical scheme
Let us now fix and let be a finite dimensional space with a discretization parameter . For a given , we introduce the time step and the temporal nodes , . Let be an approximation of element in space . We write and define
We present the following discretized version of Problem in the form of an operator inclusion problem.
Problem : Find such that for all
Now we introduce some preliminary material, namely we recall a special case of the Jensen inequality and the discrete version of Gronwall inequality. On several occasions, we will also apply the elementary inequality
(the Jensen inequality) Let be a set of positive measure and let be an integrable function. Then
In particular, for , , , we have
(the Gronwall inequality, [10, Lemma 7.25]) Let be given. For we define . Let , be two nonnegative sequences satisfying for and for all
Then there exists a constant such that
Let us now prove the following lemma.
Under assumptions , , , , and if Problem has a solution , then it is unique and satisfies
for all with a positive constant .
Proof. To simplify the notation in this proof, we write instead of . Let be a solution to Problem and let us fix . This means that there exists such that
From the definition of generalized subdifferential of we have for all ,
After reformulation, we obtain discretized version of hemivariational inequality ,
Let us now assume that Problem has two solutions and . We will prove inductively that these solutions are equal. For we get . In inequality (3.4) for a solution we set , and for a solution we set . Then adding these inequalities, we obtain
Hence, (c) and (c) yield
and consequently from we have . We now show that if for , then . Similarly, in (3.4) for a solution we set , and for a solution we set . Adding these inequalities we obtain
We observe that , hence
Again (c) and (c) yield
Under assumption , we obtain that . This equality holds for , hence a solution of is unique.
Now, in order to prove , we set in (3.4) to obtain
Using (c), we get
From Proposition 3.23(ii) in  and assumption (b) we have
Taking sufficiently small , using () and inequality (3.1), we obtain
From discrete version of the Gronwall inequality (Lemma 3) with
which concludes the proof of (3.3).
We now consider an optimization problem, which is equivalent to Problem under the stated assumptions. To this end, let us fix and assume to be known, hence also is given. Let a functional be defined for all as follows
The next lemma collects some properties of the functional .
Under assumptions , , , , and , the functional defined by (3.8) has the following properties
is locally Lipschitz continuous,
for all ,
is strictly convex,
Proof. The proof of (i) is immediate since for the functional is locally Lipschitz continuous as a sum of locally Lipschitz continuous functions with respect to .
For the proof of (ii), we observe that from , and , the functions
are strictly differentiable with
Now, using the sum and the chain rules for generalized subgradient (cf. Propositions 3.35 and 3.37 in), we obtain
which concludes (ii).
In order to prove (iii), let us fix . We take for . From (ii), there exists such that
From the equivalent condition to (c), and consequently from (2.1), we have
Hence and from (c), we obtain