Well-posedness of constrained evolutionary differential variational–hemivariational inequalities with applications

Well-posedness of evolutionary differential variational-hemivariational inequalities with applications

“…So, we may deduce that the locally Lipschitz requirement H(G)(ii) holds for (1.4). Moreover, H(G)(iii) is clear, since G(x, 0, 0) = c and meas(Γ C ) > 0, which extends the framework in [14,21]. Another application is, e.g., the slip law…”

“…On the other hand, we say (1.1b)-(1.1c) is a variational inequality if j • ≡ 0, i.e., we require the functionals to be convex. The main purpose of this paper is to extend the results from [14,21] to prove wellposedness of (1.1) with applications to rate-and-state frictional contact problems. We prove that the pair (w, α) is a solution of (1.1) in the sense of Definition 1.1 and show that the flow map depends continuously on the initial data.…”

“…We prove that the pair (w, α) is a solution of (1.1) in the sense of Definition 1.1 and show that the flow map depends continuously on the initial data. The problem setting is motivated by [18,21,23,25], and the techniques have taken inspiration from [10,14,15,25].…”

“…Special cases of (1.1) have been investigated in literature. The closest to our setting is the recent work [14], which proves well-posedness for an ordinary differential equation coupled with a variational-hemivariational inequality with applications to both viscoplastic material and viscoelasticity with adhesion. In fact, if we let ϕ be independent of M w in its third argument, then (1.1) reduces to [14].…”

“…The closest to our setting is the recent work [14], which proves well-posedness for an ordinary differential equation coupled with a variational-hemivariational inequality with applications to both viscoplastic material and viscoelasticity with adhesion. In fact, if we let ϕ be independent of M w in its third argument, then (1.1) reduces to [14]. We find that in applications involving rate-and-state friction, µ may depend both upon the slip rate | uτ | and the external state variable α and hence an extra argument of M w occurs in ϕ.…”