A variational approach via bipotentials for unilateral contact problems

Abstract

a b s t r a c tWe consider a unilateral contact model for nonlinearly elastic materials, under the small deformation hypothesis, for static processes. The contact is modeled with Signorini's condition with zero gap and the friction is neglected on the potential contact zone. The behavior of the material is modeled by a subdifferential inclusion, the constitutive map being proper, convex, and lower semicontinuous. After describing the model, we give a weak formulation using a bipotential which depends on the constitutive map and its Fenchel conjugate. We arrive to a system of two variational inequalities whose unknown is the pair consisting of the displacement field and the Cauchy stress field. We look for the unknown into a Cartesian product of two nonempty, convex, closed, unbounded subsets of two Hilbert spaces. We prove the existence and the uniqueness of the weak solution based on minimization arguments for appropriate functionals associated with the variational system. How the proposed variational approach is related to previous variational approaches, is discussed too.

Two-Field Weak Solutions for a Class of Contact Models

“…Using (96) and the fact that γ : H 1 (Ω) 3 → L 2 (Γ) 3 is a linear and continuous map, we can see that γ ϕ n − γ ϕ L 2 (Γ) 3 → 0 and therefore, ϕ n → ϕ in W as n → ∞. As ε is a linear and continuous operator, see (5), then ε( ϕ n ) → ε( ϕ) as n → ∞. This convergence, together with (97), (98), and (104), lead to µ n → µ in L 2 s (Ω) 3×3 .…”

“…The necessity of a better approximation of the solutions of physical models by using numerical methods determined the consideration of additional fields in the variational setup, leading to multifield variational formulations; see, e.g., [1][2][3] and the references therein for some variational approaches based on the saddle point theory. When the constitutive laws present in the description of the models are governed by possibly set-valued operators, then a possible approach is the one governed by bipotentials (see, e.g., [4][5][6][7]); for other relevant works devoted to bipotentials and their applicability in mechanics, we refer, for instance, to [8][9][10][11].…”

A Variational Formulation Governed by Two Bipotentials for a Frictionless Contact Model

“…In connection with the calculus of variations, the bipotential theory allowed to deliver two-field variational formulations for many boundary value problems. Such formulations were proposed for several models in contact mechanics, see [9,16,17,18,19,20,21,22], where the existence and uniqueness of the pair solutions consisting of the displacement vector and the Cauchy stress tensor have been studied.…”

Two-field variational formulations for a class of nonlinear mechanical models

“…It is well known that, in the linear elasticity literature, two-field ( u , σ ) variational formulations already exist, see, e.g., [6] for some historical traces. In this context, it is worth emphasizing that our approach applies to nonlinear models governed by nonlinear constitutive laws described by means of subdifferential inclusions, as in [7–9]. In contrast to the approach used in [7–9], where the weak formulations consist of variational systems on fixed sets, in the present paper, the model under consideration leads us to a variational system which involves a variable convex set K j ( · ; f ) .…”

“…In this context, it is worth emphasizing that our approach applies to nonlinear models governed by nonlinear constitutive laws described by means of subdifferential inclusions, as in [7–9]. In contrast to the approach used in [7–9], where the weak formulations consist of variational systems on fixed sets, in the present paper, the model under consideration leads us to a variational system which involves a variable convex set K j ( · ; f ) . In this approach, we are looking for a pair ( u , σ ) such that σ belongs to the u -dependent convex set K j ( u ; f ) . …”