PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 63 (77), 1998, 75–80

EXISTENCE RESULT FOR THE DISPLACEMENT FIELD OF ELASTIC BODY. THE CASE OF A LOCKING SUPPORT

Ivan Šestak and Boško Jovanovoć

Communicated by Gradimir Milovanović

Abstract. The problem of existence result for displacement field of elastic body in contact with locking support is considered in this paper. Mathematically the problem is refered to, so called, hemivariational inequalities [4]. The existence result is obtained by making use of the theory of pseudo–monotone operators as in

[4] or [1].

1. Introduction

Many inequality problems in mechanics are formulated not only as variational inequalities, but also in terms of, so called, hemivariational inequalities [5], [4]. Hemivariational inequalities are derived from nonconvex nondifferentiable superpotentials by making use of the generalized gradient introduced by Clarke [2].

This paper deals with the existence of solutions of problem related to hemivariational inequalities which correspond to superpotential on the boundary of the body having their nonconvex and nondifferentiable part with infinite branches on closed and convex subsets. The theory of pseudo–monotone set–valued mappings introduced by Browder and Hess [1] is the main tool of the problem under consideration.

For the reader’s convenience let us recall some definitions (all of them can be found in [4]).

We denote by V a reflexive Banach space with dual V^{∗}. The pairing over V^{∗} × V and a norm on V will be denoted by ⟨ · , · ⟩_{V} and ∥ · ∥_{V} respectively.

AMS Subject Classification (1991): Primary 49A29, 49A99, 35A15.

Key words: Hemivariational inequalities, linear elasticity, locking support.

Supported by Ministry of Science and Technology of Serbia, grant number 04M01

Let the convex function f : V → R̄ = R ∪ {+∞} be not everywhere differentiable. A vector u^{∗} ∈ V^{∗} for which f(v) − f(u) ≥ ⟨u^{∗}, v − u⟩_{V} for all v ∈ V hold, where f(v) is finite, is called a subgradient of f at u and the set of such vectors is denoted by ∂f(u).

The directional derivative f^{0}(u, v) of Clarke at u in the direction v is given for f Lipschitzian near u by the expression

f^{0}(u, v) = lim sup_{\begin{matrix}h→0 \\ μ→0+ \end{matrix}} \frac{1}{μ} [f(u + h + μ v) − f(u + h)]

and the generalized gradient ¯∂f(u) of f at u ∈ V, f(u) finite, is defined as ¯∂f(u) = {u^{∗} ∈ V^{∗} : f^{0}(u^{∗}, v − u) ≥ ⟨u, v − u⟩_{V} for all v ∈ V }. If f is convex, then ¯∂f( ·) = ∂f( ·) .

Let T be a mapping from a real reflexive Banach space V into 2^{V^{∗}}. Then T is said to be pseudo–monotone if the following conditions hold:

a) The set T u is nonempty, bounded, closed and convex for all u ∈ V .

b) T is upper semicontinuous for each finite dimensional subset F of V to the weak topology of V^{∗}.

c) If {u_{i}} is a sequence in V converging weakly to u, and if u_{i}^{∗} ∈ T u_{i} is such that lim sup ⟨u_{i}^{∗}, u_{i} −u⟩_{V} ≤ 0 , then to each element v ∈ V there exists u^{∗}(v) ∈ T u with the property that lim inf ⟨u_{i}^{∗}, u_{i} − u⟩_{V} ≥ ⟨u^{∗}(v), u − v⟩_{V} .

Let T be a mapping from V into V^{∗}. Then T is said to be quasi–bounded if for each M > 0 there exists K(M) > 0 such that, whenever (u, u^{∗}) lies in the graph G(T) = {(v, v^{∗}) ∈ V × V^{∗} : v^{∗} ∈ T v} of T and ⟨u^{∗}, u⟩_{V} ≤ M ∥u∥_{V} , ∥u∥_{V} ≤ M , then ∥u^{∗}∥_{V^{∗}} ≤ K(M) .

Let T be a mapping from V into V^{∗}. Then T is said to be strongly quasi– bounded if for each M > 0 there exists K(M) > 0 such that for all (u, u^{∗}) ∈ G(T) with ⟨u^{∗}, u⟩_{V} ≤ M , ∥u∥_{V} ≤ M we have ∥u^{∗}∥_{V^{∗}} ≤ K(M) .

2. Classical Formulation

Let Ω be an open, bounded and connected subset of ℝ^{n}, n = 2 or 3, occupied by a linear elastic body in its undeformed state. The body is refered to an orthogonal Cartesian coordinate system. The boundary Γ of Ω is assumed to be Lipschitzian.

In the framework of linear elasticity and small deformations, the following relations hold

(1)  ÷ σ + f = 0  in Ω
(2)  2 ε(u) = ∇u + (∇u)^{T}  in Ω
(3)  σ = ℂ ε(u)  in Ω

where σ = {σ_{ij}} (resp. ε = {ε_{ij}}), i, j = 1, …, n, is the stress (resp. strain) tensor and ℂ is Hooke’s elasticity tensor fulfilling the well known ellipticity and symmetry properties [4]. Moreover, let u = {u_{i}} and f = {f_{i}}, i = 1, …, n be the displacement and volume force respectively. Further, let Γ = Γ̄_{U} ∪ Γ̄_{F} ∪ Γ̄_{S} with properties: Γ_{U} ∩ Γ_{F} ∩ Γ_{S} = ∅ and meas (Γ_{U}) > 0 .

We assume that

(4)  u = 0  on Γ_{U} ,
(5)  σ n = F  on Γ_{F} ,

where n = {n_{i}}, i = 1, …, n, is the outward unit normal vector to Γ_{F}.

Let K be a given convex and closed subset of displacement vector field of points on Γ_{S}, and let I_{K}( ·) be indicator function for the displacement vectors on Γ_{S}, i.e., I_{K}(v) = 0 if v ∈ K and I_{K}(v) = +∞ if v ∉ K.

Here the nonmonotone multivalued reaction–displacement law with infinite branches on K will be defined by a nonmonotone superpotential j(x, u); then the boundary conditions on Γ_{S} is given by [4]:

(6)  −S ∈ ¯∂j( · , u) + ∂I_{K}(u)  on Γ_{S} .

The relation (6) describes the adhesive contact with a locking support, e.g. a rubber support with limited compressibility [4].

Now we can formulate the classical problem (P) for displacement field as: For given f, ℂ, F, j and K find the displacement field u(x), x ∈ Ω̄, such that the relations (1)–(6) will be satisfied.

3. Variational Formulation

To give the variational, i.e. hemivariational formulation of the classical problem (P) we introduce the kinematically admisible space

(7)  V = {v ∈ (H^{1}(Ω))^{n} : v = 0 on Γ_{U}} .

From (1) and (2) we obtain the variational equality (by application of the Green–Gauss theorem):

(8)

∫_{Ω} σ · (ε(v) − ε(u)) dΩ = ∫_{Ω} f · (v − u) dΩ + ∫_{Γ_{F}} F · (v − u) dΓ
+ ∫_{Γ_{S}} S · (v − u) dΓ for all v ∈ V .

The relation (6) is equivalent to the following one: (9)

ξ ∈ ℝ^{n} ,  j^{0}(x, ξ; η − ξ) + I_{K}(η) − I_{K}(ξ) ≥ (−S) (η − ξ)  for all η ∈ ℝ^{n} .

Then, by (9), the equality (8) becomes an hemivariational inequality of the form:

(10) a(u, v − u) + I_{K}(v) − I_{K}(u) + ∫_{Γ_{S}} j^{0}(x, u(x); v(x) − u(x)) dΓ ≥ l(v − u)

for all v ∈ V ,

where, by definition

(11)  a(u, v) = ∫_{Ω} ℂε(u) · ε(v) dΩ ,
(12)  l(v) = ∫_{Ω} f · v dΩ + ∫_{Γ_{F}} F · v dΓ .

If we introduce the linear operator A : V → V^{∗} as ⟨Au, v⟩_{V} = a(u, v) , and the linear functional g : V → ℝ by ⟨g, v⟩_{V} = l(v) then we can formulate the following hemivariational problem (V): For given f ∈ (L_{2}(Ω))^{n}, ℂ ∈ (L_{∞}(Ω))^{n×n×n×n}, F ∈ (L_{2}(Γ_{F}))^{n}, j and K ⊂ (L_{2}(Γ_{S}))^{N}, N ≥ 1, find the displacement field u(x), x ∈ Ω̄, such that

(13) ⟨Au − g, v − u⟩_{V} + I_{K}(v) − I_{K}(u) + ∫_{Γ_{S}} j^{0}(x, u(x); v(x)−u(x)) dΓ ≥ 0

for all v ∈ V .

The functional J_{S} : L_{2}(Γ_{S}; ℝ^{N}) → ℝ, N ≥ 1, indicated in (13), defined by

(14)  J_{S}(v) = ∫_{Γ_{S}} j(x, v(x)) dΓ ,

to be locally Lipschitz on L_{2}(Γ_{S}; ℝ^{N}), for the function j : Γ_{S} × ℝ^{N} → ℝ, the following conditions are introduced [2]:

(i) for all ξ ∈ ℝ^{N} the function x → j(x, ξ) is measurable on Γ_{S};

(ii) for almost all x ∈ Γ_{S} the function ξ → j(x, ξ) is locally Lipschitz on ℝ^{N};

(iii) the function j( · , 0) is finitely integrable on Γ_{S}, i.e., j( · , 0) ∈ L_{1}(Γ_{S});

(iv) for almost all x ∈ Γ_{S} and each ξ ∈ ℝ^{N} : −S ∈ ¯∂j(x, ξ) ⇒ |S| ≤ c (1 + |ξ|)^{p−1} for some constant c > 0 not depending on x ∈ Γ_{S}. (In our example p = 2).

Moreover, we suppose that

(v) for almost all x ∈ Γ_{S} and each ξ ∈ ℝ^{N} : j^{0}(x, ξ; −ξ) ≤ β(x) (1 + |ξ|^{s}) , where 0 ≤ s < 2 and β( ·) is a nonnegative function from L_{q}(Γ_{S}) with q = p/(p − s) . (In our example p = 2).

Then the hemivariational inequality (13) can be presented in the form:

(15) ⟨Au − g, v − u⟩_{V} + I_{K}(v) − I_{K}(u) + J_{S}^{0}(i u; i v − i u) ≥ 0  for all v ∈ V ,

where i is the compact injection from Y (Γ_{S}) = (H^{1/2}(Γ_{S}))^{n} into L_{2}(Γ_{S}; ℝ^{N}).

The dual of Y (Γ_{S}) is denoted by Y^{∗}(Γ_{S}), i.e. Y^{∗}(Γ_{S}) = (H^{−1/2}(Γ_{S}))^{n}.

4. Existence

The hemivariational inequality (15) is equivalent to the inclusion [4]:

(16)  g ∈ Au + ∂I_{K}(u) + ¯∂^{i}J_{S}(u) ,

where

(17) ¯∂^{i}J_{S}(u) = {χ ∈ Y^{∗}(Γ_{S}) : J_{S}^{0}(i u, i v) ≥ ⟨χ, v⟩_{Y (Γ_{S})} for all v ∈ Y (Γ_{S})} .

The theorem below provides conditions which guarantee the existence of solution to the problem (V). This theorem is a slight modification of theorems 4.28 and 4.26 in [4].

Theorem 1. Let A be a pseudo–monotone operator from the reflexive Banach space V into V^{∗}. Let us suppose that the injection Y (Γ_{Q}) ⊂ L_{p}(Γ_{Q}; ℝ^{N}), N ≥ 1, Γ_{Q} ⊂ Γ, is compact for some 2 ≤ p < ∞, and that j : Γ_{Q} × ℝ^{N} → ℝ fulfills the requirements (i)–(v). Further, assume that the functional φ_{Q} : Y (Γ_{Q}) → ℝ̄ = ℝ ∪ {+∞} is convex, lower semicontinuous and proper. Suppose that the following hypotheses hold:

(H1) u_{0} ∈ Dom (∂φ_{Q}) ;

(H2) either A_{u_{0}} (A_{u_{0}}v = A(v + u_{0})) is quasibounded or ∂φ_{Q u_{0}} is strongly quasibounded;

(H3) there exists a function c : ℝ^{+} → ℝ with c(r) ≈ r as r → ∞, such that for all v ∈ V and χ ∈ ¯∂^{i}F_{Q}(v) :

⟨Av, v − u_{0}⟩_{V} + ⟨χ, v − u_{0}⟩_{Y (Γ_{Q})} ≥ c(∥v∥_{V}) ∥v∥_{V} ,

where the functional F_{Q} : L_{p}(Γ_{Q}; ℝ^{N}) → ℝ, N ≥ 1, is defined by F_{Q}(v) = ∫_{Γ_{Q}} j(x, v) dΓ , and ¯∂^{i}F_{Q}(v) as ¯∂^{i}J_{S}(u) by (17).

Then the hemivariational inequality

(18) ⟨Au − g, v − u⟩_{V} + φ_{Q}(v) − φ_{Q}(u) + F_{Q}^{0}(i u; i v − i u) ≥ 0  for all v ∈ V

has at least one solution.

Proof: Hemivariational inequality (18) can be written equivalently as g ∈ Au + ∂φ_{Q}(u) + ¯∂^{i}F_{Q}(u) . Thus the problem is reduced to the question whether g belongs to the range of A + ∂φ_{Q} + ¯∂^{i}F_{Q}. The operator ∂φ_{Q} is maximal monotone by properties of the functional φ_{Q} [3]. Similarly as in theorem 4.23 in [4] we can prove that the operator ¯∂^{i}F_{Q} is pseudo–monotone. Since A + ¯∂^{i}F_{Q} is coercive, A + ¯∂^{i}F_{Q} + ∂φ_{Q} is coercive too, because of the existence of an affine minorant of φ_{Q}. Then theorem 2.12 in [4] implies that the range of A + ¯∂^{i}F_{Q} + ∂φ_{Q} coincides with the whole V^{∗}. This establishes the existence of solution of (18). □

All conditions of Theorem 1 for the problem (V) are fulfilled. Obviously Γ_{Q} = Γ_{S}, φ_{Q} = I_{K}, p = 2 and F_{Q} = J_{S}, and it remains to verify the hypotheses (H2) and (H3). The operator A_{0} = A is quasi–bounded (u_{0} = 0, for example) because the bilinear form a(u, v) is bounded. By the ellipticity, Korn’s inequality and the estimate

⟨χ, v⟩_{Y (Γ_{S})} ≤ ∥χ∥_{Y^{∗}(Γ_{S})} ∥v∥_{Y (Γ_{S})} ≤ c ∥v∥_{V} ,

the hypothesis (H3) is fulfiled.

Then the problem (V) has at least one solution.

References

[1] Browder, F.E.; Hess, P.: Nonlinear mappings of monotone types in Banach spaces. J. Funct. Anal. 11 (1972), 251–294.

[2] Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York 1983.

[3] Moroşanu, G.: Nonlinear Evolution Equations and Applications. Series: Mathematics and Applications, D. Reidel, Dordrecht, Boston, Lancaster, Tokyo 1988.

[4] Naniewicz, Z.; Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, 1995.

[5] Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Basel, Boston, Stuttgart 1985.

Ivan Šestak (Received 25 08 1997) Tehnički fakultet 19210 Bor, p.p. 50 Yugoslavia

Boško Jovanović

Matematički fakultet

11001 Beograd, p.p. 550

Yugoslavia