Unilateral contact problem for two inclined elastic bodies

European Journal of Mechanics A/Solids 27 (2008) 365–377

Unilateral contact problem for two inclined elastic bodies Alexander Khludnev a,∗ , Atusi Tani b,1 a Lavrentyev Institute of Hydrodynamics of the Russian Academy of Sciences, Novosibirsk 630090, Russia b Keio University, Department of Mathematics, Yokohama 223-8522, Japan

Received 29 November 2006; accepted 6 August 2007 Available online 11 August 2007

Abstract The paper concerns an analysis of unilateral contact problems between two inclined elastic plates and between a plate and a beam. Considered problems are characterized by a contact set having a dimension less than one of that of a domain. This property leads to a new class of free boundary problems with inequality type boundary conditions. The main attention is paid to a suitable description of boundary conditions along the contact zone. Asymptotic properties of solutions are established provided that elasticity parameters of the contacting bodies are changing, in particular, in the case when a stiffness of the elastic body goes to infinity. © 2007 Elsevier Masson SAS. All rights reserved. Keywords: Thin elastic obstacle; Elastic plate; Inequality type boundary condition; Crack

1. Introduction Let Ω ⊂ R2 be a bounded domain with smooth boundary Γ . We assume that Ω corresponds to the middle surface of an elastic plate. A unit external normal vector to Γ is denoted by q = (q1 , q2 ). Another elastic plate with a middle surface D is situated at angle α with respect to the first one, α ∈ (0, π2 ]. The domain D is assumed to be bounded, and its boundary is denoted by ∂D, Ω ∩ D = ∅, Ω ∩ ∂D = ∅. Denote γ0 = (∂D) \ Ω. In this case ∂D = γ ∪ γ¯0 . Let ν = (ν1 , ν2 ) be a unit normal vector to γ located in the plane Ω. By n = (n1 , n2 ) we denote a unit normal external vector to ∂D located in the plane D (see Fig. 1). Also assume that ∂D is a smooth curve, γ is a connected set, and γ does not intersect the boundary Γ of the domain Ω. The first part of the paper is concerned with analysis of a unilateral contact between the two inclined elastic plates described above. We consider only vertical (normal) displacements of each plate, i.e. displacements in direction orthogonal to D, Ω, respectively. Since Ω ∩ ∂D = ∅ the contact may occur along the line γ . Consequently, the first (horizontal) plate has a thin elastic obstacle along γ . Simultaneously, the second plate is in contact with an elastic structure on its boundary. To describe a behavior of the first plate we consider an equilibrium equation in a cracked domain Ωγ = Ω \ γ¯ , and impose inequality type boundary conditions on γ . It is known that crack models with * Corresponding author. Fax: +7 (383) 333 1612.

E-mail addresses: [email protected] (A. Khludnev), [email protected] (A. Tani). 1 Fax: +81 (45) 566 1642.

0997-7538/$ – see front matter © 2007 Elsevier Masson SAS. All rights reserved. doi:10.1016/j.euromechsol.2007.08.001

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possible contact between crack faces contain inequality type boundary conditions on the crack faces (Khludnev and Kovtunenko, 2000; Khludnev and Sokolowski, 2000; see also Leblond, 2000). In our case γ can be interpreted as a crack, but the boundary conditions on γ are different from those in Khludnev and Kovtunenko (2000). We focus on the proper description of a complete set of boundary conditions appearing on γ . A passage to a limit when a stiffness of the plates tends to infinity is also analyzed in the paper. Remark that Signorini type contact problems, i.e. problems with nonpenetration conditions of inequality type, can be found in a huge number of publications (see Khludnev and Sokolowski, 1997; Eck et al., 2005). The first results for these problems can be found in Fichera (1972). A unilateral contact for an elastic plate with rigid obstacles was analyzed in Caffarelli and Friedman (1979), Caffarelli et al. (1982), Dal Maso and Paderni (1988), Khludnev and Sokolowski (1997); rigid and elastic thin obstacles for plates are considered in Schild (1986) and Khludnev et al. (2006), respectively. In the second part of the paper we analyze a contact problem between an elastic plate and elastic beam situated at angle α with respect to the plate (see Fig. 3). Variational and differential formulations of this problem are presented. In this case a contact between the plate and the beam may occur at a given point. The equilibrium equation for the plate is fulfilled in a domain with the removed point. Analysis of equivalent settings of the problem is provided. A complete system of inequality type boundary conditions is found, and suitable relations are obtained to specify a sense in which these conditions hold. Also, a passage to the limit is justified in the case when a stiffness of the beam goes to infinity. Contact problems between plates and beams are important in applications in describing the behavior of flexible structures. In the paper we consider exemplary situations, the contact of two such structural elements. In principle, it is possible to analyze more complicated contact problems, in particular, to take into account horizontal (tangential) displacements of the plates. In this case the system of differential equations would be too big to demonstrate main ideas in a quite simple form. We restrict ourselves to vertical displacements of the plates, and consider both vertical and horizontal displacements of the beam. Underline that in both cases we have to deal with free boundary problems and unknown contact sets describing a contact of an elastic structure with a thin elastic obstacle. Important peculiarity of the analyzed problems is that a dimension of contact sets is less than one of that of solution domains, thus providing new type boundary problems with non-standard inequality type boundary conditions. To describe properly these boundary conditions it is necessary to use suitable weighted Sobolev spaces. It is turned out that the boundary conditions are fulfilled in terms of inequalities for elements of dual spaces. 2. Contact between two plates We start with differential formulation of the contact problem between the two inclined plates (Fig. 1). It is necessary to find functions v(x), w(y), x = (x1 , x2 ) ∈ D, y = (y1 , y2 ) ∈ Ωγ , such that

Fig. 1.

A. Khludnev, A. Tani / European Journal of Mechanics A/Solids 27 (2008) 365–377

2 v = g

in D,

w=f 2

367

(1)

in Ωγ ,

(2)

v = vn = 0 on γ0 ,

(3)

w = wq = 0

(4)

on Γ,  ν v cos α − w  0, t (w) (v cos α − w) = 0 on γ ,   m(w) = 0 on γ , [w] = [wν ] = 0,  ν  ν t (w) cos α = t n (v) on γ . t (w)  0, m(v) = 0,

(5) (6) (7)

Here [u] = u+ − u− is a jump of a function u on γ , where u± correspond to the positive and negative (with respect to ν) faces γ ± , respectively, vn =

∂v , ∂n

wq =

∂w , ∂q

wν =

∂ 2w m(w) = κ1 w + (1 − κ1 ) 2 , ∂ν

∂w , ∂ν

  ∂ ∂ 2w w + (1 − κ1 ) 2 , t (w) = ∂ν ∂s ν

where κ1 is the Poisson ratio of the horizontal plate, (s1 , s2 ) = (−ν2 , ν1 ). The values m(v), t n (v) are introduced similar to those of m(w), t ν (w),   ∂ 2w ∂ ∂ 2v m(v) = κ2 v + (1 − κ2 ) 2 , t n (v) = v + (1 − κ2 ) 2 ∂n ∂n ∂τ with the Poisson ratio κ2 for the inclined plate, (τ1 , τ2 ) = (−n2 , n1 ). The functions g ∈ L2 (D), f ∈ L2 (Ω) are given. Remark that (1)–(2) are equilibrium equations; functions v, w describe vertical (normal) displacements of the plates; m(w), t ν (w) are the bending moment and transverse force respectively for the first plate; similar notations, m(v), t n (v), are used for the second plate. Relations (3)–(4) describe the plates clamping on γ0 and Γ , respectively. The first inequality in (5) guarantees a mutual nonpenetration between the two plates. If we have a strict inequality in the first relation of (5), i.e. no contact at a given point, the last relation of (5) implies that a jump of transverse forces is zero. On the other hand, a strict inequality in the first relation of (7) necessarily yields that a contact occurs at a given point, i.e. we have an equality in the first relation of (5). The last equality of (7) provides a relation between transverse forces for the two plates at γ . We should underline that mechanical hypothesis used to derive the model (1)–(7) assumes that in-plane stiffness of plates is much larger than the out-of-plane stiffness. In what follows we discuss an existence of a solution to problem (1)–(7). To this end, a variational formulation of this problem is provided. All relations (1)–(7) will be derived from the variational formulation of the problem. Introduce the following Sobolev spaces   Hγ20 (D) = v ∈ H 2 (D) | v = vn = 0 on γ0 ,   H02 (Ω) = w ∈ H 2 (Ω) | w = wq = 0 on Γ and the set of admissible displacements   K = (v, w) | v ∈ Hγ20 (D), w ∈ H02 (Ω), v cos α − w  0 on γ . Denote



¯ = BΩ (w, w)



w,11 w¯ ,11 + w,22 w¯ ,22 + κ1 (w,11 w¯ ,22 + w,22 w¯ ,11 ) + 2(1 − κ1 )w,12 w¯ ,12 ,

Ω

 BD (v, v) ¯ =



v,11 v¯,11 + v,22 v¯,22 + κ2 (v,11 v¯,22 + v,22 v¯,11 ) + 2(1 − κ2 )v,12 v¯,12 ,

D

v,i =

∂v ∂xi ,

w,i =

∂w ∂yi ,

i = 1, 2.

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Fig. 2.

We introduce the energy functional 1 1 Π(v, w) = BD (v, v) + BΩ (w, w) − 2 2



 gv −

D

fw Ω

and consider the minimization problem inf

(v,w)∈K

(8)

Π(v, w)

which is equivalent to the variational inequality (v, w) ∈ K,



BD (v, v¯ − v) + BΩ (w, w¯ − w) −

g(v¯ − v) − D

(9)

 f (w¯ − w)  0,

∀(v, ¯ w) ¯ ∈ K.

(10)

Ω

is weakly closed in Hγ20 (D) × H02 (Ω), and the functional Π

Note that the set K is coercive and weakly lower semicontinuous on this space. Consequently, the minimization problem (8) has a solution satisfying the variational inequality (9)–(10). This solution is unique. Now we derive (1)–(7) from (9)–(10) and clarify in what sense the boundary conditions (5)–(7) are fulfilled. First observe that Eqs. (1), (2) follow from (10) in the distributional sense. Indeed, we can substitute into (10) test functions (v, ¯ w) ¯ = (v ± ϕ, w ± ψ), ϕ ∈ C0∞ (D), ψ ∈ C0∞ (Ωγ ) which implies (1), (2). Consider next an extension of γ to a closed curve Σ of the class C 1,1 such that Σ ⊂ Ω. In this case the domain Ω is divided into two subdomains Ω1 , Ω2 with boundaries Σ and Σ ∪ Γ , respectively (see Fig. 2). The vector ν = (ν1 , ν2 ) is considered on Σ as external with respect to Ω1 . We choose (v, ¯ w) ¯ = (v, w + ψ) as test functions in (10), ψ  0 on γ , ψ ∈ H02 (Ω). This provides  BΩ (w, ψ) − f ψ  0. (11) Ω

Introduce the space   V = u ∈ H 2 (Ω1 ) | 2 u ∈ L2 (Ω1 ) . Then for u ∈ V we can define m(u) ∈ H −1/2 (Σ), t ν (u) ∈ H −3/2 (Σ), and the following Green’s formula holds (Khludnev and Kovtunenko, 2000; Temam, 1983)    ψ 2 u = BΩ1 (ψ, u) + t ν (u), ψ 3/2,Σ − m(u), ψν 1/2,Σ , ∀ψ ∈ H 2 (Ω1 ). (12) Ω1

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369

Here ·,· i/2,Σ means the duality pairing between the space H −i/2 (Σ) and its dual H i/2 (Σ), i = 1, 3. This Green’s formula together with (2) allows us to derive from (11) the following inequality     − m(w) , ψν 1/2,Σ + t ν (w) , ψ 3/2,Σ  0. Since ψν is arbitrary on Σ it follows   m(w) = 0 in the sense of H −1/2 (Σ),   ν t (w) , ψ 3/2,Σ  0 ∀ψ ∈ H02 (Ω), ψ  0 on γ .

(13) (14)

Let us take (v, ¯ w) ¯ = (v + ϕ, w) as test functions in (10), ϕ ∈ Hγ20 (D), ϕ  0 on γ . This yields  BD (v, ϕ) − gϕ  0.

(15)

D

Taking into account equilibrium equation (1) and the formula like (12) for the domain D we derive   m(v), ϕn 1/2,∂D − t n (v), ϕ 3/2,∂D  0. 3/2

1/2

1/2

(16)

3/2

In our case ϕ = ϕn = 0 on ∂D \ γ . Hence ϕ ∈ H00 (γ ), ϕn ∈ H00 (γ ). The spaces H00 (γ ), H00 (γ ) are defined as follows (Grisvard, 1985)

   v2  1/2 H00 (γ ) = v ∈ H 1/2 (γ )  0. We have to find a unique solution v δ , w δ to the following variational inequality (v δ , w δ ) ∈ K,



1 BD (v δ , v¯ − v δ ) + BΩ (w δ , w¯ − w δ ) − δ

g(v¯ − v δ ) −

(v, ¯ w) ¯ = 2(v δ , w δ )

1 BD (v δ , v δ ) + BΩ (w δ , w δ ) − δ





gv δ − D

f (w¯ − w δ )  0,

∀(v, ¯ w) ¯ ∈ K.

(24)

Ω

D

Substitution of (v, ¯ w) ¯ = (0, 0),

(23)



in (24) implies

f w δ = 0.

(25)

Ω

This equality yields two estimates 1 δ 2 w H 2 (Ω)  c1 , δ 0

v δ 2H 2

γ0 (D)

 c2

(26)

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371

with constants c1 , c2 being uniform in δ. Assume that for a subsequence uδ , w δ , with the previous notation, we have as δ → 0 w δ → 0 strongly in H02 (Ω),

(27)

v →v

(28)

δ

0

weakly in

Hγ20 (D).

Since v δ cos α − w δ  0 on γ , the limit function v 0 satisfies the inequality v 0 cos α  0 on γ .

(29)

Let us take test functions of the form (v, ¯ 0) in (24), where v¯ ∈ Hγ20 (D), v¯ cos α  0 on γ . This provides the relation   1 BD (v δ , v¯ − v δ )  BΩ (w δ , w δ ) + g(v¯ − v δ ) − f w δ . δ Ω

D

Since 1 lim inf BΩ (w δ , w δ )  0, δ→0 δ from the above inequality it follows v 0 ∈ L,

(30)



BD (v , v¯ − v ) − 0

g(v¯ − v )  0,

0

0

∀v¯ ∈ L,

(31)

D

where   L = v ∈ Hγ20 (D) | v cos α  0 on γ . It is clear that for α ∈ (0, π2 ) we can define L in an equivalent form   L = v ∈ Hγ20 (D) | v  0 on γ , and variational inequality (30)–(31) corresponds to the following boundary value problem. Find v 0 such that 2 v 0 = g v

0

= vn0 0

in D,

= 0 on γ0 ,

m(v ) = 0,

v 0  0,

t n (v 0 )  0,

v 0 · t n (v 0 ) = 0

on γ .

In this case the limit problem (30)–(31) describes an equilibrium state for the second plate having a rigid obstacle on γ . If α = π2 there is no restriction for v 0 since L = Hγ20 (D), and (30)–(31) reduces to v 0 ∈ Hγ20 (D),  0 BD (v , v) ¯ − g v¯ = 0,

(32) ∀v¯ ∈ Hγ20 (D).

D

Differential formulation of the problem (32)–(33) is to find v 0 such that 2 v 0 = g v

0

= vn0 0

in D,

= 0 on γ0 ,

m(v ) = 0,

t n (v 0 ) = 0 on γ .

In fact, we can improve the convergences (27), (28). First note, that (25) implies

(33)

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   1 lim sup BΩ (w δ , w δ ) = lim sup −BD (v δ , v δ ) + gv δ + f w δ δ→0 δ δ→0 Ω D     δ δ δ  lim sup −BD (v , v ) + lim sup gv + lim sup f w δ δ→0

δ→0

  −BD (v 0 , v 0 ) +

D

δ→0

Ω

gv 0 . D

On the other hand, from (31) it follows  BD (v 0 , v 0 ) = gv 0 .

(34)

D

Hence, the above arguments provide the relations 1 1 0  lim inf BΩ (w δ , w δ )  lim sup BΩ (w δ , w δ )  0 δ→0 δ δ→0 δ which proves the following convergence as δ → 0 1 BΩ (w δ , w δ ) → 0, δ i.e., (27) is improved in the sense

(35)

1 √ w δ → 0 strongly in H02 (Ω). δ Now, instead of (28), the convergence vδ → v0

strongly in Hγ20 (D)

(36)

will be established. Since the weak convergence of v δ to v 0 is already proved, it suffices to state as δ → 0 BD (v δ , v δ ) → BD (v 0 , v 0 ). From (25) it follows 1 BD (v δ , v δ ) = − BΩ (w δ , w δ ) + δ

(37) 

 gv δ +

D

f wδ . Ω

By (27), (35), the right-hand side here has a limit equal to  δ δ lim BD (v , v ) = gv 0 .



D gv

0,

i.e., as δ → 0

D

Taking into account (34), we derive (37) which proves the strong convergence (36). 4. Plate–beam contact problem In this section we consider a contact problem between an elastic plate and an inclined elastic beam. The middle surface of the plate is denoted by Ω. The properties of Ω are described in Section 1. The beam is situated at angle α ∈ (0, π2 ] with respect to the plate (see Fig. 3). The plane x1 0x2 is orthogonal to the plane y1 0y2 . The beam has both vertical v and tangential u displacements along axes x2 , x1 , respectively. Like in Sections 1–3 the plate has only the vertical displacement w. Let (0, 0) ∈ Ω be a point of possible contact between the plate and the beam. A middle line of the beam is denoted by σ . We assume that σ is the interval (0, 1), and the point x = 0 is a contact one for the beam. Here and in what follows x1 is denoted by x. The end point x = 1 of the beam is clamped. The boundary Γ of the plate is also clamped.

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373

Fig. 3.

We start with a variational formulation of the problem. Consider the Sobolev spaces   H˜ 1 (σ ) = u ∈ H 1 (σ ) | u = 0 at x = 1 ,   H˜ 2 (σ ) = v ∈ H 2 (σ ) | v = vx = 0 at x = 1 and introduce the energy functional on the space H = H˜ 1 (σ ) × H˜ 2 (σ ) × H02 (Ω),      1 1 1 2 E(u, v, w) = u2x − hu + vxx − gv + BΩ (w, w) − f w, 2 2 2 σ

∈ L2 (Ω), h, g

where f

σ

σ

σ

Ω

∈ L2 (σ )

are given functions. Consider the set of admissible displacements   P = (u, v, w) ∈ H | u(0)p1 + v(0)p2  w(0)

with p1 = sin α, p2 = cos α. We can find a unique solution of the minimization problem inf

(u,v,w)∈P

(38)

E(u, v, w).

The solution of this problem satisfies the variational inequality (u, v, w) ∈ P ,        ux (u¯ x − ux ) − h(u¯ − u) + vxx (v¯xx − vxx ) − g(v¯ − v) + BΩ (w, w¯ − w) − f (w¯ − w)  0, σ

σ

∀(u, ¯ v, ¯ w) ¯ ∈ P.

(39)

Ω

(40)

It is possible to give a differential formulation of the problem (39), (40). To this end, choose a closed curve Σ of the class C 1,1 , Σ ⊂ Ω, such that 0 ∈ Σ. Denote by ν = (ν1 , ν2 ) a unit normal vector to the curve Σ . In this case the domain Ω is divided into two subdomains Ω1 and Ω2 with boundaries Σ and Σ ∪ Γ , respectively. Assume that ν is oriented towards Ω2 (see Fig. 2). Denote Ω0 = Ω \ {0}. We have to find functions u(x), v(x), w(y), x ∈ σ, y = (y1 , y2 ) ∈ Ω0 , such that −uxx = h

in σ,

(41)

vxxxx = g

in σ,

(42)

w=f

in Ω0 ,

(43)

2

w = wq = 0

on Γ,

(44)

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u = v = vx = 0

at x = 1,

vxx = 0

at x = 0,

(45)

u(0)p1 + v(0)p2  w(0), ux (0)p2 = −vxxx (0)p1 ,

ux (0)  0, ux (0) w(0) − u(0)p1 − v(0)p2 = 0,   ν  1 m(w) = 0, t (w) = ux (0)δΣ on Σ. p1

(46) (47) (48)

Here δΣ is a distribution on Σ defined by the formula δΣ (ξ ) = ξ(0). It is important that Σ is an arbitrary curve with the above properties. Now we derive relations (41)–(48) from the variational inequality (39), (40) and demonstrate in what sense boundary conditions (48) are fulfilled. First note that equilibrium equations (41)–(43) follow from (39), (40) in the distributional sense. We can next take test functions (u, ¯ v, ¯ w) ¯ = (u, v, w +ϕ) in (40), ϕ ∈ H02 (Ω), ϕ(0)  0. This provides the inequality  BΩ (w, ϕ) − f ϕ  0. Ω

Applying the Green’s formula like (12) for the subdomains Ω1 , Ω2 , and taking into account (41)–(44), we derive     − m(w) , ϕν 1/2,Σ + t ν (w) , ϕ 3/2,Σ  0, ∀ϕ ∈ H02 (Ω), ϕ(0)  0 which gives   m(w) = 0 in the sense of H −1/2 (Σ),  ν  t (w) , ϕ 3/2,Σ  0, ∀ϕ ∈ H02 (Ω), ϕ(0)  0.

(49) (50)

From (50) it follows [t ν (w)], ϕ 3/2,Σ = 0 provided that ϕ(0) = 0. Hence the value [t ν (w)], ϕ 3/2,Σ depends only on ϕ(0), and consequently  ν  t (w) , ϕ 3/2,Σ = kϕ(0), k = const. (51) The constant k will be found below. Substitute next in (40) test functions of the form (u, ¯ v, ¯ w) ¯ = (u + ψ, v, w), where ψ ∈ H˜ 1 (σ ), ψ(0)  0. We obtain   ux ψx − hψ  0. σ

σ

Whence, by (41), the inequality ux (0)  0 follows. Now we substitute in (40) test functions of the form (u, ¯ v, ¯ w) ¯ = (u, v + ξ, w), ξ ∈ H˜ 2 (σ ), ξ(0)  0. It implies   vxx ξxx − gξ  0, σ

σ

hence, by (42), the inequality vxxx (0)ξ(0) − vxx (0)ξx (0)  0 follows which, in its own turn, implies vxxx (0)  0, vxx (0) = 0. The next step consists in substituting (u, ¯ v, ¯ w) ¯ = (u, v, w) ± (ψ, ξ, ϕ) as test functions in (40), where (ψ, ξ, ϕ) ∈ H, ψ(0)p1 + ξ(0)p2 = ϕ(0). This substitution provides the identity      ux ψx − hψ + vxx ξxx − gξ + BΩ (w, ϕ) − f ϕ = 0. σ

σ

σ

σ

Ω

Integrating by parts here, in view of (12), (49), (41)–(44), we derive  ν  t (w) , ϕ 3/2,Σ + ux ψ|10 − vxxx ξ |10 = 0

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375

or, by (51), kϕ(0) − ux (0)ψ(0) + vxxx (0)ξ(0) = 0.

(52)

This relation holds for all (ψ, ξ, ϕ) ∈ H, ψ(0)p1 + ξ(0)p2 = ϕ(0). Take ψ(0) = 0 in (52). In this case ξ(0)p2 = ϕ(0). It gives kϕ(0) + vxxx (0)ξ(0) = 0, hence kp2 = −vxxx (0).

(53)

If we take ξ(0) = 0, i.e. ψ(0)p1 = ϕ(0), the equality (52) implies k=

1 ux (0). p1

(54)

Relations (53), (54) prove the second equality (46). Moreover, by (51), we have ν  1 t (w) = ux (0)δΣ . p1 Now substitute (u, ¯ v, ¯ w) ¯ = (u, v, w) + (ψ, ξ, ϕ) as test functions in (40), (ψ, ξ, ϕ) ∈ P . It implies the inequality      ux ψx − hψ + vxx ξxx − gξ + BΩ (w, ϕ) − f ϕ  0 (55) σ

σ

σ

σ

Ω

which, by (12), (49), (41)–(44), can be rewritten as   ν t (w) , ϕ 3/2,Σ − ux (0)ψ(0) + vxxx (0)ξ(0)  0,

∀(ψ, ξ, ϕ) ∈ P .

(56)

Inequality (56) together with (49) and (58) below contains a full information on boundary conditions (45)–(48). Substitutions of (u, ¯ v, ¯ w) ¯ = (0, 0, 0), (u, ¯ v, ¯ w) ¯ = 2(u, v, w) in (40) yield kw(0) − ux (0)u(0) + vxxx (0)v(0) = 0.

(57)

The constant k can be taken from (54) and substituted in (57). This gives ux (0)w(0) − p1 ux (0)u(0) + p1 vxxx (0)v(0) = 0. Meanwhile, we have ux (0)p2 = −vxxx (0)p1 , whence, ux (0)(w(0) − u(0)p1 − v(0)p2 ) = 0

(58)

which coincides with the first relation of (47). To summarize, we see that all relations (41)–(48) are derived from (39), (40). In fact, we can give one more differential formulation of the problem (39), (40) equivalent to (41)–(48). In this case instead of Ω0 the smooth domain Ω is used. Namely, we have to find functions u(x), v(x), w(y), x ∈ σ, y = (y1 , y2 ) ∈ Ω, such that −uxx = h

in σ,

(59)

vxxxx = g

in σ, 1 2 w = f + ux (0)δ0 p1 w = wq = 0 on Γ,

(60) in Ω,

u = v = vx = 0 at x = 1,

(61) (62) vxx = 0

at x = 0,

ux (0)p2 = −vxxx (0)p1 , u(0)p1 + v(0)p2  w(0),

ux (0)  0, ux (0) w(0) − u(0)p1 − v(0)p2 = 0, where δ0 is the Dirac measure, i.e., δ0 (ξ ) = ξ(0), ξ ∈ C0∞ (Ω). To derive (61) we can take ψ(0)p1 + ξ(0)p2 = ϕ(0) in (55), (ψ, ξ, ϕ) ∈ H . This implies

(63) (64) (65)

376

A. Khludnev, A. Tani / European Journal of Mechanics A/Solids 27 (2008) 365–377



 ux ψx − σ

 hψ +

σ

 vxx ξxx −

σ

 gξ + BΩ (w, ϕ) −

σ

f ϕ = 0.

(66)

Ω

Integrating by parts in (66) we obtain  BΩ (w, ϕ) − f ϕ − ux (0)ψ(0) + vxxx (0)ξ(0) = 0.

(67)

Ω

Since ψ(0) =

ϕ(0) − ξ(0)p2 , p1

vxxx (0) = −

we arrive at the relation  ux (0)ϕ(0) BΩ (w, ϕ) − f ϕ = , p1

ux (0)p2 , p1

∀ϕ ∈ C0∞ (Ω)

Ω

which means a fulfillment of (61). On the other hand, variational inequality (39)–(40) can be derived from (59)–(65) as well as from (41)–(48). To simplify our considerations we did not consider any parameters in the model (41)–(48), but in practice the model should include a number of parameters. Like in the previous section we consider a passage to the limit when a rigidity of a contacting body goes to infinity. To specify a case, assume that a rigidity of the inclined beam is increasing. In this case instead of (59), (60) the following equilibrium equations are assumed to be fulfilled 1 1 vxxxx = g in σ − uxx = h, δ δ with a positive parameter δ. We are interesting in a passage to the limit as δ → 0. First, consider the variational formulation of this problem. We have to find functions uδ , v δ , w δ such that (uδ , v δ , w δ ) ∈ P ,     1 δ 1 δ δ ux (u¯ x − uδx ) − h(u¯ − uδ ) + vxx (v¯xx − vxx ) − g(v¯ − v δ ) δ δ σ σ  ¯ v, ¯ w) ¯ ∈ P. + BΩ (w δ , w¯ − w δ ) − f (w¯ − w δ )  0, ∀(u,

(68)

(69)

Ω

Taking in (69) test functions of the form (u, ¯ v, ¯ w) ¯ = (0, 0, 0), (u, ¯ v, ¯ w) ¯ = 2(uδ , v δ , w δ ) we derive      1 1 δ 2 (uδx )2 − huδ + (vxx ) − gv δ + BΩ (w δ , w δ ) − f w δ = 0. δ δ σ

σ

σ

σ

Ω

Hence the following estimate holds 1 δ 2 1 u H˜ 1 (σ ) + v δ 2H˜ 2 (σ ) + w δ 2H 2 (Ω)  c δ δ 0 with a constant c independent of δ. We can assume that as δ → 0 the following convergence takes place wδ → w0

weakly in H02 (Ω), v δ → 0 strongly in H˜ 2 (σ ), uδ → 0 strongly in H˜ 1 (σ ). Note that the limit function w 0 satisfies the condition w 0 (0)  0. Let us take test functions of the form (0, 0, w) ¯ in (69), w¯ ∈ H02 (Ω), w(0) ¯  0. We obtain

(70) (71) (72)

A. Khludnev, A. Tani / European Journal of Mechanics A/Solids 27 (2008) 365–377

 BΩ (w , w¯ − w ) − δ

δ

1 f (w¯ − w )  δ

Ω



 (uδx )2

δ

σ

−

1 hu + δ

σ



 δ 2 (vxx )

δ

377

σ

−

gv δ . σ

Passing to the lower limit here we derive a variational inequality for finding the function w 0 . Namely, w 0 ∈ H02 (Ω),

w 0 (0)  0,  BΩ (w 0 , w¯ − w 0 ) − f (w¯ − w 0 )  0,

(73) ∀w¯ ∈ H02 (Ω), w(0) ¯  0.

(74)

Ω

The problem (73), (74) describes a contact of the plate with a rigid punch at the point y = 0. It is possible to improve (70)–(72). Namely, the following convergence takes place w δ → w 0 strongly in H02 (Ω), 1 √ v δ → 0 strongly in H˜ 2 (σ ), δ 1 δ √ u → 0 strongly in H˜ 1 (σ ). δ We omit the details. Acknowledgement This work was completed during the visit in 2006 of the first author to Keio University whose support is appreciated very much. It was also supported by the Russian Fund for Basic Research (06-01-00209). References Caffarelli, L.A., Friedman, A., 1979. The obstacle problem for the biharmonic operator. Ann. Sc. Norm. Super. Pisa Ser. IV 6, 151–184. Caffarelli, L.A., Friedman, A., Torelli, A., 1982. The two-obstacle problem for the biharmonic operator. Pacific J. Math. 103 (3), 325–335. Dal Maso, G., Paderni, G., 1988. Variational inequalities for the biharmonic operator with varying obstacles. Ann. Mat. Pura Appl. 153, 203–227. Eck, Ch., Jarusek, J., Krbec, M., 2005. Unilateral Contact Problems. Variational Methods and Existence Theorems. Chapman and Hall/CRC, Boca Raton, Fl. Fichera, G., 1972. Boundary value problems of elasticity with unilateral constraints. In: Handbuch der Physik, Band 6a/2. Springer-Verlag, Berlin. Grisvard, P., 1985. Elliptic Problems in Nonsmooth Domains. Pitman, Boston. Khludnev, A.M., Kovtunenko, V.A., 2000. Analysis of Cracks in Solids. WIT Press, Southampton. Khludnev, A.M., Sokolowski, J., 1997. Modelling and Control in Solid Mechanics. Birkhäuser, Basel. Khludnev, A.M., Sokolowski, J., 2000. Griffith formulae for elasticity systems with unilateral conditions in domains with cracks. Eur. J. Mech. A Solids 19, 105–119. Khludnev, A.M., Hoffmann, K.-H., Botkin, N.D., 2006. The variational contact problem for elastic objects of different dimensions. Siberian Mat. J. 47 (3), 584–593. Leblond, J.B., 2000. Basic results for elastic fracture mechanics with frictionless contact between crack lips. Eur. J. Mech. A Solids 19, 633–647. Schild, B., 1986. On the coincidence set in biharmonic variational inequalities with thin obstacles. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 13 (4), 559–616. Temam, R., 1983. Problemes mathematiques en plasticite. Gauthier-Villars, Paris.