A SOLID-FLUID MODEL WITH UNILATERAL CONTACT CONDITIONS

G. Chichua

I.Vekua Institute of Applied Mathematics I.Javakhishvili Tbilisi State University

Let Ω = Ω^{s}∪Ω^{f} ∪Σ′′ be a bounded domain in ℝ^{3} with a Lipschitz boundary ∂\overline{Ω}, where Ω^{s} and Ω^{f} are the subdomains occupied by the elastic solid and viscous fluid respectively, with the Lipschitz interface Σ (see Fig.1).

[figure: Ω^{s}; Ω^{f}; n; n; n^{f}(n); n^{s}(n); N]

Σ′ = \{x ∈ Σ|[u_{i}N_{i}] = 0\}, u(x) := \{u_{i}\}_{i=1}^{3} = \{ \begin{matrix}u_{i}^{s} if x ∈ Ω^{s}, \\ u_{i}^{f} if x ∈ Ω^{f}, \end{matrix}

u_{i}^{s} and u_{i}^{f} denote components of the displacement vectors of the solid and fluid respectively, by the symbol [φ] we denote the jump of a function [φ] = = φ|_{f} − φ|_{s} on the interface Σ, Σ′′ ⊂ Σ′.

In Ω^{s} we have the equations of the linear theory of elasticity:

ρ^{s} \frac{∂^{2}u_{i}^{s}}{∂t^{2}} = \frac{∂σ_{ij}^{s}}{∂x_{j}} + f_{i}, x ∈ Ω^{s} t ∈ [0, T],  (1)

where

σ_{ij}^{s} = a_{ijkh}^{s}e_{kh}(u^{s}), i, j = 1, 2, 3, e_{kh}(u^{s}) := \frac{1}{2} (\frac{∂u_{k}^{s}}{∂x_{h}} + \frac{∂u_{h}^{s}}{∂x_{k}} ) , k, h = 1, 2, 3, (2)

σ_{ij}^{s} is the stress tensor, f is a given force and the Einstein summation convention has been used (the Latin indices run the values 1,2,3). The elastic coefficients a_{ijkh}^{s} are constants and they satisfy the conditions of symmetry and positive definiteness:

a_{ijkh}^{s} = a_{jikh}^{s} = a_{jihk}^{s} = a_{khij}^{s},  (3)
a_{ijkh}^{s}e_{ij}e_{kh} ≥ αe_{ij}e_{ij}, α > 0, e_{ij} = e_{ji}.  (4)

In Ω^{f} we have the following equations

ρ^{f} \frac{∂^{2}u_{i}^{f}}{∂t^{2}} = \frac{∂σ_{ij}^{f}}{∂x_{j}} + f_{i}, x ∈ Ω^{s}, t ∈ [0, T],  (5)

σ_{ij}^{f} = −δ_{ij}p + (ηδ_{ij}δ_{kh} + 2μδ_{ik}δ_{jh})e_{kh} (\frac{∂u^{f}}{∂t} ) ,  (6)

where σ_{ij}^{f} is the component of the stress tensor, η and μ are coefficients of viscosity satisfying the condition as follows:

μ > 0, η/μ > −(2/3)α; 0 < α < 1.

δ_{ij} is Kronecker delta.

The symbols ρ^{s} and ρ^{f} in (1) and (5) are respectively the densities of solid and fluid in the position of a rest. Let us assume that they are constants.

p is a perturbation of pressure with respect to the position of a rest. It is connected with the perturbation of the density ρ by the following expression:

p = c^{2}ρ,  (7)

where c > 0 is the velocity of sound spreading in the position of a rest of fluid.

Besides, ρ and v^{f} := \frac{∂u^{f}}{∂t} are connected with each other by the continuity equation, which in the linear case has the following form:

\frac{∂ρ}{∂t} + ρ^{f}divv^{f} = 0,

Hence taking into account, that ρ = 0 when u^{f} = 0, after integration we obtain:

ρ + ρ^{f}divu^{f} = 0.

Finally, the expresion (7) takes the following form:

p = −c^{2}ρ^{f}divu^{f}.  (8)

From expression (6) using (8) we get

σ_{ij}^{f} = δ_{ij}c^{2}ρ^{f}divu^{f} + (ηδ_{ij}δ_{kh} + 2μδ_{ik}δ_{jh})e_{kh} (\frac{∂u^{f}}{∂t} ) ,  (9)

Therefore σ_{ij}^{f} is a function of the vector u^{f} as well as of the velocity vector

\frac{∂u^{f}}{∂t} .

Let us consider the following boundary and initial conditions:

u = 0 on ∂Ω  (10)
u = \frac{∂u}{∂t} = 0 in Ω when t = 0.  (11)

On the interface Σ we set conditions as follows:

[u_{i}N_{i}] ≥ 0  on Σ;  (12)
σ_{ij}^{s}n_{j}^{s} = σ_{NN}^{s}N_{i}; σ_{ij}^{f}n_{j}^{f} = −σ_{NN}^{s}N_{i}, i = 1, 2, 3;  (13)
σ_{NN}^{s} ≤ 0  (14)
and σ_{NN}^{s} = 0 if [u_{i}N_{i}] > 0,  (15)

where (see Fig.1) N is a unit normal vector to the interface Σ with an external direction with respect to Ω^{s}, n is a unit outer normal vector to ∂(Ω̄\Σ), φ, σ_{NN} is the projection on the direction N of the stress vector acting on the surface Σ.

The relation (12) means that on the interface Σ the quantity of the projection u_{N}^{f} of the vector u^{f} on the direction N can not be less than u_{N}^{s}; the relations (13) denote that the inner force acting on Σ is normal to Σ and the action is opposite to the reaction, (14) means that there is compression (but not traction!) on Ω^{s} through Σ; (15) means that if the crack is open at a point, the acting force is zero at this point.

Let us introduce the following spaces:

V := \{u|u_{i} ∈ H^{1}(Ω^{s} ∪ Ω^{f} ∪ Σ′′); u_{i}|_{∂\overline{Ω}} = 0\}  (16)

and

K := {u|u_{i} ∈ V; [u_{i}N_{i}] ≥ 0 a.e. on Σ},  (17)

where H^{1}(Ω^{s} ∪ Ω^{f} ∪ Σ′′) is the usual Sobolev space.

Theorem 1. K is the closed convex set.

Proof. Let us first prove that K is closed. Let v^{m} ∈ K, m = 1, 2, ..., and v^{m} → v_{,} if m → ∞, by the norm of the space V. Let us prove that v ∈ K. Obviously,

v_{N}^{m}|_{f} − v_{N}^{m}|_{s} ≥ 0  (18)

According to the trace theorem, we have

v_{N}^{m}|_{f} → v_{N}|_{f}, and v_{N}^{m}|_{s} → v_{N}|_{s},

so 0 ≤ v_{N}^{m}|_{f} − v_{N}^{m}|_{s} → v_{N}|_{f} − v_{N}|_{s}, that means, that v_{N}|_{f} − v_{N}|_{s} ≥ 0, hence v ∈ K.

Now let us prove the convexity of the set K. Let u, v ∈ K. Now, we shall prove that

αu + βv ∈ K, ∀α, β = const > 0, α + β = 1.

We have

u_{N}|_{f} − u_{N}|_{s} ≥ 0, and v_{N}|_{f} − v_{N}|_{s} ≥ 0

i.e.

(αu_{N})|_{f} − (αu_{N})|_{s} ≥ 0 and (βv_{N})|_{f} − (βv_{N})|_{s} ≥ 0,

from which we obtain that

(αu_{N} + βv_{N})|_{f} − (αu_{N} + βv_{N})|_{s} ≥ 0.

Hence

αu + βv ∈ K.

Let H := L^{2}(Ω).

Definition. A vector-function u is called a generalized solution of the problem (1)-(6), (10)-(15) , if u satisfies the following conditions

u, u′ ∈ L^{∞}(0, T; V), u′′ ∈ L^{∞}(0, T; H), u′ ∈ K, (ρu′′(t), v − u′(t)) + a(u(t), v − u′(t)) + b(u′(t), v − u′(t)) ≥ (f(t), v − u′(t)) ,

∀v ∈ K, ∀t ∈ [0, T],

u = 0 and u′ = 0 when t = 0,

where L^{∞}(0, T; V) := {f(t)|f(t) : [0, T] → V, is measurable and ess sup_{t∈[0,T]}∥f(t)∥_{v} =

∥f∥_{L^{∞}(0,T,V)} < ∞}, by the symbol (.,.) we denote the scalar product in H,

ρ = ρ(x) = \{ \begin{matrix}ρ^{s} if x ∈ Ω^{s}, \\ ρ^{f} if x ∈ Ω^{f}; \end{matrix}

\begin{matrix}a(u, w) = ∫_{Ω} a_{ijkh}(x)e_{kh}(u)e_{ij}(w)dx, \\ b(v, w) = ∫_{Ω} b_{ijkh}(x)e_{kh}(v)e_{ij}(w)dx; \end{matrix}

a_{ijkh}(x) = \{ \begin{matrix}a_{ijkh}^{s} if x ∈ Ω^{s}, \\ a_{ijkh}^{f} ≡ ρ^{f}c^{2}δ_{ij}δ_{kh} if x ∈ Ω^{f}; \end{matrix}

b_{ijkh}(x) = \{ \begin{matrix}b_{ijkh}^{s} = 0 if x ∈ Ω^{s}, \\ b_{ijkh}^{f} ≡ 2μδ_{ik}δ_{jh} + ηδ_{ij}δ_{kh} if x ∈ Ω^{f}. \end{matrix}

Theorem 2. The classical solution is also the generalized solution of the problem (1)-(6), (10)-(15).

Proof. Let u be the classical solution of problem (1)-(6), (10)-(15) and let v := {v_{i}} ∈ K. If we multiply both sides of equations (1) and (5) by

v_{i} = \{ \begin{matrix}v_{i}^{s} if x ∈ Ω^{s}, i = 1, 2, 3, \\ v_{i}^{f} if x ∈ Ω^{f}, \end{matrix}

and use the formula of integration by parts respectively on Ω^{s} and Ω^{f}, we get the relations as follows

∫_{Ω^{s}} ρ^{s} \frac{∂u_{i}^{s}}{∂t^{2}} v_{i}^{s}dx − ∫_{Σ} n_{j}σ_{ij}^{s}v_{i}^{s}dΣ = ∫_{Ω^{s}} f_{i}v_{i}^{s}dx − ∫_{Ω^{s}} σ_{ij}^{s} \frac{∂v_{i}^{s}}{∂x_{j}} dx  (19)

and

∫_{Ω^{f}} ρ^{f} \frac{∂u_{i}^{f}}{∂t^{2}} v_{i}^{f}dx − ∫_{Σ} n_{j}σ_{ij}^{f}v_{i}^{f}dΣ = ∫_{Ω^{f}} f_{i}v_{i}^{f}dx − ∫_{Ω^{f}} σ_{ij}^{f} \frac{∂v_{i}^{f}}{∂x_{j}} dx  (20)

Let us sum the relations (19) and (20), and take into account the interface conditions (13), then we obtain:

∫_{Ω} ρ\frac{∂^{2}u_{i}}{∂t^{2}} v_{i}dx + ∫_{Σ} σ_{NN}^{s}[v_{i}N_{i}]dΣ = ∫_{Ω} f_{i}v_{i}dx − ∫_{Ω} σ_{ij} \frac{∂v_{i}}{∂x_{j}} dx,  (21)

where ρ = \{ \begin{matrix}ρ^{s} if x ∈ Ω^{s}, \\ ρ^{f} if x ∈ Ω^{f}, \end{matrix} σ_{ij}(x) = \{ \begin{matrix}σ_{ij}^{f} if x ∈ Ω^{s}, \\ σ_{ij}^{s} if x ∈ Ω^{f}, \end{matrix} u_{i} = \{ \begin{matrix}u_{i}^{s} if x ∈ Ω^{s}, \\ u_{i}^{f} if x ∈ Ω^{f}, \end{matrix}

i = 1, 2.3.

From (21) and (17) we get

∫_{Ω} ρ\frac{∂^{2}u_{i}}{∂t^{2}} v_{i}dx + ∫_{Ω} σ_{ij} \frac{∂v_{i}}{∂x_{j}} dx ≥ ∫_{Ω} f_{i}v_{i}dx.

Substituting here v − u′ instead of v the proof is fulfilled.

The following theorems can also be proved:

Theorem 3. If the generalized solution is smooth enough then it will be also a classical solution of the problem (1)-(6), (10)-(15).

Theorem 4. There exists the unuque generalized solution of the problem (1)-(6), (10)-(15).

R e f e r e n c e s

[1] Sanches-Palencia E., Non-homogeneous Media and Vibration Theory. Lecture Notes in Physics 127. Springer-Verlag, New-York, 1980, (Russian).

[2] Lions J.L., Quelqes Methodes de Resolution de Problems aux Limites Non- lineares. Dunod, Gauthier-Villars, Paris, 1969, (Russian).