\underline{30  Bulletin of TICMI}  +

ON A CYLINDRICAL BENDING OF A PRISMATIC SHELL WITH TWO CUSPED EDGES UNDER ACTION OF AN IDEAL FLUID

N. Chinchaladze, G. Jaiani

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

1. Some General Remarks on Solid-Fluid Transmission Conditions (G.Jaiani).

In the three-dimensional Solid-Fluid models usually the following transmission conditions are used [1]:

[u] := u^{s} − u^{f} = 0, [σ_{ij}n_{j}] := σ_{ij}^{s}n_{j} − σ_{ij}^{f}n_{j} = 0,  (1)

where u^{s}, σ_{ij}^{s} and u^{f}, σ_{ij}^{f} are displacement vector and stress tensor in the solid and fluid parts correspondingly, n is outward normal of ∂(Ω\I), Ω = Ω^{s} ∪ Ω^{f}, I is interface between solid part Ω^{s} and fluid part Ω^{f}.

The Solid-Fluid model with unilateral transmission conditions as follows:

[u_{i}N_{i}] ≥ 0, σ_{ij}^{f}n_{j} = σ_{NN}^{f}N_{i},  (2)
σ_{ij}^{s}n_{j} = −σ_{NN}^{f}N_{i},  (3)
σ_{NN}^{f} ≤ 0,  (4)
and σ_{NN}^{f} = 0 if [u_{i}N_{i}] > 0  (5)

has been investigated in [2]. (2) and (3) mean that stresses on interface I are normal (tangential components are equal to zero, σ_{NN} is a normal component, N is an outward normal to Ω^{f} on I) and balancing each other, (4) means that we have compression, (5) means that if crack between solid and fluid parts is open than at that point normal force equals zero.

Our aim is to construct transmission conditions between three dimensional fluid part, and solid part of plate or shell type, e.g. based on the classical bending theory of plates. Since in this case all mechanical quantities in solid part can be calculated by deflection, as transmission conditions can be used either (1) or (2)-(5). Another possibility of transmission conditions. If middle plane of the plate lies in the plane 0x_{1}x_{2} and flow of moving fluid is parallel to 0x_{3}, it could also have the form:

\begin{matrix}σ_{33}^{f} (x_{1}, x_{2}, h^{(+)}(x_{1}, x_{2}), t) − σ_{33}^{f} (x_{1}, x_{2}, h^{(−)}(x_{1}, x_{2}), t) = q(x_{1}, x_{2}, t), \\ v_{3} (x_{1} − h^{(+)}(x_{1}, x_{2})w,_{1} (x_{1}, x_{2}, t), x_{2} − h^{(+)}(x_{1}, x_{2})w,_{2} (x_{1}, x_{2}, t), h^{(+)}(x_{1}, x_{2})+ \end{matrix}

+w(x_{1}, x_{2}, t), t) = v_{3} (x_{1} − h^{(−)}(x_{1}, x_{2})w,_{1} (x_{1}, x_{2}, t), x_{2}−  (6)

− h^{(−)}(x_{1}, x_{2})w,_{2} (x_{1}, x_{2}, t), h^{(−)}(x_{1}, x_{2}) − w(x_{1}, x_{2}, t), t) = \frac{∂w(x_{1}, x_{2}, t)}{∂t},

+  \underline{Volume 2, 1998  31}

+ (the first of the last pair of equalities is valid since deflection of plate w is independent of x_{3}), where (x_{1}, x_{2}) ∈ Ω, Ω is projection of plate before deformation, q(x_{1}, x_{2}, t) is intensivity of lateral load, v := (v_{1}, v_{2}, v_{3}) is a velocity vector of fluid (compare (6) with the transmission conditions considered in [3], where motion of fluid in elastic pipe has been investigated). On ∂Ω admissible boundary conditions of plate theory should be given.

2. Title Problem (N. Chinchaladze).

Let the projection on the plane 0x_{1}x_{2} of plate under consideration be

Ω = {(x_{1}; x_{2}) : −∞ < x_{1} < +∞; 0 ≤ x_{2} ≤ b}.

Let us consider the interface problem of the interaction of the plate, whose flexural rigidity is given by the equation

D = D_{0}x_{2}^{α}(b − x_{2})^{β}, D_{0}, α, β = const > 0,

caused by the thickness

2h(x_{2}) = h_{0}x_{2}^{α/3}(b − x_{2})^{β/3}, h_{0} = const > 0,

and of a flow of the incompressible ideal fluid.

Let us consider the case of the flow of the fluid independent of x_{1} and parallel to 0x_{3} (directed upwards), with the following conditions at infinity

v_{3}(x_{2}, x_{3}, t)|_{x_{2}^{2}+x_{3}^{2}→∞} = v_{3∞}(t), v_{2}(x_{2}, x_{3}, t)|_{x_{2}^{2}+x_{3}^{2}→∞} = 0,

p(x_{2}, x_{3}, t)|_{x_{2}^{2}+x_{3}^{2}→∞} = p_{∞}(t).

The transmission conditions in a view of smallness of deflection and of smallness of plate’s thickness, taking into account (6), are

v_{3}(x_{2}, 0, t) = \frac{∂w(x_{2}, t)}{∂t},  (7)

−p(x_{2}, h^{(+)}(x_{2}), t) + p(x_{2}, h^{(−)}(x_{2}), t) = q(x_{2}, t).  (8)

In case of the potential motion of the flow there exists a complex function Φ = ψ+iφ, where

\frac{∂φ}{∂x_{2}} = \frac{∂ψ}{∂x_{3}} = v_{2}, \frac{∂φ}{∂x_{3}} = −\frac{∂ψ}{∂x_{2}} = v_{3}.  (9)

Pressure is given by the equation

p(x_{2}, x_{3}, t) = ρ^{f} [\frac{v_{∞}^{2}}{2} + \frac{p_{∞}}{ρ} + \frac{∂φ_{∞}}{∂t} − \frac{∂φ}{∂t} − \frac{1}{2}(v_{2}^{2} + v_{3}^{2})] .  (10)

+

In case under consideration for w(x_{2}, t), in virtue of (8), we get the following differential equation [4]

\begin{matrix}\frac{d^{2}}{dx_{2}^{2}} [x_{2}^{α}(b − x_{2})^{β}w′′_{x_{2}}(x_{2}, t)] = \frac{2ρ^{s}h}{D_{0}} \frac{∂^{2}w(x_{2}, t)}{∂t^{2}}+ \\ +\frac{−p (x_{2}, \frac{1}{2}h_{0}x_{2}^{α/3}(b − x_{2})^{β/3}, t) + p (x_{2}, −\frac{1}{2}h_{0}x_{2}^{α/3}(b − x_{2})^{β/3}, t)}{D_{0}}. \end{matrix}  (11)

For Φ′_{x_{2}}(x_{2}, x_{3}, t) = −v_{3} +iv_{2}, in view of (7), we get the following expression (see[5])

Φ′_{x_{2}} = −\frac{1}{πi\sqrt{(x_{2} + ix_{3})(x_{2} + ix_{3} − b)}} ∫_{x_{0}}^{x_{2}} \frac{\sqrt{(τ_{2} + ix_{3})(τ_{2} + ix_{3} − b)}}{(τ_{2} − x_{2}) − ix_{3}} \frac{∂w(τ_{2}, t)}{∂t}dτ_{2}+

+v^{3∞} \frac{x_{2} + ix_{3} − b/2}{\sqrt{(x_{2} + ix_{3})(x_{2} + ix_{3} − b)}}.  (12)

Let

w(x_{2}, t) = e^{iωt}w_{0}(x_{2}), p(x_{2}, x_{3}, t) = e^{iωt}p_{0}(x_{2}, x_{3}), ψ(x_{2}, x_{3}, t) = e^{iωt}ψ_{0}(x_{2}, x_{3}),

v_{2}(x_{2}, x_{3}, t) = e^{iωt}v_{2}^{0}(x_{2}, x_{3}), v_{3}(x_{2}, x_{3}, t) = e^{iωt}v_{3}^{0}(x_{2}, x_{3}).  (13)

If velocity of the flow is too small, then, in virtue of (10), p can be represented in the form (compare [3], p. 14)

p(x_{2}, x_{3}, t) = ρ^{f} [\frac{p_{∞}}{ρ} + \frac{∂φ_{∞}}{∂t} − \frac{∂φ}{∂t}] .  (14)

From (12), we have expression for v_{3}. By means of the latter, in view of (9), we can calculate φ which we have to substitute in (14).

Finally, taking into account (14), and (13) from (11) after four times integration, for w_{0}(x_{2}) we get the following integral equation

w_{0}(x_{2}) + \frac{ω^{2}}{D_{0}π} ∫_{x_{2}^{0}}^{x_{2}}K(τ_{2}, x_{2})w_{0}(τ_{2})dτ_{2} = f(x_{2}),  (15)

where

\begin{matrix}f(x_{2}) = c_{3}(x_{2} − x_{2}^{0}) + c_{4} + ∫_{x_{2}^{0}}^{x_{2}}\frac{(c_{1} + c_{2})y(x_{2} − y)}{y^{α}(b − y)^{β}}dy− \\ −\frac{ω^{2}ρ^{f}}{πD_{0}} ∫_{x_{2}^{0}}^{x_{2}}\frac{(x_{2} − y)dy}{y^{α}(b − y)^{β}} ∫_{x_{2}^{0}}^{y}dτ ∫_{− h^{(+)}(τ)}^{h^{(+)}(τ)} \{(τ − b/2)cos\frac{ϕ(τ, τ_{3})}{2} + τ_{3}sin\frac{ϕ(τ, τ_{3})}{2}\} \frac{v_{3∞}^{0}dτ_{3}}{\sqrt{r(τ, τ_{3})}}, \end{matrix}

+

\begin{matrix}K(τ_{2}, x_{2}) = ∫_{τ_{2}}^{x_{2}}\frac{x_{2} − y}{y^{α}(b − y)^{β}} \{−2(y − τ_{2})πρ^{s}h(τ_{2}) + ρ^{f} ∫_{x_{2}^{0}}^{τ_{2}}(y − τ)dτ ∫_{− h^{(+)}(τ)}^{h^{(+)}(τ)} \begin{matrix}√ & \overline{r(τ_{2}, τ_{3})} \\ √ & \overline{r(τ, τ_{3})} \end{matrix}× \\ ×\frac{(τ − τ_{2})cos\{(ϕ(τ_{2}, τ_{3}) − ϕ(τ, τ_{3}))/2\} + τ_{3}sin\{(ϕ(τ_{2}, τ_{3}) − ϕ(τ, τ_{3}))/2\}}{(τ_{2} − τ)^{2} + τ_{3}^{2}}dτ_{3}\} dy, \end{matrix}

cosϕ(x_{2}, x_{3}) = (x_{2}^{2} − x_{3}^{2} − bx_{2})/r(x_{2}, x_{3}),
r(x_{2}, x_{3}) = \sqrt{(x_{2}^{2} − x_{3}^{2} − bx_{2})^{2} + (2x_{2} − b)x_{3}^{2}}.

The integral equation (15) can be solved by method of successive approximations, and then constants c_{i} (i = 1, ..., 4) will be defined from the admissible boundary value conditions [4]:

1.1. w_{0}(0) = \frac{dw_{0}(0)}{dy} = w_{0}(b) = \frac{dw_{0}(b)}{dy} = 0, when 0 < α, β < 1;

1.2. w_{0}(0) = \frac{dw_{0}(0)}{dy} = \frac{dw_{0}(b)}{dy} = Q_{2}(b) = 0, when 0 < α, β < 1;

1.3. w_{0}(0) = \frac{dw_{0}(0)}{dy} = w_{0}(b) = M_{2}(b) = 0, when 0 < α < 1, 0 < β < 2;

1.4. w_{0}(0) = \frac{dw_{0}(0)}{dy} = Q_{2}(b) = M_{2}(b) = 0, when 0 < α < 1, β > 0;

2.1. \frac{dw_{0}(0)}{dy} = Q_{2}(0) = w_{0}(b) = \frac{dw_{0}(b)}{dy} = 0, when 0 < α, β < 1;

2.2. \frac{dw_{0}(0)}{dy} = Q_{2}(0) = w_{0}(b) = M_{2}(b) = 0, when 0 < α < 1, 0 < β < 2;

3.1. w_{0}(0) = M_{2}(0) = w_{0}(b) = \frac{dw_{0}(b)}{dy} = 0, when 0 < α < 2, 0 < β < 1;

3.2. w_{0}(0) = M_{2}(0) = \frac{dw_{0}(0)}{dy} = Q_{2}(b) = 0, when 0 < α < 2, 0 < β < 1;

3.3. w_{0}(0) = M_{2}(0) = w_{0}(b) = M_{2}(b) = 0 when 0 < α, β < 2;

4.1. M_{2}(0) = Q_{2}(0) = w_{0}(b) = \frac{dw_{0}(b)}{dy} = 0, when α > 0, 0 < β < 1,

where M_{2} is the bending moment, Q_{2} is the intersecting force.

In the stationary case (i.e. \frac{∂w}{∂t} = 0) symmetrical prismatic shell ( h^{(+)}(x_{2}) = = − h^{(−)}(x_{2}))

acts as absolutely rigid (q(x_{2}) ≡ 0).

R e f e r e n c e s

[1] Sanches-Palencia E., Non-homogenous Media and Vibration Theory. Springer-Verlag, 1980.

[2] Chichua G., On a Boundary-Contact Problem for a Solid-Fluid Model. Reports of Enlarged Session of the Seminar of I. Vekua Institute of Applied Mathematics, 1995, v.10, N\underline{0} 1, pp.18-20.

[3] Wollmir A., Shells on the Flow of Fluid and Gas. Problems of Hydro-elasticity. Moscow, 1981 (in Russian).

[4] Chinchaladze N., Cylindrical Bending of the Prismatic Shell with Two Sharp Edges in Case of a Strip. Reports of Enlarged Session of the Seminar of I.Vekua Institute of Applied Mathematics, 1995, v. 10, N\underline{0} 1, pp.21-23.

[5] Muskhelishvili N., Singular Integral Equations. Noordhoff, 1953.

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