N. Chinchaladze

I. Vekua Institute of Applied Mathematics Tbilisi, Georgia

CYLINDRICAL BENDING OF SHELLS WITH CUSPED EDGES

The main purpose of this paper is to study the Initial Boundary Value Problems (IBVPs) for an elastic cusped prismatic shell (see [1], [2]), with the projection Ω = {(x_{1}, x_{2}) : −∞ < x_{1} < ∞, 0 ≤ x_{2} ≤ ℓ}.

The equation of the cylindrical bending of a prismatic shell has the form

(D(x_{2})w,_{22} (x_{2}, t))_{22} = q(x_{2}, t) − 2ρh(x_{2}) \frac{∂^{2}w(x_{2}, t)}{∂t^{2}} , 0 ≤ x_{2} ≤ ℓ, (1)

where w(x_{2}, t) is a deflection of the shell, q(x_{2}, t) is an intensivity of a load, ρ is a density, 2h(x_{2}) = h_{0}x_{2}^{α/3}(ℓ − x_{2})^{β/3}, h_{0}, α β = const > 0, is the thickness of the shell, D(x_{2}) is a flexural-rigidity (see [3]).

In the case under consideration, bending moment and interecting force have the following forms M_{2}(x_{2}, t) := −D(x_{2})w,_{22} (x_{2}, t), Q_{2}(x_{2}, t) := M_{2,2}(x_{2}, t).

At the points 0 and ℓ all above quantities are defined as the corresponding limits when x_{2} → 0_{+} and x_{2} → ℓ_{−}. On the cusped edge x_{2} = 0 (x_{2} = ℓ) admissible are only four different pairs of the boundary data (see [2]).

Let w(·, t) ∈ C^{4}(]0, ℓ[), w(x_{2}, ·) ∈ C^{1}(t ≥ 0)∩C^{2}(t > 0), w(x_{2}, t) ∈ C(t ≥ 0, 0 ≤ x_{2} ≤ ℓ).

\begin{matrix}w(·, t) \\ (w,_{2} (·, t)) \end{matrix} \begin{cases}∈ C([0, ℓ]) when 0 ≤ α, β < 2, (0 ≤ α, β < 1) \\ \{ \begin{matrix}∈ C([0, ℓ[), \\ = O(1), x_{2} → ℓ_{−}, \end{matrix} & \begin{matrix}when 0 ≤ α < 2, β ≥ 2, \\ (0 ≤ α < 1, β ≥ 1) \end{matrix} \\ \{ \begin{matrix}∈ C(]0, ℓ]), \\ = O(1), x_{2} → 0_{+}, \end{matrix} & \begin{matrix}when α ≥ 2, 0 ≤ β < 2, \\ (α ≥ 1, 0 ≤ β < 1) \end{matrix} \end{cases} (2)

We assume that initial datas are satisfying same boundary conditions, which we have for w, and they are from the class (2).

Equation (1) under usual initial and above mentioned admissible boundary conditions has a unique solution. The IBVPs can be reduced to the integro-differential equations with symmetric kernel, which we solve by the Fourier method. The convergence of appropriate series are proved.

References

1. G. Jaiani, Elastic bodies with non-smooth boundaries-cusped plates and shells. Zeitschrift fuer Angewandte Mathematik und Mechanik 76(1996), Supplement 2, 117–120.

2. N. Chinchaladze, 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 10(1995), No. 1, 21–23.

3. S. Timoshenko, S. Woinowsky-Krieger, Theory of plates and shells. Mcgrawhill Book Company, INC, New York–Toronto–London, 1959.