Preamble

In 1967, the study of boundary value problems for differential equations remained a central focus in mechanics, particularly in the theory of elasticity and shells. Following the foundational work in \cite{1}, we consider the complex-valued function $W(x, y)$ satisfying the equation:
$$\Delta \Delta W(x, y) - ik^2 \Delta W(x, y) = 0$$
where the boundary conditions are defined by functions $a(s)$ and $b(s)$ along the contour $L$. Specifically, the equilibrium of the system requires that:
$$\int_L [a(s) dx + b(s) dy] = 0$$
The function $W(x, y)$ is related to the physical displacement $w(x, y)$ and the stress function $\phi(x, y)$ of a shallow spherical shell. According to the theory established in \cite{1}, we define:
$$W(x, y) = w(x, y) + iC\phi(x, y)$$
where the constant $C$ depends on the shell geometry and material properties:
$$C = \frac{\sqrt{12(1 - \nu^2)}}{Eh^2}, \quad k^2 = \frac{\sqrt{12(1 - \nu^2)}}{Rh}$$
Here, $R$ is the radius of curvature, $h$ is the thickness, $E$ is Young's modulus, and $\nu$ is Poisson's ratio.

II. Stress and Displacement Relations

The internal forces and moments within the shell can be expressed through the derivatives of $W(x, y)$. Following the formulations in \cite{2, 3}, the stress components $\sigma_x, \sigma_y, \tau_{xy}$ and the displacements $u, v$ are integrated across the thickness $z$. We define the auxiliary functions $u_0, v_0$ and the rotation $\chi_0$ such that the boundary conditions on the contour $L$ can be represented in terms of the complex potential.

The relationship between the physical components and the complex function $W$ is given by:
$$\begin{aligned}
2\mu h C u_1(x, y) &= \text{Im} \left[ \frac{1+ \nu}{2\mu h C} \int ( \dots ) dx - ik^2 \int W dy \right] \
2\mu h C u_2(x, y) &= \text{Im} \left[ \dots \right]
\end{aligned}$$
where $\mu$ is the shear modulus. By introducing the constants $A, B_1, B_2$, we account for the rigid-body displacements of the shell. The boundary conditions on the arc length $s$ ($0 < s < l$) for the generalized displacement and force functions $a(s)$ and $b(s)$ are then derived.

III. Integral Representations and Kernels

To solve the governing equation (1), we utilize fundamental solutions and potential theory. Let $(\xi, \eta)$ be a point in the domain $D$. The fundamental solution involves terms of the form $\ln(r)$ and modified Bessel functions. Specifically, we define the kernels $N_1$ and $N_2$ as:
$$\begin{aligned}
N_1(u, v, \chi) &= -\frac{\partial}{\partial n} \ln r + \dots \
N_2(u, v, \chi) &= \dots
\end{aligned}$$
As shown in \cite{1}, the asymptotic behavior of these kernels near the singularity $r \to 0$ involves terms like $\ln(i\alpha r) {1 + O(\alpha^2 r^2)}$.

We represent the solution $W(x, y)$ as an integral over the boundary $L$ using density functions $p_0(s)$ and $q_0(s)$:
$$W(x, y) = \int_L [u^ p_0(s) - v^ q_0(s)] ds$$
where $u^$ and $v^$ are derived from the fundamental solutions of the operator $\Delta(\Delta - ik^2)$. The densities $p_0(s)$ and $q_0(s)$ are related to the physical boundary values through a system of singular integral equations.

The geometric kernels $G_1, G_2, G_3, G_4$ are introduced to simplify the expressions for the stresses and displacements. These kernels depend on the relative coordinates and the parameter $\alpha = \sqrt{i}k$. For instance:
$$G_1(\rho, t) = -\int \frac{\rho^2 \sqrt{1 - \rho^2} \cos \psi}{\dots} d\rho$$
These functions allow us to construct the influence coefficients $Q_{nm}$ and $T_{nm}$ that form the matrix of the integral equation system.

IV. System of Integral Equations

The boundary value problem is ultimately reduced to a system of four integral equations for the unknown densities $p_\nu(s)$ ($\nu = 1, 2, 3, 4$):
$$x_n p_n(s_0) + \int_L \sum_{m=1}^4 K_{nm}(s, s_0) p_m(s) ds = f_n(s_0)$$
where $x_n$ are constants (e.g., $x_1 = x_3 = 1+\nu$). The right-hand side functions $f_n(s_0)$ are determined by the prescribed boundary displacements and forces.

The kernels $K_{nm}$ contain both logarithmic singularities and regular parts. The existence and uniqueness of the solution $P(s)$ are analyzed using the Fredholm alternative and the properties of the operator in the space $C(L)$. For a shell with a fixed boundary, the system can be written in operator form as:
$$(I + \lambda^2 B) P = F$$
where $B$ is a compact operator. As demonstrated in \cite{4, 5}, for sufficiently small $\lambda$, the solution can be obtained via successive approximations or by direct inversion of the singular part. The condition for the existence of a single-valued displacement field is ensured by the integral constraint:
$$\int_L [p_1(s) x'(s) + p_4(s) y'(s)] ds = 0$$
which corresponds to the global equilibrium of the shell segment.

References

1. Bakushinsky, S. M., & Isakov, M. A. (1965). Journal of Computational Mathematics and Mathematical Physics, 5(2), 219–226.
2. Timoshenko, S. P., & Woinowsky-Krieger, S. (1963). Theory of Plates and Shells.
3. Muskhelishvili, N. I. (1935). Some Basic Problems of the Mathematical Theory of Elasticity.
4. Kantorovich, L. V., & Akilov, G. P. (1959). Functional Analysis in Normed Spaces.
5. Smirnov, V. I. (1957). A Course of Higher Mathematics, Vol. IV.