Abstract Generated abstract
The paper treats the displacement boundary-value problem for the Lamé system in a homogeneous isotropic elastic layer with prescribed displacements on the two bounding planes. Using invariance under the group of plane-parallel motions, the solution is expanded in minimal invariant subspaces expressed through cylindrical harmonics and Bessel functions, reducing the partial differential system to a system of ordinary differential equations in the transverse coordinate. This system is solved by a residue method, after which the boundary data determine the remaining constants through algebraic equations involving Fourier-Bessel transforms. Under stated regularity, periodicity, and integrability assumptions on the boundary functions, the resulting series-integral representation is asserted to converge and provides the required solution.
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MATHEMATICAL PHYSICS
M. D. MARTYNENKO
THE SECOND BOUNDARY-VALUE PROBLEM OF THE THEORY OF ELASTICITY FOR A HOMOGENEOUS ISOTROPIC LAYER
(Presented by Academician A. Yu. Ishlinskii, February 14, 1964)
In this paper, by the group method, the second boundary-value problem of the theory of elasticity (the displacement problem) is solved for an elastic homogeneous isotropic layer.
- The system of equilibrium equations of a homogeneous elastic isotropic body in displacements (the Lamé system) can be written in the following form:
\[ \Delta u(x)+\tau \partial \partial' u(x)=0, \tag{1} \]
where \(\tau=(\lambda+\mu)/\mu\), \(\lambda,\mu\) are elastic constants, \(\Delta\) is the Laplace operator,
\[ u(x)= \begin{pmatrix} u_1(x)\\ u_2(x)\\ u_3(x) \end{pmatrix}, \qquad \partial= \begin{pmatrix} \dfrac{\partial}{\partial x_1}\\[2mm] \dfrac{\partial}{\partial x_2}\\[2mm] \dfrac{\partial}{\partial x_3} \end{pmatrix}, \qquad x= \begin{pmatrix} x_1\\ x_2\\ x_3 \end{pmatrix}, \]
and the prime denotes transposition.
For system (1) the following problem is posed:
Determine a solution of system (1), twice continuously differentiable inside the layer \(0<x_3<a\), and satisfying on the boundary of the layer the conditions
\[ \begin{aligned} u(x_1,x_2,x_3)\big|_{x_3=0}&=f^{(1)}(x_1,x_2),\\ u(x_1,x_2,x_3)\big|_{x_3=a}&=f^{(2)}(x_1,x_2). \end{aligned} \tag{2} \]
Here \(f^{(1)}\) and \(f^{(2)}\) are prescribed twice continuously differentiable bounded functions (columns) satisfying the conditions:
1) \(f^{(1)}(r,\varphi)\) and \(f^{(2)}(r,\varphi)\) are periodic functions of the angle with period \(2\pi\) \((r,\varphi\) are the polar coordinates of the point \((x_1,x_2))\).
2) \(\partial f^{(1)}/\partial\varphi\), \(\partial f^{(2)}/\partial\varphi\) are finite.
3) The integrals exist
\[ \int_{-\pi}^{\pi}\int_0^\infty \left|f^{(i)}(r,\varphi)\sqrt{r}\right|\,dr\,d\varphi<+\infty; \]
\[ \int_{-\pi}^{\pi}\int_0^\infty \left| \frac{\partial^2 f^{(i)}(r,\varphi)}{\partial \varphi^2}\sqrt{r} \right|\,dr\,d\varphi<+\infty, \qquad (i=1,2). \]
4) \(f^{(i)}(r,\varphi)\) \((i\le 2)\) in any finite interval \(a\le r\le b\), for any \(\varphi\), have a finite number of zeros and extrema.
- It is known that the equations of the theory of elasticity, and consequently their solutions as well, are invariant with respect to the group of plane-parallel motions. Therefore it is natural to seek the solution among elements of minimal invariant subspaces. The elements of a minimal invariant subspace for the group of plane-parallel motions are repre-
are represented as follows:
\[
u_{l,\lambda}(r,\varphi,x_3)=X_1(x_3)J_{l+1}(\lambda r)e^{i(l+1)\varphi}(e_1-ie_2)+
\]
\[
+X_2(x_3)J_{l-1}(\lambda r)e^{i(l-1)\varphi}(e_1+ie_2)+X_3(x_3)J_l(\lambda r)e^{il\varphi}e_3,
\tag{3}
\]
where
\[ e_1=\begin{pmatrix}1\\0\\1\end{pmatrix},\qquad e_2=\begin{pmatrix}0\\1\\0\end{pmatrix},\qquad e_3=\begin{pmatrix}0\\0\\1\end{pmatrix}, \]
\(J_l(r)\) is the Bessel function of order \(l\); \((r,\varphi,x_3)\) are the cylindrical coordinates of the point \(x\); \(X_i(x_3)\) \((i\leqslant 3)\) are unknown functions, for the determination of which the following system of three ordinary differential equations of the second order is obtained:
\[
X_1''-\frac{2+\tau}{2}\lambda^2X_1+\frac{\tau}{2}\lambda^2X_2-\frac{\tau}{2}\lambda X_3'=0,
\]
\[
X_2''+\frac{\tau}{2}\lambda^2X_1-\frac{2+\tau}{2}\lambda^2X_2-\frac{\tau}{2}\lambda X_3'=0,
\tag{4}
\]
\[
(1+\tau)X_3''+\lambda\tau X_1'-\lambda\tau X_2'-\lambda^2X_3=0.
\]
This system is written in matrix form as
\[ \begin{pmatrix} 1&0&0\\ 0&1&0\\ 0&0&1+\tau \end{pmatrix}X'' + \begin{pmatrix} 0&0&-\dfrac{\tau}{2}\lambda\\ 0&0&-\dfrac{\tau}{2}\lambda\\ \lambda\tau&-\lambda\tau&0 \end{pmatrix}X' + \begin{pmatrix} \dfrac{2+\tau}{2}\lambda^2&\dfrac{\tau}{2}\lambda^2&0\\ \dfrac{\tau}{2}\lambda^2&-\dfrac{2+\tau}{2}\lambda^2&0\\ 0&0&-\lambda^2 \end{pmatrix}X=0, \tag{4*} \]
where
\[ X(x_3)= \begin{pmatrix} X_1(x_3)\\ X_2(x_3)\\ X_3(x_3) \end{pmatrix}. \]
Solving system \((4^*)\) by the residue method, we obtain
\[
X(x_3,\lambda)=\frac{1}{\lambda}e^{\lambda x_3}
\left\{
E+\frac{\tau}{4}
\begin{pmatrix}
3&1&0\\
1&3&0\\
0&0&2
\end{pmatrix}
+
\frac{\tau\lambda x_3}{4}
\begin{pmatrix}
1&-1&1\\
-1&1&-1\\
-2&2&-2
\end{pmatrix}
\right\}C^{(1)}-
\]
\[
-\frac{1}{\lambda}e^{-\lambda x_3}
\left\{
E+\frac{\tau}{4}
\begin{pmatrix}
3&1&0\\
1&3&0\\
0&0&2
\end{pmatrix}
-
\frac{\tau\lambda x_3}{4}
\begin{pmatrix}
1&-1&-1\\
-1&1&1\\
2&-2&-2
\end{pmatrix}
\right\}C^{(2)},
\tag{5}
\]
where
\[ C^{(i)}= \begin{pmatrix} C_1^{(i)}\\ C_2^{(i)}\\ C_3^{(i)} \end{pmatrix} \quad (i\leqslant 2), \qquad C^{(1)},\ C^{(2)} \text{ are constant column matrices.} \]
Then (3) is written in the following form:
\[ u_{l,\lambda}(r,\varphi,x_3) = e^{il\varphi} \begin{pmatrix} e^{i\varphi}&e^{-i\varphi}&0\\ -ie^{i\varphi}&ie^{-i\varphi}&0\\ 0&0&1 \end{pmatrix} \begin{pmatrix} J_{l+1}(\lambda r)&0&0\\ 0&J_{l-1}(\lambda r)&0\\ 0&0&J_l(\lambda r) \end{pmatrix} X(x_3,\lambda). \tag{6} \]
- Representing the solution of the required problem in the form
\[ u(r,\varphi,x_3)=\sum_{l=-\infty}^{\infty}\int_0^\infty u_{l,\lambda}(r,\varphi,x_3)\,d\lambda, \tag{7} \]
we obtain, for determining the unknown matrices \(C^{(1)}\) and \(C^{(2)}\), the following system of algebraic equations:
\[ A(0,\lambda)C^{(1)}+B(0,\lambda)C^{(2)}=\psi_l^{(1)}(\lambda), \]
\[ A(a,\lambda)C^{(1)}+B(a,\lambda)C^{(2)}=\psi_l^{(2)}(\lambda), \]
where
\[ A(x_3,\lambda)=\frac{1}{\lambda}e^{\lambda x_3} \left\{ E+\frac{\tau}{4} \begin{pmatrix} 3&1&0\\ 1&3&0\\ 0&0&2 \end{pmatrix} +\frac{\tau\lambda x_3}{4} \begin{pmatrix} 1&-1&1\\ -1&1&-1\\ -2&2&-2 \end{pmatrix} \right\}, \]
\[ B(x_3,\lambda)=-\frac{1}{\lambda}e^{-\lambda x_3} \left\{ E+\frac{\tau}{4} \begin{pmatrix} 3&1&0\\ 1&3&0\\ 0&0&2 \end{pmatrix} +\frac{\tau\lambda x_3}{4} \begin{pmatrix} 1&-1&-1\\ -1&1&1\\ 2&-2&-2 \end{pmatrix} \right\}, \]
\[ \psi_l^{(1)}= \begin{pmatrix} \dfrac{\tilde f_{1,l+1}^{(1)}(\lambda)+i\tilde f_{2,l+1}^{(1)}(\lambda)}{2}\\[1.2em] \dfrac{\tilde f_{1,l-1}^{(1)}(\lambda)-i\tilde f_{2,l-1}^{(1)}(\lambda)}{2}\\[1.2em] \dfrac{\tilde f_{3,l}^{(1)}(\lambda)}{2} \end{pmatrix}, \]
\[ \psi_l^{(2)}= \begin{pmatrix} \dfrac{\tilde f_{1,l+1}^{(2)}(\lambda)+i\tilde f_{2,l+1}^{(2)}(\lambda)}{2}\\[1.2em] \dfrac{\tilde f_{1,l-1}^{(2)}(\lambda)-i\tilde f_{2,l-1}^{(2)}(\lambda)}{2}\\[1.2em] \dfrac{\tilde f_{3,l}^{(2)}(\lambda)}{2} \end{pmatrix}, \]
\[ \tilde f_l^{(i)}(\lambda)= \begin{pmatrix} \tilde f_{1,l}^{(i)}\\ \tilde f_{2,l}^{(i)}\\ \tilde f_{3,l}^{(i)} \end{pmatrix}_{\lambda} = \int_0^\infty \left\{ \int_0^{2\pi} e^{-il\theta} f^{(i)}(r,\theta)\,d\theta \right\} J_l(\lambda r)\,r\,dr \qquad (i=1,2). \]
Having determined the constant matrices \(C^{(1)}, C^{(2)}\) (columns) and substituting them into (7), we obtain
\[ u(r,\varphi,x_3)= \sum_{l=-\infty}^{\infty} \begin{pmatrix} e^{i\varphi}&e^{-i\varphi}&0\\ -ie^{i\varphi}&ie^{-i\varphi}&0\\ 0&0&1 \end{pmatrix} e^{il\varphi} \int_0^\infty \begin{pmatrix} J_{l+1}(\lambda r)&0&0\\ 0&J_{l-1}(\lambda r)&0\\ 0&0&J_l(\lambda r) \end{pmatrix} \begin{pmatrix} X_1(x_3,\lambda)\\ X_2(x_3,\lambda)\\ X_3(x_3,\lambda) \end{pmatrix} \,d\lambda . \]
The resulting series converges under the restrictions imposed on the functions \(f^{(1)}(r,\varphi)\), \(f^{(2)}(r,\varphi)\).
Lviv State University
named after Ivan Franko
Received
10 I 1964