Reports of the Academy of Sciences of the USSR
1958. Vol. 118, No. 4

THEORY OF ELASTICITY

Yu. A. Dem’yanov

SELF-SIMILAR PROBLEMS OF DYNAMIC BENDING OF PLATES

(Presented by Academician L. I. Sedov on 12 VIII 1957)

The fundamental equation of bending of plates has the form \((^{1})\)

\[ \frac{\partial^{2}M_{1}}{\partial x^{2}} +2\frac{\partial^{2}M_{12}}{\partial x\,\partial y} +\frac{\partial^{2}M_{2}}{\partial y^{2}} -\mu\frac{\partial^{2}w}{\partial t^{2}}=0. \tag{1} \]

Here \(w(t,x,y)\) is the displacement perpendicular to the initial position of the plate; \(\mu\) is the mass of a cylinder with base \(dx\,dy\) on the middle surface; \(M_{1}, M_{12}, M_{2}\) are bending moments, which in the general case are certain functions of the curvatures
\(\kappa_{1}=\partial^{2}w/\partial x^{2}\),
\(\kappa_{12}=\partial^{2}w/\partial x\,\partial y\),
\(\kappa_{2}=\partial^{2}w/\partial y^{2}\), determined by the properties of the plate material.

Let us consider a class of self-similar solutions of equation (1) depending on two variables:
\(\xi=x/\sqrt{t}\), \(\eta=y/\sqrt{t}\), i.e.,
\(w=t\Phi_{0}(\xi,\eta)\). Since in this case
\(\kappa_{1}=\partial^{2}\Phi_{0}/\partial \xi^{2}\),
\(\kappa_{12}=\partial^{2}\Phi_{0}/\partial \xi\,\partial \eta\),
\(\kappa_{2}=\partial^{2}\Phi_{0}/\partial \eta^{2}\), the functions
\(M_{1}, M_{12}, M_{2}\) depend only on the variables \(\xi\) and \(\eta\).

Consequently, equation (1) can be transformed to the form

\[ \frac{\partial^{2}M_{1}}{\partial \xi^{2}} +2\frac{\partial^{2}M_{12}}{\partial \xi\,\partial \eta} +\frac{\partial^{2}M_{2}}{\partial \eta^{2}} +\frac{\mu}{4}\left[ \xi\frac{\partial}{\partial \xi} \left( 2\Phi_{0}-\xi\frac{\partial \Phi_{0}}{\partial \xi} -\eta\frac{\partial \Phi_{0}}{\partial \eta} \right) +\eta\frac{\partial}{\partial \eta} \left( 2\Phi_{0}-\xi\frac{\partial \Phi_{0}}{\partial \xi} -\eta\frac{\partial \Phi_{0}}{\partial \eta} \right) \right]=0. \tag{2} \]

A solution of the form under consideration is possessed, for example, by a number of problems on the impact upon a plate by a body moving with constant velocity.

In the particular case of impact on a plate, unbounded in both directions, by a body having one point \((x=y=0)\) of contact with the plate, a further simplification of the solution is possible.

In this case equation (1), in polar coordinates \(r,\theta\), assumes the form:

\[ \frac{1}{r}\frac{\partial}{\partial r} \left( r\frac{\partial M_{1}}{\partial r} \right) +\frac{1}{r}\frac{\partial}{\partial r}(M_{1}-M_{2}) -\mu\frac{\partial^{2}w}{\partial t^{2}}=0, \tag{3} \]

where the bending moments \(M_{1}\) and \(M_{2}\) depend only on the curvatures

\[ \kappa_{1}=\frac{\partial^{2}w}{\partial r^{2}}, \qquad \kappa_{2}=\frac{1}{r}\frac{\partial w}{\partial r}. \]

By virtue of the symmetry of this problem, the single variable on which the function \(\Phi_{0}\) depends will be
\(\zeta=r/\sqrt{t}\), whence equation (3) becomes the ordinary differential equation

\[ \left[ \frac{d}{d\zeta} \left( \zeta\frac{dM_{1}}{d\zeta} \right) +\frac{d}{d\zeta}(M_{1}-M_{2}) +\frac{\mu\zeta^{2}}{4}\left(\Phi_{0}^{\prime\prime}-\zeta(\Phi_{0}^{\prime})\right) \right]=0 \tag{4} \]

of the third order with respect to the function \(\Phi_{0}^{\prime}\) (since
\(\kappa_{1}=\Phi_{0}^{\prime\prime}\),
\(\kappa_{2}=\Phi_{0}^{\prime}/\zeta\)).

For a linear dependence between stresses and strains, equation (4) is linear and can be integrated in elementary functions \((^{2})\).

For the case of elastic-plastic deformations, equation (4), being nonlinear, has a different form in the regions of loading and unloading. We note that the boundary of these regions is determined by the equation $\zeta=\mathrm{const}$; moreover, on it the condition of continuity of velocities, displacements, and bending moments must be satisfied, by analogy with the corresponding problem of impact on a beam ($^3$).

Received
10 VIII 1957.

CITED LITERATURE

$^1$ A. A. Ilyushin, Plasticity, Moscow—Leningrad, 1948.
$^2$ A. I. Lur’e, Operational Calculus in Application to Problems of Mechanics, 1932.
$^3$ P. E. Duwer, D. S. Clark, N. F. Bohnenblust, J. Appl. Mech., 17, No. 1 (1950) (transl. collected volume Mechanics, vol. 3, 1950).