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).