UDC 539.376+517.433

THEORY OF ELASTICITY

E. I. GOL’DENGERSHEL

ON THE LIMITING DEFLECTION OF A VISCOELASTIC BEAM

(Presented by Academician Yu. N. Rabotnov on 18 V 1970)

1°. We shall consider a viscoelastic beam of variable cross-section and finite length \(l\), lying on an elastic foundation of Winkler type.

As is known \((^{5,8})\), the deflection \(y(x,t)\) of its axis is described by the following boundary-value problem:

\[ \frac{\partial^2}{\partial x^2}\left(EI(x)\frac{\partial^2 y}{\partial x^2}\right) +Cy(x,t)+C\int_0^t K(t,\tau)y(x,\tau)\,d\tau = \]

\[ = p(x,t)+\int_0^t K(t,\tau)p(x,\tau)\,d\tau; \qquad 0\le x\le l;\quad 0\le t<\infty, \tag{1} \]

\[ U_i[y]=0, \tag{2} \]

where (2) are the boundary conditions describing the nature of the fastening of the beam at its ends, \(p(x,t)\) is the external transverse load, \(K(t,\tau)\) is the kernel characterizing the hereditary properties of the material of the beam, \(I(x)\) is the moment of inertia of the beam cross-section relative to the axis, \(C\) is the foundation coefficient, and \(E\) is the instantaneous modulus of elasticity. We shall assume that \(E\) and \(C\) are constants.

Let \(\alpha(t)\) be some continuous function, positive for \(t\ge 0\). We shall assume that

a)
\[ \sup_{0\le t<\infty}\int_0^t |K(t,\tau)|\frac{\alpha(t)}{\alpha(\tau)}\,d\tau<\infty; \]

b)
\[ \lim_{t\to\infty}\int_E K(t,\tau)\frac{\alpha(t)}{\alpha(\tau)}\,d\tau=0 \]
for any measurable bounded set \(E\subset[0,\infty)\);

c)
\[ l_K=\lim_{t\to\infty}\int_0^t K(t,\tau)\frac{\alpha(t)}{\alpha(\tau)}\,d\tau \]
exists.

Let us note that if \(K(t,\tau)=K_0(t-\tau)\) and

\[ \alpha(t)=e^{-\theta t}, \tag{3} \]

where \(\theta\) is some real number, then conditions a), b), c) are equivalent to the following:

\[ \int_0^\infty |K_1(t)|e^{-\theta t}dt<\infty. \tag{4} \]

We are interested in the conditions under which the existence of the limit, uniform in \(x\),

\[ (\mathrm{L}p)(x)=\lim_{t\to\infty}p(x,t)\alpha(t) \tag{5} \]

implies the existence of the limit, uniform in \(x\),

\[ (\mathrm{L}y)(x)=\lim_{t\to\infty}y(x,t)\alpha(t) \tag{5″} \]

and how to find this latter limit.

In particular, we are interested in the case when the equality

\[ (Lp)(x)\equiv 0 \tag{6} \]

implies the equality

\[ (Ly)(x)\equiv 0. \tag{6''} \]

\(2^\circ\). Let us denote by \(Q(x,\xi)\) the Green’s function of the differential operator \(\Lambda_x\):

\[ \Lambda_x[y]=\frac{d^2}{dx^2}\left(I(x)\frac{d^2y}{dx^2}\right)+\frac{C}{E}y \]

under the boundary conditions (2), and by \(Q\) the Fredholm operator acting in the Banach space \(C[0,l]\):

\[ (Qg)(x)=\int_0^l Q(x,\xi)g(\xi)\,d\xi . \tag{7} \]

With the aid of Tauberian theorems of the type of Paley—Wiener—Gelfand \((^{1-4})\), the following propositions can be proved.

Theorem 1. If

\[ |E/Cq_0|>\lim_{s\to\infty}\ \sup_{s\le t<\infty}\int_s^t |K(t,\tau)|\frac{\alpha(t)}{\alpha(\tau)}\,d\tau, \tag{8} \]

where \(q_0\) is the eigenvalue of the operator \(Q\) largest in modulus, then the existence of the limit (5) implies the existence of \((5'')\). Moreover,

\[ (Ly)(x)=\frac{1}{C}\left(l_kQ+\frac{E}{C}I\right)^{-1}Q(Lp+LVp)(x). \tag{9} \]

Condition (8) is also sufficient for (6) to imply \((6'')\).

Theorem 2. Let \(\alpha(t)\) have the form (3) and

\[ K(t,\tau)=K_0(t-\tau)+\widetilde K(t,\tau), \tag{10} \]

where \(K_0(t)\) satisfies condition (4), and \(\widetilde K(t,\tau)\) satisfies conditions a), b), c) and also the condition

\[ \lim_{s\to\infty}\ \sup_{s\le t<\infty}\int_s^t |\widetilde K(t,\tau)|e^{-\theta(t-\tau)}\,d\tau=0. \tag{11} \]

Then the condition

\[ -\frac{E}{C}\ne q_i k_0(W);\qquad \operatorname{Re}W\ge 0;\qquad q_i\in\sigma(Q), \tag{12} \]

where \(k_0(W)\) is the Laplace transform of the function \(K_0(t)\), and \(\sigma(Q)\) is the spectrum of the operator \(Q\) (7) in \(C[0,l]\), is sufficient for the existence of the limit (5) to imply the existence of the limit \((5'')\). In this case formula (9) holds.

If

\[ k_0(W)+1\ne 0;\qquad \operatorname{Re}W\ge 0, \tag{13} \]

then condition (12) is not only sufficient but also necessary for this.

Theorem 3. Let

1) \(\alpha(0)=1;\quad \alpha(t+\tau)\le \alpha(t)\cdot\alpha(\tau);\quad \theta=\lim_{t\to\infty}\frac{\ln\alpha(t)}{-t}<\infty;\)

2) \(K(t,\tau)\) be representable in the form (10);

3) \(\widetilde K(t,\tau)\) satisfy conditions a), b), c) and

\[ \lim_{s\to\infty}\ \sup_{t<t\le\infty}\int_s^t |\widetilde K(t,\tau)|\frac{\alpha(t)}{\alpha(\tau)}\,d\tau=0; \]

4) \[ \int_0^\infty |K_0(t)|\,\alpha(t)\,dt<\infty . \]

Then condition (12) is sufficient for (6) to imply (6″).

Analogous results are obtained when the problem of buckling of a viscoelastic rod is investigated by our method \({}^{6}\).

I take this opportunity to express my sincere gratitude to Yu. N. Rabotnov, L. Kh. Papernik, V. S. Ekelchik, and V. N. Rivkind for their useful discussion of the present work.

Moscow State University
named after M. V. Lomonosov

Received
19 I 1970

REFERENCES

\({}^{1}\) N. Wiener, R. Paley, Fourier Transform in the Complex Domain, Nauka, 1964.
\({}^{2}\) I. Gelfand, Mat. sbornik, 9 (51), 51 (1941).
\({}^{3}\) I. Goldenblat, DAN, 129, No. 5, 971 (1959).
\({}^{4}\) E. I. Goldenblat, Mat. sbornik, 64, No. 1, 115 (1964).
\({}^{5}\) Yu. M. Rabotnov, Creep of Structural Elements, Nauka, 1966.
\({}^{6}\) J. N. Distefano, J. Structural Division. Proc. of the Am. Soc. of Civil Engineers, 91, No. 3, 127 (1965).
\({}^{7}\) A. R. Rzhanitsyn, Theory of Creep, 1968.