UDC 532.135

THEORY OF ELASTICITY

B. E. POBEDRYA

ON THE RELATION BETWEEN STRESSES AND STRAINS IN NONLINEAR VISCOUS ELASTICITY

(Presented by Academician L. I. Sedov, 18 VI 1966)

Let us fix some Lagrangian coordinate system \(\xi^k\) and consider the stress tensor \(P^{ij}(\tau,\xi^k)\) and the strain tensor \(E_{ij}(\tau,\xi^k)\) in the form of abstract functions \(P(\tau)\) and \(E(\tau)\), defined on the interval \([0,t]\) in the Hilbert spaces \(H_P\) and \(H_E\), respectively.

Suppose that there exists a sufficiently smooth operator \(P=F(E)\), mapping the space \(H_E\) into \(H_P\), with \(F(0)=0\). For example, let all Fréchet derivatives at zero exist. Then one can write

\[ P=\sum_{n=1}^{\infty}\Gamma_n E^n, \tag{1} \]

where

\[ \Gamma_n E^n=\int_0^t\cdots\int_0^t \Gamma_n(t,\tau_1,\tau_2,\ldots,\tau_n)E(\tau_1)\cdots E(\tau_n)\,d\tau_1\cdots d\tau_n . \]

Denote by \(K_1\) the linear operator inverse to \(\Gamma_1\).

It can be shown that equations (1) are solvable:

\[ E=\sum_{n=1}^{\infty}K_n P^n, \tag{2} \]

and, for fixed \(n\), each kernel \(K_i\) \((i=2,3,\ldots,n)\) is expressed explicitly in terms of the known kernels \(\Gamma_j\) \((j=2,3,\ldots,i)\) and the kernel \(K_1\). For example:

\[ K_2(t,\tau_1,\tau_2) = -\int_0^t K_1(t,\xi)\,d\xi \int_{\tau_1}^{\xi}\int_{\tau_2}^{\xi} \Gamma_2(\xi,\xi_1,\xi_2)K_1(\xi_1,\tau_1)K_1(\xi_2,\tau_2)\,d\xi_1d\xi_2; \tag{3} \]

\[ \begin{aligned} K_3(t,\tau_1,\tau_2,\tau_3) ={}&-\int_0^t K_1(t,\xi)\,d\xi \int_{\tau_1}^{\xi}\int_{\tau_2}^{\xi}\int_{\tau_3}^{\xi} \Gamma_3(\xi,\xi_1,\xi_2,\xi_3)K_1(\xi_1,\tau_1)\times \\ &\times K_1(\xi_2,\tau_2)K_1(\xi_3,\tau_3)\,d\xi_1d\xi_2d\xi_3 +2\int_0^t K_1(t,\xi)\,d\xi\int_0^{\xi}d\xi_1 \int_{\tau_3}^{\xi}\Gamma_2(\xi,\xi_1,\xi_2)\times \\ &\times K_1(\xi_2,\tau_3)\,d\xi_2 \int_0^{\xi_1}K_1(\xi_1,\eta)\,d\eta \int_{\tau_1}^{\eta}\int_{\tau_2}^{\eta} \Gamma_2(\eta,\eta_1,\eta_2)K_1(\eta_1,\tau_1)K_1(\eta_2,\tau_2)\,d\eta_1d\eta_2 . \end{aligned} \tag{4} \]

And so on. Owing to this, the specification of the equations of state of the medium in the form (1) and (2) is completely equivalent.

Each kernel \(\Gamma_n(t,\tau_1,\ldots,\tau_n)\) is a tensor of rank \(2(n+1)\) and, in the case of an isotropic medium, is expressed in the form of a sum of tensors, compos—

composed of products of the metric tensors \(\delta_{ij}\) and certain scalars. In this case equations (4) can be written in the form

\[ \begin{aligned} E_{ij}(t)={}& \delta_{ij}\int_0^t K_{11}(t,\tau_1)P_k^k(\tau_1)\,d\tau_1 +\int_0^t K_{12}(t,\tau_1)P_{ij}(\tau_1)\,d\tau_1+\\ &+\delta_{ij}\int_0^t\int_0^t K_{21}(t,\tau_1,\tau_2)P_k^k(\tau_1)P_l^l(\tau_2)\,d\tau_1d\tau_2 +\delta_{ij}\int_0^t\int_0^t K_{22}(t,\tau_1,\tau_2)P^{kl}(\tau_1)\times\\ &\qquad\qquad\qquad\qquad\qquad\qquad{}\times P_{lk}(\tau_2)\,d\tau_1d\tau_2 +\int_0^t\int_0^t K_{23}(t,\tau_1,\tau_2)P_k^k(\tau_1)P_{ij}(\tau_2)\,d\tau_1d\tau_2+\\ &+\int_0^t\int_0^t K_{24}(t,\tau_1,\tau_2)P_i^k(\tau_1)P_{kj}(\tau_2)\,d\tau_1d\tau_2+\\ &+\delta_{ij}\int_0^t\int_0^t\int_0^t K_{31}(t,\tau_1,\tau_2,\tau_3)P_k^k(\tau_1)P_l^l(\tau_2)P_m^m(\tau_3)\,d\tau_1d\tau_2d\tau_3+\\ &+\int_0^t\int_0^t\int_0^t K_{32}(t,\tau_1,\tau_2,\tau_3)P_k^k(\tau_1)P_m^l(\tau_2)P_{lm}(\tau_3)\,d\tau_1d\tau_2d\tau_3+\\ &+\int_0^t\int_0^t\int_0^t K_{33}(t,\tau_1,\tau_2,\tau_3)P_k^k(\tau_1)P_l^l(\tau_2)P_{ij}(\tau_3)\,d\tau_1d\tau_2d\tau_3+\\ &+\int_0^t\int_0^t\int_0^t K_{34}(t,\tau_1,\tau_2,\tau_3)P^{kl}(\tau_1)P_{lk}(\tau_2)P_{ij}(\tau_3)\,d\tau_1d\tau_2d\tau_3+\\ &+\int_0^t\int_0^t\int_0^t K_{35}(t,\tau_1,\tau_2,\tau_3)P_k^k(\tau_1)P_i^l(\tau_2)P_{lj}(\tau_3)\,d\tau_1d\tau_2d\tau_3+\\ &+\delta_{ij}\int_0^t\int_0^t\int_0^t K_{36}(t,\tau_1,\tau_2,\tau_3)P_l^k(\tau_1)P_m^l(\tau_2)P_k^m(\tau_3)\,d\tau_1d\tau_2d\tau_3+\\ &+\int_0^t\int_0^t\int_0^t K_{37}(t,\tau_1,\tau_2,\tau_3)P_i^k(\tau_1)P_k^l(\tau_2)P_{lj}(\tau_3)\,d\tau_1d\tau_2d\tau_3+\ldots \end{aligned} \tag{5} \]

In exactly the same form, equations (1) can be written, and the kernels of these equations are expressed through the kernels of equations (5) by formulas of the type (3) and (4).

If one requires that the equations of state (1) and (2) be invariant with respect to the choice of the time origin, then all the kernels become kernels of difference type.

Moscow State University
named after M. V. Lomonosov

Received
17 VI 1966