THEORY OF ELASTICITY

M. I. ROZOVSKII

STRESSES IN A SYMMETRICALLY HEATED SPHERICAL SHELL WHOSE MECHANICAL PROPERTIES DEPEND ON TIME AND TEMPERATURE

(Presented by Academician S. L. Sobolev, 13 I 1958)

1. Let a closed spherical shell be given, with external and internal radii \(a_1\) and \(a_2\), free of external forces and initial stresses, at a uniformly distributed temperature. Suppose that, beginning at some moment, the temperature in the shell becomes nonuniformly distributed, and assume that the temperature \(T = T(r, t)\) is known. Then stresses and strains of a definite magnitude will arise in it.

We determine the radial displacement \(u(r,t)\) and the stresses arising in the body in question in the case where the Lamé coefficients \(\lambda\) and \(\mu\) and the relaxation characteristics depend on temperature, and also taking into account aging of the material, i.e., the variability of \(\lambda\) and \(\mu\) with time \(t\).

Generalizing in the appropriate way the Volterra dependences \((^1)\), we write the stress components in spherical coordinates

\[ \sigma_r=\lambda\theta+2\mu\frac{\partial u}{\partial r} -\int_{t_0}^{t}\left[\varphi(t,\tau;r)\,\theta +2\psi(t,\tau;r)\frac{\partial u}{\partial r}\right]\,d\tau-\alpha\beta T, \]

\[ \sigma_\theta=\sigma_\varphi=\lambda\theta+2\mu\frac{u}{r} -\int_{t_0}^{t}\left[\varphi(t,\tau;r)\,\theta +2\psi(t,\tau;r)\frac{u}{r}\right]\,d\tau-\alpha\beta T, \tag{1} \]

where \(\theta(r,t)\) is the volume expansion; \(\varphi(t,\tau;r)\) and \(\psi(t,\tau;r)\) are the relaxation kernels; \(\lambda\), \(\mu\), and \(\beta=3\lambda+2\mu\), as well as the coefficient of linear expansion \(\alpha\), depend on the coordinate \(r\) and time \(t\), since, by assumption, the temperature of the shell is considered a known function of \(r\) and \(t\).

Substituting \(\sigma_r\), \(\sigma_\theta=\sigma_\varphi\) from (1) into the usual equilibrium equation, we obtain the initial integro-differential equation, which we present in the form

\[ \frac{\partial}{\partial r} \left[ (\lambda+2\mu)\frac{1}{r^2}\frac{\partial}{\partial r}(r^2u) \right] -\frac{4}{r}\frac{\partial\mu}{\partial r}u -\frac{\partial}{\partial r}(\alpha\beta T) +\frac{4}{r}\int_{t_0}^{t} \frac{\partial\psi(t,\tau,r)}{\partial r}u(r,\tau)\,d\tau = \]

\[ = \int_{t_0}^{t} \frac{\partial}{\partial r} \left\{ [\varphi_1(t,\tau;r)+2\psi(t,\tau;r)] \frac{1}{r^2}\frac{\partial}{\partial r}(r^2u) \right\}\,d\tau . \tag{2} \]

Put

\[ \frac{\partial}{\partial r} \left[ \frac{1}{r^2}\frac{\partial}{\partial r}(r^2u) \right] =z(r,t). \tag{3} \]

From (3) it follows that

\[ u(r,t)=\frac{1}{3r^2}\int_{a_1}^{r}(r^3-\rho^3)z(\rho,t)\,d\rho +c_1(t)r+r^{-2}c_2(t). \tag{4} \]

where \(c_1\) and \(c_2\) are arbitrary functions of time, which are determined from the boundary conditions

\[ \left\{ \lambda\left(\frac{\partial u}{\partial r}+\frac{2u}{r}\right)+2\mu\frac{\partial u}{\partial r}-\alpha\beta T - \int_{t_0}^{t} \left[ \varphi(t,\tau;r)\left(\frac{\partial u}{\partial r}+\frac{2u}{r}\right) +2\psi(t,\tau;r)\frac{\partial u}{\partial r} \right]\,d\tau \right\}_{r=a_k}=0 \tag{5} \]

\((k=1,2)\) after \(z(r,t)\) has been found.

Substituting (3) and (4) into equation (2), after the corresponding transformations we obtain the mixed two-dimensional integral equation

\[ z(r,t)+\int_{a_1}^{r} K_1(t;r,\rho) z(\rho,t)\,d\rho -\int_{t_0}^{t} K_2(t,\tau;r) z(r,\tau)\,d\tau = \]

\[ = \int_{t_0}^{t}\int_{a_1}^{r} K_3(t,\tau;r,\rho) z(\rho,\tau)\,d\rho\,d\tau +f(r,t,c_1,c_2), \tag{6} \]

where

\[ K_1(t;r,\rho)= \frac{1}{\lambda+2\mu} \left[ \frac{\partial}{\partial r}(\lambda+2\mu) -\frac{4(r^3-\rho^3)}{3r^3}\frac{\partial\mu}{\partial r} \right], \]

\[ K_2(t,\tau;r)= \frac{\varphi(t,\tau;r)+2\psi(t,\tau;r)}{\lambda+2\mu}, \]

\[ K_3(t,\tau;r,\rho)= \frac{1}{\lambda+2\mu} \left\{ \frac{\partial}{\partial r}\left[\varphi(t,\tau;r)+2\psi(t,\tau;r)\right] -\frac{4(r^3-\rho^3)}{3r^3}\frac{\partial\psi(t,\tau;r)}{\partial r} \right\}, \]

\[ f(r,t;c_1,c_2)= \frac{1}{\lambda+2\mu} \left\{ \frac{4}{r}\left(c_1r+\frac{c_2}{r^2}\right)\frac{\partial\mu}{\partial r} -\frac{4}{r}\int_{t_0}^{t} \left(c_1r+\frac{c_2}{r^2}\right) \frac{\partial\psi(t,\tau;r)}{\partial r}\,d\tau \right. \]

\[ \left. -3c_1\frac{\partial}{\partial r}(\lambda+2\mu) +3\int_{t_0}^{t} c_1(\tau) \frac{\partial}{\partial r}\left[\varphi(t,\tau;r)+2\psi(t,\tau;r)\right]\,d\tau \right\}. \]

Determining \(z\) from equation (6) and then substituting into (4), after the corresponding transformations we obtain the formula for the displacement

\[ u=c_1 A_1(t,r) -\int_{t_0}^{t}\left[A_2(t,\tau;r)c_1(\tau)-A_3(t,\tau;r)c_2(\tau)\right]\,d\tau +c_2 r^{-2}, \tag{7} \]

where

\[ A_1(t,r)= r+\frac{1}{3r^2}\int_{a_1}^{r} \frac{D(t;r,\rho)}{\lambda+2\mu} \left[ 4\frac{\partial\mu}{\partial\rho} -3\frac{\partial}{\partial\rho}(\lambda+2\mu) \right]\,d\rho, \]

\[ D(t;r,\rho)= r^3-\rho^3-\int_{\rho}^{r}(r^3-\rho_1^3)R_1(t;\rho_1,\rho)\,d\rho_1, \]

\[ A_2(t,\tau;r)= \frac{1}{3r^2}\int_{a_1}^{r} \frac{D(t;r,\rho)}{\lambda+2\mu} \left[ 4\frac{\partial\psi(t,\tau;r)}{\partial\rho} -3\frac{\partial}{\partial\rho}\bigl(\varphi(t,\tau;\rho)+2\psi(t,\tau;\rho)\bigr) \right]\,d\rho, \]

\[ A_3(t,\tau;r)= \frac{4}{3r^2}\int_{a_1}^{r} \frac{\rho^{-3}}{\lambda+2\mu} \left[ E(t,\tau;r,\rho)\frac{\partial\mu}{\partial\rho} -\int_{\tau}^{t} E(t,\tau_1;r,\rho) \frac{\partial\psi(\tau_1,\tau;\rho)}{\partial\rho}\,d\tau_1 \right], \]

\[ E(t,\tau;r,\rho)= (r^3-\rho^3)R_2(t,\tau;\rho) +\int_{\rho}^{r}(r^3-\rho_1^3)R_3(t,\tau;\rho_1,\rho)\,d\rho_1, \]

\[ R_3(t,\tau;r,\rho)=R(t,\tau;r,\rho)+\int_{\tau}^{t}R_2(t,\tau_1;r)\,R(\tau_1,\tau;r,\rho)\,d\tau_1- \]

\[ -\int_{\rho}^{r}R_1(t;r,\rho_1)\,R(\tau_1,\tau;\rho_1,\rho)\,d\rho_1- \]

\[ -\int_{\tau_1}^{t}d\tau_1\int_{\rho}^{r}R_1(\tau_1;r,\rho_1)\,R_2(t,\tau_1;\rho_1)\,R(\tau_1,\tau;\rho_1,\rho)\,d\rho_1; \]

\(R_1(t;r,\rho)\) and \(R_2(t,\tau;r)\) are the resolvents of the kernels \(K_1(t;r,\rho)\) and \(K_2(t,\tau;r)\), respectively; \(R(t,\tau;r,\rho)\) is the resolvent of the kernel

\[ \left[ Q(t,\tau;r,\rho)+\int_{\tau}^{t}Q(t,\tau_1;r,\rho)\,R_2(\tau_1,\tau;\rho)\,d\tau_1 \right]; \]

\[ Q(t,\tau;r,\rho)=K_3(t,\tau;r,\rho)-K_2(t,\tau;r)R_1(\tau;r,\rho) -\int_{\rho}^{r}K_3(t,\tau;r,\rho_1)R_1(\tau;\rho_1,\rho)\,d\rho_1 . \]

The functions \(c_1(t)\) and \(c_2(t)\) appearing in (7) are determined from the system of ordinary integral equations

\[ \sum_{i=1}^{2}\left[ \alpha_{ik}c_k(t)-\int_{t_0}^{t}\Phi_{ik}(t,\tau)c_k(\tau)\,d\tau \right]=F_k, \tag{8} \]

where

\[ \alpha_{ik}= \left\{ \frac{1-(-1)^i}{2} \left[ (\lambda+2\mu)\frac{\partial A_1}{\partial r} +2\lambda\frac{A}{r} \right] -2\mu\frac{(-1)^i+1}{r^3} \right\}_{r=a_k} \quad (i,k=1,2); \]

\[ F_k=\alpha_k\beta_kT_k; \qquad \Phi_{ik}(t,\tau)= \left\{ \frac{1-(-1)^i}{2}(\varphi+2\psi)\frac{\partial A_1}{\partial r} -2\psi\frac{(-1)^i+1}{r^3} +\right. \]

\[ \left. +\frac{(-1)^k+1}{2} \left[ (\lambda+2\mu)\frac{\partial A_{i+1}}{\partial r} -\frac{2\lambda}{r}A_{i+1} -\int_{\tau}^{t} \left( (\varphi+2\psi)_{\tau_1}\frac{\partial A_{i+1}}{\partial r} +\frac{2\varphi}{r}A_{i+1} \right)d\tau_1 \right] \right\}_{r=a_k}. \]

System (8) has a unique solution.
From the first equation of system (8) it follows that

\[ c_1(t)=\frac{F_1}{a_{11}} +\int_{t_0}^{t}\frac{H(t,\tau)}{a_{11}(\tau)}F_1(\tau)\,d\tau -\frac{a_{21}}{a_{11}}c_2(t) +\int_{t_0}^{t}H_1(t,\tau)c_2(\tau)\,d\tau, \tag{9} \]

where

\[ H_1(t,\tau)=\alpha_{11}^{-1}(\tau) \left[ \Phi_{21}(t,\tau)-\alpha_{21}H(t,\tau) +\int_{\tau}^{t}H(t,s)\Phi_{21}(s,\tau)\,ds \right], \]

\(H(t,\tau)\) is the resolvent of the kernel \(\Phi_{11}(t,\tau)/\alpha_{11}\).

Substituting (9) into the second equation of system (8), we obtain an integral equation with one unknown \(c_2\). Determining \(c_2\) from the equation thus obtained and then again using (9), we obtain formulas determining \(c_k\).

Knowing the displacement \(u\), it is easy to determine \(\sigma_r,\ \sigma_\theta=\sigma_\varphi\).

1. The obtained results are considerably simplified if it is assumed that \(\lambda=\mu=\lambda_0e^{-mT}\), \(\varphi(t,\tau;r)=\psi(t,\tau;r)=\varphi_0(t,\tau)e^{-mT}\), where \(m\) is a constant for the given material; \(\lambda_0\) and \(\varphi_0(t,\tau)\) are the Lamé coefficient and the relaxation kernel at \(T=0\); the temperature \(T\) still depends on \(r\) and \(t\).

The decrease of the physico-mechanical characteristics is proportional to many-

... for a viscosity that decreases exponentially with increasing temperature, is confirmed experimentally \((^2)\).

The substitution (3) leads to an ordinary integral equation with respect to \(z\), with a fairly simple kernel.

As a result we shall have

\[ \begin{gathered} u(r,t)=\int_{a_1}^{r} N_1(t;r,\rho)\,T_*(\rho,t)\,d\rho +c_1\left[r+\frac{5m}{3}\int_{a}^{r} N_1(t;r,\rho)\frac{\partial T}{\partial \rho}\,d\rho\right]+ \\ +c_2\left[\frac{1}{r^2}-\frac{4m}{3}\int_{a_1}^{r} N_1(t;r,\rho)\frac{1}{\rho^3}\frac{\partial T}{\partial \rho}\,d\rho\right], \qquad T_* = T_1+\int_{t_0}^{t} P(t,\tau)T_1\,d\tau, \end{gathered} \tag{10} \]

\[ N_1(t;r,\rho)=\frac{1}{3r^2}\left[r^3-\rho^3+\int_{\rho}^{r}(r^3-\rho_1^3)N(t;\rho_1,\rho)\,d\rho_1\right]; \]

\(P(t,\tau)\) and \(N(t;r,\rho)\) are the resolvents of the kernels \(\dfrac{\varphi_0(t,\tau)}{\lambda_0}\) and
\[ \frac{m}{9}\left(5+\frac{4\rho^3}{r^3}\right)\frac{\partial T}{\partial r}, \]
respectively;
\[ T_1=5\left[\frac{\partial}{\partial r}(\alpha T)-m\alpha T\frac{\partial T}{\partial r}\right]. \]

The functions \(c_1(t)\) and \(c_2(t)\) appearing in (10) are expressed by the formulas

\[ c_1=\frac{a_1^3 B_2\Phi_{a_1}-a_2^3\Phi_{a_2}}{a_2^3B_1-a_1^2B_2}, \qquad c_2=\frac{5a_1^3a_2^3(\Phi_{a_2}-B_1\Phi_{a_1})}{4(a_2^3B_1-a_1^2B_2)}, \]

\[ B_1=1+\frac{m}{3}\int_{a_1}^{a_2}\frac{\partial T}{\partial \rho} \left[\frac{\partial N_1(t;r,\rho)}{\partial r}+\frac{N_1(t;r,\rho)}{a_2}\right]_{r=a_2}d\rho, \tag{11} \]

\[ B_2=1+ma_2^3\int_{a_1}^{a_2}\frac{1}{\rho^3}\frac{\partial T}{\partial \rho} \left[\frac{\partial N_1(t;r,\rho)}{\partial r}+\frac{2N_1(t;r,\rho)}{3a_2}\right]_{r=a_2}d\rho; \]

\(\Phi_{a_1}\) and \(\Phi_{a_2}\) are the values of the function
\[ \Phi(r,t)=\alpha T+\int_{t_0}^{t}P(t,\tau)\alpha T\,d\tau \]
at \(r=a_1\) and \(r=a_2\), respectively.

Formulas (11) are obtained as the result of solving the system of algebraic equations formed by using the boundary conditions (5).

As an illustration, let us consider a steel hollow ball with the following parameters:
\(m=0.0008\ {\rm deg}^{-1}\) (according to the experimental data of Zaikov \((^2)\)),
\(a_1=30\) cm, \(a_2=33\) cm,
\(T=9900r^{-1}+670\),
\(T(a_1)=1000^\circ\),
\(T(a_2)=970^\circ\),
\(\alpha=11\cdot10^{-6}\ {\rm deg}^{-1}\),
\[ P(s)=0.0175(s/33)^{0.75}\exp\left(-\sqrt[4]{s/33}\right) \]
(according to \((^3)\)). Approximately we obtain
\[ u=(0.1165-0.00762\exp(-\sqrt[4]{t/33})) (0.9853-0.04097r-17.85r^{-1}-525.5r^{-2}-2253r^{-3} +1898r^{-4}+46r^{-3}\ln r+3.536r^{-2}\ln r). \]
In particular, for example, the steady-state value of the strain at the outer surface will be
\[ \varepsilon=0.00507. \]

In conclusion I express my gratitude to Yu. A. Ponomarenko and L. I. Paikova for assistance in the computations connected with the reduction to concrete examples.

Dnepropetrovsk Mining Institute
named after Artem

Received
31 V 1957

CITED LITERATURE

\(^{1}\) V. Volterra, Theory of Functionals and of Int.-differential Equations, London, 1931.
\(^{2}\) M. A. Zaikov, ZhTF, 18, no. 6 (1948).
\(^{3}\) A. P. Bronskii, Prikl. matem. i mekh., 5, no. 1 (1941).