THEORY OF ELASTICITY

M. A. ZADOYAN

A PARTICULAR SOLUTION OF THE EQUATIONS OF THE THEORY OF IDEAL PLASTICITY IN CYLINDRICAL COORDINATES

(Presented by Academician L. I. Sedov on 6 II 1964)

The equations of the theory of ideal plasticity under the Huber—Mises plasticity criterion in cylindrical coordinates have the form:

\[ \frac{\partial \sigma_r}{\partial r} +\frac{1}{r}\frac{\partial \tau_{r\theta}}{\partial \theta} +\frac{\partial \tau_{rz}}{\partial z} +\frac{\sigma_r-\sigma_\theta}{r}=0, \]
\[ \frac{\partial \tau_{r\theta}}{\partial r} +\frac{1}{r}\frac{\partial \sigma_\theta}{\partial \theta} +\frac{\partial \tau_{\theta z}}{\partial z} +\frac{2\tau_{r\theta}}{r}=0, \tag{1} \]
\[ \frac{\partial \tau_{rz}}{\partial r} +\frac{1}{r}\frac{\partial \tau_{\theta r}}{\partial \theta} +\frac{\partial \sigma_z}{\partial z} +\frac{\tau_{rz}}{r}=0; \]

\[ (\sigma_r-\sigma_\theta)^2+(\sigma_\theta-\sigma_z)^2+(\sigma_z-\sigma_r)^2 +6\left(\tau_{r\theta}^{\,2}+\tau_{\theta z}^{\,2}+\tau_{rz}^{\,2}\right)=6k^2; \tag{2} \]

\[ \varepsilon_r=\frac{\partial u}{\partial r} =\lambda(2\sigma_r-\sigma_\theta-\sigma_z), \]
\[ \varepsilon_\theta=\frac{u}{r}+\frac{1}{r}\frac{\partial v}{\partial \theta} =\lambda(2\sigma_\theta-\sigma_z-\sigma_r), \tag{3} \]
\[ \varepsilon_z=\frac{\partial w}{\partial z} =\lambda(2\sigma_z-\sigma_r-\sigma_\theta); \]

\[ 2\gamma_{r\theta} =\frac{\partial v}{\partial r}-\frac{v}{r}+\frac{1}{r}\frac{\partial u}{\partial \theta} =6\lambda\tau_{r\theta}, \]
\[ 2\gamma_{\theta z} =\frac{\partial v}{\partial z}+\frac{1}{r}\frac{\partial w}{\partial \theta} =6\lambda\tau_{\theta z}, \tag{4} \]
\[ 2\gamma_{rz} =\frac{\partial u}{\partial z}+\frac{\partial w}{\partial r} =6\lambda\tau_{rz}. \]

From relations (3)—(4) we have

\[ u(r,\theta,z)=u_0(r,z) -\int_{0}^{\theta}\left(r\frac{\partial v}{\partial r}-v\right)d\theta +2r\int_{0}^{\theta}\gamma_{r\theta}\,d\theta; \tag{5} \]

\[ v(r,\theta,z)=v_0(r,z) -\int_{0}^{\theta}u\,d\theta -r\int_{0}^{\theta}(\varepsilon_r+\varepsilon_z)\,d\theta; \tag{6} \]

\[ w(r,\theta,z)=w_0(r,z) -r\int_{0}^{\theta}\frac{\partial v}{\partial z}\,d\theta +2r\int_{0}^{\theta}\gamma_{\theta z}\,d\theta, \tag{7} \]

where \(u_0,\ v_0,\ w_0\) are arbitrary functions of the coordinates \(r\) and \(z\).

Assuming that the strain-rate tensor does not depend on \(r\) and \(z\), we obtain

\[ \varepsilon_r=\frac{\partial u_0}{\partial r}-r\frac{\partial^2 v_0}{\partial r^2}\theta +2\int_0^\theta \gamma_{r\theta}\,d\theta =A_0+A_1\theta+2\int_0^\theta \gamma_{r\theta}\,d\theta; \tag{8} \]

\[ \gamma_{rz}=\frac12\left(\frac{\partial u_0}{\partial z}+\frac{\partial w_0}{\partial r}\right) -r\frac{\partial^2 v_0}{\partial r\,\partial z}\theta +\int_0^\theta \gamma_{\theta z}\,d\theta =C_0+C_1\theta+\int_0^\theta \gamma_{\theta z}\,d\theta; \tag{9} \]

\[ \varepsilon_z=\frac{\partial w_0}{\partial z} -r\int_0^\theta \frac{\partial^2 v}{\partial z^2}\,d\theta, \tag{10} \]

where \(A_0, A_1, C_0, C_1\) are arbitrary constants. From the preceding relations it follows that

\[ \frac{\partial^2 v}{\partial z^2} +\int_0^\theta d\theta\int_0^\theta \frac{\partial^2 v}{\partial z^2}\,d\theta = \frac{\partial^2 v_0}{\partial z^2} -\frac{\partial^2 u_0}{\partial z^2}\theta, \tag{11} \]

and, consequently,

\[ \frac{\partial^2 v}{\partial z^2} = \frac{\partial^2 v_0}{\partial z^2}\cos\theta -\frac{\partial^2 u_0}{\partial z^2}\sin\theta. \tag{12} \]

Substituting (12) into (10), we find

\[ \varepsilon_z = \frac{\partial w_0}{\partial z} +r\frac{\partial^2 u_0}{\partial z^2} -r\frac{\partial^2 v_0}{\partial z^2}\sin\theta -r\frac{\partial^2 u_0}{\partial z^2}\cos\theta = B_0+B_1\sin\theta+B_2\cos\theta;\quad B_0,B_1,B_2=\mathrm{const}. \tag{13} \]

Comparison of relations (8), (9), and (13) leads to the conclusion that the arbitrary constants \(B_1, B_2, C_1\) are equal to zero and to the formulas

\[ u_0(r,z)=A_0r+G_1z+G_0; \tag{14} \]

\[ v_0(r,z)=(A_1+D_1)r-A_1r\ln r+E_1z+E_0; \tag{15} \]

\[ w_0(r,z)=(2C_0-G_1)r+B_0z+D_0, \tag{16} \]

where \(D_1, D_0, E_1, E_0, G_1, G_0\) are new arbitrary constants. Determining from (6) the value of \(r\,\partial v/\partial r-v\) and substituting into (5), we obtain

\[ u+\int_0^\theta d\theta\int_0^\theta u\,d\theta = u_0-\left(r\frac{\partial v_0}{\partial r}-v_0\right)\theta +r\int_0^\theta d\theta\int_0^\theta \varepsilon_r\,d\theta +2r\int_0^\theta \gamma_{r\theta}\,d\theta. \tag{17} \]

Hence

\[ u(r,\theta,z) =(A_0+A_1\theta)r+(E_1z+E_0)\sin\theta +(G_1z+G_0)\cos\theta +2r\int_0^\theta \gamma_{r\theta}\,d\theta. \tag{18} \]

Substitution of (18) into (6) gives

\[ \begin{aligned} v(r,\theta,z) &=(A_1+D_1)r-A_1r\ln r-(2A_0+B_0)r\theta-A_1r\theta^2 \\ &\quad +(E_1z+E_0)\cos\theta-(G_1z+G_0)\sin\theta -4r\int_0^\theta \gamma_{r\theta}(\xi)(\theta-\xi)\,d\xi. \end{aligned} \tag{19} \]

Determining from (6) and (18) that \(\partial v/\partial z=E_1\cos\theta-G_1\sin\theta\), we shall have

\[ w(r,\theta,z) = 2C_0r+B_0z+D_0-(E_1\sin\theta+G_1\cos\theta)r +2r\int_0^\theta \gamma_{r\theta}\,d\theta. \tag{20} \]

With the aid of relations (3)—(4), the stress components are represented in the form

\[ \sigma_r=\sigma_\theta+\frac{2\varepsilon_r+\varepsilon_z}{\gamma_{r\theta}}\tau_{r\theta}, \qquad \sigma_z=\sigma_\theta+\frac{\varepsilon_r+2\varepsilon_z}{\gamma_{r\theta}}\tau_{r\theta}, \qquad \tau_{rz}=\frac{\gamma_{rz}}{\gamma_{r\theta}}\tau_{r\theta}. \tag{21} \]

Substituting (21) into the equilibrium equations (1) and assuming that \(\tau_{r\theta}\) does not

depends on \(r\) and \(z\), we obtain

\[ r\frac{\partial\sigma_\theta}{\partial r} +\frac{\partial\tau_{r\theta}}{\partial\theta} +\sigma_r-\sigma_\theta=0, \]

\[ \frac{\partial\sigma_\theta}{\partial\theta}+2\tau_{r\theta}=0,\qquad r\frac{\partial\sigma_\theta}{\partial z} +\frac{\partial\tau_{\theta z}}{\partial\theta} +\tau_{rz}=0. \tag{22} \]

From the second equation (22) we determine

\[ \sigma_\theta(r,\theta,z)=H(r,z)-2\int_0^\theta \tau_{r\theta}(\theta)\,d\theta, \tag{23} \]

where \(H\) is an arbitrary function of \(r\) and \(z\). Substituting the result into the first and third of equations (22), we obtain \(H=M\ln r+N\), and

\[ \frac{d\tau_{r\theta}}{d\theta} +\frac{2\varepsilon_r+\varepsilon_z}{\gamma_{r\theta}}\tau_{r\theta} +M=0,\qquad \frac{d\tau_{\theta z}}{d\theta} +\frac{\gamma_{rz}}{\gamma_{\theta z}}\tau_{\theta z}=0, \tag{24} \]

where \(M\) and \(N\) are arbitrary constants. From equations (24) and the yield condition (2) the following formulas result:

\[ \gamma_{r\theta}=\frac{\sqrt{3}}{2}B_0\frac{\tau_{r\theta}}{\Omega},\qquad \gamma_{\theta z}=\frac{\sqrt{3}}{2}B_0\frac{\tau_{\theta z}}{\Omega}, \tag{25} \]

\[ \Omega(\theta)= \sqrt{k^2-\tau_{r\theta}^2-\tau_{\theta z}^2-\frac14(\tau'_{r\theta}+M)^2-\tau_{rz}^2}. \tag{26} \]

Using relations (8), (9), (13), and (25), from (24) we obtain

\[ B_0\Omega\frac{d}{d\theta} \left[ \frac{1}{\Omega}\left(\frac{d\tau_{r\theta}}{d\theta}+M\right) \right] +\frac{4A_1}{\sqrt{3}}\Omega +2B_0\tau_{r\theta}=0, \tag{27} \]

\[ \Omega\frac{d}{d\theta} \left[ \frac{1}{\Omega}\frac{d\tau_{\theta z}}{d\theta} \right] +\tau_{\theta z}=0. \tag{28} \]

We also have

\[ \tau'_{r\theta}(0) =-M-(2A_0+B_0) \sqrt{ \frac{k^2-Q_1^2-Q_2^2} {A_0^2+A_0B_0+B_0^2+C_0^2} }, \]

\[ \tau'_{\theta z}(0) =-C_0 \sqrt{ \frac{k^2-Q_1^2-Q_2^2} {A_0^2+A_0B_0+B_0^2+C_0^2} }, \tag{29} \]

where \(Q_1=\tau_{r\theta}(0)\) and \(Q_2=\tau_{\theta z}(0)\) are also arbitrary constants.

Thus, the problem has been reduced to the construction of solutions of a system of two ordinary differential equations. The solution obtained contains 14 arbitrary constants. From it, by taking \(\tau_{\theta z}=\tau_{rz}=B_0=0\), we obtain the known case of plane deformation of a plastic mass between inclined rough plates, investigated by A. Nadai \((^1)\). Setting \(A_1=D_1=E_1=G_1=G_0=E_0=0\), we obtain certain cases of spatial flow of a plastic mass between inclined rigid plates, when the plates rotate with a prescribed velocity about the line of intersection of the contact surfaces. The boundary conditions, analogously to \((^7)\), are satisfied in the integral sense. A somewhat different particular solution in cylindrical coordinates, obtained in this way, is found in the article \((^8)\).

Other particular solutions of spatial problems in cylindrical coordinates having a different structure are given in works \((^{2-6})\).

Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR

Received
3 II 1964

CITED LITERATURE

\(^1\) A. Nadai, Zs. Phys., 30 (1924).
\(^2\) A. Yu. Ishlinskii, Prikl. matem. i mekh., 8, issue 3 (1944).
\(^3\) R. Hill, The Mathematical Theory of Plasticity, Oxford, 1950.
\(^4\) R. T. Shield, Proc. Roy. Soc., A 233, 267 (1955).
\(^5\) D. D. Ivlev, Prikl. matem. i mekh., 22, issue 5 (1958).
\(^6\) D. D. Ivlev, DAN, 123, No. 6 (1958).
\(^7\) M. A. Zadoyan, Izv. AN ArmSSR, ser. fiz.-matem. nauk, No. 3 (1964).
\(^8\) M. A. Zadoyan, Dokl. AN ArmSSR, 39, No. 5 (1964).