THEORY OF ELASTICITY

D. D. IVLEV

APPROXIMATE SOLUTION OF ELASTOPLASTIC PROBLEMS OF THE THEORY OF IDEAL PLASTICITY

(Presented by Academician A. I. Nekrasov, 12 X 1956)

We shall seek the solution of the elastoplastic problem in the form of series in powers of a certain parameter \(\delta\):

\[ \sigma_{ij}=\sum_{n=0}\delta^n\sigma_{ij}^{(n)},\qquad \sigma_{ii}\equiv\sigma_\rho,\qquad \sigma_{jj}\equiv\sigma_\theta,\qquad \sigma_{ij}\equiv\sigma_{ji}\equiv\tau_{\rho\theta}. \tag{1} \]

We shall confine ourselves to consideration, in the theory of plane strain, of the plasticity conditions of Mises and Saint-Venant \((^1)\), which are essentially coincident:

\[ \frac14(\sigma_\rho-\sigma_\theta)^2+\tau_{\rho\theta}^2=1; \tag{2} \]

in the theory of plane stress—the Saint-Venant plasticity condition \((^1)\):

\[ \frac14(\sigma-\sigma_\theta)^2+\tau_{\rho\theta}^2= \left[1-\frac12(\sigma_\rho+\sigma_\theta)\right]^2 \quad \text{for } \sigma_\theta>\sigma_\rho>0. \tag{3} \]

In (1), (2), (3) all quantities are dimensionless. Substituting (1) into (2), with \(\tau_{\rho\theta}^0=0\), we obtain:

\[ \sigma_\rho'-\sigma_\theta'=0,\qquad (\sigma_\rho''-\sigma_\theta'')\mu+\tau_{\rho\theta}'^{\,2}=0,\qquad \frac12(\sigma_\rho'''-\sigma_\theta''')\mu+\tau_{\rho\theta}'\tau_{\rho\theta}''=0, \tag{4} \]

where \(\mu=\operatorname{sign}(\sigma_\rho^0-\sigma_\theta^0)\).

Substituting (1) into (3) with \(\tau_{\rho\theta}^0=0\), we obtain:

\[ \sigma_\theta'=0,\qquad (1-\sigma_\rho^0)\sigma_\theta''+\tau_{\rho\theta}'^{\,2}=0,\qquad (1-\sigma_\rho^0)\sigma_\theta'''-\sigma_\rho'\sigma_\theta''+2\tau_{\rho\theta}'\tau_{\rho\theta}''=0. \tag{5} \]

If on the boundary \(L\) of the body under consideration

\[ \sigma_{\mathrm n}=P_{\mathrm n},\qquad \tau_{\mathrm n}=P_\tau, \tag{6} \]

then, assuming the equation of the contour \(L\) to be given in the form

\[ \rho=\sum_{n=0}\delta^n\rho_n(\theta), \]

we obtain the linearized boundary conditions (6) in the form

\[ \sigma_\rho' + \frac{d\sigma_\rho^0}{d\rho}\rho_1 = \frac{dP_{\mathrm n}}{d\rho}\rho_1;\qquad \tau_{\rho\theta}'-(\sigma_\theta^0-\sigma_\rho^0)\dot R_1 = \frac{dP_\tau}{d\rho}\rho_1; \]

\[ \sigma_\rho''+ \frac{d\sigma_\rho'}{d\rho}\rho_1+ \frac{d^2\sigma_\rho^0}{d\rho^2}\frac{\rho_1^2}{2!} + \frac{d\sigma_\rho^0}{d\rho}\rho_2 + (\sigma_\theta^0-\sigma_\rho^0)\dot R_1^{\,2} - 2\tau_{\rho\theta}'\dot R_1 = \]

\[ = \frac{d^2P_{\mathrm n}}{d\rho^2}\frac{\rho_1^2}{2!} + \frac{dP_{\mathrm n}}{d\rho}\rho_2; \]

\[ \begin{gathered} \tau_{\rho\theta}^{\prime\prime} -(\sigma_\theta^0-\sigma_\rho^0)(\dot R_2-R_1\dot R_1) -(\sigma_\theta'-\sigma_\rho')\dot R_1 +\frac{d}{d\rho}\left[\tau_{\rho\theta}'-(\sigma_\theta^0-\sigma_\rho^0)\dot R_1\right]\rho_1 =\\ =\frac{d^2P_\tau}{d\rho^2}\frac{\rho_1^2}{2!} +\frac{dP_\tau}{d\rho}\rho_2; \end{gathered} \tag{7} \]

\[ \begin{gathered} \sigma_\rho^{\prime\prime\prime} +\frac{d\sigma_\rho^{\prime\prime}}{d\rho}\rho_1 +\frac{d^2\sigma_\rho'}{d\rho^2}\frac{\rho_1^2}{2!} +\frac{d^3\sigma_\rho^0}{d\rho^3}\frac{\rho_1^3}{3!} +2(\sigma_\theta^0-\sigma_\rho^0)(\dot R_1\dot R_2-R_1\dot R_1^2)+\\ +\frac{d(\sigma_\theta-\sigma_\rho)}{d\rho}\dot R_1^2\rho_1 -(\sigma_\theta'-\sigma_\rho')\dot R_1^2 -2\tau_{\rho\theta}'(\dot R_2-R_1\dot R_1)-\\ -2\frac{d\tau_{\rho\theta}}{d\rho}\dot R_1\rho_1 -2\tau_{\rho\theta}^{\prime\prime}\dot R_1 +\frac{d\sigma_\rho'}{d\rho}\rho_2 +\frac{d^2\sigma_\rho^0}{d\rho^2}\rho_1\rho_2 +\frac{d\sigma_\rho^0}{d\rho}\rho_3=\\ =\frac{d^3P_\Pi}{d\rho^3}\frac{\rho_1^3}{3!} +\frac{d^2P_\Pi}{d\rho^2}\rho_1\rho_2 +\frac{dP_\Pi}{d\rho}\rho_3; \end{gathered} \]

\[ \begin{gathered} \tau_{\rho\theta}^{\prime\prime\prime} -2\tau_{\rho\theta}'\dot R_1^2 -(\sigma_\theta^0-\sigma_\rho^0)(\dot R_3-R_1\dot R_2+R_1^2\dot R_1-\dot R_1R_2-\dot R_1^3)-\\ -(\sigma_\theta'-\sigma_\rho')(\dot R_2-R_1\dot R_1) -(\sigma_\theta^{\prime\prime}-\sigma_\rho^{\prime\prime})\dot R_1 +\frac{d}{d\rho}\left[\tau_{\rho\theta}^{\prime\prime}-(\sigma_\theta^0-\sigma_\rho^0)(\dot R_2-R_1\dot R_1)-\right.\\ \left.-(\sigma_\theta'-\sigma_\rho')\dot R_1\right]\rho_1 +\frac{d^2}{d\rho^2}\left[\tau_{\rho\theta}'-(\sigma_\theta^0-\sigma_\rho^0)\dot R_1\right]\frac{\rho_1^2}{2!} +\frac{d}{d\rho}\left[\tau_{\rho\theta}'-(\sigma_\theta^0-\sigma_\rho^0)\dot R_1\right]\rho_2=\\ =\frac{d^3P_\tau}{d\rho^3}\frac{\rho_1^3}{3!} +\frac{d^2P_\tau}{d\rho^2}\rho_1\rho_2 +\frac{dP_\tau}{d\rho}\rho_3, \end{gathered} \]

where \(R_i=\rho_i/\rho_0\).

Since on the boundary of the plastic region \(L_s\) the solutions for the plastic and elastic regions are matched continuously (1):

\[ [\sigma_\rho]=[\sigma_\theta]=[\tau_{\rho\theta}]=0 \quad \text{on } L_s, \tag{8} \]

then, representing the equation of the contour \(L_s\) in the form

\[ \rho_s=\sum_{n=0}\delta^n\rho_{sn}(\theta), \tag{9} \]

we obtain the linearized matching conditions (8) in the form

\[ \begin{gathered} \left[\sigma_{ij}'+\frac{d\sigma_{ij}^0}{d\rho}\rho_{s1}\right]=0,\qquad \left[\sigma_{ij}^{\prime\prime} +\frac{d\sigma_{ij}'}{d\rho}\rho_{s1} +\frac{d^2\sigma_{ij}^0}{d\rho^2}\frac{\rho_{s1}^2}{2!} +\frac{d\sigma_{ij}^0}{d\rho}\rho_{s2}\right]=0;\\ \left[\sigma_{ij}^{\prime\prime\prime} +\frac{d\sigma_{ij}^{\prime\prime}}{d\rho}\rho_{s1} +\frac{d^2\sigma_{ij}'}{d\rho^2}\frac{\rho_{s1}^2}{2!} +\frac{d^3\sigma_{ij}^0}{d\rho^3}\frac{\rho_{s1}^3}{3!} +\frac{d\sigma_{ij}^0}{d\rho}\rho_{s3} +\frac{d\sigma_{ij}'}{d\rho}\rho_{s2}+\right. \tag{10}\\ \left. +\frac{d^2\sigma_{ij}^0}{d\rho^2}\rho_{s1}\rho_{s2} \right]=0\ \text{etc.} \end{gathered} \]

We note that from the equilibrium equations and conditions (8) it follows that

\[ \left[\frac{d\sigma_\rho^0}{d\rho}\right]=0 \quad \text{on } L_s . \]

Consequently, the matching conditions (10) for \(\sigma_\rho\) and \(\tau^{\rho\theta}\) play the role of boundary conditions for determining the stresses in the elastic region, while the matching condition for \(\sigma_\theta\) serves to determine \(\rho_{sn}\). The greatest difficulty and interest in such problems is the determination of the equation of the boundary of the plastic zone \(L_s\). We shall present several approximate solutions.

1. Biaxial tension of a thick plate with a circular hole of radius \(a\) by forces \(P_1\) and \(P_2\). Denoting

\[ \delta=\frac{|P_1-P_2|}{2k}, \tag{11} \]

where \(k=\frac14\sigma_s\) according to Saint-Venant and \(k=\frac13\sigma_s\) according to Mises, one may obtain:

\[ \rho_s=1+\delta\cos 2\theta-\frac34\delta^2(1-\cos 4\theta)+\frac58\delta^3(-\cos 2\theta+\cos 6\theta)+ \]
\[ +\frac{7}{64}\delta^4(-1-4\cos 4\theta+5\cos 8\theta)+\cdots \tag{12} \]

Expansion (12) coincides exactly with the expansion of the equation of an ellipse with semiaxes \((1+\delta)\), \((1-\delta)\), which, as shown in \((^2)\), is the exact solution of this problem.

1. Biaxial tension of a thin plate with a circular hole of radius \(a\) by forces \(P_1\) and \(P_2\). Introducing \(\delta\) (11), one may obtain:

\[ \rho_s=1+4\delta^*\cos 2\theta-8\delta^{*2}(1-2\cos 4\theta)-80\delta^{*3}(\cos 2\theta-\cos 6\theta)+ \]
\[ +32\delta^{*4}(1-16\cos 4\theta+14\cos 8\theta)+\cdots, \tag{13} \]

where \(\delta^*=\delta/\alpha;\ \alpha=a/r_s^0;\ r_s^0\) is the size of the radius of the plastic zone for \(\delta=0\). The first approximation (13) was obtained in \((^3)\).

1. Biaxial tension of a thin plate with an elliptical hole by forces \(P_1d_2\) and \(P_2d_2\), directed at an angle \(\theta_0\) to the principal central axes of the ellipse. Introducing \(\delta\) (11), one may obtain:

\[ \rho_s=1+\delta^*(4d_2\cos 2(\theta-\theta_0)+3\alpha d_1\cos 2\theta)+\delta^{*2}\{d_1^2(\alpha^2/4-8\alpha^4)- \]
\[ -(18d_1d_2\alpha\cos 2\theta_0+8d_2^2)+[-d_1^2(15/4\,\alpha^2-8\alpha^3-3/4\,\alpha^4)+ \]
\[ +(18d_1d_2\alpha\cos 2\theta_0+16d_2^2\cos 4\theta_0)]\cos 4\theta+ \]
\[ +[18d_1d_2\alpha\sin 2\theta_0+16d_2^2\sin 4\theta_0]\sin 4\theta\}+\cdots, \]

where the equation of the ellipse of the hole is represented in the form

\[ \rho=\alpha+\delta\alpha d_1\cos 2\theta-\delta^2\frac{3\alpha d_1^2}{4}(1-\cos 4\theta)+\cdots . \]

For \(d_1=0,\ \theta_0=0,\ d_2=1\) there is the case of biaxial tension of a thin plate with a circular hole; for \(d_2=0,\ d_1=1\), the case of uniform tension of a thin plate with an elliptical hole.

1. An eccentric tube under the action of internal pressure \(p_0\). Referring all linear quantities to the outer radius of the tube \(b\), denote \(\delta=c/b\), where \(c\) is the eccentricity of the tube. One may obtain:

\[ \rho_s=\beta_0-\delta\frac{2\beta_0^4}{1-\beta_0^4}\cos\theta+\delta^2\left\{\frac{\beta_0^3(2-\beta_0^4-\beta_0^6)}{(1-\beta_0^2)(1-\beta_0^4)^2}+\frac{2\beta_0^7}{(1-\beta_0^4)^2}+\right. \]
\[ +\frac{1}{N}\left[-\frac{(1-\beta_0^2)(1-3\beta_0^4)}{\beta_0} +\frac{(1-\beta_0^2)^2(5+3\beta_0^4)\beta_0}{(1-\beta_0^4)}-\right. \]
\[ \left.\left.-\frac{\beta_0^3}{(1-\beta_0^4)^2}\bigl[(1+\beta_0^2)^4+4\beta_0^4(2-\beta_0^2)^2-4(1+4\beta_0^4)\bigr] +\frac{2\beta_0^3(1-\beta_0^2)^2}{(1+\beta_0^2)^2}\right]\cos 2\theta\right\}, \]

where \(\beta_0=r_s^0/b,\ N=(\beta_0-1/\beta_0)^4\).

Moscow State University
named after M. V. Lomonosov

Received
9 X 1956

References

1. V. V. Sokolovskii, Theory of Plasticity, Moscow, 1950.
2. L. A. Galin, Applied Mathematics and Mechanics, vol. 3 (1946).
3. A. P. Sokolov, DAN, vol. 60, No. 1 (1948).
4. G. N. Savin, Stress Concentration around Holes, Moscow, 1951.