Full Text
Reports of the Academy of Sciences of the USSR
1964. Vol. 156, No. 1
THEORY OF ELASTICITY
M. A. ZADON
ON ONE PARTICULAR SOLUTION OF THE EQUATIONS OF THE THEORY OF IDEAL PLASTICITY
(Presented by Academician L. I. Sedov on 6 I 1964)
A particular solution of the general (three-dimensional) equations of the theory of ideal plasticity is found. Several particular problems correspond to this solution—the pure bending of a rectangular plate, the spatial flow of a plastic material between rough plates, the triaxial compression of a rectangular prism, etc.
The general relations of the theory of ideally plastic flow under the Huber–Mises condition in Cartesian coordinates have the form
\[ \frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{xy}}{\partial y} + \frac{\partial \tau_{xz}}{\partial z} =0,\qquad \frac{\partial \tau_{xy}}{\partial x} + \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_{yz}}{\partial z} =0,\qquad \frac{\partial \tau_{xz}}{\partial x} + \frac{\partial \tau_{yz}}{\partial y} + \frac{\partial \sigma_z}{\partial z} =0; \tag{1} \]
\[ (\sigma_x-\sigma_y)^2+(\sigma_y-\sigma_z)^2+(\sigma_z-\sigma_x)^2 +6(\tau_{xy}^2+\tau_{yz}^2+\tau_{xz}^2)=6k^2; \tag{2} \]
\[ \begin{aligned} \varepsilon_x&=\frac{\partial u}{\partial x} =\lambda(2\sigma_x-\sigma_y-\sigma_z),\\ \varepsilon_y&=\frac{\partial v}{\partial y} =\lambda(2\sigma_y-\sigma_z-\sigma_x),\\ \varepsilon_z&=\frac{\partial w}{\partial z} =\lambda(2\sigma_z-\sigma_x-\sigma_y); \end{aligned} \tag{3} \]
\[ 2\gamma_{xy} = \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} =6\lambda\tau_{xy},\qquad 2\gamma_{yz} = \frac{\partial v}{\partial z} + \frac{\partial w}{\partial y} =6\lambda\tau_{yz},\qquad 2\gamma_{xz} = \frac{\partial u}{\partial z} + \frac{\partial w}{\partial x} =6\lambda\tau_{xz}. \tag{4} \]
From relations (3)–(4) we have
\[ \begin{aligned} u(x,y,z)&=u_0(x,y)-\int \frac{\partial w}{\partial x}\,dz +2\int \gamma_{xz}\,dz,\\ v(x,y,z)&=v_0(x,y)-\int \frac{\partial w}{\partial y}\,dz +2\int \gamma_{yz}\,dz,\\ w(x,y,z)&=w_0(x,y)-\int(\varepsilon_x+\varepsilon_y)\,dz, \end{aligned} \tag{5} \]
where \(u_0, v_0\), and \(w_0\) are arbitrary functions of \(x\) and \(y\).
Assuming that the strain-rate tensor does not depend on \(x\) and \(y\), we find
\[ \begin{aligned} \varepsilon_x&=\frac{\partial u_0}{\partial x} -\frac{\partial^2 w_0}{\partial x^2}\,z =A_0+A_1z,\\ \varepsilon_y&=\frac{\partial v_0}{\partial y} -\frac{\partial^2 w_0}{\partial y^2}\,z =B_0+B_1z,\\ \gamma_{xy}&=\frac12\left( \frac{\partial u_0}{\partial y} + \frac{\partial v_0}{\partial x} \right) -\frac{\partial^2 w_0}{\partial x\,\partial y}\,z =C_0+C_1z. \end{aligned} \tag{6} \]
Here \(A_0, B_0, C_0, A_1, B_1, C_1\) are arbitrary constants.
Further, from (6) it follows that
\[ \begin{aligned} u_0&=A_0x+Dy+E,\qquad v_0=(2C_0-D)x+B_0y+F,\\ w_0&=-\frac{A_1}{2}x^2-\frac{B_1}{2}y^2-C_1xy-Gx-Hy-Q, \end{aligned} \tag{7} \]
where \(D, E, F, G, H, Q\) are also arbitrary constants.
From relations (3)—(4) we obtain
\[ \sigma_x=\sigma_z+\frac{2\varepsilon_x+\varepsilon_y}{\gamma_{xz}}\tau_{xz},\qquad \sigma_y=\sigma_z+\frac{\varepsilon_x+2\varepsilon_y}{\gamma_{xz}}\tau_{xz},\qquad \tau_{xy}=\frac{\gamma_{xy}}{\gamma_{xz}}\tau_{xz}. \tag{8} \]
Assuming that \(\tau_{xz}\) does not depend on \(x\) and \(y\), and substituting (8) into the equilibrium equations (1), we shall have
\[ \frac{\partial \sigma_z}{\partial x}+\frac{\partial \tau_{xz}}{\partial z}=0,\qquad \frac{\partial \sigma_z}{\partial y}+\frac{\partial \tau_{yz}}{\partial z}=0,\qquad \frac{\partial \sigma_z}{\partial z}=0, \tag{9} \]
and hence
\[ \sigma_z=-a_1x-b_1y-c_0,\qquad \tau_{xz}=a_1z+a_0,\qquad \tau_{yz}=b_1z+b_0. \tag{10} \]
Here \(a_0,b_0,c_0,a_1,b_1\) denote new arbitrary constants.
Determining from relations (2) and (8) the value of \(\gamma_{xz}\) (and consequently also \(\gamma_{yz}\)) and substituting into (8) and (5), we finally obtain
\[ \sigma_x=\sigma_z+(2\varepsilon_x+\varepsilon_y) \sqrt{\frac{k^2-\tau_{xz}^2-\tau_{yz}^2} {\varepsilon_x^2+\varepsilon_x\varepsilon_z+\varepsilon_y^2+\gamma_{xy}^2}}; \tag{11} \]
\[ \sigma_y=\sigma_z+(\varepsilon_x+2\varepsilon_y) \sqrt{\frac{k^2-\tau_{xz}^2-\tau_{yz}^2} {\varepsilon_x^2+\varepsilon_x\varepsilon_y+\varepsilon_y^2+\gamma_{xy}^2}}; \tag{12} \]
\[ \tau_{xy}=\gamma_{xy} \sqrt{\frac{k^2-\tau_{xz}^2-\tau_{yz}^2} {\varepsilon_x^2+\varepsilon_x\varepsilon_y+\varepsilon_y^2+\gamma_{xy}^2}}; \tag{13} \]
\[ u=2\int \sqrt{\frac{\varepsilon_x^2+\varepsilon_x\varepsilon_y+\varepsilon_y^2+\gamma_{xy}^2} {k^2-\tau_{xz}^2-\tau_{yz}^2}}\, \tau_{xz}\,dz+A_1xz+C_1yz+A_0x+Dy+Gz+E; \tag{14} \]
\[ v=2\int \sqrt{\frac{\varepsilon_x^2+\varepsilon_x\varepsilon_y+\varepsilon_y^2+\gamma_{xy}^2} {k^2-\tau_{xz}^2-\tau_{yz}^2}}\, \tau_{yz}\,dz+C_1xz+B_1yz+(2C_0-D)x+ \]
\[ \qquad\qquad +B_0y+Hz+F; \tag{15} \]
\[ w=\frac{A_1}{2}x^2-\frac{B_1}{2}y^2-\frac{A_1+B_1}{2}z^2-C_1xy-Gx-Hy-(A_0+B_0)z-Q. \tag{16} \]
The solution obtained, (10)—(16), contains 17 arbitrary constants. When \(A_1\ne0\) and \(B_1\ne0\), while all the remaining constants are equal to zero, we have the case of pure bending of a rectangular plate. When only \(A_0,E,c_0,a_1\) are nonzero, we obtain the case of plane deformation of a layer compressed by rough plates (Prandtl’s problem \((^1)\)). Taking \(A_1=B_1=C_1=G=H=Q=a_0=b_0=0\), we shall have the case of spatial flow of material between rough plates \((^{11})\), etc.
We note that some other particular solutions of the spatial problem of the theory of plasticity have been obtained in works \((^{2-10})\).
Institute of Mathematics and Mechanics
Academy of Sciences of the Armenian SSR
Received
4 I 1964
CITED LITERATURE
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