Abstract Generated abstract
This paper derives a particular three-dimensional solution of the equations of ideal plastic flow under the Huber-Mises yield condition in Cartesian coordinates. By assuming that selected components of the strain-rate tensor are independent of two spatial coordinates and that a shear stress component has corresponding restricted dependence, the author reduces the equilibrium and plasticity relations to explicit formulae for stresses, velocities, and displacements. The resulting family of solutions contains seventeen arbitrary constants and includes, as special cases, pure bending of a rectangular plate, plane deformation of a layer compressed by rough plates, spatial flow between rough plates, and triaxial compression of a rectangular prism.
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
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