Machine Interpretation of Results of the Polarization-Optical Method for Stress Analysis
Abstract Generated abstract
A computational procedure is presented for interpreting polarization-optical stress analysis data without directly measuring transverse model deformations. The method derives hydrostatic pressure at grid nodes by numerically integrating equilibrium equations expressed through fringe order, fringe value, and isocline angle, then calculates the stress components from these quantities. The algorithm was programmed for the Minsk-1 computer for a triangular region with a load-free contour and applied to contact stresses in sheet rolling. In the reported example, the rolling force obtained from the calculated normal stress distribution agreed with the measured force.
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Proceedings of the Tomsk Order of the October Revolution and Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov
Vol. 225 1972
Machine Interpretation of Results from the Polarization-Optical Method of Stress Analysis
V. D. Del
(Presented by the scientific seminar of the Laboratory of Plastic Deformations)
The difficulty of measuring the transverse deformations of a model when studying stresses by the polarization-optical method makes it expedient to determine stresses directly from the patterns of fringes and isoclines. The latter is usually associated with the numerical integration of the differential equations of equilibrium.
Because of the great labor intensity of such a calculation, a procedure has been developed that is convenient for implementation on digital computers.
The hydrostatic pressure at the nodes of the coordinate grid imposed on the region under investigation is determined by numerical integration of the differential equations:
\[ \frac{\partial \sigma}{\partial x} = -\frac{1}{3} \left[ \left( \frac{\partial \tau}{\partial y} - 2\tau \frac{\partial \theta}{\partial x} \right)\sin 2\theta + \left( \frac{\partial \tau}{\partial x} + 2\tau \frac{\partial \theta}{\partial y} \right)\cos 2\theta \right], \]
\[ \frac{\partial \sigma}{\partial y} = -\frac{1}{3} \left[ \left( \frac{\partial \tau}{\partial x} + 2\tau \frac{\partial \theta}{\partial y} \right)\sin 2\theta + \left( 2\tau \frac{\partial \theta}{\partial x} - \frac{\partial \tau}{\partial y} \right)\cos 2\theta \right]. \tag{1} \]
obtained by substituting the known relations
\[ \sigma_x=\frac{3}{2}\sigma+\frac{\tau}{2}\cos 2\theta,\quad \sigma_y=\frac{3}{2}\sigma-\frac{\tau}{2}\cos 2\theta,\quad \tau_{xy}=\frac{\tau}{2}\sin 2\theta \tag{2} \]
into the differential equations of equilibrium
\[ \frac{\partial \sigma_x}{\partial x} + \frac{\partial \tau_{xy}}{\partial y} =0;\quad \frac{\partial \sigma_y}{\partial y} + \frac{\partial \tau_{xy}}{\partial x} =0. \tag{3} \]
In equations (1) and (2), the maximum shear stress is
\[ \tau=\frac{\sigma_1-\sigma_2}{2}, \]
where \(\sigma_1\) and \(\sigma_2\) are the principal stresses \((\sigma_1>\sigma_2)\), determined from the fringe order \(k\) and the fringe value \(\sigma_0\). The angle \(\theta\) between the direction of \(\sigma_1\) and the \(x\)-axis is found from the isocline pattern. By integrating equations (1), the hydrostatic pressure \(\sigma\) is determined; then, from relations (2), the stresses \(\sigma_x\), \(\sigma_y\), \(\tau_{xy}\) are calculated.
The stress calculation was programmed on the “Minsk-1” digital computer for the case of determining stresses in a triangular region (Fig. 1, a), poi-
Fig. 1. Contact stresses in sheet rolling
…to which belongs the contour free of external load. The calculation flowchart is presented in Fig. 2 ($n$ is the number of nodes on the ray $AB$).
The calculation begins from point $A$. By integrating the first of equations (1), the values at the nodes of the ray $AB$ are calculated, taking into account its rotation relative to the $X$ axis. Integration of the second equation then gives the values of $\sigma$ at the remaining nodes of the coordinate grid.
When performing the numerical integration, a difference scheme of the form
\[ \sigma_1=\sigma_0+f_{01}, \]
was used, where
$f_{01}$ is the right-hand side of equations (1), calculated for the point lying between the points with indices 0 and 1.
Initial data: $n$, $\theta_0$; $R_{k,e}$, $\theta^{0}_{kp}$
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Conversion to the binary system, formation of variables
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Calculation of $k$ and conversion of $\theta^{0}$ to radians
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Calculation of $\sigma$ at the boundary point; calculation of $\tau_{xy}$, $\sigma_x$, $\sigma_y$ and printing
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Calculation of $\sigma$ at the nodes of the boundary row; calculation and printing of stresses
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Calculation and printing of $\sigma$, $\tau_{xy}$, $\sigma_x$, $\sigma_y$, $\theta$ at the nodes of the next row
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Check for completion of the computations
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Stop
Fig. 2. Flowchart for calculating stresses from optical-method data.
The developed procedure was used to investigate the stress state of rolls during sheet rolling. In Fig. 1b an epure of contact stresses is shown for the rolling of a lead strip 1.7 mm thick between rolls made of an optically active material. The reduction of the strip was 25%. The rolling force calculated from the epure $\sigma_y$ coincided with the actual value.