Method for Determining the Stress State by Measuring Hardness in Plates with Holes
Abstract Generated abstract
This paper presents a procedure for determining stresses in the plastic region near the free contour of holes in stretched brass plates using hardness measurements. Hardness values measured on a square grid are converted to stress intensity by a calibration curve, then combined with boundary conditions, equilibrium equations in finite difference form, and the stress intensity relation to compute the stress components successively across the plastic zone. The method was applied to a triangular region adjacent to the hole contour and calculated on a Minsk-1 digital computer. Comparison with stress diagrams obtained from hardness and transverse deformation measurements, and with the applied tensile force under an assumption of uniform through-thickness deformation, showed agreement within about 4 percent for the force check.
Full Text
METHOD FOR DETERMINING THE STRESS STATE BY HARDNESS MEASUREMENT IN PLATES WITH HOLES
B. P. Chebaevskii, V. D. Del
(Presented by the scientific seminar of the Department of Strength of Materials)
The article sets forth a method for determining stresses in the plastic region by measuring hardness in plates with holes. The investigation was carried out on brass plates stretched by an external force (Fig. 1). The contour \(AB\) is free of loads. The \(x\)-axis is the axis of symmetry. It is required to determine the stresses in the plastic region adjacent to the contour \(AB\).
Let us superpose on the indicated region a square grid with spacing \(\Delta = 1\) mm. Having measured the hardness at the grid nodes, with the aid of the calibration curve “hardness—stress intensity” [1] we determine the stress intensity \(\sigma_i\). Along \(AB\) the stress normal to the contour is equal to 0, and that directed along the contour is equal to \(\sigma_i\). For definiteness we set \(\sigma_1 = 0,\ \sigma_2 = \sigma_i\). Let us determine the stresses in the column nearest to the contour. At the point \(0.0\), \(\sigma_x = 0,\ \sigma_y = \sigma_i,\ \tau_{xy} = 0\). To determine the stresses at the subsequent points we proceed as follows. At the point \(0.0\)
\[ \frac{\partial \sigma_1}{\partial S_1} + \frac{\sigma_1 - \sigma_2}{\rho_2} = 0,\qquad \left(\frac{\partial \sigma_1}{\partial S_1}\right)_{0.0} = \left(\frac{\sigma_i}{\rho_2}\right)_{0.0}, \tag{1} \]
where \(S_1\) is the coordinate measured along the trajectory \(\sigma_1\);
\(\rho_2\) is the radius of curvature at the point \(0.0\), then
\[ (\sigma_x)_{0.1} = (\sigma_1)_{0.0} + \left(\frac{\partial \sigma_1}{\partial S_1}\right)_{0.0}\cdot \Delta . \tag{2} \]
To determine \((\sigma_2)_{0.1}\) we use the equation
\[ \sigma_1^2 + \sigma_2^2 - \sigma_1\cdot\sigma_2 = \sigma_i^2 . \tag{3} \]
\[ (\sigma_y)_{0.1} = (\sigma_2)_{0.1} = \frac{(\sigma_1)_{0.1} \pm \sqrt{4(\sigma_i)^2_{0.1} - 3(\sigma_1)^2_{0.1}}}{2}. \tag{4} \]
(The choice of sign is made in accordance with the specific conditions.) The stresses in the system of axes \(x, y\) are determined from the known relations:
\[ \sigma_x = \frac{\sigma_1 + \sigma_2}{2} + \frac{\sigma_1 - \sigma_2}{2}\cdot \cos 2\Theta, \]
\[ \sigma_y = \frac{\sigma_1 + \sigma_2}{2} - \frac{\sigma_1 - \sigma_2}{2}\cos 2\Theta,\qquad \tau_{xy} = \frac{\sigma_1 - \sigma_2}{2}\cdot \sin 2\Theta . \tag{5} \]
Let us express the stresses at the points of the zeroth column through the stresses at the points of the first column and the stresses on the contour \(AB\), calculated by formulas (5). For point 1,0, for example, we can write
\[ (\sigma_x)_{1.0} \simeq (\sigma_x)_k + \left(\frac{\partial \sigma_x}{\partial x}\right)_k \cdot \delta, \]
\[ (\sigma_y)_{1.0} \simeq (\sigma_y)_k + \left(\frac{\partial \sigma_y}{\partial y}\right)_k \cdot \delta, \tag{6} \]
\[ (\tau_{xy})_{1.0} \simeq (\tau_{xy})_k + \left(\frac{\partial \tau_{xy}}{\partial x}\right)_k \cdot \delta. \]
Fig. 1. Computational scheme
Or, in finite differences,
\[ (\sigma_x)_{1.0} \simeq (\sigma_x)_k + \frac{(\sigma_x)_{1.1} - (\sigma_x)_k}{\Delta + \delta}\cdot \delta, \]
\[ (\sigma_y)_{1.0} \simeq (\sigma_y)_k + \frac{(\sigma_y)_{1.1} - (\sigma_y)_k}{\Delta + \delta}\cdot \delta, \]
\[ (\tau_{xy})_{1.0} \simeq (\tau_{xy})_k + \frac{(\tau_{xy})_{1.1} - (\tau_{xy})_k}{\Delta + \delta}\cdot \delta. \]
To determine the stresses at point 1.1, we write the equilibrium equations for point \(a\) (Fig. 1)
\[ \frac{\partial \sigma_x}{\partial x}+\frac{\partial \tau_{xy}}{\partial y}=0, \tag{7} \]
\[ \frac{\partial \sigma_y}{\partial y}+\frac{\partial \tau_{xy}}{\partial x}=0, \tag{8} \]
or, in finite differences,
\[ (\sigma_x)_{1.1}-(\sigma_x)_{1.0}+(\sigma_x)_{0.1}-(\sigma_x)_{0.0} +(\tau_{xy})_{1.0}+(\tau_{xy})_{1.1}-\tau_{xy0.0}-(\tau_{xy})_{0.1}=0, \]
\[ (\sigma_y)_{1.1}+(\sigma_y)_{1.0}-(\sigma_y)_{0.1}-(\sigma_y)_{0.0} +(\tau_{xy})_{1.1}+(\tau_{xy})_{0.1}-(\tau_{xy})_{0.0}-(\tau_{xy})_{0.0}=0. \]
Introducing the equation
\[ (\sigma_y)_{1.1}^{2}+(\sigma_x)_{1.1}^{2} -(\sigma_y)_{1.1}(\sigma_x)_{1.1} +3(\tau_{xy})_{1.1}^{2}=(\sigma_i)_{1.1}^{2} \tag{9} \]
and solving equations (7), (8), (9) simultaneously, we determine the stresses at point 1.1:
\[ (\tau_{xy})_{1.1}=\frac{B-aA}{a^2-1}, \qquad (\sigma_y)_{1.1}=\frac{A-aB}{a^2-1}, \]
\[ (\sigma_x)_{1.1}= \frac{(\sigma_y)_{1.1}\pm \sqrt{4(\sigma_i)_{1.1}^{2}-12(\tau_{xy})_{1.1}^{2}-3(\sigma_y)_{1.1}^{2}}}{2}. \tag{10} \]
Here
\[ A=\{(\sigma_x)_{0.1}-(\sigma_x)_k-(\sigma_x)_{0.0} +(\tau_{xy})_k-(\tau_{xy})_{0.0}-(\tau_{xy})_{0.1} +\delta[(\sigma_x)_{0.1}- \]
\[ -(\sigma_x)_{0.0}-(\tau_{xy})_{0.0}-(\tau_{xy})_{0.1}]\}, \]
\[ B=\{(\sigma_y)_k-(\sigma_y)_{0.1}-(\sigma_y)_{0.0} +(\tau_{xy})_{0.1}-(\tau_{xy})_k-(\tau_{xy})_{0.0} +\delta[(\tau_{xy})_{0.1}- \]
\[ -(\sigma_y)_{0.1}-(\sigma_y)_{0.0}-(\tau_{xy})_{0.0}]\}. \]
\[ a=1+2\delta. \]
In the same way we determine the stresses at all points of the first column, and, using formulas (6), we determine the stresses at the points of the zero column.
To determine the stresses \((\sigma_x)\) at the points of the second column, we write the equilibrium equation (7) in finite differences for the corresponding points of the first column. For point 0.1 it will be
\[ (\sigma_x)_{0.2}-(\sigma_x)_{0.0}+2(\tau_{xy})_{1.1}=0. \]
Hence
\[ (\sigma_x)_{0.2}=(\sigma_x)_{0.0}-2(\tau_{xy})_{1.1}. \]
Using formulas (4), for point 0.2 we determine \((\sigma_y)\). To determine the stresses \((\sigma_y)\), \((\tau_{xy})\) at point 1.2, we write equation (7) in finite differences for point \(b\) (Fig. 1):
\[ (\tau_{xy})_{1.2}-(\tau_{xy})_{0.2}+(\tau_{xy})_{1.1}-(\tau_{xy})_{0.1} +(\sigma_x)_{1.2}+(\sigma_x)_{0.2}-(\sigma_x)_{1.1}-(\sigma_x)_{0.1}=0. \tag{11} \]
Hence
\[ (\tau_{xy})_{1.2}=C-(\sigma_x)_{1.2}, \]
where
\[ C=(\tau_{xy})_{0.2}-(\tau_{xy})_{1.1}+(\tau_{xy})_{0.1} -(\sigma_x)_{0.2}+(\sigma_x)_{1.1}+(\sigma_x)_{0.1}. \]
Solving equation (9) for point 1.2, we obtain
\[ (\sigma_y)_{1.2}= \frac{(\sigma_x)_{1.2}\pm \sqrt{4(\sigma_i)_{1.2}^{2}-12(\tau_{xy})_{1.2}^{2}-3(\sigma_x)_{1.2}^{2}}}{2}. \]
Further, according to the scheme given, we determine the stresses at the remaining points of the plastic region.
By this procedure, stresses in the triangular region \(ABC\) (Fig. 1) were calculated on the Minsk-1 digital computer. The resulting stress diagrams along the \(x\)-axis were compared with analogous diagrams obtained by measuring hardness and transverse deformation [2]. Both diagrams are shown in Fig. 2.
Fig. 2. Diagrams of the principal stresses along the \(x\)-axis
Assuming the deformation to be uniform through the thickness of the plate, we checked the obtained \(\sigma_2\) diagram against the tensile force. The difference between the calculated and actual tensile forces does not exceed 4%.
REFERENCES
- G. D. Del’, “Hardness of Deformed Metal,” Izv. AN SSSR, Metally, No. 4, 1967.
- G. D. Del’, “Determination of the Stress State of a Material by Hardness Measurement,” Trudy metrologicheskikh institutov SSSR, issue 91, 1967.