USE OF ELECTRONIC DIGITAL COMPUTERS IN A SEMI-EXPERIMENTAL STUDY OF STRESSES IN THE PLASTIC REGION

V. D. Del, G. D. Del

(Presented by the Department of Strength of Materials)

In view of the fact that analytical and experimental determination of stresses in the plastic region is often impossible, semi-experimental methods are finding ever wider application [1, 2, 3]. Determination of stresses by these methods is associated with the numerical integration of the differential equilibrium equations. In this connection, the use of modern means of computing technology in this area appears promising.

The present work sets the task of developing a methodology for studying the plane stress state in the plastic region with the use of an electronic digital computer. The problem is solved on the basis of the method for determining stresses from the distribution of hardness and transverse strain [3].

To determine the stresses \(\sigma_x\), \(\sigma_y\), \(\tau_{xy}\) in the general case, we have the following system of equations:

\[ \left. \begin{aligned} \sigma_x+\sigma_y &= 3\sigma;\\ \sigma_x^2+\sigma_y^2-\sigma_x\sigma_y+3\tau_{xy}^2 &= \sigma_i^2;\\ \frac{\partial\sigma_x}{\partial x}+\frac{\partial\tau_{xy}}{\partial y} &= 0;\\ \frac{\partial\tau_{xy}}{\partial y}+\frac{\partial\sigma_y}{\partial y} &= 0. \end{aligned} \right\} \tag{1} \]

To solve it, it is necessary to know the value of the hydrostatic pressure \(\sigma\).

In the case of a plane stress state,

\[ \sigma=-\frac{2}{3}\,\frac{\sigma_i}{l_i}\,l_z . \tag{2} \]

The stress intensity \(\sigma_i\) and strain intensity \(e_i\) entering into this expression are determined under the assumption of a single flow curve from hardness; the transverse strain is determined under the assumption of its homogeneity,

\[ l_z=\ln\frac{h}{h_0}. \]

Here \(h_0\) and \(h\) are the transverse dimension before and after deformation, respectively.

The number of equations in system (1) is greater than the number of unknowns; therefore, various methods of solving it are possible. In choosing a method, the determining factors are the reliability of the boundary conditions required for the calculation, the accuracy with which the experimental data are determined, and the efficiency of use of the computing machine.

A comparative analysis of calculation methods was carried out as applied to the radial upsetting of a disk. A disk of St. 3 steel, 160 mm in diameter and 50 mm thick, was subjected to upsetting. Figure 1 shows a quarter of the deformed disk.

The boundary \(BC\) is free of loads; the straight lines \(OA\) and \(AB\) are segments of axes of symmetry. A grid with a spacing of 5 mm was applied to the specimen; at the nodes of this grid the hardness and the magnitude of the transverse deformation were measured.

Fig. 1

From these data the quantities necessary for calculating the hydrostatic pressure are determined.

Let us determine the stresses at the grid nodes.

On the axes of symmetry \(\tau_{xy}=0\); therefore \(\sigma_x\) and \(\sigma_y\) can be determined by the simultaneous solution of the first two equations of system (1). This gives the boundary conditions for the calculation.

Since the system given above contains four equations with three unknowns, four solution paths are possible (the number of combinations of four taken three at a time). Analysis showed that the following is the most acceptable of them.

We use the equations:

\[ \left. \begin{aligned} \frac{\partial\sigma_x}{\partial x}+\frac{\partial\tau_{xy}}{\partial y}&=0;\\ \frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\sigma_y}{\partial y}&=0;\\ \sigma_x^2+\sigma_y^2-\sigma_x\sigma_y+3\tau_{xy}^{\,2}&=\sigma_i^{\,2}. \end{aligned} \right\} \tag{3} \]

For the node with coordinates \(k-1, l\), the first equation in finite differences is written in the following form:

\[ \frac{(\sigma_x)_{k-1,l+1}-(\sigma_x)_{k-1,l-1}}{2\Delta} + \frac{(\tau_{xy})_{k,l}-(\tau_{xy})_{k-2,l}}{2\Delta} =0. \]

Hence

\[ (\tau_{xy})_{k,l}=(\tau_{xy})_{k-2,l}-(\sigma_x)_{k-1,l+1}+(\sigma_x)_{k-1,l-1}. \]

Since at points symmetric with respect to the axis of symmetry the tangential stresses are equal in magnitude but opposite in sign, for the nodes of the first row we obtain

\[ (\tau_{xy})_{1,l}=-\frac{(\sigma_x)_{0,l+1}+(\sigma_x)_{0,l-1}}{2}. \]

From the second equation,

\[ (\sigma_y)_{k,l}=(\sigma_y)_{k-2,l}-(\tau_{xy})_{k-1,l+1}+(\tau_{xy})_{k-1,l-1}. \]

\(\sigma_x\) is now determined from the third equation.

The method described is comparatively easily translated into machine language. Along with the enormous saving of time and increase in accuracy, an advantage of the machine method of solution here is the elimination of random errors, inevitable in manual calculation, as well as the elimination of the subjective factor in the computations.

Using the method set forth, stresses were calculated for radial compression of a disk by 10%. Near the horizontal axis of symmetry, multiplying the diagram of \(\sigma_y\) by the thickness of the disk gave, with sufficient accuracy, the magnitude of the deforming force. However, as one approaches the contact area, the error in determining \(\sigma_y\) increases rapidly. This is explained, first, by the influence of friction in the contact, which disturbs the homogeneity of the transverse strain, and, second, by the accumulation of error in the course of numerical integration.

At smaller strains the influence of friction is less, and therefore the stresses can be estimated with greater accuracy. However, at small strains (up to 5–7%) a considerable part of the section is deformed elastically; therefore the method set forth is inapplicable here.

The following method is proposed.

If the half-difference of the principal stresses at a point is denoted by \(\tau\),

\[ \tau=\frac{\sigma_1-\sigma_2}{2}, \]

then for the stresses at any point one may write [4]

\[ \sigma_x=\frac{3}{2}\sigma+\tau\cdot\cos 2\vartheta;\quad \sigma_y=\frac{3}{2}\sigma-\tau\cdot\cos 2\vartheta;\quad \tau_{xy}=\tau\cdot\sin 2\vartheta, \tag{4} \]

where \(\vartheta\) is the angle between the direction of \(\sigma_1\) and the \(x\)-axis.

The hydrostatic pressure is calculated by the method described above. By solving jointly the first two equations of system (1) with respect to the principal stresses, we obtain

\[ \tau=\sigma_i\sqrt{\frac{1}{3}-\frac{1}{3}\left(\frac{e_z}{e_i}\right)^2}. \]

Substituting (4) into the differential equations, we find

\[ \frac{\partial\vartheta}{\partial x} = \frac{1}{2\tau} \left( \frac{3}{2}\frac{\partial\sigma}{\partial x}\sin 2\vartheta - \frac{3}{2}\frac{\partial\sigma}{\partial y}\cos 2\vartheta + \frac{\partial\tau}{\partial y} \right), \]

\[ \frac{\partial\vartheta}{\partial y} = -\frac{1}{2\tau} \left( \frac{3}{2}\frac{\partial\sigma}{\partial x}\cos 2\vartheta + \frac{3}{2}\frac{\partial\sigma}{\partial y}\sin 2\vartheta + \frac{\partial\tau}{\partial x} \right). \]

In finite differences the expressions obtained are written as follows:

\[ \vartheta_{\kappa,l}=\vartheta_{\kappa,l-2} +\frac{1}{2\tau_{\kappa,l-1}} \left[ \frac{3}{2}\left(\sigma_{\kappa,l}-\sigma_{\kappa,l-2}\right) \sin 2\vartheta_{\kappa,l-1} -\right. \]

\[ \left. -\frac{3}{2}\left(\sigma_{\kappa+1,l-1}-\sigma_{\kappa-1,l-1}\right) \cos 2\vartheta_{\kappa,l-1} +\tau_{\kappa+1,l-1}-\tau_{\kappa-1,l-1} \right]. \tag{5} \]

\[ \vartheta_{\kappa,l}=\vartheta_{\kappa-2,l} -\frac{1}{2\tau_{\kappa-1,l}} \left[ \frac{3}{2}\left(\sigma_{\kappa-1,l+1}-\sigma_{\kappa-1,l-1}\right) \cos 2\vartheta_{\kappa-1,l} +\right. \]

\[ \left. +\frac{3}{2}\left(\sigma_{\kappa,l}-\sigma_{\kappa-2,l}\right) \sin 2\vartheta_{\kappa-1,l} +\tau_{\kappa-1,l+1}-\tau_{\kappa-1,l-1} \right]. \tag{6} \]

Taking into account that at points symmetric with respect to the axis of symmetry the values of \(\vartheta\) are equal in magnitude but opposite in sign, from (5) we obtain, for the points of the first column,

\[ \vartheta_{\kappa,1} = \frac{1}{4\tau_{\kappa,0}} \left[ -\frac{3}{2}\left(\sigma_{\kappa+1,0}-\sigma_{\kappa-1,0}\right) +\tau_{\kappa+1,0}-\tau_{\kappa-1,0} \right]. \tag{7} \]

This procedure is readily amenable to automation. It was tested in the following experiment.

A disk of the same dimensions as before was compressed in the radial direction by 5%. The transverse deformation \(\Delta h\) was measured with an indicator at the grid nodes. Hardness was measured by the Brinell process with a ball of diameter \(10\ \mathrm{mm}\) under a load of 3 t. The points for measuring hardness in the different quarters of the disk were chosen so that, when reduced to one quarter, they gave the hardness at all nodes of the grid. The left half of Fig. 2 shows the distribution of hardness over the cross section of the disk; on the right are lines of equal transverse deformation \(e_z\) (in %).

Fig. 2

To reduce the influence of friction, the experiment was carried out with lubrication of the contact surface.

The stresses were calculated on the Minsk-1 electronic digital computer. The resulting diagram of compressive stresses \(\sigma_y\) in the contact is shown in Fig. 2. The area of this diagram, multiplied by the thickness of the specimen, gives the value

of the compressive force with an error of about 5%. The tangential stresses at the contact are close to zero.

The flow chart of the solution is shown in Fig. 3.

Thus, the only initial data for the calculation are the experimental results. All calculations and intermediate checks are performed by the machine.

After computing \(e_z\), \(\sigma\), and \(\tau\), the values of \(\vartheta\) are first calculated at the nodes of the first column from expression (7). Then, using expression (5), the machine successively calculates \(\vartheta\) at the nodes of the rows of the grid up to

Input of initial data
ε_i, ℓ_i, Δh
        ↓
Calculation of e_z
        ↓
Calculation of σ and τ at the grid nodes
        ↓
Calculation of values of ϑ at the grid nodes
        ↓
Computation of stresses σ_x, σ_y, τ_xy
at the grid nodes and printing of results

Fig. 3

the contact area and, by (6), determines \(\vartheta\) at the contact area. Now, from the computed \(\sigma\), \(\tau\), and \(\vartheta\), the stresses at the grid nodes are calculated.

The proposed method makes it possible to determine stresses along any ray, provided that the direction of the principal stresses is known at least at one of its points.

REFERENCES

1. E. G. Thomsen and J. T. Lapsley. Experimental Stress Determination within a Metal during Plastic Flow, Proceeding Soc. Exp. Stress Analysis, 11, 1954.

2. A. Ya. Aleksandrov, M. Kh. Akhmetzyanov. Investigation of elastic-plastic problems by the method of photoelasticity. Proceedings of the Second All-Union Congress on Theoretical and Applied Mechanics. Mechanics of Solids. Publishing House of the Academy of Sciences of the USSR, Moscow, 1966.

3. G. D. Del. Determination of the stressed state in the plastic region by hardness distribution. Proceedings of the Institutes of the State Committee of Standards, Measures, and Measuring Instruments. Leningrad, 1967.

4. V. I. Fedos’ev. Strength of Materials. Nauka Publishing House, Moscow, 1964.