ON THE QUESTION OF DETERMINING FORCES BY THE CRUSHER METHOD

L. G. TSEKHANOVA

(Presented by the scientific seminar of the Department of Strength of Materials)

The simplest method for determining large forces is the crusher method. Cylindrical crushers made of annealed copper are usually used; these undergo plastic deformation in the axial direction. As is known, the dependence between force and deformation in such crushers is curvilinear. On the compression curve two regions are distinguished: the doupolytropic and the polytropic. Representing the change in force with deformation in logarithmic coordinates, we obtain a linear dependence for the polytropic part of the compression curve. The magnitude of the polytropic exponent depends on many factors, including the dimensions of the crusher. However, under certain conditions this dependence can be used to determine the force from the amount of deformation of the crusher. It is more convenient to use crushers that have undergone a large degree of preliminary plastic deformation,

Fig. 1.

since in this case the doupolytropic part of the compression curve disappears. But this entails a substantial drawback, namely that small loads cannot be measured and, moreover, excessively small deformations reduce the accuracy of determining the forces [1].

In the present work it is proposed to use, as crushers, cylindrical steel specimens deformed in the radial direction (Fig. 1).

We conducted experiments with such specimens. The specimens were made of mild steel, grade St. 3, with various ratios of diameter \(d\) to length \(l\), with the range of variation of volume \(v\) from \(0.065\) to \(4.72\ \mathrm{cm}^3\). The compressive load was applied in stages, followed by unloading and measurement of the area of the imprint on the lateral surface of the specimen. At small loads the imprint has the form of a rectangle, the measurement of which presents no difficulty. As the load increases, the imprint acquires complex outlines, and its area was measured by us by planimetry. For this purpose the imprint was transferred to paper. When the specimen is compressed, two strain-hardening cores are visible on its end surfaces [2], the boundaries of which expand as the load increases. At large forces the cores approach one another and flatten out.

Fig. 2.

As the tests showed, the dependence between the compressive load \(P\) and the imprint area \(F\), up to a certain limit, proves to be linear (Fig. 2). This initial dependence is characterized by the value \(\tan \alpha\), where \(\alpha\) is the angle between the straight line of deformation and the horizontal axis of imprint areas \(F\). Thus, from Fig. 2 it follows that

\[ \tan \alpha = \frac{P}{F} = 8000\ \mathrm{kg/cm^2}. \]

The investigations showed that the angle of inclination of the straight lines of deformation is the same for specimens of different volumes.

It is known that the ratio of the force \(P\) to the area \(F\) is the stress \(\sigma\). Consequently, the plastic deformation of a cylindrical specimen of steel grade St. 3, deformed up to a certain limit in the radial direction, occurs at a constant stress \(\sigma = 8000\ \mathrm{kg/cm^2}\) and does not depend on the dimensions of the specimen.

The greatest force at which the rectilinear dependence between the load and the imprint area is preserved was called by us the limit of plastic proportionality \(P_{\mathrm{pp}}\). From the graph in Fig. 3 it follows that the larger the volume \(V\) of the specimen, the longer the linear dependence between the load and the imprint area is preserved, i.e., the higher the limit of plastic proportionality \(P_{\mathrm{pp}}\). For example, for a specimen with volume \(V = 0.5\ \mathrm{cm^3}\) the limit of plastic proportionality is \(6000\ \mathrm{kg}\), while for a specimen with volume \(V = 4.72\ \mathrm{cm^3}\), \(P_{\mathrm{pp}} = 27000\ \mathrm{kg}\).

On the basis of the experimental results, a graph was constructed for the dependence between the limit of plastic proportionality \(P_{\mathrm{pp}}\) and the volume \(V\) of the specimen, Fig. 4. From the graph presented it is evident that for specimens with volume less than \(V = 1\ \mathrm{cm^3}\) this dependence is curvilinear, while for specimens with larger volumes it is rectilinear. The rectilinear part of the graph in Fig. 4 (for \(V > 1\ \mathrm{cm^3}\)) gives the empirical dependence:

\[ P_{\mathrm{pp}} = a + kV, \tag{1} \]

where: \(a = 6000\ \mathrm{kg}\), \(K = 4500\ \mathrm{kg/cm^3}\).

If the volume of a cylindrical specimen is known, then with the aid of the formula given above one can determine the load-carrying capacity of the latter. But specimens of one and the same volume can be obtained by choosing different values of the length \(l\) and diameter \(d\). As tests have shown, in specimens whose length is greater than the diameter, the area of the imprint in the compression process retains the shape of a rectangle for a longer time. This circumstance simplifies the measurement of the imprint area. Since cylindrical specimens made of raw steel of grade St. 3, when compressed in the radial direction, deform plastically up to a certain limit at constant stress, they therefore can be used as crush gauges for determining the magnitude of the acting force. For this it is necessary to multiply the imprint area \(F\) in \(\text{cm}^2\) by the stress \(\sigma = 8000\ \text{kg}/\text{cm}^2\).

\[ P = F \cdot 8000\ \text{kg}. \]

Moreover, it should be borne in mind that the result of calculating the force will correspond to the actual value if it is at least somewhat smaller than the limit of plastic proportionality \(P_{\text{pp}}\), found from the graph in Fig. 4 from the volume of the given crush gauge, or smaller than the limit of plastic proportionality calculated by formula (1), if the volume of the crush gauge is greater than \(V = 1\ \text{cm}^3\).

Fig. 3.

Fig. 4.

REFERENCES

1. Kuznetsov V. D. Physics of the Solid State, Vol. II, “Krasnoe znamya,” 1941.
2. Konyakhin I. R. The hardening core, Scientific Reports of Higher Education Institutions, Mechanical Engineering and Instrument Making, No. 3, 1958.