DETERMINATION OF THE DEFORMED STATE IN THE PLASTIC REGION FROM THE HARDNESS DISTRIBUTION

G. D. Del

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

The deformed state in the plastic region is most often investigated by the method of dividing grids [1]. This method, while possessing a number of indisputable advantages, is not free of shortcomings, of which the most serious are the difficulty of investigating relatively small plastic deformations (up to 10%), as well as of determining the deformed state in the internal regions of a body. In this respect, the method of dividing grids is, in a number of cases, well supplemented by the method of measuring hardness [2, 3].

In studying plastic deformation by measuring hardness, a calibration graph is first constructed: “stress intensity $\sigma_i$—hardness $H$—logarithmic strain intensity $e_i$.” The calibration graph is constructed by testing the material under investigation under conditions of the simplest stress states (usually compression, torsion, or tension). The plastic-deformation process under investigation is modeled. For this purpose a number of models are made, which are subjected to plastic deformation of varying degree. At various points of the deformed models the hardness is measured. To do this, it is sometimes necessary to cut the models or remove layers of material. From the calibration graph, the corresponding value of the intensities of stresses and strains is determined from the hardness.

The method is based on the assumption that the relation between hardness, stress intensity, and strain intensity is the same for different stress states. As a result of testing this assumption it has been confirmed [4] that the relation between $H$ and $\sigma$ is the same. The relation between strain intensity and hardness is the same under the assumption of a single flow curve $\sigma_i(e_i)$. This means that the form of the curve $e_i(H)$ depends to some extent on the type of stress state. Therefore calibration should be carried out under conditions close to the process under investigation.

Other difficulties also arise in determining strain intensity from hardness. The calibration graph is very sensitive to the properties of the material; therefore, calibration specimens should be made from the same rod, from the same billet, from which the models are made. The initial hardness of the calibration specimens and of the models must be identical.

The sensitivity of hardness to the increment of plastic deformation, \(\dfrac{dH}{de_i}\), is significant in the region of small deformations. As the plastic deformations increase, it decreases. The quantity \(\dfrac{dH}{de_i}\) depends to a considerable extent on the hardenability of the material. Therefore, the accuracy of determining deformations from the hardness distribution can vary greatly from point to point in the model.

Let it be required to determine the value of the strain intensity with an accuracy of \(\pm \Delta e_i\). For this purpose the hardness must be measured with a relative error not exceeding \(\pm \dfrac{dH}{de_i}\dfrac{\Delta e_i}{H}\). The scatter of hardness follows the normal distribution law; therefore such accuracy can be obtained with probability \(f\) if the hardness is determined as the mean of \(n\) measurements [5]:

\[ n=\frac{f}{\Phi\left(\dfrac{dH}{de_i}\,\dfrac{\Delta e_i}{m\sqrt{2}}\right)}, \tag{1} \]

where \(\Phi\) is the Laplace function of the argument in parentheses, and \(m\) is the standard deviation. The standard deviation is practically independent of the magnitude of deformation (Fig. 1).

Fig. 1.

The results presented in Fig. 1 were obtained by torsion tests of steels Kh18N9T and steel 45. At different degrees of deformation, 10 Rockwell hardness measurements, \(HRD\), were made on the cylindrical surface of the specimens.

Consequently, the value of \(m\) can be determined from calibration results by the relation:

\[ m \simeq \frac{\sqrt{\displaystyle \sum_{i=1}^{\kappa}(H_i-H_t)}}{\kappa}, \tag{2} \]

where \(H_i\) is the result of measuring hardness at the specimen deformation \(e_i\),
\(H\) is the hardness value found from the calibration graph for \(e_i\),
\(k\) is the number of measurements made during calibration (usually about 100).

If we set

\[ \eta_e=\frac{dH}{de_i}\frac{e_i}{m}, \tag{3} \]

then it follows from (1) that the number of hardness measurements required in order to determine, with probability \(f\), the value \(e_i\) with a relative error lying within

\[ \pm \frac{\Delta e_i}{e_i}, \]

independently of the material and the method of hardness testing, is completely determined by the value of the parameter \(\eta_e\). For a given number of hardness measurements, this parameter characterizes the accuracy of determining \(e_i\).

Reasoning analogously, for the stress intensity we obtain

\[ \eta_\sigma=\frac{dH}{d\sigma_i}\frac{\sigma_i}{m}. \tag{4} \]

Figure 2 shows the change in the parameters \(\eta_e\) and \(\eta_\sigma\) in the process of plastic deformation of stainless steel when measuring hardness by the Rockwell method, \(HRD\). This figure shows that the accuracy of determining \(e_i\) from hardness is much lower than the accuracy of determining \(\sigma_i\). It should

Fig. 2.

also be borne in mind that, while the value \(\eta_\sigma\) depends only weakly on the hardenability of the material, \(\eta_e\) depends on it very strongly, decreasing as hardenability decreases. Practice shows that the determination of deformations is reliable only in those cases where \(\eta_e > 10\). The required value of \(\eta_e\) is sometimes achieved by an appropriate choice of hardness-testing method (the highest \(\eta_e\) in Brinell hardness testing, lower in Vickers testing, and in Rockwell testing).

Thus, by measuring hardness it is possible to determine the distribution of deformation intensity. But such a possibility does not often satisfy the investigator; methods must be developed for determining all components of the strain tensor from the hardness distribution.

In plane deformation, the magnitude of the principal strains can be determined from the hardness. Indeed, from the system

\[ \frac{\sqrt{2}}{3}\sqrt{(e_1-e_2)^2+e_1^2+e_2^2}=e_i,\qquad e_1+e_2=0 \tag{5} \]

we find

\[ e_1=-e_2=\frac{\sqrt{3}}{2}e_i. \tag{6} \]

(The second of equations (5) follows from the condition of incompressibility.) To determine the directions of \(e_1\) and \(e_2\), additional conditions are required.

Using relations (6), one can determine the deformed state along axes of symmetry. In this case one of the strains is directed along the axis, and the other is perpendicular to it.

In studying axisymmetric deformation, we have two equations analogous to (5):

\[ \left. \begin{aligned} \frac{\sqrt{2}}{3}\sqrt{(e_r-e_\varphi)^2+(e_r-e_z)^2+(e_\varphi-e_z)^2+\frac{3}{2}\gamma_{rz}^2}&=e_i,\\ e_r+e_z+e_\varphi&=0. \end{aligned} \right\} \tag{7} \]

The problem is considered in a cylindrical coordinate system; the \(z\) axis is made to coincide with the axis of symmetry.

Along the \(z\) axis, from the condition of symmetry,

\[ e_r=e_\varphi,\qquad \gamma_{rz}=0, \tag{8} \]

therefore

\[ e_r=e_\varphi=-\frac{1}{2}e_z=\pm \frac{1}{2}e_i. \tag{9} \]

Axial symmetry imposes a definite restriction on the deformed state. Indeed, a circle of radius \(r_0\) with its center on the \(z\) axis, drawn in the plane \(z=\mathrm{const}\), is deformed by virtue of axial symmetry into a circle with radius \(r_1\).

The circumferential strain is

\[ e_\varphi=\ln\frac{r_1}{r_0}. \tag{10} \]

The mean radial strain is equal to this same quantity. Consequently, we have the condition

\[ e_\varphi=\frac{1}{r}\int_0^r e_r\,dr. \tag{11} \]

If we differentiate (11), we obtain

\[ \frac{de_\varphi}{dr}=\frac{1}{r}(e_r-e_\varphi). \tag{12} \]

Thus, in the general case of axisymmetric deformation, in order to determine the four unknowns \(e_r, e_\varphi, e_z, \gamma_{rz}\), one equation is lacking.

If the body has a plane of symmetry \(z=\mathrm{const}\), then in this plane

\[ \gamma_{rz}=0 \tag{13} \]

and the strains are determined from the system (7), (12). From (7) we obtain

\[ e_r^2+e_\varphi^2+e_r e_\varphi=\frac{3}{4}e_i^2 . \tag{14} \]

Solving (12) and (14) jointly, we determine the strains \(e_r\) and \(e_\varphi\). For this it is necessary to have initial conditions (it is necessary to know \(e_r\) and \(e_\varphi\) at least at one point of the section under consideration). If the body is solid, then as the initial condition one may use the strains on the \(z\)-axis, determined from relations (9). If the contour of the section under consideration is free, then from the relation

\[ e_\varphi=\ln \frac{R}{R_0}, \tag{15} \]

where \(R_0\) and \(R\) are the radius of the body in this section before and after deformation, one can determine the circumferential strain on the contour, and from (14) the radial strain.

Let \(e_\varphi\) and \(e_r\) be specified at point 1. At point 2, located at a distance \(\Delta r\) from point 1, according to (12)

\[ e_{\varphi 2}=e_{\varphi 1}\pm \frac{1}{2} \left( \frac{e_{r1}-e_{\varphi 1}}{r_1} + \frac{e_{r2}-e_{\varphi 2}}{r_2} \right)\Delta r . \tag{16} \]

Here and below (with the exception of (19)) the plus sign refers to the case when \(r_1<r_2\), and the minus sign to the case when \(r_1>r_2\). From (11) we obtain

\[ e_\varphi=B+Ce_r, \tag{17} \]

where

\[ B=\frac{r_2\left[2r_1 e_{\varphi 1}\pm \Delta r\,(e_{r1}-e_{\varphi 1})\right]} {r_1(2r_2\pm \Delta r)}, \qquad C=\pm \frac{\Delta r}{2r_2\pm \Delta r}. \tag{18} \]

Substituting (17) into (14) and solving the resulting equation, we determine

\[ e_r=\frac{-B(2C+1)\pm 1.73\sqrt{\,e_i^2(1+C^2+C)-B^2\,}} {2(1+C+C^2)} . \tag{19} \]

The sign before the radical should be chosen in such a way that no discontinuity arises in the strains \(e_r\).

Fig. 3. Strains under axial compression.

Using the developed method, strain curves were determined along the axis of symmetry, perpendicular to \(z\), under axial compression of a cylindrical-

of a specimen made of stainless steel Kh18N9T with diameter and height of 20 mm. The hardness was measured by the Rockwell method, \(HRD\). As initial conditions, the strains on the \(z\)-axis were used. The epures obtained are shown as solid lines in Fig. 3.

The points plotted on these curves were obtained as follows. Two cylindrical specimens were made, the diameters of which were equal and twice their height. Along the radius of one of the end faces, before deformation, points were scribed. During compression, the second specimen was placed on this end face. The amount of upsetting was 20%. Before and after deformation, the distances from the points to the axis of the specimen, as well as between the points, were measured. From the measurement results the strains \(e_r\) and \(e_\varphi\) were determined; these are also plotted in Fig. 3. The agreement of the results obtained by the two methods is fairly good. These results show that at the periphery of the specimen the radial strain is not equal to the circumferential strain, which casts doubt on the possibility of determining the stress state during cylinder upsetting under the Haar–Kármán assumption.

REFERENCES

1. Ya. B. Fridman, T. K. Zilova, N. I. Demkin, Study of Plastic Deformation and Fracture by the Method of Scribed Grids. Oborongiz, 1962.

2. G. A. Smirnov-Alyaev. Resistance of Materials to Plastic Deformation. Mashgiz, 1961.

3. G. D. Del. Investigation of plastic deformation by hardness measurement. Izv. TPI, vol. 198, 1965.

4. G. D. Del. Relation between hardness and stresses in the plastic region. Izv. VUZov—Fizika, No. 3, 1966.

5. E. S. Venttsel. Probability Theory. Fizmatgiz, 1958.