SYNTHESIS OF EXPERIMENTAL METHODS FOR INVESTIGATING THE STRESS STATE IN THE PLASTIC REGION

G. D. DEL

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

By an experimental determination of the stress–strain state is meant such a determination in which, as a result of an experiment, the distribution throughout the volume of a body is established for one or several functions of the unknown stresses \(\sigma_x, \tau_{xy}, \ldots\) and strains \(\varepsilon_x, \gamma_{xy}, \ldots\):

\[ f_n(\sigma_x,\tau_{xy},\ldots,\varepsilon_x,\gamma_{xy},\ldots)=\psi_n(x,y,z),\quad n=1,2,3\ldots \tag{1} \]

To determine the stress–strain state, i.e., to interpret the experimental results, equations (1) are solved together with the differential equations of equilibrium and the constitutive equations; the use of the latter is less desirable, since they are obtained under assumptions that are not always confirmed by experiment.

The most widespread methods have been those that make it possible to determine the strain state and, from it, the stress state of the body (these are, above all, the grid method and the method of optically sensitive coatings), and methods that make it possible experimentally to establish the distribution of the yield limit increased as a result of strain hardening (the hardness-measurement method, the magnetic-permeability measurement method, etc.).

In evaluating various methods for the experimental determination of the stress state in the plastic region, in our view, one should be guided by the following criteria:

1. The possibility of determining, as a result of the experiment, all components of the stress state.
2. The set of boundary conditions required for interpretation.
3. The rigor of the method. The assumptions under which the stress state is determined are meant.
4. The range of application of the method.
5. The laboriousness of the experiment. The complexity and laboriousness of the interpretation.

The grid method (as well as the method of optically active coatings) makes it possible to determine the strain state; from the strains the intensity of strains \(\varepsilon_i\) is determined, and from the latter

under the assumption of a single flow curve—and the stress intensity $\sigma_i$. Then the components of the stress deviator are determined:

\[ \left. \begin{aligned} s_x &= \frac{2}{3}\,\frac{\sigma_i}{\varepsilon_i}\,\varepsilon_x,\ldots\\[6pt] \tau_{xy} &= \frac{\sigma_i}{3\varepsilon_i}\,\gamma_{xy},\ldots \end{aligned} \right\} \tag{2} \]

Here

\[ s_x=\sigma_x-\sigma, \tag{3} \]

where $\sigma$ is the hydrostatic pressure. The normal stresses are obtained by integrating the differential equations of equilibrium [1]. For example,

\[ \sigma_x=\int\left(-\frac{\partial \tau_{xy}}{\partial y}-\frac{\partial \tau_{xz}}{\partial z}\right)\,dx+\varphi(y,z). \tag{4} \]

To determine the constants of integration it is sufficient to know the position of the free contour. The stresses $\sigma_y$ and $\sigma_z$ are determined from relations (3).

In investigating a biaxial stress state the interpretation is considerably simplified. Having determined, from (2), the components of the stress deviator, and from the equation

\[ \sigma_z=\sigma+s_z=0 \tag{5} \]

the hydrostatic pressure, in this case the stresses $\sigma_x$, $\sigma_y$ can be determined directly from (3).

By the method of grid lines, the stress state is determined under the following assumptions:

1. The flow curve is assumed to be unique.
2. The directions of the principal stresses are assumed to coincide with the directions of the principal strains.
3. The components of the stress and strain deviators are assumed to be proportional.
4. The material is assumed to be continuous, homogeneous, and isotropic.

The interpretation can be performed in the general case.

The complexity of determining strains in internal regions considerably limits the field of application of the method. For this purpose, composite specimens are often tested. The grid is applied to the internal surface. In order that the associated violation of continuity of the specimen not affect the stress state, the following conditions must be satisfied: $\tau_{xz}=\tau_{yz}=0$, $\sigma_z\leqslant 0$ (the $z$ axis is directed along the normal to the surface with the grid). The stresses $\sigma_z$ must be balanced by external forces.

The interpretation can be performed without using relations (2). Thus, papers [2, 3] describe a method for determining stresses on the surface of a deforming body and, in plane strain, from a known grid of trajectories of the principal strains (it is assumed that the directions of the principal strains and principal stresses coincide). In this case assumptions 1 and 3 are abandoned. The material may be inhomogeneous.

The labor involved in interpretation thereby increases, and the necessary set of boundary conditions is expanded.

Below we shall discuss the grid method. However, all the results of the investigation will apply equally to the method of optically active coatings, since the information on the stress–strain state obtained by these methods is the same.

The hardness-measurement method [4] makes it possible to determine the distribution of the stress intensity. Consequently, equations (1) in this case have the form:

\[ (\sigma_x-\sigma_y)^2+(\sigma_y-\sigma_z)^2+(\sigma_z-\sigma_x)^2 +6(\tau_{xy}^2+\tau_{yz}^2+\tau_{xz}^2)=2\sigma_i^2(x,y,z). \tag{6} \]

The fact that hardness is directly related to the stress intensity and, only to within the accuracy of the assumption of a single flow curve, to the strain intensity (as has already been pointed out [5]) is confirmed by results obtained by us under conditions in which a unique relation between \(\sigma_i\) and \(e_i\) is not observed.

A method for interpreting the results of hardness measurements has been developed for the torsion of noncircular bars, plane deformation, and axisymmetric deformation [6]. In the investigation of axisymmetric deformation, the interpretation is carried out under the assumption that the circumferential stress is equal to one of the principal stresses in meridional planes.

The interpretation is laborious and requires knowledge of the boundary conditions along the entire contour. The stress state is determined under the assumptions of continuity and isotropy of the material.

Let us consider the possibilities that arise as a result of combining the grid method with the hardness-measurement method (or with any other method that makes it possible to determine the distribution of the stress intensity).

Even a simple combination of these methods, when the deformed state is determined by the grid method and the stress intensity by hardness measurement, makes it possible to abandon the assumption of a single flow curve and material homogeneity. The complication of the experiment is insignificant: by means of a torsion, compression, or tension test and a parallel hardness measurement, a calibration plot “hardness—stress intensity” is constructed. Then the hardness is measured in the cells of the deformed grid, and the corresponding stress intensity is determined from the calibration plot. The interpretation is carried out in the same way as in the case of investigating the stress state by the grid method. An example of such an investigation will be given below.

Let us consider a procedure for determining the stress state when the field of principal directions (determined by the grid method) is known and the distribution of the stress intensity, established by hardness measurement, is known.

The stress state is then determined under the following assumptions:

1. The directions of the principal stresses are considered to coincide with the directions of the principal strains.

2. The material is considered continuous and isotropic. As will be shown, in a number of practically important cases the interpretation is simplified. To carry it out, it is sufficient to know the position of the free contour.

Thus, the synthesis of the indicated methods makes the interpretation more rigorous in comparison with the grid method when used—

using relations (2), and less laborious in comparison with the same method without using relations (2) and with the hardness-measurement method.

Plane stress state. Equation (6) in this case takes the form:

\[ (\sigma_x-\sigma_y)^2+\sigma_x^2+\sigma_y^2+6\tau_{xy}^2=2\sigma_i^2(x,y). \tag{7} \]

By the grid method we obtain the principal directions:

\[ \frac{2\tau_{xy}}{\sigma_y-\sigma_x}=\operatorname{tg}2\vartheta(x,y), \tag{8} \]

where \(\vartheta\) is the angle of inclination of the tangent to the trajectory of the principal stress \(\sigma_1\), measured in the positive direction from the \(x\)-axis.

For the deciphering we use the differential equations of equilibrium:

\[ \frac{\partial\sigma_x}{\partial x}+\frac{\partial\tau_{xy}}{\partial y}=0,\qquad \frac{\partial\tau_{xy}}{\partial x}+\frac{\partial\sigma_y}{\partial y}=0. \tag{9} \]

By the relations valid for any symmetric tensor:

\[ \sigma_x=\frac{\sigma_1+\sigma_2}{2}+\frac{\sigma_1-\sigma_2}{2}\cos2\vartheta, \]

\[ \sigma_y=\frac{\sigma_1+\sigma_2}{2}-\frac{\sigma_1-\sigma_2}{2}\cos2\vartheta, \tag{10} \]

\[ \tau_{xy}=\frac{\sigma_1-\sigma_2}{2}\sin2\vartheta, \]

we pass to new variables—the principal stresses \(\sigma_1\) and \(\sigma_2\). Then, putting \(\vartheta=0\), we pass to a local coordinate system formed by tangents to the trajectories of the principal stresses. The direction of the trajectories \(\sigma_1,\sigma_2\) is fixed in such a way that they form a right-handed coordinate system. Any one of the principal stresses may be denoted by \(\sigma_1\).

Equation (8) becomes an identity, and (7) and (9) take the following form:

\[ \sigma_1^2+\sigma_2^2-\sigma_1\sigma_2=\sigma_i^2, \tag{11} \]

\[ \frac{\partial\sigma_1}{\partial s_1}+(\sigma_1-\sigma_2)\frac{\partial\vartheta}{\partial s_2}=0, \tag{12} \]

\[ \frac{\partial\sigma_2}{\partial s_2}+(\sigma_1-\sigma_2)\frac{\partial\vartheta}{\partial s_1}=0. \tag{13} \]

Here \(\dfrac{\partial}{\partial s_1}\), \(\dfrac{\partial}{\partial s_2}\) are derivatives along the trajectories \(\sigma_1,\sigma_2\).

Let us consider a procedure for determining the principal stresses along the trajectory \(\sigma_1\). Suppose that on the boundary \(L\) (Fig. 1) \(\sigma_1\) and \(\sigma_2\) are specified (if \(L\) is a free boundary, the principal stress normal to the contour is equal to zero, and the second stress is equal to \(\pm\sigma_i\)). We divide the trajectory \(\sigma_1\) into a number of sufficiently small segments of length \(\Delta s_1\). Through the ends of these segments we draw trajectories \(\sigma_2\) and, along them, lay off on both sides of the trajectory \(\sigma_1\) segments of length \(\Delta s_2\).

At point 1, according to (12),

\[ \left(\frac{\partial \sigma_1}{\partial s_1}\right)_1 = -(\sigma_1-\sigma_2)_1\frac{\vartheta_{11}-\vartheta_{21}}{2\Delta s_2}. \tag{14} \]

At point 2

\[ (\sigma_1)_2=(\sigma_1)_1+\left(\frac{\partial \sigma_1}{\partial s_1}\right)_1\Delta s_1, \tag{15} \]

\[ (\sigma_2)_2=\frac{(\sigma_1)_2 \pm \sqrt{4(\sigma_i)_2^2-3(\sigma_1)_2^2}}{2}. \tag{16} \]

Let us write equation (12) for point 2 in finite differences:

\[ \frac{(\sigma_1)_3-(\sigma_1)_1}{2\Delta s_1} +(\sigma_1-\sigma_2)_2\frac{\vartheta_{12}-\vartheta_{22}}{2\Delta s_2}=0. \tag{17} \]

Hence we have

\[ (\sigma_1)_3=(\sigma_1)_1-(\sigma_1-\sigma_2)_2(k_1)_2, \tag{18} \]

where

\[ (k_1)_2=(\vartheta_{12}-\vartheta_{22})\frac{\Delta s_1}{\Delta s_2}. \tag{19} \]

Fig. 1

The stress \(\sigma_2\) at point 3 is found from a formula analogous to (16). The principal stresses for subsequent points are determined in the same way as for point 3, i.e., by the formulas

\[ \left. \begin{aligned} (\sigma_1)_{n+1} &=(\sigma_1)_{n-1}-(\sigma_1-\sigma_2)_n(k_1)_n,\\ (\sigma_2)_{n+1} &= \frac{(\sigma_1)_{n+1}\pm \sqrt{4(\sigma_i)_{n+1}^2-3(\sigma_1)_{n+1}^2}}{2}, \end{aligned} \right\} \tag{20} \]

where

\[ (k_1)_n=(\vartheta_{1n}-\vartheta_{2n})\frac{\Delta s_1}{\Delta s_2}. \tag{21} \]

The sign before the radical should be chosen so that no discontinuity of stresses arises. If this condition is satisfied for different signs (when \(\sigma_2 \approx \dfrac{\sigma_1}{2}\)), then additional conditions are required for choosing the solution.

Along the trajectory \(\sigma_2\), the principal stresses are determined analogously:

\[ \left. \begin{aligned} (\sigma_2)_{l+1} &=(\sigma_2)_{l-1}-(\sigma_1-\sigma_2)_l(k_2)_l,\\ (\sigma_1)_{l+1} &= \frac{(\sigma_2)_{l+1}\pm \sqrt{4(\sigma_i)_{l+1}^2-3(\sigma_2)_{l+1}^2}}{2}, \end{aligned} \right\} \tag{22} \]

where

\[ (k_2)_l=(\vartheta_{3l}-\vartheta_{4l})\frac{\Delta s_2}{\Delta s_1}, \tag{23} \]

Knowing the magnitudes \(\sigma_1\) and \(\sigma_2\), the stresses \(\sigma_x, \sigma_y, \tau_{xy}\) can be determined from (10).

To reduce the error in determining \(k_1\) and \(k_2\), decoding should begin with the construction of a graph of the variation of these quantities along the trajectories of the principal stresses.

Since the axes of symmetry are trajectories of the principal stresses, the proposed procedure is especially convenient for determining the stress state along these axes.

Plane strain. The stress state under plane strain is characterized by the stresses $\sigma_x, \sigma_y, \sigma_z, \tau_{xy}$ (the $z$ axis coincides with the direction in which the strain is zero). From the incompressibility condition it follows [7] that

\[ \sigma_z=\frac{1}{2}(\sigma_x+\sigma_y). \tag{24} \]

Under plane strain, the maximum shear stress is proportional to the stress intensity:

\[ \tau_{\max}=\frac{1}{\sqrt{3}}\,\sigma_i \tag{25} \]

and therefore can be established by measuring hardness. In addition, the directions of the principal stresses are known, being established by the split-grid method. Consequently, for determining the stresses $\sigma_x, \sigma_y, \tau_{xy}$ we have at our disposal the same information as when investigating a plane stress state in the elastic region by the photoelastic method. The equilibrium equations in these two cases coincide. Therefore, in reducing the experimental results in the case under consideration, all the techniques developed for the method of photoelasticity can be used.

Let us consider the procedure for determining the stress state along axes of symmetry. Under plane strain,

\[ \tau_{\max}=\sigma_1-\sigma_2, \tag{26} \]

where $\sigma_1$ and $\sigma_2$ are the principal stresses in the plane of strain, with $\sigma_1>\sigma_2$. Therefore, from equations (12) and (13), which are also valid for plane strain, the following relations are obtained along the trajectory $\sigma_1$:

\[ \left. \begin{aligned} \sigma_1&=-\frac{1}{\sqrt{3}}\int \sigma_i\,\frac{\partial \vartheta}{\partial s_2}\,ds_1+\varphi_1(s_2),\\ \sigma_2&=\sigma_1-\frac{1}{\sqrt{3}}\,\sigma_i, \end{aligned} \right\} \tag{27} \]

and along the trajectory $\sigma_2$:

\[ \left. \begin{aligned} \sigma_2&=-\frac{1}{\sqrt{3}}\int \sigma_i\,\frac{\partial \vartheta}{\partial s_1}\,ds_2+\varphi_2(s_1),\\ \sigma_1&=\sigma_2+\frac{1}{\sqrt{3}}\,\sigma_i. \end{aligned} \right\} \tag{28} \]

The constants of integration are determined from the boundary conditions. Using relations (27) and (28), one can determine the stress state in the region under investigation, but they are most convenient for determining stresses along axes of symmetry. Let $AB$ (Fig. 2) be a segment of an axis of symmetry along which the stress $\sigma_1$ acts. Construct along it the curve

\[ \left(\frac{\partial \vartheta}{\partial s_2}\right)_n = \frac{\vartheta_{1n}-\vartheta_{2n}}{2\Delta s_2}, \tag{29} \]

then we construct the graph of \(\sigma_i \dfrac{\partial \vartheta}{\partial s_2}\). The stresses at point \(B\) are determined from the relations

\[ \left. \begin{aligned} (\sigma_1)_B&=(\sigma_1)_A-\frac{1}{\sqrt{3}}\int_A^B \sigma_i\,\frac{\partial \vartheta}{\partial s_2}\,ds_1,\\[4pt] (\sigma_2)_B&=\frac{1}{\sqrt{3}}(\sigma_i)_B-(\sigma_1)_B. \end{aligned} \right\} \]

The value of the integral can be determined graphically, as the area enclosed under the curve \(\sigma_i \dfrac{\partial \vartheta}{\partial s_2}\) between points A and B.

Example. By the developed procedure, the stressed state was determined for the tension of a strip with circular notches (Fig. 3). On the surface of a plate made of steel Kh13, before deformation, a division grid from a system of circles with a base of 1 mm was rolled on. Then the strip was stretched to a strain intensity in the section \(y=0\), \(\varepsilon_i \approx 0.35\). After deformation, on a toolmaker’s microscope, the angle of inclination of the major axes of the ellipses was measured in sections \(y=\pm 2.5\) mm. The measurement results were graphically averaged. Then, along the \(x\)-axis, the hardness was measured. It proved to be constant, which indicates the constancy of \(\sigma_i\) along this axis. The stresses were determined by formulas (20).

Fig. 2

Fig. 3. Tension of a strip with notches

In parallel, a decoding was carried out using relations (2). For this purpose, along the lines \(y=\pm 2.5\) mm, envelopes of the tangential stresses determined by formulas (2) were constructed, in which

\[ \gamma_{xy}=\operatorname{tg}\vartheta . \tag{30} \]

Then, along the \(x\)-axis, a graph of the variation of \(\dfrac{\partial \tau_{xy}}{\partial y}\) was constructed. The stresses—

the stresses \(\sigma_x\) were determined by graphical integration using the formulas

\[ (\sigma_x)_B = (\sigma_x)_A - \int_A^B \frac{\partial \tau_{xy}}{\partial y}\, dx. \]

The stresses \(\sigma_y\) were determined from the known \(\sigma_i\). The stress diagrams obtained by these methods practically coincided.

As already indicated, along the \(x\)-axis \(\sigma_i\) was constant. However, this is still not a sufficient condition for applying, in the case under consideration, the apparatus of the theory of ideally plastic bodies, since along the \(y\)-axis an intensive change in \(\sigma_i\) was observed. Nevertheless, the results obtained are in good agreement with the results obtained analytically for a non-hardening material [7].

In conclusion, we shall point out the following circumstance. Often, investigations by the grid-division method and by the optical method are performed on materials with little strain hardening (for example, lead). In this case one may set \(\sigma_i = \mathrm{const}\) throughout the entire plastic region. This makes it possible to apply the proposed interpretation procedure without measuring hardness. The procedure is also considerably simplified in the investigation of a plane stress state with axial symmetry. In this case the directions of the principal stresses are known—one of them coincides with the radial direction, the other with the circumferential direction.

REFERENCES

1. A. A. Pozdeev, V. I. Tarnovskii. Izvestiya vysshikh uchebnykh zavedenii, Ferrous Metallurgy, No. 11, 1962.

2. A. E. Aleksandrov, M. Kh. Akhmetzyanov, PMTF, No. 6, 1961.

3. P. I. Polukhin, V. K. Vorontsov. Izvestiya vysshikh uchebnykh zavedenii. Ferrous Metallurgy, No. 11, 1962.

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

5. A. M. Rozenberg, A. N. Eremin. Elements of the Theory of the Metal-Cutting Process, Mashgiz, 1956.

6. G. D. Del. Izvestiya TPI, Vol. 138, No. 1, 1965.

7. L. M. Kachanov. Fundamentals of the Theory of Plasticity, GITTL, 1956.