Determination of the Magnitude of the Bauschinger Effect with Increasing Degree of Plastic Deformation
Abstract Generated abstract
This paper examines how the Bauschinger effect changes with increasing plastic deformation in plastically hardening metals. It proposes a mechanical model that treats a metal beyond the elastic limit as a composite of elastic and ideally plastic subelements with differing yield limits, deriving a relation for the reverse yield stress after prior tension. Experiments on low-carbon steel, brass, and duralumin are compared with the proposed formula and with Masing’s relation, showing that the new expression agrees more satisfactorily as plastic strain increases, while Masing’s formula is mainly adequate only at very small plastic strains. The resulting estimate is also connected to a theoretical form of the loading surface for plasticity models with translational hardening.
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DETERMINATION OF THE MAGNITUDE OF THE BAUSCHINGER EFFECT WITH INCREASING DEGREE OF PLASTIC DEFORMATION
G. A. Doshchinsky, N. V. Sitnikova
(Presented by the scientific seminar of the Department of Strength of Materials)
The development of modern mechanics of materials is characterized by a striving toward a more complete accounting for the properties of real materials and toward the expansion and improvement of their computational models. Classical phenomenological models of elastic and plastic media, despite their broad use in engineering applications, often fail to include essential phenomena observed in the deformation processes of real metallic materials. The greatest discrepancies of this kind are found in the behavior of metals beyond the elastic limit. The accumulating experimental data on the study of the behavior of real metals serve as a basis for improving both physical theories and mechanical-mathematical models of elastoplastic media.
In the present work, some possibilities are considered for bringing theoretical description closer to the actual behavior of metals by taking into account the interrelated phenomena of a plastically hardening material: hardening anisotropy, the Bauschinger effect, and changes in the shape of the yield surface.
As early as the 1890s, Bauschinger’s investigations of low-carbon steels noted a substantial decrease in the yield limit in compression after prior tension. Later this phenomenon, which received the name of the Bauschinger effect, was studied in other metals as well and was recognized as a characteristic property of plastic deformation in hardening bodies.
A highly probable cause of such a phenomenon may be assumed to be the emergence of residual stresses in the metal after prior tension which, being superimposed on the working stresses, lead to a reduction of the yield limit during subsequent compression. However, the model of an elastoplastic hardening body does not imply, upon unloading of a homogeneous medium, any residual stresses. The possibilities of a theoretical predetermination of the Bauschinger effect within mechanical models can be obtained by accounting for the possible inhomogeneity of a plastically deformed body.
A physical picture of deformation of a polycrystalline metal more consistent with reality is not the condition of an integral homogeneous state of all crystals, usually adopted by theory, but a different one.
state of individual grains, some of which may still remain in the elastic state, while others have acquired irreversible distortions of various kinds. Such an approach introduces substantial difficulties into the mathematical representation of the local conditions of compatibility of deformations. However, it may be assumed that the compatibility conditions are in this case ensured by group behavior in certain volumes that include a definite number of subelements, in the form of elastically or plastically deformed crystal grains. Attempts at such an approach, in any case, have already been made in a number of works [1], etc.
Fig. 1
We shall consider a metal beyond the elastic limit as a composite consisting of elastic and plastic subelements. We shall take the behavior of the former to correspond to the model of an ideally elastic body, and that of the latter to an ideally plastic body with different yield limits for the individual elements. Let us introduce the concept of concentration $\xi$ as the ratio of the volume of grains affected by plastic deformation to the total volume of the metal. For a homogeneous (uniform) mixture, the quantity $\xi$ will also determine the fraction of plastic elements in the cross section of the bar. For the calculation scheme (Fig. 1) one may, as a first approximation (which is not essential for the derivation), assume a linear distribution of yield limits in the various grains, from the value $\sigma_{\mathrm{t}}$, corresponding to the yield limit of the weakest grain, to the value $2\sigma_{\mathrm{t}}$, corresponding to the attainment by the mean stress in the metal $\sigma$ of the ultimate strength $\sigma_{\mathrm{B}}$ (at $\xi=1$ all grains will be affected by plastic deformation, $d\sigma/de=0$).
The mean stress in the metal, from the equilibrium condition, is determined by the expression
\[ \sigma=\xi\sigma_{\mathrm{p}}+(1-\xi)\sigma_{\mathrm{y}}, \tag{1} \]
where $\sigma_{\mathrm{p}}$ is the mean stress in the plastic part of the section;
$\sigma_{\mathrm{y}}$ is that in the elastic part.
Since the elastic elements attain larger elastic deformations than the plastic grains currently deviating from elastic resistance, upon unloading this difference in elastic deformations will lead to the occurrence of residual compressive stresses $\sigma$ in the plastic grains and tensile stresses $\sigma_{p}^{0}$ in the elastic part. The largest value of the compressive residual stress $\sigma-\sigma_{\mathrm{t}}$ will occur in the grains with the smallest yield limit; the smallest, obviously, is equal to zero. With equal probability of the different values of the yield limits in the plastic zone, the mathematical expectation will determine the mean residual stress by the value
\[ \sigma_{\text{c}}^{0}=\frac{\sigma-\sigma_{\text{T}}}{2}. \tag{2} \]
Under loading by the subsequent compressive stress \(\sigma_{\text{c}}\), further development of plastic deformation may be expected after the compressive stresses, together with the residual stresses in the plastic part, reach the value of the stress at which the process of plastic deformation by the preliminary tension was stopped,
\[ \sigma_{\text{c}}+\sigma_{\text{c}}^{0}>\sigma, \]
whence
\[ \sigma_{\text{c}}>\sigma-\sigma_{\text{c}}^{0} =\sigma-\frac{\sigma-\sigma_{\text{T}}}{2} =\frac{\sigma+\sigma_{\text{T}}}{2}. \]
Replacing the inequality by an equality, the magnitude of the Bauschinger effect can be determined by the expression
\[ \frac{\sigma_{\text{c}}}{\sigma} =\frac{\sigma+\sigma_{\text{T}}}{2\sigma} =\frac{1}{2} =\frac{\sigma_{\text{T}}}{2\sigma}. \tag{3} \]
Repeated loading after unloading, on the basis of the scheme considered, gives
\[ \sigma_{\text{p}}>\sigma, \]
i.e., further development of plastic deformation begins after the tensile stress exceeds the value of the preliminary stress.
The best-known theoretical solution, obtained under similar assumptions and corresponding to Masing’s principle [3], [6], determines the Bauschinger effect by the expressions
\[ \sigma_{\text{c}}=2\sigma_{\text{T}}-\sigma;\qquad \frac{\sigma_{\text{c}}}{\sigma} =\frac{2\sigma_{\text{T}}-\sigma}{\sigma} =\frac{2\sigma_{\text{T}}}{\sigma}-1. \tag{4} \]
In experiments verifying Masing’s formula [3], close agreement with reality was observed only in the region of very small plastic strains.
In the present work we give the results of experiments we carried out to determine the magnitude of the Bauschinger effect for three metals: low-carbon steel, brass, and duralumin. The experiments were conducted on a Gagarin press. Tensile diagrams were first obtained for all the materials. From specimens stretched to different degrees of deformation, 2–3 duplicate specimens for compression were cut out. The yield limits were determined from the intersection points of the elastic- and plastic-deformation lines, according to the method proposed by L. M. Kachanov in [4].
The results of the investigations carried out are presented in Fig. 2, where the points show the experimental values of the Bauschinger effect, characterized by the ratio \(\sigma_{\text{c}}/\sigma\). The calculated curves, constructed on the basis of the tensile diagrams of the tested metals according to the proposed calculation formula (3) and Masing’s formula (4), are also given there. The experiments show a satisfactory approximation of the calculated values based on the proposed relation for theoretically predicting the magnitude of the Bauschinger effect. At the same time, Masing’s formula, which approaches the experimental values at the initial sta-
of plastic deformation, as the latter increases, leads to substantial deviations. And at degrees of deformation of several percent, Masing’s principle leads to the assertion that plastic deformation will begin already during unloading, and that for the subsequent compression there is no elastic resistance. This proposition is in obvious contradiction with experiment, which, as a rule, shows elastic behavior of a compressed specimen after preliminary tension and unloading.
Fig. 2
If the tensile diagram is approximated by a power-law function of the form
\[ \sigma = A e^m, \]
then the dependence of the Bauschinger effect on the degree of deformation is determined by the expression
\[ \frac{\sigma_{\text{comp}}}{\sigma} = \frac{1}{2} + \frac{\sigma_{\text{t}}}{2 A e^m}, \tag{5} \]
which shows asymptotic approach to the value
\[ \frac{1}{2}\left(1+\frac{\sigma_{\text{t}}}{\sigma_{\text{B}}}\right). \]
The tendency toward stabilization of the Bauschinger effect has been noted in the experiments of many researchers.
The Bauschinger effect is often considered in the theory of plastic deformation as the most vivid expression of the anisotropy of the hardening process. This phenomenon was the starting point for the hypothesis of translational hardening—the displacement of the yield surface without rotations in the direction of loading. Therefore, the result obtained may find application in theories of plasticity with translational hardening.
In our investigations [5], on the basis of studying the regularities of plastic deformation under complex loadings caused by a change in the type of stress state, an equation was obtained for the yield surface after simple loading (in good agree-
... agreeing with experiment), which in polar coordinates of A. A. Ilyushin’s vector space [2] takes the form
\[ \rho = S - \frac{\Delta S_{\mathrm{B}}}{\pi}\,\varphi, \tag{6} \]
where \(\Delta S_{\mathrm{B}}=\sqrt{\frac{2}{3}}\,(\sigma-\sigma_{\mathrm{cж}})\) was assumed to be the experimental value of the Bauschinger effect. The calculated value of the magnitude of the Bauschinger effect obtained in the present work makes it possible, on this basis, to give also a theoretical equation for the loading surface
\[ \rho=\sqrt{\frac{2}{3}}\left(\sigma-\frac{\sigma-\sigma_{\mathrm{T}}}{2\pi}\,\varphi\right). \tag{7} \]
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