On the Experimental Verification of the Law of Similarity of Deviators in the Theory of Plasticity
Abstract Generated abstract
The paper examines experimental checks of the plasticity hypothesis that stress and strain deviators are similar, commonly tested through equality of Lode parameters. Assuming a constitutive relation that exactly satisfies this similarity condition, it analyzes how standard processing of thin-walled tube and mixed stress experiments can generate apparent deviations when one strain component is inferred from incompressibility, elastic strain components are retained, volumetric compressibility is present, or “true” stresses and logarithmic strains are used. The derived relations show that such deviations can arise from measurement and data reduction procedures rather than from failure of the deviator similarity law. The paper thus argues that some previously reported discrepancies in Lode and Taylor-Quinney type experiments may support, rather than weaken, the reliability of the similarity law.
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ON THE EXPERIMENTAL VERIFICATION
OF THE SIMILARITY LAW FOR DEVIATORS
IN THE THEORY OF PLASTICITY
G. A. Doshchinskii
(Presented by the scientific seminar of the Department of Strength of Materials)
The most commonly used form for assessing the hypothesis of the similarity of the deviators of stresses and strains in the theory of plasticity is the comparison of the parameters $\mu$ and $\nu$, introduced by Lode.
\[ \mu=\frac{2\sigma_2-\sigma_1-\sigma_3}{\sigma_1-\sigma_3},\qquad \nu=\frac{2e_2-e_1-e_3}{e_1-e_3}. \tag{1} \]
Equality of these parameters corresponds to the similarity of Mohr’s circles for the stressed and strained states, i.e., it indicates proportionality between the maximum shear stresses and the maximum shears.
Experimental verification, carried out by Lode, Taylor–Quinney, and a number of other researchers [1], [2], [3], showed close agreement of this condition with the behavior of materials in the plastic region. However, in these studies some deviations from equality of $\mu$ and $\nu$ were also noted. This allowed a number of researchers to assert that the relationship between $\mu$ and $\nu$ is of a more complex character and that the indicated equality can be accepted as approximate. Some authors believe that these deviations are in fact only a consequence of the anisotropy of the specimen materials.
Let us indicate some of the possible causes of experimentally obtained deviations in individual studies of this kind that are due to the experimental technique.
The relation between stresses and strains in the theory of small elastic-plastic strains can be represented for the principal directions in the form [3], [4], [5].
\[ e_1=\frac{1}{E(\lambda)}\left[\sigma_1-m(\sigma_2+\sigma_3)\right] \quad \begin{array}{c} 1\\[-0.3em] \swarrow\ \searrow\\[-0.3em] 2\longleftarrow 3 \end{array} \tag{2} \]
Here $m=m(\lambda)$ is the coefficient of total (elastic-plastic) strain. It is easy to verify that this very form corresponds to the similarity of deviators. Substituting into $\nu$ the strain values determined from (2), we obtain an expression for $\mu$.
\[ \nu=\frac{2e_2-e_1-e_3}{e_1-e_3} = \frac{ \frac{1}{E(\lambda)} \left\{2[\sigma_2-m(\sigma_1+\sigma_3)]-[\sigma_1-m(\sigma_2+\sigma_3)]- [\sigma_3-m(\sigma_1+\sigma_2)]\right\} }{ \frac{1}{E(\lambda)} \left\{[\sigma_1-m(\sigma_2+\sigma_3)]-[\sigma_3-m(\sigma_1+\sigma_2)]\right\} } = \frac{2\sigma_2-\sigma_1-\sigma_3}{\sigma_1-\sigma_3} =\mu . \]
Let us suppose that, under elastic-plastic deformation, materials obey law (2) exactly. The principal means for the experimental study of elastic-plastic deformation is the testing of thin-walled tubular specimens, in which a plane stress state is realized; for this state, neglecting \(\sigma_\rho \approx 0\), we have
\[ e_\theta=\frac{1}{E}[\sigma_\theta-m\sigma_z],\qquad e_z=\frac{1}{E}[\sigma_z-m\sigma_\theta], \]
\[ e_\rho=\frac{1}{E}[-m(\sigma_\theta+\sigma_z)],\qquad E=E(\lambda),\quad m=m(\lambda). \tag{3} \]
A characteristic feature of such experimental work is that in the experiments only two principal strains are measured, while the third is calculated from the condition of incompressibility of the material.
\[ \theta=e_\theta+e_z+e_\rho=0,\qquad \frac{1}{E}[\sigma_\theta-m\sigma_z+\sigma_z-m\sigma_\theta]+e_\rho=0, \]
whence
\[ e_\rho=\frac{m-1}{E}(\sigma_\theta+\sigma_z). \]
Using this value in representing the experimental results, we shall have
\[ \mu=\frac{2\sigma_\theta-\sigma_z}{\sigma_z}=2n-1,\qquad \nu=\frac{3(n-m)}{2-m+n(1-2m)},\qquad n=\frac{\sigma_\theta}{\sigma_z}. \tag{4} \]
Often, in experiments beyond the elastic range, the elastic components are not separated out from the measured strains. Although small, by their presence they nevertheless substantially affect the value of \(m\), making it always less than 0.5. Therefore, when experimental results are presented using the incompressibility condition in their processing, the dependence between \(\mu\) and \(\nu\), determined by (4), gives deviations from the diagonal straight line of the \(\mu\)—\(\nu\) graph.
The character of these deviations (for example, for \(m=0.45\)) is shown by the dashed line in Fig. 1. The results of the experiments of Lode and Taylor–Quinney, obtained using the indicated procedure, are also presented there.
The occurrence of such deviations under the noted processing of the experimental data is evidence of the reliability of the law of similarity of deviators, and not of its approximate nature.
Similarly, it can be shown that in a mixed (tension—compression, \(M—P—q\) experiments) stress state
\[ \sigma_1=\sigma_p,\qquad \sigma_2=0,\qquad \sigma_3=\sigma_c, \]
Fig. 1
\[ \mu=\frac{\sigma_c-\sigma_p}{\sigma_p+\sigma_c}, \]
\[ \nu=\frac{3(1-m)(\sigma_c-\sigma_p)}{(1+m)(\sigma_p+\sigma_c)} \quad \text{when determining } e_2 \text{ from the incompressibility condition,} \]
\[ \nu=\frac{3m(\sigma_c-\sigma_p)}{(2-m)\sigma_p+3m\sigma_c} \quad \text{when determining } e_3 \text{ from the incompressibility condition.} \]
The corresponding character of the deviations that should follow from the experiments is shown in Fig. 2, a.
If irreversible volumetric compressibility occurs in the material, then analogous deviations will be obtained even if the elastic components of deformation are excluded.
In investigations, the concepts of “true” stresses and strains are often used.
\[ \bar e_k=\ln(1+e_k),\qquad \bar\sigma_k=\sigma_k(1+e_k)\quad (k=1,2,3). \tag{5} \]
Let us note that the very form of the expressions \(\bar{\sigma}_k\) already presupposes the condition \(\bar e_1+\bar e_2+\bar e_3=0\); the \(\bar e_k\) are not, in this case, directly measurable quantities.
Assuming, as before, that the materials obey condition (2), let us determine how the specific content of the “true” stresses and strains may affect the processing of experimental results by means of these expressions in the region of small elastic-plastic strains.
Fig. 2.
a) \(1\)—\(\mu=\nu\); \(2\)—\(\nu=f(\mu)\) when \(e_2\) is determined from the condition \(\Theta=0\);
\(3\)—\(\nu=f(\mu)\) when \(e_3\) is determined from the condition \(\Theta=0\).
b) \(1\)—\(\mu=\nu\); \(2\)—\(m=0.45;\ e_1=0.01\); \(3\)—\(m=0.45;\ e_1=0.1\); \(4\)—\(m=0.3;\ e_1=0.1\)
For a plane stress state we shall have:
\[ \bar{\sigma}_1=\sigma_1(1+e_1),\qquad \bar{\sigma}_2=\sigma_2(1+e_2)=n\sigma_1(1+e_2),\qquad \bar{\sigma}_3=0, \]
where
\[ n=\frac{\sigma_2}{\sigma_1}. \]
\[ \frac{e_2}{e_1} = \frac{\dfrac{1}{E}\,[\sigma_2-m\sigma_1]} {\dfrac{1}{E}\,[\sigma_1-m\sigma_2]} = \frac{n-m}{1-mn}, \]
whence
\[ e_2=e_1\,\frac{n-m}{1-mn}. \]
\[ \bar{\mu} = \frac{2\bar{\sigma}_2-\bar{\sigma}_1}{\bar{\sigma}_1} = 2n\,\frac{1+e_2}{1+e_1}-1 = \frac{2n-2mn^2+2n^2e_1-1+mn-e_1}{1-mn+e_1-e_1mn}. \]
\[ \bar{\nu} = \frac{3\bar e_2}{2\bar e_1+\bar e_2} = \frac{3\ln(1+e_2)}{2\ln(1+e_1)+\ln(1+e_2)} = \]
\[ = \frac{3\ln(1-mn+ne_1-me_1)-3\ln(1-mn)} {2\ln(1+e_1)+\ln(1-mn+ne_1-me_1)-\ln(1-mn)}. \]
It is evident that the relationship between \(\bar{\mu}\) and \(\bar{\nu}\) for the conditions considered here depends on the degree of deformation. In Fig. 2, б, the dependence \(\bar{\mu}-\bar{\nu}\) is shown graphically for \(m=0.3;\ 0.45\) and deformations of order \(l=0.01;\ 0.1\). It follows from the graph that the deviations become noticeable as the order of magnitude of the deformation increases.
Thus, the presence of deviations of this kind when processing an experiment using the concepts of “true” stresses and strains may be predetermined by the specific nature of the latter, although the material will satisfy the condition \(\mu=\nu\) and, correspondingly, the dependencies (2).
If, however, the law governing the behavior of the materials is the form in “true” stresses and strains, then an influence of the opposite character on the dependence \(\mu=\nu\) may be expected.
REFERENCES
- A. A. Ilyushin. Plasticity. GITTL, 1948.
- Collection Theory of Plasticity, ed. by Yu. N. Rabotnov. IL, 1948.
- A. Nadai. Plasticity and Fracture of Solids. IL, 1954.
- Yu. N. Rabotnov. Strength of Materials. GIFML, 1962.
- A. R. Rzhanitsyn. Calculation of Structures Taking Account of the Plastic Properties of Materials. Moscow, 1954.