Stress distribution in the plastic region around stress concentrators
Abstract Generated abstract
This study investigates stress distributions in plastically deformed regions around stress concentrators in tensile steel strips, focusing on conditions approaching crack initiation. Experiments on strips with holes, sharp notches, and rounded notches used fine reference grids and thickness measurements to determine principal strains and calculate principal stresses along critical sections at several loading stages. The results show that, in strain-hardening material, the maximum longitudinal principal stress occurs at the concentrator and that plastic stress concentration is substantially lower than elastic concentration and varies with load. The findings indicate significant differences from predictions for ideally plastic bodies, suggesting that such theory is of doubtful applicability for estimating concentration factors under ordinary temperature and loading-rate conditions.
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PROCEEDINGS
OF THE TOMSK POLYTECHNIC INSTITUTE
NAMED AFTER S. M. KIROV, HOLDER OF THE ORDER OF THE OCTOBER REVOLUTION
AND THE ORDER OF THE RED BANNER OF LABOR
Vol. 188 1974
STRESS DISTRIBUTION IN THE PLASTIC REGION AROUND STRESS CONCENTRATORS
B. P. CHEBAEVSKII, G. D. DEL
(Presented by the scientific seminar of the Department of Strength of Materials)
Analytical and experimental studies of stress concentration in the elastic region [1] have shown that the stresses near concentrators are 3–10 times higher than the average stresses over the cross section. In this connection, the material in this zone is very often deformed plastically. In a number of cases—for example, in designing structures of minimum weight—if the material of the part is sufficiently ductile, it is expedient deliberately to allow certain plastic deformations, since this eliminates overloading of the main material by a factor of 3–10. At the same time, however, the danger arises of crack formation. In order to avoid cracks, it is necessary to know the stressed state in the plastic region under comparatively large plastic deformations. Meanwhile, in earlier studies [2], [3], the stress distribution was determined only for small plastic deformations.
The aim of the present investigation is an experimental study of the stressed state in the plastic region during tension of strips with stress concentrators, and an assessment of the possibility of determining the stresses in this case by methods of the theory of ideally plastic bodies.
Strips of steel grade 3 with a transverse section of \(50 \times 5\) mm were tested. Stress concentration was studied at holes of different diameters, at sharp notches with different depths and varying opening angles, and at notches with a rounded base (Figs. 2, 3). The stresses were determined at several loading stages up to the appearance of cracks. After each loading stage, diagrams of \(\sigma_1\) and \(\sigma_2\) were constructed along the critical section.
The principal stresses were calculated by the formula [4]
\[ \sigma_{1,2}=\frac{\sigma_i}{e_i} \left[ -e_3 \pm \frac{\sqrt{e_i-e_3}}{\sqrt{3}} \right], \]
where
\[
\sigma_i
\]
is the stress intensity, \(e_i\) is the strain intensity, and \(e_3\) is the transverse strain.
To calculate the stresses \(\sigma_1, \sigma_2\) at any point of the axis of symmetry, it is sufficient to know two principal strains at that point. These strains can be determined experimentally, and the third strain can be calculated from the condition of constancy of volume. To determine the longitudinal strain \(e_1\), a reference grid was rolled onto the plate—
as the matrix, a raster cliché with a base of 0.376 mm was used [5]. The measurement was carried out over a base of six grid cells (2.26 mm). The measurements were made with an instrumental microscope. The measurement accuracy was 0.01 mm. The use of an image of a raster cliché as the grid has major advantages: the grid base is exceptionally precise and sufficiently small, which makes it possible to investigate small regions of plastic deformation. The method makes it possible to determine plastic strains of 1–1.5%.
Fig. 1. Stress concentration at holes
The strain \(e_2\) (Fig. 1) is comparatively small, while the gradient of its variation along the axis is large. This limits the base over which the measurement is made and, consequently, reduces the accuracy of determining this strain. It is therefore expedient to determine, along with \(e_1\), the transverse strain \(e_3\). In determining it we assume that the deformation in the transverse direction is homogeneous and, by virtue of this,
\[ e_3 = \ln \frac{h}{h_0}, \]
where
\(h_0\) and \(h\) are the thicknesses of the strip before and after deformation. In our experiments the thickness of the plate was measured with a dial-type indicator with an accuracy of up to 0.01 mm. First the surface topography was recorded; then, from these data, the thickness of the plate at points along the axis of symmetry was determined and the strain \(e_3\) was calculated.
The stress diagrams obtained at holes of different diameters are shown in Fig. 1. The character of the diagrams does not change qualitatively when the hole diameter is varied. For a hole 3 mm in diameter, stress diagrams are given for three loading stages.
The stress distribution in plates with notches is shown in Figs. 2, 3. The diagrams correspond to the moment when a visible crack appears at the stress concentrator and are practically independent of the die angle or the radius of rounding. At different loading levels, the stress diagrams are identical to those presented. However, the stress concentration factor, defined as
\[ \alpha=\frac{\sigma_{\max}}{\sigma_{\text{nom}}}, \]
where
\[ \sigma_{\max} \]
is the maximum stress in the stress diagram \(\sigma_1\), and \(\sigma_{\text{nom}}\) is the nominal stress over the section, is different for each loading level.
Fig. 2. Stress concentration at sharp notches
The accuracy of determining the principal stress \(\sigma_1\) over the section was assessed by comparing the calculated and actual deformation force. The mean discrepancy between the calculated and actual forces in 34 comparisons was \(6\%\).
The stress \(\sigma_1\) reaches its maximum value near the stress concentrator. In an ideally plastic body, \(\sigma_1\) is minimal at the stress concentrator. This is explained as follows. The Mises plasticity condition for a plane stress state has the form
\[ \sigma_1^2+\sigma_2^2-\sigma_1\cdot\sigma_2=\sigma_i^2, \]
where
\(\sigma_1\) and \(\sigma_2\) are the principal stresses, and \(\sigma_i\) is the yield limit, constant throughout the volume. Since \(\sigma_1>\sigma_2\ge 0\), it follows that \(\sigma_1\ge\sigma_i\). At the boundary the stress state is linear, and therefore \(\sigma_1=\sigma_i\). For an ideally plastic material this is the minimum value of \(\sigma_1\).
Fig. 3. Stress concentration at notches with a rounded root
In a real strain-hardening material, \(\sigma_i\) varies over the volume of the part and depends on the deformation. As the distance from the concentrator increases, \(\sigma_2\) increases, while \(\sigma_i\) decreases, since the deformation at the concentrator is much greater than at any other point of the plate. The increase in \(\sigma_1\) caused by the growth of \(\sigma_2\) is insignificant, whereas the decrease in \(\sigma_1\) caused by the decrease in \(\sigma_i\) is very large. As a result, the stress \(\sigma_1\) proves to be maximal at the concentrator. The investigations carried out make it possible to draw the following conclusions:
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Stress concentration in the elastic and plastic regions has certain common features. In both cases the principal stresses \(\sigma_1\) and \(\sigma_2\) are positive, and the longitudinal stresses \(\sigma_1\) are maximal at the concentrator.
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The stress concentration under plastic deformation of a strain-hardening material differs substantially from the stress concentration in ideally plastic bodies. The greater the strain hardening of the material, the greater this difference. Therefore, the possibility of applying the theory of ideally plastic bodies to calculating the concentration factor under ordinary temperature–rate conditions is doubtful.
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The stress concentration factor in the plastic region is considerably smaller than in the elastic region and depends on the load.
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Further experimental investigations are necessary for
establishing the dependence of the concentration coefficient on the relative dimensions of the concentrator, the load, and the mechanical properties of the material.
References
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G. Neuber. Stress Concentration. OGIZ, Moscow, 1947.
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A. I. Kodaev. Stress concentration in the plastic region. Transactions of the N. E. Zhukovsky Academy, issue 316, 1949.
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M. Kh. Akhmetzyanov. Investigation of stress concentration in the plastic region by means of photoelastic coatings. Bulletin of the Academy of Sciences of the USSR, Division of Engineering, Mechanics and Machine Building, No. 1, 1963.
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G. D. Del. Determination of the stress state in the plastic region by hardness measurement. Proceedings of the Institutes of the State Committee for Standards, Measures, and Measuring Instruments, 1966.
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G. D. Del, M. G. Goldschmidt. Investigation of plastic deformation on models made from extension dies. Factory Laboratory, No. 4, 1966.