Stress-Strain State under Shear
Abstract Generated abstract
This paper investigates the stress and strain state in double shear during the stage of developed plastic deformation preceding fracture. Using hardness measurements and division grids on metal specimens, the authors show that principal strain directions remain essentially fixed at 45 degrees to the separation line, allowing shear stresses along this line to be inferred from hardness as octahedral shear stresses. The results indicate that shear stress distributions become nearly uniform with increasing punch penetration, especially in ductile materials, while disturbed zones near tool edges depend on specimen thickness. The study also relates strain rate to punch velocity and sheet thickness, and proposes a way to estimate deforming force from shear and torsion diagrams, with experimental deviations reported within about 12 percent.
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STRESS–STRAIN STATE DURING SHEARING
L. M. Sedokov, G. D. Del
(Recommended for publication by the scientific seminar of the Department of Strength of Materials)
Shearing is understood as the displacement of one part of a material relative to another under forces applied in the immediate vicinity of the separation surface. Material undergoes this type of deformation in a number of stamping operations and in shear tests. Numerous fastening elements work in shear.
The shearing process is usually divided into three stages: 1) elastic and incipient plastic deformation, 2) developed plastic deformation, 3) the appearance of cracks and fracture. The stress–strain state in the first stage has been studied quite well by the optical method [1, 2]. The deformed state in the region of deep plastic deformations during punching and blanking was investigated by V. D. Golovlev [3].
Below are presented the results of an investigation of the stress–strain state during double shear at the stage of developed plastic deformation preceding fracture. The investigation was carried out by methods of measuring hardness and division grids. Hardness was measured with a diamond pyramid under a load of 20 kg. The base of the division grid was 0.208 mm.
Observations of the deformation of the division grid during penetration of the punch into the specimen showed that along the entire separation line, with the exception of a disturbed zone of insignificant dimensions (about 5% of the thickness) adjoining directly the tips of the punch and die, there is no rotation of the directions of the principal strains. The principal directions, independently of the specimen thickness and the amount of punch penetration, remain constant and form angles of 45° with the separation line. It follows from this that the separation line practically coincides with the shear slip line.
This circumstance considerably simplifies the investigation of the stress state along the separation line by measuring hardness. As is known, the stress state in plane deformation is characterized by the superposition of hydrostatic pressure on the tangential stress of pure shear, equal to the octahedral shear stresses and directed along the slip lines.
Thus, during shearing the envelope of the shear stresses along the separation line coincides with the envelope of the octahedral shear stresses, which can be established by measuring hardness [4].
In specimens of precipitation-hardening stainless steel 1Kh18N9T, 5 mm thick and 20 mm wide, the punch of a double-shear fixture was driven in to depths of 12 and 15%. The specimen was then cut along the mid-plane of its width. The resulting section was polished. Along the separation line the hardness was measured. Using formula [4],
\[ \tau = 0.19H \]
the shear stresses were determined.
The stress diagrams obtained (Fig. 1) show that, as the penetration increases, the shear-stress diagram becomes more uniform, and already at a penetration of about 15% the shear stresses are distributed practically uniformly along the separation line. Similar results were also obtained on specimens of another thickness. In less work-hardening materials, the leveling of the diagrams occurs still earlier.
Fig. 1. Diagrams of shear stresses in shearing.
Hence, first, it follows that strength calculations in shearing based on the assumption of a uniform distribution of shear stresses are more approximate for brittle materials and less so for ductile materials.
Second, in shear tests of ductile materials, the shear strength is revealed; this is confirmed by the coincidence of the shear and torsional strength limits in ductile materials [5].
An investigation of deformations along the separation line on specimens 5 to 15 mm thick showed that the shear diagrams of specimens of different thickness coincide over the entire height, with the exception of the disturbed zone. As for the disturbed zone, the deformations in it depend to a considerable degree on the specimen thickness: at identical relative punch penetrations, the shears are greater the greater the thickness. This, in particular, explains a certain decrease in the relative penetration and in the average shear stresses at the moment of appearance of sliding cracks with increasing thickness.
The dependence of the width of the deformed layer on the depth of penetration was studied on specimens 3 mm thick by V. D. Golovlev. We additionally studied the dependence of this width on the specimen thickness. It was established that this dependence is not linear: with increasing thickness, the rate of increase of the width of the deformed layer decreases.
Figure 2 presents the dependence of the shears on the relative penetration and on the dimensionless coordinate
\[ h = \frac{y}{s-i}, \]
where \(s\) is the specimen thickness and \(i\) is the penetration (specimen \(10 \times 10\) mm). This figure shows that the maximum shear along the separation line is in a linear dependence on the relative penetration:
\[ g = k(h)\frac{i}{s}. \]
Consequently, the strain rate is
\[ \frac{de}{dt}=\frac{\kappa(h)}{2s}\cdot \frac{di}{dt}=\frac{\kappa(h)}{2}\frac{v}{s}, \tag{1} \]
where \(v\) is the punch velocity. It follows from (1) that the strain rate in shearing is proportional to the relative velocity of displacement of the punch and varies over the height (Fig. 2).
The angular coefficients \(\kappa(h)\) in the disturbed zone depend on the thickness of the specimen. Therefore, the diagrams of strain rates for specimens of different thickness at equal relative velocities of punch displacement
Fig. 2. Shears and strain rate in shearing.
diverge somewhat at the extreme points. However, this divergence is insignificant and, to some approximation, may be neglected.
From Fig. 2 we have
\[ \frac{de}{dt}=(1.7\div 3)\frac{v}{s}. \]
Hence, for example, it follows that in punching with a punch moving at a velocity of \(150\ \text{mm/sec}\), in a sheet of thickness \(1\ \text{mm}\), the strain rate will be \(250\)—\(450\ \text{s}^{-1}\). At such a strain rate the mechanical properties undergo substantial changes. For example, the hardness of mild steels under these conditions increases by \(10\)—\(15\%\) [6].
Investigations performed on various metals have shown that the deformed state in shearing is practically independent of the properties of the material.
According to Fig. 2, for given thickness \(s\) and penetration \(i\), one can construct a shear diagram. From the shear diagram of the given material one can construct the corresponding diagram of tangential stresses and, by integrating it, obtain the magnitude of the deforming force. Thus, one can relate the torsion diagram to the shear diagram for a
in the region of substantial indentations. A check on 1Kh18N9T steel and copper showed that, in the range from 7 to 21% indentation, the deviation of the diagram constructed in this way from the actual one, obtained by oscillography, does not exceed 12%.
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