Strength Assessment of Disks under Radial Upsetting
Abstract Generated abstract
This paper examines fracture risk in disks made from ductile and low-ductility metals during radial upsetting, with attention to cracks that form in internal regions during forging. The authors combine limiting-ductility diagrams with measurements of strain fields on upset disks of duralumin, steel, and brass, calculating strain intensity and the stress-state rigidity coefficient from deformation and flow-theory relations. The study identifies the central region of the disk section as the most dangerous zone, where deformation is greatest and the stress state is close to simple shear, and shows that fracture occurs when the stress-strain region intersects the limiting-plasticity diagram. An empirical relation is proposed between upsetting degree and strain intensity at the dangerous point, allowing approximate estimation of safe upsetting limits from torsion-test plasticity data.
Full Text
STRENGTH ASSESSMENT OF DISKS UNDER RADIAL UPSETTING
V. A. Ogorodnikov, V. D. Pantsiulius
(Presented by the scientific seminar of the Department of Strength of Materials)
Radial upsetting of strain-hardening materials models the forging operation. In forging disks made of low-ductility steels, cracks often arise in the internal regions. In order to select optimal parameters of the forging process, it is necessary to know the amount of upsetting at which the material of the blanks fails.
In the present work, the strength of disks made of ductile and low-ductility metals under radial upsetting is investigated, and a parameter is sought by which one can judge the capacity of a metal for deformation.
To assess the strength of a blank, it is necessary to know the dependence of the limiting ductility of the material on the type of stress state. In works [1, 2] it is proposed that this dependence be characterized by a limiting-ductility diagram relating the intensity of logarithmic strain at the moment of crack initiation, \(e_i\), to the stress-state rigidity coefficient
\[ \eta = \frac{\sigma_1 + \sigma_2 + \sigma_3}{\sigma_i} = \frac{3\sigma}{\sigma_i}, \tag{1} \]
where
\[ \begin{aligned} \sigma_1,\ \sigma_2,\ \sigma_3 &\text{ are the principal stresses,}\\ \sigma &\text{ is the hydrostatic pressure,}\\ \sigma_i &\text{ is the stress intensity.} \end{aligned} \]
Figure 1 shows a limiting-ductility diagram for duralumin D-1, constructed according to the procedure proposed in [3]. At \(\eta = 1\), the diagram was constructed from the results of tensile testing of a solid specimen; in the range \(1 > \eta > -0.6\), solid cylindrical specimens subjected to a torque and an axial force were tested (tensile for \(\eta > 0\) and compressive for \(\eta < 1\)).
To assess the strength of blanks, it is necessary to know the range of the stress-strain state in the process of radial upsetting of disks. During upsetting of disks with \(H_0/D_0 = 0.25\), a plane stress state is realized. The parameters \(\eta\), \(e_i\) for the plane stress state can be calculated from the relations of deformation theory
\[ \sigma_3 - \sigma = \frac{2}{3}\frac{\sigma_i}{e_i}\,(e_3 - e), \tag{2} \]
where
\[ \sigma_3 \]
is the principal stress (it is equal to zero for the plane stress state).
\(e_3\) — the principal strain in the direction perpendicular to the diametral section of the disk,
\(e\) — the mean strain.
Fig. 1. Plasticity diagram of duralumin and the regions \(\eta - e_i\) during upsetting of disks
For an incompressible material we have
\[ e=-\frac{2}{3}\frac{\sigma_i}{e_i}e_3. \tag{3} \]
Substituting (3) into (1), we obtain
\[ \eta=2\frac{e_1+e_2}{e_i}, \tag{4} \]
where
\[ e_i=\frac{2}{\sqrt{3}}\sqrt{e_1^2+e_2^2+e_1e_2}, \tag{5} \]
\(e_1, e_2\) — the principal strains in the diametral section of the disk during its upsetting.
The stiffness coefficient of the stress state can also be calculated with the aid of the relations of flow theory
\[ \eta=-2\frac{\partial e_3}{\partial e_i}. \tag{6} \]
For the investigation, disks were made with diameter \(D_0 = 80\) mm and height 20 mm, from D-1 duralumin, steel 40, and LS-59-1 brass. The end surface of the disks was polished, and on one of the quarters a dividing grid was applied, consisting of a system of circles with diameter \(d_0 = 3.5\) mm. The duralumin disks were upset in stages by 8, 14, 18, and 20%, with unloading until visible cracks appeared on the end surface. During upsetting, a lubricant made from a mixture of colloidal graphite and glycerin was used, with gaskets made of lead foil.
At each stage of deformation the dimensions of the ellipses into which the initial circles had deformed were measured. From the known relations, the principal strains \(e_1\), \(e_2\), and, from (4), (5), \(\eta\) and \(e_i\), were calculated.
The parameters \(\eta\) and \(e_i\) were preliminarily smoothed by drawing isolines. Figure 2 shows the levels of constant \(\eta\) and \(e_i\) for a disk upset by 20%. It follows from the figure that the distribution of strains during
Fig. 2. Isolines of \(\eta\) and \(e_i\) for a duralumin disk upset by 20%
upsetting is nonuniform. At the center of gravity of the section the strain intensity reaches \(e_i = 0.38\), and the stress-state rigidity coefficient is \(\eta = +0.15\). In the indicated region fracture is most probable—here there is the greatest deformation and an unfavorable stress-state scheme, close to simple shear \((\eta = 0)\).
Figure 1 shows the regions of the stress-strain state considered for the cases investigated. The upper boundaries of the regions of the stress-strain state indicate the required plasticity. For example, disks upset by 8, 14, and 18% did not fail. Their regions of \(\eta\) and \(e_i\) were located below the limiting-plasticity diagram. In the disk upset by 20%, a crack appeared at the center of gravity of the section. The upper boundary of the region of \(\eta\) and \(e_i\) intersected the limiting-plasticity diagram. It should be noted that in all the cases investigated, the magnitude of the required plasticity can be established by testing the metal in torsion, since the parameter \(\eta\) at the dangerous points is close to zero. The regions of \(\eta\) and \(e_i\) for the brass disk upset by 20% were calculated from relations of deformation theory and flow theory. The satisfactory agreement of the upper boundaries of the regions of \(\eta\) and \(e_i\) makes it possible to use deformation theory for the calculation.
Figure 3 presents a curve of the dependence of the degree of upsetting of D-1 duralumin on the magnitude of \(e_i\) at the center of gravity of the disk sections. The obtained
the graph makes it possible to determine the amount of upsetting as a function of the strain intensity at the dangerous point, which can be determined by testing the material in torsion. Approximating the experimentally found function, we obtained the formula
\[ \frac{\Delta D}{D}=-1.67\,(\bar e_i)^2+1.13\,\bar e_i, \tag{7} \]
where \(\Delta D=(D-H)\),
\(H\) is the height of the disk after upsetting,
\(D\) is the disk diameter,
\(\bar e_i\) is the magnitude of the strain intensity at the dangerous point.
Fig. 3. Dependence of the degree of upsetting on the value of \(e_i\) at the dangerous point
The value \(\bar e_i\) can be determined by testing the material in torsion. Specifying the plasticity safety factor
\[ \kappa=\frac{\bar e_i}{e_i}, \tag{8} \]
where \(e_i\) is the required plasticity, one can determine the amount of upsetting at which the disk is deformed without fracture.
In [5] it is shown that the distribution of strains, other conditions being equal—the amount of upsetting, dimensions, and contact conditions—practically does not depend on the properties of the material. This apparently makes it possible to use formula (7) for an approximate determination of the safe amount of upsetting for other materials as well. This is confirmed by the experimentally obtained regions of the stress–strain state calculated for disks made of brass L59-1 and duralumin D-1, steel 40, upset by 20%. The upper boundaries of the regions \(\eta-e_i\) practically coincide for disks made of the indicated materials.
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