In this article we will discuss about the geometrical properties of slip lines.
Both the theorems of Hencky are geometrical properties of slip lines. The first theorem may be stated as follows.
If a pair of lines of one type (say α) is intersected by lines of second type (β) then the angle between the tangents to α-lines at the points of intersection with any β line remains constant.
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Consider (Fig. 5.6) in which two α-lines are intersected by two β-lines at points A, D with one β-line and at B and C with second β-line. According to the theorem angle between tangents at points A and D is equal to the angle between the tangents at points B and C, i.e. ∠AOD = ∠BO’C. The proof of the theorem is as below-
2. Hencky’s Second Theorem:
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Hencky’s second theorem states that if we move along a slip line (say α-line) the radii of curvature of the other lines (β) at the points of intersections change equal to the distance traversed, i.e.
Example of a Slip Line Solution:
In order to illustrate the application of slip lines in metal forming problems, we take the case of extrusion with 50% reduction (Fig. 5.8) with frictionless cylinder walls and dies. The solution illustrated in Fig. 5.8(a) was given by Hill, Figure 5.8(b) gives the velocity diagram or hodograph.
The slip line field consists of two circular quadrants with circular and radial lines as slip lines. The slip lines meet the central axis and surface of container at 45°. The triangular zones ADC and RSQ (shown hatched) are dead zones. The deformation and slip line field is symmetrical about the center line and hence we need consider only the upper half region.
During the extrusion process, the material particles moving parallel to container wall on L.H.S. of point A have a velocity V which is represented by line EF in Fig. 5.8(b). The material particles during crossing the line AB which is α-line, suffer a velocity discontinuity. At the point A the direction of discontinuity is along tangent to the curve AB i.e. at 45° to central line. It is shown in Fig. 5.8(b) by the line FA’.
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Similarly the particles coming at the point M suffer discontinuity along FM as they cross AB. So is the case with particles coming at the point N. They suffer discontinuity along FN’. But the magnitude of the discontinuity is same at all these points. So the final velocities at the points A, M, and N after crossing AB are given by vectors EA’, EM’ and EN’ respectively. Similarly for the particles coming to point B have velocity EB’ after crossing AB.
As these particles cross the deformation zone and come to line BC they again suffer velocity discontinuity along BC when they cross this line, so that on exit from the deformation zone their velocity becomes parallel to the center line.
The velocity discontinuity along BC is shown by line B’G in Fig. 5.8(b). The magnitude of discontinuity along BC can be determined by drawing line B’G at 45° to EF from B’ to intersect the line EF at G. Thus B’G is the magnitude of the discontinuity. The final velocity, therefore, is given by EG which is equal to 2V. For 50% area reduction this should be the final velocity. Therefore, this is an acceptable slip line field.
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For determining the punch pressure let us consider the stresses acting on the line BC. One is the shear stress K acting along BC and the other is hydrostatic pressure P in Fig. 5.9(a). The directions of these stresses are such that net stress along the central axis is zero. Since BC is at 45° to the central-axis we get
The solution is an intuitive solution. Since stresses on the boundary are not known so the constructed slip line field is verified by comparing the velocity of material particles entering the deformation zone with the velocity of particles getting out of the deformation zone. If the ratio is in accordance with the area reduction then the solution is an acceptable one. It may not be exact solution. It can be regarded as one of the upper bound solutions. There are a large number of slip lines solutions for different plastic deformation problems in literature.