Slip lines may be constructed in the body which is undergoing plastic deformation or is at the yield point, provided the stresses, i.e. normal stress and shear stress, are completely known on its entire boundary. The construction gives an orthogonal network of slip lines covering the cross section of the deforming body.
We can reach any point inside the body through this network. We go along α and β-lines of the network, gives the state of stress at the desired point in the body. In this way we get an exact solution of the problem. Two types of boundaries may be encountered.
(i) The boundary is a free line. A free line is one which is not a slip line. In this case the directions of α and β-slip lines may be determined from the stresses prescribed on the boundary and the solution may be extended into the deforming zone limited by the intersecting slip lines emanating from end points of the free line.
(ii) If only one slip line is given as boundary the solution cannot be extended. Therefore, boundary must consists of two intersecting slip lines α and β. Solution may be extended within the region bounded by the two given slip lines and two intersecting slip lines passing through the end points of the given slip lines.
Direction of Slip Lines:
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For construction of slip lines first of all we have to determine the direction of α and β-slip lines on the boundary points from the stresses prescribed on it. Four types of stress conditions may be encountered on the boundary.
These are as follows:
(i) Stress free surface not contacting tool.
(ii) Frictionless interface between tool and deforming material.
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(iii) Interface with Coulomb friction condition.
(iv) Interface with sticking friction condition.
In this case there is no shear or normal stress acting on the surface. Since there is no shear stress, the plane tangent to the surface is a principal plane and normal to it is the direction of one of the principal stresses. The other two principal stresses lie in the tangential plane. Let σ1, σ2 and σ3 be the principal stresses. Let us take σ3 along the normal to the tangential plane. Let σ2 be the middle stress. We know that in plane strain case σ2 is given as follows-
This shows that plastic deformation is caused by σ1 which may be compressive or tensile.
These two cases are discussed below:
(a) σ1-Compressive:
The directions of principal stresses are given below:
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(i) Normal to the tangential plane, i.e. direction of σ3 which is zero.
(ii) A direction in tangential plane, i.e. direction of σ1
(iii) Normal to the above two directions and in tangential plane, i.e. direction of σ2
Since plastic deformation takes place because of σ1 and σ3 and σ3 = 0, therefore,
In this case σ3 is algebraically the major principal stress. The slip lines are at 45° to the tangent to the boundary. The directions of slip lines are as shown in Fig. 5.10(a). The α-line is inclined at 45° to σ3 in clockwise direction and β line is at 90° to α -line in counter clock direction.
(b) σ1-Tensile:
In this case σ1 = 2K and hence σ2 = K and hydrostatic pressure P = – K. Therefore σ1 is the major principal stress and α slip line is inclined at 45° in clockwise direction to major principal stress in Fig. 5 10(b).
On a frictionless interface, there is no shear stress. Thus normal to interface at the desired point is the direction of a principal stress. Let σ3 be this principal stress. This case differs from the above case in that σ3 is now not equal to zero. In real problems the stress σ3 can only be compressive.
Now if σ1 is largest compressive stress then σ3 is the smallest compressive stress or largest algebraic stress. In this case α-line and β-lines are shown in Fig. 5.11(a). Now if σ3 is the compressive stress, and σ1 is tensile or algebraically the largest stress, α-line is at 45° to σ1 in clockwise direction in Fig. 5.11 (b).
(iii) Interface with Coulomb Friction:
Let σ3 be the stress normal to the surface. The frictional shear stress is equal to μσ3. Let one of the slip line in Fig. 5.12(a) be inclined at θ with the free surface. Figure 5.12(b) shows the Mohr’s stress circle. Take radius CE such that EF = μσ3 and ∠ECG = 2θ. Then,
(iv) Interface with Sticking Friction:
The sticking friction occurs when the shear stress at the interface reaches the value K = yield strength of material in shear. And this is also the maximum shear stress.
The coefficient of friction becomes irrelevant because friction stress cannot exceed K.
Therefore, one of the slip lines is parallel to the surface and other is perpendicular to it. Friction occurs only when σ3 is compressive. The directions of or α and β lines are shown in Fig. 5.13.
Constructing Slip Lines from a Free Line as Boundary:
A free line is one which is not a slip line. Let OR (Fig. 5.14) be the part of the boundary of plastically deforming zone of the body on which normal and shear stresses are known. Let OR be an interface between tool and deforming region. Now if σxx and τxy are given on all points of OR, we may determine σyy from the yield condition.
With this data one can determine the directions of α and β slip lines at all points on OR. Let us divide OR into small segments by taking points A, B, C etc. on the line. The accuracy of solution is increased if the distances OA, AB, BC etc. are small. We may draw the directions of α and β-lines on the chosen points. Let α-line from A and β-line from O intersect at the point D.
Let OX and OY be the reference coordinate axes. The third coordinate OZ is perpendicular to these two, making a right handed co-ordinate system. Let θo, θd and θa be the inclinations of α-lines with OX at points O, D and A respectively. Let Po, Pd and Pa be the values of hydrostatic pressures at these points. New for segment AD of α-line.
Now at the point D we know θd, and Pd from Eqns. (5.44 and 5.45). We can determine all the three components of stress state. In short we know the stress state at D. Now we have to determine the location of D in the real body.
Let us take that the small segments AD and OD of α and β lines respectively can be approximated as straight segments. Because the points O and A are near to each other, the error due to this assumption would be negligible. Therefore, we may draw AD at the mean of angles of α-line at A and D.
The intersection of the two lines gives the location of D. Following the above procedure we can establish other points inside the body. In this way the solution is extended to entire region bounded by ORSO. We may also follow an analytical method for calculation of co-ordinates of point D in terms of co-ordinates of A and O and inclinations of α-line and β-line.
Construction of Slip Lines when Two Slip Lines are given:
Let OA be an α-slip line (Fig. 5.15) and OB be a β-slip line intersecting α-line at O. The normal stress-hydrostatic pressure and shear stress which is equal to K are known at all the points on OA and OB. Either slip line may be a part of interface between tool and the deforming body, on which there is sticking friction, i.e. the magnitude of frictional stress is equal to K.
In order to construct slip lines in the region bounded by two slip lines, we take a number of points on the two lines, such as (0,1), (0,2) (0,3) etc. on β line and (1,0), (2,0), (3,0) etc. on α line. Let OX and OY be the reference rectangular coordinates.
Let us denote the inclination of α-line with respect to OX at (1, 0), (2, 0), etc., by θ10, θ20 and so on. Similarly the inclination of α-line at the points (0, 1), (0, 2), (0, 3) are denoted by θ01, θ02, θ03 etc. The values of hydrostatic pressures are also denoted similarly.
Let us take that α-line from point (0, 1) and β-line from point (1, 0) intersect at point (1, 1). Let P11 be the pressure and θ11 be the inclination of α-line at the point (1, 1). We may write for segment (0, 1) – (1, 1) on α-line as,
Proceeding in this way we can know the stress state at the points (1, 1), (1, 2), (1, 3) etc. as well as locate them in the body bounded by the given two slip lines. The slip lines from at the extreme points B and A intersect at D. The region OADBO is the region of the solution. Beyond this region the solution cannot be extended.