Slip-Line Field Analysis

1. INTRODUCTION
Slip-line field theory is based on analysis of a deformation field that is both geometrically self-consistent and statically admissible. Slip lines are planes of maximum shear stress and are therefore oriented at 45? to the axes of principal stress. It is assumed that
1. The material is isotropic and homogeneous,
2. The material is rigid – ideally plastic (i.e., no strain hardening),
3. Effects of temperature and strain rate are ignored,
4. Plane-strain deformation prevails, and
5. The shear stresses at interfaces are constant: usually frictionless or sticking friction.
Figure 1 shows the very simple slip line for indentation where the thickness, t, equals the width of the indenter, b. The maximum shear stress occurs on line DEB and CEA. The material in triangles DAE and CEB is rigid. As the indenters move closer together the field must change. However, for now, we are concerned with calculating the force when the geometry is as shown. The stress ?y must be zero because there is no restraint to lateral movement. The stress ?z must be intermediate between ?x and ?y. Figure 2 shows the Mohr’s circle for this condition. The compressive stress necessary for this indentation is ?x = ?2k. Few slip-line fields are composed of only straight lines. More complicated fields will be considered.
2. GOVERNING STRESS EQUATIONS
With plane strain, all of the flow is in the x–y plane. This means that d?y =?d?x and d?z = 0 so ?z = ?2 = (?x + ?y)/2. Therefore, according to the von Mises criterion, ?z is always the mean or hydrostatic stress.
?2 = (?1 + ?2 + ?3)/3 = ?mean ………………………………………...…. (1)

Fig. 1: A slip-line field for frictionless plane-strain indentation.
and
?1 = ?2 + k, ?3 = ?2 ? k ……………………………………………….(2)
Thus plane-strain deformation can be considered as pure shear with a superimposed hydrostatic stress, ?2.
Planes of maximum shear stress are mutually perpendicular. The projections of these planes form a series of orthogonal lines called slip lines. Figure 3 illustrates a section of a field of slip lines. The shear stress acting on these lines is k, while the mean stress, ?2, acts perpendicular to the slip lines. The slip lines are rotated at some angle ? to the x and y axes.

Fig. 2 : Mohr’s stress circle for frictionless plane-strain indentation in Fig. 9.1.

Fig. 3: Stresses acting on a curvilinear element.
To develop the necessary equations it is necessary to adopt a convention for slipline identification. The families of slip lines are labeled either ? and ?. The convention is that the largest principal stress (most tensile) lies in the first quadrant formed by ? and ? lines as illustrated in Figure 4. If all of the stresses are compressive, the least negative is ?1.
For plane strain, ?xy and ?zx are zero, so the equilibrium equations (equation 1.40) reduce to
??x/?x + ??yx/?y = 0
and ??y/?y + ??xy/?x = 0. ………………………………………………(3)
From the Mohr’s stress circle diagram, Figure 5,
?x = ?2 ? k sin 2?,
?y = ?2 + k sin 2?,
?xy = k cos 2? …………………………………………………………..(4)
Differentiating equations 4 and substituting into equations 3,
??2/?x ? 2k cos 2? ??/?x ? 2k sin 2? ??/?y = 0
??2/?y + 2k cos 2? ??/?y ? 2k sin 2? ??/?x = 0. ……………………….(5)

Fig. 4: The 1-axis lies in the first quadrant formed by the ?- and ?-lines.

Fig. 5: (a) Mohr’s stress and (b) strain-rate circle for plane strain.