Yield and Plastic Flow David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge MA 02139 October 15 2001 Introduction In our overview of the tensile stress strain curve in Module 4 we described yield as a permanent molecular rearrangement that begins at a su ciently high stress denoted Y in Fig 1 The yielding process is very material dependent being related directly to molecular mobility It is often possible to control the yielding process by optimizing the materials processing in a way that in uences mobility General purpose polystyrene for instance is a weak and brittle plastic often credited with giving plastics a reputation for shoddiness that plagued the industry for years This occurs because polystyrene at room temperature has so little molecular mobility that it experiences brittle fracture at stresses less than those needed to induce yield with its associated ductile ow But when that same material is blended with rubber particles of suitable size and composition it becomes so tough that it is used for batting helmets and ultra durable children s toys This magic is done by control of the yielding process Yield control to balance strength against toughness is one of the most important aspects of materials engineering for structural applications and all engineers should be aware of the possibilities Figure 1 Yield stress Y as determined by the 0 2 o set method Another important reason for understanding yield is more prosaic if the material is not allowed to yield it is not likely to fail This is not true of brittle materials such as ceramics that fracture before they yield but in most of the tougher structural materials no damage occurs before yield It is common design practice to size the structure so as to keep the stresses in the elastic range short of yield by a suitable safety factor We therefore need to be able to predict 1 when yielding will occur in general multidimensional stress states given an experimental value of Y Fracture is driven by normal stresses acting to separate one atomic plane from another Yield conversely is driven by shearing stresses sliding one plane along another These two distinct mechanisms are illustrated n Fig 2 Of course bonds must be broken during the sliding associated with yield but unlike in fracture are allowed to reform in new positions This process can generate substantial change in the material even leading eventually to fracture as in bending a metal rod back and forth repeatedly to break it The plastic deformation that underlies yielding is essentially a viscous ow process and follows kinetic laws quite similar to liquids Like ow in liquids plastic ow usually takes place without change in volume corresponding to Poisson s ratio 1 2 Figure 2 Cracking is caused by normal stresses a sliding is caused by shear stresses b Multiaxial stress states The yield stress Y is usually determined in a tensile test where a single uniaxial stress acts However the engineer must be able to predict when yield will occur in more complicated real life situations involving multiaxial stresses This is done by use of a yield criterion an observation derived from experimental evidence as to just what it is about the stress state that causes yield One of the simplest of these criteria known as the maximum shear stress or Tresca criterion states that yield occurs when the maximum shear stress reaches a critical value max k The numerical value of k for a given material could be determined directly in a pure shear test such as torsion of a circular shaft but it can also be found indirectly from the tension test as well As shown in Fig 3 Mohr s circle shows that the maximum shear stress acts on a plane 45 away from the tensile axis and is half the tensile stress in magnitude then k Y 2 In cases of plane stress Mohr s circle gives the maximum shear stress in that plane as half the di erence of the principal stresses max p1 p2 2 2 1 Figure 3 Mohr s circle construction for yield in uniaxial tension Example 1 Using p1 pr b and p2 z pr 2b in Eqn 1 the shear stress in a cylindrical pressure vessel with closed ends is1 1 pr pr pr 2 b 2b 4b where the z subscript indicates a shear stress in a plane tangential to the vessel wall Based on this we might expect the pressure vessel to yield when max z max z k Y 2 which would occur at a pressure of 4b max z 2b Y r r However this analysis is in error as can be seen by drawing Mohr s circles not only for the z plane but for the r and rz planes as well as shown in Fig 4 pY Figure 4 Principal stresses and Mohr s circle for closed end pressure vessel The shear stresses in the r plane are seen to be twice those in the z plane since in the r plane the second principal stress is zero pr 1 pr 0 2 b 2b Yield will therefore occur in the r plane at a pressure of b Y r half the value needed to cause yield in the z plane Failing to consider the shear stresses acting in this third direction would lead to a seriously underdesigned vessel max r Situations similar to this example occur in plane stress whenever the principal stresses in the xy plane are of the same sign both tensile or both compressive The maximum shear stress 1 See Module 6 3 which controls yield is half the di erence between the principal stresses if they are both of the same sign an even larger shear stress will occur on the perpendicular plane containing the larger of the principal stresses in the xy plane This concept can be used to draw a yield locus as shown in Fig 5 an envelope in 1 2 coordinates outside of which yield is predicted This locus obviously crosses the coordinate axes at values corresponding to the tensile yield stress Y In the I and III quadrants the principal stresses are of the same sign so according to the maximum shear stress criterion yield is determined by the di erence between the larger principal stress and zero In the II and IV quadrants the locus is given by max 1 2 2 Y 2 so 1 2 const this gives straight diagonal lines running from Y on one axis to Y on the other Figure 5 Yield locus for the maximum shear stress criterion Example 2 Figure 6 a Circular shaft subjected to simultaneous twisting and tension b Mohr s circle construction A circular shaft is subjected to a torque of half that needed to cause yielding as shown in Fig 6 we now ask what tensile stress could be applied simultaneously without causing yield 4 A Mohr s circle is drawn with shear stress k 2 and unknown tensile stress Using