50 
One hypothesis due to Mohr states that plastic yielding commences when the 
maximum shear stress reaches a definite value which is independent of the 
nature of the stress system. Since the maximim shear stress is half the 
greatest principal stress difference, this hypothesis may be written as 
6,- 6, = constant =8, ach meee MeO (51) 
in which the constant is chosen to agree with the notation of equation 50 
for the special case of simple tension, This hypothesis is often referred 
to as the maximum stress-difference or maximum shear stress hypothesis. 
133. An alternative hypothesis is that yielding commences when 
2 2 2 
G6 ba.) + ( 6-63) + ( 6,- 6) = constant = 2 Sota (5p) 
in which the constant is again conveniently expressed in terms of the yield 
stress 8, for simple tension by substituting from equation 50 in equation 51. 
This hypothesis oan be given a physical significance by considering the 
elastic deformation prior to yielding. The energy U stored per unit 
volumes due to a purely elastic deformation obeying Hooks's law can be 
written if E is Young's Modulus and AL is Poisson's ratio in the form 
v GEL 6 66+ GP Deed | (6-6) + (C63) + (6-668) 
in which the first term represents the work done in increasing or decreasing 
volume whilst the second term is the remaining energy stored in change of 
Shape. Henos, equation 52 states that yielding will commence when the 
energy involved in change of shape, that is distortion, reaches a certain 
definite magnitude independent of the stress system, This hypothesis is 
attributed variously to von Mises, Hencky,and Huber and is usually known 
as the maximum energy of distortion’hypothesis or the maximum shear strain 
energy theory. 
134. Both equations 51 and 52 refer only to the condition for the 
commenesment of yielding and the constant 8, refers to varying stress 
systems rather than to varying strain after yielding. For use in the 
present problem the oonstant in either case will be taken as holding 
throughout yielding and since elastic strain is to be neglected, equation 51 
or equation 52 will then be assumed to hold from zero strain up to failure. 
Experimental evidence indicates that the assumption of equation 52 is more 
accurate than that of equation 51. However, equation 52 is a non-linear 
relation which frequently leads to intractable mathematics and it is 
necessary to use equation 51 or even more drastic approximations, So far as 
snergy absorption is concerned, equations 51 and 52 differ quantitatively 
at the most by about 15% whilst for qualitative deductions the two hypotheses 
are unlikely to lead to any essential differences. It may be noted that 
both depend essentially on yielding due to shear stresses since when 
6: = 62 = 6s and there are no shear stresses, neither hypothesis oan be 
satisfied; that is, both indicate no plastic yielding under hydrostatic 
tension or compression, In general, neither equation 51 nor equation 52 is 
sufficient to determine what happens after yielding and two further basic 
assumptions are necessary. 
A55- The second basic assumption, known as the "Law of Yielding" relates 
the principal stress-differences and principal strain-differences by a 
proportionality rule, namely 
O.=62. 6-63 G- 61 > 9 rae mee (54) 
3 
0, -@n €,- 64 63 = by 
There is little evidence to substantiate the validity of this law of yielding 
but experiments have shown that, while not exact, it is at least a fair 
overall approximation in the range of stresses covered by these experiments, 
