The problem of deducing mathematical
relationships for predicting the conditions at which plastic yielding begins
when a material is subjected to a complex state of stress is an important
consideration in the field of plasticity. In uniaxial loading, plastic
flow begins at the yield stress, and it is to be expected that yielding under a situation of combined stresses is related to some particular combination of the principal stresses. A yield criterion can be expressed in the general form F (σ1, σ2, σ3, K1 . . .) = 0, but there is at present no theoretical way of calculating the relationship between the stress components to correlate yielding in a three-dimensional state of stress with yielding in the uniaxial tension test. The yielding criteria are therefore essentially empirical relationships. At present, there are two generally accepted theories for predicting the onset of yielding in ductile metals.
flow begins at the yield stress, and it is to be expected that yielding under a situation of combined stresses is related to some particular combination of the principal stresses. A yield criterion can be expressed in the general form F (σ1, σ2, σ3, K1 . . .) = 0, but there is at present no theoretical way of calculating the relationship between the stress components to correlate yielding in a three-dimensional state of stress with yielding in the uniaxial tension test. The yielding criteria are therefore essentially empirical relationships. At present, there are two generally accepted theories for predicting the onset of yielding in ductile metals.
1. Maximum shear stress Theory :
The maximum shear stress theory, sometimes
called the Tresca yield criterion, states that yielding will occur when the
maximum shear stress reaches a critical value equal to the shearing yield
stress in a uniaxial tension test. The maximum shear stress is given by
τmax = (σ1
– σ3)/2
where, σ1 is the algebraically largest and σ3
is the algebraically smallest principal stress.
For uniaxial tension σ1 = σ0, σ2
= σ3 = 0, where σ0 is the yield strength in simple
tension. Therefore, the shearing yield stress for simple tension
τ0 = σ0/2
Substituting these values into the equation for maximum shear
stress results in
τmax = (σ1
– σ3)/2 = τ0 = σ0/2 (or)
σ1 – σ3
= σ0
This is sometimes written as
σ1 – σ3
= σ1’– σ3’ = 2k
where σ1’ and σ3’ are the deviators of
the principal stresses and k is the yield stress for pure shear, i.e., the
stress at which yielding occurs in torsion, where σ1 =
σ3.
The maximum shear stress theory is in good agreement with
experimental results, being slightly on the safe side, and is widely used by
designers for ductile metals. It has replaced the older and far less accurate
maximum shear stress theory, Rankine’s theory.
2. Von Mises or Distortion-energy Theory :
A somewhat better fit with
experimental results is provided by the yield criterion,
σ0 = 1/√2 [(σ1
σ2)2
+ (σ2
σ3)2
+ (σ3
σ1)2 ]1/2
According to this criterion, yielding
will occur when the differences between the principal stresses expressed by the
right-hand side of the equation exceed the yield stress in uniaxial tension, σ0.
Von Mises proposed this criterion in the invariant form given by primarily
because it was mathematically simpler than the invariant form of the
maximum-shear-stress theory. Subsequent experiments showed that provides better
over-all agreement with combined stress-yielding data than the
maximum-shear-stress theory.
J2 – k2
= 0 (or) J2 = k2
where J2 is the second invariant of the stress
deviator, and k is the yield stress in pure shear.
A number of attempts have been made to
provide physical meaning to the Von Mises yield criterion. One commonly
accepted concept is that this yield criterion expresses the strain energy of
distortion. On the basis of the distortion-energy concept, yielding will occur
when the strain energy of distortion per unit volume exceeds the strain energy
of distortion per unit volume for a specimen strained to the yield stress in
uniaxial tension or compression. The strain energy of distortion will be based
on the stress deviator. It represents only the strain energy associated with
changing the shape of the specimen and neglects the strain energy associated
with changes in volume.
