Summary of von Mises Yield Criterion
Murat Ocalan
July 8, 2009
Statement of yield criterion
Von Mises yield condition is 1:
IIσ = k 2 (1)
where IIσ is the second invariant of deviatoric stress and k is a constant.
With this statement, we have enough information to relate k to a presumably known
material property yield stress. In the next sections there is a review of relevant tensor
algebra, formulae for second order tensor invariants, evaluation of yield criterion for a
few examples and description of yield surface for a 2-D stress condition.
Invariants of a stress tensor1
Principal stresses of a symmetric tensor T can be found by:
T − λI = 0 (2)
This equation is equivalent to the cubic equation:
λ3 − I T λ2 − II T λ − III T = 0 (3)
where,
I T = Tii = trT
1
d
II T = Tij Tij − I T
2
2
i (4)
III T = det T
As the name suggests, the invariants of a tensor do not vary with the coordinate system
selected.
Determination of yielding
As the criterion is on deviatoric stress, the mean normal stress (or equivalently pressure)
component should be removed from the total stress tensor T :
σ ij = Tij − σ hδ ij (5)
b
where δ ij is Kronecker delta and mean hydrostatic stress σ h = T11 + T22 + T33 / 3 . The g
second deviatoric stress invariant is:
IIσ = σ 12 2 + σ 232 + σ 132 − σ 11σ 22 − σ 22σ 33 − σ 11σ 33 (6)
(5) and (6) can be combined to find this invariant in terms of total stress components
IIσ =
1
6
b
T11 − T22 g + bT
2
22 − T33 g + bT
2
33 − T11 g 2
+ T12 + T23 + T13
2 2 2
(7)
Now, we can calculate yielding for some examples.
Uniaxial stress: ( T11 = σ y + C , T22 = T33 = C , T12 = T13 = T23 = 0 )
In this example, the material is loaded in one direction until yield stress, σ y above the
hydrostatic stress. Hydrostatic stress (or pressure) for an incompressible medium is
indeterminate. Therefore, the constant C is introduced in the diagonal terms. Evaluated at
this stress condition (7) reduces to:
σ y2
IIσ = = k2 (8)
3
Here, we see that k = σ y / 3 .
Simple shear: ( T12 = τ y , T11 = T22 = T33 = C , T13 = T23 = 0 )
IIσ = τ y = k 2
2
(9)
From this example we see that k is yield stress in simple shear.
Plane stress: ( T33 = T13 = T23 = 0 )
For convenience we choose a coordinate system coinciding with the principal axes
( T12 = 0 ) where T11 = σ 1 , T22 = σ 2 . The second invariant of the deviatoric stress is:
IIσ =
3
d
1 2
i
σ 1 − σ 1σ 2 + σ 2 2 = k 2 (10)
Tresca yield criterion1
Tresca condition for yield is where maximum shear stress reaches yield stress. In terms of
stress components this can be stated as:
Tmax − Tmin = σ y (11)
Yield surface
(10) can be simplified as:
σ 12 − σ 1σ 2 + σ 2 2 = σ y 2 (12)
In graphical form the criterion is an ellipse in the principal stress plane as shown below.
σ2/σy
x
1
y
-1 1
σ1/σy
-1
von Mises
Tresca
We can make a coordinate transformation to convert the ellipse equation into a more
familiar form:
x y
σ1 = −
2 2
(13)
x y
σ2 = +
2 2
The equation in x-y coordinates is:
x2 y2
+
2σ y
2
F 2σ I = 1
2
(14)
GH 3 JK
y
As it can be seen from the equation, the semimajor and the semiminor axes of the yield
curve are 2σ y and 2 3σ y respectively. There are also two good resources on
analytical algebra on ellipses available on the web2,3.
References
1
Malvern, L.E., “Introduction to the Mechanics of a Continuous Medium,” Prentice Hall
1969
2
Weisstien, Eric W., “Ellipse” From Mathworld – A Wolfram web resource.
http://mathworld.wolfram.com/Ellipse
3
Wikipedia, “Ellipse,” http://en.wikipedia.org/wiki/Ellipse
