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Equivalent stresses
The "equivalent" stress output from LUSAS and MODELLER (also
known as the "effective" stress) represents an envelope of the direct and shear
stress components and is based upon classical failure criteria theorems. There are a
number of such theorems, each of which caters for the failure characteristics of different
materials. In this section, the von Mises failure criterion is in focus, but the general
points made apply equally to other yield functions such as Tresca, MohrCoulomb, etc.
When using the von Mises material models in LUSAS, the equivalent stress is computed
from equations based upon the distortionenergy theorem (also known as the shearenergy or
von MisesHencky theory). This yield criteria has been shown to be particularly effective
in the prediction of failure for ductile materials such as metals.
In terms of the principal deviatoric stresses (the stress tensor less the hydrostatic
pressure component), the von Mises stress is computed from
s_{E} = Ö 3(J_{2})^{½}
where J_{2} is the second deviatoric stress invariant of the stress tensor
defined by
J_{2} = 1/6 [(s_{1}
 s_{2})^{2} + (s_{2} 
s_{3})^{2}
+ (s_{3}  s_{1})^{2}]
The equivalent stress may also be expressed in terms of direct stress components as
s_{E} = [(s_{x} 
s_{y})^{2} + (s_{y} 
s_{z})^{2} + (s_{z} 
s_{x})^{2}
+ 6(s_{xy}^{2} +
s_{yz}^{2} +
s_{zx}^{2})]^{½
}/ Ö 2
When expanded, this becomes
s_{E} = [s_{x}^{2}
+ s_{y}^{2} + s_{z}^{2}
s_{x}s_{y} 
s_{y}s_{z }_{
}s_{z}s_{x} + 3(s_{xy}^{2} +
s_{yz}^{2} + s_{zx}^{2})]^{½}
and the corresponding equation for equivalent strain is
e_{E} = [e_{x}^{2}
+ e_{y}^{2} + e_{z}^{2} 
e_{x}e_{y} 
e_{y}e_{z}
 e_{z}e_{x} + 0.75(g_{xy}^{2} +
g_{yz}^{2} + g_{zx}^{2})]^{½}
The definitions of equivalent stress and strain for stress resultant output may
be obtained by simply replacing the stress components (s_{x}, etc.) with their counterparts (N_{x}, etc.).
The specific equations two and three dimensional stress and strain are
as follows:
Two dimensional, three component strain
e_{E} = [e_{X}^{2}
+ e_{Y}^{2}  e_{X}e_{Y} +
0.75g_{XY}^{2}]^{½}
Two dimensional, three component stress
s_{E} = [s_{X}^{2}
+ s_{Y}^{2}  s_{X}s_{Y} +
3t_{XY}^{2}]^{½}
Two dimensional, four component strain
e_{E} = [e_{X}^{2}
+ e_{Y}^{2} + e_{Z}^{2}
 e_{X}e_{Y}  e_{Y}e_{Z}  e_{Z}e_{X} +
0.75g_{XY}^{2}]^{½}
Two dimensional, four component stress
s_{E} = [s_{X}^{2}
+ s_{Y}^{2} + s_{Z}^{2}
s_{X}s_{Y}  s_{Y}s_{Z}  s_{Z}s_{X} +
3t_{XY}^{2}]^{½}
Three dimensional strain
e_{E} = [e_{X}^{2}
+ e_{Y}^{2} + e_{Z}^{2}
 e_{X}e_{Y}  e_{Y}e_{Z}  e_{Z}e_{X} +
0.75(g_{XY}^{2} + g_{YZ}^{2}
+ g_{ZX}^{2})]^{½}
Three dimensional stress
s_{E} = [s_{X}^{2}
+ s_{Y}^{2} + s_{Z}^{2}
s_{X}s_{Y} 
s_{Y}s_{Z
}_{
}s_{Z}s_{X} + 3(t_{XY}^{2} +
t_{YZ}^{2} + t_{ZX}^{2})]^{½}
Confusion can occur in the equivalent strain equations
because of the definition of shear strain. LUSAS and MODELLER
output g as the shear strain
(commonly termed the Engineering Strain) which is not the
same as the shear strain tensor component, e.
So that the format of the last term could be written as
1.5*e or 0.75*g
