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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, Mohr-Coulomb, etc.

When using the von Mises material models in LUSAS, the equivalent stress is computed from equations based upon the distortion-energy theorem (also known as the shear-energy or von Mises-Hencky 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

sE = 3(J2)

where J2 is the second deviatoric stress invariant of the stress tensor defined by

J2 = 1/6 [(s1 - s2)2 + (s2 - s3)2 + (s3 - s1)2]

The equivalent stress may also be expressed in terms of direct stress components as

sE = [(sx - sy)2 + (sy - sz)2 + (sz - sx)2 + 6(sxy2 + syz2 + szx2)] / 2

When expanded, this becomes

sE = [sx2 + sy2 + sz2- sxsy - sysz - szsx + 3(sxy2 + syz2 + szx2)]

and the corresponding equation for equivalent strain is

eE = [ex2 + ey2 + ez2 - exey - eyez - ezex + 0.75(gxy2 + gyz2 + gzx2)]

The definitions of equivalent stress and strain for stress resultant output may be obtained by simply replacing the stress components (sx, etc.) with their counterparts (Nx, etc.).

The specific equations two and three dimensional stress and strain are as follows:

Two dimensional, three component strain

eE = [eX2 + eY2 - eXeY + 0.75gXY2]

Two dimensional, three component stress

sE = [sX2 + sY2 - sXsY + 3tXY2]

Two dimensional, four component strain

eE = [eX2 + eY2 + eZ2 - eXeY - eYeZ - eZeX + 0.75gXY2]

Two dimensional, four component stress

sE = [sX2 + sY2 + sZ2 -sXsY - sYsZ - sZsX + 3tXY2]

Three dimensional strain

eE = [eX2 + eY2 + eZ2 - eXeY - eYeZ - eZeX + 0.75(gXY2 + gYZ2 + gZX2)]

Three dimensional stress

sE = [sX2 + sY2 + sZ2- sXsY - sYsZ - sZsX + 3(tXY2 + tYZ2 + tZX2)]

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

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