What Is Lam´e’s Ellipsoid?

There are always three orthogonal principal directions in a stress state. It is therefore always possible to choose a rectangular Cartesian reference system which coincides with the three principal directions. In this case, the shearing components of the stress tensor vanish and it takes the form

In an inclined facet, with a semi-normal defined by the direction cosines l,m, n, the relation between the components of the stress vector and the principal stresses may be deduced from expression 9, yielding

Since the direction cosines must obey the condition l2+m2+n2 = 1, expression gives

If we consider a Cartesian reference system T1, T2, T3, this expression represents the equation of an ellipsoid, whose principal axes are the reference system and where the points on the ellipsoid are the tips P of the stress vectors
OP (T1, T2, T3) acting in facets containing the point with the stress state defined by expression 24 (point O, Fig. 9)

This ellipsoid is a complete representation of the magnitudes of the stress vectors in facets around point O. It allows an important conclusion about the stress state: the magnitude of the stress in any facet takes a value between the maximum principal stress σ1 and the minimum principal stress σ3.

It must be mentioned here that this conclusion is only valid for the absolute value of the stress, since in expression 26 only the squares of the stresses are considered.

From Fig. 9 we conclude immediately that if the absolute values of two principal stresses are equal the ellipsoid takes a shape of revolution around the third principal direction and if the three principal stresses have the same absolute value the ellipsoid becomes a sphere.

In the first case, the stress→T acting in facets, which are parallel to the third principal direction have the same absolute value. Besides, if these two principal stresses have the same sign, we have an axisymmetric stress state.

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