The objective of this classroom note is to recall the relation between Mohr’s three circles
and Cauchy’s stress definition of a (symmetric) tensor by its projection onto a unit
vector
.
In the case of Cauchy stress, the resulting projection vector is called
(surface) traction and acts on the section plane with outward normal
. The normal component
of the projection vector
points in direction
normal to the section plane, and its shear component
lies within the section
plane. In Mohr’s normal
and shear
component plane, the 2.5-dimensional coordinate rotations about principal axes
reflect Mohr’s three circles, and the tensor projections onto arbitrary unit vectors
cover
the interspace between the three. The tensor projections and Mohr’s three circles can
mathematically be applied to all symmetric second-order tensors. But only a
few tensors (like the Cauchy stress or the stretch tensors) have a physical
interpretation with respect to tensor projections. The most relevant of these stress or
deformation tensors are discussed and presented with their associated stress
or deformation (reciprocal) ellipsoids. In this context, the Cauchy–Green
deformation tensors (i.e., the stretch tensors to the second power) have no
physical interpretation in terms of tensor projections or additive trace-deviator
separations.