5.4 Geometric Interpretation of Symmetric Tensors 67
Fig. 5.1 Perspective view of
uniaxial (left) and biaxial
(right) ellipsoids generated
by linear mappings with
S(^1 )=S(^2 )= 0 .8,
S(^3 )= 1 .4andS(^1 )= 1 .4,
S(^2 )= 0 .6,S(^3 )=1,
respectively
when the tensor is isotropic. In Fig.5.1, uniaxial (left) and biaxial (right) ellipsoids
are presented in a perspective view. The principal axes of the ellipsoids are parallel
to the edges of the boxes, their lengths are proportional to the semiaxes.
In the uniaxial case whereS(^1 )=S(^2 )=S(^3 )the ellipsoids generated by the
linear mapping are cigar-like whenS(^3 )>S(^1 )holds true, and disk-like when one
hasS(^3 )<S(^1 ). They are referred to asprolateandoblateellipsoids, respectively.
Now a geometric interpretation can be given to the decomposition of the sym-
metric tensorSaccording to (3.4) into an isotropic part and the symmetric traceless
(anisotropic) partS. In the mapping (5.37), the isotropic part of tensorSyields a
sphere with its radius equal to the mean value^13 (S(^1 )+S(^2 )+S(^3 ))of the principal
values. The symmetric traceless partScharacterizes the deviation of the ellipsoid
generated bySfrom that sphere.
For the uniaxial case depicted on the left hand side in Fig.5.1, the cross section
of the ellipsoid in the 1–3-plane and of the pertaining sphere are shown on the left
side of Fig.5.2as thick and dashed curves. The area between them is a measure for
the anisotropic (symmetric traceless) part of the tensorS. The cross section of the
ellipsoid in the 2–3-plane has the same appearance as in the 1–3-plane whereas it is
a circle in the 1–2-plane.
The cross sections of the biaxial ellipsoid on the right hand side of Fig.5.1are
ellipses in all three planes containing two of the coordinate axes. On the right hand
side of Fig.5.2, the cross sections of the ellipsoid (and the sphere pertaining to the
isotropic part ofS) are shown for the 1–3-plane (thick curve) and the 2–3-plane
(thin curve) in the upper diagram. The lower diagram is for the 1–2-plane. Again,
the area between the ellipses and the dashed circles is a measure for the deviation of
the tensorSfrom being isotropic.
5.4.3 Volume and Surface of an Ellipsoid
The volumeVand the surface areaAof they-ellipsoid generated by the linear
mapping of a unit sphere inx-space according to (5.37), as discussed above for
tensors with non-negative principal values, are given by