86 7 Fields, Spatial Differential Operators
Fig. 7.3 Uniaxial
squeeze-stretch field
(iv) Uniaxial Squeeze-stretch Field
The vector field
vμ= 3 eμeνrν−rμ= 3 zeμ−rμ, (7.21)withz=eνrνcan be used to describe a stretching in the direction parallel to the unit
vectoreand a squeezing in both directions perpendicular toe. This vector field is
the gradient of the potential function
Φ=
3
2
(eμrμ)^2 −1
2
r^2 =1
2
(
2 z^2 −x^2 −y^2)
, (7.22)
where the components ofrperpendicular toeare denoted byxandy(Fig.7.3).
(v) Planar Squeeze-stretch Field
Leteandube two orthogonal unit vectors,e·u=0. The vector field
vμ=eμuνrν+uμeνrν=yeμ+xuμ, (7.23)withx =eνrνandy=uνrν, is of the form needed to describe the deformation
of an elastic solid with stretching and squeezing, within thex–y-plane, under 45◦
and 135◦with respect to thex-axis, cf. Fig.7.4. There is no deformation in the third
direction.
Again, the vector field can be obtained as the gradient of a potential function, in
this case one has
Φ=eμrμuνrν=xy. (7.24)When the coordinate axis are rotated by 45◦, the potential reads
Φ=( 1 / 2 )
(
(eνrν)^2 −(uνrν)^2)
=( 1 / 2 )(x^2 −y^2 ). (7.25)