. Derive results equivalent to for the case of hydrostatic compression σ 11 = σ 22 = σ 33 = − p, σ 12 = σ 23 = σ 31 = 0, where p is pressure.   . Relative to axes (x, y, z) , let u = w = 0 and v = kz...

.
Derive results equivalent to for the case of hydrostatic compression

σ11 =
σ22 =
σ33 = −p, σ12 =
σ23 =
σ31 = 0, where
p
is pressure.

.
Relative to axes
(x, y, z), let
u
=
w
= 0 and
v
=
kz, where
k
is a constant. Show that

this case (simple shear) differs from that of pure shear by a rigid-body rotation.

.
The volumetric strain or dilatation
e
is equal to the strain invariant
J1.

What is the dilatation for the cases of (a) simple tension and (b) pure shear?