.
For a problem in small-displacement theory, the strain components are given by
_x
=
A(x
−
z), _y
=
A(x
−
y), _z
=
A(y
+
z), γxy
=
γyz
=
γxz
= 0, where
A
= constant.
Determine the (x, y, z)
displacement components (u, v,w)
in terms of (x, y, z), where
u
=
v
=
w
=
ω23 =
ω31 =
ω12 = 0, for
x
=
y
=
z
= 0.
.
Let the displacement
uα
be defined by the equations
uα
=
Cαβxβ, where
Cαβ
are constants
and
xβ
denotes rectangular Cartesian coordinates. Is it possible to select the
Cαβ
so that the components
_αβ
of the strain tensor consist only of quadratic terms in
Cαβ
?
Explain. Assuming that it is possible, discuss the significance of this result in the process
of approximating the strain components
_αβ
by their small-displacement approximations
eαβ
.