Appendix E
Miscellaneous derivations
E.1. Relation between the
areas of the faces of a tetrahedron
For the tetrahedron shown,
we prove that:
where
is the area of the face with normal n, and is the face with normal
Note that
Note also that we can compute the area of the face with
normal n by taking cross products of
the vectors defining the sides of the face:
so that
as
required.
E.2. Relation between area elements before and after
deformation

|
Consider an element of area with normal in a deformable solid. Suppose the solid is deformed, and let denote the components of the deformation
gradient tensor. The area element deforms with the solid, and has a new area and normal n. We plan to prove that the
deformed area element is related to its undeformed area through
where
Start by noting that the area before deformation can be
computed by taking the cross product of two infinitesimal vectors bounding the area element in the undeformed
configuration
Note that the infinitesimal vectors map to ,
in the deformed configuration. Therefore
Let denote the inverse of the deformation gradient
tensor, i.e. .
Then, we could write
Now,
recall the identity
so that
where ,
giving the required result.
E.3. Time derivatives of integrals over volumes within
a deforming solid

|
Consider a deformable solid.
Let denote the mass density of the solid in the
original configuration, and let denote the mass density in the deformed
configuration. (Both and can vary with position in the solid). Let denote a closed region within the undeformed
solid, and let V be the same region of the solid in the deformed
configuration. Suppose that
denotes the velocity field
within the body. We shall show that
and also that
At
first glance, this looks obvious just take the derivative under the integral
sign. You can’t do this, however,
because the volume V changes with
time, as the solid is deforming. In
addition, the mass density varies with time, because of the deformation, so
even if we could take the time derivative under the integral, we’d end up with
an additional term. To do the derivative properly, we first need to change
variables so the integral is evaluated over the undeformed volume (which is independent of time). Thus
where
and we have recalled a
result from the Kinematics section
Now,
we can happily differentiate. The mass
density in the undeformed configuration does not vary with time, so that
The
last expression was obtained by changing variables in the integral back to the
deformed configuration. This is the
first result we wanted.
To show the second result,
follow exactly the same procedure, until you obtain
Now, observe that
(the
cross product of two parallel vectors is zero) so substituting this into the
preceding equation and changing variables in the integral as before gives the
required result.
E.4. Time Derivatives of the Curvature Vector for
Deforming Rods
Consider
a deforming rod, as shown in the figure.
Let denote the position vector of a material
particle on the axis of the undeformed rod;
Let denote the arc-length coordinate of this
particle after deformation.
Define basis
vectors attached to the deformed rod, following the
convention described in Section 10.2.
Define an angular velocity vector and curvature vector through
We shall show that the gradient of the angular
velocity vector characterizing the rotation of the rod’s cross-section is
related to the time derivative of the curvature vector by
To see this, start by
differentiating the definitions of the angular velocity vector and the
curvature vector
and, similarly,
where with held fixed. The preceding two results show
that
Next, note that we can expand
the triple cross-products (see Appendix A) as
Hence, we conclude that
This result must hold for all
three vectors ,
and therefore
as stated.