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.

 

 

 

(c) A.F. Bower, 2008
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