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




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




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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