Continuum mechanics/Reynolds transport theorem
Let be a region in Euclidean space with boundary . Let be the positions of points in the region and let be the velocity field in the region. Let be the outward unit normal to the boundary. Let be a vector field in the region (it may also be a scalar field). Show that
This relation is also known as the Reynold's Transport Theorem and is a generalization of the Leibniz rule. Content of example.
Let be reference configuration of the region . Let the motion and the deformation gradient be given by
Let . Then, integrals in the current and the reference configurations are related by
The time derivative of an integral over a volume is defined as
Converting into integrals over the reference configuration, we get
Since is independent of time, we have
Now, the time derivative of is given by (see Gurtin: 1981, p. 77)
where is the material time derivative of . Now, the material derivative is given by
Using the identity
we then have
Using the divergence theorem and the identity we have
- M.E. Gurtin. An Introduction to Continuum Mechanics. Academic Press, New York, 1981.
- T. Belytschko, W. K. Liu, and B. Moran. Nonlinear Finite Elements for Continua and Structures. John Wiley and Sons, Ltd., New York, 2000.