Continuum mechanics/Reynolds transport theorem

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Reynolds transport theorem[edit]

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.

Proof:

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)

Therefore,

where is the material time derivative of . Now, the material derivative is given by

Therefore,

or,

Using the identity

we then have

Using the divergence theorem and the identity we have

References

  1. M.E. Gurtin. An Introduction to Continuum Mechanics. Academic Press, New York, 1981.
  2. T. Belytschko, W. K. Liu, and B. Moran. Nonlinear Finite Elements for Continua and Structures. John Wiley and Sons, Ltd., New York, 2000.