Reynolds transport theorem
edit
|
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
- 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.