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Introduction to Mechanics of Continua 23

2.5 The Cauchy Motion Equations
Cauchy’s theorem allows us to rewrite in a different form the principle
of the momentum torsor variation, that means of the linear momentum

and of the angular momentum variation.
Precisely, it is known that

D
a | evan = [irinda + [ pede

D Ss D
and

D
ay | ex evde = fr x (Inda+ | x x ptdv.

D S D

Obviously, in the conditions of the continuous motions (which cor-
respond to the parameters field of class C'(D)), by using the exten-
sion of Green’s formulas for tensors of order greater than 1 [Appendix
A] together with the fundamental lemma, from the (linear) momentum
theorem one gets

PQ; = Tij,4 + pfi, (4 = 1, 2,3),

relations known as Cauchy’s equations or “the first Cauchy’s law (theo-
rem)”.
These equations could be established under different forms too. Thus,
starting with the formulas for the total derivative of both the momentum
(pv) = g (pv) + (v- V) pv and the volume (depending on time)

integral, we have

O
/ 5; (pv) + (v- V) pv + pv div v| dv = / (div[(T] + pf) dv.

D D

As (v- V) pv+pv div v = div (pv ® v) ,the symbol @designating the
dyadic product [Appendix A], the above equation could be rewritten in
the form

[| Flora + | ov ®v — [T]) nda = | etav,

D S D

known also as the transport equation of (linear) momentum and which

could be used, in fluid dynamics, for evaluation of the global actions
exerted on the immersed bodies.