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Introduction to Mechanics of Continua 15

Obviously a rigid surface & (for instance a wall), which is in contact
with a moving continuum, is a particles locus i.e., it is a material surface.
Using the above criterium we will have, on such a surface of equation
f(r,t) = 0, the necessary condition v-n\y = — of and when the
\gradfl|s.
rigid surface is fixed then, v - n|;, = 0, so that the continuum velocity is
tangent at this surface.

The Euler theorem establishes that the total derivative of the motion
Jacobian J = det(gradr), is given by J = Jdivv.
The proof of this result uses the fact that the derivative of a determi-
nant J is the sum of the determinants J; which are obtained from J by
the replacement of the “z” line with that composed by its derivative vs.
the same variable.

In our case, for instance,

Ov, Ov, Ou Ov; Oz; Ov, Ox; Ov, Oz;

a OM OM
Oz ; Wy Ox J,

because Jj; = J and Jjq = Ji3 = O.
Hence, by identical assessments of J2 and J3, we get the result we
were looking for J = Jdivv. Using this result together with the known
relation between the elemental infinitesimal volumes from D and Do,
i.e., dv = JdV, we can calculate the total derivative of the elemental
infinitesimal volume, at the moment ¢ (that means from PD). Precisely

we have

dv = JdV + JdV = Jdiv vdV = dudivv

(dV being fixed in time).
Reynolds’ (transport) theorem is a quantizing of the rate of change
of an integral of a scalar or vectorial function F(r,t), integral evaluated
on a material volume D(t). As the commutation of the operators of
total time derivative and of integration will not be valid any more, the
integration domain depending explicitly on time, we have to consider,

first, a change of variables which replaces the integral material volume
D(t), depending on time, by a fixed integral domain Dp and so the

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