Page 48 - Basics of Fluid Mechanics and Introduction to Computational Fluid Dynamics

Introduction to Mechanics of Continua 33

of the motion equations of particles within the microscopic theory and

which, in our phenomenological approach are given by experience as
physical laws, will be designated as constitutive or behaviour laws or
simply “stresses-deformations” relations (in fact they are functional
dependences between the stress tensor and the mecanical and thermo-
dynamical parameters which are associated to the motion, between the
quantities which characterize the deformation and the stresses which
arise as a reaction to this deformation).

Noll has given a set of necessary conditions, in the form of general
principles, which should be fulfilled by any constitutive law. By using the
necessary conditions, some general dependences between the mechanical
and thermodynamical parameters will be “filtered” and thus a screening
of real candidates among different “stresses-deformations” relations is

performed [95].
In what follows we will present, in short, the most important of such
principles (the details could be found in [95]).
The first principle is that of dependence on “the history” of the ma-
terial. According to this principle the stress state at a certain point

of the deformable continuum and at a given moment, depends on the
whole “history” of the evolution (from the initial to the given considered
moment) of the entire material system. In other words, this principle
postulates that the stress at a point of continuum and at a certain mo-
ment is determined by a sequence of all the configurations the continuum
has passed through from the initial moment till the considered moment

(included).
A second principle which is in fact a refinement of the previous prin-
ciple is that of spatial localization. According to this principle, to de-
termine the stress state at a certain point and at a certain moment f,
not the whole history of the entire continuum is required but only the

history of a certain neighborhood of the considered point.
Finally, the most powerful (by its consequences) principle would be
that of objectivity or material frame indifference. | According to this
principle a constitutive law should be objective and so frame invariant
which agrees with the intrinsic character of such a law.

An important consequence of this objectivity principle is the impos-
sibility of the time to appear explicitly in such a law.
If in a constitutive law the point where the stress is evaluated does
not appear explicitly, the respective medium is called homogeneous. The
homogeneity is also an intrinsic property of the medium. It can be
shown then if there is a reference configuration where the medium is

homogeneous that it will keep this quality in any other configuration
[150].