  Form-invariance (covariance) and frame-indifference (objectivity) are two
notions in classical continuum mechanics which have attracted much attention
and controversy over the past decades. Particularly in turbulence modelling it
seems that there still is a need for clarification. The aim and purpose of this
study is fourfold: (i) To achieve consensus in general on definitions and
principles when trying to establish an invariant theory for modelling
constitutive structures and dynamic processes in mechanics, where special focus
is put on the principle of Material Frame-Indifference (MFI). (ii) To show that
in constitutive modelling MFI can only be regarded as an approximation that
needs to be reduced to a weaker statement when trying to advance it to an axiom
of nature. (iii) To convince that in dynamical modelling, as in turbulence, MFI
may not be utilized as a modelling guideline, not even in an approximative
sense. Instead, its reduced form has to be supplemented by a second,
independent axiom that includes the microscopic (fluctuating) description of
the dynamical processes. Concerning Navier-Stokes turbulence, the axiom of
Turbulent Frame-Indifference (TFI) is stated in which turbulence has to be
modelled consistently with the invariant properties of the deterministic
Navier-Stokes equations, and finally (iv) to propose a novel invariant
modelling ansatz both for constitutive and dynamical modelling that allows to
include the (mean) velocity field as an own independent modelling variable,
however, not in an absolute but only in the relative sense as a velocity
difference; a result that would systematically improve current modelling
procedures in extended thermodynamics and turbulence theory to be more
consistent with physical observations.
