Objectivity frame-indifference

In more complicated material models we modify oruse further constitutive principles determinism is enlarged for densities (mass concentrations) in mixtures (cf. Sects. 2.4,3.5,4.5), and the definition of fluid used in this principle is in fact the result of constitutive principle of symmetry (see Rem. 30 in Chap. 3). Another constitutive principle is the objectivity (frame indifference) principle. Here it is trivially satisfied because motion is neglected and all quantities are objective (see Sects. 3.2,3.5). In nonuniform systems the influence of neighborhood is described in the principle of local action (cf. Sect. 3.5). In mixtures, the property of mixture invariance [32] may also be used as a constitutive principle [33]. 

Transformation properties of some objects (mostly derivatives useful in the following chapters) formed from scalar a, vector a, tensor A which are objective (frame indifferent) (3.30)-(3.32), will be discussed now. We must realize that these objective conditions must be valid at any x, t transforming by (3.25), (3.26) to X, t (the same event seen from different frames passing at the same particle X) therefore ... 

As a primitive we assign to each particle X of this body the (mass) density p— positive and (assuming) objective (frame indifferent) scalar. Mass of the body or its arbitrary part with material volume V is then... 

Similarly as for the single substance in Sect. 3.4 we postulate that scalars <, q, Q are objective (frame indifferent) as well as ba, T . 

Moreover, because q and (arbitrary) n are objective (see end of Sect. 3.2), then from (4.78) follows (similarly as below (3.101)) that the heat flux q in mixture is objective (frame indifferent) vector cf. also Rem. 21 in Chap. 3. 

We also note that mixture properties (4.90)-(4.92) are objective (frame indifferent). For later applications, it is useful to define the total stress T by... 

The principle of material frame indifference (objectivity) states that variables, for which constitutive equations are needed, cannot depend on the coordinate frame (i.e.. Euclidean space, plus time) in which the variables are expressed. 

However, just after this paper was published, Acrivos pointed out in a personal communication with Drew and Lahey that a fundamental error had been made in the derivation. Therefore, in a corrigendum by Drew and Lahey [34] the conclusions regarding objectivity were modified to apply only approximately to a fluid with very small vorticity. In practice, the sum of the added mass and lift forces is thus not necessarily frame-indifferent when applied to reactor simulations. 

Many quantities used in the following considerations are called objective ot frame-indifferent, if they are invariant in the change of frame (3.25), (3.26) as follows (because this change contains rotations and/or inversions of corresponding Cartesian systems as a very special case (cf. Fig. 3.1), the following definition is motivated by (b), (c) of Rem. 4) ... 

Objective or frame-indifferent scalar a, vector a and (second order) tensor A transform by the change of frame on scalar a, vector a and tensor A as follows ... 

Applying the change of frame (3.25), (3.26) to the above definitions of Sect. 3.1 and, using these precepts, we can decide about objectivity or nonobjectivity (frame indifference or not) of the following quantities (more detailed proofs of some of them are written in the footnote-sized script below) the remainder from the next sections may be proved analogously. 

Deformation gradient F (3.10) is not a frame-indifferent (objective) tensor, because it transforms as... 

At the end of these Sects. 3.3 and 3.4 we note that energy balance and entropy inequality motivated by procedures like those in Chap. 1 together with generalization of frame indifference (plausible objectivity is postulated not only for motion (Sect. 3.2) but also, e.g., for power of surface and body forces or heating) permit to deduce balances in Sect. 3.3 (i.e., for mass, linear and angular momentum), internal energy, entropy and their objectivity, etc. For details see, e.g., [1, 22, 42, 43] and other works on modern thermomechanics [7, 8, 18, 20, 41]. 

Further reduction of constitutive equations (3.119) may be achieved by the constitutive principle of frame indifference or the principle of objectivity, the material properties and therefore also constitutive equations must be independent of the choice of frame. This principle is a generalization of common experience with mate-... 

In a special choice Q = 1 and arbitrary b, c, c, Q, these transformations must give the same values of responses in (3.119) and (3.120) (because T is the same in both frames) and this is possible (change from (3.119) to (3.120) is valid for any values of independent variables) only ifresponses are independent of variables x, t, v and W. This means that two observers with a shift in origins of time and space and with different velocities of translation and rotation must obtain the same responses. Therefore, the constitutive equations (3.119) must be reduced by the principle of frame indifference (or objectivity) to the form... 

Muschik W Objectivity and frame indifference. Arch. Mech. 50, 541-547 (1998)... 

Now we restrict such constitutive equations—responses (4.120) as functions of (4.126)—by the principle of objectivity (or (material) frame indifference), cf. Sect. 3.5 constitutive equations cannot depend explicitly on (non-objective) x and t... 

The principle of frame indifference is sometimes called objectivity. 

Liu IS (2004) On Euclidean objectivity and the principle of material frame-indifference. Continuum Mech Thermodyn 16 177-183... 

Time derivatives play a central role in rheology. As seen above, the upper and lower convected derivatives fall out naturally from the deformation tensors. The familiar partial derivative, 8/9t, corresponds to an observer with a fixed position. The total derivative, d/dt, allows the observer to move freely in space, while if the observer follows a material point we have the material , or substantial derivative, denoted variously by the symbols d(m)/dr, D/Dr or ( ). We could expect that these different expressions could find their way into constitutive relations (see Section 5) as time rates of change of quantities that are functions of spatial position and time. However, only certain rate operations can be used by themselves in constitutive relations. This will depend on how two different observers who are in rigid motion with respect to each other measure the same quantity. The expectation is that a valid constitutive relation should be invariant to such changes in observer. This principle is called material frame indifference or material objectivity , and constitutes one of the main tests that a proposed constitutive relation has to pass before being considered admissible. 

A fundamental principle of classical physics is that material properties must be independent of the frame of reference or observer. This axiom is commonly called the principle of material frame-indifference or objectivity [270, pp.41-44]. This principle states that constitutive equations (discussed in greater detail in Section 4.2.3) must be invariant under changes of frame of reference. Under the motion defined by... 

We examine the transformations under the motion (4.13) for the quantities Vij, Aij, Wijy hiy A/i, and proceed to show that of these only Aij and Ni are frame-indifferent, that is, they are objective. This result will be of crucial importance in the construction of suitable constitutive equations. Define Q by... 

Therefore A j satisfies the principle of material frame-indifference, that is, A is objective. Similar reasoning leads to the result... 

To summarise, we have from equations (4.22), (4.23), (4.25) and (4.26) that the quantities v, Vv, W and h do not satisfy the principle of material frame-indifference, while from equations (4.18), (4.24) and (4.27) we see that the director n, the rate of strain tensor A and the co-rotational time flux N are frame-indifferent and therefore objective. For later ease of reference, we record that... 

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

The objectivity of the spatial stress rate relation (5.154) may be investigated by applying the coordinate transformation (A.50) representing a rotation and translation of the coordinate frame. The spatial strain and its convected rate are indifferent by (A.58) and (A.62). So are the stress and its Truesdell rate. It is readily verified from (5.151), (5.152), and the fact that K has been assumed to be invariant, that k and its Truesdell rate are also indifferent. Using these facts together with (A.53) in (5.154) with c and b given by (5.155)... 

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