Principle of material frame-indifference

In consideration of the principle of material frame indifference the fundamental laws for the multi-component mixture in the Eulerian description are formulated in the following local forms. 

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. 

Equations 9.8 and 9.10b are written for rectilinear flows and infinitesimal deformations. We need equations that apply to finite, three-dimensional deformations. Intuitively, one might expect simply to replace the strain rate dy/dt by the components of the symmetric deformation rate tensor (dv /dy -y dvy/dx, etc.) to obtain a three-dimensional formulation, as in Section 2.2.3, and dr/dt by the substantial derivative D/Dt of the appropriate stress components. The first substitution is correct, but intuition would lead us badly astray regarding the second. Constitutive equations must be properly invariant to changes in the frame of reference (they must satisfy the principle of material frame indifference), and the substantial derivative of a stress or deformation-rate tensor is not properly invariant. The properly invariant... 

The simplest generalization of the linear Equation 9.10 that satisfies the principle of material frame indifference is as follows ... 

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

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... 

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... 

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... 

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. 

The frame indifference principle states that constitutive equations must exhibit coordinate indifference, that is, the properties of a material must be independent of the reference frame. 

We now invoke the fundamental principle of classical physics that material properties are indifferent to the frame of reference or the observer. Hence the constitutive equations should be invariant under proper orthogonal transformations. It is seen that rif and do not transform as tensors. The parameters (3.1.12) must therefore be replaced by... 

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]. 

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-... 

Principle of local action The stress at a point is not influenced by far-field motions. Principle of frame indifference The response of a material must be described under the frame indifference (see Sect. 2.2.2). 

Indeed, carbon black-filled rubber, when loaded with time-dependent external forces, suffers a state of stress which is the superposition of two different aspects a time independent, long-term, behavior (sometimes improperly called hyperelastic ) opposed to a time dependent, short-term, response. Step-strain relaxation tests suggest that short term stresses are larger than the long term or quasi-static ones [117]. Moreover, oscillatoiy (sinusoidal) tests indicate that dissipative anelastic effects are significant, which leads to the consideration of a constitutive relation which depends not only on the current value of the strain but on the entire strain history. This assumption must be in accordance with some principles which restrict the class of rehable constitutive equations. These restrictions can be classified as physical and constitutive . The former are restrictirMis to which every rational physical theory must be subjected to, e.g., frame indifference. The latter, on the other hand, depends upon the material under consideration, e.g., internal symmetries. 

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