Principle of material objectivity

As a result of simultaneous introduction of elastic, viscous and plastic properties of a material, a description of the actual state functions involves the history of the local configuration expressed as a function of the time and of the path. The restrictions, which impose the second law of thermodynamics and the principle of material objectivity, have been analyzed. Among others, a viscoplastic material of the rate type and a strain-rate sensitive material have been examined. 

One physical restriction, translated into a mathematical requirement, must be satisfied that is that the simple fluid relation must be objective, which means that its predictions should not depend on whether the fluid rotates as a rigid body or deforms. This can be achieved by casting the constitutive equation (expressing its terms) in special frames. One is the co-rotational frame, which follows (translates with) each particle and rotates with it. The other is the co-deformational frame, which translates, rotates, and deforms with the flowing particles. In either frame, the observer is oblivious to rigid-body rotation. Thus, a constitutive equation cast in either frame is objective or, as it is commonly expressed, obeys the principle of material objectivity . Both can be transformed into fixed (laboratory) frame in which the balance equations appear and where experimental results are obtained. The transformations are similar to, but more complex than, those from the substantial frame to the fixed (see Chapter 2). Finally, a co-rotational constitutive equation can be transformed to a co-deformational one. 

There are two proper explanations, one based on physical intuition and the other based on the principle of material objectivity. The latter is discussed in many books on continuum mechanics.19 Here, we content ourselves with the intuitive physical explanation. The basis of this is that contributions to the deviatoric stress cannot arise from rigid-body motions -whether solid-body translation or rotation. Only if adjacent fluid elements are in relative (nonrigid-body) motion can random molecular motions lead to a net transport of momentum. We shall see in the next paragraph that the rate-of-strain tensor relates to the rate of change of the length of a line element connecting two material points of the fluid (that is, to relative displacements of the material points), whereas the antisymmetric part of Vu, known as the vorticity tensor 12, is related to its rate of (rigid-body) rotation. Thus it follows that t must depend explicitly on E, but not on 12 ... 

The above equation is generalized to three dimensions by replacing the stress component T by the stress matrix and the strain component 7 by the strain matrix. This procedure works well as long as one confines oneself to small strains. For large strains, the time derivative of the stress requires special treatment to ensure that the principle of material objectivity(78) is not violated. This principle requires that the response of a material not depend on the position or motion of the observer. It turns out that one can construct several different time derivatives all of which satisfy this requirement and also reduce to the ordinary time derivative for infinitesimal strains. By experience over many years, it has been found (see Chapter 3 of Reference 79) that the Oldroyd contravariant derivative also called the codeformational derivative or the upper convected derivative, gives the most realistic results. This derivative can be written in Cartesian coordinates as(79)... 

Applying the principle of material objectivity plus the dissipation inequality, we are led to the constitutive equation for an incompressible elastic solid... 

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. 

Summary. A procedure really specific for the rational thermodynamics is introduced in this section in the form of several principles put forward to derive the thermodynamically consistent constitutive equations. In their most general form, the constitutive equations were proposed as functions (3.118) on the basis of the principles of determinism, local action, differential memory, and equipresence. They were further reduced to the form (3.121) considering the same material throughout the body and applying the principle of objectivity. Because of our interest in fluids only, the constitutive equations were further modified to this material type by means of the principle of material symmetry giving the final form (3.127). Two special types of fluid were defined by (3.129) and (3.130). 

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... 

Life-cycle analysis, in principle, allows an objective and complete view of the impact of processes and products on the environment. For a manufacturer, life-cycle analysis requires an acceptance of responsibility for the impact of manufacturing in total. This means not just the manufacturers operations and the disposal of waste created by those operations but also those of raw materials suppliers and product users. 

Here the principles of constructing a 3D structure model from several HREM images of projections of inorganic crystals will be presented. Some of the principles may also be applied to non-periodic objects. A complex quasicrystal approximant v-AlCrFe is used as an example (Zou et al., 2003). Procedures for ab initio structure determination by 3D reconstruction are described in detail. The software CRISP, ELD. Triple and 3D-Map are used for 3D reconstruction. The 3D reconstruction method was demonstrated on the silicate mineral (Wenk et al. 1992). It was also applied to solve the 3D structures of a series mesoporous materials (Keneda etal. 2002). 

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