Constitutive equations coordinate invariance

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. 

This is the hypoelastic constitutive equation considered by Truesdell (see Truesdell and Noll [20]). In large deformations, this equation should be independent of the motion of the observer, a property termed objectivity, i.e., it should be invariant under rigid rotation and translation of the coordinate frame. In order to investigate this property, a coordinate transformation (A.50) is applied. If the elastic stress rate relation is to be unchanged in the new coordinate system denoted x, then... 

It is expected that constitutive equations should be invariant to relative rigid rotation and translation between the material and the coordinate frame with respect to which the motion is measured, a property termed objectivity. In order to investigate this invariance, the coordinate transformation... 

The nonlinear viscoelastic models (VE), which utilize continuum mechanics arguments to cast constitutive equations in coordinate frame-invariant form, thus enabling them to describe all flows steady and dynamic shear as well as extensional. The objective of the polymer scientists researching these nonlinear VE empirical models is to develop constitutive equations that predict all the observed rheological phenomena. 

It is important to emphasize that the mathematical constraint imposed by coordinate invariance addresses only the selection of an allowable form of a constitutive equation, given the physical assumption, based on an educated guess, that there is a linear relationship between q and VO. Whether the resulting constitutive equation captures the behavior of any real material is really a question of whether the physical assumption of linearity is an adequate approximation. In fact, in the generalized Fourier heat conduction model, Eq. (2-65), there are several additional physical assumptions that must be satisfied, besides linearity between q and V0 ... 

A mathematical statement of the property of isotropy is that the constitutive equation must be completely invariant to orthogonal rotations of the coordinate system. For the constitutive form (2-65), it can be shown that this condition will be satisfied if and only if... 

Purely viscous constitutive equations, which account for some of the nonlinearity in shear but not for any of the history dependence, are commonly used in process models when the deformation is such that the history dependence is expected to be unimportant. The stress in an incompressible, purely viscous liquid is of the form given in equation 2, but the viscosity is a function of one or more invariant measures of the strength of the deformation rate tensor, [Vy - - (Vy) ]. [An invariant of a tensor is a quantity that has the same value regardless of the coordinate system that is used. The second invariant of the deformation rate tensor, often denoted IId, is a three-dimensional generalization of 2(dy/dy), where dy/dy is the strain rate in a one-dimensional shear flow, and so the viscosity is often taken to be a specific function-a power law, for example-of (illu). ]... 

It is called the first invariant of the tensor T, IIt the second invariant, and IIIt the third invariant. They are called invariants because no matter what coordinate systems we choose to express T, they will retain the same value. We will see that this property is particularly helpful in writing constitutive equations. Note that other combinations of 7 j can be used to define invariants (cf. Bird et al., 1987a, p. 568). 

Munera and Guzman [56] obtained new explicit noncyclic solutions for the three-dimensional time-dependent wave equation in spherical coordinates. Their solutions constitute a new solution for the classical Maxwell equations. It is shown that the class of Lorenz-invariant inductive phenomena may have longitudinal fields as solution. But here, these solutions correspond to massless particles. Hence, in this framework a photon with zero rest mass may be compatible with a longitudinal field in contrast to that Lehnert, Evans, and Roscoe frameworks. But the extra degrees of freedom associated with this kind of longitudinal solution without nonzero photon mass is not clear, at least at the present state of development of the theory. More efforts are needed to clarify this situation. 

Equations (6.99) and (6.105) are the final result. It gives the canonically invariant expression of the tunneling splitting, and actual computations can be carried out in any convenient system of coordinates. Once the instanton trajectory is found, the only thing remaining is to solve the system of the first-order differential equations. Equation (6.97), which does not constitute any numerical problem. One useful remark can be made about the form of kinetic-energy operator. In general, it may differ from the canonical form in Equation (6.82) by first-order derivative terms, which leads only to evident modification of A in the transport equation. Equation (6.87). 

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