Constitutive equations deformation

A constitutive equation is a relation between the extra stress (t) and the rate of deformation that a fluid experiences as it flows. Therefore, theoretically, the constitutive equation of a fluid characterises its macroscopic deformation behaviour under different flow conditions. It is reasonable to assume that the macroscopic behaviour of a fluid mainly depends on its microscopic structure. However, it is extremely difficult, if not impossible, to establish exact quantitative... 

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

All of the described differential viscoelastic constitutive equations are implicit relations between the extra stress and the rate of deformation tensors. Therefore, unlike the generalized Newtonian flows, these equations cannot be used to eliminate the extra stress in the equation of motion and should be solved simultaneously with the governing flow equations. 

Many industrially important fluids cannot be described in simple terms. Viscoelastic fluids are prominent offenders. These fluids exhibit memory, flowing when subjected to a stress, but recovering part of their deformation when the stress is removed. Polymer melts and flour dough are typical examples. Both the shear stresses and the normal stresses depend on the history of the fluid. Even the simplest constitutive equations are complex, as exemplified by the Oldroyd expression for shear stress at low shear rates ... 

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 referential formulation is translated into an equivalent current spatial description in terms of the Cauchy stress tensor and Almansi strain tensor, which have components relative to the current spatial configuration. The spatial constitutive equations take a form similar to the referential equations, but the moduli and elastic limit functions depend on the deformation, showing effects that have misleadingly been called strain-induced hardening and anisotropy. Since the components of spatial tensors change with relative rigid rotation between the coordinate frame and the material, it is relatively difficult to construct specific constitutive functions to represent particular materials. 

The deformation may be viewed as composed of a pure stretch followed by a rigid rotation. Stress and strain tensors may be defined whose components are referred to an intermediate stretched but unrotated spatial configuration. The referential formulation may be translated into an unrotated spatial description by using the equations relating the unrotated stress and strain tensors to their referential counterparts. Again, the unrotated spatial constitutive equations take a form similar to their referential and current spatial counterparts. The unrotated moduli and elastic limit functions depend on the stretch and exhibit so-called strain-induced hardening and anisotropy, but without the effects of rotation. 

While the above equations are approximations to the equations of Section 5.2 when deformations are small, it may now be asked under what circumstances may they be used as constitutive equations in their own right when deformations are large. 

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... 

The behavior of constitutive equations may be investigated by prescribing particular deformations. Consider tbe simple homogeneous shear deformation... 

The referential constitutive equations for an inelastic material may be set into spatial terms. Casey and Naghdi [14] did so for their special case of finite deformation rigid plasticity discussed by Casey [15], Using the spatial (Almansi) strain tensor e and the relationships of the Appendix, it is possible to do so for the full inelastic referential constitutive equations of Section 5.4.2. 

Casey, J. and Naghdi, P.M., Constitutive Results for Finitely Deforming Elastic-Plastic Materials, in Constitutive Equations Macro and Computational Aspects (edited by K.J. Wiliam), ASME, 1984, pp. 53-71. 

Naghdi, P.M. and Trapp, J.A., Restrictions on Constitutive Equations of Finitely Deformed Elastic-Plastic Materials, Quart. J. Mech. Appl. Math. 28, Part 1,25-46(1975). 

Although elastic strain and plastic deformation are expressed as numbers and have the same units (length/length), since they are physically different entities, they cannot be mixed in arithmetic operations. That is, mixtures of them cannot be added, subtracted, multiplied, or divided. Therefore, separated equations should describe them. Constitutive equations that combine them into a single equation are physically meaningless. A consequence is that elastic... 

Note 2 For small deformations, the constitutive equation may be written... 

Equation relating stress and deformation in an incompressible viscoelastic liquid or solid. Note 1 A possible general form of constitutive equation when there is no dependence of stress on amount of strain is... 

In order to use the preceding equations, one requires a constitutive equation that relates the resin viscosity to temperature, degree of cure, and the deformation rate. If the resin can be considered Newtonian, then usually,... 

This section summarizes results of the phenomenological theory of viscoelasticity as they apply to homogeneous polymer liquids. The theory of incompressible simple fluids (76, 77) is based on a very general set of ideas about the nature of mechanical response. According to this theory the flow-induced stress at any point in a substance at time t depends only on the deformations experienced by material in an arbitrarily small neighborhood of that point in all times prior to t. The relationship between stress at the current time and deformation history is the constitutive equation for the substance. 

If no volume or internal energy changes accompany deformation, the network is neo-Hookean (258), obeying the constitutive equation... 

Although appealing from an engineering perspective, the analyses based on linear thermoelasticity do not address the action of defects and dislocations created by microscopic yield phenomena below the CRSS and of those that are incorporated in the crystal at the solidification front. In the previous works cited (104-108), the authors assume that no defects exist at the melt-crystal interface and that the stresses on this surface are zero. Constitutive equations incorporating models for plastic deformation in the crystal due to dislocation motion have been proposed by several authors (109-111) and have been used to describe dislocation motion in the initial stages of... 

The constitutive equation here is the relation between the shear stress, r and the rate of deformation 7. We can define the shear stress, n, for system i using... 

Jimenez-Melendo, M., Dominguez-Rodriguez, A., and Bravo-Leon, A., Superplastic flow in fine-grained yttria-stabilized zirconium poly crystals constitutive equation and deformation mechanisms , J. Am. Ceram. Soc., 1998, 81, 2761-76. 

The title of the book, Optical Rheometry of Complex Fluids, refers to the strong connection of the experimental methods that are presented to the field of rheology. Rheology refers to the study of deformation and orientation as a result of fluid flow, and one principal aim of this discipline is the development of constitutive equations that relate the macroscopic stress and velocity gradient tensors. A successful constitutive equation, however, will recognize the particular microstructure of a complex fluid, and it is here that optical methods have proven to be very important. The emphasis in this book is on the use of in situ measurements where the dynamics and structure are measured in the presence of an external field. In this manner, the connection between the microstructural response and macroscopic observables, such as stress and fluid motion can be effectively established. Although many of the examples used in the book involve the application of flow, the use of these techniques is appropriate whenever an external field is applied. For that reason, examples are also included for the case of electric and magnetic fields. 

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