Constitutive isotropic

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. 

In the classical theory of plasticity, constitutive equations for the evolution of the isotropic and kinematic hardening parameters are usually expressed as... 

This is the condition (A.85) that the elastic limit function / be an isotropic tensor functions of its arguments. By analogy with the hypoelastic constitutive equation, the name hypoinelasticity suggests itself for this formulation. 

It is the dependence of the spatial constitutive functions on the changing current configuration through F that renders the spatial constitutive equations objective. It is also this dependence that makes their construction relatively more difficult than that of their referential counterparts. If this dependence is omitted, then the spatial moduli and elastic limit functions must be isotropic to satisfy objectivity, and the spatial constitutive equations reduce to those of hypoinelasticity. Of course, there are other possible formulations for the spatial constitutive functions which are objective without requiring isotropy. One of these will be considered in the next section. 

Atluri, S.N., On Constitutive Relations at Finite Strain Hypo-Elasticity and Elasto-Plasticity with Isotropic or Kinematic Hardening, Comput. Methods Appl. Mech. Engrg. 43, 137-171 (1984). 

The Plate Constitutive equations can be used for curved plates provided the radius of curvature is large relative to the thickness (typically r/h > 50). They can also be used to analyse laminates made up of materials other than unidirectional fibres, eg layers which are isotropic or made from woven fabrics can be analysed by inserting the relevant properties for the local 1-2 directions. Sandwich panels can also be analysed by using a thickness and appropriate properties for the core material. These types of situation are considered in the following Examples. 

The refractive index is an important quantity for characterizing the structure of polymers. This is because it depends sensitively on the chemical composition, on the tacticity, and - for oligomeric samples - also on the molecular weight of a macromolecular substance. The refractive indices (determined using the sodium D line) of many polymers are collected in the literature. In order to characterize a molecule s constitution one requires knowledge of the mole refraction, Rg. For isotropic samples, it can be calculated in good approximation by the Lorentz-Lorenz equation ... 

The mass conservation equation only relates concentration variation with flux, and hence cannot be used to solve for the concentration. To describe how the concentrations evolve with time in a nonuniform system, in addition to the mass balance equations, another equation describing how the flux is related to concentration is necessary. This equation is called the constitutive equation. In a binary system, if the phase (diffusion medium) is stable and isotropic, the diffusion equation is based on the constitutive equation of Pick s law ... 

A special anisotropic particle scattering problem has been treated by Roth and Dignam (1973), who considered an isotropic sphere coated with a uniform film with constitutive relations... 

This specter should once and for all be laid to rest. The Curie principle as it applies to the system at hand (I will not state it in its most general form) forbids, in the linear regime, coupling between a vectorial process such as a flow and a scalar process such as a chemical reaction in an isotropic space. However, active transport does not occur in an isotropic or symmetrical system. Clearly, the protein constituting the pump is uniquely oriented within the membrane. 

Numerous studies by other workers (I, 10) have shown that the releases of iodine and the noble-gas fission products from pyrolytic carbon-coated fuel particles are controlled by diffusion of these nuclides through grain boundaries, cracks, and defects in the isotropic pyrolytic carbon coating. When coatings are intact, however, the release of these fission product nuclides is low. However, the pyrolytic carbon coating constitutes only a delaying barrier to the metallic nuclides barium and strontium through which they diffuse with diffusion coefficients of the order of 10 9 cm.2/sec. (at — 1400°C.). The steady-state release of these metallic nuclides is controlled instead by diffusion out of the fuel kernel,... 

The wavelength dependence in Eq. 14.13 can be used for experimental measurements of the surface and kinetic coefficients that constitute Bs. If an array of evenly spaced parallel grooves is introduced on a surface, the spacing dependence of the grooves amplitude-decay factor can be measured [6]. An analysis for flattening of an isotropic surface by bulk diffusion as in Fig. 3.7 is presented in Exercise 14.1. 

The fundamental quantities in elasticity are second-order tensors, or dyadicx the deformation is represented by the strain thudte. and the internal forces are represented by Ihe stress dyadic. The physical constitution of the defurmuble body determines ihe relation between the strain dyadic and the stress dyadic, which relation is. in the infinitesimal theory, assumed lo be linear and homogeneous. While for anisotropic bodies this relation may involve as much as 21 independent constants, in the euse of isotropic bodies, the number of elastic constants is reduced lo two. 

Both lyotropic and thermotropic liquid-crystalline synthetic polymers have been widely studied. Aromatic polyamides constitute the most important class forming liquid-crystalline solutions the solvents are either powerfully protonating acids such as 100% sulphuric acid, chloro-, fluoro- or methane-sulphonic acid, and anhydrous hydrogen fluoride, or aprotic dipolar solvents such as dimethyl acetamide containing a small percentage, usually 2-5 %, of a salt such as lithium chloride or calcium chloride. Such solutions constitute a nematic phase within certain limits. Some criteria for formation of a nematic instead of an isotropic phase are ... 

So far we have not considered the influence of the constitution of the polymer main chain on the formation of the nematic phase. If the same mesogenic group is linked to different backbones, the nematic phase can be preserved, as shown for one example in Table 3. Owing to the different flexibilities of the backbones, the nematic state is shifted with respect to the temperature. With falling flexibility of the main chain, as indicated by the increasing glass transition temperature, the phase transformation temperatures nematic to isotropic are shifted towards higher temperatures. This clearly indicates that the restriction of motions, due to the polymer-fixation, directly reflects on the phase transformation temperature. If this restriction... 

The quantities h, (p0 and /, v(/o constitute (to within a constant multiplier) two pairs of the canonically conjugate arbitrary constants. Therefore we may choose them as the phase variables while averaging over T in (27). We note that these quantities refer to a local phase space corresponding to any chosen direction of the symmetry axis. Hence, integration performed in the overall phase space dTls corresponding to an isotropic fluid should additionally include averaging over all possible inclinations 0 of the symmetry axis C to the a.c. field vector E. Thus,... 

Some researchers have used approximate microscopic descriptions to develop more rigorous macroscopic constitutive laws. A microstructural model of AC [5] linked the directionality of mechanical stiffness of cartilage to the orientation of its microstructure. The biphasic composite model of [6] uses an isotropic fiber network described by a simple linear-elastic equation. A homogenization method based on a unit cell containing a single fiber and a surrounding matrix was used to predict the variations in AC properties with fiber orientation and fiber-matrix adhesion. A recent model of heart valve mechanics [8] accounts for fiber orientation and predicts a wide range of behavior but does not account for fiber-fiber interactions. 

For an isotropic medium without optical rotation, use of the constitutive relations (1.4) yields... 

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