Ideal elastic systems

Generalized Strain-Stress Relationships for Ideal Elastic Systems 170... 

BETWEEN THE STRESS AND STRAIN TENSORS IN IDEAL ELASTIC SYSTEMS... 

The system with ideal elastic energy (t) = 0, ro = 1), in which the reversible work W is totally transformed into the internal energy AU. Deformation of such systems is not accompanied by thermal effects. The classical theory of elasticity treats deformation of elastic systems from this point of view. 

The ideal entropy-elastic system (ri = — 1, to = 0), in which the work exchanged is totally transformed into a change in entropy. A well-known example of such systems is the ideal gas. 

Elastic unshifted scattering Particles Tyndall (Hie) Characterization of particle distribution Information can be difficult to interpret for non-ideal particle systems... 

The complex relationship between the configurational distortion produced by a perturbation field in polymers and the Brownian motion that relaxes that distortion make it difficult to establish stress-strain relationships. In fact, the stress at a point in the system depends not only on the actual deformation at that point but also on the previous history of deformation of the material. As a consequence the relaxation between the stress and strain or rate of strain cannot be expressed by material constants such as G or /, as occurs in ideal elastic materials, but rather by time-dependent material functions, G t) and J t). It has been argued that the dynamics of incompressible liquids may be characterized by a function of the evolution of the strain tensor from the beginning up to the present time. According to this criterion, the stress tensor would be given by (3,4)... 

Under an applied stress ideally solid materials may deform, but they do not flow. When the force is released the deformation relcixes. The energy, supplied to the material in order to obtain the deformation, remcilns stored upon terminating the stress, this energy.is fully and immediately released. Such a system is called (ideally) elastic. 

In the rheological structure of most food systems there is a viscous element present, and the deformation curves are often highly influenced by the rate of the imposed strain. This is due to the fact that the material relaxes (or flows) while tested under compression and the resultant deformation of this flow is dependent on the nature of the viscous element (Szczesniak, 1963 Peleg and Bagley, 1983). In the viscoelastic food systems, where during processing it is caused to oscillate sinusoidally, the strain curve may or may not be a sine wave. In cases when a periodic oscillatory strain is applied on a food system like fluid material, oscillating stress can be observed. The ideal elastic solid produces a shear stress wave in phase with... 

Figure 20.1 Idealized rheological systems, (a) A system with two parts in each part, the only behavior shown is viscous (no energy recoverable), (b) A system with two parts in each part, behavior is the sum of a viscous element and an elastic element. In both systems, the load is imposed in such a way that the displacements in the two parts are equal, (c) A possible loading history for the system in (b). (d) The loads in the two parts of the system as they would change with time. Equations and a numerical example are given in Appendix 20A.

Polymeric liquids have a microstructure that is like springs representing the linear chain. Restoration of these springs to their equilibrium state is through the elastic energy that is stored during the deformation process. But polymeric fluids are not ideal elastic materials, and they also have a dissipative reaction to deformation, which is the viscous dissipation. For small deformations, the response of the system is linear, meaning that the response is additive effect of sum of two small deformations is equal to the sum of the two individual responses. 

The elastic properties of an ideally elastic crystal may be characterized corrrpletely by the nrrmerical valrres of either the elastic stiffnesses c,yw or cxfi or else of its compliartces syu or These nrrmerical values, however, are referred to a specific coordirrate system arrd are, therefore, dependent on its choice. Usually they are reproduced in tables as referred to a Cartesian coordinate system with axes parallel to irrrportartt crystallographic directiorrs. It is chosen in a way that makes the representation of crystal properties as sirrtple as possible, as was shown already in Chap. 2, and is called the crystal coordinate system. 

In ideally elastic solids, the deformation energy will be stored within the solid without any losses, whereas in ideally viscous liquids the deformation energy will be completely transformed into frictional heat In the real world however, the elastic constants connecting the tensorial components of stress and strain, including the modulus of elasticity E, the shear modulus G, the Poisson number v, the compressibility k, and other parameters, are not time-invariant. Consequently, even after releasing very small stresses, both residual and time-dependent deformations remain in engineering solids that require a certain relaxation time to restore the equilibrium state of the system (e.g., see Meyers and Chawla, 2009). 

The relatively low value of the exponent n compared with an expected value of 3 according to beam theory may be explained by the fact that the system is not ideally elastic and the thickness of the specimen is quite considerable. Hoagland O has observed that n... 

Figure A2.2.3. Planck spectral density fimction as a fimction of the dimensionless frequency /)oi/(/rj 7). A2.2.4.7 APPLICATION TO IDEAL SYSTEMS ELASTIC WAVES IN A SOLID...

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