Elasticity, phenomenological approach

There are three different approaches to a thermodynamic theory of continuum that can be distinguished. These approaches differ from each other by the fundamental postulates on which the theory is based. All of them are characterized by the same fundamental requirement that the results should be obtained without having recourse to statistical or kinetic theories. None of these approaches is concerned with the atomic structure of the material. Therefore, they represent a pure phenomenological approach. The principal postulates of the first approach, usually called the classical thermodynamics of irreversible processes, are documented. The principle of local state is assumed to be valid. The equation of entropy balance is assumed to involve a term expressing the entropy production which can be represented as a sum of products of fluxes and forces. This term is zero for a state of equilibrium and positive for an irreversible process. The fluxes are function of forces, not necessarily linear. However, the reciprocity relations concern only coefficients of the linear terms of the series expansions. Using methods of this approach, a thermodynamic description of elastic, rheologic and plastic materials was obtained. 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

In a recent series of papers, Kilian 9,50 52) proposed a new phenomenological approach to rubber elasticity and suggested a molecular network might be considered as a formelastic fluid the conformational abilities of which were adequately characterized by the model of a van der Waals conformational gas with weak interaction. The ideal network is treated as an ideal conformational gas. According to... 

Abstract Contribution of the Jahn-Teller system to the elastic moduli and ultrasonic wave attenuation of the diluted crystals is discussed in the frames of phenomenological approach and on the basis of quantum-mechanical theory. Both, resonant and relaxation processes are considered. The procedure of distinguishing the nature of the anomalies (either resonant or relaxation) in the elastic moduli and attenuation of ultrasound as well as generalized method for reconstruction of the relaxation time temperature dependence are described in detail. Particular attention is paid to the physical parameters of the Jahn-Teller complex that could be determined using the ultrasonic technique, namely, the potential barrier, the type of the vibronic modes and their frequency, the tunnelling splitting, the deformation potential and the energy of inevitable strain. The experimental results obtained in some zinc-blende crystals doped with 3d ions are presented. 

Fig. 6). The details of this lateral repulsion are not known. A simple elastic model based on a phenomenological approach [31] is unable to gain precise quantitative information on this interaction.

A macroscopic theory of strength is based on a phenomenological approach. No direct reference to the mode of deformation and fi acture is made. Essentially, this approach employs the mathematical theories of elasticity and tries to establish a yield or failure criterion. Among the most popular strength theories are those based on maximum stress, maximum strain, and maximum work. 

The main phenomenological approach for rubber elasticity at small deformations is due to Langley The shear modulus is assumed to be the sum of two terms... 

Phenomenological approaches to describe large elastic deformation were proposed by Moone3 and Rivlin. The Mooney model was recendy extended to consider very high... 

The phenomenological approach to rubber-like elasticity is based on continuum mechanics and symmetry arguments rather than on molecular concepts [2, 17, 26, 27]. It attempts to fit stress-strain data with a minimum number of parameters, which are then used to predict other mechanical properties of the same material. Its best-known result is the Mooney-Rivlin equation, which states that the modulus of an elastomer should vary linearly with reciprocal elongation [2],... 

Oseen s elastic theory was developed further by Zocher [3] and re-examined by Frank [4] in a phenomenological approach. 

It is necessary therefore to consider the deformation of polymers as affected by temperature and rate of loading as well as of the stress imposed. Unlike the elasticity of the rubbery state, for which a reasonable theoretical treatment exists dependent on fundamental physical constants and molecular concepts, the glassy state can only be treated by a phenomenological approach. 

The simple statistical mechanical models, i.e. affine and phantom, or the phenomenological approach cannot reproduce the maximum in the plot of Aln(ai/a ) vs. n6,28i jjjjg maximum is predicted, however, by the constrained junction theory but, as shown in Figure 19, agreement with experiment is only qualitative.This has to lead to questioning of the separability of mixing and elastic contributions in equation (145). This postulate may be valid only in the limit of swelling equilibrium. ... 

From the viewpoint of the mechanics of continua, the stress-strain relationship of a perfectly elastic material is fully described in terms of the strain energy density function W. In fact, this relationship is expressed as a linear combination erf the partial derivatives of W with respect to the three invariants of deformation tensor, /j, /2, and /3. It is the fundamental task for a phenomenologic study of elastic material to determine W as a function of these three independent variables either from molecular theory or by experiment. The present paper has reviewed approaches to this task from biaxial extension experiment and the related data. The results obtained so far demonstrate that the kinetic theory of polymer network does not describe actual behavior of rubber vulcanizates. In particular, contrary to the kinetic theory, the observed derivative bW/bI2 does not vanish. 

The parameters measured in an ultrasonic experiment are the amplimde and phase of the signal. They are determined by attenuation and phase velocity of a wave. In turn, the attenuation and phase velocity are associated with material constants. In our case they are elastic coefficients (or elastic moduli). These constants can be calculated using quanmm-mechanical approach. Finally, we will obtain the expressions for the measured (phenomenological) parameters in terms of the microscopic ones. In the present section we will discuss the basics of the phenomenological elasticity theory and the microscopic description of the Jahn-Teller contribution to the elastic moduli will be discussed later. 

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