Phenomenology of Elasticity

In general, Hooke s law is the basic constitutive equation giving the relationship between stress and strain. Generahzed Hooke s law is often expressed in the following form [20,108]  

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It is not particularly difficult to introduce thermodynamic concepts into a discussion of elasticity. We shall not explore all of the implications of this development, but shall proceed only to the point of establishing the connection between elasticity and entropy. Then we shall go from phenomenological thermodynamics to statistical thermodynamics in pursuit of a molecular model to describe the elastic response of cross-linked networks. 

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. 

This is the fundamental for phenomenologic analyses of elastic bodies. 

This consists of experimental measurements of stress-strain relations and analysis of the data in terms of the mathematical theory of elastic continua. Rivlin7-10 was the first to pply the finite (or large) deformation theory to the phenomenologic analysis of rubber elasticity. He correctly pointed out the above-mentioned restrictions on W, and proposed an empirical form... 

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. 

As an alternative to the molecular approach of the three models described above, a phenomenological model of elasticity may be used. In such a... 

From a multiple scale modeling perspective, the presence of phenomenological parameters in various effective theories provides an opportunity for information passage in which one theory s phenomenological parameters are seen as derived quantities of another. We have already seen that although the linear theory of elasticity is silent on the particular values adopted by the elastic moduli (except for important thermodynamic inequalities), these parameters may be deduced on the basis of microscopic analysis. The advent of reliable models of material behavior makes it possible to directly calculate these parameters, complementing the more traditional approach which is to determine them experimentally. 

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 phenomenological Griineisen parameter electron-phonon coupling for the heavy-fermion compounds is very successful in describing the thermal expansion and temperature dependence of elastic constants. The parameters deduced from the different experiments agree with each other and they agree also with the parameters used in the microscopic description, sect. 4.3. 

Figure 2.18. Energies are shown that can be inter-converted by means of elastic-contractile model proteins capable of exhibiting inverse temperature transitions functioning by means of the competition for hydration between oil-like and charged groups called an apolar-polar repulsive free energy of hydration. See Chapter 5 for a more complete development of the phenomenology and physical basis and Chapter 8 for details of the molecular process.

The consilient mechanism was bom out of controlling the hydrophobic association-dissociation of elastic-contractile model proteins to achieve the possibility of some 18 classes of pairwise energy conversions (see Chapter 5, section 5.6). In the process a set of five Axioms became the phenomenology out of which the consilient mechanism arose. For the first time a common groundwork of explanation was able to perform the diverse energy conversions of biology. 

We complete this synopsis with a list of substantive predictions and their realizations that flow from the hydrophobic and elastic consilient mechanisms when confronted with the structure and phenomenology of the Fi-ATPase ... 

We have discussed the nexus of hydrophobic and elastic consilient mechanisms in Complex III at the intersection of electron flow and proton translocation (electro-chemical transduction) that was unimaginable, absent the detailed analysis of structure. The hydrophobic and elastic consilient mechanisms, however, when applied to the general structure and phenomenology of the Fi-motor of ATP synthase (chemo-chemical transduction), gave rise to a host of successful predictions, and, when applied to the structure and phenomenology of the myosin II motor (chemo-mechanical transduction), resulted in a half-dozen realized expectations. These findings do much to substantiate the relevance of the hydrophobic and elastic consilient mechanisms to the protein-based machines of biology. 

The Phenomenology of the Linear Theory of Viscoelasticity. One of the powers of the linear viscoelasticity theory is that it is predictive. The constitutive law that comes from Boltzmann superposition theory requires simply that the material functions discussed above be known for a given material. Then, for an arbitrary stress or deformation history, the material response can be obtained. In addition, the elastic-viscoelastic correspondence principle can be used so that boundary value problems such as beam bending, for which an elastic solution exists, can be solved for linear viscoelastic materials as well. Both of these subjects are treated in this section. 

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