Strain-energy function approach

When using the generalised Hooke s law strain energy function there are a number of possible strain definitions that can be used depending on the situation. When material deformation is very small the infinitesimal strain approach is a valid approximation with the strain defined as... 

We shall develop the concept of strain energy using a small strain approach. Because we will wish to follow the phenomenological treatment by one based on statistical mechanics, it is important at the outset to examine the different types of strain energy function that can be defined, depending on experimental conditions. This introduces thermodynamic considerations. 

The strain energy function 77 is a function of the components of some measure of strain, such as the stretch V or the Cauchy-Green measure C. 77 is a physical quantity with a numerical value upon which all observers will agree - it is independent of the axis set. On the other hand, the eomponents of V and C are entirely dependent on the axis set. Unless a function of these eomponents is chosen with care, it will itself be dependent on the axis set and so will be inadmissible as a strain energy function. This places restrictions on the form of 77, which can be approached in two ways ... 

In this section, we give examples of applications using the approach to the strain energy function listed as (1) in Section 3.4.2. We shall restrict the discussion to incompressible materials, so that only the invariants 7i and h will be of concern. 

More recently, Kawabata et al. [12] applied a similar approach to an isoprene mbber vulcanisation. They showed that the strain energy function of Equation (3.61) was applicable to this material. They modelled temperature dependence satisfactorily by modifying the first term to be proportional to the absolute temperature. 

The phenomenological theories following the approach of Rivlin and others were based on empirically developed strain energy functions. These have been overtaken by molecular network models notably by Edwards and colleagues, Boyce and colleagues and Stepto and colleagues. 

Preloads are sometimes considered in developing resistance functions. Preloads are any dead or live loads which cause a deformation in the member and thereby use up some of the available strain energy. Effects of preload on equivalent SDOF system analyses are sometimes handled by reducing the calculated available resistance by the amount of the preload. Another approach is to simply superimpose the preload on top of the blast load. 

In the present article, we summarize typical approaches to the evaluation of the strain energy density function from biaxial extension experiments and illustrate some intportant data. This article is not a review in the ordinary sense, as it deals to a large extent with a series of experiments carried out in our laboratory. By this we do not mean to bias or ignore any of the many important contributions by other authors. 

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. 

An alternative approach to modeling the L-M-L angles is to set the force constants to zero and include nonbonded 1,3-interactions between the ligand atoms. In most force fields, 1,3-interactions are not explicitly included for any atoms, instead they are taken up in the force constants for the valence angle terms. This is an approximation because the 1,3-interactions are most often repulsive and thus the function used to calculate the strain energy arising from valence angle deformation should be asymmetric. It was shown that the nonbonded 1,3-interactions around the metal atom are in many cases a major determinant of the coordination... 

Equations of the form of Equation (4) form the basis of the analysis of strain and elasticity reviewed in this chapter. The issues to be addressed are (a) the geometry of strain, leading to standard equations for strain components in terms of lattice parameters, (b) the relationship between strain and the driving order parameter, and (c) the elastic anomalies which can be predicted on the basis of the resulting free energy functions. The overall approach is presented as a series of examples. For more details of Landau theory and an introduction to the wider literature, readers are referred to reviews by Bruce and Cowley (1981), Wadhawan (1982), Toledano et al. (1983), Bulou et al. (1992), Salje (1992a,b 1993), Redfern (1995), Carpenter et al. (1998a), Carpenter and Salje (1998). 

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