Stored energy function

Shen 39) has also considered the thermoelastic behaviour of another widely used phenomenological equation of state, the so-called Valanis-Landel equation. Valanis and Landel40) have postulated that the stored energy function W should be expressible as the sum of three independent functions of principle extension ratios. This hypothesis leads to the following equation of state... 

In such cases W will be a function of position as well as of temperature and the coordinates of the deformation gradient tensor. Finally, most materials, in particular polymers, are anelastic. Energy is dissipated in them during a deformation and the stored energy function W cannot be defined. It is still of value, however, to consider ideal materials in which W does exist and to seek its form since such ideal materials may approximate quite closely to the real ones. 

W stored energy function Ze Doi-Edwards equilibration time... 

If entanglements acted like ordinary crosslinks (vN/2) per unit volume) the stored energy function would be given by the usual expression for tetrafunctional phantom networks with the spatial fluctuations of junctions suppressed ... 

As was discussed in some detail in chap. 2, the notion of an elastic solid is a powerful idealization in which the action of the entirety of microscopic degrees of freedom are subsumed into but a few material parameters known as the elastic constants. Depending upon the material symmetry, the number of independent elastic constants can vary. For example, as is well known, a cubic crystal has three independent elastic moduli. For crystals with lower symmetry, the number of elastic constants is larger. The aim of the present section is first to examine the physical origins of the elastic moduli and how they can be obtained on the basis of microscopic reasoning, and then to consider the nonlinear generalization of the ideas of linear elasticity for the consideration of nonlinear stored energy functions. 

As one of our central missions is to uncover the relation between microscopic and continuum perspectives, it is of interest to further examine the correspondence between kinematic notions such as the deformation gradient and conventional ideas from crystallography. One useful point of contact between these two sets of ideas is provided by the Cauchy-Bom rule. The idea here is that the rearrangement of a crystalline material by virtue of some deformation mapping may be interpreted via its effect on the Bravais lattice vectors themselves. In particular, the Cauchy-Bom mle asserts that if the Bravais lattice vectors before deformation are denoted by Ej, then the deformed Bravais lattice vectors are determined by e = FEj. As will become evident below, this mle can be used as the basis for determining the stored energy function W (F) associated with nonlinear deformations F. 

It is a straightforward exercise to show that Eq. 39 is the expression for stress components in an elastic incompressible solid with a stored energy function W (energy/volume) given by... 

The statistical theory will also describe the response under stress of elastomers swollen by solvents and, in general, it is found that the greater the degree of swelling, the better the agreement between theory and experiment. The modifications necessary for the treatment of swollen networks are relatively straightforward, and the stored energy function becomes... 

This equation represents the answer to the problem posed at the beginning of this section of deriving the stored energy function (Agi) for a deformed rubber network. Note that this equation contains only the extension ratio (a) terms for characterizing the network structure. Special cases of equation 7.39 are considered in the following sections. 

A/ifiiast = the Stored energy function for a deformed rubber network p = the sample density,... 

Hon et introduced a micromechanical model to predict the stiffness of a filled elastomer. They evaluated a range of 2D and 3D models to predict the stiffness of both medium thermal (MT) and high abrasion furnace (HAF)-filled elastomers. They used a simple Mooney stored energy function to characterize... 

We can therefore define a strain energy function or stored energy function that defines the energy stored in the body as a result of the strain. Here we have performed an analysis involving energy per unit volume under the simplifying assumption that the conditions are adiabatic. For other conditions, other forms of energy are appropriate, as discussed by Baker [7] and Houlsby and Puzrin [8]. 

This assumption is based on a more fundamental assumption eoneeming the elastie energy (stored energy function). If the elastic energy, which is a potential function for the stress tensor, vanishes in the unstrained state and can be expressed by a symmetric quadratic form, then the stiflness matrix is symmetric, i.e. the elasticity tensor is fully symmetric. 

Multiaxial response. The stress-strain response of Galcit I to multiaxial stress fields has also been investigated (15). These measurements were not made to try to determine the form of the stored energy function, W, but rather to test possible forms, especially in the region where X-d.. The experimental procedure used was combined torsion and tension/compensation on cylindrical rods. 

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