Elastic behavior strain energy density

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. 

It is convenient to express the elastic behavior of the network in terms of the strain energy density W per unit of unstrained volume. The strain energy w for a single chain is obtained from Eq. (1.1) as... 

There is an extensive body of literature describing the stress-strain response of rubberlike materials that is based upon the concepts of Finite Elasticity Theory which was originally developed by Rivlin and others [58,59]. The reader is referred to this literature for further details of the relevant developments. For the purposes of this paper, we will discuss the developments of the so-called Valanis-Landel strain energy density function, [60] because it is of the form that most commonly results from the statistical mechanical models of rubber networks and has been very successful in describing the mechanical response of cross-linked rubber. It is resultingly very useful in understanding the behavior of swollen networks. 

Another point to keep in mind here is that, in most models, the description of rubber elasticity given from statistical mechanical models results in a Valanis-Landel form of strain energy density function. This will be important in the following developments. We now look at some common representations of the strain energy density function used to describe the stress-strain behavior of crosslinked rubber. 

Various mathematical models were suggested for the analysis of left ventricular diastole (Ghista et al.y 1969 Mirsky, 1973 Rabkin and HSU 1975, Ghista and Hamid 1977, Ghista and Ray 1980, a, b Moskowitz, 1980). These models are directed toward either a part of diastole or toward analytical expressions for the cardiac elastic properties. Different parameters are used to describe the time varying behavior discussed earlier. They can be the general stiffness parameters, or parameters defined only for the need of the analysis, as for example the left ventricular medium s strain energy density (Mirsky, 1973) or others. 

While we do not want to give a sophisticated model including all the effects found in the mechanical behavior of polymers, we restrict ourselves to the simplest case, namely to an elastic small-strain model at constant temperature. Therefore, the governing variables are the linear strain tensor [Eq. (13)] derived from the spatial gradient of the displacement field u, and the microstructural parameter k and its gradient. The free energy density is assumed to be a function of the form of Eq. (14). 

In the numerical calculations, an elastic-perfectly-plastic ductile rod stretching at a uniform strain rate of e = lO s was treated. A flow stress of 100 MPa and a density of 2700 kg/m were assumed. A one-millimeter square cross section and a fracture energy of = 0.02 J were used. These properties are consistent with the measured behavior of soft aluminim in experimental expanding ring studies of Grady and Benson (1983). Incipient fractures were introduced into the rod randomly in both position and time. Fractures grow... 

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