Strain energy function isotropic materials

In order to point out the essential difference between deformations induced by body forces and surface tractions we recall Ericksen s theorem [88]. The theorem states that homogeneous deformations are the only deformations that can be achieved by the application of surface tractions alone, considering a homogeneous and isotropic material characterized by an arbitrary strain-energy function. In other words, we cannot induce diverse non-homogeneous deformations with surface tractions. In contrast to surface tractions, application of fields that act as body forces leads to non-homogeneous deformations without any additional constraints for the material. 

As with all functions, a general form of the strain energy function for an isotropic material can be formed by a Taylor expansion. The result is ... 

The isotropic part of this expression has also been used to model the myocardium of the embryonic chick heart during the ventricular looping stages, with coefficients of 0.02 kPa during diastole and 0.78 kPa at end-systole, and exponent parameters of 1.1 and 0.85, respectively [107]. Another related transversely isotropic strain-energy function was used by Guccione et al. [108] and Omens et al. [109] to model material properties in the isolated mature rat and dog hearts ... 

Usually a strain energy function is defined as W = poU. For an incompressible, isotropic material, the relation becomes... 

The phenomenological theory of rubberlike elasticity is based on continuum mechanics. It provides a mathematical structure from which, in principle, the deformation produced within a vulcanized elastomer by applied surface and bulk forces can be calculated. In the theory, the material is idealized by the assumption that it is perfectly elastic, isotropic in the undeformed state and incompressible. The most general form of the strain energy function (which vanishes at... 

Incoherent Clusters. As described in Section B.l, for incoherent interfaces all of the lattice registry characteristic of the reference structure (usually taken as the crystal structure of the matrix in the case of phase transformations) is absent and the interface s core structure consists of all bad material. It is generally assumed that any shear stresses applied across such an interface can then be quickly relaxed by interface sliding (see Section 16.2) and that such an interface can therefore sustain only normal stresses. Material inside an enclosed, truly incoherent inclusion therefore behaves like a fluid under hydrostatic pressure. Nabarro used isotropic elasticity to find the elastic strain energy of an incoherent inclusion as a function of its shape [8]. The transformation strain was taken to be purely, dilational, the particle was assumed incompressible, and the shape was generalized to that of an... 

Isotropic hyperelastic materials For this model, the strain energy density function is written in terms of the principal stress invariants I, h, h). Equation 1 becomes... 

Hence, for isotropic materials the strain energy density function takes the form ... 

The strain energy density function V for a compressible isotropic neo-Hookean material (see Attard Hunt (2004)) is given by ... 

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