Direct tensor expression by Eulerian approach for constitutive relations based on strain invariants in transversely isotropic green elasticity - finite extension and torsion

It has been proven by J.C.Criscione that constitutive relations(mixed approach) based on a set of five strain invariants (Beta-1, Beta-2, Beta-3, Beta-4, Beta-5) are useful and stable for experimentally determining response terms for transversely isotropic material. On the other hand, Rivlin?s classical model is an unsuitable choice for determining response terms due to the co-alignment of the five invariants (I1, I2, I3, I4, I5). Despite this, however, a mixed (Lagrangian and Eulerian) approach causes unnecessary computational time and requires intricate calculation in the constitutive relation. Through changing the way to approach the derivation of a constitutive relation, we have verified that using an Eulerian approach causes shorter computational time and simpler calculation than using a mixed approach does. We applied this approach to a boundary value problem under specific deformation, i.e. finite extension and torsion to a fiber reinforced circular cylinder. The results under this deformation show that the computational time by Eulerian is less than half of the time by mixed. The main reason for the difference is that we have to determine two unit vectors on the cross fiber direction from the right Cauchy Green deformation tensor at every radius of the cylinder when we use a mixed approach. On the contrary, we directly use the left Cauchy Green deformation tensor in the constitutive relation by the Eulerian approach without defining the two cross fiber vectors. Moreover, the computational time by the Eulerian approach is not influenced by the degree of deformation even in the case of computational time by the Eulerian approach, possibly becoming the same as the computational time by the mixed approach. This is from the theoretical thought that the mixed approach is almost the same as the Eulerian approach under small deformation. This new constitutive relation by Eulerian approach will have more advantages with regard to saving computational time as the deformation gets more complicated. Therefore, since the Eulerain approach effectively shortens computational time, this may enhance the computational tools required to approach the problems with greater degrees of anisotropy and viscoelasticity.