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

摘要

It has been proven by J.C.Criscione that constitutive relations(mixed approach) basedon a set of five strain invariants (Beta-1, Beta-2, Beta-3, Beta-4, Beta-5) are useful and stable for experimentallydetermining response terms for transversely isotropic material. On the otherhand, Rivlin?s classical model is an unsuitable choice for determining response termsdue 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 timeand requires intricate calculation in the constitutive relation. Through changing theway to approach the derivation of a constitutive relation, we have verified that usingan Eulerian approach causes shorter computational time and simpler calculation thanusing a mixed approach does. We applied this approach to a boundary value problemunder specific deformation, i.e. finite extension and torsion to a fiber reinforced circularcylinder. The results under this deformation show that the computational timeby Eulerian is less than half of the time by mixed. The main reason for the differenceis that we have to determine two unit vectors on the cross fiber direction from theright Cauchy Green deformation tensor at every radius of the cylinder when we use amixed approach. On the contrary, we directly use the left Cauchy Green deformationtensor in the constitutive relation by the Eulerian approach without defining the twocross fiber vectors. Moreover, the computational time by the Eulerian approach is not influenced by the degree of deformation even in the case of computational timeby the Eulerian approach, possibly becoming the same as the computational time bythe mixed approach. This is from the theoretical thought that the mixed approachis almost the same as the Eulerian approach under small deformation. This newconstitutive relation by Eulerian approach will have more advantages with regardto saving computational time as the deformation gets more complicated. Therefore,since the Eulerain approach effectively shortens computational time, this may enhancethe computational tools required to approach the problems with greater degrees ofanisotropy and viscoelasticity.