On the use of universal relations in the modeling of transversely isotropic materials

A material is of coaxial type   if the Cauchy stress tensor TT and the strain tensor BB are coaxial for all deformations. Clearly a hyperelastic material is of coaxial type if and only if it is isotropic. Here we present a weaker definition of materials of coaxial type. Anisotropic materials may be of a coaxial type in a weak sense if for a given specific  BB we have that TB=BTTB=BT. We denote these materials BB-coaxial. We show that for transverse isotropic materials weak coaxial constitutive equations may be characterized using universal relations. We discuss the impact of BB-coaxial materials in the modeling of soft tissues. We conclude that BB-coaxial materials are a strong evidence that in real world materials two anisotropic invariants are always necessary to model in a meaningful and correct way single fiber reinforced materials.