Direct Eulerian Formulation of Anisotropic Hyperelasticity

Many soft materials and biological tissues comprise isotropic matrix reinforced by fibers in the characteristic directions. Hyperelastic constitutive equations for such materials are usually formulated in terms of a Lagrangian strain tensor referred to the initial configuration and Lagrangian structure tensors defining characteristic directions of anisotropy. Such equations are "pushed forward" to the current configuration. Obtained in this way, Eulerian constitutive equations are often favorable from both theoretical and computational standpoints. In the present note, we show that the described two-step procedure is not necessary, and anisotropic hyperelasticity can be introduced directly in terms of an Eulerian strain tensor and Eulerian structure tensors referring to the current configuration. The newly developed constitutive equation is applied to the particular case of the transverse isotropy for the sake of illustration.