On Eulerian constitutive equations for modeling growth and residual stresses in arteries

Recently Volokh and Lev (2005) argued that residual stresses could appear in growing arteries because of the arterial anisotropy. This conclusion emerged from a continuum mechanics theory of growth of soft biological tissues proposed by the authors. This theory included Lagrangian constitutive equations, which were formulated directly with respect to the reference configuration. Alternatively, it is possible to formulate Eulerian constitutive equations with respect to the current configuration and to 'pull them back' to the reference configuration. Such possibility is examined in the present work. The Eulerian formulation of the constitutive equations is used for a study of arterial growth. It is shown, particularly, that bending resultants are developed in the ring cross-section of the artery. These resultants may cause the ring opening or closing after cutting the artery in vitro as it is observed in experiments. It is remarkable that the results of the present study, based on the Eulerian constitutive equations, are very similar to the results of Volokh and Lev (2005), based on the Lagrangian constitutive equations. This strengthens the authors' argument that anisotropy is a possible reason for accumulation of residual stresses in arteries. This argument appears to be invariant with respect to the mathematical description.