Different computational approaches in the modeling of wrinkling of biological membranes

In previous papers the authors examined the problem of the wrinkling of plane isotropic biological membranes following the approach of Pipkin and treating the out of plane geometric nonlinearities as constitutive nonlinearities through a modification of the elastic potential. The problem has been solved within the framework of finite strain hyperelasticity for a material characterized by a Fung type constitutive law in biaxial tension. All assumptions of classical Tension Field theory emerge as a result of such formulation. In this paper the model is applied to simulate procedures of reconstructive surgery where wrinkling of the skin occurs during and after the suture of the wound edges leading to unaesthetic scars with dog-ear formations. The effects of the natural tension of the skin on the wrinkling extension and final stress field are highlighted. The assumption of zero bending stiffness of the model formulated by the authors does not allow for detailed predictions of the deformation fields of real membranes where wrinkles with finite magnitude and wavelength develop. A critical evaluation of such assumption is then made by considering the problem of a linear elastic annular membrane subject to inner and outer uniform tractions. The results of the Tension Field theory are compared with the results of a buckling analysis. In the problem studied and under certain loading conditions, the extension of the wrinkled region and the number of circumferential waves predicted by the buckling analysis prove to be independent of the elastic constants and the bending stiffness of the membrane. This allows for a quantitative comparison with the results of the Tension Field theory. The comparison shows analogies and differences between the onset of membrane instability and the simplified description of the post-buckling configuration given by the Tension Field theory.