Isogeometric Kirchhoff-Love shell formulations for biological membranes

Computational modeling of thin biological membranes can aid the design of better medical devices. Remarkable biological membranes include skin, alveoli, blood vessels, and heart valves. Isogeometric analysis is ideally suited for biological membranes since it inherently satisfies the C¹-requirement for Kirchhoff-Love kinematics. Yet, current isogeometric shell formulations are mainly focused on linear isotropic materials, while biological tissues are characterized by a nonlinear anisotropic stress-strain response. Here we present a thin shell formulation for thin biological membranes. We derive the equilibrium equations using curvilinear convective coordinates on NURBS tensor product surface patches. We linearize the weak form of the generic linear momentum balance without a particular choice of a constitutive law. We then incorporate the constitutive equations that have been designed specifically for collagenous tissues. We explore three common anisotropic material models: Mooney-Rivlin, May Newmann-Yin, and Gasser-Ogden-Holzapfel. Our work will allow scientists in biomechanics and mechanobiology to adopt the constitutive equations that have been developed for solid three-dimensional soft tissues within the framework of isogeometric thin shell analysis.