Isogeometric Kirchhoff–Love shell formulations for biological membranes

Abstract

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.

Immersogeometric cardiovascular fluid–structure interaction analysis with divergence-conforming B-splines

“…This can be construed as a thin-shell approximation of the “shifter tensor” defined by [71, (63)]. Alternatively, the thin shell formulations of [69] and [72] do not make this approximation. The FSI analysis techniques of this paper are independent of how exactly the shell subproblem is formulated.…”

The Incompatibility of Living Systems: Characterizing Growth-Induced Incompatibilities in Expanded Skin

“…27 Particularly, isogeometric analysis has been found suitable for thin shell descriptions of biological membranes such as skin. 14 Furthermore, a three-dimensional B-spline surface patch is defined explicitly over a two-dimensional parametric domain, inherently providing the same parameterization for every configuration of the tissue as it is expanded. Here we employ the isogeometric concept to retrieve the deformation maps and deformation gradients imposed by tissue expansion.…”

Mathematical and computational modelling of skin biophysics: a review

“…Isogeometric finiteelement formulations [192][193][194] are therefore superior to those based on Lagrange polynomial interpolation. Identifying critical conditions that trigger surface instabilities is also a challenging endeavour but progress in this area is steadily moving forward [195].…”