A discrete fiber dispersion model with octahedral symmetry quadrature for mechanical analyses of skin corrective surgeries

Advanced simulations of the mechanical behavior of soft tissues frequently rely on structure-based constitutive models, including smeared descriptions of collagen fibers. Among them, the so-called Discrete Fiber Dispersion (DFD) modeling approach is based on a discrete integration of the fiber-strain energy over all the fiber directions. In this paper, we review the theoretical framework of the DFD model, including a derivation of the stress and stiffness tensors required for the finite element implementation. Specifically, their expressions for incompressible plane stress problems are obtained. The use of a Lebedev quadrature, built exploiting the octahedral symmetry, is then proposed, illustrating the particular choice adopted for the orientation of the integration points. Next, the convergence of this quadrature scheme is assessed by means of three numerical benchmark tests, highlighting the advantages with respect to other angular integration methods available in the literature. Finally, using the implemented model, we analyze the mechanical properties of the Z-plasty, a technique commonly used in reconstructive skin surgery, considering multiple geometrical configurations, orientations of the fibers, and levels of skin prestress. The results are presented in the form of mechanical quantities relevant to surgical practice.