A Quasi-Incompressible and Quasi-Inextensible Finite Element Implementation of Fibrous Soft Biological Tissues

Gültekin, Osman
Rodoplu, Burak
Dal, Hüsnü
Fibrous soft biological tissues exhibit nearly incompressible exponentially stiffening mechanical response (J–like stress–strain curve), making the development of efficient and robust finite element formulations at the quasi-incompressible and quasi-inextensible limit particularly important for the numerical analysis of such materials. The present contribution addresses the quasi–inextensible and quasi–incompressible finite hyperelastic behavior of fibrous tissues such as arterial walls. The formulation features an additive decomposition of the free energy function into isotropic and anisotropic parts. The unsplit deformation gradient is employed on both parts. While the pressure– dilatation conjugate variables are the cornerstones of the Q1P0 element formulation rendering quasi– incompressibility, similar conjugate pairs, namely the volume averaged fiber stretches and stresses variables engender the quasi–inextensible behavior of the the material reinforced with a number of fiber families, see e.g. [1]. Euler-Lagrange equations and the corresponding Q1P0F0 finite element formulation are the result of the extended Hu–Washizu variational principle with the respective potential. Following [2] and [3], the numerical implementation exploits the underlying variational structure leading to a canonical symmetric structure. The numerical performance of the Q1P0F0 element formulation are demonstrated first via series of dual clamped patch tests where the results are compared with those of Q1 and Q1P0 elements. A more representative case follows from a three–layered hollow cylinder reinforced with two families of fibers subjected to an internal pressure. Results indicate a plausible physical and numerical reasons in the replacement of all Q1P0 element by the new approach called Q1P0F0. Bibliography [1] G. A. Holzapfel, T. C. Gasser and R. W. Ogden, A new constitutive framework for arterial wall mechanics and a comparative study of material models, J. Elasticity 61, 1-48, 2000. [2] H. Dal, A quasi-incompressible and quasi-inextensible element formulation for transversely isotropic materials, Int. J. Numer. Meth. Eng. 117, 118-140, 2019. [3] O. Gültekin, H. Dal and G. A. Holzapfel, On the quasi-incompressible finite element analysis of anisotropic hyperelastic materials, Comp. Mech. 63, 443-453, 2019.
Citation Formats
O. Gültekin, B. Rodoplu, and H. Dal, “A Quasi-Incompressible and Quasi-Inextensible Finite Element Implementation of Fibrous Soft Biological Tissues,” presented at the Beyond 2019: Computational Science and Engineering Conference, (9 - 11 September 2019), Ankara, Turkey, 2019, Accessed: 00, 2021. [Online]. Available: http://files.iam.metu.edu.tr/beyond2019/booksofabstract_beyond2019.pdf#page=41.