A quasi-incompressible and quasi-inextensible finite element analysis of fibrous soft biological tissues.

The contribution presents an extension and application of a recently proposed finite element formulation for quasi-inextensible and quasi-incompressible finite hyperelasticity to fibrous soft biological tissues and touches in particular upon computational aspects thereof. In line with theoretical framework presented by Dal (Int J Numer Methods Eng 117:118-140, 2019), the mixed variational formulation is extended to two families of fibers as often encountered while dealing with fibrous tissues. Apart from that, the purely Eulerian setting features the additive decomposition of the free energy function into volumetric, isotropic and anisotropic parts. The multiplicative split of the deformation gradient and all the outcomes thereof, e.g., unimodular invariants, are simply dispensed with in the three element formulations investigated, namely Q1, Q1P0 and the proposed Q1P0F0. For the quasi-incompressible response, the Q1P0 element formulation is briefly outlined where the pressure-type Lagrange multiplier and its conjugate enter the variational formulation as an extended set of variables. Using the similar argumentation, an extended Hu-Washizu-type mixed variational potential is introduced where the volume averaged squares of fiber stretches and associated fiber stresses are additional field variables. The resulting finite element formulation called Q1P0F0 is very attractive as it is based on mean values of the additional field variables at element level through integration over the element domain in a preprocessing step, earning the model vast utilization areas. The proposed approach is examined through representative boundary value problems pertaining to fibrous biological tissues. For all the cases studied, the proposed Q1P0F0 formulation elicits the most compliant mechanical response, thereby outperforming the standard Q1 and Q1P0 element formulations through mesh-refinement analyses. Results prompt further experimental investigations as to true deformation fields under biologically relevant loading conditions which would make the assessment of Q1P0 and Q1P0F0 more based on physical grounds.