A new formulation of slight compressibility for arterial tissue and its Finite Element implementation

In order to avoid the numerical difficulties in locally enforcing the incompressibility constraint using the displacement formulation of the Finite Element Method, slight compressibility is typically assumed when simulating the mechanical response of arterial tissue. The current standard method of accounting for slight compressibility of hyperelastic soft tissue assumes an additive decomposition of the strain-energy function into a volumetric and a deviatoric part. This has been shown, however, to be inconsistent with the linear theory and results in cubes retaining their cuboid shape under hydrostatic tension and compression, which seems at variance with the reinforcement of arterial tissue with two families of collagen fibres. A remedy for these defects is proposed here, a solution which generalises the current standard model of slight compressibility to include two additional terms, one of which is quadratic in the I4,I6$ I_4, I_6 $ invariants and the other quadratic in I8$ I_8 $. Experimental data are used to motivate typical values for the associated material constants of these additional terms. Some simulations are performed to allow contrasts and comparisons to be made between the current standard model of slight compressibility and its generalisation proposed here.