Study on Compressibility Control of Hyperelastic Material for Homogenization Method Using Mixed Finite Element Analysis

It is well known that the compressibility or incompressibility of biological tissue stems from its microscopic structure, which is generally composed of material with varied compressibility, including incompressibility. This paper proposes a framework for a homogenization method in which the compressibility/incompressibility of the macrostructure properly reflects that of the microstructure. The formulation is based on the mixed variational principle with a perturbed Lagrange-multiplier. It is shown that the rate of volumetric change of the macrostructure can be controlled through the homogenization procedure by introducing the constraint on the microstructure only. A couple of numerical examples are given to demonstrate the validity of the proposed method. By comparing the numerical results with theoretical solutions, the method is also confirmed to be free from locking.