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Implementation and numerical verification of an incompressible three-parameter Mooney-Rivlin model for large deformation of soft tissues

The interest in simulating the behavior of soft tissues has grown over the past two decades. Intracranial aneurysms, for example, represents a dangerous disease and can be fatal. The evolution and outcome of this disease is related to the hemodynamics of blood flowing inside the vessels. However, to correctly characterize the hemodynamics environment, the complete fluid-solid interaction (FSI) problem must be solved, and some works suggest that it is essential to use correct models for the wall tissue. For that matter, robust codes are necessary since the mechanical constitutive response of arterial and aneurysm wall material is highly non-linear and occur in finite-strain regime, typically represented by hyperelastic laws. For example, a commonly used law in this field is the Mooney-Rivlin model, which has been used extensively to characterize the behavior of rubber-like materials, and also some soft tissues. However, the non-lineariaties of the model are sometimes hard to handle numerically, specially if the material is considered to be incompressible, as is the case for the majority of soft tissues. In this work, we present an implementation of the three-parameter incompressible Mooney-Rivlin model using a penalty function to enforce the incompressibility constraint in the solids4foam library, which presents a suitable framework that also allows for fluid-solid interaction simulations. The incompressibility constraint is dealt with by using a penalty method, which adds a numerical parameter that resembles the nulk modulus of the material. The numerical examples presented shows a good agreement between the results found with the current implementation and analytical and experimental solution as well as other numerical solutions.