On the Modeling of Active Deformation in Biological Transversely Isotropic Materials

Many biological materials exhibit the ability to actively deform, essentially due to a complex chemical interaction involving two proteins, actin and myosin, in the myocytes (the muscle cells). While the mathematical description of passive materials is well-established, even for large deformations, this is not the case for active materials, since capturing its complexities poses significant challenges. This paper focuses on the mathematical modeling of active deformation of biological materials, guided by the important example of skeletal muscle tissue. We will consider an incompressible and transversely isotropic material within a hyperelastic framework. Our goal is to design constitutive relations that agree with uniaxial experimental data whenever possible. Finally, we propose a novel model based on a coercive and polyconvex elastic energy density for a fiber-reinforced material; in this model, active deformation occurs solely through a change in the reference configuration of the fibers, following the mixture active strain approach. This model assumes a constant active parameter, preserving the good mathematical features of the original model while still capturing the essential deformations observed in experiments.