Mechanical modeling of growth considering domain variation. Part I: constitutive framework

The growth of biological tissues is described as the volumetric production of mass within tissue elements, considered as one component potential sites for tissue renewal/resorption at a mesoscopic scale of description. The growth is characterized by a growth transformation gradient, which is accompanied by an additional accommodation transformation, so that the total transformation gradient defines a compatible global displacement field. The continuous change of the domain occupied by each tissue element during growth is considered, hence the mechanical balance laws are written, accounting for the additional terms due to the domain variation. The principle of virtual power is then expressed, considering that the power of internal forces originates from both volumic and surface potentials. The surface potential is thought to express configurational forces tied to the interface motion, extent and orientation. The equilibrium equations of the growing tissue element then follow, associated to surface and line boundary conditions. The writing of the second principle of thermodynamics for a tissue element continuously receiving matter due to transport phenomena allows to identify the driving forces linked to growth, and provides the evolution laws for the growth velocity. The large strains compatibility conditions that characterize the accommodation tensor are further considered as an important aspect of growth. A first insight into the numerical solutions of the compatibility conditions is given, envisaging the situation of radial growth.