Active Stress vs. Active Strain in Mechanobiology: Constitutive Issues

Many biological tissues exhibit a non-standard continuum mechanics behavior: they are able to modify their placement in absence of external loads. The activity of the muscles is usually represented in solid mechanics in terms of an active stress, to be added to the standard one. A less popular approach is to introduce a multiplicative decomposition of the tensor gradient of deformation in two factors: the passive and the active one. Both approaches should satisfy due mathematical properties, namely frame indifference and ellipticity of the total stress. At the same time, the constitutive laws should reproduce the observed physiological behavior of the specific living tissue. In this paper we focus on cardiac contractility. We review some constitutive examples of active stress and active strain taken from the literature and we discuss them in terms of precise mathematical and physiological properties. These arguments naturally suggest new possible models.