Summary: Classical engineering materials typically seek to maintain or attain a relaxed, stress-free state, if admitted by the boundary conditions. By contrast, living tissues seek to establish and maintain a certain non-zero target stress, a behavior that is often referred to as tensional homeostasis. Although a growing body of experimental evidence is underpinning the importance of tensional homeostasis for the biomechanics of living tissues, its theoretical implications remain poorly understood. Motivated by tissue culture studies, this paper demonstrates that tensional homeostasis has significant impact on the way how the notion of free energy should be conceptualized in living tissues. It demonstrates that standard variational approaches from continuum mechanics and engineering mechanics such as the principle of minimal total mechanical potential are in practice often insufficient to capture even the qualitative mechanical behavior of living tissues. To overcome this problem, this paper introduces, under assumptions that reflect the situation in tissue culture experiments, the novel concept of mechanobiological free energy. Mechanobiological free energy is shown to share the essential favorable properties of other free energy concepts such as Helmholtz free energy or strain energy but to generalize them in a way suitable to understand mechanobiological remodeling in collagenous tissues inhabited by living cells. In particular, mechanobiological free energy gives rise to a simple variational approach to tensional homeostasis in tissue equivalents.
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