Statistical mechanics of soft collagenous tissues: a constitutive model with damage for cerebral bridging veins

Abstract

A novel constitutive material model for the collagenous tissue of bridging veins is presented in this article. The walls of blood vessels are structurally reinforced by the collagen fibers, which are interconnected with an elastin matrix. Collagen fibers are oriented according to a specific probability distribution, with a greater abundance of fibers along the longitudinal axis. It is possible to analyze the collective mechanical behavior of collagen fibers using statistical mechanics techniques. The article describes how to derive the macroscopic behavior of collagenous tissue using the formalism of the canonical ensemble of statistical mechanics. One of the innovations of this formulation is the incorporation of a damage variable that increases significantly under excessive stretching. This formalism enables the derivation of a constitutive material model that is thermodynamically consistent, in which entropy increases irreversibly with the level of damage. In the model, an increase in damage results in a permanent increase in entropy. Furthermore, for a given level of damage, an increase in strain induces organization in the fibers, which is consistent with the behavior of an entropically elastic material. In addition, the thermodynamic properties of the model account for specific empirical observations, such as the slight contraction that occurs in a blood vessel when it is heated. The model’s mechanical suitability was assessed by subjecting it to uniaxial tensile tests on bridging veins. The findings provide strong evidence that the proposed constitutive model accurately reproduces the stress–strain curves (\(R^2_{adj}> 0.99\)).