Kinematics framework optimized for deformation, growth, and remodeling in vascular organs

Abstract

A basic tenant of constitutive theory is that phenomenological relations can be derivable from phenomenological behavior or material tests; and yet, conventional representation formulas, such as those of Rivlin and Fung, fail in this regard because of the choice of kinematical variables. Granted, with these representation formulas a particular constitutive relation may be guessed that fits data, but if the relation is non-unique and cannot be derived de novo from actual and/or hypothetical tests, then such a relation is indeterminable. The representation formula of Rivlin is indeterminable because of excessive covariance or coalignment in the kinematical variables. The representation formula of Fung is indeterminable because the incompressibility constraint is not utilized to reduce the kinematical variables a priori. The proposed kinematics framework succeeds in achieving determinability for hyperelastic materials because, primarily, the kinematical variables have minimal coalignment and dilatation and distortion are separated. Determinability is discussed and demonstrated in the context of hyperelasticity. However, any representation formula, whether it is for visco-elasticity or remodeling or etcetera, will be indeterminable when kinematical variables are highly coaligned and/or are subject to a non-reducible constraint. In other words, conventional kinematical frameworks are non-starters for experimentally determining constitutive representations for soft tissues. For the sake of determinability and/or validity of continuum models of vascular tissue, the proposed framework is needed. Moreover, this framework is optimized to simplify the balance equations for tubular structures.