The mathematical formulation of a generalized Hooke's law for blood vessels

Abstract

It is well known that the stress–strain relationship of blood vessels is highly nonlinear. To linearize the relationship, the Hencky strain tensor is generalized to a logarithmic–exponential (log–exp) strain tensor to absorb the nonlinearity. A quadratic nominal strain potential is proposed to derive the second Piola–Kirchhoff stresses by differentiating the potential with respect to the log–exp strains. The resulting constitutive equation is a generalized Hooke's law. Ten material constants are needed for the three-dimensional orthotropic model. The nondimensional constant used in the log–exp strain definition is interpreted as a nonlinearity parameter. The other nine constants are the elastic moduli with respect to the log–exp strains. In this paper, the proposed linear stress–strain relation is shown to represent the pseudoelastic Fung model very well.