A Constitutive Framework for Tubular Structures that Enables a Semi-inverse Solution to Extension and Inflation

Abstract

Traditional constitutive frameworks for high-strain materials are ill-suited to solve extension and inflation, one of the simplest problems involving tubes, or more complicated problems. Moreover, it is experimentally necessary to minimize the covariance amongst constitutive response functions. We sought, hence, a constitutive framework that minimizes covariance and simplifies the balance equations for tubes, hoses, and arteries. Central to this theory are six objective scalars or strain attributes that decouple dilatation and distortion and succinctly define the strain. Because there is a one-to-one relationship between them and the components of the Right Cauchy–Green deformation tensor, these six strain attributes can be used to define the strain energy density function for hyperelastic materials. This approach yields mostly orthogonal response terms for high strain deformation (14 of the 15 inner products vanish). For infinitesimal deformation, the response terms are fully orthogonal. Further utility is demonstrated by showing how the governing equations are simplified for tubular structures and how response functions can be determined for the first time from the extension and inflation of thick-walled tubes composed of a homogeneous material with incompressible, hyperelastic behavior. This solution is applicable for materials with orthotropic behavior, and using the chain rule, this solution can be used for materials with isotropic behavior.