Anisotropy in hypoelastic soft-tissue mechanics, I: Theory

Abstract

A hypoelastic constitutive theory is derived with an eye towards modeling the passive response of soft anisotropic biological tissues. Anisotropy is handled through a material tensor whose construction is independent of the constitutive formulation. Anisotropy tensors are provided for tissues that have a single dominant fiber family with an elliptic fiber projection onto the transverse plane, and their limiting cases. The tissue is comprised of two constituents: a matrix phase and a fiber phase. The theory is derived in the polar configuration, for ease of handling the derivatives, and then mapped into the commonly used Eulerian configuration. Kirchhoff stress and its conjugate strain-rate are the state variables.

Anisotropy in hypoelastic soft-tissue mechanics, II: Simple extensional experiments

“…Hypoelastic materials establish stress-rate via a linear function in strain-rate [Truesdell 1955], and have the ability to reach states of stress that are path dependent, in accordance with recent experimental observations [Criscione et al 2003]. In [Freed 2008], a special class of hypoelastic materials was introduced where stress-rate is given by a potential function in strain-rate. Such potentials must be quadratic in strain-rate so as to remain compatible with Truesdell's general definition of a hypoelastic material.…”

“…Part I of this paper [Freed 2008] departed from longstanding tradition by considering a hypoelastic construction for soft tissues, instead of adopting a more conventional hyperelastic formulation. In this second part, we extend and apply this theory to a variety of axially loaded, soft-tissue experiments that have been published in the literature.…”

“…This paper begins with a synopsis of the isochoric, hypoelastic, constitutive model for isotropic soft tissues derived in [Freed 2008]. Solutions for this material model are provided for the BCs of uniaxial, equibiaxial, laterally constrained, and pure-shear extensions.…”

“…The next section extends our isotropic theory to one that is suitable for anisotropic materials through the introduction of an anisotropy tensor, which is a material tensor that was given a geometric interpretation in [Freed 2008]. Invariant theory is called upon to construct a potential function, through which an anisotropic, isochoric, hypoelastic, constitutive equation is derived.…”

Constitutive modelling of arteries

“…More recently models based on invariants have been adopted including those of Freed et al (2005), Holzapfel et al (2005a,b) and Gasser & Holzapfel (2006a). The model of Freed et al (2005), for example, uses a structure tensor based on the Gaussian distribution of fibre directions for both two and three dimensions, and the model is applied to bioprosthetic valves; see also Freed (2008).…”