The Holzapfel-Gasser-Ogden (HGO) model for anisotropic hyperelastic behaviour of collagen fibre reinforced materials was initially developed to describe the elastic properties of arterial tissue, but is now used extensively for modelling a variety of soft biological tissues. Such materials can be regarded as incompressible, and when the incompressibility condition is adopted the strain energy Ψ of the HGO model is a function of one isotropic and two anisotropic deformation invariants. A compressible form (HGO-C model) is widely used in finite element simulations whereby the isotropic part of Ψ is decoupled into volumetric and isochoric parts and the anisotropic part of Ψ is expressed in terms of isochoric invariants. Here, by using three simple deformations (pure dilatation, pure shear and uniaxial stretch), we demonstrate that the compressible HGO-C formulation does not correctly model compressible anisotropic material behaviour, because the anisotropic component of the model is insensitive to volumetric deformation due to the use of isochoric anisotropic invariants. In order to correctly model compressible anisotropic behaviour we present a modified anisotropic (MA) model, whereby the full anisotropic invariants are used, so that a volumetric anisotropic contribution is represented. The MA model correctly predicts an anisotropic response to hydrostatic tensile loading, whereby a sphere deforms into an ellipsoid. It also computes the correct anisotropic stress state for pure shear and uniaxial deformation. To look at more practical appli- cations, we developed a finite element user-defined material subroutine for the simulation of stent deployment in a slightly compressible artery. Significantly higher stress triaxiality and arterial compliance are computed when the full anisotropic invariants are used (MA model) instead of the isochoric form (HGO-C model).Keywords: Anisotropic, Hyperelastic, Incompressibility, Finite element, Artery, Stent Nomenclature I -identity tensor Ψ -Helmholtz free-energy (strain-energy) function Ψ vol -volumetric contribution to the free energy Ψ aniso -anisotropic contribution to the free energy Ψ iso -isotropic contribution to the isochoric free energy Ψ aniso -anisotropic contribution to the isochoric free energy σ -Cauchy stress σ -deviatoric Cauchy stress q -von Mises equivalent stress σ hyd -hydrostatic (pressure) stress F -deformation gradient J -determinant of the deformation gradient; local ratio of volume change C -right Cauchy-Green tensor I 1 -first invariant of C I 4,6 -anisotropic invariants describing the deformation of reinforcing fibres F -isochoric portion of the deformation gradient C -isochoric portion of the right Cauchy-Green deformation tensor I 1 -first invariant of C I 4,6 -isochoric anisotropic invariants a 0i , i = 4, 6 -unit vector aligned with a reinforcing fibre in the reference configuration a i , i = 4, 6 -updated (deformed) fibre direction (= Fa 0i ) κ 0 -isotropic bulk modulus µ 0 -isotropic shear modulus k i , i = 1, 2 -anisotropic material co...

We introduce a fundamental restriction on the strain energy function and stress tensor for initially stressed elastic solids. The restriction applies to strain energy functions W that are explicit functions of the elastic deformation gradient F and initial stress τ , i.e. W := W (F, τ ). The restriction is a consequence of energy conservation and ensures that the predicted stress and strain energy do not depend upon an arbitrary choice of reference configuration. We call this restriction initial stress reference independence (ISRI). It transpires that almost all strain energy functions found in the literature do not satisfy ISRI, and may therefore lead to unphysical behaviour, which we illustrate via a simple example. To remedy this shortcoming we derive three strain energy functions that do satisfy the restriction. We also show that using initial strain (often from a virtual configuration) to model initial stress leads to strain energy functions that automatically satisfy ISRI. Finally, we reach the following important result: ISRI reduces the number of unknowns of the linear stress tensor of initially stressed solids. This new way of reducing the linear stress may open new pathways for the non-destructive determination of initial stresses via ultrasonic experiments, among others.

An initial stress within a solid can arise to support external loads or from processes such as thermal expansion in inert matter or growth and remodelling in living materials. For this reason, it is useful to develop a mechanical framework of initially stressed solids irrespective of how this stress formed. An ideal way to do this is to write the free energy density Ψ in terms of initial stress τ and the elastic deformation gradient F , so we write Ψ = Ψ ( F , τ ). In this paper, we present a new constitutive condition for initially stressed materials, which we call the initial stress symmetry (ISS). We focus on two consequences of this condition. First, we examine how ISS restricts the possible choices of free energy densities Ψ = Ψ ( F , τ ) and present two examples of Ψ that satisfy the ISS. Second, we show that the initial stress can be derived from the Cauchy stress and the elastic deformation gradient. To illustrate, we take an example from biomechanics and calculate the optimal Cauchy stress within an artery subjected to internal pressure. We then use ISS to derive the optimal target residual stress for the material to achieve after remodelling, which links nicely with the notion of homeostasis.

For over 70 years it has been assumed that scalar wave propagation in (ensembleaveraged) random particulate materials can be characterised by a single effective wavenumber. Here, however, we show that there exist many effective wavenumbers, each contributing to the effective transmitted wave field. Most of these contributions rapidly attenuate away from boundaries, but they make a significant contribution to the reflected and total transmitted field beyond the low-frequency regime. In some cases at least two effective wavenumbers have the same order of attenuation. In these cases a single effective wavenumber does not accurately describe wave propagation even far away from boundaries. We develop an efficient method to calculate all of the contributions to the wave field for the scalar wave equation in two spatial dimensions, and then compare results with numerical finite-difference calculations. This new method is, to the authors' knowledge, the first of its kind to give such accurate predictions across a broad frequency range and for general particle volume fractions.

Effective medium theory aims to describe a complex inhomogeneous material in terms of a few important macroscopic parameters. To characterize wave propagation through an inhomogeneous material, the most crucial parameter is the effective wavenumber . For this reason, there are many published studies on how to calculate a single effective wavenumber. Here, we present a proof that there does not exist a unique effective wavenumber; instead, there are an infinite number of such (complex) wavenumbers. We show that in most parameter regimes only a small number of these effective wavenumbers make a significant contribution to the wave field. However, to accurately calculate the reflection and transmission coefficients, a large number of the (highly attenuating) effective waves is required. For clarity, we present results for scalar (acoustic) waves for a two-dimensional material filled (over a half-space) with randomly distributed circular cylindrical inclusions. We calculate the effective medium by ensemble averaging over all possible inhomogeneities. The proof is based on the application of the Wiener–Hopf technique and makes no assumption on the wavelength, particle boundary conditions/size or volume fraction. This technique provides a simple formula for the reflection coefficient, which can be explicitly evaluated for monopole scatterers. We compare results with an alternative numerical matching method.