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The collagen diffraction patterns of human arteries under uniaxial tensile test conditions have been investigated by time resolved synchrotron small angle X-ray diffraction [1]. Different types of arteries were chosen according to their clinical interest and have been studied after dissection into their major layers (intima, media, adventitia). Using a recently designed tensile testing device [2], the orientation and d-spacing of the collagen fibers in the layers have been measured in situ under physiological conditions, together with the macroscopic force and sample deformation. This allows reconstruction of true stresses and strains and the fitting of this data to a non linear mechanical model [3]. The results show a relation between the orientation/extension of the collagen fibers on the nanoscopic level and the macroscopic stress and strain. This is attributed first to a straightening, second to a reorientation of the collagen fibers, and third to an uptake of the increasing loads by the collagen fibers.

In this paper, we first of all review the morphology and structure of the myocardium and discuss the main features of the mechanical response of passive myocardium tissue, which is an orthotropic material. Locally within the architecture of the myocardium three mutually orthogonal directions can be identified, forming planes with distinct material responses. We treat the left ventricular myocardium as a non-homogeneous, thick-walled, nonlinearly elastic and incompressible material and develop a general theoretical framework based on invariants associated with the three directions. Within this framework we review existing constitutive models and then develop a structurally based model that accounts for the muscle fibre direction and the myocyte sheet structure. The model is applied to simple shear and biaxial deformations and a specific form fitted to the existing (and somewhat limited) experimental data, emphasizing the orthotropy and the limitations of biaxial tests. The need for additional data is highlighted. A brief discussion of issues of convexity of the model and related matters concludes the paper.

When a rubber test piece is loaded in simple tension from its virgin state, unloaded and then reloaded, the stress required on reloading is less than that on the initial loading for stretches up to the maximum stretch achieved on the initial loading. This stress softening phenomenon is referred to as the Mullins effect. In this paper a simple phenomenological model is proposed to account for the Mullins effect observed in filled rubber elastomers. The model is based on the theory of incompressible isotropic elasticity amended by the incorporation of a single continuous parameter, interpreted as a damage parameter. This parameter controls the material properties in the sense that it enables the material response to be governed by a strain-energy function on unloading and subsequent submaximal loading different from that on the primary (initial) loading path from the virgin state. For this reason the model is referred to as pseudo-elastic and a primary loading-unloading cycle involves energy dissipation. The dissipation is measured by a damage function which depends only on the damage parameter and on the point of the primary loading path from which unloading begins. A specific form of this function with two adjustable material constants, coupled with standard forms of the (incompressible, isotropic) strain-energy function, is used to illustrate the qualitative features of the Mullins effect in both simple tension and pure shear. For simple tension the model is then specialized further in order to fit Mullins effect data. It is emphasized that the model developed here is applicable to multiaxial states of stress and strain, not just the specific uniaxial tests highlighted.

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This review article is concerned with the mathematical modelling of the mechanical properties of the soft biological tissues that constitute the walls of arteries. Many important aspects of the mechanical behaviour of arterial tissue can be treated on the basis of elasticity theory, and the focus of the article is therefore on the constitutive modelling of the anisotropic and highly nonlinear elastic properties of the artery wall. The discussion focuses primarily on developments over the last decade based on the theory of deformation invariants, in particular invariants that in part capture structural aspects of the tissue, specifically the orientation of collagen fibres, the dispersion in the orientation, and the associated anisotropy of the material properties. The main features of the relevant theory are summarized briefly and particular forms of the elastic strain-energy function are discussed and then applied to an artery considered as a thickwalled circular cylindrical tube in order to illustrate its extension-inflation behaviour. The wide range of applications of the constitutive modelling framework to artery walls in both health and disease and to the other fibrous soft tissues is discussed in detail. Since the main modelling effort in the literature has been on the passive response of arteries, this is also the concern of the major part of this article. A section is nevertheless devoted to reviewing the limited literature within the continuum mechanics framework on the active response of artery walls, i.e. the mechanical behaviour associated with the activation of smooth muscle, a very important but also very challenging topic that requires substantial further development. A final section provides a brief summary of the current state of arterial wall mechanical modelling and points to key areas that need further modelling effort in order to improve understanding of the biomechanics and mechanobiology of arteries and other soft tissues, from the molecular, to the cellular, tissue and organ levels.