Ray W. Ogden

A new Constitutive Framework for Arterial Wall Mechanics and a Comparative Study of Material Models

In this paper we develop a new constitutive law for the description of the (passive) mechanical response of arterial tissue. The artery is modeled as a thick-walled nonlinearly elastic circular cylindrical tube consisting of two layers corresponding to the media and adventitia (the solid mechanically relevant layers in healthy tissue). Each layer is treated as a fiber-reinforced material with the fibers corresponding to the collagenous component of the material and symmetrically disposed with respect to the cylinder axis. The resulting constitutive law is orthotropic in each layer. Fiber orientations obtained from a statistical analysis of histological sections from each arterial layer are used. A specific form of the law, which requires only three material parameters for each layer, is used to study the response of an artery under combined axial extension, inflation and torsion. The characteristic and very important residual stress in an artery in vitro is accounted for by assuming that the natural (unstressed and unstrained) configuration of the material corresponds to an open sector of a tube, which is then closed by an initial bending to form a load-free, but stressed, circular cylindrical configuration prior to application of the extension, inflation and torsion. The effect of residual stress on the stress distribution through the deformed arterial wall in the physiological state is examined.The model is fitted to available data on arteries and its predictions are assessed for the considered combined loadings. It is explained how the new model is designed to avoid certain mechanical, mathematical and computational deficiencies evident in currently available phenomenological models. A critical review of these models is provided by way of background to the development of the new model.

Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

Constitutive relations are fundamental to the solution of problems in continuum mechanics, and are required in the study of, for example, mechanically dominated clinical interventions involving soft biological tissues. Structural continuum constitutive models of arterial layers integrate information about the tissue morphology and therefore allow investigation of the interrelation between structure and function in response to mechanical loading. Collagen fibres are key ingredients in the structure of arteries. In the media (the middle layer of the artery wall) they are arranged in two helically distributed families with a small pitch and very little dispersion in their orientation (i.e. they are aligned quite close to the circumferential direction). By contrast, in the adventitial and intimal layers, the orientation of the collagen fibres is dispersed, as shown by polarized light microscopy of stained arterial tissue. As a result, continuum models that do not account for the dispersion are not able to capture accurately the stress–strain response of these layers. The purpose of this paper, therefore, is to develop a structural continuum framework that is able to represent the dispersion of the collagen fibre orientation. This then allows the development of a new hyperelastic free-energy function that is particularly suited for representing the anisotropic elastic properties of adventitial and intimal layers of arterial walls, and is a generalization of the fibre-reinforced structural model introduced by Holzapfel & Gasser (Holzapfel & Gasser 2001 Comput. Meth. Appl. Mech. Eng. 190, 4379–4403) and Holzapfel et al. (Holzapfel et al. 2000 J. Elast. 61, 1–48). The model incorporates an additional scalar structure parameter that characterizes the dispersed collagen orientation. An efficient finite element implementation of the model is then presented and numerical examples show that the dispersion of the orientation of collagen fibres in the adventitia of human iliac arteries has a significant effect on their mechanical response.

A pseudo-elastic model for the Mullins effect in filled rubber

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.

Constitutive modelling of passive myocardium: a structurally based framework for material characterization

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.

Constitutive modelling of arteries

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 thick-walled 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.

Nonlinear elasticity : theory and applications

We elaborate on a blended continuum/atomistic theoretical picture of the nonlinear elastic properties of nanostructured materials, looking at diverse aspects such as dispersions of inhomogeneities within a matrix, random or graded nanograined materials, two-dimensional atomic sheets. In particular, we discuss the possible onset of length-scale effects and we establish the limits and merits of continuum versus atomistics. While most situations here discussed correspond to model systems, the main conclusions have a paradigmatic relevance and indeed apply to most nanomaterials of current interest. This article was invited by S Washburn.Preface 1. Elements of the theory of finite elasticity R. W. Ogden 2. Hyperelastic Bell materials: retrospection, experiment, theory M. F. Beatty 3. Universal results in finite elasticity G. Saccomandi 4. Equilibrium solutions for compressible nonlinearly elastic materials C. O. Horgan 5. Exact integrals and solutions for finite deformations of the incompressible Varga elastic materials J. M. Hill 6. Shear Ph. Boulanger and M. Hayes 7. Elastic membranes D. M. Haughton 8. Elements of the theory of elastic surfaces D. J. Steigmann 9. Singularity theory and nonlinear bifurcation analysis Y.-C. Chen 10. Perturbation methods and nonlinear stability analysis Y. B. Fu 11. Nonlinear dispersive waves in a circular rod composed of a Mooney-Rivlin material H.-H. Dai 12. Strain-energy functions with multiple local minima: modeling phase transformations using finite thermo-elasticity R. Abeyaratne, K. Bhattacharya and J. K. Knowles 13. Pseudo-elasticity and stress softening R. W. Ogden.

Comparison of a Multi-Layer Structural Model for Arterial Walls With a Fung-Type Model, and Issues of Material Stability

The goals of this paper are (i) to re-examine the constitutive law for the description of the (passive) highly nonlinear and anisotropic response of healthy elastic arteries introduced recently by the authors, (ii) to show how the mechanical response of a carotid artery under inflation and extension predicted by the structural model compares with that for a three-dimensional form of Fung-type strain-energy function, (iii) to provide a new set of material parameters that can be used in a finite element program, and (iv) to show that the model has certain mathematical features that are important from the point of view of material and numerical stability.

ELASTIC SURFACE-SUBSTRATE INTERACTIONS

A theory for three–dimensional finite deformations of elastic solids with conforming elastic films attached to their bounding surfaces is described. The Gurtin–Murdoch theory incorporating elastic resistance of the film to strain is generalized to account for the effects of intrinsic flexural resistance. This modification yields a model that can be used to describe equilibrium deformations in the presence of compressive–surface stress fields. An associated variational theory is given and material symmetry considerations are discussed. The theory is illustrated by examples.

Mechanics of biological tissue

Tissue Growth and Remodelling.- Towards a Theory of Vascular Growth and Remodeling.- Complementary Roles of Theoretical Modeling and Computer-controlled Experimentation in Vascular Growth and Remodeling.- On the Modeling of Growth and Adaptation.- Growth in Soft Biological Tissue and Residual Stress Development.- Characterization and Modeling of Growth and Remodeling in Tendon and Soft Tissue Constructs.- Modeling and Simulation of Remodeling in Soft Biological Tissues.- Anisotropic Remodelling of Biological Tissues.- A Mechanobiological Formulation of Bone Healing.- Continuum Models of Growth with Emphasis on Articular Cartilage.- Micromechanics, Cells and Matrix.- Tensile Properties and Local Stiffness of Cells.- Microfluid-dynamics in Three-dimensional Engineered Cell Systems.- Nonlinear Constitutive Models for Cochlear Outer Hair Cells.- Prediction of Changes in Cell-substrate Contact under Cyclic Substrate Deformation Using Cohesive Zone Modelling.- Micromechanics and Macromechanics of the Tensile Deformation of Nacre.- Arteries in Health and Disease.- Mechanical Properties of Atherosclerotic Tissues.- Towards a Computational Methodology for Optimizing Angioplasty Treatments with Stenting.- Computational Modeling of Stented Arteries: Considerations for Evolving Stent Designs.- Simulation of In-stent Restenosis for the Design of Cardiovascular Stents.- Material Property Alterations with Early Atheroma in an Animal Model.- Microscopic Analysis of Residual Stress and Strain in the Aortic Media Considering Anisotropy of Smooth Muscle Layer.- Parameter Identification in Arteries Using Constraints.- Collagen Organization and Biomechanics of the Arteries and Aneurysms of the Human Brain.- Image-based Simulation of Blood Flow and Arterial Wall Interaction for Cerebral Aneurysms.- Biological Tissues.- A Framework for Soft Tissue and Musculo-skeletal Modelling: Clinical Uses and Future Challenges.- Invariant Formulation for Dispersed Transverse Isotropy in Tissues of the Aortic Outflow Tract.- Mathematical Modelling of Cardiac Mechanoenergetics.- Creep and Relaxation in Ligament: Theory, Methods and Experiment.- Viscoelastic Constitutive Law Based on the Time Scale of the Mechanical Phenomena.- A Coupled FE Analysis of the Intervertebral Disc Based on a Multiphasic TPM Formulation.- Is the Free Energy of Hydrogel the Sum of Elastic Energy and Ionic Energy?.- In Vivo Experiments to Characterize the Mechanical Behavior of the Human Uterine Cervix.- Viscoelastic Response of Vocal Fold Tissues and Scaffolds at High Frequencies.- An Alternative Fabric-based Yield and Failure Criterion for Trabecular Bone.- Image-based Analysis.- Functional Micro-imaging at the Interface of Bone Mechanics and Biology.- Strain Measurement Using Deformable Image Registration.- Image-based Hierarchical Analysis and Design of Tissue Engineering Scaffolds.

Plane deformations of elastic solids with intrinsic boundary elasticity

In this paper, a nonlinear theory of elastic boundary coating (or reinforcement) of an elastic solid is developed for plane strain deformations. The coating consists of a material curve endowed with intrinsic elastic properties associated with extensibility and bending stiffness bonded to part, or all, of the bounding curve of the elastic body. The equations describing the equilibrium of the coated body when subject to finite deformation are derived using a variational method. The incremental equations describing a small departure from an equilibrium configuration are then derived and used to investigate the stability of a deformed configuration and the possibility of bifurcation. The theory is applied to the analysis of the equilibrium of a finitely deformed half-plane consisting of compressible elastic material coated along its edge. The influence of the coating on the bifurcation behaviour of the half–plane is assessed against known results for an uncoated half–plane. Numerical results are used to illustrate the influence of certain material parameters on bifurcation.