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.

https://royalsocietypublishing.org/doi/pdf/10.1098/rsif.2005.0073

Hyperelastic modelling of arterial layers with distributed collagen fibre orientations

Fibroblasts can condense a hydrated collagen lattice to a tissue-like structure 1/28th the area of the starting gel in 24 hr. The rate of the process can be regulated by varying the protein content of the lattice, the cell number, or the con- centration of an inhibitor such as Colcemid. Fibroblasts of high population doubling level propagated in vitro, which have left the cell cycle, can carry out the contraction at least as efficiently as cycling cells. The potential uses of the system as an immu- nologically tolerated "tissue" for wound hea ing and as a model for studying fibroblast function are discussed.

Production of a tissue-like structure by contraction of collagen lattices by human fibroblasts of different proliferative

We present a theory of the kinetics of the swelling of a gel. The characteristic time of swelling is proportional to the square of a linear dimension of the gel and is also proportional to the diffusion coefficient of the gel network, which is defined as D=E/f where E is the longitudinal bulk modulus of the network, and f is the coefficient of friction between the network and the gel fluid. This constitutes an essential difference between the present theory and the previous theory which is based on the assumption that the swelling time is determined by the diffusion coefficient of the fluid molecules. Experimental data are shown for spheres of 5% polyacrylamide gels and are analyzed using the present theory. The value of the diffusion coefficient obtained from the macroscopic swelling experiments shows excellent agreement with that obtained microscopically using laser light scattering spectroscopy.

Kinetics of swelling of gels

Linear elasticity is one of the more successful theories of mathematical physics Its pragmatic success in describing the small deformations of many materials is uncontested The origins of the three-dimensional theory go back to the beginning of the 19th century and the derivation of the basic equations by Cauchy, Navier, and Poisson The theoretical development of the subject continued at a brisk pace until the early 20th century with the work of Beltrami, Betti, Boussinesq, Kelvin, Kirchhoff, Lame, Saint-Venant, Somigliana, Stokes, and others These authors established the basic theorems of the theory, namely compatibility, reciprocity, and uniqueness, and deduced important general solutions of the underlying field equations In the 20th century the emphasis shifted to the solution of boundary-value problems, and the theory itself remained relatively dormant until the middle of the century when new results appeared concerning, among other things, Saint-Venant’s principle, stress functions, variational principles, and uniqueness

The Linear Theory of Elasticity

https://hal.archives-ouvertes.fr/hal-00773015/document

A review on the Mullins effect

1. Matrix Algebra 2. Linear Anisotropic Elastic Materials 3. Antiplane Deformations 4. The Lekhnitskii Formalism 5. The Stroh Formalism 6. The Structures and Identities of the Elasticity Matrices 7. Transformation of the Elasticity Matrices and Dual Coordinate Systems 8. Green's Functions for Infinite Space, Half-space, and Composite Space 9. Particular Solutions, Stress Singularities, and Stress Decay 10. Anisotropic Matrials with an Elliptic Boundary 11. Anisotropic Media with a Crack or a Rigid Line Inclusion 12. Steady State Motion and Surface Waves 13. Degenerate and Near Degenerate Materials 14. Generalization of the Stroh Formulism 15. Three-Dimensionsal Deformations

Anisotropic Elasticity: Theory and Applications

This chapter provides an overview of the recent developments concerning Saint-Venant's principle The task of determining, within the framework of the linear theory of elasticity, the stresses and displacements in an elastic cylinder in equilibrium, under the action of loads that arise solely from tractions applied to its plane ends has come to be called Saint- Venant's problem Saint-Venant's construction does not permit the arbitrary preassignment of the point-by-point variation of the end tractions giving rise to these forces and moments; indeed, this variation is essentially determined as a consequence of the special assumptions made in connection with his so-called semi-inverse procedure The early work of Saint-Venant and Boussinesq furnished the seeds from which grew a large number of more general assertions, most referring to elastic solids of arbitrary shape and many being rather imprecise, concerning the effect on stresses within the body of replacing the tractions acting over a portion of its surface by statically equivalent ones Such propositions usually went by the name of Saint-Venunt's principle, despite the fact that Saint-Venant's original conjecture was intended to apply only to cylinders This chapter discusses in detail about flow in a cylinder, a representation for the exact solution, and energy decay for other linear elliptic second-order problem Linear elastostatic problems are also stated in the chapter

Recent Developments Concerning Saint-Venant's Principle

Saint-Venant’s principle plays a central role in the theory and applications of elasticity. A comprehensive review of contemporary research on this topic was given by Horgan and Knowles (1983), and updated by Horgan (1989a). Since these articles were written, several developments in the subject area have taken place. In his second update, we review some of this progress, including a discussion of analogous issues arising in entry flow problems for viscous incompressible fluids.

Recent Developments Concerning Saint-Venant’s Principle: An Update

The purpose of this research is to investigate the effects of material inhomogeneity on the response of linearly elastic isotropic hollow circular cylinders or disks under uniform internal or external pressure. The work is motivated by the recent research activity on functionally graded materials (FGMs), i.e., materials with spatially varying properties tailored to satisfy particular engineering applications. The analog of the classic Lame problem for a pressurized homogeneous isotropic hollow circular cylinder or disk is considered. The special case of a body with Young"s modulus depending on the radial coordinate only, and with constant Poisson"s ratio, is examined. It is shown that the stress response of the inhomogeneous cylinder (or disk) is significantly different from that of the homogeneous body. For example, the maximum hoop stress does not, in general, occur on the inner surface in contrast with the situation for the homogeneous material. The results are illustrated using a specific radially inhomogeneous material model for which explicit exact solutions are obtained.

Cornelius O. Horgan

Papers

Anisotropic Elasticity: Theory and Applications

Recent Developments Concerning Saint-Venant's Principle

Recent Developments Concerning Saint-Venant’s Principle: An Update

The Pressurized Hollow Cylinder or Disk Problem for Functionally Graded Isotropic Linearly Elastic Materials

An introduction to the theory of elasticity