Nonlinear Continuum Mechanics and Modeling the Elasticity of Soft Biological Tissues with a Focus on Artery Walls
This chapter provides a detailed summary of the background from the nonlinear theory of continuum mechanics that is required in the modeling of the elastic properties of soft biological tissues. In particular, it highlights methods for including the fibro
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Abstract This chapter provides a detailed summary of the background from the nonlinear theory of continuum mechanics that is required in the modeling of the elastic properties of soft biological tissues. In particular, it highlights methods for including the fibrous structure of such tissues within the constitutive description of the material properties at the macroscopic level. Of particular relevance in this connection are the so-called preferred directions associated with fibers and the structure tensors and associated deformation invariants that are needed for taking these fibers and their dispersed directions into consideration. These are incorporated into the material models and the effect of fiber structure on the material response is then illustrated with several basic examples. Generalizations of structure tensors are also used for including within the theory the important residual stresses that are evident in unloaded tissues such as arteries and the myocardium, and the influence of residual stresses on the material response is illustrated by considering the extension and inflation of a thick-walled circular cylindrical tube.
1 Introduction This chapter is based on lectures given at the Summer School on ‘Biomechanics: Trends in Modeling and Simulation’ in Graz, Austria, in September 2014, but includes additional material that was not presented in the lectures. Effective modeling of the mechanics of soft biological tissues, such as the layered walls of arteries, the myocardium and skin, requires a sophisticated application of the nonlinear theory of continuum mechanics. Within the structure of these tissues a key component is the protein collagen, which endows the material with anisotropic properties because of its significant stiffness relative to the surrounding material within which it is embedded. We refer to the surrounding (less stiff) material as the matrix, which, depending on the tissue under consideration, includes elastin fibers, proteoglycans, and smooth R.W. Ogden (B) University of Glasgow, Glasgow, UK e-mail: [email protected] © Springer International Publishing Switzerland 2017 G.A. Holzapfel and R.W. Ogden (eds.), Biomechanics: Trends in Modeling and Simulation, Studies in Mechanobiology, Tissue Engineering and Biomaterials 20, DOI 10.1007/978-3-319-41475-1_3
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muscle cells, for example. Tissues have a naturally fibrous structure, which has a strong influence on their mechanical response. Thus, from the mechanical perspective, it is important to be able to understand the influence of the fiber structure on the overall mechanical response of the composite materials of which the fibers are constituents, and nonlinear continuum mechanics provides the vehicle for analyzing this response. Consider, for example, a length of artery, which may be idealized as a circular cylindrical tube, as illustrated in Fig. 1. Typically, in the simplest terms, an artery contains two families of collagen fibers that are helically arranged and symmetrically disposed
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