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Erschienen in:
Introduction to Modelling in Bioengineering Kapitel lesen Erstes Kapitel lesen

2017 | OriginalPaper | Buchkapitel

8. Solid Mechanics

verfasst von : Socrates Dokos

Erschienen in: Modelling Organs, Tissues, Cells and Devices

Verlag: Springer Berlin Heidelberg

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Abstract

This chapter provides an overview of solid mechanics with applications in biomechanics. It begins wih a coverage of tensors, including the tensor transformation law and tensor invariants, before proceeding to the basic mechanical concepts of stress and strain. Stress and strain are then related via material constitutive laws, the most basic being that of linear elasticity. The concepts of hydrostatic stresses and strains are also introduced, along with the notion of von Mises stress as a representative scalar stress indicator. Other material laws relevant to biological tissues are presented, including linear viscoelasticity and hyperelasticity, as well as anisotropic hyperelastic material laws. Detailed examples of models solved in COMSOL include a strap tension testing device for a respirator mask, as well simulations of simple shear experiments in myocardial tissue. The chapter ends with a set of theoretical and computational COMSOL problems, with fully-worked answers provided in the solution section of the text.

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Vorheriges Kapitel Models of Diffusion and Heat Transfer
Nächstes Kapitel Fluid Mechanics
Anhänge
Nur mit Berechtigung zugänglich
Fußnoten
1
The descriptions here and in the following sections pertain to 3D space by default, however these descriptions are readily generalizable to lower dimensions. For example in 2D, vectors have two components, second-order tensors have four components, and third-order tensors have eight components.
 
2
This notation was introduced by Albert Einstein (1879–1955) in his 1916 paper on the General Theory of Relativity [5].
 
3
Augustin-Louis Cauchy (1789–1857), prolific French mathematician who contributed numerous works in mathematical analysis and the theory of elasticity.
 
4
Co-named after George Green (1793–1841), British mathematical physicist. We can also define the left Cauchy–Green deformation tensor as \(\mathbf {B} = \mathbf {F}\mathbf {F}^{\mathrm {T}}\).
 
5
Robert Hooke, 1635–1703, English scientist, mathematician and architect who made numerous contributions to the fields of physics, biology, astronomy and mechanics.
 
6
Named after the French mathematician Gabriel Léon Jean Baptiste Lamé (1795–1870), who made important contributions to the mathematical theory of elasticity.
 
7
Thomas Young (1773–1829), English scientist who made contributions to many fields, including solid mechanics.
 
8
Siméon Denis Poisson (1781–1840), French mathematician and physicist.
 
9
Named after Richard Edler von Mises (1883–1953), German scientist and mathematician who made important contributions to the fields of solid mechanics, fluid mechanics, statistics and probability theory.
 
10
After James Clerk Maxwell who, in addition to formulating his famous equations of electromagnetism (Eqs. 6.​1–6.​4), also made important contributions to structural mechanics.
 
11
Named after American rheologists Melvin Mooney (1893–1968) and Ronald Samuel Rivlin (1915–2005).
 
12
It should also be noted that in some descriptions, the invariants of Eq. 8.41 refer to the standard first and second invariants of the Cauchy–Green deformation tensor.
 
13
So named because this modified deformation preserves local volume. Defining https://static-content.springer.com/image/chp%3A10.1007%2F978-3-642-54801-7_8/309301_1_En_8_IEq279_HTML.gif , where \(J = \det \mathbf {F}\), then \(\det \bar{\mathbf {F}} =1\), and https://static-content.springer.com/image/chp%3A10.1007%2F978-3-642-54801-7_8/309301_1_En_8_IEq282_HTML.gif .
 
14
Many of these expressions will appear orange-coloured in COMSOL when first entered, indicating that some terms have not yet been defined. This is because we have not yet added the hyperelastic material description, which we will do in the next step.
 
15
Many of COMSOL’s solid mechanics in-built variables include the term ‘el’, which is short for elastic. COMSOL separates deformations into elastic and inelastic components, with inelastic components arising from non-elastic effects such as thermal expansion and plasticity. For the models presented in this chapter, there are no inelastic effects, hence COMSOL terms such as the elastic isochoric right Cauchy Green tensor are equivalent to the standard definitions presented.
 
Literatur
1.
Atkin RJ, Fox N (2005) An introduction to the theory of elasticity, Dover edn. Dover, MineolaMATH
2.
Chadwick P (1999) Continuum mechanics: concise theory and problems, Dover edn. Dover, Mineola
3.
Dokos S, Smaill BH, Young AA, LeGrice IJ (2002) Shear properties of passive ventricular myocardium. Am J Physiol Heart Circ Physiol 283:H2650–H2659CrossRef
4.
Eglit ME, Golublatnikov AN, Kamenjarzh JA, Karlikov VP, Kulikovsky AG, Petrov AG, Shikina IS, Sveshnikova EI (1996) In: Eglit ME, Hodges DH (eds) Continuum mechanics via problems and exercises. World scientific series on nonlinear science. Series A, vol 19. World Scientific, Singapore
5.
Einstein A (1916) The foundation of the general theory of relativity (Tranlated by Engel A). In: Kox AJ, Klein MJ, Schulmann R (eds) The collected papers of Albert Einstein. The Berlin years: Writings, 1914–1917, vol 6, Princeton University press, Princeton, pp 146–200 (1997)
6.
Fung YC (1984) Biomechanics: circulation, 2nd edn. Springer, New York
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Fung YC (1994) A first course in continuum mechanics, 3rd edn. Prentice-Hall, Upper Saddle River
8.
Fung YC, Tong P (2001) Classical and computational solid mechanics. World Scientific, SingaporeCrossRefMATH
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Holzapfel GA (2000) Nonlinear solid mechanics: a continuum approach for engineering. Wiley, ChichesterMATH
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Holzapfel G, Gasser T, Ogden R (2000) A new constitutive framework for arterial wall mechanics and a comparative study for material models. J Elast 61:1–48MathSciNetCrossRefMATH
11.
Schmid H, Wang YK, Ashton J, Ehret AE, Krittian SBS, Nash MP, Hunter PJ (2009) Myocardial material parameter estimation: a comparison of invariant based orthotropic constitutive equations. Comput Meth Biomech Biomed Eng 12:283–295CrossRef
12.
Spencer AJM (1980) Continuum mechanics. Longman, Burnt MillMATH
Metadaten
Titel
Solid Mechanics
verfasst von
Socrates Dokos
Copyright-Jahr
2017
Verlag
Springer Berlin Heidelberg
Buch
Modelling Organs, Tissues, Cells and Devices

Print ISBN: 978-3-642-54800-0

Electronic ISBN: 978-3-642-54801-7

Copyright-Jahr: 2017

DOI
https://doi.org/10.1007/978-3-642-54801-7_8

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