Brief Review of Continuum Mechanics Theories

Corina Drapaca, Siv Sivaloganathan

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

The classical theory of continuum mechanics has its roots in the nineteenth century, in the foundational work of Augustin-Louis Cauchy, although its rigorous, modern development has been built upon Noll’s axiomatic framework which allows for a unified study of deformable materials. In the mathematical description of a material’s response to mechanical loading there are two important basic assumptions which form the foundation of continuum mechanics: (1) the mechanical stress at a given material point at time t is determined by the past history of the deformation of a neighborhood of the considered point (the principle of determinism and local action), and (2) the response of a material is the same for all observers (the principle of material objectivity). These principles are however too general to properly characterize the nature of specific materials and further simplifications of the relationship between mechanical stress and deformation are necessary. Such simplifications arise, for instance, from assumptions of infinitesimal deformations or for finite deformations that a material is simple, homogeneous, non-aging, has preferred directions of deformation, and experiences internal constraints, (like incompressibility, inextensibility, rigidity). In this chapter we provide a brief review of these concepts, as well as specific constitutive laws that have been used in brain research. In addition, we will present some modern theories that generalize classical continuum mechanics and may prove very useful in future studies of brain biomechanics.

Original languageEnglish (US)
Title of host publicationFields Institute Monographs
PublisherSpringer New York LLC
Pages5-37
Number of pages33
DOIs
StatePublished - Jan 1 2019

Publication series

NameFields Institute Monographs
Volume37
ISSN (Print)1069-5273
ISSN (Electronic)2194-3079

Fingerprint

Continuum Mechanics
Mechanical Stress
Simplification
Biomechanics
Determinism
Incompressibility
Finite Deformation
Constitutive Law
Classical Mechanics
Review
Rigidity
Cauchy
Observer
Roots
Internal
Generalise
Necessary

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Drapaca, C., & Sivaloganathan, S. (2019). Brief Review of Continuum Mechanics Theories. In Fields Institute Monographs (pp. 5-37). (Fields Institute Monographs; Vol. 37). Springer New York LLC. https://doi.org/10.1007/978-1-4939-9810-4_2
Drapaca, Corina ; Sivaloganathan, Siv. / Brief Review of Continuum Mechanics Theories. Fields Institute Monographs. Springer New York LLC, 2019. pp. 5-37 (Fields Institute Monographs).
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Drapaca, C & Sivaloganathan, S 2019, Brief Review of Continuum Mechanics Theories. in Fields Institute Monographs. Fields Institute Monographs, vol. 37, Springer New York LLC, pp. 5-37. https://doi.org/10.1007/978-1-4939-9810-4_2

Brief Review of Continuum Mechanics Theories. / Drapaca, Corina; Sivaloganathan, Siv.

Fields Institute Monographs. Springer New York LLC, 2019. p. 5-37 (Fields Institute Monographs; Vol. 37).

Research output: Chapter in Book/Report/Conference proceedingChapter

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Drapaca C, Sivaloganathan S. Brief Review of Continuum Mechanics Theories. In Fields Institute Monographs. Springer New York LLC. 2019. p. 5-37. (Fields Institute Monographs). https://doi.org/10.1007/978-1-4939-9810-4_2