### 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 language | English (US) |
---|---|

Title of host publication | Fields Institute Monographs |

Publisher | Springer New York LLC |

Pages | 5-37 |

Number of pages | 33 |

DOIs | |

State | Published - Jan 1 2019 |

### Publication series

Name | Fields Institute Monographs |
---|---|

Volume | 37 |

ISSN (Print) | 1069-5273 |

ISSN (Electronic) | 2194-3079 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*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

}

*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.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - Brief Review of Continuum Mechanics Theories

AU - Drapaca, Corina

AU - Sivaloganathan, Siv

PY - 2019/1/1

Y1 - 2019/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85072803136&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85072803136&partnerID=8YFLogxK

U2 - 10.1007/978-1-4939-9810-4_2

DO - 10.1007/978-1-4939-9810-4_2

M3 - Chapter

AN - SCOPUS:85072803136

T3 - Fields Institute Monographs

SP - 5

EP - 37

BT - Fields Institute Monographs

PB - Springer New York LLC

ER -