## Contents |

## spring 17 - me333c - mechanics/continuum mechanics

**continuum mechanics 17**

ellen kuhl spring 2017 |

## objectives

although the basic concepts of continuum mechanics have been established more than five decades ago, the 21st century faces many new and exciting potential applications of continuum mechanics that go way beyond the standard classical theory. when applying continuum mechanics to these challenging new phenomena, it is critical to understand the main three ingredients of continuum mechanics: the kinematic equations, the balance equations and the constitutive equations. after a brief summary of the relevant equations in tensor algebra and analysis, we will introduce the basic concepts of finite strain kinematics. we will then discuss the concept of stress, followed by the balance equations for mass, momentum, moment of momentum, energy and entropy. while all these equations are general and valid for any kind of material, the last set of equations, the constitutive equations, specifies particular subclasses of materials. we will discuss constitutive equations for hyperelastic materials, both isotropic and anisotropic, and for inelastic materials with internal variables. last, we will address these considerations in the context of variational principles.

## continuum mechanics of the heart

special thanks to wolf and john for the hands on wet lab related to homework 02

## grading

- 30 % homework - 3 homework assignments, 10% each

- 50 % final - closed book, closed notes, one single page cheat sheet

- 20 % final project - written evaluation of a manuscript and its discussion in class

## syllabus

... this is the book we will use in class...

holzapfel ga: nonlinear solid mechanics, a continuum approach for engineering, john wiley & sons, 2000

day | date | topic | chapters | pages | notes | |
---|---|---|---|---|---|---|

tue | apr | 04 | why continuum mechanics? | syllabus | s01 | |

thu | apr | 06 | introduction to vectors and tensors - tensor algebra | 1.1-1.5 | 1-32 | s02 |

tue | apr | 11 | introduction to vectors and tensors - tensor analysis | 1.6-1.9 | 32-55 | s03 |

thu | apr | 13 | kinematics - motion | 2.1-2.3 | 55-69 | s04 |

tue | apr | 18 | kinematics - deformation gradient | 2.4,2.6 | 70-94 | s05 |

thu | apr | 20 | kinematics - strain | 2.5,2.7,2.8 | 76-108 | s05 |

tue | apr | 25 | concept of stress | 3.1-3.4 | 109-131 | s06 |

thu | apr | 27 | balance principles - mass and momentum | 4.1-4.3 | 131-151 | s07 |

tue | may | 02 | balance principles - energy and entropy | 4.4-4.7 | 152-179 | s08 |

thu | may | 04 | aspects of objectivity | 5.1-5.4 | 179-205 | s09 |

tue | may | 09 | hyperelasticity - general remarks | 6.1-6.6 | 205-264 | s11 |

thu | may | 11 | hyperelasticity - isotropy | 6.2 | 212-221 | s12 |

tue | may | 16 | hyperelasticity - incompressibility | 6.3,6.4 | 222-234 | s13 |

thu | may | 18 | hyperelasticity - examples | 6.5 | 235-251 | s14 |

tue | may | 23 | hyperelasticity - transverse isotropy | 6.7 | 265-277 | s15 |

thu | may | 25 | final prep | 1.1-6.11 | 1-304 | s16 |

tue | may | 30 | inelasticity - viscoelasticity | 6.9, 6.11 | 282-294 | s17 |

thu | jun | 01 | final | |||

tue | jun | 06 | inelasticity - damage | 6.9, 6.11 | 278-304 | s19 |

fri | jun | 09 | written projects due |

## final project

the final project of later year's class was the review of a recently published manuscript that discusses constitutive models for ultrasoft materials. if you would like to read more about it, here is some additional reading material.

mihai la, chin l, janmey pa, goriely a. a comparison of hyperelastic constitutive models applicable to brain and fat tissues. journal of the royal society interface, 2015;12:20150486.

## suggested reading

... here are some other cool books for additional reading...

murnaghan fd: finite deformation of an elastic solid, john wiley & sons, 1951

eringen ac: nonlinear theory of continuous media, mc graw-hill, 1962

truesdell c, noll, w: the non-linear field theories of mechanics, springer, 1965

eringen ac: mechanics of continua, john wiley & sons, 1967

malvern le: introduction to the mechanics of a continuous medium, prentice hall, 1969

oden jt: finite elements of nonlinear continua, dover reprint, 1972

chadwick p: continuum mechanics - concise theory and problems, dover reprint, 1976

ogden, rw: non-linear elastic deformations, dover reprint, 1984

maugin ga: the thermodynamics of plasticity and fracture, cambridge university press, 1992

spencer ajm: continuum mechanics, dover reprint, 1992

robers aj: one-dimensional introduction to continuum mechanics, world scientific, 1994

bonet j, wood rd: nonlinear continuum mechanics for fe analysis, cambridge university press, 1997

silhavy m: the mechanics and thermodynamics of continuous media, springer, 1997

haupt p: continuum mechanics and theory of materials, springer, 2000

podio-guidugli p: a primer in elasticity, kluwer academic press, 2000

liu is: continuum mechanics, springer, 2002

reddy jn: an introduction to continuum mechanics, cambridge university press, 2007