## Contents |

## me338A - continuum mechanics

ellen kuhl,
serdar goktepe, winter 2010 |

## goals

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 important to understand the main three ingredients of continuum mechanics: the kinematic equations, the balance equations and the constitutive equations. after a brief repetition of the relevant equations in tensor algebra and analysis, this class will introduce the basic concepts of small strain kinematics. we will then discuss 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. towards the end of the quarter, we will illustrate how the three sets of equations will change in nonlinear continuum mechanics. throughout this class, we will discuss the mechanics of the heart with its strains, fiber stretches, stresses, forces, and constitutive equations, to illustrate how continuum mechanics can be applied to relevant clinical problems.

## 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

- 40 % midterm - closed book, closed notes, one single page cheat sheet

- 30 % final project - written evaluation of a manuscript and its discussion in class

## syllabus

day | date | topic | notes | slides | hw | |
---|---|---|---|---|---|---|

tue | jan | 05 | tensor calculus I - vector algebra | n01 | s01 | |

thu | jan | 07 | tensor calculus II - tensor algebra | n02 | s02 | |

tue | jan | 12 | tensor calculus III - tensor analysis | n03 | s03 | |

thu | jan | 14 | tensor calculus IV - tensor analysis | n04 | s04 | h01 |

tue | jan | 19 | kinematics I - motion | n05 | ||

thu | jan | 21 | kinematics II - strain | n06 | s06 | |

tue | jan | 26 | balance equations I - contact fluxes | n07 | ||

thu | jan | 28 | balance equations II - concept of stress | n08 | s08 | h02 |

tue | feb | 02 | balance equations III - mass, momentum | n09 | s09 | |

thu | feb | 04 | balance equations IV - angular momentum, energy | n10 | ||

tue | feb | 09 | balance equations V - entropy, master balance law | n11 | s11 | |

thu | feb | 11 | constitutive equations I - linear equations | n12 | s12 | h03 |

tue | feb | 16 | constitutive equations II - hyperelasticity | n13 | s13 | |

thu | feb | 18 | constitutive equations III - isotropic elasticity | n14 | s14 | project |

tue | feb | 23 | midterm prep | n15 | ||

thu | feb | 25 | midterm | |||

tue | mar | 02 | intro to nonlinear continuum mechanics I - finite strain kinematics | n15 | ||

thu | mar | 04 | intro to nonlinear continuum mechanics II - stress measures | n16 | ||

tue | mar | 09 | intro to nonlinear continuum mechanics III - constitutive equations | n17 | ||

thu | mar | 11 | journal club - final project discussion | n18 |

## final project

the final project of this class is the review of a recently published manuscript that introduces a new constitutive model for passive cardiac muscle tissue. if you would like to read more about it and learn how it can be applied to a realistic heart geometry, here is some additional reading material.

holzapfel ga, ogden rw: constitutive modelling of passive myocardium. a structurally-based framework for material characterization, philosophical transactions of the royal society a, 2009;367:3445-3475.

## suggested reading

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

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

... and 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