## Contents |

## me338A - continuum mechanics

**continuum mechanics**

ellen kuhl,
serdar goktepe,
gilwoo choi spring 2008 |

## goals

although the basic concepts of continuum mechanics have been established more than five decades ago, the 21 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 finite deformation kinematics. within the framework of large deformations, 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. in particular, we will focus on isotropic and anisotropic hyperelasticity and on viscoelasticity and elastodamage. finally, we will briefly address variational principles that characterize the governing equations.

## grading

- 50 % homework - 3 homework assignments, 16.7% each

- 30 % midterm - open book, open notes

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

## syllabus

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

tue | apr | 01 | introduction - why potatos? | n01 | |

thu | apr | 03 | vectors & tensors - vector algebra | n02 | |

tue | apr | 08 | vectors & tensors - tensor algebra | n03 | h01 |

thu | apr | 10 | vectors & tensors - tensor analysis | n04 | |

tue | apr | 15 | kinematics - configurations, deformation | n05 | |

thu | apr | 17 | kinematics - temporal derivatives | n06 | h01 solution |

tue | apr | 22 | kinematics - spatial derivatives | n07 | h02 |

thu | apr | 24 | kinematics - strain measures | n08 | |

tue | apr | 29 | kinematics - examples, midterm evaluation | n09 | evaluation |

thu | may | 01 | balance equations - concept of stress | n10 | h02 due |

tue | may | 06 | balance equations - mass, momentum | n11 | h03 |

thu | may | 08 | balance equations - energy, entropy, master balance law | n12 | |

tue | may | 13 | constitutive equations - 2nd law, objectivity, symmetry | n13 | |

thu | may | 15 | constitutive equations - isotropic elasticity | n14 | h03 solution |

tue | may | 20 | midterm | project | |

thu | may | 22 | constitutive equations - iterative computation of stress-stretch response | n15 | |

tue | may | 27 | constitutive equations - nearly and strictly incompressible elasticity | n16 | |

thu | may | 29 | constitutive equations - transversely isotropic elasticity | n17 | |

tue | jun | 03 | journal club - final project discussion |

## final project

the final project is a paper review of your choice, you can choose between rubber mechanics, biomechanics, geomechanics and growth mechanics

(1) boyce mc, arruda em: constitutive models of rubber elasticity: a review, rubber chemistry and technology 73, 504-533, 2000

(2) holzapfel ga: biomechanics of soft tissues, in: handbook of material behavior, academic press, 2000

(3) jeremic b, runesson k, sture s: finite deformation analysis of geomaterials, international journal for numerical and analytical methods in geomechanics 25, 809-840, 2001

(4) rodriguez ek, hoger a, mc culloch ad: stress-dependent finite growth in soft elastic tissues, journal of biomechanics 27, 455-467, 1994

## 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