Nonlinear Continuum Mechanics

Vertiefungsfach Master Materials Science and Engineering (MSE)
(3 SWS, 5 ECTS, winter term, Modul-Nr. MW2368, Prof. M.W. Gee)

The understanding of material behavior under environmental influences is the key for the development or optimization of countless technical systems. At the same time, materials show a high variety of different behavior patterns, which makes sophisticated modeling of materials very complex. Nonlinear Continuum Mechanics is a mathematical theory that has proven its worth in the description of material behavior to environmental influences in case of real world engineering problems.

This course provides an introduction in the mathematical theory of continuum mechanics. Starting from the description of the motion of a continuum, measures for stretch and strain as well as the corresponding rates are derived consistently. The concept of stress is introduced and applied in the derivation of balance laws for mass, momentum, energy and entropy. The relation between stress and strain is established by the derivation of constitutive laws for hyperelastic, viscoelastic and plastic material behavior, so that the presented mathematical theory is self-contained and serves as a basis for further specification or even direct application in numerical simulations.

Contents

  • Introduction of the terms continuum and continuum mechanics 
  • Brief review of tensor calculus
  • Kinematics (deformation map, deformation gradient, stretch and strain, strain rate)
  • Stress (Piola-Kirchhoff stress, Cauchy stress, stress states)
  • Balance laws (mass, momentum, energy, entropy)
  • Constitutive laws (hyperelasticity, viscoelasticity, plasticity)

Language

  • English

Prerequisites 

  • Knowledge of the axioms of Newtonian mechanics and thermodynamics, tensor calculus in cartesian bases, basic knowledge of at least linearized continuum mechanics, basic knowledge of linear ordinary and partial differential equations as well as linear algebra

Lecture

Prof. Gee 
Mi, 09:00-10:30, 8102.03.108
Office hours Prof. Gee: by appointment

Exercise

Mr. Rinderer
Mi, 10:45-11:30, 8102.03.108
Office hours Mr. Rinderer: by appointment