Continuum Mechanics

In mechanics of continuous media, the theories of solid and fluid mechanics are presented in a unified framework. Basic notions of tensor algebra and analysis are covered in the first part of the course. Subsequently, a general description of deformations is introduced, from which the distinct measures of deformation and rates of deformation used in geometrically-nonlinear theories are obtained. Special emphasis is given to indicate the differences between the Eulerian and Lagrangian approaches as well as their connections, which are relevant for the proper description of fluid-structure interaction problems in the presence of large structural deformations. Fundamental principles (balance equations) are developed in both Lagrangian and Eulerian formulations, which naturally leads to the definition of different stress measures. The last part of the course is devoted to the constitutive theory where typical models for fluids and solids are presented.

Topics include:

1. Introduction to tensor algebra and tensor analysis.

2. Kinematics: bodies and configurations. Description of deformations and motions. Lagrangian and Eulerian descriptions of a continuum. Material and spatial time derivatives. Theory of admissible deformations. Deformation and strain tensors. Polar decomposition (stretch and rotation tensors). Rates of deformation (stretching and spin tensor).

3. Kinetics: Mass and mass balance. Reynolds transport theorem. Body and contact forces. Balance of linear and angular momentum in Eulerian and Lagrangian formulations. Cauchy and Piola-Kirchhoff stresses.

4. Introduction to constitutive relations. Constitutive relations in elasticity and Newtonian fluid mechanics.