Mechanics of a Continuous Medium

Tensor analysis. The Rayleigh transport theorem. The deformation gradient. The polar decomposition theorem. Rotations and stretches. Lagrangian and Eulerian description of deformation metrics. Mass conservation. Conservation of linear momentum. Conservation of angular momentum. The stress tensors: Cauchy, 1st and 2nd Piola-Kirchhoff. Objective deformation measures. The velocity gradient tensor. Decomposition to strain rate and spin. Principal stretches and principal directions. Invariants of symmetric tensors. Orthogonal tensors. Equilibrium equations and the Virtual Work theorem. Constitutive equations in elasticity and fluid mechanics. Anisotropy. Hyperelasticity. Internal constrains: incompressibility, inextensibility. The first thermodynamic theorem. The second thermodynamic theorem. Objective stress rates. Objective deformation rates. Mechanical power and work conjugate stresses and deformation tensors. Jump conditions and discontinuities. Problems of large deformation elasticity. Problems of fluid mechanics.