Hydrodynamics and Elasticity: Class 12
Stress and strain: elastic equilibria 1. Given the stress tensor
σ = 2µU + λ(Tr U )1, (1)
obtain the deformation tensor U in terms of the stress tensor.
2. Find the deformation of a cylindrical rod stretched axially by a force F and attached to a wall at one end.
3. From the mechanical equilibrium conditions in the presence of a body force f , find the Cauchy’s equation for deformation
f + µ∇2u + (λ + µ)∇(∇ · u) = 0, (2)
or
2(1 + ν)
E f + ∇2u + 1
1 − 2ν∇(∇ · u) = 0. (3)
4. A cylindrical rod with cross-sectional surface area A and density ρ is attached at z = 0 in a gravitational field. Find the deformation of this rod under its own weight.
