3H: Mechanics of Rigid and Deformable Bodies

Course Aims
The aim of the course Modelling Rigid and Deformable Bodies is to extend the work on the one dimensional motion of particles covered in 2C, first, to the planar motion of particles acted on by a central force, second, to the motion of rigid body using the Lagrangian methodology, third, to the one-dimensional motion of deformable bodies, and fourth, to the three-dimensional motion of deformable bodies using Cartesian tensors. Conservation laws of linear momentum, angular momentum and energy will be considered for rigid and deformable bodies. The concept of entropy will be introduced and used to develop constitutive relations for fluids and solids leading to the solution of simple one-dimensional problems for these classes of materials.

Intended Learning Outcomes of Course
By the end of this course students will be able to:
1. derive conservation laws of linear momentum, angular momentum and energy for systems of particles interacting along their line of centres;
2. solve two-dimensional motions involving central forces and identify what quantities are conserved in the motions;
3. derive the vectorial relationship between linear and angular velocity for the motion of the particles in a rigid body;
4. calculate inertia tensors for various simple rigid bodies;
5. prove and apply the parallel-axis and perpendicular-axis theorems;
6. construct the Lagrangian function for simple motions of particles and rigid bodies;
7. formulate and solve the Lagrangian equations for simple motions of particles and rigid bodies;
8. derive the equations expressing conservation of mass, linear momentum and energy for a deformable body undergoing motion described by a single spatial variable;
9. derive the entropy inequality and use it to particularise constitutive equations;
10. define Helmholtz free energy, internal energy and specific entropy and establish expressions for these functions given an equation of state for a perfect fluid;
11. derive the Rankine-Hugoniot conditions and apply them to the propagation of shocks in a motion described by a single spatial variable;
12. solve simple one-dimensional problems involving the behaviour of fluids and beams;
manipulate Cartesian tensors and use them to derive the conservation equations for a three-dimensional deformable body.