As5950: Continuum mechanics
No. Date Lecture topic Textbook reference
1. 10 Jan Introduction. Matrices and Determinants. Spencer Chap 1; §2.1
2. 11 Jan Index notation. Orthogonal matrices. Relationship between det A, δij, and eijk § 2.2.
3. 13 Jan Eigenvalues and eigenvectors. Diagonalization of a symmetric real matrix. § 2.3.
4. 16 Jan The Cayley-Hamilton and the polar decomposition theorems. § 2.4, 2.5.
5. 17 Jan Vectors and cartesian tensors, an introduction. Coordinate transformation of vectors. § 3.1, 3.2.
6. 18 Jan Equivalence of dot product, and dyadic product in all coordinate systems. Cartesian second order tensors. § 3.3, 3.4.
7. 20 Jan Isotropic tensors. Outer product of tensors. § 3.5, 3.6
8. 23 Jan Contraction. Inner product of tensors. Symmetric positive definite orthogonal tensors. Eigenvalues of a tensor. Positive definite tensors. § 3.6, 3.8
9. 24 Jan Invariants of a tensor. Deviatoric parts of a tensor. Tensor calculus. § 3.9, 3.10.
10. 25 Jan Particle kinematics. Bodies and their configuration. Preservation of particle identity hypothesis. Reference and current configurations. Material and spatial description of problems. Displacement in the reference and spatial configurations. § 4.1, 4.2.
11. 27 Jan Velocity of particles. Meaning of velocity at a spatial location. Rate of change of fields, material and spatial desciptions. Material derivative. § 4.3.
12. 30 Jan Acceleration. Steady motion. Path lines and stream lines. § 4.4, 4.5.
13. 31 Jan Prob. 1. § 4.6
14. 1 Feb Prob. 4 § 4.6
15. 3 Feb Rigid body motion. Translation and rotation. § 6.1
16. 6 Feb General rigid body motion. Deformation. Stretch ratio of material line elements, as defined in the reference configuration. § 6.1, 6.2.
17. 7 Feb Stretch ratio of material line elements, in the current configuration. The deformation gradient tensor, and its inverse. § 6.2, 6.3.
18. 8 Feb The right and left Cauchy-Green deformation tensor. Lagrangian and Eulerian strain tensors. Connection with small deformation tensors. § 6.4.
19. 10 Feb Problems from Chap 6: 2, 3, 5. § 6.11
20. 15 Feb Quiz 1: on everything covered up to 10 Feb. Question paper; solutions
21. 17 Feb Some simple finite deformations. § 6.5.
22. 20 Feb Homogeneous deformations. § 6.5.
23. 21 Feb Infinitesimal strain, and rotation. § 6.6, 6.7.
24. 22 Feb The material rate of the stretch ratio, (D λ/Dt)/ λ. § 6.8
25. 24 Feb Velocity gradient, deformation rate, and vorticity. Application to simple flows. § 6.9, 6.10.
26. 27 Feb Problems 9, 10, and 11. § 6.11.
27. 28 Feb Traction, stress tensor, equilibrium equations, principal stresses, principal axes, and deviatoric stresses § 5.1-5.9.
28. 1 Mar dv / dV = det F; Conservation of mass: Lagrangian and Eulerian forms. § 7.1, 7.2.
29. 3 Mar Material rate of change of a spatial integral. Conservation of linear momentum. Conservation of angular momentum. § 7.3, 7.4, 7.5.
30. 6 Mar Conservation of energy. Principle of virtual work. § 7.6, 7.7
31. 7 Mar Problems 1, 2, and 3. § 7.8
32. 8 Mar Problem 4. Linear constitutive laws. Requirements thereof: dimensional consistency, objectivity, independence of rigid rotations. § 7.8, 8.1.
33. 10 Mar Material symmetries: rotational and reflectional. § 8.2
34. 14 Mar Linear elastic solids. Minor and major symmetries of Cijkl. § 8.3.
35. 15 Mar The effect of material symmetries on Cijkl. The effect of reflectional symmetry about a plane. Newtonian viscous fluids. § 8.3, 8.4.
36. 17 Mar Linear viscoelasticity. Isotropic case. Relaxation function, and creep function. Problem 1. § 8.5, 8.6.
37. 20 Mar Problems 2-7. § 8.6.
38. 21 Mar Problem 8. Deformation of a surface element. § 8.6, 9.1.
39. 22 Mar Quiz 2. Question paper; Solutions
40. 24 Mar Polar decomposition of the deformation gradient. Principal stretches and principal axes of deformation. § 9.2, 9.3.
41. 29 Mar Alternative stress measures. § 9.5.
42. 3 Apr Exercises. End of syllabus for the end-semester exam. § 9.6.
43. 4 Apr Non-linear constitutive equations. Non-linear elasticity. The case of isotropy. § 10.1, 10.2.
44. 5 Apr Non-linear elastic constitutive relations between T and B. Non-linear incompressible viscous fluids. § 10.2, 10.3.
45. 10 Apr The Reiner-Rivlin fluid. Plasticity. § 10.3, 10.5.
46. 26 Apr Final examination.