1. Review (2 nd order tensors): Tensor – Linear mapping of a vector onto another vector Tensor components in a Cartesian basis (3x3 matrix): Basis change formula for tensor components Dyadic vector product General dyadic expansion

2. Routine tensor operations Addition Vector/Tensor product Tensor product

3. Routine tensor operations Transpose Trace Inner product Outer product Determinant Inverse Invariants (remain constant under basis change) Eigenvalues, Eigenvectors (Characteristic Equation – Cayley-Hamilton Theorem)

4. Recipe for computing eigenvalues of symmetric tensor

5. Special Tensors Symmetric tensors Have real eigenvalues, and orthogonal eigenvectors Skew tensors Have dual vectors satisfying Proper orthogonal tensors Represent rotations – have Rodriguez representation Polar decomposition theorem

6. Polar Coordinates Basis change formulas

7. Gradient operator

8. Review: Deformation Mapping Eulerian/Lagrangian descriptions of motion Deformation Gradient

9. Review Sequence of deformations Lagrange Strain

10. Review Volume Changes Area Elements Infinitesimal Strain Approximates L-strain Related to ‘Engineering Strains’

11. Review Principal values/directions of Infinitesimal Strain Infinitesimal rotation Decomposition of infinitesimal motion

12. Left and Right stretch tensors, rotation tensor U,V symmetric, so principal stretches Review Left and Right Cauchy-Green Tensors

13. Review Generalized strain measures Eulerian strain

14. Review Velocity Gradient Stretch rate and spin tensors

15. Vorticity vector Spin-acceleration-vorticity relations Review

16. Review: Kinetics Surface traction Body Force Internal Traction Resultant force on a volume

17. Restrictions on internal traction vector Review: Kinetics Newton II Newton II&III Cauchy Stress Tensor

18. Other Stress Measures Kirchhoff Nominal/ 1 st Piola-Kirchhoff Material/2 nd Piola-Kirchhoff

19. Review – Reynolds Transport Relation

20. Review – Mass Conservation Linear Momentum Conservation Angular Momentum Conservation

21. Rate of mechanical work done on a material volume Conservation laws in terms of other stresses Mechanical work in terms of other stresses

22. Review: Thermodynamics Specific Internal Energy Specific Helmholtz free energy Temperature Heat flux vector External heat flux First Law of Thermodynamics Second Law of Thermodynamics Specific entropy

23. Conservation Laws for a Control Volume R is a fixed spatial region – material flows across boundary B Mass Conservation Linear Momentum Conservation Angular Momentum Conservation Mechanical Power Balance First Law Second Law

24. Review: Transformations under observer changes Transformation of space under a change of observer All physically measurable vectors can be regarded as connecting two points in the inertial frame These must therefore transform like vectors connecting two points under a change of observer Note that time derivatives in the observer’s reference frame have to account for rotation of the reference frame

25. Some Transformations under observer changes

26. Constitutive Laws General Assumptions: 1.Local homogeneity of deformation (a deformation gradient can always be calculated) 2.Principle of local action (stress at a point depends on deformation in a vanishingly small material element surrounding the point) Restrictions on constitutive relations: 1. Material Frame Indifference – stress-strain relations must transform consistently under a change of observer 2. Constitutive law must always satisfy the second law of thermodynamics for any possible deformation/temperature history. Equations relating internal force measures to deformation measures are known as Constitutive Relations