Class Information

Course Description

Cartesian tensors, transformation laws, basic continuum mechanics concepts; stress, strain, deformation, constitutive equations. Conservation laws for mass, momentum, energy. Applications in solid and fluid mechanics.


Class Information

  • Location: UTK:  Doherty 406                    UTSI: Main Academic Building E110

Office Hours

By appointment.

Office: B203, Main academic building, UTSI
Phone: (931) 393-7334
Skype username: rpabedi


  • No class on 08/19 and 08/24 (two make up classes will be held during the semester).
  • HW1: link. Due 09/16/2015 – Extra credit: Parts c & d, problem 4.
  • HW2: link. Due 09/24/2015
  • HW3: link, link to Matlab file. Due 10/08/2015
  • HW4: link, Due 10/15/2015
  • HW5: link, Due 11/09/2015
  • HW6: link, Due 11/19/2015
  • HW on Jump Conditions: : link, Due: FYI, do not return.
  • Make up class Tuesday 11/24 11:40 am -2:10 pm EST (right after the regular class time).
  • Presentation day: Friday 12/04 10-12 am  EST (Those of you who cannot attend the presentation day should contact me, if not have already done, beforehand so I can arrange your presentations in one of our regular class hours).
  • HW7: link, Due 12/11/2015 (No late submissions due to the deadline for reporting the final grades).


Class timeline

  1. 08/25/2015 Lecture: notes          Topics: Indicial and direction notations.
  2. 08/27/2015 Lecture: notes          Topics: Indicial and direction notations.
  3. 09/01/2015 Lecture: notes          Topics: Inner-product and change of coordinate.
  4. 09/03/2015 Lecture: notes          Topics: Vector space, Inner-product and norm vector spaces; Linear operators.
  5. 09/08/2015 Lecture: notes          Topics: Second order tensors (1.11.1-1.11.4) – Please read 1.11.5-1.11.10 at home.
  6. 09/10/2015 Lecture: notes          Topics: Second order tensors (1.11.5-1.11.10) – Please read higher order tensors 1.12 at home.
  7. 09/15/2015 Lecture: notes          Topics: Higher order tensors 1.12, vector & triple products (1.13, 1.14), Special second order tensors (1.15.1)
  8. 09/17/2015 Lecture: notes          Topics: Special second order tensors (1.15.1) symmetric tensors
  9. 09/22/2015 Lecture: notes          Topics: Special second order tensors (1.15.1) symmetric tensors and positive definite tensors (1.15.2 & 3)
  10. 09/24/2015 Lecture: notes          Topics: Special second order tensors (1.15.1) symmetric tensors – Mohr’s circle; 1.16 Tensor fields (part I)
  11. 09/29/2015 Lecture: notes          Topics: Tensor fields (part II). Tensor calculus in curvilinear coordinate systems. (please to useful resources below for more information).
  12. 10/01/2015 Lecture: notes          Topics: Tensor fields (part III). Tensor calculus in curvilinear coordinate systems, 2.1 Kinematics: Mathematical preliminaries, Bodies.
  13. 10/06/2015 Lecture: notes          Topics: 2.3 Kinematics: Finite deformation to 2.3.2 Rigid deformation.
  14. 10/08/2015 Lecture: notes          Topics: Change of length, angle, surface, and volume elements by finite deformation.
  15. 10/13/2015 Lecture: notes          Topics: Different strain measures / Finite deformation 2.3
  16. 10/20/2015 Lecture: notes          Topics: Relation between strain and stretch (good reference is Abeyaratne II: 2.7, 2.8), Infinitesimal deformation gradient (2.4).
  17. 10/22/2015 Lecture: notes          Topics: Relation between strain and stretch (part 2), Infinitesimal deformation gradient, including Cesaro Line Integral (CLI);  (2.4.2). For motivations on the uses of CLI and derivation of displacement from strain refer to fracture 2015, or fracture 2014 lectures 6,7 (and 8 for applications).
  18. 10/27/2015 Lecture: notes          Topics: 2.5 & 2.6: Eulerian and Lagrangian representations; Raynold’s transport theorem (part 1).
  19. 10/29/2015 Lecture: notes          Topics: 2.6 Raynold’s transport theorem (part 2); Balance laws (part 1) Resource: Overview of balance laws for steady and dynamic problems expressed for spacetime domains: link.
  20. 11/03/2015 Lecture: notes          Balance laws
  21. 11/05/2015 Lecture: notes          Balance law examples: 3.2 Conservation of mass; Balance of linear momentum, Piola-Kirchhoff Stress tensors (3.3-3.7) part 1.
  22. 11/10/2015 Lecture: notes (continuation of last time notes)    Balance law examples: momentum, Piola-Kirchhoff Stress tensors (part 1); Balance of energy  (3.3-3.7).
  23. 11/12/2015 Lecture: notes (continuation of last time notes)   Kinetics: Stress tensor and traction vector (3.3, 3.4, 3.6).
  24. 11/17/2015 Lecture:  notes          Constitutive equations; 4.2 Elastic Response function; 4.3 Principle of frame-invariance (objectivity)
  25. 11/24/2015 Lecture:  notes           4.3 Principle of frame-invariance (objectivity); 4.4 Material Symmetry; Isotropy; 4.5 Hyperelasticity.
  26. 12/02/2015 Lecture:  notes           4.6 Elastic response to infinitesimal motions; 5. Linearized Elasticity

Selected Bibliography

  • (GUR) Morton E. Gurtin, Eliot Fried, and Lallit Anand, The Mechanics and Thermodynamics of Continua, Cambridge University Press, 2010.
  • (SPE) A. J. M. Spencer, Continuum Mechanics, Dover Publishing, 2004.
  • (MAL) L. E. Malvern, Introduction to the Mechanics of a Continuous Medium, Englewood Cliffs (NJ), Prentice-Hall, 1969.
  • (CHA) P. Chadwick, Continuum Mechanics: Concise Theory and Problems, Dover Publishing, 1999 (first edition: Wiley, 1976).
  • (WU) H.C. Wu, Continuum Mechanics and Plasticity, Chapman and Hall/CRC, 2004 (Solids, Plasticity).
  • (DIM) Y. I. Dimitrienko, Nonlinear Continuum Mechanics and large Inelastic Deformations, Springer, 2011 (Solids).
  • (CHA) E.W.V. Chaves, Notes on Continuum Mechanics, Springer, 2013 (Solids, Plasticity, Damage mechanics).
  • (LAI) W.M. Lai, D. Rubin, Erhard Krempl, Introduction to Continuum Mechanics, Elsevier, 4th edition, 2009 (Fluids).
  • (BOW) R. M. Bowen, Introduction to Continuum Mechanics for Engineers, Plenum Press, 1989. (Thermodynamics).
  • (TAD) E.B. Tadmor, R.E. Miller, R.S. Elliot, Continuum Mechanics and Thermodynamics, Cambridge University Press, 2012 (Thermodynamics).
  • (TRU) C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Springer, 3rd edition, 2004 (Mathematics).
  • (TAL) Y.R. Talpaert, Tensor Analysis and Continuum Mechanics, Springer, 2003 (Mathematics).
  • (ROM) G. Romano, R Barretta, Continuum Mechanics on manifolds, 2009 (Mathematics, Exterior Calculus).

Useful Resources:

  • Curvilinear coordinate systems:
    • Appendix C: “Continuum Mechanics” (link) course notes from Professor Zdenek Martinec has a very good overview of this topic (I follow the same notations as these notes).
    • This short document  (link) posted by Professor Piaras Kelly  has a good explanation of the meaning of gradient operator. It also discussed possible confusions that can arise with the uses of nabla operator in the definitions of grad/div operators (see also here).
    • For further discussion on this topic “Curvilinear Analysis in a Euclidean Space” (link) by Professor Rebecca Brannon (University of Utah) is an excellent reference.