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.

Syllabus

Class Information

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

Office Hours

By appointment.

Office: B203, Main academic building, UTSI
Skype username: rpabedi

Announcements

  • HW1: link. Due 09/14/2017 – Extra credit: Parts c & d, problem 6.
  • HW2: link. Due 09/27/2017.
  • HW3: link, link to Matlab file. Due 10/10/2017.
  • HW4: link, Due 10/17/2017.
  • HW5: link, Due 11/09/2017
  • HW6: link, Due 11/19/2017
  • HW on Jump Conditions: : link, Due: FYI, do not return.
  • 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 12/11/2017 (No late submissions due to the deadline for reporting the final grades).

Resources

Class timeline

  1. 08/24/2017 Lecture: notes          Topics: Indicial and direction notations (TAM551, sections 1.1 to 1.3).
  2. 08/29/2017 Lecture: notes          Topics: Delta Kronecker and alternating symbols (TAM551, sections 1.4 to 1.6).
  3. 08/31/2017Lecture: notes          Topics: Vector space and inner product (introduction) (TAM551, sections 1.7 to 1.10).
  4. 09/05/2017 Lecture: notes          Topics: Basis, vector components, and coordinate transformation (TAM551, sections 1.8 and 1.9).
  5. 09/07/2017 Lecture: notes          Topics: Vector space, inner product, and norm (abstract notion) (TAM551, sections 1.10).
  6. 09/12/2017 Lecture: notes          Topics: Linear operators, second order tensors (part 1) (1.11.1-1.11.4) – Please read 1.11.5-1.11.10 at home.
  7. 09/14/2017 Lecture: notes          Topics: Second order tensors (part 2) (1.11.3,5,6) – Please read 1.11.10-1.11.15 at home.
  8. 09/19/2017 Lecture: notes          Topics: Second order tensors (part 2) (1.11.6-1.11.10), higher order tensors (1.12) – part 1.
  9. 09/21/2017 Lecture: notes          Topics: Higher order tensors (1.12); vector and scalar triple product, skew symmetric tensors (1.13-1.15.1)
  10. 09/26/2017 Lecture: notes          Tensor fields (part I). Tensor calculus in curvilinear coordinate systems. (please to useful resources mentioned in class).
  11. 09/28/2017 Lecture: notes, video       Tensor calculus in curvilinear coordinate systems (part II) (references: mentioned in course notes).
  12. 10/03/2017 Lecture: notes          Tensor calculus in curvilinear coordinate systems (part III), Eigendecomposition (part I); for eigendecomposition please refer to wikipedia (click here) and read ahead of time symmetric matrix eigendecomposition (TAM551, section 1.15.2).
    FYI: being careful (nabla vs. grad): This was discussed in class, but if you want to read more refer to here.
    FYI: Interesting relation of coordinate transformation and PML: Refer to here.
  13. 10/05/2017 Lecture: notes          Topics: Eigen decomposition: special second order tensors (1.15.1) symmetric tensors.
  14. 10/10/2017 Lecture: notes          Topics: Positive definite tensors, polar decomposition theorem(1.15.1).
  15. 10/12/2017 Lecture: notes          Topics: Kinematics (2.1, 2.2, 2.3.1).
  16. 10/17/2017 Lecture: notes          Topics: Kinematics, Rigid deformation and change in segment, area, and volume (2.3.2 and 2.3.3).

09/22/2015 Lecture: notes          Topics: Special second order tensors (1.15.1) symmetric tensors and positive definite tensors (1.15.2 & 3)
09/24/2015 Lecture: notes          Topics: Special second order tensors (1.15.1) symmetric tensors – Mohr’s circle; 1.16 Tensor fields (part I)
09/29/2015 Lecture: notes          Topics: Tensor fields (part II). Tensor calculus in curvilinear coordinate systems. (please to useful resources below for more information).
10/01/2015 Lecture: notes          Topics: Tensor fields (part III). Tensor calculus in curvilinear coordinate systems, 2.1 Kinematics: Mathematical preliminaries, Bodies.
10/06/2015 Lecture: notes          Topics: 2.3 Kinematics: Finite deformation to 2.3.2 Rigid deformation.
10/08/2015 Lecture: notes          Topics: Change of length, angle, surface, and volume elements by finite deformation.
10/13/2015 Lecture: notes          Topics: Different strain measures / Finite deformation 2.3
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).
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).
10/27/2015 Lecture: notes          Topics: 2.5 & 2.6: Eulerian and Lagrangian representations; Raynold’s transport theorem (part 1).
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.
11/03/2015 Lecture: notes          Balance laws
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.
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).
11/12/2015 Lecture: notes (continuation of last time notes)   Kinetics: Stress tensor and traction vector (3.3, 3.4, 3.6).
11/17/2015 Lecture:  notes          Constitutive equations; 4.2 Elastic Response function; 4.3 Principle of frame-invariance (objectivity)
11/24/2015 Lecture:  notes           4.3 Principle of frame-invariance (objectivity); 4.4 Material Symmetry; Isotropy; 4.5 Hyperelasticity.
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. http://www1.mengr.tamu.edu/rbowen/ (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.