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

  • Hours: Mondays and Wednesdays 9:45 – 11:05 am EST (8:45 – 10:05 am  CST)
  • Location: UTK:  Dougherty 406                    UTSI: Main Academic Building E110

Office Hours

By appointment. [one_third last=”no”]Office: B203, Main academic building, UTSI[/one_third] [one_third last=”no”][/one_third] [one_third last=”yes”]Skype username: rpabedi[social_links colorscheme=”” linktarget=”_self” rss=”” facebook=”” twitter=”” dribbble=”” google=”” linkedin=”” blogger=”” tumblr=”” reddit=”” yahoo=”” deviantart=”” vimeo=”” youtube=”” pinterest=”” digg=”” flickr=”” forrst=”” myspace=”” skype=”skype:rpabedi?add”][/one_third]

Announcements

  • HW1: link. Due 09/19/2023 – Extra credit: Parts c & d, problem 6.
  • HW2: link. Due 10/02/2023.
  • HW3: link, link to Matlab file. Due 10/11/2023.
  • HW4: link, Due 10/25/2023. (problem 7 is extra credit and you do not need to solve it; if you return problem 7, you can return your HW4 on 10/23).
  • HW5: link, Due 11/08/2023; usefule Matlab function: link.
  • HW6: link, Due 11/15/2023.
  • HW7: link, Due 11/29/2023.
  • HW8: link, Due 12/06/2023.
  • HW9: link, Due 12/09/2023.
  • Term project 1: Includes (About equal weights are allocated to each part)
    • 1) An up to 4 pages paper/proposal(including references if any) on a topic related to continuum mechanics. The format of the document is either that of a
      • Research article mostly focusing on introducing a topic of interest and presenting related results. Suggested sections are abstract, introduction, formulation, results (can present results from existing literature, doesn’t need to be from your own research), conclusion.
      • Research proposal that basically introduces a problem, discusses current state of the art and research gaps, and finally proposes a new approach to address the mentioned research gaps. Suggested sections are (abstract), introduction (why this problem is important and what is the main contribution of the proposed work), background (state of the art and what are the existing gaps and challenges), objective (describing the goal and objectives of the research), research tasks (what is proposed to be done). Some optional sections are intellectual merits and broader impacts as often required in research proposals.
    • 2) Presentation of the article on the “Presentation day”. Each student will have about 15 minutes to present the material in the article (and related to it) to the entire class.
    • Notes:
      • The choice between research article or proposal is up to the student. The topic can be related to your own research work (as long as it is related to continuum mechanics) or any other topic related to the course that is of interest to you. I can help you in choosing a topic if needed. Please confirm your research topic by the end of 11/29/2023. Some proposed topics are:
        • Mathematical background:
          • Vectors vs. covectors, tensors and cotensors / differential form notation.
          • Curvilinear and non orthonormal coordinate systems.
        • Kinematics:
          • Eulerian versus Lagrangian strains.
          • Arbitrary Lagrangian Eulerian (ALE) formulations.
          • Objective rates of deformation.
        • Balance laws, forces / stress:
          • Balance laws in spacetime.
          • Jump condition (Rankine-Hugoniot jump conditions); shocks, expansion waves, contact discontinuity.
          • Thermodynamic laws (in relation to the course content).
        • Constitutive Equations (possibly in combination with kinematics / balance laws):
          • Constitutive equations for various types of fluids.
          • Gradient elasticity theory (formulations that use beyond strain value in the constitutive equation) – topic for solid mechanics.
          • Thermodyanmically motivated damage / phase field models for solid materials.
          • Constitutive equations (and if needed kinematics / balance laws) for specific group of materials:
            • Dispersive materials: viscoelasticity, dynamic metamaterials, etc.
            • Any other type of so-called mechanical metamaterials (light weight, auxetic, pentamode, origami, etc.).
            • 3D printed materials.
            • Granular materials.
            • Foams, soft material, etc..
    • If you choose the proposal format, your presentation will be on the general topic of your proposal not actually on selling your idea to the class (that is done in the proposal).

Resources

  • Equation sheet: Credit to my colleague Dr. Scott Miller for this material. The formulation sheet will be updated throughout the course.
  • Matlab file for computing various kinematic quantities for a given F tensor: link.

Class timeline

  1. 08/23/2023 Lecture: notes ,video         Topics: Indicial and direction notations (TAM551, sections 1.1 to 1.3).
  2. 08/28/2023 Lecture: notes ,video         Topics: Delta Kronecker and alternating symbols (TAM551, sections 1.4 to 1.6).
  3. 08/30/2023 Lecture: notes ,video         Topics: Alternating symbol and determinant; Vector space and inner product (introduction) (TAM551, sections 1.7 to 1.10).
  4. 09/06/2023 Lecture: notes, video         Topics: Vector space and inner product (TAM551, sections 1.7 to 1.10).
  5. 09/08/2023 Lecture: notes, video         Topics: Inner product and norm vector spaces, basis, and coordinate system (TAM551, sections 1.7 to 1.10), (1.11.1-1.11.4).
  6. 09/11/2023 Lecture: notes, video         Topics: Basis, vector components, and coordinate transformation (TAM551, sections 1.8 and 1.9).
  7. 09/13/2023 Lecture: notes, video         Topics: Linear operators, second order tensors (part 1) (1.11.1-1.11.4) – Please read 1.11.5-1.11.10 at home.
  8. 09/18/2023 Lecture: notes, video         Topics: Topics: Second order tensors (part 2) (1.11.3,5,6) – Please read 1.11.10-1.11.15 at home.
  9. 09/25/2023 Lecture: notes ,video         Topics: : Second order tensors (part 3) (1.11.3,5,6) – Please read 1.11.10-1.11.15 at home; Higher order tensors 1.12.
  10. 09/27/2023 Lecture: notes ,video         Topics: Higher order tensors 1.12, vector & triple products (1.13, 1.14), Special second order tensors (1.15.1).
  11. 10/02/2023 Lecture: notes ,video         Topics: Special second order tensors (1.15.1) symmetric tensors and positive definite tensors (1.15.2 & 3) (part 1).
  12. 10/04/2023 Lecture: notes ,video         Topics: Special second order tensors (part 1), (1.15.1).
  13. 10/11/2023 Lecture: notes ,video         Topics: Special second order tensors (1.15.1) positive definite tensors (1.15.2 & 3) (part 2), polar decomposition theorem.
  14. 10/16/2023 Lecture: notes ,video         Topics: Special second order tensors (1.15.1) Tensor fields: Tensor calculus in Cartesian and curvilinear coordinate systems. (please to useful resources below for more information).
  15. 10/18/2023 Lecture: notes ,video         Topics: Curvilinear coordinate systems (FYI); 2.3 Kinematics: Finite deformation to 2.3.2 Rigid deformation. Change of line segment.
  16. 10/23/2023 Lecture: notes ,video         Topics: 2.3 Kinematics: Finite deformation to 2.3.2 Rigid deformation; Change of length, angle, surface, and volume elements by finite deformation.
  17. 10/25/2023 Lecture: notes ,video         Topics: 2.3 Kinematics: Finite deformation, definition of stretch tensor.
  18. 10/30/2023 Lecture: notes ,video         Topics: Different strain measures / Finite deformation 2.3 (part 1).
  19. 11/01/2023 Lecture: notes ,video         Topics: Different strain measures / Finite deformation 2.3 (part 2). Different strain measures / Finite deformation 2.3, Relation between strain and stretch (good reference is Abeyaratne II: 2.7, 2.8), Infinitesimal deformation gradient (2.4).
  20. 11/06/2023 Lecture: notes ,video         Topics: Relation between strain and stretch (good reference is Abeyaratne II: 2.7, 2.8), Infinitesimal deformation gradient (2.4), part 2. Mohr circle.
  21. 11/08/2023 Lecture: notes ,video         Topics:Infinitesimal deformation gradient (2.4), part 2. Mohr circle; Motions, Lagrangian vs. Eulerian representation (2.5)
  22. 11/13/2023 Lecture: notes ,video         Topics: Raynold’s transport theorem and Balance laws (part 1) Resource: Overview of balance laws for steady and dynamic problems expressed for spacetime domains: link
  23. 11/15/2023 Lecture: notes ,video         Topics: Raynold’s transport theorem and Balance laws (part 1) Resource: Overview of balance laws for steady and dynamic problems expressed for spacetime domains: link
  24. 11/20/2023 Lecture: notes ,video         Topics: Balance laws (part 2): General form of balance laws, PDEs, and jump conditions. Resource link, Balance of mass (Lagrangian & Eulerian). Balance of linear momentum.
  25. 11/27/2023 Lecture: notes ,video         Topics: Balance laws (part 3): Balance of mass (Lagrangian & Eulerian); balance of linear momentum (Eulerian and Lagrangian); balance of energy.
  26. 11/29/2023 Lecture: notes ,video         Topics: Kinetics: Stress tensor and traction vector (3.3, 3.4, 3.6), Constitutive equations (part 1).
  27. 12/04/2023 Lecture: notes ,video         Topics: Constitutive equations (part 2): 4.2 Elastic Response function; 4.3 Principle of frame-invariance (objectivity); 4.4 Material Symmetry; Isotropy.
  28. 12/06/2023 Lecture: notes ,video         Topics: Hyperelasticity; 4.6 Elastic response to infinitesimal motions; 5. Linearized Elasticity; Voigt notiation.
  29. Pre-recorded lecture: notes ,video (a cleaner write-up: link)        Topics: An identity used for proof of elasticity tensor form for hyperelastic material.
  30. From 2019 Lecture: notes          Topics: Abeyaratne chapter 12. Constitutive equations for fluids: Compressible elastic fluid.

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:
    • A short write-up for a lecture on 10/05/2023: link.
    • 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.

rezaabedi.com/…M2023-ME536/CM20231004.mp4