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. Zoom: https://tennessee.zoom.us/j/4441721751

Announcements

  • HW1: link. Due 09/8/2025 – Extra credit: Parts c & d, problem 6.
  • HW2: link. Due 09/25/2025.
  • HW3: link, link to Matlab file. Due 10/8/2025.
  • HW4: link, Due 10/20/2025.
  • HW5: link, Due 11/05/2025; useful Matlab function: link.
  • HW6: link, Due 11/10/2025.
  • HW7: link, Due 11/17/2025.
  • HW8: link, Due 12/01/2025.
  • HW9: link, Due 12/10/2023 :optional; it’s only an extra credit assignment. This assignment adds a maximum of 3.5% to the grade. For example, if you get 70% on this assignment and 80% from regular assignments and the final exam, 80% is increased to 80% + 3.5% * 70% = 82.45%.
  • Optional Term project. If it improves your grade, 8% will be allocated to the term project, and the final exam % changes from 30% to 22%. 
    • The optional term project only involves a presentation file (PowerPoint, etc.) and should be designed for a 10-13 minute presentation. I may ask you to present it to me either in person or in a Zoom meeting. Please confirm the topic of the presentation with me beforehand.
    • 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.

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

Link to all notes

  1. 08/18/2025: notes ,video         Topics: Indicial and direction notations (TAM551, sections 1.1 to 1.3).
  2. 08/20/2025: notes ,video         Topics: Delta Kronecker and alternating symbols (TAM551, sections 1.4 to 1.6).
  3. 08/25/2025: notes ,video         Topics: Alternating symbol and determinant
  4. 08/27/2025: notes ,video         Topics: Vector space and inner product (introduction) (TAM551, sections 1.7 to 1.10); Vector space and inner product (TAM551, sections 1.7 to 1.10).
  5. 09/03/2025: 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. 09/08/2025: notes ,video         Topics: Basis, vector components, and coordinate transformation (TAM551, sections 1.8 and 1.9).
  7. 09/10/2025: 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. 09/15/2025: notes ,video         Topics: Second order tensors (part 2) (1.11.3,5,6) – Please read 1.11.10-1.11.15 at home.
  9. 09/17/2025: 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. 09/22/2025: notes ,video         Topics: Higher order tensors 1.12, vector & triple products (1.13, 1.14), Special second order tensors (1.15.1).
  11. 09/24/2025: notes ,video         Topics: Special second order tensors (1.15.1), orthogonal & skew symmetric tensors.
  12. 09/29/2025: notes ,video         Topics: Special second order tensors (1.15.1) symmetric tensors and positive definite tensors (1.15.2 & 3) (part 1).
  13. 10/01/2025 : notes ,video         Topics: Special second order tensors (1.15.1) symmetric tensors and positive definite tensors (1.15.2 & 3) (part 2).
  14. 10/08/2025: notes ,video         Topics: Special second order tensors (1.15.1) positive definite tensors (1.15.2 & 3) (part 2), polar decomposition theorem.
  15. 10/13/2025: 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).
  16. 10/15/2025: notes ,video Topics: 2.3 Kinematics: Finite deformation to 2.3.2 Rigid deformation. Change of line segment, line length, and angle.
  17. 10/20/2025: notes ,video Topics: 2.3 Kinematics: Finite deformation to 2.3.2 Rigid deformation. Change of surface, and volume; definition of stretch tensor.
  18. 10/22/2025: notes ,video Topics: Different strain measures / Finite deformation 2.3 (part 1).
  19. 10/27/2025: 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. 10/29/2025: 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. 11/03/2025: notes ,video Topics:Infinitesimal deformation gradient (2.4), part 2. Mohr circle; Motions, Lagrangian vs. Eulerian representation (2.5).
  22. 11/05/2025: notes ,video Topics: Reynold’s transport theorem and Balance laws (part 1) Resource: Overview of balance laws for steady and dynamic problems expressed for spacetime domains: link
  23. 11/10/2025: 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. 11/12/2025: notes ,video  Topics: Balance laws (part 2): General form of balance laws, PDEs, and jump conditions. Resource link, Balance of mass.
  25. 11/17/2025: notes ,video  Topics: Balance laws (part 3): Balance of mass (Lagrangian & Eulerian); balance of linear momentum (Eulerian and Lagrangian); 
  26. 11/19/2025: notes ,video  Topics: Balance of energy, Kinetics: Stress tensor and traction vector (3.3, 3.4, 3.6), Constitutive equations (part 1).
  27. 11/24/2025: 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. 12/01/2025: notes ,video  Topics: Hyperelasticity; 4.6 Elastic response to infinitesimal motions; 5. Linearized Elasticity; Voigt notation. Certain symmetries (brief discussion: cubic, orthotropic, transverse isotropic, isotropic)
  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 parts of chapters 5 and 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.