# 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.
- 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
**10/31/2017**. 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..

- Dispersive materials: viscoelasticity, dynamic metamaterials,

- Mathematical background:
- 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).

- 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

- 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
- HW5: link, Due 11/09/2017
- HW6: link, Due 11/16/2017
- HW on Jump Conditions: : link, Due: FYI, do not return (If returned there is a maximum of 50 extra credit for this assignment).
- Presentation day: Thursday 12/07 9:40 am -12:40 pm EST (8:40-11:40 am CT). 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 12/12/2017 (No late submissions due to the deadline for reporting the final grades).

## Resources

**Equation sheet**: Credit to my colleague Dr. Scott Miller for this material. The formulation sheet will be updated throughout the course.

## Class timeline

- 08/24/2017 Lecture: notes Topics: Indicial and direction notations (TAM551, sections 1.1 to 1.3).
- 08/29/2017 Lecture: notes Topics: Delta Kronecker and alternating symbols (TAM551, sections 1.4 to 1.6).
- 08/31/2017Lecture: notes Topics: Vector space and inner product (introduction) (TAM551, sections 1.7 to 1.10).
- 09/05/2017 Lecture: notes Topics: Basis, vector components, and coordinate transformation (TAM551, sections 1.8 and 1.9).
- 09/07/2017 Lecture: notes Topics: Vector space, inner product, and norm (abstract notion) (TAM551, sections 1.10).
- 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.
- 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.
- 09/19/2017 Lecture: notes Topics: Second order tensors (part 2) (1.11.6-1.11.10), higher order tensors (1.12) – part 1.
- 09/21/2017 Lecture: notes Topics: Higher order tensors (1.12); vector and scalar triple product, skew symmetric tensors (1.13-1.15.1)
- 09/26/2017 Lecture: notes Tensor fields (part I). Tensor calculus in curvilinear coordinate systems. (please to useful resources mentioned in class).
- 09/28/2017 Lecture: notes, video Tensor calculus in curvilinear coordinate systems (part II) (references: mentioned in course notes).
- 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. - 10/05/2017 Lecture: notes Topics: Eigen decomposition: special second order tensors (1.15.1) symmetric tensors.
- 10/10/2017 Lecture: notes Topics: Positive definite tensors, polar decomposition theorem(1.15.1).
- 10/12/2017 Lecture: notes Topics: Kinematics (2.1, 2.2, 2.3.1).
- 10/17/2017 Lecture: notes Topics: Kinematics, Rigid deformation and change in segment, area, and volume (2.3.2 and 2.3.3).
- 10/19/2017 Lecture: notes Topics: Kinematics: Definition of strain (2.3.3).
- 10/24/2017 Lecture: notes Topics: 2.4 Innitesimal Deformation Theory(2.4). Also see Abeyaratne, Vol II, section 2.7 as an external reference.
- 10/26/2017 Lecture: notes Topics: Analysis of innitesimal strain / rigid motion (2.4.2), Motions, Lagrangian vs. Eulerian representation (2.5).
- 10/31/2017 Lecture: notes Topics: Motions, Lagrangian vs. Eulerian representation (2.5), 2.6 Raynold’s transport theorem and Balance laws (part 1) Resource: Overview of balance laws for steady and dynamic problems expressed for spacetime domains: link.
- 11/02/2017 Lecture: notes Topics: Balance laws (part 2): General form of balance laws, PDEs, and jump conditions. Resource link.
- 11/07/2017 Lecture: notes Topics: Balance laws (part 3): Balance of mass (Lagrangian & Eulerian); balance of linear momentum (Eulerian).
- 11/14/2017 Lecture: notes Topics: Balance laws (part 4): Balance of linear momentum (Lagrangian), Piola-Kirchhoff Stress tensors (3.3-3.7); balance of energy.
- 11/16/2017 Lecture: notes Topics: Kinetics: Stress tensor and traction vector (3.3, 3.4, 3.6), Constitutive equations (part 1).
- 11/21/2017 Lecture: notes Topics: Constitutive equations; 4.2 Elastic Response function; 4.3 Principle of frame-invariance (objectivity)
- 11/28/2017 Lecture: notes Topics: 4.4 Material Symmetry; Isotropy.
- 11/30/2017 Lecture: notes Topics: 4.5 Hyperelasticity.
- 12/05/2017 Lecture: notes Topics: 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.