Continuum Mechanics

Course Information

• Course Title: Continuum Mechanics
• Course Code: 2014276-01
• Credits: 3
• Level: M.Sc., Biomedical Engineering (Biomechanics)
• Schedule: Monday 14:00-16:00 & Tuesday 14:00-16:00
• Location: Class 32
• Instructor: Seyed Sadjad Abedi-Shahri
  • Email: AbediSadjad@gmail.com
• Lecture Materials: Provided weekly in LMS.

Course Overview

This course introduces the principles of nonlinear continuum mechanics and their application to biomechanics and biological soft tissues. It is the foundational mechanics course for graduate students in biomechanics, covering the mathematical and physical framework required to analyze deformable biological materials undergoing finite deformation. The course develops tensor analysis, the kinematics of finite deformation, stress measures and the balance laws, and the constitutive modeling of hyperelastic and fiber-reinforced materials, and closes with the canonical boundary value problems used to characterize soft tissue.

Learning Objectives

By the end of this course, students will be able to:

1. Tensor Analysis
   • Manipulate vectors and second-order tensors in direct and index notation
   • Compute invariants, eigenvalues, and the spectral decomposition
   • Apply differential and integral operators to tensor fields
2. Kinematics of Finite Deformation
   • Distinguish material and spatial descriptions of motion
   • Compute the deformation gradient and extract stretch, rotation, and volume change
   • Select and interpret an appropriate finite strain measure
3. Stress and Balance Laws
   • Apply Cauchy’s theorem and convert among the alternative stress measures
   • Derive the local balance equations from their integral forms
   • Explain the role of the entropy inequality in constraining constitutive equations
4. Constitutive Modeling
   • Derive stress from a strain-energy function
   • Select and justify a hyperelastic model for a given soft tissue
   • Impose incompressibility and represent fiber reinforcement
5. Applications
   • Formulate and solve the canonical finite-deformation boundary value problems
   • Relate these solutions to standard soft tissue testing protocols

Syllabus

1. Mathematical Foundations
2. Kinematics of Finite Deformation
3. Stress and Balance Laws
4. Constitutive Theory and Hyperelasticity
5. Canonical Problems

References

1. [BON] Nonlinear Solid Mechanics for Finite Element Analysis by Javier Bonet and Richard D. Wood (primary reference for the mechanics core; Farsi translation available)
2. [TAB-M] Continuum Modeling in Mechanobiology by Larry A. Taber (primary reference for the biomechanics material)
3. [MAS] Continuum Mechanics for Engineers [2nd ed.] by George T. Mase and George E. Mase (problems and worked examples; Farsi translation in circulation)
4. [HOL] Nonlinear Solid Mechanics: A Continuum Approach for Engineering by Gerhard A. Holzapfel (objectivity, strain-energy functions, and fiber-reinforced materials)
5. [CAP] Continuum Mechanics: Constitutive Modeling of Structural and Biological Materials by Franco M. Capaldi
6. [TAB-E] Nonlinear Theory of Elasticity: Applications in Biomechanics by Larry A. Taber (further depth on the derivations)
7. [RED] Principles of Continuum Mechanics by J. N. Reddy (supporting reference for index notation and vector calculus)
8. [SHA] Computational Continuum Mechanics by Ahmed A. Shabana

Evaluation Scheme

To be announced.

Session Outline

Additional Information

Prerequisites

• Strength of Materials

Policies

1. Regular attendance is strongly recommended to stay on track with course material and acquire continuous evaluation score
2. Students are expected to arrive on time. Late arrivals may disrupt the class and could impact participation evaluation.
3. Collaboration on assignments, exercises, and projects is encouraged. However, all submissions must reflect individual understanding and adhere to academic integrity policies. Plagiarism or copying will not be tolerated.