This course focuses on nonlinear solid mechanics. Topics include mathematical preliminaries –Cartesian tensors and tensor algebra; analysis of deformation and motion; concept of stress and balance principles.
The fundamentals of nonlinear solid mechanics is important and is essential for all branches of engineering and applied mechanics. Students learn the basics of nonlinear solid mechanics and will be able to solve some problems in engineering.
By completing this course, students will be able to:
1) Understand analysis of deformation and motion.
2) Understand concept of stress.
3) Understand balance principles.
Euclidean vector space - orthonormal bases and components, and change of basis, second-order tensors, tensor algebra, tensor fields, deformation gradient, polar decomposition, push-forward and pull-back operations, deformation and strain rates, concepts of stress, balance principles
✔ Specialist skills | Intercultural skills | Communication skills | ✔ Critical thinking skills | ✔ Practical and/or problem-solving skills |
Most of time in the class is devoted to fundamentals and the rest to advanced contents or applications. To allow students to get a good understanding of the course contents and practical applications, problems related to the contents of this course are given as homework assignments.
Course schedule | Required learning | |
---|---|---|
Class 1 | Introduction of the course. Euclidean vector space - orthonormal bases and components, and change of basis. | Review sections 1.1.1-1.1.2 of class notes. |
Class 2 | Cartesian tensors | Review sections 1.2.1-1.2.5 of class notes. |
Class 3 | Tensor algebra - second-order tensors, eigenvalues and eigenvectors of a second-order tensor. | Review sections 1.3.1-1.3.2 of class notes. |
Class 4 | Symmetric second-order tensors, antisymmetric second-order tensors and orthogonal second-order tensors | Review section 1.3.3-1.3.5 of class notes. |
Class 5 | Tensor fields | Review sections 1.5.1 and 1.5.2 of class notes. |
Class 6 | The deformation gradient; and deformation of volume and surface elements | Review sections 2.2.1-2.2.2 of class notes. |
Class 7 | Strain, stretch, extension and shear, and polar decomposition of the deformation gradient | Review sections 2.2.3-2.2.4 of class notes. |
Class 8 | Examples of deformations | Review section 2.2.6 of class notes. |
Class 9 | Push-forward and pull-back operations | Review section 2.2-A2 of class notes. |
Class 10 | Deformation and strain rates | Review sections 2.3.1 of class notes. |
Class 11 | Traction vectors and stress tensors; extremal stress values | Review sections 3.1-3.2 of class notes |
Class 12 | Examples of states of stress; alternative stress tensors | Review sections 3.3-3.4 of class notes |
Class 13 | Conservation of Mass | Review sections 4.1 of class notes. |
Class 14 | Momentum balance principles | Review sections 4.3 of class notes. |
Class 15 | Balance of mechanical energy | Review sections 4.4 of class notes. |
Holzapfel, G. A., 2001, “Nonlinear solid mechanics”, John Wiley, Chichester.
Class notes are available in the Instructor’s HP.
Ogden, R. W., 1997, “Non-linear elastic deformations”, Dover publications, New York.
Students' knowledge of nonlinear solid mechanics and their ability to apply them to problems will be assessed.
Midterm exam 30%, Final exam 50%, exercise problems 20%.
Basic solid mechanics