Special topics in civil and environmental engineering offered occasionally by permanent or visiting faculty members.
This course will present the Foundations of Continuum Mechanics for Master's and Ph.D. level students.
* Week 1: Kinematics of a general motion; tangent vectors (line segments), deformation gradient, Green-Lagrange strain and Almansi strain tensors, push-forward and pull-back, Lie-derivative. * Week 2: Equilibrium and Principle of virtual displacements in Eulerian and Lagrangian form; Cauchy stress, 1st and 2nd Piola-Kirchhoff stress, Kirchhoff stress tensor, Piola transform. * Week 3: Constitutive relations; Strain energy and free energy; Objectivity; Isotropic versus Anisotropic materials. * Week 4: Variational principles in mechanics (Minimum of potential energy, Hellinger-Reissner functional, Hu-Washizu principle); Euler equations; Balance laws. * Week 5: Wrap up.
* Marsden & Hughes: Mathematical Foundation of Elasticity, Dover * Holzapfel, G.: Nonlinear Solid Mechanics: A Continuum Approach for Engineering, Springer
Student learning goals
Develop skills necessary for reading and understanding top papers in the field. The focus is on communicating between notation and writing styles of both engineering sciences and applied mathematics.
Develop a deeper understanding of large deformation kinematics of continua. You will also get some basic insight into theory of manifolds which governs large deformation continuum mechanics.
Understand definitions and relations between alternative stress measures; Understand their similarities and differences.
Gain basic understanding of objectivity and its implication to material models. This is key for understanding complex constitutive models for materials such as metals, soils, concrete, composites, or biological tissue.
Understand how to develop and use alternative variational principles to find solutions to problems in nonlinear continuum mechanics.
General method of instruction
The course will be taught in a seminar style setting. A usual session will last 2 hours, including presentation of new material and discussion.
Review you previous mechanics of materials and/or continuum course notes. You should be familiar with the representation of stress and strain as tensors. Basic knowledge of material equations (Hooke's law in 2D/3D) is highly recommended.
Class assignments and grading
There will be four assignments/discussion sessions (20% each) and a final discussion session (20%).