TAM 545: Advanced Continuum Mechanics

Class Description:

This course is intended to introduce students to a broad spectrum of theories of contemporary continuum mechanics and thermodynamics. Following a review of classical continuum mechanics, a range of more advanced thermomechanics theories (e.g., internal variables, micropolar, non-local) are outlined from a broad perspective. These are then used to construct multifarious non-classical, multi- and/or coupled-field theories. The student may then go more in depth into any particular model. The primary focus of the course is on the construction of constitutive laws, with the leitmotif being that each continuum theory, and its various spin-offs, offers its own pros and cons. Some aspects of solution methods of initial-boundary value problems are also discussed. Mathematical concepts (e.g., elements of group theory, Legendre transforms, functionals) are introduced as needed. 
Every advanced model/theory can be presented in terms of simple and self-explanatory examples. The objective is to emphasize physical examples and applications to help students gain proper understanding of various phenomena so as to develop a panoramic view of continuum mechanics. 
This course is intended to introduce students to a broad spectrum of theories of contemporary continuum mechanics and thermodynamics. Following a review of classical continuum mechanics, a range of more advanced thermomechanics theories (e.g., internal variables, micropolar, non-local) are outlined from a broad perspective. These are then used to construct multifarious non-classical, multi- and/or coupled-field theories. The student may then go more in depth into any particular model. The primary focus of the course is on the construction of constitutive laws, with the leitmotif being that each continuum theory, and its various spin-offs, offers its own pros and cons. Some aspects of solution methods of initial-boundary value problems are also discussed. Mathematical concepts (e.g., elements of group theory, Legendre transforms, functionals) are introduced as needed. 
 
Every advanced model/theory can be presented in terms of simple and self-explanatory examples. The objective is to emphasize physical examples and applications to help students gain proper understanding of various phenomena so as to develop a panoramic view of continuum mechanics. 
 

Textbook:
O. Gonzalez & A.M. Stuart (2008), A First Course in Continuum Mechanics, Cambridge University Press.

 
Reference books:
H. Ziegler (1983), An Introduction to Thermomechanics, North-Holland.
J. Ignaczak and M. Ostoja-Starzewski (2009), Thermoelasticity with Finite Wave Speeds, Oxford University Press.
G.A. Maugin (1998), The Thermomechanics of Nonlinear Irreversible Behaviours: An Introduction, World Scientific. 
W. Nowacki (1986), Theory of Asymmetric Elasticity, Pergamon Press. 
A.C. Eringen (1999), Microcontinuum Field Theories I, II, Springer. 
G.A. Maugin (2017), Non-Classical Continuum Mechanics: A Dictionary, Springer. 
R. Temam & A. Miranville (2002), Mathematical Modeling in Continuum Mechanics, Cambridge University Press.
 

Topics:

Classical and rational continuum mechanics (12 hours)
kinematics (review); stress (review); balance laws (review); balance laws via invariance of energy; constitutive equations: axioms, restrictions/constraints; memory functionals; rational and extended continuum theories
 
Introduction to thermomechanics with internal variables (12 hours)
free energy and dissipation functionals; Legendre transformations; from non-Newtonian fluids to visco-plasticity of metals and soils; thermodynamic orthogonality; non-Fourier heat conduction; primitive thermomechanics; damage thermomechanics
 
Introduction to generalized continuum theories (20 hours) 
classical versus generalized thermoelasticity theories; Cosserat-type (micro-continuum) models of solids and fluids; granular media, lattices, helices and chiral media; strain-gradient, stress-gradient; non-local models; deterministic versus stochastic fields in fluids and solids; fractional calculus, fractal media; violations of second law of thermodynamics
 
Coupled fields (8 hours)
visco-thermoelasticity; permeability, poromechanics, thermodiffusion; electromagnetism; magnetoelasticity, piezoelectricity, …
 
Dimensional analysis and similarity theories(4 hours)
 
Singular surfaces and waves (4 hours)
acceleration waves; shock waves

All Courses

ME

TAM