TAM 545

Unified treatment of modern continuum mechanics: mathematical preliminaries; review of kinematics and general balance laws; general theory of mechanical constitutive equations, including material constraints and material symmetry. Course Information: 4 graduate hours. No professional credit. Prerequisite: TAM 445 OR TAM 551.

Introduction to a broad spectrum of theories of contemporary continuum mechanics and thermodynamics. First, basic continuum mechanics is outlined from a modern perspective. This provides a stepping-stone to several more advanced theories: rational continuum mechanics, thermomechanics with internal variables, generalized continuum theories (micropolar, micromorphic, non-local…), multi-scale, and/or coupled-field theories (hyperbolic thermoelasticity, poromechanics, interactions with electromagnetic fields…). 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 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.

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

5/11/2020