|Contact Info|| Leonid Berlyand, 337 McAllister Bldg,
telephone: 863-9683, email: firstname.lastname@example.org
|Office hours||Thursday: 2pm-3pm and by appointment|
|Academic Integrity |
|All Penn State Policies regarding ethics and honorable behavior apply to this course.|
|Textbook|| Primary text: Theory and Problems of Continuum Mechanics,
by George E. Mase,
Schaum's Outline Series, ISBN 9780070406636 |
Secondary text: The Mechanics and Thermoynamics of Continua, M. Gurtin, E. Fried and L. Anand.
- P. Ciarlet, Mathematical Elasticity
- R. Temam, Navier-Stokes Equations
- L. Evans, PDEs, chapter 8 on Calculus of Variations
- V. Berdichevsky, Variational Principles of Continuum Mechanics
- S.G. Michlin, The Problem of the Minimum of a Quadratic Functional. Holden-Day (1965)
Review of mathematical foundations: General tensors, metric tensors,
tensor fields. Review of some basic notions of classical mechanics: statics, kinematics and dynamics of a rigid body.
Deformation and strain: Lagrangian and Eulerian descriptions, compatibility conditions.
Motion and flow: materials derivatives, rate of deformation, vorticity
Fundamental laws of continuum mechanics: Conservation of mass, continuity equations, momentum principle, frame-indifference principle, conservation of energy. Equation of state, second law of thermodynamics. Ergodicity and mixing in Hamiltonian systems, concepts of temperature and entropy. Thermomechanical continuum.
Analysis of stress: Cauchy's theorem for existence of stress, the continuum concept, equations of equilibrium and principal of virtual work.
Elastic materials and their constitutive equations: Linear elasticity, Hooke's law, isotropy/anisotropy, elastic symmetries, existence theorems in elasticity, Korn's inequality. Some models of nonlinear elasticity and hyperelasticity.
Mechanics of fluids: Ideal, Stokesian and Newtonian fluids. Euler's and Navier-Stokes equations. Viscous dissipation of energy. In the Spring semester an advanced course on fluids will be offered.
Variational Principles of Continuum Mechanics for solids and fluids: Basic concepts of Calculus of Variations: first variation, Euler-Lagrange equation, second variation. Existence of minimizers: coercivity, lower semiconitinuity, convexity. Variational formulations for Stokes and Elasticity equations.
Composite materials and homogenization theory: If time permits, basics of the mechanics of composite materials and solids (multiphase mixtures) and mathematical techniques of homogenization theory (multiscale analysis) will be presented.
Plasticity: If time permits, basics of the Plasticity theory will be presented.
|Classroom||We meet in 309 Boucke.|
|Prerequisites||MATH 412; MATH 401 or MATH 403|
|Grading||The course grade will be based on homeworks and the project.|