| Homework problems |
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| Contact Info | Leonid Berlyand, 337 McAllister Bldg, telephone: 863-9683, email: berlyand@math.psu.edu |
| Office hours | Thursday: 2pm-3pm and by appointment |
| Academic Integrity Statement | 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. Additional texts: - 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) |
| Course content |
Course Topics: 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. |