ME 6770: Energy and Variational Methods in Elasticity and Plasticity

Offered Every Fall


Credit Hours: 3-0-3
Prerequisites: MATH 2403 and (AE3120 or ME 3201)
Catalog Description: Applications in energy and variational methods in engineering mechanics to elastic, plastic and dynamical behavior of deformable media. Crosslisted with AE 6770.
Textbooks: J. N. Reddy, Energy and Variational Methods in Applied Mechanics, 1st Edition, John Wiley, 1984.
Instructors: Chris Lynch (ME), David McDowell (ME), Richard Neu (ME), Jianmin Qu (ME), Min Zhou (ME), Olivier A. Bauchau (AE)
Mechanics of Materials Research Group
Audience: First and second year graduate students in ME, AE, CE and MSE.
Goals: This class will emphasize the applications of energy and variational methods in engineering mechanics. Several energy and numerical methods that are derived from the variational principles will be introduced and their use will be illustrated through various examples including elasticity, plasticity, dynamics, etc.
Topics:
  • Review of Fundamental Equations
    • Stresses, strains and their relationships
    • Equations of equilibrium
    • Compatibility conditions
    • Boundary conditions
  • Review of variational calculus
    • Functional and its first variation
    • Euler equations
    • Natural and essential boundary conditions
    • Constraint conditions and Lagrange multiplier
  • Virtual work
    • Principle of virtual work
    • Principle of complimentary virtual work
  • Variational Principles
    • Principle of minimum potential energy
    • Principle of minimum complimentary potential energy
    • Hellinger-Reissner principle
    • Hu-Washizu principle
  • Energy theorems in mechanics
    • Castigliano's theorems
    • Rayleigh-Betti reciprocity theorem
  • Approximate methods
    • Rayleigh-Ritz method
    • Galerkin method
    • Weighted residual methods
    • Finite element method
    • Boundary element method
  • Applications
    • Elasticity
      • bending of beams and plates
      • torsion of prismatic bars (St. Venant problem)
      • buckling and natural frequencies (eigenvalue problems)
    • Elastoplasticity
      • deformation theory
      • flow theory
      • limit analysis
    • Composite materials