ME458 - BME487 NONLINEAR FINITE ELEMENT ANALYSIS
date 
lect.#
topics 
    Primary references:
Cook4th:= R.D.Cook, D.S. Malkus, M.E. Plesha, R.J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed., Wiley, 2001.
Cook3rd:= R.D.Cook, D.S. Malkus, M.E. Plesha, Concepts and Applications of Finite Element Analysis, 3rd ed., Wiley, 1989.
Bathe:= K-J Bathe, Finite Element Procedures, Prentice Hall 1996.
Sahmes&Dym:=I.H. Shames and C.L. Dym, Energy and Finite Element Methods in Structural Mechanics, Taylor and Francis, 2003.
Bonet&Wood:=J. Bonet and R.D. Wood, Nonlinear Continuum Mechanics for Finite Elemnt Analysis, Cambridge U. Press, 1997.
Srinivasan &Perucchio:= R. Srinivasan and R. Perucchio, "Finite element analysis of anisotropic nonlinear incompressible elastic solids by a mixed model,” IJNME, 37, 3075-3092, 1994.
1/18/07 1 1-D linear elastic problem: equilibrium eq.s and boundary conditions, exact solution, finite difference discretization, variational formulation,  Rayleigh-Ritz approximation.
Ref: Cook4th: Ch. 4:4.1,4.2,4.3, 4.4 (focus on bar under axial load), 4.5, 4.6.
1/23/07
2
 1-D linear elastic problem: variational formulation, Rayleigh-Ritz approximation with polynomial expansion, principle of virtual works, R-R approximation based on domain discretization.
Ref: Cook4th: Ch. 4:4.1,4.2,4.3, 4.4 (focus on bar under axial load), 4.5, 4.6.
1/25/07
3
 1-D linear elastic problem: derivation of finite element (FE) formulation from Rayleigh-Ritz, equilibrium eq.s in FE form, evaluation of [K] for the linear 1-D element.
Ref: Cook4th: Ch2:2.1, 2.2; Ch.3:3.2.
1/30/07 4 1-D linear elastic problem: lagrangian shape functions (linear and quadratic); isoparametric formulation of bar element, assembly of global [K], partitioning, boundary conditions, solution.
Ref: Cook4th: Ch. 4:4.8.
Ref: Cook3rd: Ch2:2.1-2.5.
2/1/07 5
 1-D linear elastic problem:  bar element in 2-D Euclidean space: transformation from local to global co-ord.s, transformation of the stiffness matrix.
Ref: Cook3rd: Ch7: 7.1,7.2, 7.4, 7.5.
Continuum mechanics: Cartesian tensors: vector transformations, definition of tensors, symmetry and skew symmetry, inner product, cross product.
Ref: Shames&Dym: Appendix 1.
       Bathe: Ch2:2.3, 2.4.
2/6/07 6
 Continuum mechanics: Cartesian tensors: field operators, Gauss theorem, Green's formula, Kronecker delta, ex: tensorial  nature of [K].
Continuum mechanics:  Stress: force distributions, stress, Cauchy formula.
Ref: Shames&Dym: Appendix 1. Ch1: 1.1-1.4.
       Bathe: Ch2:2.3, 2.4.
2/7/07 7
 Continuum mechanics:  Stress: equations of motion, stress symmetry, stress is a second-order tensor, principal stresses and principal directions, tensor invariants.
Ref: Shames&Dym:  Ch1: 1.4-1.6.
2/8/07 8
 Continuum mechanics:  Strain: Green and Almansi strains, infinitesimal strains, engineering strains, physical interpretation of strain components, variation of volume as a function of normal strains.
Ref: Shames&Dym:  Ch1: 1.7-1.8.
2/12/07 9 Continuum mechanics:  Strain: deformation gradient, rotation tensor, transformation of strains, compatibility equations.
Ref: Shames&Dym:  Ch1: 1.9-1.11.
2/13/07 10
 Continuum mechanics:  Constitutive laws: strain energy density function, mechanical work during deformation, elastic behavior, complementary strain energy.
Ref: Shames&Dym:  Ch1: 1.13. 
L
2/14/07 11 Continuum mechanics:  Constitutive laws: Hooke's law, Lame' constants for homogeneous and isotropic linear elastic materials, Young's modulud and Poisson's ratio.
Ref: Shames&Dym:  Ch1: 1.13.
Linear finite elements: Derivation of the principle of virtual work.
Ref: Cook4th: Ch3:3.1- 3.8.
3/06/07 12 Linear finite elements: Isoparametric formulation: 1-D element, 2-D element, Jacobian matrix, [B] matrix, numerical integration of [K].
Ref: Cook4th: Ch6:6.1- 6.3.
       Bathe: Ch5:5.1-5.2 .
3/1/07 13 Linear finite elements: Isoparametric formulation: 1-D element, 2-D element, Jacobian matrix, [B] matrix, numerical integration of [K].
Ref: Cook4th: Ch6:6.1- 6.3.
       Bathe: Ch5:5.1-5.2 .
3/6/07 14 Linear finite elements: Isoparametric formulation: element distortions, quarter-point element for linear elastic fracture mechanics; stress calculation at Gauss points, smoothed stress field built by extrapolation.
Ref: Cook4th: Ch6:6.10 - 6.11.
       Bathe: Ch5:5.5.4, 5.5.5
3/20/07 15 FE Nonlinear Analysis: classification of nonlinear analysis, ex pr: elastic-plastic bar, ex pr.: contact.
Ref: Bathe: Ch6:6.1
3/21/07 16 FE Nonlinear Analysis: Ex pr: snap-through. Incremental solution algorithms, Newton-Raphson iterations, modified Newton-Raphson iterations, load increments vs. displacement increments.
Ref: Bathe: Ch6:6.1
3/22/07 17
MIDTERM EXAM
3/27/07 18 Review and discussion of exam.
3/29/07 19 Continuum mechanics: kinematics: Lagrangian formulation, deformation gradient, mass density, computation of the deform. gradient for isoparametric formulation, computation of the defor. gradient for Lagrangian element.
Ref: Bathe: Ch6:6.2.2
4/03/07 21 Continuum mechanics: kinematics:  Cauchy-Green deformation tensor, stretch, angle between line elements, polar decomposition of the defor. gradient.
Ref: Bathe: Ch6:6.2.2
4/05/07
cancelled
4/10/07 22 Continuum mechanics: kinematics:  right and left stretch, velocity gradient L, decomposition L=D+W
Ref: Bathe: Ch6:6.2.2
4/12/07 23 Continuum mechanics: kinematics:   Green-Lagrange strain tensor, Hencky strain tensor, Rate of change of G-L strais.
Ref: Bathe: Ch6:6.2.2
4/17/07 24 Continuum mechanics: kinematics: relationship between G-L strains and infinitesimal strains. Second Piola-Kirchhoff stress tensor.Second Piola-Kirchhoff stress tensor: properties. IIP-K stresses and G-L strains are invariant under rigid body motions (objective measures).
FE Nonlinear Formulation: transformation of internal virtual work, Total Lagrangian (TL) and Updated Lagrangian (UL) formulations.
Ref: Bathe: Ch6:6.2.2-6.2.3
4/19/07 25 FE Nonlinear Formulation: incremental TL: eq. of motion, incremental decomposition of stresses and strains, linearization, linearized incremental eq. of motion; incremental UL: eq. of motion, incremental decomposition of stresses and strains, linearization, linearized incremental eq. of motion. Error due to linearization. Newton-Raphson iterations for the T.L. formulation.
Ref: Bathe: Ch6:6.2.3
4/24/07 26 FE Nonlinear Formulation: loading: deformation dependent; inertial forces. Materially-nonlinear-only analysis. Displacement-based isoparametric continuum formulation: linearization of V.W. w.r.t. FE variables. Matrix equations: T.L. formulation, U.L. formulation, Materially-nonlinear-only formulation
Ref: Bathe: Ch6:6.3.1-6.3.2
4/26/07 27 FE Nonlinear Formulation: FE matrices; 2-D element (plane strain/stress): Jacobian transformation, U.L. formulation, incremental strains, linear and nonlinear B matrices.
Ref: Bathe: Ch6:6.3.2-6.3.4
5/01/07 28 FE Nonlinear Formulation: solution process, elastic material models, generalized Hooke's law, Almansi strain tensor and its properties. Incompressibility: isochoric condition based on the Cauchy-Green strain tensor, augmented hyperelastic strain energy density function.
Ref: Bathe: Ch6:6.6 - 6.6.1
5/03/07 29 ORAL EXAM




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updated 19 April 2007 By Renato Perucchio