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16.225

Computational Mechanics of Materials

3-0-9 units · Graduate

Formulation of numerical (finite element) methods for the analysis of the nonlinear continuum response of materials. The range of material behavior considered includes finite deformation elasticity and inelasticity. Numerical formulation and algorithms include variational formulation and variational constitutive updates; finite element discretization; constrained problems; time discretization and convergence analysis. Strong emphasis on the (parallel) computer implementation of algorithms in programming assignments. The application to real engineering applications and problems in engineering science are stressed throughout. Experience in either C++, C, or Fortran required.

Fall 2003

Topics

  • Elastic Solids; Legendre Transformation; Isotropy; Equilibrium; Compatibility; Constitutive Relations; Variational Calculus; Example of a Functional: String; Extrema - Calculus of Variations; Local Form of Stationarity Condition
  • Vainberg Theorem; Hu-Washizu Functional
  • Specialized (Simplified) Variational Principles; Hellinger-Reissner Principle; Complementary Energy Principle; Minimum Potential Energy Theorem; Approximation Theory; Rayleigh - Ritz Method
  • Weighted - Residuals / Galerkin; Principle of Virtual Work; Geometrical Interpretation of Galerkin’s Method; Galerkin Weighting; Best Approximation Method; The Finite Element Method
  • Sobolev Norms; Global Shape Function; Computation of K and f ext ; Isoparametric Elements
  • Higher Order Interpolation; Isoparametric Triangular Elements; Numerical Integration; Gauss Quadrature
  • Error Estimation, Convergence of Finite Element Approximations; Error Estimates From Interpolation Theory
  • Linear Elasticity; Numerical Integration Errors; Basic Error Estimates; Conditions for Convergence; Patch Test
  • Incompressible Elasticity; Hooke’s Law; Governing Equations; “B”-Matrix; Volumetric and Deviatoric Components of “K h ”
  • Constraints Ratio; Variational Principle of Incompressible Elasticity; Saddle Point Problem; Constrained Minimization Problem; Reduced Selective Integration; Penalty Formulation
  • Assumed Strain Methods; Euler Equations; Mean Dilatation Method; General Expression for Anisotropic Elasticity; Mixed Methods; Discretized Lagrangian
  • Finite Elasticity; Metric Changes; State of Stress; Field Equations: Linear Momentum Balance, Angular Momentum Balance, Energy Balance; Nonlinear Elastic Solid
  • Variational Formulation; Minimum Potential Energy Principle; Finite Element Approximations; Rayleigh - Ritz Method; Galerkin Approach
  • Newton-Raphson Solution Procedure; Continuation Method; Iteration Process; Computation of Tangent Stiffness; Spatial Formulation
  • Isoparametric Elements; Commutative Diagram; Tangent Stiffness; Calculation of Tangent Stiffness (continued); Material Frame Indifference; Lagrangian Moduli
  • Material Formulation; Specific Material Models; Isotropic Elasticity; Stress-strain Relations; Cayley-Hamilton Theorem; Examples of Constitutive Relations for Finite Elasticity; Saint-Venant / Kirchhoff Model; Mooney-Riulin Model; Neo-Hookean Model Extended to Compressible Range; Computation of Tangent Moduli
  • Time Dependent Problems; Nonlinear Elastodynamics (Hyperbolic); Nonlinear Heat Conduction (Parabolic); Initial Boundary Value Problem (IBVP); Finite Element (semi) Discretization
  • Constitutive Relations: Fourier Law of Heat Conduction; Finite Element Discretization (Spatial); Time-stepping Algorithms; Newmark Predicators; Newmark Correctors; Convergence Check; Explicit Dynamics
  • Trapezoidal Rule - Heat Conduction; Trapezoidal Predictor; Equivalent Static Problem; Trapezoidal Correctors; Convergence Check
  • Connection Between Newmark Algorithm and Multistep Methods; Mass Humping; Consistent Mass; Nodal Quadrature; Row (Column) Sum Method; Algorithms Analysis; General Initial Value Problem (IVP)
  • Energy Conservation / Dissipation; Abstract Algorithms; Convergence; Conditions of Convergence; Consistency
  • Examples: Trapezoidal Rule; Newmark’s Algorithm; Stability; Trapezoidal Rule, Scalar Problem
  • Multidimensional Case; Spectral Radius, Lax Equivalence Theorem
  • Stability Properties of Trapezoidal Rule; Eigenprojections; Choice of time step; Stability of Newmark’s Algorithm; Iron’s Bounding Principle
  • Nonlinear Algorithms; Small-strain Plasticity; Kuhn-Tucker Form; Elastic-plastic Moduli; Isotropic-kinematic Hardening
  • Time-stepping Algorithms for Constitutive Relations; Numerical Quadrature; Newton-Raphson Solution Procedure; Backward Euler; Geometrical Interpretation; Closest Point Projection Algorithms; J 2 -isotropic Hardening

Readings

  • Marsden, J. E., and T. J. R. Hughes. Mathematical Foundations of Elasticity. Prentice-Hall, 1983.
    Listed reading
  • Malvern, L. E. Introduction to the Mechanics of a Continuous Medium. Prentice-Hall, 1969.
    Listed reading
  • Gurtin, M. E. An Introduction to Continuum Mechanics. Academic Press, 1981.
    Listed reading
  • Bathe, K. J. Finite Element Procedures. Prentice Hall, 1996.
    Listed reading
  • Hughes, T. J. R. The Finite Element Method, Linear Static and Dynamic Finite Element Analysis. Dover.
    Listed reading
  • Zienkiewicz, O. C., and R. L. Taylor. The Finite Element Method. Mc-Graw Hill, 1989.
    Listed reading

Prerequisites & course information

Same subject as 2.099[J] Prereq: Permission of instructor Acad Year 2027-2028: Not offered Acad Year 2028-2029: G (Spring) 3-0-9 units

Sources