MIT OpenCourseWare

Browse materialsMIT 16.225 · Fall 2003

Computational Mechanics of Materials

Subjects: Engineering · Science

Art: Claude Louis Desrais, Montgolfiere 1783, 18th… · Public domain

Instructor: Prof. Raúl Radovitzky
Term: Fall 2003
Level: Graduate
What's inside: 1 lecture note · 31 other files · 3 web pages
Materials open: MIT OCW

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About this course

Source: MIT OpenCourseWare 16.225 Computational Mechanics of Materials, Fall 2003, Prof. Raúl Radovitzky — original course page on ocw.mit.edu ↗. License: CC BY-NC-SA 4.0. Study order: MIT's own listing order, section by section. Taken from MIT's published course files, downloaded from ocw.mit.edu on September 23, 2026. How RedLectern is built.

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Lecture Notes

26 files

Assignments

MIT OpenCourseWare, Prof. Raúl Radovitzky, 16.225 Computational Mechanics of Materials, Fall 2003.

License: CC BY-NC-SA 4.0 · Edition: Fall 2003 · Open the original course on OCW ↗

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Computational Mechanics of Materials

Before you start

What's here, and what isn't

Materials by section

Lecture Notes

Assignments

Lecture 1: 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

Lecture 2: Vainberg Theorem; Hu-Washizu Functional

Lecture 3: Specialized (Simplified) Variational Principles; Hellinger-Reissner Principle; Complementary Energy Principle; Minimum Potential Energy Theorem; Approximation Theory; Rayleigh - Ritz Method

Lecture 4: Weighted - Residuals / Galerkin; Principle of Virtual Work; Geometrical Interpretation of Galerkin’s Method; Galerkin Weighting; Best Approximation Method; The Finite Element Method

Lecture 5: Sobolev Norms; Global Shape Function; Computation of K and feˣt; Isoparametric Elements

Lecture 6: Higher Order Interpolation; Isoparametric Triangular Elements; Numerical Integration; Gauss Quadrature

Lecture 7: Error Estimation, Convergence of Finite Element Approximations; Error Estimates From Interpolation Theory

Lecture 8: Linear Elasticity; Numerical Integration Errors; Basic Error Estimates; Conditions for Convergence; Patch Test

Lecture 9: Incompressible Elasticity; Hooke’s Law; Governing Equations; “B”-Matrix; Volumetric and Deviatoric Components of “Kh”

Lecture 10: Constraints Ratio; Variational Principle of Incompressible Elasticity; Saddle Point Problem; Constrained Minimization Problem; Reduced Selective Integration; Penalty Formulation

Lecture 11: Assumed Strain Methods; Euler Equations; Mean Dilatation Method; General Expression for Anisotropic Elasticity; Mixed Methods; Discretized Lagrangian

Lecture 12: Finite Elasticity; Metric Changes; State of Stress; Field Equations: Linear Momentum Balance, Angular Momentum Balance, Energy Balance; Nonlinear Elastic Solid

Lecture 13: Variational Formulation; Minimum Potential Energy Principle; Finite Element Approximations; Rayleigh - Ritz Method; Galerkin Approach

Lecture 14: Newton-Raphson Solution Procedure; Continuation Method; Iteration Process; Computation of Tangent Stiffness; Spatial Formulation

Lecture 15: Isoparametric Elements; Commutative Diagram; Tangent Stiffness; Calculation of Tangent Stiffness (continued); Material Frame Indifference; Lagrangian Moduli

Lecture 17: Time Dependent Problems; Nonlinear Elastodynamics (Hyperbolic); Nonlinear Heat Conduction (Parabolic); Initial Boundary Value Problem (IBVP); Finite Element (semi) Discretization

Lecture 18: Constitutive Relations: Fourier Law of Heat Conduction; Finite Element Discretization (Spatial); Time-stepping Algorithms; Newmark Predicators; Newmark Correctors; Convergence Check; Explicit Dynamics

Lecture 19: Trapezoidal Rule - Heat Conduction; Trapezoidal Predictor; Equivalent Static Problem; Trapezoidal Correctors; Convergence Check

Lecture 20: Connection Between Newmark Algorithm and Multistep Methods; Mass Humping; Consistent Mass; Nodal Quadrature; Row (Column) Sum Method; Algorithms Analysis; General Initial Value Problem (IVP)

Lecture 21: Energy Conservation / Dissipation; Abstract Algorithms; Convergence; Conditions of Convergence; Consistency

Lecture 22: Examples: Trapezoidal Rule; Newmark’s Algorithm; Stability; Trapezoidal Rule, Scalar Problem

Lecture 23: Multidimensional Case; Spectral Radius, Lax Equivalence Theorem

Lecture 24: Stability Properties of Trapezoidal Rule; Eigenprojections; Choice of time step; Stability of Newmark’s Algorithm; Iron’s Bounding Principle

Lecture 25: Nonlinear Algorithms; Small-strain Plasticity; Kuhn-Tucker Form; Elastic-plastic Moduli; Isotropic-kinematic Hardening

Lecture 26: Time-stepping Algorithms for Constitutive Relations; Numerical Quadrature; Newton-Raphson Solution Procedure; Backward Euler; Geometrical Interpretation; Closest Point Projection Algorithms; J₂-isotropic Hardening