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Computational Mechanics of Materials >> Content Detail



Lecture Notes



Lecture Notes

The course instructor worked from these notes when presenting each lecture. After the lecture, the notes for the day were posted online for students to access.

LEC #TOPICS
1Elastic Solids; Legendre Transformation; Isotropy; Equilibrium; Compatibility; Constitutive Relations; Variational Calculus; Example of a Functional: String; Extrema - Calculus of Variations; Local Form of Stationarity Condition (PDF)
2Vainberg Theorem; Hu-Washizu Functional (PDF)
3Specialized (Simplified) Variational Principles; Hellinger-Reissner Principle; Complementary Energy Principle; Minimum Potential Energy Theorem; Approximation Theory; Rayleigh - Ritz Method (PDF)
4Weighted - Residuals / Galerkin; Principle of Virtual Work; Geometrical Interpretation of Galerkin's Method; Galerkin Weighting; Best Approximation Method; The Finite Element Method (PDF)
5Sobolev Norms; Global Shape Function; Computation of K and fext; Isoparametric Elements (PDF)
6Higher Order Interpolation; Isoparametric Triangular Elements; Numerical Integration; Gauss Quadrature (PDF)
7Error Estimation, Convergence of Finite Element Approximations; Error Estimates From Interpolation Theory (PDF)
8Linear Elasticity; Numerical Integration Errors; Basic Error Estimates; Conditions for Convergence; Patch Test (PDF)
9Incompressible Elasticity; Hooke's Law; Governing Equations; "B"-Matrix; Volumetric and Deviatoric Components of "Kh" (PDF)
10Constraints Ratio; Variational Principle of Incompressible Elasticity; Saddle Point Problem; Constrained Minimization Problem; Reduced Selective Integration; Penalty Formulation (PDF)
11Assumed Strain Methods; Euler Equations; Mean Dilatation Method; General Expression for Anisotropic Elasticity; Mixed Methods; Discretized Lagrangian (PDF)
12Finite Elasticity; Metric Changes; State of Stress; Field Equations: Linear Momentum Balance, Angular Momentum Balance, Energy Balance; Nonlinear Elastic Solid (PDF)
13Variational Formulation; Minimum Potential Energy Principle; Finite Element Approximations; Rayleigh - Ritz Method; Galerkin Approach (PDF)
14Newton-Raphson Solution Procedure; Continuation Method; Iteration Process; Computation of Tangent Stiffness; Spatial Formulation (PDF)
15Isoparametric Elements; Commutative Diagram; Tangent Stiffness; Calculation of Tangent Stiffness (continued); Material Frame Indifference; Lagrangian Moduli (PDF)
16Material 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 (PDF)
17Time Dependent Problems; Nonlinear Elastodynamics (Hyperbolic); Nonlinear Heat Conduction (Parabolic); Initial Boundary Value Problem (IBVP); Finite Element (semi) Discretization (PDF)
18Constitutive Relations: Fourier Law of Heat Conduction; Finite Element Discretization (Spatial); Time-stepping Algorithms; Newmark Predicators; Newmark Correctors; Convergence Check; Explicit Dynamics (PDF)
19Trapezoidal Rule - Heat Conduction; Trapezoidal Predictor; Equivalent Static Problem; Trapezoidal Correctors; Convergence Check (PDF)
20Connection Between Newmark Algorithm and Multistep Methods; Mass Humping; Consistent Mass; Nodal Quadrature; Row (Column) Sum Method; Algorithms Analysis; General Initial Value Problem (IVP) (PDF)
21Energy Conservation / Dissipation; Abstract Algorithms; Convergence; Conditions of Convergence; Consistency (PDF)
22Examples: Trapezoidal Rule; Newmark's Algorithm; Stability; Trapezoidal Rule, Scalar Problem (PDF)
23Multidimensional Case; Spectral Radius, Lax Equivalence Theorem (PDF)
24Stability Properties of Trapezoidal Rule; Eigenprojections; Choice of time step; Stability of Newmark's Algorithm; Iron's Bounding Principle (PDF - 1.1 MB)
25Nonlinear Algorithms; Small-strain Plasticity; Kuhn-Tucker Form; Elastic-plastic Moduli; Isotropic-kinematic Hardening (PDF)
26Time-stepping Algorithms for Constitutive Relations; Numerical Quadrature; Newton-Raphson Solution Procedure; Backward Euler; Geometrical Interpretation; Closest Point Projection Algorithms; J2-isotropic Hardening (PDF)

 








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