Nonlinear Dynamics of Structures

Introduction.- Thermodynamic Basis of the Motion Equation.- Introduction.- Kinematics of the Deformable Bodies.- Basic definitions of tensors describing the kinematics of a point in the space.- Strain Measurements.- Mechanical variables relations.- The Objective Derivative.- Velocity.- Stress Measurements.- Thermodynamics Basis.- First Law of Thermodynamics.- Second Law of Thermodynamics.- Lagrangian local form of Mechanical Dissipation.- Internal Variables.- Dynamic Equilibrium Equation for a Discrete Solid.- Different types of Nonlinear Dynamic Problems.- Materials.Nonlinearity.- Solution of the Motion Equation.- Introduction.- Explicit-implicit solution.- Implicit solution.- Equilibrium at time (t + Δt).- Equilibrium solution in time –implicit methods.- Newmark´s procedure.- Houbolt´s procedure.- Solution of the nonlinear-equilibrium equations system.- Newton-Raphson Method.- Modified Newton-Raphson Method.- Convergence accelerators.- Aitken accelerator or extrapolation algorithm.-B.F.G.S Algorithms.- Secant-Newton algorithms.- “Line-Search”algorithms.- Solution control algorithms – “Arc-Length”.- Ecuación de control de desplazamiento – Superficie esférica.- Convergence Analysis of the dynamic solution.- Introduction.- Reduction to the linear elastic problem.- Solution of second-order linear symmetric systems.- The dynamic equilibrium equation and its convergence-consistency and stability.- Solution stability of second –order linear symmetric systems.- Stability analysis procedure.- Determination of A and L for “Newmark”.- Determination of A and L for central differences- Newmark´s explicit form.- Solution stability of second-order non-linear symmetric systems.- Stability of the linearized equation.- Energy conservation algorithms.- APPENDIX - 1.- APPENDIX - 2.- Time-independent models.- Introduction.- Elastic behavior.- Invariant of the tensors.- Non-linear Elasticity.- Introduction.- Non-linear hyper-elastic model.- Stress based hyper-elastic model.- StabilityPostulates.- Plasticity in small deformations.- Introduction.- Discontinuity behavior or plastic yield criterion.- Elasto-Plastic behavior.- Levy-Mises theory.- Prandtl-Reus theory.- The classic plasticity theory.- Plastic unit or Specific work.- Plastic loading surface. Plastic hardening variable.- Isotropic hardening.- Kinematic hardening.- Stress-Strain relation. Plastic consistency and Tangent rigidity.- Drucker´s stability postulate and maximum plastic dissipation.- Stability condition.- Local stability.- Global stability.- Condition of Unicity of Solution.- Kuhn-Tucker. Loading-unloading condition.- Yield or plastic discontinuity classic criteria.- Rankine criterion of maximum tension stress.- Tresca criterion of maximum shear stress.- Von Mises criterion of octahedral shear stress.- Mohr-Coulomb criterion of octahedral shear stress.- Drucker-Prager criterion.- Geomaterials plasticity.- Basis of the plastic-damage model.- Mechanical behavior required for the constitutive model formulation.- Some characteristics of the plastic damage model.- Main variables of the plastic-damage model.- Definition of the plastic damage variable.- Definition of the law of evolution of cohesion c -κp.- Definition of the variable φ internal friction angle.- Variable definition ψ, dilatancy angle.- Generalization of the damage model with stiffness degradation.- Introduction.- Elasto-plastic constitutive equation with stiffness degradation.- Tangent constitutive equation for stiffness degradation processes.- Particular yield functions.- Mohr-Coulomb modified function.- Drucker-Prager Modified function.- Isotropic Continuous Damage – Introduction.- Isotropic damage model.- Helmholtz´s free energy and constitutive equation.- Damage threshold criterion.- Evolution law of the internal damage variable.- Constritutive tensor of tangent damage.- Particularization of the damage criterion.- General Softening.- Exponential softening.- Linear softening.- Particularization of the stress threshold function.- Simo -Ju. Model.- Setting of A parameter for Simo-Ju. Model.- Lemaitre and Mazars Model.- General model for different damage surfaces.- Setting of A parameter.- Time-dependent Models.- Introduction.- Constitutive equations based on spring-damping analogies.- Kelvin simplified model.- Maxwell simplified model.- Kelvin generalized model.- Kelvin multiple generalized model.- Maxwell generalized model.- Maxwell multiple generalized model.- Dissipation Evaluation.- Multiaxial generalization of the viscoelastic constitutive laws.- Multiaxial form of viscoelastic models.- Numerical solution of the integral and algorithms.- Kelvin model in dynamic problems.- Kelvin model dissipation.- Equation of the dynamic equilibrium for Kelvin model.- Stress considerations. Rayleigh vs. Kelvin model.- Dissipation considerations. Rayleigh vs. Kelvin model.- Cantilever beam.- Frame with rigid beam and lumped mass.- Viscoplasticity.- Limit states of viscoplasticity.- Over stress function.- Integration algorithm for the viscoplastic constitutive equation.- Particular case of the Duvaut-Lyon model a Von Mises viscoplastic material.