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The Resource The finite element method for solid and structural mechanics, by Olek C. Zienkiewicz, Robert L. Taylor, David A. Fox

The finite element method for solid and structural mechanics, by Olek C. Zienkiewicz, Robert L. Taylor, David A. Fox

The finite element method for solid and structural mechanics
The finite element method for solid and structural mechanics
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by Olek C. Zienkiewicz, Robert L. Taylor, David A. Fox
The Finite Element Method for Solid and Structural Mechanics is the key text and reference for engineers, researchers and senior students dealing with the analysis and modeling of structures, from large civil engineering projects such as dams to aircraft structures and small engineered components. This edition brings a thorough update and rearrangement of the book's content, including new chapters on: Material constitution using representative volume elements Differential geometry and calculus on manifolds Background mathematics and linear shell theory Focusing on the core knowledge, mathematical and analytical tools needed for successful structural analysis and modeling, The Finite Element Method for Solid and Structural Mechanics is the authoritative resource of choice for graduate level students, researchers and professional engineers. A proven keystone reference in the library of any engineer needing to apply the finite element method to solid mechanics and structural design. Founded by an influential pioneer in the field and updated in this seventh edition by an author team incorporating academic authority and industrial simulation experience. Features new chapters on topics including material constitution using representative volume elements, as well as consolidated and expanded sections on rod and shell models
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The finite element method for solid and structural mechanics, by Olek C. Zienkiewicz, Robert L. Taylor, David A. Fox
The finite element method for solid and structural mechanics, by Olek C. Zienkiewicz, Robert L. Taylor, David A. Fox
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  • Machine generated contents note: 1.1.Introduction -- 1.2.Small deformation solid mechanics problems -- 1.2.1.Strong form of equation: Indicial notation -- 1.2.2.Matrix notation -- 1.2.3.Two-dimensional problems -- 1.3.Variational forms for nonlinear elasticity -- 1.4.Weak forms of governing equations -- 1.4.1.Weak form for equilibrium equation -- 1.5.Concluding remarks -- References -- 2.1.Introduction -- 2.2.Finite element approximation: Galerkin method -- 2.2.1.Displacement approximation -- 2.2.2.Derivatives -- 2.2.3.Strain-displacement equations -- 2.2.4.Weak form -- 2.2.5.Irreducible displacement method -- 2.3.Numerical integration: Quadrature -- 2.3.1.Volume integrals -- 2.3.2.Surface integrals -- 2.4.Nonlinear transient and steady-state problems -- 2.4.1.Explicit Newmark method -- 2.4.2.Implicit Newmark method -- 2.4.3.Generalized midpoint implicit form -- 2.5.Boundary conditions: Nonlinear problems -- 2.5.1.Displacement condition -- 2.5.2.Traction condition -- 2.5.3.Mixed displacement/traction condition -- 2.6.Mixed or irreducible forms -- 2.6.1.Deviatoric and mean stress and strain components -- 2.6.2.A three-field mixed method for general constitutive models -- 2.6.3.Local approximation of p and u -- 2.6.4.Continuous u-p approximation -- 2.7.Nonlinear quasi-harmonic field problems -- 2.8.Typical examples of transient nonlinear calculations -- 2.8.1.Transient heat conduction -- 2.8.2.Structural dynamics -- 2.8.3.Earthquake response of soil structures -- 2.9.Concluding remarks -- References -- 3.1.Introduction -- 3.2.Iterative techniques -- 3.2.1.General remarks -- 3.2.2.Newton's method -- 3.2.3.Modified Newton's method -- 3.2.4.Incremental-secant or quasi-Newton methods -- 3.2.5.Line search procedures: Acceleration of convergence -- 3.2.6."Softening" behavior and displacement control -- 3.2.7.Convergence criteria -- 3.3.General remarks: Incremental and rate methods -- References -- 4.1.Introduction -- 4.2.Tensor to matrix representation -- 4.3.Viscoelasticity: History dependence of deformation -- 4.3.1.Linear models for viscoelasticity -- 4.3.2.Isotropic models -- 4.3.3.Solution by analogies -- 4.4.Classical time-independent plasticity theory -- 4.4.1.Yield functions -- 4.4.2.Flow rule (normality principle) -- 4.4.3.Hardening/softening rules -- 4.4.4.Plastic stress-strain relations -- 4.5.Computation of stress increments -- 4.5.1.Explicit methods -- 4.5.2.Implicit methods: Return map algorithm -- 4.6.Isotropic plasticity models -- 4.6.1.Isotropic yield surfaces -- 4.6.2.J2 model with isotropic and kinematic hardening (Prandtl-Reuss equations) -- 4.6.3.Plane stress -- 4.7.Generalized plasticity -- 4.7.1.Nonassociative case: Frictional materials -- 4.7.2.Associative case: J2 generalized plasticity -- 4.8.Some examples of plastic computation -- 4.8.1.Perforated plate: Plane stress solutions -- 4.8.2.Perforated plate: Plane strain solutions -- 4.8.3.Steel pressure vessel -- 4.9.Basic formulation of creep problems -- 4.9.1.Fully explicit solutions -- 4.10.Viscoplasticity: A generalization -- 4.10.1.General remarks -- 4.10.2.Implicit solution -- 4.10.3.Creep of metals -- 4.10.4.Soil mechanics applications -- 4.11.Some special problems of brittle materials -- 4.11.1.The no-tension material -- 4.11.2."Laminar" material and joint elements -- 4.12.Nonuniqueness and localization in elasto-plastic deformations -- 4.13.Nonlinear quasi-harmonic field problems -- 4.14.Concluding remarks -- References -- 5.1.Introduction -- 5.2.Governing equations -- 5.2.1.Kinematics and deformation -- 5.2.2.Stress and traction for reference and deformed states -- 5.2.3.Equilibrium equations -- 5.2.4.Boundary conditions -- 5.2.5.Initial conditions -- 5.2.6.Constitutive equations: Hyperelastic material -- 5.3.Variational description for finite deformation -- 5.3.1.Reference configuration formulation -- 5.3.2.First Piola-Kirchhoff formulation -- 5.3.3.Current configuration formulation -- 5.4.Two-dimensional forms -- 5.4.1.Plane strain -- 5.4.2.Plane stress -- 5.4.3.Axisymmetric with torsion -- 5.5.A three-field, mixed finite deformation formulation -- 5.5.1.Finite element equations: Matrix notation -- 5.6.Forces dependent on deformation: Pressure loads -- 5.7.Concluding remarks -- References -- 6.1.Introduction -- 6.2.Isotropic elasticity -- 6.2.1.Isotropic elasticity: Formulation in invariants -- 6.2.2.Isotropic elasticity: Formulation in modified invariants -- 6.2.3.Isotropic elasticity: Formulation in principal stretches -- 6.2.4.Plane stress applications -- 6.3.Isotropic viscoelasticity -- 6.4.Plasticity models -- 6.5.Incremental formulations -- 6.6.Rate constitutive models -- 6.7.Numerical examples -- 6.7.1.Necking of circular bar -- 6.7.2.Adaptive refinement and localization (slip-line) capture -- 6.8.Concluding remarks -- References -- 7.1.Introduction -- 7.2.Coupling between scales -- 7.2.1.RVE with specified boundary displacements -- 7.2.2.Kirchhoff and Cauchy stress forms -- 7.2.3.Periodic boundary conditions -- 7.2.4.Small strains -- 7.3.Quasi-harmonic problems -- 7.4.Numerical examples -- 7.4.1.Linear elastic properties -- 7.4.2.Uniformly loaded plate: Cylindrical bending -- 7.4.3.Moment-curvature: Elastic-plastic response -- 7.5.Concluding remarks -- References -- 8.1.Introduction -- 8.2.Node-node contact: Hertzian contact -- 8.2.1.Geometric modeling -- 8.2.2.Contact models -- 8.3.Tied interfaces -- 8.3.1.Surface-surface tied interface -- 8.4.Node-surface contact -- 8.4.1.Geometric modeling -- 8.4.2.Contact modeling: Frictionless case -- 8.4.3.Contact modeling: Frictional case -- 8.5.Surface-surface contact -- 8.5.1.Frictionless case -- 8.6.Numerical examples -- 8.6.1.Contact between two disks -- 8.6.2.Contact between a disk and a block -- 8.6.3.Frictional sliding of a flexible disk on a sloping block -- 8.6.4.Upsetting of a cylindrical billet -- 8.7.Concluding remarks -- References -- 9.1.Introduction -- 9.2.Pseudo-rigid motions -- 9.3.Rigid motions -- 9.3.1.Equations of motion for a rigid body -- 9.3.2.Construction from a finite element model -- 9.3.3.Transient solutions -- 9.4.Connecting a rigid body to a flexible body -- 9.4.1.Lagrange multiplier constraints -- 9.5.Multibody coupling by joints -- 9.5.1.Translation constraints -- 9.5.2.Rotation constraints -- 9.5.3.Library of joints -- 9.6.Numerical examples -- 9.6.1.Rotating disk -- 9.6.2.Beam with attached mass -- 9.6.3.Biofidelic rear impact dummy -- 9.6.4.Sorting of randomly sized particles -- 9.7.Concluding remarks -- References -- 10.1.Introduction -- 10.2.Basic notation and differential calculus -- 10.2.1.Calculus in several variables -- 10.2.2.Differential calculus: Frechet derivative -- 10.2.3.Tangent spaces and tangent maps -- 10.2.4.Parameterizations, curvilinear coordinates, and the Jacobian transformation -- 10.3.Parameterized surfaces in R3 -- 10.3.1.Surfaces and tangent vectors -- 10.3.2.First fundamental form and arc length -- 10.3.3.Surface area measure -- 10.4.Vector form of three-dimensional linear elasticity -- 10.4.1.Notation -- 10.4.2.Three-dimensional linear elasticity -- 10.4.3.Cartesian vector expressions for linear elasticity -- 10.4.4.Curvilinear vector expressions for linear elasticity -- 10.5.Linear shell theory -- 10.5.1.Shell description and parameterization -- 10.5.2.Shell resultant momentum balance equations -- 10.5.3.Component expressions, balance of angular momentum, and the effective resultants -- 10.5.4.Shell kinematic assumption -- 10.5.5.Linear shell boundary value problem -- 10.5.6.Stress power theorem and the shell strain measures -- 10.5.7.Elastic shells -- 10.5.8.Variational formulation of the linear shell equations -- 10.6.Finite element formulation -- 10.6.1.Interpolation of the reference geometry -- 10.6.2.Galerkin approximation: Element interpolations for the displacements and variations -- 10.6.3.Discrete weak form and matrix expressions -- 10.6.4.Treatment of membrane strain -- 10.6.5.Transverse shear treatment -- 10.7.Numerical examples -- 10.7.1.Cylindrical bending of a strip -- 10.7.2.Barrel vault -- 10.7.3.Spherical cap -- 10.8.Concluding remarks -- References -- 11.1.Introduction -- 11.2.Differential calculus on manifolds -- 11.2.1.Differentiable manifolds and coordinate charts -- 11.2.2.Tangent spaces: Tangent map -- 11.3.Curves in R3: Some basic results -- 11.3.1.Basic definitions: Tangent map -- 11.3.2.The Frenet-Serret frame -- 11.3.3.The Gauss equation: Linear connection on a surface -- 11.3.4.Parallel vector fields along a curve -- 11.3.5.Geodesics -- 11.3.6.Curvature -- 11.4.Analysis on manifolds and Riemannian geometry -- 11.4.1.Vector fields and Lie bracket -- 11.4.2.Linear connections on a manifold -- 11.4.3.Tangent vectors to a curve, parallel vectors, and geodesics -- 11.4.4.Riemannian manifolds -- 11.5.Classical matrix groups: Introduction to Lie groups -- 11.5.1.Notation and basic concepts -- 11.5.2.The orthogonal group -- 11.5.3.The special orthogonal group -- References -- 12.1.Introduction -- 12.2.Bodies, configurations, and placements -- 12.3.Configuration space parameterization -- 12.4.Motions: Velocity and acceleration fields -- 12.5.Stress tensors: Momentum equations -- 12.6.Concluding remarks -- References -- 13.1.Introduction -- 13.2.Restricted rod model: Basic kinematics -- 13.2.1.Mathematical model -- 13.2.2.Motions: Basic kinematic relations -- 13.2.3.Velocity and acceleration fields -- 13.3.The exact momentum equation in stress resultants -- 13.3.1.Parameterization: Cross-sections and normal fields -- 13.3.2.Stress resultants and stress couples: Definitions from the three-dimensional theory -- 13.3.3.Stress power and conjugate strain measures: Basic kinematic assumption -- 13.3.4.Balance laws and constitutive equations -- 13.3.5.Hyperelastic constitutive equations -- 13.3.6.Particular forms of the balance equations: Basic kinematic assumption -- 13.4.The variational formulation and consistent linearization -- 13.4.1.Space of kinematically admissible variations --
  • Note continued: 13.4.2.Variational form of the momentum balance equations -- 13.4.3.Consistent linearization: Tangent operator -- 13.5.Finite element formulation -- 13.5.1.Configuration and stress update algorithm -- 13.6.Numerical examples -- 13.6.1.Circular ring -- 13.6.2.Cantilever L-beam -- 13.6.3.Cantilever beam with co-linear end force and couple -- 13.7.Concluding remarks -- References -- 14.1.Introduction -- 14.2.Shell balance equations -- 14.2.1.Geometric description of the shell -- 14.2.2.Deformation, velocity fields, and linear and angular momenta -- 14.2.3.Shell momentum balance equations -- 14.2.4.Three-dimensional derivation and parameterized form of the shell balance equations -- 14.2.5.Balance of angular momentum and the effective stress resultants -- 14.2.6.Stress power and the shell strain measures -- 14.3.Conserved quantities and hyperelasticity -- 14.3.1.Conservation laws: Momentum maps -- 14.3.2.Hyperelastic constitutive equations -- 14.3.3.Hamiltonian formulation and conservation of energy -- 14.4.Weak form of the momentum balance equations -- 14.4.1.Variations and the weak form of the momentum equations -- 14.4.2.Momentum conservation and the weak form -- 14.4.3.Multiplicative decomposition of the director field and invariance under drill rotation -- 14.4.4.Component and matrix formulation of the weak form -- 14.5.Finite element formulation -- 14.5.1.Galerkin approximation: Element interpolations for the configuration and variations -- 14.5.2.Discrete weak form and matrix expressions -- 14.5.3.Interpolation and configuration updates -- 14.5.4.Linearization: Tangent operator -- 14.5.5.Treatment of membrane strain -- 14.5.6.Transverse shear treatment -- 14.6.Numerical examples -- 14.6.1.L-beam: Shell model -- 14.6.2.Pinched hemisphere -- 14.6.3.Buckling of skin-stringer panel -- 14.6.4.Car crash -- References -- 15.1.Introduction -- 15.2.Solution of nonlinear problems -- 15.3.Eigensolutions -- 15.4.Restart option -- 15.5.Concluding remarks -- References
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