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Nonlinear computational solid mechanics

Author: J Ghaboussi; D A W Pecknold; Xiping Wu
Publisher: Boca Raton, FL : CRC Press, Taylor & Francis Group, [2017] ©2017
Edition/Format:   eBook : Document : EnglishView all editions and formats
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Genre/Form: Electronic books
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Ghaboussi, J.
Nonlinear computational solid mechanics
(DLC) 2016059346
(OCoLC)967417700
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: J Ghaboussi; D A W Pecknold; Xiping Wu
ISBN: 9781498746137 1498746136 9781351682633 1351682636 9781523114221 1523114223
OCLC Number: 993587952
Description: 1 online resource (xvi, 378 pages)
Contents: 1.1. Linear computational mechanics --
1.2. Nonlinear computational mechanics --
1.3. Nonlinear behavior of simple structures --
2.1. Motion and deformation of line elements --
2.2. Deformation of volume and area elements --
2.2.1. Volume elements --
2.2.2. Area elements --
2.3. Strains --
2.3.1. Lagrangian (Green) strain --
2.3.1.1. Interpretation of Green strain components --
2.3.1.2. Diagonal components of Green strain --
2.3.1.3. Off-diagonal components of Green strain --
2.3.1.4. Simple shear --
2.3.2. Eulerian (Almansi) strain --
2.3.2.1. Uniaxial stretching --
2.3.2.2. Uniaxial stretching and rigid body rotation --
2.3.2.3. Relation between Green strain and Almansi strain --
2.4. Objectivity and frame indifference --
2.4.1. Objectivity of some deformation measures --
2.4.1.1. Deformation gradient --
2.4.1.2. Metric tensor --
2.4.1.3. Strain tensors --
2.5. Rates of deformation --
2.5.1. Velocity and velocity gradient --
2.5.2. Deformation and spin tensors. 2.5.3. Deformation gradient rates --
2.5.4. Rate of deformation of a line element --
2.5.5. Interpretation of deformation and spin tensors --
2.5.6. Rate of change of a volume element --
2.5.7. Rate of change of an area element --
2.6. Strain rates --
2.6.1. Lagrangian (Green) strain rate --
2.6.2. Eulerian (Almansi) strain rate --
2.7. Decomposition of motion --
2.7.1. Polar decomposition --
2.7.2. Polar decomposition of deformation gradient --
2.7.3.Computation of polar decomposition --
2.7.3.1. Right stretch tensor --
2.7.3.2. Left stretch tensor --
2.7.4. Strains --
2.7.5. Strain and deformation rates --
2.7.5.1. Green strain rate --
2.7.5.2. Material rotation rate --
2.7.6. Simple examples --
2.7.6.1. Plate stretched and rotated --
2.7.6.2. Simple shear --
3.1. Traction vector on a surface --
3.2. Cauchy stress principle --
3.3. Cauchy stress tensor --
3.4. Piola-Kirchhoff stress tensors --
3.5. Stress rates --
3.5.1. Material rates of stress. 3.5.2. Jaumann rate of Cauchy stress --
3.5.3. Truesdell rate of Cauchy stress --
3.S.4. Unrotated Cauchy stress and the Green-Naghdi rate --
3.5.4.1. Unrotated Cauchy stress --
3.5.4.2. Green-Naghdi rate --
3.6. Examples of stress rates for simple stress conditions --
3.6.1. Uniaxial extension of an initially stressed body --
3.6.2. Rigid body rotation of an initially stressed body --
3.6.3. Simple shear of an initially stressed body --
3.6.3.1. Jaumann stress rate --
3.6.3.2. Truesdell stress rate --
4.1. Divergence theorem --
4.2. Stress power --
4.2.1.2PK stress power --
4.2.2. Unrotated Cauchy stress power --
4.3. Virtual work --
4.4. Principle of virtual work --
4.4.1. Internal virtual work --
4.4.2. Lagrangian form of internal virtual work --
4.5. Vector forms of stress and strain --
5.1. Introduction --
5.2. Linear elastic material models --
5.3. Cauchy elastic material models --
5.3.1. Characteristic polynomial of a matrix --
5.3.2. Cayley-Hamilton theorem. 5.3.3. General polynomial form for isotropic Cauchy elastic materials --
5.4. Hyperelastic material models --
5.4.1. Strain energy density potential --
5.4.2. Deformation invariants --
5.4.3. Rubber and rubberlike materials --
5.4.3.1. Ogden hyperelastic model --
5.4.3.2. Balloon problem --
5.4.4. Soft biological tissue --
5.4.4.1. Fung exponential hyperelastic model --
5.5. Hypoelastic material models --
5.5.1. Hypoelastic grade zero --
5.5.2. Hypoelastic grade one --
5.5.3. Lagrangian versus hypoelastic material tangent stiffness --
5.5.3.1. Truesdell stress rate --
5.5.3.2. Jaumann stress rate --
5.5.4.Comparison of linear isotropic hypoelastic models in simple shear --
5.5.4.1. Truesdell rate hypoelastic model --
5.5.4.2. Jaumann rate hypoelastic model --
5.5.4.3. Green-Naghdi rate hypoelastic model --
5.6. Numerical evaluation of a linear isotropic hypoelastic model --
5.6.1. Natural coordinate system for triangle --
5.6.2. Kinematics. 5.6.3. Nodal displacement patterns --
5.6.4. Strain cycles --
5.6.5. Calculation procedure --
5.6.6. Numerical results --
6.1. Introduction --
6.2. Volumetric and deviatoric stresses and strains --
6.3. Principal stresses and stress invariants --
6.4. Alternative forms of stress invariants --
6.5. Octahedral stresses --
6.6. Principal stress space --
6.7. Standard material tests and stress paths --
6.7.1. Representations of stress paths --
6.7.2. Uniaxial tests --
6.7.2.1. Uniaxial tension --
6.7.2.2. Uniaxial compression --
6.7.3. Biaxial tests --
6.7.4. Isotropic compression tests --
6.7.5. Triaxial tests --
6.7.6. True triaxial tests --
6.7.6.1. Pure shear test --
7.1. Introduction --
7.2. Behavior of metals under uniaxial stress --
7.2.1. Fundamental assumptions of classical plasticity theory --
7.3. Inelastic behavior under multiaxial states of stress --
7.3.1. Yield surface --
7.4. Work and stability constraints --
7.4.1. Work and energy --
7.4.2. Stability in the small. 7.4.3.Complementary work --
7.4.4.Net work --
7.5. Associated plasticity models --
7.5.1. Drucker's postulate --
7.5.2. Convexity and normality --
7.5.2.1. Normality of the plastic strain increment --
7.5.2.2. Convexity of the yield surface --
7.6. Incremental stress-Strain relations --
7.6.1. Equivalent uniaxial stress and plastic strain --
7.6.2. Loading and unloading criteria --
7.6.3. Continuum tangent stiffness --
7.6.4. Elastic-perfectly plastic behavior --
7.6.5. Interpretation of incremental stresses --
7.6.5.1. Elastic-perfectly plastic --
7.6.5.2. Hardening plasticity --
7.7. Yield surfaces in principal stress space --
7.7.1. Material isotropy and symmetry requirements --
7.7.2.von Mises yield surface --
7.7.2.1. Biaxial (plane) stress --
7.7.2.2. Tension-torsion test --
7.7.3. Tresca yield surface --
7.8. Hardening plasticity models --
7.8.1. Determination of hardening parameter from uniaxial test --
7.8.2. Isotropic hardening. 7.8.3. Kinematic hardening and back stress --
7.8.4.Combined isotropic and kinematic hardening --
7.9. Stress update --
7.9.1.von Mises plasticity in simple shear problem --
7.9.2. Stress update algorithm --
7.9.2.1. Calculation of updated stress --
7.9.2.2. Elastic trial stress at step (n + 1) --
7.9.2.3. Stress correction --
7.9.2.4. Elastoplastic response --
7.10. Plasticity models for frictional and pressure-Sensitive materials --
7.10.1. Mohr-Coulomb yield surface --
7.10.2. Drucker-Prager yield surface --
7.10.3. Model refinements --
7.10.3.1. Cap models --
7.10.3.2. Refined (q, theta) shape in octahedral plane --
8.1. Introduction --
8.2. Finite element discretization --
8.2.1. Shape functions --
8.2.1.1. Serendipity elements --
8.2.1.2. Simplex elements --
8.2.2. Isoparametric mapping --
8.2.2.1. Numerical (Gaussian) quadrature --
8.2.2.2. Shape function derivatives --
8.3. Total Lagrangian formulation --
8.3.1. Geometric nonlinearity --
8.3.1.1. Green strain. 8.3.1.2. Green strain rate --
8.3.1.3. Tangent stiffness matrix --
8.3.2. Nonlinear material behavior --
8.3.2.1. Elastoplastic material behavior --
8.3.2.2. Consistent tangent stiffness --
8.3.2.3.von Mises yield criterion --
8.3.2.4. Discussion --
8.3.2.5.2PK stress rate versus Green strain rate for TL formulation --
8.3.3. Plane stress --
8.3.3.1. Partial inversion and condensation of material stiffness --
8.3.3.2. Nonlinear material properties --
8.3.3.3. Expansion of strain increment and stress update --
8.3.3.4. Condensed tangent stiffness and internal resisting force vector --
8.3.4. Pressure loading --
8.3.4.1. Load stiffness for 2D simplex element --
8.3.4.2. Secant load stiffness --
8.3.4.3. Tangent load stiffness --
8.4. Updated Lagrangian methods --
8.4.1. Basic UDL formulation --
8.4.1.1. Virtual work in deformed configuration --
8.4.1.2. Coordinate systems --
8.4.1.3. Hughes-Winget incremental update --
8.4.1.4. Element geometry updating --
9.1. Introduction. 9.2. Finite rotations in three dimensions --
9.2.1. Rotation matrix R --
9.2.1.1. Euler's theorem --
9.2.1.2. Rodrigues's formula --
9.2.1.3. Extraction of the axial vector from R --
9.2.1.4. Eigenstructure of the spin matrix --
9.2.1.5. Matrix exponential --
9.2.1.6. Exponential map --
9.2.2. Cayley transform --
9.2.3.Composition of finite rotations --
9.2.3.1. Infinitesimal rotations --
9.2.3.2. Two successive finite rotations --
9.2.3.3. Update of finite nodal rotations --
9.3. Element local coordinate systems --
9.3.1. Euler angles --
9.3.2. Orienting plate and shell elements in three dimensions --
9.4. Corotational finite element formulation --
9.4.1. Corotational coordinate systems --
9.4.1.1. Notation --
9.4.1.2. Element triads --
9.4.1.3. Nodal triads --
9.4.1.4. Nodal rotational freedoms --
9.4.2. Incremental nodal degrees of freedom --
9.4.2.1. Incremental nodal displacements --
9.4.2.2. Incremental nodal rotations --
9.4.3. Incremental variation of element frame. 9.4.4. Incremental variation of the nodal triad --
9.4.4.1. Corotated incremental nodal rotations --
9.4.5. Corotated incremental element freedoms combined --
9.4.5.1. Geometric stiffness --
9.4.5.2. Variation of LambdaTaui contracted with a nodal moment vector mi --
9.4.5.3. Tangent stiffness summary --
9.4.6. Discussion of a CR versus UDL formulation --
10.1. Introduction --
10.2. Modeling of shell structures --
10.3.3D Structural element formulations --
10.3.1. Kirchhoff beam, plate, and shell finite elements --
10.3.1.1. Flat plate and shell elements --
10.3.1.2. General form of geometric stiffness --
10.3.1.3. Remarks --
10.3.1.4. Synthesis of space frame stiffness matrices --
10.3.1.5. Space frame geometric stiffness --
10.3.1.6. Planar bending --
10.3.1.7. Plane frame stiffness matrices --
10.3.1.8. Remarks --
10.3.1.9. Elastic tangent stiffness matrices --
10.3.2. Mindlin plate theory --
10.3.3. Degeneration of isoparametric solid elements. 10.3.4. Mindlin plate and flat shell finite elements --
10.3.4.1. Bending stiffness --
10.3.4.2. Shear stiffness --
10.3.4.3. In-plane response --
10.3.4.4. Geometric stiffness --
10.3.4.5. Membrane stiffness --
10.3.4.6. Performance of Mindlin elements --
10.3.4.7. Shear locking, zero-energy modes, and hourglass control --
10.3.4.8. Membrane locking --
10.4. Isoparametric curved shell elements --
10.4.1. Normal rotation in classic thin-shell theory --
10.4.2. Isoparametric modeling of curved shell geometry --
10.4.3. Nodal surface coordinate system --
10.4.4. Jacobian matrix --
10.4.5. Lamina Cartesian coordinate system --
10.4.6. Summary of coordinate systems and transformations --
10.4.7. Displacement interpolation --
10.4.8. Approximations --
10.4.9. Lamina kinematics --
10.4.10. Lamina stresses --
10.4.11. Virtual work density --
10.4.12. Rotational freedoms --
10.4.13.Comments on the UDL formulation --
10.5. Nonlinear material behavior. 10.5.1. Through-thickness numerical integration --
10.5.2. Layered models --
11.1. Introduction --
11.2. Pure incremental (Euler-Cauchy) method --
11.3. Incremental method with equilibrium correction --
11.4. Incremental-Iterative (Newton-Raphson) method --
11.5. Modified Newton-Raphson method --
11.6. Critical points on the equilibrium path --
11.6.1. Characterization of critical points --
11.6.2. Monitoring the incremental solution on the primary path --
11.6.3. Determinant of the tangent stiffness --
11.6.4. Current stiffness parameter --
11.7. Arc-length solution methods --
11.7.1. Crisfield's spherical method --
11.7.2. Ramm's normal plane method --
11.8. Lattice dome example --
11.8.1. Load-deflection response of hexagonal dome --
11.9. Secondary solution paths --
11.10. Structural imperfections --
11.10.1. Application to hexagonal dome example --
12.1. Introduction --
12.2. Multilayer neural networks --
12.2.1. Training of neural networks. 12.3. Hard computing versus soft computing --
12.4. Neural networks in material modeling --
12.5. Nested adaptive neural networks --
12.6. Neural network modeling of hysteretic behavior of materials --
12.7. Acquisition of training data for neural network material models --
12.8. Neural network material models in finite element analysis --
12.9. Autoprogressive algorithm --
12.9.1. Autoprogressive algorithm in modeling composite materials --
12.9.2. Autoprogressive algorithm in structural mechanics and in geomechanics --
12.9.3. Autoprogressive algorithm in biomedicine.
Responsibility: Jamshid Ghaboussi, David A. Pecknold, Xiping Steven Wu.

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"An excellent introduction to non-linear computational mechanics addressed to Masters students interested in the mechanics of structures."-- Laurent Delannay, Universite Catholique de Louvain"...a Read more...

 
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<http://www.worldcat.org/oclc/993587952> # Nonlinear computational solid mechanics
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   schema:about <http://experiment.worldcat.org/entity/work/data/4011791640#Topic/mechanics_applied_mathematics> ; # Mechanics, Applied--Mathematics
   schema:about <http://experiment.worldcat.org/entity/work/data/4011791640#Topic/solids_mathematical_models> ; # Solids--Mathematical models
   schema:about <http://experiment.worldcat.org/entity/work/data/4011791640#Topic/nonlinear_mechanics_matematics> ; # Nonlinear mechanics--Matematics
   schema:author <http://experiment.worldcat.org/entity/work/data/4011791640#Person/wu_xiping_1962> ; # Xiping Wu
   schema:author <http://experiment.worldcat.org/entity/work/data/4011791640#Person/pecknold_d_a_w> ; # D. A. W. Pecknold
   schema:author <http://experiment.worldcat.org/entity/work/data/4011791640#Person/ghaboussi_j> ; # J. Ghaboussi
   schema:bookFormat schema:EBook ;
   schema:datePublished "2017" ;
   schema:description "10.3.4.Mindlin plate and flat shell finite elements -- 10.3.4.1.Bending stiffness -- 10.3.4.2.Shear stiffness -- 10.3.4.3.In-plane response -- 10.3.4.4.Geometric stiffness -- 10.3.4.5.Membrane stiffness -- 10.3.4.6.Performance of Mindlin elements -- 10.3.4.7.Shear locking, zero-energy modes, and hourglass control -- 10.3.4.8.Membrane locking -- 10.4.Isoparametric curved shell elements -- 10.4.1.Normal rotation in classic thin-shell theory -- 10.4.2.Isoparametric modeling of curved shell geometry -- 10.4.3.Nodal surface coordinate system -- 10.4.4.Jacobian matrix -- 10.4.5.Lamina Cartesian coordinate system -- 10.4.6.Summary of coordinate systems and transformations -- 10.4.7.Displacement interpolation -- 10.4.8.Approximations -- 10.4.9.Lamina kinematics -- 10.4.10.Lamina stresses -- 10.4.11.Virtual work density -- 10.4.12.Rotational freedoms -- 10.4.13.Comments on the UDL formulation -- 10.5.Nonlinear material behavior."@en ;
   schema:description "1.1.Linear computational mechanics -- 1.2.Nonlinear computational mechanics -- 1.3.Nonlinear behavior of simple structures -- 2.1.Motion and deformation of line elements -- 2.2.Deformation of volume and area elements -- 2.2.1.Volume elements -- 2.2.2.Area elements -- 2.3.Strains -- 2.3.1.Lagrangian (Green) strain -- 2.3.1.1.Interpretation of Green strain components -- 2.3.1.2.Diagonal components of Green strain -- 2.3.1.3.Off-diagonal components of Green strain -- 2.3.1.4.Simple shear -- 2.3.2.Eulerian (Almansi) strain -- 2.3.2.1.Uniaxial stretching -- 2.3.2.2.Uniaxial stretching and rigid body rotation -- 2.3.2.3.Relation between Green strain and Almansi strain -- 2.4.Objectivity and frame indifference -- 2.4.1.Objectivity of some deformation measures -- 2.4.1.1.Deformation gradient -- 2.4.1.2.Metric tensor -- 2.4.1.3.Strain tensors -- 2.5.Rates of deformation -- 2.5.1.Velocity and velocity gradient -- 2.5.2.Deformation and spin tensors."@en ;
   schema:description "2.5.3.Deformation gradient rates -- 2.5.4.Rate of deformation of a line element -- 2.5.5.Interpretation of deformation and spin tensors -- 2.5.6.Rate of change of a volume element -- 2.5.7.Rate of change of an area element -- 2.6.Strain rates -- 2.6.1.Lagrangian (Green) strain rate -- 2.6.2.Eulerian (Almansi) strain rate -- 2.7.Decomposition of motion -- 2.7.1.Polar decomposition -- 2.7.2.Polar decomposition of deformation gradient -- 2.7.3.Computation of polar decomposition -- 2.7.3.1.Right stretch tensor -- 2.7.3.2.Left stretch tensor -- 2.7.4.Strains -- 2.7.5.Strain and deformation rates -- 2.7.5.1.Green strain rate -- 2.7.5.2.Material rotation rate -- 2.7.6.Simple examples -- 2.7.6.1.Plate stretched and rotated -- 2.7.6.2.Simple shear -- 3.1.Traction vector on a surface -- 3.2.Cauchy stress principle -- 3.3.Cauchy stress tensor -- 3.4.Piola-Kirchhoff stress tensors -- 3.5.Stress rates -- 3.5.1.Material rates of stress."@en ;
   schema:description "10.5.1.Through-thickness numerical integration -- 10.5.2.Layered models -- 11.1.Introduction -- 11.2.Pure incremental (Euler-Cauchy) method -- 11.3.Incremental method with equilibrium correction -- 11.4.Incremental-Iterative (Newton-Raphson) method -- 11.5.Modified Newton-Raphson method -- 11.6.Critical points on the equilibrium path -- 11.6.1.Characterization of critical points -- 11.6.2.Monitoring the incremental solution on the primary path -- 11.6.3.Determinant of the tangent stiffness -- 11.6.4.Current stiffness parameter -- 11.7.Arc-length solution methods -- 11.7.1.Crisfield's spherical method -- 11.7.2.Ramm's normal plane method -- 11.8.Lattice dome example -- 11.8.1.Load-deflection response of hexagonal dome -- 11.9.Secondary solution paths -- 11.10.Structural imperfections -- 11.10.1.Application to hexagonal dome example -- 12.1.Introduction -- 12.2.Multilayer neural networks -- 12.2.1.Training of neural networks."@en ;
   schema:description "9.2.Finite rotations in three dimensions -- 9.2.1.Rotation matrix R -- 9.2.1.1.Euler's theorem -- 9.2.1.2.Rodrigues's formula -- 9.2.1.3.Extraction of the axial vector from R -- 9.2.1.4.Eigenstructure of the spin matrix -- 9.2.1.5.Matrix exponential -- 9.2.1.6.Exponential map -- 9.2.2.Cayley transform -- 9.2.3.Composition of finite rotations -- 9.2.3.1.Infinitesimal rotations -- 9.2.3.2.Two successive finite rotations -- 9.2.3.3.Update of finite nodal rotations -- 9.3.Element local coordinate systems -- 9.3.1.Euler angles -- 9.3.2.Orienting plate and shell elements in three dimensions -- 9.4.Corotational finite element formulation -- 9.4.1.Corotational coordinate systems -- 9.4.1.1.Notation -- 9.4.1.2.Element triads -- 9.4.1.3.Nodal triads -- 9.4.1.4.Nodal rotational freedoms -- 9.4.2.Incremental nodal degrees of freedom -- 9.4.2.1.Incremental nodal displacements -- 9.4.2.2.Incremental nodal rotations -- 9.4.3.Incremental variation of element frame."@en ;
   schema:description "5.6.3.Nodal displacement patterns -- 5.6.4.Strain cycles -- 5.6.5.Calculation procedure -- 5.6.6.Numerical results -- 6.1.Introduction -- 6.2.Volumetric and deviatoric stresses and strains -- 6.3.Principal stresses and stress invariants -- 6.4.Alternative forms of stress invariants -- 6.5.Octahedral stresses -- 6.6.Principal stress space -- 6.7.Standard material tests and stress paths -- 6.7.1.Representations of stress paths -- 6.7.2.Uniaxial tests -- 6.7.2.1.Uniaxial tension -- 6.7.2.2.Uniaxial compression -- 6.7.3.Biaxial tests -- 6.7.4.Isotropic compression tests -- 6.7.5.Triaxial tests -- 6.7.6.True triaxial tests -- 6.7.6.1.Pure shear test -- 7.1.Introduction -- 7.2.Behavior of metals under uniaxial stress -- 7.2.1.Fundamental assumptions of classical plasticity theory -- 7.3.Inelastic behavior under multiaxial states of stress -- 7.3.1.Yield surface -- 7.4.Work and stability constraints -- 7.4.1.Work and energy -- 7.4.2.Stability in the small."@en ;
   schema:description "7.8.3.Kinematic hardening and back stress -- 7.8.4.Combined isotropic and kinematic hardening -- 7.9.Stress update -- 7.9.1.von Mises plasticity in simple shear problem -- 7.9.2.Stress update algorithm -- 7.9.2.1.Calculation of updated stress -- 7.9.2.2.Elastic trial stress at step (n + 1) -- 7.9.2.3.Stress correction -- 7.9.2.4.Elastoplastic response -- 7.10.Plasticity models for frictional and pressure-Sensitive materials -- 7.10.1.Mohr-Coulomb yield surface -- 7.10.2.Drucker-Prager yield surface -- 7.10.3.Model refinements -- 7.10.3.1.Cap models -- 7.10.3.2.Refined (q,theta) shape in octahedral plane -- 8.1.Introduction -- 8.2.Finite element discretization -- 8.2.1.Shape functions -- 8.2.1.1.Serendipity elements -- 8.2.1.2.Simplex elements -- 8.2.2.Isoparametric mapping -- 8.2.2.1.Numerical (Gaussian) quadrature -- 8.2.2.2.Shape function derivatives -- 8.3.Total Lagrangian formulation -- 8.3.1.Geometric nonlinearity -- 8.3.1.1.Green strain."@en ;
   schema:description "8.3.1.2.Green strain rate -- 8.3.1.3.Tangent stiffness matrix -- 8.3.2.Nonlinear material behavior -- 8.3.2.1.Elastoplastic material behavior -- 8.3.2.2.Consistent tangent stiffness -- 8.3.2.3.von Mises yield criterion -- 8.3.2.4.Discussion -- 8.3.2.5.2PK stress rate versus Green strain rate for TL formulation -- 8.3.3.Plane stress -- 8.3.3.1.Partial inversion and condensation of material stiffness -- 8.3.3.2.Nonlinear material properties -- 8.3.3.3.Expansion of strain increment and stress update -- 8.3.3.4.Condensed tangent stiffness and internal resisting force vector -- 8.3.4.Pressure loading -- 8.3.4.1.Load stiffness for 2D simplex element -- 8.3.4.2.Secant load stiffness -- 8.3.4.3.Tangent load stiffness -- 8.4.Updated Lagrangian methods -- 8.4.1.Basic UDL formulation -- 8.4.1.1.Virtual work in deformed configuration -- 8.4.1.2.Coordinate systems -- 8.4.1.3.Hughes-Winget incremental update -- 8.4.1.4.Element geometry updating -- 9.1.Introduction."@en ;
   schema:description "7.4.3.Complementary work -- 7.4.4.Net work -- 7.5.Associated plasticity models -- 7.5.1.Drucker's postulate -- 7.5.2.Convexity and normality -- 7.5.2.1.Normality of the plastic strain increment -- 7.5.2.2.Convexity of the yield surface -- 7.6.Incremental stress-Strain relations -- 7.6.1.Equivalent uniaxial stress and plastic strain -- 7.6.2.Loading and unloading criteria -- 7.6.3.Continuum tangent stiffness -- 7.6.4.Elastic-perfectly plastic behavior -- 7.6.5.Interpretation of incremental stresses -- 7.6.5.1.Elastic-perfectly plastic -- 7.6.5.2.Hardening plasticity -- 7.7.Yield surfaces in principal stress space -- 7.7.1.Material isotropy and symmetry requirements -- 7.7.2.von Mises yield surface -- 7.7.2.1.Biaxial (plane) stress -- 7.7.2.2.Tension-torsion test -- 7.7.3.Tresca yield surface -- 7.8.Hardening plasticity models -- 7.8.1.Determination of hardening parameter from uniaxial test -- 7.8.2.Isotropic hardening."@en ;
   schema:description "3.5.2.Jaumann rate of Cauchy stress -- 3.5.3.Truesdell rate of Cauchy stress -- 3.S.4.Unrotated Cauchy stress and the Green-Naghdi rate -- 3.5.4.1.Unrotated Cauchy stress -- 3.5.4.2.Green-Naghdi rate -- 3.6.Examples of stress rates for simple stress conditions -- 3.6.1.Uniaxial extension of an initially stressed body -- 3.6.2.Rigid body rotation of an initially stressed body -- 3.6.3.Simple shear of an initially stressed body -- 3.6.3.1.Jaumann stress rate -- 3.6.3.2.Truesdell stress rate -- 4.1.Divergence theorem -- 4.2.Stress power -- 4.2.1.2PK stress power -- 4.2.2.Unrotated Cauchy stress power -- 4.3.Virtual work -- 4.4.Principle of virtual work -- 4.4.1.Internal virtual work -- 4.4.2.Lagrangian form of internal virtual work -- 4.5.Vector forms of stress and strain -- 5.1.Introduction -- 5.2.Linear elastic material models -- 5.3.Cauchy elastic material models -- 5.3.1.Characteristic polynomial of a matrix -- 5.3.2.Cayley-Hamilton theorem."@en ;
   schema:description "5.3.3.General polynomial form for isotropic Cauchy elastic materials -- 5.4.Hyperelastic material models -- 5.4.1.Strain energy density potential -- 5.4.2.Deformation invariants -- 5.4.3.Rubber and rubberlike materials -- 5.4.3.1.Ogden hyperelastic model -- 5.4.3.2.Balloon problem -- 5.4.4.Soft biological tissue -- 5.4.4.1.Fung exponential hyperelastic model -- 5.5.Hypoelastic material models -- 5.5.1.Hypoelastic grade zero -- 5.5.2.Hypoelastic grade one -- 5.5.3.Lagrangian versus hypoelastic material tangent stiffness -- 5.5.3.1.Truesdell stress rate -- 5.5.3.2.Jaumann stress rate -- 5.5.4.Comparison of linear isotropic hypoelastic models in simple shear -- 5.5.4.1.Truesdell rate hypoelastic model -- 5.5.4.2.Jaumann rate hypoelastic model -- 5.5.4.3.Green-Naghdi rate hypoelastic model -- 5.6.Numerical evaluation of a linear isotropic hypoelastic model -- 5.6.1.Natural coordinate system for triangle -- 5.6.2.Kinematics."@en ;
   schema:description "9.4.4.Incremental variation of the nodal triad -- 9.4.4.1.Corotated incremental nodal rotations -- 9.4.5.Corotated incremental element freedoms combined -- 9.4.5.1.Geometric stiffness -- 9.4.5.2.Variation of LambdaTaui contracted with a nodal moment vector mi -- 9.4.5.3.Tangent stiffness summary -- 9.4.6.Discussion of a CR versus UDL formulation -- 10.1.Introduction -- 10.2.Modeling of shell structures -- 10.3.3D Structural element formulations -- 10.3.1.Kirchhoff beam, plate, and shell finite elements -- 10.3.1.1.Flat plate and shell elements -- 10.3.1.2.General form of geometric stiffness -- 10.3.1.3.Remarks -- 10.3.1.4.Synthesis of space frame stiffness matrices -- 10.3.1.5.Space frame geometric stiffness -- 10.3.1.6.Planar bending -- 10.3.1.7.Plane frame stiffness matrices -- 10.3.1.8.Remarks -- 10.3.1.9.Elastic tangent stiffness matrices -- 10.3.2.Mindlin plate theory -- 10.3.3.Degeneration of isoparametric solid elements."@en ;
   schema:description "12.3.Hard computing versus soft computing -- 12.4.Neural networks in material modeling -- 12.5.Nested adaptive neural networks -- 12.6.Neural network modeling of hysteretic behavior of materials -- 12.7.Acquisition of training data for neural network material models -- 12.8.Neural network material models in finite element analysis -- 12.9.Autoprogressive algorithm -- 12.9.1.Autoprogressive algorithm in modeling composite materials -- 12.9.2.Autoprogressive algorithm in structural mechanics and in geomechanics -- 12.9.3.Autoprogressive algorithm in biomedicine."@en ;
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