Abaqus Theory Manual.pdf

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Abaqus 6.10 PDF Documentation

Abaqus Theory Manual

Legal Notices

Locations

Preface

Contents

1. Introduction and Basic Equations

1.1 Introduction

1.1.1 Introduction: general

1.2 Notation

1.2.1 Notation

Basic quantities

Basic operations

Components of a vector or a matrix in a coordinate system

Components of a derivative

Virtual quantities

Initial and current positions

Nodal variables

1.3 Finite rotations

1.3.1 Rotation variables

Quaternion parametrization

Compound rotations

Rotation vector extraction

Director and rotation field updates

Variations of the rotation field

Velocity and acceleration

Coupling of rotations: constant velocity joint

Reference

1.4 Deformation, strain, and strain rates

1.4.1 Deformation

Reference

1.4.2 Strain measures

Strain in one dimension

Strain in general three-dimensional motions

Reference

1.4.3 Rate of deformation and strain increment

Reference

1.4.4 The additive strain rate decomposition

Reference

1.5 Equilibrium, stress, and state storage

1.5.1 Equilibrium and virtual work

Reference

1.5.2 Stress measures

Reference

1.5.3 Stress rates

Stress rates used in Abaqus

Reference

1.5.4 State storage

Reference

1.5.5 Energy balance

References

2. Procedures

2.1 Overview

2.1.1 Procedures: overview and basic equations

Basic finite element equations

Reference

2.2 Nonlinear solution methods

2.2.1 Nonlinear solution methods in Abaqus/Standard

References

2.2.2 Quasi-Newton solution technique

Reference

2.2.3 Direct cyclic algorithm

Reference

2.3 Buckling and postbuckling

2.3.1 Eigenvalue buckling prediction

Reference

2.3.2 Modified Riks algorithm

Basic variable definitions

Reference

2.4 Nonlinear dynamics

2.4.1 Implicit dynamic analysis

Reference

2.4.2 Intermittent contact/impact

Reference

2.4.3 Subspace dynamics

Reference

2.4.4 Equivalent rigid body dynamic motion

Reference

2.4.5 Explicit dynamic analysis

Stability

Bulk viscosity

Rotational bulk viscosity for shell elements

Reference

2.5 Modal dynamics

2.5.1 Eigenvalue extraction

Eigenvalue extraction for symmetric systems

Subspace iteration—the basic algorithm

Lanczos eigensolver

The Householder method with quarter rotation

Complex eigenvalue extraction

Subspace projection method

References

2.5.2 Variables associated with the natural modes of a model

Generalized mass

Modal participation factors

Modal effective mass

Composite modal damping

References

2.5.3 Linear dynamic analysis using modal superposition

Reference

2.5.4 Damping options for modal dynamics

Critical damping factors

Rayleigh damping

Composite modal damping

Structural damping

Reference

2.5.5 Modal dynamic analysis

Solution to the uncoupled system

Solution to the coupled system

Integral coefficients

Damping less than critical

Damping equal to critical

Damping higher than critical

Rigid body mode with damping

Rigid body mode without damping

Structural damping in modal dynamic analysis

Response of nodal and element variables

Initial conditions

Base motion definition

Reference

2.5.6 Response spectrum analysis

Reference

2.5.7 Steady-state linear dynamic analysis

Reference

2.5.8 Random response analysis

Statistical measures

Correlation

Fourier transforms

Cross-spectral density

Random response analysis

The frequency response function

Response development

Decibel conversion

von Mises stress computation

Reference

2.5.9 Base motions in modal-based procedures

Primary base motions

Secondary base motions

References

2.6 Complex harmonic oscillations

2.6.1 Direct steady-state dynamic analysis

Reference

2.6.2 Subspace-based steady-state dynamic analysis

Output

Reference

2.7 Steady-state transport analysis

2.7.1 Steady-state transport analysis

Kinematics of steady-state rolling

Inertia

Performing a harmonic analysis about a nonlinear base state

Contact conditions

Reference

2.8 Analysis of porous media

2.8.1 Effective stress principle for porous media

References

2.8.2 Discretized equilibrium statement for a porous medium

References

2.8.3 Constitutive behavior in a porous medium

Liquid response

Grains response

Liquid entrapment

Effective strain

Reference

2.8.4 Continuity statement for the wetting liquid phase in a porous medium

Constitutive behavior

Volumetric strain in the liquid and grains

Saturation

Jacobian contribution

References

2.8.5 Solution strategy for coupled diffusion/deformation

Reference

2.9 Coupled fluid-solid analysis

2.9.1 Coupled acoustic-structural medium analysis

Acoustic equations

Physical boundary conditions in acoustic analysis

Formulation for direct integration transient dynamics

Variational statement

The discretized finite element equations

Time integration

Summary of additional approximations of the direct integration transient formulation

Formulation for steady-state response using nodal degrees of freedom

Variational statement

The discretized finite element equations

Formulation for eigenvalue extraction and mode-based procedures

Lanczos formulation

Frequency-domain solution using projections onto modal spaces

Using uncoupled Abaqus/AMS modes

Volumetric drag and fluid viscosity

Acoustic output quantities

Acoustic contribution factors

Impedance and admittance at fluid boundaries

Radiation boundary conditions

An improvement on radiation boundary conditions for plane waves

References

2.10 Piezoelectric analysis

2.10.1 Piezoelectric analysis

Equilibrium and flux conservation

Constitutive behavior: material coupling

Kinematics

System equations

Reference

2.11 Heat transfer

2.11.1 Uncoupled heat transfer analysis

Energy balance

Constitutive definition

Boundary conditions

Spatial discretization

Time integration

Reference

2.11.2 Shell heat conduction

Reference

2.11.3 Convection/diffusion

Thermal equilibrium equation

Reference

2.11.4 Cavity radiation

Thermal radiation

Spatial interpolation

Cavity radiation flux and Jacobian contributions

Reference

2.11.5 Viewfactor calculation

Viewfactor between elementary areas

Discretization

Radiation blocking

Radiation symmetries

Viewfactor checking

Radiation to ambient

References

2.12 Coupled thermal-electrical analysis

2.12.1 Coupled thermal-electrical analysis

Governing equations

Constitutive behavior

Thermal energy balance

Thermal energy due to electrical current

Surface conditions

Spatial discretization

Jacobian contributions

Reference

2.13 Mass diffusion

2.13.1 Mass diffusion analysis

Governing equations

Constitutive behavior

Discretization and time integration

Jacobian contribution

Reference

2.14 Substructuring

2.14.1 Substructuring and substructure analysis

Large-rotation substructures

Fixed direction gravity loading

Reference

2.15 Submodeling

2.15.1 Submodeling analysis

Node-based submodeling

Interpolation procedure and tolerance checking

Driven variables for shell-to-solid submodeling

Surface-based submodeling

Interpolation procedure and tolerance checking

Stress solution smoothing

Surface traction determination

References

2.16 Fracture mechanics

2.16.1 J -integral evaluation

J -integral in two dimensions

J -integral in three dimensions

Reference

2.16.2 Stress intensity factor extraction

Interaction integral method

Reference

2.16.3 T -stress extraction

Reference

2.16.4 Prediction of the direction of crack propagation

Maximum tangential stress criterion

Maximum energy release rate criterion

K II = 0 criterion

Reference

2.17 Stress linearization

2.17.1 Stress linearization

Computing the stress components

Three-dimensional structures

Axisymmetric structures

Meridional stress

Hoop stress

Thickness stress

Shear stress

Curvature correction in nonaxisymmetric structures

Computing the local coordinate system

Axisymmetric case

Three-dimensional case

Reference

2.18 Design sensitivity analysis

2.18.1 Design sensitivity analysis

DSA for static stress analysis

Total displacement DSA formulation for nonlinear equilibrium problems

Incremental displacement DSA formulation for history-dependent equilibrium problems

Computational approach

DSA for frequency analysis

Distinct eigenvalue case

Repeated eigenvalue case

Computational approach

Choosing the finite difference interval

Reference

3. Elements

3.1 Overview

3.1.1 Element library: overview

Reference

3.2 Continuum elements

3.2.1 Solid element overview

Reference

3.2.2 Solid element formulation

Reference

3.2.3 Hybrid incompressible solid element formulation

Reference

3.2.4 Solid isoparametric quadrilaterals and hexahedra

Interpolation

Integration of homogeneous solids

Fully integrated first-order isoparametric elements

Integration of composite solids

Reference

3.2.5 Continuum elements with incompatible modes

Geometrically linear formulation

Geometrically nonlinear formulation

Reference

3.2.6 Triangular, tetrahedral, and wedge elements

Interpolation

Integration

Reference

3.2.7 Generalized plane strain elements

References

3.2.8 Axisymmetric elements

Kinematic description

Parametric interpolation and integration

Deformation gradient

Virtual work

Stiffness in the current state

Reference

3.2.9 Axisymmetric elements allowing nonlinear bending

Interpolation

Integration

Deformation gradient

Strain and rotation increments

Virtual work

Stiffness in the current state

Hourglass control

Pressure loads and load stiffness

Mass matrix

Hybrid and pore pressure elements

Pore pressure gradient

Reference

3.3 Infinite elements

3.3.1 Solid infinite elements

Static analysis

Dynamic response

Reference

3.3.2 Acoustic infinite elements

Transient formulation

Steady-state analysis

Lateral face terms due to impedance conditions

Coupling between acoustic and solid infinite elements

References

3.4 Membrane and truss elements

3.4.1 Membrane elements

Equilibrium

Jacobian

Thickness change

Total deformation

Reference

3.4.2 Truss elements

Interpolation

Strain measure

First variation of strain

Second variation of strain

Integration

Virtual work contribution

Mixed (hybrid) forms

Reference

3.4.3 Axisymmetric membranes

Kinematic description

Parametric interpolation and integration

Deformation gradient

Virtual work

Stiffness in the current state

Reference

3.5 Beam elements

3.5.1 Beam element overview

Reference

3.5.2 Beam element formulation

Beam element in undeformed configuration

Change of position, warping, and normal direction

Change of curvature and twist

First variations

Solution with Newton's algorithm

Second variations

Strains

Virtual work

The rate of change of virtual work

Section integration

Beams with unconstrained warping

Beams with elastic torsion and constrained warping

Circular section

Solid noncircular sections

Closed, thin walled cross-sections

Thin walled open sections

Reference

3.5.3 Euler-Bernoulli beam elements

Strain measures

Internal virtual work rate Jacobian

First variations of strains

Second variations of strains

Interpolation

Integration

Reference

3.5.4 Hybrid beam elements

Axial and bending behavior

Transverse shear

Reference

3.5.5 Mass and inertia for Timoshenko beams

3.5.6 Meshed beam cross-sections

Overview

Defining the shear center and warping function

3.6 Shell elements

3.6.1 Shell element overview

Thickness change

Reference

3.6.2 Axisymmetric shell elements

Interpolation and integration

Theory

Reference

3.6.3 Shear flexible small-strain shell elements

Notation

Surface measures

Displacements

Interpolation

Strains

Penalties

First variations of strain

Second variations of strains

Internal virtual work rate

Pressure load stiffness

Reference

3.6.4 Triangular facet shell elements

Kinematics

First variations of strain

Second variations of strain

Internal virtual work rate

Reference

3.6.5 Finite-strain shell element formulation

Geometric description

Parametric interpolation

Membrane deformation and curvature

Orientation update

Curvature change

Deformation gradient

Membrane strain increment

Curvature increment

Virtual work

The rate of virtual work

Second variation of the membrane strain

Second variation of the curvature

Transverse shear treatment

Construction of the assumed strain field

Constitutive relations

Initial stress stiffness

One point integration plus stabilization

Comments on stabilization

In-plane displacement hourglass control

Rotational hourglass control

Degenerate elements

Rotary inertia scaling for explicit dynamics

Rotary inertia scaling

Rotational bulk viscosity for explicit dynamics

Fully integrated finite-membrane-strain shell formulation

Patch test

References

3.6.6 Small-strain shell elements in Abaqus/Explicit

Corotational coordinate system

Velocity strain formulation

Small-strain element S4RS

Small-strain element S4RSW

Small-strain element S3RS

Hourglass control

Reference

3.6.7 Axisymmetric shell element allowing asymmetric loading

Geometric description

Interpolations

Virtual work

Orthonormal surface coordinate system and coordinate transformation

Strain displacement operators

Corotational coordinate system

Consistent linearization

Incremental degrees of freedom: interpolations and configuration updates

Strain increment and stress resultant update

Pressure loads and load stiffness

Penalty constraints: transverse shear and drill rotation

Transverse shear

Drill rotation

Zero radius: collapsed edge

Mass matrix

Reference

3.6.8 Transverse shear stiffness in composite shells and offsets from the midsurface

Transverse shear stiffness

Finite-strain shells

Small-strain shells

Offset: reference surface versus midsurface

Reference

3.6.9 Rotary inertia for 5 degree of freedom shell elements

Reference

3.7 Rebar

3.7.1 Rebar modeling in two dimensions

Reference

3.7.2 Rebar modeling in three dimensions

Reference

3.7.3 Rebar modeling in shell, membrane, and surface elements

Reference

3.8 Hydrostatic fluid elements

3.8.1 Hydrostatic fluid elements

Hydraulic fluid with thermal expansion

Ideal gas

Volume calculation

Fluid link element

Negative eigenvalues

Reference

3.9 Special-purpose elements

3.9.1 Elbow elements

Geometric definitions

Interpolation

Pipe axis quantities

Cross-sectional deformation quantities

Strains and constraint

Some simplifying approximations

Reference

3.9.2 Frame elements with lumped plasticity

Degrees of freedom on the element

The elastic formulation

Lumped plasticity model

The hardening model

Plastic interaction surface

Large-displacement and large-rotation formulation

Additional data

Reference

3.9.3 Buckling strut response for frame elements

Buckling prediction and the ISO equation

Marshall strut envelope

References

3.9.4 Tube support elements

Geometry and kinematics

Forces in the element

Reference

3.9.5 Line spring elements

Virtual work contribution

Interpolation

Elasticity

Plasticity

Reference

3.9.6 Flexible joint element

Kinematics

Virtual work

Reference

3.9.7 Rotary inertia element

Reference

3.9.8 Distributing coupling elements

Load distribution

Constraint expression

Finite motion

Virtual work contribution

Initial stress stiffness terms

Mass

References

4. Mechanical Constitutive Theories

4.1 Overview

4.1.1 Mechanical constitutive models

Reference

4.2 Plasticity overview

4.2.1 Plasticity models: general discussion

Reference

4.2.2 Integration of plasticity models

Tangent matrix

Reference

4.3 Metal plasticity

4.3.1 Metal plasticity models

Reference

4.3.2 Isotropic elasto-plasticity

Material model definition

Plane stress

Uniaxial stress

Material stiffness

Plane stress

Uniaxial stress

Reference

4.3.3 Stress potentials for anisotropic metal plasticity

Reference

4.3.4 Rate-dependent metal plasticity (creep)

User subroutine CREEP

Reference

4.3.5 Models for metals subjected to cyclic loading

Strain rate decomposition

Elastic behavior

Plastic behavior

Linear kinematic hardening model

Nonlinear isotropic/kinematic hardening model

Reference

4.3.6 Porous metal plasticity

Yield condition

Flow rule

Evolution of the plastic strains and f

Integration of the elastoplastic equations

Computing the linearization moduli

Reference

4.3.7 Cast iron plasticity

Strain rate decomposition

Elastic behavior

Yield condition

Flow rule

Hardening

Reference

4.3.8 ORNL constitutive theory

ORNL plasticity theory of stainless steel

ORNL creep theory for stainless steel

Reference

4.3.9 Deformation plasticity

One-dimensional model

Multiaxial generalization

Strain energy density

Stress solution

The uniaxial case (the only nonzero stress component is one direct stress)

All strain components defined by the kinematic solution

Plane stress

Reference

4.3.10 Heat generation caused by plastic straining

References

4.4 Plasticity for non-metals

4.4.1 Porous elasticity

Reference

4.4.2 Models for granular or polymer behavior

Available yield criteria

Hardening, rate dependence, and creep

Strain rate decomposition

Elastic behavior

Linear Drucker-Prager model

Flow rule

Hyperbolic and general exponent models

Flow rule

Creep models

Creep behavior

Creep flow rule

Reference

4.4.3 Critical state models

The strain rate decomposition

Elastic behavior

Plastic behavior

Reference

4.4.4 Drucker-Prager/Cap model for geological materials

Strain rate decomposition

Elastic behavior

Plastic behavior

Flow rule

Creep model

Creep behavior

Creep flow rule

Cohesion creep

Consolidation creep

Reference

4.4.5 Mohr-Coulomb model

Strain rate decomposition

Elastic behavior

Yield behavior

Flow rule

Reference

4.4.6 Models for crushable foams

The strain rate decomposition

Elastic behavior

Plastic behavior

Crushable foam model with volumetric hardening

Yield surface

Flow potential

Hardening

Crushable foam model with isotropic hardening

Yield surface

Flow potential

Hardening

Reference

4.5 Other inelastic models

4.5.1 An inelastic constitutive model for concrete

Elastic-plastic model for concrete

Strain rate decomposition

Compression yield

Hardening

Flow

Crack detection and damaged elasticity

Strain rate decomposition

Yield

Flow

Hardening

Damaged elasticity

Cracking

Integration of the model

Reference

4.5.2 Damaged plasticity model for concrete and other quasi-brittle materials

Overview

Strain rate decomposition

Stress-strain relations

Hardening variables

Yield function

Flow rule

Summary

Damage and stiffness degradation

Uniaxial conditions

Uniaxial cyclic conditions

Multiaxial conditions

Elastic stiffness degradation

Yield condition

Flow rule

Viscoplastic regularization

Integration of the model

4.5.3 A cracking model for concrete and other brittle materials

Smeared cracking assumption

Crack direction assumptions

Elastic-cracking model for concrete

Strain rate decomposition

Crack direction transformations

Elasticity

Crack detection

Cracking conditions

Cracking relation

Rate constitutive equations

Tension softening models

Cracked shear models

Reference

4.5.4 Constitutive model for jointed materials

Joint system definitions

Strain rate decomposition

Elasticity and joint opening/closing

Plastic behavior of joint systems

Plastic behavior of bulk material

Integration of the model

Reference

4.6 Large-strain elasticity

4.6.1 Hyperelastic material behavior

Definitions and basic kinematic results

Rate of change of the internal virtual work

Particular forms of the strain energy potential

Polynomial form and particular cases

Ogden form

Arruda-Boyce form

Van der Waals form

Strain energy potential for highly compressible elastomers

Subroutine UHYPER

References

4.6.2 Fitting of hyperelastic and hyperfoam constants

Stress-strain relations for the polynomial strain energy potential

Uniaxial mode

Equibiaxial mode

Planar (pure shear) mode

Volumetric mode

Stress-strain relations for the reduced polynomial strain energy potential

Uniaxial mode

Equibiaxial mode

Planar (pure shear) mode

Volumetric mode

Stress-strain relations for the hyperelastic Ogden strain energy potential

Uniaxial mode

Equibiaxial mode

Planar (pure shear) mode

Volumetric mode

Stress-strain relations for the hyperelastic Arruda-Boyce strain energy potential

Uniaxial mode

Equibiaxial mode

Planar (pure shear) mode

Volumetric mode

Stress-strain relations for the hyperelastic Van der Waals energy potential

Uniaxial mode

Equibiaxial mode

Planar (pure shear) mode

Volumetric mode

Stress-strain relations for the hyperfoam strain energy potential

Uniaxial mode

Equibiaxial mode

Planar mode

Simple shear mode

Volumetric mode

Least squares fit

Linear least squares fit for the polynomial model

Linear least squares fit for the reduced polynomial model

Nonlinear least squares fit

Nonlinear least squares fit for the Ogden model

Nonlinear least squares fit for the Arruda-Boyce model

Nonlinear least squares fit for the Van der Waals model

Drucker stability check

Polynomial form

Ogden form

Arruda-Boyce form

Van der Waals form

Hyperfoam

References

4.6.3 Anisotropic hyperelastic material behavior

Strain-based formulation

Invariant-based formulation

Particular forms of the strain energy potential

Generalized Fung form

Holzapfel-Gasser-Ogden form

User-defined form: strain-based

User-defined form: invariant-based

Reference

4.7 Mullins effect and permanent set

4.7.1 Mullins effect

Material behavior

Idealized material behavior

Basic framework for the model

Primary hyperelastic behavior

Augmented strain energy density function

Stress computation

Damage function and damage variable

Material tangent stiffness

Damage dissipation

Reference

4.7.2 Permanent set

Approach used

Reference

4.8 Viscoelasticity

4.8.1 Viscoelasticity

Reduced states of stress

Automatic time stepping procedure

Reference

4.8.2 Finite-strain viscoelasticity

Integral formulation

Implementation

Integration of the hydrostatic stress

Integration of the deviatoric stress

Rate equation

Cauchy versus Kirchhoff stress

Reference

4.8.3 Frequency domain viscoelasticity

Reference

4.9 Hysteresis

4.9.1 Hysteresis

Reference

5. Interface Modeling

5.1 Contact modeling

5.1.1 Small-sliding interaction between bodies

Identification of tangent plane based on nearest neighbor interaction

Contact formulation as a constrained variational principle

Formulation for two-dimensional small-sliding deformable contact

Formulation for three-dimensional small-sliding deformable contact

Formulation for two-dimensional small-sliding rigid contact

Formulation for three-dimensional small-sliding rigid contact

Reference

5.1.2 Finite-sliding interaction between deformable bodies

Two-dimensional and axisymmetric slide line elements

Tube-tube interface elements

Slide line smoothing

Self contact

References

5.1.3 Finite-sliding interaction between a deformable and a rigid body

Reference

5.2 Surface interactions

5.2.1 Contact pressure definition

Hard contact

Softened contact defined with an exponential pressure-overclosure relationship

Softened contact defined with a tabular pressure-overclosure relationship

Softened contact defined with a linear pressure-overclosure relationship

Softened contact implementation

Viscous damping option

Reference

5.2.2 Pressure and fluid flow in pore pressure contact

The pore fluid constraints on the contact interface

The transient equations

Closed contact

The steady-state equations

Small sliding

Reference

5.2.3 Coulomb friction

Elastic stick formulation

Frictional stress due to forced sliding in static analysis

Friction contributions for complex eigenvalue extraction analysis

Viscous stick formulation for steady-state transport

Exact stick formulation

Anisotropic friction

Viscous damping

Reference

5.2.4 Thermal interface definition

Conduction

Radiation

Jacobian matrix

Reference

5.2.5 Heat generation caused by frictional sliding

References

5.2.6 Heat generation caused by electrical current

References

5.2.7 Surface-based acoustic-structural medium interaction

The equations on the contact interface between the structure and the acoustic medium

Solid/structural master, fluid slave

Fluid master, solid/structural slave

Reference

6. Loading and Constraints

6.1 Dynamic loading

6.1.1 Centrifugal, Coriolis, and rotary acceleration forces

Reference

6.1.2 Baseline correction of accelerograms

Reference

6.2 Abaqus/Aqua loading

6.2.1 Drag, inertia, and buoyancy loading

Continuously distributed drag and inertia loading

Point drag and inertia loading

Distributed buoyancy

Point buoyancy

Load stiffness

Reference

6.2.2 Airy wave theory

Reference

6.2.3 Stokes wave theory

Fluid particle velocities and accelerations for Stokes 5th order wave

Reference

6.3 Incident wave loading

6.3.1 Loading due to an incident dilatational wave field

Scattering formulation

Virtual mass formulation

Scattering load amplitude

Steady-state dynamics

Effect of structural motion

Underwater explosion—bubble load amplitude

Geers-Hunter model

“Waveless” model

Reflections outside the computational domain

Spherical waves

“Soft” and “hard” limits

Planar waves

Reference

6.4 Pressure penetration loading

6.4.1 Pressure penetration loading with surface-based contact

Reference

6.5 Load stiffness

6.5.1 Pressure load stiffness

Pressure load stiffness on a surface in three-dimensional space

Pressure load stiffness on a surface in two-dimensional space

Reference

6.5.2 Load stiffness for beam elements

Reference

6.5.3 Pressure loadings on elbow elements

Reference

6.6 Multi-point constraints

6.6.1 Sliding constraint

Reference

6.6.2 Shell to solid constraint

Reference

6.6.3 Revolute joint

Reference

6.6.4 Universal joint

Reference

6.6.5 Local velocity constraint

Reference

6.6.6 Kinematic coupling

Rotations

Translations

Linearized form

Second-order form

Reference

7. References

7.1 References

7.1.1

Abaqus Theory Manual Abaqus Version 6.10 ID: Printed on:

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Preface This section lists various resources that are available for help with using Abaqus Uniﬁed FEA software. Support Both technical engineering support (for problems with creating a model or performing an analysis) and systems support (for installation, licensing, and hardware-related problems) for Abaqus are offered through a network of local support ofﬁces. Regional contact information is listed in the front of each Abaqus manual and is accessible from the Locations page at www.simulia.com. SIMULIA Online Support System The SIMULIA Online Support System (SOSS) provides a knowledge database of SIMULIA Answers. The SIMULIA Answers are solutions to questions that we have had to answer or guidelines on how to use Abaqus, SIMULIA SLM, Isight, and other SIMULIA products. You can also submit new requests for support in the SOSS. All support incidents are tracked in the SOSS. If you are contacting us by means outside the SOSS to discuss an existing support problem and you know the incident number, please mention it so that we can consult the database to see what the latest action has been. To use the SOSS, you need to register with the system. Visit the My Support page at www.simulia.com to register. Many questions about Abaqus can also be answered by visiting the Products page and the Support page at www.simulia.com. Anonymous ftp site To facilitate data transfer with SIMULIA, an anonymous ftp account is available on the computer ftp.simulia.com. Login as user anonymous, and type your e-mail address as your password. Contact support before placing ﬁles on the site. Training All ofﬁces and representatives offer regularly scheduled public training classes. We also provide training seminars at customer sites. All training classes and seminars include workshops to provide as much practical experience with Abaqus as possible. For a schedule and descriptions of available classes, see www.simulia.com or call your local ofﬁce or representative. Feedback We welcome any suggestions for improvements to Abaqus software, the support program, or documentation. We will ensure that any enhancement requests you make are considered for future releases. If you wish to make a suggestion about the service or products, refer to www.simulia.com. Complaints should be addressed by contacting your local ofﬁce or through www.simulia.com by visiting the Quality Assurance section of the Support page. Abaqus Version 6.10 ID: Printed on:

Abaqus Version 6.6 ID: Printed on:

CONTENTS 1.1.1 1.2.1 1.3.1 1.4.1 1.4.2 1.4.3 1.4.4 1.5.1 1.5.2 1.5.3 1.5.4 1.5.5 2.1.1 2.2.1 2.2.2 2.2.3 2.3.1 2.3.2 2.4.1 2.4.2 2.4.3 Contents 1. Introduction and Basic Equations Introduction Introduction: general Notation Notation Finite rotations Rotation variables Deformation, strain, and strain rates Deformation Strain measures Rate of deformation and strain increment The additive strain rate decomposition Equilibrium, stress, and state storage Equilibrium and virtual work Stress measures Stress rates State storage Energy balance 2. Procedures Overview Procedures: overview and basic equations Nonlinear solution methods Nonlinear solution methods in Abaqus/Standard Quasi-Newton solution technique Direct cyclic algorithm Buckling and postbuckling Eigenvalue buckling prediction Modiﬁed Riks algorithm Nonlinear dynamics Implicit dynamic analysis Intermittent contact/impact Subspace dynamics v Abaqus ID:stm-toc Printed on: Thu March 18 -- 13:49:15 2010

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