ABAQUS Theory Manual

ABAQUS Theory Manual

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ADAMS is a registered United States trademark of Mechanical Dynamics, Inc. ADAMS/Flex and ADAMS/View are trademarks of Mechanical Dynamics, Inc. CATIA is a registered trademark of Dassault Systmes. C-MOLD is a registered trademark of Advanced CAE Technology, Inc., doing business as C-MOLD. Compaq Alpha is registered in the U.S. Patent and Trademark Office. FE-SAFE is a trademark of Safe Technology, Ltd. Fujitsu, UXP, and VPP are registered trademarks of Fujitsu Limited. Hewlett-Packard, HP-GL, and HP-GL/2 are registered trademarks of Hewlett-Packard Co. Hitachi is a registered trademark of Hitachi, Ltd. IBM RS/6000 is a trademark of IBM. Intel is a registered trademark of the Intel Corporation. NEC is a trademark of the NEC Corporation. PostScript is a registered trademark of Adobe Systems, Inc. Silicon Graphics is a registered trademark of Silicon Graphics, Inc. SUN is a registered trademark of Sun Microsystems, Inc.TEX is a trademark of the American Mathematical Society.

UNIX and Motif are registered trademarks and X Window System is a trademark of The Open Group in the U.S. and other countries. Windows NT is a registered trademark of the Microsoft Corporation. ABAQUS/CAE incorporates portions of the ACIS software by SPATIAL TECHNOLOGY INC. ACIS is a registered trademark of SPATIAL TECHNOLOGY INC. This release of ABAQUS on Windows NT includes the diff program obtained from the Free Software Foundation. You may freely distribute the diff program and/or modify it under the terms of the GNU Library General Public License as published by the Free Software Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA. This release of ABAQUS/CAE includes lp_solve, a simplex-based code for linear and integer programming problems by Michel Berkelaar of Eindhoven University of Technology, Eindhoven, the Netherlands. Python, copyright 1991-1995 by Stichting Mathematisch Centrum, Amsterdam, The Netherlands. All Rights Reserved. Permission to use, copy, modify, and distribute the Python software and its documentation for any purpose and without fee is hereby granted, provided that the above copyright notice appear in all copies and that both that copyright notice and this permission notice appear in supporting documentation, and that the names of Stichting Mathematisch Centrum or CWI or

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Corporation for National Research Initiatives or CNRI not be used in advertising or publicity pertaining to distribution of the software without specific, written prior permission. All other brand or product names are trademarks or registered trademarks of their respective companies or organizations.

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General conversion factors (to five significant digits) Quantity U.S. unit SI equivalent Length 1 in 0.025400 m 1 ft 0.30480 m 1 mile 1609.3 m 2 Area 1 in 0.64516 10-3 m2 1 ft2 0.092903 m 2 1 acre 4046.9 m2 Volume 1 in3 0.016387 10-3 m3 3 1 ft 0.028317 m 3 1 US gallon 3.7854 10-3 m3 Quantity Density Energy Force Mass Power Pressure, Stress Conversion factors for stress analysis U.S. unit SI equivalent 1 slug/ft3 = 1 lbf s2/ft4 515.38 kg/m3 1 lbf s2/in4 10.687 106 kg/m3 1 ft lbf 1.3558 J (N m) 1 lbf 4.4482 N (kg m/s2) 2 1 slug = 1 lbf s /ft 14.594 kg (N s2/m) 175.13 kg 1 lbf s2/in 1 ft lbf/s 1.3558 W (N m/s) 2) 1 psi (lbf/in 6894.8 Pa (N/m2)

Conversion factors for heat transfer analysis Quantity U.S. unit SI equivalent Conductivity 1 Btu/ft hr F 1.7307 W/m C 1 Btu/in hr F 20.769 W/m C Density 1 lbm/in3 27680. kg/m3 Energy 1 Btu 1055.1 J Heat flux density 1 Btu/in 2 hr 454.26 W/m2 Power 1 Btu/hr 0.29307 W Specific heat 1 Btu/lbm F 4186.8 J/kg C Temperature 1 F 5/9 C Temp F 9/5 Temp C + 32 9/5 Temp K - 459.67 Constant Absolute zero Acceleration of gravity Atmospheric pressure Stefan-Boltzmann constant Important constants U.S. unit -459.67 F 32.174 ft/s 2 14.694 psi 0.1714 10-8 Btu/hr ft2 R4 where R = F + 459.67 SI unit -273.15 C 9.8066 m/s2 0.10132 106 Pa 5.669 10-8 W/m2 K4 where K = C + 273.15

Approximate properties of mild steel at room temperature Quantity U.S. unit SI unit Conductivity 28.9 Btu/ft hr F 50 W/m C 2.4 Btu/in hr F Density 15.13 slug/ft3 (lbf s2/ft4) 7800 kg/m3 0.730 10-3 lbf s2/in4

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Elastic modulus Specific heat Yield stress

0.282 lbm/in 3 30 106 psi 0.11 Btu/lbm F 30 103 psi

207 109 Pa 460 J/kg C 207 106 Pa

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UNITED STATES Hibbitt, Karlsson & Sorensen, Inc. 1080 Main Street Pawtucket, RI 02860-4847 Tel: 401 727 4200 Fax: 401 727 4208 E-mail: [email protected], [email protected] http://www.abaqus.com Hibbitt, Karlsson & Sorensen (West), Inc. 39221 Paseo Padre Parkway, Suite F Fremont, CA 94538-1611 Tel: 510 794 5891 Fax: 510 794 1194 E-mail: [email protected] AC Engineering, Inc. 1440 Innovation Place West Lafayette, IN 47906-1000 Tel: 765 497 1373 Fax: 765 497 4444 E-mail: [email protected] ARGENTINA KB Engineering S. R. L. Florida 274, Of. 37 (1005) Buenos Aires, Argentina Tel: +54 11 4393 8444 Fax: +54 11 4326 2424 E-mail: [email protected]

Hibbitt, Karlsson & Sorensen (Michigan), Inc. 14500 Sheldon Road, Suite 160 Plymouth, MI 48170-2408 Tel: 734 451 0217 Fax: 734 451 0458 E-mail: [email protected]

ABAQUS Solutions Northeast, LLC Summit Office Park, West Building 300 Centerville Road, Suite 209W Warwick, RI 02886-0201 Tel: 401 739 3637 Fax: 401 739 3302 E-mail: [email protected]

AUSTRIA VOEST-ALPINE STAHL LINZ GmbH Department WFE Postfach 3 A-4031 Linz Tel: 0732 6585 9919 Fax: 0732 6980 4338 E-mail: [email protected] CHINA Advanced Finite Element Services Department of Engineering Mechanics Tsinghua University Beijing 100084, P. R. China Tel: 010 62783986

AUSTRALIA Compumod Pty. Ltd. Level 13, 309 Pitt Street Sydney 2000 P.O. Box A807 Sydney South 1235 Tel: 02 9283 2577 Fax: 02 9283 2585 E-mail: [email protected] http://www.compumod.com.au BENELUX ABAQUS Benelux BV Huizermaatweg 576 1276 LN Huizen The Netherlands Tel: +31 35 52 58 424 Fax: +31 35 52 44 257 E-mail: [email protected] CZECH REPUBLIC AND SLOVAK REPUBLIC ASATTE Technick 4, 166 07 Praha 6 Czech Republic Tel: 420 2 24352654 Fax: 420 2 33322482 0-6

Fax: 010 62771163 E-mail: [email protected] FRANCE ABAQUS Software, s.a.r.l. 7, rue de la Patte d'Oie 78000 Versailles Tel: 01 39 24 15 40 Fax: 01 39 24 15 45 E-mail: [email protected] ITALY Hibbitt, Karlsson & Sorensen Italia, s.r.l. Viale Certosa, 1 20149 Milano Tel: 02 39211211 Fax: 02 39211210 E-mail: [email protected]

E-mail: [email protected] GERMANY ABACOM Software GmbH Theaterstrae 30-32 D-52062 Aachen Tel: 0241 474010 Fax: 0241 4090963 E-mail: [email protected] JAPAN Hibbitt, Karlsson & Sorensen, Inc.

3rd Floor, Akasaka Nihon Building 5-24, Akasaka 9-chome Minato-ku Tokyo, 107-0052 Tel: 03 5474 5817 Fax: 03 5474 5818 E-mail: [email protected] KOREA MALAYSIA Hibbitt, Karlsson & Sorensen Korea, Inc. Compumod Sdn Bhd Suite 306, Sambo Building #33.03 Menara Lion 13-2 Yoido-Dong, Youngdeungpo-ku 165 Jalan Ampang Seoul, 150-010 50450 Kuala Lumpur Tel: 02 785 6707/8 Tel: 3 466 2122 Fax: 02 785 6709 Fax: 3 466 2123 E-mail: [email protected] E-mail: [email protected] NEW ZEALAND POLAND Matrix Applied Computing Ltd. BudSoft Sp. z o.o. P.O. Box 56-316, Auckland 61-807 Pozna Courier: Unit 2-5, 72 Dominion Road, Sw. Marcin 58/64 Mt Eden, Auckland Tel: 61 852 31 19 Tel: +64 9 623 1223 Fax: 61 852 31 19 Fax: +64 9 623 1134 E-mail: [email protected] E-mail: [email protected] SINGAPORE SOUTH AFRICA Compumod (Singapore) Pte Ltd Finite Element Analysis Services (Pty) Ltd. #17-05 Asia Chambers Suite 20-303C, The Waverley 20 McCallum Street Wyecroft Road Singapore 069046 Mowbray 7700 Tel: 223 2996 Tel: 021 448 7608 Fax: 226 0336 Fax: 021 448 7679 E-mail: E-mail: [email protected] [email protected] SPAIN SWEDEN Principia Ingenieros Consultores, S.A. FEM-Tech AB Velzquez, 94 Pilgatan 8 28006 Madrid SE-721 30 Vsters

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Tel: 91 209 1482 Fax: 91 575 1026 E-mail: [email protected] TAIWAN APIC 7th Fl., 131 Sung Chiang Road Taipei, 10428 Tel: 02 25083066 Fax: 02 25077185 E-mail: [email protected]

Tel: 021 12 64 10 Fax: 021 18 12 44 E-mail: [email protected] UNITED KINGDOM Hibbitt, Karlsson & Sorensen (UK) Ltd. The Genesis Centre Science Park South, Birchwood Warrington, Cheshire WA3 7BH Tel: 01925 810166 Fax: 01925 810178 E-mail: [email protected]

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This section lists various resources that are available for help with using ABAQUS, including technical and systems support, training seminars, and documentation.

SupportHKS offers both technical (engineering) support and systems support for ABAQUS. Technical and systems support are provided through the nearest local support office. You can contact our offices by telephone, fax, electronic mail, or regular mail. Information on how to contact each office is listed in the front of each ABAQUS manual. Support information is also available by visiting the ABAQUS Home Page on the World Wide Web (details are given below). When contacting your local support office, please specify whether you would like technical support (you have encountered problems performing an ABAQUS analysis) or systems support (ABAQUS will not install correctly, licensing does not work correctly, or other hardware-related issues have arisen). We welcome any suggestions for improvements to the support program or documentation. We will ensure that any enhancement requests you make are considered for future releases. If you wish to file a complaint about the service or products provided by HKS, refer to the ABAQUS Home Page.

Technical supportHKS technical support engineers can assist in clarifying ABAQUS features and checking errors by giving both general information on using ABAQUS and information on its application to specific analyses. If you have concerns about an analysis, we suggest that you contact us at an early stage, since it is usually easier to solve problems at the beginning of a project rather than trying to correct an analysis at the end. Please have the following information ready before calling the technical support hotline, and include it in any written contacts: The version of ABAQUS that are you using. - The version numbers for ABAQUS/Standard and ABAQUS/Explicit are given at the top of the data (.dat) file. - The version numbers for ABAQUS/CAE and ABAQUS/Viewer can be found by selecting Help->On version from the main menu bar. - The version number for ABAQUS/CAT is given at the top of the input ( .inp) file as well as the data file. - The version numbers for ABAQUS/ADAMS and ABAQUS/C-MOLD are output to the screen. - The version number for ABAQUS/Safe is given under the ABAQUS logo in the main window. The type of computer on which you are running ABAQUS.

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The symptoms of any problems, including the exact error messages, if any. Workarounds or tests that you have already tried. When calling for support about a specific problem, any available ABAQUS output files may be helpful in answering questions that the support engineer may ask you. The support engineer will try to diagnose your problem from the model description and a description of the difficulties you are having. Frequently, the support engineer will need model sketches, which can be faxed to HKS or sent in the mail. Plots of the final results or the results near the point that the analysis terminated may also be needed to understand what may have caused the problem. If the support engineer cannot diagnose your problem from this information, you may be asked to send the input data. The data can be sent by means of e-mail, tape, or disk. Please check the ABAQUS Home Page at www.abaqus.com for the media formats that are currently accepted. All support calls are logged into a database, which enables us to monitor the progress of a particular problem and to check that we are resolving support issues efficiently. If you would like to know the log number of your particular call for future reference, please ask the support engineer. If you are calling to discuss an existing support problem and you know the log number, please mention it so that we can consult the database to see what the latest action has been and, thus, avoid duplication of effort. In addition, please give the receptionist the support engineer's name (or include it at the top of any e-mail correspondence).

Systems supportHKS systems support engineers can help you resolve issues related to the installation and running of ABAQUS, including licensing difficulties, that are not covered by technical support. You should install ABAQUS by carefully following the instructions in the ABAQUS Site Guide. If you encounter problems with the installation or licensing, first review the instructions in the ABAQUS Site Guide to ensure that they have been followed correctly. If this does not resolve the problems, look on the ABAQUS Home Page under Technical Support for information about known installation problems. If this does not address your situation, please contact your local support office. Send whatever information is available to define the problem: error messages from an aborted analysis or a detailed explanation of the problems encountered. Whenever possible, please send the output from the abaqus info=env and abaqus info=sys commands.

ABAQUS Web serverFor users connected to the Internet, many questions can be answered by visiting the ABAQUS Home Page on the World Wide Web athttp://www.abaqus.com

The information available on the ABAQUS Home Page includes: Frequently asked questions ABAQUS systems information and machine requirements

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Benchmark timing documents Error status reports ABAQUS documentation price list Training seminar schedule Newsletters

Anonymous ftp siteFor users connected to the Internet, HKS maintains useful documents on an anonymous ftp account on the computer ftp.abaqus.com. Simply ftp to ftp.abaqus.com. Login as user anonymous, and type your e-mail address as your password. Directions will come up automatically upon login.

Writing to technical supportAddress of HKS Headquarters: Hibbitt, Karlsson & Sorensen, Inc. 1080 Main Street Pawtucket, RI 02860-4847, USA Attention: Technical Support Addresses for other offices and representatives are listed in the front of each manual.

Support for academic institutionsUnder the terms of the Academic License Agreement we do not provide support to users at academic institutions unless the institution has also purchased technical support. Please see the ABAQUS Home Page, or contact us for more information.

TrainingAll HKS offices offer regularly scheduled public training classes. The Introduction to ABAQUS/Standard and ABAQUS/Explicit seminar covers basic usage and nonlinear applications, such as large deformation, plasticity, contact, and dynamics. Workshops provide as much practical experience with ABAQUS as possible. The Introduction to ABAQUS/CAE seminar discusses modeling, managing simulations, and viewing results with ABAQUS/CAE. "Hands-on" workshops are complemented by lectures. Advanced seminars cover topics of interest to customers with experience using ABAQUS, such as engine analysis, metal forming, fracture mechanics, and heat transfer. We also provide training seminars at customer sites. On-site training seminars can be one or more days in duration, depending on customer requirements. The training topics can include a combination of material from our introductory and advanced seminars. Workshops allow customers to exercise ABAQUS on their own computers.

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For a schedule of seminars see the ABAQUS Home Page, or call HKS or your local HKS representative.

DocumentationThe following documentation and publications are available from HKS, unless otherwise specified, in printed form and through our online documentation server. For more information on accessing the online books, refer to the discussion of execution procedures in the user's manuals. In addition to the documentation listed below, HKS publishes two newsletters on a regular schedule: ABAQUS/News and ABAQUS/Answers. ABAQUS/News includes topical information about program releases, training seminars, etc. ABAQUS/Answers includes technical articles on particular topics related to ABAQUS usage. These newsletters are distributed at no cost to users who wish to subscribe. Please contact your local ABAQUS support office if you wish to be added to the mailing list for these publications. They are also archived in the Reference Shelf on the ABAQUS Home Page.

Training ManualsGetting Started with ABAQUS/Standard: This document is a self-paced tutorial designed to help new users become familiar with using ABAQUS/Standard for static and dynamic stress analysis simulations. It contains a number of fully worked examples that provide practical guidelines for performing structural analyses with ABAQUS. Getting Started with ABAQUS/Explicit: This document is a self-paced tutorial designed to help new users become familiar with using ABAQUS/Explicit. It begins with the basics of modeling in ABAQUS, so no prior knowledge of ABAQUS is required. A number of fully worked examples provide practical guidelines for performing explicit dynamic analyses, such as drop tests and metal forming simulations, with ABAQUS/Explicit. Lecture Notes: These notes are available on many topics to which ABAQUS is applied. They are used in the technical seminars that HKS presents to help users improve their understanding and usage of ABAQUS (see the "Training" section above for more information about these seminars). While not intended as stand-alone tutorial material, they are sufficiently comprehensive that they can usually be used in that mode. The list of available lecture notes is included in the Documentation Price List.

User's ManualsABAQUS/Standard User's Manual: This volume contains a complete description of the elements, material models, procedures, input specifications, etc. It is the basic reference document for ABAQUS/Standard. ABAQUS/Explicit User's Manual: This volume contains a complete description of the elements, material models, procedures, input specifications, etc. It is the basic reference document for ABAQUS/Explicit.

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ABAQUS/CAE User's Manual: This reference document for ABAQUS/CAE includes three comprehensive tutorials as well as detailed descriptions of how to use ABAQUS/CAE for model generation, analysis, and results evaluation. ABAQUS/Viewer User's Manual: This basic reference document for ABAQUS/Viewer includes an introductory tutorial as well as a complete description of how to use ABAQUS/Viewer to display your model and results. ABAQUS/ADAMS User's Manual: This document describes how to install and how to use ABAQUS/ADAMS, an interface program that creates ABAQUS models of ADAMS components and converts the ABAQUS results into an ADAMS modal neutral file that can be used by the ADAMS/Flex program. It is the basic reference document for the ABAQUS/ADAMS program. ABAQUS/CAT User's Manual: This document describes how to install and how to use ABAQUS/CAT, an interface program that creates an ABAQUS input file from a CATIA model and postprocesses the analysis results in CATIA. It is the basic reference document for the ABAQUS/CAT program. ABAQUS/C-MOLD User's Manual: This document describes how to install and how to use ABAQUS/C-MOLD, an interface program that translates finite element mesh, material property, and initial stress data from a C-MOLD analysis to an ABAQUS input file. ABAQUS/Safe User's Manual: This document describes how to install and how to use ABAQUS/Safe, an interface program that calculates fatigue lives and fatigue strength reserve factors from finite element models. It is the basic reference document for the ABAQUS/Safe program. The theoretical background to fatigue analysis is contained in the Modern Metal Fatigue Analysis manual (available only in print). Using ABAQUS Online Documentation: This online manual contains instructions on using the ABAQUS online documentation server to read the manuals that are available online. ABAQUS Release Notes: This document contains brief descriptions of the new features available in the latest release of the ABAQUS product line. ABAQUS Site Guide: This document describes how to install ABAQUS and how to configure the installation for particular circumstances. Some of this information, of most relevance to users, is also provided in the user's manuals.

Examples ManualsABAQUS Example Problems Manual: This volume contains more than 75 detailed examples designed to illustrate the approaches and decisions needed to perform meaningful linear and nonlinear analysis. Typical cases are large motion of an elastic-plastic pipe hitting a rigid wall; inelastic buckling collapse of a thin-walled elbow; explosive loading of an elastic, viscoplastic thin ring; consolidation under a footing; buckling of a composite shell with a hole; and deep drawing of a metal sheet. It is generally useful to look for relevant examples in this manual and to review them when embarking on a new class of problem.

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ABAQUS Benchmarks Manual: This volume (available online and, if requested, in print) contains over 200 benchmark problems and standard analyses used to evaluate the performance of ABAQUS; the tests are multiple element tests of simple geometries or simplified versions of real problems. The NAFEMS benchmark problems are included in this manual. ABAQUS Verification Manual: This online-only volume contains more than 5000 basic test cases, providing verification of each individual program feature (procedures, output options, MPCs, etc.) against exact calculations and other published results. It may be useful to run these problems when learning to use a new capability. In addition, the supplied input data files provide good starting points to check the behavior of elements, materials, etc.

Reference ManualsABAQUS Keywords Manual: This volume contains a complete description of all the input options that are available in ABAQUS/Standard and ABAQUS/Explicit. ABAQUS Theory Manual: This volume (available online and, if requested, in print) contains detailed, precise discussions of all theoretical aspects of ABAQUS. It is written to be understood by users with an engineering background. ABAQUS Command Language Manual: This online manual provides a description of the ABAQUS Command Language and a command reference that lists the syntax of each command. The manual describes how commands can be used to create and analyze ABAQUS/CAE models, to view the results of the analysis, and to automate repetitive tasks. It also contains information on using the ABAQUS Command Language or C++ as an application programming interface (API). ABAQUS Input Files: This online manual contains all the input files that are included with the ABAQUS release and referred to in the ABAQUS Example Problems Manual, the ABAQUS Benchmarks Manual, and the ABAQUS Verification Manual. They are listed in the order in which they appear in the manuals, under the title of the problem that refers to them. The input file references in the manuals hyperlink directly to this book. Quality Assurance Plan: This document describes HKS's QA procedures. It is a controlled document, provided to customers who subscribe to either HKS's Nuclear QA Program or the Quality Monitoring Service.

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Introduction and Basic Equations

1. Introduction and Basic Equations1.1 Introduction 1.1.1 Introduction: generalThe ABAQUS system includes ABAQUS/Standard, a general-purpose finite element program; ABAQUS/Explicit, an explicit dynamics finite element program; and ABAQUS/Viewer, an interactive postprocessing program that provides displays and output lists from output database files written by ABAQUS/Standard and ABAQUS/Explicit. This manual describes the theories used in ABAQUS. Many sections in this manual apply to both ABAQUS/Standard and ABAQUS/Explicit. Certain sections obviously apply only to either ABAQUS/Standard or ABAQUS/Explicit; for example, all sections in the chapter on procedures apply to ABAQUS/Standard, except the section discussing the explicit dynamic integration procedure, which applies to ABAQUS/Explicit. If it is not obvious to which program a section applies, it is clearly indicated. ABAQUS/Standard includes several added-cost options. The ABAQUS/Aqua option includes features specifically designed for the analysis of beam-like structures installed underwater and subject to loading by water currents and wave action. The ABAQUS/Design option enables the user to parametrize input file quantities and write Python scripts to perform parametric studies. The ABAQUS/USA option allows the Underwater Shock Analysis program originally developed by Lockheed's Research Laboratory and supported by Unique Software Applications to be used within ABAQUS/Standard to study the coupled problem of acoustic shock wave loading of underwater structures. Certain aspects of the theory behind these options are described in this manual. The options are available only if the user's license includes them. The objective of this manual is to define the theories used in ABAQUS that are generally not available in the standard textbooks on mechanics, structures, and finite elements but are well known to the engineer who uses ABAQUS. The manual is intended as a reference document that defines what is available in the code. Nevertheless, it is written in such a way that it can also be used as a tutorial document by a reader who needs to obtain some background in an unfamiliar area. The material is presented in a way that should make it accessible to any user with an engineering background. Some of the theories may be relatively unfamiliar to such a user; for example, few engineering curricula provide extensive background in plasticity, shell theory, finite deformations of solids, or the analysis of porous media. Yet ABAQUS contains capabilities for all of these models and many others. The manual is far from comprehensive in its coverage of such topics: in this sense it is only a reference volume. The user is strongly encouraged to pursue topics of interest through texts and papers. Chapter 7, "References," at the end of this manual lists references that should provide a starting point for obtaining such information. (HKS does not supply copies of papers that have appeared in publications other than those of HKS. EPRI reports can be obtained from Research Reports Center ( RRC), Box 50490, Palo Alto, CA 94303.) Chapter 1, "Introduction and Basic Equations," discusses the notation used in the manual, some basic concepts of kinematics and mechanics--such as rotations, stress, and equilibrium--as well as the basic

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equations of nonlinear finite element analysis. Chapter 2, "Procedures," describes the various analysis procedures (nonlinear static stress analysis, dynamics, eigenvalue extraction, etc.) that are available in ABAQUS. Chapter 3, "Elements," describes the element formulations. Chapter 4, "Mechanical Constitutive Theories," describes the mechanical constitutive theories. Chapter 5, "Interface Modeling," discusses the most important aspects of the contact/interaction formulation in ABAQUS/Standard. Chapter 6, "Loading and Constraints," describes the formulation of some of the more complicated load types and multi-point constraints. If you are reading this book through the online documentation, it is recommended that you enlarge the book window so that the equations and figures are clearly visible. Refer to Chapter 3, "Printing from an online book," of Using ABAQUS Online Documentation for instructions on printing from the online documentation. Be sure to toggle on Print graphics and equations, or the graphics and equations will not appear in the printed copy. The equations in this manual may appear different in the printed output from the way they appear online; bold terms are sometimes output incorrectly (see the Status Reports on the HKS Home Page, http://www.abaqus.com, for details). To obtain a bound printed copy of this manual, contact your local HKS office or representative.

1.2 Notation 1.2.1 NotationNotation is often a serious obstacle that prevents an engineer from using advanced textbooks; for example, general curvilinear tensor analysis and functional analysis are both necessary in some of the theories used in ABAQUS, but the unfamiliar notations commonly used in these areas often discourage the user from pursuing their study. The notation used in most of this manual (direct matrix notation) may be unfamiliar to some readers; but it is not difficult or time consuming to gain enough familiarity with the notation for it to be useful, and it is definitely worthwhile. This notation is commonly used in the modern engineering literature--it is a shorthand version of the familiar matrix notation used in many older engineering textbooks. The notation is appealing--once it is understood--because it allows the equations to be developed concisely, and the physical ideas can be perceived without the distraction of the complexities that arise from the choice of the particular basis system that will eventually be used to express the same concepts in component form. Because the notation has become so standard in the literature, the user who wishes or needs to read textbooks and papers that are related to the use of ABAQUS will find that familiarity with this notation is desirable. Both direct matrix notation and component form notation are used in the manual. Both notations are described in this section. Direct matrix notation is used whenever possible. However, vectors, matrices, and the higher-order tensors used in the theories must eventually be written in component form to store them as a set of numbers on the computer. Thus, both ways of writing these quantities will be needed in the manual.

Basic quantitiesThe quantities needed to formulate the theory are scalars, vectors, second-order tensors (matrices),

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and--occasionally--fourth-order tensors (for example, the stress-strain transformation for linear elasticity). In direct matrix notation these are written as: a a scalar value a vector a or bac aT or fag with the transpose a second-order tensor or matrix a or [a] aT or [a]T with the transpose and A a fourth-order tensor Vectors and second-order tensors (matrices) are written in the same way: they are distinguished by the context. In direct matrix notation there is generally no need to indicate that a vector must be transposed. The context determines whether a vector is to be used as a "column" vector a or as a "row" vector aT . In this case the transpose superscript is only used to improve the readability of an expression. On the other hand, for second-order nonsymmetric tensors the addition of a transpose superscript will change the meaning of an expression. This representation of vectors and tensors is very general and convenient for developing the theory so that the equations can be understood easily in terms of their physical meaning. However, in actual computations we have to work with individual numbers, so vectors and tensors must be expressed in terms of their components. These components are associated with an axis system that defines a set of base vectors at each point in space. The simplest axis system is rectangular Cartesian, because the base vectors are orthogonal unit vectors in the same direction at all points. Unfortunately, we need more generality than this because we will be dealing with shells and beams, where stress, strain, etc. are most conveniently described in terms of directions on the surface of the shell (or associated with the axis of the beam), and these usually change as we move around on the surface. To retain this necessary generality and express vectors and matrices in component form, we introduce a general set of base vectors, e , = 1; 2; 3 , which are not necessarily orthogonal or of unit length but are sufficient to define the components of a vector (for this purpose they must not be parallel or have zero length). A vector a can then be written a = a1 e1 + a2 e2 + a3 e3 ; where the numbers a1 , a2 , and a3 are the components of a associated with e1 , e2 , and e3 . In actual cases the e are chosen for convenience (for example, see ``Conventions,'' Section 1.2.2 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual, for a description of how base vectors are chosen for surface elements in ABAQUS), and then the a are obtained. To save writing, we adopt the usual summation convention that a repeated index is summed--in this case over the range 1 to 3--so that the above equation is written a = a e : Likewise, the component form of a matrix will be a = e a e = a e e ;

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or, written out, a =e1 a11 e1 + e1 a12 e2 + e1 a13 e3 e2 a21 e1 + e2 a22 e2 + e2 a23 e3 e3 a31 e1 + e3 a32 e2 + e3 a33 e3 : Similarly, a fourth-order tensor can be written in component form as A = A e e e e : While we will need such completely general base vectors for describing the stresses and strains on shells and beams, in many cases it is convenient to use rectangular Cartesian components so that the e are orthogonal unit vectors. To distinguish this particular case, we will use Latin indices instead of Greek indices. Thus, e are a set of general base vectors; while ei are rectangular Cartesian base vectors; and a is the component of the vector a along a general base vector, while ai , i = 1; 2; 3 , is the component of a along the ith Cartesian direction. Vector and tensor concepts and their representation are discussed in many textbooks--see Flugge (1972), for example.

Basic operationsThe usual matrix and vector operators are indicated in this manual as follows: Dot product of two vectors: a=bc (The dot symbol defines this operation completely, regardless of whether b or c is transposed--i.e., b c = bT c:) Cross product of two vectors: a=bc Matrix multiplication: a=bc (It is implicitly assumed that b and c are dimensioned correctly, as needed for the operation to make 6 sense; in addition, if b is a nonsymmetric tensor, bT c = b c:) Scalar product of two matrices: a=b:c 1-18


This operation means that corresponding conjugate components of the two matrices are multiplied as pairs and the products summed. Thus, for instance, if b is the stress matrix, , and c the conjugate rate " " of strain matrix, d", then : d" would give the rate of internal work per volume, dW I . It is also necessary to define the dyadic product of two vectors: a = bcor

a = bcT

This operation creates a second-order tensor (or dyad) out of two vectors. In component notation this notation is equivalent to aij = bi cj . A matrix of derivatives, @a ; @b means da = @a @b db:

Throughout this manual it will be assumed implicitly that, when a derivative is taken with respect to time, we mean the material time derivative; that is, the change in a variable with respect to time whilst looking at a particular material particle. When this is not the case for a particular equation, it will be stated explicitly when the equation appears. Provided that we are careful about interpreting (@a=@b) in the manner illustrated above, standard concepts of elementary calculus clearly hold; for example, if a is a vector-valued function of the vector-valued function b, which in turn is a vector-valued function of c, that is a = a(b(c)) , then da = @a @b dc; @b @c

or, if a(b; c) : da = @a @a db + dc: @b @c

Due to these properties many useful results can be obtained quickly and expressed in a compact, easily understood, form.

Components of a vector or a matrix in a coordinate systemIn the previous section we introduced the idea that a vector a or a matrix a can be written in terms of components associated with some conveniently chosen set of base vectors, e . We now show how the

1-19


components a (or a ) are obtained. We can do so using the dot product. For each of the three base vectors, e , we define a conjugate base vector e , as follows. Choose e1 as normal to e2 and e3 , such that the dot product e1 e1 = 1. Similarly, choose e2 normal to e3 and e1 , such that e2 e2 = 1; and e3 normal to e1 and e2 , such that e3 e3 = 1. Thus, e1 e1 = 1 ; e1 e2 = 0 ; e1 e3 = 0 e2 e1 = 0 ; e2 e2 = 1 ; e2 e3 = 0 e3 e1 = 0 ; e3 e2 = 0 ; e3 e3 = 1 We can write this compactly as e e = ; where = 1 if = , and = 0, otherwise. ( is called the "Kronecker delta.") In matrix notation is the unit matrix I: we can also write the above equation defining e1 , e2 , and e3 in matrix form as

so that, if one set of base vectors--ei , say--is known, the others are easily obtained. With this additional set of base vectors, we can immediately obtain the components of a vector or a matrix as follows. Consider a vector a. Then a e = a e e (writing a in component form, using the basis vectors e ), and since e e = = 1 , only if = , a e = a e e = a = a :

8 1 9 < be c = be2 c :bfe1 g fe2 g fe3 gc = I; : 3 ; be c

In exactly the same way we could have written a = a e by expressing a as components associated with the e base vectors, a = a e . Similarly, for a matrix, a = (e )T a e = e a e ;

1-20


and a = (e )T a e = e a e : These component definitions are particularly convenient for calculating the dot product of two vectors, for we can write a b = (a e ) (b e ) = a b e e ; = a b ;

= a b ;

which is a b = a1 b1 + a2 b2 + a3 b3 : Similarly, the scalar product of two matrices is a : b = a b ; that is, we simply multiply corresponding entries in the a and b arrays, arranged as matrices, and then sum the products. Finally, on the computer we need to store only one form of component: a , a or a , a . We can always go from one to the other using the "metric tensor," g , and its inverse, g , which are defined as g = e e ; and g = e e : For a = a e= a g (from above), (expressing a in component form) ; (by the denition of g ) :

= a e e

Thus, a = g a ; similarly a = g a , and, by extension, for matrices, a = g g a

1-21


and a = g g a : The metric tensor and its inverse are symmetric: g = e e = e e = g : The two sets of base vectors and components of vectors or matrices associated with them are named as follows: e are covariant base vectors, e are contravariant base vectors, a (or a ) are covariant components of a vector (or matrix), a (or a ) are contravariant components of a vector (or matrix). Thus, the contravariant components are those associated with the covariant base vectors, a = a e , and vice versa. The simplest case is when the basis is a set of orthogonal unit vectors (a rectangular Cartesian system) because then--from the definition e e = --we see that e = e , and so

a = a and we need not distinguish the type of component. Whenever possible a rectangular Cartesian system is chosen, so the type of component need not be distinguished. This system is discussed in more detail in the sections on beam elements and shell elements.

Components of a derivativeConsider a vector-valued function, b, which is expressed in component form on a basis system, e . Let the vector-valued function a depend on b: a(b). Then da = @a @b e db

so that the component of da associated with a change db is @a @b e ;

which we write, for convenience, as @a @b ;

meaning da = @a @b db :

1-22


Now suppose a is written on a different basis--E , say--so that we store a as the components da = da E : Then da = E @a db : @b

Typically we would then write da = H db ; where H = E @a @a e : = E @b @b

Readers who are familiar with general curvilinear tensor analysis will recognize H as the covariant derivative of a with respect to b , often written as aj . The advantage of the direct matrix notation is clear: because we can imagine a and b as vectors in space, we have a physical understanding of what we mean by @a=@b; it is the change in the vector-valued function a as a function of another vector-valued function b. For computations we must express a and b in component form. Then H = aj = E @a e @b

provides the necessary components once we have chosen convenient basis systems: e for b and E for a. Typically e and E will both be the simple rectangular Cartesian bases e1 = (1; 0; 0) e2 = (0; 1; 0) e3 = (0; 0; 1) everywhere. But sometimes we must use more complicated basis systems--examples are when we need quantities associated with the surface of a general shell and when the symmetry of the geometry and, possibly, of the deformation makes it convenient to work in an axisymmetric system. The careful projection of the general results written in direct matrix notation onto the chosen basis system allows us to implement the theory for computation. As an example, consider the usual expression for strain rate,1 _ "= 2

T ! _ _ @u @u + ; @x @x

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_ _ which requires the matrix @ u=@x to be evaluated, where u is the velocity of the material currently _ flowing through the point x in space. Let us now derive the components of " when the basis system for _ both u and x is the cylindrical system that we usually choose for axisymmetric problems, with the basis vectors

e1 (radial) = (cos ; sin ; 0) e2 (axial) = (0; 0; 1) e3 (circumferential) = ( sin ; cos ; 0) (in ABAQUS for axisymmetric cases we always take the components in this order--radial, axial, circumferential). These basis vectors are orthogonal and of unit length, so that e = e : We consider position to be defined by the coordinates (r; z; ), with dx = dr e1 + dz e2 + r d e3 ; so that dx1 = dr; Thus,_ @u = @x

dx2 = dz;

and

dx3 = r d:

_ _ _ @u @u 1 @u ; ; @r @z r @

;

where_ u = u r e1 + u z e2 + u e3 ; _ _ _

so that_ @u @ ur _ @ uz _ @ u _ e1 + e2 + e3 = @r @r @r @r _ @u @ ur _ @ uz _ @ u _ e1 + e2 + e3 = @z @z @z @z _ 1 @ ur 1 @ uz 1 @ u 1 @e1 1 @e3 1 @u _ _ _ e1 + e2 + e3 + u r : = _ + u _ r @ r @ r @ r @ r @ r @

We know that @e1 = ( sin ; cos ; 0) = e3 ; @ so thatand

@e3 = ( cos ; sin ; 0) = e1 ; @

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_ 1 1 @u = r @ r

@ ur _ u _ @

1 @ uz 1 _ e2 + e1 + r @ r

@ u _ + ur _ @

e3 ;

and thus,_ @ ur =@r @ u _ 4 @ uz =@r = _ @x @ u =@r _

2

@ ur =@z _ @ uz =@z _ @ u =@z _

3 (1=r )(@ ur =@ u ) _ _ 5: (1=r )@ uz =@ _ (1=r )(@ u =@ + ur ) _ _

The components of the strain rate are thus "rr _ @ ur _ ; = @r "zz _ @ uz _ ; = @z " _1 = r _ @ u + ur _ @

;

rz = 2"rz = 2"zr _ _ _ and

@ ur _ @ uz _ ; = + @z @r

r = 2"r = 2"r _ _ _

1 = r

_ @ ur u _ @

+

@ u _ ; @r

z = 2"z = 2"z = _ _ _

1 @ uz _ @ u _ : + r @ @z

_ _ For the case of purely axisymmetric deformation, u = 0 and @ ur =@ = @ uz =@ = 0 , so these results _ simplify to the familiar expressions

"rr = _

@ ur _ ; @r

"zz = _

@ uz _ ; @z

" = _

_ ur r

rz = _

@ ur _ @ uz _ ; + @z @r

r = z = 0: _ _

In summary, direct matrix notation allows us to obtain all our fundamental results without reference to any particular choice of coordinate system. Careful application of the concept of the covariant derivative then allows these general results to be projected into component form for computation.

Virtual quantitiesThe concepts of virtual displacements and virtual work are fundamental to the development. Virtual quantities are infinitesimally small variations of physical measures, such as displacement, strain, velocity, and so on. The virtual variation of a scalar quantity a is indicated by a; of a vector or matrix a by a. We extend this notation to such expressions as

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" " = sym

@v @x

;

which is the symmetric part of the spatial gradient of a virtual vector field v. This notation corresponds to the virtual rate of deformation (a measure of strain rate) if v is a virtual velocity field.

Initial and current positionsMost structural problems concern the description of the way a structure behaves as it is loaded and moves from its reference configuration. Thus, we often compare positions of a point in the current (deformed) configuration and a reference configuration that is usually chosen as the configuration when the structure is unloaded or, in the case of geotechnical problems, when the model is subject only to geostatic stresses. To distinguish these configurations, we use lowercase type ( x) to indicate the current position and uppercase type (X) to indicate the initial position of the same material point in the same spatial coordinate frame. In ABAQUS we almost always store the rectangular Cartesian components of X and x. The exception is in axisymmetric structures, where radial ( r) and axial (z) components are stored.

Nodal variablesSo far we have discussed quantities that are considered to be associated with all points in a model. The finite element approximation is based on assuming interpolations, by which displacement, position, and--often--other variables at any material point are defined by a finite number of nodal variables. In this manual we use uppercase superscripts to refer to individual nodal variables or nodal vectors and adopt the summation convention for these indices. Hence, the interpolation can be written quite generally as a = NN aN ; where a is some vector-valued function at any point in the structure; NN (g ) , N = 1; 2 : : : up to the total number of variables in the problem, is a set of N vector interpolation functions (these are functions of the material coordinates, g ); and aN , N = 1; 2 : : : is a set of nodal variables. In some sections in this manual we need to describe operations on nodal variables for the complete system of finite element equations. In these sections we use the classical matrix-vector notation. In this notation fag represents a column vector containing nodal variables, bac represents a row vector, and a matrix is written as [A]. Common operations are the scalar product between two vectors, c = bac fbg (which is equivalent to c = aN bN in index notation) and the matrix-vector product fcg = [A] fbg

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(which is equivalent to cM = AM N bN in index notation).

1.3 Finite rotations 1.3.1 Rotation variablesSince ABAQUS contains such capabilities as structural elements (beams and shells) for which it is necessary to define arbitrarily large magnitudes of rotation, a convenient method for storing the rotation at a node is required. The components of a rotation vector are stored as the degrees of freedom 4, 5, and 6 at any node where a rotation is required. This method of storing rotation is described in ``Conventions,'' Section 1.2.2 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual. The finite rotation vector consists of a rotation magnitude = k k and a rotation axis or direction in space, p = =k k. Physically, the rotation is interpreted as a rotation by radians around the axis p. To characterize this finite rotation mathematically, the rotation vector is used to define an ^ orthogonal transformation or rotation matrix. To do so, first define the skew-symmetric matrix associated with by the relationships^ = 0 ^ and v = v for all vectors

v:

^ is called the axial vector of the skew-symmetric matrix . In matrix components relative to the standard Euclidean basis, if = f 1 2 3 gT , then 0 ^ ] = 4 3 [ 2

2

3 0 1

3 2 1 5 : 0

^ In what follows, a will be used to denote the skew-symmetric matrix with axial vector a. ^ A well-known result from linear algebra is that the exponential of a skew-symmetric matrix is an orthogonal (rotation) matrix that produces the finite rotation . Let the rotation matrix be C, such that C1 = CT . Then by definition, ^ ^ C = exp[ ] = I + + 1 ^2 + : 2!

However, the above infinite series has the following closed-form expression Equation 1.3.1-1 :

^ C = exp[ ] = cos k kI +

sin k k k k

^ +

(1cos k k) k k2

In components, Cij = cos ij + (1 cos )pi pj + sin ikj pk ; 1-27


where p = f p1

p2

p3 gT and ijk is the alternator tensor, defined by 132 = 213 = 321 = 1; all other ijk = 0:

123 = 231 = 312 = 1;

It is this closed-form expression that allows the exact and numerically efficient geometric representation of finite rotations.

Quaternion parametrizationEven though ABAQUS stores and outputs the rotation vector, quaternion parameters prove to be an efficient and convenient way to treat finite rotations computationally. Let q0 2 R be a scalar, and let q 2 R3 be a vector field. The quaternion q is simply the pairing q = (q0 ; q) : To associate q with the finite rotation vector , define the following: Equation 1.3.1-2 :

q0 = cos k=2k

and

q=

sin k=2k k k

^ By trigonometric identities it follows that the orthogonal matrix exp[ ] in Equation 1.3.1-1 is given in terms of q as

Equation 1.3.1-32 ^ ^ exp[ ] = (2q0 1)I + 2q0 q + 2qq :

^ By the convention introduced above, q is the skew-symmetric matrix with axial vector q.

For a more detailed discussion of quaternion algebra and its relation to other representations of finite rotations, see the discussion by Spring (1986).

Compound rotationsA compound rotation is the successive application of two or more rotation fields. In geometrically linear problems compound rotations are obtained simply as the linear superposition of the individual ^ (linearized) rotation vectors. This fact follows directly from the series expansion for exp[ ]. Let 1 and ^ ^ ^ ^ 2 be infinitesimal rotations. Thus, exp[ 1 ] I + 1 , exp[ 2 ] I + 2 , and^ ^ ^ ^ ^ ^ exp[ 1 ] exp[ 2 ] exp[ 2 ] exp[ 1 ] I + 1 + 2 :

In geometrically nonlinear analysis compound rotations are no longer additive. Furthermore, they are not commutative; that is, the order of application is important. A significant exception occurs when the multiple rotations share the same rotation axis. This special case is investigated further below. A

1-28


detailed example of a finite compound rotation is given in ``Conventions,'' Section 1.2.2 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual. Let Ci be the orthogonal transformation representing the compound rotation defined as the product of a set of individual or incremental rotations Cp , for p = 1; 2; : : : ; i. (For the case of specified boundary conditions Ci is the final product after i steps of all the specified rotations Cp ; for the iterative numerical solution procedure Ci is the total rotation after i increments, where Cp , for p = 1; : : : ; i, is the converged rotation field solution at each increment.) By definition, the compound rotation is the product Ci = Ci Ci1 C1 ; or equivalently by the recursion relation, Ci = Ci Ci1 : It is important to note that C C, which is interpreted as the finite rotation C superposed on the finite rotation C, is different from C C, which is interpreted as the finite rotation C superposed on the finite rotation C. Although compound rotations are defined in terms of orthogonal matrices, in a numerical context the rotation vectors (or equivalently the quaternion parameters) associated with the rotation matrices are the degrees of freedom. Compound rotations are performed as follows: Given a quaternion parametrization q = (q0 ; q) and an incremental (finite) rotation q = (q0 ; q) , where q is defined in terms of an incremental rotation vector by Equation 1.3.1-2, the total or compound rotation is given by the quaternion r = (r0 ; r) , which is calculated as r = q q : Here denotes the quaternion product and is defined as Equation 1.3.1-4q q = (q0 q0 q q ; q0 q + q0 q + q q) :def

Equation 1.3.1-4 allows for the update of rotation fields without ever calculating the orthogonal matrix from the quaternion and without performing a matrix multiplication. Furthermore, all operations are singularity free regardless of the magnitude of the incremental rotation field . The final (total) rotation vector can be calculated from the quaternion r by inverting Equation 1.3.1-2. For the special case when compound rotations share the same rotation axis, the compound rotation reduces to an additive form. Let q and q have the same rotation axis p. Then q = (cos k =2k; sin k =2kp) , q = (cos k =2k; sin k =2kp) , and q q = (cos k =2k cos k =2k sin k =2k sin k =2k ; cos k =2k sin k =2kp + cos k =2k sin k =2kp) ; 1-29


which reduces toq q = (cos k( + )=2k; sin k( + )=2kp) :

Rotation vector extractionFor output purposes it is necessary to extract the rotation vector corresponding to a given quaternion. The extraction procedure is as follows: Let r = (r0 ; r) be the quaternion, and let be the rotation vector. Thus, h i Equation 1.3.1-5and =

kk = 2 tan1

krk r0

r k k krk

:

It is important to note that the extraction of the rotation vector from the quaternion is not unique. The magnitude kk is determined only up to the addition of 2n, n = : : : ; 1; 0; 1; : : : ABAQUS will always choose that rotation vector such that kk < 2.

Director and rotation field updatesAs an example of the utility of the quaternion parameters, consider the incremental update of a director field for either a beam or shell analysis. At some stage of the solution the director field ti , the quaternion parametrization of the rotation field q i , and the incremental rotation field i are known at increment i. To update the director field by the incremental rotation to increment i + 1, proceed as follows: First calculate the quaternion parametrization of the incremental rotation:i q0

= cos k =2k and

i

sin k i =2k q = i : i k ki i i

c c The director field at i + 1 is then defined as ti+1 = exp[ ] ti , where exp[ ] is calculated with Equation 1.3.1-3. Thus, the director is calculated directly from the quaternion asi i ti+1 = (2(q0 )2 1)ti + 2q0 qi ti + 2(qi ti )qi :

Furthermore, the update of the rotation field is obtained by quaternion multiplication q i+1 = q i q i and is defined byi+1 i i i i q i+1 = (q0 ; qi+1 ) = (q0 q0 qi qi ; q0 qi + q0 qi + qi qi ) :

Variations of the rotation fieldIn the development of the balance equations, it is necessary to calculate the variation of the rotation field. Consider the vector field a, which is obtained by rotation of the reference vector field A: 1-30


a = C A: Variations a in this field are obtained as a = C A ; where C is the linearized rotation matrix; that is, the variation of the orthogonal tensor C. On the other hand, the variation can be defined in terms of the linearized rotation field : b b a = a = a = C A :

Consequently, it follows that b C = C :

It is important to note that the linearized rotation , which is analogous to the angular velocity in dynamics, is not the variation of the rotation vector . By a straightforward (but involved) calculation, it can be shown that the variation of the rotation vector ( ) is related to the linearized rotation by Equation 1.3.1-6 = H( ) ;

where 1 sin k k 1 cos k k ^ 1 I + + : H() = 2 2 2 k k k k k k kk The inverse of H() is H()1

1 k k sin k k 1 ^ I = + 1 : 2 k k2 2(1 cos k k) k k2

Let d represent an infinitesimal change in the rotation field. A direct calculation of the variation of d , which is equivalent to calculation of the second variation of either C or a, leads to an expression that is not symmetric in the variations and the changes d . However, it is shown in Simo (1992) that the correct definition of the Hessian operator--that is, the "covariant" derivative of the weak form of the balance equations--requires only the symmetric part (with respect to the variations) of the second variation. Thus, without loss of generality, we can writed(C) = 1 b c c b ( d C + d C) : 2

Similarly, the second variation of the vector field rotated by C can be written as

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d(a) =

1 (d a) + d ( a) 2 1 = ( d )a + ( a)d + (d a) : 2

Velocity and accelerationTaking the time derivative of the rotation matrix, we find with the same arguments as used in the calculation of the variations that_ b C = ! C; b _ b b C = ! C +! ! C;

b where ! is the angular velocity matrix. Equivalently, the first and second time derivative of a are written as_ a = ! a; _ a = ! a + ! (! a) :

The instantaneous angular velocity vector ! is related to the time rate of change of the rotation vector by the relation_ ! = H( ) ;

where H() is given by Equation 1.3.1-6. In the linearization of the dynamic balance equations, it is necessary to calculate the variation of the angular velocity, d! . This quantity, however, can be calculated only by linearizing the specific algorithm used for the time integration of the dynamic equations.

Coupling of rotations: constant velocity jointNext, a more rigorous treatment of the two-dimensional constant velocity joint described in ``MPC,'' Section 23.2.13 of the ABAQUS/Standard User's Manual, is presented. This derivation exemplifies some of the issues associated with the treatment of finite rotations. ``Uniform collapse of straight and curved pipe segments,'' Section 1.1.5 of the ABAQUS Benchmarks Manual, deals with a different finite rotation constraint and tackles additional complications. Let a, b, c (see Figure 1.3.1-1) be the nodes making up the joint, with a the dependent node.

Figure 1.3.1-1 Nonlinear MPC example--constant velocity joint.

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The joint is operated by prescribing an axial rotation c = c ex at c and an out-of-plane rotation b = b ez at b. The compounding of these two prescribed rotation fields will determine the total rotation at a. We can formally write this constraint as follows: f ( a ; b ; c ) = a b c = 0: The constraint can be written in terms of the rotation matrices as Equation 1.3.1-7 C( ) C( ) C( ) = 0: With the previously defined variational expressions, the constraint can be linearized as c c d a C(a ) b C( b ) C( c ) C( b ) c C( c ) = 0:a b c

This expression can be simplified by right-multiplying the expression by CT ( a ) and by making use of the constraint Equation 1.3.1-7, which yields d c c a b C( b ) c CT ( b ) = 0; which can be written in vector form as a b C( b ) c = 0: Sincecos b b C( ) = @ sin b 0

0

sin b cos b 0

1 0 0A; 1

the linearized constraint is indeed identical to the one derived based on simple linear considerations in the ABAQUS/Standard User's Manual.

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The linearized constraint is used for the calculation of equilibrium. It can also be used for the recovery of the dependent rotation, a , as was done in the ABAQUS/Standard User's Manual. The resulting rotation will satisfy the constraint approximately (unless one of the angles b or c is constant, in which case the constraint is linear and the recovery is exact). For an exact enforcement of the constraint, user subroutine MPC must define the components of the b total rotation vector a exactly. To do so, a must be updated based on the current values of and c . This is most easily accomplished with the aid of the quaternion parameters. Let q b = (cos(b =2); sin(b =2)ez ) and q c = (cos(c =2); sin(c =2)ex ) be the quaternion b parameterizations associated with the finite rotation vectors and c , respectively. The total a compound rotation a is given by the quaternion q a = (q0 ; qa ) , wherea q0 = cos(b =2) cos(c =2) ;

qa = cos(b =2) sin(c =2)ex + sin(b =2) sin(c =2)ey + cos(b =2) sin(c =2)ez ; according to the quaternion compound formula Equation 1.3.1-4. The rotation vector a is extracted from the quaternion q a as follows: qa = kqa ka a

with

= 2 tan

a

1

kqa k a q0

;

where kqa k is the norm of the vector qa . ``MPC,'' Section 23.2.13 of the ABAQUS/Standard User's Manual, shows the implementation of the linearized form of the constraint in user subroutine MPC. The implementation of the exact nonlinear constraint is shown below:SUBROUTINE MPC(UE,A,JDOF,MDOF,N,JTYPE,X,U,UINIT,MAXDOF,LMPC, * KSTEP,KINC,TIME,NT,NF,TEMP,FIELD) C INCLUDE 'ABA_PARAM.INC' C DIMENSION UE(MDOF), A(MDOF,MDOF,N), JDOF(MDOF,N), X(6,N), * U(MAXDOF,N), UINIT(MAXDOF,N), TIME(2), TEMP(NT,N), * FIELD(NF,NT,N) PARAMETER( SMALL = 1.E-14 ) C IF ( JTYPE .EQ. 1 ) THEN A(1,1,1) = 1. A(2,2,1) = 1. A(3,3,1) = 1. A(3,1,2) = -1. A(1,1,3) = -COS(U(6,2)) A(2,1,3) = -SIN(U(6,2)) C

1-34


JDOF(1,1) JDOF(2,1) JDOF(3,1) JDOF(1,2) JDOF(1,3) C CPHIB SPHIB CPHIC SPHIC C QA0 QAX QAY QAZ C = = = = = = = =

= = = = =

4 5 6 6 4

COS(0.5*U(6,2)) SIN(0.5*U(6,2)) COS(0.5*U(4,3)) SIN(0.5*U(4,3))

CPHIB*CPHIC CPHIB*SPHIC SPHIB*SPHIC CPHIB*SPHIC

QAMAG = SQRT( QAX*QAX + QAY*QAY + QAZ*QAZ ) IF ( QAMAG .GT. SMALL ) THEN PHIA = 2.*ATAN2( QAMAG , QA0 ) UE(1) = PHIA*QAX/QAMAG UE(2) = PHIA*QAY/QAMAG UE(3) = PHIA*QAZ/QAMAG ELSE UE(1) = 0. UE(2) = 0. UE(3) = 0. END IF END IF C RETURN END

1.4 Deformation, strain, and strain rates 1.4.1 DeformationIn any structural problem the analyst describes the initial configuration of the structure and is interested in its deformation throughout the history of loading. The material particle initially located at some position X in space will move to a new position x: since we assume material cannot appear or disappear, there will be a one-to-one correspondence between x and X, so we can always write the history of the location of a particle as Equation 1.4.1-1 x = x(X; t)

1-35


and this relationship can be inverted--we know X when we know x and t. Now consider two neighboring particles, located at X and at X + dX in the initial configuration. In the current configuration we must have Equation 1.4.1-2 dx =@x @X

dX

using the "mapping" Equation 1.4.1-1. The matrix Equation 1.4.1-3 F=@x @X

is called the deformation gradient matrix, and Equation 1.4.1-2 is written Equation 1.4.1-4 dx = F dX: As the material behavior depends on the straining of the material and not on its rigid body motion, those parts of the motion in the vicinity of a material point must be distinguished. Looking at an infinitesimal gauge length dX emanating from the particle initially at X, we can measure its initial and current lengths as dL2 = dXT dXand

dl2 = dxT dx;

so the "stretch ratio" of this gauge length is q Equation 1.4.1-5dxT dx dXT dX

=

dl dL

=

:

If = 1, there is no strain of this infinitesimal gauge length--it has undergone rigid body motion only. Now using Equation 1.4.1-4, dxT dx = dXT FT F dX; so that, from Equation 1.4.1-5, Equation 1.4.1-6 2 = p dX dXTT

dX T T = N F F N;

FT F p

dX dXT dX

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where N is a unit vector in the direction of the gauge length dX. Equation 1.4.1-6 shows how to measure the stretch ratio associated with any direction, N, at any material point defined by X (or by x). Useful results are obtained when we vary the direction defined by N at a particular material point and look for stationary values of the stretch ratio, . Since N must always be a unit vector, stationary values of 2 are obtained by solving the constrained variational equation NT FT F N p (NT N 1) = 0; NT N = 1: Taking the variation gives back the constraint (conjugate to p) and, conjugate to N, gives Equation 1.4.1-7(F F pI) N = 0:T

where p is a Lagrange multiplier, introduced to retain the constraint

Taking the dot product of the left-hand side of this equation with N and comparing with Equation 1.4.1-6 identifies p = 2 , so Equation 1.4.1-7 is Equation 1.4.1-8(F F I) N = 0:T 2

This problem is an eigenvalue one that can be solved for the three extreme values of 2 . Since is always real and positive (and nonzero), 2 > 0, and hence FT F must be positive definite. Equation 1.4.1-8 thus gives three real, positive eigenvalues, I , II , III , the "principal stretches," with three corresponding eigenvectors, NI , NII , NIII , which will be orthogonal, by Equation 1.4.1-8, if the corresponding eigenvalues are different, and can be orthogonalized otherwise. The NI are the principal directions of strain. Now let nI , nII , nIII be unit vectors corresponding to NI , NII , NIII , but in the current configuration, so that, using Equation 1.4.1-4, nI = Then1 F NI ; etc: I

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Introduction and Basic Equations nI T nII =1 NI T FT F NII I II 1 = 2 NI T NII I II II =0

by the orthogonality results just mentioned. Thus, nI , nII , and nIII will also be an orthogonal set. Since each is a unit vector, nI = R NI ;

nIII = R NIII ; where R is the same pure rigid body rotation matrix in each of these equations. A pure rigid body motion matrix has the property that its inverse is its transpose: RT = R1 . Comparing the principal stretch directions in the current and original configurations, therefore, isolates the rigid body rotation and the stretch. Finding the principal stretch ratios and their directions thus provides one solution to the problem of isolating straining motion and rigid body motion in the vicinity of a material point. Now consider a gauge length in the reference configuration, dXI , directed along NI . The same infinitesimal material line in the current configuration will be along nI and will be stretched by I , so that dxI = I R dXI : Similarly, along the other principal directions, dxII = II R dXII and dxIII = III R dXIII : Since (NI , NII , NIII ) is an orthonormal set of base vectors in the reference configuration, any infinitesimal material line (gauge length) dX at X can be written in terms of its components in this basis: dX = dXI + dXII + dXIII ; where dXI = NI NI T dX;etc.

nII = R NII ;

Each of the vectors dXI moves and stretches to the corresponding dxI , as defined above. Thus, the current gauge length, dx, is

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dx = dxI + dxII + dxIII= I R dXI + II R dXII + III R dXIII = I R NI NI T + II R NII NII T + III R NIII NIII T dX = I nI NI T + II nII NII T + III nIII NIII T dX = I nI nI T + II nII nII T + III nIII nIII T R dX

which we write as

Equation 1.4.1-9 dx = V R dX; where Equation 1.4.1-10 V = (I nI nI + II nII nIIT T

+ III nIII nIII )

T

is the "left stretch" matrix, which is the sum of three dyadic products. Comparison with the definition of the deformation gradient, Equation 1.4.1-4, shows that Equation 1.4.1-11 F = V R; which is the polar decomposition theorem--that any motion can be represented as a pure rigid body rotation, followed by a pure stretch of three orthogonal directions. The polar decomposition theorem is important because it allows us to distinguish the straining part of the motion from the rigid body rotation. Specifically, F completely defines the relative motions of material particles in the infinitesimal neighborhood of the material particle that was at X in the reference configuration; and the left stretch matrix, V, completely defines the deformation of the material particles at X. The rotation matrix R defines the rigid body rotation of the principal directions of strain ( NI in the reference configuration; nI in the current configuration). R represents only the rigid body rotation of the material at the point under consideration in some average sense: in a general motion, each infinitesimal gauge length emanating from a material particle has a different amount of rotation. This distinction between the rotation of the principal directions of strain, R, and the rotations of individual directions in the material becomes significant when we must discuss large deformations of nonisotropic materials. Nevertheless, we have established an important result: if F = R only, we know there is no deformation of the material in the immediate neighborhood of the point originally at X and currently at x , since in this case V = I and so I = II = III = 1. V I must be nonzero for there to be any deformation of the material at the point in question: in this sense V I (and, hence, V itself) is sufficient to define the deforming part of the motion (it contains complete information about all except pure rigid body rotation of the point). For this reason--so that, later in the development, we will be able to link the kinematics to the stressing of the material--we will need to be

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able to isolate V from F. It is easy to obtain V V, for F FT = V R RT VT= V V;

since RT = R1 and V is symmetric. Since we originally defined V from the principal stretches and their principal directions in the current configuration as V = I nI nI T + II nII nII T + III nIII nIII T ; then Equation 1.4.1-12 F F = V V = I nI nI + II nII nIIT 2 T 2 T

+ III nIII nIII :

2

T

We see that the eigenvalues of F FT , are I 2 , II 2 , and III 2 , and the corresponding eigenvectors are nI , nII , and nIII . We can then construct V. The deformation at the point is, thus, readily obtained by multiplying a 3 3 matrix with its transpose (F FT ) and solving the real eigenproblem for the resulting (symmetric) matrix. We can then obtain the rotation R as R = V1 F: Since V has been constructed from its eigenvalues and eigenvectors, its inverse is immediately available: V1 =1 1 1 nI nT + nII nT + nIII nT : I II III I II III

So far we have written the results quite generally, without reference to any particular coordinate system. To perform computations we must choose a basis system to express these results as arrays of individual numbers. We now do so with some generality with respect to the choice of basis system. The justification for retaining generality at this stage is twofold: as an exercise, to provide a little more familiarity in the notation system we have chosen to use in this manual, and because we do need some--but, as it turns out, not all--of the generality when we have to deal with shell elements, where it is undesirable to use the rectangular Cartesian base vectors of the global, spatial system because the natural orientation of the shell reference surface causes us to prefer to choose two of the base vectors to be tangent to the shell's reference surface and the other to be normal to this surface. This preference causes us to need two basis systems: one associated with the body in its current configuration, when the point in question is at x, and one associated with the body in its reference configuration, when the same point was at X, because the orientation of the shell's reference surface--which determines our choice of basis vectors--will be quite different in these two configurations. We will write e , = 1; 2; 3 , as the basis vectors chosen to write components associated with the current configuration

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(so that any vector a associated with the current configuration is written as a = a e ) and E , = 1; 2; 3 , as the basis at the same material point but in the reference configuration. (Since we assume that both of these basis systems are adequate to express any vector-valued function by its components in the basis system--that is, the basis vectors are not linearly dependent--either would, in principal, serve for both configurations. We introduce two distinct systems by preference, because each is chosen as particularly suitable for a particular configuration.) Since we do not yet impose any particular restrictions on the e or the E (except for the requirement that the vectors must not be linearly dependent), we cannot assume that they will be orthogonal or of unit length: we will, therefore, need to use the corresponding contravariant vectors defined by e e =

and

E E = ;

and the contravariant metric tensors g = e eand

G = E E :

We can express the deformation gradient, F, numerically by projecting it onto the bases: Equation 1.4.1-13 F = e F E : Recall the definition of F: @x dX : dx = F dX = @X Since the components of dx along e are dx = dx e and we can write dX = dX E , dx = e F E dX @x = e dX : @X Thus, writing dx = F dX defines F = e F E = e @x : @X

We must continue to bear in mind that the first index of F is associated with a component of F along a base vector in the current configuration ( e in this case), while its second index is associated with a component of F along a base vector in the reference configuration (E ). From Equation 1.4.1-13 we can write

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F FT = e F E E F e= e F G F e ;

where G is the contravariant metric of the basis system that we have chosen in the reference configuration. The eigenproblem for the squared principal stretch ratios and their directions is solved by finding the eigenvalues of the matrix of numbers F G F . The eigenvectors will appear as the components (nI ) along the e base vectors in the current configuration. Since we have defined the left stretch on the current configuration as V = I nI nI T + II nII nII T + III nIII nIII T ; we will write its components on the basis in the current configuration as V = I nI nI + II nII nII + III nIII nIII = e V e ; and, since V1 =1 1 1 nI nI T + nII nII T + nIII nIII T ; I II III

(V 1 ) =

1 1 1 nI nI + nII nII + nIII nIII : I II III

The polar decomposition gives R = V1 F

= e (V 1 ) e e F E = e (V 1 ) g F E ;

so R = (V 1 ) g F ; where g is the contravariant metric tensor of the basis system we have chosen to use in the current configuration and--as with F --we see that the first index of R is associated with the contravariant base vector e in the current configuration, while the second index is associated with the contravariant base vector E in the reference configuration. We should take care to understand the distinction between the direct matrix expression of the rigid body rotation of the principal directions of strain of the material, R, and the components of R expressed on a particular basis. Suppose, for example, that the rigid body rotation at a point is zero

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(that is, R = I) but we, nevertheless, have chosen different basis systems e and E . In this case R = e R E = e I E = e E . This implies that, even though R is a unit matrix (in the sense that operating on any vector with this matrix makes no change in that vector), the numerical values we have chosen to store the matrix--the R --do not form a unit matrix of numbers unless the e and the E are coincident and orthonormal. Thus, our choice of quite general basis systems that are not the same in the current and reference configurations (introduced as being "natural" for writing results for shells) somewhat complicates the interpretation of the numbers we store. In the previous few paragraphs we have chosen to explore the expression of the basic results we have derived so far for the kinematics of the total motion in terms of quite general basis systems, e and E . In ABAQUS we wish to express results as simply and directly as possible, and we can do so by choosing particular sets of basis vectors that offer the most convenience for our purposes. First, we take the e (and, by extension, the E , since these are just the e at the beginning of the motion) to be a local, orthonormal system at each point. Although it is not possible to construct a Cartesian system with orthonormal base vectors over a general shell surface, we can always project the general results onto such a system when that system is chosen specifically at each point where we need to make the projection--typically at the integration points of the elements. The choice of which system is used as this local orthonormal basis is made in ABAQUS at two levels: we distinguish continuum ("solid") elements from structural (shell and beam) elements, and we distinguish the default choice of directions from the particular choice of directions indicated by the user when the *ORIENTATION option is included. For continuum elements the default E are unit vectors along the axes of the global Cartesian system chosen for the problem. At points where the *ORIENTATION option is invoked, the E are those specified in that option. For shells (and membranes) we take E1 and E2 tangent to the shell's reference surface and E3 normal to that surface at the point under consideration. Without *ORIENTATION E1 is the projection of the global x-axis onto the reference surface or, if the global x-axis is almost normal to that surface at the point, E1 is the projection of the global z-axis onto the surface. With *ORIENTATION E1 and E2 are the projections of two axes specified in the *ORIENTATION option onto the reference surface at the point. In all cases E3 is normal to the shell's reference surface. For beams E1 is along the beam axis, with E2 and E3 defined from the beam section definition option and beam normals given as part of the nodal coordinate definition. For continuum elements without *ORIENTATION the same schemes are applied to define the basis system in the current configuration. For continuum elements with the *ORIENTATION option invoked at the point and in all cases for shells, beams, and membranes, the e are defined by e = R E : These schemes all have the same property: at any point in time the e are orthonormal vectors: e e = , so e = e and, thus, g = g = g = , and--in particular--E E = and, thus, G = . This simplifies the understanding of all the quantities we write, since the components of any tensor T::: are always the physical projections of that tensor-valued quantity on the local orthogonal basis system e and we need not distinguish covariant and contravariant components as we did in the general development above. In practical terms the only price we must pay for this simplicity is in shells when we have to use a separate basis system at each point under study, since we cannot construct a single system with the orthonormal property on a general curved surface. 1-43


(In an axisymmetric system we also have to use dx3 = r d to ensure that the e3 base vector is a unit vector, but this is a minor point.) The simplifications are valuable and, from our perspective of studying finite element formulations, they are bought at modest cost, since we generally only consider a single integration point at a time. Throughout the rest of this manual, whenever we need to write down particular components of a tensor, we shall assume that the basis on which they are written has the orthonormal property e e = . The material also undergoes rigid body translation, but this is not important in the development since we need consider only relative motion of neighboring points because we are interested in the deformation of the material to link the kinematics of the motion to the material's constitutive behavior. Numerically, rigid body translation is significant only for two reasons. One is that the spatial discretization must allow rigid body translation without giving strain, which is important in choosing interpolation functions for the finite elements. The other is that care must be exercised to ensure that the strain and rotation are calculated accurately when the rigid body motion is large, since then the strain and rotation depend on the difference between two very large motions.

1.4.2 Strain measuresStrain measures used in general motions are most simply understood by first considering the concept of strain in one dimension and then generalizing this to arbitrary motions by using the polar decomposition theorem just derived.

Strain in one dimensionWe already have a measure of deformation--the stretch ratio . In fact, is itself an adequate measure of "strain" for a number of problems. To see where it is useful and where not, first notice that the unstrained value of is 1.0. A typical soft rubber component (such as a rubber band) can change length by a large factor when it is loaded, so the stretch ratio would often have values of 2 or more. In contrast, a typical structural steel component will be designed to respond elastically to its working loads. Such a material has an elastic modulus of about 200 103 MPa (30 106 lb/in2) at room temperature and a yield stress of about 200 MPa (30 103 lb/in2), so the stretch at yield will be about 1.001 in tension, 0.999 in compression. The stretch ratio is an unsatisfactory way of measuring deformation for this case because the numbers of interest begin in the fourth significant digit. To avoid this inconvenience, the concept of strain is introduced, the basic idea being that the strain is zero at = 1, when the material is "unstrained." In one dimension, along some "gauge length" dX, we define strain as a function of the stretch ratio, , of that gauge length: " = f (): The objective of introducing the concept of strain is that the function f is chosen for convenience. To see what this implies, suppose " is expanded in a Taylor series about the unstrained state: Equation 1.4.2-1 " = f (1) + ( df 1) d

+

1 ( 2!

2 1)2 d f d2

+

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We must have f (1) = 0 , so " = 0 at = 1 (this was the main reason for introducing this idea of "strain" instead of just using the stretch ratio). In addition, we choose df =d = 1 at = 1 so that for small strains we have the usual definition of strain as the "change in length per unit length." This ensures that, in one dimension, all strain measures defined in this way will give the same numerical value to the order of the approximation when strains are small (because then the higher-order terms in the Taylor series are all negligible)--regardless of the magnitude of any rigid body rotation. Finally, we require that df =d > 0 for all physically reasonable values of (that, is for all > 0) so that strain increases monotonically with stretch; hence, to each value of stretch there corresponds a unique value of strain. (The choice of df =d > 0 is arbitrary: we could equally well choose df =d < 0 , implying that the strain is positive in compression when < 1. This alternative choice is often made in geomechanics textbooks because geotechnical problems usually involve compressive stress and strain. The choice is a matter of convenience. In ABAQUS we always use the convention that positive direct strains represent tension when > 1. This choice is retained consistently in ABAQUS, including in the geotechnical options.) With these reasonable restrictions ( f = 0 and df =d = 1 at = 1, and df =d > 0 for all > 0), many strain measures are possible, and several are commonly used. Some examples areNominal strain (Biot's strain):

f () = 1:

In a uniformly strained uniaxial specimen, where l is the current and L the original gauge length, this strain is measured as (l=L) 1. This definition is the most familiar one to engineers who perform uniaxial testing of stiff specimens.Logarithmic strain:

f () = ln :

This strain measure is commonly used in metal plasticity. One motivation for this choice in this case is that, when "true" stress (force per current area) is plotted against log strain, tension, compression and torsion test results coincide closely. Later we will see that this strain measure is mathematically appropriate for such materials because, for these materials, the elastic part of the strain can be assumed to be small.Green's strain:

f () =

1 2 1 : 2

This strain measure is convenient computationally for problems involving large motions but only small strains, because, as we will show later, its generalization to a strain tensor in any three-dimensional motion can be computed directly from the deformation gradient without requiring solution for the principal stretch ratios and their directions. All of these strains satisfy the basic restrictions. Obviously many strain functions are possible: the choice is strictly a matter of convenience. Since strain is usually the link between the kinematic and the constitutive theories, the convenience of this choice in the context of finite elements is based on two considerations: the ease with which the strain can be computed from the displacements, since the latter

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are usually the basic variables in the finite element model, and the appropriateness of the strain measure with respect to the particular constitutive model. For example, as mentioned above, it appears that log strain is particularly appropriate to plasticity, while large-strain elasticity analysis (for rubbers and similar materials) can be done quite satisfactorily without ever using any "strain" measure except the stretch ratio .

Strain in general three-dimensional motionsHaving defined the basic concept of "strain" in one dimension, we now generalize the idea to three dimensions. In ``Deformation,'' Section 1.4.1, we established that the deforming part of the motion in the immediate neighborhood of a material point is completely characterized by six variables: the three principal stretch ratios (I , II , and III ) and the orientation of the three principal stretch directions in the current (or in the reference) configuration. This immediately gives the generalization of the one-dimensional strain function introduced above. We first choose the function f that will be used as the strain measure. "I = f (I ) will be the strain along the first principal direction, nI ; "II = f (II ) will be the strain along nII ; and "III = f (III ) will be the strain along nIII . The matrix Equation 1.4.2-2 " = "I nI nI + "II nII nIIT T

+ "III nIII nIII

T

completely characterizes the state of strain at the material point. Notice the resemblance to the definition of the stretch matrix, Equation 1.4.1-10: we might consider " to be defined by the matrix function " = f (V); where we understand a matrix function to mean that the two matrices have the same principal directions with their principal values related by the definition of f , which is a convenient shorthand way of indicating a relationship between two matrices. In Equation 1.4.2-2 we have written the matrix " by using the principal strain directions in the current configuration. We could equally have begun with the polar decomposition into a stretch followed by rotation of the principal directions of stretch: " would be defined in a similar way and would then be associated with its principal directions in the reference configuration. ABAQUS generally reports strains referred to directions in the current configuration. There is no obvious reason for this choice: either approach would suffice so long as the user knows which is being used. The strain measures reported by ABAQUS are enumerated in ``Conventions,'' Section 1.2.2 of the ABAQUS/Standard User's Manual and the ABAQUS/Explicit User's Manual. In a finite element code the deformation gradient F is usually computed at each material calculation point from the displacement solution at the nodes of each element and the interpolation function chosen for the element. We now need an algorithm to obtain ", given a choice of strain measure. This algorithm is available immediately from Equation 1.4.1-12: the eigenvalues and eigenvectors of the 3 3 matrix F FT are I 2 ; II 2 and III 2 ; and nI , nII ; and nIII . We can then calculate 1-46


"I = f (I ) , etc. for the function f chosen as the strain measure and, thus, construct " = "I nI nI T + "II nII nII T + "III nIII nIII T : This algorithm also gives principal strain and stretch values--often a useful output because they give a concise description of the state of deformation at a point. However, the algorithm requires computation of the eigenvalues and eigenvectors of a 3 3 matrix at each of many points in the model at each of many iterations, which involves some computational cost. Thus, it would be useful if " could be computed less expensively from F, which is possible only for certain choices of the strain measure, f (). We now consider one such possibility. The unit matrix I can be written as I = nI nI T + nII nII T + nIII

ABAQUS Theory Manual

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