LS-DYNA
THEORETICAL MANUAL
May 1998
Copyright © 1991-1998
LIVERMORE SOFTWARE
TECHNOLOGY CORPORATION
All Rights Reserved
Compiled By
John O. Hallquist
Mailing Address:
Livermore Software Technology Corporation
2876 Waverley Way
Livermore, California 94550-1740
Support Address:
Livermore Software Technology Corporation
97 Rickenbacker Circle
Livermore, California 94550-7612
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WEBSITE: www.lstc.com
Copyright © 1991-1998 by Livermore Software Technology Corporation
All Rights Reserved
LS-DYNA Theoretical Manual Table of Contents
iii
TABLE OF CONTENTS
ABSTRACT ......................................................................................................................... 1.1
1. INTRODUCTION.............................................................................................................. 1.1
1.1 History of LS-DYNA ................................................................................................... 1.1
1.2 Acknowledgments....................................................................................................... 1.10
2. PRELIMINARIES............................................................................................................. 2.1
2.1 Governing Equations .................................................................................................... 2.1
3. SOLID ELEMENTS .......................................................................................................... 3.1
3.1 Volume Integration ...................................................................................................... 3.3
3.2 Hourglass Control........................................................................................................ 3.4
3.3 Fully Integrated Brick Elements...................................................................................... 3.9
3.4 Fully Integrated Brick Element With 48 Degrees-of-Freedom............................................... 3.10
3.5 Tetrahedron Element With 12 Degrees-of-Freedom............................................................. 3.13
3.6 Fully Integrated Tetrahedron Element With 24 Degrees-of-Freedom....................................... 3.14
3.7 Integral Difference Scheme as Basis For 2D Solids............................................................ 3.17
3.8 Rezoning With 2D Solid Elements................................................................................. 3.23
4. BELYTSCHKO BEAM ...................................................................................................... 4.1
4.1 Co-Rotational Technique ............................................................................................. 4.1
4.2 Belytschko Beam Element Formulation.......................................................................... 4.5
4.2.1 Calculation of Deformations................................................................................ 4.6
4.2.2 Calculation of Internal Forces.............................................................................. 4.7
4.2.3 Updating the Body Coordinate Unit Vectors............................................................ 4.9
5. HUGHES-LIU BEAM ........................................................................................................ 5.1
5.1 Geometry.................................................................................................................. 5.1
5.2 Fiber Coordinate System ............................................................................................. 5.6
5.2.1 Local Coordinate System.................................................................................... 5.7
5.3 Strains and Stress Update............................................................................................. 5.8
5.3.1 Incremental Strain and Spin Tensors ..................................................................... 5.8
5.3.2 Stress Update.................................................................................................... 5.8
5.3.3 Incremental Strain-Displacement Relations............................................................. 5.9
5.3.4 Spatial Integration............................................................................................ 5.10
6. BELYTSCHKO-LIN-TSAY SHELL ..................................................................................... 6.1
6.1 Co-Rotational Coordinates............................................................................................. 6.1
6.2 Velocity-Strain Displacement Relations ........................................................................... 6.3
6.3 Stress Resultants and Nodal Forces.................................................................................. 6.5
6.4 Hourglass Control (Belytschko-Lin-Tsay)......................................................................... 6.6
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6.5 Hourglass Control (Englemann and Whirley)..................................................................... 6.8
6.6 Belytschko-Wong-Chiang Improvement .......................................................................... 6.10
7. C
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TRIANGULAR SHELL................................................................................................. 7.1
7.1 Co-Rotational Coordinates............................................................................................. 7.1
7.2 Velocity-Strain Relations .............................................................................................. 7.2
7.3 Stress Resultants and Nodal Forces.................................................................................. 7.6
8. MARCHERTAS-BELYTSCHKO TRIANGULAR SHELL....................................................... 8.1
8.1 Element Coordinates..................................................................................................... 8.1
8.2 Displacement Interpolation ............................................................................................ 8.3
8.3 Strain-Displacement Relations........................................................................................ 8.5
8.4 Nodal Force Calculations............................................................................................... 8.6
9. HUGHES-LIU SHELL ....................................................................................................... 9.1
9.1 Geometry ................................................................................................................... 9.1
9.2 Kinematics ................................................................................................................. 9.4
9.2.1 Fiber Coordinate System...................................................................................... 9.5
9.2.2 Lamina Coordinate System................................................................................... 9.8
9.3 Strains and Stress Update............................................................................................... 9.9
9.3.1 Incremental Strain and Spin Tensors....................................................................... 9.9
9.3.2 Stress Update .................................................................................................... 9.10
9.3.3 Incremental Strain-Displacement Relations ............................................................. 9.10
9.4 Element Mass Matrix................................................................................................. 9.12
9.5 Accounting for Thickness Changes............................................................................... 9.13
9.6 Fully Integrated Hughes-Liu Shells............................................................................... 9.13
10. EIGHT-NODE SOLID SHELL ELEMENT.......................................................................... 10.1
11. TRUSS ELEMENT......................................................................................................... 11.1
12. MEMBRANE ELEMENT ................................................................................................ 12.1
12.1 Co-Rotational Coordinates......................................................................................... 12.1
12.2 Velocity-Strain Displacement Relations ....................................................................... 12.1
12.3 Stress Resultants and Nodal Forces.............................................................................. 12.2
12.4 Membrane Hourglass Control..................................................................................... 12.2
13. DISCRETE ELEMENTS AND MASSES........................................................................... 13.1
13.1 Orientation Vectors................................................................................................. 13.2
13.2 Dynamic Magnification “Strain Rate” Effects .............................................................. 13.4
13.3 Deflection Limits in Tension and Compression............................................................ 13.4
13.4 Linear Elastic or Linear Viscous................................................................................ 13.5
13.5 Nonlinear Elastic or Nonlinear Viscous ...................................................................... 13.5
13.6 Elasto-Plastic With Isotropic Hardening...................................................................... 13.6
LS-DYNA Theoretical Manual Table of Contents
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13.7 General Nonlinear ................................................................................................. 13.7
13.8 Linear Visco-Elastic .............................................................................................. 13.8
13.9 Seat Belt Material ................................................................................................. 13.9
13.10 Seat Belt Elements.............................................................................................. 13.10
13.11 Sliprings........................................................................................................... 13.11
13.12 Retractors.......................................................................................................... 13.11
13.13 Sensors............................................................................................................. 13.15
13.14 Pretensioners...................................................................................................... 13.16
13.15 Accelerometers ................................................................................................... 13.16
14. SIMPLIFIED ARBITRARY LAGRANGIAN-EULERIAN ..................................................... 14.1
14.1 Mesh Smoothing Algorithms................................................................................... 14.3
14.1.1 Equipotential Smoothing of Interior Nodes...................................................... 14.3
14.1.2 Simple Averaging .................................................................................... 14.11
14.1.3 Kikuchi’s Algorithm................................................................................. 14.11
14.1.4 Surface Smoothing................................................................................... 14.12
14.1.5 Combining Smoothing Algorithms............................................................. 14.12
14.2 Advection Algorithms........................................................................................... 14.12
14.2.1 Advection Methods in One Dimension ......................................................... 14.13
14.2.2 Advection Methods in Three Dimensions...................................................... 14.16
14.3 The Manual Rezone.............................................................................................. 14.24
15. STRESS UPDATE OVERVIEW....................................................................................... 15.1
15.1 Jaumann Stress Rate.............................................................................................. 15.1
15.2 Jaumann Stress Rate Used With Equations of State ..................................................... 15.2
15.3 Green-Naghdi Stress Rate ....................................................................................... 15.3
15.4 Elastoplastic Materials........................................................................................... 15.5
15.5 Hyperelastic Materials............................................................................................ 15.9
15.6 Layered Composites ............................................................................................ 15.11
15.7 Constraints on Orthotropic Elastic Constants........................................................... 15.14
15.8 Local Material Coordinate Systems in Solid Elements............................................... 15.15
15.9 General Erosion Criteria For Solid Elements............................................................ 15.16
15.10 Strain Output to the LS-TAURUS Database ............................................................ 15.17
16. MATERIAL MODELS.................................................................................................... 16.1
Model 1: Elastic............................................................................................................ 16.7
Model 2: Orthotropic Elastic ........................................................................................... 16.7
Model 3: Elastic Plastic With Kinematic Hardening............................................................. 16.8
Plane Stress Plasticity..................................................................................... 16.12
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