LS-DYNA® THEORY MANUAL March 2006 Compiled By John O. Hallquist Livermore Software Technology Corporation 7374 Las Positas Road Livermore, California 94551 Fax: 925-449-2507 Tel: 925-449-2500 Email: [email protected] Website: www.lstc.com Copyright © 1991-2006 Livermore Software Technology Corporation. All Rights Reserved. ISBN 0-9778540-0-0 LS-DYNA is a registered trademark of Livermore Software Technology Corporation. LS-DYNA Theory Manual Table of Contents TABLE OF CONTENTS ABSTRACT......................................................................................................................1.1 1. INTRODUCTION...................................................................................................1.1 1.1 History of LS-DYNA...................................................................................1.1 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 Puso Hourglass Control .............................................................................3.10 3.4 Fully Integrated Brick Elements ................................................................3.16 3.5 Four Node Tetrahedron Element................................................................3.18 3.6 Six Node Pentahedron Element .................................................................3.19 3.7 Fully Integrated Brick Element With 48 Degrees-of-Freedom..................3.19 3.8 Fully Integrated Tetrahedron Element With 24 Degrees-of-Freedom .......3.22 3.9 Integral Difference Scheme as Basis For 2D Solids ..................................3.25 3.10 Rezoning With 2D Solid Elements............................................................3.31 4. BELYTSCHKO BEAM ..........................................................................................4.1 4.1 Co-rotational Technique..............................................................................4.1 4.2 Belytschko Beam Element Formulation ......................................................4.4 4.2.1 Calculation of Deformations.........................................................4.5 4.2.2 Calculation of Internal Forces.......................................................4.6 4.2.3 Updating the Body Coordinate Unit Vectors................................4.8 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. WARPED BEAM ELEMENTS IN LS-DYNA ......................................................6.1 6.1 Resultant Warped Beam ..............................................................................6.1 6.1.1 Green-Lagrange Strains/Deformational Displacements................6.1 6.1.2 Deformational Displacements After Large Rotations...................6.1 iii Table of Contents LS-DYNA Theory Manual 6.1.3 Centroidal Displacements and Angular Rotations........................6.2 6.1.4 Strain Energy.................................................................................6.3 6.1.5 Displacement Field.......................................................................6.4 6.1.6 Strain Energy in Matrix Form.......................................................6.5 6.1.7 Internal Nodal Force Vector..........................................................6.6 6.2 Integrated Warped Beam..............................................................................6.7 6.2.1 Kinematics....................................................................................6.7 6.2.2 Kinetics.........................................................................................6.9 6.2.3 Penalty on Twist..........................................................................6.11 6.3 Generalization to Large Displacements .....................................................6.12 7. BELYTSCHKO-LIN-TSAY SHELL......................................................................7.1 7.1 Co-rotational Coordinates............................................................................7.1 7.2 Velocity-Strain Displacement Relations......................................................7.3 7.3 Stress Resultants and Nodal Forces .............................................................7.5 7.4 Hourglass Control (Belytschko-Lin-Tsay)...................................................7.6 7.5 Hourglass Control (Englemann and Whirley)..............................................7.7 7.6 Belytschko-Wong-Chiang Improvements....................................................7.9 8. TRIANGULAR SHELLS........................................................................................8.1 8.1 C0 Trinagular Shell ......................................................................................8.1 8.1.1 Co-rotational Coordinates.............................................................8.1 8.1.2 Velocity-Strain Relations..............................................................8.1 8.1.3 Stress Resultants and Nodal Forces ..............................................8.5 8.2 Marchertas-Belytschko Triangular Shell .....................................................8.7 8.2.1 Element Coordinates.....................................................................8.7 8.2.2 Displacement Interpolation...........................................................8.9 8.2.3 Strain-Displacement Relations....................................................8.11 8.2.4 Nodal Force Calculations............................................................8.12 9. TYPE 16: FULLY INTEGRATED SHELL............................................................9.1 9.1 Introduction ...............................................................................................9.1 9.2 Hu-Washizu Three Field Principle ..............................................................9.1 9.3 In-plane Assumed Strain Field.....................................................................9.2 9.4 Transverse Shear Assumed Strain Field ......................................................9.3 9.5 Rigid Body Motion ......................................................................................9.4 9.6 Belytschko-Leviathan Projection.................................................................9.5 10. HUGHES-LIU SHELL..........................................................................................10.1 iv LS-DYNA Theory Manual Table of Contents 10.1 Geometry....................................................................................................10.1 10.2 Kinematics.................................................................................................10.4 10.2.1 Fiber Coordinate System................................................................10.5 10.2.2 Lamina Coordinate System............................................................10.7 10.3 Strains And Stress Update .........................................................................10.8 10.3.1 Incremental Strain And Spin Tensors............................................10.8 10.3.2 Stress Update.................................................................................10.9 10.3.3 Incremental Strain-Displacement Relations ..................................10.9 10.4 Element Mass Matrix...............................................................................10.10 10.5 Accounting For Thickness Changes ........................................................10.11 10.6 Fully Integrated Hughes-Liu Shells .........................................................10.12 11. TRANSVERSE SHEAR TREATMENT FOR LAYERED SHELL ....................11.1 12. EIGHT-NODE SOLID SHELL ELEMENT .........................................................12.1 13. EIGHT-NODE SOLID ELEMENT FOR THICK SHELL SIMULATIONS .......13.1 13.1 Abstract......................................................................................................13.1 13.2 Introduction................................................................................................13.1 13.3 Element Formulations................................................................................13.1 13.3.1 Strain Operator...............................................................................13.1 13.3.2 Corotational Coordinate System....................................................13.7