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 FAX: 925-449-2507 TEL: 925-449-2500 EMAIL: [email protected] WEBSITE: www.lstc.com Copyright © 1991-1998 by Livermore Software Technology Corporation All Rights Reserved LS-DYNA Theoretical Manual Table of Contents 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 iii Table of Contents LS-DYNA Theoretical Manual 6.5 Hourglass Control (Englemann and Whirley)..................................................................... 6.8 6.6 Belytschko-Wong-Chiang Improvement ..........................................................................6.10 7. C0 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 iv LS-DYNA Theoretical Manual Table of Contents 13.7 General Nonlinear .................................................................................................13.7