Non-Linear Fracture Mechanics in LS-DYNA and LS-Prepost ∫
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Elastic Plastic Fracture Mechanics Elastic Plastic Fracture Mechanics Presented by Calvin M
Fracture Mechanics Elastic Plastic Fracture Mechanics Elastic Plastic Fracture Mechanics Presented by Calvin M. Stewart, PhD MECH 5390-6390 Fall 2020 Outline • Introduction to Non-Linear Materials • J-Integral • Energy Approach • As a Contour Integral • HRR-Fields • COD • J Dominance Introduction to Non-Linear Materials Introduction to Non-Linear Materials • Thus far we have restricted our fractured solids to nominally elastic behavior. • However, structural materials often cannot be characterized via LEFM. Non-Linear Behavior of Materials • Two other material responses are that the engineer may encounter are Non-Linear Elastic and Elastic-Plastic Introduction to Non-Linear Materials • Loading Behavior of the two materials is identical but the unloading path for the elastic plastic material allows for non-unique stress- strain solutions. For Elastic-Plastic materials, a generic “Constitutive Model” specifies the relationship between stress and strain as follows n tot =+ Ramberg-Osgood 0 0 0 0 Reference (or Flow/Yield) Stress (MPa) Dimensionaless Constant (unitless) 0 Reference (or Flow/Yield) Strain (unitless) n Strain Hardening Exponent (unitless) Introduction to Non-Linear Materials • Ramberg-Osgood Constitutive Model n increasing Ramberg-Osgood −n K = 00 Strain Hardening Coefficient, K = 0 0 E n tot K,,,,0 n=+ E Usually available for a variety of materials 0 0 0 Introduction to Non-Linear Materials • Within the context of EPFM two general ways of trying to solve fracture problems can be identified: 1. A search for characterizing parameters (cf. K, G, R in LEFM). 2. Attempts to describe the elastic-plastic deformation field in detail, in order to find a criterion for local failure. -
2 Review of Stress, Linear Strain and Elastic Stress- Strain Relations
2 Review of Stress, Linear Strain and Elastic Stress- Strain Relations 2.1 Introduction In metal forming and machining processes, the work piece is subjected to external forces in order to achieve a certain desired shape. Under the action of these forces, the work piece undergoes displacements and deformation and develops internal forces. A measure of deformation is defined as strain. The intensity of internal forces is called as stress. The displacements, strains and stresses in a deformable body are interlinked. Additionally, they all depend on the geometry and material of the work piece, external forces and supports. Therefore, to estimate the external forces required for achieving the desired shape, one needs to determine the displacements, strains and stresses in the work piece. This involves solving the following set of governing equations : (i) strain-displacement relations, (ii) stress- strain relations and (iii) equations of motion. In this chapter, we develop the governing equations for the case of small deformation of linearly elastic materials. While developing these equations, we disregard the molecular structure of the material and assume the body to be a continuum. This enables us to define the displacements, strains and stresses at every point of the body. We begin our discussion on governing equations with the concept of stress at a point. Then, we carry out the analysis of stress at a point to develop the ideas of stress invariants, principal stresses, maximum shear stress, octahedral stresses and the hydrostatic and deviatoric parts of stress. These ideas will be used in the next chapter to develop the theory of plasticity. -
Ls-Dyna Theoretical Manual
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 ..................................................................................................... -
Fracture Mechanics - Calculations
Fracture Mechanics - Calculations • Theory for calculations of severity of crack. • FEM – J integrals for plane problems. • Special Finite Elements + video of crack process • FEM – J integrals for 3-D problems. 1 Conclusion of continuum exercise • At sharp edges there is a stress singularity • The stresses go to infinity • The strain energy density is limited • The magnitude of the stress singularity determines the strain energy density or the stress intensity factor K • Stress intensity factors should not be confused with stress concentration factors. 2 Stress Intensity Factor for a small crack with a length of a: For a very small 3 The crack singularity - polar coordinates Stress state around crack (Continuum Mechanics) 4 Mode I - Opening mode Stresses depend on inverse of squareroot of r - singularity Singularity dominates at the crack tip - C11 depends on external load Symmetric solution. 5 Mode II - Shearing mode Stresses depend on inverse of squareroot of r - singularity Singularity dominates at the crack tip - C21 depends on external load Antimetric solution. 6 Reformulation of parameters The D1 is the case with out-of-plane shear (torsion mode). 7 How to find the singularity? Structure with crack is analysed by means of superposition 8 Structure without crack + stresses on crack Crack is closed and stresses will be finite. The stresses on the crack is easily calculated (FEM – hand calculation) 9 Calculation of singularity 10 Singular part - Infinite stresses The infinite stress state around the crack can be found by Analytically solution for a crack in an infinite plate subjected to - 2 opposite point forces (normal or shear direction) Integration of analytically solution In this way many analytical or semianalytical solutions have been found. -
LS-DYNA Theory Manual Table of Contents
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 -
The Applic · F the J Integral to Fracture I Under Mixed~Mode Loading
UCRL-53182 - The Applic · f the J Integral to Fracture I under Mixed~Mode Loading Robert Allen Riddle (PH.D. Thesis) • June 1981 . DISTIHBUTION OF THIS OGCUMENT IS UNLIMfT£0 DISCLAIMER This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency Thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof. DISCLAIMER Portions of this document may be illegible in electronic image products. Images are produced from the best available original document. DISCLAIMER This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, com pleteness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial products, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. -
APPENDIX F Exercises
540 APPENDIX F Exercises 1. Chapter 1: Objectives and Methods of Solid Mechanics 1.1. Defining a problem in solid mechanics 1.1.1. For each of the following applications, outline briefly: • What would you calculate if you were asked to model the component for a design application? • What level of detail is required in modeling the geometry of the solid? • How would you model loading applied to the solid? • Would you conduct a static or dynamic analysis? Is it necessary to account for thermal stresses? Is it necessary to account for temperature variation as a function of time? • What constitutive law would you use to model the material behavior? 1.1.1.1. A load cell intended to model forces applied to a specimen in a tensile testing machine 1.1.1.2. The seat-belt assembly in a vehicle 1.1.1.3. The solar panels on a communications satellite. 1.1.1.4. A compressor blade in a gas turbine engine 1.1.1.5. A MEMS optical switch 1.1.1.6. An artificial knee joint 1.1.1.7. A solder joint on a printed circuit board 1.1.1.8. An entire printed circuit board assembly 1.1.1.9. The metal interconnects inside a microelectronic circuit 1.2. What is the difference between a linear elastic stress-strain law and a hyperelastic stress-strain law? Give examples of representative applications for both material models. 1.3. What is the difference between a rate-dependent (viscoplastic) and rate independent plastic constitutive law? Give examples of representative applications for both material models. -
Ju L 2 0 2004
Three-dimensional Mixed-Mode Linear Elastic Fracture Mechanics Analysis Using Domain Interaction Integrals by Ekrem Alp Esmen Bachelor of Science in Mechanical Engineering, Bachelor of Science in Economics, University of Michigan at Ann Arbor, MI (2000) Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June 2004 @ Massachusetts Institute of Technology 2004. All rights reserved. Apt A Author............................. Department of cal Engineering . May 26440 //, C ertified by ............................... David M. Parks Professor of Mechanical Engineering ' p upervisor Accepted by ............................. %06A.Sonin Chairman, Department Committee on Graduate Students MASSACHUSETTS INSTITUTE, OF TECHNOLOGY JU L 2 0 2004 BARKER LIBRARIES Room 14-0551 77 Massachusetts Avenue Cambridge, MA 02139 Ph: 617.253.2800 M ITLibraries Email: [email protected] Document Services http://libraries.mit.edu/docs DISCLAIMER NOTICE The accompanying media item for this thesis is available in the MIT Libraries or Institute Archives. Thank you. 5..5%n..:-st.VeTal..ri--i-4..,-:-!-Iru. s1M:Wrne'1HrFNei49--.--- ra-matur.ge:. -r-.-en:w.-.:..--o-Aus-:e....e,:--.. .,.& - v.ia-t--.il-'g:4ma.e= 4..'*-*%.:;'5- A 3 <K/s/2,>*'~ 5- 'S 4 St I Three-dimensional Mixed-Mode Linear Elastic Fracture Mechanics Analysis Using Domain Interaction Integrals by Ekrem Alp Esmen Submitted to the Department of Mechanical Engineering on May 25, 2004, in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Abstract Three-dimensional mixed-mode linear elastic fracture mechanics analysis is presented using domain interaction integrals.