1. Finite Element Methods and Crack Growth Simulations Materials Simulations Physics 681, Spring 1999 David (Chuin-Shan) Chen Postdoc, Cornell Fracture Group david@lager.cfg.cornell.edu www.cfg.cornell.edu

2. Tentative Syllabus Part I: Finite Element Analysis and Crack Growth Simulation Introduction to Crack Growth Analysis Demo: Crack Propagation in Spiral-bevel Gear Introduction to Finite Element Method Stress Analysis: A Simple Cube Crack Growth Analysis: A Simple Cube with A Crack Part II: Finite Element Fundamentals Basic Concepts of Finite Element Method Case Study I: A 10-noded Tetrahedron Element Case Study II: A 4-noded Tetrahedron Element

3. Motivation: Why we are interested in Computational Fracture Mechanics Cracking Is a Worldwide-Scale Problem: –> $200B per year cost to U.S. national economy –Energy, Defense and Life Safety Issues Simulation of Crack Growth Is Complicated and Computationally Expensive: –An evolutionary geometry problem –Complex discretization problem –Many solutions of mega-DOF finite element problems We Were at An Impasse: –Needed better physics--required larger problems –Larger problems impossible/impractical

4. Crack Propagation in Gear Simulation Based on Fracture Mechanics Compute Fracture Parameters (e.g., Stress Intensity Factors) from Finite Element Displacements Determine Crack Shape Evolution crack growth direction from SIFs user specified maximum crack growth increment Initial Crack Final Crack Configuration (29 Propagation Steps)

5. Crack Growth Simulation Need: Life Prediction in Transmission Gears U.S. Army OH-51 Kiowa Allison 250-C30R Engine Fatigue Cracks in Spiral Bevel Power Transmission Gear Project: NASA Lewis NAG3-1993

6. A LIFE-SAFETY ISSUE Crack Growth Simulation Need :

7. A NATIONAL DEFENSE ISSUE The combined age of the 3 frontline aircraft shown here is over 85 years. Defense budget projections do not permit the replacement of some types for another 20 or more years. Crack Growth Simulation Need :

8. Projects: NASA NLPN 98-1215, NASA NAG 1-2069, AFOSR F49620-98-1

9. KC-135 Blow-out!

10. Finite Element Method A numerical (approximate) method for the analysis of continuum problems by: –reducing a mathematical model to a discrete idealization (meshing the domain) –assigning proper behavior to “elements” in the discrete system (finite element formulation) –solving a set of linear algebra equations (linear system solver) used extensively for the analysis of solids and structures and for heat and fluid transfer

11. Finite Element Concept  Differential Equations : L u = F General Technique: find an approximate solution that is a linear combination of known (trial) functions x y Variational techniques can be used to reduce the this problem to the following linear algebra problems: Solve the system K c = f

12. 3D tetrahedron element

13. Crack Propagation on Teraflop Computers Software Framework: Serial Test Bed 1 FRANC3D Life Prediction Crack Propagation Fracture Analysis Boundary Conditions Introduce Flaw(s) Solid Model Volume Mesh Finite Element Formulation Iterative Solution