Teraflop Crack Growth Simulation Need: Hydraulic Fracturing

							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
               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
    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
             Crack Propagation in Gear

   • Simulation Based on Fracture Mechanics

                                      Determine Crack Shape Evolution
   Compute Fracture Parameters
                                      • crack growth direction from SIFs
   (e.g., Stress Intensity Factors)
                                      • user specified maximum crack growth
from Finite Element Displacements
                                        increment




                                      Initial Crack
                                                      Final Crack Configuration
                                                        (29 Propagation Steps)
         Crack Growth Simulation Need:
      Life Prediction in Transmission Gears
Project: NASA Lewis NAG3-1993     Allison 250-C30R Engine




   U.S. Army OH-51 Kiowa




 Fatigue Cracks in Spiral Bevel
   Power Transmission Gear
         Crack Growth Simulation Need:
     A LIFE-SAFETY ISSUE
The National Aging Aircraft Problem
April 28, 1988. Aloha Airlines Flight 243
levels off at 24,000 feet...




The Impetus


...The plane, a B-737-200,
                                                      r its
has flown 89,680 flights, an average of 13 per day ove 19 year
lifetime. A “high time” aircraft has flown 60,000 fli
                                                   ghts.
              Crack Growth Simulation Need:
           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.
   The KC-135 Fleet Will Be Operating for
           More Than 70 Years
         Corrosion and Fatigue Can Become aProblem




                                               ture
              The Residual Strength of the Struc
              with Both Present Must be Predictable

Projects: NASA NLPN 98-1215, NASA NAG 1-2069, AFOSR F49620-98-1
KC-135 Blow-out!
        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
       Finite Element Concept                                W
Differential Equations : L u = F             y
                                                        W

                                              x
General Technique: find an approximate solution that is a linear
combination of known (trial) functions
                                  n
                   u * ( x , y)   ci  i ( x , y)
                                 i 1
Variational techniques can be used to reduce the this problem to
the following linear algebra problems:
                Solve the system K c = f
            K ij    i (L j ) dW       f i    i F dW
                  W                              W
3D tetrahedron element
Crack Propagation on Teraflop Computers
 Software Framework: Serial Test Bed 1
                                        Solid
                                        Model


                        FRANC3D
        Life           Crack         Boundary
     Prediction     Propagation      Conditions

             Fracture                Introduce
             Analysis                 Flaw(s)



Iterative           Finite Element       Volume
Solution             Formulation          Mesh