Finite Element Methods and Crack Growth Simulations

Materials Simulations

Physics 681, Spring 1999

David (Chuin-Shan) ChenPostdoc, Cornell Fracture Group

david@lager.cfg.cornell.eduwww.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

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 CrackFinal Crack Configuration

(29 Propagation Steps)

Crack Growth Simulation Need: Life Prediction in Transmission Gears

U.S. Army OH-51 Kiowa

Allison 250-C30R Engine

Fatigue Cracks in Spiral BevelPower Transmission Gear

Project: NASA Lewis NAG3-1993

The National Aging Aircraft Problem

The Impetus

...The plane, a B-737-200, has flown 89,680 flights, an average of 13 per day over its 19 yearlifetime. A “high ti me” aircraft has flown 60,000 fli ghts.

April 28, 1988. Aloha Airlines Flight 243levels off at 24,000 feet...

A LIFE-SAFETY ISSUE

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 sometypes for another 20 or more years.

Crack Growth Simulation Need:

The KC-135 Fleet Will Be Operating forMore Than 70 Years

Corrosion and Fatigue Can Become a Problem

The Residual Strength of the Struc turewith 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

Differential Equations : L u = F

General Technique: find an approximate solution that is a linearcombination of known (trial) functions

x

y

)y,x(c)y,x(* i

n

1ii

u

Variational techniques can be used to reduce the this problem to the following linear algebra problems:

Solve the system K c = f

d )L(K jiij

d Ff ii

3D tetrahedron element

Crack Propagation on Teraflop ComputersSoftware Framework: Serial Test Bed 1

FRANC3D

LifePrediction

CrackPropagation

FractureAnalysis

BoundaryConditions

IntroduceFlaw(s)

SolidModel

VolumeMesh

Finite ElementFormulation

IterativeSolution