Finite Elements and Fracture Mechanics

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Finite Elements and Fracture Mechanics

• Leslie Banks-Sills
• The Dreszer Fracture Mechanics Laboratory
• Department of Solid Mechanics, Materials and
  Systems
• Tel Aviv University

ISCM-15, October, 2003
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Outline

• Introduction to fracture mechanics (homogeneous
  material).
• The finite element method.
• Methods for calculating stress intensity factors.
• Interface fracture mechanics.

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Dreszer Fracture Mechanics Laboratory
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(No Transcript)
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Liberty Ships-World War II

• The hulls of Liberty Ships fractured without
  warning, mainly in the North Atlantic.
• There were 2,751 Liberty Ships manufactured
  between 1941- 1945. Cracks propagated in 400 of
  these ships including 145 catastrophic failures
  only 2 exist today which are sea- worthy.

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Liberty Ships-(continued)

• The low temperatures of the North Atlantic caused
  the steel to be brittle.
• These are the first ships mass produced with
  welds.
• Fractures occurred mainly in the vicinity of
  stress raisers.

• The problem may be prevented by employing
  higher quality steels and improvement of the
  design of the ship.

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The Aloha Boeing 737 Accident
On April 28, 1988, part of the fuselage of a
Boeing 737 failed after 19 years of service. The
failure was caused by fatigue (multi-site damage).
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The Aloha Boeing 737 Accident
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Modes of Fracture
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Asymptotic Stress Field in Mode I
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Stress Intensity Factor
m I, II, III
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Fracture Toughness
ASTM 399 Standard
compact tension specimen
material parameter, depends on environment
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J -- integral
strain energy density
tractions
J is a conservative integral
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Griffiths Energy G
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J vs G
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The Finite Element Method
For a static problem
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The Element
Lagrangian shape functions for a four noded
element
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The Element (continued)
isoparametric element
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Special Crack Tip Elements
quarter-point elements
Henshell and Shaw, 1975, quadrilateral
elements Barsoum, 1974,1976, triangular elements
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Special Crack Tip Elements
quarter-point elements
Henshell and Shaw, 1975, quadrilateral
elements Barsoum, 1974,1976, triangular elements
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Special Crack Tip Elements
quarter-point elements
Henshell and Shaw, 1975, quadrilateral
elements Barsoum, 1974,1976, triangular elements
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Special Crack Tip Elements
quarter-point elements
Henshell and Shaw, 1975, quadrilateral
elements Barsoum, 1974,1976, triangular elements
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Eight Noded Isoparametric Element
shape functions
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Eight Noded Isoparametric Element
shape functions (continued)
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Square-Root Singular Element
Banks-Sills and Bortman (1984)
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Methods of Calculating KI

• Direct Methods
• Stress extrapolation
• Displacement extrapolation
• Indirect Methods
• J integral
• Griffiths energy
• Stiffness derivative

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Displacement Extrapolation
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Displacement Extrapolation (continued)
for plane strain
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Displacement Extrapolation (continued)
for
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J -- integral
strain energy density
tractions
J is a conservative integral
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J -- integral (continued)
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Area J -- integral
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Griffiths Energy
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Stiffness Derivative Technique
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Results (central crack)
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Results (edge crack)
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Mixed modes M integral
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Auxiliary Solutions
solution (2a)
solution (2b)
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Interface Fracture Mechanics
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Interface Fracture Mechanics (continued)
phase angle or mode mixity
energy release rate
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Interface Fracture Mechanics (continued)
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M integral
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Auxiliary Solutions
solution (2a)
solution (2b)
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Results
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Summary

• Accurate methods have been presented for
  calculating stress intensity factors based on
  energy methods.
• The best methods are the area J integral,
  stiffness derivative and area M integral for
  mixed modes and interface cracks.
• The J and M integrals can be extended for
  thermal stresses, body forces and tractions along
  the crack faces.
• Conservative integrals have been derived for
  homo- geneous notches and bimaterial wedges
  including thermal stresses.
• Student wanted for extending these methods to
  piezo-electric materials