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Equations for Linear Elastic Fracture Mechanics and Fatigue
EAE 135, Winter 2006
Classes on 03/06/2006, 03/08/2006
This document provides further information with respect to the presentation file
fracture_fatigue_lecture.pdf
.
1. Linear Elastic Fracture Mechanics
LEFM is a methodology that allows you to predict, study and measure fracture toughness.
Fracture toughness characterizes the resistance of a material to cracking, and it depends on a
variety of factors such as temperature, environment, loading rate etc.
Out of
the three crack opening modes (see presentation file), the one that is analyzed is Mode I,
which is typically more critical from the design standpoint (lower fracture toughness with respect
to the Mode II and Mode III fracture toughness. Note that mixed mode is not addressed in this
document, but exists).
a
S
F
K
I
π
⋅
⋅
=
(
1
)
Equation 1 shows the Mode I stress intensity factor, which characterizes the behavior of the
material when a crack of length 2a is present in it. F is a factor that depends on the geometry of
the specimen and the crack itself, S is the nominal stress. When the stress intensity factor reaches
a critical value, the crack grows.
The document attached at the end provides some (nonfocused) tables that show how F (called Y
in the document) changes with respect to the specimen’s geometry and crack length (divided by
the width of the specimen).
It is possible to express the stress at a distance r ahead of a crack in terms of
r
K
I
. As r
Æ
0, the
stress should theoretically be infinite, but this is not possible in nature. The material will yield, so
there will be a plastic zone at the crack tip. The size of this plastic zone depends on whether, for
example, the specimen is subject to a plane strain or a plane stress case. If the specimen has
dimensions x, y, z, plane strain corresponds to the case
ε
z = 0 (for example, a beam with width z
where there is no deformation in the width direction) and plane stress to the case
σ
z
= 0 (for
example, a (x,y) plate where there is no perpendicular loading). In a plane stress problem, the
plastic region is found to be larger. However, the plastic region could be even larger than in the
case of plane stress if the material has yielded in a great part of the specimen.
Linear elastic fracture mechanics (LEFM) is used when the plastic zone ahead of the crack is
much smaller than the dimensions of the crack or of the specimen. When this is no longer the
case (for example, in a cracked specimen with 80% yielding present), one has to use Elastic
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Plastic Fracture Mechanics (EPFM), which is not based on the concepts of stress intensity factors
and fracture toughness, but on other more complicated concepts (crack tip opening displacement,
Jintegral, covered in an advanced Fracture Mechanics course).
KIC is the Mode I, plane strain fracture toughness. It is best to use this parameter because this is
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