EN222: Mechanics of Solids
Homework 5: Cracks and Contacts
Due Thursday March 16, 2006
Division of Engineering
Brown University
1. The figure shows a periodic array of edge dislocations
with
in an infinite solid.
Show that the
vertical stress along
can be calculated as
12
0,
bb
==
b
2
0
x
=
1
22
2
2
2
11
4(
1
)
x
Eb
22
1
2(
1
)
n
Eb
x
nL
x
πν
=+
−−
−
σ
∞
=
∑
L
L
x
1
x
2
Sum the series analytically (MAPLE will do it).
L
2a
x
1
x
2
2. Using the results of problem 1, set up an integral equation to calculate the burgers vector
distribution that represents the solution to an infinite array of parallel cracks with length 2
a
and
spacing
L
, which is loaded by a uniform stress
22
∞
far from the crack.
Show that the integral
equation can be expressed as
1(
)
(,)
)
)
bd
kx
b
d
gx
x
ξξ
ξ
πξ
+
(=
(
−
∫∫
where
2
22
( )
4(1
)
/
( ,
)
(
/
)cot(
(
)
/
)
1/(
(
))
E
a L
xa L
x
νσ
∞
=−
=
−
−
−
Use the numerical procedure discussed in class to solve the integral equation for
2
()
/ 1
b
φ
The stress intensity factors for a mode I crack can be extracted from the solution for
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 HUAJIANGAO
 Force, stress intensity factors

Click to edit the document details