UNIVERSITY THE OF IOWA Department of Mechanical & Industrial Engineering Fracture Mechanics 58:159 Homework #1 Total Points: 20 Assigned: January 28, 2009 Due: February 09, 2009
Problem 1: Consider a plate containing a circular hole of radius a, as shown in the figure below. The plate, which has geometric dimensions large enough to be idealized as an infinite plate, is subjected to a uniform far-field tensile stress . In the class, we showed that the linear-elastic stress field in this body is
r r
2
a 2 3a 4 4a 2 1 2 1 4 2 cos 2 r r r 2 4 a 3a 1 2 1 4 cos 2 2 r r
2
3a 4 2a 2 1 4 2 sin 2 r r
where r and are polar coordinates with the center of the hole as the origin. (1) (2) Plot r and as a function of r at the horizontal (line AB) and vertical (line CD) planes through the center of the Plot hole. as a function of (0 2) at the free surface of the hole (r = a).
D
a 1 in, 1 ksi, W H 20 in
2H
C y
r
x A
B
2a
2W
Problem 2: Consider an infinite plate, shown in the figure below, containing an elliptical hole with 2a and 2b as the lengths of its major and minor axes. If the plate is subjected to a uniform far-field tensile stress , we showed in the class that the normal stress at the tip of the ellipse (i.e., at point A) under linear-elastic condition is a y 1 2 . x a , ...