Fracture Mechanics Homework.pdf - Finite Element Analysis...

Finite Element Analysis II (ENME 6333) Professor H. Waisman
Spring 2022 Columbia University
Homework set 2 (due 03/10/2022)
Fracture Mechanics
Consider a square plate with an edge crack, as shown in Figure 1. The dimension of
this square plate is L = 1 and the crack length is a = 0.5. Young’s modulus is E =
3e7 and Poisson’s ratio is ν = 0.3. All units are assumed to be consistent and the
plate is assumed to be in plane stress conditions.
Figure 1: A square plate with an edge crack
Exact displacements, from William ’ s asymptotic solution, corresponding to mode I
fracture with the stress intensity factor (SIF) K
I
= 100, are imposed on the external
boundaries (not on crack surfaces), as follows:
?
?
=
𝐾
𝐼
2?
√
?
2𝜋
cos
𝜃
2
[𝜅 − 1 + 2sin
2
𝜃
2
]
?
?
=
𝐾
𝐼
2?
√
?
2𝜋
sin
𝜃
2
[𝜅 + 1 − 2cos
2
𝜃
2
]

where ? is the shear modulus and 𝜅 the Kolosov constant ( κ = (3 − ?)/(1 + ?) for
plane stress problems). Note that the range of the polar coordinate 𝜃 is from – 𝜋
(lower crack surface) to 𝜋 (upper crack surface) and K
I
= 100 is the reference
solution.
Assignment:
Part 1: Linear Elastic Fracture Mechanics