OCTOBER 19-20, 2012 - YMCA University of Science & Technology

Proceedings of the National Conference on
Trends and Advances in Mechanical Engineering,
YMCA University of Science & Technology, Faridabad, Haryana, Oct 19-20, 2012
ν
⎡ 1
0
1 − ν
⎤
⎢ ν
⎥
1 − ν ⎢ 1 0 ⎥
D =
(1 + ν)(1 − 2ν) ⎢1 − ν
⎥ Eq. (3)
⎢
(1 − 2ν)
0 0
⎥
⎢
2(1 − ν) ⎥
⎣
⎦
When the closed crack splits, ∆u 1 (∆a — r, t) and∆u 2 (∆a — r, t) represents, respectively the crack sliding and
opening displacement components at t with the following form [8]:
∆u 1(∆a — r, t) = u 1 (∆a − r, π, t) − u 1 (∆a − r, −π, t)
Eq. (4)
∆u 2 (∆a — r, t) = u 2 (∆a − r, π, t) − u 2 (∆a − r, −π, t)
Where ∆a is the virtual extended length of the crack and r is the radial crack length.
3. Numerical Example
In order to validate the effectiveness and accuracy of the current method for a cracked linear viscoelastic body, a
fracture example is studied and numerical results are compared to available reference solutions. For the
following example the material is assumed to be a three-parameter linear viscoelastic solid as shown in Figure 1.
Two groups of assumed material properties are used in the validation case studies shown in Table 1.
A viscoelastic strip having an edge crack with remotely applied tension as shown in Figure 2 is investigated. The
height of strip is H=4W, where W=1 is the width of the strip. The crack depth is a=0.3W. Poisson’s ratio ν=0.49.
The unit step loading is applied and the total loading time is T=200. COD {u 2 (r, π, t)—u 2 (r,—π, t)}is computed
at a pair of reference points A and B as shown in Figure 3.
The formulation of displacement field at crack tip under remotely applied tension is [8]
u 2 (r, θ, t) = Fσ 0 √πa(1 + ν) r
k + 1 − 2π 2cos2 θ sin θ J(t)Eq. (5)
2 2
where u 2 is the y-direction displacement component in Cartesian coordinate system located at crack tip. r and θ
are the polar coordinates defined around the crack tip. F is a nondimensional
coefficient for stress intensity factor with the value 1.6599, J (t) is the creep compliance, and k
is the Kolosov constant takes the value of k= (3- 4ν) for plain strain state andσ0 is assumed to be 1 [8].
TABLE I
MATERIAL PARAMETERS OF THE LINEAR VISCOELASTIC SOLID.
Set E ∞ E 1 η 1
A 1.0 1.0 5.0
B 1.0 2.0 5.0
Figure 1: The three-parameter linear viscoelastic solid.
496