Crack Paths 2009

higher or lower than those predicted by the plane theory of elasticity, i.e. the usual stress

concentration factor of 3.

o'

/

2h

Z

1%

2h / x ‘ L g ;i Z

.

.

x

IIIIIIIIIII

T

x

.

T

0-

+1 R +

}_

+ | R+

Fig. la. Plate with a circular

Fig. lb. Thin plate: failure Figlb. Thick plate: failure

hole

location

location

In order to obtain a more accurate solution, many three-dimensional studies have

been undertaken in the past including Green’s solution in an infinite series form [6],

approximate solution given by Stemberg and Sadowsky[7] and numerical solutions by

Alblas [8]. In 1962, Reiss [9] using a perturbation analysis was able to obtain a solution,

which yielded three-dimensional corrections to those of the plane stress. The results,

which are valid for small values of thickness to radius ratios, substantiate the findings

by Alblas. Kotousov and W a n g[10] utilized the first order plate theory and obtained an

exact analytical solution of the problem within this theory. Finally, an analytical

solution utilizing Kantorovish and Krylov [11] and Fourier transform methods were

derived by Folias and W a n g[12]. Results obtained by Folias and W a n gfor the

m a x i m u mstress concentration factor distribution across the thickness at the critical

locationxI i Rand y I 0, see Figla, are shownin Fig. 2, where SCFmax I omax /o.

S C F m a x

S C F m a x

3.1 -

3 —

.

R / h : 4 , 2 , 1

R/h:0.5,0.3,0.05

V203

2.8-

vI0.3

2.8 —

-

l

l

l

l

l

l

l

|

l

_

I

|

I

|

l

l

l

l

l

I

0

0.2

0.4

06 0.8

Z/h

0

0.2

0.4

0.6

0.8

Z/h

Fig.2a. Thin plate

Fig. 2b. Thick plate

Dependenceof the stress concentration factor (SCF) across the plate thickness

at critical location (x I i R and y I 0, see Figla)

These theoretical results provide a convincing explanation of the phenomenon

associated with the preferable crack formation region as for relatively thin plates IUh >1

the maximumof stress is located on the mid-plane (z I 0) and the failure is expected to

193