ON THE STRUCTURE OF THE 3D STRESS FIELD AND ITS 3D STRESS SINGULARITY
IN A 3D GRIFFITH CRACK
by
E. S. Folias
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
December 2006
Keywords: three-dimensional stress singularity, vertex singularity, three-dimensional corner singularity,
surface cracks, three-dimensional Griffith crack, three-dimensional stress analysis
ABSTRACT
The author in this paper investigates the analytical three-dimensional stress field in the neighborhood of
the intersection of a crack and a free surface. Utilizing the form of a general three-dimensional solution,
which the author constructed in a previous paper, he recovers the explicit displacement and stress fields
in this neighborhood. The analysis shows the stresses to be proportional to l<<i
Folias, 1976) and which, he subsequently cast into a much more convenient form (Folias 1988a),
and (ii) by the author’s paper on the solution of the stress field at the intersection of a hole and a
free surface (Folias, 1987). Thus, we may assume the homogeneous displacementfield to be of
the form, where x,y, h z, and r, h z, 0 now reflect the local coordinates along the crack front
—
—
(i,) in rectangular cartesian coordinates:
h)
7
(
(h)
+ 1) +
(2v
f
0
2
=
m .2_
(J)’
nz20yk‘2’ oJ+
-
(la)-(3a)
-
—
“
—
—
—2v ‘1+
m—2z’
or
(ii
in cylindrical coordinates:
m ±
m—2 i
—
—
,(h)
+)+
(2v
f
0
in— 2
=
ao
--
flY
=
f and
=
ög
(ib) - (3b)
‘
+ CL)
(—2v
f
0
m—2 z
where the functions
—
‘2’
g satisi’ the 3D Laplace’s equation, and
1—v, m= -r ; and fr,h—z),l, gr,h—z,U
(4)
(5)
To this, one may also add, if and when it is subsequently needed, the corresponding
particular solution which, at this time for the sake of convenience, we chose to write in
rectangular cartesian coordinates. The reason being that later, or in the future, one may have to
use it in that form in order to deal with the boundary condition -r, at y 0
—
—
—
8x
—
.)
L.
—
tn-i-I
y
,,2 cIq
__
n,+1
—
—
—
q
j.L.
&zI-,n+1
E
)‘
a
tn-i-I
Jq
,,
a
2
2
ax
I
where in the above we have also adopted the additional definitions
=
1
—
v ; v = Poisson’s ratio
(9)
January 10, 2014
APPENDIX B
A POINT OF CLARIFICATION:
The main goal of the above paper was to show only what the prevailing strength of the
stress singularity is, at such corners. As a result, eqs. (20)-(21) reflect only the first term of many
others of a similar nature. The reader, however, should realize that up to this point of the analysis
0 is still largely arbitrary. Consequently, the general form of the function
the parameter b = v
us:
2
11 [r
A,
]d
2
+(h_z)
)
{p2b+k(J
L4)+p2h÷k(_J
2!z).)} cos(a 0)
which for k 0 reduces to eq. (20). But, for purposes of showing what the order of the stress
singularity is, one really needs to consider only the first term.
Similarly, the general form of the inner expansion of the solution may be written in the
form:
*
nO
1 (—b +
F
AA[n] (h _z)2b() 2
+ a,
—b +
+ a,
1 +a, _(hrZ)2) cosa0.
a/h
radius / half thickness
Fig. 1 Maximum stress concentration factor versus radius to half thickness ratio.
0
cr
t :i t I
H2OH
Fig. 2 Infinite plate with a circular hole.
LL
+
-c
1
Fig. 3 Infinite plate with a crack.
—
-_-.
Fig. 4 Local geometly at the corner point.
r,x