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The order of singularities and the stress intensity factors near corners of regular polygonal holes
0020-7225/87$3.00+ 0.00 Copyright 0 1987 Pergamon Journals Ltd
1~. J. Engng Sci. Vol. 25, No. 7, pp. 821-832, 1987 Printed in Great Britain. All rights reserved
THE ORDER OF SINGULARITIES AND THE STRESS INTENSITY FACTORS NEAR CORNERS OF REGULAR POLYGONAL HOLES P. S. THEOCARIS
and L. PETROU
Department of Theoretical and Applied Mechanics, The National Technical University of Athens, Athens GR-157 73, Greece Abstract-A new method was developed for the calculation of the order of singularity (SO) and the respective stress intensity factor @IF) near corners of regular polygonal holes perforated in elastic plates submitted to an overall tension at infinity. For the solution of the problem the Muskhelishvili potential function cp(z)was determined for an infinite plate weakened by a respective regular polygonal hole with rounded-off corners under tension. For each rounded off corner a virtual singular point was defined inside the hole along the bisector of the angle of the apex and at a distance depending on the radius of curvature of the corner. Then, the stress concentration factors (SCF) at the apices of these rounded-off corners were evaluated when their respective radii of curvature tended to zero. The corresponding stress-singularity and the SIF were afterwards derived, by correlating the SCF and the distance of the virtual singular point for each corner. By conveniently selecting the number of terms of the series development of q(z) the method can approach the real values for SO and SIF to any desired degree of accuracy. A comparison of the stress field around and near a corner of a polygonal hole in an infinite plate when this corner is submitted to mode-1 deformation and a V-notch in an infinite plate under tension indicated the similarities between the two affine problems.
INTRODUCTION
The stress field at the close vicinity of a singular point may be defined in a simple way by using the idea of the stress intensity factor (SIF) and the order of singularity (SO). Especially for edge V-notched infinite plates submitted to simple tension at infinity Williams [1,2] gave the SO at the bottom of the notch as the first eigenvalue of the eigenequation satisfying the boundary conditions of the plate. When the SO is known, the SIF can be calculated exactly by its definition given by Sih et al. [3]. This definition consisted in equating expressions for the sum of principal stresses in plane cracked plates given in terms of the SIF and of the derivative q’(z) of the Muskhelishvili potential function. In the vicinity of the crack tip q’(z) is expressed as a constant divided by (z - z,)* (see Ref. [4] pp. 496-498), where z is the complex coordinate and z1 the coordinate of the crack-tip. In this way the SIF may be defined as the limit for z -+ z1 of the expression (z - zJf$(z). Another possibility developed in the literature is to derive the SIF as the limiting value of the SCF at the apex of an equivalent notch with its corner rounded-off multiplied by the square root of the radius of curvature of the rounded apex, when this radius tends to zero [S]. Applications of this method were given in Refs. [6-81 and reasonable values for the SIFs are given. On the other hand, Sinclair and Kondo [9] have studied the possibility of determining in a realistic procedure the peak stresses at the bottoms of sharp re-entrant corners with zero curvature at their apices. In this method they approximated SCFs by defining the applied far-field stress at fracture of a plain plate divided by the applied stress for an identical plate with the stress raiser. Although the method yields, in general, working engineering estimates, it points out, in a general context, the necessity to seek realistic methods to define the exact local stress concentrations at sharp notches. Another method developed in Ref. [IO], where a relative problem of a triangular internal hole in a plate under tension was solved, was based in the following procedure. The plate with the triangular hole was conformally mapped into a perforated plate and the potential function q(z), truncated to a certain term, represented an equivalent triangular hole with its corners rounded off. The number of terms in the series expansion of q(z) defined the radius of curvature of the rounded apieces, with this radius diminishing as the number of terms was increased. The points along the bisectors of angles of the apices, where the 821
P. S. THEOCARIS
822
and L. PETROU
derivatives of the mapped function w(i) in the transformed plane were equal to zero, constituted the virtual singular points of the rounded vertices. A new coordinate system was introduced with its origin on the virtual singular point and the Ox-axis coinciding with the bisector of the corner. Then the SIF and the SO are taken immediately from the new function q(z) corresponding to the new coordinate system. In this paper the SCFs at the apices of the rounded corners were now combined with the distances r, of the virtual singular points from the apices. Then, the SIFs and SOS were derived when the values of r, tended to zero. This method was applied to any regular polygonal hole, whose apices were submitted to a combined in-plane loading creating mode-1 and -11 deformations at some apices. The conformal mapping of the regular polygon to a unit circle was achieved by using the Schwarz-Christoffel mapping function. The number of terms in the series expansion of the mapping function was selected to yield a virtual radius of curvature, r,, combined with its respective SCF for the rounded corner by using a least-square curve fitting with a correlation coefficient approximately equal to unity. Then, the respective SIF for the sharp corner was derived for the case when rs tends to zero. The SIFs for the corners of various regular polygons under a mode-1 deformation were compared with the respective sharp V-notched plates existing in the literature [ 11~ 131 and possessing the same values of orders of singularity as those given by the more accurate theory of Ref. [14]. Interesting results were derived from the confrontation of the two cases of stress raisers in flat plates. THE CONFORMAL MAPPING FUNCTION The stress distribution around a regular polygonal hole in an infinite plate induced to simple tension, as it is indicated in Fig. la, may be solved by conformally mapping the field outside the polygon on the interior of the unit circle by using the most efficient method of Schwarz-Christoffel [lS]. The appropriate function maps every polygon made up of a closed broken line to the unit circle. The appropriate relationship is expressed by [16]:
&p-l)
z = co,([) = R,
where the coefficients
fii are given by: 8, = 2(1 - k)(2 - 2k). . . (2 - (i - 1)k) I (1.2.3 .....i)k’(ki - 1)
and i = polygon polygon. origin 0
l,n,
(1)
with n tending mapped. The real If it is desired that and radius R, (Fig.
to infinity and k being the number of constant R, is a factor characterizing all the regular polygons are inscribed la) the real constant R, may take the
R.=Rdjl +ilBi}
(2) sides or corners of the the magnitude of the to the same circle with values:
(3)
Equation (1) maps a regular polygon with sharp corners if n tends to infinity. If a finite number of terms n in the series expansion of relation (1) is considered the corners of the polygon are rounded-off. In this case their radius of curvature pc depends on the number of terms n and it is given by the well known relationship: (.p P cn =
,xlyIl
+
p)3/2 _
,,,y’,
(4)
The order of singularities and the stress intensity factors near comers of regular polygonal holes
823
Fig. la. An infinite plate submitted to simple tension and weakened by any regular polygonal hole of any number of sides.
where x, y are the coordinates of each corner in the complex plane z = (x + iy) which are expressed by:
x =
/&cos(ki - 1)0
R, cost3 + f i=l
y = R, sin0 + i /?,sin(ki - 1)0
(5)
i=l
and the primes and double-primes denote differentiation with respect to the angle 8. The radius of curvature for the apex placed at 8 = 0” becomes:
R,
(ki -
t
l)& - 1
i=l PC”
(6)
= $1
W
-
l)Di
+
11
When n tends to infinity the pzi-radius becomes equal to zero and the corner becomes a sharp one. Figure la presents an infinite plate induced in tension and weakened by a regular polygonal hole A&Ct.. . C;B;A,, where k denotes the number of corners of each polygon, inscribed always inside the same circle of radius R,. Figure lb presents the details of an arbitrary corner of the polygon, in this case the corner A, of an equilateral triangular hole, mapped into the unit circle by eqn (1) with a finite number of terms n varying between n = 2 and 24. The increase of sharpness of the corner angle of the polygon as the number of terms n is increasing, as well as the good adaptation between the side of the
P. S. THEOCARIS and L. PETROU
Fig. lb. The rounded-off
A, of an equilateral triangular hole yielded from the Schwarz-Christoffel transformation with a finite number n of terms.
corner
Table 1. The normalized to the radius R, values of the radius of curvature pc the virtual singular distance rr of the rounded-off corner and the values of stress concentration factors there, for polygonal holes with number of sides k = 3, 4, 6 and 9, for parametric values of n (n = 2 + 24) k=3
v 2 3 4 5 6 12 24
PJR, 0.0126 0.0067 0.0042 0.0029 0.0021 0.0007 0.0002
k=4
rJR,
SCF
0.0060 0.0032 0.0020 0.0014 0.0010 0.0003 O.ooOl
17.39 23.95 30.03 35.81 41.37 71.93 125.66
k=6
k=9
q
PJR,
rJR,
SCF
q
PJK
‘JR<
SCF
q
PJR,
rJR,
SCF
2 3 4 5 6 12 18
0.352 0.201 0.135 0.098 0.076 0.027 0.015
0.0159 0.0091 0.0061 0.0044 0.0034 0.0012 O.ooO7
11.21 14.60 17.62 20.40 23.00 36.51 47.95
2 3 4 5 6 9 12
0.0737 0.0457 0.0321 0.0243 0.0193 0.0115 0.0080
0.0294 0.0182 0.0127 0.0097 0.0077 0.0046 0.0031
7.28 8.83 10.15 11.32 12.38 15.13 17.46
2 3 4 5 6 7 8
0.0640 0.0462 0.0357 0.0289 0.0240 0.0205 0.0178
0.0373 0.0242 0.0176 0.0137 0.0111 0.0093 0.0079
5.45 6.26 6.92 7.48 7.98 8.42 8.83
triangle and the rounded-off corner is apparent, indicating that the Schwarz-Christoffel transformation is the most suitable for such mappings. The values of the radii of curvature pen normalized to the radius R, of the circumscribed circle are given in Table 1 for k = 3, 4, 6 and 9. In the same table the normalized values of the virtual distances r,, and the respective stress concentration factors are also included. The respective virtual points inside the hole and along the bisectors of the angles of the apices are defined by the value of rsn satisfying the equation: I sn =
Rc- w,(P*)
(74
where p* is yielded from equation:
P-4
IwxP*)l= 0
It may be observed that the radii of curvature of the corners of a regular triangle become practically zero when n takes higher values than twelve (n 2 12). For a polygon with a k-number of corners its apex angle Bk is given by:
d,=
(1-i > II
(8)
The order of singularities and the stress intensity factors near comers of regular polygonal holes
825
When k tends to infinity, the polygon tends to coincide with its circumscribed circle. As k is increasing a larger number of terms n is needed in the series expansion to attain sharp corners.
EVALUATION
OF
STRESS
CONCENTRATION
FACTOR
plate submitted to simple tensile stresses at infinity, p, and perforated with a regular polygonal hole whose center coincides with the origin of the coordinate Oxy-system, in the case when the hole is free from externally applied loads, the complex potential function q(z) is given by [4]: For
an infinite
7 + q*(z)
cp(z)=
Function
q(z) in the mapped
plane [ is expressed
(9)
by:
di) = $cw(m + cpo(i) In this relation
cpO(c)is a function
expressed
(10)
by:
(11)
cpo(i)= f a,rl = f (Rear + iZ,a,)i r=l
r=1
where a, are complex constants. The boundary condition of the basic stress state is given by:
f(: + if! = -
![w(cr) - eZi%o(b)]
(12)
where c( denotes the angle subtended between the loading axis and the Ox-axis and r~ represents the points along the circumference of the unit circle in the transformed plane. For the determination of the unknown cp,([)-function use was made of the relationship [16]:
(13)
where l now is a point inside the unit circle y. If relations (1) and (11) are introduced relation (10) yield the following expression for the function (p,(c):
into
(km-1) +
c
r=l
Moreover,
the integral
of the right-hand
4
(14)
side of eqn (13) yields:
(15)
826
whereas
P. S. THEOCARIS
the left-hand
and L. PETROU
side of eqn (13) becomes:
Rear, + iIma,)l:
(16)
where P(i) is given by: p(i) =
-P(n-i+l)
-
P(1)&_,[2(i
‘. - P(i - 1)/3(l)(k - 1)
- 1)k - 21
with: P(l) = -B, Comparing now the respective real and imaginary coefficients of the respective terms of [ in eqns (15) and (16) we obtain a system of 2(kn - 1)-equations, which when solved yields the values of the real and imaginary parts of a,. Since along the boundaries of the polygonal hole the normal to these boundaries component of the normal stresses is zero it remains from the first stress invariant only the g,-stress parallel to the boundaries of the hole. Then, it is valid that [4]:
(17)
This value for a9 becomes equal to infinity for w’(i) polygonal hole. For rounded-off apices of the hole tending to infinity. This maximum value normalized p, yields the SCF at the respective corner A,. This
SCF, = 4Re
= 0, that is at the sharp corners of the this stress becomes maximum without to the applied tensile stress at infinity, SCF is given by:
cplxl) 1 40) ~
L
(18)
where:
- 1 + 2 (ki - l)pi i=l
o;(l)
= R,
+ ““i”
r{Rea, + iIma,}
(19)
r=l
- 1+ i
(ki - l)fli
(20)
i=l
Equation (19) indicates that the q’(l)-function becomes a real number when the (Ima,)term is equal to zero. This happens for loading angles c( = 7c/2 and a = 0”. In these two cases the A,-corner is subjected to a mode-1 deformation. For any other loading angle, a, the A,-corner is subjected to a mixed mode deformation except for some special values of the a-angle for which the real part of q’(l) becomes zero, when the boundary at this particular point is deformed exclusively under mode II. Let us assume that a regular polygonal perforation in an infinite plate which is submitted to a simple tension load has the bisector of the angle of its A,-corner subtending an angle a = 71/2 with the loading axis. The SCF corresponding to the respective rounded-off comer, normalized to p, is given in Table 1. When the number of terms n is increased the radius ofcurvature of the apex is decreased and the SCF increased. Figure 2 presents the variation
The order of singularities and the stress intensity factors near corners of regular polygonal holes
827
of the SCF, versus the virtual singular distance r, normalized to the radius R, of the circumscribed circle, when the quantity k of the corners in the polygon is increased from three to ten. It may be observed from this plotting that for the same virtual singular distance r, the SCF is decreasing when k is increased.
0
4
8
12
16
20
24
28
rSIR, (~10~1 e Fig. 2. The SCF of the rounded-off corner A, of the polygonal hole with k = 3, 4,. , , 10 in an infinite plate under tension, when the direction of loading a = 90”, versus the distance rS of the virtual singular point normalized to the radius of the circumscribed circle R,.
EVALUATION
OF SIF AND
THE ORDER
OF SINGULARITY
If we consider two identical infinite plates weakened by identical regular polygonal holes and submitted to the same loads p at infinity, if the one plate has its perforation with sharp corners and the other with rounded-off corners having a virtual singular distance rs of the corner Ak, it has been already established [lo] that the tangential a,-stresses at the apex of the rounded-off corner A, are equal to the sum of normal stresses (a, + a,) of the sharp corner appearing at the virtual singular point, that is along the bisector of the angle of the apex and at a distance from it equal to I,. Although this equality of stresses was proved only for a regular triangular perforation, its validity may be extended by the same procedure as this developed in Ref. [lo] for any regular polygon, since for the proof of this equality there was not used any particular property exclusively related to the triangular hole. The sum of the normal stresses (a, + [T,+J for a polar angle cp = 0” is related to the stress intensity factor KI and the order of singularity ;I by:
car +%)= 7K, =
PG (r/R,)”
(21)
Using relation (21) and the definition for SCF as related to the (6, + cr,)-sum of stresses it may be readily derived that:
K:
SCF =(r,/R,)"
(22)
828
P. S. THEOCARIS and L. PETROU
The variation of the SCF versus the virtual singular distance rs was given in Fig. 2 for parametric values of k. From these plottings and by using the method of least square curve-fitting the values of K: and A may be determined. This method is applied to a regular polygonal hole with k corners and for n-terms in the series expansion of the mapping equation. From these values the respective K:‘“’ and A(“)were determined. The exact values for Kf and 1 were derived by extending the expansion to n + cc. 0.275r
I
I
1
-----WM.__
k=lO
OJOO .-w____, ,9
0
16
8
r, IR, (~10~) Fig.3. The order of singularity I” at the close vicinity of the rounded-off corner of a polygonal hole with k = 3, 4,. . , 10, versus the virtual singular distance r,, normalized to R,.
2.1 r
I
I
I
1 krl0
--
-9
t
..-
0
8
16
24
28
r,/R, (~10~) + Fig.4. The SIF KY = K;/p(R,)’ at the close vicinity of the rounded-off corner of the polygonal hole with k = 3, 4,. . , 10, versus the virtual singular distance rs, normalized to R,.
Figures 3 and 4 present the values of K:” and 1” versus the virtual singular distance, rs, corresponding to n-terms and for k varying between three and ten. When n tends to infinity and (r,/R,) tends to zero the values of K,” and A” were derived by the tangential
The order of singularities
and the stress intensity
factors
near comers
of regular
polygonal
holes
829
Table 2. The values of the stress intensity factor and the order of singularity for a sharp corner of a regular polygonal hole and the respective sharp edge notch and their ratio for a number of sides k varvina between 3 and 10 k
SC’
K:
A
Icy
60 90 108 120 128.6 135 140 144
1.450 1.700 1.805 1.870 1.925 1.970 1.995 2.035
0.488 0.458 0.422 0.388 0.358 0.332 0.308 0.287
2.171 2.438 2.710 2.972 3.222 3.463 3.692 3.912
AN 0.488 0.456 a419 0.384 0.354 0.325 0.305 0.282
Ky/K: 1.497 1.434 1.501 1.647 1.674 1.751 1.869 1.922
continuation of the KY = fi(rs) and 2” = f&J curves and definition of their intercepts with the appropriate axes. Table 2 contains the values for K, and A, as derived by the above-described method. The values of the orders of singularity are in good agreement with the respective orders of singularities for the corresponding sharp V-notched plates under simple tension having the same opening angle [17]. The orders of singularities in sharp notches were determined also by the simple relationship given for the first time by Williams [2] expressed by: sin(A” - 1)(27c- 6,) = *(AN - l)sin(27r - 6,)
(23)
The coincidence of the values of the orders of singularities for the notched and perforated plates was expected since the SO is independent of the mode of loading of plate and depends only on the close to the singularity stress field. In these relations N-superscripts correspond to quantities referred to notches. On the other hand the SIFs for notched plates with depth equal to R, are given by the simple relationship [ 143: KTN = 2RfN sec4/‘(6,/2)
(24)
The KFN-values for the respective notched plates are included in Table 2. It is clear from these values that always KfN > K:. This inequality may be readily explained by the fact that, while the faces of the edge notch are free to undergo large opening displacements, the respective angular wedges in the polygonal hole are restrained to freely deform from the adjacent sides of the polygonal hole. In the general case when the direction of the external loading of the plate is oblique (IX# O,n/2) in all corners of the polygonal holes mixed-mode SIFs are operative. In this case we have: K = K, - iK,, =
[co(() - o(l),%
Along the bisector of the corner A, of the polygon, where the polar angle from the Oxaxis equals zero and the complex number i = p, the sum of the polar components of stresses is only depending on K, and the shearing stresses are depending on K,,. In a small distance I, along the bisector, in front of the singular point, the following relation exists: K, = @(a, + c,J = Re ~~ [w(p) - w(l)]‘%
1
(26)
Also K, is given by: K, = lim ri4ReE “-+C.X n
(27)
830
P. S. THEOCARIS and L. PETROU
From eqns (26), (27) it is obtained:
(28) Then: 21m F_“:[o(p) - w(l)]‘$j
= Im ?i_li 4r:$ n
(29)
and the normalized SIF K* may be written:
(30) where the order of singularity ,l is the one defined by the procedure already developed and is given in Fig. 5, versus the angle ?I,,of the corners of the polygonal hole. The K,- and K,,-components of SIF are derived from the real and imaginary parts of the right-hand side of eqn (26). Figure 6 presents the values of K: and K$SIFs versus the angle of inclination of the loading axis, ct, for polygonal holes with k = 3,4, 6 and 10 sides inscribed inside the same circle.
0
30
60
90
polygon
120 angle
150
180
(“1 ---)
Fig. 5. The order of singularity I at the close vicinity of a sharp corner of a polygonal hole, versus the polygonal angle 6,.
RESULTS
AND
DISCUSSION
A new method was presented in this paper for evaluation of the stress intensity factors and the orders of singularity, valid at the corners of regular polygonal holes perforated inside infinite plates submitted in simple tension at infinity. According to this method the real apices of the polygons were approximated by smoothened curves so that their corners presented high curvatures and therefore small radii of curvature. In this way the shape of the approximated polygons with rounded-off corners resembled closely the respective shape of a canonic hole with sharp corners and straight sides, the only perturbations appearing closely to their apices. This representation was achieved by using the Schwarz-Christoffel transformation, where a polygon with rounded-off corners was mapped into the unit circle. From the known mapping relationship the Muskhelishvili potential q(z) was determined. By increasing the number of terms in the series expansion of the potential function we can achieve any degree of similarity between the approximate representation and the exact one of the polygonal hole.
The order of singularities and the stress intensity factors near corners of regular polygonal holes
831
2.5
Fig. 6. The SIFs Kf and Kg near the close vicinity of a sharp corner of a regular polygonal hole, versus the loading angle GI,corresponding to this corner.
When an apex of the polygon was submitted to mode-1 deformation a correlation between the SCF existing at the rounded-off corner and the virtual singular distance, rs, and on the other hand, the SIF at the sharp corner and its order of singularity was established. This distance r, was defined along the bisector between the apex of the comer and the point inside the hole for which the mapped potential function in the transformed plane presented a zero first derivative. This was because it has been previously shown for the triangular hole [lo] that the first stress invariant, at a point placed at a distance I from the apex of a polygonal hole and along its bisector is equal to the only existing tangential stress, rs9, on the apex of the rounded-off corner of the same apex, for which the virtual singular distance r, becomes equal to r. The correlation coefficient for the method of least square curve-fitting was found to be of the order of 0.99995 that is almost equal to unity, fact which indicates that the correlation introduced is valid for every regular polygonal hole. The method introduced presents the advantage that with a few terms in the series expansion of the mapping function the K:‘“’ = f,(r,/R,) and A(“)= f,(r,/R,) may be graphically evaluated with high accuracy. Indeed, as the number of corners k of the polygonal hole is increasing, these curves tend to become straight lines (see Figs 3 and 4). Then, the number of terms n in the series expansion may be reduced considerably and the values of K: and 1 may be readily found with high accuracy. In the common case when the bisectors of the angles of the corners subtend an angle a different than zero or n/2 with the loading axis of the plate, both mode-1 and mode-II SIFs appear and these are evaluated by the real and imaginary parts of a convenient relationship. Figure 6 presents the variation of K: and Ka versus the angle c1of inclination of loading. It may be observed that as the z-angle is increasing from zero to IL/~ the respective corner is submitted first to compression and then to a mixed compression and shear mode. At a particular angle a, the compression term disappears and the corner is submitted to pure shear. For larger values of tl, tension is combined with shear, which for LY= 7r/2 disappears completely leaving the corner under mode-1 deformation. While all the Kg = f(a) curves present a parabolic shape with a maximum located at a = n/4 and values of Ksm,, decreasing as the number k of the sides of the polygon is
P. S. THEOCARIS
832
and L. PETROU
increasing, the K: = f(a) curves present a sigmoid shape, starting from negative values of K: for c1between zero and 30”, passing through zero at the zone CI= 15” and LY= 30” and then they become positive. All these K: = f(a) curves for parametric values of k lie between a lower bound corresponding to a sharp crack and an upper bound corresponding to a perfect circle, the straight parts of these sigmoid curves increasing in slope as k is increasing. The limiting value for the crack presents a zone between cx= 0” and CI= 15” with a compression type of loading which makes the lips of the crack to be completely closed in concordance to each other for simple compression and in discordance to each other lip when shear is also present. In this case the well known stick-slip phenomenon is developed which is typical in problems of geomechanics. A detailed study of the compression and shear loading modes of a crack was presented in a comparison study by the senior author
E171. The upper bound for the circular hole presents the case when the stress concentration factor and the respective stress intensity factor coincide and the sigmoid K: = f(a) curve passes from K: = - 1.0 for a = 0” to K: = 0 at a = 30” and K: = 3.0 for c1= n/2. For the circle the order of singularity becomes equal to zero. On the other hand the K,,-value is everywhere equal to zero, as it was expected. From the respective values of Table 2 a relationship may be derived for the respective SIFs for a regular polygonal hole inscribed inside a circle of radius R, and the corresponding edge notch of depth R, and the same angle 6, at its corner. The ratio of the respective stress intensity factors KTN/KT for various values of k is given in this table and indicates the stiffness which is developed behind the particular corner under mode I of the polygonal hole by the closing of the plate behind this apex, whereas in the respective V-edge notch the flanks of the notch are free to move. It is well known [lS] that this ratio for an edge crack and an internal one is given by KFN/K: = 1.1215. This means that the external loading for the internal crack, convenient to develop the same stress field as in the respective edge notch, should be 1.1215 times the load of the notched plate. This ratio for polygonal holes with increasing k is increasing, becoming almost equal to 2.0 for a regular decagonal hole. This high increase of the KFN/K:-ratio is due to the fact that a great part of the elastic strain energy is absorbed by the other singular corners of the hole. For example for the square hole with the diagonals of the square parallel and normal to the loading axis, the corners A, and C, (Fig. la) are submitted to simple tension where the B, and Bk-corners to simple compression. Then, the method developed in this paper yields the mixed-mode stress intensity factors K: and Kg at each corner of a regular polygonal hole in an infinite plate submitted to tension in an easy and accurate manner. REFERENCES [l] r2] c31 r41 L
.
[S] [6] [7] [S] [9] [lo] [11] [12] [13] [14] [15] [16] [17] [18]
M. L. WILLIAMS, J. Appl. Mech. 24, 109 (1957). M. L. WILLIAMS, J. Appl. Mech. lA, 526 (1952). G. SIH, P. PARIS and F.- ERDOGAN, J. Appl. Mech. 29, 306 (1962). N. I. MUSKHELISHVILI. Some Basic Problems of Mathematical Theory. of_ Elasticity. Noordhoff, Groningen Holland (1953). N. HASEBE and Y. KUTANDA, J. Engng Fracture Mech. 10, 215 (1978). G. R. IRWIN, Structural Mechanics. Pergamon Press Oxford (1960). P. C. PARIS and G. C. SIH, ASTM STP 381, 34 (1965). D. 0. HARRIS, ASME, Ser. D. 89-1, 49 (1967). G. B. SINCLAIR and M. KONDO, Inc. J. Mech. Sci. 26, 477 (1984). P. S. THEOCARIS and L. PETROU, Int. J. Fracture, 31, 135 (1986). B. CROSS and A. MENDELSON, Int. J. Fracture Mech. 8, 267 (1972). B. CROSS, Some plane problem elastostatic solution for plate having a V-notch. Ph.D. Thesis Case Western Reserve University (1970). D. J. HART, J. Engng Fracture Mech. 8, 747 (1976). P. S. THEOCARIS and N. I. IOAKIMIDES, J. Engng Fracture Mech. 10, 685 (1978). V. I. SMIRNOV, Course in Higher Mathematics 111. Costekhteoretizdat (1983). G. N. SAVIN, Stream Concentration Around Holes. Pergamon Press (1961). P. S. THEOCARIS, Engng Fract. Mech. 26, in press (1987). G. C. SIH, Mechanics of Fracture 1, Chap. 1. Noordhoff Press (1973). (Received
6 May 1986)
1~. J. Engng Sci. Vol. 25, No. 7, pp. 821-832, 1987 Printed in Great Britain. All rights reserved
THE ORDER OF SINGULARITIES AND THE STRESS INTENSITY FACTORS NEAR CORNERS OF REGULAR POLYGONAL HOLES P. S. THEOCARIS
and L. PETROU
Department of Theoretical and Applied Mechanics, The National Technical University of Athens, Athens GR-157 73, Greece Abstract-A new method was developed for the calculation of the order of singularity (SO) and the respective stress intensity factor @IF) near corners of regular polygonal holes perforated in elastic plates submitted to an overall tension at infinity. For the solution of the problem the Muskhelishvili potential function cp(z)was determined for an infinite plate weakened by a respective regular polygonal hole with rounded-off corners under tension. For each rounded off corner a virtual singular point was defined inside the hole along the bisector of the angle of the apex and at a distance depending on the radius of curvature of the corner. Then, the stress concentration factors (SCF) at the apices of these rounded-off corners were evaluated when their respective radii of curvature tended to zero. The corresponding stress-singularity and the SIF were afterwards derived, by correlating the SCF and the distance of the virtual singular point for each corner. By conveniently selecting the number of terms of the series development of q(z) the method can approach the real values for SO and SIF to any desired degree of accuracy. A comparison of the stress field around and near a corner of a polygonal hole in an infinite plate when this corner is submitted to mode-1 deformation and a V-notch in an infinite plate under tension indicated the similarities between the two affine problems.
INTRODUCTION
The stress field at the close vicinity of a singular point may be defined in a simple way by using the idea of the stress intensity factor (SIF) and the order of singularity (SO). Especially for edge V-notched infinite plates submitted to simple tension at infinity Williams [1,2] gave the SO at the bottom of the notch as the first eigenvalue of the eigenequation satisfying the boundary conditions of the plate. When the SO is known, the SIF can be calculated exactly by its definition given by Sih et al. [3]. This definition consisted in equating expressions for the sum of principal stresses in plane cracked plates given in terms of the SIF and of the derivative q’(z) of the Muskhelishvili potential function. In the vicinity of the crack tip q’(z) is expressed as a constant divided by (z - z,)* (see Ref. [4] pp. 496-498), where z is the complex coordinate and z1 the coordinate of the crack-tip. In this way the SIF may be defined as the limit for z -+ z1 of the expression (z - zJf$(z). Another possibility developed in the literature is to derive the SIF as the limiting value of the SCF at the apex of an equivalent notch with its corner rounded-off multiplied by the square root of the radius of curvature of the rounded apex, when this radius tends to zero [S]. Applications of this method were given in Refs. [6-81 and reasonable values for the SIFs are given. On the other hand, Sinclair and Kondo [9] have studied the possibility of determining in a realistic procedure the peak stresses at the bottoms of sharp re-entrant corners with zero curvature at their apices. In this method they approximated SCFs by defining the applied far-field stress at fracture of a plain plate divided by the applied stress for an identical plate with the stress raiser. Although the method yields, in general, working engineering estimates, it points out, in a general context, the necessity to seek realistic methods to define the exact local stress concentrations at sharp notches. Another method developed in Ref. [IO], where a relative problem of a triangular internal hole in a plate under tension was solved, was based in the following procedure. The plate with the triangular hole was conformally mapped into a perforated plate and the potential function q(z), truncated to a certain term, represented an equivalent triangular hole with its corners rounded off. The number of terms in the series expansion of q(z) defined the radius of curvature of the rounded apieces, with this radius diminishing as the number of terms was increased. The points along the bisectors of angles of the apices, where the 821
P. S. THEOCARIS
822
and L. PETROU
derivatives of the mapped function w(i) in the transformed plane were equal to zero, constituted the virtual singular points of the rounded vertices. A new coordinate system was introduced with its origin on the virtual singular point and the Ox-axis coinciding with the bisector of the corner. Then the SIF and the SO are taken immediately from the new function q(z) corresponding to the new coordinate system. In this paper the SCFs at the apices of the rounded corners were now combined with the distances r, of the virtual singular points from the apices. Then, the SIFs and SOS were derived when the values of r, tended to zero. This method was applied to any regular polygonal hole, whose apices were submitted to a combined in-plane loading creating mode-1 and -11 deformations at some apices. The conformal mapping of the regular polygon to a unit circle was achieved by using the Schwarz-Christoffel mapping function. The number of terms in the series expansion of the mapping function was selected to yield a virtual radius of curvature, r,, combined with its respective SCF for the rounded corner by using a least-square curve fitting with a correlation coefficient approximately equal to unity. Then, the respective SIF for the sharp corner was derived for the case when rs tends to zero. The SIFs for the corners of various regular polygons under a mode-1 deformation were compared with the respective sharp V-notched plates existing in the literature [ 11~ 131 and possessing the same values of orders of singularity as those given by the more accurate theory of Ref. [14]. Interesting results were derived from the confrontation of the two cases of stress raisers in flat plates. THE CONFORMAL MAPPING FUNCTION The stress distribution around a regular polygonal hole in an infinite plate induced to simple tension, as it is indicated in Fig. la, may be solved by conformally mapping the field outside the polygon on the interior of the unit circle by using the most efficient method of Schwarz-Christoffel [lS]. The appropriate function maps every polygon made up of a closed broken line to the unit circle. The appropriate relationship is expressed by [16]:
&p-l)
z = co,([) = R,
where the coefficients
fii are given by: 8, = 2(1 - k)(2 - 2k). . . (2 - (i - 1)k) I (1.2.3 .....i)k’(ki - 1)
and i = polygon polygon. origin 0
l,n,
(1)
with n tending mapped. The real If it is desired that and radius R, (Fig.
to infinity and k being the number of constant R, is a factor characterizing all the regular polygons are inscribed la) the real constant R, may take the
R.=Rdjl +ilBi}
(2) sides or corners of the the magnitude of the to the same circle with values:
(3)
Equation (1) maps a regular polygon with sharp corners if n tends to infinity. If a finite number of terms n in the series expansion of relation (1) is considered the corners of the polygon are rounded-off. In this case their radius of curvature pc depends on the number of terms n and it is given by the well known relationship: (.p P cn =
,xlyIl
+
p)3/2 _
,,,y’,
(4)
The order of singularities and the stress intensity factors near comers of regular polygonal holes
823
Fig. la. An infinite plate submitted to simple tension and weakened by any regular polygonal hole of any number of sides.
where x, y are the coordinates of each corner in the complex plane z = (x + iy) which are expressed by:
x =
/&cos(ki - 1)0
R, cost3 + f i=l
y = R, sin0 + i /?,sin(ki - 1)0
(5)
i=l
and the primes and double-primes denote differentiation with respect to the angle 8. The radius of curvature for the apex placed at 8 = 0” becomes:
R,
(ki -
t
l)& - 1
i=l PC”
(6)
= $1
W
-
l)Di
+
11
When n tends to infinity the pzi-radius becomes equal to zero and the corner becomes a sharp one. Figure la presents an infinite plate induced in tension and weakened by a regular polygonal hole A&Ct.. . C;B;A,, where k denotes the number of corners of each polygon, inscribed always inside the same circle of radius R,. Figure lb presents the details of an arbitrary corner of the polygon, in this case the corner A, of an equilateral triangular hole, mapped into the unit circle by eqn (1) with a finite number of terms n varying between n = 2 and 24. The increase of sharpness of the corner angle of the polygon as the number of terms n is increasing, as well as the good adaptation between the side of the
P. S. THEOCARIS and L. PETROU
Fig. lb. The rounded-off
A, of an equilateral triangular hole yielded from the Schwarz-Christoffel transformation with a finite number n of terms.
corner
Table 1. The normalized to the radius R, values of the radius of curvature pc the virtual singular distance rr of the rounded-off corner and the values of stress concentration factors there, for polygonal holes with number of sides k = 3, 4, 6 and 9, for parametric values of n (n = 2 + 24) k=3
v 2 3 4 5 6 12 24
PJR, 0.0126 0.0067 0.0042 0.0029 0.0021 0.0007 0.0002
k=4
rJR,
SCF
0.0060 0.0032 0.0020 0.0014 0.0010 0.0003 O.ooOl
17.39 23.95 30.03 35.81 41.37 71.93 125.66
k=6
k=9
q
PJR,
rJR,
SCF
q
PJK
‘JR<
SCF
q
PJR,
rJR,
SCF
2 3 4 5 6 12 18
0.352 0.201 0.135 0.098 0.076 0.027 0.015
0.0159 0.0091 0.0061 0.0044 0.0034 0.0012 O.ooO7
11.21 14.60 17.62 20.40 23.00 36.51 47.95
2 3 4 5 6 9 12
0.0737 0.0457 0.0321 0.0243 0.0193 0.0115 0.0080
0.0294 0.0182 0.0127 0.0097 0.0077 0.0046 0.0031
7.28 8.83 10.15 11.32 12.38 15.13 17.46
2 3 4 5 6 7 8
0.0640 0.0462 0.0357 0.0289 0.0240 0.0205 0.0178
0.0373 0.0242 0.0176 0.0137 0.0111 0.0093 0.0079
5.45 6.26 6.92 7.48 7.98 8.42 8.83
triangle and the rounded-off corner is apparent, indicating that the Schwarz-Christoffel transformation is the most suitable for such mappings. The values of the radii of curvature pen normalized to the radius R, of the circumscribed circle are given in Table 1 for k = 3, 4, 6 and 9. In the same table the normalized values of the virtual distances r,, and the respective stress concentration factors are also included. The respective virtual points inside the hole and along the bisectors of the angles of the apices are defined by the value of rsn satisfying the equation: I sn =
Rc- w,(P*)
(74
where p* is yielded from equation:
P-4
IwxP*)l= 0
It may be observed that the radii of curvature of the corners of a regular triangle become practically zero when n takes higher values than twelve (n 2 12). For a polygon with a k-number of corners its apex angle Bk is given by:
d,=
(1-i > II
(8)
The order of singularities and the stress intensity factors near comers of regular polygonal holes
825
When k tends to infinity, the polygon tends to coincide with its circumscribed circle. As k is increasing a larger number of terms n is needed in the series expansion to attain sharp corners.
EVALUATION
OF
STRESS
CONCENTRATION
FACTOR
plate submitted to simple tensile stresses at infinity, p, and perforated with a regular polygonal hole whose center coincides with the origin of the coordinate Oxy-system, in the case when the hole is free from externally applied loads, the complex potential function q(z) is given by [4]: For
an infinite
7 + q*(z)
cp(z)=
Function
q(z) in the mapped
plane [ is expressed
(9)
by:
di) = $cw(m + cpo(i) In this relation
cpO(c)is a function
expressed
(10)
by:
(11)
cpo(i)= f a,rl = f (Rear + iZ,a,)i r=l
r=1
where a, are complex constants. The boundary condition of the basic stress state is given by:
f(: + if! = -
![w(cr) - eZi%o(b)]
(12)
where c( denotes the angle subtended between the loading axis and the Ox-axis and r~ represents the points along the circumference of the unit circle in the transformed plane. For the determination of the unknown cp,([)-function use was made of the relationship [16]:
(13)
where l now is a point inside the unit circle y. If relations (1) and (11) are introduced relation (10) yield the following expression for the function (p,(c):
into
(km-1) +
c
r=l
Moreover,
the integral
of the right-hand
4
(14)
side of eqn (13) yields:
(15)
826
whereas
P. S. THEOCARIS
the left-hand
and L. PETROU
side of eqn (13) becomes:
Rear, + iIma,)l:
(16)
where P(i) is given by: p(i) =
-P(n-i+l)
-
P(1)&_,[2(i
‘. - P(i - 1)/3(l)(k - 1)
- 1)k - 21
with: P(l) = -B, Comparing now the respective real and imaginary coefficients of the respective terms of [ in eqns (15) and (16) we obtain a system of 2(kn - 1)-equations, which when solved yields the values of the real and imaginary parts of a,. Since along the boundaries of the polygonal hole the normal to these boundaries component of the normal stresses is zero it remains from the first stress invariant only the g,-stress parallel to the boundaries of the hole. Then, it is valid that [4]:
(17)
This value for a9 becomes equal to infinity for w’(i) polygonal hole. For rounded-off apices of the hole tending to infinity. This maximum value normalized p, yields the SCF at the respective corner A,. This
SCF, = 4Re
= 0, that is at the sharp corners of the this stress becomes maximum without to the applied tensile stress at infinity, SCF is given by:
cplxl) 1 40) ~
L
(18)
where:
- 1 + 2 (ki - l)pi i=l
o;(l)
= R,
+ ““i”
r{Rea, + iIma,}
(19)
r=l
- 1+ i
(ki - l)fli
(20)
i=l
Equation (19) indicates that the q’(l)-function becomes a real number when the (Ima,)term is equal to zero. This happens for loading angles c( = 7c/2 and a = 0”. In these two cases the A,-corner is subjected to a mode-1 deformation. For any other loading angle, a, the A,-corner is subjected to a mixed mode deformation except for some special values of the a-angle for which the real part of q’(l) becomes zero, when the boundary at this particular point is deformed exclusively under mode II. Let us assume that a regular polygonal perforation in an infinite plate which is submitted to a simple tension load has the bisector of the angle of its A,-corner subtending an angle a = 71/2 with the loading axis. The SCF corresponding to the respective rounded-off comer, normalized to p, is given in Table 1. When the number of terms n is increased the radius ofcurvature of the apex is decreased and the SCF increased. Figure 2 presents the variation
The order of singularities and the stress intensity factors near corners of regular polygonal holes
827
of the SCF, versus the virtual singular distance r, normalized to the radius R, of the circumscribed circle, when the quantity k of the corners in the polygon is increased from three to ten. It may be observed from this plotting that for the same virtual singular distance r, the SCF is decreasing when k is increased.
0
4
8
12
16
20
24
28
rSIR, (~10~1 e Fig. 2. The SCF of the rounded-off corner A, of the polygonal hole with k = 3, 4,. , , 10 in an infinite plate under tension, when the direction of loading a = 90”, versus the distance rS of the virtual singular point normalized to the radius of the circumscribed circle R,.
EVALUATION
OF SIF AND
THE ORDER
OF SINGULARITY
If we consider two identical infinite plates weakened by identical regular polygonal holes and submitted to the same loads p at infinity, if the one plate has its perforation with sharp corners and the other with rounded-off corners having a virtual singular distance rs of the corner Ak, it has been already established [lo] that the tangential a,-stresses at the apex of the rounded-off corner A, are equal to the sum of normal stresses (a, + a,) of the sharp corner appearing at the virtual singular point, that is along the bisector of the angle of the apex and at a distance from it equal to I,. Although this equality of stresses was proved only for a regular triangular perforation, its validity may be extended by the same procedure as this developed in Ref. [lo] for any regular polygon, since for the proof of this equality there was not used any particular property exclusively related to the triangular hole. The sum of the normal stresses (a, + [T,+J for a polar angle cp = 0” is related to the stress intensity factor KI and the order of singularity ;I by:
car +%)= 7K, =
PG (r/R,)”
(21)
Using relation (21) and the definition for SCF as related to the (6, + cr,)-sum of stresses it may be readily derived that:
K:
SCF =(r,/R,)"
(22)
828
P. S. THEOCARIS and L. PETROU
The variation of the SCF versus the virtual singular distance rs was given in Fig. 2 for parametric values of k. From these plottings and by using the method of least square curve-fitting the values of K: and A may be determined. This method is applied to a regular polygonal hole with k corners and for n-terms in the series expansion of the mapping equation. From these values the respective K:‘“’ and A(“)were determined. The exact values for Kf and 1 were derived by extending the expansion to n + cc. 0.275r
I
I
1
-----WM.__
k=lO
OJOO .-w____, ,9
0
16
8
r, IR, (~10~) Fig.3. The order of singularity I” at the close vicinity of the rounded-off corner of a polygonal hole with k = 3, 4,. . , 10, versus the virtual singular distance r,, normalized to R,.
2.1 r
I
I
I
1 krl0
--
-9
t
..-
0
8
16
24
28
r,/R, (~10~) + Fig.4. The SIF KY = K;/p(R,)’ at the close vicinity of the rounded-off corner of the polygonal hole with k = 3, 4,. . , 10, versus the virtual singular distance rs, normalized to R,.
Figures 3 and 4 present the values of K:” and 1” versus the virtual singular distance, rs, corresponding to n-terms and for k varying between three and ten. When n tends to infinity and (r,/R,) tends to zero the values of K,” and A” were derived by the tangential
The order of singularities
and the stress intensity
factors
near comers
of regular
polygonal
holes
829
Table 2. The values of the stress intensity factor and the order of singularity for a sharp corner of a regular polygonal hole and the respective sharp edge notch and their ratio for a number of sides k varvina between 3 and 10 k
SC’
K:
A
Icy
60 90 108 120 128.6 135 140 144
1.450 1.700 1.805 1.870 1.925 1.970 1.995 2.035
0.488 0.458 0.422 0.388 0.358 0.332 0.308 0.287
2.171 2.438 2.710 2.972 3.222 3.463 3.692 3.912
AN 0.488 0.456 a419 0.384 0.354 0.325 0.305 0.282
Ky/K: 1.497 1.434 1.501 1.647 1.674 1.751 1.869 1.922
continuation of the KY = fi(rs) and 2” = f&J curves and definition of their intercepts with the appropriate axes. Table 2 contains the values for K, and A, as derived by the above-described method. The values of the orders of singularity are in good agreement with the respective orders of singularities for the corresponding sharp V-notched plates under simple tension having the same opening angle [17]. The orders of singularities in sharp notches were determined also by the simple relationship given for the first time by Williams [2] expressed by: sin(A” - 1)(27c- 6,) = *(AN - l)sin(27r - 6,)
(23)
The coincidence of the values of the orders of singularities for the notched and perforated plates was expected since the SO is independent of the mode of loading of plate and depends only on the close to the singularity stress field. In these relations N-superscripts correspond to quantities referred to notches. On the other hand the SIFs for notched plates with depth equal to R, are given by the simple relationship [ 143: KTN = 2RfN sec4/‘(6,/2)
(24)
The KFN-values for the respective notched plates are included in Table 2. It is clear from these values that always KfN > K:. This inequality may be readily explained by the fact that, while the faces of the edge notch are free to undergo large opening displacements, the respective angular wedges in the polygonal hole are restrained to freely deform from the adjacent sides of the polygonal hole. In the general case when the direction of the external loading of the plate is oblique (IX# O,n/2) in all corners of the polygonal holes mixed-mode SIFs are operative. In this case we have: K = K, - iK,, =
[co(() - o(l),%
Along the bisector of the corner A, of the polygon, where the polar angle from the Oxaxis equals zero and the complex number i = p, the sum of the polar components of stresses is only depending on K, and the shearing stresses are depending on K,,. In a small distance I, along the bisector, in front of the singular point, the following relation exists: K, = @(a, + c,J = Re ~~ [w(p) - w(l)]‘%
1
(26)
Also K, is given by: K, = lim ri4ReE “-+C.X n
(27)
830
P. S. THEOCARIS and L. PETROU
From eqns (26), (27) it is obtained:
(28) Then: 21m F_“:[o(p) - w(l)]‘$j
= Im ?i_li 4r:$ n
(29)
and the normalized SIF K* may be written:
(30) where the order of singularity ,l is the one defined by the procedure already developed and is given in Fig. 5, versus the angle ?I,,of the corners of the polygonal hole. The K,- and K,,-components of SIF are derived from the real and imaginary parts of the right-hand side of eqn (26). Figure 6 presents the values of K: and K$SIFs versus the angle of inclination of the loading axis, ct, for polygonal holes with k = 3,4, 6 and 10 sides inscribed inside the same circle.
0
30
60
90
polygon
120 angle
150
180
(“1 ---)
Fig. 5. The order of singularity I at the close vicinity of a sharp corner of a polygonal hole, versus the polygonal angle 6,.
RESULTS
AND
DISCUSSION
A new method was presented in this paper for evaluation of the stress intensity factors and the orders of singularity, valid at the corners of regular polygonal holes perforated inside infinite plates submitted in simple tension at infinity. According to this method the real apices of the polygons were approximated by smoothened curves so that their corners presented high curvatures and therefore small radii of curvature. In this way the shape of the approximated polygons with rounded-off corners resembled closely the respective shape of a canonic hole with sharp corners and straight sides, the only perturbations appearing closely to their apices. This representation was achieved by using the Schwarz-Christoffel transformation, where a polygon with rounded-off corners was mapped into the unit circle. From the known mapping relationship the Muskhelishvili potential q(z) was determined. By increasing the number of terms in the series expansion of the potential function we can achieve any degree of similarity between the approximate representation and the exact one of the polygonal hole.
The order of singularities and the stress intensity factors near corners of regular polygonal holes
831
2.5
Fig. 6. The SIFs Kf and Kg near the close vicinity of a sharp corner of a regular polygonal hole, versus the loading angle GI,corresponding to this corner.
When an apex of the polygon was submitted to mode-1 deformation a correlation between the SCF existing at the rounded-off corner and the virtual singular distance, rs, and on the other hand, the SIF at the sharp corner and its order of singularity was established. This distance r, was defined along the bisector between the apex of the comer and the point inside the hole for which the mapped potential function in the transformed plane presented a zero first derivative. This was because it has been previously shown for the triangular hole [lo] that the first stress invariant, at a point placed at a distance I from the apex of a polygonal hole and along its bisector is equal to the only existing tangential stress, rs9, on the apex of the rounded-off corner of the same apex, for which the virtual singular distance r, becomes equal to r. The correlation coefficient for the method of least square curve-fitting was found to be of the order of 0.99995 that is almost equal to unity, fact which indicates that the correlation introduced is valid for every regular polygonal hole. The method introduced presents the advantage that with a few terms in the series expansion of the mapping function the K:‘“’ = f,(r,/R,) and A(“)= f,(r,/R,) may be graphically evaluated with high accuracy. Indeed, as the number of corners k of the polygonal hole is increasing, these curves tend to become straight lines (see Figs 3 and 4). Then, the number of terms n in the series expansion may be reduced considerably and the values of K: and 1 may be readily found with high accuracy. In the common case when the bisectors of the angles of the corners subtend an angle a different than zero or n/2 with the loading axis of the plate, both mode-1 and mode-II SIFs appear and these are evaluated by the real and imaginary parts of a convenient relationship. Figure 6 presents the variation of K: and Ka versus the angle c1of inclination of loading. It may be observed that as the z-angle is increasing from zero to IL/~ the respective corner is submitted first to compression and then to a mixed compression and shear mode. At a particular angle a, the compression term disappears and the corner is submitted to pure shear. For larger values of tl, tension is combined with shear, which for LY= 7r/2 disappears completely leaving the corner under mode-1 deformation. While all the Kg = f(a) curves present a parabolic shape with a maximum located at a = n/4 and values of Ksm,, decreasing as the number k of the sides of the polygon is
P. S. THEOCARIS
832
and L. PETROU
increasing, the K: = f(a) curves present a sigmoid shape, starting from negative values of K: for c1between zero and 30”, passing through zero at the zone CI= 15” and LY= 30” and then they become positive. All these K: = f(a) curves for parametric values of k lie between a lower bound corresponding to a sharp crack and an upper bound corresponding to a perfect circle, the straight parts of these sigmoid curves increasing in slope as k is increasing. The limiting value for the crack presents a zone between cx= 0” and CI= 15” with a compression type of loading which makes the lips of the crack to be completely closed in concordance to each other for simple compression and in discordance to each other lip when shear is also present. In this case the well known stick-slip phenomenon is developed which is typical in problems of geomechanics. A detailed study of the compression and shear loading modes of a crack was presented in a comparison study by the senior author
E171. The upper bound for the circular hole presents the case when the stress concentration factor and the respective stress intensity factor coincide and the sigmoid K: = f(a) curve passes from K: = - 1.0 for a = 0” to K: = 0 at a = 30” and K: = 3.0 for c1= n/2. For the circle the order of singularity becomes equal to zero. On the other hand the K,,-value is everywhere equal to zero, as it was expected. From the respective values of Table 2 a relationship may be derived for the respective SIFs for a regular polygonal hole inscribed inside a circle of radius R, and the corresponding edge notch of depth R, and the same angle 6, at its corner. The ratio of the respective stress intensity factors KTN/KT for various values of k is given in this table and indicates the stiffness which is developed behind the particular corner under mode I of the polygonal hole by the closing of the plate behind this apex, whereas in the respective V-edge notch the flanks of the notch are free to move. It is well known [lS] that this ratio for an edge crack and an internal one is given by KFN/K: = 1.1215. This means that the external loading for the internal crack, convenient to develop the same stress field as in the respective edge notch, should be 1.1215 times the load of the notched plate. This ratio for polygonal holes with increasing k is increasing, becoming almost equal to 2.0 for a regular decagonal hole. This high increase of the KFN/K:-ratio is due to the fact that a great part of the elastic strain energy is absorbed by the other singular corners of the hole. For example for the square hole with the diagonals of the square parallel and normal to the loading axis, the corners A, and C, (Fig. la) are submitted to simple tension where the B, and Bk-corners to simple compression. Then, the method developed in this paper yields the mixed-mode stress intensity factors K: and Kg at each corner of a regular polygonal hole in an infinite plate submitted to tension in an easy and accurate manner. REFERENCES [l] r2] c31 r41 L
.
[S] [6] [7] [S] [9] [lo] [11] [12] [13] [14] [15] [16] [17] [18]
M. L. WILLIAMS, J. Appl. Mech. 24, 109 (1957). M. L. WILLIAMS, J. Appl. Mech. lA, 526 (1952). G. SIH, P. PARIS and F.- ERDOGAN, J. Appl. Mech. 29, 306 (1962). N. I. MUSKHELISHVILI. Some Basic Problems of Mathematical Theory. of_ Elasticity. Noordhoff, Groningen Holland (1953). N. HASEBE and Y. KUTANDA, J. Engng Fracture Mech. 10, 215 (1978). G. R. IRWIN, Structural Mechanics. Pergamon Press Oxford (1960). P. C. PARIS and G. C. SIH, ASTM STP 381, 34 (1965). D. 0. HARRIS, ASME, Ser. D. 89-1, 49 (1967). G. B. SINCLAIR and M. KONDO, Inc. J. Mech. Sci. 26, 477 (1984). P. S. THEOCARIS and L. PETROU, Int. J. Fracture, 31, 135 (1986). B. CROSS and A. MENDELSON, Int. J. Fracture Mech. 8, 267 (1972). B. CROSS, Some plane problem elastostatic solution for plate having a V-notch. Ph.D. Thesis Case Western Reserve University (1970). D. J. HART, J. Engng Fracture Mech. 8, 747 (1976). P. S. THEOCARIS and N. I. IOAKIMIDES, J. Engng Fracture Mech. 10, 685 (1978). V. I. SMIRNOV, Course in Higher Mathematics 111. Costekhteoretizdat (1983). G. N. SAVIN, Stream Concentration Around Holes. Pergamon Press (1961). P. S. THEOCARIS, Engng Fract. Mech. 26, in press (1987). G. C. SIH, Mechanics of Fracture 1, Chap. 1. Noordhoff Press (1973). (Received
6 May 1986)