- Email: [email protected]
Stress intensity factors of a rhombic hole with symmetric cracks under uniform transverse thin plate bending
Engineering Fracture Mechanics 156 (2016) 16–24
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Stress intensity factors of a rhombic hole with symmetric cracks under uniform transverse thin plate bending Jiro Iida a, Norio Hasebe b,⇑ a b
Nagoya Highway Corporation, Marunouchi 1, Nakaku, Nagoya 460, Japan Department of Civil Engineering and System Management, Nagoya Institute of Technology, Gokisocho, Showaku, Nagoya 466, Japan
a r t i c l e
i n f o
Article history: Received 30 November 2015 Received in revised form 12 January 2016 Accepted 1 February 2016 Available online 6 February 2016 Keywords: Stress intensity factor Kirchhoff thin plate Transverse bending Rhombic hole Crack Mapping function Complex stress function Approximate equation
a b s t r a c t A thin plate bending problem (Kirchhoff thin plate) is solved analytically for an infinite plate with a rhombic hole with symmetric cracks at the both corners. The loading is a uniform transverse bending at infinity. The stress distribution and the stress intensity factors of the bending are obtained for a range of corner angles and crack lengths of the rhombic hole. Approximate expressions of the stress intensity factor are proposed for some corner angles and Poisson’s ratios using the stress distribution expressed by the roots of the characteristic equation of the sharp notched plate before the crack initiation. The precision of the expressions are investigated. Characteristic factors for the approximate expressions are stated. A rational mapping function of a sum of fractional expressions and complex stress functions are used for the stress analysis. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Fractures of plate structure cause under in plane loading, out of bending and twisting, and their combined loadings. The crack tip stress fields combined in plane and out of plane loadings are a superposition of the plane stress and plate theory fields [1]. Then stress intensity factor of Mode I, K I , is assumed to be large enough to eliminate crack closure. Therefore, the contact of the crack surface is not considered. Hui and Zehnder [2] studied the fracture of thin plates subjected to bending and twisting moments. Zehnder and Viz [3] also reviewed fracture mechanics of thin plates and shell combined membrane, bending and twisting loadings. They stated that fracture mechanics of the Kirchhoff thin plate theory for small deflection is valid outside the small scale yielding. Therefore, a linear fracture mechanics is useful for thin plate with cracks subjected to flexural deformation. Many crack problems of a classical thin plate (Kirchhoff thin plate) for various shapes and boundary conditions, for example, mixed boundary of stress and displacement, have been solved, and the stress intensity factors were tabulated and also the references were described in [4–8]. In practical application, an approximate expression to obtain the stress intensity factor is useful, and it is desired to be simple and available. The results of this article were developed for the plane elastic problem by Lukas and Klesnil [9] using the radius of curvature at the end of the notch, by Kobayashi [10] using equivalent length of the crack and the stress
⇑ Corresponding author. Tel.: +81 52 876 5015. E-mail address: [email protected] (N. Hasebe). http://dx.doi.org/10.1016/j.engfracmech.2016.02.002 0013-7944/Ó 2016 Elsevier Ltd. All rights reserved.
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
17
Nomenclature a b A D FB F 0B Mx ; My M xy M0 SIF kB ; kS m1 ; m2 f h ; f h2 b d
half-length of a diagonal of a rhombic hole crack length factors of Eqs. (19) and (20) flexural rigidity non-dimensional stress intensity factor non-dimensional stress intensity factor of an approximate expression bending moments torsional moment uniform bending moment at infinity stress intensity factor stress intensity factors for bending and torsional moments roots of characteristic Eq. (18) coefficients of stress distribution near corner (Eq. (17)) coordinate regarding corner angle on a unit circle parameter of corner angle of a rhombic hole m Poisson’s ratio /ðfÞ; wðfÞ stress functions xðfÞ mapping function
distribution before the crack initiation, by Nisitani et al. [11] using the values of stress concentration at the end of the notch and the stress distribution before the crack initiation, and by Hasebe and Iida [12,13] using the stress distribution before the crack initiation at the angular corner. Approximate expressions for the crack initiating at the angular corner under out-ofplane bending seem not to be proposed many. Hasebe and Iida [14] reported the approximate expressions for the triangular notch with a crack on the rim of the semi-infinite plate. In the present paper, the stress distribution and the stress intensity factors are analyzed for the symmetric cracks initiating from a rhombic hole in an infinite plate for some corner angles and Poisson’s ratios. The loading condition is uniform out-of-plane bending at infinity. A mapping function of a sum of fractional expressions [15,16] and complex stress functions [17] are used for the stress analysis. Thin plates with a rhombic hole are one of typical thin plate structures and are found in many structures. Approximate expressions for the crack initiating at the angular corner of the rhombic hole in the out-ofplane bending problem are proposed. The approximate expressions are represented as the product of a certain coefficient and the stress distribution before the crack initiation. The stress distributions for the corner angles are expressed in the general form of an infinite series using the roots obtained by the characteristic equation. Then the approximate expressions of the stress distribution are derived by using the first few terms of the series and are compared with the analytical values, and the range of the application is investigated. The relationships between the stress concentration at the notch tip before the crack initiation and the stress intensity factors are also investigated. 2. Mapping function and method of analysis An infinite region with symmetric cracks at the both corners of a rhombic hole is mapped to the infinite region outside a unit circle (see Fig. 1). The mapping function can be obtained by the Schwartz–Christoffel’s transformation:
Fig. 1. Rhombic hole with cracks and unit circle.
18
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
Z z ¼ xðfÞ ¼ k
ðf 1Þd ðf þ 1Þd ðf iÞðf þ iÞ ðf þ
d=2 ebi Þ ðf
ebi Þ
d=2
ðf þ ebi Þ
d=2
ðf ebi Þ
d=2 2
f
ð1Þ
df
where d and b are parameters depending on the corner angle of the rhombic hole and the crack length, respectively. When b ¼ p=2, the hole is a rhombic hole without crack. ‘‘k” before the integral symbol is a parameter depending on the expansion or reduction of the size of the rhombic hole. Using (1), a rational mapping function is formed as a sum of fractional expressions. Eq. (1) is divided into a fast convergent term and slow ones:
z=k ¼
ð1d=2Þ ð1d=2Þ ð1d=2Þ A1 ebi B1 ebi C 1 ebi 1 þ þ 1 þ 1 þ ð1 d=2Þebi f ð1 d=2Þebi f ð1 d=2Þebi f 8 9 d d ð11=fÞ ð1þ1=fÞ ðfiÞðfþiÞ Z < ð1d=2Þ = d=2 d=2 d=2 d=2 D1 ebi ð1þebi =fÞ ð1ebi =fÞ ð1þebi =fÞ ð1ebi =fÞ f2 þ 1 þ df bi A1 B1 C 1 D1 : ; ð1 d=2Þe f d=2 d=2 d=2 d=2 f
2
ð1þebi =fÞ
f
2
ð1ebi =fÞ
f
2
ð1þebi =fÞ
2
f
ð2Þ
ð1ebi =fÞ
where
A1 ¼
1 e2bi
1 þ e2bi d=2
C1 ¼
d
2d=2 ð1 e4bi Þ d 1 e2bi 1 þ e2bi 2d=2 ð1 e4bi Þ
d=2
;
B1 ¼
1 e2bi
d=2
;
d 1 þ e2bi
D1 ¼
2d=2 ð1 e4bi Þ d 1 e2bi 1 þ e2bi 2d=2 ð1 e4bi Þ
ð3a; b; c; dÞ
d=2
The first term in (2) is expressed by the following fractional expression:
! ð1d=2Þ 12 X ebi F j 1þ ¼1þ þ Fj f 1 þ f j ebi =f j¼1
ð4Þ
The procedure to determine F j and f j is described in [15,16,18]. The second, third, and fourth terms in (2) can be expressed by using the same values F j and f j (see (6)). The fifth integral term in (2) is expressed by the following expression:
The fifth integral term in ð2Þ ¼
n X Gj g j =f j¼1
ð5Þ
1 g j =f2
The procedure to determine Gj and g j is also described in [15,16,18]. Consequently, the rational mapping function is derived as follows:
2
(
12 X
F j 1þf j ebi =f
A1 6 ð1d=2Þe bi 6
)
(
12 X
F j 1þf j ebi =f
)
3
7 þ 1þ þ Fj 1þ þ Fj 7 j¼1 j¼1 7 6 6 ( ) ( )7 48þ2n 12 12 6 7 X Ek X X F F 7 6 C 1 D1 j j z ¼ xðfÞ ¼ k6 þ ð1d=2Þe 1þ 1þ þ ð1d=2Þe 7 E0 f þ bi bi =f þ F j bi bi =f þ F j 1þf e 1þf e j j 7 6 fk f k¼1 j¼1 j¼1 7 6 7 6 n 7 6 X G g =f j j 5 4þ þf 1g =f2 j¼1
B1 ð1d=2Þebi
ð6Þ
j
In the present paper, the number of n in (6) is adopted for n = 11, 12, 13, and 14. The radius of curvature q at the crack tip is not equal to zero, because (6) is a rational function. However, they are very small and the ratios with the crack length are
q=b ¼ 107 —109 . The stress analysis is carried out by using the rational mapping function (6) and complex stress functions. Since the analytical method was described in [17,14,16], the results are only shown here. Denoting regular complex stress functions outside the unit circle by /ðfÞ and wðfÞ, the bending moments M x ; M y and the torsional moment M xy , and the shear forces N x ; N y are expressed as follows:
M x þ M y ¼ 4Dð1 þ mÞRe½/0 ðfÞ=x0 ðfÞ h i. 0 M y M x þ 2iMxy ¼ 2Dð1 mÞ xðfÞf/0 ðfÞ=x0 ðfÞg þ w0 ðfÞ x0 ðfÞ 0
ð7a; b; cÞ
0
Nx iN y ¼ 4Df/ ðfÞ=x0 ðfÞg =x0 ðfÞ where D ¼ Eh =½12ð1 m2 Þ = flexural rigidity, E = Young’s modulus, h = the thickness of the plate, and v = Poisson’s ratio. Re [.] indicates the real part of [.], and the bar of xðfÞ does the conjugate function. The stress components in the orthogonal curvilinear coordinates expressed by the mapping function (6) are 3
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
19
Mr þ Mh ¼ Mx þ My M h M r þ 2iMrh ¼ e2ic ðMy Mx þ 2iMxy Þ
ð8a; b; c; dÞ
Nr iNh ¼ eic ðNx iNy Þ eic ¼ fx0 ðfÞ=jfx0 ðfÞj
Uniform transverse bending M 0 applies at infinity in the x-axis, and the traction is free on the boundary. Then the following complex stress functions /ðfÞ and wðfÞ can be derived [16]: 48þ2n X C k Ak mþ3 M0 xðfÞ E0 f E0 =f /ðfÞ ¼ m1 f f 2Dð1 m Þ k k¼1
wðfÞ ¼
48þ2n X k¼1
0
ð9a; bÞ
C k Ak f02 xð1=fÞ 0 M0 k 0 xðfÞ E0 f E0 =f / ðfÞ þ x ðfÞ 2Dð1 mÞ f f0k
ðf0k Þ
where / Ak ; C k Ek =x0 ðf0k Þ with f0k 1=fk ðk ¼ 1; 2; 3; . . . ; 48 þ 2nÞ. In order to determine the unknown coefficients Ak , (9a) is differentiated once and f ¼ f0k is substituted, then the real and imaginary parts of Ak can be determined by solving 2 (48 + 2n) simultaneous equations, using /0 ðf0k Þ Ak . When a traction free boundary exists, wðfÞ can be obtained by analytic continuation [19]:
wðfÞ ¼
mþ3 xð1=fÞ 0 /ð1=fÞ 0 / ðfÞ m1 x ðfÞ
ð10Þ
Fig. 2 shows an example of the stress distribution for the corner angle dp ¼ 90 , the crack length b=a ¼ 0:5 and Poisson’s ratio m ¼ 0:25. M h is the bending moment along the boundary. 3. Stress intensity factor For the thin plate bending problem, the bending moments M x ; M y and the torsional moment M xy near the crack tip are expressed as follows [5]:
2 h h M x þ M y ¼ pffiffiffiffiffi kB cos kS sin 2 2 2r 1 pffiffiffiffiffi fð7 þ mÞkB þ ið5 þ 3mÞkS gei h=2 þ ð1 mÞðkB ikS Þe5i h=2 M y M x þ 2iMxy ¼ 2ð1 þ mÞ 2r
ð11a; bÞ
where r and h are the polar coordinates measured from the crack tip; kB and kS are the stress intensity factors (SIF) for bending and shearing, respectively, and are calculated by using the complex stress function /ðfÞ and the mapping function xðfÞ [14,15],
kB ikS ¼ 2Dð1 þ mÞeik=2 /0 ðr1 Þ
.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x00 ðr1 Þ
Fig. 2. Stress distributions for dp ¼ 90 , b/a = 0.5 and
ð12Þ
v = 0.25.
20
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
where r1 is the coordinates corresponding to the crack tip on the unit circle; k is the angle between the crack and the x-axis. In the present paper, r1 ¼ i; k ¼ p=2 for two crack tips and kS ¼ 0 due to the symmetry. The stress components rx ; ry ; sxy are expressed by
12t
12t 12t M x ; ry ¼ 3 My ; sxy ¼ 3 M xy 3 h h h 2 2 3ðh 4t 2 Þ 3ðh 4t2 Þ ¼ Nx ; syz ¼ Ny 3 3 2h 2h
rx ¼ sxz
ð13a; b; c; d; e; fÞ
where the parameter t is a value between the limits h=2 6 t 6 þh=2 and represents the distance of the point from the midplane in the un-deformed states. The following non-dimensional SIF is defined:
FB ¼
3þm kB pffiffiffiffiffiffiffiffiffiffiffi 1 þ m M0 a þ b
ð14Þ
The values of F B are calculated for dp ¼ 60 ; 90 and 120°, and m ¼ 0; 0:25 and 0.5 and their values are shown in Table 1 and Fig. 3. As the crack exists on the y axis in the present paper, the bending moments in the vicinity of the crack can be represented by
rffiffiffiffiffiffiffiffiffiffiffi aþb 2r rffiffiffiffiffiffiffiffiffiffiffi m 1 kB m 1 aþb pffiffiffiffiffi ¼ My ¼ F M m þ 1 2r m þ 3 B 0 2r
Mx ¼
3 þ m kB pffiffiffiffiffi ¼ F B M 0 1 þ m 2r
ð15Þ
When dp ¼ 0 , the crack with the length 2(a þ bÞ is obtained and then F B is
F B ¼ 1:0
ð16Þ
The hole does not affect for the long crack and F B approaches asymptotically to the value of the crack length 2(a þ bÞ. The asymptotic tendency is faster as the corner angle is smaller. Therefore, when the crack length becomes to some degree, the values of F B are approximated by value of the crack length 2(a þ bÞ. From Table 1 and Fig. 3, it is clear that F B becomes larger as Poisson’s ratio m is larger for the same crack length. However, the values of F B are almost the same as those of the long crack, and the effect of Poisson’s ratio on the F B values is little. The effect of Poisson’s ratio m on the value F B depends on the factor ð1 þ mÞ=ð3 þ mÞ in (14). The value of F B for an arbitrary m is obtained by interpolation of values in Table 1 with sufficient accuracy, since the relationship between F B and m is almost linear [14,15]. 4. Approximate expression of stress intensity factor The following approximate expression of the SIF is proposed by using the stress distribution before the crack initiation,
kB ¼
pffiffiffi 1þm AMðbÞ b 3þm
ð17Þ
where A is a coefficient depending on dp and m; and b is the crack length. MðbÞ is the bending moment before the crack initiates at the position that corresponds to the crack length b. The factor ð1 þ mÞ=ð3 þ mÞ in (17) is multiplied in order that the coefficient A has smaller effect on Poisson’s ratio. Approximate expressions for SIF of a type of (17) were proposed for a crack
Table 1 Values of F B obtained by Eq. (13). b=a ða=bÞ
dp ¼ 60
dp ¼ 90
dp ¼ 120
v = 0.0
v = 0.25
v = 0.5
v = 0.0
v = 0.25
v = 0,5
v = 0.0
v = 0.25
v = 0.5
0.100 0.200 0.400 0.600 0.800 1.000 (0.800) (0.600) (0.400) (0.0)
0.875 0.929 0.970 0.984 0.991 0.994 0.997 0.998 1.000 1.000
0.885 0.935 0.972 0.985 0.991 0.994 0.997 0.998 1.000 1.000
0.894 0.940 0.974 0.986 0.992 0.995 0.997 0.998 1.000 1.000
0.771 0.861 0.933 0.963 0.978 0.986 0.991 0.996 0.999 1.000
0.788 0.872 0,938 0.966 0.979 0.986 0.992 0.996 0.999 1.000
0.803 0.882 0.943 0.969 0.981 0.987 0.992 0.996 0.999 1.000
0.634 0.756 0.866 0.917 0.946 0.964 0.978 0.989 0.996 1.000
0.655 0.772 0.876 0.923 0.950 0.967 0.978 0.989 0.996 1.000
0.674 0.786 0.885 0.929 0.954 0.969 0.980 0.989 0.996 1.000
21
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
Fig. 3. Non-dimensional stress intensity factors F B .
from a triangular notch on the rim of the half plane for the thin plate bending problem [14] and for the plane elastic problem of the same shape [12]. The expression of MðbÞ is derived as follows. If r is the distance from the tip of the sharp corner, then the bending moment MðrÞ to the symmetric axis (y axis in Fig. 1 in the present paper) is represented by the following equation [20]:
MðrÞ ¼ f h r m1 þ f h2 r m2 þ f h3 r m3 þ
ð18Þ
where f h ; f h2 ; f h3 . . . are the coefficients determined by the boundary condition. m1 ; m2 ; m3 ; . . . are the roots of the following equation for the stress state symmetric to the bisector of the corner:
ð3 þ mÞ sin½2ðm þ 1Þð2p dpÞ ðm þ 1Þð1 mÞ sinð2p dpÞ ¼ 0
ð19Þ
The values of m1 and m2 for some dp and m are listed in Table 2 and also the values were listed in [21,13]. Since the uniform out-of-plane bending at infinity on the x-axis is considered, then the stress distribution becomes symmetric to the y-axis. Therefore, the distribution of Mx on the y-axis is used. Generally, it is difficult to get the accurate values f h of the first term in (18). In the present paper, the values f h were achieved by the method described in [21] and are shown in Table 2. The first term in (18) is singular for small r, since the values of m1 are negative. Therefore, the stress distribution in the vicinity of the crack tip can be approximated by only the first term. If we use until the second term, the stress distribution is expressed more accurately. Thus the following non-dimensional approximate expressions of SIF are proposed from (17) and (18) by using the rhombic hole size of the diagonal length 2a as standard of the length,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b=ða þ bÞ
ð20Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m b=ða þ bÞ F 0B ¼ A f h ðb=aÞ 1 þ f h2 ðb=aÞ 2
ð21Þ
F 0B ¼ Afh ðb=aÞ
m1
where f h2 is determined by using the value of stress at an appropriate point near the corner before the crack initiation. For example, when dp ¼ 90 , the values of f h2 ¼ 0:0208; 0:0427 and 0.0615 are obtained for m ¼ 0, 0.25 and 0.5, respectively. Table 3 and Fig. 4 show the values of A for m ¼ 0:25 in (20) and (21). The values of A in (20) and (21) are the same values and depend on the corner angle, Poisson’s ratio and the crack length. The values A for a crack initiated from a triangular notch on a rim of a half plane under out-of-plane bending were listed for m ¼ 0 and 0.5 in [14]. These values can be applied for the calculation of SIF for the crack of the rhombic hole for m ¼ 0 and 0.5. Table 4 shows the values of F 0B and F B for dp ¼ 90 and m ¼ 0:25 obtained by (20), (21) and (16) and the errors to the analytical results of Table 1. It is noticed from Table 4 that
Table 2 Values of m1 and m2 , and f h for the rhombic holes. dp
20° 40° 60° 80° 90° 100° 120° 140°
v = 0.0
v = 0.25
v = 0.5
ml
m2
fh
ml
m2
fh
ml
m2
fh
0.4602 0.4150 0.3648 0.3104 0.2821 0.2531 0.1939 0.1332
0.0388 0.0830 0.1362 0.2027 0.2427 0.2884 0.4006 0.5483
0.795 0.893 0.993 1.089 1.130 1.166 1.211 1.217
0.4635 0.4221 0.3761 0.3257 0.2988 0.2710 0.2123 0.1488
0.0449 0.0958 0.1555 0.2278 0.2702 0.3176 0.4308 0.5750
0.787 0.875 0.968 1.058 1.099 1.135 1.192 1.211
0.4662 0.4281 0.3855 0.3381 0.3125 0.2855 0.2271 0.1618
0.0502 0.1068 0.1722 0.2496 0.2939 0.3427 0.4567 0.5980
0.781 0.862 0.948 1.033 1.074 1.112 1.175 1.205
22
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
Table 3 Values of A in Eqs. (19) and (20) for
v = 0.25.
m ¼ 0:25
b=a
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5
dp ¼ 20
40°
60°
80°
90°
100°
12000
140°
1.35 1.33 1.31 1.30
1.30 1.30 1.28 1.27 1.26 1.25
1.26 1.25 1.25 1.24 1.23 1.23 1.22 1.21
1.22 1.22 1.21 1.21 1.21 1.20 1.20 1.19 1.19
1.19 1.19 1.19 1.19 1.19 1.19 1.18 1.18 1.18 1.17
1.17 1.17 1.17 1.17 1.17 1.16 1.16 1.16 1.16 1.16
1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12
1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09
Fig. 4. Values of A versus b=a for Poisson’s ratio 0.25.
Table 4 Approximate values F 0B and F B using Eqs. (19), (20) and (15) for dp ¼ 90 and b=a ða=bÞ
Eq. (19)
Error (%)
0.100 0.200 0.400 0.600 0.800 1.000 (0.800) (0.600) (0.400) (0.0)
0.785 0.864 0.919 0.933
0.4 0.9 2.0 3.4
Eq:ð19ÞEq:ð13Þ Eq:ð13Þ
v = 0.25 and the precision.
Eq. (20)
Error (%)
0.793 0.877 0.940 0.960 0.956
0.6 0.6 0.2 0.6 2.3
Eq:ð20ÞEq:ð13Þ Eq:ð13Þ
Eq. (15)
Error (%)
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
6.6 3.5 2.1 1.4 0.8 0.4 0.1 0.0
Eq:ð15ÞEq:ð13Þ Eq:ð13Þ
the approximate values are accurately obtained by (20) for short cracks, by (21) for longer cracks and by (15) for very long cracks. We can estimate the range of the application of (20), (21) and (16) from Table 4. The precision for other dp and m are almost the same, but they are not shown in the present paper. The relationship between A and m is almost linear shown in [14]. Without any question, if the expression of the stress distribution is represented by using more terms in (18), then the accuracy of the approximate expression can be improved. However, taking account of the convenience, (20) and (21) seem enough to approximate. Inversely, if the stress intensity factors have been known for some crack lengths, coefficients f h ; f h2 ; f h3 . . . of the first few terms can be calculated by using (17) and (18). Accordingly, the stress distribution before the crack initiation can be derived. This fact seems to be useful in engineering because the stress distribution before the crack initiation can be obtained. The values of A in (20) and (21) must be applied for SIF of Mode I of a crack on the bisector of a V shaped notch in any structures, even if the stress states are not symmetric to the bisector of the shaped notch. It seems also to be interesting to investigate the stress concentration at the round corner before a crack initiation. The round corner shown in Fig. 5a is considered, where the symbol q denotes the radius of curvature at the tip of the round corner. The maximum stress at the notch tip can be generally expressed by the following expression [22,21]:
Mmax ¼ kq qm1 þ kq2 qm2 þ kq3 qm3 þ where m1 ; m2 ; m3 ; . . . are roots of (19) and the same as those of (18).
ð22Þ
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
δπ
δπ
ηπ
ηπ
(a)
23
ηπ
b ηπ
(b)
Fig. 5. (a) Round corner with the radius of curvature q, corner angle is gp ¼ ðp dp=2Þ and (b) crack initiating at the sharp corner with the length b.
The coefficients of the first terms of (18) and (22) are related by the following expression [21]:
pffiffiffi 3þm kq 2f h ¼ C 1þm
ð23Þ
The values of the factor C are listed in [21]. Therefore, the values of kq for a rhombic hole without crack are derived from (23) using the values of f h in Table 2. The maximum values at the tip of the round corner can be obtained for small values of q using of the first term of (22), that is, the stress concentration factor before the crack initiation can be obtained. Inversely, if the values of kq are known, the values of f h are calculated; therefore, the stress intensity factors for short cracks (see Fig. 5b) can be achieved, for example, see (20) and the values of A in Table 3 can be applied. 5. Conclusions The stress distribution and the stress intensity factors were analyzed for the rhombic hole with symmetric cracks at the both corners in an infinite plate under uniform out-of-plane bending. The closed stress functions for the rational mapping function (6) expressed by a sum of fractional expression can be obtained; therefore, the precious stress distributions are archived. The accurate stress intensity factors are also calculated by using (6), because the radii of curvature at the tip of the crack are very small. For antisymmetric cracks, the stress analysis can also be carried out by using the corresponding mapping function. The approximate expressions of the stress intensity factor using the stress distribution before the crack initiation were proposed. They are significant in the engineering work, since they are convenient for use and have wide applications. For short cracks, the stress intensity factors with sufficient accuracy can be obtained by the approximate expression of (19) and (20). The range of the application of (19), (20) and (15) can be estimated from Table 4. For longer cracks, we can also calculate by using more terms of stress distribution before the crack initiation. The values of A of (19) and (20) shown Fig. 4 and Table 3 must be applied for calculating Mode I SIF of a crack on the bisector of a V shaped notch in any structures, and must be the characteristic values. Those of m ¼ 0 and 0.5 were listed in [14]. If the stress intensity factors are known for some crack lengths initiating from the sharp corner, then the stress distribution before the crack initiation can be derived by (19) and (20). This fact seems to be useful in engineering because the stress distribution before the crack initiation can be observed. When the values of f h are known, the values of kq are obtained from (22); therefore, the values of stress concentration at the tip of the round corner can be achieved. When the values of the stress concentration are known, the values of f h can be obtained; therefore, the stress intensity factors for the short crack length can be achieved by an expression corresponding (19). References [1] Sih GC, Paris PC, Erdogan F. Crack-tip stress intensity factors for plane extension and plate bending problems. J Appl Mech 1962;29:306–12. [2] Hui CY, Zehnder AT. A theory for fracture of thin plates subjected to bending and twisting moments. Int J Fract 1993;61:211–29. [3] Zehnder AT, Viz MJ. Fracture mechanics of thin plates and shells under combined membrane, bending and twisting loads. Appl Mech Rev 2005;58:37–48. [4] Sih GC. Handbook of stress intensity factors. Lehigh University; 1973. [5] Isida M. Plates and shells with cracks. Sih GC, editor. Mechanics of fracture, vol. 3. Noordhoff; 1977. Chap. 1. [6] Murakami YCh, editor. Stress intensity factors handbook, vol. 2. Pergamon Press; 1987. [7] Murakami YCh, editor. Stress intensity factors handbook, vol. 3. The Society of Materials Science, Japan, and Pergamon Press; 1992. [8] Murakami YCh, editor. Stress intensity factors handbook, vol. 5. The Society of Materials Science, Japan, and Pergamon Press; 2001. [9] Lukas P, Klesnil M. Fatigue limit of notched bodies. Mater Sci Engng 1978;34:61–6. [10] Kobayashi AS. Computational methods in the mechanics of fracture. In: Atluri SN, editor. Computational methods in mechanics. North-Holland; 1986. Chap. 2. [11] Nisitani H, Chen D, Isida M. An approximate method for calculating K I and K II of various edge cracks emanating from the apex of an elliptic hole. Trans Jpn Soc Mech Eng 1984;50:341–50. [12] Hasebe N, Iida J. A crack originating from a triangular notch on a rim of a semi-infinite plate. Engng Fract Mech 1978;10:773–82. [13] Hasebe N, Iida J, Nakamura T. Notch mechanics for plane and thin plate bending problems. Engng Fract Mech 1990;37(1):87–99.
24
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
[14] Hasebe N, Iida J. A crack originating from a triangular notch on a rim of a semi-infinite plate under transverse bending. Engng Fract Mech 1979;11:645–52. [15] Hasebe N. Bending of strip with semi-elliptic notches or cracks. J Engng Mech ASCE 1978;104:1433–50. [16] Hasebe N, Inohara S. Stress intensity factor at a bilaterally bent crack in the bending problem of thin plate. Engng Fract Mech 1981;14:607–16. [17] Savin GN. Stress concentration around holes. Pergamon Press; 1961. [18] Wang XF, Hasebe N. Formulation of the rational mapping function. Research gate, Norio Hasebe, Dataset; 2014. [19] Muskhelishvili NI. Some basic problems of mathematical theory of elasticity. Noordhoff; 1963. [20] Williams M.L. Surface stress singularities resulting from various boundary conditions in angular corners of plates under bending. In: 1st U.S. national congress of appl mech; 1951. p. 325–29. [21] Hasebe N, Iida J. Intensity of corner and stress concentration factor. J Engng Mech 1983;109:346–56. [22] Hasebe N. Stress analyses of a semi-infinite plate with a triangular notch or mound. Trans Jpn Soc Civil Eng 1971;194:29–40 [In Japanese].
Contents lists available at ScienceDirect
Engineering Fracture Mechanics journal homepage: www.elsevier.com/locate/engfracmech
Stress intensity factors of a rhombic hole with symmetric cracks under uniform transverse thin plate bending Jiro Iida a, Norio Hasebe b,⇑ a b
Nagoya Highway Corporation, Marunouchi 1, Nakaku, Nagoya 460, Japan Department of Civil Engineering and System Management, Nagoya Institute of Technology, Gokisocho, Showaku, Nagoya 466, Japan
a r t i c l e
i n f o
Article history: Received 30 November 2015 Received in revised form 12 January 2016 Accepted 1 February 2016 Available online 6 February 2016 Keywords: Stress intensity factor Kirchhoff thin plate Transverse bending Rhombic hole Crack Mapping function Complex stress function Approximate equation
a b s t r a c t A thin plate bending problem (Kirchhoff thin plate) is solved analytically for an infinite plate with a rhombic hole with symmetric cracks at the both corners. The loading is a uniform transverse bending at infinity. The stress distribution and the stress intensity factors of the bending are obtained for a range of corner angles and crack lengths of the rhombic hole. Approximate expressions of the stress intensity factor are proposed for some corner angles and Poisson’s ratios using the stress distribution expressed by the roots of the characteristic equation of the sharp notched plate before the crack initiation. The precision of the expressions are investigated. Characteristic factors for the approximate expressions are stated. A rational mapping function of a sum of fractional expressions and complex stress functions are used for the stress analysis. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Fractures of plate structure cause under in plane loading, out of bending and twisting, and their combined loadings. The crack tip stress fields combined in plane and out of plane loadings are a superposition of the plane stress and plate theory fields [1]. Then stress intensity factor of Mode I, K I , is assumed to be large enough to eliminate crack closure. Therefore, the contact of the crack surface is not considered. Hui and Zehnder [2] studied the fracture of thin plates subjected to bending and twisting moments. Zehnder and Viz [3] also reviewed fracture mechanics of thin plates and shell combined membrane, bending and twisting loadings. They stated that fracture mechanics of the Kirchhoff thin plate theory for small deflection is valid outside the small scale yielding. Therefore, a linear fracture mechanics is useful for thin plate with cracks subjected to flexural deformation. Many crack problems of a classical thin plate (Kirchhoff thin plate) for various shapes and boundary conditions, for example, mixed boundary of stress and displacement, have been solved, and the stress intensity factors were tabulated and also the references were described in [4–8]. In practical application, an approximate expression to obtain the stress intensity factor is useful, and it is desired to be simple and available. The results of this article were developed for the plane elastic problem by Lukas and Klesnil [9] using the radius of curvature at the end of the notch, by Kobayashi [10] using equivalent length of the crack and the stress
⇑ Corresponding author. Tel.: +81 52 876 5015. E-mail address: [email protected] (N. Hasebe). http://dx.doi.org/10.1016/j.engfracmech.2016.02.002 0013-7944/Ó 2016 Elsevier Ltd. All rights reserved.
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
17
Nomenclature a b A D FB F 0B Mx ; My M xy M0 SIF kB ; kS m1 ; m2 f h ; f h2 b d
half-length of a diagonal of a rhombic hole crack length factors of Eqs. (19) and (20) flexural rigidity non-dimensional stress intensity factor non-dimensional stress intensity factor of an approximate expression bending moments torsional moment uniform bending moment at infinity stress intensity factor stress intensity factors for bending and torsional moments roots of characteristic Eq. (18) coefficients of stress distribution near corner (Eq. (17)) coordinate regarding corner angle on a unit circle parameter of corner angle of a rhombic hole m Poisson’s ratio /ðfÞ; wðfÞ stress functions xðfÞ mapping function
distribution before the crack initiation, by Nisitani et al. [11] using the values of stress concentration at the end of the notch and the stress distribution before the crack initiation, and by Hasebe and Iida [12,13] using the stress distribution before the crack initiation at the angular corner. Approximate expressions for the crack initiating at the angular corner under out-ofplane bending seem not to be proposed many. Hasebe and Iida [14] reported the approximate expressions for the triangular notch with a crack on the rim of the semi-infinite plate. In the present paper, the stress distribution and the stress intensity factors are analyzed for the symmetric cracks initiating from a rhombic hole in an infinite plate for some corner angles and Poisson’s ratios. The loading condition is uniform out-of-plane bending at infinity. A mapping function of a sum of fractional expressions [15,16] and complex stress functions [17] are used for the stress analysis. Thin plates with a rhombic hole are one of typical thin plate structures and are found in many structures. Approximate expressions for the crack initiating at the angular corner of the rhombic hole in the out-ofplane bending problem are proposed. The approximate expressions are represented as the product of a certain coefficient and the stress distribution before the crack initiation. The stress distributions for the corner angles are expressed in the general form of an infinite series using the roots obtained by the characteristic equation. Then the approximate expressions of the stress distribution are derived by using the first few terms of the series and are compared with the analytical values, and the range of the application is investigated. The relationships between the stress concentration at the notch tip before the crack initiation and the stress intensity factors are also investigated. 2. Mapping function and method of analysis An infinite region with symmetric cracks at the both corners of a rhombic hole is mapped to the infinite region outside a unit circle (see Fig. 1). The mapping function can be obtained by the Schwartz–Christoffel’s transformation:
Fig. 1. Rhombic hole with cracks and unit circle.
18
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
Z z ¼ xðfÞ ¼ k
ðf 1Þd ðf þ 1Þd ðf iÞðf þ iÞ ðf þ
d=2 ebi Þ ðf
ebi Þ
d=2
ðf þ ebi Þ
d=2
ðf ebi Þ
d=2 2
f
ð1Þ
df
where d and b are parameters depending on the corner angle of the rhombic hole and the crack length, respectively. When b ¼ p=2, the hole is a rhombic hole without crack. ‘‘k” before the integral symbol is a parameter depending on the expansion or reduction of the size of the rhombic hole. Using (1), a rational mapping function is formed as a sum of fractional expressions. Eq. (1) is divided into a fast convergent term and slow ones:
z=k ¼
ð1d=2Þ ð1d=2Þ ð1d=2Þ A1 ebi B1 ebi C 1 ebi 1 þ þ 1 þ 1 þ ð1 d=2Þebi f ð1 d=2Þebi f ð1 d=2Þebi f 8 9 d d ð11=fÞ ð1þ1=fÞ ðfiÞðfþiÞ Z < ð1d=2Þ = d=2 d=2 d=2 d=2 D1 ebi ð1þebi =fÞ ð1ebi =fÞ ð1þebi =fÞ ð1ebi =fÞ f2 þ 1 þ df bi A1 B1 C 1 D1 : ; ð1 d=2Þe f d=2 d=2 d=2 d=2 f
2
ð1þebi =fÞ
f
2
ð1ebi =fÞ
f
2
ð1þebi =fÞ
2
f
ð2Þ
ð1ebi =fÞ
where
A1 ¼
1 e2bi
1 þ e2bi d=2
C1 ¼
d
2d=2 ð1 e4bi Þ d 1 e2bi 1 þ e2bi 2d=2 ð1 e4bi Þ
d=2
;
B1 ¼
1 e2bi
d=2
;
d 1 þ e2bi
D1 ¼
2d=2 ð1 e4bi Þ d 1 e2bi 1 þ e2bi 2d=2 ð1 e4bi Þ
ð3a; b; c; dÞ
d=2
The first term in (2) is expressed by the following fractional expression:
! ð1d=2Þ 12 X ebi F j 1þ ¼1þ þ Fj f 1 þ f j ebi =f j¼1
ð4Þ
The procedure to determine F j and f j is described in [15,16,18]. The second, third, and fourth terms in (2) can be expressed by using the same values F j and f j (see (6)). The fifth integral term in (2) is expressed by the following expression:
The fifth integral term in ð2Þ ¼
n X Gj g j =f j¼1
ð5Þ
1 g j =f2
The procedure to determine Gj and g j is also described in [15,16,18]. Consequently, the rational mapping function is derived as follows:
2
(
12 X
F j 1þf j ebi =f
A1 6 ð1d=2Þe bi 6
)
(
12 X
F j 1þf j ebi =f
)
3
7 þ 1þ þ Fj 1þ þ Fj 7 j¼1 j¼1 7 6 6 ( ) ( )7 48þ2n 12 12 6 7 X Ek X X F F 7 6 C 1 D1 j j z ¼ xðfÞ ¼ k6 þ ð1d=2Þe 1þ 1þ þ ð1d=2Þe 7 E0 f þ bi bi =f þ F j bi bi =f þ F j 1þf e 1þf e j j 7 6 fk f k¼1 j¼1 j¼1 7 6 7 6 n 7 6 X G g =f j j 5 4þ þf 1g =f2 j¼1
B1 ð1d=2Þebi
ð6Þ
j
In the present paper, the number of n in (6) is adopted for n = 11, 12, 13, and 14. The radius of curvature q at the crack tip is not equal to zero, because (6) is a rational function. However, they are very small and the ratios with the crack length are
q=b ¼ 107 —109 . The stress analysis is carried out by using the rational mapping function (6) and complex stress functions. Since the analytical method was described in [17,14,16], the results are only shown here. Denoting regular complex stress functions outside the unit circle by /ðfÞ and wðfÞ, the bending moments M x ; M y and the torsional moment M xy , and the shear forces N x ; N y are expressed as follows:
M x þ M y ¼ 4Dð1 þ mÞRe½/0 ðfÞ=x0 ðfÞ h i. 0 M y M x þ 2iMxy ¼ 2Dð1 mÞ xðfÞf/0 ðfÞ=x0 ðfÞg þ w0 ðfÞ x0 ðfÞ 0
ð7a; b; cÞ
0
Nx iN y ¼ 4Df/ ðfÞ=x0 ðfÞg =x0 ðfÞ where D ¼ Eh =½12ð1 m2 Þ = flexural rigidity, E = Young’s modulus, h = the thickness of the plate, and v = Poisson’s ratio. Re [.] indicates the real part of [.], and the bar of xðfÞ does the conjugate function. The stress components in the orthogonal curvilinear coordinates expressed by the mapping function (6) are 3
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
19
Mr þ Mh ¼ Mx þ My M h M r þ 2iMrh ¼ e2ic ðMy Mx þ 2iMxy Þ
ð8a; b; c; dÞ
Nr iNh ¼ eic ðNx iNy Þ eic ¼ fx0 ðfÞ=jfx0 ðfÞj
Uniform transverse bending M 0 applies at infinity in the x-axis, and the traction is free on the boundary. Then the following complex stress functions /ðfÞ and wðfÞ can be derived [16]: 48þ2n X C k Ak mþ3 M0 xðfÞ E0 f E0 =f /ðfÞ ¼ m1 f f 2Dð1 m Þ k k¼1
wðfÞ ¼
48þ2n X k¼1
0
ð9a; bÞ
C k Ak f02 xð1=fÞ 0 M0 k 0 xðfÞ E0 f E0 =f / ðfÞ þ x ðfÞ 2Dð1 mÞ f f0k
ðf0k Þ
where / Ak ; C k Ek =x0 ðf0k Þ with f0k 1=fk ðk ¼ 1; 2; 3; . . . ; 48 þ 2nÞ. In order to determine the unknown coefficients Ak , (9a) is differentiated once and f ¼ f0k is substituted, then the real and imaginary parts of Ak can be determined by solving 2 (48 + 2n) simultaneous equations, using /0 ðf0k Þ Ak . When a traction free boundary exists, wðfÞ can be obtained by analytic continuation [19]:
wðfÞ ¼
mþ3 xð1=fÞ 0 /ð1=fÞ 0 / ðfÞ m1 x ðfÞ
ð10Þ
Fig. 2 shows an example of the stress distribution for the corner angle dp ¼ 90 , the crack length b=a ¼ 0:5 and Poisson’s ratio m ¼ 0:25. M h is the bending moment along the boundary. 3. Stress intensity factor For the thin plate bending problem, the bending moments M x ; M y and the torsional moment M xy near the crack tip are expressed as follows [5]:
2 h h M x þ M y ¼ pffiffiffiffiffi kB cos kS sin 2 2 2r 1 pffiffiffiffiffi fð7 þ mÞkB þ ið5 þ 3mÞkS gei h=2 þ ð1 mÞðkB ikS Þe5i h=2 M y M x þ 2iMxy ¼ 2ð1 þ mÞ 2r
ð11a; bÞ
where r and h are the polar coordinates measured from the crack tip; kB and kS are the stress intensity factors (SIF) for bending and shearing, respectively, and are calculated by using the complex stress function /ðfÞ and the mapping function xðfÞ [14,15],
kB ikS ¼ 2Dð1 þ mÞeik=2 /0 ðr1 Þ
.pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x00 ðr1 Þ
Fig. 2. Stress distributions for dp ¼ 90 , b/a = 0.5 and
ð12Þ
v = 0.25.
20
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
where r1 is the coordinates corresponding to the crack tip on the unit circle; k is the angle between the crack and the x-axis. In the present paper, r1 ¼ i; k ¼ p=2 for two crack tips and kS ¼ 0 due to the symmetry. The stress components rx ; ry ; sxy are expressed by
12t
12t 12t M x ; ry ¼ 3 My ; sxy ¼ 3 M xy 3 h h h 2 2 3ðh 4t 2 Þ 3ðh 4t2 Þ ¼ Nx ; syz ¼ Ny 3 3 2h 2h
rx ¼ sxz
ð13a; b; c; d; e; fÞ
where the parameter t is a value between the limits h=2 6 t 6 þh=2 and represents the distance of the point from the midplane in the un-deformed states. The following non-dimensional SIF is defined:
FB ¼
3þm kB pffiffiffiffiffiffiffiffiffiffiffi 1 þ m M0 a þ b
ð14Þ
The values of F B are calculated for dp ¼ 60 ; 90 and 120°, and m ¼ 0; 0:25 and 0.5 and their values are shown in Table 1 and Fig. 3. As the crack exists on the y axis in the present paper, the bending moments in the vicinity of the crack can be represented by
rffiffiffiffiffiffiffiffiffiffiffi aþb 2r rffiffiffiffiffiffiffiffiffiffiffi m 1 kB m 1 aþb pffiffiffiffiffi ¼ My ¼ F M m þ 1 2r m þ 3 B 0 2r
Mx ¼
3 þ m kB pffiffiffiffiffi ¼ F B M 0 1 þ m 2r
ð15Þ
When dp ¼ 0 , the crack with the length 2(a þ bÞ is obtained and then F B is
F B ¼ 1:0
ð16Þ
The hole does not affect for the long crack and F B approaches asymptotically to the value of the crack length 2(a þ bÞ. The asymptotic tendency is faster as the corner angle is smaller. Therefore, when the crack length becomes to some degree, the values of F B are approximated by value of the crack length 2(a þ bÞ. From Table 1 and Fig. 3, it is clear that F B becomes larger as Poisson’s ratio m is larger for the same crack length. However, the values of F B are almost the same as those of the long crack, and the effect of Poisson’s ratio on the F B values is little. The effect of Poisson’s ratio m on the value F B depends on the factor ð1 þ mÞ=ð3 þ mÞ in (14). The value of F B for an arbitrary m is obtained by interpolation of values in Table 1 with sufficient accuracy, since the relationship between F B and m is almost linear [14,15]. 4. Approximate expression of stress intensity factor The following approximate expression of the SIF is proposed by using the stress distribution before the crack initiation,
kB ¼
pffiffiffi 1þm AMðbÞ b 3þm
ð17Þ
where A is a coefficient depending on dp and m; and b is the crack length. MðbÞ is the bending moment before the crack initiates at the position that corresponds to the crack length b. The factor ð1 þ mÞ=ð3 þ mÞ in (17) is multiplied in order that the coefficient A has smaller effect on Poisson’s ratio. Approximate expressions for SIF of a type of (17) were proposed for a crack
Table 1 Values of F B obtained by Eq. (13). b=a ða=bÞ
dp ¼ 60
dp ¼ 90
dp ¼ 120
v = 0.0
v = 0.25
v = 0.5
v = 0.0
v = 0.25
v = 0,5
v = 0.0
v = 0.25
v = 0.5
0.100 0.200 0.400 0.600 0.800 1.000 (0.800) (0.600) (0.400) (0.0)
0.875 0.929 0.970 0.984 0.991 0.994 0.997 0.998 1.000 1.000
0.885 0.935 0.972 0.985 0.991 0.994 0.997 0.998 1.000 1.000
0.894 0.940 0.974 0.986 0.992 0.995 0.997 0.998 1.000 1.000
0.771 0.861 0.933 0.963 0.978 0.986 0.991 0.996 0.999 1.000
0.788 0.872 0,938 0.966 0.979 0.986 0.992 0.996 0.999 1.000
0.803 0.882 0.943 0.969 0.981 0.987 0.992 0.996 0.999 1.000
0.634 0.756 0.866 0.917 0.946 0.964 0.978 0.989 0.996 1.000
0.655 0.772 0.876 0.923 0.950 0.967 0.978 0.989 0.996 1.000
0.674 0.786 0.885 0.929 0.954 0.969 0.980 0.989 0.996 1.000
21
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
Fig. 3. Non-dimensional stress intensity factors F B .
from a triangular notch on the rim of the half plane for the thin plate bending problem [14] and for the plane elastic problem of the same shape [12]. The expression of MðbÞ is derived as follows. If r is the distance from the tip of the sharp corner, then the bending moment MðrÞ to the symmetric axis (y axis in Fig. 1 in the present paper) is represented by the following equation [20]:
MðrÞ ¼ f h r m1 þ f h2 r m2 þ f h3 r m3 þ
ð18Þ
where f h ; f h2 ; f h3 . . . are the coefficients determined by the boundary condition. m1 ; m2 ; m3 ; . . . are the roots of the following equation for the stress state symmetric to the bisector of the corner:
ð3 þ mÞ sin½2ðm þ 1Þð2p dpÞ ðm þ 1Þð1 mÞ sinð2p dpÞ ¼ 0
ð19Þ
The values of m1 and m2 for some dp and m are listed in Table 2 and also the values were listed in [21,13]. Since the uniform out-of-plane bending at infinity on the x-axis is considered, then the stress distribution becomes symmetric to the y-axis. Therefore, the distribution of Mx on the y-axis is used. Generally, it is difficult to get the accurate values f h of the first term in (18). In the present paper, the values f h were achieved by the method described in [21] and are shown in Table 2. The first term in (18) is singular for small r, since the values of m1 are negative. Therefore, the stress distribution in the vicinity of the crack tip can be approximated by only the first term. If we use until the second term, the stress distribution is expressed more accurately. Thus the following non-dimensional approximate expressions of SIF are proposed from (17) and (18) by using the rhombic hole size of the diagonal length 2a as standard of the length,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b=ða þ bÞ
ð20Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m m b=ða þ bÞ F 0B ¼ A f h ðb=aÞ 1 þ f h2 ðb=aÞ 2
ð21Þ
F 0B ¼ Afh ðb=aÞ
m1
where f h2 is determined by using the value of stress at an appropriate point near the corner before the crack initiation. For example, when dp ¼ 90 , the values of f h2 ¼ 0:0208; 0:0427 and 0.0615 are obtained for m ¼ 0, 0.25 and 0.5, respectively. Table 3 and Fig. 4 show the values of A for m ¼ 0:25 in (20) and (21). The values of A in (20) and (21) are the same values and depend on the corner angle, Poisson’s ratio and the crack length. The values A for a crack initiated from a triangular notch on a rim of a half plane under out-of-plane bending were listed for m ¼ 0 and 0.5 in [14]. These values can be applied for the calculation of SIF for the crack of the rhombic hole for m ¼ 0 and 0.5. Table 4 shows the values of F 0B and F B for dp ¼ 90 and m ¼ 0:25 obtained by (20), (21) and (16) and the errors to the analytical results of Table 1. It is noticed from Table 4 that
Table 2 Values of m1 and m2 , and f h for the rhombic holes. dp
20° 40° 60° 80° 90° 100° 120° 140°
v = 0.0
v = 0.25
v = 0.5
ml
m2
fh
ml
m2
fh
ml
m2
fh
0.4602 0.4150 0.3648 0.3104 0.2821 0.2531 0.1939 0.1332
0.0388 0.0830 0.1362 0.2027 0.2427 0.2884 0.4006 0.5483
0.795 0.893 0.993 1.089 1.130 1.166 1.211 1.217
0.4635 0.4221 0.3761 0.3257 0.2988 0.2710 0.2123 0.1488
0.0449 0.0958 0.1555 0.2278 0.2702 0.3176 0.4308 0.5750
0.787 0.875 0.968 1.058 1.099 1.135 1.192 1.211
0.4662 0.4281 0.3855 0.3381 0.3125 0.2855 0.2271 0.1618
0.0502 0.1068 0.1722 0.2496 0.2939 0.3427 0.4567 0.5980
0.781 0.862 0.948 1.033 1.074 1.112 1.175 1.205
22
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
Table 3 Values of A in Eqs. (19) and (20) for
v = 0.25.
m ¼ 0:25
b=a
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5
dp ¼ 20
40°
60°
80°
90°
100°
12000
140°
1.35 1.33 1.31 1.30
1.30 1.30 1.28 1.27 1.26 1.25
1.26 1.25 1.25 1.24 1.23 1.23 1.22 1.21
1.22 1.22 1.21 1.21 1.21 1.20 1.20 1.19 1.19
1.19 1.19 1.19 1.19 1.19 1.19 1.18 1.18 1.18 1.17
1.17 1.17 1.17 1.17 1.17 1.16 1.16 1.16 1.16 1.16
1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12 1.12
1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09 1.09
Fig. 4. Values of A versus b=a for Poisson’s ratio 0.25.
Table 4 Approximate values F 0B and F B using Eqs. (19), (20) and (15) for dp ¼ 90 and b=a ða=bÞ
Eq. (19)
Error (%)
0.100 0.200 0.400 0.600 0.800 1.000 (0.800) (0.600) (0.400) (0.0)
0.785 0.864 0.919 0.933
0.4 0.9 2.0 3.4
Eq:ð19ÞEq:ð13Þ Eq:ð13Þ
v = 0.25 and the precision.
Eq. (20)
Error (%)
0.793 0.877 0.940 0.960 0.956
0.6 0.6 0.2 0.6 2.3
Eq:ð20ÞEq:ð13Þ Eq:ð13Þ
Eq. (15)
Error (%)
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
6.6 3.5 2.1 1.4 0.8 0.4 0.1 0.0
Eq:ð15ÞEq:ð13Þ Eq:ð13Þ
the approximate values are accurately obtained by (20) for short cracks, by (21) for longer cracks and by (15) for very long cracks. We can estimate the range of the application of (20), (21) and (16) from Table 4. The precision for other dp and m are almost the same, but they are not shown in the present paper. The relationship between A and m is almost linear shown in [14]. Without any question, if the expression of the stress distribution is represented by using more terms in (18), then the accuracy of the approximate expression can be improved. However, taking account of the convenience, (20) and (21) seem enough to approximate. Inversely, if the stress intensity factors have been known for some crack lengths, coefficients f h ; f h2 ; f h3 . . . of the first few terms can be calculated by using (17) and (18). Accordingly, the stress distribution before the crack initiation can be derived. This fact seems to be useful in engineering because the stress distribution before the crack initiation can be obtained. The values of A in (20) and (21) must be applied for SIF of Mode I of a crack on the bisector of a V shaped notch in any structures, even if the stress states are not symmetric to the bisector of the shaped notch. It seems also to be interesting to investigate the stress concentration at the round corner before a crack initiation. The round corner shown in Fig. 5a is considered, where the symbol q denotes the radius of curvature at the tip of the round corner. The maximum stress at the notch tip can be generally expressed by the following expression [22,21]:
Mmax ¼ kq qm1 þ kq2 qm2 þ kq3 qm3 þ where m1 ; m2 ; m3 ; . . . are roots of (19) and the same as those of (18).
ð22Þ
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
δπ
δπ
ηπ
ηπ
(a)
23
ηπ
b ηπ
(b)
Fig. 5. (a) Round corner with the radius of curvature q, corner angle is gp ¼ ðp dp=2Þ and (b) crack initiating at the sharp corner with the length b.
The coefficients of the first terms of (18) and (22) are related by the following expression [21]:
pffiffiffi 3þm kq 2f h ¼ C 1þm
ð23Þ
The values of the factor C are listed in [21]. Therefore, the values of kq for a rhombic hole without crack are derived from (23) using the values of f h in Table 2. The maximum values at the tip of the round corner can be obtained for small values of q using of the first term of (22), that is, the stress concentration factor before the crack initiation can be obtained. Inversely, if the values of kq are known, the values of f h are calculated; therefore, the stress intensity factors for short cracks (see Fig. 5b) can be achieved, for example, see (20) and the values of A in Table 3 can be applied. 5. Conclusions The stress distribution and the stress intensity factors were analyzed for the rhombic hole with symmetric cracks at the both corners in an infinite plate under uniform out-of-plane bending. The closed stress functions for the rational mapping function (6) expressed by a sum of fractional expression can be obtained; therefore, the precious stress distributions are archived. The accurate stress intensity factors are also calculated by using (6), because the radii of curvature at the tip of the crack are very small. For antisymmetric cracks, the stress analysis can also be carried out by using the corresponding mapping function. The approximate expressions of the stress intensity factor using the stress distribution before the crack initiation were proposed. They are significant in the engineering work, since they are convenient for use and have wide applications. For short cracks, the stress intensity factors with sufficient accuracy can be obtained by the approximate expression of (19) and (20). The range of the application of (19), (20) and (15) can be estimated from Table 4. For longer cracks, we can also calculate by using more terms of stress distribution before the crack initiation. The values of A of (19) and (20) shown Fig. 4 and Table 3 must be applied for calculating Mode I SIF of a crack on the bisector of a V shaped notch in any structures, and must be the characteristic values. Those of m ¼ 0 and 0.5 were listed in [14]. If the stress intensity factors are known for some crack lengths initiating from the sharp corner, then the stress distribution before the crack initiation can be derived by (19) and (20). This fact seems to be useful in engineering because the stress distribution before the crack initiation can be observed. When the values of f h are known, the values of kq are obtained from (22); therefore, the values of stress concentration at the tip of the round corner can be achieved. When the values of the stress concentration are known, the values of f h can be obtained; therefore, the stress intensity factors for the short crack length can be achieved by an expression corresponding (19). References [1] Sih GC, Paris PC, Erdogan F. Crack-tip stress intensity factors for plane extension and plate bending problems. J Appl Mech 1962;29:306–12. [2] Hui CY, Zehnder AT. A theory for fracture of thin plates subjected to bending and twisting moments. Int J Fract 1993;61:211–29. [3] Zehnder AT, Viz MJ. Fracture mechanics of thin plates and shells under combined membrane, bending and twisting loads. Appl Mech Rev 2005;58:37–48. [4] Sih GC. Handbook of stress intensity factors. Lehigh University; 1973. [5] Isida M. Plates and shells with cracks. Sih GC, editor. Mechanics of fracture, vol. 3. Noordhoff; 1977. Chap. 1. [6] Murakami YCh, editor. Stress intensity factors handbook, vol. 2. Pergamon Press; 1987. [7] Murakami YCh, editor. Stress intensity factors handbook, vol. 3. The Society of Materials Science, Japan, and Pergamon Press; 1992. [8] Murakami YCh, editor. Stress intensity factors handbook, vol. 5. The Society of Materials Science, Japan, and Pergamon Press; 2001. [9] Lukas P, Klesnil M. Fatigue limit of notched bodies. Mater Sci Engng 1978;34:61–6. [10] Kobayashi AS. Computational methods in the mechanics of fracture. In: Atluri SN, editor. Computational methods in mechanics. North-Holland; 1986. Chap. 2. [11] Nisitani H, Chen D, Isida M. An approximate method for calculating K I and K II of various edge cracks emanating from the apex of an elliptic hole. Trans Jpn Soc Mech Eng 1984;50:341–50. [12] Hasebe N, Iida J. A crack originating from a triangular notch on a rim of a semi-infinite plate. Engng Fract Mech 1978;10:773–82. [13] Hasebe N, Iida J, Nakamura T. Notch mechanics for plane and thin plate bending problems. Engng Fract Mech 1990;37(1):87–99.
24
J. Iida, N. Hasebe / Engineering Fracture Mechanics 156 (2016) 16–24
[14] Hasebe N, Iida J. A crack originating from a triangular notch on a rim of a semi-infinite plate under transverse bending. Engng Fract Mech 1979;11:645–52. [15] Hasebe N. Bending of strip with semi-elliptic notches or cracks. J Engng Mech ASCE 1978;104:1433–50. [16] Hasebe N, Inohara S. Stress intensity factor at a bilaterally bent crack in the bending problem of thin plate. Engng Fract Mech 1981;14:607–16. [17] Savin GN. Stress concentration around holes. Pergamon Press; 1961. [18] Wang XF, Hasebe N. Formulation of the rational mapping function. Research gate, Norio Hasebe, Dataset; 2014. [19] Muskhelishvili NI. Some basic problems of mathematical theory of elasticity. Noordhoff; 1963. [20] Williams M.L. Surface stress singularities resulting from various boundary conditions in angular corners of plates under bending. In: 1st U.S. national congress of appl mech; 1951. p. 325–29. [21] Hasebe N, Iida J. Intensity of corner and stress concentration factor. J Engng Mech 1983;109:346–56. [22] Hasebe N. Stress analyses of a semi-infinite plate with a triangular notch or mound. Trans Jpn Soc Civil Eng 1971;194:29–40 [In Japanese].