- Email: [email protected]
Numerical solution of multiple hole problem by using boundary integral equation
THEORETICAL & APPLIED MECHANICS LETTERS 1, 031005 (2011)
Numerical solution of multiple hole problem by using boundary integral equation Yizhou Chena) Division of Engineering Mechanics, Jiangsu University, Zhenjiang 212013,China
(Received 12 December 2010; accepted 26 March 2011; published online 10 May 2011) Abstract This paper studies a numerical solution of multiple hole problem by using a boundary integral equation. The studied problem can be considered as a supposition of many single hole problems. After considering the interaction among holes, an algebraic equation is formulated, which is then solved by using an iteration technique. The hoop stress around holes can be finally determined. c 2011 The Chinese Society of Theoretical One numerical example is provided to check its accuracy. ⃝ and Applied Mechanics. [doi:10.1063/2.1103105] Keywords boundary integral equation, iteration, multiple holes, stress concentration factors The stress concentration problems for holes have attracted much attention of researchers. A variety of methods were suggested to solve the hole problem in an infinite plate.1−9 However, most of these methods were limited to the circular or elliptic hole. For a single hole problem, the boundary integral equation (BIE) for its exterior region was suggested.10 This paper studies a numerical solution of multiple hole problem by using the BIE, where the problem can be considered as a supposition of many single hole problems. After considering the interaction among holes, an algebraic equation is formulated, which is then solved by using an iteration technique. The hoop stress around holes can be finally determined. The shape of holes can be arbitrary. One numerical example is provided at the end to check its accuracy. Recently, a boundary integral equation for an exterior region using complex variable was suggested10 ∫ ( U (to ) κ−1 − B1 i U (t)dt− 2 Γ t − to ) L1 (t, to )U (t)dt + L2 (t, to )U (t)dt ∫ = −B2 i (2κ ln |t − to | Q(t)dt+ Γ ) t − to Q(t)dt¯ , (to ∈ Γ ), (1) t¯ − t¯o where Γ denotes the boundary of hole (Fig. 1), and U (t) = u(t) + iv(t), (2) Q(t) = σN (t) + iσN T (t), (t ∈ Γ ), 1 1 , B2 = , (3) 2π(κ + 1) 4πG(κ + 1) { } t−τ 1 dt¯ 1 d ln + , =− L1 (t, τ ) = − dt t−τ t¯ − τ¯ t¯ − τ¯ dt { } d t−τ 1 t − τ dt¯ L2 (t, τ ) = = − . (4) dt t¯ − τ¯ t¯ − τ¯ (t¯ − τ¯)2 dt B1 =
a) Corresponding
author. Email: [email protected].
Fig. 1. Superposition method for the solution of perturbation field.
Here G denotes the shear modulus of elasticity, κ = 3 − 4ν in plane strain case, and ν is the Poisson’s ratio. In Eq. (2), Q(t) = σN (t) + iσN T (t) denotes the traction applied on the boundary and U (t) = u(t)+iv(t) denotes the boundary displacement (Fig. 1). In addition, the traction at a domain point τ in Fig. 1 can be evaluated by10 ∫ 1 1 Q(τ ) = 2B1 i U (t)dt+ 2G (t − τ )2 Γ
031005-2 Y. Z. Chen
Theor. Appl. Mech. Lett. 1, 031005 (2011)
∫
∫ M1 (t, τ )U (t)dt − B1 i M2 (t, τ )U (t)dt − ∫ Γ κ−1 Q(t)dt−B2 i κK1 (t, τ ) · Q(t)dt − t−τ Γ
B1 i ∫Γ B2 i ∫
Γ
K2 (t, τ )Q(t)dt¯,
B2 i
(5)
Γ
where d K1 (t, τ ) = dτ
( ) t−τ 1 1 d¯ τ ln =− + , t¯ − τ¯ t−τ t¯ − τ¯ dτ
d K2 (t, τ ) = − dτ
(
t−τ t¯ − τ¯
) =
(t − τ ) d¯ 1 τ − , 2 ¯ ¯ t − τ¯ (t − τ¯) dτ (6)
d [K1 (t, τ )] dt [ ( )] d d t−τ =− ln dτ dt t¯ − τ¯ 1 1 dt¯ d¯ τ =− + , 2 2 ¯ (t − τ ) (t − τ¯) dt dτ
M1 (t, τ ) = −
Bkj can be evaluated by using Eqs. (1)–(8). In computation, the following iteration scheme is suggested (i+1)
Qk
˜k − =Q
M ∑
′
(i)
Bkj Qj .
(10)
j=1 (1)
Generally, we take Qj = 0 (j = 1, 2, ..., M ) in the first round computation. An example is presented to examine the achieved accuracy for the suggested method (Fig. 2). In the example, we assume that two elliptic holes are in series. The elliptic hole has half-axis “a” and “b” and the spacing between the two holes is denoted by “c”. The remote loading is denoted by σy∞ = p. In the solution of BIE, 96 divisions are used for the discretization of elliptic contour. It is known that for the single hole case, the maximum hoop stress at the crown point is ( ) 2a pc = 1 + p. For the following cases: b/a= 0.25, b 1/3, 0.5 and 1.0 and c/a=0.1, 0.2, . . . , 1.0, the hoop stresses at the point “H” and “F ” are denoted by σT ,H = sH (b/a, c/a)pc ,
[ ( )] d d d t−τ M2 (t, τ ) = − {K2 (t, τ )} = dt dτ dt t¯ − τ¯ ( ) 1 dt¯ d¯ τ 2(t − τ ) dt¯ d¯ τ = + − . 2 3 ¯ ¯ (t − τ¯) dt dτ (t − τ¯) dt dτ (7)
σ T ,F = sF (b/a, c/a)pc , with pc =
( 1+
2a b
) p.
(11)
It is assumed that the remote loading is σy∞ = p, and many holes are placed in an infinite plate. The original stress field can be decomposed into a uniform field and a perturbation field. In fact, we need only to solve the boundary value problem for the perturbation field, as shown in Fig. 1(a). The boundary condition can be written as ˜ k , (k = 1, 2, · · · , M ), Qk = Q
(8)
˜ k (k = 1, 2, . . . , M ) is traction vector applied on where Q ˜ N (t) the k-th hole, which is composed of components σ ˜ N T (t) along the boundary. and σ The multiple hole problem shown in Fig. 1(a) can be considered as a superposition of many single hole problems with undetermined traction on the individual hole (Fig. 1(b)). The traction vector applied on the j-th hole is denoted by Qj . Using the principle of superposition, we can get the following algebraic equation Qk +
M ∑
′
˜ k, Bkj Qj = Q
(9)
j=1
where
M ∑
′
denotes that the term j = k should be ex-
j=1
cluded in the summation. Clearly, the influence matrix
Fig. 2. Two elliptic holes in series under the remote tension of σy∞ = p.
The computed non-dimensional stress concentration factors sH (b/a, c/a) and sF (b/a, c/a) are listed in Table 1. It is found that in the case of b/a=0.25 and c/a=0.1, or a rather narrow spacing case, the stress concentration factor can reach a huge value, e.g. σT ,F = 1.908pc = 17.172p (pc = 9p in case of b/a=0.25). In addition, for the case of b/a=1/3, the computed results are found in good agreement with those obtained by other researcher.6 Some advantages can be found from the suggested formulation. First, the formulation can be applied to arbitrary hole configuration. Secondly, the suggested BIE in conjunction with iteration can provide accurate results for stress concentration factors, as shown in the example.
031005-3 Numerical solution of multiple hole problem
Theor. Appl. Mech. Lett. 1, 031005 (2011)
Table 1. Non-dimensional stress concentration factors sH (b/a, c/a) at point H and sF (b/a, c/a) (at point F ) under the remote loading of σy∞ = p (see Fig. 2 and Eq. (10)) b/a
sH (b/a, c/a)
sF (b/a, c/a)
0.25 1/3 1/3∗ 0.50 1.00 0.25 1/3 1/3∗ 0.50 1.00
c/a 0.1 1.152 1.155 1.150 1.162 1.171 1.908 2.049 2.060 2.338 2.864
0.2 1.119 1.121 1.120 1.126 1.137 1.480 1.523 1.510 1.656 2.028
0.3 1.100 1.100 1.090 1.104 1.116 1.327 1.339 1.330 1.403 1.674
0.4 1.086 1.086 1.080 1.089 1.100 1.247 1.247 1.240 1.276 1.474
0.5 1.076 1.076 1.070 1.077 1.087 1.197 1.192 1.180 1.203 1.345
0.6 1.068 1.067 1.065 1.068 1.077 1.164 1.156 1.150 1.156 1.258
0.7 1.062 1.061 1.060 1.061 1.069 1.139 1.131 1.130 1.125 1.195
0.8 1.056 1.055 1.050 1.055 1.062 1.120 1.112 1.110 1.103 1.150
0.9 1.052 1.050 1.045 1.050 1.056 1.106 1.097 1.090 1.087 1.116
1.0 1.048 1.046 1.040 1.045 1.050 1.094 1.086 1.080 1.074 1.090
Note: ∗ from Ref. 6 1. N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity (Noordhoff, Groningen, 1963). 2. G. N. Savin, Stress Concentration around Holes (Pergamon Press, New York, 1961). 3. H. Nisitani, Mechanics of Fracture, (Noordhoof, Netherlands, 1978). 4. M. Isida, and H. Igawa, Int. J. Solids Struct. 27, 849 (1991). 5. M. Denda, and I. Kosaka, Int. J. Numer. Meth. Engrg. 40, 2857 (1997).
6. I. Tsukrov, and M, Kachanov, Int. J. Solids Struct. 34, 2887 (1997). 7. K. Ting, K. T. Chen, and W. S. Yang, Int. J. Solids Struct. 36, 533 (1999). 8. K. Ting, K. T. Chen, and W. S. Yang, Int. J. Pres. Ves. Pip. 76, 503 (1999). 9. J. Wang, S. L. Crouch, and S. G. Mogilevskaya, Eng. Anal. Boun. Elem. 27, 789 (2003). 10. Y. Z. Chen, and X. Y. Lin, Eng. Anal. Boun. Elem. 34, 834 (2010).
Numerical solution of multiple hole problem by using boundary integral equation Yizhou Chena) Division of Engineering Mechanics, Jiangsu University, Zhenjiang 212013,China
(Received 12 December 2010; accepted 26 March 2011; published online 10 May 2011) Abstract This paper studies a numerical solution of multiple hole problem by using a boundary integral equation. The studied problem can be considered as a supposition of many single hole problems. After considering the interaction among holes, an algebraic equation is formulated, which is then solved by using an iteration technique. The hoop stress around holes can be finally determined. c 2011 The Chinese Society of Theoretical One numerical example is provided to check its accuracy. ⃝ and Applied Mechanics. [doi:10.1063/2.1103105] Keywords boundary integral equation, iteration, multiple holes, stress concentration factors The stress concentration problems for holes have attracted much attention of researchers. A variety of methods were suggested to solve the hole problem in an infinite plate.1−9 However, most of these methods were limited to the circular or elliptic hole. For a single hole problem, the boundary integral equation (BIE) for its exterior region was suggested.10 This paper studies a numerical solution of multiple hole problem by using the BIE, where the problem can be considered as a supposition of many single hole problems. After considering the interaction among holes, an algebraic equation is formulated, which is then solved by using an iteration technique. The hoop stress around holes can be finally determined. The shape of holes can be arbitrary. One numerical example is provided at the end to check its accuracy. Recently, a boundary integral equation for an exterior region using complex variable was suggested10 ∫ ( U (to ) κ−1 − B1 i U (t)dt− 2 Γ t − to ) L1 (t, to )U (t)dt + L2 (t, to )U (t)dt ∫ = −B2 i (2κ ln |t − to | Q(t)dt+ Γ ) t − to Q(t)dt¯ , (to ∈ Γ ), (1) t¯ − t¯o where Γ denotes the boundary of hole (Fig. 1), and U (t) = u(t) + iv(t), (2) Q(t) = σN (t) + iσN T (t), (t ∈ Γ ), 1 1 , B2 = , (3) 2π(κ + 1) 4πG(κ + 1) { } t−τ 1 dt¯ 1 d ln + , =− L1 (t, τ ) = − dt t−τ t¯ − τ¯ t¯ − τ¯ dt { } d t−τ 1 t − τ dt¯ L2 (t, τ ) = = − . (4) dt t¯ − τ¯ t¯ − τ¯ (t¯ − τ¯)2 dt B1 =
a) Corresponding
author. Email: [email protected].
Fig. 1. Superposition method for the solution of perturbation field.
Here G denotes the shear modulus of elasticity, κ = 3 − 4ν in plane strain case, and ν is the Poisson’s ratio. In Eq. (2), Q(t) = σN (t) + iσN T (t) denotes the traction applied on the boundary and U (t) = u(t)+iv(t) denotes the boundary displacement (Fig. 1). In addition, the traction at a domain point τ in Fig. 1 can be evaluated by10 ∫ 1 1 Q(τ ) = 2B1 i U (t)dt+ 2G (t − τ )2 Γ
031005-2 Y. Z. Chen
Theor. Appl. Mech. Lett. 1, 031005 (2011)
∫
∫ M1 (t, τ )U (t)dt − B1 i M2 (t, τ )U (t)dt − ∫ Γ κ−1 Q(t)dt−B2 i κK1 (t, τ ) · Q(t)dt − t−τ Γ
B1 i ∫Γ B2 i ∫
Γ
K2 (t, τ )Q(t)dt¯,
B2 i
(5)
Γ
where d K1 (t, τ ) = dτ
( ) t−τ 1 1 d¯ τ ln =− + , t¯ − τ¯ t−τ t¯ − τ¯ dτ
d K2 (t, τ ) = − dτ
(
t−τ t¯ − τ¯
) =
(t − τ ) d¯ 1 τ − , 2 ¯ ¯ t − τ¯ (t − τ¯) dτ (6)
d [K1 (t, τ )] dt [ ( )] d d t−τ =− ln dτ dt t¯ − τ¯ 1 1 dt¯ d¯ τ =− + , 2 2 ¯ (t − τ ) (t − τ¯) dt dτ
M1 (t, τ ) = −
Bkj can be evaluated by using Eqs. (1)–(8). In computation, the following iteration scheme is suggested (i+1)
Qk
˜k − =Q
M ∑
′
(i)
Bkj Qj .
(10)
j=1 (1)
Generally, we take Qj = 0 (j = 1, 2, ..., M ) in the first round computation. An example is presented to examine the achieved accuracy for the suggested method (Fig. 2). In the example, we assume that two elliptic holes are in series. The elliptic hole has half-axis “a” and “b” and the spacing between the two holes is denoted by “c”. The remote loading is denoted by σy∞ = p. In the solution of BIE, 96 divisions are used for the discretization of elliptic contour. It is known that for the single hole case, the maximum hoop stress at the crown point is ( ) 2a pc = 1 + p. For the following cases: b/a= 0.25, b 1/3, 0.5 and 1.0 and c/a=0.1, 0.2, . . . , 1.0, the hoop stresses at the point “H” and “F ” are denoted by σT ,H = sH (b/a, c/a)pc ,
[ ( )] d d d t−τ M2 (t, τ ) = − {K2 (t, τ )} = dt dτ dt t¯ − τ¯ ( ) 1 dt¯ d¯ τ 2(t − τ ) dt¯ d¯ τ = + − . 2 3 ¯ ¯ (t − τ¯) dt dτ (t − τ¯) dt dτ (7)
σ T ,F = sF (b/a, c/a)pc , with pc =
( 1+
2a b
) p.
(11)
It is assumed that the remote loading is σy∞ = p, and many holes are placed in an infinite plate. The original stress field can be decomposed into a uniform field and a perturbation field. In fact, we need only to solve the boundary value problem for the perturbation field, as shown in Fig. 1(a). The boundary condition can be written as ˜ k , (k = 1, 2, · · · , M ), Qk = Q
(8)
˜ k (k = 1, 2, . . . , M ) is traction vector applied on where Q ˜ N (t) the k-th hole, which is composed of components σ ˜ N T (t) along the boundary. and σ The multiple hole problem shown in Fig. 1(a) can be considered as a superposition of many single hole problems with undetermined traction on the individual hole (Fig. 1(b)). The traction vector applied on the j-th hole is denoted by Qj . Using the principle of superposition, we can get the following algebraic equation Qk +
M ∑
′
˜ k, Bkj Qj = Q
(9)
j=1
where
M ∑
′
denotes that the term j = k should be ex-
j=1
cluded in the summation. Clearly, the influence matrix
Fig. 2. Two elliptic holes in series under the remote tension of σy∞ = p.
The computed non-dimensional stress concentration factors sH (b/a, c/a) and sF (b/a, c/a) are listed in Table 1. It is found that in the case of b/a=0.25 and c/a=0.1, or a rather narrow spacing case, the stress concentration factor can reach a huge value, e.g. σT ,F = 1.908pc = 17.172p (pc = 9p in case of b/a=0.25). In addition, for the case of b/a=1/3, the computed results are found in good agreement with those obtained by other researcher.6 Some advantages can be found from the suggested formulation. First, the formulation can be applied to arbitrary hole configuration. Secondly, the suggested BIE in conjunction with iteration can provide accurate results for stress concentration factors, as shown in the example.
031005-3 Numerical solution of multiple hole problem
Theor. Appl. Mech. Lett. 1, 031005 (2011)
Table 1. Non-dimensional stress concentration factors sH (b/a, c/a) at point H and sF (b/a, c/a) (at point F ) under the remote loading of σy∞ = p (see Fig. 2 and Eq. (10)) b/a
sH (b/a, c/a)
sF (b/a, c/a)
0.25 1/3 1/3∗ 0.50 1.00 0.25 1/3 1/3∗ 0.50 1.00
c/a 0.1 1.152 1.155 1.150 1.162 1.171 1.908 2.049 2.060 2.338 2.864
0.2 1.119 1.121 1.120 1.126 1.137 1.480 1.523 1.510 1.656 2.028
0.3 1.100 1.100 1.090 1.104 1.116 1.327 1.339 1.330 1.403 1.674
0.4 1.086 1.086 1.080 1.089 1.100 1.247 1.247 1.240 1.276 1.474
0.5 1.076 1.076 1.070 1.077 1.087 1.197 1.192 1.180 1.203 1.345
0.6 1.068 1.067 1.065 1.068 1.077 1.164 1.156 1.150 1.156 1.258
0.7 1.062 1.061 1.060 1.061 1.069 1.139 1.131 1.130 1.125 1.195
0.8 1.056 1.055 1.050 1.055 1.062 1.120 1.112 1.110 1.103 1.150
0.9 1.052 1.050 1.045 1.050 1.056 1.106 1.097 1.090 1.087 1.116
1.0 1.048 1.046 1.040 1.045 1.050 1.094 1.086 1.080 1.074 1.090
Note: ∗ from Ref. 6 1. N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity (Noordhoff, Groningen, 1963). 2. G. N. Savin, Stress Concentration around Holes (Pergamon Press, New York, 1961). 3. H. Nisitani, Mechanics of Fracture, (Noordhoof, Netherlands, 1978). 4. M. Isida, and H. Igawa, Int. J. Solids Struct. 27, 849 (1991). 5. M. Denda, and I. Kosaka, Int. J. Numer. Meth. Engrg. 40, 2857 (1997).
6. I. Tsukrov, and M, Kachanov, Int. J. Solids Struct. 34, 2887 (1997). 7. K. Ting, K. T. Chen, and W. S. Yang, Int. J. Solids Struct. 36, 533 (1999). 8. K. Ting, K. T. Chen, and W. S. Yang, Int. J. Pres. Ves. Pip. 76, 503 (1999). 9. J. Wang, S. L. Crouch, and S. G. Mogilevskaya, Eng. Anal. Boun. Elem. 27, 789 (2003). 10. Y. Z. Chen, and X. Y. Lin, Eng. Anal. Boun. Elem. 34, 834 (2010).