Journal of Solid Mechanics and Materials Engineering

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1 and Materials Engineering Analysis of In-Plane Problems for an Isotropic Elastic Medium with Many Circular Holes or Rigid Inclusions Mutsumi MIYAGAWA, Jyo SHIMURA, Takuo SUZUKI and Takanobu TAMIYA Dept.of Creative Manufacturing Tokyo Metropolitan College of Industrial Technology. Arakawa Campus , Minami-senju, Arakawa-ku, Tokyo , Japan Tokyo National College of Technology , Kunugida-machi, Hachioji-shi, Tokyo , Japan Abstract In this paper, we derive the general solutions for many cylindrical holes or rigid inclusions perfectly bonded to an elastic medium (matrix) of infinite extent, under In-Plane deformation. These many holes or rigid inclusions have different radii and different central points. The matrix is subjected to arbitrary loading like uniform stresses at infinity. The solution is obtained, via iterations of Möbius transformation as a series with an explicit general term involving the complex potential functions of the corresponding homogeneous problem. This procedure has been termed heterogenization. Using these solutions, several numerical examples are shown by graphical representation. Key words : Isotropic Elasticity, In-Plane Problem, Many Cylindrical Holes, Many Rigid Inclusions, Uniform Stress Loading 1. Introduction *Received 4 July, 2013 (No. T ) Japanese Original : Trans. Jpn. Soc. Mech. Eng., Vol. 77, No. 774, A (2011), pp (Received 12 July., 2010) [DOI: /jmmp.7.540] Copyright 2013 by JSME A number of studies have examined the problems associated with disturbances around a single hole or rigid circular inclusion under in-plane loading, such as loading due to uniform stresses or a concentrated force at an arbitrary point. Therefore, these problems have many applications in engineering fields. These inclusion problems have proved to be very useful for mechanical analysis. These problems have been developed further in order to observe the interacting disturbances for multiple circular holes. However, these techniques have been applied using different numerical analysis methods such as the finite element method (FEM) or the boundary element method (BEM). So, for example, if one engineer is an expert in FEM analysis of a model, while another engineer is not, their results will not be the same. The purpose of the present study is to apply the reflection principle of Moriguchi (1), who investigated a single hole in in-plane problems, and the techniques of Honein (2) and Hirashima (4) (6) to consider anti-plane multi-hole problems. We obtained general solutions (7)(8) for up to two circular inclusions. Using these techniques, we expanded these problems to cases involving many circular holes or rigid inclusions. In the present study, these holes or rigid inclusions have arbitrary arrangements and sizes inside the matrix. Using this explicit general solution, we present several numerical examples under uniform stresses at infinity. 2. Fundamental Equation and General Solution 2.1. Formulation of elastostatics for in-plane problems In this section, we review the fundamental formulation of in-plane elastostatics and 540

2 present the notation used herein. We consider the complex region z = x + iy, where i is the imaginary unit (i = 1), to be infinite. Under in-plane deformations, there exist displacements u x and u y and stresses σ x, σ y, and τ xy, which are obtained in Cartesian coordinates only. The formulation used to find the stresses and displacements is satisfied by the complex potential functions φ(z) and ψ(z), which are also used in the techniques of Moriguchi (1). u x iu y = 1 2G M [ κm φ(z) { zφ (z) + ψ (z) } ]. (1) P y ip x = φ(z) + { zφ (z) + ψ (z) }. (2) where a prime indicates differentiation with respect to the complex variables z and κ M as follows: { (3 νm )/(1 + ν M ) Plane Stress κ M = (3) 3 4ν M Plane Strain where G M and ν M are the shear modulus and Poisson s ratio for the matrix, respectively. P x and P y indicate the resultant forces that act from right to left along an arbitrary course from point A to point B in the matrix. B ( P x = σx dy τ xy dx ) B (, P y = τxy dy σ y dx ). (4) A Hence, the stresses are obtained as follows: σ x = 2Re [ φ (z) ] Re [ z φ (z) + ψ (z) ], σ y = 2Re [ φ (z) ] + Re [ z φ (z) + ψ (z) ], τ xy = Im [ z φ (z) + ψ (z) ], A where Re[ ] and Im[ ] are the real and imaginary parts, respectively, of the complex function in parentheses, and the overbar indicates complex conjugation General solution in the presence of a single circular hole We first investigate the problem in the presence of a single circular hole or a rigid inclusion disturbing the uniform stresses σ x, σ y, and τ xy at infinity. If the circular boundary is a hole, then the boundary is referred to as a free boundary, and if the circular boundary is a rigid inclusion, then the boundary is referred to as a fixed boundary. We consider the heterogeneous problem of the j th elastic circular inclusion perfectly bonded to an elastic matrix of infinite extent. This matrix is given by the complex potential functions ϕ(z) and ψ(z). The matrix and the boundary produce an in-plane deformation, as shown in Fig. 1. We set the general boundary conditions of the tractions and displacements on the boundary L j ( i.e., z = z j + a j e iθ) between the j th hole and the matrix, where a j and z j ( = (0, 0)) are the radius and origin, respectively, of the j th hole. In this subsection, we use j = 1 because there is only one inclusion. Required continuity of the tractions and displacements along the circular interface. Hole (Free boundary): P ( j) x = 0, P ( j) y = 0. (6) Rigid inclusion (Fixed boundary): u ( j) x = 0, u ( j) y = 0. (7) After subsection 2.4, it is not difficult to imagine that the center point of each hole will be moved by the interacting disturbances for many holes under the in-plane loading. These rigid movements of the holes or the inclusions would need a new supposition other than the boundary conditions defined in Eqs. (6) and (7). So, we consider that such rigid movements do not happen in this paper. (5) 541

3 Fig. 1 Geometry of an infinite elastic medium with a single hole or rigid inclusion. The most general complexes for this problem may be written as follows: φ M (z) = φ(z) + ˆ f (z). (8) χ M (z) = χ m (z) + ĝ(z). (9) Based on Eqs. (1) and (2), we set χ(z) as an auxiliary function in the following. χ M (z) = zφ (z) + ψ (z) (10) The matrix is an isotropic material and the central point of the circular boundary is the origin and its radius is a 1. Using the mirror projection of a point z that was described by Moriguchi (1) on the boundary, we have a 2 1 /z. Hence, the auxiliary function χ m(z) may be replaced by χ m (z) = a2 1 z φ (z) + ψ (z), (11) where, f and g are arbitrary functions, after we establish f ˆ(z) and ĝ(z) in the following equation using the principle of mirror projection: ˆ f (z) = f ( a 2 1 /z), ĝ(z) = g ( a 2 1 /z). (12) Equation (12) is obtained when the center of the j th hole coincides with the origin. This problem reduces to finding f and g such that the continuities given by Eqs. (1) and (2) are satisfied. Imposition of the boundary conditions Eqs. (6) and (7) yields f (z) = 1 χ m (z), g(z) = φ(z). (13) Hence, we obtain the general solutions in this subsection in the following forms: φ M (z) = φ(z) + 1 χ m (a 2 K 1 /z), (14) M χ M (z) = χ m (z) + φ(a 2 1 /z). (15) where { 1 Hole = Rigid Inclusion κ M (16) 542

4 Fig. 2 Infinite elastic medium with a single hole or rigid inclusion under uniform stresses Problem in the presence of a single circular hole under uniform shear stresses In this section, we consider the problem of disturbing the uniform stresses σ x, σ y, and τ xy at infinity in Fig. 2. The fundamental complex potential functions ϕ(z), ψ (z) are given at infinity z = as where φ(z) = τ z, ψ (z) = 2τ z, (17) τ = σ x + σ y 4, τ = σ y σ x 4 + i τ xy 2. (18) These functions do not have a singularity inside the region a 1 < z <. We consider the functions obtained by substituting Eq. (17) into Eqs. (14) and (15), and coinciding with the initial condition σ x, σ y, and τ xy at infinity. The auxiliary function χ(z) defined in Eq. (11) at infinity z = reduces to χ m (z) 2τ z. (19) The functions obtained by substituting Eq. (19) into Eqs. (14) and (15) are general solutions in this subsection, and we obtain the following: φ M (z) = τ z + 2τ a 2 1 z, (20) χ M (z) = 2τ z + τ a2 1 z. (21) These solutions coincide with the solution of Moriguchi (1) and Hirashima (4)(5). When the central point of the boundary L 1 moves to the arbitrary point z 1, Eqs. (20),(21) can be reduced to φ M (z) = φ(z) + 1 χ m a2 1 + z 1. (22) z z 1 χ M (z) = χ m (z) + φ a2 1 + z 1. (23) z z 1 543

5 Fig. 3 Geometry of an infinite elastic medium with many holes or rigid inclusions General solution in the presence of many circular holes In this section, we use the solution presented in the previous section as a starting point for obtaining the solution of many circular holes that have radii a j and origins z j ( j = 1, 2,, Q), as shown in Fig. 3. In order to analyze the problem of rigid inclusions, we need only change the coefficient from 1 (Hole) to κ M (Rigid inclusion). This matrix is given by the complex potential functions ϕ(z) and ψ(z). We set the general boundary conditions, given by Eqs. (6) and (7), of the tractions and displacements. The proposed method can be regarded as an extension of the Schwarz alternating method, which, in principle, permits a solution to be obtained to any desired degree of accuracy. However, in the present study, the analysis is carried a step further. By exploiting the Möbius transformation, the general term of the series is obtained, and thus the general solution is written as a rapidly convergent series with an explicit general term. To this end, we define A j z = a2 j + z j, ( j = 1, 2,, Q). (24) z z j Moreover, A j specifies the operator with respect to the complex variable z. Normally, the left-hand side of Eq. (24) would be written as A j (z). However, for convergence, in the present paper, we denote A j z as in Eq. (24). Thus, A i A j z, for example, is expressed as follows: [ A i A j z = A i A j (z) ] a 2 i a 2 i = + z i = A j (z) z i a 2 + z i. (25) j + z j z z z i j For the purpose of the multi-hole problem, we first considered the problem with three holes (i.e. Q = 3 ) as a simple case. After that, we produced the general solution for many holes (Q is arbitrary number). Using the same technique, we could satisfy the continuities given by Eqs. (1) and (2) for the boundary L j. We first set fˆ 1 (z) and ĝ 1 (z) on L 1 as follows: φ M (z) = φ(z) + fˆ 1 (z). (26) χ M (z) = χ m (z) + ĝ 1 (z). (27) These functions reduce to finding f 1 and g 1 such that the continuities on L 1 are satisfied. The following are obtained using sets fˆ 2 (z) and ĝ 2 (z) on L 2 : φ M (z) = φ(z) + 1 χ m (A 1 z) + fˆ 2 (z). (28) 544

6 χ M (z) = χ m (z) + φ(a 1 z) + ĝ 2 (z). (29) These functions reduce to finding f 2 and g 2 such that the continuities on L 2 are satisfied. The following are obtained using sets fˆ 3 (z) and ĝ 3 (z) on L 3 : φ M (z) = φ(z) + 1 χ m (A 1 z) + 1 χ m (A 2 z) + φ(a 1 A 2 z) + ˆ f 3 (z). (30) χ M (z) = χ m (z) + φ(a 1 z) + φ(a 2 z) + χ m (A 1 A 2 z) + ĝ 3 (z). (31) Applying the continuity on L 3, we obtain fˆ 3 (z) and ĝ 3 (z). Note that the boundary condition on L 1 is not satisfied for L 2 and L 3 by the previous steps. For this reason, we may set fˆ 4 (z) and ĝ 4 (z) to satisfy the continuity on L 1. φ M (z) = φ(z) + 1 χ m (A 1 z) + 1 χ m (A 2 z) + φ(a 1 A 2 z) + 1 χ m (A 3 z) + φ(a 1 A 3 z) + φ(a 2 A 3 z) + 1 χ m (A 1 A 2 A 3 z) + ˆ f 4 (z). (32) χ M (z) = χ m (z) + φ(a 1 z) + φ(a 2 z) + χ m (A 1 A 2 z) + φ(a 3 z) + + χ m (A 1 A 3 z) + χ m (A 2 A 3 z) + φ(a 1 A 2 A 3 z) + ĝ 4 (z). (33) We applied the continuity on L 1, repeating the previous steps and obtaining these additional terms each time. In this way, we could obtain the following explicit solution of the in-plane problem in the presence of many circular holes or rigid inclusions. To this end, we used φ M (z) = φ(z) + + n=1 φ(m p (n) q (n)z) χ m (A q (0) M p K (n) q (n)z). (34) M n=0 χ M (z) = χ m (z) + + χ m (M p (n) q (n)z) + n=1 + n=0 φ(a q (0) M p (n) q (n)z). (35) The coefficients in the above expressions are given as follows. The arguments p (i), q (i) indicate different arguments from the index i and p (i), q (i) have values from 1 to Q. In addition, δ p(i) is q (i) Kronecker delta. Now, the right side functions of Eqs.(34) and (35) are shown by φ(m p (n) q (n)z) = n i=1 p (i) =1 q (i) =1 χ m (A q (0) M p (n) q (n)z) = δn 0 +(1 δ n 0 ) n q (0) =1 {1 (1 δ i 1 )δq(i 1) χ m (A q (0)z) i=1 q (0) =1 p (i) =1 q (i) =1 p (i) (1 δ q(i 1) We note that these functions have the following relations: φ(m p (n) q (n)z) = Q p (n) =1 q (n) =1 χ m (A q (0) M p (n) q (n)z) = p (n) =1 q (n) =1 (1 δ q(n 1) p (n) (1 δ q(n 1) p (n) p (i) }(1 δ p(i) q (i) )φ(a p (i) A q (i)z). (36) )(1 δ p(i) q (i) )χ m (A q (0)A p (i)a q (i)z). (37) )(1 δ p(n) )φ(m q (n) p (n 1) q (n 1) A p (n)a q (n)z). (38) )(1 δ p(n) )χ q (n) m (A q (0) M p (n 1) q (n 1)A p (n)a q (n)z). (39) 545

7 The above coefficients are given by A p (n) A q (n)z set = AI nz + Bn I Cnz I + Dn I. (n 1) An I = a 2 zq q (n) (n) 2 + z p (n)z q (n), Bn I = (a 2 z q (n) q (n) 2 )z p (n) (a 2 z p (n) p (n) 2 )z q (n), Cn I = z q (n) z p (n), Dn I = a 2 zq q (n) (n) 2 + z p (n)z q (n). where we set (40) = AII n 1 z + BII n 1 Cn 1 II z +. (41) DII n 1 We obtained the following relations from the above recursions. set M p (n 1) q (n 1)z M p (n) q (n)z = M p (n 1) q (n 1) A p (n) A set q (n)z = AII nz + Bn II Cn II z + Dn II. (42) where, An, II Bn, II Cn II and Dn II are expressed as the following recursions. A0 II = DII 0 = 1, BII 0 = CII 0 = 0. An II = (An 1 II AI n + Bn 1 II CI n), Bn II = (An 1 II BI n + Bn 1 II DI n), Cn II = (Cn 1 II AI n + Dn 1 II CI n), Dn II = (Cn 1 II BI n + Dn 1 II DI n). (n 1) (43) and A q (0)z set = AIII 0 z + BIII 0 C0 III z +, DIII 0 A0 III = z q (0), BIII 0 = a2 z q (0) q (0) 2, C0 III = 1, DIII 0 = z q (0). From the above relations, we obtain (44) A q (0) M p (n) q (n)z = A q (0) M p (n 1) q (n 1) A p (n)a set q (n)z = AIV n z + Bn IV Cn IV z + Dn IV. (45) where An IV Bn IV Cn IV Dn IV = A III 0 AII n + B III = A III 0 BII n + B III = C III 0 AII n + D III = C III 0 BII n + D III 0 CII n, 0 DII n, 0 CII n, 0 DII n. (n 1) In addition, An, I Bn, I Cn, I Dn, I An, II Bn, II Cn II, Dn, II A0 III, BIII 0, CIII 0, DIII 0, AIV n, Bn IV, Cn IV and Dn IV are complex constants using the index n, which means the calculation of n-count, and they are known constants because they satisfy the above recursions. The sign set = means that the sign makes a connection with the complex variables and complex coefficients on the right-hand side of the equation. From the above results, we obtained the external theoretical solutions. After that, we show the analysis solutions in a concrete example General solution in the presence of many circular holes under uniform stresses In this section, we consider the problem in the presence of many circular holes or rigid inclusions disturbing the uniform stresses σ x, σ y and τ xy at infinity in Fig. 4. In this problem, these functions do not have a singularity inside the region a j < z < ( j = 1, 2,, Q). Therefore, the fundamental complex potential functions ϕ(z) and ψ (z) (46) 546

8 Fig. 4 Infinite elastic medium with many holes or rigid inclusions under uniform stresses. are given at infinity z =, as shown in Eq. (17) through (19). The functions obtained by substituting the above equation into Eqs. (34) and (35) are general solutions to this problem, and are obtained as follows: φ M (z) = τ z + τ + n=1 χ M (z) = 2τ z + 2τ + τ + M p (n) q (n)z + 2 A q (0) M p K (n) q (n)z. (47) M n=0 n=1 M p (n) q (n)z + τ + n=0 A q (0) M p (n) q (n)z. (48) These solutions coincide with the solution of Moriguchi (1) and Hirashima (4)(5) for reduction to the single-hole problem. 3. Numerical Examples Then, we must account for the convergence of Eqs. (34), (35). Generally, when many inclusions are near or tangential to each other, the relative error may be large; that is to say, the convergences of the series tend to be large. We use n with a tolerance of relative error within 1% under the n and n 1 counts about the displacement u x, u y and stresses σ x, σ y, τ rθ in the matrix. As an example, we produce the convergence properties of numerical examples in Fig.7. We consider a geometry where the problem has three circular holes that have distance D/a 1 = 0.1 in the x-axis. In the plane stress problem, we denote the convergence properties of the problem under uniform stress σ y. Table 1 Table of a relative error [RE] u x and σ θ at θ = 0 in calculation of n-count for 2 nd Inclusion. n u x /a RE [%] σ θ /σ y RE [%]

9 Fig. 5 Graph of a relative error [RE] in calculation of n-count. Table.1 shows the relative error [E r ] of stresses σ θ /σ y and displacements u x /a 1 on the boundary as the value of n changes, and Fig.5 is given by the Table1. From this figure, we can obtain a very precise value when we set it at a higher level. In this example, we set n such that [E r ] is lower than 1%. Specifically, the tolerable value is confirmed to be sufficiently satisfied when we perform a general analysis using n = Problems under uniform shear stresses Fig. 6 Graph of σ θ around L 2 under σ y, when L 1, L 3 approaches. In this section, we show the stresses and displacements under a uniform stress σ y in the plane stress state using Eqs. (47) and (48) given in Section 2.5. Three holes that have the same radii (a 1 = a 2 = a 3 ) are arranged on the x-axis in the matrix. We observe the disturbances of the 2 nd hole, when the other 1 st and 3 rd hole approaches the 2 nd hole from a great distance. 548

10 Fig. 7 Distribution of τ max for the case of three holes under σ y. Fig. 6 shows the stress σ θ on the boundary L 2. When D/a 1 = 10, these results were in complete agreement with the results σ θ = 3σ y at θ = 0, 180 σ θ = σ y at θ = 90, 270 reported by Moriguchi (1). We can thus find the interacting disturbances of the holes on each other about D/a 1 = 2. Fig. 7 shows the distribution of τ max for the case of a single hole (top figure) and three holes (bottom figure) under σ y, when D/a 1 = 0.1. Three rigid inclusions that are the same shape were arranged on the x-axis in the matrix. We observed the disturbances of the 2 nd inclusion, when the 1 st and 3 rd inclusions approach the 2 nd inclusion from a great distance. Fig. 8 shows the stresses σ θ, σ r, τ rθ on the boundary L 2. Fig. 9 shows the distribution of τ max for the case of a single rigid inclusion (top figure) and three rigid inclusions (bottom figure) under σ y, when D/a 1 = 0.1. Fig. 8 Graph of σ θ, σ r, τ rθ around L 2 under σ y, when L 1, L 3 approaches. Fig. 10 shows the distribution of τ max for the case of four holes (left figure) and five holes (right figure) under σ y. In the case of five holes, these radii are all same and the interference 549

11 Fig. 9 Distribution of τ max for the case of three rigid inclusions under σ y. D/a 1 with each other is 0.1. And the geometry in the case of four holes is the same as in the case of five holes. From the disturbance of five holes, we can observe the concentrated stress at ±45 at the 5 th hole at center. Subsequently, Fig. 11 shows the same geometry for the rigid inclusions. Fig. 10 Distribution of τ max for the case of four and five holes under σ y. Fig. 12 shows the distribution of τ max for the case of seven holes (left figure) and seven rigid inclusions (right figure) under σ y. These radii are all same and the interference D/a 1 with each other is 0.1. In the case of holes, we can observe the concentrated stress horizontally (0, 180 ). And in the case of rigid inclusions, we can observe it as ±60 for each of the 7 th inclusions. 550

12 Fig. 11 Distribution of τ max for the case of four and five rigid inclusions under σ y. Fig. 12 Distribution of τ max for the case of seven holes or rigid inclusions under σ y. 551

13 4. Concluding Remarks In the present paper, we examined the in-plane problem of a two-dimensional isotropic matrix containing many circular holes or rigid inclusions subjected to arbitrary loading and produced the general solution to find the stresses and displacements. The purpose of this characteristic study was to apply the Moriguchi s reflection principle. Using these solutions, several numerical examples were presented graphically. These problems have been solved using different numerical analysis methods such as the finite element method (FEM) and the boundary element method (BEM). However, our studies were developed in order to observe the interacting disturbances for many circular holes with high precision. Our next project is to produce the general solution for many elastic inclusions inside a matrix. References ( 1 ) Moriguchi, S., Two Dimensional Elastic Theory (in Japanese), (1956), pp. 1-77, Iwatani Ltd. (I.S.Sokolnikoff:Mathematical Theory of Elasiticity, McGraw-Hill,(1956), Chap5) ( 2 ) Honein, E., Honein, T., and Herrmann, G., ON TWO CIRCULAR INCLUSIONS IN HARMONIC PROBLEMS, QUARTERLY OF APPLIED MATHEMATICS, Vol. L, No. 3 (1992), pp ( 3 ) Hamada, et al., A Numerical Method for Stress Concentration Problems of Infinite Plates with Multiholes Subjected to Uniaxial Tension - 1st Report, Transactions of the Japan Society of Mechanical Engineers, Vol. 36, No. 288 (1970), pp ( 4 ) Hirashima, K., and Sugisaka, N., Analytical Solution for Out-of-Plane Problems with Two Circular Elastic Inclusions, Transactions of the Japan Society of Mechanical Engineers, Series A, Vol. 60, No. 575 (1994), pp ( 5 ) Kimura, K., Hirashima, K., and Hirose, Y., Analytical Solutions for In-Plane and Outof-Plane Problems with Elliptic Hole or Elliptic Rigid Inclusion and Their Applications Transactions of the Japan Society of Mechanical Engineers, Series A, Vol. 58, No. 555 (1992), pp ( 6 ) Hirashima, K., Miyagawa, M., and Nakane, S., Analysis of Antiplane Problems with Singular Disturbances for Isotropic Elastic Medium Having Many Circular Elastic Inclusions, Transactions of the Japan Society of Mechanical Engineers, Series A, Vol. 64, No. 623 (1998), pp ( 7 ) Miyagawa, M., Suzuki,T., and Shimura, J., Analysis of In-Plane Problems with Singular Disturbances for Isotropic Elastic Medium Which Has Two Circular Holes or Rigid Inclusions, Transactions of the Japan Society of Mechanical Engineers, Series A, Vol. 75, No. 750 (2009), pp ( 8 ) Miyagawa, M., Tamiya, T., Shimura, J., and Suzuki, T., Analysis of In-Plane Problems for Isotropic Elastic Medium Which Has Two Circular Elastic Inclusions, Transactions of the Japan Society of Mechanical Engineers, Series A, Vol. 76, No. 762 (2010), pp