Smulaton of 2D Elastc Bodes wth Randomly Dstrbuted Crcular Inclusons Usng the BEM Zhenhan Yao, Fanzhong Kong 2, Xaopng Zheng Department of Engneerng Mechancs 2 State Key Lab of Automotve Safety and Energy Tsnghua Unversty Bejng, 00084 Chna Abstract Based on the Rzzo s drect boundary ntegral equaton formulaton for elastcty problems, elastc bodes wth randomly dstrbuted crcular nclusons are smulated usng the boundary element method. The gven numercal examples show that the boundary element method s more accurate and more effcent than the fnte element method for such type of problems. The presented approach can be successfully appled to estmate the equvalent elastc propertes of many composte materals. Introducton As composte materals are appled more and more to many mportant engneerng projects, researchers have pad much more attenton to the smulaton of the composte materals and to the estmaton of ther equvalent elastc propertes. To estmate the equvalent elastc propertes of composte materals, many theoretcal models have been developed, such as the composte cylnder model [], the dlute or non-nteractng soluton [2], the self-consstent method [3], the generalzed self-consstent method [4~7] and the Mor-Tanaka method [8~]. At the same tme, only few data obtaned from numercal smulatons can be found n lterature. A numercal procedure, based on the seres expanson of complex potentals, was proposed n Ref. [2], but only several knds of perodc arrays of holes were consdered. A sprng force model has been used to smulate a sheet contanng crcular holes arranged as trangular and hexagonal arrays [3]. A numercal equvalent ncluson method was presented [4], whch can be appled to analyze stress felds n and around nclusons of varous shapes by the fnte element method. But ths approach can only be appled to smulate the elastc body wth only one ncluson. The smulaton of the sheet wth randomly or normally dstrbuted crcular holes was nvestgated n our group several years ago [5]. Based on Rzzo s drect BEM [6] for elastcty problem, a new BEM approach for the smulaton of elastc bodes wth randomly dstrbuted crcular nclusons s proposed n ths paper. Several numercal examples are presented to menstruate the advantages of ths new BEM approach over the doman-based FEM approach. 270
BEM for smulaton of 2D elastc body wth a crcular ncluson The model of 2D elastc body wth a crcular ncluson s shown n Fgure, where Ω ΙΙ denote the doman of matrx and ncluson respectvely, Ω Ι, Γ the matrx-ncluson nterface boundary, and t u ΓI, Γ I ndcate the gven tracton part and gven dsplacement part of the outer boundary of matrx materal doman Γ Ι. Fg.. Model of 2D elastc body wth a crcular ncluson The boundary ntegral equatons can be wrtten for the matrx and ncluson subdoman respectvely: Ι I Γ I+Γ Γ I+Γ ΙΙ II = (, ) d Γ (, ) dγ Γ Γ C p u p = U p, q t q d Γ T p, q u q dγ αβ β αβ β αβ β C p u p U p q t q T p q u q αβ β αβ β αβ β Where superscrpts Ι and ΙΙ ndcate matrx and ncluson subdoman respectvely, p and q stand for the source pont of fundamental soluton and feld pont on the boundary respectvely, Cαβ ( p) s a free term determned from the shape of the Ι ΙΙ boundary at source pont p, Uαβ ( p, q), Tαβ ( p, q) and Tαβ ( p, q) are fundamental solutons of 2D elastcty problem, and uβ, tβ are boundary dsplacement and boundary tracton respectvely. After dscretzaton usng lnear or quadrc boundary elements, equatons () can be rewrtten nto matrx form as follows: () I I I I I I I I A A2 A 3 U B B2 B 3 T I I I I I I I I A2 A22 A23 T = B2 B22 B23 U I I I I I I I A3 A32 A33 U B3 B32 B33 T (2a) II II II GT = HU (2b) 27
Ι where U and T Ι represent the unknown nodal dsplacement vector and the gven t Ι nodal tracton vector respectvely on the gven tracton boundary Γ I, T and U Ι, the unknown nodal tracton vector and the gven nodal dsplacement vector on the gven u dsplacement boundary Γ I, U, the unknown nodal dsplacement vector on the Ι matrx-ncluson nterface boundary Γ. Whle T and T ΙΙ stand for the unknown nodal tracton vector on the matrx-ncluson nterface boundary for the matrx and ncluson materal respectvely. For the matrx materal, the subdoman s a multply connected doman, the nodal number and the correspondng sequence of boundary varables can be arranged sequentally n the postve drecton of the boundary. When advancng n the postve drecton of boundary, the nner doman surrounded by the boundary s always at the left sde. For the ncluson, the nodal number of the matrx-ncluson nterface should keep n lne wth that of matrx materal subdoman. So the nodal number and the correspondng sequence of boundary varables should be arranged sequentally n the negatve drecton of the boundary for the ncluson tself. In equatons (2), the dsplacement contnuty on the matrx-ncluson nterface has been taken nto account. The nterface condton for the tractons can be wrtten as: II I T = T (3) Substtutng equaton (3) nto the equaton (2b), we can obtan the relaton between the tractons and dsplacements of the matrx on the nterface: I II II T = G H U (4) Substtutng equaton (4) nto equaton (2a), we can obtan the fnal system of equatons for the 2D elastc body wth a crcular ncluson as follows: I I I I II II A A2 A3 + B3 G H I I I U B B 2 I I I I I II II I I I T A2 A22 A23 + B23 G H T = B2 B22 I I I I I I I II U II U B3 B32 A 3 A32 A33 + B33 G H BEM for smulaton of 2D elastc body wth randomly dstrbuted crcular nclusons The model of 2D elastc body wth randomly dstrbuted dentcal crcular nclusons s shown n Fgure 2. Where Ω 0 s the subdoman of matrx materal, Ω, Ω2, L, Ω, L, Ωn, the subdomans of ncluson materal, Γ the matrx-ncluson nterface boundares, and Γ 0 the outer boundary of the matrx materal subdoman. (5) 272
Fg. 2. Model of 2D elastc body wth randomly dstrbuted dentcal crcular nclusons If the conventonal subdoman boundary element method s adopted, n + equaton systems for the n + subdomans should be solved. As the number of nclusons ncreases, the computng tme wll ncrease sgnfcantly. If we notce that the relatons between the tractons and dsplacements of each dentcal ncluson are just the same and smlar as shown n equaton (4), we can reduce the full computaton to the soluton of the equaton for the matrx materal doman wth nner boundary condtons smlar to equaton (4), whch can be descrbed as follows: where n A A2 3 L 3 L 3 U B B 2 n A2 A22 23 23 L L 23 T B2 B22 n A 3 A32 33 L 33 L U 33 B3 B 32 T M M M O M O M M = M M n A U 3 A32 33 L 33 L U 33 B 3 B 32 M M M O M O M M M M n n n n nn n A3 A32 33 L 33 L n n 33 U B3 B32 j j j 33 33 33 3 = A3 + B3 G H 23 = A23 + B23 G H = A + B G H The frst and the second subscrpt ndcate the boundary where the source pont node p and the feld pont node q located, and, 2, 3 denote the tracton gven part, dsplacement gven part of outer boundary and the nner boundares respectvely. To dstngush dfferent nner boundares, the superscrpts are used. The frst and the second (f there s a second one) superscrpt ndcate the number of nner boundary where the (6) (7) 273
source pont node p and the feld pont node q located. Matrces G and H n equaton (7) are coeffcent matrxes for the ncluson materal subdomans. As all the randomly dstrbuted crcular nclusons are dentcal, t needs to form the coeffcent matrxes G and H for a certan ncluson only one tme. In equaton (6), U, T and U ndcate the unknown dsplacement vector on the tracton gven part of outer boundary, the unknown tracton vector on the dsplacement gven part of outer boundary and the unknown dsplacement vector on the -th nner nterface boundares respectvely. On the other hand, T and U stand for the gven tracton vector and the gven dsplacement vector on the outer boundary respectvely. For the case of 2D elastc body wth randomly dstrbuted crcular nclusons of dfferent sze, the above-presented approach can be generalzed no dffculty, provded the number of dfferent sze s much less than the number of nclusons. In such case, the equaton (7) should be modfed as follows: k () 3 = 3 + 3 k () k() 23 = 23 + 23 k() j j j k () 33 = 33 + 33 k A B G H A B G H () A B G H where k =, 2, L, m denote dfferent ncluson sze, k() can be also a random functon, and for the ncluson of each sze the matrces G and H should be computed once. Numercal examples ) A square sheet wth a crcular ncluson at the center subjected to unform tenson on two opposte edges (8) Fg. 3. Model of a square sheet wth a crcular ncluson at the center subjected to unform tenson on two opposte edges 274
Fgure 3 shows the computatonal model. The sde length a = 00 mm, the radus of the crcular ncluson r = 2 mm, the tracton q = 0 MPa, the materal propertes of the matrx E = 0 MPa, v = 0.3, and that of the ncluson E2 = ke and v 2 = 0.3. The rgd body dsplacement s constraned properly. In the computaton, the matrx-ncluson nterface s dvded nto 0 quadratc elements. Fgure 4 shows the absolute value of crcumferental stress σ θ on the matrxncluson nterface obtaned by BEM n comparson wth the analytcal soluton, for the specal case of crcular hole. The sold lne s the analytcal soluton, and the sold dots show the BEM results. As to the case of an nfnte sheet wth a crcular hole at the center, the analytcal soluton can be wrtten as: σθ = q 2cos2θ (9) It s obvous that the maxmum of the crcumferental stress σ θ s 3q = 30MPa when θ s equal to π 2 or 3π 2, and the mnmum s zero when θ s equal to π 6, 5π 6, 7π 6 or π 6. It can be found that the present numercal results agree wth the analytcal soluton very well. The maxmum error of BEM results s less than 0.02%. σ θ (MPa) 35 30 25 Theoretcal soluton BEM soluton 20 5 0 5 0-5 0 60 20 80 240 300 360 θ( ) Fg. 4. Comparson of the crcumferental stress σ θ on the matrx- ncluson nterface obtaned by the BEM and the analytcal soluton 275
3.5 Stress concentraton factor 3 2.5 2.5 0.5 0 2 3 4 5 6 7 8 9 0 Modulus rato of ncluson and matrx materals Fg. 5. Relaton between stress concentraton factor and the ncluson-matrx modulus rato obtaned from BEM scheme Fgure 5 shows that the stress concentraton factor obtaned by BEM vares wth the ncluson-matrx modulus rato, the sold dots show the numercal results, and the sold lne s nterpolated curve. It can be found from Fgure 5 that the stress concentraton factor decreases quckly wth the ncrease of ncluson-matrx modulus rato when ncluson materal s softer than matrx materal. Furthermore, stress concentraton factor ncreases slowly wth the ncrease of the ncluson-matrx modulus rato when ncluson materal s harder than matrx materal. As to the analytcal soluton of an nfnte sheet wth a crcular hole or a crcular rgd core at the center, the stress concentraton factor s equal to 3.0 or.5 from elastc theory respectvely. The correspondng results obtaned by BEM are 3.0005 and.5002 respectvely. It s obvous that the present numercal results agree wth the elastc theory very well. 2) A square sheet wth two very close nclusons subjected to unform dsplacement on one edge Fgure 6 shows the computatonal model. The sde length a = 00 mm, the radus of two very close crcular nclusons R = 5 mm, the mnmum dstance between the matrx-ncluson nterface boundares b = 0.5 mm, the gven unform dsplacement on the rght edge d =.0 mm, the materal property of the matrx E = 0 MPa, v = 0.3, and that of the ncluson E2 = ke, v 2 = 0.3. It s obvous that the stress concentraton factor ncreases to the maxmum when the ncluson-matrx modulus rato approaches zero. So the stress gradent around the matrx-ncluson nterface wll reach the maxmum for the case of two crcular holes. To ensure hgh accuracy, t s necessary to take fner mesh around the two nclusons for ether the BEM or FEM computaton. Fgure 7 shows the varaton of Von-Mses stress on one quarter of the matrx-ncluson 276
nterface by usng 0 quadratc boundary elements and 20 quadratc boundary elements. It can be found that the varaton s very small, whch ndcates a convergent soluton has been obtaned by usng only 0 quadratc boundary elements n BEM computaton. a=00mm R=5mm E =0 v =0.3 b=0.5mm E 2 =ke v 2 =0.3 d=.0mm Fg. 6. The model of a square sheet wth two very close nclusons under gven unform dsplacement on one edge Mses stress 0.3 0.25 0.2 0.5 0. 0 quadrc elements 20 quadrc elements 0.05 0-0.05-20 0 20 40 60 80 00 20 40 60 80 200 Fg. 7. Comparson of the Von-Mses stress on the quarter of the matrx-ncluson boundares by dfferent boundary element number θ 277
Mses stress 0.3 0.26 0.22 0.8 0.4 0. 0.06 0.02 BEM(0 quadrc elements) MSC.Marc(46356 quadrc elements) MSC.Marc(96 quadrc elements) -0.02-30 0 30 60 90 20 50 80 20 θ Fg. 8. Comparson of Von-Mses stress on the quarter of the matrx-ncluson nterface boundary obtaned from BEM and the famous MSC.Marc software Fgure 8 shows the comparson of Von-Mses stress on one quarter of matrx-ncluson nterface obtaned by the presented BEM and the FEM usng MSC/Marc software, for the case of two crcular holes. It can be found that the Von-Mses stress ncreases slowly to the results obtaned by BEM wth the mmense ncrease of fnte elements. It s obvous that the accuracy reached by MSC/Marc usng 46356 quadratc elements s far lower than by BEM usng 0 quadratc boundary elements on the nterface. The presented BEM s much more effectve than FEM for such knd of problems. Furthermore, for 2D elastc body wth randomly dstrbuted crcular nclusons, there wll be plenty of very close nclusons. Thus, the presented scheme of BEM s very sutable to such smulaton problems, and t has obvous advantage over the FEM. 3) A square sheet wth 00 randomly dstrbuted dentcal crcular nclusons subjected to unform tenson on the opposte edges Fgure 9 shows the computatonal model. The sde length of ths square sheet s 00mm, the tracton on the opposte edges s 0Mpa, the thckness of ths square sheet s mm, and the volume rato of all 00 nclusons s 0.4. Then the radus of the crcular ncluson can be determned automatcally. The model s taken as a plane stress problem. In addton, doman mesh s only needed for plottng results. After BEM computaton, we can obtan the deformaton pattern and stress dstrbuton. As examples, Fgure 0 shows the deformaton pattern for the case of 00 crcular holes, E 2 = 0, Fgure and Fgure 2 shows the Von-Mses stress dstrbuton of a square sheet wth 00 randomly dstrbuted dentcal crcular hole under the unform tenson on two opposte edges for the case of E2/ E = 0.5. 278
Fg. 9. Model of a square sheet wth 00 randomly dstrbuted dentcal crcular nclusons under unform tenson on two opposte edges Fg. 0. Deformaton pattern of a square sheet wth 00 randomly dstrbuted dentcal crcular hole under the unform tenson on two opposte edges (E 2 = 0) 279
Fg.. Deformaton pattern of a square sheet wth 00 randomly dstrbuted dentcal crcular nclusons under the unform tenson on two opposte edges ( E2 E = 0.5 ) Fg.2. Von-Mses stress dstrbuton of a square sheet wth 00 randomly dstrbuted dentcal crcular holes under unform tenson on two opposte edges ( E2/ E = 0.5 ) 280
Concludng Remarks ) A scheme of the BEM for the smulaton of 2D elastc bodes wth randomly dstrbuted crcular nclusons has been presented n ths paper. The gven numercal examples ndcate ts hgh accuracy and hgh effcency. 2) As for the elastc bodes wth randomly dstrbuted dentcal crcular nclusons, the presented BEM scheme has a dstnctve advantage over the FEM due to hgh stress gradent resulted from the presence of many very close nclusons. 3) The presented BEM scheme can be generalzed wthout dffculty to the elastc bodes wth randomly dstrbuted nclusons of dfferent geometrcal szes, dfferent shapes (ellptcal ncluson wth dfferent shape and prncpal drecton, cracks wth dfferent drecton, etc.) and dfferent elastc modulus. 4) The presented BEM scheme can be appled to estmate the equvalent elastc propertes of correspondng composte materals. 5) The presented BEM can be combned wth some knd of fast multpole algorthms. It s possble to effcently smulate the elastc bodes wth much more dfferent nclusons wth such fast algorthms. Snce ths manuscrpt was submtted two years ago, some related nvestgatons n the authors group have been publshed [7-20]. In those nvestgatons, a large number of numercal examples of the 2D elastc solds wth randomly dstrbuted nclusons, usng repeated smlar sub-doman BEM, have shown that ths method has hgher accuracy and hgher effcency, and provdes an effcent tool for the numercal smulatons of correspondng composte materals. Ths method can be appled to smulate not only the 2D solds wth nclusons of dfferent shapes, szes and materals, but also the nclusons wth nterphases layers. By applyng the fast multpole BEM nto ths feld, the scale of the computaton can be ncreased. In a prelmnary nvestgaton, the number of nclusons smulated was ncreased from 00 to 600. Further nvestgatons wll be carred out n two drectons: on the one hand, t wll be developed from 2D to 3D problems; on the other hand from the smulaton of effectve elastc modul to smulatons of the falure process of such composte materals. Acknowledgements Fnancal support for the project from the Natonal Natural Scence Foundaton of Chna under grant No. 9772025 and No. 072053 s gratefully acknowledged. References [] Hashn Z. The elastc modul of heterogeneous materals. J Appl Mech., 962; 29: 43~50. [2] Budansky Y. On the elastc modul of heterogeneous materals. J Mech Phys Solds, 965; 3: 223~227. 28
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