Universities of Leeds, Sheffield and York

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1 promoting access to White ose research papers Uniersities of Leeds, Sheffield and Yor This is an athor prodced ersion of a paper accepted for pblication in International Jornal of Solids and Strctres. White ose esearch Online UL for this paper: Paper: Ni, Z, Cheng, CZ, Ye, JQ and echo, N 009 A new bondary element approach of modeling singlar stress fields of plane V-notch problems. International Jornal of Solids and Strctres, White ose esearch Online eprints@whiterose.ac.

2 Accepted Manscript A new bondary element approach of modeling singlar stress fields of plane V-notch problems Zhongrong Ni, Changzheng Cheng, Jianqiao Ye, Naman echo PII: S DOI: 0.06/j.ijsolstr eference: SAS 6657 To appear in: International Jornal of Solids and Strctres eceied Date: 3 October 008 eised Date: 30 March 009 Accepted Date: 30 March 009 Please cite this article as: Ni, Z., Cheng, C., Ye, J., echo, N., A new bondary element approach of modeling singlar stress fields of plane V-notch problems, International Jornal of Solids and Strctres 009, doi: 0.06/ j.ijsolstr This is a PDF file of an nedited manscript that has been accepted for pblication. As a serice to or cstomers we are proiding this early ersion of the manscript. The manscript will ndergo copyediting, typesetting, and reiew of the reslting proof before it is pblished in its final form. Please note that dring the prodction process errors may be discoered which cold affect the content, and all legal disclaimers that apply to the jornal pertain.

3 ACCEPTED MANUSCIPT Sbmitted to < International Jornal of Solids and Strctres > on October 3, 008 eised by the athors on Febrary 4, 009 Second reision on March 30, 009 A new bondary element approach of modeling singlar stress fields of plane V-notch problems Zhongrong Ni a,, Changzheng Cheng a,c, Jianqiao Ye a,b, Naman echo c a School of Ciil Engineering, Hefei Uniersity of Technology, Hefei 30009, P China b School of Ciil Engineering, The Uniersity of Leeds, Leeds, LS 9JT, UK c EMESS/ EPF-Ecoles d Ingéniers, Sceax and LaMI/Uni.B.Pascal, Clermont Fd. France Abstract: In this paper, a new bondary element BE approach is proposed to determine the singlar stress field in plane V-notch strctres. The method is based on an asymptotic expansion of the stresses in a small region arond a notch tip and application of the conentional BE in the remaining region of the strctre. The ealation of stress singlarities at a notch tip is transformed into an eigenale problem of ordinary differential eqations that is soled by the interpolating matrix method in order to obtain singlarity orders degrees and associated eigen-fnctions of the V-notch. The combination of the eigen-analysis for the small region and the conentional BE analysis for the remaining part of the strctre reslts in both the singlar stress field near the notch tip and the notch stress intensity factors SIFs. Examples are gien for V-notch plates made of isotropic materials. Comparisons and parametric stdies on stresses and notch SIFs are carried ot for arios V-notch plates. The stdies show that the new approach is accrate and effectie in simlating singlar stress fields in V-notch/crac strctres. Keywords: linear elasticity, V-notch, stress intensity factor, bondary element method, asymptotic expansion.introdction V-notches of bonded dissimilar materials are freqently encontered in engineering applications, sch as in welded connections and strctres strengthened with fiber reinforce polymer plates. In sch cases, the stress concentration near the sharp notch and the interface Corresponding athor. Tel.: address: ni-zr@hft.ed.cn

4 ACCEPTED MANUSCIPT end is ery high. In particlar, the pea stress at the notch tip is singlar according to the theory of elasticity. For ealating fatige strength of the strctres, the stress distribtion arond the notch tip and the notch/generalized stress intensity factors GSIFs are important mechanical qantities to be determined. Since there exist mltiple stress singlarities at a general V-notch strctre, it is difficlt to model the singlar stress field by conentional methods. Analytical soltions of the notch SIFs hae been fond only for a few special cases. Extensie stdies hae been carried ot to deal with the singlar stress fields of V-notch problems. Gross et al. 964; 97 and Carpenter 984 obtained the GSIFs for plane V-notch problems by a bondary collocation method. Chen 995 deried the notch SIFs of some plane V-notch problems by sing the body force method, in which the density fnctions associated to the body forces had to be configred in adance. By applying Lehnitsii formalism and Stroh formalism, Ting 996 stdied the soltions of the stress fields of general anisotropic elastic materials and composites. For the two-dimensional V-notches, Szabó and Yosibash 996 obtain first a conentional finite element method soltion for the entire domain, which was then sed as bondary conditions of a small sb-domain arond the notch tip. Then a special finite element formlation in conjnction with the assmption of the asymptotic expansion of the displacement field in the sb-domain, where the singlar eigenpairs hae been obtained by other methods, is sed to extract the GSIFs. Eidently, the efficiency of the method depends on the accracy of the preliminary FE soltion. Matsmoto et al. 000 deried a bondary integral expression of the interaction energy release rate for a craced body and compted the stress intensity factors of bimaterial interfacial cracs. Based on the analytical expressions of the local stress field near a V-notch tip, Atzori et al. 00 estimated the fatige strength of welded joints with notches in alminm alloys at a gien nmber of cyclic loading. With the soltions of stress singlarities, Labossiere and Dnn 999 and Hw and Ko 007 compted the stress intensity factors throgh the near tip displacement and stress fields from the conentional finite element analysis as well as the calclation of the path-independent H-integral Stern et al., 976. Based on a critical interface corner stress intensity factor, eedy and Gess 993 gae a failre criterion for the strength of the adhesiely bonded btt tensile joint. Gómez and Elices 003 presented a simple approximate formlation of fractre criterion for sharp V-notches of some materials, sch as steel, alminm PMMA and PVC. Utilizing the reslts of the GSIFs, Liieri and Lazzarin 005 ealated the fatige strength of steel and alminm welded joints as a type of V-notched strctres by stdying on a large amont of experimental data. By sing coherent gradient sensing CGS method, Yao et al. 006 experimentally stdied mode-і stress singlarity and fractre behaiors of V-notch tip. At present, the most commonly sed nmerical methods to determine stress distribtions

5 ACCEPTED MANUSCIPT of a V-notch are the finite element and bondary element methods BEM. The conentional FEM Pagea et al., 995; Mohammed and Liechti, 00 and BEM Tan et al., 99; X et al., 999; Ioa et al., 007 sally se ery fine meshes in the icinity of the notch tip to simlate the singlar stress field. Then the stresses or displacements at some mesh points near the notch tip are sed to determine the GSIFs by the extrapolation techniqe. Howeer, any frther increase of element nmber has ery limited impact on improing the accracy of the approaches. This is becase the conentional elements near the notch tip can not reflect the essence of the stress singlarity in the tip region. It is well nown in the finite element analysis that the qarter-point element Henshel and Shaw, 976;.S. Barsom, 976 at crac tip can efficiently model the singlar stress field near the crac tip, in which the mid-side nodes near the crac tip are placed at the qarter point. Sbseqently the idea was introdced to the BEM by Blandford et al. 98. In fact, the shape fnctions of the qarter-point element can only represent stress singlarity of order -/ that is correct for line cracs of isotropic materials. Unfortnately the qarter-point element is not sitable to model the stress field near a V-notch tip. esearchers hae been trying for some time to constrct a singlar element for modeling stress fields in the notch tip region. To the athors best nowledge, it has not been ery sccessfl. On the basis of asymptotic expansion of the stresses near a V-notch tip, a special finite element method was recently sed to deal with some V-notch problems. For V-notches and line cracs, according to the singlar stress fields of the type r, Seweryn 00 ij f ij too two or three dominating stress eigen-fnctions as analytical constrains in the notch tip region. Then the reslting elements in terms of the analytical constrains were applied to model the stress field in the core region arond the notch tip. The remaining area of the notched strctre can be modeled sing the conentional FEM. This approach reqires predefined analytical stress eigen-fnctions. For the V-notch of isotropic materials sbjected to arios bondary conditions, Seweryn and Molsi 996 gae the analytical soltions of the eigen-fnctions, and explicit closed-form eigen eqation, from which the eigenales can be determined by an iteration techniqe. Howeer the analytical stress eigen-fnctions are not always acqirable for general V-notch problems of bonded dissimilar mlti-material. To address this problem, some approximate stress eigen-fnctions were proposed to constrct the stress fnctions. For example, sing the FEM, Carpinteri et al. 006 calclated the first leading singlar exponent and the ModeⅠ stress amplitde for a mlti-layered beam with a crac or re-entrant corner that forms symmetrically a bimaterial interface. Chen and Sze 00 proposed a hybrid finite element with asymptotic expansion to determine the stress 3

6 ACCEPTED MANUSCIPT exponents and stress distribtions of bonded bimaterial V-notches. On the basis of this wor, Sze et al. 00 and Chen and Ping 007 deeloped the approach frther and sed it for the analyses of singlar electro-elastic fields arond a V-notch of piezoelectric material. By the same method, Ping et al. 008 analyzed the singlar stress fields of V-notched anisotropic plates. In this stdy, a new BE approach is proposed to determine the singlar stress fields of plane V-notch strctres in conjnction with asymptotic expansions of the stress distribtion near the notch tips. To find the stress field near a tip, a small sector arond the notch tip is isolated from the V-notch strctre as a free body. Based on the expression of the asymptotic expansion, the ealation of stress singlarities in the notch tip region is transformed into an eigenale problem of ordinary differential eqations ODEs. The eigenale problem is then soled by the interpolating matrix method Ni, 993 to obtain the singlarity orders and the associated eigenectors of the V-notch. Frther to the eigen-analysis of the small sector, the bondary integral eqation is applied to treat the remaining region of the notched strctre. Ths the displacement and stress fields of the V-notch strctre and the notch SIFs are finally determined.. The eigen-analysis of the stress singlarities of the plane V-notch problems x O x Fig.. A plane V-notch problem. Let s consider a plane V-notch of isotropic material with an opening angle as shown in Fig.. Define a polar coordinate system,, with an origin at the notch tip. Two radial edges of the notch are denoted by and. A small sectorial region within the range of a radis of the V-notch tip is ct ot from the solid region of the V-notch strctre as shown in Fig. b. The remaining part of the notch strctre is referred to as region See Fig. a. There are, 4

7 ACCEPTED MANUSCIPT, i,. i i i a b x x O x O x Fig.. A plane V-notch. a The remaining part after a sector arond the notch tip being remoed. b The sector taen arond the notch tip In the linear elasticity, it has been erified that the asymptotic displacement field in the notch tip region can be expressed as a series expansion with respect to the radial coordinate originating from the notch tip Yosibash and Szabó, 996. A typical term of the series can be written in the form where the exponent, A, A is called stress singlarity order; and are the associated eigen-fnctions which represent the displacement components in the - and -directions, respectiely; A is the amplitde coefficient of each term of the asymptotic expansion called also the generalized/notch stress intensity factor. Sbstitting Eq. into the strain-displacement relations of the linearly elastic theory yields the strain components. Then from Hooe s law of plane stress problems, the components of the plane stress tensor are expressed as E, A [ E, A [ E, A ] ] where d / d ; E is the Yong s modls and the Poisson s ratio. If the stress eigen-fnctions, and are defined as follows 5

8 ACCEPTED MANUSCIPT 6 ] [ ] [ E E E 3 The three plane stresses are,,, p p A A A Note that the eigen-pairs in Eq. do not depend on load conditions. In the small sectorial region, the body forces can be neglected and the eqilibrim eqations are Introdcing Eq. into Eq. 4 and remoing the common factor A from the eqations lead to a set of ordinary differential eqations ODEs as follows,, 0 ] [ 0 5 The appearance of in Eq. 5 reslts in a nonlinear eigen problem that is difficlt to sole. Here an alternatie approach is adopted to transfer the eqations into a linear eigenale problem. To this end, two new field ariables are introdced as below, g g 6 Hence, by introdcing g and g into Eq. 5, one has, 0 ] [ 0 g g 7 Assme that all the tractions on the two edges, and, near the notch tip are zero. That is

9 ACCEPTED MANUSCIPT Sbstittion of Eq. into Eq. 8 yields the bondary conditions as follows 0 0 and 9 If the edges are fixed along either or, the bondary condition can easily be expressed as 0, or 0a 0, or 0b By the aboe deriation, the ealation of the singlarity orders and the associated eigen-fnctions and near the V-notch tip is transformed to the soltion of a linear eigenale problem of the ODEs goerned by Eqs. 6 and 7 sbjected to the bondary conditions of Eq. 9 or 0. Throgh Eq. 3, and can be sed to determine the stress eigen-fnctions, and in the icinity of the notch tip. Frthermore, for the analysis of plane V-notch problems of bonded dissimilar mlti-materials, inclding orthotropic and anisotropic materials, the same dedction process as shown aboe can be implemented to compte eigen soltions of the associated ODEs. It is obios that Eqs. 6 and 7 are alid for each sb-domain with a single material for analyzing the stress singlarity orders near the interface tip of the dissimilar mlti-material. This process will prodce a set of ODEs that are similar to Eqs. 6, 7, 9 and 0. By soling the eigenale problem of the ODEs, the singlarity orders and the associated eigen-fnctions near the notch tip of the dissimilar mlti-materials are then determined. To determine the stress singlarities of the V-notch as described aboe, the ODEs need to be soled first. Ni 993 proposed a nmerical soltion called interpolating matrix method that can sole the eigenale problem of ODEs effectiely. By the interpolating matrix method, an interal, ] is diided into n sbinterals with n diisions. All the [ nnown fnctions in the ODEs are approximated with piecewise polynomial interpolation within each of the sbinterals. The highest deriaties at the diision points, appearing in the ODEs, are chosen as the basic nnowns of the discrete ODEs system. The interpolating matrix method has two distinct adantages: All fnctions and their deriaties appearing in the BVPs of ODEs are simltaneosly obtained with the same degree of accracy. This featre is particlarly beneficial to the calclation of stress field that reqires the first 7

10 ACCEPTED MANUSCIPT deriatie of the displacement fnctions; It can sole a general eigenale problems of ODEs and is conenient to write a general-prpose compter program. Generally, in the case of <80 0, there exist one or seeral stress singlarity orders in the range of e, 0 for a V-notch problem of dissimilar mlti-material. 3. The BE analysis of the stress fields and SIFs for the plane notch strctres For a plane notch strctre, the displacement components in the icinity of the notch tip can be expressed with the following expansion as Yosibash and Szabó, 996,, N N A A where,,, N are the associated amplitde coefficients with dimension mm ; N A is the trncated nmber of the eigenales. The e,0 A corresponding to the eigen-pairs of are eqialent to the generalized stress intensity factors of the V-notch. In general, the more terms Eq. taes, the bigger is the region arond the notch tip that is alid for the displacement soltion described by Eq.. The eigenales and the associated A,,,, N are sally complex and can be obtained by soling Eqs. 6, 7, 9, 0. The soltions can be expressed by ii A A iai i i I I where i ; sbscripts and I represent the real and imaginary parts of a complex ariable or fnction, respectiely. After introdcing Eq. into Eq., the displacement components,,, are obtained by taing the real parts as follows,, N I { A[ cos I ln sin I ln ] I I AI[ sin I ln cos I ln ]} I 3 Considering Eqs. and 3, the stress components in the icinity of the notch tip are denoted also by the asymptotic expansion as follows 8

11 ACCEPTED MANUSCIPT,,, N N N A A A 4 Inserting Eq. into Eq. 3 yields E i I {[ ] I I i[ I I I I ]} E {[ i I ] I I i [ I I I I ]} E [ i I I I i I I I ] 5 Sbstitting Eqs. and 5 into Eq. 4, then taing the real parts of,,, and,, respectiely, one obtains the following stress components:,,, N I { A[ cos ln sin ln ] I I I I I [ A sin ln I cos ln } I I I I 6 In order to combine the aboe soltion with the soltion in the region, the aboe displacement and stress components are transformed into a global Cartesian coordinate system. On the bondary See Fig. b, i, t i i, are sed to denote the displacement and traction components in the Cartesian coordinate system, ox x, respectiely. The transformation of the displacements and tractions from the polar coordinate system to the Cartesian coordinate system is shown below. cos sin sin cos t t cos sin sin t cos t cos sin sin cos 7 8 9

12 ACCEPTED MANUSCIPT Comparing Fig. a with Fig. b, it can be easily seen that same interface. Assming that components on interface yield i, t i i, in the coordinate system, ox x i i, ti ti and share the are defined as the displacement and traction, the conditions of the perfectly bonded,, on 9 Sbstitting Eqs. 3 and 6 into Eqs. 7 9, i and t i on are gien as below t t N K N K cos sin { A[ cos I ln sin cos cos I I sin sin I ln ] I sin I cos cos sin AI [ sin I ln sin cos cos I I sin cos I ln ]} I sin I cos cos sin { A[ cos I ln sin cos cos I KI sin sin I ln ] I sin I cos cos sin AI [ sin I ln sin cos cos I I sin cos I ln ]} I sin I cos 0 Ths, the displacements i and tractions ti on the inner bondary of the remaining region See Fig. a are represented by the soltions of the eigen-analysis of the ODEs in Section. To determine the displacement and stress fields of the plane V-notched strctre, the new techniqe proposed here reqires diision of the regions and Firstly, the soltions of the stress singlarities in the small sector as shown in Fig.. are ealated by analyzing the eigenale problem of the ODEs shown by Eqs. 6, 7 and the bondary conditions of Eqs. 9 and 0. In the present paper, the interpolating matrix method Ni, 993 is adopted to sole the ODEs, which can proide the soltions of the top N dominant singlarities, inclding,,,, and,, N. 0

13 ACCEPTED MANUSCIPT Note that A and A I,,, N determined in the next step. in Eqs. 0, are nnown qantities that will be Let s consider now the remaining region See Fig. a. There is no stress singlarity in. Therefore the conentional BEM is capable of analyzing the part described by withot other special treatment. The bondary is diided into meshes. The bondary integral eqation is written as follows of C y y ij j * * U x, y t x d T x, y x d ij j ij j * U x, y b x d ij j y where y is the sorceload point; x the field point; bj is the body force per nit olme; j and t i, are, respectiely, the displacements and tractions on j. U * ij x, y and x, y T * ij are the Kelin displacement and traction fndamental soltions. d denotes the Cachy principal ale integral. The Cij y are nown coefficients, the ales of which depend on the geometric configration and material property arond a sorce point y, The coefficients are calclated by C ij y sin sin 4 cos cos 4 cos cos 4 sin sin 4 where the angles,, are defined in Fig. 3; C y = ij C ij y = 0.5 ij for smooth bondary points; ij ij for internal points and is the Kronecer delta. For a plane strain problem of an isotropic homogeneos medim, the fndamental soltions are x y x Fig. 3. Geometric configration arond a sorce point y on bondary.

14 ACCEPTED MANUSCIPT T * ij * U ij 3 4 lnr ij r,i r, j 8 G r,i n j r, jni r 4 r where G is the shear modls; yi and xi and x, respectiely. can write and r ni,n [ ij r r ],i, j 3 4 denote the Cartesian coordinate components of y denotes the nit otward normal ector on the bondary. Hence we r ri r, r, n r,i ni, ri xi yi, i x r n r r i r 5, i i Note that for the region, is the oter bondary of the notch strctre is an inner bondary which arises from the separation of the regions and M nodes are generated on are chosen on. from the diision for the BE analysis and n points for the interpolating matrix method. The nodal displacements and tractions at all the nodes on ector are expressed, respectiely, by the displacement T U x x x x x M x M and the traction ector T t x t x tx t x T t x M tx M. The displacements and tractions at the n interpolation points on hae been gien by Eqs. 0,. The and the ales of,,, and,, N at the n points on hae been obtained from the preios eigen-soltions of Eqs. 6, 7 and 9, 0 by sing the interpolating matrix method with the A, A I,, N remaining as nnowns. The amplitde coefficients in Eq. T are the components of ector A A A A A. A I I N A NI It is well nown that the BEM is a techniqe to transform the bondary integral eqation into a set of algebraic eqations. After Eq. is discretized by the bondary elements on, any nnown fnctions in Eq., sch as i x, t i x, U * ij x, y and T * ij x, y, can be approximated with piecewise polynomial shape fnctions mltiplied by nodal ales of the fnctions along the bondary. At each load point y, the integration of Eq. with respect to x oer all the elements leads to two algebraic eqations for a -dimesional problem. Letting each of the M nodes on act as the load point in applying Eq. for the region,, in trns, the following M algebraic eqations are obtained where H bb, H bp, Gbb and Gbp U T bb b bb b 6 A A H H G G are constant coefficient matrices. In a well-posed problem either the displacement, or the traction, or a combination of the two at the xi -direction of each node is prescribed on the bondary. Therefore in the ectors U and T of Eq. 6

15 ACCEPTED MANUSCIPT there are M nown and M nnown qantities, respectiely. Considering that all the components of A are also nnown qantities, N additional algebraic eqations are reqired to determine the ectors A, U and T. To this end, the N points from the niform diision on the inner bondary are chosen as the sorce points to apply the bondary integral eqation Eq. for the region. Ths the following N algebraic eqations are established: U T b b 7 A A H H G G where H b, H p, G b and G p are also constant coefficient matrices. The combination of Eqs. 6 and 7 gies H H bb b H H bρ U G A G bb b G G b T A 8 Eq. 8 is a set of M+N algebraic eqations with M+N nnown qantities. The soltions of Eq. 8 gie all the nodal displacements + and the amplitde coefficients A, A I,, N. Next, by sbstitting A and AI i and tractions into Eqs. 3, 6, the displacement and stress distribtions in the icinity of the notch tip are fond. The displacements and stresses at any inner point y in the region can be calclated, respectiely, by Eq. and the following stress integral eqation: * * Wij x, y t x Sij x, y x]d * y [ W b xd, y in 9 ij where the ernel fnctions * Wij and * Sij ij ti on are linear combinations of the deriaties of U * * ij x, y and Tij x, y with respect to y. Finally, from the obtained amplitde coefficients A, A I,, N and the stress soltions of Eq. 6 or stress eigen-fnctions ij, the notch stress intensity factors Ki can be determined as follows. When and are real nmbers, K K ΙΙ Ι lim, 0 A 0 0 lim, 0 A a 30b When is a complex nmber ice, 988, 3

16 ACCEPTED MANUSCIPT K ik Ι II lim 0 A [ A i [ A ii ia I [, i [ 0 I 0 0 I 0 A I, ] 0 i I 0 A I I 0 0 I 0 0 0] I 0] 0] 3 The notch SIFs represent the strength of the singlarity at the sharp notch tip and are the important parameters of interest. 4. Nmerical examples Example. Consider a V-notch plate of isotropic material nder niaxial tension as shown in Fig. 4a. a b x c 6 4 x l O x h O x w Fig. 4. A symmetrical V-notched plate nder niaxial tension load. a Geometry of the notch plate. b Sb-domain of radis arond the notch tip. c BE mesh diision along the bondary of the remaining strctre. The plate is in plane stress state with h 00mm, w 40mm, Yong s modls 9 E Pa, Poisson s ratio 0. 5 and load MPa. The opening angle and the depth l of the notch are ariable. First, the rotine IMMEI of the interpolating matrix method is sed to ealate the stress singlarity orders and the associated eigen-fnctions arond the notch tip. eferring to Eqs. 6,7 and Fig. 4b where and are a pair of fictitios bondaries, interal [, ] is diided into n niform sb-interals. The 0 eigenales are complex and are expressed by i. For 60, Tables and present the first 3 eigenales whose e, i.e.,, are larger than or eqal to -. Note that the conjgate ales and are not listed in the tables. The eigenales in Tables 4

17 ACCEPTED MANUSCIPT and are, respectiely, corresponding to the mode I opening and mode II sliding eigen-fnctions. Comparisons are made with F and Long s reslts 998 that were obtained from the sbregion accelerated Müller method for the characteristic eqation deried by Williams 95. The reslts of Seweryn 00, obtained sing a special FEM in conjnction with an analytical stress expression, are also shown in the tables for comparisons. To obtain Seweryn s reslts, half of the notch strctre, de to symmetry, was discretized with 5 six-node trianglar elements. Table The eigenales of the mode I eigen-fnctions, 0 60 Methods F Seweryn IMMEI, n = IMMEI, n = IMMEI, n = Table The eigenales of the mode II eigen-fnctions 0, 60 Methods F Seweryn IMMEI, n = IMMEI, n = IMMEI, n = It can be seen in Tables and that the eigenales obtained sing the present method approach the reslts of F and Long 998 as n increases. The two eigenales in the range of - 0 for n 80 hae conerged p to the fifth significant figre. In addition, the associated eigenectors,,, and are simltaneosly obtained with the same degree of accracy by the IMMEI. The BEM is adopted next to analyze the remaining strctre Fig. 4c exclding the small sector arond the notch tip Fig. 4b. The bondary of the remaining strctre is diided by 80 iso-parametric qadratic elements with 60 nodes. The fictitios bondary is diided by 48 iso-parametric qadratic elements with 96 nodes, where the displacements and tractions at all the nodes on hae been represented by Eqs. 0 and throgh Eq. 9. To test the effectieness of the present approach, the notch plate is 5

18 ACCEPTED MANUSCIPT analyzed by choosing different radis and nmber of terms, N, in the expansion. By fixing the notch depth ratio, depth ratio, l w, to 0., the soltions are obtained for ariable radis to l, between 0.% and.% and ariable from 0 0 to By performing the BE analysis, the displacements and tractions on + and the amplitde coefficients whole region of A A A are obtained. Conseqently, the displacement and stress distribtions in the ia I are compted by Eqs. 3, 6 and Eqs., 9. Table 3 shows the ales for 60, l w 0., l 0.% and N 8. Table 3 The stress amplitdes A corresponding to when 60, l w 0., l A mm A I mm E E-.3E E E E-06.3E E E E E-0-8.4E-09.37E E E E-04 Table 4 The notch SIF K N mm iz different and N 60, l w 0. Ι K N Ι l % % % % % % % % % % % Note that K = N mm from Chen 995. I Sbstitting the A into Eq. 30 yields the notch stress intensity factors. Table 4 shows K I of the notch with 60 and l w 0. for different radis and term nmber N. The reslts are compared with Chen s soltion 995 that was obtained by sing the body force method. De to symmetry, mode II notch SIF does not exist in this problem. This can be seen from Table 3 where the ale of A, which is associated to K, is nearly zero in II 6

19 ACCEPTED MANUSCIPT comparison with other A. It is obsered in Table 4 that for any fixed ale of N, the SIFs are all in good agreement with the existing reslts Chen, 995. This demonstrates that the present approach is nmerically stable for ealating the stress fields of notch strctres in terms of the radis. In Table 4 we can also see that all the compted K by taing nmber N I are ery close to those obtained by taing N 4. It shows that the first two terms in the asymptotic expansion of Eq. dominate the singlar stress field in the notch tip region for the homogeneos and isotropic material. Althogh higher order terms in Eq. do not mae significant contribtions to the notch SIF of the first order, the inclsion of them in the soltion is still important and can improe the accracy of the stress soltions in the tip region. Especially, the higher order terms are becoming more significant in the stress expression Eq. 6 as increases. In addition, notch SIFs of higher orders exist in many notch strctres and hae to be determined by the soltions of higher order terms. It is worth to mention that in the past stdies on notch strctres, only a few researchers hae mentioned and dealt with higher order terms becase of the analytical difficlties. Table 4 shows also that the largest relatie difference of all the ales of obtained sing the present BEM with N 4 soltion K =7.067 N mm I K I is 0.7% in comparison with the reference Chen, 995. The relatie difference is calclated by Soltion Present Soltion ef. % 00 3 Soltion ef. In order to compare the accracy of the present BEM with that of the conentional BEM for analyzing notch problems, the notch plate of Fig. 4a with 60, l / w 0. is again considered. Withot the eigen-analysis, the conentional BEM CBEM reqires ery fine meshes on the bondary of the region near the tip to model the stress distribtion. The displacement and stress distribtions are achieed, respectiely, with 40 qadratic elements and 0 qadratic elements on the bondary of the half strctre. Along the x -axes, the stresses at the mesh points of distance measred from the tip O are sbstitted into Eq. 30a to determine the notch SIF. The reslts are plotted in Fig. 5. It can be seen in Fig. 5 that the compted K I from the conentional BEM aries significantly as the ale of changes. Only in the range of 3.75% / l 0% reslts of K I are reliable since the conentional BEM cannot model the stress singlarities -- in the icinity of the notch tip. For the problem soled in Table 4, where K I = N mm Chen, 995, the best soltion of K I from the conentional BE analysis is 7.390N mm at which l is 6.5% see Fig. 5 and the relatie difference the 7

20 ACCEPTED MANUSCIPT between the two soltions increases to =.08%. Frther tests, which are not shown here, confirmed that a refinement of the meshes in the tip region helped little in improing the accracy of the reslts shown in Fig. 5. K I N mm CBEM with 40 elements CBEM with 0 elements mm Fig. 5. The notch SIF compted by CBEM when 60, l / w 0.. Table 5 The notch SIF K N mm with different l w Ι and l w Present Chen995 % Present Chen995 % 0.05 method method Clearly, since the assmption of asymptotic expansion of the stress field can reflect the essential stress singlarities near a V-notch tip, the stress distribtions from the present BEM are more accrate than those from the conentional BEM. It is obsered that the compted notch SIF is nearly a fixed ale that does not depend on the chosen radis of the small sector. Frthermore the present BEM obtains all the amplitde coefficients terms. The A A of the first N can be sed to determine the notch SIF of the -th order together with the -th order eigen-fnctions. Moreoer, mltiple notch SIFs can be calclated by the new method, 8

21 ACCEPTED MANUSCIPT while the conentional BEM and FEM hae difficlties in dealing with this. Table 5 shows the notch SIFs obtained sing the present BEM for ariable depth l and opening angle. N and mm are taen in the calclation. It can be seen that the relatie differences between the compted SIFs and Chen s soltions 995 for all the cases are less than.3%. Example. Consider a finite plate with a central crac nder niaxial tension as shown in Fig. 6a. a b c 3 x h x a o x O x x x h h b b h Fig. 6. A plate with a central crac nder niaxial tension load. a Geometry of the crac plate. b Sb-domain of radis arond the right crac tip. c BE mesh diision along the bondary of the remaining strctre. The standard mode-i crac in an isotropic material is sed to show the accracy of the present method. The plate is in plane stress state with b.0m, h 3.0m, a 0.4m. The material parameters and load are E 0Pa, 0. 3 and MPa, respectiely. De to symmetry, only one half of the plate is considered in the analysis, as shown in Fig.6b,c. After the stress singlarity orders and eigenectors of the small sector of radis are obtained by the interpolating matrix method, the new BEM techniqe is adopted to ealate the displacement and stress fields of the crac plate which is diided by two sb-domians, as shown in Fig. 6c. In the calclation, a.3%, and n 96 are taen. The bondaries of the two sb-domains are diided by 58 qadratic elements with 36 nodes. Table 6 shows the compted SIF K iz different term nmber N of the eigenales by sing the present method. For the finite plate with mode I central-crac, an analytic soltion of the SIF K is.063 a The AIC, 993. Compared with the analytical soltion, the relatie errors of all the reslts in the table are less than 0.33%. 9

22 ACCEPTED MANUSCIPT The idea of the present method to analyze the V-notch strctres is similar to one of Chen and Sze 00. Chen and Sze 00 sed the fll-field finite elements to analyze the V-notch/crac strctres. In the similar case as Fig. 6a, Chen and Sze 00 too b.0m, h.0m, a 0.m as an infinite plate with a central crac while other parameters are the same as aboe. The exact soltion of the SIF for the crac plate is KΙ.0000 a. For one half of the crac plate, Chen and Sze 00 applied 404 fie-parameter 4-node elements a special hybrid FE, one sper 9-notch crac-tip element and 3 eigenales to ealate the stress field of the plate and obtained K I.03 a. The relatie error is 0.3%. It can be seen that the present BEM is to apply less node nmber nnown qantities than the hybrid FEM for analyzing the notch/crac strctres with the similar way. Table 6 The SIF K / a of the central-crac plate N 8 0 Present BEM Analytic soltion % 0.099% 0.30% 0.33% Example 3. Consider a plate with an inclined V-notch sbjected to niaxial tension load as shown in Fig. 7 x l x h w Fig. 7. A plate with an inclined V-notch nder niaxial tension load. The dimensions of the plate are h 00mm, w 40mm. is the inclined notch angle. 0

23 ACCEPTED MANUSCIPT is the angle between the bisector of the notch angle and the x -axis. The material parameters and load are 9 E Pa, and MPa, respectiely. For the sae of testing the accracy of the reslts in terms of different geometric configrations, the angles and and the notch depth l are taen ariable. Since the notch is inclined, there exist the notch SIFs of both mode I and mode II. After the stress singlarity orders and eigenectors of the small sector of radis are obtained by the interpolating matrix method, the new BEM techniqe is adopted to ealate the displacement and stress fields of the notch plate. In the calclation, N = and n mm, are taen. As done in Example, the oter bondary of the remaining region is diided into 80 qadratic elements. Table 7 shows the compted notch SIFs for 30, 60, l w 0. and different. Chen s soltions 995 shown in the table were obtained from the body force method. The table shows that an increase of reslts in an increase of the mode II notch SIF, while a decrease of the mode I notch SIF. All the relatie differences between the present reslts and Chen s soltions 995 for than.% and.6%, respectiely. K I and K II are less Table 7 The notch SIFs of the inclined V-notch plate l w 0. / K N mm K N mm Ι ΙΙ Present method Chen995 % Present method Chen995 % 30/ / 30/ / / / / 60/ / / Conclsions In conjnction with an asymptotic expansion of the stress distribtion near the notch tip, a new BEM has been proposed in this paper to determine the singlar stress fields of plane V-notch strctres. For a plane V-notch strctre, the new method treats it as an assembly of two parts, i.e., a small sector of material arond the notch tip and the remaining part of the strctre. The stresses and displacements of the small sector are represented by asymptotic expansions. On

24 ACCEPTED MANUSCIPT the basis of the liner elasticity theory, the ealations of the stress singlarity orders and the associated displacement/stress eigen-fnctions were transformed into an eigenale problem of ordinary differential eqations. The interpolating matrix method was applied to soling the eigenale problem with both the eigenales and the eigenectors being obtained simltaneosly. The displacements and tractions along the arc of the sector were expressed as a linear combination of the terms from the series expansions with different singlarity orders. In fact, the small sector can be thoght as a sper singlar element arond the notch tip, and the displacement/stress eigen-fnctions obtained throgh the eigen-analysis are essentially eqialent to a series of shape fnctions of the singlar element for modeling the displacement/stress distribtion. Since the remaining part has no stress singlarities, the conentional BE is sfficiently accrate to predict the displacement and stress distribtions. A combination of the two soltions finally proided the stress fields of the sector and the remaining strctre as well as the notch SIFs. In contrast to the conentional BEM and FEM, the proposed BE approach does not reqire fine meshes near a V-notch tip. Another adanced featre of the new method is that, both the basic and higher order stress singlarities can be reealed simltaneosly throgh the asymptotic expansion and eigen-analysis. Using the present method, the fractre analysis of complex geometric and load conditions becomes possible and accrate. Three nmerical examples hae been gien to show the application of the new method on V-notch/crac plates made of isotropic materials. For V-notch plates with different geometry, the stress fields and the notch SIFs were compted for different ales of the radis and the expansion term nmber N. Throgh comparisons with alternatie soltions aailable in the literatre, it can be conclded that the new method is an accrate and effectie tool for modeling singlar stress fields in V-notch strctres. The present techniqe can be frther deeloped to ealate the stress distribtions and the generalized SIFs of jnctions, inclsions and V-notches in bonded dissimilar mlti-material. In these cases, high order stress singlarities are needed and can be obtained by the present method. This will mae it possible for fractre criteria associated to these complex cases to be established. Ftre wor is in progress in order to establish new fractre criteria on the basis of eqialent energy release rate of different V-notch angles. Acnowledgements The first two athors acnowledge the financial spport from the Natral Science Fondation of Anhi Proince of China The athors are ery gratefl to the anonymos reiewers for their constrctie comments and sggestions.

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