Blucher Mechancal Engneerng Proceedngs May 2014, vol. 1, num. 1 www.proceedngs.blucher.com.br/evento/10wccm Numercal Nonlnear Analyss wth the Boundary Element Method E. Pneda 1, I. Vllaseñor 1 and J. Zapata 1 1 Insttuto Poltécnco Naconal, ESIA-UZ, Undad Profesonal Adolfo López Mateos Méxco D.F. (epnedal@pn.mx) Abstract. Ths paper presents a new formulaton of the Boundary Element Method to vscoplastc problems n a two-dmensonal analyss. Vsco-plastc stresses and strans are obtaned untl the vsco-plastc stran rate reaches the steady state condton. A perfect vscoplastc analyss s also carred out n lnear stran hardenng (H =0) materals. Part of the doman, the part that s susceptble to yeld s dscretzed nto quadratc, quadrlateral contnuous cells. The loads are used to demonstrate tme effects n the analyss carred out. Numercal results are compared wth soluton obtaned from the Fnte Element Method (FEM). Keywords: Boundary Element, Vsco-plastcty, Boundary Integral Equatons 1. INTRODUCTION In the case of problems wth hgh temperature gradents where nelastc stran rates are proportonal to hgh power of stress, regons wth stran rate concentraton provde nearly all the nelastc contrbuton to the stress rates [1]. The man reason for the success of the BEM (boundary element method) n any problem s the ablty to model hgh stress concentraton felds accurately and effcently. A comprehensve revew of the hstorc development of the BEM for nelastcty can be found n the work of Alabad [2]. An alternatve methodology based on the use of the Kelvn fundamental solutons was presented n [3], [4] and[5]. Recently, the DBEM (dual boundary element method) has been developed as a very effectve numercal tool to model general fracture problems wth numerous applcatons to lnear elastc and non-elastc fracture problems [6]. BEM has been appled to elastoplastc problems snce the early seventes wth the work of Swedlow and Cruse [7] and Rchardella [8] who mplemented the von Mses crteron for 2D problems usng pecewse constant nterpolaton for the plastc strans. Later, Telles and Brebba [9] and others had, by the begnnng of the eghtes, developed and mplemented BEM formulatons for 2D and 3D nelastc, vscoplastc and elastoplastc problems (see [10] for further detals). In recent years, Alabad and co-workers [11] have ntroduced a new generaton of boundary element method for soluton of fracture mechancs problems. The method whch was orgnally proposed for lnear elastc problems[12], [13] and [14] has snce been extended to many other felds ncludng problems nvolvng nonlnear materal and geometrc behavor [15].
In the present paper applcatons of the DBEM to vsco-plastcty are presented. The specmens analyzed are three dfferent plates. The boundary was dscretzed wth quadratc contnuous and sem-dscontnuous elements, but the doman wth nne nodes nternal cells. In vsco-plastcty only the part susceptble to yeldng was dscretzed. The von Mses yeld crteron was appled so the materal used for these sort of analyss were metals. 2. VISCO-PLASTICITY THEORY In order to explan the theory of vsco-plastcty t s convenent to analyze the onedmensonal rheologcal model see Error! No se encuentra el orgen de la referenca. for more detals. A unaxal yeld stress governs the onset of the vsco-plastc deformaton. Once vsco-plastcty begns the stress level for contnung vsco-plastc flow depends on the stran hardenng characterstcs of the materal ( ). After applyng Hook s law and boundary condtons, t s possble to obtan: (1) Expresson (1) s the vsco-plastc stran rate n terms of the stresses for the unaxal case n whch (.) denotes the dervatve wth respect to the tme,. From the vsco-plastc model the stran response wth tme can be represented by two cases. The frst case s the perfectly vsco-plastc materal n whch. In ths case the vscoplastc deformaton contnues at a constant stran rate. The second case s the lnear hardenng case ( ), where after the ntal elastc response, the vsco-plastc stran rate s exponental and reaches the steady state condton when ths value becomes zero. On the other hand, for a perfectly vsco-plastc materal there s always an mbalance of stress n the system whch does not reduce and consequently the steady state condton can not be acheved. 3 BOUNDARY INTEGRAL EQUATIONS The boundary condtons n terms of rates are; for dsplacements u u and for tractons t and the equaton representng the tracton boundary condtons s, t t 2 a 2 e n u l, l n u, u, n (2) 1 2 1 2 Equaton (2) s for three dmensonal problems. In order to work wth two dmensonal problems for the plane stress state t s necessary to remove the stran n z drecton, so a 33 0. The soluton of the equaton (2) leads to the followng boundary Integral representaton of the boundary dsplacements when the ntal stran approach for the soluton of nelastc problems s used
c u p t ' u d u ' t d ' kd (3) k In a smlar way the boundary ntegral equaton of the nternal stresses s expressed by Dkt d Sku d k a k d f (4) Where s a Cauchy ntegral, D k and Sk are terms contanng the dervatve of the dsplacements and tractons, f s the free term and s the fundamental soluton for the doman. 3.1 Boundary Integral Formulaton for Vsco-plastcty In the vsco-plastc analyss lke plastcty, the ntal stran approach wll be appled and the ntegral equaton to calculate the dsplacement on the boundary s bascally the same, the only dfference s that the plastc stran s replaced wth the vsco-plastc stran rate. So the dsplacement equaton can be rewrtten as: c p k x') u ( x') t' ( x', x) u ( x') d u' ( x', x) t ( x') d ', x' z vp zd ( (5) vp Where u, t and are the dsplacement, tracton and vsco-plastc stran rates respectvely. t ', u' and ' are the dsplacement, tracton and thrd order fundamental solutons, respectvely, whch are functons of the postons of the collocaton pont x and the feld pont x whch belong to the boundary, or the nternal pont z and the materal propertes. In order to llustrate the results obtaned wth the Boundary Element Method three dfferent plates were analyzed as s shown below. 4 EXAMPLES 4.1. The Notched plate An alumnum plate wth a notch and geometry as llustrated n fgure 1, t s consdered n ths case. The plate s constraned n X and Y drecton on the edge of the notch and t s assumed to have the followng materal propertes: Young s modulus, E =70000 MPa; Posson s rato ν= 0.2; Appled stress a =140 Mpa., y = 243 Mpa. wth γ=0.01 and t = 0.01 s
5.625 6.25 6.875 7.5 8.125 8.75 9.375 10 36 mm. Fgure 1 Geometry n a plate wth a notch σy (N/mm.²) 450 400 350 300 250 200 150 100 50 0 FEM BEM Dstance from the notch to the edge of the plate n X drecton (mm.) Fgure 2. Stresses n Y drecton for a notched plate.
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 Dsplacement (mm. 0.12 0.1 0.08 0.06 0.04 FEM BEM 0.02 0 Dstance from the notch to the edge of the plate n X drecton (mm.) Fgure 3. Dsplacements n Y drecton for a notched plate. 4.2 Plate wth a hole A perforated tenson specmen wth dmensons and geometry as llustrated n fgure 4, t s consdered n ths example. The plate s constraned n X and Y drecton on the edge and was subected to the same tensle load, materal propertes and constants as the example 4.1. 36 mm.
5 5.5 6 6.6125 7.225 7.9 8.575 9.2875 10 Fgure 4 Geometry for a plate wth a hole σy (N/mm.²) 400 350 300 250 200 150 100 50 0 FEM BEM Coordnate from the center of the crcule to the edge n x drecton. (mm.) Fgure 5 Stresses n the center of the plate n Y drecton from the center of the hole. 5 CONCLUSIONS In ths paper the BEM was appled to the analyss of non-elastc tme dependent problems. It has been demonstrated here that ths method s an accurate and effcent method for analyzng and modelng vsco-plastc problems. The vsco-plastc stresses and dsplacements obtaned of the Boundary Element Method are n good agreement wth the ones calculated n the Fnte Element Method. The ncrement of tme plays a very mportant role for the accuracy of the results, f we make a good selecton the effcency of the program and the results wll be guaranteed. The vsco-plastc behavour s represented by a plastc stran feld over a regon, susceptble to yeld, dscretzed wth quadrlateral quadratc contnuous and dscontnuous nternal cells. REFERENCES [1] Provdaks, C.P., Vscoplastc BEM Fracture Analyss of Creepng Metallc Cracked Structures n Plane Stress usng Complex Varable Technque. Engneerng Fracture Mechancs, 70, 707-720.,2003 [2] Alabad, M.H., Boundary Element Methods n Fracture Mechancs. Appl Mech Rev,, 50, 83-96.,1997 [3] Tan, C.L., Lee, K.H. Elastc-Plastc Stress Analyss of a Cracked thck-walled Cylnder, Journal of Stran Analyss, 50-57., 1983 [4] Yong, L., and Guo, W.G., The calculaton of J I Integrals of Thck-Walled Tubes wth One and Two Symmetrck Cracks by Elastoplastc BEM, Int. J. Pres. Ves. & Pppng, 51, 143-154. 1992 [5] Hantschel T., Busch, M., Kuna, M., and Maschke, H.G., Soluton of Elastc-Plastc Crack Problems by an Advanced Boundary Element Method, n Proceedngs of the 5th. Internatonal Conference on Numercal Methods n Fracture Mechancs, A.R..
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