Mode III fracture mechanics analysis with Fourier series method

Size: px

Start display at page:

Download "Mode III fracture mechanics analysis with Fourier series method"

Abigayle Sims
5 years ago
Views:

$Mode III fracture mechancs analyss wth Fourer seres method F W M Kwok 3 L C Kang, and C R Steele Dvson of ppled Mechancs, Stanford Unversty, Stanford C 94305 numercal scheme based on the Fourer seres$

1 Mode III fracture mechancs analyss wth Fourer seres method F W M Kwok 3 L C Kang, and C R Steele Dvson of ppled Mechancs, Stanford Unversty, Stanford C numercal scheme based on the Fourer seres method s developed for the soluton of Laplace's equaton on a polygonal doman. Ths scheme s appled to the soluton of the pure mode III fracture mechancs problem. For the sngle edge cracked strp problem under out-of-plane loadng, the value of the J-Integral evaluated from the Fourer soluton was found to be n good agreement wth classcal soluton. The real power of the scheme les n ts applcablty to general polygonal shapes. Ths capablty s llustrated by three sample problems that nvolve strps wth a slanted crack, multple cracks and branchng cracks under mode III loadng. INTRODUCTION The current study was ntated to develop an effcent computatonal approach to solve boundary value problems n domans of complex shape. The frst phase of the study centers on the soluton of second order ellptc partal dfferental equaton under two-dmensonal domans. Wth the excepton of smple geometres, classcal approaches often fal to gve the desred results. Numerous numercal schemes have been developed to handle ths stuaton. These nclude the fnte element method, fnte dfference method, and the boundary element method. However, these methods often nvolve prohbtvely hgh computatonal andor user set-up tmes. These problems are exaggerated n certan applcatons such as fracture mechancs. n example s the complcated mesh patterns requred to capture the stress varatons around a sngle crack. The tme and sklls necessary to perform ths task are sgnfcant. It s the purpose of ths study to allevate these dffcultes. The Fourer seres method s taken as our approach. By usng a kernel functon that s the soluton of a half-plane problem governed by the equaton to be solved, nfluence coeffcents can be derved. These coeffcents relate the Fourer seres coeffcents of all sdes of the doman to each other. The superposton prncple s used to sum up the nfluences from all sdes of the doman and thus buld up a system of lnear equaton. The boundary value problem s essentally reduced to the soluton of ths lnear system of equatons. Due to the characterstc of the kernel functon, the restrcton on the doman s that t must be convex. However, an effectve scheme for combnng several convex domans nto concave and multply-connected domans has been developed. Wth no numercal ntegratons nvolved, ths approach s very effcent from a computatonal standpont. nother major advantage of the method s the ease of settng up problems. No mesh generaton s requred. The user smply has to subdvde the doman nto a number of convex regons. Ths contrasts wth the amount of work nvolved n settng up fnte element meshes for complcated geometres. In ths paper, the Fourer seres method s specfcally appled to the mode III fracture mechancs problem. The governng equaton of whch s the Laplace's equaton. For verfcaton purposes, the J-Integral was calculated for the classcal case of the sngle edge cracked strp. Fourer results were compared wth analytcal results. few examples are also provded to llustrate the versatlty of the Fourer seres method. PPROCH For ths phase of the study n [1], the governng equaton to be solved s the Laplace's equaton gven as V 2 0=O n Q (1) (f>=fts) ondo (2) 4>,v = g(s) on 0O2 (3) where O represents a polygonal doman and do U dd2 = dq represents the boundary of the doman. In other words, the boundary condton of our problem can be of the Neumann type, Drchlet type, or mxed. n example of the type of problem to be solved s gven n Fg 1. For arbtrary shapes, analytcal solutons are mpossble to obtan. Instead of attackng the problem drectly, a smpler problem, whch has a readly avalable soluton, s approached and then related to the desred problem. Ths smpler problem s the half-plane problem under Eq (1). For an arbtrary nput f(x) on lne p (Fg 2), whch s located along the edge of the half-plane, the Fourer representaton of whch s a=o Lp (4) ppl Mech Rev vol 44, no 11, part 2, Nov 1991 S166 Copyrght 1991 mercan Socety of Mechancal Engneers Downloaded From: on Terms of Use:

2 ppl Mech Rev vol 44, no 11, part 2, Nov 1991 Kwok et al: Mode III fracture mechancs analyss S167 <>v = *,v, 4) = <f> y v.*. <f> v = <J>v <f> = 4>, FIG 1. n example of the type of problem solved by the Fourer seres method. Each lne segment of the arbtrary two-dmensonal polygon may have ether a prescrbed value or a prescrbed slope n the normal drecton. FIG 2. The related half-plane problem of the twodmensonal polygonal doman under Laplace's equaton. Lnes p and q are taken as two separate lne segments of the orgnal polygonal doman. where C% are the Fourer cosne coeffcents, and Lp s the length of lne p. The soluton of ths half-plane problem s Introducng the notaton (p p (x)= J_ G p cosq exp(- ^L) n= 0 -Lp Lp (5) qp = 2 Lq [ cos ' 1!^Lex, Lp :_rmim{z) )]cos!mlds (9) The next step nvolves determnng the response of ths soluton on an arbtrary lne, named lne q, n the half-plane. For a pont on lne q, ts coordnate can be wrtten as z{s) = zcj + se'y, where 2D s the coordnate of the startng pont of lne q, s s the dstance measured from zo along q, and Y s the angle between lnes p and q (Fg 2). The response on lne q due to the nput ^(x) on lne p can be expressed as 4,0%)= Y c p ncos'^m?l exp.'hm±) n-0 Lp Lp (5) gan usng a Fourer cosne expanson, ths response can be expanded n the form Eq (8) can be reduced to Bg p _ mn -n m= 0,1,..., n-0 (10) ffn are the nfluence coeffcents that relate the Fourer seres coeffcents of the nput on one lne to the coeffcents of the response functon on another lne. local coordnate system s defned for each boundary lne segment of a polygonal doman. The above procedure can be contnued to construct the relatonshps between all the boundares of ths doman. Thus, the soluton on each boundary can be vewed as the summaton of the nfluences of all the other sdes of the doman. For the Drchlet problem, ths can be expressed n a system of lnear equatons as mjs )7=0 Lq (7) 171= 0,1,..., where p= 1 p= l n-0 q=l,...,n (11) I& P =^ Lc,pQ%)cos ds m=0,l,..., Lq (8) where Vs the number of sdes of the polygonal doman. To facltate a numercal soluton of the problem, a fnte number of harmoncs s taken. Eq (11) reduces to Downloaded From: on Terms of Use:

3 S168 MECHNICS PN-MERIC 1991 ppl Mech Rev 1991 Supplement m rf^> XL '"? m=0 1 nhaa f p-1 P-l-0 q=l,...,n (12) where jap represents the number of harmoncs on sde p, and nfag represents the number of harmoncs on sde q. Eq (12) can be put nto matrx form F(m* Fowl)J Qpr) (njx. j) C&Fl) L Mm*r) (& JLC^DJ (13) where the superscrpts denote the number of the boundares, and the subscrpts denote the number of harmoncs on the correspondng boundares. For brevty, Eq (13) can be wrtten as PPLICTION TO MODE II! FRCTURE MECHNICS Wth <p taken as U3, the out-of-plane dsplacement, Eq (1) becomes the governng equaton of the antplane problem of lnear elastcty V Z U3=0 (17) Thus, the Fourer seres method proposed above s applcable to the class of antplane problems under mode III fracture mechancs. For verfcaton purposes, the frst problem solved by the Fourer seres method s the classcal case consstng of an nfntely long sngle edge cracked strp wth unt wdth (Fg 3). The crack length s taken as half the strp wdth, and the shear stress (732) normalzed by the shear modulus (u) has a value of ten. F = C (14) where F s the vector contanng the Fourer cosne coeffcents of the boundary condtons of the problem, s the matrx contanng the nfluence coeffcents, whch can be obtaned from Eq (9), and fnally, C s the vector of undetermned Fourer coeffcents. In essence, the boundary value problem has been reduced to the soluton of ths matrx equaton. fter solvng for C, the soluton of any pont n the polygonal doman can be obtaned by y lsp fa>y> 2. 1 c " cos p-1fl-0 nnxp nnyp-j exp- 17 Lp (15) where (x,y) s the global coordnate of the nteror pont and (xpj'p) s the coordnate of the pont that bounds to the p* boundary. For Neumann and mxed boundary value problems, the procedure s the same as above wth the excepton that <p!v s expanded n Eq (7) nstead of 4> 9P. Ths can be easly obtaned by replacng the expresson n the square brackets n Eq (9), whch we call the kernel functon K, wth 6K _ 6K dx de dy dv dx dv dy dv (16) where v represents the outer normal drecton of the boundary. Due to the presence of the exponental term n the kernel functon, the technque descrbed so far works only n the case of convex domans. For concave domans, exponentally ncreasng terms wll be present n some parts of the doman. To overcome ths problem, the concave regon s sub-dvded nto a number of convex domans through the ntroducton of nternal boundares. The contnuty condtons mposed on these nternal boundares can be used as the addtonal condtons requred to solve ths problem. fter the soluton of the Fourer coeffcents on the nternal boundares, the whole problem can then be decoupled nto ndependent problems for each convex polygon. X2 T 32 ]U = j^ggl 0.5- u 1.0 H X1 FIG 3. Infntely long sngle edge cracked strp under outof-plane loadng. good measure of the accuracy of the Fourer soluton s to evaluate the J-Integral [2] for an arbtrary contour around the crack-tp. From the J-Integral, the mode III stress ntensty factor can be easly obtaned wth = M 2J (18) nalytcal soluton [3] yelds a value of for Km'fJ- pproxmatng the nfnte strp as fnte strps wth lengthto-wdth ratos of one, two and three gave results of 14.32, 14.28, and 14.23, respectvely. These calculatons were performed wth seventeen Fourer harmoncs on each lne segment. Thus, good agreements wth percentage dfferences smaller than 1.3% were obtaned between Fourer and classcal solutons of KHJJ. Downloaded From: on Terms of Use:

4 ppl Mech Revvo 44, no 11, part 2, Nov 1991 Kwok et al: Mode II! fracture mechancs analyss S169 The next step of the verfcaton nvolves the path ndependence of the J-Integral. Three dfferent rectangular contours were used n the calculaton of the J-Integral, wth the largest contour gong around the outer boundary of the strp. Neglgble dfferences were found between the J- Integral values from the varous contours, wth the largest percentage dfference less than 0.1%. Thus, path ndependence of the J-Integral s assured. Wth encouragng results from the above test case, the Fourer seres method was appled to a number of ncreasngly complcated mode III fracture mechancs problems. The frst problem s a strp wth a slanted crack under out-of-plane loadng (Fg 4). The doman was subdvded nto four convex domans (Fg 5). Seventeen Fourer cosne harmoncs were used on each lne segment to generate the nofj plot n Fg 6. The second problem nvolves a multply cracked strp under out-of-plane loadng (Fg 7). Only eght elements were requred to "mesh" the strp (Fg 8). Wth seventeen harmoncs on each lne segment of the convex polygons, the calculated mn s gven n Fg 9. * ^32 O \? o- 30 V I -l.o- FIG 4. Strp wth slanted crack under out-of-plane loadng. 32 ement No. ne Segment FIG 5. Modelng detals of strp wth slanted crack. FIG 6. Dstrbuton of T u. DOOOOOO T 32 "t o k t d -H 0.3 <- H^-Q.4-~^" ( «^ 0.3 v l < 1.0 >- T 32 FIG 7. Strp wth multple cracks under out-of-plane loadng. Element No. «Lne Segment FIG 8. Modelng detals of multply-cracked strp FIG 9. Dstrbuton of T 32 U. Downloaded From: on Terms of Use:

5 S170 MECHNICS PN-MERIC 1991 ppl Mech Rev 1991 Supplement The last example nvolves branchng cracks under antplane loadng (Fg 10). The doman was dvded nto nne convex regons (Fg 11). Seventeen Fourer harmoncs were used n the calculaton of the T$JJ plot shown n Fg 12. CONCLUSIONS Fourer seres method has been successfully developed for the soluton of Laplace's equaton n arbtrary twodmensonal polygonal shapes. s one example, ths approach s applcable to the soluton of problems n mode III fracture mechancs. For the sngle edge cracked strp, good agreements between analytcal and Fourer results were obtaned. The path ndependence of the J-Integral was also checked. s demonstrated by the sample problems, extremely smple meshes are requred, even n the presence of stress sngulartes at the crack-tps. Ths translates to a very short user set-up tme. No specal crack-tp elements are nvolved. The method s also hghly effcent. ll the sample cases n ths paper requre less than one CPU mnute of computatonal tme on a CONVEX manframe. The next phase of ths research nvolves developng a Fourer soluton scheme for the bharmonc equaton n arbtray two-dmensonal polygonal domans. Ths wll address problems n plate bendng, and plane stress and plane stran problems n elastcty. CKNOWLEDGEMENT The fnancal support granted to FMW Kwok by the Natural Scences and Engneerng Research Councl of Canada s deeply apprecated. REFERENCES 1. Kang LC, pplcaton of Fourer seres method to boundary value problems wth complex domans, PhD Thess n preparaton, Stanford Unversty, Stanford C, Rce JR, Mathematcal analyss n the mechancs of fracture, n Fracture-n dvanced Treatse, Vol. II, H Lebowtz (ed), cademc, New York, 1968, Sh GC, External cracks under longtudnal shear, J Frankln Insttute 280(2), (1965). n 0 «^, I 3! J >^ ^ I > H I 5 \ 9 'X m D Element No.» Lne, Segment FIG 11. Modelng detals of strp wth branchng cracks. L^ o g _^^L^ Q v >< 2.0 > FIG 10. Strp wth branchng cracks under out-of-plane loadng. 32 FIG 12. Dstrbuton of T 3 -,u. Downloaded From: on Terms of Use:

An Accurate Evaluation of Integrals in Convex and Non convex Polygonal Domain by Twelve Node Quadrilateral Finite Element Method

An Accurate Evaluation of Integrals in Convex and Non convex Polygonal Domain by Twelve Node Quadrilateral Finite Element Method Internatonal Journal of Computatonal and Appled Mathematcs. ISSN 89-4966 Volume, Number (07), pp. 33-4 Research Inda Publcatons http://www.rpublcaton.com An Accurate Evaluaton of Integrals n Convex and