Fracture analysis of FRP composites using a meshless finite point collocation method

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1 Forth Internatonal Conference on FRP Compostes n Cvl Engneerng (CICE008) -4Jly 008, Zrch, Swtzerland Fractre analyss of FRP compostes sng a meshless fnte pont collocaton method M. Shahverd & S. Mohammad MSc stdent, Unversty of Tehran & Tazand Consltng Engneers, Tehran, Iran. Assocate professor, Hgh Performance Comptng Lab, School of Cvl Engneerng, Unversty of Tehran, Tehran, Iran (correspondng athors). ABSTRACT: Wth the wde applcaton of FRP compostes for retrofttng and strengthenng of exstng strctres, accrate and effcent analyses of models become very mportant as the stdy of nterface crac growth s essental to ensre the relablty of strctres nder statc and cyclc loadng condtons. In the absence of exact analytcal soltons for complex plate problems, nmercal methods sch as the fnte element method, the bondary element method, and recently meshless methods have been freqently sed and extended to nclde nonlnear effects. In the past decade, meshless methods have been wdely developed and mplemented to varos engneerng problems. Krysl and Belytscho have extended the element free Galern (EFG) method for statc analyss of thn plates and shells. All EFG formlatons reqre a bacgrond mesh of cells for ntegraton. In a dfferent meshless approach, the proposed trly meshless Fnte Pont Method (FPM) ses a movng least sqare approxmaton wthn a collocaton strong form for solvng the governng eqlbrm eqaton. In FPM, mposton of bondary condtons s not drectly performed. Instead, a procedre smlar to other nternal nodes s followed. In ths paper, the fractre analyss of two dmensonal FRP compostes s consdered. Enrched bass fnctons are ntrodced n the meshless formlaton to captre the snglarty at potental crac tps. A nmber of problems are solved and the reslts are compared wth avalable analytcal soltons and other nmercal technqes to assess the performance of the proposed approach. INTRODUCTION The needs of modelng orthotropc materals have been recently revved wth great nterest, havng enormos applcatons n varos strctral systems le aerospace and atomoble ndstres. The man advantages of sng these materals can be attrbted to ther hgh stffness and low rato of weght to strength. Some analytcal nvestgatons have been reported on the fractre behavor of sch materals sch as the poneerng one by (Mselshvl 95) and (Sh & Pars & Irwn 965 ; Tpholme 974 ; Tng 996 ; Asadpore & Mohammad 007). Nmercal methods, however, have been wdely tlzed for solvng dfferent mechancal problems. Despte the fact that the fnte element method s more convenent and applcable becase of ts ablty n modelng complex behavors, meshless methods have been ncreasngly adapted for smlaton of complex problems sch as crac ntaton and propagaton problems. For elastcty problems(oñate & Perazzo & Mqel 00) presented a fnte pont method sng the FIC technqe n order to overcome the nstablty of the reslts obtaned from the conventonal verson of FPM. (Boroomand & Tabatabae & Oñate 005) proposed a mappng scheme to redce the problem of ll-condtonng of the coeffcent matrx whch sometmes occrs de to nonsotropc arrangement of the ponts. Ths stablzaton procedre dramatcally mproves the convergence and accracy of the method. (Btaraf & Mohammad 006) solvng the chlorde dffson eqaton for predcton of servce lfe of concrete strctres and ntaton tme of corroson of renforcements. In ths stdy, the trly meshless fnte pont method (FPM) s combned wth prncples of partton of nty and the extended fnte element method to accrately smlate snglar felds arond a crac tp and the dscontnos feld along the crac. FPM s a strong form solton and can be regarded as a contnos extenson of the fnte dfference method based on the movng least sqare approxmaton. - -

2 FRACTURE MECHANICS FOR D ORTHOTROPIC MATERIALS The stress-stran law n an arbtrary lnear elastc materal can be wrtten as (,,, l,,3) ε = slσl = () where ε and σ l are lnear stran and stress tensors, respectvely, and s l s a forth-order complance tensor. By ntrodcng the contracted form (Lehnts 963), the stress-stran relaton for a plane stress problem can be re-wrtten as εα = a αβσ β ( α, β =,,6) () Now assme an ansotropc body s sbected to arbtrary forces wth general bondary condtons and a crac. Global Cartesan co-ordnate ( X, X ), local Cartesan co-ordnate ( x, y ) and local Polar co-ordnate ( r, θ ) defned on the crac-tp are llstrated n Fgre. A forthorder partal dfferental eqaton wth the followng characterstc eqaton can be obtaned sng eqlbrm and compatblty condtons(lehnts 963) 4 3 a μ a μ + a + a μ a μ+ a = 0 (3) ( ) Fgre. An arbtrary orthotropc craced body sbected to tracton t. The two dmensonal dsplacement feld n the vcnty of the crac-tp have been prevosly derved by (Sh & Pars & Irwn 965) by means of analytcal fnctons and complex varables, z = x + μ y, ( =,). The dsplacement components for pre mode I are: r = K I Re { μ p + μ μ p + μ } (4) π μ μ r = K I Re { μq + μ μq + μ } (5) π μ μ where p and q are defned as: a p = aμ + a a6 μ, q = aμ + a (6) 6 μ (Lehnts 963) showed that the roots of (Eq. 3) are always complex or prely =,, x + y = ) and occr n congate pars as μ, μ and μ, magnary ( μ μ μ ( ) μ. 3 ENRICHED FINITE POINT METHOD In ths secton an overvew s gven to the approxmaton sed to constrct the shape fnctons of the proposed FPM. - -

3 3. Weghedt least sqare method Least sqare methods are among the effcent procedres freqently sed for approxmaton of nnown fnctons n meshless methods. Assme that a fncton (x) s to be approxmated n a doman Ω wth n nodal ponts. The approxmate fncton n sb-doman Ω n the vcnty of the th node wth n neghborng nodes may be wrtten as ^ m ( x) ( x) = p ( x) α (7) = where p(x) s a vector of base monomals and α s a vector of coeffcents. Assmng that generally n s greater than m, then satsfacton of (7) reqres a least sqare procedre, whch leads to mnmzaton of the followng dscrete norm: J n n T = w ( x )[ ( x )] = w ( x )[ p ( x ) α] (8) = = where w (x) s a stable weghtng fncton for the th sb-doman. Note that for each doman a local co-ordnate system s defned; the orgn of whch s located at the master node of the doman (node nmber ). The dscrete norm n Eqaton (8) s mnmzed as J = 0 (9) α whch yelds to the followng system of eqatons: (0) A α = B where A n = = T w ( x ) p( x ) p ( x ) and B = [ w ( x ) p( x),..., w ( xn ) p( xn ) () Solton of (0) gves (3) α = A B The approxmaton for the th sb-doman s obtaned by sbstttng Eqaton (3) nto (7) T (4) ( x) = p ( x) A B = N( x) where N(x) s a matrx contanng the shape fnctons for each doman. 3. Crac tp enrchment One way to enrch the FPM formlaton for modelng snglar stress felds arond a crac tp s to nclde the leadng terms of the near-tp asymptotc expanson for the dsplacement feld n the bass fncton. In the fll ntrnsc enrchment of FPM approxmaton for orthotropc fractre problems, the entre near-tp asymptotc dsplacement terms are added to the lnear terms of the bass: θ θ, x, y, r cos g( θ ), r cos g( θ ), p T ( x) = (5) θ θ r sn g ( ), sn ( ) θ r g θ where (6) g ( θ ) = ( + μ ) + ( μ ),( =,) x y () - 3 -

4 μy θ = arctg,( =,) cos sn θ + μx θ (7) 4 NUMERICAL SIMULATION 4. Crac analyss n an sotropc plate A closed form solton for a crac can be constrcted by sng the well-nown near- tp feld n a doman arond the crac tp and prescrbng the dsplacement along the bondares accordngly. Ths can be consdered as a patch test for snglar felds. A sqare patch wth sde length of d and a crac length d s consdered. The exact soltons for the stress feld are gven by: K θ θ 3θ σ = I K θ θ 3θ yy cos( ) + sn( )sn( ) πr, σ = I cos( ) sn( )sn( ) (8) xx πr Also, the qartc sphercal weght fncton wth a rads of d ml s adopted: 3 4 ( 6r + 8r 3r ) for > r >= 0, (9) w( xi, x) = w( r) =, r = xi x / dml 0 for r >=. Fgre llstrates the nodal dstrbton and selecton of spport domans for a typcal node and a crac tp. The total nmber of nodes s 87, and each clod ncldes abot nodes. (a) Fgre. a) nodal dstrbton, b) spport doman of node x p Fgre 3 depcts the mode I stress contors over the craced plate obtaned by the proposed approach n comparson wth the exact solton (Eq.8). The low level of generated error shows the qalty of the nmercal predctons by the proposed enrched fnte pont method. (a) - 4 -

5 (c) (d) Fgre 3. Stress contors: a) σ xx (FPM), b) σ xx (exact), c) σ yy (FPM), d) σ yy (exact). 4. Orthotropc FRP plate A sqare patch test wth a sde length d and a crac length d s consdered. The plate s composed of a graphte-epoxy materal wth the followng orthotropc propertes: E = 4.8 GPa, E =.7 GPa, G = 9.66 GPa, υ = 0. Fgre 4 and fgre 5 depcts the mode I dsplacement contors over the craced plate obtaned by the proposed approach n comparson wth the exact solton (Eq.4 and Eq.5). Smlar to sotropc plate, the total nmber of nodes s 87 and each clod ncldes at least nodes. (a) Fgre 4. Dsplacements contors: a) x (FPM), b) x (exact)

6 (a) Fgre 5. Dsplacements contors: a) y (FPM), b) y (exact). 5 CONCLUSIONS A trly meshless enrched fnte pont approach has been presented for solvng the governng eqatons of FRP craced plates. FPM s a strong form solton based on the movng least sqare approxmaton for the dsplacement feld varable. Enrched bass fnctons have been ntrodced n the meshless approxmaton to captre the snglarty of the stress feld at crac tps. Nmercal reslts have shown good agreement wth avalable analytcal soltons. 6 REFRENCES Asadpore, A. and Mohammad, S Developng new enrchment fnctons for crac smlaton n orthotropc meda by the extended fnte element method, Internatonal Jornalfor nmercal methods n engneerng; 69:50-7. Btaraf, M. and Mohammad, S Analyss of chlorde dffson n concrete strctres for predcton of ntaton tme of corroson sng a new meshless approach, Constrcton and Bldng Materals. Boroomand, B. and Tabatabae, A.A. and Oñate, E Smple modfcatons for stablzaton of the fnte pont method, Internatonal Jornalfor nmercal methods n engneerng; 63: Lehnts, SG. 963.Theory of an ansotropc elastc body. Holden-Day: San Francsco. Mselshvl, N.I. 95. Some basc problems on the mathematcal theory of elastcty, Gronngen. Oñate, E. and Perazzo, F. and Mqel, J. 00. A fnte pont method for elastcty problems. Compters and Strctres; 79:5 63. Oñate, E. and Perazzo, F. Mqel, J Advances n the stablzed fnte pont method for strctral mechancs. Internatonal Center for Nmercal Methods n Engneerng (CIMNE). Sh, G.C. and Pars, P.C. and Irwn, G.R On cracs n rectlnearly ansotropc bodes, Int. J. Fract. Mech., Tpholme, GE A stdy of cracs n orthotropc crystals sng dslocatons layers. Jornal of Engneerng and Mathematcs. 8: Tng, TC Ansotropc Elastcty, Theory and Applcatons. Oxford Press: New Yor and Oxford