The Weak Patch Test for Nonhomogeneous Materials Modeled with Graded Finite Elements

Transcription

1 Th Wak Patch Tst for Nonhomognous Matrials Modld with... Glaucio H. Paulino Univrsit of Illinois at Urbana Champaign Dpt. of Civil and Environmntal Eng. Urbana, IL 680. USA Jong-Ho Kim Univrsit of Conncticut Dpt. of Civil and Environmntal Eng. Storrs, CT USA Th Wak Patch Tst for Nonhomognous Matrials Modld with Gradd Finit Elmnts Functionall gradd matrials hav an additional lngth scal associatd to th spatial variation of th matrial proprt fild which compts with th usual gomtrical lngth scal of th boundar valu problm. B considring th lngth scal of nonhomognit, this papr prsnts th wak patch tst (rathr than th standard on) of th gradd lmnt for nonhomognous matrials to assss convrgnc of th finit lmnt mthod (FEM). Both consistnc (as th siz of lmnts approach zro, th FEM approimation rprsnts th act solution) and stabilit (spurious mchanisms ar avoidd) conditions ar addrssd. Th spcific gradd lmnts considrd hr ar isoparamtric quadrilatrals (.g. 4, 8 and 9-nod) considring two dimnsional plan and aismmtric problms. Th finit lmnt approimat solutions ar compard with act solutions for nonhomognous matrials. Kwords: finit lmnt mthod (FEM), patch tst, wak patch tst, functionall gradd matrial (FGM), gradd lmnt Introduction Th patch tst was originatd b th mmorabl Bruc Irons and coworkrs (Irons, 966; Bazl t al., 966; Irons and Razzaqu, 97). Th concpt is so important that it can b asil found in man ttbooks in finit lmnts, ithr classical (Hughs, 987; Cook t al., 00; Bath, 995) or mor rcnt (Bltschko t al., 000; Zinkiwicz and Talor, 000) ons, and it is ndd to nsur rliabilit of th finit lmnt mthod (FEM) (Babuška and Strouboulis, 00). Th original patch tst providd a ncssar consistnc condition and thus turnd out to b vr usful for assssing convrgnc of finit lmnts analsis, including nonconforming lmnts (Wilson t al., 973; Talor t al., 976; Talor t al., 986). An arl mathmatical tratmnt was givn b Strang (97), and Strang and Fi (973). For an lmnt which appars to b convrgnt but fails th Iron's patch tst, th wak patch tst is an altrnativ tst, as suggstd b Talor t al. (986). In addition, Bltschko and Lasr (988) hav studid th bhavior of a distortd lmnt with a fractal patch tst, which is also valid in th wak patch tst sns. Th patch tst has bn applid to man problm-tps including, for ampl, mid displacmnt-prssur finit lmnt formulations (Talor t al., 986; Zinkiwicz t al., 986; Razzaqu, 986; Wu and Chn, 997; Zinkiwicz and Talor, 997), thr-dimnsional (3-D) solid lmnts (Loikkann and Irons, 984), plat bnding lmnts (Samulsson t al., 987; Zinkiwicz and Lfbvr, 988; Zhifi, 993; Zinkiwicz t al., 993; Auricchio and Talor, 993; Aricchio and Talor, 994; Martins and Sabino, 997; Park and Choi, 997), and shll lmnts (Hrrmann, 989). Th patch tst has also bn usd as a fundamntal tool to crat nw lmnts or to improv isting ons (Ju and Sin, 996; Chung t al., 00; Piltnr and Talor, 000). Th patch tst has bcom a widl usd procdur, which can b numricall prformd, in ordr to chck th validit of a finit lmnt formulation and coding. It is th ncssar and sufficint condition for finit lmnt analsis convrgnc (Zinkiwicz and Talor, 000). For sufficinc, at last on intrnal lmnt boundar is rquird to vrif that consistnc of a patch solution is maintaind btwn lmnts. To nsur convrgnc, both consistnc and stabilit conditions must b vrifid. Th consistnc rquirmnt nsurs that as th siz of th lmnts h tnds to zro, th finit lmnt approimation rprsnts th act Papr accptd Ma, 006. Tchnical Editor: Paulo E. Miagi. solution. Th stabilit condition is a rquirmnt that an lmnt admits no zro-nrg mod dformation stats whn adquatl supportd against rigid-bod motion, which mans that th lmnt stiffnss matri ( κ ) must b non-singular. Stabilit is usuall chckd b nsuring that th stiffnss matri is of appropriat rank, and thus dosn't allow for apparanc of spurious mchanisms. Modling of functionall gradd matrials (FGMs) b th FEM can b accomplishd b using ithr gradd or homognous lmnts (Santar and Lambros, 000; Kim and Paulino 00a), as illustratd b Fig.. Part (a) of this figur shows an ampl of an ponntiall gradd matrial and part (b) illustrats an L-shapd domain mad of this matrial. Th gradd lmnt (s Fig. (c)) incorporats th matrial proprt gradint at th siz scal of th lmnt, whil th homognous lmnt (s Fig. (d)) producs a stp-wis constant approimation to a continuous matrial proprt fild. Th patch tst has bn usd to vrif convrgnc of convntional homognous lmnts (Fig. (d)). In ordr to assss convrgnc of th gradd lmnts, th must b patch-tstd in th contt of th wak patch tst. θ E ( ) E(,)=E β +γ 0, E( ) = E 0 β = δ cos θ, γ = δ sin θ δ (a) (b) (c) (d) Figur. FEM modling of FGMs: (a) nonhomognous mdium; (b) gnric rgion,.g. L-shapd domain; (c) gradd lmnt; (d) homognous lmnt. Th proprt of th homognous lmnts ma b takn as th actual proprt at th cntroid of th lmnt (cf. (c) and (d)). Hr th smbol mans a nodal point, and th smbol mans a Gauss sampling point. Th smbol ο indicats th location for matrial proprt sampling (s parts (a) and (c)). Various finit lmnt invstigations of gradd matrials hav bn conductd using ithr commrciall availabl (.g. ABAQUS, ANSYS) or rsarch-orintd cods. A sampling (which is, b no mans, haustiv) of publishd paprs includ a broad rang of applications such as lasticit (Santar and Lambros, 000; Kim and Paulino 00a); linar lastic fractur mchanics (Eischn, 987; Gu t al., 999; Anlas t al., 000; Kim and Paulino, 00b, 003, 004, 005; Paulino and Kim, 004); nonlinar fractur mchanics J. of th Braz. Soc. of Mch. Sci. & Eng. Copright 007 b ABCM Januar-March 007, Vol. XXIX, No. / 63

2 Glaucio H. Paulino and Jong-Ho Kim (Carpntr t al., 999; Kim t al., 997); cohsiv zon lmnts for fractur of FGMs (Jin t al., 00; Zhang and Paulino, 005); notch ffct on FGM spcimns (Lin and Miamoto, 000); tribolog (Stphns t al., 000; Jitcharon t al., 998); thrmal strsss (Giannakopoulos t al., 995; Cho and Odn, 000; Cho and Ha, 00; Noda, 999); rsidual strsss (L and Erdogan, 995; Williamson t al., 993; Bckr t al., 000; Khor and Gu, 000); various aspcts of micromchanical modling (Grujicic and Zhang, 998; Dao t al., 997; Shn, 998); numrical homognization (Schmaudr and Wbr, 00; Lβl t al., 999); snsitivit analsis and optimization (Tanaka t al., 996); valuation of th so calld highr ordr thor (Pindra and Dunn, 997); and functionall gradd pizolctric actuators (Almajid t al., 00; Carbonari t al, 006a, 006b). In particular, th us of gradd finit lmnts is of dirct rlvanc to this work. Such lmnts wr usd b Santar and Lambros (000) to modl th bhavior of nonhomognous lastic matrials, and b L and Erdogan (995) to invstigat rsidual/thrmal strsss in FGMs and thrmal barrir coatings. Both authors (Santar and Lambros, 000; L and Erdogan, 995) usd th Gauss point sampling of matrial proprtis. Gradd lmnts wr also usd b Kim and Paulino (003, 004, 005) and Paulino and Kim (004) to invstigat fractur mchanics of FGMs and to modl nonhomognous isotropic and orthotropic matrials, howvr, th hav mplod a gnralizd isoparamtric formulation (Kim and Paulino, 00a). Th goal of th rmaindr of this papr is to dvlop a comprhnsiv prsntation of th wak patch tst for nonhomognous matrials modld with gradd finit lmnts, and assss convrgnc rat of gradd lmnts. This papr is organizd as follows. First, w provid som act solutions for nonhomognous lasticit that will b usd as rfrnc solutions for numrical ampls. Thn w prsnt th gradd lmnt formulations, and various ampls on th wak patch tst, and also assss convrgnc rat. Finall w addrss stabilit considrations for gradd finit lmnts followd b conclusions of th prsnt invstigation. Eact Solutions for Nonhomognous Elasticit: Rfrnc Solutions This sction rvisits a fw closd-form solutions for nonhomognous lasticit problms. Two classs of problms ar considrd: plan and aismmtric. In th first class, w considr an infinitl long plat, gradd along its finit width, undr smmtric loading conditions (fid grip, tnsion, and bnding) and also a simpl shar problm. In th scond class, w considr an aismmtric problm, gradd along th radial dirction, undr aismmtric loading conditions. Ths closd-form solutions will b usd as rfrnc solutions for th wak patch tst. Plan Problms Erdogan and Wu (997) drivd act solutions for strsss to plan lasticit problms involving functionall gradd plats of infinit lngth and finit width undr smmtric loading conditions such as fid grip, tnsion, and bnding awa from th cntr rgion of th spcimn (s Fig. ). Kim and Paulino (00a) tndd th work to orthotropic FGMs, and providd act solutions for displacmnts. Lt's considr th gradd plat illustratd b Fig., and lt's assum th Poisson's ratio as constant. Th shar modulus is givn b W L L σ ( ) σ σ (a) (b) (c) (d) Figur. A gradd strip: (a) gomtr th shadd rgion indicats th smmtric rgion of th plat usd in th prsnt FEM analss (u (a,0) = u (a,0) = 0 with a = 0 whr a dnots th coordinat.); (b) fid-grip loading; (c) tnsion; (d) bnding. E μ( ) = μ β μ =,, ( + ν ) whr /β is th lngth scal of th nonhomognit, which is charactrizd b E log β = W E with E=E() whr E = E(=0) and E = E(=W). Fid Grip Loading For fid grip loading (Fig. (b)) with ε (, ) ε 0 strss is givn b σ ( ) ( ) 8μ = ε 0, + κ t b () () ± =, th whr κ = 3 4 ν : plan strain κ = ( 3 ν) ( + ν) : plan strss. Using th strain-displacmnt rlations and appling th following boundar conditions (,0) = 0, (,0) = 0, (4) u a u whr th paramtr a dnots a rfrnc point for th displacmnt boundar condition, on obtains th following displacmnt filds κ 3 u a (, ) = ε 0 ( + ) + κ (, ) = ε. u 0 Notic that th displacmnt filds ar linar and thus strains ar constant; howvr, strsss var ponntiall. (3) (5) 64 / Vol. XXIX, No., Januar-March 007 ABCM

3 Th Wak Patch Tst for Nonhomognous Matrials Modld with... Tnsion and Bnding For tnsion (Fig. (c)) and bnding (Fig. (d)) loads applid prpndicular to matrial gradation, th applid strsss ar dfind b bw N = σ tw, M = σ, 6 whr N is a mmbran rsultant applid along th = W lin, and M is th bnding momnt. An infinitl long strip undr ths two loading cass can b considrd as on-dimnsional problm whr σ = σ = 0, and 0. Thus th compatibilit condition ε = 0 givs σ ( ) ( ) ( ) 8 = μ A + B, + κ whr th constants A and B ar dtrmind from W σ W ( ), σ ( ) d = N d = M 0 0 b assuming M = NW for tnsion and N = 0 for bnding. (9) Thus, for tnsion load, th constants A and B ar: ( ) βn + κ Wβ β + Wβ + β A = 6 βn ( κ) W β Wβ W Wβ B = 6 βw βw, βw βw βw μ β W + βw βw βw βw βw βw, μ β W + rspctivl. For bnding load, th constants A and B ar: βw ( + κ) β ( ) β M A =, βw βw βw 8μ β W + βw βw β M ( + κ) βw + B =, βw βw βw 8μ β W + (5) (7) (8) (0) () τ τ E()=E β L W Figur 3. A functionall gradd plat undr constant shar (σ = τ =.0). μ( ) = μ β. (3) Bcaus σ =.0 vrwhr, th shar strain distribution bcoms ε = = μ ( ) μ. β τ (4) Using th strain-displacmnt rlations and appling th boundar conditions: ( ) ( ) u,0 = u,0 = 0, on obtains th following displacmnt fild β u(, ) = ( ), u(, ) = 0. μβ Aismmtric Problm (5) Horgan and Chan (999) providd act solutions for strsss for a hollow circular clindr subjctd to uniform prssur p i and p on th innr (r 0 i = a) or outr (r 0 = b) surfacs, rspctivl (s Fig. 4). Th assumd powr-law variation of Young's modulus givn b rspctivl. Using th strain-displacmnt rlations and appling th boundar conditions givn b Eq.(4), on obtains th following displacmnt fild κ 3 A A A u ( ) B a Ba + k u, A B., = +, ( ) = ( + ) () r θ z σ z Simpl Shar Figur 3 shows a gradd plat undr uniform simpl shar load, i.. σ = τ =.0. Assum that th Poisson's ratio is constant, and th shar modulus varis in th dirction as follows: τ zr σ θ σ r Figur 4. An aiall smmtric hollow clindr or disk. J. of th Braz. Soc. of Mch. Sci. & Eng. Copright 007 b ABCM Januar-March 007, Vol. XXIX, No. / 65

4 Glaucio H. Paulino and Jong-Ho Kim r E( r) = E a with E E( a) n (6) = and th powr n bing a dimnsionlss constant. Th displacmnts ar givn b: ( + ) ( + ) ( ) = n k n k u r Cr + C r k = ( n + n ) /, (7), 4 4 ν whr C and C ar constants which can b dtrmind b appling th aismmtric boundar conditions s th papr b Horgan and Chan (999). Gradd Finit Elmnt Formulation as Displacmnts for an isoparamtric finit lmnt can b writtn m u = N u i= i i (8) whr N i ar shap functions, u i is th nodal displacmnts corrsponding to nod i of lmnt, and m is th numbr of nods in an lmnt. For ampl, for a Q4 lmnt, th standard shap functions ar = ( + )( + ) 4, i=,...,4 (9) N ξξ ηη i i i whr (ξ,η) dnot intrinsic coordinats in th intrval [-,] and (ξ i,η i ) dnot th local coordinats of nod i. Strains ar obtaind b diffrntiating displacmnts as ε = Bu, (0) whr B is th strain-displacmnt matri of shap function drivativs. Th strain-strss rlations ar givn b σ = D ( ) ε, () whr D () is th constitutiv matri, which is a function of spatial position, i.. D ( ) = D (, ). () Th principl of virtual work ilds th following finit lmnt stiffnss quations (Hughs, 987): ( ) ku = f, k = T B D BdΩ Ω (3) whr f is th load vctor, k is th lmnt stiffnss matri, and Ω is th domain of lmnt (). For th gradd lmnt, th D () matri varis spatiall within th lmnt. Th polnomial ordr of th matri will influnc th numbr of Gauss intgration points rquird for th rducd and full intgrations. This bhavior is invstigatd in th numrical ampls sction using two sts of Gauss intgration points. A sstm of algbraic quations is assmbld such that n n ij ij i i = = K u = F, K = k, F = f, (4) whr n is th numbr of lmnts. Th linar sstm and th drivativs (.g. strains and strsss) ar rcovrd using standard procdurs (Cook t al., 00). Two kinds of FEM formulations ar usd for gradd lmnts: dirct Gaussian intgration formulation and gnralizd isoparamtric formulation (GIF). Ths approachs diffr on th location that th matrial proprtis ar sampld in th lmnt: Gauss sampling points for th dirct Gaussian formulation (Fig. 5(a)) and nodal sampling points for th GIF (Fig. 5(b)). In this work, w slctivl us both formulations. P(,) P P(,) Figur 5. Gradd finit lmnts: (a) Dirct Gaussian intgration formulation; (b) Gnralizd isoparamtric formulation (Kim and Paulino, 00a). P dnots a gnric proprt. Dirct Gaussian Intgration Formulation Th intgral of Eq.(3) is valuatd b Gaussian quadratur, and th matri D () can b dirctl spcifid b mploing th Young's modulus and th Poisson's ratio at ach Gaussian intgration point (s Fig. 5(a)). Thus, for D problms, th rsulting intgral bcoms T ( ξ ) k = B D B tjww i j, i j ( i, j) (5) whr i and j indicats th corrsponding Gauss sampling points in th lmnt, ξ = (ξ,η), t dnots thicknss, J is th dtrminant of th Jacobian matri, i.. J = dt(j), and W i is th wight of ach Gauss point. Gnralizd Isoparamtric Formulation (GIF) Th displacmnts (u,v) = (u, u ) ar intrpolatd for -D problms as u = Nu, v= N v i i i i i i (6) whr th summation is don ovr th lmnt nodal points. Similarl, th spatial coordinats (,) ar intrpolatd as = N %, = N % (7) i i i i i i Matrial proprtis can also b intrpolatd from th lmnt nodal valus b mans of shap functions, as illustratd b Fig. 5(b). For instanc, th Young's modulus E = E() and Poisson's ratio ν = ν() ar givn b E = NE, v= Nv ˆ i i i i i i (8) rspctivl, whr i N and ˆ i N ar appropriat shap functions, which ma b distinct from ach othr. Th gnralizd isoparamtric formulation (GIF) concpt lads to 66 / Vol. XXIX, No., Januar-March 007 ABCM

5 Th Wak Patch Tst for Nonhomognous Matrials Modld with... N = N % = N = N ˆ. (9) E= Shadd Elmnt In this approach, matrial proprtis at Gaussian intgration points ar intrpolatd from th nodal matrial proprtis of th lmnt using isoparamtric shap functions, which ar th sam shap functions as spatial coordinats and displacmnts. Numrical Eampls In ordr to assss convrgnc and convrgnc rats of gradd finit lmnts b mans of th wak patch tst, a st of problms in plan and aismmtric stats ar invstigatd undr msh rfinmnt using both an in-hous MATLAB cod and th commrcial softwar ABAQUS. Plan Problms A fw problms in plan strss stat ar considrd whr Young's modulus is a function of, i.. E = E(), whil th Poisson's ratio is constant. Th modulus is assumd to var ponntiall, i.. E ( ) = E β (30) whr E = E(0) and /β is th lngth scal of th nonhomognit charactrizd b Eq.(). Th applid loading involvs fid-grip, tnsion, and bnding cass. Th GIF is usd for this stud. Patch Tst for Gradd Elmnt: Standard or Wak? Figur 6 shows a 5-lmnt patch of isoparamtric, 4-nod (Q4), 8-nod (Q8) Srndipit (not shown) and 9-nod (Q9) Lagrangian (shown) quadrilatral lmnts undr fid-grip loading. Th applid loading corrsponds to constant normal strains, i.. (,) = ε 0 E β whr ε 0 =.0, E =.0, and β = log()/. This strss distribution was obtaind b appling nodal forcs along th right dg of th finit lmnt msh. Th displacmnt boundar conditions ar prscribd such that u = 0 in th rgion 0 along = 0 lin and, in addition, u = 0 at ithr top (for Q4) or middl (for Q8 and Q9) nod on th lft dg (s Fig. 6). W = L = A C A B C E= G4 G4 Gauss Sampling points 33 Gauss Sampling points Figur 6. Th patch tst with 5 gradd lmnts for 4-nod, 8-nod (not shown) and 9-nod (shown) isoparamtric quadrilatrals. Th applid load corrsponds to (,) = ε 0 E β (ε 0 =, E =.0, β = log()) for th fid grip cas. Th quivalnt nodal loads ar shown in th figur. Th following data wr usd for th finit lmnt analsis: E =.0, v = 0.3, plan strss, and 3 3 Gauss quadraturs (3) Tabl compars th FEM rsults for strsss and displacmnts with th analtical solutions givn b Erdogan and Wu (997) and in th Sction for Eact Solutions (Kim and Paulino, 00a), rspctivl. Th nodal displacmnts ar calculatd at nods A, B, and C and strsss ar computd at th and 3 3 Gaussian intgration points as spcifid in Fig. 6. Tabl shows that th gradd Q4, Q8 and Q9 lmnts provid slightl inaccurat displacmnts and strsss with th givn lvl of msh rfinmnt. On can conclud from th abov patch tst that th prformanc of gradd lmnts dpnds on th dgr of msh rfinmnt corrsponding to matrial gradation and loading conditions. Thus, th patch tst nds to b prformd for th gradd lmnts b subdividing th msh. This ampl justifis th nd for th wak rathr than th standard patch tst for th gradd lmnts. Tabl. Th patch tst with a constant-strain condition using fiv gradd lmnts (Q4, Q8 and Q9) subjctd to fid-grip loading (s Figur 6). Th displacmnts ar u = u and υ = u. Gnralizd isoparamtric formulation (GIF) is usd for sampling matrial proprtis. Eact solutions for th displacmnts for th Q4 lmnt ar diffrnt from thos for th Q8 and Q9 lmnts du to diffrnt displacmnt boundar conditions. Th numric prcision is O(0 4 ). Loading Cas Fid-grip Displacmnts 4-nod 8-nod 9-nod Eact & Strsss (Q4) 3 3 ua va ub vb uc vc G J. of th Braz. Soc. of Mch. Sci. & Eng. Copright 007 b ABCM Januar-March 007, Vol. XXIX, No. / 67

6 Glaucio H. Paulino and Jong-Ho Kim Wak Patch Tst: Q4 Figur 7 shows a non-homognous bam with ponntiall gradd modulus subjctd to applid load at th right nd. Th msh discrtization consists of 4, 8 4, and 6 8 patchs of 4-nodd isoparamtric quadrilatral lmnts undistortd or distortd according to th gomtrical distortion paramtr d. Th applid loading corrsponds to (,0) = ε 0 E β for fid grip, (,0)=.0 for tnsion, and (,0)= + for th bnding cas, whr ε 0 =.0, E =.0, and β = log(4)/. This strss distribution was obtaind b appling nodal forcs along th right dg of th finit lmnt msh. Th displacmnt boundar condition is prscribd such that u = 0 in th rgion 0 along = 0 lin and, in addition, u = 0 for th nod in th middl of th lft hand sid (s Figur 7). E = W = L = 0 E = 4 4 B Fid grip Tnsion Bnding d A B C 0.5d d 0.5d Fid grip Tnsion Bnding A B C 0.5d d 0.5d Fid grip Tnsion Bnding Shadd Elmnt Gauss Sampling points Figur 7. Wak patch tst with M N lmnts (4,8 4,6 8) for 4-nod isoparamtric quadrilatrals (Q4). Th mshs ar distortd according to th gomtric distortion paramtr d. Th quivalnt nodal loads ar shown at th right-hand-sid of th corrsponding mshs. 68 / Vol. XXIX, No., Januar-March 007 ABCM

7 Th Wak Patch Tst for Nonhomognous Matrials Modld with... Th following data ar usd for th finit lmnt analsis: E =.0, v = 0.3, plan strss, Gauss quadratur (3) Tabl compars th FEM rsults for 4, 8 4, and 6 8 undistortd (d = 0) mshs of th gradd Q4 lmnt with th act solutions. Th nodal displacmnts ar calculatd at nods A, B, and C, and strsss ar computd at th Gaussian intgration points as spcifid in Figur 7. Notic that, for all loading cass, msh rfinmnt is ndd for acquiring a dsird accurac and it incrass th accurac of FEM rsults. Th accurac of ach individual msh is bttr for fid-grip loading than tnsion and bnding loading cass du to th diffrnc in th natur of boundar conditions. Th distortd msh ( d 0 ) givs wors rsults than undistortd mshs (d = 0), and onl th cas d = 0 is invstigatd in this ampl. Distortd mshs ( d 0 ) ar considrd in latr in this papr. Tabl. A wak patch tst using 4, 8 4, and 6 8 mshs of undistortd (d = 0) Q4 gradd lmnts (s Figur 7). Th displacmnts ar u = u and υ = u. Gnralizd isoparamtric formulation (GIF) is usd for sampling matrial proprtis. Th numric prcision is O(0 4 ). Loading Displacmnt 4-nod Eact Cas & Strss ua va ub vb Fid-grip uc vc Tnsion Bnding ua va ub vb uc vc ua va ub vb uc vc Wak Patch Tst: Q8 and Q9 Figur 8 shows a nonhomognous bam with ponntiall gradd modulus subjctd to applid load at th right nd. Th msh discrtization consists of, 4, and 8 4 patchs of 8-nod (Q8) Srndipit (not shown) or 9-nod (Q9) Lagrangian (shown) quadrilatral lmnts. Th msh is distortd according to th gomtrical distortion paramtr d. Th applid loading corrsponds to (,0)= ε 0 E β for fid grip, (,0)=.0 for tnsion, and (,0)= + for bnding whr ε 0 =.0, E =.0, and β= log(4)/. This strss distribution was obtaind b appling nodal forcs along th right dg of th finit lmnt msh. Th displacmnt boundar condition is prscribd such that u = 0 in th rgion 0 along = 0 lin and, in addition, u = 0 for th nod in th middl of th lft hand sid (s Figur 8). J. of th Braz. Soc. of Mch. Sci. & Eng. Copright 007 b ABCM Januar-March 007, Vol. XXIX, No. / 69

8 Glaucio H. Paulino and Jong-Ho Kim E = W = L = 0 E = 4 4 Fid grip Tnsion Bnding A d A 0.5d d 0.5d Fid grip Tnsion Bnding A 0.5d d 0.5d Fid grip Tnsion Bnding Shadd lmnt Gauss Sampling points 33 Gauss Sampling points Figur 8. Wak patch tst with M N lmnts (, 4, 8 4) for 8-nod (not shown) and 9-nod (shown) isoparamtric quadrilatrals. Th mshs ar distortd according to th gomtric distortion paramtr d. Th quivalnt nodal loads ar shown at th right-hand-sid of th corrsponding mshs. Th following data wr usd for th finit lmnt analsis: E =.0, v = 0.3, plan strss, and 3 3 Gauss quadraturs (33) Tabls 3, 4, and 5 compar th FEM rsults for th, 4, and 8 4 mshs, rspctivl. Th act solutions includ strsss obtaind b Erdogan and Wu (997), and displacmnts drivd b th authors. Th nodal displacmnts ar calculatd at nod A and strsss ar computd at th and 3 3 Gaussian intgration points as spcifid in Figur / Vol. XXIX, No., Januar-March 007 ABCM

9 Th Wak Patch Tst for Nonhomognous Matrials Modld with... For fid grip loading cas, in gnral 3 3 Gauss quadratur for Q8 and Q9 lmnts givs bttr rsults than quadratur for Q8 lmnt, and th accurac incrass with msh rfinmnt. For tnsion and bnding loading cass, and 3 3 Gauss quadraturs for Q8 and 3 3 Gauss quadratur for Q9 lmnts giv wors rsults than for thos fid grip loading; howvr, th accurac is improvd with msh rfinmnt. Tabls 3 to 5 show that both Q8 and Q9 gradd lmnts bhav rlativl wll with distortd mshs. In gnral, th ffct of th distortion as masurd b th paramtr d, is rducd with msh rfinmnt. For th tnsion cas, w obsrv that a msh with 6 8 lmnts, although not prsntd hr, lads to th act rsults for Q8 lmnts with both and 3 3 intgration ruls within O(0 4 ) accurac. For Q9 lmnts, th rsults for th tnsion cas convrg for th 8 4 msh (cf. Tabl 5). Tabl 3. A wak patch tst with gradd Q8 and Q9 lmnts (s Figur 8). Th displacmnts ar u = u and υ = u. Gnralizd isoparamtric formulation (GIF) is usd for sampling matrial proprtis. Th numric prcision is O(0 4 ). Distortion d=0 d= Loading Displacmnts 8-nod 9-nod Eact Cas & Strsss ua va Fid-grip Tnsion Bnding Fid-grip Tnsion Bnding ua va ua va ua va ua va ua va J. of th Braz. Soc. of Mch. Sci. & Eng. Copright 007 b ABCM Januar-March 007, Vol. XXIX, No. / 7

10 Glaucio H. Paulino and Jong-Ho Kim Tabl 4. A wak patch tst with 4 gradd Q8 and Q9 lmnts (s Figur 8). Th displacmnts ar u = u and υ = u. Gnralizd isoparamtric formulation (GIF) is usd for sampling matrial proprtis. Th numric prcision is O(0 4 ). Distortion d=0 d= Loading Displacmnts 8-nod 9-nod Eact Cas & Strsss ua va Fid-grip Tnsion Bnding Fid-grip Tnsion Bnding ua va ua va ua va ua va ua va / Vol. XXIX, No., Januar-March 007 ABCM

11 Th Wak Patch Tst for Nonhomognous Matrials Modld with... Tabl 5. A wak patch tst with 8 4 gradd Q8 and Q9 lmnts (s Figur 8). Th displacmnts ar u = u and υ = u. Gnralizd isoparamtric formulation (GIF) is usd for sampling matrial proprtis. Th numric prcision is O(0 4 ). Distortion d=0 d= Loading Displacmnts 8-nod 9-nod Eact Cas & Strsss ua va Fid-grip Tnsion Bnding Fid-grip Tnsion Bnding ua va ua va ua va ua va ua va Highr-Ordr Wak Patch Tst Figur 9 compars th ratio of numrical ( υ = u ) and analtical (υ (act) ) displacmnts vrsus Poisson's ratio ( v ) for th bnding loading cas (s Fig. (d)). Both rgular (d = 0) and distortd (d = ) mshs discrtizd with Q4, Q8 and Q9 lmnts ar considrd using a patch of 4 lmnts. Th nodal displacmnts ar valuatd at location B in Fig. 7, or location A in Fig. 8. Figur 9 shows that both Q8 and Q9 lmnts convrg to th act solutions indpndnt of th Poisson's ratio, rul of Gauss quadratur, and distortion, whil th rgular and distortd Q4 lmnts giv significantl inaccurat rsults. Th bhavior of Q4 lmnts can b improvd with an incompatibl lmnt such as th Q6 and th radr is rfrrd to rfrncs (Wilson t al., 973; Talor t al., 976; Cook t al., 00). J. of th Braz. Soc. of Mch. Sci. & Eng. Copright 007 b ABCM Januar-March 007, Vol. XXIX, No. / 73

12 Glaucio H. Paulino and Jong-Ho Kim Figur 9. Msh with 4 lmnts for Q4, Q8, and Q9 lmnts. Th bullt in th insrt dnots th point whr th displacmnts ar calculatd; rgular msh (d = 0) and distortd msh (d = ). Aismmtric Problms Considr th aismmtric problm of a hollow circular clindr or disk with th innr radius (a = ) and th outr radius (b = ) subjctd to uniform prssur on th innr surfac. Patchs of 4 4 and 8 8 isoparamtric lmnts ar considrd hr. Figur 0 shows 4-nod (Q4) quadrilatral lmnts and Figur illustrats 8-nod (Q8) Srndipit or 9-nod (Q9) Lagrangian lmnts for diffrnt distortion factors d (d = 0 or d = 0.). Th applid loading corrsponds to σ rr (,z)=.0 along 0 z whr z dnots th vrtical ais from which th innr (a) and outr (b) radii ar dfind in Figurs 0 and (s also Figur 4). This strss distribution was obtaind b appling nodal forcs along th lft dg of th finit lmnt msh. Th displacmnt boundar condition is prscribd such that u z = 0 for th nods on th top and bottom dgs (s Figs. 0 and ). Elmnt Gauss Sampling points a = P = W = A B C 0.5d d 0.5d b = (a) H= r P = W = A B C 0.5d d 0.5d Figur 0. A wak patch tst with 4 4 and 8 8 lmnts for 4-nod isoparamtric quadrilatrals for aismmtric problm. (b) Elmnt Gauss 3 3 Gauss Sampling points a = P = b = W = A B C 0.5d d 0.5d (a) H= r P = W = A B C 0.5d d 0.5d Figur. A wak patch tst with 4 4 and 8 8 lmnts for 8-nod (not shown) and 9-nod (shown) isoparamtric quadrilatrals for aismmtric problm. Young's modulus is an ponntial function of r as givn b Eq. (6), in which n is th nonhomognit paramtr. Th Poisson's ratio is assumd constant. Th dirct Gaussian formulation is usd for ths aismmtric problms. Th following data ar usd for th FEM analss: E =.0, n =, v = 0.0,0.3, plan strss, and 3 3 Gauss quadraturs (b) (34) Tabls 6, 7 ( v = 0 ) and 8 ( v 0 ) compar th FEM rsults with th act solutions providd b Horgan and Chan (999). Th nodal displacmnts ar calculatd at nods A, B, and C, and strsss ar computd at th or 3 3 Gaussian intgration points as spcifid in Figs. 0 and. 74 / Vol. XXIX, No., Januar-March 007 ABCM

13 Th Wak Patch Tst for Nonhomognous Matrials Modld with... Tabl 6. A wak patch tst with 4 4 quadrilatral lmnts for aismmtric cas (u = u r,ν= 0)- s Figurs 0 and. Dirct Gaussian intgration mthod is usd for sampling matrial proprtis. Th numric prcision is O(0 4 ). Distortion d=0 d=0. Displacmnts 4-nod 8-nod 9-nod Eact & Strsss ua ub uc σ rr σ θθ σ rr σ θθ σ rr σ θθ ua ub uc σ rr σ θθ σ rr σ θθ σ rr σ θθ Tabl 7. A wak patch tst with 8 8 quadrilatral lmnts for aismmtric cas (u = u r, ν = 0)- s Figurs 0 and. Dirct Gaussian intgration mthod is usd for sampling matrial proprtis. Th numric prcision is O(0 4 ). Distortion d=0 d=0. Displacmnts 4-nod 8-nod 9-nod Eact & Strsss ua ub uc σ rr σ θθ σ rr σ θθ σ rr σ θθ ua ub uc σ rr σ θθ σ rr σ θθ σ rr σ θθ J. of th Braz. Soc. of Mch. Sci. & Eng. Copright 007 b ABCM Januar-March 007, Vol. XXIX, No. / 75

14 Glaucio H. Paulino and Jong-Ho Kim Tabl 8. A wak patch tst with 8 8 quadrilatral lmnts for aismmtric cas (u = u r, ν = 0.3)- s Figurs 0 and. Dirct Gaussian intgration mthod is usd for sampling matrial proprtis. Th numric prcision is O(0 4 ). Distortion d=0.0 d=0. Displacmnts 4-nod 8-nod 9-nod Eact & Strsss ua ub uc σ rr σ θθ σ rr σ θθ σ rr σ θθ ua ub uc σ rr σ θθ σ rr σ θθ σ rr σ θθ A comparison of th rsults of Tabls 6 and 7 indicat that th 4 4 discrtization of Figs. 0(a) and (a) ar too coars to achiv an accurat solution for this problm. For v = 0.0 and 8 8 msh (Tabl 7 and Fig. 0(b)), th Gauss quadratur for th Q8 givs act rsults for all strsss rgardlss of distortion. Also, th 3 3 Gauss quadratur of th Q8 and Q9 lmnts producs act rsults for hoop strsss rgardlss of distortion, but lads to slightl incorrct rsults for th radial strss σ rr (s Tabl 7 and Figs. 0(b) and (b)). For v = 0.3 and 8 8 msh (Tabl 8 and Figs. 0(b) and (b)), th Gauss quadratur of th Q8 givs act rsults for all strsss onl for th d = 0 cas. Th 3 3 Gauss quadratur of th Q8 and Q9 lmnts producs slightl incorrct rsults for all strsss. Th bhavior of th and 3 3 Gauss quadraturs on th radial strsss is consistnt with th prvious obsrvation about th strss for th tnsion loading applid paralll to th matrial gradation (Kim and Paulino, 00a). Convrgnc Rats This ampl invstigats convrgnc rats of Q4 and Q8 lmnts considring an FGM strip undr far-fild tnsion loads. Figur (a) shows gomtr and boundar conditions for a strip with infinit lngth undr far-fild tnsion, and Figur (b) shows a plat with finit lngth undr tractions quivalnt to th far-fild tnsion (s Eqs.(7) and (0)). L N=Wσ t W L=W σ () Figur. Problm dfinition: (a) plat with infinit lngth undr far-fild constant tnsion; (b) plat with finit lngth undr act distribution of tractions that ar quivalnt to th far-fild tnsion applid to th gradd strip (L/W = ). Th applid load is prscribd on th uppr dg with normal strss: () givn b Eq.(7) with appropriat constants A and B. Th displacmnt boundar condition is spcifid such that u = 0 along th lowr dg, and u = 0 for th nod at th lft-bottom cornr. Young's modulus is an ponntial function of as givn b Eq.(30). Th Poisson's ratio is assumd constant. Th following data ar usd for th FEM analss: W 76 / Vol. XXIX, No., Januar-March 007 ABCM

15 Th Wak Patch Tst for Nonhomognous Matrials Modld with... W =, L =, E =.0, E ( 3, 5, 7,0), = v = 0.3, plan strss, and 3 3 Gauss quadraturs (35) Th discrtization rror can b quantifid b th rror in th nrg norm dfind as T = ( ε εfe ) D( )( ε εfe ) dω Ω, (36) whr ε and ε FE ar th act and finit lmnt strain filds, D() is th constitutiv matri of FGMs, and Ω is th domain of th problm. W us msh subdivision for assssing convrgnc rats of Q4 and Q8 lmnts with th following lmnt discrtization along th width (W) and lngth (L), i.. 0 0, 0 0, and lmnts. Th GIF is usd for this stud. Figur 3 shows th rror in th nrg norm calculatd b considring th whol plat for E /E = 3,5,7 and 0, and also givs usful information on th convrgnc rat. Th notation h dnots th siz of th squar lmnt usd. Th 8-nod (Q8) quadrilatral lmnts with 3 3 and Gauss quadratur provid highr accurac and convrgnc rats than thos for 4-nod (Q4) quadrilatral lmnts. rfrnc, Figur 4 shows imposd displacmnts for diffrnt dformation mods for homognous matrials (β = 0), whil Figs. 5 and 6 illustrat th dformations for β =.0, 0.5,.0. In ordr to rprsnt tnsion, shar, and pur bnding dformation in FGMs (β 0), th act solutions to displacmnts ar proportionall factord to giv Δ = 0. as th singl displacmnt at th nod indicatd in Figurs 4 to 6. Not that dformation-quivalnt loads ar diffrnt for homognous and FGM cass. Figur 4 rprsnts wll-known and pctd rsults for homognous matrials (β = 0). In Figs. 4 and 6, notic that th matrial gradation β has a significant influnc on th dformation mod. Figur 5 shows that for th tnsion cas, if Δ = 0. is imposd at th lft-top cornr nod, thn th dformd shap scals with β (cf. th first two clls of th first row) and is significantl changd if β changs sign, i.. rvrsal of gradation (cf. compar th first two clls with th third cll). If Δ = 0. is imposd at th righttop cornr, th dformd shaps ar consistnt with thos just dscribd abov, i.. th first and scond rows of clls of Figur 5. Notic that if β > 0 th rfrnc displacmnt (Δ) is in th wak matrial sid, whil if β < 0 it is on th strong matrial sid. A comparison btwn th first cll of th first row of Figur 4(a) (β = 0) and thos clls of Figur 5 (β 0) indicat that th dformd shaps in FGMs can b countr-intuitiv as bnding-tp dformation dvlops for purl applid tnsion load Q4 log Q8 (a) E constant (b) Q8 33 E /E = log h 0 Figur 3. Error in th nrg norm of th problm for E /E = 3, 5, 7, and 0. Th nrg norm is calculatd considring all th lmnts (L/W =.0). Stabilit Considrations Two kinds of invstigations ar mad rgarding th stabilit analsis. First, basic dformation mods (tnsion, bnding, and shar) ar studid, and thn, an ignanalsis is prformd at th lmnt lvl (ignvalu tst). Th lattr tst can dtct zronrg dformation mods, lack of invarianc (with rspct to gomtrical orintation), and absnc of rigid bod motion capabilit; and can also provid an stimation of th rlativ qualit of compting lmnts. In ths invstigations, th gradd lmnt is compard with th homognous lmnt. Th dirct Gaussian formulation is usd for this stud (c) Figur 4. Imposd displacmnt vctors for homognous matrials (ν = 0): (a) tnsion, (b) pur shar, (c) pur bnding for Q4, and (d) bnding for Q8. 0. E E() (d) Singl Elmnt Tst - Basic Dformation Mods Th strain nrg stord in th Q4 and Q8 lmnts is compard for homognous and gradd lmnts considring tnsion, pur shar, and pur bnding dformation mods. For th sak of β=.0 β=0.5 β=.0 Figur 5. Imposd displacmnt vctors to giv Δ = 0. at th nod indicatd for tnsion applid prpndicular to th matrial gradation (β =.0, 0.5, -.0 and ν = 0). J. of th Braz. Soc. of Mch. Sci. & Eng. Copright 007 b ABCM Januar-March 007, Vol. XXIX, No. / 77

16 Glaucio H. Paulino and Jong-Ho Kim Figur 6 illustrats various dformation mods for FGMs (β 0) considring ν = 0. Th clls in th first row illustrats dformation mods undr tnsion loading for E=E(). Diffrntl from th configuration obsrvd in Figur 5, thr is no bnding dformation in th clls of Figur 6(a), indpndnt of th β valu. This diffrnc is du to th orintation of matrial gradation in ach cas. Figur 6(b) shows dformd shaps for th pur bnding loading considring E=E() and β =.0, 0.5, and -.0. Th rfrnc dformation is Δ = 0. at th bottom-right cornr. It is intrsting to obsrv th shift in th middl point of th bottom-dg (cf. solid and hollow bullts) as a function of th matrial gradation paramtr β. Th hollow dot rprsnts a point in th homognous matrial, which dos not mov upon bnding dformation (cf. Fig. 4(d)) and thus indicats th nutral ais location for this configuration. Th solid dot rprsnts th FGM cas and, diffrntl from th homognous cas, th point shifts as a function of th matrial gradation. For ampl, according to Fig. 6(b) β > 0, u > 0, u < 0. β < 0, u > 0, u > 0. Notic that th rfrnc point shifts down for β > 0 and it shifts up for β < 0. Figur 6(c) shows dformd shaps for pur shar loading considring E=E() and β=.0, 0.5, and -.0. Notic that th curvatur of th dgs that wr vrtical in th original configuration changs whn β changs sign. E E() E E() 0.9 E E() 0. β=.0 β= (a) β=0.5 β= (b) β= β= β=.0 β=0.5 β=.0 (c) Figur 6. Imposd displacmnt vctors to giv Δ = 0. at th point indicatd for β =.0, 0.5, -.0 and matrial gradation as shown abov (ν = 0). (a) tnsion; (b) bnding; (c) shar. Tabl 9 summarizs th strain-nrg valus inducd b th abov actl imposd displacmnts. Imposd displacmnts for Q4 and Q8 lmnts ar diffrnt from th act imposd displacmnts bcaus of approimating function charactristics of lmnts. In othr words, nodal displacmnts for both lmnts ar imposd actl, but displacmnts among nods insid an lmnt ar intrpolatd using shap functions. All th rsults ar normalizd with rspct to th strain nrg for th homognous matrial (β = 0). This tabl shows that th ordr of intgration can hav a significant impact in th FEM rsults (cf. vrsus for Q4 and vrsus 3 3 for Q8). Thus rducd intgration for gradd lmnts should b usd with grat car. Nglcting th rducd intgration for th Q4, w obsrv that, for th bnding cas, th strain nrg for th FGM (β 0) is alwas lowr than or qual to that for th homognous matrial (β = 0). Th rsults ar functions of th matrial gradation paramtr β. Tabl 9. Strain nrg ratio of th FEM rsults for th FGM with rspct to th act solution for th homognous matrial inducd b th imposd displacmnts (Cas(a): displacmnt Δ = 0. imposd at th top-lft cornr; Cas(b): displacmnt Δ = 0. imposd at th top-right cornr). Cas Tnsion E() Tnsion E() Cas (a) Tnsion E() Cas (b) Shar E() Bnding E() β 4-nod 8-nod Spctral Analsis - Eignvalu Tst Th ignvalu tst is prformd for th singl lmnt stabilit chck. Th tst can dtct zro nrg dformation mods (both rigid bod and spurious mods). Th lmnt is not constraind so that th lmnt stiffnss matri κ is th complt matri. Thus thr indpndnt rigid-bod motions ist in th plan, and thr of th ignvalus should b zro for a plan lmnt. In addition, zro-nrg or spurious singular mods also ild zro ignvalus. Th lmnt is squar, and its lngth and width ar.0 with th origin ( = = 0) at th lft-bottom-cornr nod. In th lmnt, Young's modulus is givn b Eq.(30) with E =.0 (normalizd), and β = 0.0 (homognous) and.0 (nonhomognous matrial). Th Poisson's ratio is assumd to b constant, i.. ν = 0.3. Figurs 7 to illustrat th rsults of th spctral analsis for th Q4, Q8, and Q9 considring β=0 (homognous matrial) and 78 / Vol. XXIX, No., Januar-March 007 ABCM

17 Th Wak Patch Tst for Nonhomognous Matrials Modld with... β=.0 (FGM). A comparison btwn th rsults for homognous matrials and FGMs lads to th following obsrvations: As pctd, th numbr of rigid-bod mods (thr) is th sam for both homognous and gradd lmnts Figur 0. Eign-analsis for Q8 ( Gauss quadratur) with β = (FGM). Th numbrs indicat th ignvalus (λ i ). Compar with Figur 9. Figur 7. Eign-analsis for Q4 ( Gauss quadratur) with β = 0 (homognous matrial). Th numbrs indicat th ignvalus (λ i ). Compar with Figur Figur 8. Eign-analsis for Q4 ( Gauss quadratur) with β = (FGM). Th numbrs indicat th ignvalus (λ i ). Compar with Figur Figur. Eign-analsis for Q9 (3 3 Gauss quadratur) with β = 0 (homognous matrial). Th numbrs indicat th ignvalus (λ i ). Compar with Figur Figur 9. Eign-analsis for Q8 ( Gauss quadratur) with β = 0 (homognous matrial). Th numbrs indicat th ignvalus (λ i ). Compar with Figur Figur. Eign-analsis for Q9 (3 3 Gauss quadratur) with β = (FGM). Th numbrs indicat th ignvalus (λ i ). Compar with Figur. J. of th Braz. Soc. of Mch. Sci. & Eng. Copright 007 b ABCM Januar-March 007, Vol. XXIX, No. / 79

18 Glaucio H. Paulino and Jong-Ho Kim Th numbr of spurious dformation (zro nrg) mods is also th sam for both homognous and gradd lmnts, i.. two spurious mods for Q4 with ordr of quadratur, on for Q8 with Gauss intgration, and thr for Q9 with Gauss intgration Smmtr, as prssd b th dformation mods (ignvctors), is brokn for gradd lmnts, i.. thr ar no rpatd ignmods or rpatd ignvalus as in th homognous lmnt. Th total nrg (U i = λ i /, i=,...,ndofs) incrass for th FGM with β > 0 in comparison with that for th homognous matrial. Hr NDOFs indicats th numbr of dgrs of frdom in th lmnt. Conclusions Onc an lmnt passs th patch tst with a consistnc and a stabilit chck, convrgnc is assurd as th siz of lmnts tnd to zro. Th original patch tst considrs constant strain or strss stat for convntional homognous finit lmnts. Howvr, for nonhomognous matrials, consistnc and stabilit of gradd finit lmnts ar vrifid in th contt of th wak patch tst. In th prcding sctions, convrgnc and convrgnc rats of th Q4, Q8, and Q9 gradd lmnts (for both plan and aismmtric problms) to th act solutions hav bn studid undr subdivision of finit lmnts. This stud indicats that, in gnral, 3 3 Gauss Quadratur for Q8 and Q9 lmnts shows bttr prformanc than Gauss Quadratur for Q8, and that Q4 lmnts nd to b usd with car du to low convrgnc rats. Morovr, th stabilit invstigation rvals that on should b vr carful whn using homognous lmnts with picwis constant matrial proprtis to modl nonhomognous matrials. This papr has shown that th dformation mods (and associatd strain nrg) for gradd lmnts (Figur (c)) ar quit diffrnt from thos for homognous lmnts (Figur (d)). Thus poor numrical rsults ma b obtaind whn homognous lmnts ar usd instad of gradd lmnts, spciall whn rlativl coars mshs ar usd to modl FGMs. Acknowldgmnts W gratfull acknowldg th support from th National Scinc Foundation (NSF) undr grant No. CMS (Mchanics and Matrials Program) and from th NASA Ams Rsarch Cntr (NAG -44) to th Univrsit of Illinois at Urbana-Champaign. At NASA, Dr. Tina Panontin srvs as th projct Tchnical Monitor. Th scond author acknowldgs th start-up support from Univrsit of Conncticut. An opinions prssd hrin ar thos of th writrs and do not ncssaril rflct th viws of th sponsors. 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