Gradient Elasticity Theory for Mode III Fracture in Functionally Graded Materials Part II: Crack Parallel to the Material Gradation

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1 Youn-Sha Chan Department of Computer an Mathematial Sienes, University of Houston-Downtown, One Main Street, Houston, TX 77 Glauio H. Paulino Department of Civil an Environmental Engineering, University of Illinois, 9 Newmark Laboratory, 5 North Mathews Avenue, Urbana, IL 6 Albert C. Fannjiang Department of Mathematis, University of California, Davis, CA 9566 Graient Elastiity Theory for Moe III Frature in Funtionally Grae Materials Part II: Crak Parallel to the Material Graation A Moe-III rak problem in a funtionally grae material moele by anisotropi strain-graient elastiity theory is solve by the integral equation metho. The graient elastiity theory has two material harateristi lengths an, whih are responsible for volumetri an surfae strain-graient terms, respetively. The governing ifferential equation of the problem is erive assuming that the shear moulus G is a funtion of, i.e., GGG e, where G an are material onstants. A hypersingular integroifferential equation is erive an isretize by means of the olloation metho an a Chebyshev polynomial epansion. Numerial results are given in terms of the rak opening isplaements, strains, an stresses with various ombinations of the parameters,, an. Formulas for the stress intensity fators, K III, are erive an numerial results are provie. DOI:.5/.9933 Introution This work is a ontinuation of the paper on Graient Elastiity Theory for Moe III Frature in Funtionally Grae Materials Part I: Crak Perpeniular to the Material Graation by Paulino et al. hereinafter referre to as Part I. In Part I, the authors onsiere a plane elastiity problem in whih the meium ontains a finite rak on the y= plane an the material graation is perpeniular to the rak. In Part II, the material graation is parallel to the rak see Fig.. In Part I, the shear moulus G that rules the material graation is a funtion of y only, G Gy=G e y ; while in Part II, it is a funtion of, i.e., G G=G e. An immeiate onsequene of the ifferene in geometry, whih is iniate in Fig., is that the loation of the rak in Part I is rather irrelevant to the problem an thus an be shifte so that the enter is at the origin point,. On the other han, if the material graation is parallel to the rak, then the loation of the rak is pertinent to the solution of the problem. The metho of solution is essentially the same in both Parts I an II, i.e., the integral equation metho. However, beause of ifferenes in the geometrial onfigurations, some hanges are epete. For instane, in Part I, the rak opening isplaement profile is symmetri with respet to the y-ais, while in Part II, the symmetry of the rak profiles no longer eists. Thus, some interesting questions arise. How are the rak opening isplaement profiles affete by the graient elastiity an the graation of the material? How are the stresses influene uner the graient elastiity? How are the stress intensity fators SIFs alulate? How o the results ompare to the lassial linear elasti frature mehanis LEFM? We will aress all the above questions. The remainer of the paper is organize as follows. First, the onstitutive equations of anisotropi graient elastiity for nonhomogeneous materials subjete to antiplane shear eformation are given. Then, the governing partial ifferential equations PDEs are erive, an the Fourier transform metho is introue an applie to onvert the governing PDE into an orinary ifferential equation ODE. Afterwar, the rak bounary value problem is esribe, an a speifi omplete set of bounary onitions is given. The governing hypersingular integroifferential equation is erive an isretize using the olloation metho. Net, various relevant aspets of the numerial isretization are esribe in etail. Subsequently, numerial results are given, onlusions are inferre, an potential etensions of this work are isusse. One appeni, proviing the hierarhy of the PDEs an the orresponing integral equations, supplements the paper. Constitutive Equations of Graient Elastiity A shemati emonstration of ontinuously grae mirostruture in funtionally grae materials FGMs is illustrate by Fig.. The linkage between graient elastiity an grae materials within the framework of frature mehanis an its relate work has been aresse in Part I. For the sake of ompleteness, the notation an onstitutive equations of graient elastiity for an antiplane shear rak in a FGM are briefly given in this setion an partiularize to the ase of an eponentially grae material along the -iretion. For an antiplane shear problem, the relevant isplaement omponents are as follows: u = v =, w = w,y an the nontrivial strains are as follows: z = w, yz = w y The onstitutive equations of graient elastiity for FGMs are, as follows: Contribute by the Applie Mehanis Division of ASME for publiation in the JOURNAL OF APPLIED MECHANICS. Manusript reeive February 3, 7; final manusript reeive February 7, 7; publishe online August,. Review onute by Robert M. MMeeking. ij = kk ij +G ij ij k k ll ij k G k ij ij = kk ij +G ij + k ij k G + G k ij 3 4 Journal of Applie Mehanis Copyright by ASME NOVEMBER, Vol. 75 / 65- Downloae 4 Aug 9 to Reistribution subjet to ASME liense or opyright; see

2 y y G=G e β -a a Material graation perpeniular to the rak. y Fig. 3 Geometry of the rak problem an material graation yz =G yz / yz =G z /y z Material graation parallel to the rak. Fig. A geometri omparison of the material graation with respet to the rak loation kij = k G ij + G k ij 5 where is the harateristi length of the material responsible for volumetri strain-graient terms, is responsible for surfae strain-graient terms, ij is the stress tensor, an ijk is the ouple-stress tensor. The Lamé mouli an GG are assume to be funtions of. Moreover, k =/ k. The parameter is assoiate with surfaes an k, k k =, is a iretor fiel equal to the unit outer normal n k on the bounaries. For a Moe-III problem, the onstitutive equations above beome Cerami phase y = yy = zz =, y = z =G z z G z yz =G yz yz G yz Cerami matri with metalli inlusions z =G z / Transition region Metalli matri with erami inlusions Metalli phase Fig. A shemati illustration of a ontinuously grae mirostruture in FGMs yyz =G yz /y yz 6 If G is onstant, i.e., the material is homogeneous, then the onstitutive equations 3,4 are as follows: = yy = zz =, y = z =G z z yz =G yz yz z =G z / yz =G yz / yz =G z /y z yyz =G yz /y yz 7 It is worth to point out that eah of the total stresses z an yz in Eq. 6 has an etra term than the ones in 7 ue to the material graation interplay with the strain-graient effet. 3 Governing Partial Differential Equation an Bounary Conitions By imposing the only nontrivial equilibrium equation z + yz y = the following PDE is obtaine: G w yg w w y w G + w + G 3 w 3 + G y 3 w = y If the shear moulus G is assume as an eponential funtion of see Fig. 3, G = G = G e then PDE 9 an be simplifie as 4 w w + w w + w = or Aoring to the geometry of the problem see Fig. 3, it is the upper half-plane that is onsiere in the formulation. The rak is sitting on the -ais, whih is on the bounary of the upper half-plane. Thus, the outwar unit normal shoul be,,, an not,,. Base on Eq. 5, or the last equation in Eq. 5 of Ref. 4, the sign in front of in the epression for both yz an yyz shoul be instea of / Vol. 75, NOVEMBER Transations of the ASME Downloae 4 Aug 9 to Reistribution subjet to ASME liense or opyright; see

3 Table + w = whih is the governing PDE solve in the present paper. It may be seen, from a viewpoint of perturbation, that PDE an be epresse in an operator form, i.e., H L w =, Governing PDEs in antiplane shear problems Cases Governing PDE Referenes =,= =,,=, Laplae equation: w= Perturbe Laplae equation: +/w= Helmholtz Laplae equation: w= Equation : + w = H =, L = + 3 where H is the perturbe Helmholtz operator, L is the perturbe Laplaian operator, an the two operators ommute H L =L H. By sening, we get the PDE 4,5 w = orhlw = 4 where the Helmholtz operator H= an the Laplaian operator L= are invariant uner any hange of variables by rotations an translations. FGM reates the perturbation an ruins the invariane. However, the perturbing term / in L, whih is not purely ause by the graation of the material, involves both the graation parameter an the harateristi length the prout of an. It an be interprete as a onsequene of the interation of the material graation an the strain-graient effet. If we let alone, then the perturbe Helmholtz ifferential operator H will beome the ientity operator, an one reues PDE to + w = 5 the perturbe Laplae equation, whih is the PDE that governs the Moe-III rak problem for nonhomogeneous materials with shear moulus G=G e 6,7. The limit of sening will lower the fourth orer PDE to a seon orer one Eq. 5, an a singular perturbation is epete. By taking both limits an, one obtains the harmoni equation for lassial elastiity. Various ombinations of parameters an with the orresponing governing PDE are liste in Table. One may notie that in the governing PDE, there is no surfae term parameter involve. However, oes influene the solution through the bounary onitions. By the priniple of virtual work, the following bounary onitions an be erive an are aopte in this paper: yz, = p, w, =,,, Stanar tetbooks Erogan 7 Varoulakis et al. 4 Fannjiang et al. 5 Zhang et al. Stuie in this paper yyz, =, + 6 The first two bounary onitions in Eq. 6 are from lassial LEFM, an the last one, involving the ouple-stress yyz, is neee for the higher orer theory. This set of bounary onitions is the same as those aopte by Varoulakis et al. 4 An alternative treatment of bounary onitions an be foun in Ref Fourier Transform Let the Fourier transform be efine by Fw = W = we i 7 Then, by the Fourier integral formula, F W = w = We i where F enotes the inverse Fourier transform. Now, let us assume that w,y = W,ye i 9 where w,y is the inverse Fourier transform of the funtion W,y. By onsiering eah term in Eq. an using Eq. 9, one obtains 4 w = 4 W,y W y + 4 W y 4 e i w = i 3 W,y i W y e i w = W,y + W y e i w,y = W,ye i 3 w,y = iw,ye i 4 Equations 4 are ae aoring to Eq., an after simplifiation, the governing ODE is obtaine: 4 y 4 +i + y + 4 +i iw = 5 5 Solutions of the ODE The orresponing harateristi equation to the ODE 5 is 4 +i i i = 6 whih an be further fatorize as +i + i = 7 Clearly, the four roots i i=,,3,4 of the polynomial 7 above an be written as = i Journal of Applie Mehanis NOVEMBER, Vol. 75 / 65-3 Downloae 4 Aug 9 to Reistribution subjet to ASME liense or opyright; see

4 Table Roots i together with the orresponing mehanis theory an type of material Cases Number of roots Roots Mehanis theory an type of material Referenes =,= Classial LEFM, homogeneous materials =, an in Eqs. Classial LEFM, an 9, respetively nonhomogeneous materials,= 4, +/ Graient theories, homogeneous materials, 4 The four roots 4 Graient theories, in Eqs. 3 nonhomogeneous materials Stanar tetbooks Erogan 7 Varoulakis et al. 4 Fannjiang et al. 5 Stuie in this paper = i = +/ + + +/ i +/ + + +/ 4 = +/ + + +/ i 3 +/ + + +/ If, then the imaginary part of eah root i i=,...,4 isappears. Thus, we have eatly the same roots foun by Varoulakis et al. 4 an Fannjiang et al. 5. The root orrespons to the solution of the perturbe harmoni equation w +w/=, an the root 3 agrees with the solution of the perturbe Helmholtz equation / w=. Various hoies of parameters an with their orresponing mehanis theories an materials are liste in Table. In ontrast to the four real roots foun in Part I, the four roots here are all omple an amit a more ompliate epression. By the symmetry of the geometry, one an only onsier the upper half-plane y. By taking aount of the far-fiel bounary onition w,y as + y + 3 one an epress the solution for W,y as W,y = Ae y + Be 3 y 33 where the nonpositive real part of an 3 has been hosen to satisfy the far-fiel onition in the upper half-plane. Aoringly, the isplaement w,y takes the form w,y = Ae y + Be 3 y e i 34 Both A an B are etermine by the bounary onitions. 6 Hypersingular Integroifferential Equation Approah By substituting Eq. 34 into Eq. 6, we have an yz,y =G yz yz G yz = G Ae y e i, y 35 yyz,y =G yz y yz = G Ae y Be 3 y e i, y 36 From the bounary onition in Eq. 6 impose on the ouplestress yyz i.e., yyz,= for, one obtains the following relationship between A an B: B = A =,A with, = 3 3 = Denote + i + + i + i +/ + + i + = w,+ = ia + Be i 3 = F ia + B 39 The seon bounary onition in Eqs. 6 an 39 implies that an =,, 4 = 4 whih is the single-valueness onition. By Eqs. 39 an 4, we obtain ia + B = e i = te it t 4 By substituting Eq. 37 into Eq. 4 above, one gets where A = te it t i+, +, = + i i +/ + + i +/ + i By replaing the A in Eq. 35 by Eq. 43, one obtains the following integral equation in the limit y + : 65-4 / Vol. 75, NOVEMBER Transations of the ASME Downloae 4 Aug 9 to Reistribution subjet to ASME liense or opyright; see

5 G lim yz,y = lim y + y + with, te t it e y e i i+, y, i+, t e G = lim y + e it t G = lim K,ye y + t it t, 45 46, K,y = i+, e y 47 Equation 46 is an epression for the stress yz,y in the limit form of y +, whih is vali for,. Note that for the first bounary onition in Eq. 6, yz,= p, is restrite to the rak surfae,. It is this bounary onition that leas to the governing hypersingular integroifferential equation see Eq. 55a below. However, when SIFs are alulate, takes values outsie of,, an the integral 46 is not singular see Eq. 55b below. We split K,y into the singular part K,y an the nonsingular part N,y: K,y = K,y + N,y 4 where K,y is the nonvanishing part of the asymptoti epansion of K,y in the powers of, as. When y is set to be, we have K, = i i + + i 49 Note that the real an the imaginary parts of K, given in Eq. 49 are even an o funtions of, respetively. In view of the following istributional onvergene, y + ie y e it 4 3 y + e y e it 5 5 ie y e it 5 y + i y + e yeit 53 e y e it 54 y + with being the Dira elta funtion, we obtain the limit lim y + = G = K,ye it tt / / + k,ttt + = 3 + = p, / / 55a + k,ttt, or 55b where = enotes the finite-part integral, an the regular kernel k,t is given by k,t N,e = it 56 Equation 55a is a Freholm integral equation of the seon kin with the ubi an Cauhy singular kernels. 7 Numerial Solution To numerially solve the unknown slope funtion t in Eq. 55a, we follow the general proeure outline in the Part I paper. For the sake of larity an ompleteness, eah step is presente below an partiularize to the problem at han see Figs. an. 7. Normalization. By the hange of variables t = s/+ + / an = r/+ + / 57 the rak surfae, an be onverte into,, an the main integral equation 55a an be rewritten in normalize form /a s r 3 3/a a s r = where + 3/a a / /a//a s r + /a + Kr,sss a/a r + r = Pr e ar++/, r G a = / = half of the rak length 5 59 r = ar + + /, Pr = par + + / 6 Kr,s = akar + + /, as + + / 6 an k,t is esribe by Eq. 56. Note that in Eq. 5, G has been written as G = G e = G e /r++/ = G e ar e a +/ where a= / an +/ are two imensionless quantities. Together with the terms /a an /a that appear in Eq. 5, the following imensionless parameters are efine: = /a, = /a an = a 6 They will be use in the numerial implementation an results. 7. Representation of the Density Funtion. To proee with the numerial approimation, a representation of s is ho- Journal of Applie Mehanis NOVEMBER, Vol. 75 / 65-5 Downloae 4 Aug 9 to Reistribution subjet to ASME liense or opyright; see

6 sen so that one an evaluate the hypersingular an the Cauhy singular integrals by finite part an Cauhy prinipal value, respetively see Refs.,. For the ubi hypersingular integral equation 5, the solution s an be represente as s = gs s 63 where g is finite an g 5. By fining numerial solution for gs, one an fin the approimate solution for s. The representation 63 of s is suggeste by the following asymptoti behavior of the solution aroun the rak tips: Displaements r 3/, strains r, stresses r 3/ 64 reporte in Refs., Chebyshev Polynomial Epansion. In view of Eq. 63 an the fat that U n s are orthogonal on, with respet to the weight funtion s, gs an be most naturally epresse in terms of the Chebyshev polynomials of the seon kin U n s. However, the orthogonality is not require in the implementation of the numerial proeures, an either Chebyshev polynomials of the first kin T n s or of the seon kin U n s may be employe in the approimation, i.e., gs = a n T n s or gs = n= A n U n s n= 65 An n= = U n s s 3 s r 3 s An n= n= An r n= n= An s U n skr,ss n +A n T n+ r + = U n s s s r s U n s s s s r r n= A n U n r = Pr, r 7 Gr We have use the running ine n that starts from see Eq Evaluation of Singular an Hypersingular Integrals. Formulas for evaluating singular integral terms in Eq. 7 are liste below: U n s s s = T n+ r, r, n Z + 7 s r The oeffiients a n s or A n s are numerially etermine by the olloation metho. As shown by Chan et al. 6, the two epansions shoul lea to onsistent numerial results. In this paper, the epansion in terms of U n s is aopte, i.e., s = s A n U n s n= where U n s is efine, as usual, by 66 U n s = sinn +os s sinos, n =,,, s Note that the single-valueness onition 4 or, equivalently, ss= implies A = 6 Thus, the running ine n in Eq. 66 an start from instea of. 7.4 Evaluation of the Derivative of the Density Funtion. The term r in Eq. 5 is evaluate using the epansion 66 an the fat that Thus r U nr r = n + T r n+r, n 69 r = r r A n U n r = n= r n= n +A n T n r Formation of the Linear System of Equations. The A n oeffiients are etermine by transforming the integral equation 55a into a system of linear algebrai equations in terms of the A n s. By replaing s in Eq. 5 by the representation 66, an using Eq. 7, one obtains the governing integral equation in isretize form = U n s s s r s = n +U n r, r, n Z + 73 = U n s s s r 3 s =, n = n + nu n+ r n +3n +U n r r 4 r, n 74 The etails of the alulation an be foun in Ref Evaluation of Nonsingular Integral. By ombining all the results obtaine so far in the numerial approimation, one may rewrite Eq. 7 in the following form: r A n n + nu n+ r n +3n +U n r n= n +A n U n r n= A n T n+ r + n= n= r n= An n +A n T n+ r + s U n skr,ss r n= A n U n r = Pr, r 75 Gr The regular kernel in Eq. 75 is atually a ouble integral, i.e., 65-6 / Vol. 75, NOVEMBER Transations of the ASME Downloae 4 Aug 9 to Reistribution subjet to ASME liense or opyright; see

7 s U n skr,ss = s U n sakar,ass K III = lim + yz, = s U n s an,sinas rs 76 The Fourier sine transform in Eq. 76 an be effiiently evaluate by applying fast Fourier transform FFT 4. The integral along, an be reaily obtaine by the Gaussian quarature metho 5. Stress Intensity Fators an In lassial LEFM, the SIFs are efine by K C III = lim yz, 77 + K C III = lim yz, 7 After normalization, the rak surfaes are loate in the interval,. The ensity funtion t is epane in terms of Chebyshev polynomials of the seon kin U n, whih, when substitute into Eq. 55b, give rise to the following formulas for r see Ref. 3: Un s s s = r r n+ r, n 79 s r r Un s s r s r s = n + r Un s s s r 3 s = r n +r n r r n r r n r, n r r r + r r r r, n r 3 The highest singularity in Eqs. 79 appears in the last term in Eq. an it behaves like r 3/ as r + or r. Motivate by suh asymptoti behavior, we efine the SIFs for strain-graient elastiity as Thus, K III = lim + yz, K III = lim yz, 3 r + + ar a yz, r = lim r + r + + =a a lim r r yz r/ r /, r =a a lim r r G e ar e +/ r + s a s r3s r 4 By using Eq. 66 in onjuntion with Eq., we obtain from Eq. 4 Similarly, K III = a G e lim r 3/ a r + N n= n n + r r n r r = ag e /a n= r r + r r K III = ag e /a n= 9 Results an Disussion n +A n r r A r 3 n n n +A n 5 6 The numerial results inlue rak surfae isplaements, strains, stresses, an SIFs. 9. Crak Surfae Displaements. The normalize rak surfae isplaements shown in Figs. 4 are obtaine by integrating the slope funtion see Eq. 7 below. Figure 4 shows a full normalize rak sliing isplaement profile for a homogeneous meium = with the strain-graient effet. The rak profile in Fig. 4 is symmetri beause the material is homogeneous. Figures 5 an 6 are for lassial LEFM. Figures 7 an are for the strain-graient theory. As, the material has larger shear moulus at the left sie of the rak than at the right sie, an thus the material is stiffer on the left an more ompliant on the right, as shown in Figs. 5 an 7. Similarly, Figs. 6 an illustrate the ase of an onfirm that the material is stiffer on the right an more ompliant on the left. The variation of the shear moulus estroys the symmetry of the isplaement profiles. The most prominent feature is the usping phenomena aroun the rak tips, as shown in Figs. 4, 7, an. The ifferene between Figs. 5 an 6 an Figs. 7 an is the usp at the rak tips. In Figs. 5 an 6, one may observe that the profiles have a tangent line with infinite slope at the rak tips, whih is a ommon rak behavior ehibite in the lassial LEFM. However, suh is not the ase in graient theory as eviene by the numerial results shown. Journal of Applie Mehanis NOVEMBER, Vol. 75 / 65-7 Downloae 4 Aug 9 to Reistribution subjet to ASME liense or opyright; see

8 .6 Normalize isplaement, w(,) Normalize rak length, Fig. 4 Full rak isplaement profile for homogeneous material = uner uniform rak surfae shear loaing yz, = p with hoie of normalize =. an = 9. Strains. We have use the strain-like fiel, the slope funtion, as the unknown ensity funtion in our integral equation formulation. The normalize version,, with various is plotte in Fig. 9. Note that = while in lassial LEFM, =. The vanishing slope is equivalent to the usping at the rak tips. The normalize rak isplaement profile wr, an be obtaine by r r N wr, ss s A n U n ss 7 = = n= As ereases, seems to onverge to the slope funtion of the lassial LEFM ase in the region away from the rak tips, where is very ifferent from its lassial ounterpart near the rak tips. 9.3 Stresses. Similar to lassial LEFM, the stress yz, iverges as approahes the rak tips along the ligament Fig. w(,)/(ap /G ) y. G() = G e β.5.5 β =..5.5 / a Fig. 6 Classial LEFM, i.e., = \. Crak surfae isplaement in an infinite nonhomogeneous plane uner uniform rak surfae shear loaing yz, = p an shear moulus G =G e. Here, a= / enotes the half rak length.. Moreover, the sign of the stress hanges, an as ereases, the interior part i.e., the region apart from the two rak tips of yz, seems to onverge to the solution of lassial LEFM. The fining of the negative near-tip stress is onsistent with the results by Zhang et al. who also investigate a Moe-III rak in elasti materials with strain-graient effets; this negative stress may be onsiere as a neessity for the rak surfae to reattah near the tips. The point worth noting here is that not all straingraient theories possess the negative-stress feature near the rak tips. For instane, the strain-graient elastiity theory for ellular materials 6 an elasti-plasti materials with strain-graient effets 7, whih fall within the lassial ouple-stress theory framework, shows a positive-stress singularity near the rak tip. On the other han, the strain-graient theory propose by Flek an Huthinson, whih oes not fall into the above framework, preits a ompressive stress near the tip of a tensile Moe-I rak 9, w(,)/(ap /G ) 6 4 y G()=G e β..4 β = /a Fig. 5 Classial LEFM, i.e., = \. Crak surfae isplaement in an infinite nonhomogeneous plane uner uniform rak surfae shear loaing yz, = p an shear moulus G =G e. Here, a= / enotes the half rak length. w(,)/(ap /G ) 6 4 G()=G e β β =. β =.5 β =. β =.5 β = /a Fig. 7 Crak surfae isplaement in an infinite nonhomogeneous plane uner uniform rak surfae shear loaing yz, = p an shear moulus G =G e with hoie of normalize =. an =.. Here, a= / enotes the half rak length. 65- / Vol. 75, NOVEMBER Transations of the ASME Downloae 4 Aug 9 to Reistribution subjet to ASME liense or opyright; see

9 .4 4 lassial LEFM w(,)/(ap /G ). G()=G e β.. β =..6 β =.5.4 β =.. β = /a σ yz (/a,)/g /a.6.. Fig. Crak surfae isplaement in an infinite nonhomogeneous plane uner uniform rak surfae shear loaing yz, = p an shear moulus G =G e with hoie of normalize =. an =.. Here, a= / enotes the half rak length. Fig. Stress yz /a, /G along the ligament for =.5, =, an various. Crak surfae, =, loateinaninfinite nonhomogeneous plane is assume to be uner uniform rak surfae shear loaing yz, = p an shear moulus G =G e. Here, a= / enotes the half rak length. 9.4 Stress Intensity Fators. Besies using as the unknown ensity funtion, one may also use isplaement w to be the unknown in the formulation of the integral equation see Appeni. By rewriting K C III in terms of the oeffiients in the epansion for w, one obtains see Ref. 6 the following. With T n epansion, φ(/a) With U n epansion, K C III G / = e K C III G / = e lassial LEFM N n a n /a Fig. 9 N a n.. Strain /a along the rak surfae, =, for =.5, =, an various in an infinite nonhomogeneous plane uner uniform rak surfae shear loaing yz, = p an shear moulus G =G e. Here, a= / enotes the half rak length.. K III G / = e K III G / = e N n n +A n N n +A n 9 Table 3 ontains the normalize SIFs for the ase of lassial LEFM by using both T n an U n epansions see Eqs. 63 an 65. The SIFs in Table 3 have been obtaine by using Eqs. an 9, an they are lose to the results reporte by Erogan 7. Table 4 ontains the SIFs for strain-graient elastiity at =. an =.. One observes that the epenene of K III an K III is similar to the lassial ase reporte in Table 3. Conluing Remarks This paper has shown that the integral equation metho is an effetive means of formulating rak problems for a FGM onsiering strain-graient effets. The theoretial framework an numerial analysis has been utilize to solve antiplane shear rak problems in FGMs by using Casal s ontinuum. The behavior of Table 3 Normalize SIFs for Moe-III rak problem in a FGM =\ U n representation K III p / K III p / T n representation K III p / K III p / Journal of Applie Mehanis NOVEMBER, Vol. 75 / 65-9 Downloae 4 Aug 9 to Reistribution subjet to ASME liense or opyright; see

10 Table 4 Normalize generalize SIFs for a Moe-III rak at =., =., an various values of the solution aroun the rak tips is affete by the strain-graient theory, an not by the graation of the materials. Also, the integral equation formulation has been foun to be an aequate tool for implementing the numerial proeures an to assess physial quantities suh as rak surfae isplaements, strains, stresses, an SIFs. Further eperiments are neee for justifying the physial aspets of the metho. Future work inlues etension of the theory to Moe-I rak problems. Aknowlegment G.H.P. woul like to thank the support from the National Siene Founation uner Grant No. CMS 5954 Mehanis an Materials Program an from the NASA-Ames Engineering for Comple Systems Program, an the NASA-Ames Chief Engineer Dr. Tina Panontin through Grant No. NAG -44. A.C.F. aknowleges the support from the USA National Siene Founation NSF through Grant No. DMS-9973 an UC Davis Chanellor s Fellowship. Y.-S.C. thanks the support from the Applie Mathematial Sienes Researh Program of the Offie of Mathematial, Information, an Computational Sienes, U.S. Department of Energy, uner Contrat No. DE-AC5-OR75 with UT-Battelle, LLC; he also thanks the grant No. W9NF-5--9 from the U.S. Department of Defense, Army Researh Offie. Organize Researh Awar 7 from University of Houston-Downtown is also aknowlege by Y.-S.C. Appeni: Hierarhy of Governing Integral Equations In this appeni, we list the type of the physial problem uner antiplane shear loaing, its governing PDE, an integral equation assoiate with the hoie of the ensity funtion. The orresponing referenes in the literature are also provie.. Classial LEFM, Homogeneous Materials GG PDE: Laplae equation w,y=. Integral equation with the ensity funtion =w,/: G t t = p, A Integral equation with the ensity funtion =w,: G = K III p / K III p / t t = p, A Many stanar tetbooks have overe the Laplae equation see, for eample, Ref... Classial LEFM, Nonhomogeneous Materials GGy =G e y PDE: Perturbe Laplae equation +/yw,y =. Integral equation with the ensity funtion =w,/: G a a + K,ttt = p, a a A3 Erogan an Ozturk have investigate this problem as bone nonhomogeneous materials with an interfae ut. 3. Classial LEFM, Nonhomogeneous Materials GG =G e PDE: Perturbe Laplae equation +/w,y =. Integral equation with the ensity funtion =w,/: G + log + Ñ,t tt = p, A4 Integral equation with the ensity funtion =w,: G + + N,t tt = p, A5 The regular kernels Ñ,t in Eq. A4 an N,t in Eq. A5 an be foun in Ref. 6. Erogan 7 has stuie this problem for bone nonhomogeneous materials. 4. Graient Elastiity, Homogeneous Materials GG PDE: Helmholtz Laplae equation w,y =. Integral equation with the ensity funtion =w,/: = 3 + / /4 + p = G + K tt A6 Fannjiang et al. 5 have stuie Eq. A6 in etail. 5. Graient Elastiity, Nonhomogeneous Materials GGy =G e y PDE: /y +/yw,y=. Integral equation with the ensity funtion =w,/: 65- / Vol. 75, NOVEMBER Transations of the ASME Downloae 4 Aug 9 to Reistribution subjet to ASME liense or opyright; see

11 G a = /+/4+ / /4 a + k,ttt + G/+ = p, a A7 This is the Part I paper by Paulino et al.. 6. Graient Elastiity, Nonhomogeneous Materials GG =G e PDE: / +/w,y=. Integral equation with the ensity funtion =w,/: = / / + k,ttt + + = p/g, This is the main governing integral equation 55a. Integral equation with the ensity funtion =w,: 6 Referenes = = p G, + k,ttt 3 + Paulino, G. H., Chan, Y.-S., an Fannjiang, A. C., 3, Graient Elastiity Theory for Moe III Frature in Funtionally Grae Materials-Part I: Crak Perpeniular to the Material Graation, ASME J. Appl. Meh., 74, pp Chan, Y.-S., Paulino, G. H., an Fannjiang, A. C., 6, Change of Constitutive Relations Due to Interation Between Strain-Graient Effet an Material Graation, ASME J. Appl. Meh., 735, pp Eaaktylos, G., Varoulakis, I., an Aifantis, E., 996, Craks in Graient Elasti Boies With Surfae Energy, Int. J. Frat., 79, pp Varoulakis, I., Eaaktylos, G., an Aifantis, E., 996, Graient Elastiity With Surfae Energy: Moe-III Crak Problem, Int. J. Solis Strut., 333, pp Fannjiang, A. C., Chan, Y.-S., an Paulino, G. H.,, Strain-Graient Elastiity for Moe III Craks: A Hypersingular Integroifferential Equation Approah, SIAM J. Appl. Math., 63, pp Chan, Y.-S., Paulino, G. H., an Fannjiang, A. C.,, The Crak Problem for Nonhomogeneous Materials Uner Antiplane Shear Loaing A Displaement Base Formulation, Int. J. Solis Strut., 37, pp Erogan, F., 95, The Crak Problem for Bone Nonhomogeneous Materials Uner Antiplane Shear Loaing, ASME J. Appl. Meh., 54, pp. 3. Zhang, L., Huang, Y., Chen, J. Y., an Hwang, K. C., 99, The Moe III Full-Fiel Solution in Elasti Materials With Strain Graient Effets, Int. J. Frat., 94, pp Georgiais, H. G., 3, The Moe III Crak Problem in Mirostruture Solis Governe by Dipolar Graient Elastiity: Stati an Dynami Analysis, ASME J. Appl. Meh., 74, pp Tithmarsh, E. C., 96, Introution to the Theory of Fourier Integrals, Chelsea, New York. Kaya, A. C., an Erogan, F., 97, On the Solution of Integral Equations With Strongly Singular Kernels, Q. Appl. Math., 45, pp. 5. Monegato, G., 994, Numerial Evaluation of Hypersingular Integrals, J. Comput. Appl. Math., 5, pp Chan, Y.-S., Fannjiang, A. C., an Paulino, G. H., 3, Integral Equations With Hypersingular Kernels-Theory an Appliation to Frature Mehanis, Int. J. Eng. Si., 4, pp Follan, G. B., 99, Fourier Analysis an Its Appliations, Wasworth an Brooks/Cole Avane Books an Software, Paifi Grove, CA. 5 Strou, A. H., an Serest, D., 966, Gaussian Quarature Formulas, Prentie-Hall, New York. 6 Chen, J. Y., Huang, Y., Zhang, L., an Ortiz, M., 99, Frature Analysis of Cellular Materials: A Strain Graient Moel, J. Meh. Phys. Solis, 465, pp Huang, Y., Chen, J. Y., Guo, T. F., Zhang, L., an Hwang, K. C., 999, Analyti an Numerial Stuies on Moe I an Moe II Frature in Elasti-Plasti Materials With Strain Graient Effets, Int. J. Frat.,, pp. 7. Flek, N. A., an Huthinson, J. W., 997, Strain Graient Plastiity, Avanes in Applie Mehanis, Vol. 33, J. W. Huthinson an T. Y. Wu, es., Aaemi, New York, pp Chen, J. Y., Wei, Y., Huang, Y., Huthinson, J. W., an Hwang, K. C., 999, The Crak Tip Fiels in Strain Graient Plastiity: The Asymptoti an Numerial Analyses, Eng. Frat. Meh., 64, pp Shi, M. X., Huang, Y., an Hwang, K. C.,, Frature in a Higher-Orer Elasti Continuum, J. Meh. Phys. Solis, 4, pp Sneon, I. N., 97, The Use of Integral Transforms, MGraw-Hill, New York. Erogan, F., an Ozturk, M., 99, Diffusion Problems in Bone Nonhomogeneous Materials With an Interfae Cut, Int. J. Eng. Si., 3, pp Journal of Applie Mehanis NOVEMBER, Vol. 75 / 65- Downloae 4 Aug 9 to Reistribution subjet to ASME liense or opyright; see