Elasto-Plastic EFGM Analysis of an Edge Crack

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1 Proceedings of te World Congress on Engineering 9 Vol WCE 9, Jly - 3, 9, London, U.. Elasto-Plastic EFGM Analysis of an Edge Crack B Aswani mar, V Sing, and V H Saran Abstract-n tis aer, elasto-lastic analysis of an edge crack as been erformed sing element free Galerkin metod. A model roblem as been solved in lane stress condition nder mode- loading. A system of nonlinear eqations as been obtained by sing incremental teory of lasticity, and te eqations are solved by assming iecewise linear aroximation. A code as been written in Matlab for elastolastic analysis of cracked comonents. e size of lastic zone is calclated arond te ti of te crack by EFGM code. eywords- Elasto-lastic, Edge crack, Element free Galerkin metod, Plane stress.. NRODUCON Now a day, Finite Element Metod (FEM) is a well establised simlation aroac, and is widely sed in many brances of engineering and sciences. However, it still as some sortcomings. e reliance of te metod on a mes leads to comlications for certain classes of roblems. Consider te modeling of large deformation rocesses; considerable loss in accracy arises de to element distortion. Examining te growt of cracks wit arbitrary and comlex ats, and te simlations of ase transformations is also difficlt. o overcome some of FEM sortcomings, a nmber of mesless metods were roosed sc as EFGM, MLPG, and PM as discssed by Li []. n a mesless metod, nlike FEM, a redefined mes is not necessary, at least for field variables interolation. wo main caracteristics of te EFGM seem to be a niqe aroximate fnction for te wole field and rater an easier kind of crack modeling. n tis aer, te EFGM as been alied to solve a single edge crack roblem aving elasto-lastic material beavior. e nonlinear eqations are solved by assming eac load as a iecewise linear.. REVEW OF EFGM n EFGM, a field variable is aroximated by moving least sqare aroximation (MLS) fnction ( x ) [], wic is given by m ( x) = ( x) a ( x) ( x) a( x) () j= j j were, (x) is a vector of basis fnctions, a(x) are nknown coefficients, and m is te nmber of terms in te basis. B Aswani mar, M.ec Stdent, ndian nstitte of ecnology Roorkee, Uttarancal, ndia. ( aswani.iitr@gmail.com). V Sing, Assistant Professor, ndian nstitte of ecnology Roorkee, Uttarancal, ndia. (Pone: , iv_sing@yaoo.com, ivsing@gmail.com). V H Saran, Assistant Professor, ndian nstitte of ecnology Roorkee, Uttarancal, ndia. ( saranfme@iitr.ernet.in) e nknown coefficients a(x) are obtained by minimizing a weigted least sqare sm of te difference between local aroximation, (x) and field fnction nodal arameters. e weigted least sqare sm L (x) can be written in te following qadratic form: L( x ) = n w( x x )[ (x)a(x) ] () = were, is te nodal arameter associated wit node at x. are not te nodal vales of ( x x ) becase (x) is sed as an aroximant and not an interolant. w( x x ) is te weigt fnction aving comact sort associated wit node, and n is te nmber of nodes wit domain of inflence containing te oint x, w( x x ). By setting L / a =, following set of linear eqation is obtained: A(x) a( x) = B( x) (3) By sbstitting Eq. (3) in Eq. (), te aroximation fnction is obtained as: n ( x ) = Φ ( x) (4) =. DSCREZED VARAONAL FORMULAON Consider two-dimensional (D) roblem wit small dislacements on te domain Ω bonded by. e governing eqilibrim eqation is given as:. σ + b = in Ω (5) wit te following essential and natral bondary conditions: = on (6) σ. n = t on t (7) were, σ is te stress tensor wic is defined as σ = D( ε ε ), D is te linear elastic material roerty matrix, ε is te strain vector, b is te body force vector, is te dislacement vector, t is te traction force and n is te nit normal, Enforcing essential bondary conditions [3] sing Lagrange mltilier aroac [4], and alying variational rincile, te following discrete eqations are obtained from Eq. (4): SBN: WCE 9

2 Proceedings of te World Congress on Engineering 9 Vol WCE 9, Jly - 3, 9, London, U.. G f = (8) G λ q were, = B D B dω (8a) J Ω f = t Φ dt + Φb dω (8b) Ω t G = Φ N d (8c) q = N d (8d) Φ, x N B = Φ y,, N = N Φ, y Φ, x ν E D = = D ν e (for lane stress) ν ( ν ) / were, E is te modls of elasticity and ν is te Poisson s ratio. V. ELASO-PLASC CONSUVE EQUAON e elasto-lastic constittive relation for a material can be modeled sing incremental teory of lasticity. e material beavior [5] is modeled in te form of an incremental stress vector d σ and incremental strain vector dε sc tat dσ = Dedε. n tis relation, D e is called elasto-lastic stiffness matrix. o find D e matrix, te following relations are assmed to be known a) otal strain increment is te sm of elastic and lastic arts i.e., dε = dε e + dε (9) b) Elastic stress strain relation in te incremental form is similar to te relation in its total form, e.g., dσ = Dedεe () c) Failre criteria is given as F ( σ ) = f ( σ ) () in wic F and f are two different forms of failre fnctions, σ is te stress tensor and σ is te eqivalent stress. d) Flow rle tat relates strain increment to oter qantities, is te gradient of a fnction called lastic otential. f one assmes tat te lastic otential fnction is te same as te failre fnction, ten one can get te following relation known as normality rle as te flow rle, dε = F.dλ () e) Plastic modls Η' is given as dσ Η'= (3) dε f) For a given strain energy δ w, and according to te definition of dε we mst ave, δw = σ.dε (4) g) According to te von Mises criteria [6], F = J, were J is te second invariant of deviatoric stress tensor. So, we mst ave f ( σ ) = σ / 3. e Eqs. (9) and () reslts in dσ D { dε dε } = e (5) After taking te derivatives from bot sides of Eq. (), one obtains F f σ w.dσ =....dε (6) σ σ w For simlicity we take F / σ = a, and of f / σ = a also by means of Eqs. (3) and (4), Eq (6) can be written in te following form a.dσ = a. Η'. σ. dε (7) σ dλ is calclated by omitting d σ between Eq. (5) and (6) and sbstitting dε from Eq. (). By sbstitting dλ in Eq. (), te final form of material matrix is obtained as, De = De D (8) were Daa D D = (9) a Η' σ a + a Da σ V. ELASO-PLASC EFGM ALGORHM As indicated earlier, te elasto-lastic constittive relations are reqired to get te incremental soltion. f in Eqs. (8) and (8a), is relaced by incremental axiliary nodal dislacement Δ and D is relaced by D e, ten a new set of eqations will be obtained wic describes incremental elastolastic beavior. Hence, te incremental form of Eq. (8) is written as G e G Δ Δf = Δλ Δq tis can be sown in more comact form as, () S e = f () for te sake of brevity, S e will be called as te stiffness matrix, as te axiliary nodal dislacement vector and f as te force vector. n tis manner, to obtain te total dislacement, te bondary conditions sold be canged gradally and related incremental eqations need to be solved and finally field qantities are obtained by smmation of incremental vales. Aart from incremental beavior, tere is anoter difference between te forms of Eqs. (8) and () i.e. in tis model (Eq.), te beavior of elasto-lastic EFGM incremental stiffness matrix is nonlinear. e stiffness matrix, S e (wic deends on material roerties) is sed to obtain te dislacement field. n oter words, it can be easily verified tat te elasto-lastic material roerty matrix D e indirectly deends on dislacement field. So, in order to obtain te nknown dislacement vector Δ in Eq. (), te nonlinear eqations are solved by assming te SBN: WCE 9

3 Proceedings of te World Congress on Engineering 9 Vol WCE 9, Jly - 3, 9, London, U.. iecewise linear beavior wit in te load. n tis metod, te total stress is alied in small increments trog varios load s in sc a way tat te beavior of te material in lastic region is assmed to be linear in eac load. V. RESULS AND DSCUSSONS A. Nmerical examle Sigma YY Vs Strain YY 3 5 Gassian oint near crack ti Gassian oint at (,) 5 A rectanglar late wit an edge crack of dimensions L x D is considered, sbjected to a traction at te free end as sown in Fig.. e roblem as been solved for te lane stress case wit te following material roerties: modls of elasticity E = x5 nit, Poisson s ratio ν =.3, tangent modls E t =x4, yield stress = nits, and te late dimensions are L = nit D = nits, lengt of te crack ( a ) =.4 nit. n eac integration cell 6 6 Gass qadratre is sed over te domain and 8 8 is sed near te crack ti region to evalate EFGM stiffness matrix. e soltions were obtained sing a linear basis fnction wit te cbicsline weigt fnction and a d max vale of.5 is sed for domain nodes and 3. is sed for te nodes along te crack. e total stress σ = 4 nits as been alied to te late in small increments in eac load Strain YY x Fig. : Stress vs. Strain lot for te Gassian oint sitated near te crack ti and te lower left corner Gassian oint.8 σo.6 y D a -.8 ` L x x Fig. 3a: Distribtion of Gassian oints and stress arond te ti of te crack at te end of First, Second & ird load s. σo Fig. : Problem geometries and teir dimensions along wit bondary conditions Fig. sows te stress vs strain lot for an evalation oint near te crack ti as well as for a oint wic is near to te lower left corner. From Fig., it is clearly seen tat te oint at te crack ti reaced te lastic region were as te base oint is still in te elastic region to te final load, and te increments in stress and strain are a very small in eac load of te lastic region leading to iecewise linear aroximation Fig. 3a-3j sow te Gassian oints and teir stress distribtion over te domain wic are in lastic region from initial load to te final load. SBN: Fig. 3b: Distribtion of Gassian oints and stress arond te ti of te crack at te end of Fort load. WCE 9

4 Proceedings of te World Congress on Engineering 9 Vol WCE 9, Jly - 3, 9, London, U.. yy Fig. 3c: Distribtion of Gassian oints and stress arond te ti of te crack at te end of Fift load 45 Fig. 3f: Distribtion of Gassian oints and stress arond te ti of te crack at te end of Eigt load Fig. 3d: Distribtion of Gassian oints and stress arond te ti of te crack at te end of Sixt load - Fig. 3g: Distribtion of Gassian oints and stress arond te ti of te crack at te end of Nint load Fig. 3e: Distribtion of Gassian oints and stress arond te ti of te crack at te end of Sevent load SBN: Fig. 3: Distribtion of Gassian oints and stress arond te ti of te crack at te end of ent load WCE 9

5 Proceedings of te World Congress on Engineering 9 Vol WCE 9, Jly - 3, 9, London, U.. 8 V. CONCLUSONS Fig. 3i: Distribtion of Gassian oints and stress arond te ti of te crack at te end of Elevent load n tis aer, te EFGM as been sed to simlate elastolastic solid mecanics roblems wit geometrical discontinity sc as crack. e elasto-lastic formlation as been derived and imlemented for an edge crack roblem. e nonlinear eqations are solved assming iecewise linear aroximation wit in te eac load. Distribtion of Gassian oints and stress arond te ti of te crack are obtained for different load s. From tis analysis, it is observed tat elasto-lastic analysis of cracked comonents can be done by taking small load s in lastic region, and wit in eac load, system of eqations can be solved by linear solvers. REFERENCES [] Li G. R. (3), Mes-free metods moving beyond te finite element metod, CRC ress, First edition, [] Belytscko,. and L, Y., (994) Element-free Galerkin metods, 8 nternational Jornal for Nmerical Metods in Engineering, 37, [3] rongaz, Y. and Belytscko,., (996) "Enforcement of essential 6 bondary conditions in mesless aroximations sing Finite 5 Elements", Comter Metods in Alied Mecanics and Engineering, 4 3-3, [4] Ngyen, P., Rabczk,., Bordas, S. and Dflot, M., (8) "Mesless metods: A review and comter imlementation asects", Matematics and Comters in Simlation, 79, [5] Owen DR, Hinton E (98) Finite Element in Plasticity. Pineridge ress: U. [6] Cakrabarty J (987) eory of Plasticity. Mc Graw Hill. [7] argarnovin M. H., ossi H. E. and Fariborz S. J., Elasto-lastic element-free Galerkin metod, Comtational Mecanics, Vol 33,. Fig. 3j: Distribtion of Gassian oints and stress arond te ti of te crack at te final load 6 4, 4. [8] Ji-fa Zang, Wen- Zang and Yao Zeng, A mes-free metod and its alications to elasto-lastic roblems, Jornal of Zejiang University Science, Vol 6A (),.48-54, 5. [9] Wei Ma Go, F Qiang Hong and Qiang Zang Yong., Generalized Plasticity, Sringer,. 3,7, Fig. 4: Stress contors of te total lastic region arond te crack ti at te end of te final load SBN: WCE 9