Isotropic. Anisotropic. Orthogonal. Isotropic orthogonal

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1 mme.modares.ac..ir - * - - mj_mahmoudi@sbu.ac.ir * / : 395 : 395 : - Numerical analysis of free corner effects in angle-ply composite laminates based on global-local method Hossein Mohammadi Roknabadi, Mohammad Javad Mahmoodi * Faculty of Mechanical and Energy Engineering, Shahid Beheshti University, Tehran, Iran. * P.O.B Tehran, mj_mahmoudi@sbu.ac..ir ARTICLE INFORMATION Original Research Paper Received 3 May 6 Accepted 6 July 6 Available Online 3 August 6 Keywords: Free corner effect Angle-ply laminates Global-local model Interlaminar stresses ABSTRACT The main purpose of this paper is modeling of the free corner effect of cross-ply and angle-ply graphite/epoxy composite laminates using finite element method based on global-local method. The global area is modeled by first order shear deformation theory and the local area, in the free cornerr vicinity, is modeledd by the Reddy's layer-wise theory. Using this method provides the possibility of analysis of thick angle-ply and cross-pland extension loading, respectively and the effects of the free edgee and free cornerr interlaminar stressess are investigated. Verification of the presented results is performed via available results in the previous studies whichh show good agreement. The present study resultss show that when the cross-ply laminate is subjected to thermal loading, the interlaminar stresses distribution is uniform in both length and width of the laminate. However, for the uni-axial extension loading, the interlaminarr stresses possess different distribution in the two directions of the laminate. Also, results demonstratee thatt in angle-ply laminates under extension loading, the free corner effect increases by increasing fiber angle and the maximum interlaminar stresses occur in 3 degree plies in the freee corner vicinity. Moreover, results prove that the effects of the free edge and the free corner are almost similar in layers with fiber angle lesss than 3 degrees. Parametric study on the thickness and stacking of the laminate layers illustrates that both parameters have a significant influence on the interlamianar stresses at the freee laminates. The cross-ply and angle-ply laminates are subjected to uniform thermal corner.. - Pleasecitethisarticleusing: : H.MohammadiRoknabadi,M.J.MahmoodiNumericalanalysisoffreecornereffectsinangle-plycompositelaminatesbasedonglobal-localmethodModaresMechanical EngineeringVol.6No.8,pp.7-76(iPersian)

2 - 9 [9]. -.. [-].[5].[6] - [7] [9] [8] -..[-]. -.. ().....[,3]. [3] -. Local-global theory delamination. [].[5-].[9-6]. [].[]..[]....[3] - [,3] [5] [] 3. [6] [7] 5. [8] 9. Free corner Free edge 3 Single layer higher-order theory Transfer matrix 5 Isotropic 6 Anisotropic 7 Orthogonal 8 Isotropic orthogonal 9 Boundary finite element

3 ..[6-5] - = -.[6-5] (5) () [6,5] -..[5,-7].. [5] (6). (,, ) = (,, ) + (,, ) x i=3 Multiple models Step by step = = = ( + ) = = + + (5) (6) [5]() (,,, ) = (,, ) + (,, ) (,,, ) = (,, ) + (,, ) (,,, ) = (,, ) w v u. y x y x z yx () - () = + = + = = + + = + + (,, ) + (,, ) + + = + () :[5] (3) - () () = () = () () () () (3) (3) (i,j= 6 5 ). xy y x. - -.[6] () (,,, ) = (,, ) () (,,, ) = (,, ) () (,,, ) = (,, ) ().[5] yx wvu I () Nz.[5] z

4 -..[5] (9) [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ { } ] { } [ ] [ ] [ ] [ ] [ ] { } [ ] [ ] [ ] [ ] [ ] { } [ ] [ ] [ ] [ ] [ ] { } { } { } { } { } = { } { } { } { } { } (9) (,, ) () (, ) (,, () (, ) (,, ) () (, ) :[5] () q p () () () (). I j W I V I U I (x,y) (x,y). = q=p. () [ ] [ ] [ ] { } { } { } [ ] [ ] [ ] { } = { } { } [ ] [ ] [ ] { } { () } { } () (9).[5] () - [ ] [ ] [ ] [ ] = } { { } [K LL ] (8) [K EE ] () (3) [K LE ] [K LE ] () :[5] u i ESL u i LWT. z y (). () (FSDT) (7) (LWT) = =, = =, = (7) (7) y x w v u. (,, ) () (, ) (,, () (, ) (,, () (, ) (,, () (, ) (,, ) () (,. FSDT.[5] (8) w j S j S j u j v j e. j (x,y) N N- Rotation U N = U N (8) [ ] =, [ ] = [ ] (3) :[5] () 3 Translation U U 3 = = + U = Fig. Superposition of a FSDT and LWT elements displacement fields LWT FSDT 86395

5 = z, = xy.3 mm. 3.. ) Fig. Geometry of the laminate; the coordinate origin coincides with point A A substrate dz. [6,5,] - - (5) =, = (5) ()) 86395

6 - [5]. [9].. [3] [3]. (FEM) Hex 7. () 9 6.[3] x x= mm zz 9 8. ( zz - x) 6 y xz yz y= x=3.77 mm.. yz = xz =8.83 MPa y yz=.67 MPa y=3.993mm yz. yz =.67 MPa x=3.993 mm xz xz yz zz (MPa) Tahani-Nosier [5] present n=7 present n=5 present n= zz Fig. 5 Interlaminar normal stresss zz versus y coordinate in the freee edge at the interface of /9.under extension loading. /9 zz 5 closed-form higher-order displacement model 3 MSC/NASTRAN delamination 6.5 y (mm ) () 7 zz =3.75MPa y Fig. 3 The modified meshing of the solution domain 3 Fig. The modified layout of the layers in the thickness direction [5[ 5] ] [3] [9]. R [9/] s CFRP. T= C.[9 93] ] (6) = 35GPa, = = GPa = = 5GPa, = 3.97GPa = = =.7, =.6 C, = = C (6)) x. y [7]. 5 5 n= [3]

7 xz (MPa) Zhen-wanji [9] - Becker-Mielstedt [5] - Becker [3] -6 present -8 FEM (MSC/NASTRAN) [3] x (mm) Fig. 8 Interlaminar shear stress xz in the vicinity of free corner in the cross-ply laminate at the 9/ interface under thermal loading 8 yz (MPa) Zhen-wanji [9] Becker-Mielstedt [5] Becker [3] present FEM (MSC/NASTRAN) [3] x 9/ y (mm) Fig. 9 Interlaminar shear stress yz in the vicinity of free corner in the cross-ply laminate at the 9/ interface under thermal loading 9 z y 9/ a Fig. Geometry of the angle-ply composite laminate b x y -. [/-] s. x = -6 /.[7,8] (7) = 37.9GPa, = =.8GPa = = = 5.86GPa = = =. (7). -. y x zz x. -. zz (MPa) Zhen-wanji [9] Becker-Mielstedt [5] Becker [3] present FEM (MSC/NASTRAN) [3] x (mm) Fig. 6 Interlaminar normal stress zz in the vicinity of free corner in the cross-ply laminate at the 9/ interface under thermal loading. 6.x 9/ = y/b=. zz =.8 MPa 5. zz =6.33 MPa 6 5 = xz xz =.3 MPa 3 xz =.7 MPa yz. xz =. MPa 6 3 yz =.3 MPa 5 y/a= yz =.9 MPa 6 yz =.5 MPa yz 3. 3 xz zz (MPa) Zhen-wanji [9] Becker-Mittelstedt [5] Becker [3] present FEM (MSC/NASTRAN) [3] y (mm) Fig. 7 Interlaminar normal stress zz in the vicinity of free corner in the cross-ply laminate at the 9/ interface under thermal loading. 7 y 9/

8 - yz (MPa) 5 5 [5/-5]s [3/-3]s [5/-5]s [6/-6]s y/b Fig. Interlaminar shear stress yz versus non dimentional coordinate in the angle-ply laminate at the /- interface under extension loading. y/b /- 5/- [5/-5] s. h=.,.3,,,.5 mm (7). = x 8-5. zz. 6.5 MPa h=.mm =..73 MPa h=.5 mm y zz 6.5 MPa y/a= h=.mm..73 MPa h=.5 mm. 8 5 h=.mm = xz. MPa h=.5 mm.87 MPa.33 h=.mm y/a= yz. h=.5 mm MPa 8-5. zz (MPa) [5/-5]s [3/-3]s [5/-5]s [6/-6]s Fig. Interlaminar normal stress zz versus non dimentional coordinate in the angle-ply laminate at the /- interface under extension loading. zz (MPa) /- [5/-5]s [3/-3]s [5/-5]s [6/-6]s Fig. Transverse normal stress zz versus y/b non dimentional at the angle-ply laminate at /- interface under extension loading. xz (MPa) y/b y/b /- [5/-5]s [3/-3]s [5/-5]s [6/-6]s Fig.3 Interlaminar shear stress xz versus non dimentional coordinate in the angle-ply laminate at the /- interface under extension loading. /

9 yz (MPa) y/b h= & h=.5 h=.3 h= h=. Fig. 8 Interlaminar shear stress yz versus y/b non-dimentional coordinate for different laminate thickness in the angle-ply laminate at the 5/-5 interface under extension loading 5/-5 8 y./b - zz (MPa) h=. h= & h=.5 h=.3 h= Fig. 5 Interlaminar normal stress zz versus non-dimentional coordinate for different laminate thickness in the angle-ply laminate at the 5/-5 interface under extension loading 5/-5 5 x./a = zz.78 MPa [5/-5] s zz..3 MPa [-5/5] s.78 MPa [5/-5] s y/b=..3 MPa [-5/5] s 9 8 [-5/5] s [5/-5] s xz.3 MPa = y/b= yz.3 MPa zz (MPa) h=. h=.3 h= & h = & h= y/b Fig. 6 Interlaminar normal stress zz versus y/b non-dimentional coordinate for different laminate thickness in the angle-ply laminate at at 5/-5 interface under extension loading 5/-5 6 xz (MPa) y./b h=.3 h= h=& h =.5 h= Fig. 7 Interlaminar shear stress xz versus non-dimentional coordinate for different laminate thickness in the angle-ply under extension loading laminate at the 5/-5 interface 5/-5 7 x./a [-5/5] s [5/-5] s. (7) /

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