Dynamic Crack kpropagation in Fiber Reinforced Composites

Transcription

1 Presened e COMSO Conference 2009 Miln COMSO CONFERENCE MIAN 2009 OCTOBER Dynmic Crck kpropgion in Fiber Reinforced Composies P. onei, C. Cruso, A. Mnn

2 DYNAMIC FRACTURE MECHANICS MONOITIC MATERIAS Crck Brncing penomen Crck speeds re limied Unknown p of e crck COMPOSITE STRUCTURES Rvi-Cndr nd Knuss, In J Frc, 1984 Hig crck speed Wek plne Crck consrined long e inerfces (Roskis, A.J., Inersonic ser crcks nd ful rupures propgion, Advnces in Pysics, 2002)

3 DYNAMIC CRACK GROWTH MODEING Coesive modeling Inerfce elemens re inroduced e crck region Dmged consiuive relionsip is required Frcure Mecnics pproces Sic nlyses: (e ime dependence is negleced priori ) w Sedy se crck grow pproces: (Moving reference sysem sem wi e ip, crck ip speed is consn) Unsedy models : Full Time dependence, ineril forces, loding re,.

4 DYNAMIC CRACK GROWTH MODEING Node relese ecnique Grdul relese of e nodl forces beind e crck ip Virul crck c closure meods Te ERR is evlued by e muul work e crck ip nd beind e crck ip Moving mes meodology Te nodes re moved o predic cnges of e geomery produced by e crck moion

5 MOTIVATION OF THE WORK AND SUMMARY AIM OF THE WORK Propose generlized modeling bsed on Frcure mecnics nd moving mes meodology o predic e dynmic bevior of composie lmined srucures SUMMARY Review e min equions of e AE formulion in view of e Dynmic Frcure Mecnics pproc Evlue e specilized expressions of e ERR by e use of e decomposiion meodology of e J-inegrl nd propose proper p mixed mode crck ougness crierion Develop e finie elemen implemenion. Propose vlidion by mens of comprisons wi experimenl d nd prmeric sudy o nlyze dynmic crck bevior (i.e. crck rres penomen, llowble ip speeds nd re dependence of e inerfcil crck grow)

6 BASICS OF MOVING MESH STRATEGY: ARBITRARY- AGRANGIAN EUERIAN grngin Approc B X x X,. ț x Eulerin Approc B E ț. : B X R B r, X r, r x r, : B B x r, E B R Arbirry grngin Eulerin Te Reference configurion is fixed nd independen of ny plcemen of e meril body

7 BASICS OF MOVING MESH STRATEGY: ARBITRARY- AGRANGIAN EUERIAN ime. ț X r, X B X 3 ime +d X 2 X 1, B X + + X r,+ Pysicl quniies: d v X, X, d Meril X d d r, r Referenil r 3 r 2 r B R Referenil configurion d f f X f X, dx Time derivive rule r 1 Pysicl fields in AE formulion u u2xuxxuxx xu X Xxu xx X xu ru J 1 Grd. rnsform. de J 0 Meril ccel. one-o-one relionsip

8 DESCRIPTION OF THE DEAMINATION MODE Muli-lyer Modeling 2D Kinemic formulion Te lmine is divided ino n memicl lyer represening e sking sequence, F, n j j-1. ț u=0,v=0 - yer j+1- yer j- yer j-1- v>0 u=0,v=0 1 yer j-2 Compibiliy equions MM: u u u 0, v v v 0, i i1 i i i1 i undelmined inerfces v v v i i1 i 0, delmined inerfces

9 DESCRIPTION OF THE DEAMINATION MODE IN THE REFERENTIA CONFIGURATION Governing Equions: Principle of d Alember n n n n udv u udv uda f udv i1 i1 i1 i1 V V V i i i i Inernl work Exernl work n n 1 1 de udv C ruj ruj J dvr ri : Jcobin i1 V i1 V i n n [ 2 r r i1 V i1 V i 1 1 u udv u u J X u J X ri ] de uj J X X uj XJ X u J dv r r r r r X 3 X 2 X. ț X 1 X r, B n n n n uda fudv u de( J ) d r fude( J ) dv i1 i1 V i1 i1 V i i ri ri r r 3 r 2 r B r 1

10 ERR RATE EVAUATION : J-INTEGRA APPROACH Revision of e J-inegrl Dec. procedure (Rigby & Alibdy, 1998, Greco & onei, 2009) Expressions of e ERR X 2 X 1 P P'. ț n u J lim 1 0 W K n ds X u J W K n 1 ds u f u u u da X P independen (Nisiok,T, 2001)\ Decomposiion of e ERR ino symmeric nd nisymmeric fields J G W K n n ds u f u u u da S u S S S S S S S S I I 1, ij j x AS AS AS AS AS AS AS AS J II GII W K n ij nj ds u f u u u da x AS u 1,

11 DYNAMIC CRACK PROPAGATION ANAYSIS: GROWTH CRITERION Crck grow crierion dx 1 c R m G G 0 G D G0 1 c VR c V G c R D m 0 c0 G c G 0 D Meril prmeer Criicl vlue ueof e ERR (Freund, 1990; Rvi-Cndr, 2004) Ryleig wve speed iniiion vlue 1) Mixed mode crck grow crierion i g f GI GII 10 GID c GIID c Meril prmeer G0I G0II GID c, GIID c m c c 1 1 VR VR m ,0 2,0 2,5 2,0 1,5 GI 1,5 1,0 GII 1,0 0,5 0,5 0,0 0,0 2,5 3,0

12 MOVING MESH METHOD: FOUNDAMENTA EQUATIONS AE formulion o describe mes moion 2 X X1 0, X1 X1r1 X 2 X 2 r2 Winslow Smooing meod 2 X X 2 0. Mes regulrizion ecnique Mes displcemens of nodes Sould be regulr Minimize e mes wrping Boundry condiions Exmple: DCB sceme X 0, X 0 on, X 0 on X 0 if g 0 on, 1 X c if g 0 on, 1 X 0 on 2 f f X 0 0, X 0 0, X 0 0, X ip 4 2

13 VARIATIONA AND FE IMPEMENTATION Wek forms: coupled equions for e AE nd PS formulions: n n r r de r [ 2 r r i1 i1 V V ri C uj uj J dv u u J X u J X ] de uj J X X uj XJ X u J dv r r r r r n r i1 i1 V ri n u de( Jd ) fude( JdV ) ri r r r r ri 1 XJ wj 1 de J dv X c i Xi J ds 0, V r r PS AE Explici equions for PS+AE Implici equion e crck grow ip 4 Crck grow crierion

14 VARIATIONA AND FE IMPEMENTATION FE pproximion by Comsol Mulipysics : Qudric grngin inerpolion funcions for displcemens, velociy nd ccelerion fields Qudric grngin inerpolion funcions for mes poins displcemens FE equions n n n n n MU i i CU i i Ki K0i K1 i K2i Ui Ti Pi 0 i1 i1 i1 i1 i1 W X QX0, Soluion Procedure Non iner Equions Sysem CHECK MESH EEMENTS QUAITY Implici ime inegrion sceme bsed on vrible-sep-size bckwrd differeniion formul Ierive-incremenl Solving procedure

15 : VAIDATION OF THE STRUCTURA MODE B X( )/ Experimenl d Proposed model B /=0.367, /B=0.1, m=0.5, G 0 /G s =0.3, mm/s DCB mode I loding sceme u u E c )/(G 0 B) (E d, c /c s E d B u /=0.367, /B=0.1, m=0.5, G 0 /G s =0.3, mm/s u u c /c s Comprisons wi experimenl d 5 E c AS Grpie/Epoxy

16 : EFFECT OF THE OADING RATE c /c s u u /=0.1, /B=0.1, m=0.5, G 0 /G s =50/ u B B u /T 1 DCB mode I loding sceme Influence of e loding re Evoluion of e crck ip speed /c s c / /=0.1, /B=0.1, m=0.5, G 0 /G s =50/ X / c s c /c

17 DEFORMED SHAPE OF THE BEAM UNDER MODE I OADING CONDITIONS Horizonl displcemen of e crck-ip fron = s = 0.12 s = 0.18 s x direcion

18 DEFORMED SHAPES OF THE BEAM UNDER MIXED MODE OADING CONDITIONS TRIANGUAR MESH EEMENTS C T CRACK - TIP = 0.10 s

19 X()/ : MODE II ENF SCHEME x x x10-9 Experimenl d Proposed model B /=0.216, /B=0.213, m=1, G 0 /G s =0.52, mm/s, (F,u) x x x x x10-8 B c /c s (F,u) B /=0.367, /B=0.1, m=1, G 0 /G s =0.52, mm/s u u ENF mode II loding sceme 0.4 Comprisons wi experimenl d S2/8553 Glss/Epoxy x x x x x x10-8

20 G / G s : MODE II ENF SCHEME /=0.25, /B=0.213, m=1, P G 0 /G s =0.52, 0 / T 1 = / T 1 = / T 1 = / T 1 = / T 1 = / T 1 = /T 1 Hig mplificions in e ERR predicion B (F,u)

21 X()/ : MIXED MODE ANAYSIS x x x10-9 B 2 1 (F,u) Proposed model Experimenl d B 2 1 (F,u) Mes ip discreizion /=0.220, H/B=0.485, 1 / 2 =0.66, m=0.5, G 0 /G s =0.526, mm/s x x x x x10-8 AS Grpie/Epoxy

22 s c /c s : MIXED MODE ANAYSIS u,g f u/u s =[1.2; 1.42; 1.74; 2.36; 3.98;5.08; 7.12] Eq. (22), g f =[1.5; 3.0; 5.0; 10; 30;50; 100] 1 u/u s T 1 u/u=7.12 s u/u=1.20 s g f = g f =1.5 /T 1 Crck rres penomenon oding curves mxx() )/) X()/ B 2 1 (F,u) u/u s =[1.2; 1.42; 1.74; 2.36; 3.98;5.08; 7.12] Eq. (22), gu/u f =[1.5; s 3.0; 5.0; 10; 30;50; 100] u/u s /T 1 mx X()/)

23 : MIXED MODE ANAYSIS Time incremenion

24 CONCUDING REMARKS A delminion model for generl loding condiions bsed on moving mes meodology nd frcure mecnics is proposed. New expressions of e ERR mode componens bsed on e J-inegrl decomposiion procedure. Comprisons wi experimenl d re proposed p o vlide e delminion modelling Te nlyzed prmeric sudy sows delminion penomen y p y p re quie influenced by e loding re, ineril effecs leding o ig mplificions in e ERR predicion nd e crck grow.