ORAL/POSTER REFERENCE : COUPLING OF DAMAGE MECHANICS AND PROBABILISTIC APPROACH FOR LIFE-TIME PREDICTION OF COMPOSITE STRUCTURES Y. Bruner, J. Renrd, D. Jeulin nd A. Thionnet Centre des Mtériux Pierre-Mrie Fourt, Ecole Ntionle Supérieure des Mines de Pris BP 87, F-913 Evry Cedex, Frnce Centre de Morphologie Mthémtique, Ecole Ntionle Supérieure des Mines de Pris 35, rue St Honoré 7735 Fontineleu Cedex, Frnce ABSTRACT We propose dmge model with proilistic pproch for lmintes mde of unidirectionl fire reinforced plies. Sttisticl informtion is collected through multiple crcking tests. The defects re considered s trnsverse mtrix crcks nd we study them y exmining, nd lmintes. Prmeters of cumultive distriution function of the filure strength re determined. Proilistic prmeters of the cumultive distriution re chosen to e independent of the ply thickness nd multixil loding to hve intrinsic vlues for descriing the ply. Proilistic prmeters found previously re introduced into finite element computtion of lmintes using Sttisticl Volume Element (SVE). As experimentl results, numericl ones present dispersion of the filure stress. KEYWORDS Cron/epoxy fric lmintes, composite lmintes, defects sttistics, crck density, trnsverse crcking, multixil loding, proilistic filure criteri. INTRODUCTION The frcture ehviour of fire reinforced composites hs n inherent vriility which results from the presence of defects in the constituents. A proilistic model including filure criteri, tking into ccount the presence of defects for predicting the sttisticl frcture ehviour is proposed for composite lminte under multixil loding. The studied composite lminte is stck of plies where ech ply is mde of unidirectionl cron fire (T3) emedded in n epoxy mtrix (914). Different stckings of composite lmintes, nd re sujected to mechnicl lods which led to dmge. The dmge is known to consist of intrlminr crcks (fire reks, xil nd trnsverse crcks) nd interlminr crcks formed y locl seprtion of plies (delmintion). We focus on trnsverse crcks which give vriility of frcture properties.
In order to predict filure stresses in composite lmintes, it is necessry to tke into ccount the proilistic nture of defects. Such n pproch consists of two prts: identifiction of popultion of defects nd simultion of the sttisticl ehviour of the mteril under multixil loding. The sttisticl spect is introduced in finite element clcultion with the Sttisticl Volume Element (SVE) descried y Bxevnkis et l. [1]. The pper is orgnized in the following wy. The first prt descries the identifiction of trnsverse crcks nd multixil frcture criterion. The second prt presents the numericl simultion of the frcture of lminte composite. TRANSVERSE CRACKING Description of the test The im of the test is to estimte the popultion of defects tht genertes trnsverse crcks during the dmge process of composite lmintes. The mteril used in this study is cron fire reinforced epoxy (T3, 914) with the ply properties reported on T. 1. TABLE 1: MECHANICAL PROPERTIES OF CARBON/EPOXY PLY. Mteril! (GP) #" (GP) $% &" (GP) '&(" )+* (MP), * (%) Ply 14 9.5 3.2.31 215 1.1 In symmetric lminte, n xil lod produces n in-plne stress stte in off-xis plies consisting of norml stresses prllel nd perpendiculr to fires nd sher stresses. Following the orienttion - nd the stcking, the stress stte vries. To hve different stress stte we consider three lmintes :,., / lmintes. Smples from the lmintes re cut nd tested in n Instron testing mchine. Axil nd trnsverse strins re mesured using strin guges. Once the edges of the specimens re polished, the specimens re loded to selected strin level. The position of every crck is mesured in situ y trveling opticl microscope (Fig. 1 ). The position nd the numer of crcks re collected for ech ply nd for ech level of deformtion until the specimen fils. Results To determine the locl filure stress stte resulting from the glol stress pplied experimentlly, we ccurtely simulte experiments y introducing crcks exctly t the positions nd t the deformtion levels they were found experimentlly. Figure 2 shows how we clculte the stresses in ech ply for deformtion level t which crck ppers. We choose the dmge vrile defined y Thionnet [2] s 13245.6 (4.7 numer of crcks per unit length nd 6!7 the thickness of the ply) nd we define qudrtic criterion )98:*;2=< ) "?>A@B &" ()C" is the trnsverse stress nd B &" is the sher stress in the locl reference nd @ is coefficient representing the effect of defects on sher) for descriing the defect popultion in ply (Fig. 3 ). These vriles re chosen to descrie the popultion of defects independently CD of the ply thickness nd multixil loding. First from the curves of Fig. 3, we determine @ 2, this vlue is consistent to wht is oserved in literture for deterministic mesoscopic Tsi-Hill criterion. Next, we gther the informtion on defects coming from different plies. This is possile only if trnsverse crcks were oserved just efore the occurrence of the delmintion ecuse our pseudo-tridimensionl clcultion does not tke into ccount this phenomenon. This is the reson why Figure 3 ) shows the ddition of two popultions in nd E; lmintes. The popultion of crcks of F; lminte is removed ecuse the delmintion ppers t the sme time s trnsverse crcks. Then, the possile model of distriution for the trnsverse crcks is sigmoidl distriution represented y the Eq. 1 : N OQPSR 1GH)+*JIK2LGM PSTCUWV I (1) CDY[Z with LX2 ^DY_[` is the mximum numer of defects (nondimensionl), )+\]2 scle prmeter (MP),
direction of trction t s length 5mm guge length 12mm length oserved 8mm totl length of the smple 22mm opticl microscope width L of the smple x2 x3 thickness t θ 2 θ 2 4 θ 2 2 Figure 1: Counting crcks on the smple during the tension. crck position of clculted stresses x3 Figure 2: Mesh of the thickness of the lminte with crcks (pseudo-tridimensionl clcultion) c 2 ddfe shpe prmeter (nondimensionl). SIMULATION OF THE FRACTURE OF A LAMINATE COMPOSITE Sttisticl Volume Element (SVE) The Sttisticl Volume Element (SVE) gives the possiility to introduce the stochstic spect in the numericl simultion. It is defined s hving one criticl defect which is in our cse crck. During experiments, we oserved tht the geometry of the crck hd the thickness of the ply (.246 mm to.738 mm) nd the width of the smple (25 mm). So two dimensions of the SVE re directly defined. The third dimension of the SVE is the intercrck spcing t sturtion. Only the lminte hs its crcks t sturtion (1.41 mm), lmintes with off-xis plies undergo delmintion. Jeulin [3] ssumes tht defects re distriuted ccording to Poisson point process nd tht SVE reks with the wekest link ssumption descried y Weiull [4] so its proility to rek is given y Eq. 2: gih N lnmpoqrtsuofv GH)+*JIj2kM R Uxwzy g}h (2) Knowing 1G{)+* I from the experimentl study (Eq. 1 ) nd using uniform rndom vrile etween nd 1, we ssocite to ech SVE frcture vlue )+*d~ otined y Eq. 3 :
g.5.45.4.35 [ nd 3 /9 3 ] s, [ 2 /6 2 /-6 2 ] s [ 2 /9 2 /45 2 /-45 2 ] s pli / 6 / pli / 45 / pli /-45 / pli / 9 6 /.5.45.4.35 /9 6 / /6 2 / /-6 4 / experiment A=.47 σ o =64.82 m=5.39.3.3.25.25.2.2.15.15.1.1.5.5 2 4 6 8 1 σ er (MP) 1 2 3 4 5 6 7 8 9 1 σ er (MP) Figure 3: ) Cumultive dmge vrile 1 1 for crcks in, ƒ for crcks in different lmintes. ) Cumultive dmge vrile lmintes. )+*d~ 2 )+\ ˆ Š M > ˆ GM L Œ ~&+ *JI V (3) nž% Dmge model for numericl simultion Figure 4 shows two meshes of plte representing the smple studied experimentlly : finite element mesh nd sttisticl mesh. The finite elements re used to clculte under clssicl lmintion theory [5] the stress nd displcement sttes. The sttisticl mesh is mde of SVE s which re introduced to give the filure stress vriility of the smple. The different gry levels represent the vlues of the frcture criterion )+*d~ + ssocited with ech SVE. The two meshes re superimposed to give the input for numericl clcultion. The ppernce of dmge in the lminte is simulted s follows. If in one SVE, the criterion )98:~&+ which is the men of the criterion )98{ 8Q of ll finite element which elong to the SVE reches )+*C~&u ccording to Eq. 4, the dmge of these elements is simulted numericlly y stiffness reduction descried y Renrd nd Thionnet [6]. 8 W)98 8 )98Q~ 9 2 )+*d~ (4) Œ+~&+ Numericl results A numericl simultion is mde on - nd? lmintes with the model descried previously, to compre it with experimentl results. For oth lmintes we hve low vriility on ehviour lws so the stress-strin curves re less interesting thn the density of crcks represented in Fig. 5, where is plotted the dmge vrile (lso clled the cumultive density 1 ) ginst the criterion )98:* ( )98:* 2š)+*d~ + for numericl results). In the cse of - lminte, the SVE reks in the sme wy s experimentlly s shown in the Fig. 5 ). In the cse of F lminte (Fig. 5 ), the experimentl nd numericl curves re not similr, since we do not rech experimentlly the sturtion, the clcultion eing done with the size of the SVE of S lminte. Numericlly, y using the size of SVE of - lminte, we introduce more crcks thn there re in relity. In ddition, the prmeters used for the sttisticl model penlize this lminte configurtion. CONCLUSION We proposed experimentl nd numericl schemes for the determintion of defects œ: popultion prmeters CD in ply. At the sme time, we defined multixil criterion )98 2 GH) " >k@[b &" I, with @ 2. More
intercrck length t sturtion = width of the SVE σ RSVE (MP) 9 84 78 72 67 61 55 49 43 38 32 26 2 x2 Finite Elements (FE) Mesh Mesh Sttisticl Volume Element (SVE) Input for numericl simultion Figure 4: Input for the numericl simultion we hve informtion on defects more the prmeters of popultion nd the prmeter @ will e precise nd relistic. Next, we uilt proilistic dmge model y introducing stochstic filure stresses with the SVE. Numericl results S gve cumultive density similr to the experimentl one. Concerning K lminte, the cumultive density ws higher ecuse of the choice of the size of the SVE. To model the cumultive density more precisely, we hve to ccount for the tridimensionl stress stte, in order to include the delmintion phenomenon so tht the clssicl lminte theory ssumptions could not e considered nymore. The ongoing work is to pply the model for high stresses grdient zones (notched pltes). Figure 6 shows the numericl input. The filure criterion is chnged. Eq. 5 tkes into ccount the fct tht we hve higher proility to rek in the vicinity of the circulr hole. ˆ )98:~& 2)+\žŸ žÿ} 8 8Q N l G P PST I V V Œ ~&9 9 )+*C~& (5) ACKNOWLEDGEMENTS The uthors wish to thnk J.-C. Tesseidre nd Y. Fvry for the technicl ssistnce in the experimentl prt of the study. This reserch ws supported y Ministry of Defence. REFERENCES 1. Bxevnkis, C., Jeulin, D. nd Renrd, J. (1995). Interntionl Journl of Frcture, 73, 149. 2. Thionnet, A. nd Renrd, J. (1993). Composites Engineering, 3(9), 851. 8H ª8 3. Jeulin, D. (199). In: Proceedings 32 Colloque de Métllurgie, pp. 99 113, volume 4, INSTN, Revue de l Métllurgie, Frnce. 4. Weiull, W. (1951). Journl of Applied Mechnics, 18, 293. 5. Tsi, S. W. nd Hnh, H. T. (198). Introduction to Composite Mterils, Technomics, CT, Stnford.
.45 [ 3 /9 3 ] s.4 [ 2 /6 2 /-6 2 ] s.4.35.35.3.25.2.15 experimentl results numericl results.3.25.2.15 experimentl results numericl results.1.1.5.5 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 σ er (MP) σ er (MP) Figure 5: Comprison of numericl nd experimentl results for ) - nd ) i«lmintes. 1.4 mm 8 mm.7 mm.69 mm 25 mm x2 σ RSVE 2 26 32 38 43 49 55 61 67 72 78 84 9 (MP) Figure 6: Exmple of reliztion of )+*C~&+ for notched pltes. 6. Renrd, J. nd Thionnet, A. (1992). In: Dmge Mechnics in Composites, pp. 31 39, Allen, D. H. nd Lgouds, D. C. (Eds.), volume 15, ASME Symposium, Anheim, USA.