ESTIMATION OF THE VISCOELASTIC PARAMETERS OF LAMINATED COMPOSITES. PART I. ANALYTICAL CONSIDERATIONS

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1 nd International Conerence danced Coposite aterials Engineering COT 8 9 October 8, Braso, Roania ESTITION OF THE VISCOELSTIC RETERS OF LINTED COOSITES. RT I. NLYTICL CONSIDERTIONS.Katouzian, S. Vlase,.V.Guian Technical Uniersity unich, unich, GERNY Transilania Uniersity, Braso, RONI, slase@unitb.ro bstract : In an eort to study the nonlinear iscoelastic behaior o neat and carbon iber-reinorced olyetheretherketone (EEK and epoxy resin, creep tests were peroed on [94]s and [+454]s lainates as well as on neat resin speciens. The lainates [94]s and [+454]s are used to study the tie dependent transerse and shear properties, respectiely. Series o -hour isotheral tensile creep tests were conducted on each lainate at our teperatures (up to 4 C or the epoxy syste and up to C or the EEK syste and dierent stress leels. n analytical procedure based on the ori-tanaka's ean ield theory but extended into iscoelastic doain is used to account or the tie dependent oerall response o the coposite body. Directed toward this, Schapery's nonlinear constitutie equation is input into the analysis which akes the proposed ethod capable o accounting or tie dependent behaior o the polyeric atrix. The nonlinearity actors in the Schapery's orulation axe ound based on the creep tests perored at dierent stress leels and test teperatures. The stress and teperature dependence o the nonlinearity actors was ealuated.using a nuerical procedure based on least squares techniques. The results or the pure resin and 9 degree speciens show good agreeent between experient and the predicted data. For the 45 degree specien, the aboe correlation is less ipressie, neertheless the general trend in the cures o the predicted data closely atch those generated by experient. The accuracy o the results will be ore iproed i the gradual change o the iber orientation specially at high alues o teperature and stress is accounted or in the analysis.. INTRODUCTION The eectie iscoelastic behaior o a two phase coposite body is dependent upon the elastic/iscoelastic properties o the constituent aterials. nuber o approaches are presented in the literature or the prediction o the bounds on elastic/iscoelastic paraeters o ulti (two phase coposites [-6]. ost o the works which hae been done to estiate the bounds on the eectie elastic/iscoelastic property o iber reinorced coposites, assue both o the phases to possess isotropic aterial behaior. Little work has been presented in the literature or those cases where or instance the reinorcing phase has anisotropic or transersely isotropic properties. n exaple o this could be a graphite/epoxy coposite where the atrix aterial is considered to be isotropic but the ibers show anisotropic behaior. The proble considered or the coputation o the elastic/iscoelastic oduli is that o parallel ibers which are long enough so that end eects can be neglected. The aterial ay be represented by a cylindrical specien whose cross section is ery large in coparison to iber cross section. The longitudinal axes o the specien coincides with the iber direction and since the end eects are neglected, the ibers ay be assued to run continuously through the entire length o the specien. It is also assued that the specien is statistically both hoogeneous and transersely isotropic. The proble to be considered is that o predicting the bounds on elastic/- iscoelastic properties o such a specien in ters o its geoetry and the elastic/iscoelastic oduli o its constituents. 6

2 . THEORETICL BCKGROUND In the ollowing, an atte will be ade to estiate the characteristic paraeters o a coposite aterial using the ori-tanaka theory [7]. The ethod utilizes a cylindrical specien containing elliic cylindrical ibers in the atrix aterial. The ibers are considered to be transersely isotropic and elastic while the atrix is isotropic with iscoelastic properties. Two cases ay be considered: one in which the ibers are onotonically aligned and uniorly dispersed which results in an orthotropic coposite specien while in the second case the ibers are randoly-oriented elliic cylinders. For the latter case, the resulting coposite is transersely isotropic. The shape o the elliic ibers is characterized by the ratio α t/w as shown in Figure. For a coposite containing circular ibers, as in the current study, the cross sectional aspect ratio α is equal to. Figure :. Scheatic representation o a coposite with onotonically aligned elliic cylindrical ibers The solution o the proble is based on Eshelby's [8] approach or an ellipsoidal inclusion in conjunction with the ori-tanaka's [7] ean-ield theory or which the results are obtained by Zhao and Weng [9]. It should be pointed out that in [9], the two constituents are considered to hae isotropic properties. In the present inestigation howeer, the ibers possess anisotropic aterial behaior. This is the author's contribution to the deelopent o the theory to ealuate the eectie oduli o a iber reinorced coposite. It is interesting to note that through this theory it is possible to obtain all the eectie oduli which are required to characterize an orthotropic coposite (in particular a transersely isotropic coposite. In suery, using the oregoing approach, the copliance atrix o a iscoelastic body can be deterined which allows one to inestigate the tie dependent response - here in or o creep cures. For the understanding o the subsequent analysis, the ori-tanaka's theory [7] will be briely reiewed. Let us irst consider a iber-reinorced coposite in which transersely isotropic ibers are uniorly distributed in the atrix aterial. Here a representatie olue eleent (RVE o the coposite and one o a coparison aterial (C ade only o the atrix aterial are introduced. Let both o the aboe RVE's be subjected to the sae boundary traction σ. Let us urther denote the elastic coeicients atrix o C by C. For the real coposite howeer, under the sae σ, the ean strain in the atrix diers ro that in the C. Let ~ represent the dierence o the two ean alues o strain. The ean alue o the stress in ~ the C is σ. On the other hand, in the atrix o the coposite, there exists a dierent ean stress σ. s a result, the ollowing obserations can be ade: In C, due to the ean strain ield and the ean stress ield σ, the stress-strain relation becoes: σ C ( 7

3 In the atrix constituent o RVE o the coposite, the ean strain and stress ields are + and σ σ + ~ σ respectiely. The stress-strain relation is thereore written in the ollowing or: σ σ + ~ σ C ( + ( In the iber o RVE o the coposite, the ean strain ield diers ro that in atrix through an additional ter and hence + + ~ +. In the sae anner, the ean stress ield is dierent ro that in atrix by an additional ter σ and thereore σ σ + ~ σ + σ. The stress-strain relation becoes: ~ ( ~ σ σ + σ + σ C + + ( It should be pointed out here that C is to represent the atrix o the elastic coeicients o the iber. Using Eshelby's equialence principle [8], one ay write the aerage stress in iber in ters o the elastic coeicients o the atrix C by introducing the ter in the aerage strain ield, i,e. ~ ( ~ ( ~ σ σ + σ + σ C + + C + + (4 where the ollowing relation holds: (5 The our-rank tensor is Eshelby's transoration tensor and has the syetry property ikjl jikl ijtk. The coponents o the aboe tensor or the cylindrical iber with an elliical cross section are gien in the ppendix. The aerage stress or the whole coposite can be written as σ σ + σ ( + σ + ( + ( σ + ~ σ + σ ~ σ + σ + ( σ + ~ σ σ + ~ σ + σ (6a which reduces to: ~ σ σ Following a siilar procedure one can obtain (6b ( ( ( I (7 where I is the unit tensor. Substitution o this last relation into (4 yields: C [ ( I + ] C ( I [ + ] (8a which can be sipliied to: [ ( ( I + + C ( ( I + I ] + ( C C C (8b This last relation can be written in an alternatie or as 8

4 [ ( + I C ( I ] + ( C C C (8c or [ ( C C + ( C C + C ] + ( C C (8d and inally as: [( C ( + I + C ] + ( C C C (8e Fro the aboe equation, one ay write the relation between ppendix and in the ollowing or: (see (9 For the shear coponents o the strain, ollowing relations can be written [9]: ( G, G ( G G ( +, + G ( ( G, G ( G G ( +, + G ( ( G, G ( G G ( +, + G ( Let us now apply the oregoing ethod to ealuate the elastic/iscoelastic paraeters o the entire coposite when it is considered as an orthotropic body. s a particular case o the aboe approach, the transersely isotropic coposite inestigated in the present study will be considered. For the deriation o the longitudinal Young's odulus and the coparison aterial are subjected to a pure tension E o an orthotropic body, the coposite σ. Then, it ollows that 9

5 σ E or the coposite, and σ E and or the coparison aterial aking use o the relations presented earlier, the aboe relations can be written in the ollowing or: + + ( + a a [ + [ a ( a + a ] + + a ( where the notation: ai i/ has been used. It ollows that: E E + + [ a ( a a ] (4 Siilar expressions can be obtained or the elastic oduli in the other directions E E [ a ( a a ] 6 (5 and E E [ a ( a a ] 8 (6 For the coputation o the shear oduli, one ay use the relations: σ G ; σ G (7 Recall that: + (8 ( G G ( +, G, G + G (9 4

6 which results in the ollowing relation or the shear odulus G G G G G + ( + G, Following a procedure siilar to that just presented, the expressions or the reaining shear oduli are obtained. These are G G and G G G G + ( + G, G G + ( + G, Finally, in order to deterine the expressions or oisson's ratio, one can use the relation: ( ; Note that: + + a ( + a + a + a + a + a (4 lso ( + a a + a (5 which can be written in a sipliied or as: [( + a a a ] [ a ( + a a ] (6 4

7 Now, substitution o the aboe relation in that o the \i yields a4 + a ( + a a 5 a a 6 (7 which can be rearranged to: [ a4 ( a5 + a6 ] [ a ( a + ] (8 + a Siilarly, it can be shown that [ a6 ( a4 + a5 ] [ a ( a + ] ( a8 and [ a7 ( a8 + a9 ] [ a ( a + ] ( + a Note that the proposed ethod is in contrast to those by which only the bounds on characteristic paraeters o the aterial are obtained. Here, the engineering constants are uniquely ealuated as a result o which, an accurate prediction o the oerall aterial response can be ade. The procedure is original in that, the ibers are not isotropic but instead possess anisotropic or transersely isotropic behaior. The coponents o the copliance atrix are ealuated based on the coputed alues o the engineering constants while tie dependence is incorporated into the analysis by using Schapery's constitutie equation or nonlinear iscoelastic aterials []. In this way, the iscoelastic response o the coposite - here in or o creep cures - can be studied.. RESULTS ND DISCUSSION In Figures through, predictions ade or the creep response o EEK and epoxy resin together with those o the [9] 4s and [±45]s lainates are plotted. The response using the proposed theoretical technique agrees ery well with the experiental data or the neat EEK and epoxy speciens under the loading conditions indicated in Figures and. Exaination o the plots or the transerse strain o the two coposites (Figures 4 through 7 indicates that a ery accurate prediction o the experiental results is possible by utilizing the proposed ethod een at relatiely high teperature o C. It can be seen ro Figure 7 that the axiu deiation o the theoretical results ro the experiental data is less than 8%. ttention should be drawn to the act that the [±45]s lainate coniguration can be expected to operate at a higher creep rate than a [9] 4s lainate. The agreeent between experiental and predicted data or the orer lainate is less ipressie in ters o accuracy. 4

8 Neertheless the general trend in the cures o the predicted data or the stress and teperature cobinations presented closely atch those generated by experient (Figures 8 through. The "scissoring action" which occurs in the [±45]s lainate can lead to intolerable large creep distortion alues. For this reason, it is eident that soe discrepancies ay occur in the prediction o the results or the [±45]s lainates inestigated in the current study. By exaining the proposed ethod, the reason or this discrepancy becoes eident. The alue or the shear odulus G obtained through the proposed ethod is based on a ixed iber orientation o 45 in the lainate. This is howeer not the case in the creep test o 45 speciens especially at high alues o stresses and/or test teperatures where the iber orientation o 45 changes with tie thus causing error in the prediction o the results. This can be aoided i the gradual change o the iber orientation is accounted or in the analysis. Figure.Coparison between experiental and theoretical prediction o creep strain s or neat EEK resin subjected to σ 6 a at C. Figure.Coparison between experiental and theoretical prediction o creep strain s or neat epoxy resin subjected to a 8.6 a at 8 C. 4

9 Figure 4. Coparison between experiental and theoretical prediction o creep strain carbon/eek subjected to σ 6 a at 8 C. Figure 5. Coparison between experiental and theoretical prediction o creep strain or carbon/eek subjected to σ 47 a at 8 C. 44

10 Figure 6. Coparison between experiental and theoretical prediction o creep strain or carbon/epoxy subjected to σ a at C. Figure 7. Coparison between experiental and theoretical prediction o creep strain or carbon/epoxy subjected to σ 9 a at C. 45

11 Figure 8. Coparison between experiental and theoretical prediction o creep distortion γ or carbon/eek subjected to τ a at 8 C. Figure 9. Coparison between experiental and theoretical prediction o creep distortion γ or carbon/eek subjected to τ 6 a at C. 46

12 Figure. Coparison between experiental and theoretical prediction o creep distortion γ or carbon/epoxy subjected to τ 9 a at C. Figure. Coparison between experiental and theoretical prediction o creep distortion γ or carbon/epoxy subjected to τ a at C. 47

13 REFERENCES [] Hashin, Z. and Shtrikan, S., "On Soe Variational rinciples in nisotropic and Nonhoogeneous Elasticity," J. ech. hys. Solids, Vol., pp. 5-4,(l96. [] Hashin, Z. and Shtrikan, S., " Variational pproach^ tojhe Theory o the Elastic Behaior o ultiphase aterials," J. ech. hys. Solids, Vol., pp. 7-4 ] (l96. [] Budiansky, B., "On the Elastic oduli o Soe Heterogeneous aterials," J. ech. hys. Solids, Vol. j pp. - 7,(l965l- [4] Chen, T., Dorak, G.J., and Beniste, Y., "ori-tanaka Estiates o the Oerall Elastic oduli o Certain Coposite aterials," Journal o pplied echanics, Vol. 59, pp ,(l99. ~~ [5] Cherkae,.V. and Gibiansky, L.V., "Coupled Estiates or the Bulk and Shear oduli o a Two-Diensional Isotropic Elastic Coposite," J. ech. hys. Solids, Vol. 4, No. 5, pp ,(l99. [6] Christensen, R.., "Viscoelastic roperties o Heterogeneous edia," J. ech. hys. Solids, Vol. n, pp. -4,(l969. [7] ori, T. and Tanaka, K., "erage Stress in the atrix and erage Elastic Energy o aterials with isitting Inclusions," cta etallurgica, Vol. _, pp , (l97 [8] Eshelby, J.D., "The Deterination o the Elastic Field o an Ellipsoidal Inclusion, and Related robles," roceeding o the Royal Society, London, Vol. 4, pp , (957. [9] Zhao, Y.H. and Weng, G.J., "Eectie Elastic oduli o Ribbon-Reinorced Coposites," Journal o pplied echanics, Vol. 57, arch(l99. [] Lou, Y.C., and Schapery, R.., "Viscoelastic Characterization o a Nonlinear Fiber-Reinorced lastics," Journal o Coposites aterials, Vol. 5,(l97l 48

14 ENDIX Eshelby's Transoration Tensor or an Elliic Cylinder The coponents o the Estielby's ijkl or the cylindrical iber with an elliical cross section are [] ( ( + α α α + + α ( ( + α ( ( + α α ( ( + α ( + α α ( + α 4 α + α + α + + α + α ( ( + α α + + α + + α + + α + ( and ; ; (- where all other.in all the aboe equations, α is the aspect ratio introduced earlier and ijkl is the oisson's ratio o the isotropic atrix. Note also that the axis is taken to be ininitely extended, axes and are along the thickness and along the width o the elliic cylinder, respectiely. When the ibers hae circular cross section as in the present inestigation, so that α t / w, the coponents o the ijki are sipliied accordingly. 49

15 s described earlier, substitution o Eq. (7 in Eq. (4 yields the relation between the noral ij ij coponents o and.these relations are: (see Re. [9] N + N N (- where ( ( ( ( ( N ( ( N ( N ( N N N + + (-4 with the ollowing expressions or N N N + ( G G ( λ λ ( λ + G ( λ λ λ ( λ λ / / 's in ters o the Lae's constants o the constituting phases N i (-5 / Now, in order to arrie at the connection between ij and ij, the atrix written aboe is inerted which in turn leads Eq. (9 rewritten here or conenienc: (-6 5