CHAPTER 5 THEORETICAL MODELING OF ELASTIC PROPERTIES

Transcription

1 123 CHAPTR 5 THORTICAL MODLING OF LASTIC PROPRTIS 5.1 INTRODUCTION lastic properties are the basic quantities that are required durin desin and optiization of a lainated structure. An orthotropic laina/lainate subected to three diensional loadin is defined by nine independent elastic properties viz., Youn s odulus in X-direction x, Youn s odulus in Y-direction y, Youn odulus in Z-direction z, Poisson s ratio in X-Y plane υ xy, Poisson s ratio in Y-Z plane υ yz, Poisson s ratio in X-Z plane υ xz, shear odulus in X-Y plane G xy, shear odulus in Y-Z plane G yz and shear odulus in X-Z plane G xz. However, when there are no out-of-plane loads (in-plane loadin), the nuber of independent elastic properties is reduced to only four viz., Youn s odulus in X-direction x, Youn s odulus in Y-direction y, aor Poisson s ratio in X-Y plane υ xy and shear odulus in X-Y plane G xy. norous aount of work has been carried out by the researchers to deterine the elastic properties for coposite lainates which are orthotropic in nature, soe of which is discussed in chapter 2. Yet, ost of the odels proposed, have been atheatically very riorous. Further, the literature in hand reveals that, neither the Classical Laination Theory (CLT) nor the odels were applied to predict the elastic properties of natural or naturallass fiber hybrid coposite, which is crucial for practical use of these aterials in nuerous structural and non-structural applications.

2 124 This chapter deals with the evaluation of in-plane elastic properties under tension for ute-lass fabric reinforced polyester hybrid coposites usin CLT toether with echanics of aterials approach, for different relative weiht fractions of ute and lass fiber. The details of lainates (1 ply) for which elastic properties are predicted were iven in Table 3.1 of chapter 3. A theoretical odel proposed based on rule of hybrid ixture (both parallel and serial behavior) for prediction of elastic properties has been presented. The analytical results obtained fro CLT and rule of hybrid ixture odel have been validated with experiental results. The extent of applicability of CLT and proposed odel to ute and ute-lass fiber hybrid coposites has been discussed. 5.2 LASTIC PROPRTIS OF UNIDIRCTIONAL JUT AND GLASS FIBR LAMINA The elastic properties of unidirectional ute fiber laina and lass fiber laina were predicted usin siple rule of ixture relationships fro the echanics of aterials approach. The properties of ute fiber, lass fiber and experientally evaluated properties of isothalic polyester resin (usin unreinforced resin saples) are presented in Table 5.1 and are used for prediction of elastic properties for ute and lass fiber laina. Assuin unifor distribution of resin in the coposite, the averae fiber volue fraction in each ply is sae as total fiber volue fraction. This has been proved for 1 ply ute lainate in Appendix 3. Table 5.1 Properties of ute fiber, lass fiber and isothalic polyester resin Property Jute fiber Glass fiber Isothalic Polyester Density, /c Youn s odulus, GPa Poisson s ratio.38 *.2.46 Shear odulus, GPa * This value is evaluated fro Poisson s ratio of ute coposite usin rule of ixture

3 125 For unidirectional ute ply, V V (5.1) l 1 V (5.2) t V 1 V (5.3) G v lt lt V G G v V v V (5.4) For unidirectional lass ply, l V V (5.5) 1 (5.6) t V V 1 (5.7) G v lt lt V G V G v V v V (5.8) The suffix and stands for ute and lass fiber respectively. The elastic properties for unidirectional ute and lass fiber plies evaluated usin equations (5.1) to (5.8) are presented in Tables 5.2(a) and 5.2(b). Table 5.2(a) lastic properties of unidirectional ute and lass fiber plies (In all ute and hybrid H1) Lainate Property Unit Jute fiber ply Glass fiber ply l GPa AJ t GPa v lt G lt GPa l GPa H1 t GPa v lt G xy GPa

4 126 Table 5.2(b) lastic properties of unidirectional ute and lass fiber plies (In hybrids H2 and H3) Lainate Property Unit Jute fiber ply Glass fiber ply l GPa t GPa H2 v lt G lt GPa l GPa t GPa H3 v lt G lt GPa LASTIC PROPRTIS OF WOVN JUT AND GLASS FABRIC LAMINA Fro the values of elastic properties deterined for unidirectional plies, the elastic properties of /9 o woven ute fabric ply and lass fabric ply, were predicted by usin the relations (Daniel Gay et al 23) (5.9) to (5.16). For woven ute ply, ) (5.9) 11 l (1 t 22 (1 ) l t (5.1) v v lt 12 (5.11) l (1 ) t G 12 G lt (5.12) where, N N N 1 2

5 127 For woven lass ply, ) (5.13) 11 l (1 t ) (5.14) 22 ( 1 ) l t v 12 lt (5.15) v ( 1 ) l t G 12 G lt (5.16) N where, 1. 5 N N 1 2 The elastic properties for woven ute and lass fabric plies evaluated usin equations (5.9) to (5.16) are presented in Table 5.3. Table 5.3 lastic properties of woven ute and lass fabric plies Lainate Property Unit Jute fiber ply Glass fiber ply 11 GPa AJ 22 GPa V G 12 GPa GPa H1 22 GPa V G 12 GPa GPa H2 22 GPa V G 12 GPa GPa H3 22 GPa V G 12 GPa

6 128 It can be seen fro Table 5.3 that for woven lass plies, Youn s odulus in warp as well as in weft direction is the sae since the lass fabric used is balanced one (equal nuber of yarns in warp and weft direction). However, for ute lainates Youn s odulus in warp direction is found reater than that in weft direction because of reater nuber of yarns in warp direction than in weft direction (unbalanced fabric). 5.4 MODLING OF LASTIC PROPRTIS USING CLASSICAL LAMINATION THORY Thickness of Jute and Glass Plies in Hybrid Lainate In ute-lass fiber inter-ply hybrid lainates considered in this study, the thickness of ute fiber ply and lass fiber ply is not sae due to difference in the thickness of the ute and lass fabric. In this section, analytical expressions have been derived to deterine the thickness of ute and lass ply in a hybrid lainate. If n and n are the nubers of ute and lass plies, respectively in a 1 ply lainate, then, n +n =1, the total nuber of layers (n). respectively, then If V p is the volue fraction of ute plies in hybrid lainate, Volue of ute plies Volue of 1 ply no. of ute plies V p = = (5.17) Volue of coposite Volue of coposite For a particular lenth (l) and width (b) of the lainate, Volue of 1 ply no. of ute plies (l b t ) n t n V p = = = (5.18) Volue of coposite (l b h) h

7 129 Total Volue of ute fibers Jute fiber volue fraction in hybrid, V c = (5.19) Volue of coposite Volue of ute fiber in 1 ply Jute fiber volue fraction in 1 ply, V = (5.2) Volue of 1 ute ply Volue of ute fiber in 1 ply Volue of 1 ute ply = (5.21) V we et, Substitutin for volue of 1 ute ply fro equation (5.21) in (5.18), Volue of ute fiber in 1 ply no. of ute plies t n = (5.22) V Volue of coposite h (Volue of ute fiber in 1 ply no. of ute plies) ives total volue of ute fibers in hybrid coposite. Therefore, equation (5.22) becoes Total volue of ute fibers 1 t n = (5.23) Volue of coposite V h Usin equation (5.19) in (5.23), we et, V c t n = (5.24) V h h V c Hence t = (5.25) n V

8 13 Siilarly, it can be proved that, h V c t = (5.26) n V The ute and lass fiber volue fractions in their respective plies ay be taken sae as total fiber volue fraction as proved in Appendix -3. Althouh volue of ute and lass fibers in their respective plies is different, volue of the plies also varies accordinly with out alterin the ratio i.e. the volue faction Classical Laination Theory -Approach 1 (CLT AP-1) In this approach, each woven ply was represented as two unidirectional plies of o and 9 o as shown in Fiure 5.1, so that 1 woven ply lainate can be treated as 2 unidirectional ply lainate. Z X Fiure 5.1 Scheatic representations of ( o / 9 o ) as non-syetric two ply lainate Thickness of o and 9 o ute and lass plies can be deterined usin equations (5.27) and (5.28) respectively. t 1 t, t 1 t (5.27) t 2 ( 1 ) t, t 2 ( 1 ) t (5.28)

9 131 where, the thickness of each woven ute and lass ply, t and t in the hybrid lainate were evaluated usin the derived equations (5.25) and (5.26). The thickness of ute and lass plies in the hybrid lainates was also easured usin optical icroscope and luary illuinated anifyin lass Model No The easured values were copared with those evaluated by usin the equations (5.25) and (5.26) and were found to be in ood areeent. The eleents of reduced stiffness atrix Q i for unidirectional ute and lass plies were evaluated by the known values of their elastic properties l, t, υ lt and G lt usin the equation (5.29). Q 11 1 v l lt v tl, Q 22 1 v t lt v tl, Q 12 vlt t 1 v v lt tl, Q66 G lt (5.29) The eleents of transfored reduced stiffness atrix Q i were evaluated usin the transforation atrix. For o unidirectional laina, [ Q ] [ Q] i i For 9 o ply, Q11 Q 22, Q22 Q 11, Q12 Q 12, Q, 66 Q 66 and for both o and 9 lainae, Q Q Fro classical laination theory, extensional stiffness atrix [A] for hybrid lainate by approach 1, can be written as [A i ]=[ Q ] (n t 1 ) + [ Q ] 9 (n t 2 ) + [ Q ] (n t 1 ) + [ Q ] 9 (n t 2 ) (5.3) i i i i

10 132 where, the suffix and indicate ute and lass plies respectively. For all ute lainate, the last two ters of equation (5.3) ay be inored. The [A] atrices evaluated for all ute (AJ) and hybrid lainates (H1, H2, and H3) by approach 1 are presented in equation (5.31) [A] AJ = , [A] H1 = [A] H2 = , [A] H3 = (5.31) The lainate elastic properties were estiated usin the relation (5.32) (Mallick 1997). x A A A , y, ha22 ha11 A A A A12 A12 v xy, v yx, A A A (5.32) h G xy 66 The thickness h of the all ute and hybrid lainates for which elastic properties are evaluated, is presented Table 3.1 of chapter 3. Table 5.4 presents the elastic properties of the lainates evaluated by approach 1. Table 5.4 Lainate elastic properties fro CLT approach 1 Lainate code x GPa y GPa Property v xy v yx G xy GPa AJ H H H

11 Classical Laination Theory-Approach 2 (CLT AP-2) In this approach, lainate was considered to be coposed of 1 woven layers directly without considerin each layer as two unidirectional lainae. All the layers were oriented at [ o /9 o ], so that [ Q ] i = [Q] i. The eleents of [Q i ] were evaluated by the known values of the elastic properties of woven layer 11, 22, υ 12 and G 12 usin the relevant equation (5.33). Q 11 11, 1 v12v21 Q 22 22, 1 v v Q 12 v 12 22, 1 v v Q66 G 12 (5.33) can be written as The extensional stiffness atrix for woven fabric hybrid lainate [A i ] = [ Q i] (n t ) + [ Q i] (n t ) (5.34) For all ute lainate, the last ter of equation (5.34) ay be inored. The [A] atrix evaluated for all ute (AJ) and hybrid lainates (H1, H2, and H3) by approach 2 are presented in equation (5.35) [A] AJ = , [A] H1 = [A] H2 = , [A] H3 = (5.35) Observation of equation (5.31) and (5.35) revealed that there is a arinal difference in the eleents of [A] atrix obtained by two approaches. This difference is due to the reason that the curvature effect due

12 134 to interlacin of ute yarns and lass fiber yarns are nelected while assuin each woven layer as two unidirectional layers one in o and other in 9. The lainate elastic properties were estiated usin the relation (5.32) and presented in Table 5.5. Table 5.5 Lainate elastic properties fro CLT approach 2 Lainate Code x GPa y GPa Property v xy v yx G xy GPa AJ H H H MODLLING OF LASTIC PROPRTIS USING RUL OF HYBRID MIXTUR MODL The elastic properties of sinle woven ute and lass fabric plies were already deterined usin expressions (5.9)-(5.16). lastic properties of hybrid coposites were deterined usin rule of hybrid ixture odel, both parallel and serial behavior Parallel Behavior The proposed rule for elastic properties of hybrid lainates accordin to parallel behavior is iven by expressions (5.36) to (5.39). x 11 V p 11 V (5.36) p y 22 V p 22 V (5.37) p

13 135 vxy v12 V p v12 V (5.38) p Gxy G12 V p G12 V (5.39) p where, V p and V p are the volue fractions of ute and lass plies in hybrid lainates. These are calculated usin the expressions (quation 5.18 for ute and siilar equation for lass) and presented in Table 5.6. Table 5.6 Thickness and volue fraction of ute and lass plies in hybrid lainates Lainate t V p t V p AJ H H H Serial Behavior The proposed rule for elastic properties of hybrid lainates accordin to serial behavior is iven by expressions (5.4) to (5.43). 1 V V (5.4) p p x V V (5.41) p p y xy V p p (5.42) 12 V 12 1 V V (5.43) p p Gxy G12 G12

14 136 The inor Poisson s ratio can be deterined fro the followin well known expression y yx (5.44) xy x The lainate elastic properties evaluated by usin the proposed rule of hybrid ixture are presented in Table 5.7. Table 5.7 Lainate elastic properties evaluated by rule of hybrid ixture Lainate Code x, GPa y, GPa xy G xy, GPa Parallel Serial Parallel Serial Parallel Serial Parallel Serial AJ H H H VALIDATION OF ANALYTICAL RSULTS BY XPRIMNTAL RSULTS xperiental evaluation of elastic properties was discussed in chapter 3. In this section, theoretical values of elastic properties deterined by CLT and rule of hybrid ixture odel are copared with those with the experiental results in order to validate the evaluation procedure. The correlations between theoretical and experiental values are shown in Fiures 5.2(a) to 5.2(d), in which elastic properties are plotted as a function of lass fiber content(wt%). wt% lass fiber content is all ute lainate, followed by H1(8.2 wt% lass fiber), H2(16.5 wt% lass fiber) and H3(25.2 wt% lass fiber).

15 137 x, GPa xpt'al CLT AP-1 CLT AP-2 Model Glass fiber content, wt% a y, GPa xpt'al CLT AP-1 Model b Glass fiber content, wt% 3 vxy xpt'al CLT AP-2 CLT AP-1 Model Glass fiber content, wt% c Gxy, GPa xpt'al CLT AP-2 CLT AP-1 Y Glass fiber content, wt% d Fiure 5.2 Correlation between experiental and predicted values of elastic properties (a) x (b) y (c) ν xy (d) G xy A close look on the values of elastic properties predicted by rule of hybrid ixture presented in table 5.7 reveal that parallel behavior for x, serial behavior for y, ν xy, and ν yx ives close areeent with experiental values presented in Table 3.18 of chapter 3. Hence these behaviors are used in correlatin the results in Fiure 5.2. No sinificant chane in shear odulus G xy was noticed by parallel and serial behavior. It can be witnessed fro fiure 5.2 that the areeent between predicted values of elastic properties by two different approaches of CLT is excellent. Also ood areeent is noticed between the values predicted by CLT and rule of hybrid ixture odel (parallel for x, serial for y, υ xy, and

16 138 parallel or serial for G xy ). However, the theoretical and experiental results are found to be deviated upto 2 % except for y of all ute lainates where the deviation is as hih as 25% (Table 5.8). Table 5.8 Deviation in results between theoretical prediction and experiental Lainate Code lastic property xpt al & CLT AP-1 xpt al & CLT AP-2 Deviation (%) xpt al & Model AJ x, GPa y, GPa υ xy G xy,gpa (P * or S ** ) 24.5 (S).29 (S) (P or S) H1 x, GPa y, GPa υ xy G xy,gpa (P) 8.36(S) 5.71(S) 11.61(P or S) H2 x, GPa y, GPa υ xy G xy,gpa (P) 5.4(S) 6.67(S) 4.87(P or S) H3 x, GPa y, GPa υ xy G xy,gpa (P) -6.61(S) 1.5(S) -1.53(P or S) * P-Parallel behavior, ** S-Serial behavior The deviation between the experiental and predicted values ay be attributed to the followin reasons. The strenth of aterials approach odel used in theoretical prediction assues that, the bond between the fiber and the atrix is perfect, fibers are continuous with unifor diaeter.

17 139 Non unifority of ute yarns and poor fiber-atrix adhesion (as the fabric is untreated) ay affect the experiental results. Theoretical prediction by rule of hybrid ixture expects coplete interinlin of both types of fibers within the atrix, whereas in the present study, lass fiber plies are arraned as extree layers only. The theoretical prediction is based on an averae literature value of Youn s odulus of ute fiber, which is taken as 2 GPa (Paul Wabua et al 23). Presence of voids, if any ay also affect the experiental results. 5.7 CONCLUDING RMARS In this chapter, an attept is ade to evaluate the applicability of classical laination theory for deterinin the elastic properties of ute and ute- lass fabric reinforced isothalic polyester hybrid coposites. A rule of hybrid ixture odel with both parallel and serial behavior is also proposed for prediction of elastic properties. The aterial paraeters needed for application of classical laination theory and rule of hybrid ixture and a ethodoloy for prediction of ply properties (UD and Woven) are described in the theoretical predictions. Based on this study the followin iportant conclusions are drawn. Youn s odulus in warp and weft direction depict an increasin trend with the incorporation of lass fiber by all the three ethods of evaluation viz., CLT, rule of hybrid ixture and experiental.

18 14 Poisson s ratio of ute coposites is found to decrease with the addition of lass fiber. Shear odulus exhibit arinal decrease with the addition of lass fiber, when theoretically predicted and arinal increase with the addition of lass fiber, when easured. xcellent areeent between the predictions of elastic properties by CLT and rule of hybrid ixture odel is obtained. Fro the results, it can be concluded that the theoretical approach that is presented in this chapter, can be applied to ute and ute-lass hybrid coposites with a deviation upto about 2% with the experiental results.