Minimization of Thermal Expansion of Symmetric, Balanced, Angle Ply Laminates by Optimization of ﬁber Path Conﬁgurations ⇑ Aswath Rangarajan A, Royan J

Composites Science and Technology 71 (2011) 1105–1109

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Composites Science and Technology

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Minimization of thermal expansion of symmetric, balanced, angle ply laminates by optimization of ﬁber path conﬁgurations ⇑ Aswath Rangarajan a, Royan J. D’Mello a, Veera Sundararaghavan a, Anthony M. Waas a,b, a Dept. of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109-2140, United States b Dept. of Mechanical Engineering, University of Michigan, Ann Arbor, MI 48109-2140, United States article info abstract

Article history: Optimal fiber path configurations that minimize the sum of the coefficients of thermal expansion (CTE) Received 10 September 2010 values along the principal material directions for a class of laminates are presented. Previous studies sug- Received in revised form 12 March 2011 gest that balanced, symmetric, angle ply laminates exhibit negative CTE values along the principal direc- Accepted 25 March 2011 tions. Using the sum of the CTE values along the principal material directions as an effective measure of Available online 31 March 2011 the coefficient of thermal expansion (CTEeff), we have shown and provided a proof that the smallest value of CTEeff is rendered by straight fiber path configurations. The laminates considered are sufficiently thin Keywords: so as to neglect the thermal stresses induced through the thickness of the laminate. It is found that the A. Polymer-matrix composites minimal CTE values occur for [+45/ 45] lay-ups. This result is supported by numerical studies that A. Laminate eff ns B. Thermomechanical properties consider curvilinear fiber paths. The possibility of obtaining zero CTE values along both principal material C. Modelling directions and the conditions that render this situation are also examined. C. Deformation Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction optimum fiber path that yields the least value of CTEeff, maintain- ing the assumption of a balanced, symmetric, angle ply laminate. Failure by thermal fatigue can be mitigated by minimizing the In the analysis to follow, the possibility of obtaining a zero value L coefficient of thermal expansion (CTE) of composite laminates aX for CTEeff is investigated and conditions for obtaining such a CTEeff L and aY along the principal material directions. Early work on char- are derived. acterization of CTE values in fiber reinforced composites was due to Craft and Christensen [1], Marom and Weinberg [2], Ishikawa and Chou [3], Bowles and Tompkins [4], Sleight [5], Lommens 2. Model description et al. [6], and references therein. More recent studies have focused on laminates with straight fiber configurations [7–10]. Amongst Consider curvilinear fiber configurations in the x–y plane which these, those which are balanced, symmetric, angle ply ([+h/h]ns) are symmetric about the z-axis (Fig. 1) in the Representative Unit lay-ups have been found to exhibit anomalous mechanical Cell (RUC) of in-plane dimensions A B. The fibers are stacked par- response. Analysis of these laminates have shown the existence allel to the y-axis. Obliquely stacked configurations of the fibers are L L not considered since it reduces to the case under consideration as of negative CTE aX ; aY for certain range of ply orientations [7,9]. Zhu and Sun [10], showed that the ratio of shear modulus G12 to can be seen from Fig. 2. This would imply that for any infinitesi- the Young’s modulus E1 is an important parameter that determines mally small portion of the fiber curve with an orientation h, there the sign and magnitude of the CTE in the composite laminate. exists an infinitesimally small complementary fiber element with ⁄ However, negative CTE values along both principal material direc- orientation h (Fig. 3). For every fiber at angle +h at z =+z , there ⁄ tions (x, y) of the composite laminate were not obtained simulta- is another fiber of same orientation at z = z . Also, for every fiber ⁄⁄ ⁄⁄ neously for any ply angle h. In this study, we relax the at an angle h at z =+z , there is another fiber at z = z with requirement of fibers having straight configurations and seek the the same orientation. Hence, this configuration acts as a balanced symmetric laminate for which the moment resultants due to thermal stresses cancel out (see [11]), i.e. M ¼ 0; M ¼ 0 and M ¼ 0. ⇑ x y xy Corresponding author at: Dept. of Aerospace Engineering, University of Here, we use standard composite laminate nomenclature as given Michigan, Ann Arbor, MI 48109-2140, United States. Tel.: +1 734 764 8227; fax: in [11]. Similarly, the effective shear force resultants due to thermal +1 734 763 0578. expansion also cancel out, i.e. N 0 . Therefore, the only non-zero E-mail addresses: [email protected] (A. Rangarajan), [email protected] xy ¼ (R.J. D’Mello), [email protected] (V. Sundararaghavan), [email protected] (A.M. stress resultants present are normal stresses along the principal Waas). directions Nx and Ny in the plane of the laminate. The

Y A

Fig. 1. Profile of fibers in the Representative Unit Cell (RUC) of dimensions A B. The RUC has many overlaid symmetric fibers which renders the RUC to have a structure similar to that of a balanced, symmetric, angle ply laminate.

Fig. 2. Schematic showing the equivalence of the fiber stacking along the horizontal and oblique directions. curvilinear fiber format in the RUC is invariant along the y-direc- where tion. Hence, the compliance matrix of an infinitesimal strip of width S ¼f2ðS S ÞS g cos3 h sin h dx is a function of x alone. The continuity of fiber slopes across adja- 16 11 12 66 3 cent RUCs is ensured by the equality of slope at RUC boundaries. þf2ðS12 S22ÞþS66g cos h sin h 3 S26 ¼f2ðS11 S12ÞS66g cos h sin h 3. Mathematical formulation ð3Þ 3 þf2ðS12 S22ÞþS66g cos h sin h 3.1. Straight Fibers 2 2 S66 ¼ 2f2ðS11 þ S22 2S12ÞS66g cos h sin h

4 4 For a straight ﬁber, balanced, angle ply laminate, if the ﬁber is þ S66fcos h þ sin hg oriented at an angle h with respect to x-direction [7,10], we have For no thermal expansion, we should have aL ¼ 0 and aL ¼ 0 simul- S X Y L 2 2 16 taneously. This leads to ðaL aL Þ¼0. aX ¼ a1 cos h þ a2 sin h þ ða2 a1Þ sin 2h ð1Þ X Y S66 () !

L 2 2 S26 S16 S26 aY ¼ a1 sin h þ a2 cos h þ ða2 a1Þ sin 2h ð2Þ ða1 a2Þ cos 2h sin 2h ¼ 0 ð4Þ S66 S66 A. Rangarajan et al. / Composites Science and Technology 71 (2011) 1105–1109 1107 R R R R R R B A B A B A b S16dxdy b S26dxdy b S66dxdy S ¼ 0 0 ; S ¼ 0 0 ; S ¼ 0 0 16 AB 26 AB 66 AB ð12Þ Since the fibers are stacked along the y-direction in the RUC, there is b b b no variation of S16; S26 and S66 along the y-direction. Thus, we make the assumption of constant strain along the x-axis of the RUC. b Hence, the compliance matrix ½S terms reduce to R R R A A A b S16dx b S26dx b S66dx S ¼ 0 ; S ¼ 0 ; S ¼ 0 ð13Þ 16 A 26 A 66 A where the compliance terms of the infinitesimal segment are given by S 2 S S S c3s 2 S S S cs3 Fig. 3. Profile of a pair of complementary fibers in the Representative Unit Cell 16 ¼f ð 11 12Þ 66g þf ð 12 22Þþ 66g (RUC) of dimensions A B with fiber curve (red) modeled as a function 3 3 S26 ¼f2ðS11 S12ÞS66gcs þf2ðS12 S22ÞþS66gc s ð14Þ y = f(x), x 2 [0, A]. The complementary fiber curve (green) is also shown. The figure 2 2 4 4 shows the definition of the angle h for an infinitesimal element of the fiber. (For S66 ¼ 2f2ðS11 þ S22 2S12ÞS66gc s þ S66fc þ s g interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) The CTE in the principal material directions of the laminate are obtained as [21], R R A noR – A S16dx A In the above expression, a1 a2. We inspect the term in the paren- L 1 2 2 R0 1 aX ¼ A 0 a1c þ a2s dx A A 0 2csða1 a2Þdx thesis which is expanded as S66dx R0 ð15Þ ()R A noR A S26dx A 2 L 1 2 2 R0 1 sin hðS11 þ S22 2S12 S66Þ aY ¼ A 0 a1s þ a2c dx A A 0 2csða1 a2Þdx cos 2h 1 ¼ 0 ð5Þ S66dx 2 2 0 ðS11 þ S22 2S12Þ sin 2h þ S66 cos 2h L L Define a scaled measure of CTEeff ; G ¼ AðaX þ aY Þ. By inspection, h = p/4 is a solution to Eq. (5). Adding Eqs. (1) and (2) R 2 A we get csdx 0 ! G ¼ Aða1 þ a2Þþ4ðS22 S11Þ R ða1 a2Þð16Þ A S dx L L S16 þ S26 0 66 a þ a ¼ða1 þ a2Þþða2 a1Þ sin 2h ¼ 0 ð6Þ X Y S 66 From Eq. (16), Setting h = p/4, the above expression is written as G ¼ Afða1 þ a2ÞþK2ða1 a2Þg ð17Þ a1 þ a2 S11 S22 As S11 =1/E1, S22 =1/E2 and S12 =m12/E1, we get ¼ ð7Þ a1 a2 S11 þ S22 2S12 1 1 2m12 2 2 1 2 2 2 S66 ¼ 22 þ þ c s þ ðc s Þ > 0 ð18Þ Writing the compliance quantities in terms of elastic constants, we E1 E2 E1 G12 have S11 =1/E1, S22 =1/E2, and S12 =m12/E1. Therefore, Also, () a þ a 1 1 1 1 1 2 ¼ E1 E2 ð8Þ S S > 0: 19 1 1 m12 ð 22 11Þ¼ ð Þ a1 a2 þ þ 2 E2 E1 E1 E2 E1 We write G in the manner G = A(a + a )+K A(a a ), where Since E1 > E2, right hand side (RHS) < 0 ) (a1 a2)<0) a1 < a2. 1 2 2 1 2 R 2 Next, assuming (a1 + a2)>0 A 8 9 csdx S22 S11 0 < m = K2 ¼ 4 R ð20Þ 1 þ 12 A A a1 E1 E1 S66dx ¼: ; < 0 ð9Þ 0 a2 1 þ m12 E2 E1 From Eqs. (18) and (19) it follows that

A similar result has also been presented in Zhu and Sun [10]. Using K2 P 0 ð21Þ Eqs. (8) and (9), a1 < 0 < a2 ð10Þ 3.2.1. Inspecting the behavior of K2 We inspect if K2 <1orK2 > 1. Suppose K2 > 1, then

Z 2 Z 3.2. Curved fibers S S A A 4 22 11 csdx 4 ðS þ S 2S ÞðcsÞ2dx A 11 22 12 Now, consider a single curvilinear fiber path in the x-y plane Z 0 0 A and represent it by a function y = f(x). If h is the angle made by 2 2 2 S66ðc s Þ dx > 0 ð22Þ any infinitesimally small segment of the curve with the x-axis, 0 the sine and cosine terms for this segment can be written in terms Invoke an integral inequality, which is obtained as a special case of 0 df ðxÞ of the slope of the curve y ¼ dx ¼ tan h. the Cauchy–Schwarz inequality, 0 1 y Z 2 Z cos h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ c; sin h ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ s ð11Þ A A 1 6 2 1 y0 2 1 y0 2 f ðxÞdx ðf ðxÞÞ dx; f ðxÞ¼cs ð23Þ þð Þ þð Þ A 0 0 b The compliance matrix ½S terms are obtained from Hence, 1108 A. Rangarajan et al. / Composites Science and Technology 71 (2011) 1105–1109 Z Z A A 2 2 4ðS22 S11Þ ðcsÞ dx 4ðS11 þ S22 2S12ÞðcsÞ dx Z 0 0 A 2 2 2 S66ðc s Þ dx > 0 ð24Þ 0 Z Z A A 2 2 2 2 ð8S11 þ 8S12ÞðcsÞ dx S66ðc s Þ dx > 0 ð25Þ 0 0 Substituting the compliance terms with material constants, we get, Z Z A m þ 1 A 1 8 12 ðcsÞ2dx ðc2 s2Þ2dx > 0 ð26Þ 0 E1 0 G12

Now, assuming (m12 + 1) > 0, Eq. (26) presents a contradiction as the terms on the left hand side (LHS) are negative. Hence by reduc- tio ad absurdum, K2 6 1. From Eq. 17 we can see that G is an increasing function of a1 and dG dG a2 and > . Hence, G is minimum when K2 is maximum and da1 da2 Fig. 4. Schematic illustrating the range of sample candidate cubic polynomials for a2 > a1, or when K2 is minimum and a1 > a2. modeling fiber paths. L L For zero thermal expansion, aX ¼ aY ¼ 0 ) G ¼ 0, Now, considering a2 > a1, the value of CTEeff is minimum when K2 ða1 þ a2ÞþK2ða1 a2Þ¼0 ð27Þ is minimum. We have already shown the lower bound for K2 to be 0 a1 þ a2 (Eq. (21)). It can be seen that when y0 = 0, i.e when the fibers in all the ¼K2 6 0 ð0 6 K2 6 1Þð28Þ a1 a2 plies are in the same orientation, the minimum value of K2 occurs. ) a1 < a2 ð29Þ To validate these analytical findings, a numerical study was car- ried out by modeling the curvilinear fiber configurations using cu- a1 K2 1 ¼ < 0 ð30Þ bic polynomials. The curved fiber paths are selected so as to a2 K2 þ 1 represent the majority of fiber orientations. The RUC domain under ) a < 0 < a ð31Þ 1 2 consideration varies from 0.01 0.01 sq.m to 1 1 sq.m. A sample from the curved fiber paths used for the study is shown in Fig. 4. The numerical simulations show that the thermal expansivity 4. Optimal fiber configurations along each of the principal axis directions is the least when the fiber configuration is approximately linear, i.e. straight fiber configura- 4.1. Straight fiber tions. The material used for the numerical study is a glass polypro- 9 9 pylene composite having E1 = 34.5 10 Pa, E2 = 3.1 10 Pa, We start with an assumption a2 > a1. The value of CTEeff is min- m = 0.25 a =6 106/°C and a = 100 106/°C. The study shows imum when K is maximum. 12 1 2 2 L that the minimum value of aX is obtained for a straight fiber config- 2 2 L 4ðS22 S11Þc s uration. Similarly, the minimum value of aY is also obtained for a K2 ¼ 2 2 2 2 2 straight fiber configuration, which is complementary to the fiber ð4ðS11 þ S22 2S12Þc s þ S66ðc s Þ Þ configuration at which aL was a minimum. Furthermore, when both S S X 22 11 L and L are combined such that the area expansion is minimized ¼ 2 ð32Þ aX aY ðS11 þ S22 2S12ÞþS66cot 2h (in-plane dilatation) to find an optimal path, again an approximate p straight fiber path is obtained. These findings compare well with K2 is maximum when h ¼ 4. analytical results presented in Ito et al. [7]. Further details of our Maximum S22 S11 study is contained in Rangarajan et al. [12]. K2 ¼ ð33Þ ðS11 þ S22 2S12Þ 5. Conclusions Now, considering a2 > a1, the value of CTEeff is minimum when K2 is minimum. We have already shown the lower bound for K2 to be 0 (Eq. (21)). It can be seen that when h = 0, the minimum value We have shown and provided a proof that amongst all curvilinear fiber configurations, the fiber configuration with straight fiber of K2 occurs. paths yields the least value of CTEeff for symmetric, balanced, angle ply laminates. This configuration is independent of lamina princi- 4.2. Curved fiber pal material parameters, E1, E2, m12, G12, a1 and a2. Additionally, bounds for the values of a and a are presented which can lead We again start with the same assumption as in the previous 1 2 to the values of CTE along the principal material directions section , i.e., a2 > a1. The value of CTEeff is minimum when K2 is ðaL ; aL Þ to be zero simultaneously. This implies that there would maximum. X Y be no change in the in-plane dimensions (along both principal lam- R 2 A inate axes) of the laminates on applying thermal loads, if Eq. (9) is 4ðS11 S22Þ 0 csdx satisfied. The bounds also show that a1 has to be negative and a2 K2 ¼ R ð34Þ A 2 2 2 2 A 0 f4ðS11 þ S22 2S12ÞðcsÞ þ S66ðc s Þ gdx has to be positive in order to obtain zero thermal expansion along both the laminate principal axes simultaneously. Amongst the

Fiber paths that maximize K2 are sought. As shown in Appendix A, straight fiber path configurations, laminates with [+45/45]ns 0 among all possible paths, those that satisfy the condition y = 1 are lay-up have the least measure of CTEeff. In such cases, the laminate the paths that maximize K2. This corresponds to straight fiber paths. has isotropic values of CTE in the plane of the laminate. The degree Furthermore, alternately stacked plies are orthogonal in this to which CTEeff can be minimized is a function of the material configuration. parameters mentioned above. A. Rangarajan et al. / Composites Science and Technology 71 (2011) 1105–1109 1109

Appendix A References

[1] Craft WJ, Christensen RM. Coefficient of thermal expansion for composites A.1. Finding fiber paths to maximize K2 with randomly oriented fibers. J Compos Mater 1981;15:2. [2] Marom G, Weinberg A. The effect of the fibre critical length on the thermal R 2 expansion of composite materials. J Mater Sci 1975;10:1005–10. A [3] Ishikawa T, Chou TW. In-plane thermal expansion and thermal bending 0 csdx S22 S11 coefficients of fabric composites. J Compos Mater 1983;17:92. K2 ¼ 4 R ð35Þ A A [4] Bowles DE, Tompkins SS. Prediction of coefficients of thermal expansion for S66dx 0 unidirectional composites. J Compos Mater 1989;23:370. [5] Sleight AW. Isotropic negative thermal expansion. Ann Rev Mater Sci 0 Replacing c and s in terms of y from Eq. (11) and using the Cau- 1982;28:29–43. chy–Schwarz inequality as in Eq. (23), we get [6] Lommens P, De Meyer C, Bruneel E, De Buysser K, Van Driessche I, Hoste S. Synthesis and thermal expansion of ZrO2/ZrW2O8 composites. J Eur Ceram Soc S S 2005;25:3605–10. K ¼ 4 22 11 2 A [7] Ito T, Sugunama T, Wakashima K. Glass fiber polypropylene composite laminates with negative coefficients of thermal expansion. J Mater Sci Lett R 0 2 A y dx 1999;18:1363–5. 0 0 2 [8] Kelly A, Stern RJ, McCartney LN. Composite materials of controlled thermal 1þðy Þ ð36Þ R 2 2 expansion. Ann Rev Mater Sci 1982;28:29–43. A y0 1ðy0Þ2 f4ðS11 þS22 2S12Þg þS66 dx [9] McCartney LN, Kelly A. Effective thermal and elastic properties of [+h/h]s 0 1þðy0Þ2 1þðy0Þ2 laminates. Compos Sci Technol 2006;67:646–61. [10] Zhu RP, Sun CT. Effects of fiber orientation and elastic constants of thermal S22 S11 6 4 expansion in laminates. Mech Adv Mater Struct 2003;10:99–107. 1 [11] Herakovich Carl. Mechanics of fibrous composites. John Wiley & Sons Inc.; R 2 1998. A y0 2 dx [12] Rangarajan A, D’Mello RJ, Sundararaghavan V, Waas AM. On the thermal 0 1þðy0Þ 37 expansion of symmetric, balanced, angle ply laminates. UM Aerospace R 2 2 ð Þ A y0 1ðy0Þ2 Department report; 2011. f4ðS11 þS22 2S12Þg 2 þS66 2 dx 0 1þðy0Þ 1þðy0Þ