Bulletin of the Transilvania University of Braşov • Vol. 8 (57) No. 1 - 2015 Series I: Engineering Sciences
THERMAL EXPANSION OF COMPOSITE LAMINATES
M.N. VELEA1 S. LACHE1
Abstract: When connecting composite materials and isotropic materials, their different thermal expansion coefficients may introduce additional stress in the structure. This article shows the way the fibre composite laminates may be designed and optimized in order to fit the thermal expansion of the pair isotropic material and thus resulting similar thermal deformations in both of the materials. An optimization model is developed based on the micromechanics theory of composite laminates where the plies angles represent the design variables while the objective are represented by the thermal expansion coefficients of the laminate.
Key words: thermal expansion, lamina, laminate, composites.
1. Introduction properly design, the connection may fail under the combined thermal and mechanical Composite materials and structures may load. provide multiple benefits, first of all with This article presents a design method of the respect to weight. Their mechanical fibre reinforced composite laminates in order properties may be tailored according to the to obtain thermal expansion coefficients imposed loading conditions by changing the equal to the one of the metal that the fibre or matrix, the fibre orientation or the composite part must be in contact with. After stacking sequence. All these degrees of presenting the theoretical background a freedom allow optimizing the structure in calculation example is given in order order to obtain the best stiffness and exemplify the presented design method. strength to weight ratio [4]. In addition to the applied mechanical loads, thermal loads 2. Thermal Expansion of Lamina may also occur when a system is working within a large interval of temperatures. Having given the fibre and matrix Especially when metal parts are in contact individual properties, the overall properties with the composite parts, care must be taken of the lamina (one single ply) may be within the design process due to different determined by using the Rule of Mixture thermal behaviour of materials. Therefore, approach [5]. The thermal expansion one of the problems when joining dissimilar coefficients of the lamina in local materials is referring to thermal expansion. coordinates (1-2), α1 and α2 will therefore Connecting materials with different thermal be determined with Equation (1) for 1- expansion coefficients usually generates direction and with Equation (2) for 2- additional stress in the structure. If not direction, Figure 1.
1 Dept. of Mechanical Engineering, Transilvania University of Braşov. 26 Bulletin of the Transilvania University of Braşov • Series I • Vol. 8 (57) No. 1 - 2015
v f f 1E f 1 vmm Em for individual plies and based on a , (1) 1 predefined stacking sequence, the thermal v f E f 1 vm Em expansion coefficients of the laminate in global coordinates α , α and α may be x y xy v 1 f 12 f 1 determined using Equation (3). In Equation 2 f f 2 f (2) (3), A represents the extensional stiffness v (1 ) (v v ) , matrix, calculated using Equation (4): m m m f f 12 m m 1 where: K V K V x 1 0 A 2 1A α - thermal expansion coefficient of the 1 f1 y 0.5Α K1V0A K 2V1A , (3) fibre, in the fibre direction; xy K3V2 A αf2 - thermal expansion coefficient of the fibre perpendicular to the fibre direction;
αm - thermal expansion coefficient of the A11 A12 A16 matrix; A A12 A22 A26 . (4) vf - fibre volume fraction; A A A vm - matrix volume fraction; 16 26 66 Ef1 - Young’s modulus of the fibre, along the fibre direction; Equations (5) to (11) represent the Em - Young’s modulus of the matrix; components of the extensional stiffness νf12 - Poisson’s ratio of the fibre; matrix A: νm - Poisson’s ratio of the matrix;
ρf - density of the fibre. A11 U1V0A U2V1A U3V3A , (5)
A22 U1V0 A U2V1A U3V3A , (6) y
A12 U 4V0 A U3V3A , (7)
A66 U5V0A U3V3A , (8) x x A16 0.5U2V2 A U3V4A , (9) A26 0.5U2V2A U3V4 A , (10) 2 y y with: 1
x V0 A h , (11) Fig. 1. Local and global coordinates of the lamina n V1A tk cos 2k , (12) k1 3. Thermal Expansion of Laminate n Once the thermal expansion coefficient V2 A tk sin 2k , (13) are determined in local coordinates (1-2) k1 Velea, M.N., et al.: Thermal Expansion of Composite Laminates 27
n 1 U (Q Q 2Q 4Q ) , (20) V3A tk cos 4k , (14) 5 11 22 12 66 k1 8
n U1…U5 represents the Tsai-Pagano V4A tk sin 4k , (15) material invariants [3] and they are k1 calculated through Equations (16)-(20), where: where:
h - represents the total thickness of the E1 laminate; Q11 , n - represents the number of lamina; (1 12 21) θk - represents the orientation angle of the kth lamina. E Q 2 , 22 (1 ) 1 12 21 U (3Q 3Q 2Q 4Q ) , (16) 1 8 11 22 12 66 12 E2 Q12 , 1 (1 1221) U (Q Q ) , (17) 2 2 11 22 Q66 G12 . 1 U 3 (Q11 Q22 2Q12 4Q66 ) , (18) 8 K1, K2, K3 from Equation (3) represents the thermal material constants, defined 1 U (Q Q 6Q 4Q ) , (19) through the Tsai-Pagano constants by 4 8 11 22 12 66 Equations (21)-(23):
K1 (U1 U 4 ).(1 2 ) U2 (1 2 ) , (21)
K2 U2 (1 2 ) (U1 2U3 U 4 )(1 2 ) , (22)
K3 U2.(1 2 ) 2(U3 U5 )(1 2 ) . (23)
4. Case Study CFRP
A tube is made of Carbon Fibre Reinforced Steel R1 Plastics with the external radius R1 = 50 mm. R2 A metal component is fixed inside the shaft, having the radius R2 = 50 mm, Figure 2. y Let us consider that this assembly will operate in a range of temperatures from 20 x to 200 Celsius degrees. In terms of the z thermal expansion coefficients, thermal Fig. 2. Steel-composite connection deformations will occur in both materials, according to Equation (24): While for steel, polymer matrix and carbon fibre the thermal expansion R y RT. (24) coefficients are known, Table 1, in case of 28 Bulletin of the Transilvania University of Braşov • Series I • Vol. 8 (57) No. 1 - 2015 the CFRP laminates the thermal expansion HyperStudy software [6] was then used to coefficient will strongly be influenced by solve the optimization problem by linking the individual properties of the matrix and the Excel file to the solver. The objective fibre and also by the plies’ orientation function consist of reaching a target value angles [2], Equation (2). of 16E-6 [1/°C] for αy - thermal expansion According to the example given above, along Y direction of the composite laminate, and considering Equation (4) at 200 Celsius Figure 1, and thus to obtain the same degrees, the radius of the steel part R2 will thermal deformation as steel along the increase from 50 mm to 50.1449 mm while radial direction of the tube, Figure 2. The the radius of the CFRP part R1 will increase predefined stacking sequence is [45/-45/ from 50 mm to only 50.0071 mm. This 45/-45/45/-45]S which represents a means that the steel part will press on the symmetric laminate with 12 layers. θk CFRP part and thus generating additional represent the design variables; the laminate in-plane stress within the structure. being symmetric, we have only 6 different θ angles as independent variables. The in- Table 1 plane stiffness parameters of the laminate Thermal expansion coefficients [1] Ex and Ey are also considered within the optimization as constraints: E ≥ 10000 MPa Thermal x Carbon Epoxy expansion Steel and Ey ≥ 10000 MPa. fibre resin [1/°C] α1 16E-6 -0.6E-6 55E-6 5. Results and Conclusion α2 16E-6 8.5E-6 55E-6 Figure 3 illustrates the values obtained As it can be noticed from Table 1, the for αx and αy. As expected, while αy reach steel has a larger thermal expansion target value of 16E-6 [1/°C], αx is coefficient than the fibre but lower than the decreasing and becomes negative (the matrix. Moreover, the fibre is actually material contracts at high temperatures). contracts at high temperature, having a negative thermal expansion coefficient. In order to avoid additional stress caused by the larger thermal deformations of the steel part in comparison to the one of the CFRP part, it is therefore important to design the laminate such that it has similar thermal expansion coefficient as the steel has. This problem may be formulated as an optimization problem where the objective is to find the configuration of the laminate that will give the desired thermal expansion coefficient. Considering that Fig. 3. Thermal expansion coefficients of fibre and matrix properties are given, the laminate Table 1, the problem is actually reducing to the optimization of the orientation Figure 4 shows the values obtained for angles θk where k = 1…n, and n represents the plies’ orientation angles that allow the total number of plies. obtaining the desired value of αy. The mathematical model described in Although the desired thermal expansion section 2 was defined within Excel. behaviour is reached, the mechanical Velea, M.N., et al.: Thermal Expansion of Composite Laminates 29 properties of the composite laminate are the here-in considered case study. A trade of altered along Y-direction due to the between high stiffness and low thermal resulted staking sequence [20/-20/20/-20/ expansion coefficient is needed. However, 20/-20]S. there are situations where high stiffness and low thermal expansion are of interest.
Fig. 4. Values of the plies’ orientation Fig. 6. Thermal expansion coefficient αx of angles the laminate in terms of the laminate stiffness Ex along X-direction Figure 5 presents the values obtained for the in-plane stiffness parameters Ex and Ey Future studies may also consider the of the laminate. By looking on both Figure stiffness and strength of the laminate along 4 and Figure 5, it turs out that the laminate both in-plane directions as objectives to be stiffness increase while the thermal maximized. A trade-off solution may be expansion coefficient decrease, along each selected this way by considering both of the in-plane orthogonal direction X and Y. mechanical and thermal loads.
Fig. 7. Thermal expansion coefficient αy of Fig. 5. The resulted stiffness of the the laminate in terms of the laminate laminate stiffness Ey along Y-direction
This observation is also visible from Acknowledgements Figure 6 and Figure 7: small values for the thermal expansion coefficients imply large This work was partially supported by the values for the laminate stiffness and vice strategic grant POSDRU/159/1.5/S/137070 versa. This behaviour is disadvantageous for (2014) of the Ministry of Labour, Family 30 Bulletin of the Transilvania University of Braşov • Series I • Vol. 8 (57) No. 1 - 2015 and Social Protection, Romania, co- Using Neural Networks and Genetic financed by the European Social Fund - Algorithms. In: Composite Structures Investing in People, within the Sectorial 94 (2012), p. 3321-3326. Operational Programme Human Resources 3. Tsai, S.W., Melo, J.D.D.: An Invariant- Development 2007-2013. The support of Based Theory of Composites. In: Department of Aeronautical and Vehicle Composites Science and Technology engineering from KTH - Royal Institute of 100 (2014), p. 237-243. Technology, Sweden, it is gratefully 4. Velea, M.N., Wennhage, P., Zenkert, acknowledged. D.: Multi-Objective Optimisation of Vehicle Bodies Made of FRP Sandwich References Structures. In: Composite Structures 111 (2014), p. 75-84. 1. Åström, B.T.: Manufacturing of 5. Vinson, J.R.: Plate and Panel Polymer Composites. Nelson Thornes Structures of Isotropic, Composite and Ltd., 2002. Piezoelectric Materials, Including 2. Marín, L., Trias, D., Badalló, P., Rus, Sandwich Construction. Dordrecht. G., Mayugo, J.A.: Optimization of Springer, 2005. Composite Stiffened Panels Under 6. Altair. HyperWorks 11: Online Help Mechanical and Hygrothermal Loads and Documentation, 2012.