Determination of in-plane and through-the-thickness coefficients of thermal expansion in composite angle brackets using digital image correlation

1 Determination of in-plane and through-the-thickness coefficients of thermal expansion in composite angle brackets using digital image correlation Enrique Graciania,*, Jesús Justoa, Patricia Lucía Zumaqueroa a Grupo de Elasticidad y Resistencia de Materiales, Escuela Técnica Superior de Ingeniería, Universidad de Sevilla. * Corresponding author. Adress: Grupo de Elasticidad y Resistencia de Materiales. Escuela Técnica Superior de Ingeniería. Camino de los Descubrimientos s/n. 41092 Sevilla, Spain. E-mail addresses: [email protected] (E. Graciani), [email protected] (J. Justo), [email protected] (P.L. Zumaquero) Abstract A procedure for the simultaneous determination of in-plane and through-the-thickness coefficients of thermal expansion (CTEs) in composite angle brackets is presented. Carbon fibre/epoxy samples are heated and digital image correlation is used to determine the in-plane strains and out-of-plane displacements in one side of each sample. In-plane CTEs are obtained from the average in-plane strains measured at different temperatures. Through-the-thickness CTE is obtained from the angular distortion of the samples at different temperatures, which is determined from the out-of- plane displacements. One quasi-isotropic and two unidirectional samples are used. Unidirectional samples allow the material’s in-plane CTEs in orthotropic directions to be determined. Material’s in-plane CTEs permit predicting an in-plane CTE value for the quasi-isotropic sample that agrees with the experimental measurements. Conversely, results obtained for the through-the-thickness CTE in all samples show a significant This is an Accepted Manuscript of an article published by Elsevier in Composite Structures (Vol. 238) on April 2020, available at: https://doi.org/10.1016/j.compstruct.2020.111939 Copyright Elsevier 2020. En idUS: Licencia Creative Commons CC BY-NC-ND 2 dispersion. This fact is attributed to the manufacturing distortions present in the curved part of the samples. Keywords Thermomechanical properties; Digital Image Correlation; Anisotropy; Curved composite laminates; Manufacturing defects. 1. Introduction Manufacturing of structural parts made of composite laminates often presents problems in fulfilling the design tolerances [1]. An accurate prediction of the final shape of composite components after the manufacturing would significantly help in reducing the costs of manufacturing and assembling of large structural parts. Deformations undergone by composite laminates during the manufacturing process are, mainly, of chemical and thermomechanical nature, although some other aspects like tool/part interaction or lack of homogeneity in certain zones may also contribute [2-9]. The sample most commonly employed in the literature to show the distortions associated to manufacturing of composite laminates is an angle bracket like the one depicted in Fig. 1(a) [2-5,10-11]. After demoulding, the faces of the angle bracket form an angle typically smaller than that formed by the sides of the mould. This effect is usually called spring-in, spring-back or spring-forward. While part of the distortions undergone by the composite component are irreversible, since are associated to the curing of the resin, another part is associated to the anisotropic nature of the coefficients of thermal expansion of the material and, 3 therefore, will change when the composite component is subjected to a temperature variation [4,12]. In the present paper, the focus is set on characterizing the thermal distortions appearing on a heated composite angle bracket. In unconstrained isotropic materials, a change in temperature induces a change in size while maintaining the shape of the part. In contrast, in orthotropic materials, like composite laminas, a change in temperature induces different changes in size in different directions. Consequently, a composite laminate, either unidirectional or formed by laminas with different orientations, typically requires a distortion of the part in order to achieve compatibility [13]. In a composite angle bracket, the main distortion caused by a homogeneous change in temperature, , is a variation in the initial angle bracket corner , see Fig. 1(b). Δ 𝑇 Δ 𝛽 𝛽 (a) (b) (c) Fig. 1: (a) Angle bracket, (b) thermal deformation of the angle bracket corner (gray: initial shape, black: final shape), (c) detail of the angle bracket corner. Denoting by the longitudinal direction, by the circumferential direction and by the 𝑥 𝑦 𝑧 through-the-thickness direction in the angle bracket corner, see Fig. 1(c), the unit angular variation can be obtained as [2]: Δ 𝛽 / 𝛽 4 , (1) ∆𝛽 𝛽 = ( 𝛼 𝑦 ‒ 𝛼 𝑧 )∆𝑇 1 + 𝛼 𝑧 ∆𝑇 where and are, respectively, the circumferential and through-the-thickness 𝛼 𝑦 𝛼 𝑧 coefficient of thermal expansion (CTE). The circumferential CTE can be easily and accurately determined either experimentally [14] or analytically using Classical Laminate Theory from the stacking sequence, the stiffness constants and the lamina’s CTEs in orthotropic in-plane directions [15] (typically denoted as for the CTE in the direction of the fibres and in the direction 𝛼 1 𝛼 2 perpendicular to the fibres). In comparison, very few information can be found in the literature about the evaluation of , being in many cases assumed identical to the CTE 𝛼 𝑧 of the laminas in the through-the-thickness direction, , which, in turn, is assumed 𝛼 3 equal to [11]. 𝛼 2 Notice that, in fact, in a composite laminate with layers having different orientations, Classical Laminate Theory establishes that . For example, for a quasi-isotropic 𝛼 𝑧 ≠ 𝛼 3 laminate [16] it holds that , with being the Poisson ratio. 𝛼 𝑧 ≅(1 + 𝜈 23 ) 𝛼 2 𝜈 23 Moreover, many authors have reported lack of homogeneity in the corner part of a composite angle bracket, like fibre volume fraction gradients [17] or fibre misalignments [18,19], associated to the manufacturing procedure employed. Consequently, the apparent thermal and stiffness properties in this zone, which significantly contribute to the deformation of the component, may be considerably different to that of the planar parts of the component. Measurements of in composite angle brackets subjected to an homogeneous Δ 𝛽 / 𝛽 temperature change (from which the values of can be derived) were presented in 𝛼 𝑧 5 [2,20,21]. In these woks, the values of were determined measuring the deviation Δ 𝛽 / 𝛽 of a laser beam reflected in a mirror attached to one of the angle bracket faces. In the present paper a procedure for simultaneously measuring , and in 𝛼 𝑥 𝛼 𝑦 𝛼 𝑧 composite angle brackets is presented. The procedure uses digital image correlation (DIC) to determine the displacements in one of the angle bracket faces. The in-plane displacements measurements allow the in-plane CTEs, and , to be determined, 𝛼 𝑥 𝛼 𝑦 while the out-of-plane displacements enable the calculation of and, with the use of ∆𝛽 (1), determining . DIC has already been successfully applied to determine in-plane 𝛼 𝑧 CTEs in polymer films [22] and composites [23,24]. Notice that, with the procedure employed, and are measured in the planar part of 𝛼 𝑥 𝛼 𝑦 the angle bracket, while the measured is in fact an apparent which characterizes 𝛼 𝑧 𝛼 𝑧 the thermal distortion of the angle bracket corner. The apparent is evaluated using (1) 𝛼 𝑧 and assuming that in the corner and planar parts of the sample are identical, which 𝛼 𝑦 may not be the case in the actual material. However, for practical applications, the measured in-plane CTEs will accurately account for the change in size of the component, while the measured will accurately account for the change in shape. 𝛼 𝑧 2. Fundamentals Thermal strains of a planar orthotropic lamina, under a uniform temperature change 𝜖 𝑇 𝑖𝑗 , are given by ∆ 𝑇 , (2) 𝜖 𝑇 𝑖𝑗 = 𝛼 𝑖 𝛿 𝑖𝑗 ∆𝑇 where are the orthotropic directions (with being the direction of the fibres, 𝑖,𝑗 = 1,2,3 1 2 being the in-plane direction perpendicular to the fibres and 3 being the through-the- 6 thickness direction), is the Kronecker delta and are the CTEs in orthotropic 𝛿 𝑖𝑗 𝛼 𝑖 directions. In a typical planar composite laminate, thermal strains in each of the orthotropic laminas can be determined, in its orthotropic co-ordinate system, using (2). However, elastic strains should develop in order to achieve compatibility in all interfaces. In symmetric and balanced composite laminates, thermal strains of the laminate in a global coordinate system , can be approximately described as those of an equivalent 𝑥𝑦𝑧 homogeneous material. That is: , (3) 𝜖 𝑇 𝑘𝑙 = 𝛼 𝑘 𝛿 𝑘𝑙 ∆𝑇 where , and are the average CTEs in orthotropic directions, with being 𝑘,𝑙 = 𝑥,𝑦,𝑧 𝛼 𝑘 𝑥 the longitudinal in-plane direction, being the transverse in-plane direction and being 𝑦 𝑧 the through-the-thickness direction. Finally, in a curved symmetric and balanced composite laminate, having the shape of a hollow cylinder sector, see Fig. 1(c), with angle , strain compatibility requires both the 𝛽 presence of elastic strains and a change in shape. Consequently, in such situations, equation (1) is only approximately satisfied. A more precise, although more complicated, determination of the angular variation can be carried out analytically [13,25]. 3. Test campaign In order to obtain the CTEs of a composite lamina, two unidirectional L-shaped coupons, with stacking sequences and have been used. Additionally, in- [ 0° 8 ] [ 90° 8 ] plane equivalent CTE in a quasi-isotropic sample with stacking sequence has been determined. In all cases the direction of the fibres in the [0°, ‒ 45°,45°,90°] 𝑆 7 laminas follows the curvature of the sample, that is, it is parallel to the direction y 0° shown in Fig. 1(c). Samples were made of Hexcel AS4/8552 unidirectional prepreg laminas. A male L- shaped aluminum tool was used in the manufacturing of the samples, with a corner radius of . Hand lay-up was carried out and bending of the layers over the tool 5 mm corner was done layer by layer with the aid of nylon spatula and an air-heating device. To ensure a good compaction a first vacuum bag was used after the placement of the first four laminas. After completion of the lay-up a final vacuum was set and the material was cured in an autoclave, following the material manufacturer’s specifications [26]. Coupons were cut to their final dimensions with a diamond disc saw. Nominal dimensions of the samples are: width, thickness and 30 mm 1.6 mm 60 mm × 30 mm planar faces. The 3D stiffness matrix of an individual lamina of the material is required for the analytical calculations presented in section 4.2. Stiffness properties of the material, and stacking sequences employed are summarized in Table 1. In-plane stiffness properties , , and have been obtained following ASTM test standards [27,28]. 𝐸 11 𝐸 22 𝐺 12 𝜈 12 Transverse isotropy has been assumed in the composite laminas and, consequently, out of plane , and stiffness constants have been considered identical to , 𝐸 33 𝐺 13 𝜈 13 𝐸 22 𝐺 12 and . Finally, have been estimated from [29] and has been determined using 𝜈 12 𝜈 23 𝐺 23 . 𝐺 23 = 𝐸 23 /(2 + 2 𝜈 23 ) 8 Table 1. Stiffness properties of the material and stacking sequences of the samples. Stiffness properties Stacking sequences 𝐸 11 131.5 GPa 𝐺 12 4.82 GPa 𝜈 12 0.302 “ ” 0° [ 0° 8 ] 𝐸 22 9.23 GPa 𝐺 13 4.82 GPa 𝜈 13 0.302 “ ” 90° [ 90° 8 ] 𝐸 33 9.23 GPa 𝐺 23 3.30 GPa 𝜈 23 0.400 “QI” [0°, ‒ 45°,45°,90°] 𝑆 Samples have been heated inside a calibrated convection oven (Binder FED-115) with one of their planar faces resting horizontally while the other is facing the door of the oven, see Fig. 2. The oven has been modified to let the DIC system track the coupons, substituting its opaque door by a transparent glass, which has been tempered in order to allow it to withstand the high temperatures reached during the test and to reduce the heat loses as much as possible. Fig. 2. Experimental setup for heating of the samples and DIC measurements. Oven temperature has been increased up to and measurements have been taken at 120℃ intervals of approximately . Ambient temperature was . The temperature of 30℃ 26℃ one of the coupons has been monitored with a J-type thermocouple (placed over its 9 horizontal face) to verify that the temperature of the samples was stabilized prior to taking each measurement. The displacements of the coupons have been measured with a DIC system. Two cameras have been used in order to obtain the displacements in the three spatial directions. As can be seen in Fig. 2, a rectangular zone of the front 40 mm × 20 mm face of each sample has been speckled to let the DIC software track their movements. In order to measure the displacements with high accuracy, very high resolution cameras Teledyne Imaging Falcon 2 12M (4096 x 3072 @ 58 fps) have been used. The fourth sample that can be seen located more to the right in Fig. 2, is made of a different material and does not take part of this study. 4. Results and discussion As mentioned above, the DIC system employed allows us to determine simultaneously the in-plane and out of plane displacements of the speckled zone in the frontal face of the samples. In the following, displacement components are denoted by where the 𝑢 𝑃 𝑘 subscript indicates the component and the superscript refers to 𝑘 = 𝑥,𝑦,𝑧 𝑃 = 0, 90, QI the sample in which it is measured. The VIC-3D software employed to analyse the images is able to determine the in-plane strain components, which will be denoted as 𝜀 𝑃 𝑘𝑙 . Notice that direction in the frontal face is parallel to direction shown in 𝑘,𝑙 = 𝑥,𝑦 𝑥 𝑥 Fig. 1(c), direction is vertical and direction is normal to the face. 𝑦 𝑧 As detailed in the following subsections, the in-plane material’s CTEs ( and ) have 𝛼 1 𝛼 2 been obtained directly from the in-plane measurements in 0º and 90º coupons. The out- of-plane coefficient ( ) has been indirectly obtained from the measurements of the out𝛼 3 of-plane displacements in the same coupons. Average in-plane CTE of the quasi- 16 Table 4. Measurements of . ∆ 𝛽 𝑃 (𝑇) Temperature (31 ±0.5)°C (60 ±0.5)°C (91 ±0.5)°C (120 ±0.5)°C [·10−3] ∆𝛽 0 −36.9° ±0.5° −153.2° ±0.7° −355.1° ±0.6° −529.6° ±1.1° [·10−3] ∆𝛽 90 30.0° ±0.6° −42.6° ±0.9° −302.6° ±1.5° −438.3° ±1.8° [·10−3] ∆𝛽 𝑄𝐼 −46.7° ±0.6° −187.3° ±0.7° −479.2° ±0.8° −726.2° ±1.0° Measurements of the angular distortion suffered by the sample at each temperature have been obtained by least square fit of the individual measurements in each sample. Results obtained are summarized in Table 4. (a) (b) (c) Fig. 7. Determination of by best linear fit from : (a) sample, (b) ( 𝛼 𝑃 𝑧𝑧 ‒ 𝛼 𝑃 𝑦𝑦 ) ∆ 𝛽(𝑇) 0 90 sample, (c) sample. QI 17 Measurements of the angular distortion shown in Table 4 are plotted against temperature increment in Fig. 7(a), (b) and (c), corresponding to , and ∆𝑇 0 90 QI samples, respectively. As can be seen, in all cases, a reasonably linear dependency of ∆ with is observed, as defined by (5). Notice that is used in the vertical axis 𝛽 𝑃 ∆𝑇 ‒ ∆𝛽 in the plots shown in Fig. 7. The value of in each sample can be obtained from the slope of the least ( 𝛼 𝑃 𝑧𝑧 ‒ 𝛼 𝑃 𝑦𝑦 ) squares linear fit of the data shown in Table 4. Best fit lines are shown as discontinuous lines in Fig. 7. Considering that the initial angle is in all samples, the values of 𝛽 = 90° shown in Table 5 are obtained. ( 𝛼 𝑃 𝑧𝑧 ‒ 𝛼 𝑃 𝑦𝑦 ) Table 5. Measurements of and . ( 𝛼 𝑃 𝑧𝑧 ‒ 𝛼 𝑃 𝑦𝑦 ) 𝛼 𝑃 𝑧𝑧 𝛼 0 𝑧𝑧 ‒ 𝛼 0 𝑦𝑦 [/°C] 𝛼 90 𝑧𝑧 ‒ 𝛼 90 𝑦𝑦 [/°C] 𝛼 𝑄𝐼 𝑧𝑧 ‒ 𝛼 𝑄𝐼 𝑦𝑦 [/°C] 𝛼 0 𝑧𝑧 [/°C] 𝛼 90 𝑧𝑧 [/°C] 𝛼 𝑄𝐼 𝑧𝑧 [/°C] 61 ±5 48 ±16 83 ±9 60 ±5 85 ±16 85 ±9 Considering that in the curved zone of each sample is equal to the measured in 𝛼 𝑃 𝑦𝑦 𝛼 𝑃 𝑦𝑦 the planar faces (shown in the previous section), the values for in the curved zone of 𝛼 𝑃 𝑧𝑧 each sample shown in Table 5 are obtained. As can be seen, results for and are 𝛼 90 𝑧𝑧 𝛼 𝑄𝐼 𝑧𝑧 almost coincident and significantly higher than . Moreover, measured values of in 𝛼 0 𝑧𝑧 𝛼 𝑧 all cases are not consistent with the typical hypothesis that assumes . This fact 𝛼 2 = 𝛼 3 can be clearly seen comparing the actual measurements of in Fig. 7 with the ∆ 𝛽 𝑃 analytical predictions (shown as a thick line in the plots) obtained using in all samples the material properties shown in Table 1, the in-plane CTEs shown in Table 3 and 18 assuming . Analytical calculations have been carried out using the solution 𝛼 2 = 𝛼 3 presented in [25] and validated with finite element models with solid elements. 5. Concluding remarks A procedure for the simultaneous determination of in-plane, and , and through𝛼 𝑥 𝛼 𝑦 the-thickness, , CTEs in composite angle brackets has been presented. 𝛼 𝑧 To this end, L-shaped samples have been heated in an oven and DIC has been used to determine the in-plane strains and out-of-plane displacements in one of the faces of each sample. In-plane CTEs have been directly obtained from the best linear fit of the average in-plane strains measured at different temperatures. Through-the-thickness CTE have been indirectly obtained from the best linear fit of the angular distortion of the samples at different temperatures, which, in turn, have been obtained by best linear fit from the out-of-plane displacements. Results obtained for in-plane CTEs in unidirectional samples, with and [ 0° 8 ] [ 90° 8 ] stacking sequences, show a very good agreement between each other, and allow the inplane material’s CTEs and to be determined. Results obtained for the average in𝛼 1 𝛼 2 plane CTE in a quasi-isotropic sample, with stacking sequence, are [0°, ‒ 45°,45°,90°] 𝑆 also in a reasonably good agreement with the previously determined material’s in-plane CTEs. Results obtained for the through-the-thickness CTE in all samples are significantly higher than those predicted by the hypothesis of transversal isotropy typically carried out in composite laminas. Moreover, the through-the-thickness CTE obtained in the [ sample is substantially smaller than those obtained in the and 0° 8 ] [ 90° 8 ] samples. [0°, ‒ 45°,45°,90°] 𝑆 19 These disagreements were expected and are not associated to the technique used for the measurements, but to the fact that the angular distortion is caused by the anisotropic behaviour of the material in the corner of the angle bracket sample and that, in this zone, the fibres are wrinkled as a result of folding the prepregs during manufacturing. Different manufacturing procedures will produce different alterations to the material in the angle bracket corner in comparison with its planar parts: thickness changes, misalignments of the fibres, fibre volume fraction gradients, etc. Consequently, to properly characterize the thermal distortion of a component, an appropriate apparent value of the through-the-thickness CTE has to be determined. Aknowledgements This work was supported by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund [Projects MAT2015-71309-P and UNSE15- CE-3581, MINECO/FEDER, UE]. References [1] I. 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