Composites Science and Technology 97 (2014) 12–18
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Composites Science and Technology journal homepage: www.elsevier.com/locate/compscitech
Variable stiffness characteristics of embeddable multi-stable composites Andres F. Arrieta ⇑,1, Izabela K. Kuder 1, Tobias Waeber, Paolo Ermanni Laboratory of Composite Materials and Adaptive Structures, ETH Zurich, Leonhardstrasse 27, Zurich CH-8092, Switzerland
a r t i c l e
i n f o
Article history: Received 17 December 2013 Received in revised form 21 March 2014 Accepted 23 March 2014 Available online 1 April 2014 Keywords: A. Smart materials A. Flexible composites B. Non-linear behaviour C. Modelling
a b s t r a c t The possibility to achieve shape adaptation provides structural systems with the potential to adjust for optimal operation in a wide range of conditions. The formidable challenges posed by shape adaptation, and particularly in morphing applications, can be potentially addressed through the development of distributed compliance systems featuring highly directional structural properties. These characteristics can be further enhanced by embedding in such systems elements featuring variable stiffness. In this paper, a novel type of embeddable variable stiffness elements exploiting thermally-induced multi-stability in unsymmetrically laminated composites is presented. A tailored lay-up exhibiting spatially distirbuted stacking sequences is implemented to achieve multi-stability even when restricting two opposing edges. The difference between the structural responses leading exhibited by the multiple stable shapes of the designed composites are numerically investigated. The restoring force for each stable state is examined yielding a significant variability in the stiffness. Experimental specimens are manufactured and tested showing good agreement with the numerical results, validating the proposed variable stiffness implementation. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction The ability to operate optimally in diverse conditions has led to the idea of introducing the capability to adjust the properties of engineering systems in response to environmental changes, resulting in the field of adaptive structures [1]. A set of properties particularly difficult to be adapted, are those concerned with shape variation of structural elements. This problem is exemplified conformal change of the shape of aerospace structures, widely known in the engineering community as morphing. Morphing requires structures exhibiting highly directional properties to meet the conflicting requirements of load-carrying capabilities along the direction of the loads, high compliance in the directions of shape adaptation, while maintaining light-weight constructional characteristics [2]. Several challenges remain to be addressed for the implementation of this promising technology. Distributed compliance structural systems are a promising solution to the formidable challenges presented by shape adaptation requirements, particularly those posed by morphing applications [3,4]. A new trend to significantly augment the directionality of the structural properties of such systems is to embed elements capable of varying their stiffness [5]. The ability to change the structural response of certain ⇑ Corresponding author. Tel.: +41 0446322675; fax: +41 0446331125. 1
E-mail address:
[email protected] (A.F. Arrieta). These authors contributed equally to this work.
http://dx.doi.org/10.1016/j.compscitech.2014.03.017 0266-3538/Ó 2014 Elsevier Ltd. All rights reserved.
components in distributed compliance systems allows for a higher degree of decoupling between the compliance and stiffness requirements. Composite laminates exhibiting multiple statically stable shapes are good candidates to serve as variable stiffness elements embedded within distributed compliance structures [6–8]. Indeed, a significantly different directional structural response characterises each equilibrium configuration. The capability of adopting a number of statically stable states has brought these structures into the focus of extensive research in recent years. Several types of systems including biological [9], and synthetic [10– 12] exploit this phenomenon to achieve unattainable performances with conventional structures. Multi-stability in composite materials arises from an induced stress field that can be realised by several mechanisms, including unsymmetrical lamination [13], tailored lay-up [14], pre-stressed cylinders [10], fibre pre-stressing [15] and thickness variation [16]. Amongst multi-stable structures, multi-stable composite laminates have been considered for morphing applications due to the large deformations exhibited when changing between stable configurations [17–20]. Although such structures have shown promising results, the discrete nature of the achievable equilibrium shapes restricts the range of applicability of multi-stable components as a means to obtain large deflections in load carrying elements. The possibility offered by multistable components to achieve different structural properties for each of the attainable stable configurations, allowing for the realisation of load-carrying variable stiffness elements, has not been
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thus far explored. Taking advantage these characteristic, multi-stable elements can be designed to exhibit variable stiffness properties arising from the different structural response of the stable configurations. In this paper, a novel class of variable stiffness elements based on exploiting the significantly different structural characteristics associated to the stable configurations of multi-stable composites is introduced. The components herein presented are specifically designed to be embeddable into larger structures, thus providing the complete system with passive variable stiffness capabilities. In particular, the designed multi-stable laminates allow for integration without conventional fixation parts, which is particularly suitable to enhance the structural directional response of distributed compliance systems through the added capability of stiffness adaptation. A tailored lay-out featuring several sections with different stacking sequences is designed achieving both a significant difference in the structural response, and the capability to embed the components into distributed compliance systems. Thermal stresses induced during the cool-down process arising from sections of unsymmetric lamination are exploited to realise the desired multi-stable behaviour. The Finite Element Method (FEM) is employed to simulate and design the variability of the stiffness characteristics associated to each stable configuration. The room temperature shapes of the designed multi-stable components are calculated with the FEM showing good agreement with manufactured specimens. The structural response of the stable configurations is studied for relevant load cases both numerically and experimentally. Finite Element (FE) simulations demonstrate significant variability in the stiffnesses associated to each stable configuration. Tests on experimental specimens show a close match with the FE results, validating the numerical analyses. The realisation of purely elastic, passive, embeddable load carrying variable stiffness elements can be potentially used to significantly alter the structural response of a load-carrying system in a simple and robust fashion. The variable stiffness offered by the presented multi-stable components can be exploited to enhance the directionality of the structural characteristics of systems for which structural efficiency is of paramount importance, as for instance in morphing structures. Consequently, a reduction of the actuation requirements to obtain large deflections, or altogether passive morphing exploiting the external forces on the structure to induce desired shape adaptation, can be achieved. Furthermore, as multistability in these structures is obtained from passive means, the capability to achieve significant variation in the structural response can be realised in a robust manner. Moreover, the stiffness variability can be implemented on applications where structural hysteresis cycles are required, as for instance in energy dissipation applications.
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ever, the integration of such laminates through clamping two ends into a structural system was not achieved. Simply restraining the curved end of such laminates results in a moment being applied restricting the deflection required to achieve the second configuration, rendering it mono-stable as schematically shown in Fig. 1. Therefore, a smooth transition reducing the curvature of the edges to be clamped is necessary to maintain multi-stability when embedding the laminates. This is accomplished by designing compliant edges allowing for the introduction of boundary forces and moments to the multi-stable laminates. Starting from a rectangular bi-stable section, appropriate elastic boundaries allowing for a smooth reduction of the curvatures in both stable states are added to achieve the objective of the component embeddability. Adding symmetrically laminated sections to the short ends (parallel to the y-axis) of the rectangular unsymmetric sections reduces smoothly the curvature along the short edges. However, these sections impose high bending moments, opposing the straightening of the edges thus impeding the adoption of stable state 2 as shown in Fig. 1. Such a solution would result in a dramatically reduced parameter space for which such a lay-out would exhibit bi-stability. In fact, only a highly directional symmetric part exhibiting compliance in the y-direction of the short edge, while maintaining sufficient stiffness to smoothen the curvature in the x-direction, could result in a multi-stable element. However, the mechanical properties of such a solution would not be satisfactory for load carrying purposes. An alternative for achieving the design objectives to simply adding symmetric sections on either side of the main rectangular unsymmetric region is achieved by inserting sections having spatially distributed compliance on either side of the central unsymmetric region to the lay-out. These are designed to show curvatures in y-direction opposing the deformation in the (longitudinal) x-direction imposed by the main rectangular unsymmetric part, thus smoothening adequately the behaviour of the edges at room temperature. An example of the introduced concept is the seven-section lay-out shown in Fig. 2. Spatially distributed transi-
2. Lay-out design The lay-out configurations for the multi-stable embeddable variable stiffness elements are designed mainly to fulfil two desired characteristics. First, the resulting structural behaviour of the obtained composite components should be so that significant stiffness variability is attained. Equally important, the resulting lay-out should allow for the designed components to be integrated into structural systems, without compromising the desired variable stiffness characteristics. Furthermore, the integration should be realised such that a distributed compliance structure, requiring no conventional fixation components such as rivets or bolts, is achieved. Previously, multi-stable laminates in a cantilevered configuration, i.e. with one edge completely clamped, have been obtained through a symmetric-unsymmetric stacking lay-out [7,21]. How-
Fig. 1. Effect of clamping the free end of a cantilevered bi-stable laminate configuration. Simply adding an additional symmetrically laminated region results in a mono-stable laminate.
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tion regions on either side of the central unsymmetric section s4 , comprised by sections s2 and s3 , and s5 and s6 , respectively, are introduced to the lay-out allowing for obtaining an embeddable multi-stable laminate. The results for an FE model simulating a seven-section lay-out show two stable configuration: straight (state 1) and curved (state 2) shapes as can be seen in Fig. 3. 3. Static shapes To study the capabilities of multi-stable embeddable laminates for realising purely elastic variable stiffness components, first the existence of multi-stability for a given lay-out design needs to be investigated. For the type of multi-stable laminates herein studied, the multi-stability is mainly controlled by thermally induced stresses developed during the cool-down process from the elevated curing temperature due to unsymmetric lamination [13]. Hence, the cool-down process for the designed lay-out featuring sections with different stacking sequences, i.e. the specific lay-up of the region, is simulated. Specifically, finite element analyses (FEA) including nonlinear effects arising from large displacements are carried out in ABAQUSÒ . Shell S4R elements are used, the sides of which are chosen to be between 2.5 mm and 7.5 mm, with the aspect ratio being kept between 0.5 and 2. The width and length are divided at least into 20 and 92 finite elements, respectively. To avoid rigid body displacement the edges of the laminate during cool-down are restricted from translation. The uniform temperature field on the laminate during cool down is applied to the FE model by setting a ramp-like input from 140 to 0 °C, simulating a curing temperature of 160 °C and a room temperature of 20 °C. The material properties used for the numerical simulations and experimental tests are given in Table 1. The geometrical properties of the numerically and experimentally studied specimens in this work are given in Table 2. The cool-down shapes of specimen 1 for the straight and curved states are shown in Fig. 4(a and b), respectively. The effect of the mismatch between the coefficient of thermal expansion (CTE) nominal values in the longitudinal and transverse directions, a11 and a22 respectively, leading to the appearance of multi-stability in unsymmetrically laminated composites is diminished over time due to moisture pickup after room temperature is reached. The stress field generated by moisture absorption in composite materials is a well-known phenomenon, resulting from hygrothermal effects, leading to induced stresses analogous to those arising from thermal expansion [22,23]. In the particular case of bi-stable laminates, the stresses induced by moisture absorption reduce the field created by the thermal mismatch between fibres and matrix, ultimately lowering the out-of-plane static deflection [24]. In this work, this effect is modelled by identifying the effective thermal expansion coefficient from the resulting stable shapes from the manufactured specimens. This approach is more practical than including hygrothermal terms in the FE formulation, given to the difficulty to obtain reliable data [23]. Furthermore, measuring hygrothermal coefficients is outside the scope of this work; details for a procedure to obtained these values are detailed in Ref. [24]. Moreover, including hygrothermal terms in the FE calculations
precisely results in a reduction of the effective CTE of the laminate. Notice that such a parameter identification method does not compromise the accuracy of the modelling as only one parameter is varied, while all other quantities involved in the calculations are taken from material data provided in specification sheets of the prepreg manufacturer. The reduction in the resulting out-of-plane displacement as time progresses after curing due to the described environmental effects are signalled the black arrows in Fig. 4(a and b). The shapes of the laminates stabilise after 16 days from the time of manufacturing showing no further reduction in the out-of-plane displacement. The identification procedure for numerically simulating the adopted static shapes as a function of the elapsed time after curing is performed by lowering the value of the theoretical transverse CTE, a22 . As can be seen from Fig. 4(a and b), a very good match between numerical and experimental results is obtained by following this procedure. It is also interesting to notice from Fig. 4(a) that the straight configuration exhibits an angle, hereafter referred as the edge angle /, with respect to the x-axis described by the transition sections and the central flat part of the lay-out. This angle is a critical parameter to be controlled to maintain the multi-stability when clamping the opposing short ends. In the curved configuration, a similar angle is described by the edges with respect to the x-axis, however it is less critical due to the small value obtained, as seen from Fig. 4(b). In the next section two specimens with different lay-out configurations showing a reduced angle / for easier integration into a structural system are studied in detail to investigate the stiffness variability offered by the introduced multi-stable composite elements. 4. Structural variable stiffness characteristics To test the variability in the structural response offered by the designed embeddable multi-stable elements, numerical and experimental displacement controlled compression tests are performed for the straight and curved stable configurations. In this study, the focus is placed on investigating the variability of longitudinal stiffness exhibited by the design composite elements as these are envisioned to work mostly supporting in-plane forces when integrated into a larger structure. However, it should be noted that, although not shown here, considerable variability is also obtained for the bending stiffness of the designed variable stiffness laminates. The load case employed for the numerical calculations testing the axial variability of the longitudinal stiffness is schematically shown in Fig. 5, for which the following boundary conditions are set. After the cool-down is finished, the x-coordinates of the nodes at the left edge of the laminates are fixed at their position simulating the effect of a clamp. The axial force is introduced to the laminate following a displacement controlled procedure, using the x-direction deflection, U 1 , as parameter. As the displacement is changed, the increments in the reaction forces in x-direction, F a , of the nodes at left edge are stored and plotted. The test rig used for the experimental axial loading study is shown in Fig. 6. A Zwick/Roell Z005 testing machine combined
Fig. 2. Example of tailored lay-out exhibiting spatially distributed fibre orientation resulting in embeddable variable stiffness multi-stable elements.
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Straight state
Curved state
U, Magnitude
U, Magnitude
+8.36e+00 +7.67e+00 +6.97e+00 +6.27e+00 +5.57e+00 +4.88e+00 +4.18e+00 +3.48e+00 +2.79e+00 +2.09e+00 +1.39e+00 +6.97e−01 +0.00e+00
+4.44e+01 +4.07e+01 +3.70e+01 +3.33e+01 +2.96e+01 +2.59e+01 +2.22e+01 +1.85e+01 +1.48e+01 +1.11e+01 +7.40e+00 +3.70e+00 +0.00e+00
Z
Z Y
Y
X
X
Fig. 3. Example of stable shapes obtained with the introduced multi-region lay-out.
Table 1 Material properties for a typical ply of carbon fibre reinforced polymer prepreg used to manufacture the multi-stable experimental specimens. Nominal ply thickness of the used prepreg 0.155 mm.
Fibre vol. (%)
E11 (GPa)
E11 (GPa)
G12 (GPa)
m12 (–)
q
60
161
10
4.4
0.3
1570
kg m3
a11 ðK1 Þ
a22 ðK1 Þ
1 a4d Þ 22 ðK
1 a16d Þ 22 ðK
1.8E8
3E5
2.5E5
2.25E5
Table 2 Geometric properties of the sections composing the introduced lay-out for the studied specimens. Refer to Fig. 2 for the schematic representation of the given parameters. Specimen
L (mm)
w (mm)
L1 (mm)
L2 (mm)
L3 (mm)
L4 (mm)
L5 (mm)
L6 (mm)
L7 (mm)
1 2 3
300 270 220
64 64 64
55 55 31
30 15 15
5.0 5.0 5.0
120 120 120
5.0 5.0 5.0
30 15 15
55 55 31
with a 100 N force transducer capable of maintaining high accuracy for loads of less than 1 N is used for the measurements. The test is conducted by controlling the displacement with a test speed of 20 mm/min. The axial tests are conducted for specimens 2 and 3,
Fig. 4. Comparison between experimental and simulated shapes for embeddable multi-stable variable stiffness components, specimen 1 (see Table 2. (a) State 1 and (b) State 2. The black arrow indicates the direction of growing time elapsed from curing, and the resulting decrease in the thermally induced out-of-plane displacement of the stable configuration shapes due to the reduction in the effective CTE.
for which the corresponding geometrical parameters of the lay-out are provided in Table 2. The employed experimental rig is designed to replicate load case used in the numerical calculations described above. The static nonlinear FE results for specimen 2 are in good agreement with the obtained experimental data, as can be seen from in Fig. 7. In this case the experiments are carried out 4 days after the curing of the laminate, hence a4d 22 is used for the FE simulations. The restoring force experimentally obtained from the axial test for both stable configurations is closely matched by the FE results. The conducted FE simulations also capture quite accurately the maximum force, approximately 27 N, achieved before local buckling is observed, as shown in the added picture in Fig. 7. As the axial compression proceeds, the force carried by the multi-stable laminate drops until a snap-through to the curved state occurs (not shown in graph). The deformation exhibited by the specimen as it deflects is also closely match by the numerical calculations as can be seen from the inserted pictures in Fig. 7. The agreement between the numerical and experimental postbuckling behaviour still shows reasonable agreement. After the maximum force is reached in the tests for specimen 2, cracking noise developed. This suggests that for the chosen lay-out design, the maximum load developed close to buckling initiates damage in the multi-stable composite laminate. To address this, specimen 3 with improved geometrical properties is tested. In particular, specimen 3 exhibits a smaller angle between the transition and the main unsymmetric sections allowing for easier integration into a wider structure. The tests for specimen 3 are performed 16 days after curing, therefore a16d 22 is used for the FE simulations. The axial stiffness variability is studied following the same procedure as for specimen 2, both numerically and experimentally. Once more the numerical and experimental results for the axial loading tests show very good agreement. The
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Load
Load
Tailored transition region
Tailored transition region Unsymmetric central lamination
Unsymmetric central lamination
Fig. 5. Schematic representation of the axial loading process applied to the multi-stable components on each stable state.
Universal machine
Load cell
Multi-stable specimen
Clamping jaws
Fig. 6. Test rig used for axial loading tests of multi-stable variable stiffness laminates.
obtained results show a significant variability in the stiffness when comparing the response exhibited by the stable states, as can be seen in Fig. 8. The deformation shapes adopted by the laminate as the longitudinal stiffness tests are conducted are included in
Fig. 8. A similar behaviour as the one exhibited by specimen 2 can by observed. This features a load-bearing straight configuration for which a significant maximum load is achieved, 54 N, and compliant curved configuration for which large deflections with very low force are obtained. In the case of specimen 3 no cracking behaviour develops around the maximum load shown by the straight configuration. Furthermore, as the compression test progressed the composite element snapped-through to the curved state. The shortened overall length of the specimen, see Table 2, results in a more desirable response than that obtained for specimen 2 as no intermediate local buckling of the straight state occurs before snapping to the curved configuration. In addition to this, the maximum load carried by the straight state is significantly higher than that for specimen 2, 27 N compared to 54 N, as is expected from a shorted beam-like structure in compression. The stiffness variability of the designed multi-stable elements can be observed by conducting a linear regression about a small deflection range, 5 and 0.5 mm for the curved and straight states, respectively. For specimen 2, the slopes obtained from the linear regression lines associated to each stable configuration for the studied component result in a ratio of 47.43/0.60 = 79.1 when changing from the straight to the curved state, which amounts to a nearly 80-fold stiffness variation for the small deflection range. Studying the complete range for the restoring force shown in Fig. 7, it can be seen that the stiffness variation is amplified for
Fig. 7. Numerical and experimental restoring force plot of specimen 2 obtained from the axial loading test described in Fig. 6 for both stable configurations. The linear regression lines obtained for a range of small displacements about the stable equilibrium shapes is added to highlight the obtained stiffness variability.
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Fig. 8. Numerical and experimental restoring force plot of specimen 3 obtained from the axial loading test described in Fig. 6 for both stable configurations. The linear regression lines obtained for a range of small displacements about the stable equilibrium shapes is added to highlight the obtained stiffness variability.
larger deflections due to the softening of the axial response exhibited by the curved state. Similar large structural response variability to the one observed for specimen 2 is obtained for the longitudinal stiffness exhibited by specimen 3. In this case, the small deflection stiffness ratio is approximately 95/1. The compliant characteristics of the curved state are also observed for specimen 3, as the resulting difference in the stiffness increases even further with larger in-plane deflections. It is worth noticing from the restoring force graph that the multi-stable embeddable laminate exhibits other desirable characteristics. Namely, the straight state is capable of carrying significant amounts of loads before failure, and the curved state provides the possibility of achieving large deflections of the component with a relatively low actuation requirement. These characteristics are interrelated to the already discussed stiffness variability, thus these can be further tailored for specific purposes via optimisation techniques.
shapes for specimens 2 and 3, respectively, obtained from axial compression load case tests. On-going work focusing on the optimisation of the difference in stiffness between the states and the structural performance of the components when integrated into structural systems show encouraging preliminary results. Finally, multi-stability can be achieved by other means to induce a residual stress field, such as with fibre pre-stressing procedures, which could potentially further increase the design space for the achievable variability of the embeddable variable stiffness components. Acknowledgements The authors would like to thank the support of the ETH Research Commission and the Marie Curie Actions Cofund Program; Dr. A.F. Arrieta is funded through an ETH Postdoctoral Fellowship. References
5. Conclusions Multi-stable composite laminates exhibiting variable stiffness characteristics obtained by introducing a novel lay-out are presented. The innovative design proposed combines different sections of symmetric and unsymmetric lamination specifically designed to allow for clamping two opposite edges. This boundary constraint capability permits embedding the devised variable stiffness elements into a wider structure. The purely elastic nature of the introduced components enables simple implementation as no external devices are required to obtain the stiffness variability of the laminates. Furthermore, the purposely designed lay-out facilitates integration without the need for conventional fixation devices such as rivets or bolts. This monolithic integration potential is particularly important when embedding the presented multi-stable components into distributed compliance structures. In view of the composite materials used to obtain the passive stiffness variability, the elimination of the necessity of drilling holes in the components for fixation represents an additional advantage over conventional solutions, which introduce discontinuities and thus increased failure risk of the laminates. The available stiffness variation exhibited by the clamped configurations is studied both numerically and experimentally. Very good agreement between numerical and experimental results for the stable configuration shapes of the designed variable stiffness laminates is obtained. The results show significant stiffness variability: 80- and 95-fold change between the stiffness in the straight and curved stable
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