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Analysis of morphing, multi stable structures actuated by piezoelectric patches
Available online at www.sciencedirect.com
Computers and Structures 86 (2008) 347–356 www.elsevier.com/locate/compstruc
Analysis of morphing, multi stable structures actuated by piezoelectric patches Pedro Portela b
a,*
, Pedro Camanho a, Paul Weaver b, Ian Bond
b
a Faculdade de Engenharia da Universidade do Porto, Rua Roberto Frias s/n 4200-465 Porto, Portugal Department of Aerospace Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, United Kingdom
Available online 11 April 2007
Abstract In this article a novel morphing structure concept is studied using non-linear Finite Element Analysis (FEA). Bi-stable asymmetrical laminates can be snapped between two geometries through a buckling mechanism that is activated by an applied load. A piezoelectric Macro-Fibre Composite (MFC) actuator was chosen to provide this activation load. Bi-stable structures will maintain a given geometrical state without the need for a constant actuation force. FEA was used to predict the two stable geometries, to understand the buckling mechanism and to evaluate the feasibility of using MFC actuators for switching between states. Environmental effects like moisture absorption were also included in the analysis. 2007 Elsevier Ltd. All rights reserved. Keywords: Morphing structures; Asymmetrical laminates; Snap-through; MFC; Morphing; Multistable plates; Composites; Piezoelectric; Actuation; FE modelling
1. Introduction The expression ‘‘Morphing Structures’’ is a very broad and much underdefined subject. In this text a morphing structure will be defined as one capable of macroscopic geometric changes in order to better adapt to radically different environmental conditions. Because of the resources available in the aerospace industry and because of the clear advantages that these technologies can bring, morphing structures applications are currently under study, for example, for airplane wing geometry changes, replacement of mechanically driven control surfaces, helicopter blade control, and reliable actuators for space missions. If a morphing aircraft is defined merely as an aircraft that changes its configuration in-flight, it can be seen that a morphing aircraft is not a new concept. Extending flaps, elevators, ailerons and wing twisting can be considered geometric changes and can be technically termed as morphing.
*
Corresponding author. E-mail address: [email protected] (P. Portela).
0045-7949/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2007.01.032
However, these changes are either necessary enablers for controlled flight or contributors to the improved aerodynamic performances of an aircraft. As a result, these technologies do not necessarily allow an aircraft to perform different types of mission tasks [1]. Recently Schultz et al. [2] proposed a novel morphing concept based on asymmetrical carbon fibre reinforced plastic (CFRP) laminates. If a composite laminate does not have a symmetrical stacking sequence, thermally induced stresses develop during the curing process and cause an out of plane curvature. This curvature can be tailored by adjusting the stacking sequence and ply thickness. Depending on the plate geometry the thermally induced stresses may cause the laminate to have more than one stable configuration or shape. Morphing can be obtained by alternating between these states. The main advantage of using these structures is that they will hold one of the stable shapes without the need of a force holding them in that position. Hence, they can carry loads in both states as long as they do not reach the critical snap-through load. Schultz [2,3] used piezoelectric patches bonded to the surface of the laminate to induce snapthrough with a shear force. This type of device uses both
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active and composite material technologies in addition to interdigitated electrodes. The Macro Fibre Composite (MFC) actuator was developed by NASA Langley Research Centre and is currently commercially available through the company Smart-Material, Inc. This active element has uniaxially aligned piezoelectric fibres surrounded by polymer matrix. The fibres specifically have a rectangular cross-section [4]. Schultz successfully produced a morphing laminate based on the MFC actuator and obtained good predictions of the snap-through loads for a patch bonded in the middle of the plate. Dano [5] used Shape Memory Alloys (SMA), wires that, when heated, applied a bending moment to the plate. Hufenbach et al. [6,7] developed optimization algorithms to dimension such laminates as well as producing a SMA based morphing, bi-stable laminate. Both Schultz and Dano were concerned with a semi-analytical approach to predict the two stable shapes of such laminates as well as the required snap-through loads. There has been considerable amount of work done in the prediction of room temperature shapes of asymmetrical laminates [8,9]. Most of this work is based on semianalytical Rayleigh–Ritz methods, although non-linear Finite Element (FE) analysis has also been used successfully. In the present paper, non-linear FE analysis is used for several purposes. The first one is to reproduce the cured shapes of the asymmetrical laminates. This is not a new approach for this problem as FE predictions of asymmetrical laminates’ shapes are documented for example in [8]. The cure of [0/90], [0/45], [45/45] square laminate was simulated and their shapes and snap-through behaviour were compared to experimental data. This first task was useful to understand the snap-through phenomenon. It was also necessary to confirm the analysis procedures and simplifying assumptions typical of FE analysis (i.e. boundary conditions, applied loads and the material properties). The second purpose of the FE analysis was to simulate the combined laminate/actuator system. The model is helpful in predicting the changes in the laminate’s stiffness due to the presence of the actuator. Having accomplished this, the model also predicts the voltage required to induce the snap-through effect of the laminates. 2. Modelling the bi-stable laminate To determine the post cure shape of the laminate, a square plate with side length of 150 mm was modelled. The plate geometry and boundary conditions are depicted in Fig. 1. The central node was clamped. Although the plates’ nominal length was 150 mm, it was necessary to include some geometric imperfections into the FE model. The plate’s sketch is not a perfect square but rather a rectangle with side lengths of 149 mm by 151 mm. Using perfectly square plates in the FE model will make the analysis converge to unstable saddle shapes instead of stable cylindrical ones. This is reported by Gigliotti et al. [10]. The reason why this happens is still not clear. However, a hypothesis
Fig. 1. FE model boundary conditions.
was formulated that the inertia forces played a role in the curvature development. If instead of an implicit method an explicit method or an implicit direct-integration dynamic method is used to predict the cured shapes, the perfectly square plate will converge into a stable cylinder. An explicit model was created and verified this hypothesis [11]. All the models presented here are based on implicit methods including geometric imperfections. The reason for this choice is that the time it takes to run an explicit analysis compared to an implicit one and the accurate results that the implicit method provides regardless of the geometric imperfections introduced. Two materials were used for the initial tests and for the analysis: AS4-8552 and T800-2020 carbon fibre/epoxy pre-preg. The mechanical properties of these two pre-pregs used are listed in Table 1. Although the T800 fibre is stiffer then the AS4, the prepreg ply thickness is roughly 60% of the AS4. Since the snap-through phenomenon is achieved by applying a bending moment, it is the flexural stiffness that plays a key role. The stiffness matrix is computed by ½D ¼ P2 bending k 3 1 ½Q ðz z3k1 Þ. Small changes in ply thickness have k k¼1 3 significant effects in flexure due to the cubic dependence of ply thickness (zk). Hence, a 60% change in ply thickness will have a significant effect in the laminate’s bending stiffness. General purpose composite shell elements were used (type S4R in Abaqus). The meshes have approximately 900 elements. An initial temperature field of 180 C was imposed on the elements before starting the analysis. This represents the cure cycle’s maximum temperature which is taken as the stress-free temperature as well. Step one is Table 1 Mechanical properties of the two different pre-pregs Property
Value (AS4-8552)
Value (T800-2020)
E1 E2 m12 G12 G13 G23 a1 a2 tply
135 GPa 9.5 GPa 0.3 5 GPa 7.17 GPa 3.97 GPa 2 108 = C 3 105 = C 0.255 mm
294 GPa 9.5 GPa 0.3 5 GPa 7.17 GPa 3.97 GPa 2 108 = C 2:25 105 = C 0.180 mm
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Fig. 2. Representation of the analysis steps used.
Fig. 3. Comparison between FE shapes and those of the manufactured laminates.
Table 2 Measured and calculated principal curvature
jy
Measured (1/m)
Finite element (1/m)
Error (%)
6.51
6.65
2.1
a simple *STATIC step. Large deformation theory was used (*NLGEOM) to calculate the plate’s deformation due to a temperature change of DT ¼ 160 C, see Fig. 2. Fig. 3 pictures a qualitatively comparison between shapes obtained in the FE models and the ones observed in the manufactured laminates. Table 2 compares the curvatures measured in a [0/90] laminate to the ones obtained in the FE model. The curvature measurements were made five days after manufacturing and according to the procedure defined in [2]. It was not possible to adapt this procedure to measure curvatures in [45/45] and [0/45] laminates. Therefore, only the results for the [0/90] laminate are shown. Fig. 4 shows the contour plots and average values of the curvatures predicted by FE. The material used was the T8002020 pre-preg. 2.1. Environmental effects Composite materials absorb moisture from the environment in which they are immersed (air, water or any other fluid). Moisture absorption creates a stress field not unlike the one created by thermal stresses. In this particular application, the effect of the hygroscopic stress field is to relax
the existing residual stress field and thus reduce the plates’ curvature. This reduction in curvature, and consequently in the peak snap-through load, was included in the FE model. Abaqus does not include this effect in its constitutive models. In order to overcome this limitation a modification of the thermal expansion coefficients is required to account for hygroscopic effects. Using Voigt notation to represent the several tensors, the hygrothermoelastic constitutive equations are developed assuming the effective strain is the superposition of the three strains acting on ply k: k
k
k
feg ¼ fer g þ feT g þ feH g
k
ð1Þ
where fer g; feT g; feH g are the mechanical, thermal and hygroscopic strains. Assuming plane stress conditions and neglecting the mechanical strains: fegk ¼ fagk DT þ fbgk DM ¼ k k DM k DT ¼ fa g DT fag þ fbg DT
ð2Þ ð3Þ
where fagk are the coefficients of thermal expansion in the k ply coordinate system; fbg are the coefficients of moisture k expansion in the ply coordinate system; fa g are the equivalent moisture expansion coefficients accounting for the effects of both thermal and hygroscopic residual strains; DT is the applied temperature field; DM is the percent weight gained through moisture absorption. Assuming that DT < 0 and DM > 0 to force a coherent behaviour, the components of the a vector are:
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P. Portela et al. / Computers and Structures 86 (2008) 347–356 Table 3 Typical values for a and b [13]
T300/1034 AS/3501-5 T300/5208
a
b
0.017 0.018 0.015
1 1 1
where s is the thickness of the laminate when it is exposed on two sides to the same environment and Dx is the diffusivity of the material in the direction normal to the surface, obtained as [13]: ¼ cexpðT Þ Davg x d
Fig. 4. Contour plots of the curvatures predicted by FE.
8 9k 8 9k 8 9k > > > = = = DM < a1 > < a1 > < b1 > a2 ¼ a2 þ b2 > > > ; ; ; DT : > : > : > 0 0 0
ð4Þ
Typical values for b is 0.0 in the fibre direction and 0.005 1 in the transverse direction [12]. The percent weight gain DM is defined as DM ¼
Weight of moist material Weight of dry material Weight of dry material ð5Þ 100
The maximum moisture content, Mm, varies with relative humidity of the environment, the nature of the fluid, the matrix material and geometry. Thinner laminates reach a higher maximum moisture content faster than thicker ones. Springer [13] identifies the relation between maximum moisture content and relative humidity of the air, / as: M m ¼ a/b
ð6Þ
where a and b are constants to be determined experimentally. Springer [13] specifies the parameters a and b for three different materials exposed to humid air, as shown in Table 3. The time required for a composite material to attain 99.9% of its maximum possible moisture content is given by 0:67s2 tm ¼ Dx
ð7Þ
ð8Þ
The constants c and d depend on the material and must be determined experimentally and T is the temperature in Kelvin. Springer [13] provides experimental values for c and d. This data is combined into a single table with that of Table 3 to calculate the maximum moisture content and the time required to reach equilibrium for the three different materials (see Table 4). Table 5 shows that the maximum moisture content does not vary too much. However, the time it takes to reach equilibrium is significantly different. For the purpose of the analysis, and because there is no available data relative to the material, a value of M m ¼ 0:6 was chosen. The a vector is 9 9 8 9 8 8 8 8 > > > = 0:6 > <0 = > = < 2 10 = C > < 2 10 = C 5 6 ¼ 3:750 10 = C 2:25 10 = C 0:005 > > > > > ; 160 > : ; : : ; 0 2:25 105 = C 2:25 105 = C ð9Þ With this data inserted into the FE model the change in curvature due to moisture absorption can be determined by calculating the curvature of the moist material. The following table lists those values for the [0/90] 150 mm 150 mm laminate. The true measurement of moisture effects will become clearer in the next section, where the snap-through loads are calculated Table 6.
Table 4 Typical values for c and d [13]
T300/1034 AS/3501-5 T300/5208
c
d
0.83 6.51 0.58
5219 5722 5113
Table 5 Time it takes for the different materials to reach the maximum moisture content [13]
T300/1034 AS/3501-5 T300/5208
t (days)
Mm
66 47 70
0.68 0.72 0.6
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Table 6 Difference in principal curvatures after moisture absorption [13] 1
j1 ðm Þ j2 ðm1 Þ j12 ðm1 Þ
Dry
Moist
0 6.65 0.02
0 2.06 0.05
2.2. Modelling the snap-through Recalling Fig. 2, the snap-through step naturally followed the curvature development step. The elements’ definition and meshes are, therefore, the same. Again, the purpose of this step was to understand the buckling mechanism and calculate the critical loads. To do this Riks method was used [14]. Fig. 5 represents the FE plots of the two stable states for the different lay-ups. Fig. 2 shows the loading applied to the FE model during this step. Two concentrated forces were applied to the middle of the edges. The total reaction force can be monitored in the clamped node. The force initially increases linearly until large deformations become significant and a limit point is reached (critical load). This is the required force to induce transition. After this point the snap-through buckling will occur and the force can be removed. Potter et al. [15] provide experimental measurements of the load–displacement curve of the snap-through of a 150 mm · 150 mm square [0/90] laminate, Fig. 6. The material used was AS4/8552 pre-preg. The load was applied in the middle of the plate which was supported on a rigid table. The displacement measured is the displacement at this point. The FE curves were obtained for a plate in a dry state in a moistened state. The moisture content was considered to be 0.2%. The displacement of a point in the middle of the plate’s edge was used. Fig. 7 shows the variation of the applied load with the step time. The simulation results are in good agreement with the experimental results. In fact, the analysis was able to capture the secondary bifurcation in the unloading portion of the plot and highlighted in Fig. 7. This corresponds to a point where part of the plate buckles, while the other remains unchanged. Thus, this method is adequate to simulate the snap-through effect both qualitatively and quantitatively. Using the stacking sequence [0/90] three other
Fig. 6. Measured force [15] versus displacement.
Fig. 7. Applied force versus step time. Laminate is in the dry state.
square models with nominal lengths of 200, 250 and 300 mm were analyzed together with a [0/45] one. The critical force was measured experimentally and compared with the ones found in the FE models. A summary of these results can be seen in Table 7. Finally, the influence of the moisture content was analyzed. Using the procedure described in the previous section, the FE model was changed to reflect the hygroscopic effects. The critical load was calculated for a ½0 =90 T , 150 mm · 150 mm AS4 laminate. The following table compares both measured and FE results (see Table 8).
Table 7 Comparison of measured versus FE calculated critical loads
150 mm, 150 mm, 200 mm, 300 mm, 120 mm,
[0/90], [0/90], [0/90], [0/90], [0/45],
AS4 T800 T800 T800 T800
FE (N)
Measurement (N)
Error (%)
8.1 2.9 2.3 3.0 1.2
7.9 2.4 2.3 3.1 1.9
2.2 20.4 2.8 2.3 37.3
Fig. 5. FE plots of the two stables states for the three different laminates.
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Table 8 Critical loads in dry and moistened plates
Measured FE analysis Error (%)
Dry
Moist
% Decrease
7.9 N 8.09 N 2.4%
4.73 N 4.38 N 7.49%
67% 84% –
Experimental and FE results.
2.3. Modelling the actuator The next step in the modelling process is to include the MFC patch and its influence on the system stiffness together with its actuation effect. Experimentally, the actuators are bonded to the surface of the laminate after it has cured. This means that during the cure, the plates are allowed to assume their equilibrium shape naturally. The MFC’s nominal thickness is roughly 0.3 mm. This is slightly less than the two ply T800 laminate which is 0.36 mm thick. Considering what was said regarding the influence of the laminate’s thickness in the final curvature and snap-through characteristics, it is obvious that one must simulate the bonding process after the laminate has cooled down and developed its natural curvature and thereby recreate the change in the laminate’s curvature and stiffness. A method to solve the problem of virtually bonding the actuator to the laminate was defined and implemented. This method uses temperature dependent elastic properties for the MFC. The key is to realise that one can impose a temperature field to the laminate and the MFC model independently. The stiffness effects of the patch during cure are postponed until room temperature is reached (see Fig. 8). The initial temperature of 180 C is applied to the laminate and the patch. The patch remains relatively stress-free because of the very low stiffness values and null thermal expansion coefficients. When room temperature is reached (at about 25 C) the stress field in the patch increases significantly because of the sudden increase in stiffness. The thermal expansion coefficients remain unchanged. The redistribution of stress in the laminate/actuator system causes a slight change in curvature and shape of the laminate. This can easily be seen in Fig. 9.
Fig. 9. Development of curvature during cool down.
In this plot the final phases of the cool down process can be observed through the out of plane displacements of the middle of the plate. This is shown as a dashed red line in the inset schematic . As the stiffness of the patch starts to increase, the laminate must readjust its curvature. Although the temperature is decreasing, the laminate is actually losing some of its curvature. To simulate the MFC patch, shell elements were used and only the active area was represented. The patch was placed in the centre of the laminate and the two surfaces were separated by 0.338 mm to account for the thickness of both laminate and MFC. 1 1 0:3 tMFC þ tlam ¼ þ 0:188 ¼ 0:338 mm 2 2 2
ð10Þ
The elastic properties of the MFC patch were taken from the manufacturer’s literature and are shown in Table 9. At ‘‘high’’ temperatures (T > 25 C) these were reduced to Table 10.
Table 9 Mechanical properties of the MFC actuator Property
Value
E1 E2 m12 G12 a1
30.34 GPa 15.86 GPa 0.31 5.52 GPa 0:75 106 = C
Table 10 Mechanical properties of the MFC actuator at ‘‘high’’ temperatures
Fig. 8. Variation of the MFC’s elastic properties with temperature.
Property
Value
E1 E2 m12 G12 a1
10 Pa 10 Pa 0.001 10 Pa 0/C
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Fig. 10. FE mesh of the laminate and the active area of MFC patch.
Non-zero values were introduced to avoid numerical instabilities. The values of a do not represent the material thermal expansion coefficient (CTE), but rather the freestrain per volt values that will be used to recreate the actuation effect. The patch was connected to the laminate through a *TIE constraint. This constraints rigidly connects the nodes on the master surface (the laminate) to the ones on the slave surface (the MFC patch). The finite element mesh is shown in Fig. 10. The last step of the modelling task is to recreate the actuation force and use the results to design a working system. The MFC is a composite actuator made of piezoelectric fibres encased in an interdigited electrode (IDE) pattern. When the IDE is energized with a DC voltage, the fibers elongate or contract depending on whether the polarity is positive or negative. This creates an in-plane strain that is transmitted as shear stress to the adhesive layer and consequently to the laminate. The relation between applied voltage and strain is, for modelling purposes, assumed to be linear. The proportionality factor is called the free-strain per volt. This value, given by the manufacture of the MFC, is in the 0:75–0:9le=V range. If this value is used as the MFC’s CTE in the fibres’ direction (the material 1-direction) one can use the thermal expansion to simulate the piezoelectric effect simply by applying a temperature field. There is a one-to-one relation between applied temperature difference (DT ) and the real applied voltage (DV ) to the patch. Using the method described above to predict the stiffness change of the system and using the free-strain per volt value as the MFC’s CTE one can estimate the required voltage necessary to produce snap-through. A few combinations of material thickness, actuator size and number were modelled. Moisture effects were also considered. Fig. 11 lists the different combinations modelled. Both dried and moistened conditions were analysed separately with both small (actuator with 85 mm by 28 mm side lengths, M8528) and larger (actuator with 85 mm by 57 mm side lengths, M8557) actuators. The goal was to verify whether a given combination would perform better or worse given a specified moisture content. While intuition tells us that a moistened plate would be easier to snap because the snap-through loads are considerably lower
Fig. 11. Design tree of the FE models analyzed.
than in the dry state, such a plate would also have a reduced curvature and stiffness, thus being more sensitive to the presence of the MFC. The MFC patch can have a more significant effect on the laminate’s bi-stability to the extent of eliminating it completely. In order to accept a possible candidate, complete snapthrough had to occur. This means that the load proportionality factor (LPF) curve had to peak and drop to zero as shown in Fig. 12. For the T800 pre-preg with a M8557 actuator and for most of the moistened laminate plates, i.e. in the cases where the bending stiffness is reduced, the load curved looked like the one in Fig. 13. These plots can have different interpretations. In the first portion of the curve the MFC is clearly driving the laminate. After the critical load is reached, it is the laminate that drives the patch as is typical for a snap-through. However, at around 600 V both appear to reach an equilibrium point. Here, an inversion occurs and again the MFC starts driving the laminate. The FE code is unable to reach a solution at the second peak. Another possible interpretation of this is the occurrence of branching in the response. Secondary bifurcations have been seen in previous models
Fig. 12. Typical LPF plot for a successful snap-through.
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P. Portela et al. / Computers and Structures 86 (2008) 347–356
Fig. 13. LPF curve for the T800 fibre with M8557 actuator.
m ter af
oisture
ab so rp
tio n
Ac po tuat we or rfu is ll e not no ug h
. ..
Working combination
es los te . na ty.. mi ili La -stab bi
Chance of successful design
and even in experiments. It is possible that Abaqus is following the secondary branch in the equilibrium path. There are conflicting effects at work when a patch is bonded to the laminate. If the laminate is large (larger than 200 mm for example), even the largest MFC commercially available will not be powerful enough to induce snapthrough. But, if the laminate’s size is reduced to below 150 mm, the presence of the bonded patch will disrupt its bi-stability or bring it to a state in which snap-through can occur with little effort or even due to its own weight. It is also possible that a combination will work smoothly with a dried laminate and not work at all with a moistened one. An equilibrium compromise between these two competing effects must be found. This can be readily understood in Fig. 14. For a given laminate size there is a range of MFC sizes that will not produce enough actuation force to trigger snap-through. However, slightly larger ones will work on a dried laminate but not after is has absorbed moisture. Even larger MFC patches will completely destroy the bistability by affecting the underlying laminate’s bending stiffness. Somewhere in the middle between small and large there is an optimum size that will not significantly affect the curvature and still be powerful enough to produce snapthrough.
The first model, AS4 [0/90] with a M8528 actuator, did not suffer from loss of bi-stability in both the dry and the moistened plate. There was complete snap-through which means that the patch did not influence the laminate’s stiffness too much. However, the predicted snap-through voltage is 2052 V for both. With two patches, complete snap-through also occurs but at a significantly higher voltage of 2700 V. The MFC’s maximum operating voltage is 1500 V. The moistened model did not converge into a cylindrical shape despite the geometric imperfections. It was assumed that the two patches would probably destroy the laminate’s curvature. The larger MFC, the M8557 patch, looked more promising as the predicted activation voltage was 2000 V for the dry condition, but dropped to 1509 V in a moistened laminate. Branching was found as part of the solution for this particular case but the LPF dropped to zero (Fig. 15). This means that the patch did not destroy bi-stability and was probably an appropriate size. Using thinner prepreg, such as the T800 fibre, also looked promising as the peak voltage was 1950 V but dropped to less than 300 V after moisture effects were considered, but branching also occurred. It is important to note that the simulation of the actuation was based on a proportionality factor (free strain per volt) of 0:9le=V. The MFC’s voltage/strain characteristic curve is very non-linear. Using this first order approximation for the actuator underestimates the real applied strain. Because a higher order approximation is impossible to do with this FE approach, other models were developed in which the proportionality factor was increased to 1:5le=V. With this new value, the peak load dropped to 1100 V in the dried laminate and to less than 300 V in a moistened plate. Thus, this laminate/MFC configuration was chosen to perform experimental measurements. Fig. 16 shows the evolution of the in-plane strain in the laminate during the curing stage, and during the actuation stage. The presence of the actuator can be clearly seen. Additionally, the order or magnitude for the principal stress in the actuation zone is 40 MPa which is in accordance with the data specified by the MFC manufacturer.
Actuator size Fig. 14. Design curve of MFC actuated bi-stables plates.
Fig. 15. LPF plot for the moistened AS4 fibre laminate with an M8557 actuator.
P. Portela et al. / Computers and Structures 86 (2008) 347–356
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Fig. 16. Contour plots of the stress field caused by the MFC.
Fig. 18. Morphing plate in both states.
moistened conditions. Several effects need to be taken into account: Fig. 17. Deformed mesh of the bi-stable laminate in both states.
Finally, Fig. 17 represents the bi-stable laminate with the bonded MFC patch. State A is the laminate before snap-through. State B is the one obtained after the snapthrough induced by the MFC.
3. Experimental programme Bonding the MFC patch to the laminate is a critical step in the production of this morphing concept. Early experiments revealed that not only the adhesive type, layer thickness and curing temperature are important, but also the geometry state in which the adhesive is cured. Because the MFC produces a higher load when activated with positive polarity, it was bonded with its fibres perpendicular to one of the principal curvatures. The patch should be able to trigger snap-through from this state (A) to the second one (B). 3M’s DP460 two part epoxy resin was used and cured in a vacuum bag at 40 C for 4 h. It is very important that the adhesive layer be kept as thin as possible and that the adhesive be cured in the B state. If it is allowed to cure in the A state, the stress field in the resin will work in the opposite direction to the actuation load. If, on the other hand, it is cured in the B state it can be manually snapped back to the A state and the stress field in the resin will be favorable to the actuation load. Fig. 18 shows the morphing plate produced in both states. The morphing plate was tested at different voltage levels. Snap-through was achieved with as little as 390 V in the
(1) The MFC is a capacitor with very high resistance. The lower the voltage, the longer it takes for it to reach its maximum force (for that voltage level). (2) It is difficult to evaluate the true moisture content of the laminate. As time passes it becomes increasingly easier to snap the laminate. After about three months the laminate loses bi-stability and must be dried again. (3) The MFC voltage/strain behaviour shows hysteresis. This means it is history dependent and it is difficult to predict the actual strain that is to be expected for a given voltage level. To snap the laminate back to its original shape, a small weight was placed on top of it and the actuator’s poles were short-circuited to guarantee that it was not under any residual strain. 4. Summary and conclusions The development of the FE models presented in this work has resulted in a better understanding of the behaviour of the laminate with a thick piezoelectric patch bonded to its surface. The data obtained in this analysis was unavailable in the literature survey, although there is considerable work done in the prediction of post-cure shapes of unsymmetrical laminates. A very simple and computationally inexpensive method to design such morphing structures was developed and implemented. The snapthrough loads correlated well with experimental results even when environmental effects are included. Explicit or implicit direct integration dynamical methods can be used
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to predict post cure shapes of laminates without the need of artificial geometric imperfections. Although it is possible to design working active bi-stable laminates based on MFC actuators and to understand their behaviour, the low actuator force still limits the size of the bi-stable laminate. The effect it has on the laminate’s stiffness and curvature is also considerable. Because of this effect the laminate loses its symmetry, and it becomes much more difficult to snap it back from state B to state A manually and impossible to do so with the actuator. Tests were performed on the active laminate but with no success. Thus, it is difficult to design an MFC based actuator that is able to snap the laminate both ways, which is the ultimate objective of these studies. The most promising actuation method is one that does not influence the geometry of the laminate. Acknowledgement The financial support of the Portuguese Science Foundation (FCT) through the Grant SFRHnBMn16199n2004 is gratefully acknowledged by the first author. References [1] Jha AK, Kudva JN. Morphing aircraft concepts, classifications and challenges. Proc SPIE 2004;5388. [2] Schultz MR. Use of piezoelectric actuators to effect snap-through behavior of unsymmetric composite laminates. Ph.D. thesis, Virginia Polytechnic Institute and State University; 2003.
[3] Schultz MR, Hyer MW. A morphing concept based on unsymmetric composite laminates and piezoceramic MFC actuators. In: Proceedings of the 45th AIAA/ASME/ASCE/AHS/ASC structures, structural dynamics and materials conference; 2004. [4] Williams RB, Inman DJ. An overview of composite actuators with piezoceramic fibres, unpublished. [5] Dano ML, Hyer MW. SMA-induced snap-through of unsymmetric fiber-reinforced composite laminates. Int J Solids Struct 2003;40: 5949–72. [6] Hufenbach W, Gude M, Kroll L. Design of multistable composites for application in adaptive structures. Compos Sci Technol 2002: 2201–7. [7] Hufenbach W, Gude M. Analysis and optimisation of multistable composites under residual stresses. Compos Struct 2002;55:319–27. [8] Dano ML, Hyer MW. Thermally-induced deformation behavior of unsymmetric laminates. Int J Solids Struct 1998;35:2101–20. [9] Schlecht M, Schulte K, Hyer MW. Advanced calculation of the room-temperature shapes of thin unsymmetric composite laminates. Compos Struct 1995;32:627–33. [10] Gigliotti M, Wisnom MR, Potter KD. Loss of bifurcation and multiple shapes of thin [0/90] unsymmetric composite plates subject to thermal stress. Compos Sci Technol 2004;64:109–28. [11] Portela PM. Analysis and optimization of morphing, multi-stable structures actuated by piezoelectric patches. M.Sc. thesis, Faculdade de Engenharia da Universidade do Porto; 2005. [12] Herakovich CT. Mechanics of fibrous composites. John Wiley and Sons Inc; 1998. [13] Springer GS. Environmental effects on composite materials. Technomic Publishing Company; 1981. [14] Abaqus 6.4 User’s Manuals, Abaqus, Inc.; 2004. [15] Potter K, Weaver PM, Seman AA, Shah S. Phenomena in the snapthrough bifurcation of unsymmetric composite plates. Compos Part A 2007;38:100–6.
Computers and Structures 86 (2008) 347–356 www.elsevier.com/locate/compstruc
Analysis of morphing, multi stable structures actuated by piezoelectric patches Pedro Portela b
a,*
, Pedro Camanho a, Paul Weaver b, Ian Bond
b
a Faculdade de Engenharia da Universidade do Porto, Rua Roberto Frias s/n 4200-465 Porto, Portugal Department of Aerospace Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, United Kingdom
Available online 11 April 2007
Abstract In this article a novel morphing structure concept is studied using non-linear Finite Element Analysis (FEA). Bi-stable asymmetrical laminates can be snapped between two geometries through a buckling mechanism that is activated by an applied load. A piezoelectric Macro-Fibre Composite (MFC) actuator was chosen to provide this activation load. Bi-stable structures will maintain a given geometrical state without the need for a constant actuation force. FEA was used to predict the two stable geometries, to understand the buckling mechanism and to evaluate the feasibility of using MFC actuators for switching between states. Environmental effects like moisture absorption were also included in the analysis. 2007 Elsevier Ltd. All rights reserved. Keywords: Morphing structures; Asymmetrical laminates; Snap-through; MFC; Morphing; Multistable plates; Composites; Piezoelectric; Actuation; FE modelling
1. Introduction The expression ‘‘Morphing Structures’’ is a very broad and much underdefined subject. In this text a morphing structure will be defined as one capable of macroscopic geometric changes in order to better adapt to radically different environmental conditions. Because of the resources available in the aerospace industry and because of the clear advantages that these technologies can bring, morphing structures applications are currently under study, for example, for airplane wing geometry changes, replacement of mechanically driven control surfaces, helicopter blade control, and reliable actuators for space missions. If a morphing aircraft is defined merely as an aircraft that changes its configuration in-flight, it can be seen that a morphing aircraft is not a new concept. Extending flaps, elevators, ailerons and wing twisting can be considered geometric changes and can be technically termed as morphing.
*
Corresponding author. E-mail address: [email protected] (P. Portela).
0045-7949/$ - see front matter 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2007.01.032
However, these changes are either necessary enablers for controlled flight or contributors to the improved aerodynamic performances of an aircraft. As a result, these technologies do not necessarily allow an aircraft to perform different types of mission tasks [1]. Recently Schultz et al. [2] proposed a novel morphing concept based on asymmetrical carbon fibre reinforced plastic (CFRP) laminates. If a composite laminate does not have a symmetrical stacking sequence, thermally induced stresses develop during the curing process and cause an out of plane curvature. This curvature can be tailored by adjusting the stacking sequence and ply thickness. Depending on the plate geometry the thermally induced stresses may cause the laminate to have more than one stable configuration or shape. Morphing can be obtained by alternating between these states. The main advantage of using these structures is that they will hold one of the stable shapes without the need of a force holding them in that position. Hence, they can carry loads in both states as long as they do not reach the critical snap-through load. Schultz [2,3] used piezoelectric patches bonded to the surface of the laminate to induce snapthrough with a shear force. This type of device uses both
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active and composite material technologies in addition to interdigitated electrodes. The Macro Fibre Composite (MFC) actuator was developed by NASA Langley Research Centre and is currently commercially available through the company Smart-Material, Inc. This active element has uniaxially aligned piezoelectric fibres surrounded by polymer matrix. The fibres specifically have a rectangular cross-section [4]. Schultz successfully produced a morphing laminate based on the MFC actuator and obtained good predictions of the snap-through loads for a patch bonded in the middle of the plate. Dano [5] used Shape Memory Alloys (SMA), wires that, when heated, applied a bending moment to the plate. Hufenbach et al. [6,7] developed optimization algorithms to dimension such laminates as well as producing a SMA based morphing, bi-stable laminate. Both Schultz and Dano were concerned with a semi-analytical approach to predict the two stable shapes of such laminates as well as the required snap-through loads. There has been considerable amount of work done in the prediction of room temperature shapes of asymmetrical laminates [8,9]. Most of this work is based on semianalytical Rayleigh–Ritz methods, although non-linear Finite Element (FE) analysis has also been used successfully. In the present paper, non-linear FE analysis is used for several purposes. The first one is to reproduce the cured shapes of the asymmetrical laminates. This is not a new approach for this problem as FE predictions of asymmetrical laminates’ shapes are documented for example in [8]. The cure of [0/90], [0/45], [45/45] square laminate was simulated and their shapes and snap-through behaviour were compared to experimental data. This first task was useful to understand the snap-through phenomenon. It was also necessary to confirm the analysis procedures and simplifying assumptions typical of FE analysis (i.e. boundary conditions, applied loads and the material properties). The second purpose of the FE analysis was to simulate the combined laminate/actuator system. The model is helpful in predicting the changes in the laminate’s stiffness due to the presence of the actuator. Having accomplished this, the model also predicts the voltage required to induce the snap-through effect of the laminates. 2. Modelling the bi-stable laminate To determine the post cure shape of the laminate, a square plate with side length of 150 mm was modelled. The plate geometry and boundary conditions are depicted in Fig. 1. The central node was clamped. Although the plates’ nominal length was 150 mm, it was necessary to include some geometric imperfections into the FE model. The plate’s sketch is not a perfect square but rather a rectangle with side lengths of 149 mm by 151 mm. Using perfectly square plates in the FE model will make the analysis converge to unstable saddle shapes instead of stable cylindrical ones. This is reported by Gigliotti et al. [10]. The reason why this happens is still not clear. However, a hypothesis
Fig. 1. FE model boundary conditions.
was formulated that the inertia forces played a role in the curvature development. If instead of an implicit method an explicit method or an implicit direct-integration dynamic method is used to predict the cured shapes, the perfectly square plate will converge into a stable cylinder. An explicit model was created and verified this hypothesis [11]. All the models presented here are based on implicit methods including geometric imperfections. The reason for this choice is that the time it takes to run an explicit analysis compared to an implicit one and the accurate results that the implicit method provides regardless of the geometric imperfections introduced. Two materials were used for the initial tests and for the analysis: AS4-8552 and T800-2020 carbon fibre/epoxy pre-preg. The mechanical properties of these two pre-pregs used are listed in Table 1. Although the T800 fibre is stiffer then the AS4, the prepreg ply thickness is roughly 60% of the AS4. Since the snap-through phenomenon is achieved by applying a bending moment, it is the flexural stiffness that plays a key role. The stiffness matrix is computed by ½D ¼ P2 bending k 3 1 ½Q ðz z3k1 Þ. Small changes in ply thickness have k k¼1 3 significant effects in flexure due to the cubic dependence of ply thickness (zk). Hence, a 60% change in ply thickness will have a significant effect in the laminate’s bending stiffness. General purpose composite shell elements were used (type S4R in Abaqus). The meshes have approximately 900 elements. An initial temperature field of 180 C was imposed on the elements before starting the analysis. This represents the cure cycle’s maximum temperature which is taken as the stress-free temperature as well. Step one is Table 1 Mechanical properties of the two different pre-pregs Property
Value (AS4-8552)
Value (T800-2020)
E1 E2 m12 G12 G13 G23 a1 a2 tply
135 GPa 9.5 GPa 0.3 5 GPa 7.17 GPa 3.97 GPa 2 108 = C 3 105 = C 0.255 mm
294 GPa 9.5 GPa 0.3 5 GPa 7.17 GPa 3.97 GPa 2 108 = C 2:25 105 = C 0.180 mm
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Fig. 2. Representation of the analysis steps used.
Fig. 3. Comparison between FE shapes and those of the manufactured laminates.
Table 2 Measured and calculated principal curvature
jy
Measured (1/m)
Finite element (1/m)
Error (%)
6.51
6.65
2.1
a simple *STATIC step. Large deformation theory was used (*NLGEOM) to calculate the plate’s deformation due to a temperature change of DT ¼ 160 C, see Fig. 2. Fig. 3 pictures a qualitatively comparison between shapes obtained in the FE models and the ones observed in the manufactured laminates. Table 2 compares the curvatures measured in a [0/90] laminate to the ones obtained in the FE model. The curvature measurements were made five days after manufacturing and according to the procedure defined in [2]. It was not possible to adapt this procedure to measure curvatures in [45/45] and [0/45] laminates. Therefore, only the results for the [0/90] laminate are shown. Fig. 4 shows the contour plots and average values of the curvatures predicted by FE. The material used was the T8002020 pre-preg. 2.1. Environmental effects Composite materials absorb moisture from the environment in which they are immersed (air, water or any other fluid). Moisture absorption creates a stress field not unlike the one created by thermal stresses. In this particular application, the effect of the hygroscopic stress field is to relax
the existing residual stress field and thus reduce the plates’ curvature. This reduction in curvature, and consequently in the peak snap-through load, was included in the FE model. Abaqus does not include this effect in its constitutive models. In order to overcome this limitation a modification of the thermal expansion coefficients is required to account for hygroscopic effects. Using Voigt notation to represent the several tensors, the hygrothermoelastic constitutive equations are developed assuming the effective strain is the superposition of the three strains acting on ply k: k
k
k
feg ¼ fer g þ feT g þ feH g
k
ð1Þ
where fer g; feT g; feH g are the mechanical, thermal and hygroscopic strains. Assuming plane stress conditions and neglecting the mechanical strains: fegk ¼ fagk DT þ fbgk DM ¼ k k DM k DT ¼ fa g DT fag þ fbg DT
ð2Þ ð3Þ
where fagk are the coefficients of thermal expansion in the k ply coordinate system; fbg are the coefficients of moisture k expansion in the ply coordinate system; fa g are the equivalent moisture expansion coefficients accounting for the effects of both thermal and hygroscopic residual strains; DT is the applied temperature field; DM is the percent weight gained through moisture absorption. Assuming that DT < 0 and DM > 0 to force a coherent behaviour, the components of the a vector are:
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P. Portela et al. / Computers and Structures 86 (2008) 347–356 Table 3 Typical values for a and b [13]
T300/1034 AS/3501-5 T300/5208
a
b
0.017 0.018 0.015
1 1 1
where s is the thickness of the laminate when it is exposed on two sides to the same environment and Dx is the diffusivity of the material in the direction normal to the surface, obtained as [13]: ¼ cexpðT Þ Davg x d
Fig. 4. Contour plots of the curvatures predicted by FE.
8 9k 8 9k 8 9k > > > = = = DM < a1 > < a1 > < b1 > a2 ¼ a2 þ b2 > > > ; ; ; DT : > : > : > 0 0 0
ð4Þ
Typical values for b is 0.0 in the fibre direction and 0.005 1 in the transverse direction [12]. The percent weight gain DM is defined as DM ¼
Weight of moist material Weight of dry material Weight of dry material ð5Þ 100
The maximum moisture content, Mm, varies with relative humidity of the environment, the nature of the fluid, the matrix material and geometry. Thinner laminates reach a higher maximum moisture content faster than thicker ones. Springer [13] identifies the relation between maximum moisture content and relative humidity of the air, / as: M m ¼ a/b
ð6Þ
where a and b are constants to be determined experimentally. Springer [13] specifies the parameters a and b for three different materials exposed to humid air, as shown in Table 3. The time required for a composite material to attain 99.9% of its maximum possible moisture content is given by 0:67s2 tm ¼ Dx
ð7Þ
ð8Þ
The constants c and d depend on the material and must be determined experimentally and T is the temperature in Kelvin. Springer [13] provides experimental values for c and d. This data is combined into a single table with that of Table 3 to calculate the maximum moisture content and the time required to reach equilibrium for the three different materials (see Table 4). Table 5 shows that the maximum moisture content does not vary too much. However, the time it takes to reach equilibrium is significantly different. For the purpose of the analysis, and because there is no available data relative to the material, a value of M m ¼ 0:6 was chosen. The a vector is 9 9 8 9 8 8 8 8 > > > = 0:6 > <0 = > = < 2 10 = C > < 2 10 = C 5 6 ¼ 3:750 10 = C 2:25 10 = C 0:005 > > > > > ; 160 > : ; : : ; 0 2:25 105 = C 2:25 105 = C ð9Þ With this data inserted into the FE model the change in curvature due to moisture absorption can be determined by calculating the curvature of the moist material. The following table lists those values for the [0/90] 150 mm 150 mm laminate. The true measurement of moisture effects will become clearer in the next section, where the snap-through loads are calculated Table 6.
Table 4 Typical values for c and d [13]
T300/1034 AS/3501-5 T300/5208
c
d
0.83 6.51 0.58
5219 5722 5113
Table 5 Time it takes for the different materials to reach the maximum moisture content [13]
T300/1034 AS/3501-5 T300/5208
t (days)
Mm
66 47 70
0.68 0.72 0.6
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Table 6 Difference in principal curvatures after moisture absorption [13] 1
j1 ðm Þ j2 ðm1 Þ j12 ðm1 Þ
Dry
Moist
0 6.65 0.02
0 2.06 0.05
2.2. Modelling the snap-through Recalling Fig. 2, the snap-through step naturally followed the curvature development step. The elements’ definition and meshes are, therefore, the same. Again, the purpose of this step was to understand the buckling mechanism and calculate the critical loads. To do this Riks method was used [14]. Fig. 5 represents the FE plots of the two stable states for the different lay-ups. Fig. 2 shows the loading applied to the FE model during this step. Two concentrated forces were applied to the middle of the edges. The total reaction force can be monitored in the clamped node. The force initially increases linearly until large deformations become significant and a limit point is reached (critical load). This is the required force to induce transition. After this point the snap-through buckling will occur and the force can be removed. Potter et al. [15] provide experimental measurements of the load–displacement curve of the snap-through of a 150 mm · 150 mm square [0/90] laminate, Fig. 6. The material used was AS4/8552 pre-preg. The load was applied in the middle of the plate which was supported on a rigid table. The displacement measured is the displacement at this point. The FE curves were obtained for a plate in a dry state in a moistened state. The moisture content was considered to be 0.2%. The displacement of a point in the middle of the plate’s edge was used. Fig. 7 shows the variation of the applied load with the step time. The simulation results are in good agreement with the experimental results. In fact, the analysis was able to capture the secondary bifurcation in the unloading portion of the plot and highlighted in Fig. 7. This corresponds to a point where part of the plate buckles, while the other remains unchanged. Thus, this method is adequate to simulate the snap-through effect both qualitatively and quantitatively. Using the stacking sequence [0/90] three other
Fig. 6. Measured force [15] versus displacement.
Fig. 7. Applied force versus step time. Laminate is in the dry state.
square models with nominal lengths of 200, 250 and 300 mm were analyzed together with a [0/45] one. The critical force was measured experimentally and compared with the ones found in the FE models. A summary of these results can be seen in Table 7. Finally, the influence of the moisture content was analyzed. Using the procedure described in the previous section, the FE model was changed to reflect the hygroscopic effects. The critical load was calculated for a ½0 =90 T , 150 mm · 150 mm AS4 laminate. The following table compares both measured and FE results (see Table 8).
Table 7 Comparison of measured versus FE calculated critical loads
150 mm, 150 mm, 200 mm, 300 mm, 120 mm,
[0/90], [0/90], [0/90], [0/90], [0/45],
AS4 T800 T800 T800 T800
FE (N)
Measurement (N)
Error (%)
8.1 2.9 2.3 3.0 1.2
7.9 2.4 2.3 3.1 1.9
2.2 20.4 2.8 2.3 37.3
Fig. 5. FE plots of the two stables states for the three different laminates.
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Table 8 Critical loads in dry and moistened plates
Measured FE analysis Error (%)
Dry
Moist
% Decrease
7.9 N 8.09 N 2.4%
4.73 N 4.38 N 7.49%
67% 84% –
Experimental and FE results.
2.3. Modelling the actuator The next step in the modelling process is to include the MFC patch and its influence on the system stiffness together with its actuation effect. Experimentally, the actuators are bonded to the surface of the laminate after it has cured. This means that during the cure, the plates are allowed to assume their equilibrium shape naturally. The MFC’s nominal thickness is roughly 0.3 mm. This is slightly less than the two ply T800 laminate which is 0.36 mm thick. Considering what was said regarding the influence of the laminate’s thickness in the final curvature and snap-through characteristics, it is obvious that one must simulate the bonding process after the laminate has cooled down and developed its natural curvature and thereby recreate the change in the laminate’s curvature and stiffness. A method to solve the problem of virtually bonding the actuator to the laminate was defined and implemented. This method uses temperature dependent elastic properties for the MFC. The key is to realise that one can impose a temperature field to the laminate and the MFC model independently. The stiffness effects of the patch during cure are postponed until room temperature is reached (see Fig. 8). The initial temperature of 180 C is applied to the laminate and the patch. The patch remains relatively stress-free because of the very low stiffness values and null thermal expansion coefficients. When room temperature is reached (at about 25 C) the stress field in the patch increases significantly because of the sudden increase in stiffness. The thermal expansion coefficients remain unchanged. The redistribution of stress in the laminate/actuator system causes a slight change in curvature and shape of the laminate. This can easily be seen in Fig. 9.
Fig. 9. Development of curvature during cool down.
In this plot the final phases of the cool down process can be observed through the out of plane displacements of the middle of the plate. This is shown as a dashed red line in the inset schematic . As the stiffness of the patch starts to increase, the laminate must readjust its curvature. Although the temperature is decreasing, the laminate is actually losing some of its curvature. To simulate the MFC patch, shell elements were used and only the active area was represented. The patch was placed in the centre of the laminate and the two surfaces were separated by 0.338 mm to account for the thickness of both laminate and MFC. 1 1 0:3 tMFC þ tlam ¼ þ 0:188 ¼ 0:338 mm 2 2 2
ð10Þ
The elastic properties of the MFC patch were taken from the manufacturer’s literature and are shown in Table 9. At ‘‘high’’ temperatures (T > 25 C) these were reduced to Table 10.
Table 9 Mechanical properties of the MFC actuator Property
Value
E1 E2 m12 G12 a1
30.34 GPa 15.86 GPa 0.31 5.52 GPa 0:75 106 = C
Table 10 Mechanical properties of the MFC actuator at ‘‘high’’ temperatures
Fig. 8. Variation of the MFC’s elastic properties with temperature.
Property
Value
E1 E2 m12 G12 a1
10 Pa 10 Pa 0.001 10 Pa 0/C
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353
Fig. 10. FE mesh of the laminate and the active area of MFC patch.
Non-zero values were introduced to avoid numerical instabilities. The values of a do not represent the material thermal expansion coefficient (CTE), but rather the freestrain per volt values that will be used to recreate the actuation effect. The patch was connected to the laminate through a *TIE constraint. This constraints rigidly connects the nodes on the master surface (the laminate) to the ones on the slave surface (the MFC patch). The finite element mesh is shown in Fig. 10. The last step of the modelling task is to recreate the actuation force and use the results to design a working system. The MFC is a composite actuator made of piezoelectric fibres encased in an interdigited electrode (IDE) pattern. When the IDE is energized with a DC voltage, the fibers elongate or contract depending on whether the polarity is positive or negative. This creates an in-plane strain that is transmitted as shear stress to the adhesive layer and consequently to the laminate. The relation between applied voltage and strain is, for modelling purposes, assumed to be linear. The proportionality factor is called the free-strain per volt. This value, given by the manufacture of the MFC, is in the 0:75–0:9le=V range. If this value is used as the MFC’s CTE in the fibres’ direction (the material 1-direction) one can use the thermal expansion to simulate the piezoelectric effect simply by applying a temperature field. There is a one-to-one relation between applied temperature difference (DT ) and the real applied voltage (DV ) to the patch. Using the method described above to predict the stiffness change of the system and using the free-strain per volt value as the MFC’s CTE one can estimate the required voltage necessary to produce snap-through. A few combinations of material thickness, actuator size and number were modelled. Moisture effects were also considered. Fig. 11 lists the different combinations modelled. Both dried and moistened conditions were analysed separately with both small (actuator with 85 mm by 28 mm side lengths, M8528) and larger (actuator with 85 mm by 57 mm side lengths, M8557) actuators. The goal was to verify whether a given combination would perform better or worse given a specified moisture content. While intuition tells us that a moistened plate would be easier to snap because the snap-through loads are considerably lower
Fig. 11. Design tree of the FE models analyzed.
than in the dry state, such a plate would also have a reduced curvature and stiffness, thus being more sensitive to the presence of the MFC. The MFC patch can have a more significant effect on the laminate’s bi-stability to the extent of eliminating it completely. In order to accept a possible candidate, complete snapthrough had to occur. This means that the load proportionality factor (LPF) curve had to peak and drop to zero as shown in Fig. 12. For the T800 pre-preg with a M8557 actuator and for most of the moistened laminate plates, i.e. in the cases where the bending stiffness is reduced, the load curved looked like the one in Fig. 13. These plots can have different interpretations. In the first portion of the curve the MFC is clearly driving the laminate. After the critical load is reached, it is the laminate that drives the patch as is typical for a snap-through. However, at around 600 V both appear to reach an equilibrium point. Here, an inversion occurs and again the MFC starts driving the laminate. The FE code is unable to reach a solution at the second peak. Another possible interpretation of this is the occurrence of branching in the response. Secondary bifurcations have been seen in previous models
Fig. 12. Typical LPF plot for a successful snap-through.
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P. Portela et al. / Computers and Structures 86 (2008) 347–356
Fig. 13. LPF curve for the T800 fibre with M8557 actuator.
m ter af
oisture
ab so rp
tio n
Ac po tuat we or rfu is ll e not no ug h
. ..
Working combination
es los te . na ty.. mi ili La -stab bi
Chance of successful design
and even in experiments. It is possible that Abaqus is following the secondary branch in the equilibrium path. There are conflicting effects at work when a patch is bonded to the laminate. If the laminate is large (larger than 200 mm for example), even the largest MFC commercially available will not be powerful enough to induce snapthrough. But, if the laminate’s size is reduced to below 150 mm, the presence of the bonded patch will disrupt its bi-stability or bring it to a state in which snap-through can occur with little effort or even due to its own weight. It is also possible that a combination will work smoothly with a dried laminate and not work at all with a moistened one. An equilibrium compromise between these two competing effects must be found. This can be readily understood in Fig. 14. For a given laminate size there is a range of MFC sizes that will not produce enough actuation force to trigger snap-through. However, slightly larger ones will work on a dried laminate but not after is has absorbed moisture. Even larger MFC patches will completely destroy the bistability by affecting the underlying laminate’s bending stiffness. Somewhere in the middle between small and large there is an optimum size that will not significantly affect the curvature and still be powerful enough to produce snapthrough.
The first model, AS4 [0/90] with a M8528 actuator, did not suffer from loss of bi-stability in both the dry and the moistened plate. There was complete snap-through which means that the patch did not influence the laminate’s stiffness too much. However, the predicted snap-through voltage is 2052 V for both. With two patches, complete snap-through also occurs but at a significantly higher voltage of 2700 V. The MFC’s maximum operating voltage is 1500 V. The moistened model did not converge into a cylindrical shape despite the geometric imperfections. It was assumed that the two patches would probably destroy the laminate’s curvature. The larger MFC, the M8557 patch, looked more promising as the predicted activation voltage was 2000 V for the dry condition, but dropped to 1509 V in a moistened laminate. Branching was found as part of the solution for this particular case but the LPF dropped to zero (Fig. 15). This means that the patch did not destroy bi-stability and was probably an appropriate size. Using thinner prepreg, such as the T800 fibre, also looked promising as the peak voltage was 1950 V but dropped to less than 300 V after moisture effects were considered, but branching also occurred. It is important to note that the simulation of the actuation was based on a proportionality factor (free strain per volt) of 0:9le=V. The MFC’s voltage/strain characteristic curve is very non-linear. Using this first order approximation for the actuator underestimates the real applied strain. Because a higher order approximation is impossible to do with this FE approach, other models were developed in which the proportionality factor was increased to 1:5le=V. With this new value, the peak load dropped to 1100 V in the dried laminate and to less than 300 V in a moistened plate. Thus, this laminate/MFC configuration was chosen to perform experimental measurements. Fig. 16 shows the evolution of the in-plane strain in the laminate during the curing stage, and during the actuation stage. The presence of the actuator can be clearly seen. Additionally, the order or magnitude for the principal stress in the actuation zone is 40 MPa which is in accordance with the data specified by the MFC manufacturer.
Actuator size Fig. 14. Design curve of MFC actuated bi-stables plates.
Fig. 15. LPF plot for the moistened AS4 fibre laminate with an M8557 actuator.
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355
Fig. 16. Contour plots of the stress field caused by the MFC.
Fig. 18. Morphing plate in both states.
moistened conditions. Several effects need to be taken into account: Fig. 17. Deformed mesh of the bi-stable laminate in both states.
Finally, Fig. 17 represents the bi-stable laminate with the bonded MFC patch. State A is the laminate before snap-through. State B is the one obtained after the snapthrough induced by the MFC.
3. Experimental programme Bonding the MFC patch to the laminate is a critical step in the production of this morphing concept. Early experiments revealed that not only the adhesive type, layer thickness and curing temperature are important, but also the geometry state in which the adhesive is cured. Because the MFC produces a higher load when activated with positive polarity, it was bonded with its fibres perpendicular to one of the principal curvatures. The patch should be able to trigger snap-through from this state (A) to the second one (B). 3M’s DP460 two part epoxy resin was used and cured in a vacuum bag at 40 C for 4 h. It is very important that the adhesive layer be kept as thin as possible and that the adhesive be cured in the B state. If it is allowed to cure in the A state, the stress field in the resin will work in the opposite direction to the actuation load. If, on the other hand, it is cured in the B state it can be manually snapped back to the A state and the stress field in the resin will be favorable to the actuation load. Fig. 18 shows the morphing plate produced in both states. The morphing plate was tested at different voltage levels. Snap-through was achieved with as little as 390 V in the
(1) The MFC is a capacitor with very high resistance. The lower the voltage, the longer it takes for it to reach its maximum force (for that voltage level). (2) It is difficult to evaluate the true moisture content of the laminate. As time passes it becomes increasingly easier to snap the laminate. After about three months the laminate loses bi-stability and must be dried again. (3) The MFC voltage/strain behaviour shows hysteresis. This means it is history dependent and it is difficult to predict the actual strain that is to be expected for a given voltage level. To snap the laminate back to its original shape, a small weight was placed on top of it and the actuator’s poles were short-circuited to guarantee that it was not under any residual strain. 4. Summary and conclusions The development of the FE models presented in this work has resulted in a better understanding of the behaviour of the laminate with a thick piezoelectric patch bonded to its surface. The data obtained in this analysis was unavailable in the literature survey, although there is considerable work done in the prediction of post-cure shapes of unsymmetrical laminates. A very simple and computationally inexpensive method to design such morphing structures was developed and implemented. The snapthrough loads correlated well with experimental results even when environmental effects are included. Explicit or implicit direct integration dynamical methods can be used
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to predict post cure shapes of laminates without the need of artificial geometric imperfections. Although it is possible to design working active bi-stable laminates based on MFC actuators and to understand their behaviour, the low actuator force still limits the size of the bi-stable laminate. The effect it has on the laminate’s stiffness and curvature is also considerable. Because of this effect the laminate loses its symmetry, and it becomes much more difficult to snap it back from state B to state A manually and impossible to do so with the actuator. Tests were performed on the active laminate but with no success. Thus, it is difficult to design an MFC based actuator that is able to snap the laminate both ways, which is the ultimate objective of these studies. The most promising actuation method is one that does not influence the geometry of the laminate. Acknowledgement The financial support of the Portuguese Science Foundation (FCT) through the Grant SFRHnBMn16199n2004 is gratefully acknowledged by the first author. References [1] Jha AK, Kudva JN. Morphing aircraft concepts, classifications and challenges. Proc SPIE 2004;5388. [2] Schultz MR. Use of piezoelectric actuators to effect snap-through behavior of unsymmetric composite laminates. Ph.D. thesis, Virginia Polytechnic Institute and State University; 2003.
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