1. Introduction
The first major research into the behavior of bistable composites was done in 1981 by Michael W. Hyer [
1]. Hyer experimentally investigated the phenomenon of asymmetric composite laminates and how their cured shapes diverged from the predictions of classical lamination theory. The principal phenomena behind the bistability lies in the thermal expansion coefficients of the fibers. The thermal expansion coefficient in the fiber direction is less than the thermal expansion coefficient perpendicular to the fibers. When an asymmetric laminate is made, there then exists a mis-match in the thermal expansion coefficients about the mid plane of the laminate. An asymmetric laminate will exhibit different types of curvature post cure due to residual stresses induced by dissimilar thermal expansion coefficients.
The cured shapes of the asymmetric laminates were seen as undesirable due to their inherent extension-bending coupling; Hyer sought to motivate further research on possible advantages and uses of these shapes. He showed that the principle curvature directions of the cylindrical bistable shapes are predictable and that thicker asymmetric laminates conformed to classical lamination theory. In particular, he observed that a 150 mm × 150 mm (5.91 in. × 5.91 in.) [0
2/90
2]
T laminate exhibited snap-through behavior with two room temperature cylindrical shapes of opposite sign and perpendicular axes of curvature, as shown below in
Figure 1.
Hyer went on to investigate the reasons why certain thin asymmetric laminates do not conform to classical lamination theory [
2]. According to classical lamination theory, all asymmetric laminates are predicted to be saddle shape at room temperature [
3]. Hyer investigated [0
2/90
2]
T and [0
4/90
4]
T square graphite-epoxy laminates and found that depending on the thickness of the laminate and the length of the side of the square the laminates were not saddle shaped but cylindrical with bistable behavior. Hyer attributed the cured shapes of the thin asymmetric laminates to geometric nonlinearities known as Flöppl–von Karman geometric nonlinearities, which are partial differential equations that describe the deflection of thin flat plates. Because of the geometric non linearities, Hyer developed an analytical solution using the Rayleigh–Ritz energy method as an extension of classical lamination theory which minimized the potential energy of the laminate, including the effects of thermal expansion. The solution allowed him to make bifurcation diagrams for both [0
2/90
2]
T and [0
4/90
4]
T square graphite-epoxy laminates and showed that, at a critical side length, the laminates became bistable. Before this critical length, he showed that the laminates are saddle shaped as would be predicted by classical lamination theory. He also postulated that the effects of moisture absorption, viscoelastic relaxation, and any mechanism that changes the internal stress state of the laminate are important for laminates near the critical length [
2]. In addition, he postulated that asymmetric curing, asymmetric cooling, and asymmetric moisture absorption could result in the laminate favoring one of the bistable configurations and cause the snap through force to be greater going one direction compared to the force required to snap back to the other. Hyer continued his investigation by revising his previous theory. He computed numerical results for the in-plane residual strains of thin asymmetric laminates that cooled from curing into cylindrical shapes at room temperature [
4].
Research was then done to investigate the temperature–curvature relationship for asymmetric cross-ply laminates. Composite laminates are typically cured at elevated temperatures in a mold under vacuum, either in an autoclave or simple curing oven. Typically, they are allowed to cool in the mold remaining under vacuum such that they do not deform while cooling. Hyer and Hamamoto reheated cured samples, allowing them to deform, and measured the curvature along a range of temperature and they developed a temperature–curvature relationship for asymmetric laminates [
5].
After much research was done beginning with Hyer et al., and continuing with many other researchers, the room temperature shapes of asymmetric laminates can be well predicted based on ply orientation and thickness [
6,
7,
8]. Jun et al. showed that residual thermally induced in-plane shear strains are not insignificant for medium width-to-thickness ratio laminates that are bistable [
6]. They also investigated the effects of width-to-thickness, number of layers, and stacking sequences for asymmetric cross ply laminates. Ren et al. continued with the developed Rayleigh–Ritz energy method and showed that the cured shape of a cross-ply laminate depends on stacking sequence, radius, thickness, and size [
7]. They developed a model to predict the cured shape based on these parameters and compared the results to FEA predictions as well as experimental results, and found good correlations. Thus, they developed a model to give guidance to the manufacture of asymmetric cross-ply laminates. With cross-ply laminates thus being well characterized, Jun et al. sought to characterize asymmetric laminates with arbitrary lay-up angles [
8]. They present a formulation that predicts the curvatures and principle directions of curvature for asymmetrically laminated composites that are not cross-ply. They found that their formulation agreed well with classical lamination theory for the range of length-to-thickness ratios that they studied.
With the post-cure shape of asymmetric laminates as well as their response to thermal loading being well characterized, attention thus was turned towards improved modeling and application of bistable composites. Schlecht et al. built on Hyer’s previous work and used FEA to predict the snap through force as well as stresses and strains of asymmetric bistable laminates [
9]. Betts et al. developed a novel method of mapping the surface profiles of asymmetric laminates [
10]. They found that existing modeling techniques are quite good at predicting room temperature shapes but are also quite sensitive to imperfections. They emphasized the continued need for accurate modeling as the design of mechanisms using the behavior of bistable composites increases. Thus, Giddings et al. sought to include imperfections in the prediction of cured laminate shapes using ANSYS FEA software (version 18, ANSYS, Inc., Canonsburg, PA, USA) [
11]. They included manufacturing imperfections such as high resin areas and ply-thickness variations in their model. They achieved this by analyzing composite laminates using optical microscopy and measuring cured shapes using a Peak Motus motion analysis system. They were able to accurately predict the two equilibrium states of an imperfect laminate to within 3–7% compared to errors of 7–73% of an idealized laminate by including the previously mentioned manufacturing imperfections, such as high resin areas and ply-thickness variations. Their model also included the prediction of bistable laminate deflection under thermal loading within a temperature range of 20–110 °C. The work of Giddings et al. on characterizing the shape of laminates over a given temperature range was developed from the work of Dano et al. who developed a model to predict the out of plane displacements of flat asymmetric composite laminates as they cool from the elevated curing temperature [
12]. Tawfik et al. developed an FEA model using ABAQUS to predict the shapes of asymmetric cross-ply laminates under thermal curing stresses and to investigate the instability point of snap-through [
13]. They investigated rectangular asymmetric cross-ply laminates and investigated how the aspect (length vs. width) ratio affects bistability.
With regard to exterior boundary conditions on bistable laminates, various conditions have been used in research in characterizing bistable composite laminates thus far. The boundary conditions that have been typically applied in the research that has been conducted have included some variation of the conditions shown in
Figure 2, where the four points of the corners are fixed in z and allowed to move in x and y such that they can “float”, and a force is applied at the center to snap the laminate through to the other position.
Tawfik et al. used an air table to experimentally recreate these boundary conditions [
14]. Wang et al. used a three point bending test to measure the snap through forces of symmetric bistable composite laminates (VPPMCS) [
15]. Dano et al. fixed square laminates in the center and then used two protruding supports applied moments to snap the laminates through from one position to the other [
16]. While floating boundary conditions allow for simple characterization of the bistability of the laminates, they are not helpful in moving towards practical functional applications of bistable composites, whereby some amount of fixed boundary conditions are necessary. With regard to the boundary conditions between symmetric and asymmetric portions within a bistable laminate, some research has been done so far. Mattioni et al. studied the effects of varying the layup pattern but only considered a laminate that was half symmetric and half asymmetric. They studied a 180 mm × 360 mm 8 ply laminate where the left hand 180 mm × 180 mm section had a layup of [0
2,90
2]
S and the right hand 180 mm × 180 mm section had a layup of [0
4,90
4]
T. Mattioni et al. also did a parametric study to investigate the effects of boundary conditions. The parameters they studied were the thickness and stacking sequence of the symmetric part with the symmetric portion fixed at the short edge. The asymmetric portion was left unchanged. They compared the results to the original laminate both unrestrained and with the symmetric portion fixed. They found that fixing one edge eliminates the post cure curvature at that edge, but that, at the free edge, up to 60% of the curvature of the unrestrained laminate is retained. They also found that, for every configuration of the symmetric portion, the longitudinal or snap through curvature was 99% of the unrestrained panel. Thus, they found that snap through curvature of the asymmetric portion remained virtually unchanged, despite changes in the symmetric portion. They did not, however, change the overall size of either the symmetric or asymmetric portion.
The goal of this study is to gain more understanding on how introducing a symmetric region between a clamped edge boundary condition and an asymmetric region affects the bistability and curvature of a combined symmetric asymmetric laminate. The motivation behind introducing symmetry is that it creates a transition between the curved edge of a purely asymmetric laminate and the straight edge of a purely symmetric laminate, allowing for better implementation of the fixed edge boundary condition. The reason why creating a transition between curved and straight is critical is because a pure asymmetric laminate exhibiting bistability can lose bistability when one edge is fixed, depending on its overall geometry, and, if two opposing edges are fixed, no bistability will be exhibited as the edges cannot be deflected. Thus, having a transition zone allows for fixed external boundary conditions to be imposed on a bistable laminate while mitigating the effect of the boundary conditions on the snap through behavior. The ability to apply fixed external boundary conditions on bistable laminates that allow for the retaining of the bistability is critical for the implementation of bistable laminates in the realm of morphing and adaptive structures. Little research has been done to gain a better understanding on how external boundary conditions affect the bistability of bistable composite laminates. This study seeks to investigate how tailoring the laminate structure through the use of a symmetric region between a clamped edge boundary condition and an asymmetric region can perhaps reduce the negative effects of implementing a fixed edge boundary condition on a purely asymmetric bistable laminate.
4. Conclusions
The goal of this study was to gain more understanding on how introducing a symmetric region between a clamped edge boundary condition and an asymmetric region affect the bistability and curvature of a combined symmetric asymmetric laminate. The results showed that, as the amount of symmetry in the laminate increased, the representative snap through curvature decreased. It was discovered that laminates with up to 83% symmetry exhibited bistability. In addition, it was found that, in laminates with up to 20% symmetry, the representative snap through curvature decreased very little. This finding is significant because it shows that up to a 20% symmetric region can be implemented between a clamped edge boundary condition without appreciable loss of snap through curvature. The experimental results showed that, for the laminates with up to 20% symmetry, there was a high amount of statistical variation. The post cure curvatures remained relatively constant, regardless of the amount of symmetry. The post cure curvatures also showed a high amount of statistical variation. The variation in the curvatures of the laminate are most likely due to the affects of moisture ingress, human-error in the experimental setup, as well as the inherent variation between laminates due to the manual fabrication process.
Simulations were conducted in Abaqus 6.14 CAE to replicate the experiments. The initial simulation results showed good correlation with the experiments but with lower curvature values. Due to the known sensitivity of the curvatures to the thermal expansion coefficient in the direction perpendicular to the fibers, , a sensitivity study was done to see the effect of changing . This sensitivity study showed that changing does indeed affect the snap through curvatures, and it was found that an value of best matched the experimental results.
Certain phenomena were observed within the experimental portion as well as the simulation portion of this study that were not investigated. The physical meaning behind the coefficients of the polynomial equation fitted to the plot of snap through curvature versus symmetric length in
Figure 10 was not investigated. Further research would need to be done to derive a physical meaning and understanding of these coefficients. It is likely that the values for these coefficients are dependent on the material properties and as well as the parameters that affect bistability, such as the laminate length, width, and thickness, as well as the lamina layup orientation.