Numerical simulation of multistable flower-shaped composite laminates with axisymmetric layups

Abstract

Multistability is the phenomenon by which a material changes shape quickly between multiple stable states upon the application of an external trigger. Typically, fibre-reinforced composites assembled into laminates with [± 45°] or [0°/90°] layup exhibit bistability. These materials have commonly rectangular geometries, restricting their integration into more complex systems such as soft robotic actuators or biomimetic devices. One approach to increase the number of stable states is to locally vary the fibre orientation while tailoring the geometry of the bilayer laminate. This strategy is explored here using flower-shaped laminates as proof-of-concept. The dimensions of the flower’s petals as well as the local fibres’ orientations are varied using local and global coordinates systems. The morphing and the number of stable states are studied using the Finite Element Method (FEM) under various mechanical loading methods. The results demonstrate that multistability can be obtained by varying the geometry and the local fibre orientations. Generally, larger width-to-length ratios for the petals are also better for generating stable states. The simulated results are compared and discussed and could be used as a benchmark for exploring such systems in experiments or for designing even more complex multistable structures to meet the needs of soft robotics or other applications.

Introduction

Programmable morphing structures are promising for creating adaptable autonomous systems that respond to external stimuli such as light, pH, temperature, pressure, etc^1,2. Although in some cases it may be required to have a slow transformation, applications such as fast and precise positioning or grabbing in robotics, emergency safety systems, energy harvesters and others require a rapid and accurate change in shape combined with strong mechanical properties^3,4. Such precise morphing can be realized with multistable structures that exhibit 2 or more stable states with a snap-through mechanism between each state. Typically, bilayers made of long carbon fibre reinforced polymers (CFRP) under the form of prepregs, with a perpendicular angle between the fibre orientation in one layer to the other, exhibit bistability⁵. This bistability is a result of the internal stresses stored in the material during the curing and the shrinkage of the polymer matrix in the bilayer⁵. However, bilayers made of CFRP typically exhibit two stable states and simple square or rectangular shapes which limit their applications in more complex systems. There is therefore a need to explore other methods and strategies to generate a larger number of stable states and more sophisticated shapes in such reinforced composites.

Several approaches have been taken to create composites with a large number of stable states. One example is Kirigami composites where slits are cut in a bilayer CFRP⁶. With one slit, the shell with two stable states transforms into a material with 4 stable states. Lattice structures made from rectangular bistable CFRP composites attached together in a cell can also create structures with stable states when the number of cells in N^7,8. Origami types of structures with Miura-ori folds can also be used to create a large number of stable states within one sheet-like material⁹. Finally, architected metamaterials with cellular structures or local bistable dome patterns also exhibit a significant number of stable states^10,11. These approaches are successful in exhibiting multistability but suffer from complex fabrication methods or limited properties. Indeed, cellular approaches or dome structures have lower mechanical properties than CFRP composites which may limit their applications. For example, lower stiffness may be a disadvantage for autonomous robotic or safety-protecting systems. Also, despite the tunability in stable states, the geometry of the specimens is still often limited to rectangular shapes which may also not be desired in application. One study explored the effect of geometry on the curvature of CFRP composites by testing trapezoidal, cruciform, diamond and elliptical planform shapes, but the samples had only 2 stable states¹². Therefore, there is still a need to explore other approaches to multistability in reinforced composites with more diverse shapes.

Recently, researchers have been exploring the fabrication of reinforced composites containing small discontinuous particulates instead of long fibres. Using short reinforcements allows the exploration of more complex reinforcement patterns than for long fibres. For example, ceramic microplatelets have been used to create bistable shells with curved reinforcements to instil bistability in curved geometrical shapes^13,14. The experimental results were found to match well with computational simulations¹⁵. Some computational studies also explored how local variations of the fibre reinforcement orientations could create new morphing states¹⁶. In addition, new fabrication approaches using additive manufacturing have led to the 3D printing of CFRP laminates with curvilinear fibres¹⁷. With the 3D printing of glass-microfibre reinforced epoxy composites in a cross shape with 4 arms, 8 stable states could be created by actuating each arm of the cross independently¹⁸. A computational model of a similar cross with local reinforcement was also developed showing similar results to the experiments¹⁹. Another study explored triangular structures made of CFRP laminates with a hole of different dimensions at its centre, to generate multiple stable states²⁰. These recent advances suggest the possibility of fabricating sheet-like materials with decoupled control of their macroscopic geometry and of their local microstructure. Tuning microstructure and geometry could be a means to create several multiple stable states in reinforced composites.

In this paper, the effects of the geometrical shape and the reinforcement pattern on the multistability of flower-shaped laminar composites using computational modelling are studied. The flower-type shape is arbitrarily chosen for demonstrating the concept but has interesting applications for grabbing systems or for fabricating lattices. Computational modelling is used as the method to explore the combined microscopic and macroscopic design of the composite. First, the effect of geometrical shape on the multistability of the laminate with conventional lay-up with [± 45°] and [0°/90°] fibre orientations is studied. Then the effect of complex microstructures with local [± 45°] and [0°/90°] orientations in each petal of the flower-shaped laminates is investigated. The stable shapes are obtained by applying and releasing mechanical loading at the petal tips after the curing process using either symmetric or asymmetric loading. Erroneous stable shapes are eliminated using the history and the relative level of the strain energy. The results from this study can be leveraged to create more complex morphing systems out of reinforced composites.

Materials and method

Parametric modelling for finite element simulation of bistable states

The system studied is a bilayer CFRP laminate composite with each layer having a thickness of 0.1 mm, with material properties listed in Table 1²¹. Other thermomechanical parameters were adopted from the literature¹².

Table 1.

Thermomechanical properties used for the composites modelled in this study, where the indices 1 refers to the direction parallel to the reinforcing fibre and indices 2 and 3 to the directions perpendicular.

Symbol	E 1	E 2
Unit	N/mm2		–	N/mm2			°C−1
Value	125,000	9000	0.33	4500	4500	4500	− 0.1 ×	27 ×	27 ×

is the elastic modulus, the Poisson’s ratio, the shear modulus and the coefficient of thermal expansion.

To understand the deformation and multistability of flower-shaped laminates, ABAQUS Finite Element Method (FEM) is used to model and solve the simulations of the curing and morphing process of multistable laminates. The bilayer laminates with 4 petal sizes were simulated in a parameterized model, where the width-to-length ratio of the petals increases with the increase in size (Fig. 1).

Fig. 1.

(A) Schematics showing the 4 sizes of the petal and their shapes, the definition of the width-to-length ratio and table with the dimensions. (B) Schematics illustrating the fibre reinforcing patterns (red lines denoting the fibre orientations) in the 2 layers of the laminates, with global and local coordinates systems and in the [± 45°] and [0°/90°] configurations. The black arrows indicate the y axis in the coordinates systems. (C) Schematics showing the points where the loads are applied to trigger the shape changes between different stable states with centre of the laminate fixed.

The modelling is classified into three stages: Preprocessing, Solver and Post Processing.

• Stage 1: Preprocessing

An automated Python Script is used for the parametric modelling of two-dimensional laminates with different petal sizes. Different architectures of these flower-shaped composite laminates are considered for analysing the variations in multistable shapes, deflections, and strain energy differences. The variables considered for these architectures are (a) petal sizes of laminates, (b) different ply angles and (c) global and local coordinate systems. The dimensions of each petal are listed in Fig. 1A. Two laminate stacking sequences are studied, [0°/90°] and [± 45°]. As analysed by Cheng et al., the mechanical behaviour of composite laminates is significantly influenced by the stacking sequence lamina – 0° where fibres aligned with the length of the material (i.e. x direction), 90° where fibres perpendicular to the length (i.e. y direction), ± 45° where fibres at a 45° angle to the length, in opposite directions²². Two coordinate systems are adopted: a global coordinate system defined for the entire shape, such that the orientation of the fibres in the petals is fixed, regardless of the direction of the petal; a local coordinate system, varying for each petal with one axis of the local coordinate system oriented along the long axis of the petal (see Fig. 1B for schematics of the coordinate systems). The simulation path for obtaining stable states is summarized in Fig. 2A.

Fig. 2.

(A) Flowchart summarizing the simulation process for obtaining the stable states (B) Mesh Refinement Plot capturing the central region of the flower shaped laminate of size 2 with [0°/90°] layup in the global coordinate system.

The laminate is first cured and cooled down from the cure temperature typically used for making CFRP, and a temperature field is applied with the temperature gradient, ΔT, set to − 80 °C which mimics the cooling process and induces mismatched strains. A ‘Static’ step is implemented to predict the equilibrium configurations after curing. The laminate is fixed at the geometric centre using encastre boundary condition which restricts displacement and rotation in the x, y, and z directions. Symmetric and asymmetric loadings are applied at the tip or at the sides of each petal, where all the petals are loaded in the same or opposite directions, respectively (Fig. 1C). The displacement loading ranged from a magnitude of 1 mm to 20 mm. The load is then released which is the criterion for determining the stability of whether the laminate still maintained the second stable configuration in the absence of a force.

The meshing of laminates is done using linear quadrilateral shell. S4R element type, which is a 4-node doubly curved thin or thick shell with reduced integration, is used to reduce computational cost. Before a suitable mesh size is established for the model, a convergency analysis is conducted to ensure the accuracy of simulation results²³. The discretization of the laminate is optimized by considering the effect of changes in mesh sizing from extra coarse to extra fine on the deflections in Z direction (Fig. 2B). Fine mesh is chosen (5632 elements) by considering factors such as consistency of results as mesh size becomes smaller and time taken for simulation to be completed.

• Stage 2: Solver

In the solver stage, a nonlinear system of equations is solved in large deformation for multistable laminates. In order to ensure convergency of the solution, customization of general solution controls is needed with the number of iteration attempts () increased to 30.

• Stage 3: Postprocessing

After obtaining the solutions, the postprocessor enables the study of FEM results such as multistable shapes, deformation magnitudes, and strain energy plots to determine the effects of geometry and layup of laminate. The shape obtained after releasing the load was recorded and used to determine the stable states of each petal size. These stable states, obtained from each loading, were compared to ensure that the loading is sufficient to trigger all the stable states.

Identification of erroneous stable states

Bistable shapes do not exist if their strain energy is not at a minimum²⁴. To remove erroneous stable shapes that might appear in the simulation results (erroneous stable shapes means shapes that are not, in reality, stable), the relative strain energy variation in the load releasing process was calculated for each shape using the following equation:

1

where represents the changes in strain energy as a percentage. and are the initial and final strain energy, respectively, where initial strain energy is calculated at the beginning of the load releasing step, and final strain energy is calculated at the end of the load releasing step when the structure rests at its final shape. Here, we used , but the correct threshold value should be obtained from experiments. The threshold of 1% was chosen for strain energy changes, , to ensure that stable states are not discarded prematurely during computational analysis. Stable states with > 1% are preserved in the analysis, while minor strain energy fluctuations (i.e. below 1%) are filtered out. This lenient criterion is useful in the initial computational phase, as it casts a wide net to capture all potential stable states without imposing overly strict limits that might overlook subtle but significant equilibria. Consequently, the threshold provides a pragmatic means to differentiate between significant structural changes and trivial variations, with the understanding that experimental validation will be crucial to refine this criterion further. Once experiments are conducted, they will reinforce the aptness of the 1% value. Thus, the 1% criterion is not a definitive parameter, but a provisional filter intended to guide further.

The profile of the shapes along the vertical axis y is recorded and plotted. After conducting the simulation, the stable state is obtained, and its strain energy are recorded. The history of the strain energy change in the load releasing process is investigated to eliminate erroneous stable states. Moreover, lower displacement loads resulted in minimal relative strain energy differences, making it challenging to distinguish between stable and erroneous states. In some laminates, higher loading was necessary to achieve a distinct value, facilitating a clearer differentiation between erroneous and stable states. Two standards have to be satisfied at the same time to ensure a correct stable state: (1) the obtained strain energy is located at the minimum; (2) the relative strain energy change is large enough ( is used in these examples).

Results and discussion

Bilayer laminates with [± 45°] and [0°/90°] layups are well studied in the literature¹. Due to their processing typically using manual lay-up of prepregs, the laminates studied have generally square or rectangular shapes and exhibit bistability with two opposite stable shapes. In this section, the effects of the geometrical shape on the multistability are studied, by simulating flower-like shapes of different petal sizes and local fibre orientation. The method of loading consists in applying displacement loads at the petals and looking at the shapes obtained after releasing the loads. This method of loading is similar to that traditionally applied to rectangular laminates in other simulation works²⁴. In addition, multistability of laminates with both global and local coordinate systems with fibre orientations at [± 45°] and [0°/90°] are explored. These two fibre orientations are chosen as they are typically used in practical applications, where [± 45°] can generate twisting and [0°/90°] bending. These different coordinate systems are used to create local microstructure which is not typically realized using CFRP due to the complexity of the processing that would be needed. However, such local microstructure with local fibre orientations could now be realized using short reinforcing fibres and extrusion-based 3D printing of long fibre reinforced composites²⁵, for example. Displacement graphs have been drawn to observe deformation profiles for various stable states. Images of shapes that have , which are considered as stable states, provide visual impressions of the deformations that can be realized.

Global coordinates: effect of geometry and layup on multistability

First, global coordinates are used to design bilayer laminates with homogeneous microstructure in-plane, where the fibre orientation is constant throughout each layer. The geometrical shape of the flower is varied, and the number of stable states and their deformation is analysed (Fig. 3).

Fig. 3.

Stable states for shells of various sizes with global coordinates system - [± 45°] layup: (A) Images of stable states for increasing weight-to-length ratio and corresponding square shape. (B) Displacement profile of stable states along y direction. [0°/90°] layup: (C) Images of stable states of the flower-shape geometry with increasing petal weight-to-length ratio and corresponding square shape. (D) Displacement profile of stable states along y direction. In (A) and (C) the colours indicate the displacements along y in mm.

In the case of flower-shaped laminates with [± 45^o] layup, all laminate sizes exhibit bistability, with consistently two stable shapes despite the increasing width-to-length ratio of the petals. The stability of the shapes is also tested for symmetric and asymmetric loadings, but no other stable states with a significant percentage change in relative strain energy are found (Fig. 3A). As the width-to-length ratio increases, the sizes eventually converge to a square shape, a commonly explored geometry in literature. Hence, a square shell laminate is simulated to compare the validity of simulation results with experimental results in literature²⁶. The square shell is inclined at a 45° angle, similar to a kite shape, to resemble sizes 1 to 4 in the flower shape. As such, a [0°/90°] layup of the square shaped laminate, when rotated counterclockwise, adopts a [± 45°] layup with respect to the y-direction. It is remarkable to notice that the bistable shapes obtained in states 1 and 2 for laminates with [± 45°] layup are similar for all sizes of the flower shape and resemble the typical shapes obtained for square bistable shells with a [0°/90°] layup which is experimentally validated in literature²⁶. Deformations obtained for size 1 are slightly lesser than that of the other larger sizes. It is noticeable that the deformations obtained for states 1 and 2 of sizes 2, 3 and 4 are identical (Figs. 3B). This evidence suggests no impact of the geometry on the deformations for the shapes studied here above size 1, which is also observed in square laminates²⁷.

By changing the flower shaped laminates to a [0°/90°] layup, the laminates maintained their multistability with 3 stable shapes, except for the size 1 shape with [0°/90°] coordinates which has only 1 stable state (Fig. 3C). The loss of multistability in this laminate is likely due to the small dimensions of the central region of the laminate and the comparatively longer petal shape, characterized by the small width-to-length ratio of the petal. Indeed, it has been reported that smaller laminates could lose their bistability due to insufficient internal stress²⁷. A smaller width-to-length ratio of petals can enhance the mobility and flexibility of the laminates, resulting in a reduction of stiffness. As such, loss of multistability could also be due to very large deformations of the laminates, above 40 mm, which causes the structure to be unstable and collapse due to reduced structural stiffness.

Furthermore, it can be observed that bistable shapes obtained in states 2 and 3 for the [0°/90°] layup are similar for the sizes 2 to 4 and resemble the typical shapes obtained for square bistable shells with a [± 45°] layup (Fig. 3C). As aforementioned, a square shell is inclined at a 45° angle to resemble sizes 1 to 4. As such, a [± 45°] layup of the square shaped laminate when rotated counterclockwise adopts a [0°/90°] layup with respect to the y-direction. It is shown that when the laminate geometry converges to the typical square, the stable shapes also converge to two cylindrical shapes²⁸, which is also correct for the layup of [± 45°] illustrated in Fig. 3C. This is further confirmed with the displacement along the x and y-axis (Fig. 3D), with the maximum displacement at the two opposite petal tips while the centre of the laminate is fixed. An additional stable shape, state 1, with out-of-plane deformations along the y-axis and in-plane deformations along the x-axis of the laminate is observed for all sizes except for square shape. State 1 has deformations that resemble both state 2 and 3, being an intermediate shape. This shape is noted to resemble the post-cured shape of the laminate. State 1 disappears for the square laminate possibly due to the projected low deformation magnitudes. This is confirmed by the displacement profiles which indicate that as the petal size increases, due to increasing width-to-length ratios which causes a lack of mobility of individual petal tips, a reduction in deformation magnitude at the tips is observed (Fig. 3D).

Overall, a maximum of 3 stable states are obtained using the global coordinate systems for the [0°/90°] layup and 2 stable states for the [± 45°] layup. It can be concluded that there is a weak correlation between the size variations and the number of multistable states for the laminates with global coordinates. The potential of these flower shaped laminates is further expanded through the use of local coordinate system to vary the stable shapes obtained. This approach which involves a symmetric laminate stacking sequence, is explored in the next section.

Local coordinates: effect of geometry and layup on multistability

To explore more complex microstructures, the same flower geometries with petals of varying width-to-length ratios are simulated with local coordinates, leading to local reinforcing fibre alignment (Figs. 4 and 5). In the local coordinate system with the layup of [± 45°], as the size of laminate increases, the number of stable states rises from 5 to 8. Sizes 1 and 2, with width-to-length ratios of 0.3 and 0.5, have a constant number of stable states which is equal to 5. Three additional stable states appear for sizes 3 and 4 when the width-to-length ratios increases from 0.5 to 1.0.

Fig. 4.

Stable states for shells of various sizes in local coordinates system with [± 45°] layup: (A,B), States 1–3: Images and profile of stable states along y direction. (C,D), States 4–5: Images and profile of stable states along y direction (E,F), States 6–8: Images and profile of stable states along y direction. In (A), (C) and (E) the colours indicate the displacements along y in mm.

Fig. 5.

Stable states for shells of various sizes in local coordinates system with [0°/90°] layup: (A) Images of stable states, (B) profiles of stable states along y direction. In (A), the colours indicate the displacements along y in mm.

State 1 morphs with a maximum displacement at the petal tips, and the deformation is symmetrical about the fixed centre for all sizes of laminates (Fig. 4A, B). The petal tips are pointing upwards and inwards towards the centre due to the symmetrical deformations. Maximum displacements for state 1 for sizes 1 to 3 are close in magnitude, with difference ranging below 1 mm. Size 4 has the highest maximum with a displacement of 8.79 mm for state 1. States 2 and 3 have positive out-of-plane deformations with curvatures that are perpendicular to each other. Unlike state 1, states 2 and 3 have an asymmetrical deformation with two opposite tips of the laminates sharing equal magnitudes of displacement, different from the two remaining tips. Asymmetry lies in the centre of the laminate as well, resulting in two opposite curvatures. When compared across all sizes of laminates for states 2 and 3, size 4 laminates consistently contribute to the highest maximum deformations. Noticeably, strain energy calculations show that state 1 of size 3 has a low percentage change in strain energy with the value of 0.035% which is lower than the threshold of 1%. This could potentially result in the loss of the shape during experimental validation, resulting in the shape to be erroneous.

States 4 and 5 also share similar pattern as states 2 and 3 where their curvatures are perpendicular to each other (Fig. 4C, D). However, these curvatures vary in the direction with a 45° clockwise inclination from the centre point. Although sizes 1 to 3 share similar deformations patterns for states 4 and 5 where maximum displacements are at the petal tips, size 4 varies from this trend. Size 4 has deformations where maximum displacements do not occur at the petal tips which results in 2 bands of curvatures. While all the other sizes of laminates, from 1 to 3, have deformations that allow the petal tips to be pointed upwards, size 4 laminates have petal tips pointing downwards, in the in-plane direction.

States 6 to 8 only exist for sizes 3 and 4 with width-to-length of 0.8 and 1.0 (Fig. 4E, F). State 6 has a similar pattern as state 1 where deformations are symmetrical for each petal of the laminates, but these tips are pointing in the in-plane direction. States 7 and 8 have 2 bands of curvatures, similar to states 2 and 3 but the curvatures are in the opposite direction, in the in-plane direction. The deformation for states 1 to 5 is observed to be larger in magnitude than that of states 6 to 8. States 6 to 8 have small deformations, with less than 3 mm in the out-of-plane direction and 4 mm in the in-plane direction. As the width-to-length ratio decreases beyond 0.5 for sizes below size 2, multistability is lost as characterized by the loss of states 6 to 8. It has been noted prior in Sect. 3.1 that as the width-to-length ratio decreases with petal sizes, there is loss of internal stress coupled with higher flexibility and lower structural stiffness. Hence, when deformation magnitudes of the laminates in states 6 to 8 are too low, below 3 mm, the structure becomes unstable and is not able to retain its multistability. Therefore, is a reduction in number of multistable states size 2 and below.

For local coordinates with [0°/90°] layup, the results are very different than for the other layups, with the number of stable states increasing from 1 to 4 as width-to-length ratio increases from 0.3 to 1.0 (Fig. 5).

State 1 is consistent for all sizes, with size 1 having the maximum deformation magnitude of 29.7 mm in the positive out-of-plane direction. This likely results from its flexibility due to low width-to-length ratio. Size 4 has the smallest displacements due to a relatively large width-to-length ratio resulting from the limited mobility of the petal tips. In state 1, for all sizes, all petal tips point upwards and inwards towards the centre, in the out of plane direction. States 2 and 3 only exist for sizes 2 to 4 where their curvatures are perpendicular to each other, and they consist of 2 bands of curvatures. The maximum displacements are at the two opposite tips of the flower shaped laminates and the minimum displacements occur at about 50 mm from the top and bottom of the laminate, along the y-direction in the laminate. This deformation pattern results in 2 bands of curvatures for states 2 and 3. State 4 is consistent for sizes 3 and 4 only and has similarities to state 1 in terms of all the petal tips pointing in the same direction but varies by having deformations in the in-plane direction, thus point downwards and inwards towards the centre. As aforementioned, loss in the number of stable shapes at smaller sizes are due to insufficient strains in laminates and reduced structural stiffness coupled with low deformation magnitudes.

Overall, the local coordinate system in the flower shaped laminates induces the creation of more stable shapes as they have a strong dependency on size variations and laminate layups. This strategy of using local fibre orientation in bilayer laminates could be used to induce unique shapes for complex soft robotic applications, for example.

Discussion

The simulation results presented in this paper demonstrate that new and unusual stable shapes can potentially be obtained by using local layups in complex geometries in reinforced laminates, and by triggering morphing mechanism using various modes of loading. Using the simple transformation from global to local coordinates presented earlier, with a four-petal flower shapes, a maximum of 8 stable states could be created for a [± 45°] layup and 4 stable states for [0°/90°] layup, respectively. The correlation between geometrical shape and local reinforcement orientation can be further complexified as compared to the proof-of-concept cases studied here. In fact, there is an infinite possibility of local microstructures for each geometrical shape. To find which local layup would be needed to generate a specific deformation, reverse engineering algorithms could be investigated. A tunnelling method has been recently used for reverse design of bistable shapes²⁹.

The results also suggest that if a local coordinate system is used, multistability becomes strongly influenced by the local microstructures, coupled with size changes. For local [± 45°] layup, the number of stable states remain constant at 5, for sizes 1 and 2 and rises to 8 for sizes 3 to 4. The 3 additional shapes with deformation magnitudes lesser than 4 mm in both in-plane and out-of-plane directions, were able to maintain their bistability for laminates with larger width-to-length ratios, suggesting that the multi-stability is lost for these 3 states in the remaining sizes which have a lower width-to-length ratio. For the local [0°/90°] layup, the number of multistable states increases from 1 to 4 as the width-to-length ratio of the petals increases. State 1, for all sizes, morphs with large deformations, with the maximum deformation for size 1 close to 30 mm. However, for states 2–4, the laminates, for all sizes, have a significantly lower deformations relative to State 1, with the majority of the deformation values ranging below 5 mm with the exception of State 2 for Size 2 laminate. Hence, when projecting this pattern for smaller sizes of laminates, the low deformations coupled with a lower width-to-length ratio that results in highly flexible petals, cause the morphing shapes to be unstable and collapse, eventually leading to a loss in multistability. This explains the decrease in the number of stable states when size of laminates decreases. However, once transitioning to a global coordinate system, the multistability has no or minimal dependence on size variations of laminates. Simulation results for both global [± 45°] layup and [0°/90°] layup is further validated using square laminates that are commonly explored in literature.

Certain simulations with loadings at specific positions are time consuming, resulting in a lengthy convergence time. In such cases, the magnitude of loadings to induce bistable shapes is lowered, which produces results at a faster pace. Erroneous shapes are identified if the relative percentage change in strain energy is below a threshold value of 1%. As such, state 1 of size 3 for local coordinates [± 45°] layup is identified as an erroneous state that might not possibly exist experimentally. The scope of the study could be further expanded by investigating other types of loading, such as loading on 3 petals. Imagining a 6-petal flower shape, one can expect stable states from loading all the petals at the same time, from loading 2, 3, 4 or 5 petals at the same time, therefore multiplying the number of stable states. The reason for the increased number of stable states likely stems from the coupling of local microstructure with geometry leading to complex internal strain patterns.

Gravity was not considered in this study. Gravity both assists and hinders the snap-through effect for multistable laminates. In a paper by Jia-Jia Mao et al., gravity significantly influences how the structure snaps from one stable configuration to another³⁰. Gravity biases the deformation path by reducing the energy barrier for snap-through transitions, which helps control the order in which the different stable states are reached. This means that, when controlled effectively, gravity can be used as a design feature to ensure that the metamaterial reliably transitions between its multistable states. However, the potential effects of gravity explored by Liesbeth Arnouts et al. reveal that structures with gravity’s influence required loading for snap-through is positive, and without external loading, the structure moves back to its deployed configuration due to self-weight³¹. However, for the snap-through effect, a negative load-displacement curve is important to allow the self-stabilizing effect of snap-through. As such, when the laminate is scaled up to the meter scale, gravity could have potential effects on the multistability, and further research is required to understand how it affects the bistable shapes.

The results presented in this work are obtained by simulation only. Experiments could be conducted to seek validation. Discrepancy between the experiments and the simulated results may occur due to a higher threshold in the relative change in strain energy. The simulation conducted here are also based on traditional CFRP composites. The process of such long fibre composite and the nature of the long fibres itself would not allow easy fabrication of laminates with local orientation of the fibres. Current technology development in the field of 3D printing could potentially realise laminates with very complex microstructures and in complex geometries. However, the exact properties of the fibres and matrix may change depending on the process and materials used. The simulations conducted here could therefore be reproduced by adjusting these parameters in the model, to obtain more accurate validation, and open the way for the design of fast-morphing materials for various practical applications.

Embedding layups of [± 45°] and [0°/90°] angles experimentally, with global coordinate system in rectangular composite laminates, is well explored in academia^32,33. However, with the geometry of the laminate being modified into a flower shaped one and fibres following the local coordinate systems are relatively complex to construct, the fabrication of these layups will have to be done with the use of 3D printing to ensure the accuracy of fabrication. Curvilinear fibres have been explored and have been employed in the construction of bistable laminates—as such fibres following the local coordinate system could also use the same method to be fabricated. Zheng Zhang et al. analysed the morphing characteristics of variable stiffness laminates which was devised using 3D printing technology³⁴. The curvilinear fibre path of the variable stiffness laminate to be printed was designed in Simplify3D slicing software, and the corresponding G-code was exported to control the print nozzle movement to fabricate curvilinear fibres³⁴. One fabrication of laminates is completed, laminates are cured and cooled after which the geometric centre of the laminate is fixed using a clamp, with loading types exhibited on these laminates in the z-direction to observe the stable states. A tensile test machine is used to apply mechanical loading on the specimen to trigger snap-through and snap-back processes. The induced load is measured against the displacement of the laminate, resulting in a load-displacement curve. Discrepancies between simulation and experimental results due to manufacturing imperfections, such as resin-rich areas and ply-thickness variation should be considered.

Conclusions

This paper proposes the utilization of non-linear simulations to investigate the effect of (i) fibre orientations and (ii) sizes of petals on flower-shaped laminates on the multistability and deformations of bilayer laminates. To compare the differences in stable shapes obtained between sizes of flower-shaped laminates with the width-to-length ratios varying from 0.3 to 1.0 and fibre orientations of [± 45°] and [0°/90°], a total of 128 simulations were conducted using the FEM method. For a typical square laminate in global coordinate system, the simulation results led to two cylindrical stable shapes, consistent with literature results, thereby demonstrating the effectiveness of the proposed procedure in obtaining stable shapes.

Multistability for a global coordinate system has a weak relationship with size variations of the flower-shaped laminates. As petal sizes increase, the number of stable shapes remains at 2 for laminates with the layup of [± 45°] and 3 for laminates with the layup of [0°/90°], except size 1 which is monostable, which loses multistability due to low deformation magnitudes coupled with a low width-to-length ratio of smaller sized laminates. One additional shape is observed for laminates with the layup of [0°/90°] for all sizes and this shape resembles the post-cured shape of the laminates. However, size variations of laminates have a strong influence on the multistability of local coordinate system. As petal sizes increase, the number of stable shapes increases from 5 to 8 for laminates with the layup of [± 45°] and 1 to 4 for laminates with the layup of [0°/90°]. The reduction in the number of multistable states as laminate size decreases is primarily due to extreme deformation magnitudes—either very small (below 6 mm) or very large (above 40 mm). This factor, combined with smaller width-to-length ratios of laminates, lead to high flexibility and a significant reduction in the structural stiffness of the laminates, ultimately causing the morphing shapes to collapse due to instability.

Displacement profiles are gathered to understand the curvatures of the laminates and how low deformations affect the stability of shapes at smaller sizes with low width-to-length ratios. Deformation changes along y-axis with magnitudes close to 0 mm, might not exist experimentally due to imperfections present in the real-model laminates. Relative strain energy is also analysed, and it can be concluded that loading influences the strain energy of the laminates. Laminates with percentage change in strain energies below 1% are considered as erroneous states.

To conclude, multistable laminates explored in this paper offer significant advantages with their vast range of robotic applications. Unlike traditional actuation methods that require continuous energy input, multistable composite laminates leverage stored strain energy to achieve morphing capabilities, thus reducing power consumption in soft robotics, reconfigurable antennas, and energy harvesting devices. With results presented in this paper, the size of laminate and fibre orientation can be optimized for task specific applications such as for grabbing systems or for fabricating lattices in energy efficient robotic applications.

Author contributions

P.J.: Software, Investigation, Visualization, Writing - Review & Editing, Data Curation. Methodology, Formal Analysis. W.D.: Conceptualization, Methodology, Formal Analysis, Supervision, Project administration, Resources. H.L.F.: Visualization Writing - Original Draft, Project administration, Funding acquisition.

Data availability

Data is provided within the manuscript or can be requested to the corresponding author.

Declarations

Competing interests

The authors declare no competing interests.