## Abstract

Conventional shape-changing engineering structures use discrete parts articulated around a number of linkages. Each part carries the loads, and the articulations provide the degrees of freedom of the system, leading to heavy and complex mechanisms. Consequently, there has been increased interest in morphing structures over the past decade owing to their potential to combine the conflicting requirements of strength, flexibility and low mass. This article presents a novel type of morphing structure capable of large deformations, simply consisting of two pre-stressed flanges joined to introduce two stable configurations. The bistability is analysed through a simple analytical model, predicting the positions of the stable and unstable states for different design parameters and material properties. Good correlation is found between experimental results, finite-element modelling and predictions from the analytical model for one particular example. A wide range of design parameters and material properties is also analytically investigated, yielding a remarkable structure with zero stiffness along the twisting axis.

## 1. Introduction

### (a) Background

Morphing structures are currently receiving significant interest from the engineering community, especially within aerospace research, owing to their variable geometry, low weight and reduced overall complexity. Many concepts for morphing structures depend on continuously powered actuators to deform the structure (Ashwill *et al.* 2002; Bak *et al.* 2007, 2009; Daynes & Weaver 2011). However, if the structure possesses multiple stable states, i.e. it is multi-stable, power is needed only to change the shape, not to hold it. Several studies investigated the application of bistable structures to aircraft and rotor blade flaps, as a means of increasing flight efficiency by switching from one mode of flight to another (Mattioni *et al.* 2005, 2008; Diaconu *et al.* 2008; Schultz 2008; Daynes *et al.* 2009, 2010*a*,*b*). Deployable multi-stable structures are also an active field of research. The bistable stowable boom from Daton-Lovett (2001) is one example and has been extensively investigated by Iqbal and co-workers (Iqbal & Pellegrino 2000; Galletly & Guest 2004*a*,*b*; Kebadze *et al.* 2004; Pellegrino 2005; Guest & Pellegrino 2006; Guest *et al.* 2010; Seffen & Guest 2011). In a general approach, multi-stability is achieved by designing a structure that has multiple total potential energy minima. When using fibre-reinforced plastic (FRP), this can occur owing to several phenomena: the combination of residual stresses induced during the cure cycle and geometric nonlinearity in a non-symmetric laminated FRP lay-up (Hyer 1981, 1982; Dano & Hyer 1998; Potter & Weaver 2004), Gaussian curvature effects (Seffen 2007), fibre pre-stress (Daynes *et al.* 2008, 2010*a*) or plastic deformation (Guest & Pellegrino 2006).

This paper is concerned with a class of bistable structures able to achieve overall large deformations in length while the strains everywhere remain small and below the elastic limit. The device is made of at least two pre-stressed flanges combined using stiff connecting spokes, as shown in figure 1*a*. The pre-stress of the structure, resulting from the flattening of originally curved flanges, is captured through the change of curvature of these parts. The large deformations of the structure are analysed through the evolution of its strain energy and draws upon modelling work by Guest and co-workers. They presented several detailed analyses related to the inextensional deformation of thin isotropic and orthotropic cylindrical shells. Pre-stressed and stress-free shells were investigated. The bistability of the structures was achieved either by using the orthotropy of FRPs or by setting up an initial state of self-stress in isotropic shells (Iqbal & Pellegrino 2000; Guest & Pellegrino 2006; Guest *et al.* 2010; Seffen & Guest 2011). Here, inextensional deformations are also assumed, resulting in a compact one-dimensional analytical model. However, the design space is expanded by both applying pre-stress (through the geometry of the flanges and structure) and using anisotropic material. This paper focuses on the investigation of the combination of material properties and structure geometry, which has not been explored in a systematic way for such structures to date, and so reveals new bistable forms, enhancing the design and capabilities of morphing structures. Ultimately, novel bistable devices are shown with periodically nonlinear, yet tailorable, deformation responses.

The paper is organized as follows: §2 describes the device and the geometrical relationships between the different parameters of the structure. Section 3 details the analytical expression of the strain energy and the assumption of inextensional deformation associated with the mathematical model. The change of curvature of the structure is explained, and the expressions of the stable equilibria of the structure are derived. Four structures with different material properties are presented in §4, each with different stability characteristics. Analytical expressions for the axial force and stiffness developed by the structure are derived in §5. Twist moment and torsional stiffness are detailed in §6. The manufacture of one prototype and the experimental set-up used to test it is presented in §7, and a finite-element model (FEM) of the same structure is described in §8. Results from the analytical model, FEM and experimental work are presented and discussed in §9. Section 10 presents concluding remarks.

### (b) Definitions

Lay-up describes the arrangement of the layers (plies) of composite material constituting a laminate. It is expressed as a series of numbers between brackets defining the angles of the principal direction of each layer with regard to a reference axis. When the same layer is repeated several times, the number of layers is indicated by a subscript: [*χ*_{5}], where a Greek letter is used to indicate the ply angle. A lay-up is called symmetric when layers of the same properties, thickness and ply angle are positioned symmetrically with regard to the mid-plane of the laminate: [*λ*/*ψ*/*γ*/*γ*/*ψ*/*λ*]. On the contrary, an anti-symmetric lay-up is defined as [*λ*/*ψ*/*γ*/−*γ*/−*ψ*/−*λ*].

## 2. Multi-stable composite twisting structure

### (a) Description

In the straight configuration, shown as light grey in figure 1*a*, the structure consists of two flat flanges kept apart by a set of rigid spokes. The device can twist by applying opposite moments about the *x*-axis at the extremities of the structure, resulting in the configuration shown in black in figure 1*a*. Multi-stability is achieved by imposing a state of pre-stress to the flanges. In the present case, a distributed bending moment is introduced by manufacturing the parts on a cylindrical mould and then by flattening them, as illustrated in figure 1*b*.

### (b) Geometric properties

To ease the calculations in §§5 and 6, the following geometric relationships are introduced. Figure 2*a* shows the geometry of one flange in a twisted configuration under an axial force *F* or a twist moment *M* and where *L* is the flange's length, *d* is the projected circumferential arc length, ℓ is the projected axial arc length, *ϕ* is the angle of twist, *θ* is the angle of helix and *R* is half the height *H* of the structure. Writing the relation
2.1
yields the expression of the angle of twist *ϕ* as a function of the angle of helix *θ*,
2.2
We define an overall rate of twist for the structure given by *δ*=*ϕ*/ℓ with ; thus, the rate of twist is
2.3
The displacement Δℓ of one end of the structure from the straight configuration of the assembly is
2.4
Figure 2*b* shows the variation of equations (2.3) and (2.4) as a function of *θ* and highlights the nonlinearity of the structure's response to axial displacement.

## 3. Inextensional model

### (a) Strain energy

Inextensional deformations are purposefully designed because these correspond to flexural rather than membrane (stretch or contraction) strains. It is noted that flexure facilitates relatively large deformations for a given strain energy, a desirable trait for a morphing structure. Furthermore, because the spokes keep the flanges at a constant distance throughout the transformation, we can consider the flanges to lie on an underlying cylinder of constant diameter equal to the spoke length *H*=2*R*, as shown in figure 1*a*. Any deformations on the cylindrical surface are inextensional and are characterized by zero Gaussian curvature changes (*K*_{x}×*K*_{y}=0, where *K*_{x} and *K*_{y} are the curvature in the *x*- and *y*-directions of the surface, respectively); therefore, the strain energy of the assembly includes bending deformations only and is given by (Kollar & Springer 2003)
3.1
where Δ*κ*^{T} is the transpose of Δ** κ** modelling the change of curvature of the flanges.

**D***=

**D**−

**BA**

^{−1}

**B**is the reduced flexural stiffness matrix of the flange and is defined in classic lamination theory (Jones 1999), where

**A**,

**B**and

**D**are in-plane, coupling and flexural stiffness matrices, respectively. The dimensions

*L*and

*W*are the length and width of the flange, respectively, as shown in figure 1

*a*, and

*n*is the number of flanges.

The inextensional model used in equation (3.1) implies two major assumptions: the flanges' mid-surfaces do not stretch and the mid-surfaces of the flanges deform uniformly, following the surface of a cylinder of constant radius *R*. These assumptions allow us to use only two parameters to define every possible configuration of the structure: the curvature 1/*R* of the underlying cylinder and the orientation *θ* of the local axes (*x*,*y*) attached to each flange with respect to the axis of the cylinder, as shown in figure 3*a*. Finally, the variation of strain energy of the spokes is neglected owing to their much higher stiffness compared with the stiffness of the flanges.

Moreover, this model assumes that the flanges are free from geometrical defects. In particular, it does not include thermal effects occurring during the high-temperature cure of composite laminated parts.

### (b) Change of curvature

The evolution of the curvature of the flange, illustrated in figure 3*a*,*b*, is expressed by the second-order tensor Δ** κ** and can be found using Mohr's circle of curvature change (Calladine 1983) for any

*θ*in the (

*x*,

*y*) coordinate system, as shown in figure 3

*b*. It is worth noting that the flange is considered as a one-dimensional shell element owing to its narrow width; hence, the change of curvature in the

*y*-direction Δ

*κ*

_{y}is neglected for any angle

*θ*. For ease of analysis, we define the ratio

*α*=

*R*

_{i}/

*R*relating the manufactured radius

*R*

_{i}of the flange and the radius

*R*of the underlying cylinder. If the flange had no initial curvature,

*α*would take the value of infinity and the circle would be shifted to the right, as shown by the dashed circle in figure 3

*b*.

From figure 3*b*, the changes of curvature of the flanges with regard to their manufactured shape can be deduced. In the initial (straight) configuration of the structure, i.e. *θ*=0, the changes of curvature of each flange are Δ*κ*_{x}=−1/*αR*, Δ*κ*_{y}=0, and Δ*κ*_{xy}=0. For non-zero values of *θ*, , Δ*κ*_{y}=0, and .

Thus, the change in curvature to any configuration of a flange can be described by the tensor Δ** κ**,
3.2

### (c) Stable equilibria

The stable equilibria are found by minimizing the expression of the strain energy *U* with respect to the two variables *θ* and *α*. We expand *U* as a Taylor series and consider its variations *δU*, for small variations *δθ* and *δα* of *θ* and *α*, respectively (Riley *et al.* 2006),
3.3
Equilibrium configurations are found when the first derivatives of *U* with respect to *θ* and *α* are equal to zero; hence, the first two terms of Taylor's expansion (3.3) are
3.4

To determine whether or not the equilibria found by equations (3.4) are stable, we consider the second term of Taylor's expansion (3.3) and verify the positive definiteness of the matrix **M**,
3.5
It is worth noting that the matrix **M** is, in fact, a stiffness matrix of the structure. **M** is positive definite if the following three conditions are met:
3.6
When considering a particular value of *α*, equations (3.4) and (3.6), respectively, simplify to
3.7

## 4. Strain energy tailoring

We now consider the stability of the structure as a function of the angle of helix *θ*, the **D*** matrix of the flanges as well as the ratio *α*. The study of the **D*** matrix is divided into two general cases: the symmetric lay-up [*β*/*β*/0/*β*/*β*] and the anti-symmetric lay-up [*β*/*β*/0/−*β*/−*β*], where *β* is the ply angle with regard to the *x*-axis of the flange. The 0^{°} ply is inserted in the middle of each lay-up to ensure a minimum strength and to avoid delamination issues. Five layers are used to ensure a significant strain energy variation during structural deformation, using the material properties given in table 1. Each case is illustrated by two investigations of lay-up and variations in the ratio *α*. Three-dimensional polar representations and contour plots are used to assess the variations of *θ*, *β* and *α*. As the structure cannot physically achieve configurations for *θ*>±*π*/2, *θ* is limited to the range −*π*≤2*θ*≤*π*. The ply angle *β* ranges from 0^{°} to 180^{°} to cover the entire range of ply angles. Regarding *α*, values less than unity yield extreme levels of strain energy and values above 5 do not present much interest because the stability behaviour does not change; therefore, the range is limited to 1≤*α*≤5. For each lay-up, the particular case of *α*=2 is analytically detailed, as this has interesting stability characteristics. The general form of the strain energy expression is found by substituting (3.2) into (3.1), and then proceeding with the first and second derivatives of *U*, giving
4.1
4.2
and
4.3
where are the stiffness components of the **D*** matrix, with *i*=1, 2, 6 and *j*=1, 2, 6.

### (a) Symmetric lay-up [*β*/*β*/0/*β*/*β*]

Figure 4*a*,*b* is surface and contour polar plots, respectively, of strain energy, given by equation (4.1), for *α*=2, where the circumferential axis represents the angle of helix *θ* and the radial axis is the ply angle *β*. Note, for all numerical examples given in this article each flange has dimensions *L*=*π*×*R*_{i}=179 mm by *W*=9 mm.

Peaks on the surface plot of strain energy correspond to unstable configurations, whereas valleys indicate stable equilibria. Interestingly, the stable configurations rotate with increasing *β*, as shown in figure 4*b* where the dotted lines represent the path of the stable configurations. As such, it is possible to create stable positions at any predefined helical angle by carefully choosing *β*, a feature that may be exploited in design. Both cases of *β*=45^{°} (yielding stable configurations at points C, D) and *β*=0^{°}≡180^{°} (yielding stable configurations at points A, B and points E, F, respectively) are detailed below.

The lay-up [45/45/0/45/45] is chosen to illustrate the effect of variable *α*, as it contains a fully populated reduced bending stiffness matrix; thus, representing a general case,
4.4
The stability of the structure for values of *α* greater than 2 can be found by examining the limit of equation (4.2) when *α* tends to infinity and checking the validity of equation (4.3) for this case. This yields two stable configurations at *θ*=0^{°} and *θ*=240^{°}. Figure 5*a* shows the case of *α*=5, where stable configurations are at 2*θ*=12.4^{°} (point G) and 2*θ*=217.4^{°} (point H) for the present lay-up.

For the simplified case of *α*=2, the solutions for the equilibria are given by substituting the values from the **D*** matrix (4.4) into equation (4.1) and solving equations (4.2) and (4.3) for −180^{°}≤2*θ*≤180^{°}, yielding
4.5
and
4.6
thus, *θ*=13.2^{°} and *θ*=−76.2^{°}. These correspond to the stable positions C and D shown in figures 4*b* and 5*b*. Unstable positions I and J are solutions of equation (4.5) only.

The stability of the simple lay-up [0_{5}] is also investigated in order to compare the analytical prediction with the measurement made on the prototype, as detailed in §9. With the properties given in table 1, the **D*** matrix is
4.7
noting the zero values of and .

For the case of *α*=2 corresponding to the manufactured prototype, the equilibria are obtained from the simplified equations (4.2) and (4.3). Limiting the domain to −180^{°}≤2*θ*≤180^{°} yields the solution
4.8
and
4.9
yielding *θ*=±45^{°}. Note that for this case
4.10
The stable equilibria are labelled A and B on the polar plot (figure 4*b*). The unstable equilibria are positioned at 2*θ*=0^{°}, 2*θ*=180^{°} and are solutions of equation (4.8) only.

### (b) Anti-symmetric lay-up [*β*/*β*/0/−*β*/−*β*]

Figure 6*a*,*b* presents strain energy plots for the set of anti-symmetric lay-ups [*β*/*β*/0/−*β*/−*β*]. Figure 6*b* shows that stable equilibria can take only two positions: 2*θ*=±90^{°} (dotted lines K, L, and O, P) or 2*θ*=0^{°} and 180^{°} (dotted lines M and N). The 90^{°} shift in stable configuration, as *β* increases, is achieved after reaching a constant strain energy level for the entire deformation (shown by the dotted circles at *β*≈23^{°} and *β*≈180−23=157^{°}). Constant strain energy deformations are analytically investigated in §4*c*.

To contrast with the symmetric lay-up studied in §4*a*, the lay-up [45/45/0/−45/−45] is chosen to study analytically the stability of such a structure. This lay-up yields the reduced bending stiffness matrix
4.11
Noting and , the stability of the structure for increasing *α* is given by examining the limit of equation (4.1) when *α* tends to infinity and by solving equations (4.2) and (4.3). This yields two solutions: 2*θ*=0^{°} and 2*θ*=180^{°}. Figure 7*a* shows the two stable configurations when *α*=5, noted by points Q and R. For the case *α*=2, figure 7*b* shows stable configurations M at 2*θ*=0^{°} and N at 2*θ*=180^{°} (also shown in figure 7*a*). These are also points of the dotted lines N and M, respectively, in figure 6*b*. The analytical expression for these equilibria is given by equations (4.8) and (4.11). However, for the present lay-up, the inequality in the expression (4.10) reverses to become
4.12
Thus, equation (3.7), with these conditions, and limited to the domain −180^{°}≤2*θ*≤180^{°}, simplifies to
4.13
and
4.14
yielding *θ*=0^{°} and *θ*=±90^{°}. Points S and T in figure 7*b* are unstable equilibria and are solutions of equation (4.13) but not equation (4.14).

Examining equations (4.12) and (4.13), the stability of the anti-symmetric structure is fully captured by the ratio . Indeed, there are three distinct stability characteristics depending on the value of this ratio. First, for (equation (4.10)), the stable configurations of the structure are given by equation (4.9) and correspond to the dotted lines K, L and O, P shown in figure 6*b*. On the other hand, for (i.e. equation (4.12)), stable equilibria are represented by the dotted lines M and N, as per equation (4.14), and are at 90^{°} to the previous stable configurations. Finally, for (i.e. equation (4.16) below), zero stiffness is observed for all twist angles, corresponding to the dotted circles in figure 6*b*.

### (c) Zero-stiffness twisting structure

Figure 6*b* shows two contours of constant strain energy, resulting in zero torsional stiffness along the twist axis of the structure (as demonstrated in §6). Setting *α*=2, a constant strain energy level is given by equation (4.13) as
4.15
Hence the solution is given by the following condition on the stiffness terms:
4.16
Setting the lay-up to [*β*/*β*/0/−*β*/−*β*], the solution, as a function of *β*, is found by writing equation (4.16) in terms of the material invariants and the six lamination parameters, yielding two solutions: *β*=22.96^{°} and *β*=180^{°}−22.96^{°}=157.04^{°}. The reader is referred to the work by Bloomfield *et al.* (2009) for the use of lamination parameters and lay-up optimization techniques. Thus, one (of many) solution yields the lay-up [22.96/22.96/0/−22.96/−22.96], with stiffness matrix **D***:
4.17
The strain energy for this particular lay-up is shown in figure 8*a*, where the edge of the surface describes a flat circle denoting the constant level of strain energy for any helix angle −180^{°}<2*θ*<180^{°}. This is also observed for *α*=2 in the contour plot shown in figure 8*b* and is highlighted by the dashed circle.

### (d) Summary

The form of the strain energy expression reveals that the positions of stable equilibria are a function of the bending stiffnesses and ratio *α*. Different stability behaviours were noticed in the analysis of the symmetric [*β*/*β*/0/*β*/*β*] and anti-symmetric [*β*/*β*/0/−*β*/−*β*] lay-ups. First, when bend–twist coupling terms are present, as in a symmetric lay-up with off-axis fibres, the position of the equilibria can take any value of −180^{°}≤2*θ*≤180^{°}, as shown by the dotted curved lines in figure 4*b* representing the path of the stable configurations for the [*β*/*β*/0/*β*/*β*] lay-ups. Second, the positions of the equilibria of a structure made of an anti-symmetric lay-up [*β*/*β*/0/−*β*/−*β*] take the values of either 2*θ*=0^{°}, *π* or 2*θ*=±90^{°} by tailoring the ratio . Moreover, constant strain energy deformations are noticed between the two ‘opposing’ structural behaviours of the anti-symmetric lay-ups, as shown in figure 6*b*. Finally, it is worth noting the particular case of *α*=1, corresponding to no pre-stress in the coiled configuration; thus, always yielding a stable configuration at 2*θ*=180^{°}.

A movie showing the twisted stable equilibria, the straight and coiled stable configurations as well as the near-zero axial stiffness of initial prototypes can be found in the electronic supplementary material.

## 5. Axial force

The axial force *F* necessary to deform the structure is derived from equation (4.1). The strain energy is equal to the work *W*_{EXT} done by the external forces to deform the device (Megson 2007),
5.1
In the present case, the external force is the axial force *F* applied along the *x*-axis, as shown in figure 2*a*. Re-writing *U* as a function of the displacement Δℓ (equation (2.4)), the axial force *F* is expressed as (using Castigliano theorem)
5.2
The axial stiffness *k* of the structure is then given by
5.3

## 6. Twist moment

The structure can also be deformed under the application of a twist moment *M*, as shown in figure 2*a*. Similar to §5 for axial force, a change of variable is performed with respect to helical angle, *θ*, and the strain energy *U* is written as a function of the angle of twist *ϕ* (equation (2.2)); thus, the twist moment is given by (using Castigliano theorem)
6.1
Differentiating equation (6.1) with respect to *ϕ* yields the torsional stiffness *Γ*,
6.2

## 7. Prototype manufacture and experimental set-up

### (a) Manufacture

The manufactured prototype had a [0_{5}] lay-up, as described in §4a. The structure was made of two laminated flanges moulded over a cylindrical aluminium tool of radius *R*_{i}=57 mm. The height *H* of the structure was set to 57 mm to obtain a ratio *α*=2. The laminate consisted of five layers of unidirectional Hexcel 8552/IM7 (Hexcel 2007) pre-impregnated carbon fibre-reinforced plastic. The relevant material properties of the lamina are presented in table 1. The 0^{°} direction corresponds to the local *x*-axis shown in figure 1*b*. Each flange has dimensions *L*=*π*×*R*_{i}=179 mm by *W*=9 mm. Five holes were equally spaced along the length of the parts to accommodate the spokes. Local 10×9 mm^{2} patches of 90^{°} unidirectional Hexcel 8552/IM7 were bonded at the spokes' interface to limit fibre delamination. The spokes, made of stainless steel rods, were machined at their ends to accommodate an *M*2×0.25 screw with two washers sandwiching the flange after assembly. To allow the device to twist freely, the flanges were loosely tightened to the spokes. Figure 9*a*–*c* shows the prototype in the three different configurations: the straight (*a*) and coiled (*c*) configurations correspond to the unstable equilibria. The twisted stable equilibrium (*b*) corresponds to point A in figure 4*b*.

### (b) Experimental set-up

A fixture was developed enabling the prototype to twist while a vertical displacement was applied to one end of the assembly. The pivoting attachment was mounted between one end of the structure and the load transducer of the test machine's moving cross-head to allow for the twist to occur, as shown in figure 10*a*. The pivoting fixture is made of a fixed shaft attached to the test machine and a cage attached to the morphing structure. The rotational degree of freedom is realized with a ball bearing placed between the shaft and the cage. Friction in the ball bearing is neglected. The experimental set-up is shown at different stages of the test in figure 10.

## 8. Finite-element modelling

An FEM was developed using the ABAQUS/CAE (2008) standard. The FEM was a reproduction of the prototype described earlier; thus, material properties, lay-up and dimensions were identical. Owing to the symmetry of the structure with regard to the axis of twist, only one flange was analysed. In order to constrain the flange, the ABAQUS tool ‘connector’ was used instead of modelling the spokes and their interfaces with the flange. The flange was modelled using four node shell elements (S4R). The mesh was refined until an acceptable compromise between model fidelity and computation time was found. Hence, a 4×90 element structured mesh was used. In order to pre-stress the structure, the study was divided into two analyses. Both used a Newton–Raphson iterative process. The velocity and acceleration terms were neglected owing to the low mass of the flange and the slow motions involved in the studies. The first analysis modelled the flattening operation of the flange; the second analysis modelled the assembly and analysed the twist morphing, starting from the last increment of the first analysis. Nonlinear geometric effects were taken into account for each step in both analyses.

### (a) First analysis

The flange was modelled with an initial curvature *κ*_{x}=1/*R*_{i}=1/57; reference points (RPs) were added to the geometry in order to ease the flattening operation and the construction of the connectors in the second analysis, as shown in figure 11*a*. The RPs were positioned in the (*x*,*z*) plane at a distance equal to *R* from the flange, on the concave side and along a radius of the part. Five RPs were equally spaced along the flange length to accurately model the prototype's spoke interfaces. The RPs were then locally tied to the flange using rigid body constraints. Flattening the part was achieved by applying displacement boundary conditions to the RPs while RP3 was fixed, as shown in figure 11*b*.

### (b) Second analysis

First, the deformed mesh was imported from the previous analysis and connectors were added; *z*-axis hinges were simulated between the RPs and the flange, and cylindrical connectors were added between each of the five RPs along the *x*-axis, as shown in figure 12. In the initial step, the state of pre-stress resulting from the first analysis was applied as a pre-defined field to the mesh, and RP1 was fixed in space. A 0.3 rad (17^{°}) *x*-rotation was then applied to RP5. A helical stable shape was observed once the 0.3 rad *x*-rotation was disabled in the second step, as shown in figure 13*a*. The initial *x*-rotation was necessary in the first step to force the flange into one particular configuration. Without it, the model would attempt to proceed along an untwisted, unstable path. The ABAQUS function ‘stabilize’, which adds viscous forces to the global equilibrium equation with the specification of an artificial damping factor, was required in the second step to prevent the flange from returning to its straight configuration once the *x*-rotation was released from RP5. It is important to keep the damping factor as small as possible because artificial damping may affect the accuracy of the solution obtained or, indeed, find unrealistic solutions. It is monitored by ensuring that the viscous energy is small compared with the overall strain energy of the structure. Here, the value for the damping factor depended on the initial rotation given to RP5: the higher the *x*-rotation, the lower the damping factor. A damping factor of 1×10^{−5} N s mm^{−1} for a rotation of 0.3 rad appeared not to affect the response of the structure. Finally, the third step forced the flange to adopt the coiled configuration (second unstable equilibrium) by imposing a displacement boundary condition on RP5 along the *x*-axis, as shown in figure 13*b*.

## 9. Results and discussion

### (a) Strain energy, axial force and axial stiffness

Figure 14 shows the evolution of the strain energy, axial force and axial stiffness as a function of the angle of helix *θ*. Three sets of data are drawn in each plot, showing the analytical prediction, the data given by the FEM and the results from the experiments carried out on the prototype. It is important to characterize and predict such properties and so facilitate the design of morphing structures.

Figure 14*a* shows the evolution of the strain energy for the three sets of data, where the experimental curve is given by
9.1
with *A* given by equation (3.1) for *θ*=0. Equation (9.1) is then expressed as a function of *θ* with equation (2.4) for plotting purposes.

As seen in §4*a*, the unstable configurations of the structure are at *θ*=0^{°} (point A) and *θ*=90^{°} (point D). Point B (*θ*=45^{°}) is one of the two stable equilibria. Between points A and B, the structure tends to coil automatically to the stable twisted state; hence the axial force is negative in sign between those points and reaches a zero value at point B, as shown in figure 14*b*. Thereafter, the axial force is positive as the structure resists external load into the coiled unstable state (point D). Point C in figure 14*a* is an inflection point and corresponds to the local maximum axial force shown in figure 14*b*; thus, a zero axial stiffness point in figure 14*c*. Consequently, the structure develops positive axial stiffness between A and C then negative stiffness for increasing *θ*, with a nonlinear evolution.

The strain energy curves obtained by the analytical model and the FEM are in good agreement, with a maximum difference of 3 per cent at point B. At this particular position, a difference of approximately 10 per cent is also observed between the analytical curve and the experimental data, and is believed to be caused by the limitations of the one-dimensional assumption made for the flanges and localized effects due to the spoke connections.

In figure 14*b*, the axial force given by the FEM differs from the analytical model for the first 17^{°} because of the initial twist given to the model, as explained in §8*b*. An initial twist was also applied at the start of the experiment to force the prototype into one configuration, hence the delayed start of the experimental curve. The three curves are in good agreement, with a maximum difference of only approximately 2.5 per cent.

The axial stiffness of the assembly is shown in figure 14*c*. For the same reason as just mentioned, the curve obtained from the FEM differs from the analytical prediction for *θ*<17^{°}. The curves are in good qualitative agreement, although differences of up to approximately 7 per cent are noticed between experimental data and analytical predictions.

### (b) Twist moment and torsional stiffness

The twist moment is calculated with equation (6.1) and a change of variable is performed to plot *M* against the angle of helix *θ* in figure 15*a*. Experimental measurements of the moment produced by the prototype were unsuccessful owing to the high friction within the torque test-bench; thus, the results from the FEM are used to validate the analytical model. For this study, an *x*-axis rotation was applied to RP5 in the FEM; hence the curve starts at *θ*=0^{°}. Close agreement is found between the two analyses, with a maximum difference of approximately 2 per cent. For *θ*=0^{°}, the structure cannot sustain any twist moment, hence the initial zero value. For increasing twist angles, the twist moment becomes negative as the structure snaps to reach the stable equilibrium (*θ*=45^{°}) before becoming positive, as the structure is further twisted towards the coiled configuration.

The torsional stiffness is plotted in figure 15*b* and obtained from equation (6.2). As per the axial stiffness, the structure shows nonlinear behaviour with both negative and positive stiffnesses while deforming. Good agreement is found between the two curves with a maximum difference of 5.5 per cent at *θ*=45^{°}.

### (c) Discussion

In order to keep the analytical model simple, several assumptions and simplifications were made throughout the study. As such, the friction between the flanges and the spokes, as well as the friction within the pivoting fixture used for the experiment, were omitted. The holes in the flanges were not modelled and curvature changes across the width of the flanges were not considered. Despite these assumptions, the results shown in §9*a*,*b* are in close agreement. It is assumed that these simplifications contributed to the differences seen between the experimental results and the analytical predictions.

## 10. Conclusion and future work

A structure made of two curved flanges joined by spokes was analytically modelled using a simple model assuming inextensional deformations. The position of the equilibria, the morphing axial force and the snap-through moment were also derived from the expression of the strain energy. The influence of the geometry of both flanges and the structure was analysed. The material properties of the flanges were also investigated by varying the ply angle *β* in symmetric [*β*/*β*/0/*β*/*β*] and anti-symmetric [*β*/*β*/0/−*β*/−*β*] lay-ups. Examples were given for each lay-up, yielding different stability conditions. A remarkable combination of flange/structure geometry and anti-symmetric lay-up yielded a constant strain energy level throughout the deformation; thus, a zero-stiffness structure along the axis of twist. The case of the [0_{5}] lay-up was modelled with commercially available finite-element software. A prototype was also manufactured and tested. As analytically predicted, high nonlinearity in the structure's response was found. Twist moment and torsional stiffness were compared only between the FEM and the analytical model; however, the agreement found between the models and the experimental data from the axial test provide confidence in the validity of the analytical model.

Further development of this device includes modelling the thermal effects of complex lay-ups on the expression of the curvature change as well as evaluating the influence of the width of the flanges.

## Acknowledgements

This research was sponsored by Vestas Wind Systems as part of the ‘Morphing composites for wind turbine applications’ project, which is gratefully acknowledged.

- Received October 18, 2011.
- Accepted December 6, 2011.

- This journal is © 2012 The Royal Society