Bistability

Flexible cylindrical shell structures, like tape-springs, slit tubes, or STEMs, differ only on the subtended angle of the circular cross-section. These structures normally have one stable state, the extended configuration, and are thus monostable [Seffen and Pellegrino, 1999]. As oppose to this, bistable shell structures have also a second stable state, the coiled configuration, as shown in Figure 3.2. The deployment mechanisms for these bistable structures can then be made simpler and smaller compared to their monostable counterparts, yielding the more compact and lighter solutions that spacecraft designers seek. Not only this simplifies the way large deployable structures may be stored and extended, but also provides a baseline for new types of active structures, where an actuator (i.e. shape memory alloy (SMA) embedded in the resin matrix [Ayre et al., 2005]), can only be required to trigger the transition between stable configurations, and can be switched off at other times [Norman et al., 2008]. Bistability of curved structures has many other engineering applications apart from their use in deployable boom structures, such as in aircraft wing morphing skins [Thill et al., 2008].

Figure 3.2: Glass/PP bistable slit tube in both stables states: extended and coiled (left) [Guest and Pellegrino, 2006]; and the coordinate system and axes convention utilised (right). x, y and z are the longitudinal, transverse and out-of-plane directions of the extended shell.

Bistability in Composite Slit Tubes

Bistability in thin curved shells is another example of bio-mimicry, inspired by the closure-opening mechanism of the Venus Flytrap leaf [Ayre et al., 2005]. Bistable composite slit tubes were dis- covered by Daton-Lovett [Daton-Lovett, 2000] under the patented name “Bistable Reeled Compos-

ites” (BRC). Ever since, RolaTube Technology Ltd. and The University of Cambridge Deployable Structures Group have been working together on analytical and computational models that can accurately predict the bistable behavior.

There are two ways of making a slit tube bistable; either by altering the stiffness of the structure so that it is no longer isotropic, for instance by using a fiber reinforced composite, or by setting up an initial prestress in the structure. For the case of composites slit tubes, to achieve bistability, stiff fibers are placed at sufficient angles, ± δ◦, to the longitudinal axis. This gives the shell the correct anisotropic bending properties, so that the second stable state favours same-sense bending, which occurs when the center of curvature of the shell in both stable configurations is on the same surface side. Each of the two stable states are strain energy dwells, but one of the configuration is the lowest-energy state and, therefore, the boom will still tend to self-deploy or rolled-up from a partially unrolled configuration. This behaviour make them ideal as deployable structures, where the extension can be triggered by i.e. uncoiling the tip of the boom.

All of the analytical composite models discussed below make use of Classical Lamination Theory (CLT) in their formulation. CLT is used to assemble the different plies in the required orientation, build the laminate, and predict its properties. This theory can be found in most basic primers on mechanics of composite materials. Particularly good references are the books by Jones [Jones, 1998], Hyer [Hyer, 2008], Hull [Hull, 1981], Halpin [Halpin, 1992], Bunsell and Renard [Bunsell and Renard, 1995], and Matthews and Rawlings [Matthews and Rawlings, 1994].

Iqbal et al. [Iqbal et al., 1998] presented the first analytical model that captures key features of bistable composite cylindrical shells. It is a simple extensional bending model that calculates the total strain energy of the shell as a function of the longitudinal and transverse curvatures, and the angle subtended of the shell. The model considers uniform curvatures along the shell, and twisting is not allowed. Also, stretching-bending coupling effects are ignored when constructing the strain energy expression, such that the B matrix portion of the ABD matrix is zero. These assumptions restrict possible deformation modes, resulting in a model that is not able to distinguish between the bistable behaviours exhibited by symmetric and antisymmetric laminates. Iqbal and Pellegrino [Iqbal and Pellegrino, 2000, Iqbal, 2001, Pellegrino, 2005] presented a computational model using FEA to predict the natural coiled radius of the tubes for a given composite laminate. The computational and analytical models show remarkably close results, but are not in particularly good agreement with the experimental results found with E-glass/polypropylene bistable shells.

Galletly [Galletly, 2001] extended the previous analytical model to include all possible deforma- tion modes according to beam theory, including twist and shear, and stretching-bending coupling effects (B 6= 0). As in the aforementioned models, it considers the structures to be longitudinal uniform, so that the model is not intended to study the transition region between states, just give information about the two possible stables shapes. Firstly, a more simplistic analytical beam model was developed [Galletly and Guest, 2004b], which assumes the cross-section of the shell remains circular (constant transverse curvature), with a radius that is allowed to change. The model results do not correlate well with the computational model or experimental tests produced by Iqbal and Pellegrino, because of the assumption of constant transverse curvature. It also fails to predict a second stable state for symmetric layups. In order to solve this, a second shell model, that neg-

lects the previous assumption of linear geometry, was produced [Galletly and Guest, 2004a]. It formulates a differential equation, based on Calladine’s work [Calladine, 1983, Calladine, 1988], to derive non-constant transverse curvature of the shell, which is used to calculate the second stable state. The results are in good agreement with Iqbal’s FEA. Nevertheless, once again, there are some disparities with the experimental results, failing to predict the second stable configuration of some anti-symmetric layups. It is concluded in [Galletly, 2001], that the disparity of results are due the polypropylene matrix entering the plastic regime, which is beyond the analytical model that makes use of linear-elastic material properties for the calculations.

Guest and Pellegrino [Guest and Pellegrino, 2006] proposed another inextensional bending model that is, now, able to distinguish between symmetric and antisymmetric laminates. The analytical model makes the following assumptions: the deformation of the shell is everywhere uniform; and all possible configurations of the shell have zero Gaussian curvature (product of the two principal curvatures), and hence, they are developable [Calladine, 1983]. The model requires only two para- meters: the angle of the shell relative to the underlying cylinder about which the shell transitions from one state to the other; and the principal curvature of that cylinder. Further discussions re- garding this model will be carried out in Chapter 7 as it has been used as the basis of the model that predicts the spiral coiled shape of the BOWL composite boom.

As previously explained, in a bistable slit tube, each of the two stable states is a strain energy minimum. However, the “as manufactured” stress-free state is the lowest of the two energy minima. Bistable shell structures must undergo a transition between these two energy minima, requiring the input of energy. Once a certain transition point has been passed, the structure will follow the negative energy gradient to the other stable state. The shape of the energy gradient between the two stable states can be altered. For example, placing the transition point close to the coiled state energy minima, leads to a structure which will self-deploy from the stowed state with little external input. Several different concepts of deployable composite tubes with differing energy gradients between the stable states have been proposed in the last decade.

In [Murphey and Pellegrino, 2004] a novel concept coined the Neutrally Elastic Mechanism (NEM) was presented. It is a “neutrally stable” composite tape-spring, where both stable states, the extended and the coiled, have zero strain energy. Therefore, when the tape-spring is partially unrolled, it neither wants to extend or rolled up. Also, the forces required to roll and unroll the tape-spring diminish, so that relatively small and low force actuators embedded in the structure can be used to control the deployment. The tape-spring built is a CFRP cross-ply laminate, where each lamina has a curvature pre-stress. The pre-stress is produced by bonding the two laminae that initially had opposite curvatures in perpendicular axes. Nevertheless, due to wrinkles in the final laminate formed, the tape-springs produced a jerky motion when rolled or unrolled, yielding an impractical solution for large-scale deployable structures. An analytical method very similar to Guest and Pellegrinot’s was used to examine the behaviour of the tape-springs manufactured.

Following their discovery, CTD has developed a neutrally stable tape-spring that is now stable in every configuration: coiled, partially unrolled, or extended, as shown in Figure 3.3(left). The way to produce this effect is not by prestress of the structure, but by utilizing a [± 45◦] laminate with plain weave laminae that have a very low-stiffness resin [Schultz et al., 2008]. These structures

are best suited for applications seeking to avoid deployment shock, complicated mechanism, or even the need to fully or partially re-stow the member. Nevertheless, they do not have much axial and hoop stiffness, as the fibers are placed at significantly large angles from the longitudinal and transverse axes of the tape-spring, which considerably limits their structural applications.

Figure 3.3: CTD’s neutrally stable tape-spring (left) [Schultz et al., 2008]; and AFRL’s rollable antenna using bistable tape-springs (right) [Murphey et al., 2010].

In 2010, 1 m long CFRP bistable tape-springs were presented by AFRL researchers [Murphey et al., 2010]. These tape-springs could be used as meter-class deployable booms and antennas as shown in Figure 3.3 (right). However, these are limited in length, as the rolled-up stable configuration looses stability if long tape-springs are coiled, due to the overlapping wraps increasing the diameter of the coil beyond the acceptable limit for bistability. This problem was also reviewed in another paper by them [Jeon and Murphey, 2011], where the buckled rolled shape of a longer version of the boom is shown. This thesis author was already aware of the length limitations of these structures following early conversations with Daton-Lovett, and at that time was working towards a solution to mitigate the scalability problem of these type of bistable structures. A first scalable bistable boom was used in the concept presented in [Fernandez et al., 2011a]. A high performance version was later presented in [Fernandez et al., 2012, Fernandez et al., 2013, Fernandez et al., 2014a]. These booms will be the focus of Chapters 7, 8 , 9 and 10.

A new conductive composite tape-spring for cubesat deployable dipole antennas was proposed in [Constantine et al., 2012]. The boom is constructed using glass fibre reinforced epoxy with an embedded copper alloy conductor. Also for the SWIM (Space Weather using Spectrometers and Magnetometers) cubesat programme, KTH is currently developing a bistable tubular boom made with woven-composite fabric [Prigent, 2011, Mallol-Parera, 2013a].

Bistability in Metallic Cylindrical Shell Structures

Bistability on isotropic shells can be engendered by applying the correct amount of prestress on the shell, so that the second stable state favours opposite-sense bending, with the centre of curvature of the shell in the opposite side of the surface to that of the extended state’s one. In [Kebadze et al., 2004] this behaviour was explored on metallic tape-springs. Bistability was achieved with the lower-energy state now being the coiled configuration, as in a child’s slap bracelet. Hence, as shown in Figure 3.4 (top), if the tape-spring is partially rolled, it tends to roll up instead of extend. Personal discussions with some of the authors of [Kebadze et al., 2004] have revealed that it is

also possible to favour extension by applying an exact amount of prestress, so that the metallic tape-springs can be used as deployable structures. However, now both stable states contain strain energy and the gradient will generally be low and, therefore, self-extension with other coupled components, i.e. a sail membrane, may not be possible. Nevertheless, as shown in [Wolf et al., 2014], the latter option has been recently applied to deploy a small 2 m2 solar/drag sail at 81 km altitude on a REXUS ballistic rocket experiment. The Aachen University team is planning on deploying an 8 m2 version at 350-400 km altitude on a 3U cubesat platform using the same bi-stable, prestressed metallic tape-springs.

A qualitative analysis of the elasto-plastic forming process that sets up an appropriate distri- bution of residual bending stresses, that include strain hardening effects, is presented in [Kebadze et al., 2004]. The analytical model used is essentially the same as the one aforementioned by Guest and Pellegrino [Guest and Pellegrino, 2006]. The errors in the analytical predictions are typically less than 10% when compared to actual samples, showing really good agreement with the experimental data.

Figure 3.4: Bistable metallic shell showing the opposite-sense bending tendency(top) [Murphey and Pellegrino, 2004], and a zero-torsional stiffness shell structure (bottom) [Guest et al., 2011].

A theoretical approach to engender bistability on isotropic tape-springs was proposed in [Gentilini et al., 2008]. It consist of applying thermal gradients through the shell thickness in order to simulate the residual bending prestress. Nevertheless, computational simulations using FEA have shown that the values of the thermal gradients are in the order of 104 ◦C/mm, which makes this method unpractical.

It is well known that the stiffness of a structure changes with the applied load, so that a stable structure can become unstable when loaded, i.e. the buckling of a strut. It is then possible to design systems that when they buckle are neutrally stable for large deformations as shown in [Tarnai, 2003]. Using this principle, Seffen and Guest [Seffen and Guest, 2010, Guest et al., 2011] have presented an isotropic shell structure that is neutrally stable and has zero torsional stiffness. For this, the shell is prestressed in a same-sense way. Bistability cannot be engendered, but remarkably, for a particular value of prestress, the structure can be left without any torsional stiffness as shown in Figure 3.4 (bottom). Also, doubling that critical value of prestress applied, results in tape-springs that are stable when coiled but unstable when extended, as oppose to the traditional behaviour.

isotropic material properties is also a subject of current research. Prestressing a composite shell would result in matrix failure or the locked-in stresses disappearing over time, as the resin matrix is prone to creep. The way to effectively apply a prestress on a non-isotropic shell is by using a metallic shell and removing its isotropic behavior by forming small textured surface features in the shell as proposed in [Norman et al., 2008]. These structures are called multistable textured shells as they can have more than two stable states if residual bending stresses are applied on them. Textured shells can either be singly-corrugated (sinusoidal surface) or doubly-corrugated (surface with dimples).

Gossamer Technology Demonstration

Missions

In this chapter the several gossamer technology demonstration missions that the Surrey Space Center (SSC) is undertaking is presented. Emphasis is driven towards two of these missions, named CubeSail and DGOSS, as they set up the majority of the requirements for the technologies that were developed throughout the PhD. A description of the missions and its objectives are first introduced. Structural requirements for these two missions are then derived and summarized, as these can be readily adapted to similar missions and can thus can be applied for future reference on other projects. Finally, an overview of the designs of the nano-solar sail and the deorbiting system are shown. To conclude the chapter a brief description of two other international collaborative missions, named DeorbitSail and InflateSail, is presented, outlining the technologies derived from part of this research effort, that they will aim to validate in space in the near future.

4.1

CubeSail