Tristability of an orthotropic doubly curved shell

Composite Structures 96 (2013) 446–454

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Tristability of an orthotropic doubly curved shell Broderick H. Coburn a,⇑, Alberto Pirrera a, Paul M. Weaver a, Stefano Vidoli b a b

Advanced Composites Centre for Innovation and Science, University of Bristol, Queen’s Building, Bristol BS8 1TR, United Kingdom Dipartimento di Ingegneria Strutturale e Geotecnica, Sapienza – Università di Roma, via Eudossiana 18, 00184 Rome, Italy

a r t i c l e

i n f o

Article history: Available online 20 August 2012 Keywords: Tristability Anisotropy Finite element analysis Shell theory Gaussian curvature

a b s t r a c t In this work, the structural response of a doubly curved orthotropic shell is tailored to achieve tristable geometries. This is done by varying both material properties and the Gaussian curvature of the surface profile. Tristability is predicted analytically, verified with finite element analysis and, for the first time, demonstrated experimentally. Predicted geometries of the tristable states are shown to compare well with experiment. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction Multistable structures are elastic objects that exhibit more than one equilibrium configuration, only requiring actuation between stable states. These adaptive structures can provide stiffness and strength whilst allowing appreciable shape change (in a highly nonlinear way). Consequently, they are currently receiving great interest from, for example, the aerospace community for the use on aerodynamic surfaces and morphing structures in general. The majority of articles reported on multistability are due to thermal effects in anisotropic and non-symmetric composite structures [1–5]. Multistability, however, can be achieved through a variety, or combination, of means including pre-stress [6–8], curvature [9] and plastic deformation [10]. All methods, apart from curvature, rely on induced stresses during manufacture. Multistability due to curvature has gained recent interest due to the several unique advantages over the aforementioned means, such as negligible hygrothermal effects and the base stable state matching the shape of the tooling or mould allowing close control of manufactured tolerances. For a doubly curved shell the bending and stretching modes of deformation are coupled [11] and it is this interplay which permits multistability for certain combinations of initial curvature and material properties. The change from one stable state to another is accompanied by the build up and release of large amounts of strain energy under large deflections. The changes in Gaussian curvature that occur during deformation of a doubly curved shell cannot be achieved without mid-plane stretching requiring significant energy [9,11,12]. Consequently, curvature multistability demands ⇑ Corresponding author. Tel.: +44 7869681735. E-mail address: [email protected] (B.H. Coburn). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.08.026

significantly larger actuation loads to transition between stable states in comparison to the alternative methods mentioned above [7,8]. The use of curvature and anisotropy to achieve multistability allows control of stable states by tailoring the base geometry and material properties [9]. Guest and Pellegrino [13] first examined this phenomenon and showed that isotropic cylindrical shells (manufactured stress free) can only be stable in their base configuration, but by introducing anisotropy into the structure the cylindrical shell can have two stable states. Similarly, an isotropic shell can exhibit bistability if it is doubly curved. Seffen [9] analytically explored this combination of material properties and curvature leading to multistability by developing an extensible model for quasi-homogeneous orthotropic shells. By assuming constant curvature in the x- and y-directions, an elegant solution based on strain energy minimisation was developed to determine stable states. In-plane boundary conditions are satisfied for elliptic planforms and bending boundary layer effects were neglected. Concentrating on an averaged interior problem, an assumption commonly accepted for shells free at the boundary, Seffen was only able to show the existence of two stable states within the model. Vidoli and Maurini [14] extended the approach of Guest and Pellegrino [13], and Seffen [9] to show analytically a range of shallow shells with uniform base curvature and specific material parameters leading to tristability. This tristable range was found to be maximised by conceiving shells where the homogenised Young’s moduli ratio b = Ey/Ex and the Poisson’s ratio m = myx appffiffiffi proach the degeneracy condition m ! b. To the best knowledge of the authors, tristability of a uniformly doubly curved shell has only been shown analytically to date. The current contribution extends the work of Vidoli and Maurini [14]

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6 5 4

Hy

to design with analytical methods, verify with Finite Element Analysis (FEA) and show, for the first time with the manufacture of a demonstrator, a shell exhibiting three stable states. The following sections are structured as follows. Section 2 details the selection of a carbon fibre lay-up and base curvature combination leading to tristability and provides the analytically predicted curvatures of all stable states. This designed shell is then verified to be tristable in Section 3 with FEA and the significance of the bending boundary layer, actuation path between stable states, thermal expansion and sensitivity to base curvature error is explored. Section 4 presents the demonstrator manufacturing procedure, test method and experimental results. A comparison and discussion between results of the three methods is then presented in Section 5 and finally the paper is concluded in Section 6.

3 2 1 0 0

5

10

15

Hx

(a) 500

400

Using the method and approach developed by Vidoli and Maurini [14] a Carbon Fibre Reinforced Plastic (CFRP) stacking sequence and base curvature combination leading to tristability will be shown. Predictions of the stable states make use of Seffen’s model [9]. The shell is assumed to have constant curvature in the xand y-directions with no twist in all stable states. Throughout this section all quoted curvature values are constant over the planform.

Ky (10-6 mm-1)

Design

2. Analytical model

300

200

100

0 2500

3000

2.1. Material and stacking sequence

3500

4000

4500

Kx (10-6 mm-1)

Approaching the tristability degeneracy condition proposed by Vidoli and Maurini [14] requires stacking sequence tailoring to achieve desired laminate mechanical properties. For unidirectional plies, increasing m to its maximum value is achieved by orienting plies at ±45° to the principal directions; this also results in equal in-plane stiffness in the x- and y-direction (b = 1.00). The parameter m can be further increased, approaching unity, by using highmodulus CFRP. It is noted that the analytical model’s assumption of a quasi-homogeneous and orthotropic shell limits the design space of stacking sequences. Achieving these requirements with unidirectional ±45° plies is not possible with a conventional stacking sequence and a special balanced and anti-symmetric ply sequence of [45/452/45/45/452/45] is required as identified in [15]. This unique lay-up was first shown to be both in-plane and out-of-plane orthotropic by Bartholomew [16] and more recently optimal for flat plate in-plane and shear buckling by Weaver [17]. Material properties of the uni-directional CFRP prepreg selected for the study are provided in Table 1. The tristability parameters for the shell were calculated to be b = 1.00 and m = 0.80. For these parameters the combinations of curvatures in the x- and y-direction, jx and jy respectively, leading to tristability were determined and are illustrated in Fig. 1a where Hx and Hy are non-dimensional base curvatures. The white region represents a base curvature combination resulting in monostability, grey region bistability and dark grey tristability. The two labelled points are potential base curvatures for tristability, expressed as radius of curvature (Rx, Ry) in cm. 10% of all base curvature combinations, jx–jy, are analytically predicted to be tristable as per [14]. 2.2. Shell base geometry The selection of the base shell curvature is a compromise between having low error sensitivity and low strains during actuation

(b)

Fig. 1. (a) Stable state phase diagram for the analytical model as a function of nondimensional base curvatures, Hx and Hy (see [14]); showing monostable (white), bistable (grey) and tristable (dark grey) regions. (b) Curvature plane, jy–jx, showing analytical bounds on design curvature of base state to achieve tristability.

with ease of manufacture. With reference to the two labelled points in Fig. 1a, the larger base curvature is in a larger tristable region and will have less sensitivity to errors in analytical assumptions and manufacture. However, this larger curvature requires larger strains during transition between stable states and also presents difficulties during the lay-up of flat CFRP sheets onto the doubly curved mould. Ultimately, the base shell curvature selected to achieve tristability was 3333  106 mm1 and 341  106 mm1 for jx and jy respectively or 300 mm and 2930 mm for the radii of curvature in the x- and y-direction. This point is labelled as (30, 293) in Fig. 1a and the cross in Fig. 1b. 2.3. Predicted stable states For the selected base geometry and material properties the curvature of stable state 2 (SS2) and stable state 3 (SS3) were analytically predicted as per [9] with results provided in Table 2. The choice of planform dimensions for the analytical model are almost insignificant as bending boundary layer effects are neglected. Planform size becomes important when considering FEA and demonstrator manufacture as discussed in Sections 3 and 4, respectively. The planform area used in the analysis was a circular section with radius of 250 mm. Table 2 Analytically predicted curvatures of stable states with planform radius of 250 mm. State

Table 1 Carbon fibre prepreg mechanical properties [18]. tply (mm)

E11 (MPa)

E22 (MPa)

m12 (–)

G12 = G13 (MPa)

G23 (MPa)

0.125

161

11.38

0.32

5.17

3.98

SS1 (design/base) SS2 SS3

Curvature (mm1) 106  jx

106  jy

3333 494 922

341 2309 1115

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2.4. Sensitivity study

Fully constrained

Py

Py

A sensitivity study on the design’s base curvature was performed to determine the allowable tolerance for the shell to have three stable states. By fixing the design jy and varying design jx, the analytical range on the design jx was determined to extend from 23% and have no upper limit. Similarly the range for the design jy was determined to be 86% and +29%. The sensitivity to base curvature error is illustrated in Fig. 1b where the cross indicates the design curvatures and the dashed lines the tolerance in jx and jy. The grey region is only representative of the expected shape of the tristable region as per Ref. [14].

Px Y

X

Px

Z

(a)

3. Finite element analysis model Whilst the analytical model provides physical insight and understanding, as well as a fast method to predict stable states, it is based upon several assumptions that require validation. FEA enables: the observation of bending boundary layer and Coefficient of Thermal Expansion (CTE) effects; representation of real geometry; and the determination of the actuation path details between stable states.

(b)

(c)

3.1. Geometry The analytical model does not require specification of a surface profile, instead a constant jx and jy, and no twist, jxy are assumed. For the FEA and demonstrator manufacture, however, a surface profile is required. Approximation of a constant curvature surface is most frequently achieved, within the field of multistability, by assuming an elliptic paraboloid [2,7,9,19]. This approximation is valid for shells with large radii of curvature relative to planform size, violation of this condition results in the development of significant twist towards the boundary. In this study an alternative surface is created by sweeping a profile curve with constant radius along a sweep path curve with constant radius. The relevant surface equation is given by

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 1 1 f ðx; yÞ ¼   x2   y2 þ þ : 2 2

jx

jy

jx

jy

ð1Þ

Whilst also subject to developing twist curvature, the effect is 18% less than the paraboloid and hence is better suited to the approximations of constant curvature and no twist, as made in the analytical model. 3.2. Model setup and solution method The full shell was modelled with 3072 quadrilateral four-node reduced integration shell elements (S4R), with a total of 3137 nodes, using AbaqusÒ version 6.10–2. Mesh refinement studies showed that this mesh density provided converged results with acceptable computational time. All steps were simulated with a dynamic implicit analysis procedure using geometrically non-linear algorithms. A quasi-static application was used which adds numerical dissipation to the system to aid convergence, damps instabilities and introduces inertia effects to regulate unstable behaviour. The centre of the shell was fully constrained in translation and rotation. Actuation forces were applied at the shell edges intersecting the xz- and yz-planes as shown in Fig. 2a where the arrows indicate the point load locations and the subscript of the loads the curvature direction being actuated. All actuation forces were applied as z-direction boundary conditions (displacement controlled), ramped linearly over the step. The final z-position of the shell edge was predicted using the analytical model and assuming a swept profile (Eq. (1)) without mid-plane extension. Following actuation into the predicted stable shape, the actuating boundary

Fig. 2. FEA predicted (a) stable state 1 showing location of actuation forces and fully constrained centre, (b) stable state 2 and (c) stable state 3.

Table 3 Carbon fibre prepreg coefficient of thermal expansion properties [18].

a11 (°C1) 0.0

a22 (°C1)

a33 (°C1)

6

30  106

30  10

conditions were removed allowing the shell to relax to its equilibrium configuration. Selected FEA trials included a cool down step from 180 °C to 20 °C to simulate the CTE effects that will be experienced during the manufacture of the demonstrator. The curing of the selected CFRP is at 180 °C and the CTE properties for the material are provided in Table 3. 3.3. Results 3.3.1. Stable states Trials were initially conducted neglecting the cool down from the cure step, and all three stable states were successfully found with FEA as shown in Fig. 2 : (a) SS1; (b) SS2 and (c) SS3. In all states the shell is fully constrained at the centre which is indicated by a black dot in Fig. 2a. Unlike the analytical model, the FEA does not constrain the shape of stable states to have constant curvature and variations of the local curvature may exist. Hereafter, the terms global average curvature, local curvature and average interior curvature will be used frequently and a brief definition of each is provided to prevent confusion. The global average curvature is calculated by assuming an arc of constant radius connecting the centre of the shell and two points on the shell edge for either the x- or y-axis. The local curvature is the curvature at a single point and is calculated for FEA by assuming an arc of constant radius connecting three adjacent nodes and calculating the Menger curvature [20]. Finally, the average interior curvature is an average of the local curvature for a central region unaffected by the bending boundary layer, later discussed in Section 3.3.2. The FEA shell local curvatures were found to be approximately constant in a central interior region but increased towards the shell edge. The average interior curvatures, jx and jy, were determined to

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be 503  106 mm1 and 2223  106 mm1 for SS2, and 1014  106 mm1 and 1110  106 mm1 for SS3, respectively. 3.3.2. Bending boundary layer The rapid variation in shell local curvature along the x- and yaxes is shown in Fig. 3a and b for SS2 and SS3 respectively. Here the diamond markers are jx determined at points along the shell xz-plane and the triangle markers are jy determined at points along the shell yz-plane. The curvatures are plotted against the position along the x- or y-axis where the zero position is the shell centre. The increase in curvature towards the perimeter, evident in Fig. 3a and b, is expected and due to a bending boundary layer effect [21– 24]. This boundary layer phenomenon develops for any configuration other than the base state (SS1) and is due to a steep gradient in the bending moment, which increases from zero at the free edge to a non-zero value in the interior region. This bending moment gradient is equilibrated by out-of-plane shear stress resultants causing a change in local curvature [25]. The length of the boundary layer was calculated as a 10% difference (90% decay length) from the average interior curvature. A summary of results is provided in Table 4. The size of the boundary layer for a singly curved shell, with the same material properties and stacking sequence as the tristable shell, is related to the shell thickness, t, and curvature, j, as (see [23])

rffiffiffiffi t

ð2Þ

:

j

Here, the factor 0.7 accounts for the orthotropy of the shell. Assuming jx and jy act independently, the predicted boundary layer lengths in their respective directions can be calculated. K

K (10-6 mm-1)

K

Boundary layer length (mm) xz-Plane

SS1 SS2 SS3

yz-Plane

FEA

Proportion (%)

Pred.

FEA

Proportion (%)

Pred.

0 119 127

0 41 43

0 31.5 23.1

0 20a 130

0 8.5a 53

0 14.7 21.0

a The calculated boundary layer length for SS2 in the yz-plane is expected to be underestimated with the method used, visual observations of the profile indicate the boundary layer length is approximately 80 mm.

Results (provided in Table 4), range from 15 mm to 32 mm and underestimate the FEA observed boundary layer length by as much as a factor of 6. It is believed that the presence of the orthogonal curvature (non-zero Gaussian curvature) has a significant effect on the size of the boundary layer. For all stable states the relative size of the boundary layer in the x- and y-directions appears to obey the inverse relationship for the curvature as per Eq. (2), i.e. an increase in curvature reduces the boundary layer length. 3.3.3. Actuation and energy profile Actuation between SS1 and SS2 required only jx (Px in Fig. 2a) or jy (Py in Fig. 2a) displacement control. The FEA jx and jy path followed during actuation between stable states is shown in Fig. 4. The strain energy of the shell is indicated by the marker colour and accompanying colour bar. Actuation was performed from SS1 to the analytically predicted SS2 curvatures followed by removing actuation loads and allowing the shell to relax to the FEA SS2. Subsequently, actuation was performed from SS2 to the analytically predicted SS3, followed by relaxation to the FEA SS3. Similarly, direct actuation from SS1 to analytically predicted SS3 curvatures was performed, again followed by relaxation. Analytical and demonstrator stable state curvatures are also labelled and all curvatures were determined as a global average. Circled regions indicate local strain energy peaks as observed in the FEA. Actuating jx, the resulting jy followed a minimal energy path as predicted by Vidoli and Maurini [14]; a similar path is followed by actuating jy in place of jx as shown in Fig. 4. The maximum actuation force required was 22 N, applied equally at the edges along the xz-plane. 20

3000 SS2demonstrator

(a)

18

2500 SS2analytical

SS2

FEA

2000

K

K

16 14

K (10-6 mm-1)

−1

κ y (10−6mm )

1500 SS1demonstrator

1000

12 10

500 SS3 demonstrator SS1

0

8

Strain Energy (J)

dBL  0:7

Table 4 FEA boundary layer length and proportion of total shell length based on 10% error from the average interior curvature. Analytical predictions are made as per Eq. (2) for singly curved shells.

6

−500

4

−1000 SS3FEA −1500

2

SS3analytical

0

−2000

−2000

(b) Fig. 3. Local curvature variation along xz-plane (jx) and yz-plane (jy) for (a) SS2 and (b) SS3.

−1000

0

1000

2000

3000

4000

κx (10−6mm−1) Fig. 4. FEA strain energy along jx–jy (global average curvature) actuation path. Analytical and demonstrator stable state curvatures are also marked and labelled.

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IS1 Y

X Z

Y

IS2

Z

IS3

X Z Fig. 5. Intermediate states observed during actuation to SS3.

Upon overcoming the shallow energy peak (highlighted with a circle in Fig. 4) negative stiffness is momentarily observed in the shell. The transition to SS3 required simultaneous actuation of both jx and jy. During transition the shell curvature does not remain constant over the planform. Initially, the central region of the shell develops negative jx and jy, whilst the outer region retains positive jx and jy as shown by Intermediate State (IS) 1 in Fig. 5. The strain energy continues to increase until two buckled half waves form in the y-direction (IS2 in Fig. 5) causing a large decrease in strain energy. The negative jy half wave then propagates over

Kx (10-6 mm-1)

(a)

Kx (10-6 mm-1)

(b)

Fig. 6. Strain energy against jx (global average curvature) for the FEA models (a) without and (b) with CTE effects.

the entire shell accompanied with another large decrease in the strain energy that occurs due to the formation of three buckled half waves in the x-direction (IS3 in Fig. 5). The central half wave with negative jx propagates over the entire shell and the resulting shape is the third stable state. The maximum load required to actuate to SS3 was 530 N and 562 N actuating from SS2 and SS1 respectively, the peak energy and actuation force being dependent on the actuation path. It is expected that neither of these paths taken is the minimum energy path to SS3 and hence a lower load and peak energy may exist between SS1/SS2 and SS3. The strain energy profile against jx is provided in Fig. 6a and b for trials without and with CTE effects respectively. Here the strain energy of the shell is plotted against jx and actuation is from SS1 to SS2 to SS3. The troughs correspond to the stable states whilst IS indicates an intermediate state (Fig. 5). Due to the multiple half waves forming, indicated by IS in Fig. 6a, the curvature path followed from SS2 to SS3 should be treated as an indication of the position of the plate edge relative to the origin. The minimum energy trough at SS2 was found to be very shallow and required precise actuation displacement to find the stable state indicating its marginal existence. 3.3.4. Coefficient of thermal expansion Due to the small domain of tristability existence, CTE effects were considered to ensure tristability was not lost during cool down from cure. Mismatch of fibre and matrix CTE in composite structures can cause effective in-plane stress resultants and moments when subject to a thermal load. The in-plane-out-of-plane coupling stiffness matrix, B, for the laminate is zero and hence thermal strains induced during cool down only apply direct inplane stress resultants [26]. However, it is well-known that inplane stretching is accompanied by curvature changes in shells with non-zero Gaussian curvature. This results in warping and tolerance issues for manufactured parts. As such, the effects of the cool down from cure were included in the FEA. However, all three stable states were still found with little difference in results for the existence of tristability and curvature of stable states. During the cool down step strain energy is locked into the shell and the entire strain energy profile, as shown in Fig. 6a (CTE effects neglected) is subject to a vertical shift of 25.8 J resulting in the profile shown in Fig. 6b. By removing the initial strain energy of 25.8 J from the trial including CTE effects a comparison between the energy paths can be made. The difference in strain energy at SS2, SS3 and the peak between SS2 and SS3 was less than 2% with the model neglecting CTE having slightly larger strain energy values. The difference in final curvatures at SS2 and SS3 was on average 0.1% but never exceeded 1%. The only noticeable difference in the strain energy profile (Fig. 6) between the results occurs directly after the peak strain energy between SS2 and SS3. Indeed the model considering CTE effects experiences the two sets of buckled half waves (indicated by IS in Fig. 6b) directly after each other resulting in an apparent larger drop in the strain energy profile. Throughout all trials, the shell behaviour after this peak was found to be very sensitive to solution parameters, such as automatic step size, and the shape and order of occurrence of the temporary instabilities varied. 3.3.5. Strength check FEA ply stresses in all stable states were found to be below the failure strength of the material [27]. However, during transition between SS2 and SS3 an intermediate state caused a peak compressive stress of 1900 MPa and 129 MPa in the fibre and matrix directions respectively, both in excess of the manufactures quoted values of 1690 MPa and 111 MPa [27]. This peak stress only occurs in one of the outer plies in the centre of the shell, where it is fully constrained. It is during this time that buckled half waves in the xdirection have formed and the shell is not symmetric about the

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tolerance during the manufacture of a demonstrator; hence 250 mm planform radius was selected for the study.

500

Ky (10-6 mm-1)

400

4. Manufacturing and testing of demonstrator

g Design

300

200

Following prediction of tristability with the analytical model and verification via FEA, a demonstrator was manufactured exhibiting three stable states.

100

4.1. Manufacture

0 2500

3000

3500

4000

4500

Kx (10-6 mm-1) Fig. 7. FEA sensitivity on base state curvature to achieve tristability. Grey region only representative of expected shape of tristable region as per [14].

y-axis. The constraint at this point resists rotation of the shell resulting in the peak stress. For the actuation of the demonstrator allowing small amounts of rotation for the central (fully fixed) constraint will reduce the peak stress. 3.3.6. Sensitivity study As was done with the analytical model, the sensitivity of the shell to base (SS1) curvature variation was trialled with FEA by varying jx and jy. Increasing jy incrementally from the design curvatures a limit was reached where SS2 could not be found, likewise decreasing jy a limit was reached where SS3 was lost. The percentage tolerance for the design jy was determined to be 34% and +16%, the range being reduced by 57% compared to the analytical results. Similarly, the percentage tolerance for the design jx was determined to be 14% and +20%. The upper limit of jx was reached due to geometric reasons, as a planform size of 250 mm radius cannot be achieved with a curvature greater than 4000 mm1 (250 mm radius). Results for the sensitivity study are shown in Fig. 7, here the cross indicates the design curvatures and the thick solid lines the FEA tolerance in jx and jy. The grey region is only representative of the expected area of the tristable region. FEA trials were conducted with a reduced planform radius and the minimum radius necessary to achieve tristability was determined to be 200 mm. Although tristability was observed for the planform with 200 mm radius it is expected that the range of curvatures would be greatly reduced and not allow sufficient

The demonstrator mould was manufactured using fused deposition modelling with a Stratasys FDM 400mc. Due to machine limitations a single piece mould could not be manufactured and six separate ABS plastic sections were made and later bonded. From the rapid prototype mould a glass fibre cast was taken to yield a male mould with increased stiffness. The CFRP prepreg was draped over the mould in 60 mm wide strips as shown in Fig. 8. This ‘tape-laying’ method was employed to avoid wrinkles forming as a result of the positive Gaussian curvature; facilitating controllable overlaps which formed towards the outer regions of the shell. The shell was cured at 150 °C for 6 h, a lower cure temperature and longer dwell time than specified by the manufacturer due to concerns for the glass fibre male mould behaviour at high temperatures. Visual inspection of the cured demonstrator shell revealed geometric deviations; jx was not symmetric about the xz-plane and jy appeared to have regions of varying curvature. The shell was cured in a rectangular planform and later trimmed to the required circular shape. Prior to trimming, the rectangular planform was tested for tristability. Initially only SS1 and SS3 could be found and it was observed that a local instability in one of the corners prevented the shell becoming stable in the SS2 configuration. The instability was expected to be due to a local manufacturing imperfection such as a region of non-conforming curvatures. Upon removing the corner with the instability all three stable states were found. Following complete trimming of the shell to the circular planform (250 mm radius), SS3 no longer held its shape and unsupported would snap back to SS1. This snap-back was observed to be due to an instability located close to one of the edges which

Y

X Z

(a)

(b)

(c) Fig. 8. Manual tape-laying of CFRP prepreg onto male mould.

Fig. 9. (a) Demonstrator stable state 1, dashed line indicates circular planform prior to trimming out instability, (b) stable state 2 and (c) stable state 3. In all cases support is provided at the centre (origin).

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would propagate in the x-direction through the shell. In the rectangular planform this region was surrounded by material which was able to suppress the instability. To regain SS3 the edge at which the instability was found, was further trimmed as shown in Fig. 9a, this trimming was performed incrementally in order to remove the least amount of material as possible, given the sensitivity of SS2. The removal of 10 mm (xy-plane) in from this edge was sufficient to regain SS3 without the loss of SS2. 4.2. Test method Actuation between stable states was performed by resting the centre of the shell on a cylinder (approximate diameter 100 mm) and applying vertical loads by hand to points on the shell edges intersecting the xz-plane for SS1 to SS2 and both the xz- and yzplane for SS1 to SS3 (Fig. 2a). Emphasis was placed on applying the loads vertically and at a slow rate to provide a quasi-static response. Global average curvature results were determined for all stable states by measuring the position of the points on the shell edge intersecting the xz- and yz-plane and assuming a constant radius arc between these points and the shell centre. However, due to the presence of the bending boundary layer, the global average curvature calculated is expected to be different to the average interior curvature. For this reason, factors relating the global average curvature and average interior curvature determined from FEA results were used to estimate the demonstrator average interior curvature. Peak actuation loads required to transition between stable states were recorded using spring scales. 4.3. Results All three demonstrator stable shapes were successfully found and held their shape when centrally supported as shown in

Table 5 Demonstrator curvatures of stable states. Curvature (mm1)

State

SS1global ave. SS2global ave. SS2ave. interior SS3global ave. SS3ave. interior

106  jx

106  jy

3952 494 495 909 1019

396 2377 2406 967 1057

Table 6 Demonstrator actuation forces between stable states. Path

Load points (in Fig. 2a)

Peak load (N)

Error (N)

SS1 ? SS2 SS1 ? SS3

Px Px and Py

25.3 507

±2.0 ±30

-z

y

Fig. 10. Demonstrator in SS3, yz-plane view showing extent of boundary layer effect (curvature gradient towards edge) on jy.

Fig. 9. The calculated average curvatures, both global average and average interior, are provided in Table 5. Peak force measurements during actuation from SS1 to SS2 and SS1 to SS3 are provided in Table 6. Peak actuation forces required to transition back from SS2 to SS1 are estimated to be less than 5 N (applied at xz-plane edges), as very small perturbations would result in a snap back to SS1. For the rectangular planform, peak actuation forces required to transition back from SS3 to SS1 were of similar magnitude to the SS1 to SS3 direction, approximately 500 N. However, the circular planform in SS3 was sensitive to perturbations and required lower actuation loads to snap back from SS3 to SS1 (<100 N). This is believed to be due to only partial removal of the instability region discussed in Section 4.1, causing the snap back with lower than expected actuation forces. The bending boundary layer was visibly observed in the demonstrator as a rapid change in curvature towards the shell edge for both SS2 and SS3. The largest boundary layer being observed in the yz-plane of SS3 (Fig. 10), estimated to be occupying 50% of the length in this direction.

5. Discussion Three stable states for an orthotropic shallow shell with constant curvature were predicted with an analytical model, verified with FEA and shown for the first time with the manufacture of a demonstrator. Results showed good correlation of the stable state curvatures between the methods, with a summary provided in Table 7. Within the interior region (no boundary layer effect) of SS2 and SS3 the average interior curvature results between analytical and FEA correlate within 4%, analytical and demonstrator 6% and FEA and demonstrator 8%. The base state (SS1) displays no boundary layer and no difference between the analytical model and FEA exists. Differences in demonstrator SS1 curvatures from the design curvatures are seen only due to errors introduced during the manufacturing procedure. FEA and demonstrator results revealed that the bending boundary layer, which was neglected in the analytical model, has a significant effect on both the quantity and shape of stable states. The large boundary layer dominated and prevented tristability in FEA shells with a planform size less than 200 mm (radius), specifically the shallow energy trough of SS2 could not be found. Research and understanding to date on shell bending boundary layers is confined to singly curved shells. Results presented in this paper indicate a significantly larger boundary layer exists when considering doubly curved shells in comparison to singly curved. Assumptions neglecting the boundary layer for doubly curved shells can only be considered valid for large planforms. Within the shell interior region the boundary layer does not appear to affect results and good correlation between methods is observed. However other assumptions in the analytical model expected to have effects within the interior region include the constant Gaussian curvature, K, and planform area. FEA trials including CTE effects confirmed analytical predictions that the CTE strains only applied effective in-plane loads to the shell and had negligible difference on curvature of the stable states. The strain energy plot was subject to an initial shift when considering CTE (Fig. 6). Both the analytical model and FEA did not include the effects of the stretching and skewing required to drape the zero Gaussian curvature prepreg sheets over the positive Gaussian curvature mould. The laying-up of strips of prepreg aided to control the location of material overlaps that occurred towards the edge of the shell. The overlap regions effectively created local stiffening. The use of smaller strips reduced the size of these regions by creating more occurrences approaching a smeared stiffness increase over

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B.H. Coburn et al. / Composite Structures 96 (2013) 446–454 Table 7 Curvature comparison between analytical predictions, FEA and demonstrator in stable states. Curvature (mm1) Analytical (constant)

SS1 SS2 SS3 a

FEA (ave. interior)

Demonstrator (ave. interior)

jx

jy

jx

jy

jx

jy

3333 494 922

341 2309 1115

3333 503 1014

341 2223 1110

3952a 495 1019

396a 2406 1057

SS1 demonstrator curvatures are global average, not average interior.

500 Demonstrator (global ave.)

400

Ky (10-6 mm-1)

Design

300

200 Analytical

100 FEA

0 2500

3000

3500

Kx

(10-6

4000

4500

mm-1)

Fig. 11. Analytical and FEA tolerance on base curvature of tristable shell to remain in tristable region. Design and demonstrator curvatures (global average) are marked. Grey and dark grey regions only representative of expected shape of tristable region for analytical model and FEA respectively as per [14].

the entire shell. The demonstrator’s true fibre orientation away from the centre of the shell deviated from the design ±45° by up to an estimated ±5° at the shell edges due the curvature of the mould. The sensitivity of the base (SS1) curvatures of the shell to error is shown for the analytical model and FEA in Fig. 11, the design curvature was selected to fall within this tristable range. The FEA range was reduced by approximately a factor of two in comparison to the analytical range. This is expected to be due to neglecting the bending boundary in the analytical model. The large difference between the demonstrator’s SS1 global average and design curvature are mainly attributed to difficulties in controlling the geometry during manufacture of the shell. Other possible explanations include mismatch between the glass fibre mould and CFRP CTE or slight deviations from symmetry about the mid-, xz- and xy-planes. Although the demonstrator’s SS1 global average jx and jy are both larger than the design, they are still within the analytical and FEA tristable regions and three stable states were found. The actual local curvature of the demonstrator’s base state was not constant throughout the shell and regions consisting of locally large and small curvatures were visually observed. These non-conforming regions are thought to be the cause of observed instabilities that prevented tristability prior to being trimmed out. The manufacture of a thicker one-piece mould would provide greater stability for better dimensional control. Actuation forces between stable states varied depending on the path taken and change in K experienced during the transition. When actuating from SS1 to SS2 a maximum energy peak is reached close to the point of maximum K (Fig. 4). The K at this peak is only slightly larger than that at the base, indicating small amounts of mid-plane stretching; hence the small energy peak and low actuation force. Actuation from SS1/SS2 to SS3, however required a jx and jy sign change and consequently a zero value

of K occurs during the transition. At the point when K is momentarily zero there is a large amount of mid-plane stretching in the shell producing a large energy peak. Past this point there is a release of strain energy as K positively increases. Consequently, the peak energy and load transitioning to SS3 occurs at either a zero jx or jy. The K of all stable states is very close, indicating the large amounts of energy required for mid-plane stretching compared to bending. It is the mechanism of mid-plane stretching that requires the high actuation forces to transition to SS3. For most multistability applications it is desirable to have large actuation forces between states, enhancing the stability of each configuration and preventing unintentional transition. On the other hand, the SS2 configuration is achievable with a modest actuation force due to only small amounts of mid-plane stretching occurring, this can represent an advantage for applications where a limited amount of actuation power is available. Although not explored in this paper, the use of moderate to high modulus bi-directional woven CFRP can be used to approach the degeneracy condition. Woven CFRP with a stacking sequence of [±45n] can achieve b = 1.00 and m  0.80 for any integer number of plies. A benefit with this stacking sequence is the possibility of reducing the shell thickness whichpis expected ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi to reduce the boundary layer effect by a factor of t new =tcurrent allowing manufacture of smaller planform shells. 6. Conclusion This study has explored tristability of an orthotropic doubly curved shell. Tristability was predicted with an analytical model [9,14], verified with finite element analysis and shown, for the first time, with the manufacture of a demonstrator. The bending boundary layer length, assumed negligible in the analytical model, was evidenced in the FEA and demonstrator by a region of rapid curvature change towards the shell edge. The length of this region was shown to be up to six times larger than that estimated by singly curved shell methods. A large planform size (250 mm radius) was required to minimise this effect and allow all three stable shapes to be found. The non-zero Gaussian curvature appears to significantly increase the size of the bending boundary layer due to the interplay it causes between bending and stretching modes of deformation. Within the interior region, unaffected by the boundary layer, the analytical, FEA and demonstrator curvatures were approximately constant for SS2 and SS3 and agreement between analytical and FEA curvatures was within 4%, analytical and demonstrator within 6% and demonstrator and FEA within 8%. The manufacture of a demonstrator introduced several imperfections not accounted for in either the analytical model or FEA such as fibre orientation deviation, material overlap and regions of non-conforming geometry. The mould and demonstrator shell were qualitatively observed to have regions of varying curvature producing local instabilities in the demonstrator which were required to be trimmed out to allow all three shape configurations to be found. A machined thicker one-piece mould would provide

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the required rigidity and dimension control to remove instability regions. CTE effects were shown to be negligible to curvature results of the stable states, only locking in in-plane strains into the shell prior to actuation. This paper has demonstrated several benefits multistability due to Gaussian curvature offers. Multistability due to Gaussian curvature does not rely on induced strains during manufacture and exists purely due to the combination of geometry and material properties allowing design and dimension control of the base stable state. The coupling of bending and stretching modes of deformations offers the opportunity for both high and low actuation loads compared to other methods of multistability increasing the range of applications and effectiveness of multistable composites. Not relying on thermal effects for multistability, balanced and symmetric shells can be manufactured with multistable properties uncoupled from hygrothermal effects. Finally, the combination of Gaussian curvature and anisotropy creates a large design space for multistability and allows shells with three stable states to be designed and manufactured. Acknowledgement The authors gratefully acknowledge the support of the EPSRC under its ACCIS Doctoral Training Centre Grant, EP/G036772/1. References [1] Hyer MW. Some observations on the cured shape of thin unsymmetric laminates. J Compos Mater 1981;15(2):175–94. [2] Dano ML, Hyer MW. Thermally-induced deformation behavior of unsymmetric laminates. Int J Solids Struct 1998;35(17):2101–20. [3] Potter KD, Weaver PM. A concept for the generation of out-of-plane distortion from tailored FRP laminates. Compos Part A: Appl Sci Manuf 2004;35(12): 1353–61. [4] Pirrera A, Avitabile D, Weaver PM. Bistable plates for morphing structures: a refined analytical approach with high-order polynomials. Int J Solids Struct 2010;47(25–26):3412–25. [5] Pirrera A, Avitabile D, Weaver PM. On the thermally induced bistability of composite cylindrical shells for morphing structures. Int J Solids Struct 2012;49(5):685–700.

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