Alex Brinkmeyer1
Department of Aerospace Engineering,
Advanced Composites Centre forInnovation and Science,
University of Bristol,
Bristol BS8 1TR, UK
e-mail: alex.brinkmeyer@bristol.ac.uk

Sergio Pellegrino
Joyce and Kent Kresa Professor of Aeronautics
and Professor of Civil Engineering,
Graduate Aerospace Laboratories,
California Institute of Technology,
Pasadena, CA 91125
e-mail: sergiop@caltech.edu

Paul M. Weaver
Professor in Lightweight Structures,
Department of Aerospace Engineering,
Advanced Composites Centre forInnovation and Science,
University of Bristol,
Bristol BS8 1TR, UK
e-mail: paul.weaver@bristol.ac.uk

1

Effects of Long-Term Stowage
on the Deployment of Bistable
Tape Springs
In the context of strain-energy-deployed space structures, material relaxation effects
play a significant role in structures that are stowed for long durations, for example, in aspace vehicle prior to launch. Here, the deployment of an ultrathin carbon fiber reinforced plastic (CFRP) tape spring is studied, with the aim of understanding how longduration stowage affects its deployment behavior. Analytical modeling and experiments
show that the deployment time increases predictably with stowage time and temperature,
and analytical predictions are found to compare well with experiments. For cases wherestress relaxation is excessive, the structure is shown to lose its ability to deploy autonomously. [DOI: 10.1115/1.4031618]

Introduction

Undesirable time-dependent viscoelastic effects have been previously reported in deployable space structures that are stowed forextended durations prior to launch and subsequent deployment[1]. In thermoset composites, these effects increase as the temperature of the structure approaches the glass transition temperatureof the resin. For deployable structures that need to achieve andmaintain a precise geometry (e.g., a mirror or radar antenna), it
becomes essential to be able to understand and predict the effects
of temperature and deformation history on dimensional stability
and deployment accuracy [2].
A case in point underlining the significance of time-dependent
effects is the Mars Express mission, launched in 2003 by theEuropean Space Agency. This spacecraft, still in operation today,
carries thin-walled deployable booms forming the Mars Advanced
Radar for Subsurface and Ionosphere Sounding (MARSIS)
antenna—designed to search for water on Mars and study its
atmosphere. MARSIS is a large closed-section 40 m antenna manufactured from an S-Glass/Kevlar composite, with periodic slotted
sections (Fig. 1). In its stored state, the booms were folded into anaccordion manner. When released, the antenna was designed to
deploy following a strict latching sequence so as to avoid contact
with the spacecraft.
Telemetry flight data from the 2005 deployment of the
MARSIS antenna showed two anomalies in its deployment [1].
First, inertial measurements hinted toward the fact that the boom
had not completely locked into place. Second, data showed thatthe deployment dynamics differed significantly from the modeled
behavior. The ground testing report noted that a large reduction in
the hinge torque was observed, compared to a pristine hinge [1].
The hinge that was tested had undergone multiple thermal cycles
and had been stored for 2 yr prior to deployment, as well as being
1
Corrresponding author.
Contributed by the Applied Mechanics Division of ASME for publication in the
JOURNAL OF APPLIED MECHANICS. Manuscript received May 14, 2015; final manuscript
received September 15, 2015; published online November 9, 2015. Editor:
Yonggang Huang.

Journal of Applied Mechanics

highly stressed in its stowed configuration. In the final report, thereduction in torque was attributed to significant viscoelastic stressrelaxation in the material matrix.
The conclusions from the Mars Express mission report showedthat viscoelastic effects can indeed become significant in structures that are stowed for a long time—in this case in a launchvehicle—prior to their deployment in space. It seems therefore
crucial to understand precisely how storage affects the deployment behavior of this type of structure.
Another example of a thin-walled deployable boom is the Storable Tubular Extendable Member (STEM), a self-driven, energystoring deployable structure which has gained popularity over theyears. The STEM, which resembles a self-extending carpenter’stape measure, was invented in Canada in the 1960s by Rimrott [3]and commercialized a few years later by Astro Aerospace, anaffiliate of Northrop Grumman. It has flown successfully on
various spacecraft, such as the Voyager probe, the Hubble Space
Telescope, and the Mars Pathfinder rover, and is still in production
to this day [4].
This paper focuses, in particular, on how storage affects the
deployment of ultrathin bistable tape spring booms. In contrast to
the STEM, bistable tape springs are designed to be stable both intheir coiled and deployed state [3]. As a result, no mechanism isrequired to maintain the structure in either of these states, thus
considerably simplifying the design of the actuator or eliminating
the need for it completely. The structures considered herein are
inspired by Rolatube’s bistable reeled composites, which are commercialized as stowable masts for civil and military applications[5]. They are easily stowed in a backpack and can be readily
deployed by hand whenever needed.
Tape springs have been the object of extensive work, in particular by Refs. [6–8], who have studied separate aspects of deployment dynamics, bistability, and viscoelastic shape recovery. This
paper aims to reconcile these three aspects and provide a better
understanding of time-dependent effects in bistable tape spring
structures. Building on a preliminary study by the present authors[9], this paper seeks to enhance the understanding of long-term
stowage effects in such structures. The objectives are first to

C 2016 by ASME
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Table 1

Tape spring geometrical parameters

R (mm)r (mm)b (deg)

L (mm)

16.6

20.8

131.8

1253

Table 2E11 (GPa)
128

T800H elastic properties at room temperature [10]E22;0 (MPa)G12;0 (MPa)

 12

6.5

7.6

0.35

Fig. 1 MARSIS antenna boom structure. The boom consists ofa single tubular structure with several flatennable slotted sections which act as elastic hinges (courtesy of Astro Aerospace).

Fig. 3 Flow diagram of the tape spring analytical models

Fig. 2 Bistable tape spring geometry with (a) the side view ofpartially coiled spring, (b) the front view of fully deployedspring; r is the coiled radius and R is the deployed radius, withb the subtended angle of the deployed spring

investigate the change of deployment speed with different stowage conditions (time and temperature) using an analytical modeland experimental tests and second, to understand how the stabilityof bistable tape springs changes with stowage time.
This paper is structured as follows. The geometry of the tapespring and the stowage model are first described. The stability ofthe tape spring is then analyzed using an analytical model andverified using lamination parameters. The deployment of the tapespring is investigated for various stowage conditions and compared with experimental tests. Finally, singular deployment cases
are discussed before conclusions are drawn.

2

The antisymmetric layup [45/45/0/45/45] is used throughout, with the 0 deg direction oriented along the length of the tapespring. The layup is chosen to be antisymmetric as this eliminates
bend-twist effects during coiling, which would otherwise exist ina symmetric layup because of flexural/twist anisotropy.
The tape spring, manufactured in its extended shape, is fully
coiled and stored at a given temperature. Before any deployment
can occur, one end of the coil is snapped out and fixed in place.
This procedure is necessary to start the deployment process,because the structure is fully stable in its coiled state. The otherend of the structure is then allowed to deploy autonomously. The
position of the center of the coil is then measured with time. The
analytical model consists of three parts:(1) Stowage model: Transforms the material relaxation dataobtained at ply level to the overall laminate relaxation
using classical laminate analysis (CLA) [11].
(2) Stability model: Essentially a submodel of the stowage
model, the stability model simulates the nature of the structure’s stability during the stowage process.
(3) Deployment model: Uses the material state at the end of thestowage process to predict the dynamic deployment of thetape spring.

Geometry and Material Definition

The tape, in its partially and fully deployed configuration, isshown in Fig. 2. The geometry of the coiled structure is characterized by the radius of coiling r, while the extended configuration has transverse radius R, subtends an angle b, and has a total
length L. The transition between coiled and uncoiled regions of
the tape spring is called ploy region. It has no significant effecton the deployment behavior and is neglected throughout this paper. The measured geometrical parameters are given in Table 1[10].
The material selected for the prototype structures that arestudied in this paper is ThinPly T800H, an ultrathin unidirectional
carbon fiber prepreg produced by North Thin Ply Technology.
The ply thickness was found experimentally to be 30 6 1 lm,
using ultrasonic measurement of cured ½06 and ½906 laminates(the subscript denotes the number of plies in usual composite
notation). Additionally, the material elastic properties at room
temperature were determined experimentally using an Instronmaterials testing machine and a stereoscopic digital image correlation (DIC) Correlated Solutions imaging system. These properties are given in Table 2.
011008-2 / Vol. 83, JANUARY 2016

A synoptic flow diagram of the models is given in Fig. 3. The
models span two distinctively different time scales—the stowage
and stability models simulate behavior over several hours or days,
while the deployment model predicts dynamic behavior occurringin fractions of a second. The outputs of the stowage model (in particular, the net ejection force) are used as inputs to the deployment
model. Sections 3–6 focus on each model in detail and present theresults obtained.

3

Stowage Material Model

3.1 Characterization of Ply Relaxation. The carbon fibers
are assumed to be elastic for the observed relaxation times andtemperatures. Hence, because the deformation in the longitudinal
1–direction is dominated by the fibers, it is assumed that theYoung’s modulus of the composite lamina in the 1–direction isconstant, E11 ðtÞ ¼ E11 .
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Fig. 4 Relaxation experimental apparatus, consisting of: (a) a
DIC imaging system and (b) a thermal chamber mounted inside
an Instron testing machineStress relaxation tests on ½906 laminates were performed todetermine the viscoelastic behavior in transverse tension E22 ðtÞ.
An Instron 5500 Series machine equipped with a thermal chamber
and a thermocouple was used for the relaxation tests (Fig. 4). The
specimen and the Instron grips were warmed to achieve thermal
equilibrium prior to starting the relaxation test. The Instron crosshead displacement was then adjusted to remove any slack induced
by temperature effects, and the specimen was allowed to rest foranother 30 min. The transient thermal expansion of the rig was
found to have a significant influence on the relaxation results, as
thermal expansion strains were of the same order of magnitude as
the mechanical strains. Hence, achieving thermal equilibrium
prior to the test was crucial for accurate results.
The relaxation test consisted of applying a quasi-instantaneous
strain of e11;0 ¼ 0:1% to the specimen and holding this value for3 hrs. The load and strain were recorded, respectively, through theInstron machine load cell and the 3D DIC system. It was foundthat the calculation of the instantaneous modulus E22;0 from therelaxation test was inaccurate because of the small strains
involved. Hence, a separate tension test was used to determine
E22;0 at each temperature to complete the characterization of thematerial.
A Prony series is chosen to represent the relaxation modulus in
the transverse direction
!
N
X
t=si
eRi ð1  eÞ(1)E22 ðtÞ ¼ E22;0 1 
i¼1

where eRi are the tensile relaxation coefficients associated withthe transverse modulus E22 ðtÞ, and si are the characteristic times.
A similar expression can be defined for the shear modulus, wheregRi are the shear relaxation coefficients associated with theshear modulus G12 ðtÞ. It is assumed that the material has a timeindependent Poisson’s ratio, hence it is possible to set eRi ¼ gRi .
The Prony series is fitted to the experimental data curves using
a bespoke [check] MATLAB nonlinear optimization algorithm using
the fminsearch function [12]. Starting from an initial estimate of
the Prony series coefficients, this algorithm converges to the final
coefficients that minimize the least squares error.
Figure 5 shows the fitting of the normalized transverse modulus
E22 ðtÞ=E22;0 for stowage at 60  C. The Prony series coefficientsare listed in Table 3. It was found that four terms were sufficient
to provide an accurate nonlinear fitting, by minimization of the
fit error. Additionally, fitting of the curve is adjusted so that thenormalized modulus is equal to 1 at the start of the test.
Figure 5 shows that at this temperature the material experiencesa significant drop in stiffness, decreasing by around 35% after
Journal of Applied Mechanics

Fig. 5 Normalized transverse modulus fitted with a Prony
series for relaxation at 60  C for 3 hrs
Table 33 hrs

Prony series coefficients for relaxation at 60  C for

Term
i

eRi

si
(s)

1
2
3
4

0.013
0.062
0.094
0.219

2.64
16.3
266
4790

3 hrs. Oscillations were found to appear in the relaxation data,
growing in amplitude with increasing temperature. These fluctuations are due to periodic variations in temperature inside the thermal chamber, as the heating element turns on and off repeatedly.
Because the period of these oscillations was much smaller than
the total test time, no significant effect on the results was found.
Presmoothing of the data prior to fitting the Prony series was notdeemed to be necessary.
In this paper, it is sufficient to consider relaxation at 60  C as it
gives an indication of how the structure behaves at a higher temperature, compared to room temperature. The epoxy resin usedherein can be assumed to be rheologically simple, meaning thattime and temperature effects can be used interchangeably todescribe the behavior of the material under relaxation. Using theprinciple of time–temperature superposition, testing at a higher
temperature is equivalent to reducing the relaxation time of thematerial [13]. Obtaining the relaxation curves at two other temperatures would be required to complete the characterization of thematerial, but this is beyond the aim of this paper.
3.2 Transformation to Laminate Relaxation. The lamina
stiffnesses defined in Sec. 3.1 are transformed to laminate
stiffnesses using CLA. In the CLA theory, the ABD matrix represents the laminate stiffness matrix relating in-plane strains e0
and curvatures, j, to in-plane stress N and bending moment M,
resultants [14](2)
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D^ 66 and anticlastic stiffness D^ 12 tend to increase the bistabilityof the spring, while a larger transverse stiffness D^ 22 tends todecrease it.
It is proposed that the stability criterion is adapted to timedependent bistable structures; by allowing the nondimensional
constants to vary with time, S is computed at every point in time
during relaxation. Additionally, a reduced bending stiffness
matrix D is introduced to correct for the nonzero B coupling
matrix. Hence, the stability criterion is now written asD^ ðtÞ

S ¼ 4D^ 66 ðtÞ þ 2D^ 12 ðtÞ  2 22
D^ 12 ðtÞ(4)

where the reduced bending stiffness is given by D ¼ D
BA1 B.

Fig. 6 Evolution of the bending stiffnesses Dij for relaxation at
60  C. The D11, D22, and D66 terms decrease with relaxation time,
while the D12 term increases. The D16 and D26 components are
zero and are therefore not shown.

The [45/45/0/45/45] layup considered herein is balanced, as
it has equal number of positive and negative angled plies. As aconsequence, A16 ¼ A26 ¼ 0, i.e., the layup has zero extensionshear coupling. Because the layup is also clearly antisymmetric,
the bend-twist components D16 and D26 are zero, resulting in astate of flexural orthotropy. However, the antisymmetry induces asmall coupling between the extension and bending components
B16 and B26.
As the material is viscoelastic, the laminate’s ABD matrix is
not a constant material property and changes with time as thematerial relaxes. The four computed nonzero components of thelaminate bending stiffness (D matrix) are plotted in Fig. 6 withrelaxation time, and the plot shows that D11, D22, and D66
decrease with relaxation time, while D12 increases. (The D16 andD26 components are zero and therefore are not shown.) Relaxation
causes a relatively large decrease of the longitudinal bending stiffness, with the D11 and D22 terms each decreasing by approximately 9%. On the other hand, the decrease in torsional stiffness
D66 was found to be negligible.

4

Stability Analysis

4.1 Stability Model. Elastic bistable structures contain twoenergy “wells” or minima in their elastic potential [15], with atailorable deployment required to trigger the transition between
these stable configurations. Therefore, no external force is needed
to maintain the structure in each of its stable configurations.
Such behavior is highly desirable for deployable structures. A
bistable tape spring, for example, rests in either of two stableconfigurations—coiled and fully deployed.
On the other hand, for a time-dependent structure, such conclusions can no longer be drawn because viscoelastic bistable structures do not have a uniquely defined potential surface [15].
To determine whether the structure studied herein remains bistable with time, a criterion of stability for cylindrical shells, S, is
computed during relaxation.
S ¼ 4D^ 66 þ 2D^ 12  2

D^ 22
D^ 12

Thus, using Eq. (4), the stability criterion at time t ¼ 0 is
calculated to be Sð0Þ ¼ 0:95, which is a positive value with a suitable margin of stability. Therefore, it can be concluded that the[45/45/0/45/45] layup yields a suitably bistable structure.
The variation of the three nondimensional parameters compos
ing S, i.e., D^ 12 ; D^ 22 ; D^ 66 , is now investigated. These terms areplotted in Fig. 7 for the case of stowage 60  C for 3 hrs. The plotshows that, contrary to the stiffnesses represented in Fig. 6, bothD^ 12 and D^ 66 increase with relaxation time, while D^ 22 remains
approximately constant.
By inspecting each term of Eq. (3) separately, the two positive
terms increase while the negative term decreases; therefore, S
increases with relaxation time. Plotting the stability criterion
against the relaxation time in Fig. 8 does show that S increases
from S ¼ 1.20 at the start of the relaxation to S ¼ 2.05 for a relaxation time of t ¼ 3 hrs (t  104 s).
It is noted that the behavior shown in Fig. 8 is representative ofa general stowage case; therefore, conclusions can be drawn forany stowage time and temperature. Using the principle oftime–temperature superposition, then provided that Sð0Þ > 0 at
the lower bound of temperatures studied, then Sð0Þ > 0 for any
higher temperature [13]. The unequivocal increase of the stability(3)^ is the nondimensional bending stiffness, given by
where D^ ¼ D=D11 .
DPrevious research showed that, for elastic bistable structures,the existence of the second stable equilibrium depends only on thesign of Eq. (3) [16]. When S > 0, the tape spring is bistable, andwhen S  0 the structure is monostable. Larger torsional stiffness
011008-4 / Vol. 83, JANUARY 2016

^
4.2 Stability Results. The nondimensional reduced D
matrix at room temperature and at the start of relaxation is given,
for the structure considered, by
2
3
1
0:607
0

^ ¼ 4 0:607 0:976
0 5
(5)
D
0
0
0:745

Fig. 7 Evolution of the nondimensional parameters for relaxa^  and D^  terms increase with relaxation
tion at 60  C. The D
12
66

^
time, while D 22 remains approximately constant.

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Here, Wi are the material invariants defined as1
W1 ¼ ð3Q11 þ 3Q22 þ 2Q12 þ 4Q66 Þ
81
W2 ¼ ðQ11  Q22 Þ
2
1
W3 ¼ ðQ11 þ Q22  2Q12  4Q66 Þ
81
W4 ¼ ðQ11 þ Q22 þ 6Q12  4Q66 Þ
81
W5 ¼ ðQ11 þ Q22  2Q12 þ 4Q66 Þ
8(11)

where Qi are the lamina stiffnesses in the 1–2 material coordinate
system given by
Fig. 8 Evolution of the stability criterion for relaxation at 60  C.
S increases monotonically with time, which suggests that thetape spring remains bistable and does not autonomously
unfurl.

criterion with relaxation time leads to an important conclusion—
provided the structure is initially bistable, it will remain so during
the relaxation process. In other words, the tape spring will not
self-deploy once it is stored in its coiled state, even if it isunconstrained.

5

Validation of the Stability Analysis

5.1 Flexural Stiffness Analysis. The previous stability analysis is now validated using a different analytical technique. The
flexural stiffness D can be represented by a linear combination ofmaterial invariant stiffnesses, stacking sequence effects describedby lamination parameters nD , and second moment of area by
D¼

h3
C  nD
12

nD
1

nD
2

(6)

E11
;
1  12 21
E22
Q22 ¼
;
1  12 21
Q11 ¼

n ¼ ½1

T
nD
4 

nD
3

(7)

and
12
nD
1;2;3;4 ¼ 3

ð h=2

h

ðcos 2h; cos 4h; sin 2h; sin 4hÞz2 dz

(8)

h=2

where z is the distance to the midplane, and h is the total laminate
thickness [17]. The material property matrices vector C is defined as
C ¼ ½ C0

C1

C2

C4 

C3

(9)

where

12 E221  12 21(12)

Q66 ¼ G12and E11, E22, G12,  21, and  21 are the basic ply stiffness
properties.
5.2 Application to Tape Spring Problem. The [45/45/0/
45/45] laminate considered herein is balanced, antisymmetric,
and contains a combination of 0 deg and 645 deg plies only.
Therefore by inspection of the trigonometric terms, nD
4 ¼ 0. Additionally, as has already been shown, D16 ¼ D26 ¼ 0 due to thelayup asymmetry; therefore,
D11 ¼


h3 
D
W1 þ nD
1 W2 þ n2 W3
12

D22 ¼


h3 
D
W1  nD
1 W2 þ n2 W3
12


h3 
D12 ¼
W4  nD
2 W3
12

where
D

Q12 ¼

D66 ¼

(13)


h3 
W5  nD
2 W3
12The outer 645 deg plies of the laminate contribute to most of thebending stiffness, due to their enhanced second moment of area
contribution. To simplify the analysis the inner 0 deg ply cantherefore be neglected. Thus the lamination parameters further
simplify tonD
1  0;nD
2  1(14)D(A full analysis reveals in fact that nD
1 ¼ 0:008 and n2 ¼ 0:984;
hence, the above simplification is valid.) Thus, Eq. (13) reduces toh3
ðW1  W3 Þ
12h3
D22  ðW1  W3 Þ
12h3
D12  ðW3 þ W4 Þ
12h3
D66  ðW3 þ W5 Þ
12
D11 

2

W1

6
C0 ¼ 4 W 4

W4
W1

3

2

7
0 5;

6
C1 ¼ 4 0

0

0
0 W5
2
W3 W36
C2 ¼ 4 W3 W3

0

3

7
0 5;
0
0
W3
2
3
0
0
W3
6
7
C4 ¼ 4 00
W3 5W3 W3
0

W2

0

W2

7
05

0
2
0
16
C3 ¼ 4 0
2
W2

Journal of Applied Mechanics

0

3

0

0
0

W2

3

0

7
W2 5

W2

0
(10)

(15)

It is noted that with this assumption D11  D22 . Observing Fig. 6,
it can be noticed indeed that the D11 and D22 curves follow
closely. D11 remains slightly larger than D22 because of the bending orthotropy introduced by the middle 0 deg ply.
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Substituting the material invariants Wi from Eq. (11) and simplifying, the following expressions are obtained

D12;R  D12;U
"


#

h3
E22;R
1 þ 212
 4G12;U¼
 1 E22;U
48 E22;U
1  12 21h3
D11  ðQ11 þ Q22 þ 2Q12 þ 4Q66 Þ
48
h3
D22  ðQ11 þ Q22 þ 2Q12 þ 4Q66 Þ
48
h3
D12  ðQ11 þ Q22 þ 2Q12  4Q66 Þ
48

(16)

5.3 Variational Viscoelastic Analysis. Let us now consider
the variation of the Dij coefficients with time. While it is impractical to define a closed-form viscoelastic expression, it is possible
to study instead the bounding states of the material—relaxed andunrelaxed. These bounds are denoted by subscripts “U” for theunrelaxed state and by “R” for the fully relaxed state. The
assumptions stated previously are summarized as follows:(1) The longitudinal Young’s modulus E11 does not changewith time. Due to its molecular composition, carbon fiber
experiences negligible creep. Using the rule of mixtures in
the longitudinal direction, this justifies why E11 can beassumed constant.
(2) The Poisson’s ratio  12 is time-independent.
(3) The relaxation causes a decrease of the shear modulus andYoung’s modulus with time, i.e., G12;R < G12;U and
E22;R < E22;U . This is a physical process dominated by therelaxation mechanism of the epoxy resin, whereby applying
a strain to the material causes a decrease in its apparent
modulus [13].
(4) The relaxation of the shear modulus and Young’s modulus
occurs at the same rate, i.e., G12;R =G12;U ¼ E22;R =E22;U .
For simplicity, yet without losing qualitative understanding, theB matrix coupling terms are ignored; therefore, it can be assumed,with little loss of accuracy that Dij ¼ Dij .
Using these assumptions, it is possible to draw immediate conclusions concerning Eq. (16). Assumption 1 means that Q11 isconstant with time. Additionally, assumptions 2 and 3 together
show that Q12, Q22, and Q66 decrease with time. Therefore, overall
D11 and D22 decrease with time, which is consistent with the
behavior of the curves in Fig. 6.
On the other hand, the D12 and D66 terms do not lead to such an
immediate conclusion. To understand this phenomenon better, therelative contribution of each term is examined. Substituting thematerial constants Qij from Eq. (12), the D12 term is written as


h3
E11
E22
ð1 þ 212 Þ  4G12
D12 ¼
(17)
þ
48 1  12 21 1  12 21(21)

therefore the sign of D12;R  D12;U depends on the sign of the second term. Thus, D12;R  D12;U > 0 if, and only if,
1 þ 212
E22;U
 4G12;U < 0(22)
1  12 21
leading to the relation
E22;U 4ð1  12 21 Þ<
1 þ 212
G12;U

(23)

With 12 ¼ 0:35, Eq. (23) becomes
E22;U< 2:34z
G12;U

(24)

The transverse Young’s modulus needs to be more than twice ashigh as the shear modulus for this expression to hold. Here, atroom temperature
E22;U 6:5
¼ 0:86¼
G12;U 7:6

(25)

and so verifying Eq. (23), which means that
D12;R  D12;U > 0(26)

thus proving the unusual result that D12 increases with relaxation
for the material considered. It is worth noting that Eq. (23) is conveniently expressed in terms of unrelaxed values only; in otherwords, this relation only depends on the initial material properties.
5.4 Analysis of Dimensionless Terms. To conclude this analytical study, the nondimensional terms composing S are now
studied. First of all, as has previously been shown, D12 increases
while D11 decreases with time, so it can be concluded immediately that D^ 12 ¼ D12 =D11 must rise with relaxation time. Moreover, using Eq. (15) thenW1  W3¼1
D^ 22 W1  W3(27)

Hence D^ 22 remains constant with relaxation. Finally, studying theD^ 66 term
(18)

For the materials considered herein 1  212 > 0 as
0 < 12 < 0:5. Therefore, by inspection the D66 term also
decreases with relaxation.
To establish whether D12 increases with relaxation, let us study
the sign of D12;R  D12;U .
D12;R  D12;U
"


#


h3
E22;R
1 þ 212
G12;R
4G12;U
¼
1
1
E22;U
48
E22;U
1  12 21
G12;U
(19)
011008-6 / Vol. 83, JANUARY 2016(20)

Because E22;R =E22;U < 1 then
E22;R
1<0
E22;U

h3
D66  ðQ11 þ Q22  2Q12 Þ
48

Similarly, D66 is expanded as


h3
E11
E22
ð1  212 Þ
D66 ¼
þ48 1  12 21 1  12 21

Using assumption 4 and rearranging thenW3 þ W5
D^ 66 W1  W3(28)

It is helpful to write Eq. (28) in terms of material coefficients Qij asD^ 66 Q11 þ Q22  2Q12
Q11 þ Q22 þ 2Q12 þ 4Q66

(29)

It has previously been established that both the numerator anddenominator decrease with relaxation; however, noticing that
Q11 þ Q22 þ 2Q12 þ 4Q66
¼ ðQ11 þ Q22  2Q12 Þ þ 4Q12 þ 4Q66

(30)

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it becomes clear that the denominator relaxes faster (by the extra
term 4Q12 þ 4Q66 ) than the numerator. Hence D^ 66 increases withrelaxation.
In conclusion, the D^ 66 and D^ 12 terms increase with relaxation
effects, while the D^ 22 term is approximately constant, which supports the results shown in Fig. 7. Hence, it follows from Eq. (3)that the stability criterion S increases with relaxation, confirming
the important result shown in Fig. 8, which is that the structure
remains bistable at all times.

6

Deployment Model

6.1 Analytical Model. An analytical deployment model,
based on the work in Refs. [3] and [7], is used to predict the
deployment of the spring. The model, originally formulated forelastic isotropic STEMs, has here been generalized for orthotropic
viscoelastic laminates. This model assumes that the structure immediately transitions from a coiled shape to an extended state. In
reality, however, the transition occurs smoothly within the ployregion, but this has been neglected. In the following, the x-axis
refers to the longitudinal direction of the deployed spring, and they-axis to the orthogonal direction, originally defined in Fig. 2.
The bending strain energy in a composite shell of length x,
deformed such that the curvature change is Dj is1
Ub ¼ DjT D Dj A
2

(31)

where A is the midsurface area of the shell, given by A ¼ bRx [7].
Here, it can be assumed that Djxy ¼ 0 as twist is sufficiently
small. Additionally, because D16 ¼ D26 ¼ 0, Eq. (31) reduces to1
D11 Dj2xx þ 2D12 Djxx Djyy þ D22 Dj2yy A
Ub ¼
2

(32)

As the tape spring is deployed, the curvature in the longitudinal
direction jxx increases from 0 to 1=r, and the curvature in thetransverse direction jyy decreases from 1=R to 0. Therefore, thenet curvature changes are Djxx ¼ 1=r and Djyy ¼ 1=R. Factorizing and rearranging yields
Ub ¼



D11 bx D22 2R D12 R2
þ 2
2R
r D11 rD11

(33)

6.2 Ejection Force. The ejection force Fe is the force available for deployment of the spring. Fe is equal to the restraining
force needed to keep the spring at equilibrium during deployment.
Fe is given by the rate of change of the strain energy with respect
to the extension [3]; hence,
Fe ¼

@Ub
@x

(34)

Fig. 9 Plot of the ejection force with stowage time for relaxation at 60  C. The ejection force available for deployment
decreases as the structure relaxes during stowage.

6.3 Aerodynamic Correction. For deployment in air, a
simple aerodynamic drag model is implemented to adjust the net
ejection force. The drag of the structure is given by
Faero ðtÞ ¼

1 2
q v ðtÞ CD A
2

(36)

The frontal area A is roughly equal by the projected surface of thecoiled spring and is given by A ¼ 2bRr. The drag coefficient CD
is based on the average Reynolds number Re. For this application
it can be shown that Re  1000, so CD  1:17 [18]. As the dragacts in opposite sense to the deployment direction, the corrected
ejection force becomes
F0e ðtÞ ¼ Fe  Faero ðtÞ(37)

6.4 Ejection Velocity. The present configuration is a particular case of a variable mass dynamics problem. Although many
more rigorous treatments can be found in the literature, a simple
solution to the problem is presented here, inspired from Ref. [3],based on an energy conservation approach.
The kinetic energy of the tape spring K is given by a translating
component Kt and a rotating component Kr
K ¼ Kt þ Kr

(38)

Assuming the coiled radius r does not change during deployment,
then
v2
x2
þ mð L  x Þ r 2
2
2

A correction for nonsymmetric laminates is applied using the
reduced bending stiffness D [11]. Letting the bending stiffness
constants vary with time, the ejection force is given byD ðtÞb ^ 
2R ^ 
R2
D 22 ðtÞ 
D 12 ðtÞ þ 2(35)
Fe ðtÞ ¼ 11
2R
rr

where v is the ejection velocity and x is the angular speed, and mis the mass per unit length. As x ¼ v=R, Eq. (39) simplifies to

It is assumed that the ejection force does not change during
deployment, as the deployment occurs over a much shorter timescale than relaxation; hence, the ejection force at the end of relaxation is taken as the input for the deployment model, so that
Fe ¼ Fe ðt ¼ trel Þ, where trel is the total relaxation (stowage) time.
The variation of the ejection force with stowage time can be
simulated, to predict how the ejection force would decrease withstowage time. This behavior is shown in Fig. 9 for stowage at
60  C for 3 hrs.

Rimrott showed that a frictional force R diminishes the ejection
force available for deployment. Friction does indeed occur during
the deployment process, e.g., as a result of coils sliding acrosseach other and friction with the deployment surface. The frictional
force can be written as a Coulomb force Ff ¼ lF0e , where l is afriction coefficient. The empirical value l  0:6 was found to beappropriate [3]. By setting the net ejection force F0e ð1  lÞ equal
to the kinetic energy (assuming that the work done by the force
F0e  Ff is fully converted into kinetic energy), then

Journal of Applied Mechanics

K ¼ mð L  x Þ

K ¼ mðL  xÞv2

(39)

(40)

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F0e ðtÞð1  lÞ ¼ mðL  xÞv2

(41)

The ejection velocity is therefore given bysﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
F0e ðtÞð1  lÞxvðtÞ ¼
mð L  x Þ(42)

A simple iterative method is then used to integrate the velocity toobtain the position of the tape spring with time.

7

Experimental Methods

7.1 Manufacturing Techniques and Issues. The laminate isassembled on a flat surface then molded onto a cylindrical mandrel. Surface wrinkles can appear on the laminate due to the preforming process and the crumpling of the exterior bag under theapplied vacuum. It was found that this effect represents a major
source of experimental error as the presence of wrinkles canlocally increase the laminate thickness by more than 100%. As
such, large variations arise in the local bending stiffness as well as
the coiled radius, which can affect the deployment of the spring
considerably. To mitigate the appearance of wrinkles, the mandrel
is inserted into a shrink-plastic tube. Shrinking of the tube is performed prior to cure using a heat gun. Comparison of the tapesprings with and without the shrink tube shows a considerable
improvement in the surface finish. Breather fabric and a vacuum
bag are then fitted around the mandrel and sealed (Fig. 10).
Finally, the laminate is cured at 120  C for 3 hrs using an ASC
aerospace-grade autoclave.
Although an antisymmetric layup was chosen to reduce bendtwist effects, the resulting structure still presents a significant
amount of twist (around 140 deg m1 after demolding). This effectis due to residual thermal strains appearing in the laminate uponcool-down as a result of the layup antisymmetry. The overall
small thickness of the laminate further accentuates this effect.
Although the twisting in the structure is noticeably large immediately after demolding, it reduces significantly over time, settling
to around 50 deg m1 in ambient conditions. In Fig. 10(b), thespecimen on the left, demolded immediately after cure, shows amore pronounced twisting than the specimen on the right, which
was left to rest for several days. The decrease in twist is most
likely due to moisture ingress into the laminate which counteracts
the thermal stresses, as observed in Ref. [19].

7.2 Deployment Apparatus. Three stowage cases are investigated: zero stowage time, stowage for 3 hrs at 60  C, and stowage at 100  C. Stowage consists of coiling and stowing thespecimen in the Instron thermal chamber for 3 hrs at the set temperature. A thermocouple is used to monitor the chamber temperature. At the end of the stowage period, the structure is removed
from the chamber and deployed at room temperature (20  C) on aflat, horizontal surface. This procedure is followed because it isimpractical to perform the deployment within the thermal chamber. A camcorder is used to record the deployment of the tapespring at a frame rate of 29 frames per second. The position of thecenter of the coil for each time frame is then determined using avideo editor.
Previous research showed that there is no significant difference
in deploying the structure at room temperature rather than at thetemperature at which it has been stowed [20]. This result is
expected mainly because the deployment occurs over a much
shorter time scale than the relaxation. Hence, it is possible toneglect viscoelastic effects during deployment.

8

Deployment Results

8.1 Deployment Without Stowage. Stills taken from thedeployment video show a rapid and generally smooth unrolling of
the structure (Fig. 11). However, near the end of deployment(between frames Figs. 11(c) and 11(d)), the assumption of uniform uncoiling breaks down. In this region, the tape spring starts
to unravel laterally, followed by a dynamic latching of the tip.
Therefore, it is convenient to neglect the final uncoiling of thestructure, and the deployment time is henceforth defined as thetime it takes to achieve 90% of the full deployment.
Figure 12 compares the analytical (predicted) and experimental(measured) deployment curves for both cases: without and after
stowage at 60  C for 3 hrs. The curves confirm that the overall
deployment is smooth and uniform, and that the rate of deployment increases with time. The analytical curves are shown withbounds corresponding to l ¼ 0:55, and l ¼ 0:65, indicating thesensitivity of the analysis to the choice of the friction coefficient.
The experimental results are used to validate the analyticalmodel for the case without stowage. Here, the experimental results
match closely the model predictions for l ¼ 0:6. The analyticalmodel predicts a deployment time of 0:41 s, compared with 0:40 sin the experiment, i.e., a 2.5% difference in deployment time.
This suggests that the choice of the empirical parameter l isappropriate.
8.2 Deployment After Stowage. When stowage is applied,the curves exhibit a marked decrease in the deployment rate andan increase in the deployment time. Indeed, Eq. (35) shows that adecrease in the ejection force due to relaxation causes a drop in
the deployment velocity, which is the effect observed here.
The analytical results for the stowed configuration do notmatch the experimental results as well. For the case l ¼ 0:6, theanalytical model visibly underestimates the deployment time,with the analytical model predicting deployment in 0:52 s, while

Fig. 10 Manufacturing technique and demolding issues. (a)The tape spring rests on a cylindrical mandrel and is vacuum
bagged prior to cure in autoclave. (b) Twisting of tape springs
is observed after demolding, decreasing in magnitude after
exposure to air.

011008-8 / Vol. 83, JANUARY 2016

Fig. 11 Snapshots of the deployment of the tape springwith zero stowage time, with (a) t 5 0:10 s, (b) t 5 0:28 s, (c)t 5 0:38 s, (d) t50:42 s

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Fig. 12 Comparison of the deployment without stowage andafter stowage at 60  C for 3 hrs, and sensitivity of the analysis tochanges in the l friction parameter. Stowage of the structurecauses an increase the deployment time.

Fig. 13 Sensitivity of the deployment analysis to a change inrelaxation of the material. The curves 5%, 10%, 15% represent
the additional relaxation, compared to the behavior obtained inTable 3.

the experiment shows a deployment in 0:57 s (i.e., an underestimation of 9%).
The underestimation of deployment time in the stowed configuration suggests that the ejection force is in fact less than that predicted. This could be due to errors in material characterization, or
how the Prony series approximation is represented. To verify thishypothesis, the analytical model is repeated for slightly different
viscoelastic representations of the material, by performing a sensitivity analysis on the Prony coefficients. This analysis, shown in
Fig. 13, confirms that increasing the relaxation of the material
tends to increase the deployment time of the tape spring. It isnoticed that the curve for an additional relaxation of 15% shows
the analytical curve approaching the experimental results.
Although an error in material characterization is possible, theobserved discrepancy may also suggest that a refinement of the
dynamic model is required. In the current model a single coefficient of friction l, similar to Coulomb friction, is used to represent
the loss of ejection force. Further experimental work would beneeded to understand better the role of frictional forces—or otherdissipative mechanisms—in the deployment of the tape spring.

secondary latency effects, which may arise from local effects inthe transition zone, are not captured by the analytical model.
It is hypothesized that the enhanced stability predicted in Fig. 8
can be connected to the observed latency effect. It is suggested
that additional stability states between the fully coiled anduncoiled states, due to structural and material variability, aremade evident during relaxation. As was shown, relaxation tends
to deepen these stability wells. This could explain why these secondary stability states start to influence the response during relaxation and lead to the tape spring sitting idle for a period while itslowly begins to unwind. As the geometry changes and partial
recovery of the modulus occurs, the structure finally begins to
deploy.
Finally, it is suggested that the experimental deployment technique is refined to eliminate effects due to manual release of thespring, and a gravity offloading rig is used to minimize error due
to frictional and gravitational forces.

99.2 Deployment Failure. The stowage of the tape spring at
100  C for 3 hrs represents a singular case. When stowed in these

Secondary Effects

9.1 Deployment Latency. The previous results have shownthat the stowage of the tape spring leads to an increase in the
deployment time. In most of the cases observed until now, thedeployment remains continuous and consists of a rapid dynamic
deployment of the spring once the constraint has been released.
It has been observed, however, that in certain cases the deployment of the tape spring is not continuous and consists of two distinct behaviors. The first is a latent region where no or little
deployment is observed, even though the tape spring is no longer
restrained. The second is the normal smooth deployment observed
previously. Figure 14 shows the deployment curve for a tapespring stowed at 60  C for 3 hrs for this type of deployment. The
latent region lasts in this case for around 0.45 s, before the spring
deploys dynamically in 0:55 s. Total deployment lasts around
1:0 s, which is nearly twice as much as the dynamic deployment
shown in Fig. 14. The dynamic part of the deployment is identicalfor both types of deployment. However, the latent region is subject to typically large variations, making it difficult to predict the
total deployment time.
The fact that this latent behavior appears only in tape springs
that have been subject to long storage times and high temperatures
suggests that viscoelastic relaxation effects are influential. These
Journal of Applied Mechanics

Fig. 14 Deployment after stowage at 60  C for 3 hrs. Deployment here consists of a latent region with little or no deployment, immediately followed by full deployment. The latentregion significantly increases the total deployment time;here the total deployment time is double that of a normal
deployment.

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Fig. 15 Photographs of the spring overlaid at various deployment stages. The spring remains stable at any extended
position.

conditions, the tape spring fails to deploy. Figure 15 shows anoverlay of photographs of the deployment of the spring after stowage at 100  C for 3 hrs. The spring is found to be stable at any
extended position (including, of course, the fully deployed position). This observation is coherent with the results of the stability
model, because the structure remains bistable and does not autonomously unfurl when coiled. However, contrary to the results of
the analytical model, these experiments suggests that the net ejection force of the structure has dropped to zero, and that the structure does not have sufficient stored strain energy for anautonomous deployment. Overall, the deployment failure casecould be a catastrophic outcome if the sole function of the tapespring were to be autonomous deployment. Mitigating solutions,such as the use of a deployment actuator, therefore need to be
implemented, if this stowage condition is expected to arise.
Previous research has experimentally found a similar behavior
when the matrix stiffness was selected to be low enough [21].
This suggests that it is indeed matrix relaxation that is responsible
for the deployment failure of the tape spring. However, the current
analytical model is not able to capture at present this importantchange in behavior and should be modified in future work.
9.3 Deployment Repeatability. Due to experimental constraints, only one tape spring prototype was used for the deployment tests. The test for the case of deployment without stowage
was performed several times on consecutive days to test therepeatability of the tests (Fig. 16). To allow for recovery, thespring is uncoiled and stowed at room temperature for one day.
Results reveal that the deployment time increases with the numberof the tests, and seems to settle at an asymptotic value.

These results suggest that the structure experiences some hysteretic effect. Even if the spring is allowed to rest between successive tests to reduce the hysteretic effect, the results show that thisis insufficient to restore the initial stiffness completely, which
results in a residual relaxation at the start of the next test. Thiseffect grows at higher temperatures, because the structure would
experience more relaxation.
This study has informed the test procedure for subsequent tests,
and the prototype was allowed to rest for at least one week prior
to testing at 60  C and 100  C. Because it is impossible to know
the state of residual relaxation without actually performing a
deployment test, it is possible that the tests at 60  C and 100  C
contain residual relaxation from the tests at room temperature.
This may explain the accrued discrepancy in the deployment timeobserved for the experimental deployment after stowage at 60  C
for 3 hrs.
Moisture may also have some effect on the deployment behavior. It has been observed that the residual twist in the laminate
reduces considerably with exposure time to ambient air, which
was attributed to moisture ingress. The resulting change in thegeometry may then in turn influence the recovery behavior to aslight extent. Therefore, it is unclear how hysteresis, environmental, or ageing effects precisely affect the deployment of the spring.
This could be the object of future work.

10

Conclusions

In conclusion, relaxation effects due to stowage of the tapespring in a coiled state cause significant changes in the deployment behavior of the structure. In particular, stowing the structure
at higher temperatures or for longer periods of time increases the
dynamic deployment time of the structure by decreasing thestored strained energy available for deployment. In cases where
the relaxation of the structure is significant, secondary deployment
effects have been shown. For instance, stowing the structure at
60  C for 3 hrs leads to a latency effect—i.e., a significant delay
prior to the dynamic deployment of the boom. This effect isthought to arise from local stability effects, which are not captured
at present by the analytical model. For stowage at 100  C for
3 hrs, the tape spring fails to deploy and remains stable at any
deployed position. This effect has been linked to a significant
reduction in the matrix stiffness, effectively resulting in a zeroejection force.
The deployment failure of the tape spring for high stowage configurations represents a clearly undesirable effect. In this case, thetape spring loses its ability to autonomously deploy, which would
be catastrophic for missions that rely solely on this behavior. Yet,
the tape spring still retains a structural function. Provided an actuator is used to achieve full deployment, the tape spring would still
be able to latch into its fully deployed state and resist, albeit to alesser extent, inertial and vibrational forces. The reduction in
stored energy could actually be used at an advantage to eliminate
dynamic latching effects, which in turn reduces the risk of structural failure and increases the reliability of the deployment. Moreover, such a structure needs minimal energy for deployment,
simplifying the design of payout mechanisms and leading tosmaller and lighter actuators.

Acknowledgment
A.B. and P.M.W. would like to thank the EPSRC for the financial
support of the project under the ACCIS Centre for Doctoral Training(EP/G036772/1). The authors thank John Steeves and Keith Patterson (Caltech) for their help with experimental methods, and Petros
Arakelian (Caltech) for technical advice and support.
Fig. 16 Comparison on deploying the structure without stowage on successive days. The deployment time increases withthe “age” of the specimen. This behavior is most likely attributed to hysteresis effects. It is noted that the days are notconsecutive.

011008-10 / Vol. 83, JANUARY 2016

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