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3.2.2 Morphing structures in plants

Examples of morphing structures are abundant in nature. Morphing structure isdefined as macroscopic shape change here. Shape change of a biological species is a partof sequential event in a response to environmental change for a given biological species,which is composed of sensing and actuation. Fig. 5.1 shows a good example of such

sequential events of sensing and rapid actuation (morphing) of a Venus fly trap leaves. Inthis section we will review the actuation which results in shape change while the sensingpart will be reviewed in the next section.

(a) (b) (c) (d)Fig. 5.1 Venus Fly Trap action in catching a flying insect, (a) insect touches the antennalocated in the middle of leaf, (b) rapid closure of the leaves trap the insect, (c) crosssection view before the leaf motion (d) after the leaf motion(Taya, 2007).

In the following, we will review five cases of morphing structures inherent in biology, (1)Venus fly trap, (2) Mimosa pudica, (3) tendril coiling, (4) twing of vines, (5) folding andunfolding of flowers and leaves, (6) insects eclosions ,(7) wing structures of eagles, batsand insects, and (8) plasmodal slime. After these examples, we will discus on some

bioinspired morphing structures applied to aerospace structures. It is noted that theenergy dissipations used in these actuations are minimum which is realized by the modeof shape changes, i.e., bending at localized joints, coiling for which limited shear stress isused. This demonstrates the fundamental principle of actuations associated in biologicalspecies, minimum Gibbs energy.

5.1.1. Examples of morphing structures in Nature

(1) Venus fly trap

The Venus Fly Trap leaf, Fig. 5.1, is designed to sense andtrapflying insects by rapidlyclosing the leaf-like trap when triggered by sensing antennae located in the center of the

leaf. The microscopic mechanism of the Venus Fly Trap leaf is based on controlled ion

fluxes creating osmotic movement of water molecules toward the outer-most surface ofthe leaf, resulting in the expansion of the outer surface and bending. Leaves retain a flat

shape as long as the resistance offered by upper and lower epidermis layers is equivalent,

and pressure is evenly spread throughout the internal layer. But leaves fold when the

outer epidermal cell layer suddenly expands while the inner does not. This ability to

expand rests with the osmotic pressure created by an ATP-driven proton pump located on

the plasma membrane, expels protons, creates both a very negative membrane potential (-

K+

expand

H2O

H+

epidermis

vein

Parenchyma

cells epidermis

Sensing

hair


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120 to 250 mV) and acidic external pH (Hodick and Sievers 1989; Stahlberg and Van

Volkenburgh, 1999). Let us review the work of recent studies focusing on the

mechanisms of morphing of Venus fly trap in the following.

Forterre et al (2005) studied the motion of Venus fly trap lopes experimentally recording

the shape change of lopes using high speed camera and observed the snap from concaveoutword to convex outward geometry of lopes. From this observation, they calculated

strains along x- and y-directions, see Fig. 5.2(a). By using the model of poroelastic shell

model which simulates the leaf, they proposed snap through bucking, some of such

results are shown in Fig. 5.2(b). To explain the rapid snap through motion of the lopes,they used the poroelastic shell arriving at the time of such snap through ,

t=L2/(kE) (5.1)

where is viscosity of fluid that occupies the lope cells, L is the size of lope, k is the

mean curvature and E is the Youngs modulus of dehydrolayed lope. Based on this

formula, they predicted the speed of snap through as 0.1 second which coincides with thatobserved. The speed of the actuation of Venus flytrap lopes is proportional to square of

the lope thickness based on the model proposed by Forterre et al (2005) which is

challenged recently by Volkov et al (2008) who did a series of testing using Venusflytrap lopes of different thicknesses, claiming that the formula by Forterre is not valid.

Fig. 5.2 (a) measured strains along x-direction and along y-direction,(b) smooth snapping

transition in leaf closure of Venus flytrap where mean curvature (km=(kx+ ky)/2) is

plotted as a function of kxn which is defined as 1-T where = L4k

2/h

2with L is length,

k is the initial mean curvature, h is thickness of leaf of Venus fly trap, T is no-

dimensional time (Forterre et al, 2005).


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Volkov et al (2008), performed a well-defined experiment on Venus fly trap lopes by

applying change where use of two Ag/AgCl electrodes, the positive electrode located at

the midrib and the negative at one of the lopes. They first applied electrical

stimulation(14C, 1.5 V) and measured the speed of the closure of the lopes in terms ofratio of opening (d) to the maximum opening(dmax) as a function of time in second, Fig.

5.3(a) which indicates very fast speed of the lope closure, within 0.3 second. In order toexamine the condition under which the closure takes place, they conducted another testwhere charge input is primary input and the lope opening distance (d) was measured as a

function of change, Fig. 5.3(b). Fig. 5.3(b) illustrates clearly that the lopes can

accumulate the charge until 14mC, just after that the closure of lopes started andcompleted within 0.3 second.

Fig. 5. 3 Effects of applying charge on the opening of the Venus fly trap lopes(d), (a)relation between the ratio of d to maximum d(dmax) and elapsed time in second under

initial charge of 14C applied to the lopes, (b) relation between applied charge and raio

of lope opening to its maximum, d/dmax(Volkov et al, 2008).

The experiments by Volkov et al using 200 Venus flytrap plants with different sizes of

leave

ranging from 1 cm to 5 cm reveal no dependence of the closing time on size of the leaf L.This contradicts with Eq. (5.1) which predicts a dramatic increase in the closing time for

large Venus

flytrap plants. Volkov et al also claims that the motion of Venus flytrap is smooth, notlike snap through buckling which seems to be supported by the work by Nakano (2003),see Fig.5. 6.

(2)Mimosa pudicaIf leaves and/or petiole ofMimosa pudicaare touched by a foreign object, the petiole

falls downward in a second or so. This phenomena ofMimosa pudicahas attracted strongattentions of many plant scientists, for example, Snow(1924) and Shibaoka(1966;1969).

Shibaoka (1969) reviewed extensively the actuation mechanism of action plants including

Mimosa pudica where he cited other researchers work as well as his own work. In

Shibaoka (1969) review the motor cells play key role in which the stimulus exhibit thecontractile vacuoles, not due to increase in the permeability of the plasma membrane, and


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the contractile of the vacuoles are related to mechano-chemcial reactions with ATP-

ATPase system. The propagation of action potentials are along vascular tissues

(particularly phloem and neighboring tissues). Once such action potentials arrive at themotor cells underneath the pulvinus, Toriyama (1955) observed that potassium ion

escaped from the motor cells to draw water out of the motor cells by osmosis, leading to

collapse and folding down of the pulvinus. Later, Torimaya and Sato (1968) observedunder microscope that main vacuole containing fibers and tannin vacuole which

containing large amount of potassium ions can reduce their size so that the fibers are also

contracted. Toriyama and Jaffe (1972) discovered that calcium held in the tannin

vascuoles before the pulvinus movement appear to migrate to the central vascuole wherethe fibers may contact. This is similar to the contraction of animal muscle fiber where

calcium plays a key role on the contractions of protein actomyosin (Simon, 1992).Jaffe

(1973) speculated that the contractual motion of the midrib of Venus flytrap isactomyosin-like ATPase which is similar to the contractual motion of other action plants,

pea tendril (Jeffe and Glaston, 1966; 1968) and Mimosa pudica (Lyubumova et al, 1964).

Nakano (2003) studied the motion ofMimosa pudicaby using high speed camera andmeasured the angle() between petiole and main pulvinus as a function of time (T) insecond, see Fig. 5.4 from which the initial angle of 74

0is changed to final angle of 21

0

within 2 seconds. They also measured the motor cell size change before and aftertouching of petiole ofMimosa pudica, Fig. 5.5 where the diameters of the motor cells

above and below vascular bundles are recorded with darker blue for larger size and light

blue to no color for smaller sized motor cells. Comparing Fig. 5.5 (a) and (b), we can

conclude that the motor cells below the vascular bundles (located in the center of petiole)lost its osmotic pressure or equivalently its water which is considered to be contractile

vacuoles as reviewed by Shibaoka (1969) in the earlier paragraph. However, the exact

mechanism of triggering the shrinkage of motor cells in terms of loss of potassium ions

from the motor cells( uptaking again water in the motor cells in the lower part in the longrun) is not clear.


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Fig. 5.4 Sequence of photos for movement of petiole of Mimosa pudicaafter leaves aretouched where the angles () between petiole and main pulvinus as a function of time (T)recorded (Nakano, 2003)

(a) (b)

Fig. 5.5 Motor cell diameters before (a) and after (b) the leaves ofMimosa pudicaare

touched where darker blue colored motor cells for larger diameters of the cells, and light

blue to white colored cells are for smaller cell size(Nakano, 2003).

The speed of petiole motion to change the angle from initially large to smaller angle is

compared with that of the closure of Venus flytrap leaves, Fig. 5.6 where closing leaves

of Venus flytrap is slightly faster than that of petiole ofMimosa pudica, but they arewithin 1 second or so.

.


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Fig. 7 reveals that the speed of

(3) of tendril coiling.

Shape change in moving tendrils of various climbing plants (cucumbers, grapes, passionflowers, pumpkins and gourds) in search for stable support has been studied since Darwin

(1865). In "The Movements and Habits of Climbing Plants," Darwin proposes that the

helical formation is due to the asymmetric growth rate between dorsal and ventral sidesof the tendril, i.e., the former rate exceeding the latter rate.

Rainer (2002) examined the motion of cucumber tendrils which are associated with two

coilings: primary (contact) coiling to sense stable support surface using tactile sensorsand secondary (free) coiling to draw the stem of the plant closer to the support through

the formation of a series of right and left handed helices, see Fig. 5.7. Jaffe (1977) and

Jaffe and Glastone (1966) studied the contact coiling behavior of Alaskan pea with theaim of determining what environmental parameters give rise to larger coiling. It is the

free coiling that attracted many researchers beyond botanists, in order to determine why

left-handed and right-handed helices form with a reversal in between them, see Fig. 5.8.

Fig. 5.7 Sequence of tendril morphing in order to contact a secure object, then grabbing it

by primary coiling, then moving the main body of the plant toward the secure

object(Rainer, 2002)

Fig. 5.6 Angle ratio change

(t)/maxof two different

action plants, Mimosapudica petiole and Venus

flytrap leaves where maxis

the initial angle, and (t) is

the angle at time t

(Nakano,2003)


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Junker (1977) studied the change in auxin (growth hormone) concentration located indorsal and ventral domains of Passiflora quadrangularis and discovered that there was not

a significant change during coiling. Instead, he hinted that action potentials and acid

growth along the free coiling direction may be the main mechanism. Carrington andEsnard (1989) examined the coiling of two tendril-bearing plants, water melon (Citrullus

lanatus), and passion fruit (Passiflora edulis). They observed that the contacted side of

the tendril decreased the length while the non-contact side increased during primarycoiling. They concluded that the mechanism of signal transduction from the contact site

to the growth of motor cells and/or wall property changes are still a mystery. Thus, it is

still not clear if asymmetrical motor cell growth or asymmetrical arrangement of tissuesis mainly responsible for free coiling

Mathematical models in explaining the above free coiling have been proposed. Keller

(1980) proposed an elastic strain energy concept applied to thin elastic rod with boundary

condition at support and derived that the helical shape can be predicted by theminimization of the Gibbs mechanical energy. Keller focused two cases, (i) tendril has

free-end, (ii) tendril tip is constraint, such fixed with the contact coiling around the

support. The solution of the minimization of Gibbs free energy for case (i) provides the

helical shape of the tendril with free end, while the solutions of the second case leads tofree coiling with reversals. Goriely and Tabor(1998) treated a tendril as thin elastic rod

with intrinsic curvature and used linear and nonlinear stability analysis of such thin rodunder various tension and concluded that the free coiling with a reversal can be predicted

by their model. Following this work, Domokos and Healey studied the case of finite

length with initial curvature and clamped ends, and predicted multiple reversals in free

coiling of tendrils.Thomson et al (2005) studied the tendril-bearing plant, Luffa cylindrical, and measured

the Youngs modulus of the tendril to find that the average value is 33 MPa which was

used in their finite element

Fig.5.8 Coiled tendril ofLuffacylindrica with the appearance ofmultiple reversals along the

length(Eberle et al, 2009)


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b) Tendril 2

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

Normalized position (x)

Experimentallymeasuredstrain

a) Tendril 1

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

Normalized position (x)

Experimentallymea

suredstrain

Fig. 5.9 Measured strains in dorsal (open circles) and ventral domains(filled triangles) of

Luff cylindrical. It is noted that the dorsal strains are positive while the ventral strains are

negative , and both increases with position toward the tip of the tendril( Eberle et al,

2009).

analysis (FEA). In the FEA input data set, they used the equivalent thermal strain tosimulate the growth rate of tendril dorsal and ventral parts which are set in terms of

thermal expansion coefficient, i.e. dorsal side being 0.001/C and ventral side being zero.

Upon application of torsional perturbation around the tendril axis, they could produce thehelical formation with reversal point. Following Thomson et al work, Eberle et al (2009)

modified the FEA to study growth of another species of tendril-bearing plant: Adenia

lobata. Using a set of marker painted sites, they measured total strain along the tendrilaxis for Adenia lobata and implemented this in their FEA. Fig. Fig 5.9 shows the strains

in the dorsal and ventral domains along the axial direction (x). This figure shows clearly

the asymmetry growth of tendril, expansion of dorsal domain and shrinkage of ventraldomain. Providing the growth distribution between dorsal and ventral domains with those

measured, would result in only bending of the tendril. They applied small rotation (0.05

radians per unit length) of the dorsal-ventral boundary interface about the tendril axis,

arriving at helical shape in the FEA modeling

(4) twining of vines

Silk (1989) and her co-workers (1991;2005) studied twining of vines whereby a vine

grows into helices around a support. This mechanism is different from tendril coiling

because the vine does not coil freely in air but is continuously in contact with the supportthat it encircles. Silk and her colleagues used a non-orthogonal coordinate system to

predict radial and circumferential growth using empirical data obtained from various

cross sections. They proved that axial force (Ft) in climbing morning glory is balanced

with normal force (pn) exerted on the surface of cylindrical support, where friction andshear force are assumed to be neglected. The balance equation between Ft and pn are

given by


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Ft+ pn=0 (5.2)

where is the curvature of climbing morning glory. They found experimentally the order

of Ftbeing 50 to 200 g, which is proportional to the height of climbing vine. Silk andHolbrook (2005) studied theoretically and experimentally the effects of the friction force

which was neglected in their earlier work (1991). In order to account frictional force (pt),

they used an well-known formula,

pt= pn (5.3)

where is coefficient of friction.By using the governing equations of Costello (1978) developed for climbing rope, they

obtained the axial force, Ft, given by

Ft= A es

(5.4)

where A is the tension at the top of the vine, is friction coefficient, is curvature, s isthe arc length along the vine. They used Ipomoea purpurea vine in their experimental

work, to demonstrate very large frictional force active between the above vine and

support where the friction coefficient can be as large as 3.0. It can be concluded from thisstudy that the gravitational force due to older vines located in lower part of the climbing

vine can be supported by younger vine located at the top with only few turns. It is noted

that the climbing vine is very weak under compression force, resulting in immediate

detachment of the vine from the support

(5)Unfolding and folding of leaves and flower pedals

Hydration and dehydration is a key function for all plants, adjusting environmental

change that surrounds plants. Typical plant has many fibrous elements among whichmotor cells are located, accommodating water (ionic water, too) during uptaking water

from roots, or in rainy season while in dryier environment, such motor cells dehydrate.

Plant structure is fully supported mechanically with hydration, which is called as nasticstructure.


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Fig. 5.10. How a leaf of Kentucky bluegrassPoapratense(a European

weed) can fold to exhibit large morphing using the concept of

expansion and shrinkage of motor cells, (a) zoom up of the upper partof the leaf with expanded motor cells, (b)-(e) sequence of expansion,

entire leaf showing folded in morphology by kinking at several hinges.

Fig. 5.11 Dehydrated and twisted(upper picture) and smooth

hydrated halves (lower picture)

from the same seedpod ofLathyrus japonicus, a legume.The wall is made of two hydration

layers with the fibers pointing in

different directions (angle 90

degrees). When dehydrating, this

torsion motor rips the pod apart

and releases the seeds.

Fig. 5.12 Shape change

mechanism by hydration

and dehydration ofvarious laminate made of

two different textured

laminae, where blue

colored textured sheet ishydrated to deform to

curved shapes shown on

right


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Kishimoto et al (2006) examined various folding and unfolding patterns inherent inbiological species, both plant leaves and flower booms and insect ecosion process. They

categorized the folding/unfolding patterns into four types, depending on volume change,

deployment only, and both deployment and storage:Type A: large volume change with deployment only

Type B: not much volume change with deployment only

Type C: large volume change with both deployment and storage

Type D: not much volume change with both deployment and storage

Examples of type A are bloom of dandelion and development of new leaves, while those

of type B are bloom of morning glory and eclosion of insects, and those of type D aremany : bloom of bindweed , open-close movement of sensitive and insectivorous plants,

flap-stowage of beetles hind wings and flap-stowage of bats wings. Fig. 5.13 and Fig.

5.14 (a)-(c) are examples of type A development where flowers of dandelion(a), and

sunflower(b) are composed of tubular and lligulate corolla which are used to deploy theirfolded pedals sequentially. Fig. 5. 14(d) is an example of type B development of plant

flower bloom where morning glory opens only in the morning without reversing it. Fig.

5.14 is an example of type D development of plant where bindweed flower(e) can opens

daytime while it can close the flower during nights, thus exhibiting reversible movement.

The petals of both morning glory and bindweed flowers stored as helical woundconfiguration, shown in Fig. 5.14(f).

Fig. 5.13 Sequential development of bipinnately compound leaves of alegument plantupper left to mid-left inserts, then fully opend leaves, in that order of sequential

development (Kishimoto and Natori, 2006

12

34 5

6

7

12 3 4 5 6 7


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Fig. 5.14 Examples of folding and unfolding of plant flower blooms, (a) dandelion, (b)sunflower, (c) poppy, (d) morning glory, (e) bindweed, (f) helically wound buds, morning

glory (left), bindweed (right) (Kishimoto and Natori, 2006).

(a) dandelion

(b) sunflower

(c) poppy

d mornin e bindweed

(f) helically-wound

buds:

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