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Materials Science and Engineering A 496 (2008) 323–328

Contents lists available at ScienceDirect

Materials Science and Engineering A

journa l homepage: www.e lsev ier .com/ locate /msea

icromechanism of deformation in EMC laminates

.Y. Xiong, Z.D. Wang ∗, Z.F. Li, R.N. Changnstitute of Engineering Mechanics, School of Civil Engineering,eijing Jiao-Tong University, Beijing, 100044, China

r t i c l e i n f o

rticle history:eceived 23 January 2008eceived in revised form 14 May 2008ccepted 14 May 2008

a b s t r a c t

Elastic memory composites (EMCs) are receiving more attention in deployable spacecraft industry becausethey can realize much higher packaging strains without damage and automatically recover to their originalshapes when subjected to a specific thermomechanical cycle. Experimental researches have revealed thatmicrobuckling and post-microbuckling responses of compressed fibers in the soft matrix are the primary

eywords:lastic memory compositeicrobuckle

ailure modeending

deformation mechanism of EMCs to realize higher packaging strains than traditional composites. However,a thorough understanding about the deformation mechanism of EMCs has not yet been achieved. Thispaper presents a new shear/tension microbuckling solution to study the deformation and failure processof EMC laminates under bending. The microbuckling wavelength, critical compressive stress and strain forcarbon-fiber/resin EMC laminates are calculated, and different theoretical solutions are compared withthe experimental observation. Moreover, the possible failure modes of this type of materials are discussed

it

svEf(rhldml

iasaa

as well.

. Introduction

Elastic memory composites (EMCs) are similar to traditionaligh-performance, fiber-reinforced composites except for the usef a shape memory polymer matrix system that enables achieve-ent of extraordinary packaging strains without damage. The

hermomechanical process of EMCs can be simply described as theollowing. Continuous fibers are extremely slender structures thatndividually buckle under a nearly zero compressive condition. Theow shear stiffness of the soft matrix cannot prevent fibers fromuckling as well. Hence, the fibers in compressive zone of EMC lam-

nates or longerons will buckle when subjected to a bending load,hich causes these fibers free from damage even if the nominal

ending strains are much higher than the ultimate strains of thebers. On the other hand, the microbuckling can lead to a remark-ble decrease of the effective axial stiffness in the compressive zone.ig. 1 shows the mismatch of the effective axial stiffness in tensilend compressive zones that will cause the neutral-strain surface ofhe laminate to deviate from the centre of the cross section, and theffective strain of the fibers in tensioned zone is also lower than the

ailure strain. As the temperature decreases lower than the activa-ion temperature of the materials, the stiffness-recovered matrixan “freeze” the microbuckled shape of the fibers. A second reheat-ng of the EMC laminate above their activation temperature will

∗ Corresponding author. Fax: +86 10 51682094.E-mail address: zhdwang@bjtu.edu.cn (Z.D. Wang).

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921-5093/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.msea.2008.05.029

© 2008 Elsevier B.V. All rights reserved.

nduce the microbuckling deformation to gradually straighten andhen recover to the pre-packaged geometry.

Comparing with the traditional deployable structure, EMCs haveome favorable properties, such as high folding strains, low density,ery high specific stiffness and strength. During the last 10 years,MC materials have been receiving great interest in research. Dif-erent deployable structures made of EMCs have been designede.g. laminated plates and shells, open-grid lattices, pultrudedods, and hinges [1–4]). Fig. 2 is a prototype of an EMC-basedinge/actuator structure, which consists of four parts: two EMC

aminates with embedded heaters and two end-fittings [3]. A tra-itional hinge/actuator structure, however, commonly consists ofany components (e.g. torsional spring, dampers, and mechanical

ocking devices).EMCs have significant advantages in deployable spacecraft

ndustry. However, as a relatively new and potentially revolution-ry class of materials, the deformation mechanism of EMCs istill not fully understandable. The onset of fiber microbucklingnd the post-microbuckling behavior of EMCs are complicatednd many sources of nonlinearities exist, such as temperature-ependent large-displacement/strain. The microbuckling wave-

ength of carbon-fiber/resin EMC laminates as predicted by thelassical buckling theories [6,7] are much different from the exper-

mental observation. Some new-developed models [8,9] are alsoeficient in describing such deformation process.

The present paper intends to propose a new analytical solu-ion to predict the critical parameters of the deformation, such as

icrobuckling wavelength, critical stress and strain of an EMC lam-

324 Z.Y. Xiong et al. / Materials Science and E

im

2

mir

2

e

baEci0anoop

cwlcifdh

2

it

Fig. 1. Diagram of microbuckling in EMC laminates under bending.

nate with a certain bending strain. Moreover, the possible failureodes and protecting measures are also discussed.

. Available results

In order to establish a suitable theoretical model to analyze theicromechanics of EMC laminates subjected to bending conditions,

t is better to review some reported experimental and theoreticalesults firstly.

.1. Experimental observations

Fig. 3 is the microbuckling micrographs provided by Campbellt al. [10,11]. They tested 2-ply and 4-ply laminates under different

Tbast

Fig. 2. EMC hinge/ac

Fig. 3. T300-reinforcement EMC laminates under bending [10,11

ngineering A 496 (2008) 323–328

end ratios. The laminates were reinforced with T300 carbon fibers,nd the matrix materials were two types of epoxy-based thermosetMC resins named CTD–DP7 and CTD–DP5.1, respectively. The opti-al microphotographs revealed that the fibers consistently buckledn a sinusoidal-like shape and the half-wavelength was about.4–0.6 mm during the bending process. The average microbucklingmplitude of the fibers was about 0.048–0.088 mm at a nomi-al bending strain εeff = 0.05. Additionally, the damage was onlybserved at εeff ≥ 0.05, and a “fault line” of the fiber fracture wasbserved through the entire width of the specimen on the com-ressive side of the bend (Fig. 3b).

Francis [5] also reported some experimental results about IM7arbon-fiber-reinforced EMCs under pure bending (see Fig. 4). Itas observed that the fibers consistently buckled in a sinusoidal-

ike shape with a normal wavelength of about 1 mm, which did nothange substantially from the inception of the buckles at low bend-ng strains through very high bending strains. Moreover, differentrom Campbell et al’s experimental result, both matrix failure andelamination were observed at localized regions of the extremelyigh shearing strain in this report (see Fig. 4b).

.2. Microbuckling solutions

The classical analytical solutions for fiber microbuckling werenitially developed by Rosen and Dow [6,7], and based on the solu-ion for buckling of a bar on an elastic foundation proposed by

imoshenko [12]. A key kinematical assumption in the classicaluckling theory is that all fibers buckle at the same shape andmplitude. There are two fundamental buckling modes, namelyhear-buckling mode and extension-buckling mode. According tohe classical solutions, the critical stresses at which microbuckling

tuator design.

]. (a) Sinusoidal-like microbuckling and (b) fiber damage.

Z.Y. Xiong et al. / Materials Science and Engineering A 496 (2008) 323–328 325

a) In-

w

�

�

waafii

ip[tAmwmmtct

fasmt

�

wc

fiwi(

�

ε

atoltn

mEltpdafmc

3

3

abcnabnctiz

lt

y

Fig. 4. IM7-reinforcement EMC laminates under bending [5]. (

ill take place for the two modes can be respectively expressed as

Scr = Gm

(1 − �f)+ �2Efh

2�f

12�2, (1)

Tcr = �2Efh

2�f

12

[1

�2+ 24Em�2

�4ch3Ef

], (2)

here �Scr and �T

cr denote the critical shear and tensile stresses; Em

nd Gm are Young’s modulus and shear modulus of the matrix; Efnd �f are Young’s modulus and Poisson’s ratio of the fibers; h is theber diameter; c is the half spacing between adjacent fibers; and �

s the wavelength of microbuckling.The classical fiber microbuckling solutions have been exper-

mentally confirmed to be right and generally effective inredicting the compressive strength of hard-matrix composites13,14]. However, these solutions are not accurate in predictinghe microbuckling parameters of EMC materials under bending.ccording to Eq. (1), the critical buckling stress for classical shearode will always be a minimum value when the microbucklingavelength is maximized, which indicates that the theoreticalicrobuckling half-wavelength equals to the length of the speci-en. On the other hand, the critical microbuckling wavelength of

ypical carbon-fiber/resin EMC laminates predicted by the classi-al extension mode is 0.09 mm [5], which is about one-order lowerhan the experimental observation.

To overcome the classical theoretical solutions significantly dif-erent from the experimental value, Campbell et al. [9] consideredsingle fiber embedded in a finite matrix subjected to a compres-

ive loading condition. By assuming a pure shear deformation in theatrix, they proposed a new microbuckling model and determined

he wavelength, �c, as

c = �df

4

√EfVf

GmVm, (3)

here df denotes the fiber diameter, and other parameters are coin-ident with the above defined.

By replacing the typical material parameters of carbon-ber/resin EMC laminates into Eq. (3), the calculated criticalavelength is about 1.035 mm, which is very close to the exper-

mental value. However, the derived critical stress and strain by Eq.3) are respectively as

c = 2GmVm

Vf, (4)

c = 2GmVm

EfVf. (5)

wlto

plane microbuckling and (b) matrix failure and delamination.

Eqs. (4) and (5) indicate that the critical microbuckling stressnd strain will decrease with increasing the fiber volume frac-ion, which is questionable. By assuming a linear distributionf the fiber microbuckling amplitude along the thickness of theaminate, Campbell and Maji [11] further proposed an improvedwo-dimensional microbuckling model in another report. Unfortu-ately, the similar deficiency still exists.

It should be point out that pure tension- or shear-microbucklingodel cannot correctly reveal the real deformation mechanism of

MC laminates subjected to bending conditions. The microbuck-ing response of the fibers leads to a large shear strain in the matrixo accommodate such deformation. Meanwhile, the nominal com-ressive strains of the fibers in different layers of the laminate areifferent during the bending process. Naturally, the microbucklingmplitudes of the fibers in different compressive layers are dif-erent, and a tension/compression strain definitely exists in the

atrix as well. Hence, extension/shear-mixed models should beonsidered.

. A new analytical solution

.1. Principal of virtual work

Microbuckling leads to a remarkable decrease of the effectivexial stiffness in the compressive zone of EMC laminates underending. The mismatch of the effective axial stiffness in tensile andompressive zones will cause the neutral-strain surface of the lami-ate to be deviated from the middle surface of the cross section. Fortypical 0.5 mm thickness EMC laminate subjected to 5% nominalending strain, a simple theoretical calculation shows the thick-ess of the tensile zone to be about two-order lower than that of theompressive zone. Available experimental results also revealed thathe fibers in different layers of EMC laminates consistently buckledn a sinusoidal-like shape. Therefore, the thickness of the tensileone will be neglected in the following theoretical analysis.

As schematically shown in Fig. 5, the displacement of the ith-ayer microbuckling fibers apart from the neutral-strain surface ofhe laminate can be expressed as

i = ai sin(

2m�x

l

)+ i(h + 2c) − h

2, (i = 1, 2, . . . , n) (6)

here m is the mode number (number of waves along the fiberength) and must be a positive integer to satisfy the boundary condi-ions. ai is the sinusoidal amplitude of the ith-layer. n is the numberf microbuckling layers.

326 Z.Y. Xiong et al. / Materials Science and E

Fl

b

�

wte

�

woT

�

wo

3

Mttdm�

�

�

wme

r

ε

gs

�

w

�

d

�

d

3

mf

ε

l

ε

wissteofi

a

demonstrated to be nearly uniform through the thickness of thelaminate. Hence, Eq. (19) indicates that a square-root-power rela-tion exists between microbuckling amplitude of the fiber and thedisplacement from the neutral-strain surface despite of the linearrelations commonly assumed [5,10].

Table 1Material parameters for typical carbon-fiber/resin EMC composites

Parameter Value Unit Description

Ef 280 GPa Fiber elastic modulus

ig. 5. Combination of extension- and shear-buckling modes. (a) Before microbuck-ing and (b) After microbuckling.

According to the principle of virtual work, the virtual work doney the external load, �T, should be equal to:

T = �Uf + �Um, (7)

here �Um is the strain energy stored in the matrix, and �Uf ishe strain energy stored in the microbuckling fibers, which can bexpressed as

Uf = 12

EfIf

n∑1

∫ l

0

(∂2yi

∂x2

)2

dx = 4�4EfIfl3

m4n∑1

a2, (8)

here EfIf is the flexural stiffness of the fibers. It should be pointedut that the similar virtual work and fiber strain-energy terms inimoshenko’s solution are used in this study.

The virtual work, �T, can be expressed as

T = 12

Pcr

∑n1

∫ l

0(∂yi/∂x)2dx

n= Pcr�2

lm2

∑n1ai

n, (9)

here Pcr is the applied critical load for microbuckling deformationf the laminate.

.2. Strain energy in the matrix

Fiber microbuckling will cause a large shear strain in the matrix.eanwhile, the microbuckling amplitudes of the fibers along the

hickness of the laminate are different (see Fig. 5), which causesension/compression strains exist in the matrix. Hence, the matrixeformation of EMC laminates under bending is a tension/shear-ixed model. The tension and shear energies in an EMC matrix,UT

m and �USm, can be respectively expressed as

UTm =

n∑1

�UTmi = 1

2

n∑1

∫Vm

Emε2midV (10)

USm =

n∑1

�USmi = 1

2

n∑1

∫Vm

Gm2midV (11)

here �UTmi

and �USmi

are the tension and shear energies of theatrix in the ith-layer, respectively. Vm is the matrix volume of the

ffective element.The tension- and shear-strain amplitudes in the ith-layer can be

espectively expressed as

2mi =

[(ai − ai−1) sin(2m�x/l)

]2

(12)

2C

2mi =

[dyi

dx+ h

2c

dyi

dx

]2

=(

1 + h

2c

)2(2m�

l

)2a2

i cos2(

2m�x

l

)(13)

EGhVtn

ngineering A 496 (2008) 323–328

Substituting Eqs. (12) and (13) into Eqs. (10) and (11), the inte-rating result yields the elastic foundation strain energy for apecimen with n-layer effective elements:

Um = �UTm + �US

m (14)

here

UTm=1

2

n∑1

∫V

Emεi2dV=1

2

n∑1

∫V

Em

[(ai − ai−1) sin(2m�x/l)

2c

]2

V = Eml

4

n∑1

(ai − ai−1)2, (15)

USm = 1

2

n∑1

∫V

Gm2i dV = 1

2

n∑1

∫V

Gm

[dyi

dx+ h

2c

dyi

dx

]2

V = 2Gmc2�2m2

l

(1 + h

2c

) n∑1

a2i . (16)

.3. Microbuckling amplitude

The effective strain of the fibers subjected to sinusoidal-likeicrobuckling can be approximately expressed in an one-order

unction as [8]

ieff = a2

i�2

a2i�2 + 4�2

. (17)

On the other hand, the normalized strain of the ith-layer in aaminate subjected to bending conditions is

i = yi

R=≈ i(h + 2c)

R, (18)

here R is the curvature radius of a laminate under a certain bend-ng strain. Considering that EMC laminates microbuckle at a verymall compressive strain, the axial compressive strain of the fiberubjected to sinusoidal-like microbuckling can be negligible, andhe effective strain of the fiber due to microbuckling approximatelyquals to the nominal bending strain of the laminate, substitutionf Eq. (18) into Eq. (17) yields the microbuckling amplitude of theber in the ith-layer as:

i = 2�

�

√i(h + 2c)

R − i(h + 2c)≈ 2�

�

√i(h + 2c)

R. (19)

The microbuckling wavelength has been experimentally

m 0.0085 GPa Matrix elastic modulusm 0.0029 GPa Matrix shear modulus

5.20 E−6 m Fiber diameterf 0.40 Fiber volume fraction

5.0 E−4 m Laminate thickness∼38 Theoretical layers

Z.Y. Xiong et al. / Materials Science and Engineering A 496 (2008) 323–328 327

Table 2Microbuckling variables of carbon-fiber/resin EMC laminates determined by different solutions

Experimental results Classical shear mode Classical tension mode Ref. [9] Present

� (mm) ∼1.0 [5] 0.80–1.28 [10] =la 0.0925 1.035 1.12� 111.97 8.7 40.82ε 0.040 0.0031 0.0146

3

w

P

omd

�

lbc

u[wirnofdeeta

4

mtcimflm

pcsstt

c

ε

wfip

ssaod

thmiid(bwc

5

dmctmobanation along the fiber/matrix interface in the compressive surface

c (MPa) – 4.83c (%) – 0.0017

a Variable l denotes the fiber length.

.4. Determination of critical parameters

Combining Eqs. (7)–(9), (14)–(16), and (19), the critical load athich buckling occurs in a bending laminate is determined as:

cr = 4�2EfIfm2n

l2+ Emhl2

8c�2m2

∑n1(2i − 1 − 2

√i(i − 1))

n + 1

+2Gmch(

1 + h

2c

)2

n. (20)

Eq. (20) is the critical value of the load at which buckling willccur. The real number of waves should be the number that mini-izes Pcr. Hence, the real number of waves can be determined by

ifferentiating m that minimizes Pcr and the final result is

c = l

m= 2�

4

√2EfIfc

Emh

∑n1(2i − 1 − 2

√i(i − 1))

n(n + 1). (21)

Eq. (21) provides a solution of microbuckling wavelength of EMCaminates subjected to bending conditions. The real critical load cane determined by instituting Eq. (21) into Eq. (20), and then theritical compressive stress and strain can be determined as well.

Table 1 lists the typical material parameters of a typical contin-ous carbon-fiber/resin EMC laminate with 0.5 mm in thickness5]. By replacing these material parameters, the microbucklingavelength, critical stress and strain of EMC laminates under bend-

ng can be theoretically calculated. Table 2 lists the experimentalesults and theoretical values predicted by different models. It isoticeable that the predicted wavelength of microbuckling basedn the classical shear and tension modes is significantly apartrom the experimental result. Comparatively, the theoretical valuesetermined by the present model and Ref. [9] are agreeable with thexperimental result. However, as it has been pointed out that thexpressions of the critical stress and strain deduced from the solu-ion of the microbuckling wavelength in Ref. [9] are questionablend cannot give a reasonable explanation.

. Failure modes

To satisfy an adequate design margin of packaging strain for EMCaterials in the application of deployable space-structure industry,

he failure mechanisms of EMC materials under bending should beonsidered. An EMC laminate can exhibit different types of strain-nduced failures during packaging process. Generally, the failure

echanisms can be divided into two broad categories of: (1) fiberailure, and (2) resin failure. For fiber failure, there are two keyocations: the maximum tensile-strain location; and the maximum

icrobuckle amplitude location.For maximum tensile-strain location, the above analysis has

ointed out that the depth of the tensile region is very small and

an be negligible. Therefore, the maximum tensile strain in ten-ile zone does not cause fiber failure. In the compressive zone ashown in Fig. 6, the maximum microbuckling amplitude locates athe sinusoid peak of the free surface, and the value depends onhe fiber diameter and the curvature at the sinusoid peak of the

ltdta

Fig. 6. The sinusoid profile of a microbuckled fiber.

ompressive surface layer, which can be determined as

max = h

2d2y

dx2

∣∣∣∣max

= h

2an�2

�2≈ 0.77%, (22)

hich is very close to the ultimate strain of high-quality carbonber (0.5–1.0%). Hence, it indicates that fiber breakage is one of theossible failure modes for EMC laminate under bending.

As considering the matrix failure, the axial tension/compressiontrain can be neglect since the soft matrix can endure a very largetrain. However, the shear strain in matrix cannot be neglectednd is extremely high to accommodate fiber microbuckling. Basedn Eqs. (13) and (19), the maximum shear strain in matrix can beetermined as

max = 2amax�

�

(1 + h

2c

)= 2amax�

�Vm≈ 211%. (23)

It is clear that the maximum shear strain in the matrix is morehan 40 times higher than the nominal bending strain (5%), andas been in the same order of the ultimate shear strain of the softatrix. Therefore, delaminating at the compressive surface layer

s another possible failure mode for EMC laminate under bend-ng. It should be point out that both of fiber breakage and matrixelamination have been confirmed by experiments. Moreover, Eq.23) indicates that the maximum shear strain in the matrix cane decreased by increasing the matrix volume fraction (comparedith the traditional non-EMC materials), which might provide a

lue how to avoid delamination of EMC laminates under bending.

. Conclusions

A tension/shear-mixed model is proposed to explore the basiceformation mechanisms of EMC laminates under bending. Theicrobuckling wavelength, critical stress and strain are theoreti-

ally determined. Compared with the available models, the newension/shear-mixed model is more believable and the predicated

icrobuckling wavelength is better agreeable with experimentalbservations. Theoretical results show that there exist two possi-le failure modes for EMC laminate under bending: fiber breakaget the sinusoid peak of the compressive surface layer and delami-

ayer, both of which have also been confirmed by the experimen-al observations. The analytical result also indicates that a suitableecrease of the fiber volume fraction might be a method to reducehe maximum shear strain in EMC matrix and realize higher pack-ging strain.

3 and E

A

(R

R[[

28 Z.Y. Xiong et al. / Materials Science

cknowledgement

This work was funded by Natural Science Foundations of ChinaNo 10502005, 90205007) and Ministry of Education of the People’sepublic of China (NECT).

eferences

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[

[

[

ngineering A 496 (2008) 323–328

[5] W.H. Francis, M.S. Lake, AIAA Paper No. 2006-1764, 2006.[6] B.W. Rosen, American Society for Metals, Metals Park, Ohio, 1965 (Chapter 3).[7] N.F. Dow, B.W. Rosen, NASA CR-207, April 1965.[8] T.W. Murphey, T. Meink, M.M. Mikulas, AIAA Paper No. 2001-1418, 2001.[9] D. Campbell, M.S. Lake, K. Mallick, in: 45th AIAA/ASME/ASCE/AHS/ASC Struc-

tures, Structural Dynamics and Materials Conference, Palm Springs, California,2004.

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12] S. Timoshenko, Theory of Elastic Stability, McGraw-Hill Book Co., Inc., 1936, pp.

109–112.13] R.M. Jones, Mechanics of Composite Materials, 2nd ed., Taylor & Francis, 1999,

pp. 171–185.14] E.J. Barbero, Introduction to Composite Materials Design, Taylor and Francis,

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