Developing a nite element beam theory for nanocomposite ...

Scientia Iranica B (2017) 24(1), 249{259

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Developing a �nite element beam theory fornanocomposite shape-memory polymers withapplication to sustained release of drugs

M. Baghania;�, R. Dolatabadib and M. Baniassadia

a. School of Mechanical Engineering, College of Engineering, University of Tehran, Tehran, Iran.b. Department of Drug and Food Control, Faculty of Pharmacy, Tehran University of Medical Sciences, Tehran, Iran.

Received 29 June 2015; received in revised form 19 December 2015; accepted 27 February 2016

KEYWORDSShape-memorypolymers;Finite element;Smart beam;Euler-Bernoulli beamtheory;nano/microreinforcement.

Abstract. In this paper, a thermodynamically consistent constitutive model, recentlyproposed for nanocomposite Shape-Memory Polymers (SMPs), is used as a basis fordevelopment of SMP beam element in a �nite element framework. The beam theoryutilized here is the Euler-Bernoulli beam theory with its basic assumptions. E�ectsof di�erent materials as well as the geometric structural parameters, e.g. reinforcement(nano/micro-particles) volume fraction, viscosity coe�cients, and external loads, on thethermomechanical response of the structure are studied in this work. The beam elementnumerical results are compared to those of 3D �nite element modeling to verify validityof the beam element formulation and the assumptions made therein. This beam elementprovides us with a fast and reliable tool for simulation of structures, consisting of reinforcedSMP beams. As an application, the developed nanocomposite SMP beam element could beused for numerical modeling of thermomechanical response of the drugs (e.g., theophylline)coated by �lms of SMP nanocomposites. It is shown that the numerical results are incorrespondence with those of the experiments reported for sustained release of SMP-nanocomposite based drugs.© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

Shape-memory materials o�er a promising perspectivefor materials science and engineering, providing accessto extensive unconventional functions in various classesof materials: metals, ceramics, and polymers [1].These materials are capable of returning from a �xedshape to their original shape. This class of materialshave been researched, developed, and used in a widerange of applications, such as advanced technologiesin the aerospace and medical-related industries [2-5]. The Shape-Memory E�ect (SME) is normally in-

*. Corresponding author. Tel.: +98 21 6111 9921;Fax: +98 21 6600 0021E-mail address: [email protected] (M. Baghani)

duced through external stimuli, e.g. heat, electricity,or magnetism [6,7]. The mechanisms controlling thethermally activated SME in Shape-Memory Polymers(SMPs) are di�erent from those that play roles inshape-memory alloys or ceramics. These phenomenaresult in di�erences in the thermomechanical responsesof the material [8].

However, the much smaller sti�ness of un-reinforced SMPs limits their practical applications, inwhich a larger recovery stress is necessary. To get oversuch limitations, di�erent reinforcement techniques areintroduced in the literature [9-13]. The necessity toactivate SMPs through non-thermal stimuli has led toemploying conductive �llers, e.g. conductive particles,nanocarbon particles, and conductive �bers [14-17].The SMP composites are considered as a conductive

250 M. Baghani et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 249{259

polymer with potential practical applications in the�elds of micro-sensors and actuators. Thus, di�erentchemical sensors and bio-sensors with fascinating char-acteristics could be designed and used in medicine forfast bio-diagnostic systems [15,18-20].

It should be noted that multifunctional polymerscombining two functions, such as SME and biodegrad-ability [21] or biodegradability and drug release [22],have already been introduced [23]. But, a materialthat exhibits all these three functions has not beendeeply investigated yet. This would let us use SMEto perform minimally invasive implantation of quitemassive devices, biodegradability to eliminate a sec-ond surgery for removal, and controlled drug releaseduring encountering infections [24]. In a very recentstudy, Kashif et al. [25] investigated the sustaineddrug release and hydrolytic degradation of biodegrad-able shape-memory poly("-caprolactone)/polyhedraloligomeric silsequioxane nanocomposites. In vitro drugrelease, investigations of the phosphate bu�er solution(pH = 7.4) showed that TspPOSS nanocompositenanocrystals led to sustained release of the drug forthe nanocomposites. PCL/TspPOSS nanocomposite(90 wt/10 wt) containing 10 wt% drug (Theophylline)demonstrated good thermally activated SME with ashape �xation ratio of 81% and shape recovery ratio of85%.

Some researchers also investigated the e�ect ofaddition of nano-particles into SMP matrix [26,27].Zhang and Ni [28] performed some experiments tostudy the e�ects of carbon �ber fabric reinforcement onthe mechanical behavior of SMP sheets. The resultingSMP based laminates possessed good shape recover-ability. They also showed that the bending recoveryratio was high compared to that of the SMP sheetat di�erent recovery durations [28]. Despite a varietyof SMP applications involving bending loads, bendingcharacterizations are not much extensive [29,30]. Someresearchers tried to experimentally characterize the be-havior of SMPs [11,30-33]. Up to now, characterizationof the SMP behavior has been carried out with uniaxialtests [11,31-34]; however, exural thermomechanicaltests could aid the characterization of an SMP con-stitutive model.

Proposing an elastic-predictor inelastic-corrector,Ghosh et al. [5] introduced a �nite element SMPbeam based on a constitutive model already publishedby the same authors. Baghani et al. [35] presentedanalytic expressions for de ection of an SMP beamutilizing Euler-Bernoulli beam theory in a thermo-mechanical SME cycle. They employed the consti-tutive model already proposed in [36], where time-dependent e�ects were neglected. In another work,Baghani and Taheri [37] extended this method toinvestigate the mechanical behavior of reinforced SMPbeams.

In this paper, we propose �nite element solutionsfor nanocomposite SMP beams in an arbitrary loading-displacement recovery cycle. Employing the Euler-Bernoulli beam theory, a 1D SMP beam element isproposed. This formulation is helpful in simulationof structures that consist of SMP beams. This newlyintroduced SMP beam element can be utilized foreither parametric study (material or geometrical pa-rameters) of the SMP beam behavior or design andoptimization involving a large number of simulationsin which computational cost is a critical parameter.Making use of this beam element dramatically de-creases the computational cost, compared to 3D generalelement previously proposed by Baghani et al. [38]. Wealso investigate the SME of SMP nanocomposites toqualitatively study its drug release characteristics.

The paper is organized as follows. Firstly, a 3Dconstitutive equation for SMPs is brie y reviewed inSection 2. Then, in Section 3, the 3D constitutiverelations are reduced to the cases in which Euler-Bernoulli beams undergo arbitrary bending loadings.In this section, development of the SMP beam �niteelement is discussed in detail. In Section 4, byemploying the newly introduced SMP beam element,some numerical examples are solved. It will be shownthat the numerical results are in correspondence withthose of experiments reported for sustained releaseof SMP-nanocomposite based drugs. Results of thepresent work are compared to those of 3D modelingalready published by the same authors as well as someexperiments available in the literature. Finally, inSection 5, we present a summary and draw conclusions.

2. Brief review of the SMP constitutive modeldevelopment

Since the shape-memory e�ect characteristics are dis-cussed in details in some already published papers bythe same authors, in this paper, for the sake of brevity,we avoid repeating these details and refer to [36,38-40]. In those papers, the 3D diagrams of stress-strain-temperature for both stress-free strain recovery and�xed-strain stress recovery are introduced.

Baghani et al. [36] proposed a constitutive modelfor SMPs in the small strain regime. They thenextended this model to time-dependent loading con-ditions in [38]; in this work, we employ this modelto properly account for thermomechanical responseof SMPs. They also extended this model to thedomain of large strains employing two di�erent ap-proaches [40,41]. As described in [36,38-40], it hasbeen assumed that the rubbery and glassy phasesare able to be transformed into each other, throughthe external stimuli of heat. The evolution laws forinternal variables have been derived in an arbitrarythermomechanical loading. They have decomposed the

M. Baghani et al./Scientia Iranica, Transactions B: Mechanical Engineering 24 (2017) 249{259 251

strain into four parts as:

" = �p"p + �h"h + "i + "T ; (1)

where superscripts p and h stand for the SMP andhard segments, respectively. Also, "i and "T are theirreversible and thermal strains, respectively, while "pis the strain in SMP segment. �p and �h denotethe volume fractions of the SMP and hard segments,respectively (�p + �h = 1). Besides, the strain in theSMP segment is decomposed into three parts as:

"p = 'r"r + 'g"g + "is; (2)

where subscripts r and g stand for the rubbery andglassy phases, respectively. Also, 'r and 'g are volumefractions of the rubbery and glassy phases, respectively('r + 'g = 1). It is noted that 'r and 'g aretemperature-dependent functions, while �p and �h areconstant parameters. "is is the stored strain in thematerial and evolves in the following form [38]:

_"is = _'g�k1"r + k2

"is

'g

�;

where:8><>:k1 = 1; k2 = 0; _T < 0k1 = 0; k2 = 1; _T > 0k1 = 0; k2 = 0; _T = 0

(3)

where dot denotes di�erentiation with respect to time.Moreover, to account for the viscoelastic behaviorof SMPs, some other decompositions are assumed.In the range of small strains, strains are additivelydecomposed in the glassy, rubbery, and hard phasesas:

"r="er+"ir; "g="eg+"ig; "h="eh+"ih; (4)

in which the superscripts e� and i�(� = r; g; h) denotethe elastic and inelastic (viscous)parts of the strain inall phases, respectively. Employing the mixture rule,the free-energy density function, , for an amorphousreinforced SMP is introduced as:

�"; T; �p; �h; 'g; "r; "er; "g; "eg; "h; "eh

�= �hh

�"h; "eh

�+ T (T )

+ �p ['rr ("r; "er) + 'gg ("g; "eg)]

+ ��"; T; �p; �h; 'r; 'g; "r; "g; "is; "h; "i

�; (5)

in which r, g, and h are Helmholtz free-energydensity functions of the rubbery, glassy, and hardphases, respectively. Temperature is denoted by T andthe thermal energy is represented by T . Besides, to

satisfy Eqs. (1), (2), and (4) in the formulation, usingthe method of Lagrange multipliers, the term � isadded into the free energy as:

��"; T; �p; �h; 'r; 'g; "r; "g; "is; "h; "i

�=� :

�"��p�'r"r+'g"g+"is

��h"h�"i�"T � ;(6)

in which � is the Lagrange multiplier. They alsode�ned:

��"� ; "e�

�= eq

��"��

+ neqe��"e��;

� = r; g; h; (7)

in which the superscripts `eq0 and `neq0 are the equi-librium and non-equilibrium parts of �("� ; "e�). Inthis model, it is emphasized that the internal variablesare 'g, "ir, "ig, "ih, "i, and "is. Thus, evolution lawsfor the internal variables in the context of continuumthermodynamics should be introduced.

Satisfying the second law of thermodynamics inthe form of Clausius-Duhem inequality gives:

� =@eq

r@"r

+@neq

r@"er

=@eq

g

@"g+@neq

g

@"eg

=@eq

h@"h

+@neq

h@"eh

: (8)

Furthermore, in accordance with the viscoelastic be-havior of polymers, the following evolution laws aresu�cient conditions for satisfaction of Clausius-Duheminequality [38]:

_"ir =1�r@neq

r@"er

; _"ig =1�g@neq

g

@"eg;

_"ih =1�h@neq

h@"eh

; _"i =1�i�; (9)

in which �r, �g, �h, and �i are viscosity coe�cientsof the rubbery, glassy, hard, and irreversible parts,respectively.

3. Development of an SMP Euler-Bernoullibeam element

In this section, it is assumed that equilibrium andnonequilibrium elastic strain energies are in the formsof eq

� ("�) = 12"� : Keq

� : "� and neq� ("e�) = 1

2"e� :

Kneq� : "e� , � = r; g; h, in which Keq

� 's and Kneq� 's are

fourth-order elasticity tensors corresponding to eachphase.

Following the general procedure presented byReddy [42], we employ the basic assumptions of Euler-Bernoulli beam theory, in which the bending of beams


with small strains and rotations is found from thefollowing displacement �eld:

u1 = �z dwdx; u2 = 0; u3 = w(x); (10)

where u1; u2; and u3 are the displacements of a materialpoint at the location (x; y; z). u and w are the axialand transverse displacements of the geometric centroidof the cross section on the x-axis. An additionalassumption has also been made: Transverse loads onlycause transverse displacement and no shear strain isgenerated. Thus, the following linear strain is obtained:

"xx = �z d2wdx2 : (11)

Since we aim to develop Euler-Bernoulli beam elementfor SMPs, the constitutive model presented in Section 2should be reduced to the case where only axial strain(along beam axis) exists. Moreover, we should presentthe time-discrete form of the SMP constitutive model.We subdivide the time interval of interest [0; t] in sub-increments and solve the nonlinear problem over thegeneric interval [tn; tn+1] with tn+1 > tn. For thesake of simplicity, we indicate with the subscript n aquantity evaluated at time tn and with no subscripta quantity evaluated at time tn+1. We also show thetime increment by �t. Knowing the solution and thestrain "n at time tn, as well as the strain " at time tn+1,the stress and the internal variables should be updatedfrom the deformation history. Therefore, the time-discrete implicit form of the SMP constitutive model,speci�cally in Eq. (3), yields:

"is = "isn + k1�'"r + k2'�1�'"is

) "is�1� k2'�1�'

�= "isn + k1�'"r: (12)

Moreover, discretizing Eq. (9) and recasting Eq. (12)give:8>>>>>><>>>>>>:

"i� = c�1�� + E�b�"i�n

�; � = r; g; h

"i = "in + bi�

"is = k�13�"isn + k4

�(br + 1)"ir � br"irn ��

where:(k3 = 1� k2'�1�'k4 = k1�'

(13)

b� =��

�tEneq�

; bi =�t�i; c� = E�b� +Eeq

� ;

E� = Eeq� + Eneq

� ; � = r; g; h: (14)

Eqs. (12) to (14) show how internal variables areupdated in �t. Furthermore, after some mathematicalmanipulations, the axial stress is updated by Eq. (15)as shown in Box I. Eq. (15) can be recast as:

� = �1"m + �2; (16)

in which parameters �1 and �2 are de�ned by Eq. (17)as shown in Box II. Eq. (16) in terms of the transversede ection yields:

� = �1"m + �2 = �1 (�yw00) + �2yy

= y (��1w00 + �2y) ; (18)

where prime stands for di�erentiation with respect tox. As observed in Eq. (18), the stress varies linearlywithin the beam height. Computing the moment aboutthe z-axis, we may write:

M = �ZA

�ydA = �ZA

(��1w00 + �2y) y2dA

= (�1w00 � �2y) I; (19)

in which I and M denote the moment of inertia andthe moment along z-axis, respectively. A also standsfor cross sectional area of the beam. We now recastEq. (19) as:

M = 1w00 + 2;

where:

1 = �1I; 2 = ��2yI: (20)

� =

266664��"isn + k3

�bh"ihn � bg"ign '�+ br"irn (k3('� 1)� k4)

��p +

�"in � bh"ihn � "m� k3

�crcgch

+�ch (bg + 1) bg"ign Eg�p'+ (1� �p)(bh + 1)bh"ihn Ehcg

�crk3

+(k3 + k4 � k3')(br + 1)br�p"irn Ercgch

377775(((1 + bg)�p'+ a4cg)ch + (bh + 1)(1� �p)cg)k3cr + (k3(1� ') + k4)(br + 1)�pcgch

(15)

where "m denotes the mechanical part of the total strain ("m = "� "T ).

Box I


8>>>>>>>>>>>>><>>>>>>>>>>>>>:

�1 = crcgchk3�3

�2 = �1�3

0BBB@ "

"isn + k3(bh"ihn � bg"ign ')+br"irn (k3('� 1)� k4)

#�p +

�"in � bh"ihn � k3

!crcgch

+�ch(bg + 1)bg"ign Eg�p'+ (1� �p)(bh + 1)bh"ihn Ehcg

�crk3

+(k3 + k4 � k3')(br + 1)br�p"irn Ercgch

1CCCA = �2yy

�3 = (((1 + bg)�p'+ a4cg)ch + (bh + 1)(1� �p)cg) k3cr + (k3(1� ') + k4) (br + 1)�pcgch

(17)

Box II

The weak form is the result of the principle of virtualdisplacements. It states that if a body is in equilibrium,the total virtual work done by actual internal andexternal forces in moving through the virtual displace-ments is zero. This principle over a �nite elementx 2 [0; le], is:

�W e = �WE + �W I = 0; (21)

where �W I is the internal virtual work, and �WE isthe external work done by the externally applied loadsthrough the virtual displacements. The schematics ofthe beam element with the de�ned relevant degrees offreedom are shown in Figure 1.

The virtual work principle for the Euler-Bernoullibeam theory is reduced to:

�W e=ZA

leZ0

��"dxdA�leZ

0

q(x)�wdx�4Xi=1

Qi�i; (22)

where le is length of the element, q(x) is the distributedtransverse loads measured per unit length, Qi's arethe generalized nodal forces, and �i are the virtualgeneralized nodal displacements of the element. Thus,in light of de�nition (11), the virtual work statementin Eq. (21) becomes:

Figure 1. The schematics of the beam element togetherwith the de�ned degrees of freedom.

�" = �y d2(�w)dx2

)ZA

leZ0

��"dxdA = �leZ

0

d2(�w)dx2

�M(x)z }| {ZA

�ydAdx: (23)

Substituting Eq. (23) into Eq. (22) yields:

�W e=leZ

0

d2(�w)dx2 M(x)dx�

leZ0

q(x)�wdx

�4Xi=1

Qi�i=0: (24)

In light of Eqs. (20) and (24), we arrive at:

leZ0

d2(�w)dx2

� 1d2wdx2 + 2

�dx�

leZ0

q(x)�wdx

�4Xi=1

Qi�i = 0: (25)

Employing 1D Hermite cubic interpolation functionNi, the transverse beam de ection is approximated interms of four degrees of freedom, i.e. i, as:

w =4Xi=1

Nii; �w =4Xi=1

Ni�i; (26)

where these degrees of freedom are:

1 = w(0); 2 = �dw=dxjx=0 ;

3 = w(le); 4 = �dw=dxjx=le : (27)

Substituting Eq. (26) into Eq. (25), after some mathe-matical manipulations, gives:


4Xi=1

leZ0

d2Nidx2

� 1d2wdx2 + 2

�dx

�leZ

0

Niq(x)dx�Qi!�i = 0: (28)

Assuming w in the form of Eq. (26), Eq. (28) isrewritten as follows:

leZ0

d2Nidx2

24 1

4Xj=1

d2Njdx2 j + 2

35 dx� leZ0

q(x)Nidx

�Qi = 0: (29)

In order to present the matrix format for the elementsti�ness and load vectors, Eq. (29) is �nally recast as:

[Kij ] fjg = fFig ; (30)

in which:8>>>>><>>>>>:Kij = 1

leR0

d2Nidx2

d2Njdx2 dx

Fi = � 2

leR0

d2Nidx2 dx+

leR0q(x)Nidx+Qi

(31)

For the sake of completeness, the Hermite shapefunctions are reported as follows:

N1 =2x3�3x2le+le

3

le3; N2 =

x3le�2x2le2+xle

3

le3;

N3 =�2x3 + 3x2le

le3; N4 =

x3le � x2le2

le3: (32)

It is �nally reminded that 1 and 2 are memoryparameters. These parameters are updated in each in-crement and are responsible for the history dependencyof the SMP element response, as described in Eqs. (17)and (20).

4. Numerical results

To show validity of the model in time-dependentregimes of thermomechanical loadings, we �rst re-port the simulation results according to experimentsperformed in [43] already published in [38]. At thestart of the experiment, an SMP-based syntactic foamsample is compressed under a constant stress �0 =�263 kPa at Th for 30 minutes. Keeping the stress�xed, the temperature is lowered to Tl. The coolingrate is found through the Newton's law of cooling:dTdt = 4:6� 10�5(T � 293) [38]. Once the temperaturereaches Tl (at t = 1256 min), the system is unloadedand, then, it is heated up to Th with heating rate= 0.3 K

min . Therequired material parameters aretabulated in Table 1. The strain-time behavior isshown in Figure 2(a). A good correlation is observedbetween the experiments and the numerical results. Ifthe recovery occurs in a �xed-strain stress recovery,the stress-temperature behavior is found as shown in

Figure 2. Reproduction of the shape-memory e�ect [38]: (a) Stress-free strain recovery and (b) �xed-strain stressrecovery. Experiments are reported in [43].

Table 1. Material parameters reported by Baghani et al. [38] adopted from experiments conducted in [43].

Material parameters Values UnitsEeqr ; Eeq

g ; Eeqh ; E

neqr ; Eneq

g ; Eneqh 1.3, 15,70000, 0.2, 247,1000 (MPa)

�r; �g; �h 5; 30� 106; 30� 109 (MPa.min)�r; �g; �h 0.49, 0.3, 0.3 (-)Tl; Tg; Th 296, 344, 353 (K)

"T�0:5542� 10�3T � 0:01083� 7� 10�7T 2� (-)

'g 1� 11+exp(�0:66(T�Tg)) (-)


Figure 2(b). These results show that the SMP modelis capable of predicting the main governing phenomenain di�erent SME cycles.

Now, we employ the presented SMP beam elementto investigate the bending of a cantilever SMP beam,experiencing a stress-free strain recovery cycle. AnI-shaped SMP beam is selected for presenting thenumerical results. Thickness of the beam cross sectionis 1 mm and its height and width are 10 mm and12 mm, respectively. The history of the appliedforce on the free tip of the beam, as well as thetemperature history, is shown in Figure 3. Making

Figure 3. History of the force and temperature appliedto the beam.

use of the material parameters reported in Table 1,the corresponding results are illustrated in Figure 4.In order to validate results of the present work, the3D SMP constitutive model is implemented in a �niteelement framework. For 3D simulations, we use thenonlinear �nite element software ABAQUS/Standard,implementing the described algorithm in [38] withina user-de�ned subroutine UMAT. Employing this 3D�nite element tool, the whole SME cycle is simulated.In Figure 4(a), the free tip displacement results of theSMP beam model (present work) together with thoseof the 3D modeling are depicted. As one may observe,at di�erent values of �h, an acceptable correspondenceis seen between the results of these two methods.Moreover, at larger values of �h, as expected, thestructure becomes sti�er. Hence, smaller beam tipde ections are predicted. Besides, as observed inFigure 4(a), the beam is capable of complete recoveryof the shape. It can be justi�ed considering thatin this example, �i has a very large value. As aresult, no residual irreversible strain remains in thestructure. The time-dependent viscoelatic behavior ofthe structure at temperatures higher than Th is clearlyseen in Figure 4(a).

In Figure 4(b), the variations of axial strain atupper surface of the beam root (the most critical pointof the structure) are plotted. This �gure shows thatthe maximum strain is about 3.3%. Thus, the small

Figure 4. E�ect of �h on (a) the free tip displacement results of the SMP beam model (present work) together with thoseof the 3D modeling (solid lines show the beam element results, while dashed-lines represent the 3D �nite elementpredictions), (b) variation of axial strain at upper surface of the beam root (the most critical point of the structure), (c)the free tip angle of the beam, and (d) the temporarily memorized shape of the beam (after unloading). Fmax = �0:05 N,q = 0 (N/m), and �h = 0, 0.1, 0.2, 0.3, 0.4.


strain assumption used in developing the element isnot violated. Moreover, in Figure 4(c), the free tipangle of the beam is shown. This �gure also showsthe maximum angle smaller than 16 degrees. Thus,the assumption of small rotations also stands. InFigure 4(d), the temporary shape of the beam (afterunloading) is depicted at di�erent �h's.

In order to investigate the e�ects of di�erentvalues of viscosity parameters, in Figure 5, the freetip displacement results of the SMP beam model aredepicted. It is shown that increasing �r makes thestructure need more time to react against the appliedstress. To investigate the e�ect of applying an externalforce during the heating process on the shape recovery,free tip of the beam is plotted in Figure 6. In this�gure, Fmax is the maximum force applied to freetip of the beam in the loading path and Fh is theforce applied to the beam within the heating process.As observed in Figure 6, without any external force(Fh = 0), the SMP beam fully recovers its originalshape, while increasing the external force (during theheating) results in smaller values of recovery ratio asillustrated in Figure 6. In the special case where Fh =Fmax, we almost have no recovery. It shows that afterheating recovery, stored strains completely vanish andresponses of the structure to the mechanical loadingsare similar to the �rst responses, i.e. those at Th.

The developed formulation can be used in any

Table 2. SMP behavior of theophylline-loadedPCL/TspPOSS (90/10) nanocomposites [25].

Theophyllineweight (%)

Shape�xity (%)

Shaperecovery (%)

0 85 945 82 8710 81 74

structure that undergoes bending. If we want to prop-erly model a drug in which an SMP-nanocomposite iscoated on the drug, e.g. theophylline, we should just usethe SMP beam element developed here, accompaniedby elastic/hyperelastic/viscoelastic elements (theseelements stand for the drug, e.g. theophylline parts).In Table 2, the recovery and shape �xity ratios of theSMP-nanocomposite drug are tabulated. These dataare borrowed from [25]. As one may observe in thistable, increasing the weight fraction of theophyllineleads to a decrease in the shape recovery as well as theshape �xity of the drug. This trend is qualitatively inagreement with those found in this paper.

5. Summary and conclusions

In this work, by employing a new constitutive modelrecently proposed for SMPs, a new SMP beam ele-ment was developed in a �nite element framework topredict the response of reinforced SMP beams under

Figure 5. (a) E�ect of �r on the free tip displacement results of the SMP beam model (�r = 0, 1, 2, 5, 10 MPa.min and�h = 0). (b) The rectangle in part (a) is zoomed in.

Figure 6. E�ect of applying an external force during the heating process to the shape recovery on (a) the free tipdisplacement and (b) the recovered shape (Fh=Fmax = 0:0, 0.1, 0.2, 0.5, 1.0 and �h = 0).


bending in thermo-mechanical cycles. To develop theformulations, the assumptions of Euler-Bernoulli beamtheory were employed. It was observed that withinthe cooling (shape �xation) process, the stored strainsevolved to �x the temporary shape; then, during theheating process, these strains started to relax, in orderto recover the original shape. E�ects of di�erent valuesof hard segment volume fraction on the SMP beam re-sponse for a cantilever reinforced SMP beam were alsoinvestigated. It was observed that increasing volumefraction of hard segment acted as reinforcement andmade the structure sti�er. Thus, as a conclusion, largervalues of external loads could be handled. Moreover,the time-dependent e�ects were investigated througha parametric study on the viscosity coe�cient in therubbery phase. It was shown that in larger values of�r, the structure needed more time to arrive at a steadystate. With no external forces within the heating step,the SMP beam almost fully recovered its original shape,while increasing the external force (during heating)resulted in smaller values of recovery ratio. In thespecial case that the external loadings were same as theloadings in the loading step, the structure experiencedalmost no shape recovery. Comparing the numericalresults with those of 3D �nite element modeling ensuresthe validity of the beam element developments andthe assumptions made therein. The proposed �niteelement beam formulation can be used as an e�cienttool for investigating the e�ects of changing any ma-terial or geometrical parameter on reinforced SMPbeams, where we need a large number of simulationsand 3D modeling is computationally too expensive.As an application, it was shown that the developednanocomposite SMP beam element could be used fornumerical modeling of thermomechanical response ofthe drugs (e.g., theophylline) coated by �lms of SMPnanocomposites. It was observed that the numericalresults were qualitatively in correspondence with thoseof experiments reported for sustained release of SMP-nanocomposite based drugs already available in theliterature.

References

1. Schmidt, C., Chowdhury, A.M.S., Neuking, K. andEggeler, G. \Mechanical behavior of shape memorypolymers by 1WE method: Application to teco ex",Journal of Thermoplastic Composite Materials, 24, pp.853-860 (2011).

2. Hashemi, S.M.T. and Khadem, S.E. \Modeling andanalysis of the vibration behavior of a shape memoryalloy beam", International Journal of Mechanical Sci-ences, 48, pp. 44-52 (2006).

3. Zbiciak, A. \Dynamic analysis of pseudoelastic smabeam", International Journal of Mechanical Sciences,52, pp. 56-64 (2010).

4. Lendlein, A. and Behl, M. \Shape-memory polymersfor biomedical applications", Advances in Science andTechnology, 54, pp. 96-102 (2009).

5. Ghosh, P., Reddy, J.N. and Srinivasa, A.R. \Devel-opment and implementation of a beam theory modelfor shape memory polymers", International Journal ofSolids and Structures, 50, pp. 595-608 (2012).

6. Lendlein, A. and Langer, R. \Biodegradable, elasticshape-memory polymers for potential biomedical ap-plications", Science, 296, pp. 1673-1676 (2002).

7. Gall, K., Dunn, M., Liu, Y., Stefanic, G. and Balzar,D. \Internal stress storage in shape memory polymernanocomposites", Applied Physics Letters, 85, pp. 290-292 (2004).

8. Diani, J., Liu, Y. and Gall, K. \Finite strain 3D ther-moviscoelastic constitutive model for shape memorypolymers", Polymer Engineering and Science, 46, pp.486-492 (2006).

9. Leng, J., Lan, X., Liu, Y. and Du, S. \Shape-memorypolymers and their composites: Stimulus methods andapplications", Progress in Materials Science, 56, pp.1077-1135 (2011).

10. Leng, J., Lv, H., Liu, Y. and Du, S. \Electroactivateshape-memory polymer �lled with nanocarbon parti-cles and short carbon �bers", Applied Physics Letters,91, p. 144105 (2007).

11. Xu, W. and Li, G. \Constitutive modeling of shapememory polymer based self-healing syntactic foam",International Journal of Solids and Structures, 47, pp.1306-1316 (2010).

12. Lan, X., Liu, L., Liu, Y., Leng, J. and Du, S. \Postmicrobuckling mechanics of �bre-reinforced shape-memory polymers undergoing exure deformation",Mechanics of Materials, 72, pp. 46-60 (2014).

13. Qiao, T., Liu, L., Liu, Y. and Leng, J. \Post bucklinganalysis of the shape memory polymer compositelaminate bonded with alloy �lm", Composites Part B:Engineering, 53, pp. 218-225 (2013).

14. Lan, X., Liu, Y., Lv, H., Wang, X., Leng, J. and Du,S. \Fiber reinforced shape-memory polymer compositeand its application in a deployable hinge", SmartMaterials and Structures, 18, p. 024002 (2009).

15. Lu, H., Yu, K., Liu, Y. and Leng, J. \Sensing andactuating capabilities of a shape memory polymer com-posite integrated with hybrid �ller", Smart Materialsand Structures, 19, p. 065014 (2010).

16. Mohamadpour, S., Pourabbas, B. and Fabbri, P.\Anti-scratch and adhesion properties of photocur-able polymer/clay nanocomposite coatings based onmethacrylate monomers", Scientia Iranica, 18, pp.765-771 (2011).

17. Hajiali, H., Karbasi, S., Hosseinalipour, M. and Rezaie,H. \E�ects of bioglass nanoparticles on bioactivityand mechanical property of poly (3-hydroxybutirate)sca�olds", Scientia Iranica, Transactions F, Nanotech-nology, 20, p. 2306 (2013).


18. Yanju, L., Haiyang, D., Liwu, L. and Jinsong, L.\Shape memory polymers and their composites inaerospace applications: a review", Smart Materialsand Structures, 23, p. 023001 (2014).

19. Liu, Y., Lv, H., Lan, X., Leng, J. and Du, S. \Reviewof electro-active shape-memory polymer composite",Composites Science and Technology, 69, pp. 2064-2068(2009).

20. Zhou, B., Liu, Y.J., Lan, X., Leng, J.S. and Yoon, S.H.\A glass transition model for shape memory polymerand its composite", International Journal of ModernPhysics B, 23, pp. 1248-1253 (2009).

21. Lendlein, A. and Langer, R. Biodegradable \Elas-tic shape-memory polymers for potential biomedicalapplications", Science, 296, pp. 1673-1676 (2002).

22. Mohamed, F. and Van Der Walle, C.F. \Engineeringbiodegradable polyester particles with speci�c drugtargeting and drug release properties", Journal ofPharmaceutical Sciences, 97, pp. 71-87 (2008).

23. Ne�e, A.T., Hanh, B.D., Steuer, S. and Lendlein, A.\Polymer networks combining controlled drug release,biodegradation, and shape memory capability", Ad-vanced Materials, 21, pp. 3394-3398 (2009).

24. Hetrick, E.M. and Schoen�sch, M.H. \Reducingimplant-related infections: Active release strategies",Chemical Society Reviews, 35, pp. 780-7891,1 (2006).

25. Kashif, M., Yun, B.M., Lee, K.S. and Chang,Y.W. \Biodegradable shape-memory poly (caprolac-tone)/polyhedral oligomeric silsequioxane nanocom-posites: Sustained drug release and hydrolytic degra-dation", Materials Letters, 166, pp. 125-128 (2016).

26. Lu, H., Liu, J., Zhu, S., Yang, Y. and Fu, Y.Q. \En-hanced electro-activated performance of shape memorypolymer nanocomposites with self-assembled carbonnano�bre template", Nanoscience and NanotechnologyLetters, 7, pp. 94-99 (2015).

27. Shen, H., Mark, A., Thompson, K., Xu, Y., Liang,F., Gou, J. and Mabbott, B. \Thermal modelingand coe�cient identi�cation of shape memory polymernanocomposites structure", Applied Physics Letters,106, p. 081907 (2015).

28. Zhang, C.S. and Ni, Q.Q. \Bending behavior of shapememory polymer based laminates", Composite Struc-tures, 78, pp. 153-161 (2007).

29. Liu, Y., Gall, K., Dunn, M.L. and McCluskey, P.\Thermomechanical recovery couplings of shape mem-ory polymers in exure", Smart Materials and Struc-tures, 12, pp. 947-954 (2003).

30. Tobushi, H., Hayashi, S., Hoshio, K. and Ejiri, Y.\Shape recovery and irrecoverable strain control inpolyurethane shape-memory polymer", Science andTechnology of Advanced Materials, 9, p. 015009 (2008).

31. Liu, Y., Gall, K., Dunn, M., Greenberg, A. andDiani, J. \Thermomechanics of shape memory poly-mers: Uniaxial experiments and constitutive model-ing", International Journal of Plasticity, 22, pp. 279-313 (2006).

32. Atli, B., Gandhi, F. and Karst, G. \Thermomechanicalcharacterization of shape memory polymers", Journalof Intelligent Material Systems and Structures, 20, pp.87-95 (2009).

33. Kim, J., Kang, T. and Yu, W. \Thermo-mechanicalconstitutive modeling of shape memory polyurethanesusing a phenomenological approach", InternationalJournal of Plasticity, 26, pp. 204-218 (2010).

34. Tobushi, H., Okumura, K., Hayashi, S. and Ito,N. \Thermomechanical constitutive model of shapememory polymer", Mechanics of Materials, 33, pp.545-554 (2001).

35. Baghani, M., Mohammadi, H. and Naghdabadi, R.\An analytical solution for shape-memory-polymereuler-bernoulli beams under bending", InternationalJournal of Mechanical Sciences, 84, p. 8490 (2014).

36. Baghani, M., Naghdabadi, R., Arghavani, J. andSohrabpour, S. \A constitutive model for shape mem-ory polymers with application to torsion of prismaticbars", Journal of Intelligent Material Systems andStructures, 23, pp. 107-116 (2012).

37. Baghani, M. and Taheri, A. \An analytic investi-gation on behavior of smart devices consisting ofreinforced shape memory polymer beams", Journalof Intelligent Material Systems and Structures, p.1045389X14541503 (2014).

38. Baghani, M., Naghdabadi, R., Arghavani, J. andSohrabpour, S. \A thermodynamically-consistent 3Dconstitutive model for shape memory polymers", In-ternational Journal of Plasticity, 35, pp. 13-30 (2012).

39. Baghani, M., Naghdabadi, R. and Arghavani, J. \Asemi-analytical study on helical springs made of shapememory polymer", Smart Materials and Structures,21, p. 045014 (2012).

40. Baghani, M., Naghdabadi, R. and Arghavani, J. \Alarge deformation framework for shape memory poly-mers: Constitutive modeling and �nite element im-plementation", Journal of Intelligent Material Systemsand Structures, 24, pp. 21-32 (2013).

41. Baghani, M., Arghavani, J. and Naghdabadi, R. \A�nite deformation constitutive model for shape mem-ory polymers based on Hencky strain", Mechanics ofMaterials, 73, pp. 1-10 (2014).

42. Reddy, J.N., An Introduction to Nonlinear FiniteElement Analysis: With Applications to Heat Transfer,Fluid Mechanics, and Solid Mechanics, OUP Oxford(2014).

43. Li, G. and Nettles, D. \Thermomechanical characteri-zation of a shape memory polymer based selfrepairingsyntactic foam", Polymer, 51, pp. 755-762 (2014).

Biographies

Mostafa Baghani received his BS degree in Me-chanical Engineering from University of Tehran, Iran,


in 2006 and his MS and PhD degrees in Mechan-ical Engineering from the Department of Mechani-cal Engineering at Sharif University of Technology,Tehran, Iran, in 2008 and 2012, respectively. He isnow an Assistant Professor in the School of Mechan-ical Engineering, University of Tehran. His researchinterests include solid mechanics, nonlinear �nite ele-ment method, and shape-memory materials constitu-tive modeling.

Roshanak Dolatabadi received her PharmD degreefrom Mashhad University of Medical Sciences, Iran, in2014. She is now a PhD student in Department ofDrug and Food Control, Faculty of Pharmacy, Tehran

University of Medical Sciences, Tehran, Iran. Herresearch interests are pharmaceutics and designing newtechniques for release pro�le controllability of drugsthrough methods such as extrusion-spheronization, pel-letization, coating, etc.

Majid Baniassadi is working as Assistant Professorin the Faculty of Mechanical Engineering at Universityof Tehran. He received his BS degree from IsfahanUniversity of Technology, in 2004, MSc degree fromUniversity of Tehran in 2006, and PhD from Universityof Strasbourg in 2011. His research interests are mainlyin the area of micromechanics, nano-mechanics, andmaterials design.

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