Thermo-mechanical properties of shape memory polymer nanocomposites reinforced by carbon nanotubes

Accepted Manuscript

Thermo-mechanical Properties of Shape Memory Polymer Nanocomposites Reinforced by Carbon Nanotubes Mohammad Kazem Hassanzadeh-Aghdam , Reza Ansari , Mohammad Javad Mahmoodi PII: DOI: Reference:

S0167-6636(18)30368-5 https://doi.org/10.1016/j.mechmat.2018.11.009 MECMAT 2948

To appear in:

Mechanics of Materials

Received date: Revised date:

19 May 2018 17 November 2018

Please cite this article as: Mohammad Kazem Hassanzadeh-Aghdam , Reza Ansari , Mohammad Javad Mahmoodi , Thermo-mechanical Properties of Shape Memory Polymer Nanocomposites Reinforced by Carbon Nanotubes, Mechanics of Materials (2018), doi: https://doi.org/10.1016/j.mechmat.2018.11.009

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Highlight: 

Overall thermomechanical properties of CNT/SMP nanocomposites are studied using the SUC model. Two fundamental aspects are investigated herein; namely, CNT waviness and the

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

existence of an interphase. 

Increasing CNT volume fraction leads to (i) an increment in the elastic moduli and (ii) a reduction in the CTEs of the CNT/SMP nanocomposite.

Employing the straight CNTs within the SMP matrix improves the longitudinal

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

thermomechanical properties of the nanocomposite, whereas the CNT waviness

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improves the effective transverse properties.

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Effective Thermo-mechanical Properties of Shape Memory Polymer Nanocomposites Reinforced by Carbon Nanotubes Mohammad Kazem Hassanzadeh-Aghdam1,2, Reza Ansari1,*, Mohammad Javad Mahmoodi2,*

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1- Department of Mechanical Engineering, University of Guilan, Rasht, Iran 2- Faculty of Civil, Water and Environmental Engineering, Shahid Beheshti University, Tehran,

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Iran

Abstract. This work is aimed at a comprehensive characterization of thermoelastic properties of shape memory polymer (SMP) nanocomposites containing carbon nanotubes (CNTs) via a micromechanical model. Two critical aspects affecting the CNT/polymer nanocomposite overall

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behavior including, the non-straight shape of CNTs and an interphase region formed due to nonbonded van der Waals interactions between a CNT and the SMP are considered. The influences

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of volume fraction, diameter, aspect ratio, waviness of the CNTs, size and adhesion exponent of the interphase and temperature on the elastic moduli and coefficients of thermal expansion

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(CTEs) of the CNT/SMP nanocomposites are investigated. The results indicate that the SMP

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nanocomposite elastic moduli enhance with (i) increasing both the CNT volume fraction and interphase thickness and (ii) decreasing both the CNT diameter and interphase adhesion

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exponent. Also, the longitudinal elastic modulus of SMP nanocomposites can be significantly increased by using straight CNTs, whereas the transverse elastic modulus improves by employing wavy CNTs. It is observed that the SMP nanocomposite CTEs decrease with the increase in CNT volume fraction, whereas the CNT diameter effect on the thermal expansion

*

Corresponding Authors: [email protected] (R. Ansari), [email protected] (M.J. Mahmoodi). 2

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response can be ignored. Generally, the results of the presented model are found to be in very good agreement with experiments. Keywords: Carbon nanotube; Nanocomposite; Shape memory polymer; Thermal expansion;

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Elastic property; Micromechanics.

1. Introduction

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In recent years, shape memory polymers (SMPs) have attracted much attention in various fields of engineering applications, e.g. actuators, biological micro-electro-mechanical systems, control of structures, self-healing, biomedical devices and so on [1,2]. The SMPs have the ability of fixing a transient shape and recovering to their original shape after a series of external stimuli,

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most typically thermal activation [3,4]. Lightness, low manufacturing cost, easy in processing, good biocompatibility, non-toxicity are several extraordinary properties of SMPs [2-4]. Also,

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two important advantages of SMPs are high elastic deformation and high recoverable strain

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compared to shape memory alloys (SMAs). However, low mechanical properties, such as low stiffness and strength, as well as high coefficient of thermal expansion (CTE) are the

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disadvantages of SMPs compared to SMAs [3-5]. Secondary phase, such as carbon nanotubes (CNTs) due to their high strength and Young’s modulus [6,7] as well as very good thermal

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properties [8,9] is added into the pristine SMP to overcome the disadvantages [3-5,10]. It should be noted that establishing structure-property relationships for the CNT/SMP

nanocomposites is a crucial step for a reliable and optimal design of such new systems. Several experimental investigations [5,11] have been carried out to evaluate the mechanical behavior of CNT/SMP nanocomposites. For example, Raja et al. [5] investigated the mechanical, dynamic 3

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and electroactive shape memory properties of the thermoplastic polyurethane (PU)/polylactide (PLA) nanocomposites containing multi-walled CNTs (MWCNTs). Based on the experimental observation, they concluded that the tensile strength of the PU/PLA nanocomposites increases with increasing in weight fraction of MWCNTs. Also, according to the experimental results of

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Ni et al. [12], it has been found that storage elastic modulus of CNT/SMP nanocomposites improves obviously by the increase of CNT weight fraction and subsequently, the nanocomposites show a good shape memory effect. The simulation approaches have been also suggested to determine the mechanical properties of CNT/SMP nanocomposites [3]. For

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example, Dastgerdi et al. [3] studied the influences of CNT waviness and its distribution on the elastic modulus of CNT/SMP nanocomposites by the use of Mori-Tanaka (M-T) micromechanical method. Yang et al. [13] investigated the elastic properties of CNT/SMP

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nanocomposites using M-T method. The effects of CNT volume fraction, temperature and aggregation on the longitudinal and transverse elastic moduli of CNT/SMP nanocomposites were

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examined. Apart from the volume fraction, dispersion and geometry of CNTs, the overall properties of CNT-reinforced polymer nanocomposites greatly depend on the interfacial bonding

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between the CNT and the surrounding matrix [14,15]. In fact, the performance of the

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nanocomposites is critically governed by the CNT/polymer interfacial bonding. The non-bonded electrostatic and van der Waals (vdW) interactions are two types of interaction between a CNT

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and the polymer matrix [16,17]. Because of the fact that vdW interactions contribute more considerably in three higher orders of magnitude than electrostatic energy, the electrostatic interactions can be neglected as compared to vdW interactions [18,19]. To the best of authors’ knowledge, no report exists in the case of assessing the influence of interphase features on the overall behavior of the CNT/SMP nanocomposites.

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The CNT/polymer nanocomposite structures may experience a change of temperature during operation. This leads to thermal expansion in these structures which may affect their reliability. Thus, the CTE becomes an important property for the polymer nanocomposites reinforced with CNTs. This concern becomes much more significant for SMP matrix nanocomposites due to

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their temperature-dependent properties [20]. Nevertheless, at present, no report was found that specifically analyzes the thermal expansion response of the CNT/SMP nanocomposites.

This paper deals with a comprehensive micromechanical analysis of the elastic and

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thermoelastic behaviors of the CNT/SMP nanocomposites. For this purpose, the simplified unit cell (SUC) micromechanical model [21,22] is employed. The rest of the paper is organized as follows. Section 2 presents the thermomechanical theory for SMPs. Section 3 introduces the geometry of the representative volume element (RVE) of the SUC model. The three-dimensional

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SUC micromechanics model is developed in Section 4. Also, Section 5 characterizes the constituents of the considered nanocomposite system. In Section 6, the results of the model are

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introduced to quantitatively show the effects of various factors, including volume fraction,

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diameter, aspect ratio (length/diameter), waviness factor and the number of waves of the CNTs, size and adhesion exponent of the interphase as well as temperature on the mechanical and

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thermal expansion behaviors of the CNT/SMP nanocomposite. The concluding remarks are provided in Section 7. The obtained results from the present micromechanical model could be

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useful to guide reliable design of CNT/SMP nanocomposites.

2. Thermomechanical theory of SMPs

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At certain temperatures, the SMPs possess frozen and active phases [20]. The total strain of a SMP (

) in terms of the strains in the frozen phase ( ) and active phase (

) can be given by

[20] (

(1)

is volume fraction of the frozen phase and can be defined as (

where n and

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where

)

) are material constants.

(2)

represents reference temperature. Also, T is the

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analysis temperature at which the results would be extracted. As seen in Eq. (2), temperature-dependent. Consequently, the volume fraction of the active phase is

is .

The strain in the frozen phase consists of three components including, the stored entropic

(3)

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∫ where

) which can be written as

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strain ( ), the internal energetic strain ( ) and the thermal strain (

is the volume of the frozen phase. Furthermore, the strain in the active phase consists

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of two components including, the entropic strain ( ) and thermal strain ( ) which can be

(4)

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presented by

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Substituting Eqs. (3) and (4) into Eq. (1) leads to ∫

where

(

(

)

)

(

(

)

)

(5)

is the total volume of the SMP. The first term on the right-hand side of Eq. (5) is the

stored strain ( ) which can be expressed as

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∫

∫

Also, in Eq. (5),

(6)

∫

are elastic strains. By the use of Hooke’s law,

and

and

are given as (7)

and

stand for the elastic compliance tensors related to the internal energetic

deformation and entropic deformation, respectively. Also,

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where

is the total stress of SMP. It is

assumed that the stresses in the active phase and frozen phase to be equal to the total stress [20]. Hence, the total elastic strain becomes )

(

(

)

)

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(

(8)

Also, the part of thermal strain of SMP can be written as (

)

) is given as

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Thus, the total strain of the SMP (

(9)

(10)

(

)

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(

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Using Eqs. (10) and (8), the total stress can be calculated as (11)

)

Hence, the elastic stiffness tensor of the SMP ( (

)

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(

) is given as (12)

)

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In the case of uniaxial stress state, the elastic modulus of SMP ( (

where Also,

) can be extracted as [20] (13)

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is the modulus related to entropic deformation, i.e.

.

is a material constant.

is the modulus of internal energy deformation which is assumed to be constant.

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The CTE of the SMP matrix (

) noticeably depends on temperature ( ) which can be

expressed as follows [20]

3. RVE of the model

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(14)

Fig. 1 illustrates the geometry of RVE of the SUC model consisting of c×r×h sub-cells. The number of sub-cells of the RVE along the ,

and

directions are c, r and h, respectively. Each

along the ,

and

directions specify by and

directions, respectively. The lengths of each sub-cell along the , ,

and

, respectively.

,

and

and

are the lengths of the RVE along

directions, respectively, as displayed in Fig. 1.

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the ,

denoting the location of the sub-cell

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rectangular sub-cell is labeled by , and , with , and

The consideration of non-bonded vdW interactions between a CNT and the polymer matrix is

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an important factor to estimate the effective properties of the CNT/polymer nanocomposites [18,19,22]. In the micromechanical modeling of nanocomposites, an equivalent solid continuum

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interphase is commonly considered between a CNT and the polymer matrix to characterize vdW

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interactions [15,18,21]. The RVE of the unidirectional straight CNT-reinforced polymer nanocomposites with considering the interphase region is shown in Fig. 2. A unit length in the z

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direction is considered to apply generalized plane strain assumption on the RVE of the SUC model for unidirectional composites [21,22]. As indicated in the figure, the RVE is composed of three phases, including CNT, polymer matrix and interphase region with a thickness equal to . Also, it is noted that the hollow cylindrical molecular structure of CNT is modeled as an equivalent solid fiber [15,18,21,22]. The elastic stiffness tensor ( ) and CTE vector ( ) of the interphase are assumed to satisfy the following conditions 8

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[ ]|

[

] [ ]|

{ }|

{

} { }|

where

[

] {

(15) (16)

}

, as shown in Fig. 2.

stands for the CNT diameter. Moreover, and

are the

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are the elastic stiffness tensor and CTE vector of CNT, respectively.

and

elastic stiffness tensor and CTE vector of the polymer material, respectively. The elastic stiffness tensor of the interphase at any value of r is given as [23,24]

(17)

[

](

)

(

[

) [

]

[

](

)

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[ ]|

]

where η is adhesion exponent. The adhesion exponent controls the quality of adhesion between the CNT and the polymer matrix. The value of adhesion exponent depends on the specific

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surface area of the CNT at the interface, density of the CNT, interphase strength, interphase thickness, and the matrix strength. Based on the mentioned parameters, the adhesion exponent is

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calculated by means of pull-out tests, Kellye-Tyson model and Pukanszky model [23]. The CTE

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of the interphase at any value of r is given in the following form

{ }|

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( (

(18) )

)

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The elastic properties and CTEs of the interphase vary along the radial direction ranging from CNT to polymer matrix. Thus, by averaging the varying interphase properties along the radial direction,

and

can be calculated as (19)

[ ]

∫

[ ]|

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(20) { }

∫

{ }|

It should be noted that Eqs. (19) and (20) for average thermomechanical properties of the

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interphase would be used in the micromechanical computations presented in the next section. 4. Micromechanical equations

In this section, the micromechanical equations of the SUC model are presented. One of the main assumptions for deriving the SUC model is that the displacement components to be linear, which

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results in constant stress and strain state within the sub-cells [21,22]. Furthermore, it is supposed that the applied normal stress over the RVE does not present any shear stress inside the sub-cells of the RVE [21,22]. The equilibrium conditions of the RVE shown in Fig. 1 along the normal

are given by

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within the sub-cell

) and local stresses (

(21)

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{

)

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directions between the applied global stresses (

in which the indices i, j and k act as dummy index.

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Imposing the equilibrium of the local stress components along the interfaces leads to

{

(22)

Perfect bonding conditions are applied between the sub-cells of the RVE. Thus, compatibility of the displacements within the RVE can be expressed as

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∑

∑

∑

∑

∑

̅ ̅

̅

where ̅ , ̅ and ̅ are the global strains. Also, cell

and

{ }

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[ ]

{ }

(23)

are local strains for the sub-

. The three-dimensional thermo-elastic constitutive equation for the sub-cell

as { }

,

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{

∑

is given

(24)

where { } and { } represent the stress vector and the strain vector, respectively. Also, [ ] and { } are the elastic compliance tensor and the CTE vector, respectively. ΔT is the temperature equations

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deviation from a reference temperature. Eqs. (21)-(24) form a set of system with the same number of unknown and can be written in a matrix form as { }

{ }

(25)

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[ ]

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where { } is the local stress vector in the proper form and { } is the external load vector. Also, [ ] is the coefficients matrix. Solving the set of governing equations; i.e. Eq. (25), the stress

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components of each sub-cell are calculated. Consequently, the strain components of each subcell are determined using Eq. (24). Then, by means of Eq. (23), the global normal strains of the

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RVE can be extracted. So, the elastic moduli, Poisson’s ratios and CTEs are achieved. The shear moduli are extracted in the close form relations as [25]

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∑∑

∑∑

∑

∑

where

is the shear modulus of the sub-cell

CTE vector (

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∑∑ ∑ {

. Therefore, the elastic stiffness tensor (

(26) ) and

) of the straight CNT-reinforced polymer nanocomposites considering

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interphase region can be obtained using the above SUC relations.

The current model may be improved to cover large deformation by substituting the constitutive models of SMPs into Eq. (24). For example, in Ref. [1] a novel phase-transitionbased viscoelastic model, including the time factor for SMPs was developed which has a clearer

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physical significance. To define the phase transition phenomenon of SMPs, the model describes different constitutive structures for above and below transformation temperature separately. As

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the viscoelastic model is based on multiplicative thermoviscoelasticity, it can not only be used

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for different types of SMP materials, but also can be used to treat large strain problems. Note that many models are limited to small strain (within 10%). However, this model is also able to

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simulate the behavior of large strain case [1]. Also, a three-dimensional constitutive model for SMPs has been proposed that can simulate multi-axial and large deformation behavior (up to

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200% of strain) of SMPs [4]. To derive the constitutive equation, the total deformation gradient was decomposed multiplicatively into hyperelastic, viscoelastic, viscoplastic, and shape memory strains using Helmholtz free energy and the Clausius–Duhem inequality [4]. The CNT waviness is inherent to the manufacture process of CNT-reinforced nanocomposites [3,26]. Thus, the consideration of CNT waviness is an essential issue. Fig. 3

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displays a schematic of the RVE of the unidirectional wavy CNT-reinforced nanocomposites. The wavy CNT is assumed to be as a sinusoidal solid fiber [3,26] characterized by (27) ,

and

stand for the amplitude of CNT wave, the number of CNT waves and the

linear distance between CNT ends, respectively.

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where

represents waviness factor. As illustrated

in Fig. 3, the RVE is divided into very thin slices of thickness reinforce with straight CNT. The elastic stiffness tensor (

. Any slice is assumed to

) and CTE vector (

) of the

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unidirectional wavy CNT-reinforced nanocomposites can be estimated by averaging the effective properties of these slices over the length of the RVE (

). Hence, using the values of

and

along with the appropriate transformations, the effective properties at any point of any slice of the nanocomposite where the straight CNT is oriented at an angle

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(Fig. 3) can be determined. Therefore, the elastic stiffness tensor ( ̅

with the

) and CTE vector ( ̅

any point in the nanocomposite where the straight CNT is oriented at an angle

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(1) direction ) at

with the

(1) direction can be expressed as

(29)

is the transformation tensor available in Refs. [26,27]. Consequently, the values of

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where

(28)

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̅

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̅

and

are given by the following relations ∫ ̅

(30)

∫ ̅

(31)

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5. Nanocomposite system Armchair single-walled CNT (SWCNT) and SMP matrix are the constituents of the nanocomposite material considered in this work. The material of the SMP matrix is an epoxy

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resin [20]. The diameter of armchair (10,10) CNT is equal to 1.36 nm [26,28]. The mechanical properties of CNT are given in Table 1 [26]. It should be noted that the temperature-dependent of the mechanical properties of the CNT can be neglected [26]. Due to the fact that the CTEs of CNT is temperature-dependent, the variation of CTEs of CNT with the temperature deviation is

given as [26,29] {

) and transverse (

) CTEs of the CNT can be

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considered [26,29]. The longitudinal (

(32)

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The SMP obeys the thermomechanical constitutive law. Thus, the elastic modulus and CTE

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of SMP are taken from Eqs. (13) and (14), respectively. Poisson’s ratio ( ) and other material

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constants of SMP are given in Table 2 [20].

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6. Results and discussion 6.1.Validation of the model

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The validity of the presented micromechanical model is verified against that of the experimental study on the MWCNT-reinforced epoxy nanocomposite [30,31]. Shirasu et al. experimentally evaluated both the longitudinal elastic modulus (

) [30] and CTE (

) [31] of epoxy

nanocomposite reinforced with unidirectional MWCNTs. The material properties of the constituents of the nanocomposite are given in Table 3 [28,30,31]. The values of , ɳ,

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and

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are considered to be equal to 8 nm, 15.3, 0.05 and 2 [32-34], respectively. The average diameter of MWCNTs is considered to be equal to 39 nm [31]. The comparisons of longitudinal elastic modulus and CTE of the unidirectional CNT-reinforced epoxy nanocomposite obtained from the presented SUC micromechanical model and experiment [30,31] are shown in Fig. 4 a

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and b, respectively. Generally, the results reveal that both the elastic modulus and CTE of the CNT-reinforced epoxy nanocomposite predicted by the SUC model are quite close to the experiment [30,31] validating the SUC method utilized in this study. A linear increase of the longitudinal elastic modulus is observed with increasing the CNT volume fraction, as shown in

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Fig. 4a. Also, it can be found from Fig. 4b that by increasing the CNT volume fraction, the longitudinal CTE of the CNT-epoxy nanocomposite nonlinearly decreases. 6.2.Parametric study

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In this sub-section, some parametric studies are carried out to evaluate the effects of non-straight

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shape of the CNTs, the interphase region characteristics, diameter and volume fraction of the CNTs on the elastic and thermoelastic behaviors of armchair (10,10) CNT/SMP nanocomposites

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described in Section 4. In all analyses, the change of temperature is selected to be in the range of 273-358 K. Unless otherwise stated, the CNT volume fraction, waviness factor, the number of

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CNT waves, the adhesion exponent and the interphase thickness are considered to be 10%, 0.05,

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12, 40 and 0.34 nm [26,35], respectively. 6.2.1. Effect of non-straight shape of CNTs

The variation of elastic moduli and CTEs of the CNT/SMP nanocomposites with temperature is presented in Figs. 5 and 6, respectively. Also, the results are provided for both straight and wavy CNTs to illustrate the effect of non-straight shape of CNTs on the overall properties of the SMP nanocomposites. As shown in Fig. 5a, the geometry of the CNTs plays an important role in 15

.

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The value of

of the straight CNT-reinforced SMP nanocomposite is much higher than that of

the wavy CNT-reinforced SMP nanocomposite. So, the longitudinal elastic behavior of the nanocomposites can be significantly improved using the straight CNTs within the SMP matrix. For example, the magnitude of

for the SMP nanocomposite containing wavy and straight

outcomes in Fig. 5a, the change of CNT geometry to

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CNTs is about 21.5 GPa and 109.5 GPa corresponding to a 410% enhancement. According to the with temperature can be ignored. The contribution of the

is found to be very important, as clarified in Fig. 5b. The waviness of CNTs . It is observed that the value of

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leads to a significant enhancement in the value of

slightly decreases as the temperature increases. Fig. 5c demonstrates the value of

very

of the wavy

CNT-reinforced SMP nanocomposites is greater than that of the straight CNT-reinforced SMP nanocomposites. Moreover, the difference between the two sets of results of

of the CNT/SMP nanocomposites nonlinear

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as the temperature reduces. The value of

becomes larger

decreases with the increase of temperature. Fig. 6a reveals that the longitudinal CTE of the SMP

of the wavy CNT-reinforced SMP nanocomposites linearly increases with the rise of

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of

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nanocomposites is remarkably influenced by the CNT geometry. Based on the results, the value

temperature, whereas the change of

of the straight CNT-reinforced SMP nanocomposites with

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temperature can be ignored. It is concluded from Fig. 6a that the CTE of the nanocomposites in the axial direction of the CNT alignment can be greatly decreased by using the straight CNTs

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within the SMP matrix. According to the outcomes from Fig. 6b,

of SMP nanocomposite

containing straight or wavy CNTs linearly rises with increasing the temperature. Also, it can be clearly seen that the value of

of a straight CNT-reinforced SMP nanocomposite is much

higher than that of a wavy CNT-reinforced SMP nanocomposite. Consequently, in order to decrease in the value of

, it is necessary to apply the wavy CNTs within the SMP matrix. It is

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found from Fig. 6c that the

is not dependent on the CNT geometry. The value of

linearly

rises with the increase of temperature. The effects of waviness factor on the elastic and thermoelastic responses of CNT-SMP

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nanocomposite are depicted in Figs 7 and 8, respectively. The results are extracted for three different values of waviness factor, including 0.025, 0.05 and 0.1. According to the outcomes of Fig. 7a, the reduction of waviness factor leads to an improvement in the value of

of the CNT/SMP nanocomposite, as shown in Fig. 7b. The value of

reduces with the reduction of nanocomposite when

. Fig. 7c also includes the value of

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reverse trend is found for

. However, a

of the CNT/SMP

=0.01. The waviness factor affects the value of

0.025. The decrease of the waviness factor causes a reduction in the value of clearly reveal that with the reduction of the waviness factor, the value of . The value of

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an increment is observed for the value of

when

<

. Fig. 8 a and b

decreases, whereas

of the CNT/SMP nanocomposite is

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independent of the CNT waviness factor, as illustrated in Fig. 8c. The role of number of the CNT waves in the elastic and thermoelastic behaviors of

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CNT/SMP nanocomposite is examined in Figs. 9 and 10, respectively. The results are presented for three different values of

can be enhanced by decreasing the number of the CNT waves. However, it can be

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value of

including 6, 12 and 24. According to the results of Fig. 9a, the

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seen in Fig. 9b that the increase of the number of the CNT waves yields an improvement in the value of

. The contribution of the number of the CNT waves to

of the CNT/SMP

nanocomposite can be ignored, as indicated in Fig. 9c. Fig. 10a illustrates that the

tends to

decrease with the decrease of . However, it is seen from Fig. 10b that the

tends to increase

as the value of

from the number

reduces. Fig. 10c obviously proves the independency of the

of CNT waves. 17

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6.2.2. Effect of interphase region The effect of the interphase region on the elastic and thermoelastic responses of CNT/SMP nanocomposite is indicated in Figs. 11 and 12, respectively. The elastic moduli and CTEs are predicted in the presence and the absence of the CNT/SMP interphase region. Based on the

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results of Fig. 11, the elastic moduli of the SMP nanocomposites with the interphase are higher than those of the SMP nanocomposites without the interphase. It is observed that the interphase region very negligibly affects the values of

and

outcomes of Fig. 12c indicate that the value of

, as clarified in Fig. 12 a and b. The of the nanocomposite with CNT/SMP

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interphase is slightly lower than that of the nanocomposite without CNT/SMP interphase.

Figs. 13 and 14 illustrate the predicted elastic moduli and CTEs of the CNT/SMP nanocomposite, respectively, versus temperature for different interphase thicknesses, including

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= 0.17 nm, 0.34 nm and 0.68 nm. All three elastic moduli of the nanocomposite tend to an

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increment with the rise of the interphase thickness. For example, at =298 K, the values of are equal to 1.0157 GPa, 1.1116 GPa and 1.4422 GPa when = 0.17 nm, 0.34 nm and 0.68 nm,

values of

and

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respectively. It is found from Fig. 14a and b that the effect of the interphase thickness on the of the CNT/SMP nanocomposite can be ignored. A reduction in the

is

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observed with the increase of the interphase thickness, as shown in Fig. 14c. For instance, at = 298 K, the values of

are 116.2×10-6 1/K, 114.2×10-6 1/K and 110.3×10-6 1/K for = 0.17 nm,

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0.34 nm and 0.68 nm, respectively. Figs. 15 and 16 indicate the estimated elastic moduli and CTEs of the SMP nanocomposite

containing wavy CNTs, respectively, versus temperatures for different values of ɳ, including 1, 10, 20 and 40. It can be observed from Fig. 15 a and b that the values of

and

can be

improved as the value of ɳ decreases. As the value of adhesion exponent decreases, the elastic 18

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modulus of the interphase becomes closer to the elastic modulus of the CNT. Consequently, a higher elastic modulus of the interphase leads to a higher nanocomposite elastic modulus. On the other hands, if the interphase adhesion exponent becomes greater than 20, with increasing the value of ɳ, both of the

and

decrease up to a threshold values and then saturate. Increasing

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the value of adhesion exponent leads to the interphase elastic modulus to be closer to the polymer matrix elastic modulus. Thus, a lower interphase elastic modulus can decrease the effective elastic moduli of the SMP nanocomposites. According to the predicted results of Fig. 15c, the effect of ɳ on

of the CNT/SMP nanocomposite is very negligible. The results of Fig.

value of ɳ very slightly affects

and

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16 a and b specify that if the interphase adhesion exponent to be very low, the change in the of CNT/SMP nanocomposite. Moreover, the reduction

of the adhesion exponent of the interphase region can reduce the value of

, as illustrated in

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Fig. 16c. It may be attributed to the fact that the CTE of interphase becomes closer to that of CNT as the adhesion exponent decreases. Consequently, a lower CTE of interphase can decrease

ED

the effective CTE of SMP nanocomposites. Generally, according to Eqs. (17) and (18), the reduction of ɳ leads to the material properties of the interphase to be closer to those of the CNT

CE

nanocomposites.

PT

and therefore, an improvement would be observed in the effective properties of SMP

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6.2.3. Effect of CNT volume fraction The elastic and thermoelastic responses of SMP nanocomposite versus temperature for different volume fractions (VFs) of CNTs including 5%, 10% and 20% are presented in Figs. 17 and 18, respectively. The increment of the CNT volume fraction can improve the elastic behavior of SMP nanocomposites as clarified in Fig. 17a-c. Also, the results of Fig. 17c reveal the contribution of the volume fraction to the value of 19

becomes more important as the

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temperature decreases. It can be found from Fig. 18a-c that with the increase of the CNT volume fraction, the effective CTEs of the SMP nanocomposite considerably decrease. 6.2.4. Effect of CNT diameter

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The effect of the CNT diameter on the elastic and thermoelastic properties of the SMP nanocomposite is demonstrated in Figs. 19 and 20, respectively. The results are provided for three different values of the CNT diameter, including 0.68 nm, 1.36 nm and 2.72 nm. It can be concluded from the outcomes of Fig. 19a-c that the elastic moduli of the SMP nanocomposite is

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enhance with the decrease in the CNT diameter. Moreover, the enhancement in the value of

found to be more significant with the reduction of temperature as shown in Fig. 19c. Based on the obtained results from Fig. 20 a and b, the change of the CNT diameter does not affect both and

of the SMP nanocomposite. However, the decrease of the CNT diameter

can slightly reduce the value of

as indicated in Fig. 20c.

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6.2.5. Effect of CNT aspect ratio

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the values of

The variation of elastic moduli and CTEs of the straight CNT-reinforced SMP nanocomposite

PT

with the CNT aspect ratio (λ) is illustrated in Figs. 21 and 22, respectively. The results of these

CE

figures are extracted for two different CNT volume fractions including 5% and 10%. Fig. 21 a and b also include the influence of elastic modulus of the matrix (

) on the overall elastic

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behavior of CNT/SMP nanocomposites. Moreover, a sensitivity study is made to examine the effect of the matrix CTE (

) on the overall thermal expansion response of CNT/SMP

nanocomposite as clarified in Fig. 22 a and b. It is found from Fig. 21a that the contribution of three factors, including aspect ratio and volume fraction of CNTs as well as elastic modulus of SMP matrix to the overall elastic behavior of the CNT/SMP nanocomposite is highly important. With the increase of the CNT aspect ratio, the longitudinal elastic modulus increases up to the 20

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threshold value mainly depending on the CNT volume fraction. On the other words, a critical aspect ratio is observed after which the further increase of the CNT aspect ratio does not change the longitudinal elastic modulus. At a certain volume fraction, this critical aspect ratio decreases by increasing the SMP matrix elastic modulus. For instance, at 5% volume fraction, the threshold =622.32 MPa and 5×622.32 MPa is about 10000 and 1000,

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value for the nanocomposite with

respectively. Note that at a certain volume fraction, the discrepancy between the longitudinal elastic moduli with different matrix elastic moduli becomes negligible by increasing in the CNT aspect ratio. It is clearly observed from Fig. 21b that among the CNT aspect ratio, volume

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fraction and elastic modulus of matrix, the effect of SMP matrix elastic modulus on the overall transverse elastic modulus of CNT/SMP nanocomposite is the most important. The transverse elastic modulus of the nanocomposite can be considerably improved with the increase of matrix

M

elastic modulus. The role of the CNT aspect ratio in the effective transverse elastic modulus of SMP nanocomposite can be ignored, as demonstrated in Fig. 21b. The results displayed in Fig.

ED

22a reveal that all three factors, including aspect ratio and volume fraction of CNTs as well as CTE of SMP matrix have the critical influence on the thermal expansion properties along the

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longitudinal direction. The rise of the CNT aspect ratio yields a reduction in the value of the

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longitudinal CTE of the SMP nanocomposite up to a threshold value. When λ>1000, ignoring the influence of the CNT aspect ratio on the longitudinal thermal properties is a reasonable

AC

assumption. Fig. 22b reveals that the CTE of the SMP matrix contains the maximum effect on the CNT/SMP nanocomposite transverse CTE as compared with those of the CNT aspect ratio and volume fraction.

7. Conclusion 21

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In this paper, the elastic and thermoelastic behaviors of CNT/SMP nanocomposites were comprehensively studied using an analytical unit cell-based micromechanical method. The RVE of the model consisted of three phases including CNT, SMP matrix and interphase region formed due to non-bonded interactions between the CNT and SMP. Moreover, the non-straight shape of

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CNTs was taken into account in the analysis of the nanocomposites. The influences of several important factors such as volume fraction, diameter, aspect ratio, waviness factor and the number of waves of CNT, thickness and adhesion exponent of the interphase and temperature on the overall elastic moduli and CTEs of the CNT/SMP nanocomposite were extensively explored.

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The important conclusions can be summarized below.

1) Employing the straight CNTs within the SMP matrix remarkably enhances the longitudinal elastic modulus of the nanocomposite, whereas the CNT waviness rises the

M

transverse elastic moduli. Also, the use of straight CNTs within the SMP matrix

ED

considerably decreases the longitudinal CTE. While the non-straight shape of CNT did not affect the value of

. The CNT waviness could reduce the value of

of the

PT

CNT/SMP nanocomposite.

2) Generally, the interphase negligibly affects the thermal expansion behavior of CNT/SMP

CE

nanocomposites. Also, the predicted elastic moduli of the CNT/SMP nanocomposite in the presence of the interphase were higher than those of CNT/SMP nanocomposite in the

AC

absence of the interphase.

3) The reduction of the values of

and

with temperature was observed to be very slight.

However, with increasing the temperature, the value of

nonlinearly decreases. The

effective CTEs of the wavy CNT-reinforced SMP nanocomposite linearly increase with rising temperature. 22

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4) Increasing CNT volume fraction leads to a growth in the elastic moduli and a drop in the CTEs of the CNT/SMP nanocomposite. 5) The reduction of CNT diameter can improve the elastic behavior of the CNT/SMP nanocomposite. Whereas, the contribution of CNT diameter to the thermal expansion

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response is very negligible.

6) Generally, the longitudinal effective properties of SMP nanocomposites containing straight CNT could be improved by increasing the CNT aspect ratio. Whereas, the transverse effective properties were not perceptibly influenced by the variation of the

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CNT aspect ratio. However, the SMP matrix properties significantly affects the effective properties along the transverse direction.

7) Good agreement was observed between the results of the presented micromechanical

M

model and experiment.

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Declaration of conflicting interests

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The author(s) declared no potential conflicts of interest with respect to the research, authorship,

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and/or publication of this article.

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Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article

23

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References Li, Y., He, Y., & Liu, Z. (2017). A viscoelastic constitutive model for shape memory polymers based on multiplicative decompositions of the deformation gradient. International Journal of Plasticity, 91, 300-317.

[2]

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[3]

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[4]

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[10] Li, G., & John, M. (2008). A self-healing smart syntactic foam under multiple impacts. Composites Science and Technology, 68(15), 3337-3343. [11] Li, H., Zhong, J., Meng, J., & Xian, G. (2013). The reinforcement efficiency of carbon nanotubes/shape memory polymer nanocomposites. Composites Part B: Engineering, 44(1), 508-516. [12] Ni, Q. Q., Zhang, C. S., Fu, Y., Dai, G., & Kimura, T. (2007). Shape memory effect and mechanical properties of carbon nanotube/shape memory polymer nanocomposites. Composite Structures, 81(2), 176-184.

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[13] Yang, Q. S., He, X. Q., Liu, X., Leng, F. F., & Mai, Y. W. (2012). The effective properties and local aggregation effect of CNT/SMP composites. Composites Part B: Engineering, 43(1), 33-38. [14] Herasati, S., Zhang, L. C., & Ruan, H. H. (2014). A new method for characterizing the interphase regions of carbon nanotube composites. International Journal of Solids and Structures, 51(9), 1781-1791.

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[15] Seidel, G. D., & Lagoudas, D. C. (2006). Micromechanical analysis of the effective elastic properties of carbon nanotube reinforced composites. Mechanics of Materials, 38(8), 884907. [16] Rouhi, S., Alizadeh, Y., & Ansari, R. (2016). Molecular dynamics simulations of the interfacial characteristics of polypropylene/single-walled carbon nanotubes. Proceedings of the Institution of Mechanical Engineers, Part L: Journal of Materials: Design and Applications, 230(1), 190-205.

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[17] Alian, A. R., Kundalwal, S. I., & Meguid, S. A. (2015). Interfacial and mechanical properties of epoxy nanocomposites using different multiscale modeling schemes. Composite Structures, 131, 545-555. [18] Tsai, J. L., Tzeng, S. H., & Chiu, Y. T. (2010). Characterizing elastic properties of carbon nanotubes/polyimide nanocomposites using multi-scale simulation. Composites Part B: Engineering, 41(1), 106-115.

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[19] Wei, C. (2006). Adhesion and reinforcement in carbon nanotube polymer composite. Applied Physics Letters, 88(9), 093108.

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[20] Liu, Y., Gall, K., Dunn, M. L., Greenberg, A. R., & Diani, J. (2006). Thermomechanics of shape memory polymers: uniaxial experiments and constitutive modeling. International Journal of Plasticity, 22(2), 279-313.

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[21] Hassanzadeh-Aghdam, M. K., Ansari, R., & Darvizeh, A. (2018). Micromechanical analysis of carbon nanotube-coated fiber-reinforced hybrid composites. International Journal of Engineering Science, 130, 215-229.

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[22] Hassanzadeh-Aghdam, M. K., Mahmoodi, M. J., & Ansari, R. (2018). Micromechanicsbased characterization of mechanical properties of fuzzy fiber-reinforced composites containing carbon nanotubes. Mechanics of Materials, 118, 31-43.

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[23] Kundalwal, S. I., & Meguid, S. A. (2015). Micromechanics modelling of the effective thermoelastic response of nano-tailored composites. European Journal of Mechanics A/Solids 53 (2015) 241-253. [24] Boutaleb, S., Zaïri, F., Mesbah, A., Naït-Abdelaziz, M., Gloaguen, J. M., Boukharouba, T., & Lefebvre, J. M. (2009). Micromechanics-based modelling of stiffness and yield stress for silica/polymer nanocomposites. International Journal of Solids and Structures, 46(7), 17161726. [25] Aghdam, M. M., & Dezhsetan, A. (2005). Micromechanics based analysis of randomly distributed fiber reinforced composites using simplified unit cell model. Composite structures, 71(3), 327-332. 25

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[26] Kundalwal, S. I., & Ray, M. C. (2014). Effect of carbon nanotube waviness on the effective thermoelastic properties of a novel continuous fuzzy fiber reinforced composite. Composites Part B: Engineering, 57, 199-209. [27] Kundalwal, S. I., & Ray, M. C. (2012). Effective properties of a novel composite reinforced with short carbon fibers and radially aligned carbon nanotubes. Mechanics of Materials, 53, 47-60.

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[28] Shen, L., & Li, J. (2004). Transversely isotropic elastic properties of single-walled carbon nanotubes. Physical Review B, 69(4), 045414. [29] Kwon, Y. K., Berber, S., & Tománek, D. (2004). Thermal contraction of carbon fullerenes and nanotubes. Physical review letters, 92(1), 015901.

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[30] Shirasu, K., Nakamura, A., Yamamoto, G., Ogasawara, T., Shimamura, Y., Inoue, Y., & Hashida, T. (2017). Potential use of CNTs for production of zero thermal expansion coefficient composite materials: An experimental evaluation of axial thermal expansion coefficient of CNTs using a combination of thermal expansion and uniaxial tensile tests. Composites Part A: Applied Science and Manufacturing, 95, 152-160. [31] Shirasu, K., Yamamoto, G., Tamaki, I., Ogasawara, T., Shimamura, Y., Inoue, Y., & Hashida, T. (2015). Negative axial thermal expansion coefficient of carbon nanotubes: Experimental determination based on measurements of coefficient of thermal expansion for aligned carbon nanotube reinforced epoxy composites. Carbon, 95, 904-909.

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[32] Zare, Y. (2015). Effects of interphase on tensile strength of polymer/CNT nanocomposites by Kelly–Tyson theory. Mechanics of Materials, 85, 1-6.

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[33] Yanase, K., Moriyama, S., & Ju, J. W. (2013). Effects of CNT waviness on the effective elastic responses of CNT-reinforced polymer composites. Acta Mechanica, 224(7), 1351.

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[34] Fisher, F. T., Bradshaw, R. D., & Brinson, L. C. (2003). Fiber waviness in nanotubereinforced polymer composites—I: Modulus predictions using effective nanotube properties. Composites Science and Technology, 63(11), 1689-1703.

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[35] Jiang, L. Y., Huang, Y., Jiang, H., Ravichandran, G., Gao, H., Hwang, K. C., & Liu, B. (2006). A cohesive law for carbon nanotube/polymer interfaces based on the van der Waals force. Journal of the Mechanics and Physics of Solids, 54(11), 2436-2452.

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Figures: y dh Lh dk

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d1 br

bj

ijk

b1 z

ai

a1 Lc

x

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Lr

ac

M

Fig. 1. RVE of the SUC model

a1= X

a2= t

a3= d/2

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y

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1

Polymer

CNT

CE

b3=X

Lc

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Lr

x

b2=t

b1=d/2

z

Interphase r

Fig. 2. RVE of the SUC model for a unidirectional straight CNT-reinforced nanocomposite

27

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2 θ

y (2)

1

𝑑𝑧

𝐴

CNT

CR IP T

z (1) x (3) 𝐿𝑛

M ED

40

0 0

5

PT

20

10

Experiment [31] Present model

Present model

60

E1 (GPa)

(b) 65

Experiment [30]

15

20

25

50

α1 (10-6/K)

(a) 80

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Fig. 3. A schematic sketch of a unidirectional wavy CNT-reinforced nanocomposite

35

20

Volume fraction (%)

5

30

-10

Volume fraction (%)

CE

0

5

10

15

20

25

30

AC

Fig. 4. Comparison of (a) elastic modulus and (b) CTE of CNT/epoxy nanocomposite obtained from the presented SUC model and experiment [30,31]

28

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(a)

(b)

110

30 25

Straight CNT

90

Straight CNT

70

50

Wavy CNT

15 10

30

5

10

0 285

297

309

321

333

345

T (K)

(c)

2

1.6

273

285

297

309

321

333

345

357

T (K)

Straight CNT Wavy CNT

M

1.2

0.8

ED

E3 (GPa)

357

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273

CR IP T

E2 (GPa)

E1 (GPa)

20 Wavy CNT

0.4

0

PT

273

285

297

309

321

333

345

Fig. 5. Effect of non-straight shape of the CNTs on the (a) CNT/SMP nanocomposites

AC

CE

357

T (K)

29

, (b)

and (c)

of the

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115

(a)

Straight CNT

180

90

Wavy CNT

Wavy CNT

40

140

100

60

15

-10

20 285

297

309

321

333

345

T (K)

(c)

273

285

297

309

321

333

345

357

T (K)

Straight CNT

190

Wavy CNT

150

M

α3 (10-6/K)

357

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273

CR IP T

65

α2 (10-6/K)

α1 (10-6/K)

(b)

Straight CNT

ED

110

70

PT

273

285

297

309

321

333

345

Fig. 6. Effect of non-straight shape of the CNTs on the (a) CNT/SMP nanocomposites

AC

CE

357

T (K)

30

, (b)

and (c)

of the

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(b)75

(a) 75 A/Ln=0.025

60

60

A/Ln=0.1

30

15

A/Ln=0.025 45 A/Ln=0.05 A/Ln=0.1

30

15

0

0 273

285

297

309

321

333

345

357

CR IP T

45

E2 (GPa)

E1 (GPa)

A/Ln=0.05

273

285

297

309

321

333

345

357

T (K)

(c)

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T (K) 2

A/Ln=0.01

1.6

E3 (GPa)

A/Ln=0.025

1.2

A/Ln=0.05 A/Ln=0.1

M

0.8

ED

0.4

0

PT

273

285

297

309

321

333

Fig. 7. Effect of waviness factor of the CNTs on the (a) nanocomposites

AC

CE

345

357

T (K)

31

, (b)

and (c)

of the CNT/SMP

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(a) 200

(b)200

A/Ln=0.025

A/Ln=0.025

A/Ln=0.05

160

A/Ln=0.05

A/Ln=0.1

120

80

120

40

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A/Ln=0.1

α2 (10-6/K)

α1 (10-6/K)

160

80

40

0

0 273

285

297

309

321

333

345

357

273

285

297

309

321

333

345

357

T (K)

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T (K)

(c) 250

A/Ln=0.025 A/Ln=0.05 A/Ln=0.1

170

130

M

α3 (10-6/K)

210

ED

90

50

PT

273

285

297

309

321

Fig. 8. Effect of waviness factor of the CNTs on the (a) nanocomposites

AC

CE

333

345

357

T (K)

32

, (b)

and (c)

of the CNT/SMP

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(a)

(b) 55

55

n=6

n=6

n=12

45

45

n=12

35

n=24 35

25

25

15

15

5

5 273

285

297

309

321

333

345

357

273

285

(c)

297

309

321

333

345

357

T (K)

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T (K)

CR IP T

E2 (GPa)

E1 (GPa)

n=24

2

1.6

n=6

n=24

1.2

0.8

M

E3 (GPa)

n=12

ED

0.4

0

PT

273

285

297

309

321

Fig. 9. Effect of the number of CNT waves on the (a) nanocomposites

AC

CE

333

345

357

T (K)

33

, (b)

and (c)

of the CNT/SMP

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(a) 150

(b)150

n=6

n=6

n=12

n=12

120

90

60

30

n=24 90

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n=24

α2 (10-6/K)

α1 (10-6/K)

120

60

30

0

0 273

285

297

309

321

333

345

357

273

285

297

309

321

333

345

357

T (K)

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T (K)

(c) 210

n=6

n=12 n=24

150

120

M

α3 (10-6/K)

180

ED

90

60

PT

273

285

297

309

321

Fig. 10. Effect of the number of CNT waves on the (a) nanocomposites

AC

CE

333

345

357

T (K)

34

, (b)

and (c)

of the CNT/SMP

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24 Without interphase With interphase

28 Without interphase With interphase

27

E2 (GPa)

23

E1 (GPa)

(b)

22

21

26

25

20

24 273

285

297

309

321

333

345

357

273

297

309

321

333

345

357

T (K)

(c)

2

Without interphase

1.6

E3 (GPa)

285

AN US

T (K)

CR IP T

(a)

With interphase

1.2

M

0.8

ED

0.4

0

PT

273

285

297

309

321

Fig. 11. Effect of the interphase region on the (a) nanocomposites

AC

CE

333

345

357

T (K)

35

, (b)

and (c)

of the CNT/SMP

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(a)

(b)

Without interphase

105

Without interphase 90

With interphase

With interphase

90

75

60

60

45

45

30

30 273

285

297

309

321

333

345

357

273

297

309

321

333

345

357

T (K)

220

(c)

Without interphase

190

With interphase

160

130

M

α3 (10-6/K)

285

AN US

T (K)

CR IP T

α2 (10-6/K)

α1 (10-6/K)

75

ED

100

70

PT

273

285

297

309

321

Fig. 12. Effect of the interphase region on the (a) nanocomposites

AC

CE

333

345

357

T (K)

36

, (b)

and (c)

of the CNT/SMP

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(a)

30

(b)

30

t=0.17 nm t=0.34 nm

27

24

24

CR IP T

t=0.68 nm

E2 (GPa)

E1 (GPa)

27

t=0.17 nm

21

21

t=0.34 nm t=0.68 nm

18

18 273

285

297

309

321

333

345

357

273

285

309

321

333

345

357

T (K)

(c)

2

1.6

t=0.17 nm t=0.34 nm

1.2

t=0.68 nm

0.8

M

E3 (GPa)

297

AN US

T (K)

ED

0.4

0

PT

273

285

297

309

321

Fig. 13. Effect of the interphase thickness on the (a) nanocomposites

AC

CE

333

345

357

T (K)

37

, (b)

and (c)

of the CNT/SMP

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t=0.17 nm

100

t=0.17 nm

100

t=0.34 nm t=0.68 nm

α2 (10-6/K)

α1 (10-6/K)

(b) 120

80

60

40

t=0.34 nm t=0.68 nm

80

CR IP T

(a) 120

60

40

20

20 273

285

297

309

321

333

345

357

273

285

(c)

309

321

333

345

357

T (K)

210

t=0.17 nm t=0.34 nm

180

t=0.68 nm

150

120

M

α3 (10-6/K)

297

AN US

T (K)

ED

90

60

PT

273

285

297

309

321

Fig. 14. Effect of the interphase thickness on the (a) nanocomposites

AC

CE

333

345

357

T (K)

38

, (b)

and (c)

of the CNT/SMP

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(b) 40

40

32

32

24 ɳ=1

16

24 ɳ=1

16

ɳ=10

ɳ=10 ɳ=20

8

ɳ=20

8

ɳ=40

ɳ=40 0

0 273

285

297

309

321

CR IP T

E2 (GPa)

E1 (GPa)

(a)

333

345

357

273

285

297

309

321

333

345

357

T (K)

(c)

AN US

T (K) 2

1.6

E3 (GPa)

ɳ=1

1.2

ɳ=10 ɳ=20

M

0.8

ED

0.4

ɳ=40

0

PT

273

285

297

309

321

333

345

Fig. 15. Effect of adhesion exponent of the interphase on the (a) CNT/SMP nanocomposites

AC

CE

357

T (K)

39

, (b)

and (c)

of the

ACCEPTED MANUSCRIPT

(a) 120

(b) 120

ɳ=1 ɳ=10

100

ɳ=1 ɳ=10

100

ɳ=20

60

40

ɳ=40

80

CR IP T

ɳ=40

80

α2 (10-6/K)

α1 (10-6/K)

ɳ=20

60

40

20

20 273

285

297

309

321

333

345

357

273

285

297

309

321

333

345

357

T (K)

AN US

T (K)

(c) 210

ɳ=1

ɳ=10

180

ɳ=40

150

120

M

α3 (10-6/K)

ɳ=20

ED

90

60

PT

273

285

297

309

321

333

345

Fig. 16. Effect of adhesion exponent of the interphase on the (a) CNT/SMP nanocomposites

AC

CE

357

T (K)

40

, (b)

and (c)

of the

ACCEPTED MANUSCRIPT

(a) 55

(b) 55

45

45

VF=5% VF=10%

VF=10% VF=20%

25

15

VF=20%

35

25

15

5

5 273

285

297

309

321

333

345

357

273

285

297

309

321

333

345

357

T (K)

AN US

T (K)

CR IP T

E2 (GPa)

E1 (GPa)

VF=5% 35

(c) 2

VF=5%

1.6

VF=10%

E3 (GPa)

VF=20%

1.2

M

0.8

ED

0.4

0

PT

273

285

297

309

321

Fig. 17. Effect of CNT volume fraction on the (a) nanocomposites

AC

CE

333

345

357

T (K)

41

, (b)

and (c)

of the CNT/SMP

ACCEPTED MANUSCRIPT

(a) 120

(b)120 VF=5%

VF=5% 100

VF=20% 80

60

40

VF=10% VF=20%

80

60

40

20

20 273

285

297

309

321

333

345

357

273

285

297

309

321

333

345

357

T (K)

AN US

T (K)

CR IP T

VF=10%

α2 (10-6/K)

α1 (10-6/K)

100

(c) 250

VF=5%

VF=10% VF=20%

170

130

M

α3 (10-6/K)

210

ED

90

50

PT

273

285

297

309

321

Fig. 18. Effect of CNT volume fraction on the (a) nanocomposites

AC

CE

333

345

357

T (K)

42

, (b)

and (c)

of the CNT/SMP

ACCEPTED MANUSCRIPT

(a)

30

(b)

30

d=0.68 nm d=1.36 nm

27

24

21

24 d=0.68 nm

CR IP T

d=2.72 nm

E2 (GPa)

E1 (GPa)

27

d=1.36 nm

21

d=2.72 nm

18

18 273

285

297

309

321

333

345

357

273

285

309

321

333

345

T (K)

(c)

2

1.6

d=0.68 nm d=1.36 nm

1.2

d=2.72 nm

0.8

M

E3 (GPa)

297

AN US

T (K)

ED

0.4

0

PT

273

285

297

309

Fig. 19. Effect of CNT diameter on the (a)

333

, (b)

nanocomposites

AC

CE

321

345

357

T (K)

43

and (c)

of the CNT/SMP

357

ACCEPTED MANUSCRIPT

d=0.68 nm

100

d=0.68 nm

100

d=1.36 nm d=2.72 nm

α2 (10-6/K)

α1 (10-6/K)

(b) 120

80

60

40

d=1.36 nm d=2.72 nm

80

60

40

20

20 273

285

297

309

321

333

345

357

273

(c)

297

309

321

333

345

T (K)

210

d=0.68 nm d=1.36 nm

180

d=2.72 nm

150

120

M

α3 (10-6/K)

285

AN US

T (K)

CR IP T

(a) 120

ED

90

60

PT

273

285

297

309

Fig. 20. Effect of CNT diameter on the (a)

333

, (b)

nanocomposites

AC

CE

321

345

357

T (K)

44

and (c)

of the CNT/SMP

357

ACCEPTED MANUSCRIPT

VF=10%, Em=622.32 MPa (from Eq. 13)

(a)

150

(b)

VF=5%, Em=622.32 MPa (from Eq. 13) 3.2

VF=10%, Em=5×622.32 MPa VF=5%, Em=5×622.32 MPa

E2 or E3 (GPa)

VF=10%, Em=622.32 MPa (from Eq. 13)

90

60

2.4

VF=5%, Em=622.32 MPa (from Eq. 13) VF=10%, Em=5×622.32 MPa

1.6

VF=5%, Em=5×622.32 MPa

0.8

30

0

0 10

100

1000

10000

100000

CR IP T

120

E1 (GPa)

4

10

100

1000

10000

100000

λ

Fig. 21. Variation of (a)

and (b)

AN US

λ

(or

) of the straight CNT-reinforced SMP

nanocomposites with CNT aspect ratio at =298 K

VF=10%, αm=107.2×10^-6 1/K (from Eq. 14)

(a) 60

VF=5%, αm=10.72×10^-6 1/K

24

PT

12

λ

-12

CE

0

AC

10

100

120 100

α2 or α3 (10-6/K)

ED

36

M

VF=10%, αm=10.72×10^-6 1/K

48

α1 (10-6/K)

(b)

VF=5%, αm=107.2×10^-6 1/K (from Eq. 14)

80 VF=10%, αm=107.2×10^-6 1/K (from Eq. 14) VF=5%, αm=107.2×10^-6 1/K (from Eq. 14)

60

VF=10%, αm=10.72×10^-6 1/K

40

VF=5%, αm=10.72×10^-6 1/K

20 0

1000

Fig. 22. Variation of (a)

10000

100000

10

100

1000

10000

100000

λ

and (b)

(or

) of the straight CNT-reinforced SMP

nanocomposites with CNT aspect ratio at =298 K

45

ACCEPTED MANUSCRIPT

Tables:

(GPa) 1060 , and

0.162

and

(GPa)

(GPa)

64

17

CR IP T

Table 1. Mechanical properties of (10,10) CNT [26]

are the elastic modulus, shear modulus and Poisson’s ratio, respectively. Subscribes

stand for the longitudinal and transverse directions, respectively.

(K-4)

(MPa)

358

813

(MPa/K) 2.64×10-2

4

0.3

M

2.76×10-5

(K)

AN US

Table 2. Poisson’s ratio and material constants of the SMP [20]

MWCNT

(GPa)

240

50

0.162

0.47

-12

16

2.5

2.5

0.34

0.34

70

70

AC

CE

Epoxy

(GPa)

PT

Material

ED

Table 3. Material properties of the MWCNT and epoxy matrix [28,30,31]

46

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

AN US

CR IP T

GRAPHICAL ABSTRACT

47