Effect of CNTs debonding on mode I fracture toughness of polymeric nanocomposites

    Effect of CNTs debonding on mode I fracture toughness of polymeric nanocomposites M.M. Shokrieh, A. Zeinedini PII: DOI: Reference:

S0264-1275(16)30432-4 doi: 10.1016/j.matdes.2016.03.134 JMADE 1612

To appear in: Received date: Revised date: Accepted date:

2 February 2016 15 March 2016 29 March 2016

Please cite this article as: M.M. Shokrieh, A. Zeinedini, Effect of CNTs debonding on mode I fracture toughness of polymeric nanocomposites, (2016), doi: 10.1016/j.matdes.2016.03.134

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ACCEPTED MANUSCRIPT Effect of CNTs debonding on mode I fracture toughness of polymeric nanocomposites

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M. M. Shokrieh1,*, A. Zeinedini2 1 Professor, 2Ph.D. Student

*

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[email protected]

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Composites Research Laboratory, Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, 16846-13114, Iran

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Abstract

This article deals with the effect of CNTs debonding from the surrounding matrix on the fracture

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toughness of carbon nanotubes/epoxy nanocomposites. Hence, a multiscale modeling of fracture toughness improvement of nanocomposites due to CNTs debonding was developed. This energy dissipation mechanism takes place at a zone around the crack tip. A representative volume

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element containing a carbon nanotube, its surrounding resin and the interphase was chosen in

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order to develop the multiscale model. Using experimental data available in the literature, the influence of several parameters such as weight fraction of the CNT, the interphase thickness and

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Young’s modulus were studied by the present model. It was concluded that the interphase characteristics as well as the CNT weight fraction strictly affect the fracture toughness enhancement caused by the debonding mechanism and the debonding stress around the CNT. In addition, the influence of CNTs debonding on the fracture toughness of nanocomposites was investigated.

Keywords: Carbon nanotubes; debonding; interphase; fracture toughness; nanocomposites; multiscale modeling.

1. Introduction Nowadays, improvement of mechanical properties of polymers has been received lots of interest by many researchers. There are many approaches to enhance the fracture toughness of polymers [1-3]. Among these approaches, adding nanoparticles to polymers was found out as an effective method [4-7]. The fracture toughness enhancement of nanoparticles/polymer nanocomposites is related to different dissipation energy mechanisms [8]. Ritchie [9] postulated that the toughening

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ACCEPTED MANUSCRIPT mechanisms could be categorized into the zone shielding and contact shielding mechanisms. The zone shielding mechanisms are those toughening ones that happen at the neighborhood of the nanocomposites crack tip. Among zone shielding mechanisms, nanoparticle debonding and

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subsequent plastic void growth is the most important mechanisms happen around the crack tip [8, 10]. The contact shielding toughening mechanisms occur when the crack faces the fillers.

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Nanofillers pull-out and rupture are the most important of this kind of mechanism [6].

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Researchers have proposed many models to study the effect of spherical nanoparticles on the fracture toughness of nanocomposites. Williams [10] invoked that plastic void growth taking

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place at the nano or micro-scale is the dominant energy absorbing mechanism. He expressed that the plastic properties of the pure resin controlled the plastic growth of created voids. Douglass et

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al. [11] studied the effect of adding rubber particles on polymer fracture energy. Quaresimin et al. [12] suggested multiscale models to investigate the fracture toughness of reinforced polymer nanocomposites with spherical nanoparticles. Their models are based on different toughening

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mechanisms which mainly occur in nanocomposites. Hsieh et al. [13] believed that the increase

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in the fracture toughness of polymer by adding spherical silica nanoparticles is due to the shear bond phenomenon between nanoparticles and the void growth. They validated their model for

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nanocomposites with nanoparticles volume fraction of more than 2.5%. Adachi et al. [14] investigated effects of the particle size and volume fraction on the mode I fracture toughness of epoxy composites filled with spherical silica particles using experiments and an analytical model.

Carbon nanotubes (CNTs) considerably increase the fracture toughness of polymers. There are many experimental researches on the effect of CNTs on the fracture toughness of polymers [4-7, 15-17]. Furthermore, theoretical methods as well as molecular dynamic method are other approaches to investigate this effect. However, there are fewer studies conducted in this issue by molecular dynamic approaches [17]. Mirjalili et al. [18] based on the elastic-plastic fracture mechanics developed a model to study the effect of aligned and randomly oriented CNTs on the fracture toughness of polymers under opening loading. The aim of the present paper is to develop a model to simulate the fracture toughness improvement due to the CNTs debonding from the surrounding matrix around the crack tip under mode I fracture loading. To achieve this purpose, a multiscale modeling was presented based on the relations between the stress fields at the macro, micro and nano-scales. A representative

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ACCEPTED MANUSCRIPT volume element including a CNT, a pristine polymer phase and an interphase between the CNT and the pure resin was considered. Finally, the fracture toughness enhancement normalized by the neat resin fracture toughness was expressed as a function of some parameters, such as

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mechanical properties of three phases, the CNTs volume fraction and the interphase thickness. Also, effects of the interphase thickness and its Young’s modulus on the fracture toughness

2. Relation between macro, micro and nano-scales

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increase through the energy dissipation by the CNT debonding were evaluated.

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2.1. Mean mechanical properties and stiffness tensor

In micromechanical field, constitutive equations for fiber, interphase and resin are stated as

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below [19]:

and

as well as

are stiffness tensors of fibers, interphase and matrix. Moreover, ,

and

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and

,

respectively. When a load is applied on a composite material, the stress field

and its

at a point are inhomogeneous in the micro scale. The mean stress

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corresponding strain filed

,

are fiber, interphase and matrix stresses and strains,

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where

(1)

in a typical mean volume V is described as [19]: (2)

Similarly, the mean strain is described in a similar way. The relationship between the mean strain and the mean stress can be correlated by: (3) Based on the concept suggested by Hill [20], there is a certain proportion between the fiber mean stress and the corresponding composite mean stress as: (4) where H is the global stress concentration tensor.

2.2. Dilute Eshelby model This model is introduced for dilute composites with low fiber volume fraction. The applied stress in the microscale is equal to the one applied in the macroscale [21]:

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ACCEPTED MANUSCRIPT (5) where

is the applied stress at infinity. By a combination of Eqs. (4) and (5), the relation and stress in the macro scale was found as:

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between stress in the nanoscale

(6)

In order to correlate a relation between the stress fields at the nano and macro-scales, the first

2.3. Description of representative volume element

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step is to select a representative volume element (RVE) explained in the next sub-section.

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Liu and Chen [22, 23] considered some RVEs with different cross-sections to study the effective mechanical properties of nanocomposites reinforced by CNTs. They proposed three nanoscale

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RVEs, i.e. cylindrical, square and hexagonal. They expressed that the cylindrical RVE can be used to study the CNTs that have different radii. They suggested that the square and hexagonal RVEs can be applied when the CNTs are arranged evenly in a square and in a hexagonal array,

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respectively. Furthermore, when the CNTs are relatively long with a large aspect ratio, a

without any curvature.

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continuous cylindrical RVE can be used [22]. In this RVE, CNTs are assumed solid and straight

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The properties of nanocomposites reinforced with CNTs are highly affected by the interphase. Barber et al. [24], McCarthy et al. [25], Lourie et al. [26], Jin et al. [27] and Zhu et al. [28] using different approaches observed a strong bond between CNTs and the resin. There are few evidences on confirming the weak bond between CNTs and polymers. Ajayan et al. [29] added CNTs into epoxy resin and showed that CNTs easily can be detached from the resin, instead of being ripped apart. This confirms the fact that there is weak bond between CNTs and polymers. It is worth mentioning that, Barber et al. [24] believed that the reason for weak interphase reported by Ajayan et al. [29] is the inhomogeneous dispersion of CNTs in polymer. Furthermore, the lack of post cure is one of the reasons for observing a decrease in nanocomposites properties in comparison with the resin properties. Kundalwal and Meguid [30], Shokrieh and Rafiee [31], and Seidel and Lagoudas [32] considered an equivalent solid continuum as an interphase material which characterizes the van der Waals interactions to determine the properties of CNTs/Polymer nanocomposites.

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ACCEPTED MANUSCRIPT In the present research, a RVE containing a CNT surrounded by the interphase as a solid material and the neat resin was considered in order to estimate the fracture toughness enhancement due to

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CNTs debonding mechanism.

3. Description of fracture toughness enhancement

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In this section, a new model was proposed to investigate the effect of CNTs debonding at the

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debonding process zone (DPZ) on the fracture toughness of the CNTs/polymer nanocomposites. The fracture toughness of nanocomposites

as [12, 33]:

(7)

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toughening mechanisms

and the fracture toughness enhancement caused by different

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toughness of the pure epoxy

can be assumed as a summation of the fracture

In this study, fracture toughness improvement through CNTs debonding

is determined. It

must be noticed that a uniform CNTs distribution into the polymer is assumed. In addition, the

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D

contact between carbon nanotubes and agglomeration are neglected in the present model.

y

Macroscopic crack

DPZ

Micro-scale x

A

A

CNT Matrix Interphase Section A–A

Macro-scale

RVE

Nano-scale Fig. 1. Description of different scales

In order to determine the fracture toughness improvements

due to CNTs debonding from the

surrounding components, according to [33-37], the following relation was used:

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ACCEPTED MANUSCRIPT

(8) where

is the DBR radius at

, see Fig. 1, and

is the strain energy density in the

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RVE caused by toughening mechanism which can be obtained by:

is the strain energy through the debonding phenomenon and

, rn and

are the

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where

(9)

volume fraction, the mean outer radius and the mean length of CNTs, respectively.

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Considering the debonding phenomenon, the normal stresses cause the CNTs debonding from surrounding phases. The connection between radial stress

and the macroscopic stress fields

of the nanocomposites can be correlated by the following relation (Explained in more detail in

and

where

from the crack tip with an angle of , Fig.1:

and

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distance of

are the principal stresses in the macro-scale and can be calculated in a

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,

(10)

(11)

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where

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Appendix A):

are the Poisson’s ratio of nanocomposites and the radial distance from the crack

tip, respectively. Substituting Eqs. (11) into Eq. (10) gives the relation between the radial stress at the macro scale as function of the stress intensity factor and other parameters. (12) As mentioned, the radial stress around the CNT can be determined by: (13) here

the radial part of stress concentration tensor (see section 4.3.) and

macro–scale radial stress as the average value of the radial stress

is the effective

in the DPZ [37]. Therefore,

by using Eq. (12): (14)

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ACCEPTED MANUSCRIPT Inserting Eqs. (12) and (14) into Eq. (13), the radial stress component at the neighborhood of the nanotubes can be given by:

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(15)

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Therefore, the crack tip element radius (see Fig.1) was obtained as:

(16)

At the crack initiation, the radius of the crack tip element can be expressed as [12]:

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(17)

here

is the critical stress in nano–scale which is responsible for CNTs debonding from the

(18)

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surrounding components. By further noting that:

(19)

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mechanism is then:

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From Eqs. (18) and (19), the fracture toughness improvements induced by toughening

where f is a factor calculated in detail in Appendix A. Moreover,

is the nanocomposites

Young’s modulus. As it is obvious from the above equations, the fracture toughness improvement caused by CNTs debonding can be rewritten as:

where

(20)

(21) In order to find the strain energy due to the CNTs debonding, the stress and displacement fields at the nanoscale must be determined. In addition, the radial part of concentration tensor and the critical debonding stress will be approximated in the next sections.

4. The strain energy due to debonding In this section, the strain energy due to debonding of a CNT from the surrounding interphase is estimated. According to Ref. [10], this value is given by: 7

ACCEPTED MANUSCRIPT (22) where

and

are defined as the displacement of outer surface of the pure matrix phase along

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the radial direction after and before of the CNT debonding, respectively. Moreover, P is the applied force on the outer surface of the pure matrix phase and is obtained by:

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(23)

Thus, the stress and displacement fields of the RVE at two different conditions; initial debonding

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and post debonding must be analyzed.

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4.1. Stress and displacement field analysis before CNT debonding High aspect ratio of CNTs guarantees that the radial stress can be applied only to the RVE lateral

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surface. Since the CNT (n), the interphase (i) and the matrix (m) are considered as isotropic materials along the radius direction regarding the Cauchy continuum theory, the radial stresses

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and the radial displacement in a cylindrical coordinate system (before the debonding process)

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can be written as [38]:

and

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where

(24) (25)

are the coefficients of the k-th sub-domain before CNT debonding from the

surrounding components (b). Debonding takes place when the radial stress around the CNT reaches a critical value

as observed in Fig. 2. This figure schematically depicts the applied

stress on the CNT and its surrounding components before and after debonding. Therefore, boundary conditions and displacements continuity at the initiation of CNT debonding are introduced by: =

,

=

,

=

,

=

,

=

(26) where rn, rp and rm are the CNT, the interphase and the matrix outer radii in the RVE, respectively. Since the stress in the CNT must be finite, the coefficient

should be equal to

zero. Applying the boundary conditions and additionally displacements continuity on Eqs. (24)

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ACCEPTED MANUSCRIPT and (25), all unknown coefficients can be determined. Finally, substituting the calculated coefficients

and

into Eq. (26) yields:

where

and

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(27) can be written as the product of functions of critical debonding and properties

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and

(28) (29)

are defined in detail in Appendix A.

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The functions

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of the CNT, the interphase and the pure resin.

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A

Interphase CNT

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Matrix

D

Matrix

B

Interphase

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CNT

Fig. 2. Applied stresses on polymer phase and the CNT (A) before debonding (B) after debonding

4.2. Stress and displacement field analysis after CNT debonding After CNT debonding, radial stresses and governing equations for radial displacement in the cylindrical coordinate system [38] in different materials are given by: (30) (31) here

and

are the coefficients of the kth sub-domain after CNT debonding (a). In this state,

the applied stress between the CNT and resin is equal to zero while the stress

is applied along

the radius rm. Therefore, the equilibrium and compatibility conditions can be considered as:

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ACCEPTED MANUSCRIPT = ,

=

, (32)

=

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All unknown coefficients can be determined by applying the boundary conditions and displacements continuity on Eqs. (30) and (31). Furthermore, the solution for the displacement

The functions

are defined by the following functions:

and

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and

(33)

(34) (35)

have been expressed in detail in Appendix A. Finally, by a combination

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where

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results in:

of Eqs. (8), (21) and (23), the fracture toughness enhancement due to debonding can be rewritten

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as the following normalized form:

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(36)

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4.3. Solution for radial part of stress concentration tensor, As shown before, the radial component of stress concentration tensor can be determined by the following relation:

Inserting the condition

(37) into Eq. (37) reveals: (38)

Considering Eqs. (28) and (29), radial part of the stress concentration tensor can be rewritten as a function of mechanical properties and the radii of all phases as: (39)

4.4. Solution for debonding stress,

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ACCEPTED MANUSCRIPT This section deals with the computing of due to debonding mechanism

as the critical debonding stress. The strain energy

creates a new surface of area

and hence the total

energy for detachment of a CNT from the surrounding polymer per unit area

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by:

can be calculated

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(40)

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Inserting Eqs. (28), (29) and (34-36) into the last relation gives:

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is the critical debonding stress of the RVE containing two phases, disregarding the

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where

(41)

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presence of the interphase. The normalized debonding stress, i.e., debonding stress around the CNT in the RVE with the presence of the interphase (

) divided by the debonding stress

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around the CNT in the RVE without the presence of the interphase (

), can be defined as:

(42)

These equations similarity match the solutions presented by other researchers [10,37,39] for a model of spherical nanoparticles/resin nanocomposites. Jiang et al. [40] developed an analytical model based on the van der Waals force to determine the total energy for debonding of a CNT from the surrounding polymer. The cohesive energy or debonding energy per unit area can be calculated by the following equation: (43) where

is the mass density of polymer over mass unit of –CH2– (2.3×10-26 m-2) and

CNTs area density (3.82×10-19 m-2) [41]. Moreover,

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and

is the

are Lennard–Jones parameters

ACCEPTED MANUSCRIPT which for epoxy-based nanocomposites reinforced with the CNTs are 0.0007459 nm.nN and 0.3825 nm, respectively [42].

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4.5. Estimating density and volume fraction of carbon nanotubes of nanocomposites The volume fraction of carbon nanotubes in nanocomposites is calculated using the following

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is the weight fractions of CNTs in nanocomposites,

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where

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equation [43]:

and

(44) are the CNT and the

resin density, respectively. The density of CNTs in terms of the density of graphene sheet (

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determined as follows [43]:

) is

(45)

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D

where rni stands for the mean inner radius of CNTs.

5. Case studies, results and discussion

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In this section, using the data available in the literature [4,5], effects of main variable parameters on the normalized fracture toughness improvement and normalized debonding stress have been investigated. The CNTs weight fractions and the interphase characteristics, i.e., thickness and Young’s modulus, have been considered for this purpose. At the end, the effect of CNTs debonding on the fracture toughness calculated by means of the present model has been compared with experimental results. It must be mentioned that Wolfram Mathematica software [44] was used for implementing the multi-scale model. Experimental results are available from Ayatollahi et al. [4] and Gojny et al. [5]. Ayatollahi et al. [4] investigated the effect of adding of different weight fractions of multi-walled carbon nanotubes (MWCNTs), 0.1, 0.5 and 1 wt.%, on the fracture toughness of epoxy polymer. The yield strength, the Young’s modulus and the density of this epoxy have been reported as 68.35 MPa, 3.15 GPa and 1.11 gr/cm3, respectively. The influence of loading 0.1, 0.3 and 0.5 wt.% of double–walled carbon nanotubes (DWCNTs) on epoxy-based nanocomposites has been studied by Gojny et al. [5,6]. The yield strength, the Young’s modulus and the density of pure epoxy

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ACCEPTED MANUSCRIPT polymer are 63.8 MPa, 2.6 GPa and 1.19 gr/cm3, respectively. The data, relevant references and properties used in the present analysis have been summarized in Table 1.

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Table 1. Summary of all properties of experimental results found literature and used in the

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present study.

Enc (GPa)

γi (J/m2)

3.15

---

---

3.24

0.170

757.9

3.51

0.170

757.9

0.890

3.69

0.170

757.9

0

2.60

---

---

10

0.121

2.785

0.182

812.5

10

0.362

2.885

0.182

812.5

10

0.604

2.790

0.182

812.5

CNTs

Ref.

rn (nm)

ln (µm)

Vf %

0

---

[4]

---

---

0

0.1

MWCNT

[4]

7.50

20

0.089

0.5

MWCNT

[4]

7.50

20

0.445

1

MWCNT

[4]

7.50

20

0

---

[5]

---

---

0.1

DWCNT

[5]

1.40

0.3

DWCNT

[5]

1.40

0.5

DWCNT

[5]

1.40

Yi

(MPa)

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D

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wt.%

5.1. Effect of different parameters on fracture toughness improvement

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Several papers explored the effect of various characteristics of the interphase zone on the mechanical properties of CNTs-dispersed nanocomposites [45-47]. Different papers have expressed contradictory values for the interphase thickness and stiffness. It has been reported that the interphase thickness widely varied between 0.5 to 4 times of the CNT wall thickness [32,48]. According to the paper published by Kundalwal and Meguid [30], the interphase Young’s modulus is usually higher than that of the surrounding neat resin due to the polymer chains restricted mobility. Zhang et al. [49] assumed an interphase with Young’s modulus that falls between 1 to 10 times of that of the neat polymer matrix. For this reason, the Young’s modulus if the interphase is considered to vary between 1 to 8 times of that of the pure resin. Therefore, two normalized parameters, i.e., the interphase thickness normalized by the CNTs thickness and the interphase Young’s modulus normalized by the pure polymer Young’s modulus, can be defined as follows: (46)

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ACCEPTED MANUSCRIPT (47) are the interphase normalized thickness and Young’s modulus, respectively.

where and

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The effects of the CNTs weight fraction on the normalized fracture toughness enhancement for both mentioned systems have been plotted in Figs. 3 to 6. The plots show that the occurring of

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debonding is a negligible energy dissipative mechanism at low CNTs loading. This trend has

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been observed by some researchers for the nanocomposites reinforced with spherical nanoparticles [10,38]. They proved that even though fracture toughness improvement due to

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nano-fillers debonding can be negligible but nano-fillers debonding is an ineludible state for the following toughening mechanisms like the yielding of nano-voids created by debonded particles.

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However, Figs. 3 and 4 indicate that the effect of the CNTs content on the normalized fracture toughness improvement clearly depends on the normalized interphase thickness. In addition, it is observed in Figs. 5 and 6 that the increase in fracture toughness is strongly influenced by the

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normalized Young’s modulus. As seen in these plots, in order to highlight the effect of the

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interphase region, results for the RVE without the interphase zone is also have been illustrated

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(t=0 and =1).

〖 𝐺〗_𝑑𝑏/𝐺_(𝐼𝑐,𝑚)

0.015

𝜆=2 𝛾=0.170 𝐽∕𝑚^2 𝑟_𝑛=7.5 𝑛𝑚 𝐸_𝑚=3.15 𝐺𝑃𝑎

0.012

𝑡_𝑒𝑓𝑓

0.009 0.006 0.003 0

0

0.2 0.4 0.6 0.8 MWCNTs weight fraction, wt.%

1

Fig. 3. Normalized fracture toughness enhancement due to debonding as a function of MWCNTs weight fraction (Eq. (36)); considering different normalized interphase thicknesses (t=0, 0.5, 1, 2, 3 and 4)

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ACCEPTED MANUSCRIPT

𝜆=2 𝛾=0.182 𝐽∕𝑚^2 𝑟_𝑛=1.4 𝑛𝑚 𝐸_𝑚=2.6 𝐺𝑃𝑎

𝑡_𝑒𝑓𝑓

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0.0075

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0.0060 0.0045

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〖 𝐺〗_𝑑𝑏/𝐺_(𝐼𝑐,𝑚)

0.0090

0.0030

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0.0015 0.0000

0.1 0.2 0.3 0.4 DWCNTs weight fraction, wt.%

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0

0.5

Fig. 4. Normalized fracture toughness improvement through debonding as a function of

D

DWCNTs weight fraction (Eq. (36)); considering the RVE without the presence of the interphase

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zone (t=0) or the RVE with different normalized interphase thicknesses (t=0.5, 1, 2, 3 and 4)

〖 𝐺〗_𝑑𝑏/𝐺_(𝐼𝑐,𝑚)

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0.016

𝑡_𝑒𝑓𝑓=1 𝛾=0.170 𝐽∕𝑚^2 𝑟_𝑛=7.5 𝑛𝑚 𝐸_𝑚=3.15 𝐺𝑃𝑎

0.012

𝜆=𝐸_𝑖/𝐸_𝑚

0.008

0.004

0 0

0.2 0.4 0.6 0.8 MWCNTs weight fraction, wt.%

1

Fig. 5. Normalized fracture toughness enhancement due to debonding versus the MWCNTs weight fraction (Eq. (36)). Different values of normalized Young’s modulus ( =1, 1.5, 2, 4 and 8) reveal the influence of this parameter on the normalized fracture toughness improvement.

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ACCEPTED MANUSCRIPT

𝑡_𝑒𝑓𝑓=1 𝛾=0.182 𝐽∕𝑚^2 𝑟_𝑛=1.4 𝑛𝑚 𝐸_𝑚=2.6 𝐺𝑃𝑎

𝜆=𝐸_𝑖/𝐸_𝑚

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0.0075

0.005

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〖 𝐺〗_𝑑𝑏/𝐺_(𝐼𝑐,𝑚)

0.01

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0.0025

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0

0.1 0.2 0.3 0.4 DWCNTs weight fraction, wt.%

0.5

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0

Fig. 6. Influence of the DWCNTs weight fraction on the normalized fracture toughness improvement through debonding (Eq. (36)). Different values of normalized Young’s modulus

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D

( =1, 1.5, 2, 4 and 8) denote the effect of this parameter on the normalized fracture toughness increase.

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5.2. Influence of the interphase characteristics on the debonding stress The effects of the interphase Young’s modulus and its thickness on the debonding stress around the carbon nanotubes are other important issues investigated by the present model (Eq. (42)). The normalized debonding stresses of both systems versus the normalized interphase thickness have been illustrated in Figs. 7 and 8. In addition, different values of the normalized Young’s modulus were used in order to observe the influence of this parameter on the normalized debonding stress. These plots show that the effect of the normalized debonding stress strongly depends on the normalized Young’s modulus as well as low values of the normalized interphase thickness. It can be concluded that for all values of the normalized Young’s modulus, the normalized debonding stress rapidly reaches a plateau when the normalized interphase thickness is approximately greater than two. It is worth mentioning that, when the normalized Young’s modulus is greater than 1 ( >1), the normalized debonding stress enhances.

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ACCEPTED MANUSCRIPT 3.0

𝜆=8

𝜆=4

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2.0

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1.5 1.0 𝛾

0.5

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𝜎_𝑐𝑟∕𝜎_𝑐𝑟0

2.5

𝜆=1.5

𝜆=1

7 J∕m^ , 𝑟_𝑛=7.5 nm 𝐸_𝑚=3.15 GPa [3]

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0.0 2

4 𝑡_𝑒𝑓𝑓

6

8

10

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0

𝜆=2

Fig. 7. Normalized debonding stress versus the normalized interphase thickness for a 0.5 wt.% MWCNTs/epoxy system. Different values of normalized Young’s modulus reveal the influence

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D

of this parameter on the normalized debonding stress.

𝜆=8

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3.0

𝜎_𝑐𝑟∕𝜎_𝑐𝑟0

2.5

𝜆=4

2.0

1.5

𝜆=2

1.0

𝛾

0.5

𝜆=1

𝜆=1.5

J∕m^ , 𝑟_𝑛=1.4 nm 𝐸_𝑚=2.6 GPa [5]

0.0 0

2

4 𝑡_𝑒𝑓𝑓

6

8

10

Fig. 8. Normalized debonding stress versus the normalized interphase thickness for a 0.3 wt.% DWCNTs/epoxy system. Different values of normalized Young’s modulus reveal the influence of this parameter on the normalized debonding stress.

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ACCEPTED MANUSCRIPT 5.3. Comparison of results of the present model with the experimental data The effect of CNTs debonding mechanism on the fracture toughness of CNTs-based nanocomposites was evaluated by using a series of experimental data available in the literature and t = 1. The properties

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[4, 5]. In the modeling process (Eq. (36)), it was considered that

utilized in the present analysis have been summarized in Table 1. The fracture toughness

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obtained by experiments and the present model have been mentioned in Table 2. To improve the

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capability of the model, the effect of the energy dissipated by other mechanisms occurring at the nanoscale, such as the plastic void growth must be considered as well as the CNTs debonding

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mechanism.

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Table 2. Comparison between the fracture toughness obtained by experiments [4, 5] and the present multi-scale model including the contribution of CNTs debonding. CNTs

Ref.

GIc,nc (Exp.) (J/m2)

GIc,nc (Present model) (J/m2)

Difference (%)

0.0

---

[4]

742.4

742.4

---

0.1

MWCNT

[4]

951.5

743.2

21.9

0.5

MWCNT

[4]

1056.5

746.9

29.3

1.0

MWCNT

[4]

899.5

752.0

16.4

0.0

---

[5]

144.2

144.2

---

0.1

DWCNT

[5]

184.8

145.1

21.5

0.3

DWCNT

[5]

223.2

145.6

34.7

0.5

DWCNT

[5]

230.7

146.1

36.6

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wt.%

6. Conclusions This paper presents a multi-scale model to estimate the effect of CNTs debonding from its surrounding resin and the interphase on the fracture toughness of nanocomposites. The model was developed based on a correlation between the stress fields at the macro, micro and nanoscale. An RVE was considered in order to find the stress field around the CNT. Finally, the influence of CNTs loading on the fracture toughness enhancement due to energy dissipation by the CNTs debonding was investigated. In addition, the influences of the interphase thickness and its Young’s modulus on enhancement of the fracture toughness and the debonding stress were studied. Some conclusions are summarized as follows:

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ACCEPTED MANUSCRIPT 1. The results clearly showed that the enhancement in fracture toughness is sensitive to the CNTs content.

2. It is worth noting that low values of the normalized interphase thickness (t<2) strongly

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affect the fracture toughness improvement and the debonding stress.

3. It was observed that the fracture toughness enhancement due to CNTs debonding and the

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debonding stress are strongly influenced by the interphase Young’s modulus.

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4. The obtained results sufficiently manifest that the proposed model can satisfactorily estimate the effect of CNTs debonding mechanisms on the fracture toughness of

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CNTs/epoxy nanocomposites.

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Appendix A. Calculating the radial stress causes debonding mechanism Considering a cylindrical RVE, the debonding phenomenon can be occurred by the radial stress. Assume that there is a plate with a single edge crack under mode I loading state, see Fig. A.1.

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Stress fields near the crack tip are calculated as:

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(A.1)

It is obvious that

(A.2) (A.3) (A.4)

is one of the principal stresses (

) and

. The others

can be computed as: (A.5) Substituting Eqs. (A.1), (A.2) and (A.4) into the last relation, one obtains: (A.6)

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ACCEPTED MANUSCRIPT These principal stresses are along the 1, 2 and 3 directions of a selected Cartesian coordinate system known as principal axis, see Fig. A.1. Consider a nanotube located in a 3-D space. A cylindrical coordinate system has been considered for the CNT.

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Two frames (r, , ) and (1, 2, 3) differ by orientation, as observed in Fig. A.1. Let Qij declare the cosine of the angles between the i-axis (i=r, , ξ) and the j-axis (j=1, 2, 3). The relation

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between the radial stress and principal stresses can be correlated as:

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(A.7)

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1

r

2

Fig. A.1. Principal axes and cylindrical coordinate system

The radial stress around the RVE and the principal stresses can be rewritten as:

where

,

and

(A.8) . Moreover,

and

.

Finally, it can be expressed that: (A.9) In order to compute the fracture toughness improvements, in Eq. (9)

is the DPZ radius at

. Therefore, Eq. (18) can be rewritten as: (A.10) From Eqs. (9) and (A.10), the fracture toughness improvements induced by toughening mechanisms is then:

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(A.11)

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Let us consider the following term of (A.11) as a factor:

Let the Poisson’s ratio of the nanocomposites be

and it was mentioned that

where

,

(A.13)

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then:

(A.12)

. Using mean value theorem for definite double integral, the mean value of

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this factor can be found as 2.48.

As mentioned, coefficients

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Appendix B. Determination of functions ,

,

and

and were expressed as the product of functions of

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geometrical and mechanical properties of CNTs, interphase and matrix, and critical debonding stress. These functions can be defined in detailed forms as:

(B.1)

(B.2)

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(B.4)

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(B.3)

where superscripts n, i and m denote the properties of nanotubes, interphase and matrix, and

are Lame’s constants which are described as:

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respectively. Moreover,

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(B.5)

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References

(B.6)

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Graphical abstract

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ACCEPTED MANUSCRIPT Highlights

• The effect of CNTs debonding phenomenon as a zone shielding mechanism has been

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•

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•

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•

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•

investigated using a multi-scale model. The relations between macro, micro and nano-scale were correlated in order to present the model. A representative volume element contains a CNT, interphase and matrix was selected for studying the energy dissipated by debonding mechanism. Fracture toughness improvement was formulated as a function of the strain energy, CNTs volume fraction and properties of involved materials. Effects of the CNTs weight fraction and interphase properties on the fracture toughness enhancement and debonding stress were illustrated.

27