Effects of nano-sized silica particles on the off-axis creep performance of unidirectional fiber-reinforced polymer hybrid composites

Abstract

A multi-step homogenization approach is presented to predict the off-axis creep response of hybrid polymer matrix composites (HPMCs) reinforced with unidirectional carbon fibers and silica nanoparticles. The first step deals with evaluating the viscoelastic properties of silica nanoparticle-polymer nanocomposite using the Mori-Tanaka micromechanical model. Two essential features affecting the behavior, including silica nanoparticle agglomeration and interphase region generated due to the interaction between the nanoparticle and polymer are taken into account. In the second step, the off-axis viscoelastic behavior of HPMCs is extracted from the homogenized nanocomposite and carbon fiber properties using a unit cell-based micromechanical model. Some comparative studies between the predictions and available experiment are directed to verify the homogenization process. All the model predictions are in good agreement with the experimental data. The results indicate that with increasing the fiber off-axis angle from 0° to 90°, the presence of silica nanoparticles leads to a reduction in the HPMC creep compliance. Also, the proposed multi-step homogenization approach is applied to investigate the effects of volume fraction, size and agglomeration degree of silica nanoparticles; thickness and material properties of the interphase region; and off-axis angle and volume fraction of the carbon fiber on the HPMC creep response.

Keywords

Hybrid polymer composite off-axis creep property silica nanoparticle carbon fiber homogenization approach

Introduction

Fiber-reinforced polymer matrix composites (FRPMCs) are widely used in many engineering fields such as aerospace, automotive, wind energy, and sporting goods industries due to their good mechanical properties such as high specific strength and modulus, low density, high corrosion resistance and low manufacturing costs.^1–4 FRPMCs show superior axial performance with relatively poor transverse mechanical response.^5–8 This issue is a main limitation of these advanced materials in engineering applications.

Mixture of two or more types of reinforcements in common matrices to produce hybrid polymer composite systems may generate materials which have the combined properties of individual composites. So, to resolve the problem of above limitation involved in FRPMCs, several kinds of nano-sized reinforcements have been used into them.^9–12 In this frame, the substantial attention is being paid to the performance of fiber-reinforced hybrid polymer matrix composites (HPMCs) containing silica nanoparticles at various in-service conditions. The advantage of using the nano-sized silica fillers with low concentration into the polymer-based composites is that the particles do not rise the viscosity of the resin considerably, and do not display thixotropic properties.¹³

Kinloch et al.¹⁴ have shown that mode I fracture energy of hybrid glass fiber-reinforced composite containing 10 wt.% silica nanoparticles is 1015 J/m². Naito¹⁵ evaluated the tensile properties and fracture behavior of hybrid carbon fiber-reinforced polyimide composites containing silica nanoparticles. Experimental outcomes revealed that the axial tensile modulus of hybrid carbon fiber-polyimide composites were slightly improved by adding the ceramic nanoparticles.¹⁵ Moreover, in another experimental research, improvement in energy absorption and interfacial shear strength of hybrid glass fiber-epoxy composite specimens due to the presence of silica nanoparticles was reported by Gao et al.¹⁶ Furthermore, Landowski et al.¹⁷ tested the low velocity impact behavior of silica nanoparticle enriched-epoxy matrix composites reinforced by carbon fiber for naval applications. Uddin and Sun¹⁸ revealed that 15 wt% silica nanoparticles improved the longitudinal compressive strength of hybrid E-glass fiber-reinforced composites by 30-40%. Tang et al.¹⁹ also measured the tensile strength and elastic modulus in the transverse direction as well as mode I and mode II interlaminar fracture toughness of hybrid carbon fiber-reinforced epoxy systems containing 10 wt% and 20 wt% nanoscale silica particles. Sprenger²⁰ has performed a comprehensive review in the case of mechanical properties of silica nanoparticle/fiber-reinforced HPMCs. There is not any study available for predicting the general off-axis creep response of HPMCs reinforced by unidirectional fibers and nanoscale particles.

Because of the inherent viscoelastic behavior of polymers, for polymer-based composite materials, creep is a very important phenomenon, which specifies the durability of the material even at a static loading condition.^21,22 It is mainly associated with the movement of the polymeric chains like slippage and reorientation. So, evaluation of viscoelastic behavior is an important part of the long-term mechanical properties of polymer-based heterogeneous materials.^23–25 A major challenge in the field of hybrid composite systems has been recognized to be formation of agglomeration of silica nanoparticles which deteriorates the mechanical performance.^17,20,26 Also, the mechanical response of HPMCs may be affected by the interfacial interaction between the nanoparticles and surrounding polymer matrix.^27–30 Homogenization techniques in both microscale^31–34 and nanoscale^35–38 sound promising. Overall, micromechanical models can predict the effective viscoelastic properties of heterogeneous materials based on the constituents’ material properties, volume fraction, and interactions between the phases. For example, Li et al.³⁹ estimated the linearly viscoelastic properties of carbon nanotube (CNT)-reinforced polyimide nanocomposites using Mori-Tanaka micromechanics model. Li and Zhang⁴⁰ developed a semi-analytical model using the generalized method of cell (GMC) to predict the relaxation properties of polymer composites consisting of viscoelastic matrices and transversely isotropic elastic graphite fibers. Based on the Halpin-Tsai and Nielsen micromechanical models, Shokrieh and Esmkhani⁴¹ predicted the fatigue life of hybrid glass fiber-reinforced epoxy composites as well.

The main aim of this research is predicting the creep response of unidirectional carbon fiber/silica nanoparticle-reinforced HPMCs under off-axis loading condition by a new multi-step homogenization approach. The paper is organized as follows. The linear viscoelastic theory employed to describe the polymer matrix behavior is introduced in the next section. Also, in the Multi-step homogenization approach section, the details of the multi-step homogenization approach are stated. Several cases of parametric analyses for the HPMCs are presented in the Results and discussion section. The results of the proposed multi-step homogenization approach are validated using experimental data available in the literature. Finally, the conclusions of this research are summarized in the last section.

Linear viscoelastic theory for polymer matrix

Before explaining the homogenization framework, it is key to recognize the constitutive laws of the constituent’s material of hybrid composite system. In the present work, at low stress levels the polymer matrix usually possesses linear viscoelastic behavior, while the nanoparticles and the carbon fibers behave almost elastically. The viscoelastic materials demonstrate time-dependent behavior, so their elastic response depends on the rate of the applied load.^36,39,40 Moreover, these materials can show creep in response to the fixed stress, and time-dependent stress relaxation subjected to the fixed strain. The constitutive equation for the linear viscoelastic polymer materials can be expressed in the time domain as follows^36,39

σ (t) = \int_{0}^{t} L (t - τ) \dot{ε} (τ) d τ

(1)

ε (t) = \int_{0}^{t} M (t - τ) \dot{σ} (τ) d τ

(2)

where the dot represents the differentiation with respect to time (

t

σ

and

ε

are the stress and strain vectors, respectively, which are functions of time for viscoelastic materials, and

L (t)

and

M (t)

refer to the relaxation stiffness and creep compliance tensors, respectively. The solution process for elastic materials is much easier than that of viscoelastic materials because of the presence of the time variable in the viscoelastic constitutive equations as shown in equations (1) and (2). According to the correspondence principle (CP), integral transforms can be used to convert a linear viscoelastic problem to an equivalent elastic problem.^22,36,39,42 So, the Laplace-Carson (L-C) transform given in the form below^36,39,43

L^{C} \{f (t)\} = s \int_{0}^{\infty} e^{- s t} f (t) d t

(3)

can be applied on equations (1) and (2) to represent the constitutive equations in the transformed domain similar to those of the linear elasticity as follows

\hat{σ} (s) = \hat{L} (s) \hat{ε} (s)

(4)

\hat{ε} (s) = \hat{M} (s) \hat{σ} (s)

(5)

where the hatted variables refer to the dependent transformed variables, and

s

is the independent transform parameter. It is seen from equations (4) and (5) that the inverse relationship between the stiffness and compliance tensors is well-kept. In other words, one can write

\hat{L} (s) \hat{M} (s) = \hat{M} (s) \hat{L} (s) = I

(6)

where

I

denotes the identity tensor. In accordance with the classical elasticity, and based on equations (4) and (5), it is revealed the

\hat{L}

\hat{M}

for the viscoelastic problem takes the roles of stiffness and compliance tensors, respectively.

For an isotropic polymer material which shows linear viscoelastic behavior, the creep function is given as^22,36,39

M (t) = D_{0} + D_{1} t^{n}

(7)

where

D_{0}

is the initial elastic compliance, and

D_{1}

and

n

are constants specified from experimental data. By applying the L-C transform on equation (7), it can be written as

\hat{M} (s) = D_{0} + \frac{D_{1} n!}{s^{n}}

(8)

Thus, the elastic modulus can be expressed in the transformed domain as

\hat{E} (s) = {\{D_{0} + \frac{D_{1} n!}{s^{n}}\}}^{- 1}

(9)

Multi-step homogenization approach

The first step in the homogenization analysis of HPMC is to predict the viscoelastic properties of silica nanoparticle-reinforced polymer nanocomposite. In this step, in order to obtain the effective viscoelastic properties of polymer nanocomposite precisely, the homogenization analysis needs to involve two important microstructural features, including the nanoparticle agglomeration and the interfacial zone between the nanoparticles and surrounding polymer. The Mori-Tanaka micromechanical model is used to obtain the nanocomposite viscoelastic properties in the first step. The second step of the homogenization process would provide the effective viscoelastic properties of carbon fiber-reinforced HPMC considering polymer nanocomposite as the matrix phase and unidirectional carbon fiber as the reinforcement. In this step, a unit cell-based micromechanical model will be developed to extract the creep response of HPMC subjected to off-axis loading. The idealized microstructure of HPMC is schematically shown in Figure 1.

Figure 1.

A schematic sketch of microstructure of HPMCs under consideration.

Silica nanoparticle-polymer nanocomposites

This section presents the Mori-Tanaka analytical micromechanics model to estimate the effective viscoelastic properties of silica nanoparticle-reinforced polymer composites which are required as inputs for the HPMC analysis. As mentioned earlier, the silica nanoparticles tend to agglomerate.^17,20,26,27 Consequently, consideration of the nanoparticle agglomeration into the micromechanical simulation is extremely needed. For this purpose, a two-scale micromechanical modeling is suggested to evaluate the viscoelastic behavior of polymer nanocomposites containing nanoscale silica particles. The small-scale consists of the agglomerated nanoparticle-polymer matrix which in turn serves as spherical inclusion into the pure polymer matrix to make the final polymer nanocomposite at the large scale as shown in Figure 1(b). So, the entire polymer nanocomposite is divided into two domains, i.e. the nanoparticle-free polymer matrix denoted as phase $O$ with volume fraction $c_{O}$ , and the small-scale spherical inclusions consisting of the agglomerated nanoparticles and the rest of polymer matrix known as phase $I$ with the volume fraction $c_{I}$ . Based on the defined volume fractions, it can be written

c_{I} + c_{O} = 1

(10)

Phase $I$ is itself a nanocomposite material reinforced with uniformly dispersed silica nanoparticles. The volume fraction of polymer matrix and silica nanoparticle within the small-scale nanocomposite is chosen to be equal to $c_{0}^{(I)}$ and $c_{1}^{(I)}$ . Thus, it can be expressed

c_{1}^{(I)} + c_{0}^{(I)} = 1

(11)

In addition, it is assumed that $c_{0}$ and $c_{1}$ to be the volume fraction of polymer matrix and nanoscale particle in the entire polymer nanocomposite yielding

c_{1} = c_{1}^{(I)} . c_{I}, c_{0} = c_{O} + c_{0}^{(I)} . c_{I}

(12)

where phases

I

and

O

refer to the inclusion and polymer matrix, respectively, which are used in the large-scale problem, and phases 1 and 0 are for the small-scale one. The agglomeration of nano-sized silica particles is characterized by

c_{I}

and

c_{I} =

1 when all of them are uniformly dispersed in the HPMC.

In the larger-scale problem of silica nanoparticle-reinforced nanocomposite simulation, the stress-strain relation in the transformed domain for polymer matrix containing spherical inclusions is given by

{\hat{σ}}^{LNC} (s) = {\hat{L}}^{LNC} (s) {\hat{ε}}^{LNC} (s)

(13)

in which

{\hat{L}}^{LNC}

denotes the equivalent elastic stiffness tensor of the large-scale polymer nanocomposites. The Mori-Tanaka method is employed to determine

{\hat{L}}^{LNC}

as follows^42,44

{\hat{L}}^{LNC} (s) = {[{\hat{M}}^{LNC}]}^{- 1} = {\hat{L}}^{P M} (s) + c_{I} ({\hat{L}}^{I} (s) - {\hat{L}}^{P M} (s)) . (\hat{A} (s) {. [c_{O} I + c_{I} \hat{A} (s)]}^{- 1})

(14)

\hat{A} (s) = {[I + S . {({\hat{L}}^{P M} (s))}^{- 1} ({\hat{L}}^{I} (s) - {\hat{L}}^{P M} (s))]}^{- 1}

(15)

where

{\hat{L}}^{I}

and

{\hat{L}}^{P M}

are the transformed equivalent elastic stiffness tensors for the spherical inclusion and polymer matrix, respectively, and

{\hat{M}}^{LNC}

is the equivalent compliance tensor. The general form of

\hat{L} (s)

for an isotropic medium can be expressed as

\hat{L} (s) = {[\begin{matrix} \frac{1}{\hat{E} (s)} & \frac{- v}{\hat{E} (s)} & \frac{- v}{\hat{E} (s)} & 0 & 0 & 0 \\ \frac{- v}{\hat{E} (s)} & \frac{1}{\hat{E} (s)} & \frac{- v}{\hat{E} (s)} & 0 & 0 & 0 \\ \frac{- v}{\hat{E} (s)} & \frac{- v}{\hat{E} (s)} & \frac{1}{\hat{E} (s)} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{2 (1 + v)}{\hat{E} (s)} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{2 (1 + v)}{\hat{E} (s)} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{2 (1 + v)}{\hat{E} (s)} \end{matrix}]}^{- 1}

(16)

where

v

is the Poisson’s ratio. Also, in equation (15)

S

is the Eshelby tensor which its non-zeros components for a spherical inclusion are^42,44

\begin{matrix} S_{1111} = S_{2222} = S_{3333} = \frac{7 - 5 v_{P M}}{15 (1 - v_{P M})}, \\ S_{1122} = S_{2233} = S_{3311} = \frac{5 v_{P M} - 1}{15 (1 - v_{P M})} S \\ 1212 \\ = S_{2323} = S_{3131} = \frac{4 - 5 v_{P M}}{15 (1 - v_{P M})} \end{matrix}

(17)

where

v_{P M}

is the Poisson’s ratio of polymer matrix. It is worth mentioning that

{\hat{L}}^{LNC}

is the elastic stiffness tensor of polymer nanocomposite containing silica nanoparticle agglomeration. equation (14) needs to be solved for each discrete value of

s

to determine the creep compliance of polymer nanocomposite, i.e.

{\hat{M}}^{LNC}

in the transformed domain at that value of

s

. This process is performed for an adequate range of

s

. Then, assuming that the polymer nanocomposite creep function has the same function form as that of the pure polymer according to equation (7), a nonlinear regression is used to fit a curve with the same mathematical structure based on equation (8). So, it is possible to calculate the values of

{D_{0}}^{LNC}

{D_{1}}^{LNC}

and

n^{LNC}

which will be directly associated with the creep compliance function for polymer nanocomposites containing nanoscale silica particles as follows

M^{LNC} (t) = {D_{0}}^{LNC} + {D_{1}}^{LNC} t^{n^{LNC}}

(18)

Now, the viscoelastic properties of spherical inclusion that consists of the rest of polymer matrix and the nano-sized silica particles shown in Figure 1(b) must be determined. It has been mentioned that the interphase region formed due to the interaction of nanoparticle and polymer matrix atoms at the nano scale during manufacture of the composite can significantly affect the mechanical properties.^27–30,35 In this regards, proper modeling of the interphase which has properties between those of the polymeric matrix and those of the nanoparticles is important. The interphase Young’s modulus $E^{i}$ can be expressed as²⁷

E^{i} = \frac{1}{t_{i}} \int_{\frac{d}{2}}^{\frac{d}{2} + t_{i}} [E^{P M} (\frac{\frac{d}{2} + t_{i}}{r}) + {(\frac{\frac{d}{2} + t_{i} - r}{t_{i}})}^{η} [E^{N P} - E^{P M} (\frac{\frac{d}{2} + t_{i}}{\frac{d}{2}})]] d r

(19)

where

\frac{d}{2} \leq r \leq \frac{d}{2} + t_{i}

, and

d

and

t_{i}

stand for the silica nanoparticle diameter and the interphase thickness, respectively. In equation (19), η refers to the adhesion exponent. Also,

E^{N P}

and

E^{P M}

are the Young’s moduli of nanoparticle and polymer matrix, respectively. The Poisson’s ratio of interphase (

v^{i}

) is considered to be equal to that of polymer matrix.^27,30 The elastic stiffness tensor of interphase (

{\hat{L}}^{i}

) is constructed using

E^{i}

and

v^{i}

. Herein, the silica nanoparticle enclosed by a layer of interphase is reflected with an equivalent nanofiller as displayed in Figure 1(c). The Mori-Tanaka model is applied to obtain the effective properties of nanofiller (

{\hat{L}}^{NF}

) in which the nanoscale silica particle is the reinforcement and the interphase has the role of matrix. After evaluating

{\hat{L}}^{NF}

, the Mori-Tanaka model will be again used to determine the equivalent elastic stiffness tensor of spherical inclusion (

{\hat{L}}^{I}

) in the transformed domain. In this stage, the nanofiller and polymer matrix are considered as reinforcement and matrix, respectively.

Carbon fiber-reinforced HPMCs

This section presents the simplified unit cell (SUC) micromechanics method to predict the effective viscoelastic response of carbon fiber-reinforced HPMCs under off-axis loading. The SUC model is an analytical micromechanical method to predict the bulk properties of heterogeneous materials.^45–47

As mentioned in the previous section, the HPMC system can be viewed as a composite material in which the carbon fibers are embedded in the silica nanoparticle-polymer nanocomposite matrix. Figure 2(a) illustrates a typical off-axis coupon of HPMC under consideration. The carbon fibers are placed at an angle $θ$ to the loading direction ( $x$ direction) as shown in Figure 2(b). Note that two coordinate systems, including $x$ - $y$ and 1-2 are considered such that the $x$ direction is the loading direction, and the 1 direction coincides with the carbon fiber direction and also the 2 direction is perpendicular to the carbon fiber. Generally, unidirectional composite materials consist of continuous fibers which are randomly arranged in matrix. A schematic diagram of a real HPMC cross-section is shown in Figure 2(c). However, the actual cross-section of fibrous composites is idealized as a regular arrangement of fibers in a square array packing.^48,49 It is due to the fact that the simulation of random arrangement of fibers is difficult. Figure 2(d) displays a schematic diagram of the idealized cross-section of HPMC with square array packing of carbon fibers. In the unit cell-based models,^45–48 the representative volume element (RVE) of fibrous composite is usually represented by a rectangular cross-section fiber and matrix sub-cells. Considering this assumption, the RVE of the SUC model for simulating HPMCs is constructed, and shown in Figure 2(e). It is assumed that the interfaces of the neighboring sub-cells are perfectly bonded.^{35,36,45–48}

Figure 2.

(a) A typical off-axis coupon of HPMC, (b) a carbon fiber oriented at angle θ (c) a schematic diagram of a real HPMC cross-section, (d) a schematic diagram of an idealized cross-section of HPMC with square array packing, and (e) the RVE of the SUC model.

Now, the micromechanical formulations of the SUC model are described in the transformed domain for the RVE presented in Figure 2(e). Besides the perfect bonding condition, another assumption in the SUC model is that the displacement components are linear. Also, the applied normal stress on the RVE does not lead to any shear stress inside the RVE sub-cells and vice versa.^{35,36,45–48} The stress components in the principal material coordinate are expressed from the applied stress ${\hat{S}}_{x}$ by the transformation law as follows

\{\begin{matrix} {\hat{S}}_{1} \\ {\hat{S}}_{2} \\ {\hat{S}}_{12} \end{matrix}\} = \{\begin{array}{l} \cos^{2} θ \\ \sin^{2} θ \\ - \sin θ \cos θ \end{array}\} {\hat{S}}_{x}

(20)

where

{\hat{S}}_{1}

{\hat{S}}_{2}

and

{\hat{S}}_{12}

denote the axial normal, transverse normal and axial shear stresses in the material coordinate system, respectively. From the equilibrium condition between the normal local stress component inside sub-cell ij, i.e.

{\hat{σ}}^{i j} (s)

, and the applied global stress over the RVE, i.e.

{\hat{S}}_{l}

in transformed domain, one can write

\sum_{i = 1}^{2} {\hat{σ}}_{3}^{i 1} (s) a_{i} = 0 \sum_{j = 1}^{2} {\hat{σ}}_{2}^{1 j} (s) b_{j} = {\hat{S}}_{2} (b_{1} + b_{2}) \sum_{j = 1}^{2} \sum_{i = 1}^{2} b_{j} a_{i} {\hat{σ}}_{1}^{i j} (s) = {\hat{S}}_{1} (b_{1} + b_{2}) (a_{1} + a_{2})

(21)

where

a_{i}

and

b_{i}

are the geometrical parameters of sub-cells. Imposition of the equilibrium of the local stresses along the sub-cells interfaces in the transformed domain leads to

\begin{matrix} {\hat{σ}}_{2}^{1 j} (s) = {\hat{σ}}_{2}^{i j} (s) (i > 1) \\ {\hat{σ}}_{3}^{i 1} (s) = {\hat{σ}}_{3}^{i j} (s) (j > 1) \\ {\hat{τ}}_{23}^{i j} (s) = 0, {\hat{τ}}_{13}^{i j} (s) = 0 \end{matrix}

(22)

Based on the compatibility of the local strains ${\hat{ε}}^{i j} (s)$ inside each sub-cell, it can be defined that^45–48

{\hat{ε}}_{12}^{i 1} (s) = {\hat{ε}}_{12}^{i j} (s) (j \neq 1)

(23)

Imposition of the compatibility of the displacements inside the RVE which is resulted from the perfect bonding conditions between the subcells leads to

\sum_{i = 1}^{2} a_{i} {\hat{ε}}_{2}^{i 1} (s) = \sum_{i = 1}^{2} a_{i} {\hat{ε}}_{2}^{i j} (s) = (a_{1} + a_{2}) {\hat{\bar{ε}}}_{2} (s) (j > 1) \sum_{j = 1}^{2} b_{j} {\hat{ε}}_{3}^{1 j} (s) = \sum_{j = 1}^{2} b_{j} {\hat{ε}}_{3}^{i j} (s) = (b_{1} + b_{2}) {\hat{\bar{ε}}}_{3} (s) (i > 1) {\hat{ε}}_{1}^{i j} (s) = {\hat{\bar{ε}}}_{1} (s) (i > 1, j > 1)

(24)

where ${\hat{\bar{ε}}}_{l}$ is the global strain. According to the Hooke’s law in shear loading along with equation (23), it can be expressed that

{\hat{τ}}_{12}^{i j} (s) = \frac{{\hat{τ}}_{12}^{i 1} (s)}{{\hat{G}}_{i 1} (s)} {\hat{G}}_{i j} (s) (j \neq 1)

(25)

where ${\hat{G}}_{i j}$ denotes the shear modulus of sub-cell ij. By substituting the constitutive equations into equation (24), the following relation can be derived

\sum_{i = 1}^{2} \{\frac{a_{i}}{{\hat{E}}_{T}^{i 1} (s)} {\hat{σ}}_{2}^{11} (s) - \frac{a_{i}}{{\hat{E}}_{T}^{i j} (s)} {\hat{σ}}_{2}^{1 j} (s) - \frac{a_{i}}{{\hat{E}}_{T}^{i 1} (s)} v_{T}^{i 1} {\hat{σ}}_{3}^{i 1} (s) + \frac{a_{i}}{{\hat{E}}_{T}^{i j} (s)} v_{T}^{i j} {\hat{σ}}_{3}^{i j} (s) - \frac{a_{i}}{{\hat{E}}_{L}^{i 1} (s)} v_{L}^{i 1} {\hat{σ}}_{1}^{i 1} (s) + \frac{a_{i}}{{\hat{E}}_{L}^{i j} (s)} v_{L}^{i j} σ_{1}^{i j} (s)\} = 0 (j \neq 1)

\sum_{j = 1}^{2} \{- \frac{b_{j}}{{\hat{E}}_{T}^{1 j} (s)} v_{T}^{1 j} {\hat{σ}}_{2}^{1 j} (s) + \frac{b_{j}}{{\hat{E}}_{T}^{i j} (s)} {v_{T}^{i j} \hat{σ}}_{2}^{i j} (s) + \frac{b_{j}}{{\hat{E}}_{T}^{1 j} (s)} {\hat{σ}}_{3}^{11} (s) - \frac{b_{j}}{{\hat{E}}_{T}^{i j} (s)} {\hat{σ}}_{3}^{i 1} (s) - \frac{b_{j}}{{\hat{E}}_{L}^{1 j} (s)} v_{L}^{1 j} {\hat{σ}}_{1}^{1 j} (s) + \frac{b_{j}}{{\hat{E}}_{L}^{i j} (s)} v_{L}^{i j} {\hat{σ}}_{1}^{i j} (s)\} = 0 (i \neq 1)

\{- \frac{1}{{\hat{E}}_{L}^{11} (s)} {\hat{σ}}_{2}^{11} (s) + \frac{v_{L}^{i j}}{{\hat{E}}_{L}^{i j} (s)} {\hat{σ}}_{2}^{i j} (s) - \frac{1}{{\hat{E}}_{L}^{11} (s)} {\hat{σ}}_{3}^{11} (s) + \frac{v_{L}^{i j}}{{\hat{E}}_{L}^{i j} (s)} {\hat{σ}}_{3}^{i 1} (s) + \frac{1}{{\hat{E}}_{L}^{11} (s)} {\hat{σ}}_{1}^{11} (s) + \frac{1}{{\hat{E}}_{L}^{i j} (s)} {\hat{σ}}_{1}^{i j} (s)\} = 0 (i \neq 1, j \neq 1)

(26)

Note that index L and T denote the axial and transverse directions, respectively. Combination of equation (26) with equations (21) and (22) leads to a system of ten linear equations with the same number of unknowns in the transformed domain as follows

{[\hat{A} (s)]}_{10 \times 10} {\{\hat{σ} (s)\}}_{10 \times 1} = {\{\hat{F}\}}_{10 \times 1}

(27)

in which

\{\hat{σ} (s)\}

\{\hat{F}\}

and

[\hat{A} (s)]

refer to the local stress, the external load and the coefficient tensor, respectively. At a specified value of global stress, solving equation (27) for each discrete value of

s

in the transformed domain leads to determining the local stresses of each sub-cell at that value of

s

. Next, the local strains for each sub-cell in the transformed domain can be calculated using the constitutive equations. Then, the equivalent normal strains of the RVE in the transformed domain at that value of

s

will be extracted using equation (24). This process is done for an adequate range of s. So, the creep strain in the transformed domain is obtained. Assuming that the creep function of HPMC to be similar to that of silica nanoparticle-polymer nanocomposite matrix, i.e. equation (18), a nonlinear regression is utilized to fit a curve with the same mathematical structure. Finally, the values of

{D_{0}}^{HPMC}

{D_{1}}^{HPMC}

and

n^{HPMC}

which would be directly associated with the creep compliance function of carbon fiber-reinforced HPMC are achieved. One can mark the HPMC creep compliance function as follows

M^{HPMC} (t) = {D_{0}}^{HPMC} + {D_{1}}^{HPMC} t^{n^{HPMC}}

(28)

Results and discussion

Some comparative examinations with experimental measurements existing in the open literature are performed. As a first endeavor to validate the developed homogenization approach, the predictions of the creep compliance for silica nanoparticle-polyimide nanocomposites are compared with experimental data.⁵⁰ Wang and Zhao⁵⁰ synthesized some specimens of silica polyimide nanocomposites by an improved sol-gel technique and evaluated their creep behavior. The experimental data for creep strain of polyimide nanocomposites was informed for three different weight fractions of nano-sized silica particle, including 1, 3 and 8%.⁵⁰ Based on the microstructural investigation performed on the nanocomposite specimens, at the low level of nanoparticle content, i.e. 3 wt%, there was no evidence of any the nanoparticle agglomeration within the polyimide nanocomposites. However, it was observed that at the high level of silica nanoparticle weight fraction, i.e. 8 wt%, local agglomeration of the nano-sized nanoparticles exists.⁵¹ The material properties of the polyimide matrix as a viscoelastic material and the silica nanoparticle are given in Table 1.^27,50,51 Average of the silica nanoparticle diameter is equal to 50 nm.⁵¹ Comparison between the model predictions and the experiment⁵⁰ for the creep compliance of polyimide nanocomposites reinforced with 1.53 vol% and 4.182 vol% is shown in Figure 3(a) and (b), respectively. The nanoparticle weight fraction⁵⁰ is converted into volume fraction by a nanoparticle density of 2.65 g/cm³, and polyimide of 1.33 g/cm³. The results of the proposed homogenization approach are given for two different types of the interphase conditions, including elastic interphase and viscoelastic interphase as well as without interphase. The thickness and material parameters of the interphase region are considered to be $t_{i} =$ 25 nm, $η =$ 10,²⁷ $D_{1} =$ 0.5 × 0.057 GPa⁻¹s⁻¹ and $n =$ 0.152. Also, as indicated in Figure 3(b), the results of present approach are extracted with considering a uniform distribution of silica nanoparticles and an agglomerated state of the nano-sized particles. In the micromechanical simulation of 8 wt% silica-polyimide nanocomposite, it is assumed that the value of $c_{I}$ to be 0.6. The absence of the interphase region in the micromechanical modeling reveals a noticeable disparity between the predictions and experiment,⁵⁰ which it can be confirmed from Figure 3(a). However, the predictions of the homogenization approach considering the viscoelastic interphase are in acceptable agreement with the experiment results.⁵⁰ The results of the model with the elastic interphase are slightly lower than those of experiment.⁵⁰ So, it is suggested that the viscoelastic interphase needs to be taken into account in the micromechanical modeling to have reasonable predictions. Figure 3(b) clearly shows that the polymer nanocomposite creep compliance is significantly sensitive to the dispersion of nanoparticles. The agglomeration of nano-sized silica particles seems to have a deleterious effect on the nanocomposite time-dependent mechanical properties. It can be observed the nanocomposite creep compliance increases by the nanoparticle agglomeration. Consequently, the polyimide nanocomposite creep compliance attained using the present homogenization approach with the viscoelastic interphase and the nanoparticle agglomeration is quite close to that of the experiment.⁵⁰

Table 1.

The material properties of polyimide and silica particle.^27,50,51

Material	$D_{0}$ (GPa⁻¹)	$D_{1}$ (GPa⁻¹s⁻¹)	$n$	$v$	$ρ$ (g/cm³)
Silica	1/88.7	0	0	0.23	2.65
Polyimide	0.831	0.057	0.152	0.4	1.33

Figure 3.

Comparison between the model predictions and experiment⁵⁰ for the creep compliance of silica nanoparticle-polyimide nanocomposites. The silica nanoparticle volume fractions are equal to (a) 1.53% and (b) 4.182%.

Second verification is related to a comparison between the model predictions and experimental measurements of the creep response of T300/934 graphite/epoxy composites under off-axis loading. Yancey and Pindera⁵² experimentally evaluated the creep compliance of unidirectional graphite fiber-reinforced epoxy composites for two different off-axis angles, including $θ =$ 10˚ and 45˚ as well as the transverse creep compliance which is of the 90° on-axis coupon. So, the micromechanical model estimations are compared with these experimental measurements.⁵² The material properties of T300 graphite fiber and 934 epoxy are given in Tables 2 and 3, respectively.⁵² Figure 4 displays the off-axis creep compliance of T300/934 graphite/epoxy composites. Generally, the comparison shows that the micromechanics models can favorably predict the creep properties of fiber-reinforced composites. Also, it is observed that the creep compliance of fibrous composites increases with the increase of off-axis angle. It is attributed to the fact that as the applied load is varied from directly in line with the fiber axis to normal to the fiber axis, the effective properties of fiber-reinforced composites change from fiber dominated to matrix dominated.⁵³

Table 2.

Material properties of 934 epoxy.⁵²

$D_{0}$ (GPa^-1)	$D_{1}$ (GPa⁻¹min⁻¹)	$n$	$v$
0.22	0.0135	0.17	0.311

Table 3.

The material properties of T300 graphite fiber.⁵²

$E_{L}$ (GPa)	$E_{T}$ (GPa)	$G_{L}$ (GPa)	$v_{L}$	$v_{T}$
202.82	25.3	44.12	0.443	0.39

Figure 4.

Comparison between the model predictions and experiment⁵² for the creep compliance of T300/934 graphite/epoxy composites.

The last comparison has been performed between the predictions of the micromechanical modeling approach and experimental data¹⁹ in case of the transverse elastic modulus of diglycidyl ether of bisphenol A (DGEBA) epoxy composites reinforced by unidirectional carbon fibers and nanoscale silica particles. The volume fraction of carbon fiber is 65%, and the experimental data was informed for two silica nanoparticle weight fractions, including 10% and 20%.¹⁹ The mechanical properties of carbon fiber are given in Table 4,⁵⁴ while the material properties of silica nanoparticle are taken from Table 1. The silica nanoparticles have an average particle size of about 20 nm.¹⁹ Also, the elastic modulus, Poisson’s ratio and density of epoxy resin are 3.6 GPa, 0.35 and 1.17 g/cm³.¹⁹ The values of $t_{i}$ and $η$ of the interphase region are considered to 10 nm and 10.²⁷ It is note that the elastic modulus of the interphase region has been computed from equation (19) and its Poisson’s ratio is considered to be equal to that of the polymer matrix.²⁷ In the micromechanical modeling of HPMC, in addition to extracting the transverse elastic modulus assuming a uniform dispersion of nano-sized silica particles, the model predictions are provided with considering an agglomerated state of silica nanoparticles by $c_{I}$ =0.5. Table 5 shows the comparison between the analytical results from the present homogenization approach and the experimental data.¹⁹ By comparison, it can be clearly observed that the analytical result derived from the mixed micromechanics model considering the nanoparticle agglomeration agrees well with the experimental data.¹⁹ There is a large discrepancy between the modeling results considering the uniform dispersion of nanoparticle corresponding to $c_{I}$ =1, and the results with considering the nanoparticle agglomeration. It can be concluded that the uniform dispersion of silica nanoparticles contributes to the enhancement of the effective transverse elastic modulus of HPMC. For instance, the transverse elastic modulus of the 10 wt% silica nanoparticle-reinforced HPMC with $c_{I}$ =0.5 and $c_{I}$ =1 is determined to be 11.8 GPa and 13.25 GPa, respectively, corresponding to about 12.3% enhancement. The same conclusion can be obtained for the HPMC containing 20 wt% nano-sized silica particles, which the enhancement of this condition is about 22%.

Table 4.

The material properties of carbon fiber.⁵⁴

$E_{L}$ (GPa)	$E_{T}$ (GPa)	$G_{L}$ (GPa)	$v_{L}$	$v_{T}$
230	20	25	0.3	0.42

Table 5.

Comparison between the model predictions and experiment¹⁹ for the transverse elastic modulus (GPa) of the hybrid DGEBA epoxy composites.

Silica weight fraction (%)	Experiment¹⁹	Present model
Silica weight fraction (%)	Experiment¹⁹	Uniform dispersion	Agglomeration state
10	11.4 ± 0.5	13.25	11.8
20	12.7 ± 0.8	15.35	12.58

After verifying the validity of the developed homogenization approach, several parametric studies are conducted to examine the role of critical microstructural features in the creep performance of HPMCs made of epoxy resin, unidirectional carbon fibers and nanoscale silica particles. The material properties of silica, 934 epoxy and carbon fiber are taken from Table 1,²⁷ Table 2 ⁵² and Table 4,⁵⁴ respectively. The average diameter of silica nanoparticles is chosen to be 20 nm. Also, the thickness and material parameters of the viscoelastic interphase region are considered to be $t_{i} =$ 10 nm, $η =$ 10,²⁷ $D_{1} =$ 0.5 × 0.0135 GPa⁻¹s⁻¹ and $n =$ 0.17. Moreover, the volume fractions of carbon fiber and silica nanoparticle are selected to be 62% and 3%, respectively. It is assumed that the nanoscale silica particles homogenously distributed within the HPMCs, i.e. $c_{I}$ =1, too. Unless otherwise stated, above values are used to generate the numerical results of the creep compliance of the HPMCs.

First, the predicted creep response of the HPMCs for 0° and 90°on-axis coupons as well as two off-axis coupons with off-axis angle equal to $θ =$ 10° and 45° are presented in Figure 5. It is noted that 0° and 90°on-axis coupons are corresponding to axial and transverse directions, respectively. Also, the results of the present micromechanical model for carbon fiber-reinforced 934 epoxy composites, i.e. the case of without nanoparticle, are given in Figure 5. It is found from Figure 5(a) that the addition of silica nanoparticle does not affect the axial creep performance of HPMCs. It is attributed to the dominate role of the carbon fiber properties in the axial mechanical properties of fibrous composite systems. The contribution of nano-sized silica particles to the creep properties of HPMC when $θ =$ 10° also seems to be slight as shown in Figure 5(b). The results of Figure 5(c) indicate that the effect of silica nanoparticles on the creep performance of HPMCs becomes more important when $θ =$ 45°. While the creep compliance of 45°off-axis HPMC containing silica nanoparticles is lower than that of 45°off-axis carbon fiber-reinforced epoxy composite (without nanoparticle). So, adding the nanoscale particles can have a beneficial effect on the creep performance of 45°off-axis HPMC. According to the results of Figure 5(d) as compared to those of Figure 5(a) to (c), it is confirmed that the addition of silica nanoparticles has the most influence on the transverse creep behavior of HPMCs. It is due to the fact that the transverse mechanical properties of fibrous composite systems are significantly dominated by the matrix properties. Note that the polymer matrix of HPMCs is enriched by the silica nanoparticles which leads to a reduction in its creep compliance. Overall, the HPMC creep compliance decreases with the decrease of off-axis angle from 90˚ to 0˚ because the effective properties of fibrous composite systems vary from matrix dominated to fiber dominated.

Figure 5.

The creep compliance of hybrid 934 epoxy composites under various carbon fiber orientations, including (a) 0°, (b) 10°, (c) 45° and (d) 90°.

The effect of silica nanoparticle volume fraction on the overall creep performance of the 45°off-axis HPMC and 90°on-axis HPMC is demonstrated in Figure 6(a) and (b), respectively. For this micromechanical analysis, three different volume fractions of silica nanoparticle, including 1%, 3% and 5% are selected. It is found that the nanoparticle concentration can have a significant effect on the HPMC creep properties. With the increase of nanoparticle volume fraction, a decreasing trend appears in the creep compliance of hybrid carbon fiber-reinforced 934 epoxy composites.

Figure 6.

Effect of the silica nanoparticle volume fraction on the creep compliance of hybrid 934 epoxy composites under the carbon fiber orientations, i.e. (a) 45˚ and (b) 90˚.

The effect of the variation of silica nanoparticle diameter on the effective creep response of HPMCs is explored. The variation of the diameter is taken in the range of 10 nm to 200 nm. The results of the present homogenization approach for the 45°off-axis HPMC and 90°on-axis HPMC are depicted in Figure 7(a) and (b), respectively. It is observed that the size of silica nanoparticles becomes an important controller mechanism for the time-dependent mechanical properties of hybrid unidirectional composites containing nanoscale particles. Generally, the effective creep compliance of HPMCs decreases with the reduction in nanoparticle diameter. It may be attributed to the increased effect of the interphase region which becomes larger as the nanoparticle size decreases. It is worth mentioning that the creep compliance of the interphase region is lower than that of the polymer matrix. However, the influence of variation in the nanoparticle size can be neglected after $d =$ 100 nm due to the reduced contribution of the interphase region to the HPMC viscoelastic behavior. Same conclusion has been also reported in Ref.27 for the analysis of Young’s modulus of silica nanoparticle-polymer nanocomposites which may be a confirmation for the precision of this outcome.

Figure 7.

Effect of the silica nanoparticle diameter on the creep compliance of hybrid 934 epoxy composites under the carbon fiber orientations, i.e. (a) 45˚ and (b) 90˚.

Silica nanoparticles may be agglomerated within the polymer matrix.^17,20,26,27 Therefore, a parametric study in the case of the effect of silica nanoparticle agglomeration on the creep performance of HPMCs needs to be performed. Figure 8(a) and (b) show the effect of the degree of silica nanoparticle agglomeration by selecting $c_{I} =$ 1 (corresponding to the nanoparticle uniform distribution), 0.5 and 0.25 on the 45°off-axis HPMC and 90° on-axis HPMC, respectively. The creep compliance of HPMCs with the nanoparticle agglomeration is more than that of the HPMCs assuming the uniform dispersion of silica nanoparticles. This identifies that the creep performance of the HPMCs not only are affected by the nanoparticle volume percent and size, but also are dominated by the nanoparticle distribution. It is found from Figure 8 that the creep compliance is minimum when $c_{I} =$ 1 which means that the uniform distribution of silica nanoparticles throughout the whole HPMCs leads to the optimal time-dependent mechanical properties.

Figure 8.

Effect of the silica nanoparticle agglomeration degree on the creep compliance of hybrid 934 epoxy composites under the carbon fiber orientations, i.e. (a) 45˚ and (b) 90˚.

The effect of the change of silica nanoparticle/epoxy interphase material behavior, including viscoelastic and elastic behaviors is considered, and the HPMC creep compliance is obtained using the present homogenization approach. The results for the 45° off-axis HPMC and 90° on-axis HPMC are given in Figure 9(a) and (b), respectively. Figure 9 also shows the model predictions in the absence of the interphase region. Without considering the interphase in the micromechanics model, the predictions of the HPMC creep compliance are higher than those of the HPMC with the interphase. This outcome suggests that the formation of the interphase region can improve the HPMC time-dependent mechanical properties. Moreover, when $t >$ 0, the creep compliance of HPMC with the elastic interphase is lower than that of HPMC with the viscoelastic interphase. Thus, the interphase region is a significant factor which may govern the viscoelastic behavior of hybrid fibrous composites with multiscale reinforcements.

Figure 9.

Effect of the interphase material properties on the creep compliance of hybrid 934 epoxy composites under various the carbon fiber orientations, i.e. (a) 45˚ and (b) 90˚.

The thickness of nanoparticle/polymer interphase may be dependent on some factors such as the energy of interaction between the matrix and nanoparticles, and intermolecular interaction in the polymer matrix.^27,55 The variation of the interphase thickness is incorporated to analyze the HPMC creep performance. According to the literature,^27,28 the values of the interphase thickness are chosen to be $t_{i}$ = 5 nm, 10 nm and 20 nm which is used in the micromechanical simulation. The estimations for the 45° off-axis HPMC and 90° on-axis HPMC is presented in Figure 10(a) and (b), respectively. It is observed that the increase in the interphase thickness significantly improves the bulk viscoelastic behavior of the HPMCs. It is attributed to the fact that the compliance of nanoparticle/polymer interphase region is lower than that of polymer matrix, which leads to a reduction in the overall creep compliance of HPMCs.

Figure 10.

Effect of the interphase thickness on the creep compliance of hybrid 934 epoxy composites under the carbon fiber orientations, i.e. (a) 45˚ and (b) 90˚.

Finally, the effect of the carbon fiber volume fraction on the creep performance of 45° off-axis HPMC and 90° on-axis HPMC is investigated, and the results are illustrated in Figure 11. For this analysis, three different volume fractions of the carbon fiber, including 62%, 52% and 42% are selected. As expected, the overall creep compliance of the HPMCs decreases with the increase of the carbon fiber volume fraction.

Figure 11.

Effect of the carbon fiber volume fraction on the creep compliance of hybrid 934 epoxy composites under the carbon fiber orientations, i.e. (a) 45˚ and (b) 90˚.

Conclusions

In this work, two analytical micromechanics models were speculated to develop a new multi-step homogenization approach to predict the overall creep response of hybrid fiber-reinforced polymer composites containing nano-sized particles subjected to off-axis loading. In the first step, the Mori-Tanaka model was used to estimate the viscoelastic properties of polymer matrix enriched by silica nanoparticles. In this stage, the nanoparticle agglomeration and the interphase surrounding the nanoparticles were introduced to conduct a more realistic micromechanics method. In the second step, the SUC micromechanical model was employed to extract the off-axis creep compliance of hybrid carbon fiber-reinforced epoxy composites containing silica nanoparticles. By comparison with experimental measurements, the homogenization approach was validated to provide accurate quantitative predictions on the overall creep properties of HPMC systems. The results revealed that adding silica nanoparticles causes a decrease in the HPMC creep compliance and the maximum reduction was found to be for the 90° on-axis coupon. Also, with increasing the fiber off-axis angle from 0° to 90°, the HPMC creep compliance significantly increases. It was found that increasing the concentration and decreasing the size of silica nanoparticles can contribute to improvement of the HPMC creep properties, while the nanoparticle agglomeration becomes a dominant factor in degradation of the creep behavior. A homogenous dispersion of silica nanoparticle in the whole HPMC leads to minimum level of creep compliance. The results indicated that with an increase in the thickness of silica nanoparticle/polymer interphase region, the creep properties of HPMCs is improved. The present homogenization approach with quantitative predictions on the general off-axis creep performance of hybrid fiber/nanoparticle-reinforced polymer composites is expected to be helpful for the design and optimization of HPMCs with desirable time-dependent mechanical properties.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

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

ORCID iD

MJ Mahmoodi

References

Forintos

Czigany

Multifunctional application of carbon fiber reinforced polymer composites: electrical properties of the reinforcing carbon fibers – a short review. Compos Part B Eng 2019; 162: 331–343.

Zhupanska

Sierakowski

RL.

Electro-thermo-mechanical coupling in carbon fiber polymer matrix composites. Acta Mech 2011; 218: 319–332.

Jamali

Mahmoodi

Hassanzadeh-Aghdam

, et al. A mechanistic criterion for the mixed-mode fracture of unidirectional polymer matrix composites. Compos Part B Eng 2019; 176: 107316.

Song

Xiong

, et al. Fabrication and mechanical behaviors of carbon fiber reinforced composite foldcore based on curved-crease origami. Compos Sci Technol 2019; 174: 94–105.

Abidin

MSZ

Herceg

Greenhalgh

, et al. Enhanced fracture toughness of hierarchical carbon nanotube reinforced carbon fibre epoxy composites with engineered matrix microstructure. Compos Sci Technol 2019; 170: 85–92.

Hassanzadeh-Aghdam

Mahmoodi

Ansari

Micromechanics-based characterization of mechanical properties of fuzzy fiber-reinforced composites containing carbon nanotubes. Mech Mater 2018; 118: 31–43.

Pan

Bian

Coefficients of nonlinear thermal expansion for fiber-reinforced composites. Acta Mech 2017; 228: 4341–4351.

Kundalwal

Ray

MC.

Effective properties of a novel continuous fuzzy-fiber reinforced composite using the method of cells and the finite element method. Eur J Mech A 2012; 36: 191–203.

Jiang

Zhang

Zhao

, et al. Interfacially reinforced carbon fiber composites by grafting modified methylsilicone resin. Compos Sci Technol 2017; 140: 39–45.

10.

Kundalwal

Ray

MC.

Shear lag analysis of a novel short fuzzy fiber-reinforced composite. Acta Mech 2014; 225: 2621–2643.

11.

Hassanzadeh-Aghdam

Mahmoodi

Jamali

Effect of CNT coating on the overall thermal conductivity of unidirectional polymer hybrid nanocomposites. Int J Heat Mass Transfer 2018; 124: 190–200.

12.

Oskouie

Hassanzadeh-Aghdam

Ansari

A new numerical approach for low velocity impact response of multiscale-reinforced nanocomposite plates. Eng Comput. Epub ahead of print 24 August 2019. DOI: 10.1007/s00366-019-00851-9.

13.

Ngah

Taylor

AC.

Toughening performance of glass fibre composites with core–shell rubber and silica nanoparticle modified matrices. Compos Part A Appl Sci Manuf 2016; 80: 292–303.

14.

Kinloch

Masania

Taylor

, et al. The fracture of glass-fibre-reinforced epoxy composites using nanoparticle-modified matrices. J Mater Sci 2008; 43: 1151–1154.

15.

Naito

Tensile properties of polyacrylonitrile-and pitch-based hybrid carbon fiber/polyimide composites with some nanoparticles in the matrix. J Mater Sci 2013; 48: 4163–4176.

16.

Gao

Jensen

McKnight

, et al. Effect of colloidal silica on the strength and energy absorption of glass fiber/epoxy interphases. Compos Part A Appl Sci Manuf 2011; 42: 1738–1747.

17.

Landowski

Strugała

Budzik

, et al. Impact damage in SiO2 nanoparticle enhanced epoxy–carbon fibre composites. Compos Part B Eng 2017; 113: 91–99.

18.

Uddin

Sun

CT.

Strength of unidirectional glass/epoxy composite with silica nanoparticle-enhanced matrix. Compos Sci Technol 2008; 68: 1637–1643.

19.

Tang

Zhang

, et al. Characterization of transverse tensile, interlaminar shear and interlaminate fracture in CF/EP laminates with 10 wt% and 20 wt% silica nanoparticles in matrix resins. Compos Part A Appl Sci Manuf 2011; 42: 1943–1950.

20.

Sprenger

Improving mechanical properties of fiber-reinforced composites based on epoxy resins containing industrial surface-modified silica nanoparticles: review and outlook. J Compos Mater 2015; 49: 53–63.

21.

Rafiee

Mazhari

Modeling creep in polymeric composites: developing a general integrated procedure. Int J Mech Sci 2015; 99: 112–120.

22.

Ansari

Hassanzadeh-Aghdam

MH.

Micromechanics-based viscoelastic analysis of carbon nanotube-reinforced composites subjected to uniaxial and biaxial loading. Compos Part B Eng 2016; 90: 512–522.

23.

Pan

Weng

Meguid

, et al. Interface effects on the viscoelastic characteristics of carbon nanotube polymer matrix composites. Mech Mater 2013; 58: 1–11.

24.

Khan

Muliana

Effective thermal properties of viscoelastic composites having field-dependent constituent properties. Acta Mech 2010; 209: 153–178.

25.

Jeon

Kim

Muliana

Modeling time-dependent and inelastic response of fiber-reinforced polymer composites. Comput Mater Sci 2013; 70: 37–50.

26.

Tourani

Molazemhosseini

Khavandi

, et al. Effects of fibers and nanoparticles reinforcements on the mechanical and biological properties of hybrid composite polyetheretherketone/short carbon fiber/nano‐SiO2. Polym Compos 2013; 34: 1961–1969.

27.

Boutaleb

Zaïri

Mesbah

, et al. Micromechanics-based modelling of stiffness and yield stress for silica/polymer nanocomposites. Int J Solids Struct 2009; 46: 1716–1726.

28.

Hassanzadeh-Aghdam

Ansari

A micromechanical model for effective thermo-elastic properties of nanocomposites with graded properties of interphase. Iran J Sci Technol Trans Mech Eng 2017; 41: 141–147.

29.

Zhang

Friedrich

, et al. Property improvements of in situ epoxy nanocomposites with reduced interparticle distance at high nanosilica content. Acta Materialia 2006; 54: 1833–1842.

30.

Sun

Gibson

Gordaninejad

Multiscale analysis of stiffness and fracture of nanoparticle-reinforced composites using micromechanics and global–local finite element models. Eng Fract Mech 2011; 78: 2645–2662.

31.

Mahmoodi

Hassanzadeh-Aghdam

Ansari

, et al. Damage analysis of unidirectional Ti hybrid nanocomposites containing nanoparticles. J Alloys Compds 2018; 769: 397–412.

32.

Kim

Muliana

Time-dependent and inelastic behaviors of fiber-and particle hybrid composites. Struct Eng Mech 2010; 34: 525–539.

33.

Mahmoodi

Hassanzadeh-Aghdam

Ansari

Overall thermal conductivity of unidirectional hybrid polymer nanocomposites containing SiO2 nanoparticles. Int J Mech Mater Des 2019; 15: 539–554.

34.

Kari

Berger

Gabbert

, et al. Evaluation of influence of interphase material parameters on effective material properties of three phase composites. Compos Sci Technol 2008; 68: 684–691.

35.

Fralick

Gatzke

Baxter

SC.

Three-dimensional evolution of mechanical percolation in nanocomposites with random microstructures. Prob Eng Mech 2012; 30: 1–8.

36.

Snipes

Robinson

Baxter

SC.

Effects of scale and interface on the three-dimensional micromechanics of polymer nanocomposites. J Compos Mater 2011; 45: 2537–2546.

37.

Dabbagh

Rastgoo

Ebrahimi

Static stability analysis of agglomerated multi-scale hybrid nanocomposites via a refined theory. Eng Comput. Epub ahead of print 19 January 2020. 10.1007/s00366-020-00939-7.

38.

Tsai

Tzeng

Chiu

YT.

Characterizing elastic properties of carbon nanotubes/polyimide nanocomposites using multi-scale simulation. Compos Part B Eng 2010; 41: 106–115.

39.

Gao

Roy

AK.

Micromechanical modeling of viscoelastic properties of carbon nanotube-reinforced polymer composites. Mech Adv Mater Struct 2006; 13: 317–328.

40.

Zhang

A new viscoelastic model based on generalized method of cells for fiber-reinforced composites. Int J Plast 2015; 65: 22–32.

41.

Shokrieh

Esmkhani

Fatigue life prediction of nanoparticle/fibrous polymeric composites based on the micromechanical and normalized stiffness degradation approaches. J Mater Sci 2013; 48: 1027–1034.

42.

Christensen

RM.

Mechanics of composite materials. USA: Courier Corporation, 2012.

43.

Laws

McLaughlin

Self-consistent estimates for the viscoelastic creep compliances of composite materials. Proc R Soc London A 1978; 359: 251–273.

44.

Mori

Tanaka

Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metall 1973; 21: 571–574.

45.

Aghdam

Smith

Pavier

MJ.

Finite element micromechanical modelling of yield and collapse behaviour of metal matrix composites. J Mech Phys Solids 2000; 48: 499–528.

46.

Ansari

Hassanzadeh-Aghdam

MK.

Micromechanical characterizing elastic, thermoelastic and viscoelastic properties of functionally graded carbon nanotube reinforced polymer nanocomposites. Meccanica 2017; 52: 1625–1640.

47.

Hassanzadeh-Aghdam

Ansari

Mahmoodi

MJ.

Thermo-mechanical properties of shape memory polymer nanocomposites reinforced by carbon nanotubes. Mech Mater 2019; 129: 80–98.

48.

Aboudi

Arnold

Bednarcyk

BA.

Micromechanics of composite materials: a generalized multiscale analysis approach. Oxford: Butterworth-Heinemann, 2012.

49.

Bayat

Aghdam

MM.

A micromechanics-based analysis of effects of square and hexagonal fiber arrays in fibrous composites using DQEM. Eur J Mech A 2012; 32: 32–40.

50.

Wang

Zhao

XX.

Creep resistance of PI/SiO₂ hybrid thin films under constant and fatigue loading. Compos Part A Appl Sci Manuf 2008; 39: 439–447.

51.

Wang

Zhao

XX.

Modeling and characterization of viscoelasticity of PI/SiO₂ nanocomposite films under constant and fatigue loading. Mater Sci Eng A 2008; 486: 517–527.

52.

Yancey

Pindera

MJ.

Micromechanical analysis of the creep response of unidirectional composites. J Eng Mater Technol 1990; 112: 157–163.

53.

Aghdam

Pavier

Smith

DJ.

Micro-mechanics of off-axis loading of metal matrix composites using finite element analysis. Int J Solids Struct 2001; 38: 3905–3925.

54.

Kundalwal

Ray

MC.

Effective properties of a novel composite reinforced with short carbon fibers and radially aligned carbon nanotubes. Mech Mater 2012; 53: 47–60.

55.

Odegard

Clancy

Gates

TS.

Modeling of the mechanical properties of nanoparticle/polymer composites. Polymer 2005; 46: 553–562.