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# Journal of the Mechanical Behavior of Materials

Editor-in-Chief: Aifantis, Katerina

Managing Editor: Skoryna, Juliusz

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Volume 26, Issue 3-4

## Volume 2 (1989)

Vijay Choyal
• Applied and Theoretical Mechanics Laboratory, Discipline of Mechanical Engineering, Indian Institute of Technology Indore, Simrol, Indore 453552, India
• Other articles by this author:
/ Shailesh I. Kundalwal
• Corresponding author
• Applied and Theoretical Mechanics Laboratory, Discipline of Mechanical Engineering, Indian Institute of Technology Indore, Simrol, Indore 453552, India, Tel.: +91-8830670644
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• Other articles by this author:
Published Online: 2017-11-02 | DOI: https://doi.org/10.1515/jmbm-2017-0018

## Abstract

In this work, an improved shear lag model was developed to investigate the interfacial characteristics of three-phase hybrid nanocomposite which is reinforced with microscale fibers augmented with carbon nanotubes on their circumferential surfaces. The shear lag model accounts for (i) radial and axial deformations of different transversely isotropic constituents, (ii) thermomechanical loads on the representative volume element (RVE), and (iii) staggering effect of adjacent RVEs. The results from the current newly developed shear lag model are validated with the finite element simulations and found to be in good agreement. This study reveals that the reduction in the maximum value of the axial stress in the fiber and the interfacial shear stress along its length become more pronounced in the presence of applied thermomechanical loads on the staggered RVEs. The existence of shear tractions along the RVE length plays a significant role in the interfacial characteristics and cannot be ignored.

## 1 Introduction

Carbon nanotubes (CNTs) [1] have emerged as ideal candidates for multifarious nanotechnology applications. This is due to their remarkable thermoelastic and physical properties [2], [3], [4], [5], [6], [7], [8], [9]. The quest for utilizing the exceptional thermoelastic properties of CNTs has led to the development of two-phase CNT-reinforced nanocomposites [10], [11], [12]. However, the addition of CNTs in polymer matrix does not always result in improved effective properties of the two-phase nanocomposites. Several important factors, such as agglomeration, aggregation and waviness of CNTs, and difficulty in manufacturing also play a significant role [13]. These difficulties can be alleviated by using CNTs as secondary reinforcements in a three-phase CNT-reinforced composite. In this case, CNTs are grown on the circumferential surfaces of the fiber reinforcements (see Figure 1). For the first time, Thostenson et al. [14] synthesized CNTs on the circumferential surfaces of the carbon fibers using chemical vapor deposition (CVD) technique and found that the presence of CNTs at the fiber/matrix interface improve the interfacial shear strength of the hybrid nanocomposite. Veedu et al. [15] fabricated a three-dimensional (3D) composite in which multi-walled CNTs are grown normal to the surface of the micro-fiber fabric cloth layouts. The measured fracture toughness for crack initiation, interlaminar shear sliding fracture toughness, in-plane strength, modulus, toughness, transverse thermal conductivity and damping characteristics of this 3D composite exhibit 348%, 54%, 140%, 5%, 424%, 51% and 514% enhancements, respectively, over those of the base composite. Their test results also indicated that the presence of CNTs in the transverse (i.e. thickness) direction of this 3D composite reduces the effective coefficient of thermal expansion of the composite by 62% as compared to that of the base composite. Sager et al. [16] grew CNTs on the circumferential surface of T650 carbon fiber in an epoxy matrix by using thermal CVD, and reported that the interfacial shear strength of the resulting composite improves with the addition of a CNT coating. They attributed this improvement to the increase in the interphase yield strength as well as an improvement in the interfacial adhesion as a result of the presence of CNTs. Carbon nanotubes reinforce the polymer matrix between the fibers by providing enhanced strength and toughness, as depicted in Figure 1A and B [17], [18]. The resulting laminated structure is described as a hybrid advanced composite laminate, rather than as a nanocomposite, and is perhaps best described as the fuzzy fiber reinforced composite [19], [20], [21], [22], [23], [24], [25].

Figure 1:

Fiber grown with CNTs.

The transfer of load from the surrounding matrix to the fiber is one of the fundamental micromechanical processes determining the composite strength. It is a complex process that depends on the fiber/matrix interfacial properties, constitutive behavior of matrix, geometrical arrangement of the fibers, their volume fraction and strength of the fibers [26]. However, considerable simplification comes from considering the shear lag load transfer models. Previous shear lag studies have reported that the reinforced CNTs in the nanocomposites significantly enhance the stress transfer characteristics of the CNT-reinforced composites [27], [28], [29], [30], [31], [32]. Ray et al. [33] carried out the load transfer analysis of the short carbon fiber reinforced composite in which aligned CNTs are radially grown on the circumferential surfaces of the carbon fibers. In their study, the short carbon fibers being coated with the radially grown CNTs were assumed to touch each other laterally and hence, a two-phase shear lag model was developed. In practice, CNT coated carbon fibers may not touch each other laterally and the resulting composite will be composed of three phases; namely, the carbon fiber, the CNT-reinforced polymer matrix nanocomposite (interlayer) and the polymer matrix.

Unidirectional nanocomposite structures with parallel staggered platelet reinforcements are widely observed in the natural biological materials. Previous studies indicated that the staggered micro-/nanostructures in the natural biological materials play a crucial role in their superior mechanical properties [34], [35]. However, existing shear lag studies of CNT-reinforced nanocomposites do not account for such staggering effect for the adjacent RVEs and cannot provide accurate description of the overall interfacial characteristics of the CNT-reinforced composites under thermomechanical loading [21], [22]. The schematic diagram shown in Figure 2 represents the lamina of three-phase hybrid nanocomposite made of a fiber, an intermediate CNT-reinforced composite (hereinafter the “interlayer”), and a polymer matrix. In order to analyze the complex interfacial characteristics of such hybrid nanocomposite, a more accurate and consistent shear lag model that considers the staggering effect and the 3D thermomechanical nature of the applied loads must be developed. This is indeed the motivation behind the current study. The present study is devoted to the development of a comprehensive three-phase shear lag model for analyzing the interfacial characteristics of hybrid nanocomposite incorporating the staggering effect.

Figure 2:

In-plane cross-section of hybrid nanocomposite lamina.

## 2 Shear lag model

Figure 2 illustrates the cylindrical RVE of the hybrid nanocomposite based on which the three-phase shear lag model is derived. The cylindrical coordinate (r, θ and x) system is considered in such a way that the axis of the RVE coincides with the x-axis while CNTs are aligned along the r-direction. The model is derived by dividing the RVE into three zones. The portion of the RVE in the zone −Lf≤x≤Lf consists of three concentric cylindrical phases; namely, the carbon fiber, the interlayer and the polymer matrix. The RVE of the hybrid nanocomposite has the radius R and the length 2L. The radius and the length of the carbon fiber are denoted by a and 2Lf. The inner and outer radii of the interlayer phase are a and b, respectively. The portions of the RVE in the zones −L≤x≤−Lf and Lf≤x≤L are considered to be composed of an imaginary fiber, an imaginary interlayer and the polymer matrix phase. The radius of the imaginary fiber is also denoted by a while the inner and outer radii of the imaginary interlayer phase are also represented by a and b, respectively. Thus the shear lag model derived for the zone −Lf≤x≤Lf can be applied to derive the shear lag models for the zones −L≤x≤−Lf and Lf≤x≤L.

In what follows, the shear lag model for the zone −Lf≤x≤Lf is first derived. As shown in Figure 3, a tensile stress σ is applied to the RVE along the x-direction at x=±L while the RVE is subjected to a radial normal stress qo at r=R. The governing equations for an axisymmetric RVE problem in terms of cylindrical coordinates (r, θ and x) are given by

Figure 3:

RVE of three-phase composite.

$∂σri∂r+∂σxri∂x+σri−σθir=0 and ∂σxi∂x+1r∂(rσxri)∂r=0; i=f, c and m$(1)

while the relevant constitutive relations are

$σxi=C11iϵxi+C12iϵθi+C13iϵri−λ11iΔT, σri=C13iϵxi+C23iϵθi+C33iϵri −λ22iΔT and σxri=C66iϵxri; i=f, c and m$(2)

in which

${\lambda }_{11}^{\text{i}}={\text{C}}_{11}^{\text{i}}{\alpha }_{11}^{\text{i}}+{\text{C}}_{12}^{\text{i}}{\alpha }_{22}^{\text{i}}+{\text{C}}_{13}^{\text{i}}{\alpha }_{33}^{\text{i}}\text{\hspace{0.17em}and\hspace{0.17em}}{\lambda }_{22}^{\text{i}}={\text{C}}_{12}^{\text{i}}{\alpha }_{11}^{\text{i}}+{\text{C}}_{22}^{\text{i}}{\alpha }_{22}^{\text{i}}+{\text{C}}_{23}^{\text{i}}{\alpha }_{33}^{\text{i}}$

In Eqs. (1) and (2), the superscripts f, c and m denote, respectively, the carbon fiber, the interlayer and the monolithic polymer matrix. For the i-th constituent phase, ${\sigma }_{\text{x}}^{\text{i}}$ and ${\sigma }_{\text{r}}^{\text{i}}$ represent the normal stresses in the x and r directions, respectively; ${ϵ}_{\text{x}}^{\text{i}},$ ${ϵ}_{\theta }^{\text{i}}$ and ${ϵ}_{\text{r}}^{\text{i}}$ are the normal strains along the x, θ and r directions, respectively; ${\sigma }_{\text{xr}}^{\text{i}}$ is the transverse shear stress, ${ϵ}_{\text{xr}}^{\text{i}}$ is the transverse shear strain, ${\text{C}}_{\text{ij}}^{\text{i}}$ are the elastic constants, ${\lambda }_{11}^{\text{i}}$ and ${\lambda }_{22}^{\text{i}}$ are the axial and the transverse thermal stiffness coefficients, respectively, and ${\alpha }_{\text{ij}}^{\text{i}}$ are the coefficients of thermal expansion. The strain-displacement relations for an axisymmetric problem relevant to this RVE are

$ϵxi=∂ui ∂x, ϵθi=wi r, ϵri=∂wi ∂r and ϵxri=∂ui ∂r+∂wi ∂x$(3)

in which ui and wi represent the axial and the radial displacements at any point in the i-th phase along the x and the r directions, respectively. The traction boundary conditions are given by

$σrm|r=R=q and σxrm|r=R=τ=2RσLvSCFF$(4)

and the continuity conditions are

$σrf|r=a=σrc|r=a ; σxrf|r=a=σxrc|r=a=τi ; σrc|r=b=σrm|r=b; σxrc|r=b=σxrm|r=b=τo; uf|r=a=uc|r=a; uc|r=b=um|r=b; wf|r=a=wc|r=a and wc|r=b=wm|r=b$(5)

where τi is the transverse shear stress at the interface between the carbon fiber and the interlayer while τo is the transverse shear stress at the interface between the interlayer and the polymer matrix. It may be noted here that these interfacial shear stresses can be attributed to the interactions between the adjacent staggered short carbon fibers [34], [35]. The consideration of the radial load qo on the RVE also accounts for the lateral extensional interactions between the adjacent short composite fuzzy fibers. The average axial stresses in different phases are defined as

$σ¯xf=1πa2∫0aσxf2πr dr; σ¯xc=1π(b2−a2)∫abσxc2πr dr and σ¯xm=1π(R2−b2)∫bRσxm2πr dr$(6)

Now, making use of Eqs. (1) and (4)–(6), it can be derived that

$∂σ¯xf∂x=−2aτi ; ∂σ¯xc∂x=2ab2−a2τi−2bb2−a2τo and ∂σ¯xm∂x=2bR2−b2τo$(7)

It is evident from Eq. (7) that the gradients of ${\overline{\sigma }}_{\text{x}}^{\text{c}}$ and ${\overline{\sigma }}_{\text{x}}^{\text{m}}$ with respect to the axial coordinate (x) are independent of the radial coordinate (r). Hence, as the radial dimension of the RVE is very small, it is reasonable to assume that [36]

$∂σxc∂x=∂σ¯xc∂x and ∂σxm∂x=∂σ¯xm∂x$(8)

Thus using the equilibrium equations given by Eq. (1), the transverse shear stresses in the interlayer phase and in the polymer matrix phase can be expressed in terms of the interfacial shear stresses τi and τo, respectively, as follows:

$σxrc =arτi+12r(a2−r2)∂σ¯xc∂x$(9)

$σxrm =(R2r−r)bR2−b2τo+Rrτ$(10)

Also, since the RVE is an axisymmetric problem, it is usually assumed [36] that the gradient of the radial displacements with respect to the x-direction is negligible and so, from the constitutive relations given by Eq. (2) and the strain-displacement relations given by Eq. (3) between ${\sigma }_{\text{xr}}^{\text{i\hspace{0.17em}}}$ and ${ϵ}_{\text{xr}}^{\text{i}},$ one can write

$∂uc∂r=1C66cσxrc and ∂um∂r=1C66mσxrm$(11)

Solving Eq. (11) and satisfying the continuity condition at r=a and r=b, respectively, the axial displacements of the interlayer phase and the polymer matrix phase along the x-direction can be derived as follows:

$uc=uaf +A1τi+A2τo$(12)

$um=uaf +A3τi+A4τo+RC66mτ ln(rb) and$(13)

$uaf =uf|r=a$(14)

where

$\begin{array}{l}{\text{A}}_{1}=\frac{\text{a}}{{\text{C}}_{66}^{\text{c}}\left({\text{b}}^{2}-{\text{a}}^{2}\right)}\left[{\text{b}}^{2}\text{ln}\frac{\text{r}}{\text{a}}-\frac{\left({\text{r}}^{2}-{\text{a}}^{2}\right)}{2}\right], \text{\hspace{0.17em}}\\ {\text{A}}_{2}=-\frac{\text{b}}{{\text{C}}_{66}^{\text{c}}\left({\text{b}}^{2}-{\text{a}}^{2}\right)}\left[{\text{a}}^{2}\text{ln}\frac{\text{r}}{\text{a}}-\frac{\left({\text{r}}^{2}-{\text{a}}^{2}\right)}{2}\right]\\ {\text{A}}_{3}=\frac{\text{a}}{{\text{C}}_{66}^{\text{c}}\left({\text{b}}^{2}-{\text{a}}^{2}\right)}\left[{\text{b}}^{2}\text{ln}\frac{\text{b}}{\text{a}}-\frac{\left({\text{b}}^{2}-{\text{a}}^{2}\right)}{2}\right]\\ {\text{A}}_{4}=\frac{\text{b}}{{\text{C}}_{66}^{\text{m}}\left({\text{R}}^{2}-{\text{b}}^{2}\right)}\left[{\text{R}}^{2}\text{ln}\frac{\text{r}}{\text{b}}-\frac{\left({\text{r}}^{2}-{\text{b}}^{2}\right)}{2}\right]\\ \text{ }\text{ }-\frac{\text{b}}{{\text{C}}_{66}^{\text{c}}\left({\text{b}}^{2}-{\text{a}}^{2}\right)}\left[{\text{a}}^{2}\text{ln}\frac{\text{b}}{\text{a}}-\frac{\left({\text{b}}^{2}-{\text{a}}^{2}\right)}{2}\right]\end{array}$

The radial displacements in the three constituent phases can be assumed as [37]

$wf=Afr, wc=Acr+Bcr and wm=Amr+Bmr$(15)

where Af, Ac, Bc, Am and Bm are unknown constants. Invoking the continuity conditions for the radial displacement at the interfaces r=a and r=b, the radial displacement in the interlayer phase can be augmented as

$wc=a2b2−a2(b2r−r)Af−b2b2−a2(a2r−r)Am −1b2−a2(a2r−r)Bm$(16)

Substituting Eqs. (12), (13), (15) and (16) into Eq. (3) and subsequently, employing the constitutive relations (2), the expressions for the normal stresses can be written in terms of the unknowns Af, Am and Bm as follows:

$σ¯xf=C11f∂uaf∂x+2C12fAf−λ11fΔT$(17)

$σrf=C12fC11fσ¯xf+[C23f+C33f−2(C12f)2C11f]Af−λ22fΔT$(18)

$σxc=C11cC11fσ¯xf−(2C12ca2b2−a2+2C12fC11cC11f)Af+2C12cb2b2−a2Am+2C12cb2−a2Bm+ C11cA1∂τi∂x+C11cA2∂τo∂x−λ11cΔT$(19)

$σrc=C13cC11fσ¯xf+ [C13ca2b2−a2(b2r2−1)+C33ca2b2−a2(−b2r2−1)−2C12fC13cC11f]Af+ [−C13cb2b2−a2(a2r2−1)+C33cb2b2−a2(a2r2+1)]Am−λ22cΔT+ [−C23c1b2−a2(a2r2−1)+C33c1b2−a2(a2r2+1)]Bm+ C13cA1∂τi∂x+C13cA2∂τo∂x$(20)

$σxm=C11mC11fσ¯xf−2C12fC11mC11fAf+2C12mAm+C11mA3∂τi∂x+C11mA4∂τo∂x+ RC66mτ ln(rb)C11m−λ11mΔT$(21)

$σrm=C12mC11fσ¯xf−2C12fC12mC11fAf+(C11m+C12m)Am+(C12m−C11m)Bmr2+ C12mA3∂τi∂x+C12mA4∂τo∂x+RC66mτ ln(rb)C12m−λ22mΔT$(22)

Invoking the continuity conditions ${{\sigma }_{\text{r}}^{\text{f}}|}_{\text{r}=\text{a}}={{\sigma }_{\text{r}}^{\text{c}}|}_{\text{r}=\text{a}}$ and ${{\sigma }_{\text{r}}^{\text{c}}|}_{\text{r}=\text{b}}={{\sigma }_{\text{r}}^{\text{m}}|}_{\text{r}=\text{b}},$ and satisfying the boundary condition ${{\sigma }_{\text{r}}^{\text{m}}|}_{\text{r}=\text{R}}={\text{q}}_{\text{o}},$ the following equations for solving Af, Am and Bm are obtained:

$[A11A12A13A21A22A23A31A32A33]{AfAmBm}=σ¯xfC11f{C12f−C13cC13c−C12m−C12m}+{−C13cA1C13cA1−C12mA3−C12mA3}∂τi∂x +{−C13cA2(C13cA2−C12mA4−C12mA6}∂τo∂x+{001}qo+{00−RC66m ln(Rb)C12m}τ +{λ22c−λ22fλ22m−λ22cλ22m}ΔT$(23)

Solving Eq. (23), the solutions of the constants of the radial displacements Af, Am and Bm can be expressed as:

$Af=k11σ¯xf+k12∂τi∂x+k13∂τo∂x+k14qo+k15τ+k16ΔTAm=k21σ¯xf+k22∂τi∂x+k23∂τo∂x+k24qo+k25τ+k26ΔTBm=k31σ¯xf+k32∂τi∂x+k33∂τo∂x+k34qo+k35τ+k36ΔT$(24)

Now, making use of Eqs. (19), (21) and (24) in the last two equations of (6), respectively, the average axial stresses in the interlayer phase and the polymer matrix phase are written as follows:

$σ¯xc=L1σ¯xf+L2∂τi∂x+L3∂τo∂x+L4qo+L5τ+L6ΔT$(25)

$σ¯xm=L7σ¯xf+L8∂τi∂x+L9∂τo∂x+L10qo+L11τ+L12ΔT$(26)

Now, satisfying the equilibrium of force along the axial (x) direction at any transverse cross section of the RVE, the following equation is obtained:

$πR2σ=π(R2−b2)σ¯xm+π(b2−a2)σ¯xc+πa2σ¯xf$(27)

Differentiating first and last equations of (7) with respect to x, we have

$∂τi∂x=−a2∂2σ¯xf∂x2$(28)

$∂τo∂x=R2−b22b∂2σ¯xm∂x2$(29)

Use of Eqs. (25)–(29), yields

$L13σ¯xf+L14∂2σ¯xf∂x2+L15∂2σ¯xm∂x2+L16qo+L17τ+L18ΔT−R2σ=0$(30)

Deriving the expression for $\frac{\partial {\tau }_{\text{o}}}{\partial {\text{x}}^{2}}$ from Eq. (25) and substituting the same into Eq. (26) and then using Eq. (28), the following result for ${\overline{\sigma }}_{\text{x}}^{\text{m}}$ is obtained:

$σ¯xm=(L6−L1L8L3)σ¯xf+(L8L3)σ¯xc+(a2)(L2L8L3−L7)∂2σ¯xf∂x2+ (L9−L4L8L3)qo+(L10−L5L8L3)ΔT$(31)

Differentiating Eqs. (27) and (31) twice with respect to x and using the resulting equations in Eq. (31), the governing equation for the average axial stress in the carbon fiber coated with radially grown aligned CNTs is obtained as follows:

$∂4σ¯xf∂x4+L17∂2σ¯xf∂x2+L18σ¯xf−L19σ+L20qo+L21ΔT+L22τ=0$(32)

Following the above procedure, the governing equation for the average axial stress $\text{(}{\overline{\sigma }}_{\text{x}}^{\text{pf}}\right)$ in the imaginary fiber made of the polymer material lying in the zones −L≤x≤−Lf and Lf≤x≤L can be written as

$∂4σ¯xpf∂x4+L17pf∂2σ¯xpf∂x2+L18pfσ¯xpf−L19pfσ+L20pfqo+L21pfΔT+L22pfτ=0$(33)

In the above equation, the expressions for ${\text{L}}_{17}^{\text{pf}},$ ${\text{L}}_{18}^{\text{pf}},$${\text{L}}_{19}^{\text{pf}},$ ${\text{L}}_{20}^{\text{pf}}$ and ${\text{L}}_{21}^{\text{pf}}$ are similar to those of expressions L17, L18, L19, L20 and L21, respectively. But these are to be derived by considering ${\text{C}}_{\text{ij}}^{\text{f}}={\text{C}}_{\text{ij}}^{\text{c}}={\text{C}}_{\text{ij}}^{\text{m}}.$ Solutions of Eqs. (32) and (33) are given by:

$σ¯xf=L22sinh(βx)+L23cosh(βx)+L24sinh(αx)+L25sinh(αx)+ (L19/L18)σ−(L20/L18)qo−(L21/L18)ΔT$(34)

$σ¯xpf=L22pfsinh(βpfx)+L23pfcosh(βpfx)+L24pfsinh(αpfx)+ L25pfcosh(αpfx)+(L19pf/L18pf)σ−(L20pfL18pf)qo−(L21L18)τ− (L22pfL18pf)ΔT$(35)

where

$β=1/2(−L17+(L17)2−4L18) , α=1/2(−L17−(L17)2−4L18)βpf=1/2(−L17pf+(L17pf)2−4L18pf) and αpf=1/2(−L17pf−(L17pf)2−4L18pf)$(36)

The constants L22, ${\text{L}}_{22}^{\text{pf}},$ L23, ${\text{L}}_{23}^{\text{pf}},$ L24, ${\text{L}}_{24}^{\text{pf}},$ L25 and ${\text{L}}_{25}^{\text{pf}}$ are to be evaluated from the following end conditions:

$σ¯xpf=σ at x=±L and ∂σ¯xpf∂x=0 at x=±L$(37)

$σ¯xf=σ¯xpf at x=±Lf and ∂σ¯xf∂x=∂σ¯xpf∂x at x=±Lf$(38)

Utilizing the end conditions given by Eq. (37) in Eq. (35), the constants ${\text{L}}_{22}^{\text{pf}},$ ${\text{L}}_{23}^{\text{pf}},$ ${\text{L}}_{24}^{\text{pf}}$ and ${\text{L}}_{25}^{\text{pf}}$ can be obtained. Similarly, utilizing the end conditions given by Eq. (38) in Eq. (34), the constants L22, L23, L24 and L25 can be evaluated. Substitution of Eq. (34) into the first equation of (7) yields the expression for the carbon fiber/interlayer interfacial shear stress as follows:

$τi=−a2[βL22cosh(βx)+βL23sinh(βx)+αL24cosh(αx) +αL25sinh(αx)]$(39)

Following the procedure for deriving the solutions of the average axial stresses ${\overline{\sigma }}_{\text{x}}^{\text{f}}$ and ${\overline{\sigma }}_{\text{x}}^{\text{pf}}$ in the carbon fiber and the imaginary fiber, respectively, the solutions for the average axial stresses ${\overline{\sigma }}_{\text{x}}^{\text{m}}$ and ${\overline{\sigma }}_{\text{x}}^{\text{pm}}$ in the polymer matrices surrounding the short composite fuzzy fiber and the imaginary short composite fuzzy fiber, respectively, can be derived as follows:

$σ¯xm=L39sinh(βmx)+L40cosh(βmx)+L41sinh(αmx) +L42cosh(αmx)+(L36/L35)σ−(L37L35)qo−(L38L35)τ −(L39L35)ΔT$(40)

$σ¯xpm=L39pmsinh(βpmx)+L40pmcosh(βpmx)+L41pmsinh(αpmx) +L42pmcosh(αpmx)+(L36pm/L35pm)σ−(L37pm/L35pm)τ −(L38pm/L35pm)ΔT$(41)

where

$βm=1/2(−L34+(L34)2−4L35) , αm=1/2(−L34−(L34)2−4L35)βpm=1/2(−L34pm+(L34pm)2−4L35pf) and αpm=1/2(−L34pm−(L34pm)2−4L35pm)$(42)

The constants L39, ${\text{L}}_{39}^{\text{pm}},$ L40, ${\text{L}}_{40}^{\text{pm}},$ L41, ${\text{L}}_{41}^{\text{pm}},$ L42 and ${\text{L}}_{42}^{\text{pm}}$ are to be evaluated from the boundary conditions given in Eqs. (37) and (38). Finally, substitution of Eq. (40) into the last equation of (7) yields the expression for the short composite fuzzy fiber/polymer matrix interfacial shear stress as follows:

$τo=(R2−b2b2−a2)[βmL39cosh(βmx)+βmL40sinh(βmx)+ αmL41cosh(αmx)+αmL42sinh(αmx)]$(43)

## 3 Results and discussion

Mori Tanaka model was used to determine the effective thermoelastic properties of the interlayer which are required as input to the shear lag model development; more details can be found in Ref. [38]. Unlike existing techniques, both the fiber and interlayer are assumed to be transversely isotropic and undergo radial as well as axial deformations. The geometrical parameters of the RVE are taken as 2a=10 μm, Lf/a=20, L/Lf=1.1, and R/b=1.1; the volume fractions of carbon fiber and CNTs fixed to 0.4 and 0.019, respectively. For the thermoelastic properties of constituents, the readers are referred to Ref. [38].

The shear lag model derived in Section 2 analyses the interfacial characteristics of the hybrid nanocomposite incorporating different transversely isotropic constituent phases subjected to the thermomechanical loading which have not been considered in the existing shear lag studies. However, it may be imperative to justify the validity of the shear lag model derived in Section 2 considering different transversely isotropic constituent phases and the application of the radial and the thermal loads on the RVE. For this purpose, three-phase finite element (FE) shear lag model was developed using the commercial software ANSYS 11.0 to validate the analytical shear lag model derived herein. With assumed hexagonal RVE packing array, a schematic periodic arrangement of the regularly staggered RVEs and the three-phase FE model of the RVE are presented in Figures 4 and 5, respectively. It should be noted that because of the symmetry, one half of the RVE is considered for the finite element simulations. Under the conditions of an imposed tensile stress (σ), radial normal stress (q) and thermal load (ΔT) on the RVE of the hybrid nanocomposite, the average stresses {σ̅i} in the i-th phase of the RVE can be obtained as

Figure 4:

A schematic arrangement of the RVEs in the hybrid nancomposite.

Figure 5:

Three-dimensional FE mesh of the three-phase RVE.

${σ¯i}=1Vi∫{σi}dVi ; i=f, c and m$(44)

where Vi represents the volume of the i-th phase of the RVE and the field variable with an overbar represents the average of the field variable. The comparisons of the normalized average axial stress in the carbon fiber and the interfacial shear stress along its length computed by the analytical shear lag model and the finite element shear lag model are presented in Figures 6 and 7, respectively, when the magnitude of the temperature variation, ΔT=300 K. It may be observed from Figures 6 and 7 that if the staggering effect of the adjacent RVEs is ignored (i.e. τ=0), the analytical shear lag model over-estimates the values of average axial stress in the carbon fiber and the interfacial shear stress along its length. It may also be observed that the good agreement between the two sets of results have been obtained and thus verifying the reliability of the analytical shear lag model incorporating the staggering effect and the thermal load.

Figure 6:

Analytical shear lag model validation by comparison to FE shear lag model for the axial stress in the carbon fiber along its length.

Figure 7:

Analytical shear lag model validation by comparison to FE shear lag model for the interfacial shear stress along the carbon fiber length.

The variations of the normalized axial stress in the carbon fiber and the interfacial shear stresses along its length are presented in Figures 8 and 9, respectively, for different values of applied thermal loads on the RVE. It may be observed from Figure 8 that the maximum value of axial stress in the carbon fiber is significantly decreased with the decrease in the magnitude of temperature change. This is attributed to the fact that the effective thermoelastic properties of the interlayer surrounding the fiber are improved with the decrease in the magnitude of temperature variation. On the other hand, the magnitude of temperature change does not much influence the value of interfacial shear stresses along the length of a carbon fiber. Figures 8 and 9 reveal that the values of axial stress in the carbon fiber remains uniform over 90% length of the fiber from its center while it decreases sharply near the end of the fiber. On the other hand, the interfacial shear stresses along the length of a carbon fiber reaches their maximum values near the ends of the fiber and become zero at x=±Lf.

Figure 8:

Variation of the axial stress in the carbon fiber along its length.

Figure 9:

Variation of the transverse shear stress at the interface between the interlayer and the carbon fiber along its length.

## 3 Conclusions

A shear lag model developed herein provides an explanation to the thermomechanical load transfer mechanisms in three-phase hybrid nanocomposite considering transversely isotropic constituents and staggering effect which have not been previously considered elsewhere. The current study reveals that (i) the thermal loads significantly affect the average axial stress transferred to the microscale fiber, (ii) the existence of shear tractions along the length of the RVE play a crucial role in the interfacial characteristics of nanocomposite, and (iii) CNTs allow us to exploit their remarkable thermoelastic properties to improve the thermomechanical behavior of hybrid nanocomposite structures, enabling additional functionalities not available otherwise at the microscale. The three-phase shear lag model developed in this study is capable of analyzing the mechanisms of load transfer between the orthotropic constituent phases of any hybrid composite subjected to 3-D thermomechanical loads.

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Published Online: 2017-11-02

Published in Print: 2017-12-20

Citation Information: Journal of the Mechanical Behavior of Materials, Volume 26, Issue 3-4, Pages 95–103, ISSN (Online) 2191-0243, ISSN (Print) 0334-8938,

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©2017 Walter de Gruyter GmbH, Berlin/Boston.

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