Generally cylindrical orthotropic constitutive modeling of matrix-filled carbon nanotubes: Transverse mechanical properties and responses

Abstract

The main objective of this article is to introduce exact analytical closed-form solutions for the prediction of effective transverse Young’s modulus and Poisson ratio of a matrix-filled nanotube (i.e., a representative element of nanotube-based nanocomposites), as well as its mechanical behavior, when subjected to external loads. In this work, both the nanotube and its filler were considered to be generally cylindrical orthotropic. To ensure no loss of generality, the no plane strain condition was used, and the axial strain was taken into consideration to obtain a more precise set of solutions. Analytical formulae were developed based on the well-established principles of linear elasticity and continuum mechanics, considering effective orthotropic properties for both constituents as continuum tubes. To validate and verify the accuracy of the closed-form solutions obtained from the analytical approach, a three-dimensional finite element analysis was performed, and results were compared to those obtained from the analytical exact solutions. Excellent agreement was achieved, and the analytically obtained solutions were verified.

Keywords

Carbon nanotubes nanocomposites orthotropic constituents finite element analysis analytical solutions effective transverse properties

Introduction

The discovery of nanomaterials with their extraordinary properties and diverse structural configurations has provided opportunities for design and fabrication of nanocomposite material systems with improved and tunable properties. Carbon nanotubes (CNTs), with their unique structural (i.e., single- and multiwalled) and geometrical configurations (i.e., armchair, chiral, and zigzag) have shown superior properties compared to traditional reinforcements [1 –6]. CNTs are made of hexagonal graphite honeycomb lattice of mono-atomic layers that do not always exist in perfect hexagonal structures. This means that they may have structural defects (e.g., Stone–Wales defect) that can alter their properties [7]. In addition, they can be filled or surrounded by other materials in different forms (e.g., particles, filaments, and aggregates) and states (i.e., gas, liquid, and solid).

Based on previously published theoretical and experimental reports, values of nearly 1 TPa and 63 GPa for the axial Young’s modulus and axial tensile strength of CNTs, respectively, have been agreed upon [3 –5, 7–10]. Experimental and theoretical investigations have shown that the radial elastic modulus of CNTs decreases with the increase in tube dimensions (e.g., tube diameter and number of CNT layers), and in the circumferential direction, the elastic modulus of CNTs is nearly equal to the modulus in the axial direction [10 –14]. Using atomic force microscopy (AFM) and molecular dynamics (MD) simulation, Palaci et al. [10] reported a radial compressive Young’s modulus of 30 ± 10 GPa for CNTs with external radii of less than 5 nm. Shen et al. [11] determined that the radial compressive elastic modulus and strength for multiwalled CNTs at different compression levels ranged from 9.7 to 80.0 GPa and 5.3 GPa, respectively. Furthermore, molecular structural mechanics, MD, and continuum mechanics based on plane strain theory were used to study radial deformation and elastic properties of CNTs subjected to hydrostatic external uniform pressure [12, 13]. Owing to their excellent mechanical and physical properties, CNTs are widely used as reinforcement to improve the mechanical, thermal, and electrical properties of composites [1, 2, 14, 15].

CNTs can be opened using physical and chemical techniques, and their quasi-one-dimensional cylinder-like cavity can be partially or fully filled with various atoms and molecules (e.g., Ag, Au, Pt, Fe, Ni, noble gases, and polymer) and then be closed following chemical or physical routes [16 –20]. These hybrid molecular nanoscale structures (i.e., filled or surface coated) with their tunable functionality have great potential for a wide range of applications, for example, energy and storage systems, nanoelectromechanical systems, nanosensors, optoelectronics, and structural nanocomposites [17 –20].

Nanocomposite tubes with their tunable properties are often incorporated in a host material to provide further reinforcement. Generally, the host materials must be cured and solidified, whereupon substantial shrinkage and residual compressive stresses can develop. As a result, the reinforcement materials (i.e., filled/coated nanotubes) will be subjected to compressive external stresses that will translate into complicated internal stresses, due to the materials anisotropy of the constituents that are at nanoscale. In addition, upon application of external loads on a nanocomposite materials system, the tubular nanocomposite unit cells will undergo substantial stress, whereby the loads will be transferred to the reinforcement materials (i.e., filled nanotubes) through the host matrix. Thus, to design an optimal material system for high-performance structural applications, it is essential to know the effective properties and behavior of the individual constituents (i.e., filled/encapsulated nanotubes). On the other hand, experimental studies of the behavior of filled nanotubes (subjected to various loading conditions) require very sophisticated tools and expensive test procedures that often are impractical. Therefore, theoretical and analytical approaches are very beneficial and have great importance in predicting the mechanical behavior of such anisotropic structures and material systems.

There have been many studies to analytically, numerically, and experimentally investigate the effective properties of traditional composites, nanocomposites, and their constructing unit cells [21 –39]. As a result, various micromechanical models are available to predict the macroscopic behaviors of traditional fiber-reinforced composites [e.g., 21 –26,37], some of which assume that both fiber and matrix have isotropic or transversely isotropic properties [26]. These assumptions may not be acceptable in many cases, especially for micro- and nanoscale materials and structures that show orthotropic behavior. One of the most used models is the composite cylinders model introduced by Hashin and Rosen [21] and Whitney and Riley [22] who employed the classical theory of linear elasticity. To obtain the transversal shear modulus of a fiber/matrix system, Christensen [23] used a model that was closely related to the composite cylinders model [21]. Liu et al. [24] employed an advanced boundary element method based on elasticity theory to model interphases in unidirectional fiber-reinforced composites under transverse loading. Siboni [25] used a generalized self-consistent scheme to evaluate the elastic constant of unidirectional multiphase reinforced materials.

A number of studies have used finite element analysis (FEA) and MD simulations to investigate the effective physical and mechanical properties of CNT-reinforced nanocomposites [40 –44]. Ansari and Hassanzadeh Aghdam [45, 46] developed a micromechanical model to study the effective viscoelastic response and creep recovery of the CNT-reinforced nanocomposites. Using a multiscale simulation model, Tsai et al. [47] characterized the effective elastic properties of CNT-reinforced polyimide nanocomposites. In their approach, the molecular structure of the CNT was modeled as a transversely isotropic hollow cylindrical solid. Most of these models basically assumed that the fiber, matrix, and interphase are continuous, isotropic or transversely isotropic, and homogenous. Therefore, the constitutive relations for the bulk composites were formulated based on the assumptions of continuum mechanics. Liu and Chen [27] and Chen and Liu [28] used a three-dimensional (3D) nanoscale representative volume element based on continuum mechanics to evaluate the effective properties of CNT-based nanocomposites assuming isotropic properties for the constituents. Giannopoulos et al. [48] developed a micromechanical finite element model to estimate the effective Young’s modulus of uniformly distributed and aligned single-walled CNT-reinforced composites.

Vanin and Duc [49, 50] proposed fundamental equations for theory of fibrous composites reinforced with additional spherical inclusions. Duc et al. [51 –54], Thanh et al. [55], and Van Thu and Duc [56] studied the thermo-mechanical stability, static, nonlinear dynamic, and vibration responses of functionally graded CNTs-reinforced nanocomposite plates and shells surrounded by elastic foundations. Another study have investigated the nonlinear dynamic and vibration responses of imperfect CNTs-reinforced nanocomposite shells subjected to blast loading and temperature variations [57].

Kalamkarov et al. [26] used linear elastic theory and continuum mechanics to derive exact analytical solutions for the effective axial Young’s modulus, major Poisson’s ratios, displacements, strains, and stress distributions for a two-phase composite cylinder subjected to an axial load. In their work, both constituents were considered to be cylindrically orthotropic (i.e., each with nine independent constants) and perfectly bonded at the interface.

In the work here, a theoretical approach was developed based on the principles of linear elasticity and continuum mechanics to analytically predict the transverse mechanical properties and the mechanical behavior of a two-phase nanocomposite tube (i.e., consisting of two orthotropic concentric tubes, representing an inside filled CNT) subjected to uniformly applied external radial pressure. Each orthotropic concentric tube has nine different independent materials properties (e.g., E₁₁, E₂₂, E₃₃, G₁₂, G₁₃, G₂₃, ν₁₂, ν₁₃, and ν₂₃). Note that to ensure no loss of generality, the no plane strain assumption was used in this work. At this stage of the research, it was assumed that the effective properties of the constituents are known, and perfect bonding exists at the interface of the concentric tubes. In future work, the interface will also be modeled as a separate concentric tube with distinct orthotropic properties.

To verify the analytical solutions, an identical 3D structure of a two-phase composite cylinder was modeled using FEA. Results from both analytical solutions and FEA were compared, and excellent agreement was achieved, thus validating the analytical solutions. The solutions provided in this paper are valid and applicable to any two-phase cylindrical composite structure composed of constituents with orthotropic properties. It should be mentioned that considering orthotropic properties for the constituents and assuming a no plane strain condition make this work unique. Earlier analytical solutions [22, 23] were developed for transversely isotropic or isotropic constituents based on the plane strain condition assumption.

Analytical modeling of two-phase nanocomposite cylinder

This research considered a two-phase cylindrical composite model subjected to a uniformly distributed external radial load with both ends free to expand/contract in the axial direction (see Figure 1). This means that axial strain will not be neglected or, more precisely, that the plane strain condition will not be used. Both constituents, that is, inside filler (i) and outside nanotube (o), were considered as two different generally cylindrical orthotropic materials, which will yield a set of complete orthotropic solutions. Later on, it is possible to reduce these orthotropic analytical solutions to transversely isotropic and isotropic cases. Therefore, the analytical solutions provided here are applicable to any similar two-phase structure consisting of materials with anisotropy levels up to orthotropic levels. It should be noted that although common matrix materials used in composites have isotropic properties at macroscale, the filler material of the nanotube could be in the form of nanofibers or nanorods, potentially possessing orthotropic properties at nanoscale. The outside radii of the inner filler (e.g., matrix) and the outside tube (e.g., CNT) are denoted by b and c , respectively (see Figure 1). That is, the inner black tube represents the filler, and the outside white tube represents the CNT.

Derivation of exact analytical solutions for displacement fields, strain components, and stress distributions

This is an axisymmetric problem with no dependence on the polar angle θ . The only equilibrium equation to be satisfied in this case (since all others are satisfied) is [58]

\frac{d σ_{r}}{d r} + \frac{σ_{r} - σ_{θ}}{r} = 0

(1)

Considering orthotropic properties for both cylinders, that is, inside filler (i) and outside nanotube (o), the constitutive relations can be written

{\begin{cases} σ_{x}^{(i)} = C_{11}^{(i)} ε_{x}^{(i)} + C_{12}^{(i)} ε_{θ}^{(i)} + C_{13}^{(i)} ε_{r}^{(i)} \\ σ_{θ}^{(i)} = C_{12}^{(i)} ε_{x}^{(i)} + C_{22}^{(i)} ε_{θ}^{(i)} + C_{23}^{(i)} ε_{r}^{(i)} \\ σ_{r}^{(i)} = C_{13}^{(i)} ε_{x}^{(i)} + C_{23}^{(i)} ε_{θ}^{(i)} + C_{33}^{(i)} ε_{r}^{(i)} \end{cases}

(2a)

{\begin{cases} σ_{x}^{(o)} = C_{11}^{(o)} ε_{x}^{(o)} + C_{12}^{(o)} ε_{θ}^{(o)} + C_{13}^{(o)} ε_{r}^{(o)} \\ σ_{θ}^{(o)} = C_{12}^{(o)} ε_{x}^{(o)} + C_{22}^{(o)} ε_{θ}^{(o)} + C_{23}^{(o)} ε_{r}^{(o)} \\ σ_{r}^{(o)} = C_{13}^{(o)} ε_{x}^{(o)} + C_{23}^{(o)} ε_{θ}^{(o)} + C_{33}^{(o)} ε_{r}^{(o)} \end{cases}

(2b)

Superscripts (i) and (o) represent the inside and outside tubes, respectively; and subscripts ( x ), ( θ ), and ( r ) represent the axial, circumferential, and radial directions, respectively (see Figure 1). Parameters σ_i , ε_j , and C_ij in equations (2a) and (2b) are stresses, strains, and cylindrical orthotropic elastic constants, respectively. Due to the axisymmetry of the problem, the reduced form of the strain–displacement relations can be written as [26]

{\begin{cases} ε_{r} = \frac{\partial w}{\partial r}, \\ ε_{θ} = \frac{w}{r}, \\ ε_{x} = \frac{\partial u}{\partial x} \end{cases}

(3)

where

w = w (r)

and u represent the radial and axial displacements, respectively. Since this composite cylinder is only subjected to radial uniform external pressure, the axial strain, ε_x , does not vary with the radial coordinate r , (i.e., iso-strain condition) and hence,

ε_{x}^{(i)} = ε_{x} {(o)}^{=} ε = \partial u / \partial x = a

constant for both inner and outer tubes. Substitution of the strain–displacement relations (equation (3)) into the constitutive equation (equation (2)) and then substitution of the results into the equilibrium equation (equation (1)) yields a nonhomogeneous second-order ordinary linear differential equation [25,28,57]. Equations (4) and (5) are the general solutions for radial displacement fields of both domains (see equations (31) to (35) in Appendix 1 for details)

w^{(i)} = A_{i} r^{λ_{i}} + r μ_{i} ε, where λ_{i} = \sqrt{\frac{C_{22}^{(i)}}{C_{33}^{(i)}}}, μ_{i} = \frac{C_{12}^{(i)} - C_{13}^{(i)}}{C_{33}^{(i)} - C_{22}^{(i)}}

(4)

w^{(o)} = A_{o} r^{λ_{o}} + B_{o} r^{- λ_{o}} + r μ_{o} ε, where λ_{o} = \sqrt{\frac{C_{22}^{(o)}}{C_{33}^{(o)}}}, μ_{o} = \frac{C_{12}^{(o)} - C_{13}^{(o)}}{C_{33}^{(o)} - C_{22}^{(o)}}

(5)

Note that in the inner filler region, ${w^{(i)} |}_{r = 0} = 0$ ; hence, using the inner finiteness boundary condition, B_i entering equation (4) is set to zero. Here, it is assumed that both of the concentric tubes are perfectly bonded at the interface without any interface discontinuity. The condition of $u^{(i)} (b) = u^{(o)} (b)$ has already been satisfied (i.e., $ε_{x} {(i)}^{}$ = $ε_{x} {(o)}^{}$ = $ε$ = constant). The three constants (i.e., A_i _, A_o _, and B_o ) entering the general solutions in equations (4) and (5) are to be determined from the following boundary conditions

w^{(i)} (b) = w^{(o)} (b)

(6a)

σ_{r}^{(i)} (b) = σ_{r}^{(o)} (b)

(6b)

σ_{r}^{(o)} (c) = p

(6c)

The first boundary condition in equation (6a) means that the radial displacements of both inner and outside tubes are equal at the interface (i.e., r = b). That is, the interface between the inner and outside tubes is continuous with no separation. The second boundary condition equation (6b) states that the radial stresses at the interface of the constituents are equal due to equilibrium. The third boundary condition equation (6c) states that the radial stress of the outside tube at the outer surface is equal to the radially applied uniform external pressure, which again is related to the equilibrium of the system. Substituting equations (4) and (5) into the strain relations given by equation (3) and then substituting the results into the stress relations given by equation (2), the strain and stress equations for both inside and outside tubes can be rewritten in terms of ε , C_ij , r , A_i _, A_o _, and B_o (see equations (36) to (39) in Appendix 1 for details).

Next, employing these new displacement, strain, and stress equations and the boundary conditions in equation (6) yields a system of three equations with three unknowns (see equation (40) in Appendix 1 for details) from which three unknown coefficients (i.e., A_i , A_o, and B_o ) will be obtained as

{\begin{cases} A_{i} = (L_{4} L_{3}^{- 1} b^{- λ_{i}}) ε + (L_{3}^{- 1} b^{- λ_{i}}) p c^{λ_{o} + 1} \\ A_{o} = (b^{- λ_{o}} H_{2}) ε + (b^{- λ_{o}} H_{1}) p c^{λ_{o} + 1} \\ B_{o} = (L_{4} L_{3}^{- 1} - H_{2} + (μ_{i} - μ_{o}) b) b^{λ_{o}} ε + (L_{3}^{- 1} - H_{1}) b^{λ_{o}} p c^{λ_{o} + 1} \end{cases}

(7)

To keep the equations more concise and easier to follow, new parameters (i.e., L ₁ , L ₂ , L ₃ , L ₄ , H ₁ , and H ₂ entering the above equations) were introduced in terms of C_ij , μ_i , μ_j , λ_i , λ_j , b , and c (for details, see equations (41) to (43) in Appendix 1). Therefore, all three constant coefficients A_i , A_o, and B_o were obtained in terms of constituent geometries, material properties, constant axial strain, ε , and the externally applied uniform pressure, p . Since the two-phase composite cylinder was subjected only to a radial pressure and no axial load, the following boundary condition was used to obtain the relationship between the axial strain and the radial pressure

F_{x} = 2 π [\int_{0}^{b} σ_{x}^{(i)} r d r + \int_{b}^{c} σ_{x}^{(o)} r d r] = 0

(8)

where F_x , is the total axial load in the x -direction that is zero;

σ_{x}^{(i)} and σ_{x}^{(o)}

are the axial stress components in the inside and outside tube domains, respectively; and b , c , and r are the radius of the inner tube, outer radius of the outside tube, and radial coordinate, respectively. It can be concluded that the average axial stress,

{\bar{σ}}_{x}

, for this problem is also zero

{\bar{σ}}_{x} = \frac{F_{x}}{A} = \frac{F_{x}}{π c^{2}} = 0

(9)

By substituting the stress equations from the previous steps, that is, equations (38) and (39), into the integrals in equation (8), the results can be expanded to obtain the expression for total axial load, F_x , (see equation (44) in Appendix 1). As a result, equations (8) and (9) along with equation (44) were solved to obtain the solution for axial strain, ε , as follows

ε = Δ p

(10)

\begin{array}{l} Δ = - (L_{3}^{- 1} k_{4}^{(i)} β_{1} + H_{1} k_{4}^{(o)} β_{2} b^{- λ_{o}} + (L_{3}^{- 1} - H_{1}) k_{5}^{(o)} β_{3} b^{λ_{o}}) c^{λ_{o} + 1} \\ \times (k_{6}^{(i)} / 2 + k_{6}^{(o)} (c^{2} - b^{2}) / 2 + L_{4} L_{3}^{- 1} k_{4}^{(i)} β_{1} + H_{2} k_{4}^{(o)} β_{2} b^{- λ_{o}} \\ {+ (L_{4} L_{3}^{- 1} - H_{2} + (μ_{i} - μ_{o}) b) k_{5}^{(o)} β_{3} b^{λ_{o}})}^{- 1} \end{array}

(11)

The new parameter Δ entering the axial strain solution, equation (10) is introduced in equation (11), and the new parameters entering the Δ equation (i.e., β ₁ , β ₂ , β ₃ , k ₄ , k ₅ , and k ₆ ) are defined in terms of C_ij , μ_i , μ_j , λ_i , λ_j , b , and c (for details, see equations (45) and (46) in Appendix 1). Now that the coefficients A_i , A_o, and B_o and the axial constant strain, ε , are defined in terms of tube geometries, material properties, and pressure, p , exact analytical solutions for the displacements, strains, and stress distributions within the domain of each individual concentric cylinder can explicitly be obtained. Note that this is an iso-strain problem, and axial displacements and axial strains are equal for both the concentric tubes at any cross section along the x -direction (see Figure 1). As mentioned earlier, the circumferential displacement, u_θ , was zero for both constituent materials, due to the axisymmetric nature of the problem. Radial displacements were nonlinear functions of the radial coordinate, r . The substitution of coefficients given by equation (7) into equations (4) and (5) yields the exact analytical solutions for displacement as follows

{\begin{cases} w^{(i)} = ((L_{4} Δ + c^{λ_{o} + 1}) L_{3}^{- 1} b^{- λ_{i}} r^{λ_{i}} + r μ_{i} Δ) p \\ w^{(o)} = ((H_{2} Δ + H_{1} c^{λ_{o} + 1}) b^{- λ_{o}} r^{λ_{o}} \\ + ((L_{4} L_{3}^{- 1} - H_{2} + (μ_{i} - μ_{o}) b) Δ + (L_{3}^{- 1} - H_{1}) c^{λ_{o} + 1}) b^{λ_{o}} r^{- λ_{o}} + r μ_{o} Δ) p \\ u_{x}^{(i)} = u_{x}^{(o)} = u = x Δ p \\ u_{θ}^{(i)} = u_{θ}^{(o)} = 0 \end{cases}

(12)

Similarly, equations (7) and (10) can be substituted into strain relations (i.e., equations (36) and (37) in Appendix 1) to obtain the exact analytical solutions for the strain components, that is, axial, circumferential, and radial strains, within the domain of each concentric cylinder, as

{\begin{array}{l} ε_{x}^{(i)} = ε = Δ p \\ ε_{θ}^{(i)} = ((L_{4} Δ + c^{λ_{o} + 1}) L_{3}^{- 1} b^{- λ_{i}} r^{λ_{i} - 1} + μ_{i} Δ) p \\ ε_{r}^{(i)} = ((L_{4} Δ + c^{λ_{o} - 1}) L_{3}^{- 1} b^{- λ_{i}} λ_{i} r^{λ_{i} - 1} + μ_{i} Δ) p \end{array}

(13)

{\begin{array}{l} ε_{x}^{(o)} = ε = Δ p \\ \begin{matrix} ε_{θ}^{(o)} = ((H_{2} Δ + H_{1} c^{λ_{o} + 1}) b^{- λ_{o}} r^{λ_{o} - 1} \\ + ((L_{4} L_{3}^{- 1} - H_{2} + (μ_{i} - μ_{o}) b) Δ + (L_{3}^{- 1} - H_{1}) c^{λ_{o} + 1}) b^{λ_{o}} r^{- λ_{o} - 1} + μ_{o} Δ) p \end{matrix} \\ \begin{matrix} ε_{r}^{(i)} = ((H_{2} Δ + H_{1} c^{λ_{o} + 1}) b^{- λ_{o}} λ_{o} r^{λ_{o} - 1} \\ - ((L_{4} L_{3}^{- 1} - H_{2} + (μ_{i} - μ_{o}) b) Δ + (L_{3}^{- 1} - H_{1}) c^{λ_{o} + 1}) b^{λ_{o}} λ_{o} r^{- λ_{o} - 1} + μ_{o} Δ) p \end{matrix} \end{array}

(14)

Likewise, using equations (38) and (39), solutions for the stress distributions were obtained as

{\begin{cases} σ_{x}^{(i)} = (k_{6}^{(i)} Δ + (L_{4} Δ + c^{λ_{o} + 1}) L_{3}^{- 1} b^{- λ_{i}} k_{4}^{(i)} r^{λ_{i} - 1}) p \\ σ_{θ}^{(i)} = (k_{9}^{(i)} Δ + (L_{4} Δ + c^{λ_{o} + 1}) L_{3}^{- 1} b^{- λ_{i}} k_{7}^{(i)} r^{λ_{i} - 1}) p \\ σ_{r}^{(i)} = (k_{10}^{(i)} Δ + (L_{4} Δ + c^{λ_{o} + 1}) L_{3}^{- 1} b^{- λ_{i}} k_{1}^{(i)} r^{λ_{i} - 1}) p \end{cases}

(15)

{\begin{matrix} σ_{x}^{(o)} = (k_{6}^{(o)} Δ + (H_{2} Δ + H_{1} c^{λ_{o} + 1}) b^{- λ_{o}} k_{4}^{(o)} r^{λ_{o} - 1} \\ + ((L_{4} L_{3}^{- 1} - H_{2} + (μ_{i} - μ_{o}) b) Δ + (L_{3}^{- 1} - H_{1}) c^{λ_{o} + 1}) b^{λ_{o}} k_{5}^{(o)} r^{- λ_{o} - 1}) p \\ σ_{θ}^{(o)} = (k_{9}^{(o)} Δ + (H_{2} Δ + H_{1} c^{λ_{o} + 1}) b^{- λ_{o}} k_{7}^{(o)} r^{λ_{o} - 1} \\ + ((L_{4} L_{3}^{- 1} - H_{2} + (μ_{i} - μ_{o}) b) Δ + (L_{3}^{- 1} - H_{1}) c^{λ_{o} + 1}) b^{λ_{o}} k_{8}^{(o)} r^{- λ_{o} - 1}) p \\ σ_{r}^{(o)} = (k_{10}^{(o)} Δ + (H_{2} Δ + H_{1} c^{λ_{o} + 1}) b^{- λ_{o}} k_{1}^{(o)} r^{λ_{o} - 1} \\ + ((L_{4} L_{3}^{- 1} - H_{2} + (μ_{i} - μ_{o}) b) Δ + (L_{3}^{- 1} - H_{1}) c^{λ_{o} + 1}) b^{λ_{o}} k_{2}^{(o)} r^{- λ_{o} - 1}) p \end{matrix}

(16)

The new parameters entering equation (16) (i.e., k ₇ , k ₈ , k ₉ , and k ₁₀ ) were introduced in terms of C_ij , μ_i , μ_j , and λ_i to keep the equations more concise and easier to follow. For details, see equations (45) and (46) in Appendix 1.

A close examination of the final exact solutions for displacements, strains, and stresses, that is, equations (12) to (15) show that all of them were derived in terms of material properties, tube geometry, and external uniform radial pressure. Therefore, given the properties and dimensions of the constituents, these analytical closed-form solutions can be used to exclusively obtain 3D results for displacements, strains, and stress distributions throughout any two-phase composite cylindrical tube subjected to an externally applied uniform radial pressure, p . Next, Hook’s law was used to derive the formulae for effective radial Young’s modulus and Poisson’s ratios.

Derivation of generally orthotropic effective radial Young’s modulus $(E_{r r}^{e f f})$

To derive the formulae for the effective radial Young’s modulus, $E_{r r}^{e f f}$ , a uniform radial pressure, p , was applied to the two-phase composite cylinder (shown in Figure 1). Using the strain–stress relations in a cylindrical coordinate system for an orthotropic material subjected to axisymmetric loading (i.e., shear strains and stresses are zero), the basic definition for the effective radial Young’s modulus of an orthotropic cylindrical solid can be defined as [58, 59] (for details, see equations (48) and (49) in Appendix 1)

E_{r r}^{e f f} = \frac{1}{{\bar{ε}}_{r r}} ({\bar{σ}}_{r r} - ν_{r θ}^{e f f} {\bar{σ}}_{θ θ})

(17)

where

{\bar{σ}}_{r r} a n d {\bar{σ}}_{θ θ}

are the average radial and circumferential stresses, respectively, and

{\bar{ε}}_{r r}

is the average radial strain. To proceed with the solution, the effective Poisson’s ratio,

ν_{r θ}^{e f f}

, can be obtained from the following rule of mixtures relation suggested by Tsai and Thomas Hahn [60] and Halpin and Tsai [61] and also used by Whitney and Riley [22]

ν_{r θ}^{e f f} = η_{i} ν_{r θ}^{(i)} + η_{o} ν_{r θ}^{(o)}

(18)

where

η_{i} = \frac{b^{2}}{c^{2}}

and

η_{o} = 1 - \frac{b^{2}}{c^{2}}

are volume contents of the inner material, i , and outer material, o , respectively, for which

η_{i} + η_{o} = 1

is evident. To obtain the explicit closed-form solution for the effective radial Young’s modulus, average radial and circumferential stresses (

{\bar{σ}}_{r r} and {\bar{σ}}_{θ θ}

) and average radial strain (

{\bar{ε}}_{r r}

) were derived first, using the following basic definitions [58]

{\bar{σ}}_{θ} = \frac{1}{c} \int_{0}^{c} σ_{θ} d r = \frac{1}{c} [\int_{0}^{b} σ_{θ}^{(i)} d r + \int_{b}^{c} σ_{θ}^{(o)} d r]

(19)

{\bar{σ}}_{r} = \frac{1}{c} \int_{0}^{c} σ_{r} d r = \frac{1}{c} [\int_{0}^{b} σ_{r}^{(i)} d r + \int_{b}^{c} σ_{r}^{(o)} d r]

(20)

{\bar{ε}}_{r r} = \frac{1}{c} \int_{0}^{c} ε_{r r} d r = \frac{1}{c} [\int_{0}^{b} ε_{r r}^{(i)} d r + \int_{b}^{c} ε_{r r}^{(o)} d r]

(21)

The displacements, strains, and stress solutions, that is, equations (12) to (16), which were obtained in the previous section, were substituted in the above definitions to obtain the explicit formulas for the average stresses and the average radial strain (for details, see equations (50) to (55) in Appendix 1).

Finally, the substitution of average circumferential and radial stresses and average radial strain (i.e., equations (50), (51), and (55), respectively, in Appendix 1) and equation (18) into the basic definition given by equation (17) yields

\begin{array}{l} E_{r r}^{e f f} = [((k_{10}^{(i)} - ν_{r θ}^{e f f} k_{9}^{(i)}) b + (k_{10}^{(o)} - ν_{r θ}^{e f f} k_{9}^{(o)}) (c - b)) Δ + (k_{1}^{(i)} - ν_{r θ}^{e f f} k_{7}^{(i)}) \\ \times (L_{4} Δ + c^{λ_{o} + 1}) L_{3}^{- 1} λ_{i}^{- 1} + (k_{1}^{(o)} - ν_{r θ}^{e f f} k_{7}^{(o)}) (H_{2} Δ + H_{1} c^{λ_{o} + 1}) \times (c^{λ_{o}} b^{- λ_{o}} - 1) λ_{o}^{- 1} \\ - (k_{2}^{(o)} - ν_{r θ}^{e f f} k_{8}^{(o)}) ((L_{4} L_{3}^{- 1} - H_{2} + (μ_{i} - μ_{o}) b) Δ + (L_{3}^{- 1} - H_{1}) c^{λ_{o} + 1}) λ_{o}^{- 1} (γ - 1)] \\ \times {[(H_{2} Δ + H_{1} c^{λ_{o} + 1}) (γ^{- 1} - γ) + (L_{4} Δ + c^{λ_{o} + 1}) L_{3}^{- 1} γ + ((μ_{i} - μ_{o}) γ b + c μ_{o}) Δ]}^{- 1} \end{array}

(22)

which is the closed-form exact analytical solution for the effective radial Young’s modulus of a two-phase composite cylinder consisting of two orthotropic concentric tubes having orthotropic material properties. The parameters used in the above solution are all introduced in Appendix 1 and

γ = \frac{b^{λ_{o}}}{c^{λ_{o}}}

, which is dimensionless.

Derivation of generally orthotropic effective poisson’s ratio ( $ν_{r x}^{e f f}$ )

To derive the formulae for the effective Poisson’s ratio, the following definition was used [58]

ν_{r x}^{e f f} = - \frac{{\bar{ε}}_{x x}}{{\bar{ε}}_{r r}} = - \frac{ε}{{\bar{ε}}_{r r}}

(23)

The substitution of the expression for the average radial strain (i.e., equation (55) in Appendix 1) into the equation (23) yields

ν_{r x}^{e f f} = - Δ c {[(H_{2} Δ + H_{1} c^{λ_{o} + 1}) (γ^{- 1} - γ) + (L_{4} Δ + c^{λ_{o} + 1}) L_{3}^{- 1} γ + ((μ_{i} - μ_{o}) γ b + c μ_{o}) Δ]}^{- 1}

(24)

which is the closed-form exact analytical solution for the effective Poisson’s ratio

ν_{r x}^{e f f}

of a two-phase composite cylinder consisting of two orthotropic concentric tubes.

The next section of this article presents a numerical FEA of an identical two-phase nanocomposite cylinder to examine the accuracy of the above analytical solutions.

FEA modeling of two-phase nanocomposite cylinder

ANSYS [62] finite element modeling was used to generate a 3D model (i.e., one-eighth of the full model, due to the axisymmetric nature of the problem) and to perform the load displacement, strain, and stress analysis (see Figure 3 in Appendix 2). The goal was to compare the results obtained analytically (explained in previous section) with those obtained numerically (explained in this section) to verify the correctness and accuracy of the analytical solutions. For the length ( L ) inner radius ( b ), and outside radius ( c ) of the composite cylinder, dimensions of 5 nm, 0.25 nm, and 0.27 nm, respectively, were used [4, 5, 7] (see Figures 1 and 3). It should be mentioned that these dimensions were chosen based on prior work [4, 5, 7] that was related to theoretical modeling of a single-walled carbon nanotube (SWCNT). These dimensions were chosen as an example for calculating the results from analytical solutions and comparing with FEA results. The typical diameter and length of SWCNTs can be considerably larger than the above values, and they take a much longer FEA computational time. To simulate the real testing conditions (i.e., uniform axial displacements for both concentric tubes), all nodes at both ends of the nanocomposite model (i.e., at x = 0 and x = L/2) were coupled. The SOLID45 element was chosen from the ANSYS element library, and then material properties were assigned to the inner filler matrix (i.e., E⁽ⁱ⁾ = 3.45 GPa and ν⁽ⁱ⁾ = 0.358) [59] and the outside CNT (o) [4, 5, 7, 63] as follows

{\begin{array}{l} E^{(o)} 11_{=} 1.14 (T P a); E^{(o)} 2_{2} = 0.848 (T P a); E^{(o)} 3_{3} = 23.485 (T P a) \\ G^{(o)}_{12} = 0.387 (T P a); G^{(o)} 2_{3} = 1.028 (T P a); G^{(o)}_{13} = 0.774 (T P a) \\ ν^{(o)}_{12} = 0.3; ν^{(o)} 2_{3} = 0.03; ν^{(o)}_{13} = 0.03 \end{array}

(25)

The 1–2–3 axes are shown in Figure 1. Next, the generated volumes in the FEA were meshed by considering a proper mesh element size and type. The symmetric boundary conditions (i.e., nodes at each cut surface coupled together to move/deform in plane only, with zero out-of-plane displacements) were applied at the symmetric cut surfaces of the two-phase model, and a uniform surface external radial pressure (i.e., p = 1 Pa) was applied to the outside surface boundary of the outside tube. As mentioned in the analytical section, no axial loads were applied here, and axial strain was not neglected. Finally, the generated model was solved and displacement, strain, and stress results were obtained. A mesh convergence study was also performed by decreasing the mesh size with a results convergence criterion of less than 1%. Results for the axial, circumferential, and radial displacements, strains, and stresses throughout the composite cylinder were obtained in 3D contour plot (shown in Figure 4 in Appendix 2) and 2D graph formats (shown in Figures 5(a) to 10(a) in Appendix 3) to show their variations with respect to the radial coordinates.

Results obtained from FEA

Results showed that the axial displacements were uniform at any cross section of the model, and they increased linearly along the model’s longitudinal direction, as expected. Also, the radial displacement was uniform along the length coordinate but increased in the radial direction. The circumferential displacements were zero or nearly zero throughout the entire tubular composite structure, which is consistent with the assumption made for the derivation of exact analytical solutions, as explained in earlier section, as expected [58]. Note that all shear strains and shear stresses were zero for this axisymmetric loading case. Radial displacement values at the interface (i.e., r = b) and the outer boundary (i.e., r = c) were obtained as $w |_{r = b}^{F E A} = - 3.7024 e - 12 (n m) $ and $w |_{r = c}^{F E A} = - 3.6946 e - 12 (n m)$ , respectively. Similarly, results for the remaining, axial, circumferential, and radial displacements, strains, and stresses were obtained, and presented in Figures 5(a) to 10(a) and equations (57) to (62) in Appendix 3. Due to the application of radial uniform external pressure, the much stiffer outside CNT underwent tensile axial stresses, and the inside epoxy with less stiffness experienced compressive axial stresses, as expected (see Figure 8(a)). The circumferential and radial stresses throughout the entire tubular composite structure were compressive with higher values in the domain of a much stiffer CNT (see Figure 9(a) and (b)). For the same reason, the entire structure contracted in the radial and circumferential directions and elongated in the axial direction. As expected, the value of the radial stress at the outer boundary (i.e., r = b) reached the value of the radially applied uniform load (i.e., p = 1 Pa).

In the next section, displacements, normal strains, and normal stresses will be obtained from the exact analytical solutions presented in earlier sections of this work.

Calculation of results from analytical solutions

For comparison and verification of the exact analytical solutions, the displacement, strain, and stress results were graphed and calculated (similar to those obtained from FEA) using the closed-form analytical solutions obtained in equations (13) to (16), (22), and (24). Note that the results presented in this section were obtained for a two-phase composite tube with the same dimensions, engineering material properties, and loading condition as specified in the FEA section.

The stiffness matrix, $_{C}^{i j}$ , is the inverse of the compliance matrix, the components of which include engineering material properties, that is, $_{E}^{i i},_{ν}^{i j}, and_{G}^{i j}$ . For an orthotropic material, the contracted stiffness components (i.e., $_{C}^{i j}$ ) can be calculated from equations (26) and (27), presented below [60 and 61].

{\begin{array}{l} C_{11} = \frac{1 - ν_{23} ν_{32}}{E_{22} E_{33} δ}; C_{22} = \frac{1 - ν_{31} ν_{13}}{E_{33} E_{11} δ}; C_{33} = \frac{1 - ν_{12} ν_{21}}{E_{11} E_{22} δ} \\ C_{12} = \frac{ν_{21} + ν_{31} ν_{23}}{E_{22} E_{33} δ}; C_{13} = \frac{ν_{31} + ν_{21} ν_{32}}{E_{22} E_{33} δ}; C_{23} = \frac{ν_{32} + ν_{31} ν_{12}}{E_{33} E_{11} δ} \end{array}

(26)

{\begin{cases} \frac{ν_{23}}{E_{22}} = \frac{ν_{32}}{E_{33}}; \frac{ν_{31}}{E_{33}} = \frac{ν_{13}}{E_{11}}; \frac{ν_{12}}{E_{11}} = \frac{ν_{21}}{E_{22}} \\ δ = \frac{1 - ν_{12} ν_{21} - ν_{23} ν_{32} - ν_{31} ν_{13} - 2 ν_{12} ν_{23} ν_{31}}{E_{11} E_{22} E_{33}} \end{cases}

(27)

First, the material properties were used to calculate the stiffness matrix components (i.e., $C_{i j}^{(i)} and C_{i j}^{(o)}$ ) for both constituents. Next, the stiffness constants along with dimensions of the concentric tubes were used to calculate the parameters (i.e., equations (4), (5), (41) to (43), (45) to (47), and (52) in Appendix 1) that were introduced and used in the analytical solutions (i.e., equations (12) to (16), (22), and (24)). Finally, using these values, the displacement fields, strain profiles, and stress distributions were obtained throughout the entire regions of the two-phase orthotropic composite cylinder. MAPLE 7.0 [64], a mathematical software, was used to handle these rigorous mathematical operations and to plot the exact analytical results in 2D graphs.

The axial displacement field was uniform at any cross section of the tube within the r–θ plane (see Figure 1) and increased linearly along the tube’s longitudinal direction and reached its maximum value, i.e., $u _{x} = ε L$ , at the free end (i.e., $x = L / 2$ ) of the tube. As mentioned earlier, the circumferential displacements were zero throughout the tubular structure. The radial displacement field was uniform along the tube length and increased in the tube radial direction (see Figure 2(b)). The radial displacements at the interface (i.e., r = b) and the outer boundary (i.e., r = c) were obtained as $w |_{r = b}^{A n a l y t i c a l} = - 3.7024 e - 12 (n m)$ and $w |_{r = c}^{A n a l y t i c a l} = - 3.6947 e - 12 (n m)$ , respectively.

Similarly, the results for axial, circumferential, and radial strains and stresses were obtained throughout the entire constituent tubes and are presented in Appendix 3 (see Figures 5(b) to 10(b) and equations (57) to (62)). In addition, the strain and stress values at key points were calculated, discussed, and elaborated on in detail for further verification.

Finally, using the solutions given by equations (22) and (24), the effective radial Young’s modulus, $E_{r r}^{e f f}$ , and Poisson’s ratio, $ν_{r x}^{e f f}$ , for the two-phase CNT nanocomposite cylinder were calculated as $E_{r r}^{e f f} = 19.76924 (G P a)$ and $ν_{r x}^{e f f} = 0.26018$ , respectively.

Verification of exact analytical solutions

To verify the analytical approach used here and the accuracy of the presented closed-form solutions, the displacement, strain, and stress results, obtained from both analytical and FEA techniques, for the radially loaded generally orthotropic two-phase nanocomposite cylinder were compared together. Figure 2 shows the radial displacements of the two-phase nanocomposite tube obtained from both FEA and analytical solutions. The inside and outside materials regions are marked as “i” and “o,” respectively, on the graphs. The slope of the radial displacement curve (i.e., displacement rate) is drastically reduced in the region of the much stiffer outside tube (i.e., CNTs), as expected.

Close examination and comparisons of the results (obtained from FEA and analytical solutions presented in previous sections, Figure 2, Figures 5(b) to 11(b), and equations (57) to (62) in Appendix 3) revealed that both techniques predict similar results. Table 1 summarizes a quantitative comparison between the displacements, strains, and stresses results at the interface and outside boundary obtained from both techniques. The differences are presented as an error percentage.

Table 1.

Comparison of displacements, strains, and stress results obtained from exact analytical solutions and finite element analysis at interface and outside boundary of radially loaded generally orthotropic two-phase nanocomposite cylinder.

Technique	Finite element results		Analytical results		Error percentage (%)
Locations	@ r ≤ b	@ r = c	@ r ≤ b	@ r = c	@ ≤ b	@ r = c
$w (n m)$	−3.7024 × 10⁻¹²	−3.6946 × 10⁻¹²	−3.7024 × 10⁻¹²	−3.6947 × 10⁻¹²	0.0003	0.0016
$ε$	0.5581 × 10⁻¹²	0.5581 × 10⁻¹²	0.5603 × 10⁻¹²	0.5603 × 10⁻¹²	0.3926	0.3926
$ε_{θ}$	−14.809 × 10⁻¹²	−13.684 × 10⁻¹²	−14.810 × 10⁻¹²	−13.684 × 10⁻¹²	0.0034	0.0014
$ε_{r}$	−14.809 × 10⁻¹²	+0.3505 × 10⁻¹²	−14.810 × 10⁻¹²	+0.3493 × 10⁻¹²	0.0034	0.3435
$σ_{x} (P a)$	−0.074419	+0.576987	−0.074408	+0.571917	0.0148	0.8864
$σ_{θ} (P a)$	−0.121082	−11.46921	−0.121077	−11.50516	0.0041	0.3124
$σ_{r} (P a)$	−0.121082	−0.997868	−0.121077	−1.000000	0.0041	0.2132

As can be seen in the right two columns of Table 1, the percentage differences between the results obtained from the analytical solutions and the FEA range from 0.0003% to 0.8864%. Therefore, results from both techniques are in excellent agreement, and our exact analytical solutions are verified. These solutions can be used to quantitatively and qualitatively study the effective mechanical properties and mechanical behavior of any generally orthotropic cylindrical two-phase nanocomposite structure when it is subjected to uniform radial pressure. Furthermore, these closed-form analytical solutions can be used for parametric studies [29] and for designing a composite/nanocomposite material system with desired material properties and mechanical performance.

Our previous work [29] presents a complete set of parametric studies, where the effects of CNT volume fraction and variation of the material properties (simultaneous change of nine independent material properties) on the overall effective properties of the orthotropic nanocomposite cylinder were investigated. To study the effects of volume fraction, one can vary the volume fraction of the matrix, (i.e., γ_i = b²/c²), simply by changing the diameter of the inner tube (i.e., b ) from zero (i.e., 0% matrix and 100% CNTs volume fractions) to c (i.e., 100% matrix and 0% CNTs volume fraction) to investigate the variations of the effective material properties and mechanical responses of a radially loaded two-phase generally cylindrical orthotropic nanocomposite cylinder. As expected, the effective properties and performance of nanocomposite cylinder will represent material properties and performance of an individual CNT or that of the filler matrix, when the volume content of the filler matrix (i.e., γ_i = b²/c²) is set to 0 or 1, respectively.

To investigate the influence of the outside material properties (i.e., CNT) on the effective properties of the composite cylinder, parametric studies based on the variation of the material properties of the CNT can be carried out [29]. Basically for this purpose, all nine independent orthotropic properties of CNT (i.e., three $E_{i j}^{(o)}$ , three $G_{i j}^{(o)}$ , and three $ν_{i j}^{(o)}$ ) can be simultaneously varied from their original values down to the properties of the matrix filler material, and the effective material properties and mechanical responses can be determined as a function of the SWCNT material properties variation. To carry out these parametric studies, the following matrix equation was introduced to simultaneously vary the properties of the CNT, as described above [29].

[P_{i j}^{(o)}] = [a_{i j}] + [b_{i j}] ζ; (0 \leq ζ \leq 1)

(28)

[P_{i j}^{(o)}] = [\begin{matrix} E_{11}^{(o)} & G_{12}^{o} & ν_{12}^{(o)} \\ E_{22}^{(o)} & G_{13}^{o} & ν_{13}^{(o)} \\ E_{33}^{(o)} & G_{23}^{o} & ν_{23}^{(o)} \end{matrix}]; [a_{i j}] = [\begin{matrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{matrix}]; [b_{i j}] = [\begin{matrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{matrix}]

(29)

In equations (28) and (29), [ $P_{i j}^{(o)}$ ] denotes the properties of the outside CNT material, ζ is the nondimensional variable factor, and [ a_ij ] and [ b_ij ] are constant coefficients that have to be calculated from the conditions given in equation (30).

[P_{i j}^{(o)}] = {\begin{array}{l} [P_{i j}^{(M a t r i x)}]; & ζ = 0 \\ [P_{i j}^{(C N T)}]; & ζ = 1 \end{array}

(30)

Note that [ $P_{i j}^{(M a t r i x)}$ ] and [ $P_{i j}^{(C N T)}$ ] are mechanical properties matrices of inside filler matrix material and outside CNT. Therefore, if one varies the nondimensional variable ζ from one to zero, all nine independent orthotropic properties of outside tube will simultaneously vary from CNT properties to matrix properties.

It should be mentioned that our analytical solutions can be applied to bulk CNT-reinforced nanocomposites, where matrix-filled CNTs interact/are bonded to the surrounding matrix material. More simply, one can use the analytical solutions twice and obtain effective properties for matrix-filled CNTs that are surrounded with matrix material (i.e., unit cell for bulk CNT-reinforced nanocomposites). In the first round of calculations, the analytical solutions can be used to obtain the effective properties of the matrix-filled CNTs. In the second step, the calculated effective properties of the matrix-filled CNTs will be assigned to the inner tube, and then matrix properties will be assigned to the outer tube. Results of the second step will be the effective properties of matrix-filled CNTs embedded in the matrix as a bulk nanocomposite unit cell. However, the presented solutions in this study were developed based on perfect bonding assumption between the inner resin and outside nanotube, which is in the absence of interphase layer.

In another note, it is worth mentioning that presence of vacancy defects on CNTs’ structures can alter their mechanical properties (depending on its severity), and as a result it's reinforcement effectiveness in nanocomposites can be compromised [7]. Please note that the present study does not include the effects of vacancy defects on mechanical properties and performance of two-phase generally orthotropic cylindrical nanocomposite model.

Conclusions

Nanotubes and one-dimensional nanostructures are becoming more attractive for reinforcement applications in structural composites. Thus, to optimally design the desired nanocomposite material systems for high-performance structural applications, it is essential to understand and predict the effective properties and behavior of the individual constituents (i.e., filled or coated nanotubes as representative unit elements). The structural analysis of this new class of nanocomposites materials when subjected to external loads requires more sophisticated models that consider the material’s actual anisotropy level at nanoscale (i.e., orthotropic). In this work, a two-phase composite cylinder composed of two concentric tubes (i.e., representing an inside filled nanotube or an embedded nanotube, as the unit cell of a nanotube-reinforced nanocomposite) was modeled and subjected to an externally applied uniform radial pressure. It should be mentioned that both concentric tubes were considered to be continuous with known effective properties (from discrete and atomistic/molecular modeling and experiments [1 –5, 7–14]) that are well bonded at the interface.

To ensure that there was no loss of generality and to have more accurate solutions, the no plane strain condition was used to derive the solutions. First, using the well-established principles of linear elasticity and continuum mechanics, analytical formulae were developed for the prediction of displacement fields, strain components, and stress distributions within the domain of each constituent (i.e., inside and outside tubes). Next, using the exact analytical solutions and the basic definition of strain–stress relationships in a cylindrical coordinate system for orthotropic solids, exact analytical solutions were obtained for the effective radial Young’s modulus, $E_{r r}^{e f f}$ , and the Poisson’s ratio, $ν_{r x}^{e f f}$ . The equations and parameters used in this study were examined for their dimensional consistency throughout the derivation steps to ensure the correctness of the final solutions. To further verify the analytical solutions, a 3D FEA was performed, and results were compared to those obtained from the analytical exact solutions. Excellent agreement was achieved, and the correctness and accuracy of the analytical solutions were verified. Assuming perfect bonding between the concentric cylinders, the introduced exact analytical solutions are applicable to any two-phase cylindrical structure (consisting of various orthotropic materials) and can be used for parametric study and design optimization purposes. These analytical solutions can also be readily reduced to transversely isotropic as well as isotropic cases.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Office of Naval Research (grant number N00014-05–1-0586) and the Wichita State University.

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Abstract

Keywords

Introduction

Analytical modeling of two-phase nanocomposite cylinder

Derivation of exact analytical solutions for displacement fields, strain components, and stress distributions

Derivation of generally orthotropic effective radial Young’s modulus ( E r r e f f )

Derivation of generally orthotropic effective poisson’s ratio ( ν r x e f f )

FEA modeling of two-phase nanocomposite cylinder

Results obtained from FEA

Calculation of results from analytical solutions

Verification of exact analytical solutions

Conclusions

Footnotes

Declaration of Conflicting Interests

Funding

Appendixes

References

Derivation of generally orthotropic effective radial Young’s modulus $(E_{r r}^{e f f})$

Derivation of generally orthotropic effective poisson’s ratio ( $ν_{r x}^{e f f}$ )