Computational micromechanics modeling of inherent piezoresistivity in carbon nanotube

Abstract

It has been observed that carbon nanotubes have a measurable inherent piezoresistive effect, that is to say that changes in carbon nanotube strain can induce changes in carbon nanotube resistivity, which may lead to observable macroscale piezoresistive response of carbon nanotube–polymer nanocomposites. In this article, the focus is on modeling the effect of inherent piezoresistivity of carbon nanotubes on the nanocomposite’s piezoresistive behavior using computational micromechanics techniques based on finite element analysis. Both in-plane and axial piezoresistive responses are being considered in an electromechanically coupled code. The computational results are used to estimate the magnitude of the piezoresistive coefficients of carbon nanotube needed for the piezoresistive response of macroscale nanocomposites to be comparable with experimental data in the literature. It is found that the current values for inherent piezoresistivity of the carbon nanotube are not sufficiently large enough to explain the observed macroscale piezoresistive response if inherent piezoresistive effect of carbon nanotubes is the only driving force for the piezoresistive response of the macroscale nanocomposites, and hence, additional mechanisms such as electron hopping and nanotube–nanotube contact may play important roles either individually or in a coupled fashion with inherent piezoresistivity of the carbon nanotube.

Keywords

Computational micromechanics polymer nanocomposites piezoresistivity carbon nanotube

Introduction

To date, several types of carbon nanotube (CNT)–polymer nanocomposites have been shown to have linear piezoresistive responses with small loadings of single-walled carbon nanotubes (SWCNTs) or multi-walled carbon nanotubes (MWCNTs) (Bautista-Quijano et al., 2010; Ferreira et al., 2012a, 2012b; Kang et al., 2006, 2009; Oliva-Aviles et al., 2011; Zhang et al., 2006). Zhang et al. (2006) found in the small strain range of 1% that MWCNTs/polycarbonate (PC) films loaded with 5%wt MWCNTs can yield a strain sensitivity approximately 3.5 times that of a commercial (metallic) strain gauge. Kang et al. (2006) found the gauge factor of sensors made of SWCNT/poly(methyl methacrylate) (PMMA) composites to range from about 1 to 5 when the weight percentage of SWCNTs ranges from 0.5 to 10, with the higher sensitivity being achieved using 3–10%wt SWCNTs. Kang et al. (2009) reported that for the SWCNT/polyimide composite sensors, the maximum gauge factor is achieved at a concentration just above the percolation threshold concentration (0.05%wt SWCNTs) with a magnitude of 4.21. Bautista-Quijano et al. (2010) reported the gauge factors of thin films made of polysulfone (PSF) and dispersed MWCNTs to range from 0.48 to 0.73 when the weight percentage of MWCNT is between 0.2 and 1.0. Ferreira et al. (2012b) reported that the maximum gauge factor ranges up to 6.2 for the hot pressed SWCNT/polyvinylidene difluoride (PVDF) composites when the weight percentage of SWCNT is just below 2. For aligned MWCNT/PSF films with 0.5%wt MWCNTs aligned by application of alternating current (AC) electric fields during processing, Oliva-Aviles et al. (2011) reported a gauge factor of 2.78 ± 0.42 for tensile loading in the MWCNT alignment direction. It has also been observed that at a concentration just above the CNT electrical percolation threshold, the piezoresistive stress coefficient of SWCNT/polyimide nanocomposites exceeds those of metallic materials (aluminum) by two orders of magnitude (Kang et al., 2009). Such piezoresistive properties make CNT–polymer nanocomposites very attractive in the manufacturing of high–gauge factor lightweight strain gauges. In addition, CNT–polymer nanocomposite strain gauges have the potential to be directly embedded in structural composites during composite processing to provide internal strain sensing in contrast to the customary surface strain measurements obtained from commercial strain gauges bonded to the surface of structures. While there is ample experimental evidence demonstrating the piezoresistive response of nanocomposites, the mechanisms governing piezoresistivity of nanocomposites are less clear in terms of relative magnitude and interactions. In order to aid in the design of piezoresistive nanocomposite strain gauges with tailored sensitivities, it is necessary to develop an understanding of the underlying mechanisms at the micro- and nanoscales, which govern the macroscale piezoresistive response.

At present, several mechanisms are believed to contribute to the piezoresistive response of CNT–polymer nanocomposites. For example, Kang et al. (2009) have indicated that the piezoresistive response of macroscale nanocomposite material originates from the tunneling effect (Li et al., 2007) between conducting inclusions (CNTs) at the nanoscale under compression or tension. Fernberg et al. (2009) indicated that the geometric change of the specimen will have an effect on its macroscale resistance behavior.

An additional mechanism is associated with the CNTs themselves that have been shown to have a considerable inherent effect which should not be neglected. It has been observed both experimentally and in modeling that mechanical deformation of CNTs can directly lead to significant changes in their conductance (Peng and Cho, 2000), indicating that CNTs are themselves good strain sensors due to inherent piezoresistivity. Efforts to quantify the inherent piezoresistivity of CNTs have taken the form of both characterization and modeling experiments. For example, Chen et al. employed molecular dynamics simulations and quantum transport theory to study the electrical properties of (12, 0) zigzag CNTs under different uniaxial strains and found that a 1% strain can result in a 6.4% decrease in resistance (Chen and Weng, 2007). Tight-binding and zone-folding approximations have been used to predict the strain sensitivity of a CNT, which is found to be based on its chiral indices (n, m) and varies widely with the electrical structures (Anantram et al., 1999; Cullinan and Culpepper, 2010). Experimentally, Tombler et al. (2000) applied strain to a suspended CNT using atomic force microscopy (AFM) tips and observed that the conductance of the CNT could change by as much as two orders of magnitude under the applied strains. Stampfer et al. (2006) measured the relative differential resistance sensitivity for a metallic SWCNT of up to 27.5% per nanometer of deflection from which a piezoresistive gauge factor of an SWCNT of up to 2900 was extracted.

Analytic micromechanics multiscale methods such as self-consistent (Hill, 1965), Mori–Tanaka (Benveniste, 1987; Mori and Tanaka, 1973), and composite cylinder model (Hashin and Rosen, 1964) methods have been widely used to obtain the overall elastic, thermal, and electrical properties of heterogeneous materials. However, such analytic methods are generally limited to linear properties and in order to get solutions may have strict limitations on the geometries and physical properties of the constituent materials. In contrast, computational multiscale methods such as those based on multiscale expansion methods (Bakhvalov and Panasenko, 1989; Bensoussan et al., 1978) can overcome geometry and material property limitations and have been successfully applied for obtaining the overall nonlinear material properties, for example, load- and path-dependent effective properties involving plasticity and/or damage, and for solving various multiphysics problems (Fish et al., 1997, 1999; Yu and Fish, 2002). To date, analytic and computational multiscale methods have been applied toward predicting effective properties of various nanocomposites (see, e.g. the review article by Zeng et al. (2008) or other works including Odegard et al. (2003), Li and Chou (2003), Gates et al. (2005), Spanos and Kontsos (2008), Hammerand et al. (2007), Seidel and Lagoudas (2006, 2008, 2009), and Seidel and Puydupin-Jamin (2011), which have promising applications arising from the enhanced properties of nanoparticles.

However, for the observed linear, reversible, and path-independent piezoresistive response of the CNT–polymer nanocomposites, limited modeling work is found in the literature (Grimaldi et al., 2001, 2003; Hu et al., 2012; Li and Chou, 2008; Theodosiou and Saravanos, 2010; Wichmann et al., 2009).¹ For example, in studies of Grimaldi et al. (2001, 2003) and Li and Chou (2008), the focus was on the electrical tunneling effect on the macroscale piezoresistive response, while in the work of Wichmann et al. (2009), the focus was on purely geometric effects, with none of these works considering the inherent piezoresistive response of the CNTs. In another work (Hu et al., 2012), the focus has been on the inclusion of several mechanisms, including resistor network models and inherent CNT piezoresistivity in polymer nanocomposites with randomly oriented CNTs. Theodosiou and Saravanos (2010) found that the variation of CNT resistance with strain is the dominant mechanism for nanocomposite piezoresistivity, with a significant contribution also coming from the tunneling effect. In contrast, Hu et al. (2012) found that the inherent CNT piezoresistivity was not the dominant mechanism but rather that the resistive network and tunneling effects were driving the observed macroscale nanocomposite piezoresistive response. In fact, they note that with an average CNT gauge factor of 21.18, the overall piezoresistive response of the nanocomposite was only increased by roughly 9%. To our knowledge, there has been no clear discussion and quantification as to what magnitude of inherent CNT piezoresistivity is needed to achieve macroscale nanocomposite gauge factors on the order of 2–5 if this mechanism was indeed the driving force behind the nanocomposite piezoresistive response.

In this article, a computational multiscale micromechanics model based on finite element analysis is developed and used to determine the changes in macroscale resistance (i.e. effective macroscale piezoresistance) due to changes at the microscale associated with CNT deformation and inherent piezoresistivity. The effects of CNT geometry, local volume fraction, and inherent piezoresistive coefficients on the macroscale piezoresistivity are studied parametrically. The results are discussed in the context of experimentally observed gauge factors for nanocomposites and the strength of electromechanical coupling needed to achieve observed responses.

Model description

Multiscale model for the CNT–polymer nanocomposites

In the CNT–polymer nanocomposites, the strains in the CNTs may have an effect on their resistivities; therefore, the mechanical and electrical properties of the nanocomposites are one-way coupled. In other words, the mechanical properties of the domain may influence its electrical properties, but not vice versa. Typically, the dispersion and shape of CNTs in the CNT–polymer nanocomposites may be complicated, for example, the CNTs can be curved along their axes, randomly oriented, and bundled with the other CNTs. In order to focus the present study on the potential impact of inherent piezoresistivity of CNTs on the macroscale piezoresistive response of nanocomposites, we have selected a multiscale idealization as shown in Figure 1. In this representation, the CNTs are assumed to be well dispersed, aligned, and perfectly bonded to the surrounding polymer. The governing differential equations describing mechanical and electrostatic responses of the nanocomposites are given below.

Figure 1.

The hierarchical multiscale modeling of nanocomposites in which there are well-dispersed and aligned CNTs.

Mechanical governing differential equations at the macroscale

For quasi-static conditions and omitting body forces, conservation of linear momentum at the macroscale can be written in Cartesian coordinate system as

{\tilde{σ}}_{ij, j} = 0

(1)

where the overscript “∼” denotes variables at the macroscale. Note that the summation convention is used with i, j ranging from 1 to 3 and the comma denotes spatial differentiation. The infinitesimal strain–displacement relations at the macroscale are given by

{\tilde{ε}}_{ij} = \frac{1}{2} ({\tilde{u}}_{i, j} + {\tilde{u}}_{j, i})

(2)

The macroscale constitutive relation for the mechanical boundary can be expressed as

{\tilde{σ}}_{ij} = C_{ijkl}^{Eff} {\tilde{ε}}_{kl}

(3)

where $C_{ijkl}^{Eff}$ are the effective macroscale stiffness components that are obtained from the volume-averaged stress–strain response at the microscale as

〈 σ_{ij} 〉 = C_{ijkl}^{Eff} 〈 ε_{kl} 〉

(4)

where the brackets $〈 \cdot 〉$ denote volume averaging over the appropriate microscale representative volume element (RVE). The mechanical boundary-value problem is completed through the specification of the boundary conditions given by

{\tilde{u}}_{i} = {\tilde{u}}_{i}^{0} on S_{u}

(5a)

and

{\tilde{t}}_{i} = {\tilde{n}}_{j} {\tilde{σ}}_{ij} = {\tilde{t}}_{i}^{0} on S_{σ}

(5b)

where the complete macroscale boundary is specified as the union of $S_{u}$ and $S_{σ}$ . It is noted that the macroscale effective mechanical properties obtained from the microscale boundary-value problem are independent of the macroscale boundary conditions, and thus, need only be determined once at the initiation of the piezoresistive simulations. The macroscale effective elastic properties can therefore be obtained by using either computational micromechanics method (Hammerand et al., 2007) or composite cylinder method (Seidel and Lagoudas, 2006).

Electrostatic governing differential equations at the macroscale

The steady-state conservation of charge equation at the macroscale can be written in Cartesian coordinate system as

{\tilde{J}}_{i, i} = 0

(6)

where ${\tilde{J}}_{i}$ is the current density at the macroscale. The macroscale electrical field is obtained as the negative of the gradient of the macroscale potential as

{\tilde{E}}_{i} = - {\tilde{Φ}}_{, i}

(7)

where ${\tilde{E}}_{i}$ and $\tilde{Φ}$ are the electric field and electric potential, respectively, at the macroscale. In an analogous manner as in the macroscale mechanical boundary-value problem, the constitutive relation for the macroscale electrical boundary value can be expressed as

{\tilde{E}}_{i} = ρ_{ij}^{Eff} {\tilde{J}}_{j}

(8a)

{\tilde{J}}_{i} = κ_{ij}^{Eff} {\tilde{E}}_{j}

(8b)

where $ρ_{ij}^{Eff}$ and $κ_{ij}^{Eff}$ are the effective macroscale resistivity and conductivity components, respectively, which are obtained from the volume-averaged current density–electric field response at the microscale as

〈 E_{i} 〉 = ρ_{ij}^{Eff} 〈 J_{j} 〉

(9a)

〈 J_{i} 〉 = κ_{ij}^{Eff} 〈 E_{j} 〉

(9b)

where the brackets again denote volume averaging over the appropriate microscale RVE. The electrical boundary-value problem is completed through the specification of the boundary conditions given by

\tilde{Φ} = {\tilde{Φ}}^{0} on S_{Φ}

(10)

and

{\tilde{n}}_{i} {\tilde{J}}_{i} = {\tilde{n}}_{i} {\tilde{J}}_{i}^{0} on S_{J}

(10b)

where the complete macroscale boundary is specified as the union of $S_{Φ}$ and $S_{J}$ . It is noted that the macroscale effective electrical properties obtained from the microscale boundary-value problem are independent of the macroscale electrical boundary conditions. However, due to the piezoresistivity at the microscale, the macroscale effective electrical properties do depend on the macroscale mechanical boundary conditions. As such, the effective electrical properties must be determined at each mechanical load step. The macroscale effective electrical properties must therefore be obtained in an incremental manner by using either computational micromechanics method or composite cylinder method for specific material symmetries of electrical material properties (Ren and Seidel, 2011; Seidel and Lagoudas, 2009).

Mechanical governing differential equations for the microscale RVE

The three-dimensional (3D) microscale RVE for aligned CNT nanocomposites (Figure 1) can be reduced to two 2D RVEs in order to save computational time. As seen in Figure 2(a), due to high aspect ratios (ARs) ( $> 200$ ) of CNTs, the transverse properties of the nanocomposites are typically modeled using the regular hexagonal array RVE under in-plane strain assumption. The annulus areas represent the cross sections of CNTs, and the remaining area is pure polymer matrix. Correspondingly, a plane strain periodic finite element model is constructed to model the piezoresistive response of the nanocomposites in the transverse directions. On the other hand, for cases where the loading is axisymmetric, for example, in the uniaxial tension test, the axial properties of the nanocomposites are modeled by using a 2D axisymmetric RVE that is axisymmetric about the axis (Figure 2(b)). The highlighted area is CNT, and the remaining area is pure polymer matrix. Correspondingly, an axisymmetric finite element model is constructed to model the axial piezoresistive response of the nanocomposites.

Figure 2.

Microscale RVEs for the nanocomposites with well-dispersed and aligned CNTs. (a) The transverse hexagonal microscale RVE with the volume fraction of the CNTs being 10%. (b) The axisymmetric microscale RVE with the volume fraction of the CNTs being 0.5%vol and the aspect ratio of the CNTs being 20 (the aspect ratio used here is only for demonstration purposes as the real aspect ratio used in the computations is much higher ( $\geq 200$ )). $X_{1}$ and $X_{2}$ are in the same plane as $X_{r}$ and $X_{θ}$ . $W_{1}$ is the width of the hexagonal microscale RVE in $X_{1}$ direction, and $W_{2}$ is the width of the hexagonal microscale RVE in $X_{2}$ direction. L is the length of the axisymmetric microscale RVE in $X_{z}$ (or $X_{3}$ ) direction, $L_{CNT}$ is the length of CNT, and R is the half distance between two adjacent CNTs.

For quasi-static conditions omitting body forces, conservation of linear momentum for –the microscale RVE can be written in Cartesian coordinate system as

σ_{ij, j} = 0

(11)

where under the plane strain assumptions of the transverse hexagonal RVE, $σ_{13}$ and $σ_{23}$ are zero. For the axisymmetric microscale RVE, it is more convenient to express conservation of linear momentum in cylindrical coordinates as

\frac{\partial σ_{rr}}{\partial r} + \frac{\partial σ_{rz}}{\partial z} + \frac{σ_{rr} - σ_{θ θ}}{r} = 0

(12a)

\frac{\partial σ_{rz}}{\partial r} + \frac{\partial σ_{zz}}{\partial z} + \frac{σ_{rz}}{r} = 0

(12b)

where it is noted that terms involving $θ$ shear components or differentiation with respect to $θ$ have been eliminated from the general cylindrical form. The infinitesimal strain–displacement relations for the microscale RVE are given in Cartesian coordinates by

ε_{ij} = \frac{1}{2} (u_{i, j} + u_{j, i})

(13)

where under the plane strain assumptions for the transverse hexagonal RVE, $ε_{13} = 0$ , $ε_{23} = 0$ , and $ε_{33} = 0$ . For the axisymmetric microscale RVE, the strain–displacement relations in cylindrical coordinates are given as

ε_{rr} = \frac{\partial u_{r}}{\partial r}, ε_{θ θ} = \frac{u_{r}}{r}, ε_{zz} = \frac{\partial u_{z}}{\partial z}

ε_{θ z} = 0, ε_{rz} = \frac{1}{2} (\frac{\partial u_{z}}{\partial r} + \frac{\partial u_{r}}{\partial z}), ε_{r θ} = 0

(14)

At the microscale, the CNT annulus and polymer matrix can be assumed to be isotropic linear elastic materials (Saito et al., 1998) such that their constitutive relationships can be denoted with the Lame constants as

σ_{ij} = 2 μ^{C} ε_{ij} + λ^{C} ε_{kk} δ_{ij}

(15a)

σ_{ij} = 2 μ^{M} ε_{ij} + λ^{M} ε_{kk} δ_{ij}

(15b)

where the superscripts C and M on the material properties denote CNT and polymer matrix, respectively, and the standard relationships $μ = E / 2 (1 + ν)$ and $λ = E ν / (1 + ν) (1 - 2 ν)$ hold. The boundary conditions on the microscale RVE can be expressed as either traction or displacement boundary conditions by

t_{i} = n_{j} {\tilde{σ}}_{ij} ({\tilde{X}}_{k})

(16a)

u_{i} = x_{j} {\tilde{ε}}_{ij} ({\tilde{X}}_{k})

(16b)

such that in general the macroscale and microscale RVE stresses and strains can be related to one another through microscale volume averages as

{\tilde{σ}}_{ij} ({\tilde{X}}_{k}) = 〈 σ_{ij} 〉

(17a)

{\tilde{ε}}_{ij} ({\tilde{X}}_{k}) = 〈 ε_{ij} 〉

(17b)

where ${\tilde{σ}}_{ij} ({\tilde{X}}_{k})$ and ${\tilde{ε}}_{ij} ({\tilde{X}}_{k})$ are the macroscale stress and strain at a macroscale material point ${\tilde{X}}_{k}$ , which the microscale RVE represents. Note that in equations (15)–(17), $i, j, k = 1, 2, 3$ in the Cartesian coordinate system of the transverse hexagonal microscale RVE, while $i, j, k = r, θ, z$ in the cylindrical coordinate system of the axisymmetric microscale RVE.

Electrostatic governing differential equations for the microscale RVEs

The steady-state conservation of charge equation for the microscale RVE can be denoted in Cartesian coordinates as

J_{i, i} = 0

(18)

where $J_{i}$ is the current density and where under in-plane electrostatic assumptions, $J_{3, 3} = 0$ . For the axisymmetric microscale RVE, it is again more convenient to express conservation of charge in cylindrical coordinates as

\frac{\partial J_{r}}{\partial r} + \frac{\partial J_{z}}{\partial z} + \frac{1}{r} J_{r} = 0

(19)

where it is noted that terms involving $θ$ current density components or differentiation with respect to $θ$ have been eliminated from the general cylindrical form. The microscale RVE electric field is obtained as the negative of the gradient of the microscale potential as

E_{i} = - Φ_{, i}

(20)

where $E_{i}$ $(i = 1, 2, 3)$ and $Φ$ are the electric field and electric potential, respectively, as applied in Cartesian coordinate system at the microscale and where under in-plane electrostatic assumptions, it is noted that $E_{3} = 0$ . For the axisymmetric microscale RVE, the electric field is obtained from the potential in cylindrical coordinates as

E_{r} = - \frac{\partial Φ}{\partial r}

(21a)

E_{z} = - \frac{\partial Φ}{\partial z}

(21b)

where it is noted that terms involving the differentiation with respect to $θ$ have been eliminated from the general cylindrical form. For the microscale RVEs, Ohm’s law can be written as

E_{i} = ρ_{ij}^{C} J_{j}

(22a)

E_{i} = ρ_{ij}^{M} J_{j}

(22b)

where the superscripts C and M on the resistivities denote CNT and polymer matrix, respectively, such that the inverses of equations (22a) and (22b) yield the electrical conductivities of the CNT and polymer matrix, respectively, as

J_{i} = κ_{ij}^{C} E_{j}

(23a)

J_{i} = κ_{ij}^{M} E_{j}

(23b)

Note that in equations (22) and (23), $i, j, k = 1, 2, 3$ in the Cartesian coordinate system of the transverse hexagonal microscale RVE, while $i, j, k = r, θ, z$ in the cylindrical coordinate system of the axisymmetric microscale RVE.

As the pure polymer matrix is not expected to be piezoresistive, the resistivity remains fixed under the applied strains. However, the CNTs are expected to exhibit inherent piezoresistivity such that the instantaneous resistivity, $ρ_{ij}^{C}$ , is a function of the local strain state within the CNT and can therefore be written as

ρ_{ij}^{C} = ρ_{ij}^{C 0} + Δ ρ_{ij}^{C}

(24)

in which $ρ_{ij}^{C 0}$ are the initial zero strain resistivities of the CNT and $Δ ρ_{ij}^{C}$ are the change in resistivities induced by inherent piezoresistive effect of the CNT as obtained from

Δ ρ_{ij}^{C} = g_{ijkl}^{C} ε_{kl}

(25)

The $g_{ijkl}^{C}$ are the piezoresistive strain coefficient components of the fourth-order piezoresistive coefficient tensor of the CNT, which in general has 81 components. However, due to the symmetry of both the resistivity and the strain, that is

Δ ρ_{ij}^{C} = Δ ρ_{ji}^{C}

(26a)

ε_{kl} = ε_{lk}

(26b)

one may write

g_{ijkl}^{C} = g_{jikl}^{C}

(27a)

g_{ijkl}^{C} = g_{ijlk}^{C}

(27b)

The piezoresistive constitutive equation can therefore be denoted in Voigt notation as

{\begin{matrix} Δ ρ_{1}^{C} \\ Δ ρ_{2}^{C} \\ Δ ρ_{3}^{C} \\ Δ ρ_{4}^{C} \\ Δ ρ_{5}^{C} \\ Δ ρ_{6}^{C} \end{matrix}} = [\begin{matrix} g_{11}^{C} & g_{12}^{C} & g_{13}^{C} & g_{14}^{C} & g_{15}^{C} & g_{16}^{C} \\ g_{21}^{C} & g_{22}^{C} & g_{23}^{C} & g_{24}^{C} & g_{25}^{C} & g_{26}^{C} \\ g_{31}^{C} & g_{32}^{C} & g_{33}^{C} & g_{34}^{C} & g_{35}^{C} & g_{36}^{C} \\ g_{41}^{C} & g_{42}^{C} & g_{43}^{C} & g_{44}^{C} & g_{45}^{C} & g_{46}^{C} \\ g_{51}^{C} & g_{52}^{C} & g_{53}^{C} & g_{54}^{C} & g_{55}^{C} & g_{56}^{C} \\ g_{61}^{C} & g_{62}^{C} & g_{63}^{C} & g_{64}^{C} & g_{65}^{C} & g_{66}^{C} \end{matrix}] {\begin{matrix} ε_{1} \\ ε_{2} \\ ε_{3} \\ 2 ε_{4} \\ 2 ε_{5} \\ 2 ε_{6} \end{matrix}}

(28)

in which the number of the piezoresistive strain coefficients $g_{ij}^{C}$ is reduced to 36 with the Voigt notation correspondence in Cartesian coordinates given by

Δ ρ_{1}^{C} = Δ ρ_{11}^{C}, Δ ρ_{2}^{C} = Δ ρ_{22}^{C}, Δ ρ_{3}^{C} = Δ ρ_{33}^{C}

Δ ρ_{4}^{C} = Δ ρ_{23}^{C}, Δ ρ_{5}^{C} = Δ ρ_{13}^{C}, Δ ρ_{6}^{C} = Δ ρ_{12}^{C}

ε_{1} = ε_{11}, ε_{2} = ε_{22}, ε_{3} = ε_{33}

ε_{4} = ε_{23}, ε_{5} = ε_{13}, ε_{6} = ε_{12}

(29)

Similarly, in equation (28) the Voigt notation correspondence in cylindrical components is given by

Δ ρ_{1}^{C} = Δ ρ_{rr}^{C}, Δ ρ_{2}^{C} = Δ ρ_{θ θ}^{C}, Δ ρ_{3}^{C} = Δ ρ_{zz}^{C}

Δ ρ_{4}^{C} = Δ ρ_{θ z}^{C}, Δ ρ_{5}^{C} = Δ ρ_{rz}^{C}, Δ ρ_{6}^{C} = Δ ρ_{r θ}^{C}

ε_{1} = ε_{rr}, ε_{2} = ε_{θ θ}, ε_{3} = ε_{zz}

ε_{4} = ε_{θ z}, ε_{5} = ε_{rz}, ε_{6} = ε_{r θ}

(30)

Based on CNT geometry and composition,² the CNT piezoresistive strain coefficients are at most of transversely isotropic symmetry and can therefore be expressed as

{\begin{matrix} Δ ρ_{1}^{C} \\ Δ ρ_{2}^{C} \\ Δ ρ_{3}^{C} \\ Δ ρ_{4}^{C} \\ Δ ρ_{5}^{C} \\ Δ ρ_{6}^{C} \end{matrix}} = [\begin{matrix} g_{11}^{C} & g_{12}^{C} & g_{13}^{C} & 0 & 0 & 0 \\ g_{12}^{C} & g_{11}^{C} & g_{13}^{C} & 0 & 0 & 0 \\ g_{13}^{C} & g_{13}^{C} & g_{33}^{C} & 0 & 0 & 0 \\ 0 & 0 & 0 g_{44}^{C} & 0 & 0 \\ 0 & 0 & 0 & 0 & g_{44}^{C} & 0 \\ 0 & 0 & 0 & 0 & 0 & g_{66}^{C} \end{matrix}] {\begin{matrix} ε_{1} \\ ε_{2} \\ ε_{3} \\ 2 ε_{4} \\ 2 ε_{5} \\ 2 ε_{6} \end{matrix}}

(31)

The number of independent components of the $g_{ij}^{C}$ matrix is thus reduced to 5 with $g_{66}^{C} = g_{11}^{C} - g_{12}^{C} / 2$ . However, $g_{ij}^{C} = g_{ji}^{C}$ is based on an assumption that still needs to be further verified by either laboratory or lower length scale computational experiments as current efforts in this area have only explored the axial component, $g_{33}^{C}$ . It is further noted that equation (31) can be reduced according to the assumptions regarding zero strain components for both the plane strain hexagonal and axisymmetric microscale RVEs as indicated above. The electrical boundary conditions on the microscale RVE can be expressed as either current density or potential conditions by

n_{i} J_{i} = n_{i} {\tilde{J}}_{i} ({\tilde{X}}_{k})

(32a)

Φ = x_{i} {\tilde{E}}_{i} ({\tilde{X}}_{k})

(32b)

such that in general the macroscale and microscale RVE current densities and electric fields can be related to one another through microscale volume averages as

{\tilde{J}}_{i} ({\tilde{X}}_{k}) = 〈 J_{i} 〉

(33a)

{\tilde{E}}_{i} ({\tilde{X}}_{k}) = 〈 E_{i} 〉

(33b)

where ${\tilde{J}}_{i} ({\tilde{X}}_{k})$ and ${\tilde{E}}_{i} ({\tilde{X}}_{k})$ are the macroscale current density and electric field at a macroscale material point ${\tilde{X}}_{k}$ , which the microscale RVE represents. Finally, as the local CNT resistivity according to equation (31) is now a function of the local strain in the CNT, which, in turn, is influenced by the specific mechanical boundary conditions being imposed from the macroscale, the CNT and hence the effective nanocomposite conductivities will therefore be functions of the applied mechanical loads at the macroscale.

Boundary conditions and effective properties

Mechanical boundary conditions

The macroscale material’s effective elastic material properties can be obtained by constructing strain energy equivalence between the microscale RVE and the effective homogeneous representation using homogeneous periodic mechanical boundary conditions (Bensoussan et al., 1978). In the present work, periodic displacement boundary conditions consistent with certain desired uniform macroscale strain states are applied to the microscale RVEs that are known to be good representations for aligned well-dispersed fiber arrangements. The uniform strain fields chosen here are consistent with those present in macroscale plane strain tension tests and uniaxial tension test, which could be used to get effective elastic properties for the nanocomposites from the volume-averaged stress and strain relationships provided in equation (4) (Hammerand et al., 2007). Herein the focus is on the piezoresistive effect, thus the choice of the tests are to demonstrate the magnitude of the impact of different strain states on the effective piezoresistive response of the nanocomposite material. The detailed periodic mechanical boundary conditions are listed in Table 1.

Table 1.

Boundary conditions for the 2D microscale RVEs as shown in Figure 2(a) and (b).

Type of test	Boundary conditions
Uniaxial tension test for the axisymmetric microscale RVE of Figure 2(b)	$t_{r} (R, X_{z}) = t_{z} (R, X_{z}) = 0$ , $u_{r} (X_{r}, \frac{L}{2}) - u_{r} (X_{r}, - \frac{L}{2}) = 0$ , $u_{z} (X_{r}, \frac{L}{2}) - u_{z} (X_{r}, - \frac{L}{2}) = ε_{0} L$

Plane strain uniaxial tension test for transverse hexagonal microscale RVE of Figure 2(a)	$u_{1} (\frac{W_{1}}{2}, X_{2}) - u_{1} (- \frac{W_{1}}{2}, X_{2}) = - ν'_{21} ε_{0} W_{1}$ ,^a $u_{2} (\frac{W_{1}}{2}, X_{2}) - u_{2} (- \frac{W_{1}}{2}, X_{2}) = 0$ , $u_{1} (X_{1}, \frac{W_{2}}{2}) - u_{1} (X_{1}, - \frac{W_{2}}{2}) = 0$ , $u_{2} (X_{1}, \frac{W_{2}}{2}) - u_{2} (X_{1}, - \frac{W_{2}}{2}) = ε_{0} W_{2}$

Plane strain biaxial tension test for transverse hexagonal microscale RVE of Figure 2(a)	$u_{1} (\frac{W_{1}}{2}, X_{2}) - u_{1} (- \frac{W_{1}}{2}, X_{2}) = ε_{0} W_{1}$ , $u_{2} (\frac{W_{1}}{2}, X_{2}) - u_{2} (- \frac{W_{1}}{2}, X_{2}) = 0$ , $u_{1} (X_{1}, \frac{W_{2}}{2}) - u_{1} (X_{1}, - \frac{W_{2}}{2}) = 0$ , $u_{2} (X_{1}, \frac{W_{2}}{2}) - u_{2} (X_{1}, - \frac{W_{2}}{2}) = ε_{0} W_{2}$

RVE: representative volume element.

$ν'_{21} = \frac{ν_{21}^{Eff} + ν_{31}^{Eff} ν_{23}^{Eff}}{1 - ν_{32}^{Eff} ν_{23}^{Eff}}$

It can be noted that under the in-plane strain assumption, the displacement field components $u_{1}$ and $u_{2}$ for the transverse hexagonal microscale RVE are independent of $X_{3}$ , while under the axisymmetric assumption, the displacement field components $u_{r}$ and $u_{z}$ for the axisymmetric microscale RVE are independent of $X_{θ}$ . The strain $ε_{0}$ corresponds to the strain on a macroscale material point under tension and is quasi-statically increased within the linear elastic range.

In general, the effective elastic properties can be obtained from the strain energy equivalency between the microstructural representation (i.e. the RVE) and an effective homogeneous material having the same exterior geometry and boundary conditions. The equivalency can be expressed as

〈 σ_{ij} ε_{ij} 〉 = 〈 W^{RVE} 〉 = 〈 W^{Eff} 〉 = 〈 σ_{ij}^{Eff} ε_{ij}^{Eff} 〉

(34)

where $σ_{ij}^{Eff}$ and $ε_{ij}^{Eff}$ denote the stress and strain fields in the homogenous material representation. Through the application of the Hill–Mandel theorem, the energy equivalency can be expressed as

〈 σ_{ij} 〉 〈 ε_{ij} 〉 = 〈 σ_{ij}^{Eff} 〉 〈 ε_{ij}^{Eff} 〉 = 〈 C_{ijkl}^{Eff} ε_{kl}^{Eff} 〉 〈 ε_{ij}^{Eff} 〉 = C_{ijkl}^{Eff} 〈 ε_{kl}^{Eff} 〉 〈 ε_{ij}^{Eff} 〉

(35)

As both the microstructural RVE and effective homogeneous representation are subjected to the same displacement boundary conditions, $〈 ε_{ij} 〉 = 〈 ε_{ij}^{Eff} 〉$ , and thus, equation (4) is recovered. It can be shown that $C_{ijkl}^{Eff}$ obtained in this manner under the assumptions of linear elastic perfectly bonded phases within the microstructure is independent of the strains associated with the applied displacements, that is, of $ε_{0}$ .

Electrostatic boundary conditions

In an analogous manner as in the mechanical problem, the macroscale material’s effective electrostatic material properties can be obtained by constructing electrical energy equivalence between the microscale RVE and the effective one with homogeneous periodic electrical boundary conditions applied. In the present work, periodic potential boundary conditions consistent with certain desired uniform electric field states are applied to the microscale RVEs, as listed in Table 2.

Table 2.

Electrostatic boundary conditions for obtaining the effective electrical conductivities.

Type of test	Boundary conditions
For obtaining $κ_{zz}^{Eff}$ ( $κ_{33}^{Eff}$ ) in the axisymmetric microscale RVE	$Φ (X_{r}, \frac{L}{2}) - Φ (X_{r}, - \frac{L}{2}) = E_{0} L$ , $J_{r} (R, X_{z}) = 0$
For obtaining $κ_{11}^{Eff}$ in the transverse hexagonal RVE	$Φ (X_{1}, \frac{W_{2}}{2}) - Φ (X_{1}, - \frac{W_{2}}{2}) = 0$ , $Φ (\frac{W_{1}}{2}, X_{2}) - Φ (- \frac{W_{1}}{2}, X_{2}) = E_{0} W_{1}$
For obtaining $κ_{22}^{Eff}$ in the transverse hexagonal RVE	$Φ (X_{1}, \frac{W_{2}}{2}) - Φ (X_{1}, - \frac{W_{2}}{2}) = E_{0} W_{2}$ , $Φ (\frac{W_{1}}{2}, X_{2}) - Φ (- \frac{W_{1}}{2}, X_{2}) = 0$
For obtaining $κ_{12}^{Eff}$ in the transverse hexagonal RVE	$Φ (\frac{W_{1}}{2}, X_{2}) - Φ (- \frac{W_{1}}{2}, X_{2}) = E_{0} W_{1}$ , $Φ (X_{1}, \frac{W_{2}}{2}) - Φ (X_{1}, - \frac{W_{2}}{2}) = E_{0} W_{2}$

RVE: representative volume element.

Under the in-plane electrostatic assumption, the potential field $Φ$ for the transverse hexagonal microscale RVE is independent of $X_{3}$ , while under the axisymmetric assumption, the potential field $Φ$ for the axisymmetric microscale RVE is independent of $X_{θ}$ .

In general, the electrical conductivities can be obtained from the electrical energy equivalency between the microstructural representation (i.e. the RVE) and an effective homogeneous material having the same exterior geometry and boundary conditions. The equivalency can be expressed as

〈 J_{i} E_{i} 〉 = 〈 W^{RVE} 〉 = 〈 W^{Eff} 〉 = 〈 J_{i}^{Eff} E_{i}^{Eff} 〉

(36)

where $J_{i}^{Eff}$ and $E_{i}^{Eff}$ denote the current density and electric field in the homogenous material representation. Through the application of the electrical analogy to the Hill–Mandel theorem, the energy equivalency can be expressed as

\begin{matrix} 〈 J_{i} 〉 〈 E_{i} 〉 = 〈 J_{i}^{Eff} 〉 〈 E_{i}^{Eff} 〉 = 〈 κ_{ij}^{Eff} E_{j}^{Eff} 〉 〈 E_{i}^{Eff} 〉 \\ = κ_{ij}^{Eff} 〈 E_{j}^{Eff} 〉 〈 E_{i}^{Eff} 〉 \end{matrix}

(37)

As both the microstructural RVE and effective homogeneous representation are subjected to the same potential boundary conditions, $〈 E_{i} 〉 = 〈 E_{ij}^{Eff} 〉$ , and thus equation (9b) is recovered. While it can be shown that $κ_{ij}^{Eff}$ obtained in this manner under the assumptions of linear material behavior and perfectly connected phases within the microstructure is independent of the electric fields associated with the applied potentials, that is of $E_{0}$ , it must be noted, however, that due to the inherent piezoresistivity of the CNTs, the effective conductivity is a function of the strains associated with the applied displacements, that is, $κ_{ij}^{Eff} = κ_{ij}^{Eff} (ε_{kl})$ .

Piezoresistive algorithm

An incremental algorithm for determining the piezoresistive response of the nanocomposites is developed within a finite element framework and is applied to the 2D microscale RVEs as shown in Figure 2. The Galerkin and Ritz methods are used, respectively, for the formulation of in-plane and axisymmetric piezoresistive finite element models. The mechanical and electrostatic finite element formulations are coupled together sequentially with the mechanical formulation used to first obtain the strains in the microscale RVEs, which are then used as inputs for the electrostatic formulation. The detailed piezoresistive algorithm is provided in Figure 3.

Figure 3.

The piezoresistive algorithm of the finite element model.

In step 1 of the algorithm, the electrostatic potential boundary-value problems as identified in Table 2 are applied to the initial undeformed microscale RVEs resulting in potential distributions, and the electrical energy equivalence is used to obtain the conductivities of the effective microscale RVEs. In step 2, the first mechanical load increments are applied to the microscale RVEs according to the boundary conditions identified in Table 1, resulting in the local strain distributions. In step 3, the local strains in the CNT obtained from step 2 are used in the piezoresistive constitutive relation to determine changes in the local resistivities of the CNTs. In step 4, the same electrostatic potential boundary-value problems from step 1 are applied again to the microscale RVEs with updated local electrical properties of the CNT, with the electrical energy equivalency enforced to obtain the effective conductivities $κ_{ij}^{Eff} (ε_{kl})$ of the effective microscale RVEs. Step 2 through step 4 are then repeated for subsequent loading increments. From this process, the effective electrical properties of the microscale RVEs with inherent piezoresistive effect of the CNT can be obtained as a function of the applied macroscale homogeneous strain, that is, the load-dependent but reversible macroscale effective piezoresistive response.

In order to compare with measures more commonly used in experiments, the effective resistivities $ρ_{ij}^{Eff} (ε_{kl})$ (inverse of $κ_{ij}^{Eff} (ε_{kl})$ ) obtained in the plane strain tension tests and uniaxial tension test can be converted into macroscale nanocomposite gauge factors. As shown schematically in Figure 4, the gauge factor for a uniaxial tension test can be expressed in general as

Figure 4.

Schematic representation of the calculation of macroscale gauge factors for (a) a general uniaxial tension test, (b) a uniaxial tension test in the alignment direction of an effectively transversely isotropic nanocomposite, and (c) a uniaxial tension test in the transverse direction of an effectively transversely isotropic nanocomposite.

G = \frac{\frac{Δ R}{R}}{ε} = \frac{\frac{ρ^{f} \frac{L^{f}}{A^{f}} - ρ^{0} \frac{L^{0}}{A^{0}}}{ρ^{0} \frac{L^{0}}{A^{0}}}}{ε_{0}}

(38)

where G is the gauge factor, $ε$ is the applied strain, and R is the resistance measured between two electrodes. The change in resistance can be expressed in terms of the initial resistivity, $ρ^{0}$ , the undeformed distance between electrodes, $L^{0}$ , the undeformed electrode area, $A^{0}$ , and their corresponding measures in the deformed configuration of $ρ^{f}$ , $L^{f}$ , and $A^{f}$ , respectively. For the uniaxial tension test, the deformed distance between electrodes and deformed electrode area can be expressed in terms of the applied strain as $L^{f} = L^{0} (1 + ε_{0})$ and $A^{f} = A^{0} (1 - ν_{31} ε_{0}) (1 - ν_{32} ε_{0})$ , respectively, so that G in equation (38) becomes

G = \frac{1}{ε_{0}} (\frac{ρ^{f}}{ρ^{0}} \frac{1 + ε_{0}}{(1 - ν_{31} ε_{0}) (1 - ν_{32} ε_{0})} - 1)

(39)

For the metal alloy wires commonly used in strain gauges, the initial and final resistivities are the same, so that the gauge factor is governed by the isotropic Poisson’s ratio and results in an approximately linear relationship between normalized change in resistance and strain (i.e. with G as the slope of the curve) for small strains.

However, the inherent piezoresistivity of the CNTs means that the initial and final resistivities of the nanocomposite will differ from one another, introducing additional nonlinearity in the gauge factor. As such, the effective gauge factor for the nanocomposite as obtained from a uniaxial tension test in the CNT alignment direction (i.e. from the axisymmetric RVE) would be determined as

G^{Eff} = \frac{1}{ε_{0}} (\frac{ρ_{33}^{Eff} (ε_{0})}{ρ_{33}^{Eff} (0)} \frac{1 + ε_{0}}{{(1 - ν_{31}^{Eff} ε_{0})}^{2}} - 1)

(40)

where the transverse direction is the isotropic plane such that $ν_{31}^{Eff} = ν_{32}^{Eff} = ν_{zr}^{Eff}$ , which can be obtained from the volume-averaged strain response from one of the strain increments (i.e. the first time conducting step 2 of the algorithm) or may be calculated a priori using established tools such as the Mori–Tanaka method (Benveniste, 1987; Mori and Tanaka, 1973) for microscale RVEs containing finite AR CNTs or from the composite cylinder method (Seidel and Lagoudas, 2006) for infinitely long CNTs (i.e. ARs greater than 200³ (McCullough, 1990)). For the case of plane strain uniaxial tension (i.e. uniaxial tension in the transverse microscale RVE), the nanocomposite gauge factor can be derived from equation (38) and obtained as

G^{Eff} = \frac{1}{ε_{0}} (\frac{ρ_{22}^{Eff} (ε_{0})}{ρ_{22}^{Eff} (0)} \frac{1 + ε_{0}}{1 - \frac{(ν_{21}^{Eff} + ν_{31}^{Eff} ν_{23}^{Eff}) ε_{0}}{1 - ν_{32}^{Eff} ν_{23}^{Eff}}} - 1)

(41)

Similarly, for the transverse plane strain symmetric biaxial tension case, equation (38) becomes independent of the effective Poisson’s ratios and is given by

G^{Eff} = \frac{1}{ε_{0}} (\frac{ρ_{22}^{Eff} (ε_{0})}{ρ_{22}^{Eff} (0)} - 1)

(42)

Results and discussion

The polymer matrix used in the model is the epoxy resin EPON 828, which is assumed to be isotropic linear elastic below its glass transition temperature. The detailed mechanical, electrical, and geometric parameters of the CNT and EPON 828 matrix are listed in Table 3.

Table 3.

Mechanical, electrical, and geometric parameters for the CNT and EPON 828 polymer matrix (Arris, 2000; Ebbesen et al., 1996; Hammerand et al., 2007; Schadler et al., 1998; Seidel and Lagoudas, 2009). The conductivity of the CNT listed here is the initial conductivity with zero strain.

Materials	Properties
	Geometry	Mechanical	Electrical
SWCNT	$R_{Out} = 0.85$ nm, $R_{In} = 0.51$ nm	$E = 1100 GPa$ , $υ = 0.14$	$κ = 10^{5} S / m$
EPON 828	$V_{f} = 1 - \frac{V_{CNT}}{V_{RVE}}$	$E = 3.07 GPa$ , $υ = 0.3$	$κ = 1.49 \times 10^{- 9} S / m$

SWCNT: single-walled carbon nanotubes.

Experimental efforts have demonstrated that the gauge factor of a single metallic SWCNT ( $G^{C}$ ) can be as high as 2900 (Stampfer et al., 2006), which can thus be used to determine an initial value for the CNT piezoresistive strain coefficient by rearranging equation (38) as

ρ_{zz}^{C} (ε_{C}) = ρ_{zz}^{C 0} (G^{C} ε_{C} + 1) \frac{{(1 - ν_{zr}^{C} ε_{C})}^{2}}{1 + ε_{C}}

(43)

where it has been understood that the experimental conditions are most reflective of uniaxial tension of the CNT along the CNT axis. Applying the piezoresistive constitutive equation and solving for the axial piezoresistive strain coefficient, one obtains

g_{zz}^{C} (ε_{C}) = \frac{1}{ε_{C}} (ρ_{zz}^{C 0} (G^{C} ε_{C} + 1) \frac{{(1 - ν_{zr}^{C} ε_{C})}^{2}}{1 + ε_{C}} - ρ_{zz}^{C 0})

(44)

Using the data from Table 3 and the gauge factor of 2900 and CNT strain value of $ε_{C} = 1.5 %$ obtained from Stampfer et al. (2006), a CNT axial piezoresistive strain coefficient of $g_{zz}^{C} = 2.84 E - 2 Ω m$ is obtained and used along with additional selected values in a parametric study of the piezoresistive response of CNT–polymer nanocomposites, the outcomes of which are discussed below.

Nanocomposite piezoresistive response under uniaxial tension in the CNT alignment direction

The axisymmetric microscale RVE is used to parametrically assess the influence of CNT gauge factor on the macroscale piezoresistive response of nanocomposites in tension in the CNT alignment direction. Five CNT gauge factors are considered, with the first case corresponding to the experimentally reported CNT gauge factor. The CNT gauge factors and their corresponding axial piezoresistive strain coefficients are provided in Table 4. It is noted that in order to focus on the dominant strain component’s impact on piezoresistivity, it is assumed that only the axial piezoresistive strain coefficient is nonzero.

Table 4.

The conversions from CNT gauge factor to CNT axial piezoresistive strain coefficient $g_{zz}^{C}$ , in which case AU1 is converted at the strain $ε_{C} = 1.5 %$ as in the study of Stampfer et al. (2006) and cases AU2–AU5 are converted at the strain $ε_{C} = 1 %$ . For each case, it is assumed that all other $g_{ij}^{C}$ are zero.

Case	$G^{C}$	$g_{zz}^{C}$ ( $Ω m$ )
AU1	2900	2.84E−2
AU2	1E6	9.87E0
AU3	1E11	9.87E5
AU4	1E13	9.87E7
AU5	1E17	9.87E11

The evolution of local strains and resistivities under applied macroscale strain increments are provided in Figure 5 for a CNT volume fraction of 0.20 and AR ( $AR = L_{CNT} / 2 r_{Out}$ ) of 5 and for a CNT gauge factor corresponding to case AU1. The contour plots demonstrate the direct correspondence between changes in local strain in the CNT and changes in the local CNT resistivity, while the local matrix resistivity remains unchanged despite changes in local strains. Further verification of the piezoresistive implementation within the microscale RVE is obtained by plotting the resistivity as a function of applied boundary strain for an element located in the middle of the CNT as provided in Figure 6. It is observed that the axial resistivity of the CNT element is changing as a linear function of the boundary strain, while the conductivity varies as its inverse and therefore accurately reflects the linear piezoresistive constitutive response of the CNT. Though not present in the axisymmetric uniaxial tension results provided here, it is worth noting that compressive strains of a sufficiently large value for a given CNT piezoresistive strain coefficient can lead to negative values for the conductivity. As such cases are deemed to be not of physical interest herein, tension–compression asymmetry is introduced into the piezoresistive response such that at a critical compressive strain value, the linear piezoresistive response should smoothly transition to one, which asymptotically approaches zero. For practical simplicity, in the present work a constant small resistivity value is used at the critical compressive strain value and maintained throughout further compressive strains in a manner analogous to a perfectly plastic response in mechanical loading, with the exception being that unloading follows the same path as loading in the piezoresistive case (as opposed to unloading elastically in the mechanical analog). This behavior will play a role in some of the transverse hexagonal microscale RVE cases to be discussed in section “Results for the transverse piezoresistive response of the nanocomposites” for large CNT gauge factors.

Figure 5.

The evolution of local strains and resistivities under applied macroscale strain increments for a CNT volume fraction of 0.20 and aspect ratio of 5 and for a CNT gauge factor corresponding to case AU1. A potential difference of 1 V is applied between the top and bottom edge of the microscale RVE. The results are shown in the form of contour plots of the axial strain component $ε_{zz}$ and axial resistivity component $ρ_{zz}$ in response to the applied homogeneous boundary strain that is linearly increased from 0% to 1%. The volume fraction of the CNT in the microscale RVE is 0.20, with an aspect ratio of 5, which are higher and lower, respectively, than typical CNT–polymer nanocomposite values but are selected here to better view changes within the CNT.

Figure 6.

The change of electrical properties within an element of the CNT in response to the applied boundary strain, $ε_{0}$ . The results correspond to an element near the center of the CNT shown in Figure 5 and are therefore reflective of a CNT with AR = 5 and gauge factor of that of case AU1 at a CNT volume fraction of 0.20.

Having demonstrated successful implementation of the piezoresistive algorithm, the remaining results focus on nanocomposite effective piezoresistive response for CNT having higher ARs (e.g. 50, 100, 200, 300, 1000, 3000, and infinite) and at CNT volume fractions of 0.005, 0.01, 0.05, and 0.10. It is noted that common SWCNT ARs range between 300 and 600 and, in some cases, to more than 1000. It is also noted that the vertical separation between the end of the CNT and the boundary of the axisymmetric RVE is set to be consistent with the maximum allowable separation at the CNT AR of 1 for every given volume fraction. The separation is then held constant for higher ARs so that in the limit case, the radius R becomes one of the infinite cases of which the transverse hexagonal RVE is a good approximation.

The influence of the CNT gauge factor on the change in effective axial resistivity component as a function of applied boundary strain, that is, $Δ ρ_{zz}^{Eff} = ρ_{zz}^{Eff} (ε_{0}) - ρ_{zz}^{Eff} (0)$ , is provided in Figure 7. In Figure 7(a), the change in effective axial resistivity of the nanocomposite for case AU1 (CNT gauge factor of 2900) is observed to behave in an approximately linear fashion with respect to the applied boundary strain up to 1% strain. As such, one could directly determine an effective piezoresistive strain coefficient for the nanocomposite of $g_{zz}^{Eff} = 4.26 Ω m$ from the slope of the line. However, as the CNT gauge factor is increased, the change in the effective axial resistivity of the nanocomposite as a function of applied boundary strain becomes nonlinear, as shown in Figure 7(b), for case AU4 (CNT gauge factor of 1E13). As this nonlinearity is observed at applied strains of less than 1% where the common metallic materials in strain gauges would demonstrate a linear response due strictly to geometry and Poisson’s ratio (i.e. $ρ^{f} / ρ^{0} = 1$ ), the nonlinearity is directly attributed to difference in final and initial nanocomposite conductivities. Further, as the CNT piezoresistive constitutive response remains linear throughout the range of applied boundary strains, the nanocomposite nonlinearity is therefore associated with the evolution of the local strain fields under the applied load. In Figure 7(c), it is noted, however, that the change in effective nanocomposite conductivity as a function of applied boundary strain can reach a saturation limit. This is because under sufficiently large enough applied strains, the CNT resistivity effectively becomes greater than that of a matrix, thereby switching the composite behavior from that of conductive fillers in a nonconducting matrix to a nonconducting matrix with voids. Even though the piezoresistivity of the CNT may lead to CNT resistivities that are very large compared to that of the matrix, as the volume fraction of CNT remains fixed, at the saturation point, the matrix essentially only experiences a negligibly better approximation of void from the CNT so that the effective nanocomposite resistivity no longer appreciably increases with increasing applied boundary strain. The larger the CNT gauge factor, the lower the value of applied boundary strain one can expect to achieve saturation. For the case AU5 (CNT gauge factor of 1E17), the saturation point occurs at an applied boundary strain of approximately 0.05%, after which the percent difference increase of the change of effective resistivity is no higher than 0.1%. It is worth noting, however, that the use of linear elastic field equations to obtain the local strains means that the effective nanocomposite piezoresistive response, whether linear, nonlinear, or saturated, is completely reversible upon unloading. Finally, a comparison of all five axisymmetric uniaxial tension test results is provided in Figure 7(d) in terms of the semi-log plot of the difference between final and initial nanocomposite axial resistivity. There it is observed for cases AU2 and AU3 that the orders of magnitude increases in CNT gauge factor lead to comparable orders of magnitude increases in $Δ ρ_{zz}^{Eff}$ , whereas cases AU4 and AU5 demonstrate the saturation effect associated with the effective switch from conductive inclusions to voids such that further increases in CNT gauge factor will have negligible effects on the effective nanocomposite’s piezoresistive response.

Figure 7.

Nanocomposite effective axial resistivities $ρ_{zz}^{Eff}$ obtained from the axisymmetric microscale RVE in response to the applied boundary strain $ε_{0}$ for a CNT volume fraction of 0.005 and CNT aspect ratio of AR = 300. (a) Change of the effective resistivity component $Δ ρ_{zz}^{Eff}$ as a function of the applied boundary strain of the microscale RVE for case AU1 with $G^{C} = 2900$ . (b) $Δ ρ_{zz}^{Eff}$ versus $ε_{0}$ for case AU4 with $G^{C} = 1 E 13$ . (c) $Δ ρ_{zz}^{Eff}$ versus $ε_{0}$ for case AU5 with $G^{C} = 1 E 17$ . (d) $Δ ρ_{zz}^{Eff}$ versus $ε_{0}$ in semi-log plot for the five CNT gauge factor cases listed in Table 4.

The effects on the piezoresistive response of the nanocomposite of increasing the AR of the CNTs at a fixed volume fraction of 0.01 are provided in Figure 8 for CNT ARs of 50, 100, 200, 300, 1000, and 3000 and for the infinitely long (i.e. continuous) CNT case for the CNT gauge factor of AU3. In Figure 8(a), the difference in the volume-averaged resistivity of the CNT relative to its zero strain value, that is, $Δ ρ_{zz}^{C}$ , is observed to demonstrate a quickly converging trend, with percent differences relative to the infinitely long CNT response at an applied strain of 1% of −64.8%, −46.6%, and −7.41% for ARs of 50, 100, and 1000, respectively. As all the CNTs regardless of AR have the same uniform zero strain resistivity and piezoresistive strain coefficient, the observed differences in $Δ ρ_{zz}^{C}$ are attributed to the differences in the amount of strain transferred to the CNT, where the infinitely long case directly carries the applied strain, while the CNTs with ARs of 50, 100, and 1000 carry volume-averaged strains of 35.2%, 53.4%, and 92.6% of the applied strain, respectively. However, unlike the CNT, the nanocomposite effective resistivities at zero strain vary significantly with ARs as shown in Figure 12(b). In fact, while the 100 and 1000 AR cases have zero strain effective resistivities that are 16.2% and 79.2% lower than the AR of 50, due to the large differences between the zero strain CNT resistivity and that of the matrix, all the ARs considered here remain orders of magnitude higher (∼11) than the nanocomposite effective resistivity of the infinitely long case, indicating that significantly larger ARs are needed in order to see the same convergence toward the infinitely long behavior for the effective electrical properties than was observed for the convergence of effective mechanical properties of the nanocomposite. Thus, even though all the ARs considered demonstrate CNT piezoresistive responses ( $Δ ρ_{zz}^{C}$ ) which converge over a range of a few orders of magnitude ( $10^{2}$ – $10^{4}$ ), resulting in the change in nanocomposite effective resistivity ( $Δ ρ_{zz}^{Eff} (ε_{0})$ ) also varying over a range of a few orders of magnitude ( $10^{4}$ – $10^{7}$ ), it is perhaps more meaningful to consider the change in nanocomposite effective resistivity normalized by the given AR’s zero strain resistivity, that is, $Δ ρ_{zz}^{Eff} (ε_{0}) / ρ_{zz}^{Eff} (0)$ as provided in Figure 8(c). From such a representation, one can quickly observe two key points. The first is that the AR strongly influences the sensitivity of the piezoresistive response, with higher ARs demonstrating greater sensitivity, for example, an AR of 100 yields a 0.14% increase in resistivity at 1% strain, whereas an AR of 3000 yields a 22.8% increase in resistivity at that same strain level. The second key point observed from the normalized change in resistivity is that sensitivity, like the overall effective resistivity, requires much higher ARs than considered here in order to converge to the infinitely long CNT case that has a sensitivity measured in orders of magnitude at 1% strain. Thus, AR can be viewed as a tuning parameter for having a desired uniaxial strain sensitivity at a given volume fraction.

Figure 8.

The effect of AR of the CNT on the effective electrical properties of the nanocomposites for a CNT volume fraction of 0.01 and CNT gauge factor of case AU3 ( $G^{C} = 1 E 11$ ). (a) Volume-averaged $Δ ρ_{zz}^{C} (ε_{0})$ within the CNT versus $ε_{0}$ for different ARs of the CNT. (b) The zero strain effective resistivity $ρ_{zz}^{Eff} (0)$ of the nanocomposites versus AR of the CNT. (c) $Δ ρ_{zz}^{Eff} (ε_{0}) / ρ_{zz}^{Eff} (0)$ versus $ε_{0}$ for different ARs of the CNT.

The influence of CNT volume fraction on the effective piezoresistive response of the nanocomposite to applied uniaxial strain in the CNT alignment direction is provided in Figure 9 for a fixed CNT AR of 300 and for CNT gauge factors corresponding to cases AU3 and AU4. As the strains within the CNT vary by only a small amount ( $< 1 %$ with lower volume fractions having larger strains), so that the changes in nanocomposite effective resistivity ( $Δ ρ_{zz}^{Eff} (ε_{0})$ ) are all of the same order of magnitude, it is again deemed more informative to observe the zero strain normalized response, that is, $Δ ρ_{zz}^{Eff} (ε_{0}) / ρ_{zz}^{Eff} (0)$ . As the zero strain resistivities vary by as much as 88%, with resistivities of 3.91E8, 2.78E8, 8.58E7, and $4.64 E 7 Ω m$ at volume fractions of 0.005, 0.01, 0.05, and 0.1, respectively, large differences in the sensitivity of the nanocomposite piezoresistive response can be observed with the higher volume fractions being more sensitive to the applied strain than the lower volume fractions. In Figure 9(a), the CNT gauge factor of $G^{C} = 1 E 11$ is observed to yield a linear normalized piezoresistive response at all volume fractions considered, with the volume fraction of 0.1 demonstrating much higher sensitivity than the volume fraction of 0.005 by roughly two times. The same trend in terms of higher volume fractions yielding higher sensitivities is again observed in Figure 9(b) for the CNT gauge factor of $G^{C} = 1 E 13$ ; however, the sensitivities are an order of magnitude higher and demonstrate a nonlinear response as observed in Figure 7(b) for the same CNT strains and zero strain resistivities. It is of interest to note that response for the volume fraction of 0.05 is consistently closer to the 0.1 volume fraction response than it is to the 0.01 volume fraction. While the difference in CNT strains between the 0.05 volume fraction case and the 0.01 and 0.1 volume fraction cases are 0.23% and 0.36%, respectively, the difference in zero strain resistivities between these cases are 224% and 46%, respectively, so that the sensitivity of the 0.05 volume fraction case behaves more like that of the 0.1 volume fraction. This is perhaps best understood from observing a comparison between the results obtained from the axisymmetric finite element model for the infinitely long CNT case with results from composite cylinder model where the CNT resistivity has been modified according to the piezoresistive constitutive equation based on the applied strain and a CNT gauge factor of $G^{C} = 1 E 11$ . Such a comparison in terms of nanocomposite effective axial resistivity versus CNT volume fraction is provided in Figure 10 and demonstrates excellent agreement between the finite element and composite cylinder models at all strain levels and volume fractions. The effective axial conductivity obtained from the composite cylinder model is known to be well approximated by the rule of mixtures, so that the effective resistivity varies as its inverse so that, as is observed in Figure 10, the higher volume fractions have lower overall resistivities, but greater sensitivities, such that the volume fraction of 0.05 results are closer to the 0.1 volume fraction results than to the 0.01. Thus, through a combination of AR and volume fraction, the piezoresistive response in the axial direction under uniaxial tension can be tailored toward a desired overall resistivity and sensitivity level.

Figure 9.

The relative change of the effective axial resistivities of the nanocomposites in response to the applied boundary strain $ε_{0}$ with four different volume fractions of the CNT, which are 0.005, 0.01, 0.05, and 0.1. (a) $(Δ ρ_{zz}^{Eff} / ρ_{zz}^{Eff} (0)) \times 100 %$ versus $ε_{0}$ for case AU3 with $G^{C} = 1 E 11$ . (b) $(Δ ρ_{zz}^{Eff} / ρ_{zz}^{Eff} (0)) \times 100 %$ versus $ε_{0}$ for case AU4 with $G^{C} = 1 E 13$ . The AR of the CNT is 300.

Figure 10.

Comparison between the results obtained from the axisymmetric FEM for the infinitely long CNT case with results from CCM where the CNT resistivity has been modified according to the piezoresistive constitutive equation based on the applied strain and a CNT gauge factor of $G^{C} = 1 E 11$ . It is noted that all cases converge to the matrix resistivity of $6.71 E 8 Ω m$ at a volume fraction of 0 but would lead to different values at a volume fraction of 1 due to the changing resistivity of the CNT with applied strain.

It is observed that with a higher volume fraction of the CNT, the initial effective resistivity is lower and the curve increases more slowly than the ones with a lower volume fraction of the CNT, which behaves differently than the results of the transverse piezoresistive response. This is because the initial effective resistivity is directly influenced by the volume fraction of the CNT, and with a higher volume fraction, the effective resistivity can be lower; on the other hand, the slope of the curve is influenced by the strain level within the CNT, and with a higher volume fraction, the strains within the CNT are lower, causing the curves to increase less rapidly.

By applying equation (40), one can obtain the effective gauge factors of the nanocomposites, $G^{Eff}$ , under uniaxial tension in the CNT alignment direction. Values for the effective nanocomposite gauge factors are provided in Table 5 at volume fractions of 0.005 and 0.01 for CNTs having the five CNT gauge factors considered and AR of 300. For cases AU1 and AU2, the effective nanocomposite gauge factor is observed to have a constant value of 1.6 up to 1% strain indicating (1) that the piezoresistive response is linear and (2) that the inherent piezoresistivity has not had a significant impact on the effective piezoresistive response of the nanocomposite at these strain levels and volume fractions as 1.6 corresponds to the value that would be obtained by purely geometric considerations, that is, when $ρ_{33}^{Eff} (ε_{0}) = ρ_{33}^{Eff} (0)$ in equation (40). As case AU1 was derived from the currently available data on inherent CNT piezoresistivity, these observations imply that the CNTs must have larger gauge factors if inherent CNT piezoresistivity were to be the driving mechanism behind the observed nanocomposite piezoresistivity or that additional mechanisms such as CNT network deformation and electron hopping play a much larger role than inherent CNT piezoresistivity in driving nanocomposite piezoresistivity. In terms of assessing larger CNT gauge factors, it is observed that case AU3, having a CNT gauge factor which is of eight orders larger than case AU1, yields effective nanocomposite gauge factors that retain an approximately linear piezoresistive response at values of 2.4 and 2.7 for volume fractions of 0.005 and 0.01, respectively, which are on the same order as the gauge factor measured in the axial direction of a nanocomposite made of 0.5%wt MWCNT aligned by AC electric field which has been reported as 2.78 ± 0.42 (Oliva-Aviles et al., 2011). In contrast, cases AU4 and AU5 demonstrate nonlinear effective nanocomposite piezoresistive responses, which yield effective nanocomposite gauge factors that are orders of magnitude larger than experimentally reported values and therefore can be used to set upper bounds on potential CNT gauge factors.

Table 5.

The effective nanocomposite gauge factors $G^{Eff}$ at strain levels of 0.1%, 0.4%, 0.7%, and 1.0% for CNT–polymer nanocomposites having CNT volume fractions of 0.005 and 0.01 and CNTs with CNT gauge factors corresponding to cases AU1 through AU5 for a CNT aspect ratio of 300.

		$V_{f} = 0.005$				$V_{f} = 0.01$
Case	$ε_{0} =$	0.1%	0.4%	0.7%	1%	0.1%	0.4%	0.7%	1%
AU1	$G^{Eff}$	1.60	1.60	1.60	1.61	1.60	1.60	1.60	1.61
AU2	$G^{Eff}$	1.60	1.60	1.60	1.61	1.60	1.60	1.60	1.61
AU3	$G^{Eff}$	2.44	2.44	2.45	2.45	2.74	2.75	2.76	2.76
AU4	$G^{Eff}$	76.75	58.84	47.94	40.60	107.54	88.28	75.08	65.47
AU5	$G^{Eff}$	719.39	182.05	105.28	74.53	1416.16	357.29	205.97	145.35

Results for the transverse piezoresistive response of the nanocomposites

The transverse microscale RVE is used to parametrically assess the influence of CNT inherent piezoresistivity on the macroscale piezoresistive response of nanocomposites in the transverse directions with respect to CNT alignment direction. As noted in section “Introduction,” the majority of data regarding inherent piezoresistivity of CNTs is limited to axial testing. As such, there are little data available regarding the remaining independent piezoresistive strain coefficients needed to fully characterize the transversely isotropic piezoresistive response of the CNT. As such, two reduced sets of CNT piezoresistive strain coefficients in the transverse direction are considered. In the first set, the normal strain response are equally weighted in terms of piezoresistive strain coefficients but are decoupled, that is, $g_{11}^{C} = g_{22}^{C}$ and $g_{12}^{C} = 0$ , while in the second set, normal strain responses are equally weighted and coupled, that is, $g_{11}^{C} = g_{22}^{C} = g_{12}^{C}$ . The initial value for the piezoresistive strain coefficients is taken to be the same value as applied in the axial case, with the parametric study focusing on orders of magnitude increases from this base value as reported in Table 6. Both sets of piezoresistive strain coefficients are subjected to transverse uniaxial tension and transverse biaxial tension loadings.

Table 6.

The conversions from CNT gauge factor to CNT piezoresistive strain coefficients, in which cases T1 and T6 are converted at the strain $ε_{C} = 1.5 %$ as in the study of Stampfer et al. (2006) and the remaining cases are converted at the strain $ε_{C} = 1 %$ . For each case, it is assumed that all other $g_{ij}^{C}$ are zero.

Case	$G^{C}$	Piezo coefficients ( $Ω m$ )
T1	2900	$g_{11}^{C} = g_{22}^{C} = 2.84 E - 2$
T2	1E8	$g_{11}^{C} = g_{22}^{C} = 9.87 E 2$
T3	1E14	$g_{11}^{C} = g_{22}^{C} = 9.87 E 8$
T4	1E17	$g_{11}^{C} = g_{22}^{C} = 9.87 E 11$
T5	1E20	$g_{11}^{C} = g_{22}^{C} = 9.87 E 14$
T6	2900	$g_{11}^{C} = g_{22}^{C} = g_{12}^{C} = 2.84 E - 2$
T7	1E8	$g_{11}^{C} = g_{22}^{C} = g_{12}^{C} = 9.87 E 2$
T8	1E14	$g_{11}^{C} = g_{22}^{C} = g_{12}^{C} = 9.87 E 8$
T9	1E17	$g_{11}^{C} = g_{22}^{C} = g_{12}^{C} = 9.87 E 11$
T10	1E20	$g_{11}^{C} = g_{22}^{C} = g_{12}^{C} = 9.87 E 14$

The distributions of local strain and resistivity components of the plane strain uniaxial tension in the 2-direction test are provided in Figure 11 for a CNT volume fraction of 0.60 and for CNT piezoresistive strain coefficients corresponding to cases T1 and T6, respectively.

Figure 11.

The distribution of local strains and resistivities of the plane strain uniaxial tension in the 2-direction test at $ε_{0} = 1 %$ . (a) Distribution of $ε_{11}$ and $ε_{22}$ for cases T1 and T6 at $ε_{0} = 1 %$ . (b) Distribution of $ρ_{11}$ and $ρ_{22}$ for case T1 at $ε_{0} = 1 %$ . (c) Distribution of $ρ_{11}$ and $ρ_{22}$ for case T6 at $ε_{0} = 1 %$ . The volume fraction of the CNTs in the microscale RVE is 0.60, which is higher than typical CNT–polymer nanocomposite volume fraction but is selected here to better view changes within the CNTs.

The contour plots demonstrate the direct correspondence between changes in local strains in the CNTs and changes in the local CNT resistivities, while the local matrix resistivity remains unchanged despite changes in local strains. In Figure 11(b), one can observe that $ρ_{11}^{C}$ and $ρ_{22}^{C}$ within the CNTs are decoupled and directly correspond to $ε_{11}$ and $ε_{22}$ of the CNTs, respectively, which are provided in Figure 11(a). In contrast, in Figure 11(c), it is observed that under those same strain distributions, $ρ_{11}^{C}$ and $ρ_{22}^{C}$ have identical contours due to the use of the coupling term $g_{12}^{C}$ and its being equal to the values assigned to both $g_{11}^{C}$ and $g_{22}^{C}$ . As mentioned earlier, tension–compression asymmetry is introduced in the CNT piezoresistive response to avoid having compressive CNT strains sufficiently large so as to induce resistivity components which would be less than zero. The compressive strain cutoff is demonstrated in Figure 12 using an element in compression in $ε_{11}$ obtained from the transverse uniaxial tension loading in the 2-direction for a CNT piezoresistive strain coefficient corresponding to case T1. There it is observed that the resistivity of the CNT initially follows the same linear piezoresistive response as elements in tension up until the point where the subsequent strain increment would lead to a negative resistivity. At this point, the previous strain increment’s resistivity value is instead prescribed and maintained for all subsequent compressive loading steps. The behavior is analogous to perfectly plastic response, with the exception being that unloading follows the same path as loading.

Figure 12.

The CNT resistivity component $ρ_{11}^{C}$ of an element within the CNT in response to local compressive strain component $ε_{11}$ for case T1 of the plane strain uniaxial tension test as $ε_{0}$ in the 2-direction increases from 0% to 1% for a transverse RVE with CNT volume fraction of 0.60.

Having demonstrated the implementation of the piezoresistive algorithm, the remaining results focus on nanocomposite effective piezoresistive response for CNT having lower volume fractions (e.g. 0.005, 0.01, 0.05, and 0.10). The influence of CNT piezoresistive strain coefficients on the change in effective resistivity component $ρ_{22}^{Eff}$ as a function of applied boundary strain (i.e. $Δ ρ_{22}^{Eff} = ρ_{22}^{Eff} (ε_{0}) - ρ_{22}^{Eff} (0)$ ) for the plane strain uniaxial tension in the 2-direction test is provided in Figure 13.

Figure 13.

Nanocomposite effective transverse resistivity component $Δ ρ_{22}^{Eff}$ obtained from the transverse hexagonal microscale RVE in response to the applied boundary strain $ε_{0}$ of the plane strain uniaxial tension in the 2-direction test, in which the volume fraction of the CNT is 0.005. (a) Change of the effective resistivity component $Δ ρ_{22}^{Eff}$ as a function of the applied boundary strain of the microscale RVE for cases T1 and T6 with $G^{C} = 2900$ . (b) $Δ ρ_{22}^{Eff}$ versus $ε_{0}$ for cases T4 and T9 with $G^{C} = 1 E 17$ . (c) $Δ ρ_{22}^{Eff}$ versus $ε_{0}$ in semi-log plot for cases T1–T5. (d) $Δ ρ_{22}^{Eff}$ versus $ε_{0}$ in semi-log plot for cases T6–T10.

In comparing the response in Figure 13(a) to the piezoresistive response of the nanocomposite to uniaxial tension in the axial direction provided in Figure 7(a), one can observe that $Δ ρ_{22}^{Eff}$ of the plane strain uniaxial tension test for cases T1 and T6 are six orders smaller than $Δ ρ_{zz}^{Eff}$ of the axisymmetric uniaxial tension test for case AU1 having the same $G^{C} = 2900$ and display a slightly nonlinear response. Similarly, one can compare the curves in Figure 13(b) for $Δ ρ_{22}^{Eff}$ under the plane strain uniaxial tension in the 2-direction test for cases T4 and T9 to the $Δ ρ_{zz}^{Eff}$ curve in Figure 13(c) for axisymmetric case AU5 having the same $G^{C} = 1 E 17$ and observe that both T4 and T9 are two or more orders of magnitude smaller than AU5. As a result, the transverse uniaxial tension cases have yet to have achieved the saturation limit observed in the axisymmetric uniaxial loading case. Further, in comparing the curves in Figure 13(c) and (d) to the curves in Figure 7(d), one can observe that the saturated $Δ ρ_{22}^{Eff}$ of the plane strain uniaxial tension test for cases T5 and T10 with $G^{C} = 1 E 20$ are still one to two orders smaller than $Δ ρ_{zz}^{Eff}$ for case AU5 with $G^{C} = 1 E 17$ . This is in part because in the axisymmetric uniaxial tension test with AR = 300, the CNT can take 78.3% of the applied boundary strain $ε_{0}$ in a volume-averaged sense, while in the plane strain uniaxial tension test, the CNTs can only take 0.16% and 0.83%, respectively, for $ε_{11}$ and $ε_{22}$ of the applied boundary strain $ε_{0}$ and therefore have smaller piezoresistive-induced changes in the resistivity of the CNTs. It is also due in part to the well-documented micromechanics observation that the transverse properties in aligned fiber composites are dominated by the matrix properties. For example, in the present work, the zero strain axial effective resistivity, $ρ_{zz}^{Eff} (0)$ , at a volume fraction of 0.005 is 42% below that of the matrix, while the zero strain transverse effective resistivity, $ρ_{22}^{Eff} (0)$ , at that same volume fraction is only 1% below that of the matrix, thus indicating that whatever piezoresistive changes are brought in the transverse directions of the CNTs will be more difficult to detect than in the axial direction.

In Figure 13(c) and (d), one can also observe that $Δ ρ_{22}^{Eff}$ of cases T6–T10 are consistently greater than $Δ ρ_{22}^{Eff}$ of the corresponding cases T1–T5. This is due to the combination of the selected off-diagonal piezoresistive strain coefficient $g_{12}^{C}$ for cases T6–T10 and the dependence of $Δ ρ_{22}^{Eff}$ on both $Δ ρ_{11}^{C}$ and $Δ ρ_{22}^{C}$ of the CNT. The latter may be understood by considering the annular geometry of the CNT as promoting a conductive hoop path, which would be governed by the hoop resistivity, $ρ_{θ θ}^{C}$ , and which by standard transformation is a combination of both $ρ_{11}^{C}$ and $ρ_{22}^{C}$ . From Figure 11(a), it was observed that under the plane strain uniaxial tension in the 2-direction, a significant portion of the CNT was in compression in $ε_{11}$ and in tension in $ε_{22}$ . This observation is better quantified in Figure 14(a), which provides the volume-averaged strain components, that is, $〈 ε_{11} 〉$ and $〈 ε_{22} 〉$ . As a result, the resistivity distribution within the CNT under the application of the piezoresistive strain coefficients $g_{11}^{C} = g_{22}^{C}$ becomes nonuniform and orthotropic for cases T1–T5, with the compressive 1-direction strains leading to reductions in 1-direction resistivities, that is, negative $Δ ρ_{11}^{C}$ , and the tensile 2-direction strains leading to increases in 2-direction resistivity, that is, positive $Δ ρ_{22}^{C}$ , as can be seen in the volume-averaged CNT resistivities provided for case T1 in Figure 14(b). However, the application of an off-diagonal piezoresistive strain coefficient that is equal to the diagonal coefficients in cases T6–T10 results in a CNT resistivity distribution, which is still nonuniform but is now isotropic, with the much larger tensile strains in the 2-direction resulting in large net increases in CNT resistivity, that is, positive $Δ ρ_{11}^{C} = Δ ρ_{22}^{C}$ . As can be seen in Figure 14(b), even though the volume-averaged change in CNT resistivity $〈 Δ ρ_{22}^{C} 〉$ for case T1 is 24% greater than that of T6 at 1% strain, the volume-averaged change in CNT resistivity $〈 Δ ρ_{11}^{C} 〉$ for case T1 is 124% lower than that of T6, so that net effect is a larger $〈 Δ ρ_{22}^{Eff} 〉$ for T6 than T1.

Figure 14.

(a) The volume-averaged strain components $〈 ε_{11}^{C} 〉$ and $〈 ε_{22}^{C} 〉$ and (b) the volume-averaged change of resistivity components $〈 Δ ρ_{11}^{C} 〉$ and $〈 Δ ρ_{22}^{C} 〉$ of the CNTs in response to the applied boundary strain $ε_{0}$ of the plane strain uniaxial tension in the 2-direction test for cases T1 and T6.

The influence of CNT volume fraction on the effective piezoresistive response of the nanocomposite to the applied strain in the plane strain uniaxial tension in the 2-direction test is provided in Figure 15 for CNT gauge factors corresponding to cases T8 and T9. Similar to what was observed in the axisymmetric uniaxial tension test, large differences in the sensitivity of the nanocomposite piezoresistive response can be observed, with the higher volume fractions being more sensitive to the applied strain than the lower volume fractions. In Figure 15(a), case T8 is observed to yield a seemingly linear normalized piezoresistive response at all volume fractions considered, with the volume fraction of 0.10 demonstrating much higher sensitivity than the volume fraction of 0.005 by roughly 31.7 times. The same trend in terms of higher volume fractions yielding higher sensitivities is again observed in Figure 15(b) for case T9; however, the sensitivities are three orders of magnitude higher than the T8 case and demonstrate a slightly nonlinear response. However, when compared with the sensitivities observed in the axial direction, the sensitivities observed in the transverse uniaxial tension test are orders of magnitude smaller even for CNT gauge factors, which are orders of magnitude larger and include off-diagonal terms. This is due to the matrix dominance of effective properties in the transverse direction that leads to reduced influence of changes in the CNT and to zero strain resistivities $ρ_{22}^{Eff} (0)$ , which are much larger than their axial counterparts. It is of interest to note that unlike what was observed in the axisymmetric uniaxial tension test, the difference between the curves for volume fractions of 0.05 and 0.10 is much larger than the difference between the curves for volume fractions of 0.005 and 0.01. It will be seen that this observation can be attributed to the manner in which the effective transverse resistivity increases with increasing effective CNT volume fraction for each strain level starting from a large negative slope at zero strain and approaching zero slope as the strain increases.

Figure 15.

The relative change of the effective resistivity components $ρ_{22}^{Eff}$ of the nanocomposites in response to the applied boundary strain $ε_{0}$ of plane strain uniaxial tension test for CNT volume fractions of 0.005, 0.01, 0.05, and 0.10. (a) $(Δ ρ_{22}^{Eff} / ρ_{22}^{Eff} (0)) \times 100 %$ versus $ε_{0}$ for case T8 with $G^{C} = 1 E 14$ . (b) $(Δ ρ_{22}^{Eff} / ρ_{22}^{Eff} (0)) \times 100 %$ versus $ε_{0}$ for case T9 with $G^{C} = 1 E 17$ .

The results of the plane strain biaxial tension test are shown in Figure 16. By comparing with the results of plane strain uniaxial tension test in Figure 13, one can observe that for cases T1 through T10, the changes in effective resistivity component $Δ ρ_{22}^{Eff}$ of the plane strain biaxial tension test are higher than the plane strain uniaxial tension test by at most two orders. This is because in contrast to the plane strain uniaxial tension test, both the $ε_{11}$ and $ε_{22}$ strain components in the CNT are predominantly tensile, resulting in net increases in both components of CNT resistivity, that is, both $〈 Δ ρ_{11}^{C} 〉$ and $〈 Δ ρ_{22}^{C} 〉$ are positive. Again in Figure 16 as was observed in Figure 13, the cases T1–T5 are consistently lower in $Δ ρ_{22}^{Eff}$ than cases T6–T10. However, this is understood in a more straightforward manner for the symmetric biaxial tension case as the overall tensile strain components in both the 1- and 2-directions results in positive contributions to $Δ ρ_{11}^{C}$ and $Δ ρ_{22}^{C}$ from the off-diagonal piezoresistive strain coefficient in cases T6–T10 that are not present in cases T1–T5.

Figure 16.

Nanocomposite effective transverse resistivity component $Δ ρ_{22}^{Eff}$ obtained from the transverse hexagonal microscale RVE in response to the applied boundary strain $ε_{0}$ of the plane strain biaxial tension test, in which the volume fraction of the CNT is 0.005. (a) Change of the effective resistivity component $Δ ρ_{22}^{Eff}$ as a function of the applied boundary strain of the microscale RVE for cases T1 and T6 with $G^{C} = 2900$ . (b) $Δ ρ_{22}^{Eff}$ versus $ε_{0}$ for cases T4 and T9 with $G^{C} = 1 E 17$ . (c) $Δ ρ_{22}^{Eff}$ versus $ε_{0}$ in semi-log plot for the cases T1–T5. (d) $Δ ρ_{22}^{Eff}$ versus $ε_{0}$ in semi-log plot for the cases T6–T10.

It is noted that for the plane strain biaxial tension cases, the strain field in the CNT is nearly uniform, so that for both the T1–T5 and the T6–T10 cases, the resistivity in the CNT remains nearly uniform and isotropic. As such, it is possible to use the volume-averaged resistivity of the CNT at the various strain levels in a composite cylinder model and construct a comparison with the finite element results for the change in resistivity with applied strain as provided in Figure 17 for case T9. From Figure 17(a), it is observed that as the strain level increases, the slopes of the resistivity versus volume fraction curves obtained from the composite cylinder model rapidly increase in a nonlinear fashion such that there is greater divergence between strain levels with increasing volume fraction. Converting these data into the change in resistivity as a function of applied strain and overlaying with the finite element results, a very good correspondence is obtained between the two models indicating that the nonlinearity observed in the finite element results is consistent within the micromechanics framework and therefore applicable to other loading states where tools such as the composite cylinder model cannot be used due to the nonuniformity and material symmetry distribution within the CNT, for example, the plane strain uniaxial tension or in-plane shear loadings.

Figure 17.

Comparison between the results obtained from FEM with results from CCM for case T9 of the plane strain biaxial tension test, where in the CCM model, the CNT transverse resistivity has been modified to the volume-averaged resistivity based on applied strains on the CNT. (a) The effective resistivity component $ρ_{22}^{Eff}$ obtained from the CCM model with change of CNT volume fraction at applied boundary strain levels of 0.0%, 0.2%, 0.4%, 0.6%, 0.8%, and 1.0% respectively. (b) Comparison between $Δ ρ_{22}^{Eff}$ of CCM with $Δ ρ_{22}^{Eff}$ of FEM at CNT volume fractions of 0.005, 0.01, 0.05, and 0.10, respectively.

By applying equations (41) and (42), one can obtain the effective gauge factors of the nanocomposites, $G^{Eff}$ , for the plane strain uniaxial and biaxial tension tests, respectively, in the transverse directions with respect to the CNT alignment direction. Values for the effective nanocomposite gauge factors are provided in Table 7 for a CNT volume fraction of 0.005 for each of the 10 CNT gauge factors considered in cases T1–T10. For the plane strain uniaxial tension test, the effective gauge factors of cases T1–T3 and T6–T8 are observed to have a constant value of 1.43 up to 1% strain indicating (1) that the piezoresistive response is linear and (2) that the inherent piezoresistivity has not had a significant impact on the effective piezoresistive response of the nanocomposite at these strain levels and volume fractions as 1.43 corresponds to the value that would be obtained by purely geometric considerations, that is, when $ρ_{22}^{Eff} (ε_{0}) = ρ_{22}^{Eff} (0)$ in equation (41). The same conclusions can be drawn for cases T1–T3 and T6–T8 of the plane strain biaxial tension test, with an exception being that the effective gauge factors are small enough to be considered 0. This is because, as indicated in equation (42), the effective gauge factors of the plane strain biaxial tension test do not rely on geometric effects but instead depend purely on how the inherent piezoresistivities of CNTs govern the ratio of the effective resistivity to the zero strain effective resistivity. Only when the CNT gauge factor becomes very large as in cases T4–T5 and T9–T10, does the inherent piezoresistivity become a significant influence on the transverse effective nanocomposite gauge factors, doing so in a nonlinear fashion. As case T1 was related to the currently available data on inherent CNT piezoresistivity, these observations imply that the CNTs must have larger gauge factors if inherent CNT piezoresistivity were to be the driving mechanism behind the transverse nanocomposite piezoresistivity or that additional mechanisms such as CNT network deformation and electron hopping play a much larger role than inherent CNT piezoresistivity in driving transverse nanocomposite piezoresistivity. Further, these other mechanisms must have a substantial impact on the change in effective transverse resistivity with applied strain if they are to have a noticeable effect on the transverse effective nanocomposite gauge factor. Thus, in terms of deformation sensing, it is believed that structural health monitoring (SHM) applications should focus on tailoring the gauge factor in the alignment direction of aligned CNT–polymer nanocomposites.

Table 7.

The effective nanocomposite gauge factors $G^{Eff}$ at strain levels of 0.1%, 0.4%, 0.7%, and 1.0% of the plane strain uniaxial and biaxial tension tests for CNT–polymer nanocomposites having CNT volume fractions of 0.005 and CNTs with CNT gauge factors corresponding to cases T1 through T10.

		Plane strain uniaxial tension test				Plane strain biaxial tension test
Case	$ε_{0} =$	0.1%	0.4%	0.7%	1%	0.1%	0.4%	0.7%	1%
T1	$G^{Eff}$	1.43	1.43	1.43	1.43	1E−14	1E−14	1E−14	1E−14
T2	$G^{Eff}$	1.43	1.43	1.43	1.43	1E−9	1E−9	1E−9	1E−9
T3	$G^{Eff}$	1.43	1.43	1.43	1.43	1E−3	1E−3	1E−3	1E−3
T4	$G^{Eff}$	1.43	1.43	1.44	1.44	0.95	0.83	0.74	0.66
T5	$G^{Eff}$	2.08	1.62	1.54	1.51	19.69	5.00	2.87	2.01
T6	$G^{Eff}$	1.43	1.43	1.43	1.43	2E−14	2E−14	2E−14	2E−14
T7	$G^{Eff}$	1.43	1.43	1.43	1.43	2E−9	2E−9	2E−9	2E−9
T8	$G^{Eff}$	1.43	1.43	1.43	1.43	2E−3	2E−3	2E−3	2E−3
T9	$G^{Eff}$	1.70	1.70	1.68	1.68	1.57	1.56	1.56	1.56
T10	$G^{Eff}$	18.10	5.83	3.98	3.23	19.86	5.01	2.87	2.01

Conclusion

Computational micromechanics models are constructed to assess the piezoresistive responses of aligned CNT–polymer nanocomposites in the axial and transverse directions due to the inherent piezoresistivities of CNTs, which are assumed to be well dispersed and perfectly bonded to the surrounding polymer. A parametric study in terms of CNT gauge factor, AR, and volume fraction is undertaken to study the influence of these factors on the effective nanocomposite piezoresistive response and gauge factors under uniaxial loading in the axial and transverse directions and symmetric biaxial tension in the transverse direction. The results of the axisymmetric piezoresistive model show that through a combination of AR and volume fraction, the piezoresistive response in the axial direction under uniaxial tension can be tailored toward a desired overall resistivity and sensitivity level. The obtained effective axial gauge factors suggest that in order for the macroscale axial piezoresistive response of the nanocomposites to be comparable with the experimental results in the literature, very large CNT gauge factors are needed (e.g. 1E11 in case AU3), several orders of magnitude larger than the maximum CNT gauge factors currently reported in the literature (i.e. ∼2900). In the transverse direction, still larger CNT gauge factors are needed for the inherent piezoresistivity mechanism to have a noticeable effect on the nanocomposite gauge factors in the transverse directions as the electrostatic interactions in the transverse direction are strongly governed by the matrix material. Therefore, the analysis implies that either larger CNT gauge factors (in the neighborhood of 1E11 in the axial direction and in the neighborhood of 1E17 in the transverse direction) would be required if inherent CNT piezoresistivity were the principal mechanism driving the observed nanocomposite piezoresistivity or the piezoresistive response of the CNT–polymer nanocomposites is due to some additional mechanisms, for example, electrical tunneling effect or contact resistance between two adjacent CNTs. Furthermore, these other mechanisms must have a substantial impact on the change in effective resistivity with applied strain, particularly if they are to have a noticeable effect on the transverse effective nanocomposite gauge factor. Thus, in terms of deformation sensing, it is believed that SHM applications should focus on tailoring the gauge factor in the alignment direction of aligned CNT–polymer nanocomposites by controlling CNT AR, dispersion, and volume fraction to maximize the influence of inherent CNT piezoresistivity in conjunction with tunneling and contact resistance effects.

Footnotes

Funding

This research was supported by AFOSR through MURI-18 Synthesis Characterization and Modeling of Functionally Graded Hybrid Composites for Extreme Environments (contract/grant no.: FA-9550-09-0686).

Notes

References

Anantram

Han

(1999) Band-gap change of carbon nanotubes: effect of small uniaxial and torsional strain. Physical Review B 60: 874–878.

Arris

(2000) An overview of high reliability transformer encapsulation materials. Technical Report September 1^st, 2000, Sandia National Laboratories, Albuquerque, NM.

Bakhvalov

Panasenko

(1989) Homogenisation, Averaging Processes in Periodic Media: Mathematical Problems in the Mechanics of Composite Materials. Boston, MA and Dordrecht: Kluwer Academic Publishers.

Bautista-Quijano

Aviles

Aguilar

. (2010) Strain sensing capabilities of a piezoresistive MWCNT-polysulfone film. Sensors and Actuators A: Physical 159: 135–140.

Bensoussan

Lions

Papanicolaou

(1978) Asymptotic Analysis for Periodic Structures. Amsterdam: North-Holland.

Benveniste

(1987) A new approach to the application of Mori-Tanaka’s theory in composite materials. Mechanics of Materials 6: 147–157.

Chen

Weng

(2007) Electronic properties of zigzag carbon nanotubes under uniaxial strain. Carbon 45: 1636–1644.

Cullinan

Culpepper

(2010) Carbon nanotubes as piezoresistive microelectromechanical sensors: theory and experiment. Physical Review B 82, (11): 155–428.

Ebbesen

Lezec

Hiura

. (1996) Electrical conductivity of individual carbon nanotubes. Nature 382: 54–56.

10.

Fernberg

Nilsson

Joffe

(2009) Piezoresistive performance of long-fiber composites with carbon nanotube doped matrix. Journal of Intelligent Material Systems and Structures 20: 1017–1023.

11.

Ferreira

Cardoso

Klosterman

. (2012a) Effect of filler dispersion on the electromechanical response of epoxy/vapor-grown carbon nanofiber composites. Smart Materials and Structures 21: 075008.

12.

Ferreira

Rocha

Ansn-Casaos

. (2012b) Electromechanical performance of poly(vinylidene fluoride)/carbon nanotube composites for strain sensor applications. Sensors and Actuators A: Physical 178: 10–16.

13.

Fish

Shek

Pandheeradi

. (1997) Computational plasticity for composite structures based on mathematical homogenization: theory and practice. Computer Methods in Applied Mechanics and Engineering 148: 53–73.

14.

Fish

Shek

(1999) Computational damage mechanics for composite materials based on mathematical homogenization. International Journal for Numerical Methods in Engineering 45: 1657–1679.

15.

Gates

Odegard

Frankland

. (2005) Computational materials: multi-scale modeling and simulation of nanostructured materials. Composites Science and Technology 65: 2416–2434.

16.

Grimaldi

Maeder

Ryser

. (2003) Piezoresistivity and conductance anisotropy of tunneling-percolating systems. Physical Review B 67: 014205.

17.

Grimaldi

Ryser

Strassler

(2001) Gauge factor enhancement driven by heterogeneity in thick-film resistors. Journal of Applied Physics 90(1): 322–327.

18.

Hammerand

Seidel

Lagoudas

(2007) Computational micromechanics of clustering and interphase effects in carbon nanotube composites. Mechanics of Advanced Materials and Structures 14: 277–294.

19.

Hashin

Rosen

(1964) The elastic moduli of fiber-reinforced materials. Journal of Applied Mechanics 31: 223–232.

20.

Hill

(1965) A self-consistent mechanics of composite materials. Journal of the Mechanics and Physics of Solids 13: 213–222.

21.

. (2012) Multi-scale numerical simulations on piezoresistivity of CNT/polymer nanocomposites. Nanoscale Research Letters 7(1): 402.

22.

Kang

Schulz

Kim

. (2006) A carbon nanotube strain sensor for structural health monitoring. Smart Materials and Structures 15: 737–748.

23.

Kang

Park

Scholl

. (2009) Piezoresistive characteristics of single wall carbon nanotube/polyimide nanocomposites. Journal of Polymer Science Part B: Polymer Physics 47: 994–1003.

24.

Chou

(2003) Multiscale modeling of carbon nanotube reinforced polymer composites. Journal of Nanoscience and Nanotechnology 3: 423–430.

25.

Chou

(2008) Modeling of damage sensing in fiber composites using carbon nanotube networks. Composites Science and Technology 68: 3373–3379.

26.

Thostenson

Chou

(2007) Dominant role of tunneling resistance in the electrical conductivity of carbon nanotube-based composites. Applied Physics Letters 91: 223114-1–223114-3.

27.

McCullough

(1990) Micro-models for composite materials-particulate and discontinuous fiber composites. In: Delaware Composites Design Encyclopedia Volume 2: Micromechanical Materials Modeling (ed. Whitney

J. M.

McCullough

R. L.

)., ch. 2.4, pp. 93–142, Technomic Publishing Company, Inc.

28.

Mori

Tanaka

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

29.

Odegard

Gates

Wise

. (2003) Constitutive modeling of nanotube-reinforced polymer composites. Composites Science and Technology 63: 1671–1687.

30.

Oliva-Aviles

Aviles

Sosa

(2011) Electrical and piezoresistive properties of multi-walled carbon nanotube/polymer composite films aligned by an electric field. Carbon 49: 2989–2997.

31.

Peng

Cho

(2000) Chemical control of nanotube electronics. Nanotechnology 11: 57–60.

32.

Ren

Xiang

Seidel

(2011) Analytic and computational multi-scale micromechanics models for mechanical and electrical properties of fuzzy fiber composites. In: Paper for the 52nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference. Denver, Colorado, USA, 4 – 7 April, 2011 (AIAA 2011-1923).

33.

Saito

Dresselhaus

(1998) Physical Properties of Carbon Nanotubes. London: Imperial College Press.

34.

Schadler

Giannaris

Ajayan

(1998) Load transfer in carbon nanotube epoxy composites. Applied Physics Letters 73: 3842–3844.

35.

Seidel

Lagoudas

(2006) Micromechanical analysis of the effective elastic properties of carbon nanotube reinforced composites. Mechanics of Materials 38: 884–907.

36.

Seidel

Lagoudas

(2008) A micromechanics model for the thermal conductivity of nanotube-polymer nanocomposites. Journal of Applied Mechanics 75: 041025-1–041025-9.

37.

Seidel

Lagoudas

(2009) A micromechanics model for the electrical conductivity of nanotube-polymer nanocomposites. Journal of Composite Materials 43: 917–941.

38.

Seidel

Puydupin-Jamin

(2011) Analysis of clustering, interphase region, and orientation effects on the electrical conductivity of carbon nanotube-polymer nanocomposites via computational micromechanics. Mechanics of Materials 43: 755–774.

39.

Smith

(1954) Piezoresistance effect in germanium and silicon. Physical Review 94(1): 42–49.

40.

Spanos

Kontsos

(2008) A multiscale Monte Carlo finite element method for determining mechanical properties of polymer nanocomposites. Probabilistic Engineering Mechanics 23: 456–470.

41.

Stampfer

Jungen

Linderman

. (2006) Nano-electromechanical displacement sensing based on single-walled carbon nanotubes. Nano Letters 6: 1449–1453.

42.

Theodosiou

Saravanos

(2010) Numerical investigation of mechanisms affecting the piezoresistive properties of CNT-doped polymers using multi-scale models. Composites Science and Technology 70: 1312–1320.

43.

Tombler

Zhou

Alexseyev

. (2000) Reversible electromechanical characteristics of carbon nanotubes under local-probe manipulation. Nature 405: 769–772.

44.

Wichmann

Buschhorn

Gehrmann

. (2009) Piezoresistive response of epoxy composites with carbon nanoparticles under tensile load. Physical Review B 80: 245437-1–245437-8.

45.

Fish

(2002) Multiscale asymptotic homogenization for multiphysics problems with multiple spatial and temporal scales: a coupled thermo-viscoelastic example problem. International Journal of Solids and Structures 39: 6429–6452.

46.

Zeng

(2008) Multiscale modeling and simulation of polymer nanocomposites. Progress in Polymer Science 33: 191–269.

47.

Zhang

Suhr

Koratkar

(2006) Carbon nanotube/polycarbonate composites as multifunctional strain sensors. Journal of Nanoscience and Nanotechnology 6: 960–964.

Computational micromechanics modeling of inherent piezoresistivity in carbon nanotube–polymer nanocomposites

Abstract

Keywords

Introduction

Model description

Multiscale model for the CNT–polymer nanocomposites

Mechanical governing differential equations at the macroscale

Electrostatic governing differential equations at the macroscale

Mechanical governing differential equations for the microscale RVE

Electrostatic governing differential equations for the microscale RVEs

Boundary conditions and effective properties

Mechanical boundary conditions

Electrostatic boundary conditions

Piezoresistive algorithm

Results and discussion

Nanocomposite piezoresistive response under uniaxial tension in the CNT alignment direction

Results for the transverse piezoresistive response of the nanocomposites

Conclusion

Footnotes

Funding

Notes

References