The effect of negative Poisson’s ratio on the low-velocity impact response of an auxetic nanocomposite laminate beam

Abstract

In this paper, an investigation on the low-velocity impact (LVI) response of a shear deformable beam laminated by carbon nanotube reinforced composite (CNTRC) layers is performed. The composite beam is “auxetic” due to the negative out-of-plane Poisson’s ratio (NPR) through special symmetric stacking sequences of layers that are designed based on the Classical Laminate Theory. To study the effect of the out-of-plane NPR on the LVI response of the composite beam, a newly defined Hertz model is developed. The motion equations of Kármán type for the CNTRC laminate beam are derived in the framework of the Reddy beam theory and solved by means of a two-step perturbation approach while the dynamic equation of the impactor is built on Newton’s Law. Since temperature-dependent material properties of both carbon nanotube (CNT) and matrix are employed, the thermal influence on the LVI behavior is also investigated. Moreover, a piece-wise method is employed herein to investigate the effect of functionally graded (FG) patterns of the CNT reinforcements on the impact response. Numerical results elucidating the effects of temperature, FG distribution, and CNT volume fraction on the out-of-plane Poisson’s ratio and impact response of the beam are obtained by using a Range–Kutta method and discussed in details.

Appendices

Appendix A

For the laminate with arbitrary layup, the relation between load and deformation is written as

$$ \left\{ {\begin{array}{*{20}c} {\mathbf{N}} \\ {\mathbf{M}} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {\mathbf{A}} & {\mathbf{B}} \\ {\mathbf{B}} & {\mathbf{D}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varvec{\upvarepsilon}} \\ {\varvec{\upkappa}} \\ \end{array} } \right\} $$

(A.1)

From Eq. (A.1), we have

$$ {\varvec{\upvarepsilon}} = {\mathbf{A}}^{ - 1} {\mathbf{N}} - {\mathbf{A}}^{ - 1} {\mathbf{B\varvec{\upkappa} }} $$

(A.2)

$$ {\mathbf{M}} = {\mathbf{B\varvec{\upvarepsilon} }} + {\mathbf{D\varvec{\upkappa} }} = {\mathbf{BA}}^{ - 1} {\mathbf{N}} + ({\mathbf{D}} - {\mathbf{BA}}^{ - 1} {\mathbf{B}}){\varvec{\upkappa}} $$

(A.3)

Equations (A.2) and (A.3) can be rewritten in the form of matrix and vectors

$$ \left\{ {\begin{array}{*{20}c} {\varvec{\upvarepsilon}} \\ {\mathbf{M}} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}^{ - 1} } & { - {\mathbf{A}}^{ - 1} {\mathbf{B}}} \\ {{\mathbf{BA}}^{ - 1} } & {{\mathbf{D}} - {\mathbf{BA}}^{ - 1} {\mathbf{B}}} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\mathbf{N}} \\ {\varvec{\upkappa}} \\ \end{array} } \right\} $$

(A.4)

Let ${\mathbf{A}}^{ * } = {\mathbf{A}}^{ - 1}$, ${\mathbf{B}}^{ * } = - {\mathbf{A}}^{ - 1} {\mathbf{B}}$, ${\mathbf{H}}^{ * } = {\mathbf{BA}}^{ - 1}$, ${\mathbf{D}}^{ * } = {\mathbf{D}} - {\mathbf{BA}}^{ - 1} {\mathbf{B}}$. Equation (4) can be expressed as

$$ \left\{ {\begin{array}{*{20}c} {\varvec{\upvarepsilon}} \\ {\mathbf{M}} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}^{ * } } & {{\mathbf{B}}^{ * } } \\ {{\mathbf{H}}^{ * } } & {{\mathbf{D}}^{ * } } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\mathbf{N}} \\ {\varvec{\upkappa}} \\ \end{array} } \right\} $$

(A.5)

From Eq. (A.5), we have

$$ {\varvec{\upkappa}} = - {\mathbf{D}}^{ * - 1} {\mathbf{H}}^{ * } {\mathbf{N}} + {\mathbf{D}}^{ * - 1} {\mathbf{M}} $$

(A.6)

$$ {\varvec{\upvarepsilon}} = ({\mathbf{A}}^{ * } - {\mathbf{B}}^{ * } {\mathbf{D}}^{ * - 1} {\mathbf{H}}^{ * } ){\mathbf{N}} + {\mathbf{B}}^{ * } {\mathbf{D}}^{ * - 1} {\mathbf{M}} $$

(A.7)

According to Eqs. (4), (A.6) and (A.7) can be rewritten as

$$ \left\{ {\begin{array}{*{20}c} {\varvec{\upvarepsilon}} \\ {\varvec{\upkappa}} \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}^{ * } - {\mathbf{B}}^{ * } {\mathbf{D}}^{ * - 1} {\mathbf{H}}^{ * } } & {{\mathbf{B}}^{ * } {\mathbf{D}}^{ * - 1} } \\ { - {\mathbf{D}}^{ * - 1} {\mathbf{H}}^{ * } } & {{\mathbf{D}}^{ * - 1} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\mathbf{N}} \\ {\mathbf{M}} \\ \end{array} } \right\} $$

(A.8)

If there is no bending moment applied on the laminate, then we can obtain the expression of strain vector

$$ {\varvec{\upvarepsilon}} = \left( {{\mathbf{A}}^{ * } - {\mathbf{B}}^{ * } {\mathbf{D}}^{ * - 1} {\mathbf{H}}^{ * } } \right){\mathbf{N}} = \left( {{\mathbf{A}}^{ - 1} + {\mathbf{A}}^{ - 1} {\mathbf{B}}\left( {{\mathbf{D}} - {\mathbf{BA}}^{ - 1} {\mathbf{B}}} \right)^{ - 1} {\mathbf{BA}}^{ - 1} } \right){\mathbf{N}} = {\mathbf{JN}} $$

(A.9)

where

$$ {\varvec{\upvarepsilon}} = \left\{ {\begin{array}{*{20}c} {\varepsilon_{11} } & {\varepsilon_{22} } & {\varepsilon_{33} } & {\varepsilon_{12} } \\ \end{array} } \right\}^{T} ,\quad {\mathbf{N}} = \left\{ {\begin{array}{*{20}c} {N_{1} } & {N_{2} } & {N_{3} } & {N_{12} } \\ \end{array} } \right\}^{T} , $$

(A.10)

$$ \begin{aligned} {\mathbf{A}} & = \left[ {\begin{array}{*{20}c} {A_{11} } & {A_{12} } & {A_{13} } & {A_{16} } \\ {A_{12} } & {A_{22} } & {A_{23} } & {A_{26} } \\ {A_{13} } & {A_{23} } & {A_{33} } & {A_{36} } \\ {A_{16} } & {A_{26} } & {A_{36} } & {A_{66} } \\ \end{array} } \right],\quad {\mathbf{B}} = \left[ {\begin{array}{*{20}c} {B_{11} } & {B_{12} } & {B_{13} } & {B_{16} } \\ {B_{12} } & {B_{22} } & {B_{23} } & {B_{26} } \\ {B_{13} } & {B_{23} } & {B_{33} } & {B_{36} } \\ {B_{16} } & {B_{26} } & {B_{36} } & {B_{66} } \\ \end{array} } \right] \\ & \quad {\mathbf{D}} = \left[ {\begin{array}{*{20}c} {D_{11} } & {D_{12} } & {D_{13} } & {D_{16} } \\ {D_{12} } & {D_{22} } & {D_{23} } & {D_{26} } \\ {D_{13} } & {D_{23} } & {D_{33} } & {D_{36} } \\ {D_{16} } & {D_{26} } & {D_{36} } & {D_{66} } \\ \end{array} } \right], \\ \end{aligned} $$

(A.11)

Appendix B

In Eqs. (42) and (43)

$$ g_{30} = - \gamma_{17} + m^{2} \left( {\gamma_{18} + \gamma_{19} } \right)\frac{{\gamma_{21} m^{2} - \gamma_{23} }}{{\gamma_{22} m^{2} + \gamma_{23} }} - \left( {\gamma_{29} + \gamma_{28} \frac{{\gamma_{21} m^{2} - \gamma_{23} }}{{\gamma_{22} m^{2} + \gamma_{23} }}} \right)\frac{{\gamma_{12} m^{4} }}{{\gamma_{22} m^{2} + \gamma_{23} }} $$

(B.1)

$$ \begin{aligned} g_{31} & = m^{4} \left[ {\gamma_{11} - \gamma_{12} \frac{{\gamma_{21} m^{2} - \gamma_{23} }}{{\gamma_{22} m^{2} + \gamma_{23} }}} \right] + 2m^{2} \pi \left[ {\gamma_{15} - \gamma_{14} \frac{{\gamma_{21} - \gamma_{23} }}{{\gamma_{22} + \gamma_{23} }}} \right]\varPhi \\ & \quad + 2\pi C_{1} \left[ {\gamma_{15} - \gamma_{14} \frac{{\gamma_{21} m^{2} - \gamma_{23} }}{{\gamma_{22} m^{2} + \gamma_{23} }}} \right]\varPhi \\ \end{aligned} $$

(B.2)

$$ \begin{aligned} g_{32} & = 2m^{3} \pi \left[ {\gamma_{15} - \gamma_{14} \frac{{\gamma_{21} m^{2} - \gamma_{23} }}{{\gamma_{22} m^{2} + \gamma_{23} }}} \right] \\ & \quad + \frac{3}{4}\pi^{2} C_{2} \gamma_{13} \varPhi \\ \end{aligned} $$

(B.3)

$$ g_{33} = \frac{{m^{4} \pi^{2} }}{4}\gamma_{13} $$

(B.4)

$$ g_{q} = \frac{{ 2k_{c} bL^{{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0pt} 2}}} }}{{\pi^{3} \bar{D}_{11} }}\sin \frac{m}{2}\pi $$

(B.5)

$$ g_{i} = - \frac{{k_{c} \rho_{0} L^{{{5 \mathord{\left/ {\vphantom {5 2}} \right. \kern-0pt} 2}}} }}{{\pi^{2} ME_{0} }} $$

(B.6)

In Eqs. (B.2) and (B.3), C₁ and C₂ are dependent on the value of m. When m = 1, C₁ and C₂ are both equaled to 1. In other case, C₁ = C₂ = 0.