Temperature Dependent Mechanical and Structural Properties of Uniaxially Strained Planar Graphene

TL;DR

This study investigates how temperature affects the mechanical response of planar graphene under uniaxial tension in the armchair and zigzag directions using a Hamiltonian-based MD framework with a Morse bond potential and angle bending. Stress-controlled simulations, temperature calibration via added energy density, and long-time, symplectic integration yield stress–strain curves, revealing that the 2D Young modulus $E_{2D}$ decreases almost linearly with temperature while the third-order modulus $D_{2D}$ shows direction-dependent trends. Fracture strength $\sigma_f$ and failure strain $\epsilon_f$ also decrease nearly linearly with temperature, with quantified slopes that depend on loading direction; at 300 K intrinsic strengths and fracture strains align with previous MD and experimental results. The study additionally analyzes how bond lengths and angles distribute under combined temperature and stress, showing that distributions split into two Gaussian peaks whose variances grow linearly with $T$, and provides analytical expressions that jointly describe the bond-length and bond-angle distributions. Overall, the work advances understanding of thermal effects on graphene’s mechanical properties and offers a practical Gaussian-based framework to model structural fluctuations under stress.

Abstract

Using molecular dynamics simulations in a planar graphene sheet, we investigate the temperature dependence of its mechanical behavior under uniaxial tensile stress applied either along the armchair or the zigzag direction. Stress-strain curves are calculated for different temperatures and the corresponding dependence of various elastic parameters, is discussed. Fracture stress and strain, as well as the Young modulus, decrease almost linearly with temperature, in accordance with previous investigations. An almost linear variation of the third-order elastic modulus with temperature is demonstrated, revealing opposite trends for uniaxial loadings along the armchair or the zigzag direction. The detailed dependence of the distributions of bond lengths and bond angles both on strain and temperature is presented for the first time, along with approximate analytical expressions. The latter describe accurately the numerically obtained distributions.

Figures (11)

• Figure 1: A schematic of the hexagonal graphene lattice depicting $N=42$ atoms, arranged in $N_I=6$ columns and $N_J=7$ rows. Atoms in column $i$ and row $j$ are indicated in blue and orange, respectively, and the $(i,j)$th atom is colored in gray. The $A$ and $Z$ type bonds, and similarly the $\alpha$ and $\zeta$ type angles, are respectively indicated in red and green (see Section \ref{['section-distributions']} for more details on these distinctions).
• Figure 2: Symbols represent the relation between the additional energy density $e_N$ above the relaxed $T=0$ K graphene structure subjected to uniaxial tensile stress $\sigma=2.16$ eV/Å$^2$ along the zigzag direction, and the average temperature $T_{ave}$ at thermal equilibrium, evaluated through the MD simulations by averaging over both time and the different realizations. One standard deviation of the $T_{ave}$ measurements is indicated by blue horizontal error-bars. The linear fitting of the presented data points is shown by gray solid line, providing a slope equal to $1.74\times10^{-4}$ eV/K.
• Figure 3: Time evolution of the average (over individual realizations) strain $\left\langle \epsilon_T(t)\right\rangle$, Equation \ref{['eqn-strain']}, when a stress (a)$\sigma=0.188$ eV/Å$^2$, (b)$\sigma=1.03$ eV/Å$^2$, and (c)$\sigma=1.97$ eV/Å$^2$, along the zigzag direction is applied, for different temperatures: $T=100$ K (blue curves), $T=400$ K (green curves), and $T=700$ K (red curves). The average (over realizations and time) strains $\overline{\left\langle \epsilon_T\right\rangle}$, for each temperature, are indicated by the horizontal dashed lines of the same color in each panel.
• Figure 4: Stress-strain response of planar graphene for uniaxial loads along the (a) armchair (b) zigzag direction, for different temperatures as indicated in the legend. Filled circles indicate the obtained average strain for each given stress. Solid curves represent fittings of these data with Equation \ref{['eqn-stress-strain']}, see text. For $T \ne 0$ K the strain is given as the average over time and realizations, $\overline{\left\langle\epsilon_T\right\rangle}$, and the error-bars correspond to one standard deviation. The insets in each panel depict a close-up view of the region indicated by the grey rectangle in each panel.
• Figure 5: Temperature dependence of (a) the Young modulus $E_{2D}$ and (b) the third-order elastic modulus $D_{2D}$, for applied stress along the armchair (red points) or the zigzag (blue points) direction, evaluated through fittings of the data of Figure \ref{['fig-stress-strain']} with Equation \ref{['eqn-stress-strain']} (see text).