A new efficient hyperelastic finite element model for graphene and its application to carbon nanotubes and nanocones

Abstract

A new hyperelastic material model is proposed for graphene-based structures, such as graphene, carbon nanotubes (CNTs) and carbon nanocones (CNC). The proposed model is based on a set of invariants obtained from the right surface Cauchy-Green strain tensor and a structural tensor. The model is fully nonlinear and can simulate buckling and postbuckling behavior. It is calibrated from existing quantum data. It is implemented within a rotation-free isogeometric shell formulation. The speedup of the model is 1.5 relative to the finite element model of Ghaffari et al. [1], which is based on the logarithmic strain formulation of Kumar and Parks [2]. The material behavior is verified by testing uniaxial tension and pure shear. The performance of the material model is illustrated by several numerical examples. The examples include bending, twisting, and wall contact of CNTs and CNCs. The wall contact is modeled with a coarse grained contact model based on the Lennard-Jones potential. The buckling and post-buckling behavior is captured in the examples. The results are compared with reference results from the literature and there is good agreement.properties of single-walled CNCs. A second approach is based on the quasi-continuum method [35]. Yan et al. [36] apply the quasi-continuum to simulate buckling and post-buckling of CNCs. A temperature-related quasi-continuum model is proposed by Wang et al.[37] to model the behavior of CNCs under axial compression. A third approach is based on classical continuum material models. Those are popular for graphene in the context of isotropic linear material models. Firouz-Abadi et al. [38] obtain the natural frequencies of nanocones by using a nonlocal continuum theory and linear elasticity assumptions. Their work is extended to the stability analysis under external pressure and axial loads by Firouz-Abadi et al. [39] and the stability analysis of CNCs conveying fluid by Gandomani et al. [40]. Lee and Lee [29] use the finite element (FE) method to obtain the natural frequencies of CNTs and CNCs. The interaction of carbon atoms is modeled as continuum frame elements. Graphene has an anisotropic behavior under large strains. There are several continuum material models for anisotropic behavior of graphene. Sfyris et al. [41] and Sfyris et al. [42] use Taylor expansion and a set of invariants to propose strain energy functionals for graphene based on its lattice structure. Delfani et al. [43] and Delfani and Shodja [44, 45] use a similar Taylor expansion for the strain energy and apply symmetry operators to the elasticity tensors in order to reduce the number of independent variables. Nonlinear membrane material models are proposed by Xu et al. [46] and Kumar and Parks [2]. They use ab-initio results to calibrate their models. The model of Kumar and Parks [2] is based on the logarithmic strain and the symmetry group of the graphene lattice [47, 48, 49]. This symmetry group reduces the number of parameters in the model of Xu et al. [46] by a half. The membrane model of Kumar and Parks [2...

The multiplicative deformation split for shells with application to growth, chemical swelling, thermoelasticity, viscoelasticity and elastoplasticity

“…So far, it seems that only elasto-plasticity and isotropic thermoelasticity have been analyzed with multiplicatively split isogeometric shell FE (Ambati et al, 2018;Vu-Bac et al, 2019). But the authors are currently applying the present theory to extend the hyperelastic graphene FE model of Ghaffari and Sauer (2018) to anisotropic thermoelasticity (Ghaffari and Sauer, 2019), and to study the growth of fluid films using the FE model of Sauer (2014) and Roohbakhshan and Sauer (2019).…”

Comparing quantum, molecular and continuum models for graphene at large deformations

“…Ghaffari and Sauer [1] formulated the strain energy density, per unit area of the initial configuration, based on a set of invariants \scrJ i , i.e. [1,61] W (\scrJ 1 , \scrJ 2 , \scrJ 3 ) = W \mathrm{ \mathrm{ \mathrm{ \mathrm{ (\scrJ 1 ) + W \mathrm{ \mathrm{ \mathrm{ \mathrm{ (\scrJ 2 , \scrJ 3 ; \scrJ 1 ) + W \mathrm{ (\kappa 1 , \kappa 2 ; \scrJ 1 ) ,…”

A nonlinear thermomechanical formulation for anisotropic volume and surface continua

“…The Green–Lagrange strain can be decomposed additively if F 1 ≈ 1 . The logarithmic strain facilitates the development of material models, but the numerical implementation is complicated [91]. A similar discussion can be provided for the decomposition of surface strain measures.…”

“…This connection can be used to extract membrane and shell material models from their three-dimensional counterparts. In addition, it can be used to extend available isothermal membrane and shell material models [90, 91] to nonisothermal constitutive laws.…”