Behavior of the mechanical system composed of highly deformable structural elements

Carbon nanotubes (CNTs) are attractive for many applications because they possess a unique combination of mechanical and physical properties. Horizontally aligned CNT bundles under lateral compression behave as an elastic body with highly deformable elements, since their cross sections can collapse. Mechanical properties of such systems is poorly investigated in spite of the fact that they are promising for vibration and shock protection. Here we use a chain model with a reduced number of degrees of freedom in order to study the behaviour of a CNT bundle under uniaxial and biaxial lateral compression. Stress-strain curves are obtained and evolution of the CNT bundle structure is analyzed.

Physical and mechanical properties of CNT bundles have been extensively studied with the use of computer simulation methods. Vertically aligned CNTs can be transformed into horizontally aligned forest by pressing [16,17]. The thin shell theory has been used to describe CNT ensembles of different structure [18]. In the works [19,20], the applicability of the beam, plate and shell theories to the analysis of the mechanical properties of nanomaterials has been analyzed. Importantly for the present study, CNTs having diameter above a threshold value, have two equilibrium configurations, circular and collapsed [21][22][23]. Mechanical properties of CNT bundles can be efficiently studied using the nonlinear coarse-grained potential [24]. In order to considerably reduce the number of degrees of freedom, the chain model has been developed for simulation of some sp 2 -carbon nanostructures [25]. In particular, this model was successfully used to study structure and properties of various conformations of carbon nanoribbons [25][26][27][28][29] and surface ripplocations [30]. The chain model was extended to the study of CNTs under lateral plane strain compression in [31,32]. CNT bundle as an elastic damper was analyzed in [33]. Laterally compressed CNT bundle undergoes phase transitions [34,35]. Buckling critical load for axially compressed graphene nanoribbons can be increased by  [36,37]. Propagation of solitary waves and shock waves in carbon and other nanomaterials have been studied in [30,38,39].
In this work, we demonstrate the application of the chain model to the analysis of the structure evolution of CNT bundle under uniaxial and biaxial compression. In Sec. 2 we describe the model and in Sec. 3 present simulation results. Sec. 4 concludes this work.   , y) plane. Each atom stands for a row of atoms parallel to z-axis, which moves as a rigid body. CNT diameter is D, distance between walls of neighbouring CNTs is d, and the distance between atoms in the CNT wall is a.

Simulation method
The equilibrium interatomic distance in graphene is ρ = 1.418 Å. In the zigzag CNT, the distance between neighbouring atomic rows parallel to the z-axis is a = ρ(3 1/2 )/2 =1.228 Å. Diameter of CNT is D = a/ sin(π/N), where N is the number of atoms in the cross section of CNT. If d is the distance between neighbouring CNT walls, then the distance between centres of neighbouring CNTs is A=D+d. In our simulations we take N = 30, then D = 11.75 Å and d = 3.088 Å.
Carbon nanotube bundle in the frames of the chain model can be described by the Hamiltonian H=K+UB+UA+UVdW, (1) where the first term in kinetic energy with ε = 0.00166 eV and σ = 3.61 Å. Detailed information on the chain model is given in [32,33].

Simulation results
In figure 2  As it can be seen from figure 2, the stress-strain curves can be divided into three regions. For | |<0.07, both xx and yy rapidly increase with strain, and this is the first region. Within the second region, stress components show a very slow increase with strain and even a slow decrease of yy can be seen in (b). Note that in (a) yy>xx, but in (b) and (c) xx>yy within the second range. The second range is terminated by a sharp drop of stress components. In the third range, the stress components continue to slowly increase. In figure 3, structure evolution of the CNT bundle is presented. The top, middle and bottom rows show the results for the compression along x, along y, and biaxial compression, respectively. The absolute values of volumetric strain are given for each panel. The values of strain are shown by dots on the stress-strain curves presented in figure 2.
For all modes of loading the same scenario of structure transformation is observed, reflecting the three regions observed on the stress-strain curves (see figure 2). During the first stage, crystal structure remains unchanged with slightly deformed CNT cross section, see figure 3(a), (e) and (i). In the second stage translational symmetry is still preserved but the primitive translational cells increase in size. In (b) and (c) they contain four CNTs, while in (f) and (g), as well as in (j) and (k), two CNTs. The third stage is characterised by the appearance of collapsed CNTs and the loss of crystal order. Compressive deformation during the third stage results in the increase of the fraction of collapsed CNTs.

Conclusions
Molecular dynamics modelling of the lateral compression of CNT bundle was performed with the use of the chain model with greatly reduced number of degrees of freedom under the assumption that the plane strain conditions are satisfied. The efficiency of the used model was demonstrated. Peculiarities of the stress-strain curves were described and linked to the structure transformation of the CNT bundle. Overall, a CNT bundle under lateral compression is an elastic material with unusual properties due to the high deformability of structural units. In future works, the chain model can be extended for 2D materials other than graphene [40][41][42][43].