Bending instability of few-layer graphene embedded in strained polymer matrix

Abstract

In this Letter we describe analytically and simulate numerically the softening of flexural surface acoustic waves, localized in the plane of few-layer graphene embedded in soft matrix of low-density polyethylene. The softening of surface acoustic wave is triggered by the compressive strain in the matrix, which results in compressive surface stress in the few-layer graphene. Softening of the flexural surface acoustic wave leads to spatially periodic static bending deformation (modulation) of the embedded nanolayer with the definite wave number. Few-layer graphene with different numbers of graphene monolayers is considered. We describe the different models of interlayer bonding of graphene monolayers in a few-layer graphene, which correspond to the weak and strong interlayer bonding. The considered models give substantially different scaling of the wave number of periodic bending deformation and of the threshold compressive strain in the matrix as functions of the number of graphene monolayers in the few-layer graphene. Both the wave number of periodic bending deformation and the values of the threshold compressive surface stress in the few-layer graphene and of the compressive strain in the matrix are very well confirmed by the numerical simulations. Bending instability of few-layer graphene can be used for the study of bending stiffness and two-dimensional Young modulus of the graphene nanolayers, embedded in a soft matrix.

Strongly localized wrinkling modes of single- and few-layer graphene sheets in or on a compliant matrix under compression

“…xx | [8,14], that demonstrates the non-Eulerian nature of the bending instability of the 2D elastic layer placed on or embedded in a compliant matrix [8]. In the case of weak interlayer coupling in the few-layer graphene, when D s (n) = nD s (1) and E s (n) = nE s (1) [37], Eqs.…”

“…The origin of the soliton-like envelope and localization of the sinusoidal wrinkling mode we relate with the effective attraction of the wrinkling modes in the nonlinear system. The parameters of the extended sinusoidal wrinkling mode are found from the conditions of anomalous softening of flexural surface acoustic wave (SAW) propagating along the two-dimensional elastic sheet placed in or on the soft matrix [11][12][13][14], which provide an analytical approach to get the necessary parameters of the model.…”

“…( 20)-( 23) below. Essentially all the introduced 2D moduli of elasticity λ s , µ s , D 11 and D 66 are determined by the in-plane (valence-bonds and valence-angles) interatomic interactions in the sheet and are finite even in the sheet with a monolayer thickness, see, e.g., [13][14][15]. Equation (2) implies that nonlinear elasticity of the graphene sheet embedded in a compliant matrix is determined by the 2D Lamé coefficients λ s and µ s , which substantially exceed the corresponding coefficients of the matrix (multiplied by interatomic distance).…”

“…In this case, the coefficient K B (k 1 , k 2 ) in (2) takes the form K B = Bk 0 , k 0 > 0 (proposed for the first time in Ref. [4]), and the effective elastic modulus B can be determined from the spectrum of soft flexural SAW in the system [11][12][13][14]. Substituting this ansatz for w in Eq.…”

“…( 2) in the case of anisotropic elastic matrix, we turn to the description of the softening of flexural surface acoustic wave (SAW) in the system, cf. [13,14] for the case of isotropic matrix. The long-wavelength dynamical properties of the graphene sheet can be taken into account with the use of dynamic boundary conditions for the displacements u (1,2) i and surface-projected elastic stresses σ…”