Graphene resting on substrate: Closed form solutions for the perfect bonding and the delamination case

Abstract

a b s t r a c tWe study closed form solutions for the perfect bonding and the delamination case for a monolayer graphene sheet resting on an elastic foundation. The theoretical framework we adopt is restricted to the materially and geometrically linear case. Graphene is modeled as a hexagonal 2-lattice, while the substrate is assumed to behave in an isotropic linearly elastic manner. Initially, we ignore out-of-surface motions and study the case of biaxial tension/compression and simple shear. We find the components of the shift vector by solving the equations ruling the shift vector. We then substitute this expression for the shift vector components to the momentum equation. This way we obtain conditions that the field of the internal strains, the strain constants and the material parameters should satisfy in order biaxial tension/compression and simple shear to be solutions for all equilibrium equations. For the particular case of axial strain and for the simple shear case we plot the mean stress components versus strain for three different substrates. Then, we take into account out-of-surface motions. We assume the out-of-surface displacement to be the product of a wave-like function and an unknown function, which we determine under certain conditions. These conditions are constraints that the field of the internal strains, the strain constant and the material characteristics of the substrate and graphene should satisfy in order the equilibrium equations to be satisfied. These cases pertain to the perfect bonding case. Distinguishing film's displacement from the bulk (substrate) displacement we study the case where delamination occur. We again use a semi-inverse method: we assume film's displacement to be the product of a wave-like function with an unknown function. The bulk's displacement is assumed to be different from the one of the film, in areas of delamination. We determine the unknown function present in the displacement of the film, by requiring the momentum equations to be satisfied.

Wrinkling formation in simply-supported graphenes under tension and compression loadings

“…In any case, the present results show that even for simply supported graphenes lateral wrinkling initiates at a far smaller strain than the tensile fracture strain. Such modes of failure have been studied theoretically in recent works 28,29 and as the experiments confirm here are omnipresent 30 when graphenes of micron dimensions are loaded in tension. In fact, this type of failure modes is quite common for supported thin films when the underlying substrate is relative stiff (here the modulus of the substrate is ∼3 GPa).…”

Compression behavior of simply-supported and fully embedded monolayer graphene: Theory and experiment

“…Graphene on a substrate has been modelled analytically as a hexagonal 2-lattice and studied using continuum mechanics 15,16 . Other effects such as the sequential period-doubling bifurcations for graphene bonded to a PDMS substrate were also studied numerically 17 .…”

Influence of partial blistering on the global and the local stress and couple stress field for a monolayer graphene resting on substrate

“…Dependence on the shift vector, p , at the continuum level, results from well established theories of multilattices [21–28]. If we confine ourselves to weak transformation neighbor- hoods [28] and assume validity of the CauchyBorn rule [29] graphene’s elastic energy for the geometrically and materially linear case has the form [4, 6]…”

“…where e ij f = 1 2 ( u i , j f + u j , i f ) is the in-surface strain measure of the film, u f is the displacement of the film, b f is film’s curvature, while p is graphene’s shift vector. Tensors C i , i = 1 ,… , 6 are related with the material parameters of the problem at hand (see the work by Sfyris et al [4, 6]) which are found to have nine material parameters denoted by c i , i = 1 ,… , 9 and will be discussed in Section 5. Quantities related with residual/internal strains are denoted by the superscript “0”.…”

“…The components ( p 1 , p 2 ) of the shift vector p should be evaluated from the equation ruling the shift vector, i.e. ∂ W ∂ p = 0 and we refer to the work by Sfyris et al [4] for the most generic case when internal strains are present.…”