Background: It is very important to precisely comprehend nanosheet’s mechanical properties
for their future application, and the continuum-based methods play a vital role in this research
domain. But, most of continuum models doesn’t provide a systematical theory, and just display certain
property of nanostructures. The Cauchy-Born rule provides an alternative multiscale method, the
resulted model is not only less accurate, and but also doesn’t describe the bending effect.
Methods: A nanosheet is viewed as a higher-order gradient continuum planar sheet, and the strain
energy density is thus a function of both the first- and second-order deformation gradient. The higher-
order Cauchy-Born rule is used to approximate the bond vectors in the representative cell, the
multiscale model is established by minimizing the cell energy, and the structural and mechanical
properties are thus obtained.
Results: The obtained bond lengths are respectively 0.14507 nm, 0.14489 nm, 0.1816 nm for the
graphene, boron nitride and silicon carbide hexagonal nanosheets. The elastic constants, including
Young’s modulus, shear modulus, Poisson’s ratio and bending rigidity, are calculated by analyzing
the physical meaning of the first- and second-order strain gradients. The developed model can also be
used to study the nonlinear behavior of nanosheets under some simple loading situations, such as the
uniform tension, torsion and bending. The stress-strain relationship of nanosheets is presented for the
uniform tension/compression, and the three types of nannosheets exhibit better compressive resistance
far greater than tensile resistance.
Conclusion: A reasonable multiscale model is established for the nanosheets by using the higherorder
Cauchy-Born rule that provides a good interlinking between the microscale and continuum descriptions.
It is proved that all three types of nannosheets shows the isotropic mechanical property.
The current model can be used to establish a global nonlinear numerical modeling method in which
the bending rigidity is the basic elastic constants same as the elastic modulus and Poisson’s ratio.