  This work develops, discretizes, and validates a continuum model of a
molybdenum disulfide (MoS$_2$) monolayer interacting with a periodic holey
silicon nitride substrate via van der Waals (vdW) forces. The MoS$_2$ layer is
modeled as a geometrically nonlinear Kirchhoff-Love shell, and vdW forces are
modeled by a Lennard-Jones potential, simplified using approximations for a
smooth substrate topography. The material parameters of the shell model are
calibrated by comparing small-strain tensile and bending tests with atomistic
simulations. This model is efficiently discretized using isogeometric analysis
(IGA) for the shell structure and a pseudo-time continuation method for energy
minimization. The IGA shell model is validated against fully-atomistic
calculations for several benchmark problems with different substrate
geometries. The continuum simulations reproduce deflections, strains and
curvatures predicted by atomistic simulations, which are known to strongly
affect the electronic properties of MoS$_2$, with deviations well below the
modeling errors suggested by differences between the widely-used reactive
empirical bond order and Stillinger-Weber interatomic potentials. Agreement
with atomistic results depends on geometric nonlinearity in some cases, but a
simple isotropic St. Venant-Kirchhoff model is found to be sufficient to
represent material behavior. We find that the IGA discretization of the
continuum model has a much lower computational cost than atomistic simulations,
and expect that it will enable efficient design space exploration in strain
engineering applications. This is demonstrated by studying the dependence of
strain and curvature in MoS$_2$ over a holey substrate as a function of the
hole spacing on scales inaccessible to atomistic calculations. The results show
an unexpected qualitative change in the deformation pattern below a critical
hole separation.
