Over the past few decades, the measurement and analysis of thermal undulations has provided a route to estimate the mechanical properties of membranes. Theoretically, fluctuating elastic membranes have been studied mostly by Fourier analysis coupled with perturbation theory (to capture anharmonic effects), or by computer simulations of triangulated surfaces. These techniques as well as molecular dynamic simulations have also been used to study the thermal fluctuations of graphene. Here, we present a semi-analytic approach in which we view graphene as a triangulated membrane, but compute the statistical mechanical quantities using Gaussian integrals. The nonlinear coupling of in-plane strains with out-of-plane deflections is captured using a penalty energy. We recover well-known results for the scaling of the fluctuations with membrane size, but we show that the fluctuation profile strongly depends on boundary conditions and type of loading applied on the membrane. Our method quantitatively predicts the dependence of the thermal expansion coefficient of graphene on temperature and shows that it agrees with several experiments. We also make falsifiable predictions for the dependence of thermal expansion coefficient and the heat capacity of graphene on applied loads and temperature.

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