When we talk about mechanical properties, we generally refer to response of the material on application of external load. Apart from external load, responses can be of mixed nature, such as thermo-mechanical response. In the broader sense, mechanical properties have a very wide domain, and subsets of these domains often overlap. For example, the non-linear response of material that may be elastic or plastic in nature.
Even if we focus only on elastic response, we encounter linear and non-linear behavior of material against deformation. It all depends on type of material, external stimuli, and its internal configuration. Talking about elastic properties, many research groups majorly deal with constitutive behaviour of small material domain at atomic scale. These small scale modelling results are used in higher length scale simulations to mimic the material response at the macroscopic level.
Furthermore, depending upon the symmetry of the crystal, the number of independent elastic constants which defines this constitutive behavior also varies. Here in the above paper, the author is considering average elastic behaviour of two-dimensional structure on the application of small strain. The consideration of elastic constants are based on small deformation theory and material is taken as homogeneous isotropic hookean material where Young's modulus was derived from the second derivative of the strain energy. Again other properties such as in-plane stiffness and Poisson's ratio were also calculated using this same above assumption.