We performed ab initio calculations of the elastic constants of five flexible metal-organic frameworks (MOFs): MIL-53(Al), MIL-53(Ga), MIL-47, and the square and lozenge structures of DMOF-1. Tensorial analysis of the elastic constants reveals a highly anisotropic elastic behavior, some deformation directions exhibiting very low Young's modulus and shear modulus. This anisotropy can reach a 400:1 ratio between the most rigid and weakest directions, in stark contrast to the case of nonflexible MOFs such as MOF-5 and ZIF-8. In addition, we show that flexible MOFs can display extremely large negative linear compressibility. These results uncover the microscopic roots of stimuli-induced structural transitions in flexible MOFs, by linking the local elastic behavior of the material and its multistability.

We present here a framework for the analysis of the full tensors of second-order elastic constants of metal-organic frameworks, which can be obtained by ab initio calculations. We describe the various mechanical properties one can derive from such tensors: directional Young's modulus, shear modulus, Poisson ratio, and linear compressibility. We then apply this methodology to four different metal-organic frameworks displaying a wine-rack structure: MIL-53(Al), MIL-47, MIL-122(In), and MIL-140A. From these results, we shed some light into the link between mechanical properties, geometric shape, and compliance of the framework of these porous solids. We conclude by proposing a simple criterion to assess the framework compliance, based on the lowest eigenvalue of its second-order elastic tensor.