A systematic computational calculation based on the state-of-the-art quantum mechanics mothed was carried out to study the response of mechanical properties to various strains exerted on graphene, SiC sheet, and recently predicted two-dimensional (2D) sandwiched GaP and InP binary compounds. It was found that these 2D materials undergo an elastic expansion, a structural deformation, and then a structural broken process as the strain increases. Such process strongly depends on the direction of the strain exerted on 2D materials. In particular, a phase transition occurs in 2D sandwiched GaP and InP binary compounds when the strain exerts in zigzag direction. Calculated mechanical properties show that graphene has large linear and nonlinear elastic moduli, followed by 2D SiC monolayer. While the sandwiched GaP and InP structures possess significant anisotropic and nonlinear mechanical properties. Especially, those constants in the zigzag direction are about three to nine times greater than that in the armchair direction. Compared to graphene, they are softer, even along the zigzag direction. Such results provide fundamental information at atomic level for synthesizing, designing, and fabricating nanoelectromechanical and nanoelectronic devices. Copyright © VBRI Press.