Elastic properties of [InAs/AlSb] heterostructures coherently grown on a (001) InAs substrate are investigated by the density functional theory and compared to the prediction of the linear elasticity theory. The stress-strain curves of the four involved binaries (InAs, AlAs, AlSb, and InSb) are first studied: a significant deviation to the linear elasticity theory is observed for strain above 2.5% (in absolute value). Nevertheless, the relationship between the out-of-plane and in-plane strains is in a good agreement with the prediction of the linear elasticity theory. In the heterostructures, highly strained perfect AlAs-like and InSb-like interfaces are examined. The interfacial strains calculated using the density functional theory are in a surprisingly good agreement with the prediction of the linear elasticity theory. The reduction of the layer thickness to the thinnest possible InAs or AlSb layers while keeping perfect interfaces does not change these conclusions.