It has been revealed that both elastic and plastic components of the granular soils behavior are affected by the stress induced anisotropy as a result of the history of previous shear loadings. While the influence of fabric anisotropy on the plastic elements of the elasto-plastic constitutive models has been extensively studied in the literature, the anisotropic elastic response is usually neglected mainly because of avoiding complication. Herein, a simple anisotropic elasticity theory is proposed. To this aim, the fourth order elasticity tensor is related to a second order fabric-dilatancy tensor describing magnitude and direction of induced anisotropy. Proper constitutive equations for calibration of the proposed elasticity theory using data of triaxial and simple shear tests are presented. Then, the introduced elasticity theory is implemented within an advanced sand constitutive model. The model predictions are compared with the experimental data of independent research teams. It is shown that the modification of the basic platform by the proposed anisotropic elasticity theory leads to improvement of liquefaction predictions.