In this work,the B2 partial disordered structure of the austenitic parent phase,martensitic transformation,elastic and magnetic properties of the Ni8Mn4+xTi4-x (x =0,1 and 2) Heusler alloys have been systematically investigated by the first-principles calculations.The preferential atomic occupation of B2 structure is one Ti atom exchange with the nearest neighboring Mn atom from the view of lowest energy principle.This disordered exchange sites (Mn-Ti) and the excess Mn atoms occupying the Ti sites (MnTi) could reduce the nearest Mn-Mn distance,resulting in the antiferromagnetic state in the austenitic and martensitic phases of the alloys.The total magnetic moment of the alloy decreases with the increasing Mn content;it is ascribed to the antiferromagnetic magnetic moments of the excess Mn atoms.When x =0,the alloy does not undergo martensitic transformation since the austenite has absolute phase stability.The martensitic transformation will occur during cooling process for x =1 or 2,owing to the energy difference between the austenite and the martensite could provide the driving force for the phase transformation.The elastic properties of the cubic austenitic phase for the Ni2MnTi alloy is calculated,and the results reveal the reason why Ni-Mn-Ti alloy has excellent mechanical properties.The origin of martensitic transformation and magnetic properties was discussed based on the electronic density of states.