In this paper proposed is a general framework of accounting for the rotary effect on the constitutive equation of a rotational deformable object in a general deformation setting which may contain large rotation and/or spinning. Utilizing the framework we deal with the equations of the rate‐type, the total‐type, and the incremental‐type. The total‐type equation is found valid for any kinematic environment as manifested in many applications; for example, it is generally taken for granted that the constitutive law of hyperelasticity can be used directly in the problem of large deformation/rotation of an elastic material. In dealing with the rate‐type constitutive law, the proposed framework renders among others a natural definition of the Jaumann stress rate, which in the past looked rather unusual and the intrinsic meaning of which appeared not easy to apprehend. The power of the proposed framework is highlighted in our developing the incremental‐type constitutive equation in such a way as to satisfy the principle of objectivity exactly during the whole process of loading, this being quite important both from theoretical and computational points of view. The traditional, time‐consuming steps of the time integration of the rate‐type constitutive equation is thus bypassed.