It is well known that the constitutive relation plays a key role in the nonlinear finite element analysis, and a time integration is particularly required to solve the problem whose stress response is deduced by its strain increment. The hypoplastic rate constitutive model is represented by Jaumann stress rate and deformation rate, and on the basis of this model, we develop adaptive implicit and explicit integration methods with local error control respectively. Meanwhile, the consistent tangent modulus is given for the implicit integration. Based on two different element tests, results of two types of numerical simulations of integration algorithm are compared by using ABAQUS. Moreover, a UMAT-VUMAT interface is developed to implement the transition from ABAQUS/ Standard to ABAQUS/Explicit and then the flow chart of this development is presented. Therefore, the existing UMAT subroutine can be used for large deformation or high-degree dynamic problem analysis. Finally, the accuracy of numerical results is verified by a case study.