With the rapid advancement of soft materials, significant interest has been driven towards the development of finite viscoelasticity theory. Structure-preserving integrators, known for their superior stability and accuracy in nonlinear problems, have gained prominence. Given these advantages, developing structure-preserving integrators for finite viscoelasticity models is crucial. In this presentation, we aim to propose structure-preserving integrators for a recently developed finite viscoelasticity modeling framework based on the Green-Naghdi assumption and generalized strains [2]. This model eliminates the need for an imaginary intermediate configuration and generalizes several existing finite linear viscoelastic models. First, we discuss the directionality condition of the algorithmic stress, a critical step in designing structure-preserving schemes. We separately consider the directionality condition for the equilibrium and non-equilibrium parts of the viscoelastic model. Different from the equilibrium part, the non-equilibrium part’s directionality condition can be designed flexibly. We propose three feasible directionality conditions for the non-equilibrium stress. Notably, several second-order structure-preserving integrators from the literature can be derived from these conditions. Second, to address the incompressible constraints of soft materials, we employ a velocity/pressure mixed finite element formulation with grad-div stabilization. This formulation also facilitates the design of a structure-preserving integrator for incompressible viscoelastodynamics. Additionally, we introduce the arc-length method to enhance our solver’s capability in dealing with highly nonlinear problems, where the traditional Newton-Raphson iterative method fails to capture the critical physical phenomena. Finally, we provide several numerical examples to investigate the viscous effect in nonlinear structural dynamic behaviors. We compare the algorithmic stability and accuracy of our proposed structure-preserving scheme with the existing methods in long-time simulations of nonlinear dynamic behaviors, demonstrating that our method outperforms others in simulating highly nonlinear responses of soft materials.