Understanding the mechanical behaviors and properties of brain tissue are crucial to study the mechanisms of traumatic brain injury (TBI). Such injury may be associated with high rate loading conditions and the large deformation of brain tissue. Thus, constitutive models that consider the rate dependent large deformation of brain tissue and its possible damage initiation and evolution may help uncover the related mechanisms of TBI. Motivated from this, in this paper we present a fully three-dimensional large strain viscohyperelastic-damage model with the purpose of reproducing the experimentally observed rate sensitive elastic and damage-induced stress softening behaviors of brain tissue. The parameters of the proposed model can be identified using the experimental data from simple monotonic tests such as uniaxial tension, compression and simple shear. The proposed model is validated by comparing its prediction with experimental data. Good agreement between predictive results and experimental data is achieved indicating the potential of the proposed model in characterizing the mechanical behaviors of brain tissue considering rate dependence and damage effect.