In this paper, a fully variational constitutive model of soft biological tissues is formulated in the finite strain regime. The model includes Ogden-type hyperelasticity, finite viscosity, deviatoric and volumetric plasticity, rate and microinertia effects. Variational updates are obtained via time discretization and pre-minimization of a suitable objective function with respect to internal variables. Genetic algorithms are used for model parameter identification due to their suitability for non-convex, high dimensional optimization problems. The material behavior predicted by the model is compared to available tests on swine and human brain tissue. The ability of the model to predict a wide range of experimentally observed behavior, including hysteresis, cyclic softening, rate effects, and plastic deformation is demonstrated.