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The rate-dependent behavior of natural and high damping rubbers is investigated in the compression regime. The experimental results demonstrate the prominence of the rate-dependent high initial stiffness feature in high damping rubber at low stretch level. A modified hyperelastic model is proposed to represent the rate-independent elastic responses including the high initial stiffness feature. A comparative evaluation is carried out to display the better performance of the proposed hyperelastic model than conventional ones over the strain range in representing the equilibrium and the instantaneous responses. The hyperelastic model is incorporated in a finite deformation rate-dependent model structure. A parameter identification scheme is proposed to identify the parameters for the equilibrium and instantaneous responses from the experimental data. The difficulties of direct application of infinitely fast or slow …