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Large Time Approximation for Shearing Motions

Small- and large-amplitude oscillatory shear tests are widely used by experimentalists to measure, respectively, linear and nonlinear properties of viscoelastic materials. These tests are based on the quasi-static approximation according to which the strain varies sinusoidally with time after a number of loading cycles. Despite the extensive use of the quasi-static approximation in solid mechanics, few attempts have been made to justify rigorously such an approximation. The validity of the quasi-static approximation is studied here in the framework of the Mooney--Rivlin Kelvin--Voigt viscoelastic model by solving the equations of motion analytically. For a general nonlinear model, the quasi-static approximation is instead derived by means of a perturbation analysis.