Abstract: The nonlinear boundary conditions on a bonding interface between two solids of a longitudinal or shear horizontal wave under normal incidence are explored.By applying the second-order perturbation method,the nonlinear spring models are rigorously developed in the limit of small thickness to wavelength ratio.Numerical results agree well with the exact solutions obtained by continuous boundary conditions.The nonlinear spring model for longitudinal wave is verified by measuring the nonlinearity of a wedge-shaped bonding structure of steel or aluminum.Experimental results reveal that the above model is rather accurate and as the impedance ratio of adherend to adherend increases,the model becomes more accurate.