Determination of the dynamic elastic properties of paper and paperboard from the low-frequency vibration modes of rectangular plates

The dynamic elastic constants of paper/paperboards are determined by using the vibrational resonance modes. The vibration modes can be visualised by Chladni figures and correspond to the eigen-frequencies of the plates. The values of Young's modulus derived by this method are lower than those obtained from ultrasonic measurements at higher frequencies, which is consistent with a viscoelastic behaviour in which elastic modulus depends on frequency. The in-plane shear modulus and dynamic Poisson's ratios in the MD and CD can also be derived with high reproducibility. In order to determine these elastic constants it is usually necessary to observe five low-frequency vibration modes, but it has been shown that only three modes are needed if the shapes of the nodal lines in the Chladni patterns are also taken into consideration. A simpler method is also suggested as a practical approach which also gives consistent results with other methods.