Measurement of the Elastic Modulus of Paperboard from the Low-Frequency Vibration Modes of Rectangular Plates

Abstract

Static methods (e.g. tensile testing) and ultrasonic methods have been most commonly used in the past to measure the elastic modulus of paper and paperboard. The static method, however, does not provide a dynamic value for modulus, and the accurate measurement of travel-time needed for the ultrasonic method is rather delicate and prone to error. A further problem is that the timescale (or frequency) involved in these methods is often far removed from the actual conditions experienced in the paper industry and in applications of these materials. A method using low frequency vibrations to excite out-of-plane bending vibrations, which are visualised as Chladni figures, might appear to be simple and unsophisticated, but can readily be applied to paper and paperboards in the frequency range from 10 to 1,000 Hz. Characteristic patterns are generated on rectangular or square samples at each resonant frequency. From the assumption that machine-made paper is generally orthotropic and that the boundaries are free, values of dynamic Young’s modulus and dynamic Poisson’s ratio can be calculated by identifying only a few low-frequency modes.
Values of dynamic Young’s modulus derived by this method for various core-board samples are lower than those obtained from ultrasonic measurements, which is consistent with a model for viscoelastic behaviour in which elastic modulus depends on frequency. Dynamic Poisson’s ratios in the machine direction and cross direction are also derived with high reproducibility. The vibration frequencies required to excite the modes depend on the sample sizes, but no significant differences in elastic moduli were seen for samples with sizes ranging from 75 × 75 to 200 × 200 mm ; the main restriction of the method is that the sample should be reasonably flat.
The sensitivity of the vibration method is demonstrated by the reduction in stiffness of paper which can be detected when its surface is scratched or damaged ; detection of such damage by ultrasonic measurements is much more difficult. It is suggested that the vibration method using Chladni patterns is well suited to the accurate and reliable measurement of the dynamic elastic constants of the paper⁄paper board, at frequencies appropriate to the processing and use of these materials. The elastic constants obtained by this method may be useful within the paper industry for product and process design.