Linear and Nonlinear Viscoelastic Modulus of Rubber

Empirical Models for the Viscoelastic Complex Modulus with an Application to Rubber Friction

Up-to-date predictive rubber friction models require viscoelastic modulus information; thus, the accurate representation of storage and loss modulus components is fundamental. This study presents two separate empirical formulations for the complex moduli of viscoelastic materials such as rubber. The majority of complex modulus models found in the literature are based on tabulated dynamic testing data. A wide range of experimentally obtained rubber moduli are used in this study, such as SBR (styrene-butadiene rubber), reinforced SBR with filler particles and typical passenger car tyre rubber. The proposed formulations offer significantly faster computation times compared to tabulated/interpolated data and an accurate reconstruction of the viscoelastic frequency response. They also link the model coefficients with critical sections of the data, such as the gradient of the slope in the storage modulus, or the peak values in loss tangent and loss modulus. One of the models is based on piecewise polynomial fitting and offers versatility by increasing the number of polynomial functions used to achieve better fitting, but with additional pre-processing time. The other model uses a pair of logistic-bell functions and provides a robust fitting capability and the fastest identification, as it requires a reduced number of parameters. Both models offer good correlations with measured data, and their computational efficiency was demonstrated via implementation in Persson’s friction model.

Physics of suction cups

We have developed a theory of air leakage at interfaces between two elastic solids with application to suction cups in contact with randomly rough surfaces. We present an equation for the airflow in narrow constrictions which interpolates between the diffusive and ballistic (Knudsen) air-flow limits. To test the theory, we performed experiments using two different suction cups, made from soft polyvinylchloride (PVC), in contact with sandblasted polymethylmethacrylate (PMMA) plates. We found that the measured time to detach (lifetime) of the suction cups was in good agreement with theory, except for surfaces with a root-mean-square (rms) roughness below ≈1 μm, where diffusion of plasticizer from the PVC to the PMMA surface caused blockage of critical constrictions. The suction cup volume, stiffness, and elastic modulus have a huge influence on the air leakage and hence the failure time of the cups. Based on our research we propose an improved biomimetic design of suction cups that could show improved failure times with varying degrees of roughness under dry and wet environments.

Rolling friction of elastomers: role of strain softening

We study the temperature and velocity dependency of rolling friction. Steel and PMMA cylinders are rolled on sheets of nitrile butadiene rubber (NBR), with and without filler, and fluoroelastomer (FKM) with filler. Measurements of the rolling friction are performed for temperatures between -40 °C and 20 °C, and for velocities between 5 μm s^-1 and 0.5 cm s^-1. For the unfilled NBR, a smooth rolling friction master curve is obtained using the bulk viscoelastic frequency-temperature shift factor a_T. For the filled rubber compounds, a small deviation from the bulk viscoelastic shift factor is observed at low temperatures. The experimental data are analyzed using an analytical theory of rolling friction. For the filled compounds, good agreement with theory is obtained when strain softening is included, which increases the rolling friction by a factor ∼2 for the filled FKM and ∼3 for the filled NBR compounds. For the unfilled NBR, the maximum of the rolling friction occurs at higher sliding speeds than predicted by the theory. We discuss the role of the adhesive (crack-opening) contribution to the rolling friction, and the role of frozen-in elastic deformations as the rubber is cooled down below the rubber glass transition temperature.

A Discussion on Present Theories of Rubber Friction, with Particular Reference to Different Possible Choices of Arbitrary Roughness Cutoff Parameters

Since the early study by Grosch in 1963 it has been known that rubber friction shows generally two maxima with respect to speed—the first one attributed to adhesion, and another at higher velocities attributed to viscoelastic losses. The theory of Klüppel and Heinrich and that of Persson suggests that viscoelastic losses crucially depend on the “multiscale” aspect of roughness and in particular on truncation at fine scales. In this study, we comment a little on both theories, giving some examples using Persson’s theory on the uncertainties involved in the truncation of the roughness spectrum. It is shown how different choices of Persson’s model parameters, for example the high-frequency cutoff, equally fit experimental data on viscoelastic friction, hence it is unclear how to rigorously separate the adhesive and the viscoelastic contributions from experiments.