Why Does Rubber Have Odd Temperature Dependence?

Example 1: Natural Rubber

Here is exemplar experimental data for a natural rubber [Ref]. The figure shows monotonic stress-strain curves at different temperatures. The material is stiffest at the lowest temperature, but in this case the stiffness is not the lowest at the highest temperature.  The lowest stiffness occurs at an  intermediate temperature. When seeing experimental data like this, many people will think that it is due to experimental errors, but that is not necessarily the case! The temperature-dependence of rubbers can be “odd”. Later in this article I will explain what is causing this behavior.

The same data is here plotted as stress as a function of temperature at different strain values. The figure shows that the stress increases with strain, and in most temperature regions, decreases with increasing temperature.

Example 2: Unfilled Silicone Rubber

This example is for an unfilled silicone rubber. The experimental behavior of the rubber was determined using a DMA machine, and the results are shown below [Ref]. At really low temperatures the storage modulus is very high (>300 MPa) corresponding to the glassy response of the material. As the temperature increases the stiffness drops a lot. Just like one would expect when going from a glassy to a rubbery state! The figure to the right shows the stiffness for temperatures above 0°C. In this range the stiffness increases with temperature.

Cyclic stress-strain curves at different temperatures for the same silicone rubber are shown in the figure to the right. The results are super interesting. The two main observations are: (1) the stiffness increases with temperature; (2) The amount of hysteresis decreases with temperature.

Statistical Mechanics Theory

The theory of rubber elasticity was developed a long time ago, and if you are not familiar with the theory, you may be surprised to learn that it is based on statistical mechanics and simple molecular chain models. The image below is from my book, and discusses the Freely Jointed Chain (FJC) model. If you have a freely jointed macromolecular chain molecule then the entropy of the molecule is given by the statistical mechanics equation (5.65) in the book. The probability distribution can be determined by Flory’s equation (5.66).

The Helmholtz free energy per unit reference volume, in general, can be written: $$\Psi = e_0 – \theta_0 \eta_0$$, where $$e_0$$ is the internal energy, $$\theta_0$$ is the absolute temperature, and $$\eta_0$$ the entropy. Also recall that the Cauchy stress can be directly obtained from the Helmholtz free energy. For example, if there is no dependence on the second invariant $$I_2^*$$, then the Cauchy stress is given by:

$$\boldsymbol{\sigma} = \displaystyle\frac{2}{J} \frac{\partial \Psi}{\partial I_1^*} \text{dev}[\mathbf{b}^*] + \frac{\partial\Psi}{\partial J} \mathbf{I}$$.

In other words, the Cauchy stress should be proportional to the absolute temperature for those rubber materials where the Helmholtz free energy is dominated by the entropy contribution. This mainly occurs for rubbers that are almost unfilled. Adding filler particles adds an internal energy contribution to the Helmholtz free energy. In this case the Cauchy stress will have a weaker temperature dependence, and the stress may even  decrease with increasing temperature.

Summary

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