© 2023 by Exorus Koh

Biaxial Material Characterization Jig

Material characterization refers to the quantification of the properties of a material through systematic testing. It was the need to characterize an elastic material—rubber—that motivated the construction of this mechanical jig. We will explain the origin of this need later in this article.

 

The stress-strain characteristics of a sheet of elastic material, disregarding shear and torsional aspects, can be determined through uniaxial and biaxial characterization. Uniaxial stress-strain characterization is more straightforward, since it involves stretching of the material in only a single direction. Biaxial characterization, on the other hand, requires simultaneous stretching in two directions, and hence necessitates a more involved setup. This mechanical jig was an improvisation to realize biaxial characterization without professional equipment.

Each of the four arms of the jig mounts a grooved pulley, around which a string is supported. The hanging ends of the strings are attached to weights; their upper ends connect to a square of the material to be tested. By varying the weights, or alternatively motor or linear actuator power, the material can be stretched to varying degrees in two directions simultaneously. By setting the tension to be identical, equibiaxial stress-strain characterization can conveniently performed. Changing of the material sample involves unhooking the strings and replacing the entire square assembly with another.

The square of material is adhered to rigid edge supports, to which the strings are attached. This is to provide as uniform an edge load as possible. In practice we use a camera system to obtain top-view imagery of the jig, so that the experimenter can quickly iterate through the load range without stopping to manually measure the material strains. The strains are then obtained later by analyzing the captured images, a process that can be suitably automated.

It may be of interest to some readers on why exactly was there a need to characterize rubber sheets. In late 2014 I was working to qualify for a place in a research-based competitive physics avenue known as the International Young Physicists' Tournament (IYPT). To be successful, I had to perform well at the national equivalent, the Singapore Young Physicists' Tournament (SYPT). One of the problems I was in charge of was titled "Two Balloons".

Physics of Rubber Balloons

Problem Five: Two Balloons

Two rubber balloons are partially inflated with air and connected together by a hose with a valve. It is found that depending on initial balloon volumes, the air can flow in different directions. Investigate this phenomenon. 

The phenomenon brought to attention is interesting—in particular, it is possible for a smaller balloon to inflate a larger one in certain cases, and in others, a larger balloon may inflate the smaller one. It is easy to realize this phenomenon in real life.

The explanation for this lies in the inherent properties of rubber. Rubber is not a Hookean materialits elastic force is not linearly proportional to the amount of stretching applied. In other words, it does not obey Hooke's law, at least not at the strains typical in balloons. Rubber falls into a class of materials called hyperelastic. 

The reason for this non-linearity has to do with the origins of the elasticity of rubber. Rubber can be simplistically visualized as being comprised of a large number of long molecular strands. In its unstretched state, these molecular strands are tangled and messily arranged; whereas in its stretched state, these strands are forced to disentangle and straighten, in order to elongate. This disentanglement means that thermodynamic fluctuations—say, local fluctuations in molecular kinetic energy—tend to knock these strands back to an entangled state. In other words, the thermodynamic tendency to approach a higher-entropy state forces the rubber to attempt to revert to its unstretched state. This is the fundamental origin of the elastic force of rubber, which is also appropriately known as an entropic force.

This non-linearity in the elastic force of rubber also means that the pressure of a balloon is non-monotonous as it is inflated, as illustrated in the graph above. We use λ to represent the material strain. It is now clear why a smaller balloon could inflate a larger one—because at certain strains, a smaller balloon can have a pressure greater than a larger one.

We can also proceed to construct quantitative models of this phenomenon. There are three commonly-used material models for rubber—the Neo-Hookean, Mooney-Rivlin, and Ogden models. In my opinion, the first is too simplistic to completely characterize rubber elasticity, and the third contains too many free parameters, the physical significance of which may not always be clear. We hence go with the Mooney-Rivlin model, whose basis equation is presented below.      represents the strain energy density function,      and      are the invariants of the left Cauchy-Green deformation tensor, and       and       are material constants, to be characterized. 

Rivlin and Saunders managed to verify in 1951 that        is in reality a combination of two underlying material constants. This became known as a modified Mooney-Rivlin model. With this, it is possible to derive the stress-strain and pressure-strain relations.

A simultaneous fit was carried out on sets of uniaxial and equibiaxial experiment data to determine the material constants. This is where the biaxial characterization jig comes into play—to facilitate the collection of equibiaxial stress-strain data. We show an example of such datasets below. Experiment data are presented as dots and the best-fit characteristics are presented as solid lines.

The mean best-fit material constants are used to evaluate the pressure-strain characteristic of the balloon, presented as a solid line on the graph below. Lower bounds and upper bounds are presented as dotted lines. We see that all six sets of our pressure-strain experiment data, presented as dots, fall between the predicted lower and upper bounds, indicating that our characterization using the modified Mooney-Rivlin model was indeed suitable.

Now we can quantitatively answer the two balloons problem. As intuitively understood, air will flow from one balloon to another if the pressure in one is greater than the pressure in the other; since we have the pressure-strain characteristic of the balloons, we are able to predict this airflow. This is best presented using a phase diagram. 

But we can do more. We may also be interested in the dynamical behaviour of the balloon—given an initial setup of two connected balloons, how will the balloons evolve in size? It is straightforward to compute the expected evolution of the balloons, now that we have the pressure-strain characteristic of the balloons; the transitions from initial states to final states are represented using arrows on the phase diagram below.

We notice that the two-balloon setup tends to move away from certain equilibria, namely, the central branch, and the large-strain left and right branches. This indicates that these branches are unstable. The branches that the setup approaches, in converse, are stable equilibria. There is a quick way to evaluate whether an equilibrium point is stable. Stability requires an automatic self-correction in the event of small perturbations, in other words:

Indeed, we see that the experiment data, presented as dots and arrows, largely matches our predictions on airflow and equilibria stability. The theoretical model hence seems to describe the two balloons phenomenon very well. Yet, a sharp eye would notice that there seem to be discrepancies in the large-strain regime, close to the bifurcation point. These can be attributed to material hysteresis—the elasticity of rubber in reality differs during loading and unloading processes. To account for such effects, a viscoelastic material model has to be used, with strain rate-dependent stress characteristics. This would complicate the theory significantly.

Lastly, we show a picture of the two-balloon setup in the lab: