Entropy-Audited Thermoviscoelastic Modelling of Polymer Relaxation Under Non-Isothermal Loading: A Reproducible Numerical Study

Polymers, the long-chain molecules that make up everything from plastic water bottles to the rubber in car tires, behave in a way that often confuses engineers. Unlike a simple metal spring that snaps back instantly when you let go, or a thick fluid that flows slowly and steadily, these materials sit somewhere in between. They are viscoelastic, meaning they possess both the memory of a solid and the flow of a liquid. When you stretch a piece of polymer and hold it there, the force it exerts does not stay constant; it slowly fades away over time as the tangled molecular chains rearrange themselves. This process, known as relaxation, is not just a matter of time. It is deeply sensitive to heat. Warm a polymer, and its molecules wiggle faster, rearranging themselves in a fraction of the time they would at room temperature. Cool it down, and they slow to a crawl. This dependence on temperature is a fundamental rule of the material, yet it creates a tricky problem for the people who design with it.

When engineers simulate how a polymer part will behave inside a machine or a building, they often rely on mathematical models to predict stress and strain. A common and dangerous shortcut in these simulations is to assume that the material's internal clock ticks at the same speed regardless of the temperature, even while the material itself expands or contracts due to heat. It is a bit like trying to predict how long a runner will take to finish a race while ignoring the fact that they are running on a treadmill that is speeding up and slowing down beneath them. If the simulation ignores this acceleration, the prediction of how much force the part will exert can be wildly wrong, potentially leading to designs that fail unexpectedly or are built far heavier than necessary. The question is not just whether the material gets softer when hot, but exactly how the timing of its internal changes shifts as the temperature fluctuates, and whether the computer models used to track these changes are actually obeying the fundamental laws of physics.

A new study by Connor Nitchals tackles this problem by building a rigorous, transparent computer simulation designed to test exactly how temperature changes the timing of polymer relaxation. The research does not focus on a specific brand of plastic or a real-world product; instead, it constructs a perfect, idealized version of a polymer to serve as a benchmark. The goal was to see what happens when a material is stretched, held at a fixed length, heated up, held at that high temperature, and then cooled back down. The study compares two different ways of running the simulation: one that correctly accounts for the fact that heat speeds up the molecular rearrangement, and another that keeps the molecular clock frozen at its original speed, a common simplification used in many engineering calculations. By running these scenarios side-by-side, the study reveals just how much error creeps in when the temperature dependence of time is ignored.

The simulation followed a specific, controlled history. A piece of the virtual material was stretched by one percent and then held at that exact length for 800 seconds. During this time, the temperature started at a comfortable room temperature, rose steadily by 40 degrees, stayed hot for a while, and then cooled back down. The researchers watched how the internal stress—the force the material pushed back with—changed over time. When the model correctly accounted for the fact that heat accelerates the relaxation process, the stress dropped significantly faster during the hot phase. The molecules rearranged themselves quickly to match the new, slightly shorter mechanical length caused by thermal expansion. However, when the researchers ran the same scenario using the simplified model that ignored this acceleration, the material appeared to hold onto its stress much longer than it should have.

The difference between the two models was not a small, subtle error. At the end of the period where the material was held at the high temperature, the simplified model predicted a stress that was 68.0 percent higher than the more accurate, fully coupled model. This means the simplified version thought the material was holding on to nearly twice as much force as it actually was. Even more telling, this error did not disappear when the material cooled back down. After the temperature returned to its starting point and the material was allowed to rest for another 150 seconds, the simplified model still predicted a stress level that was 28.8 percent too high. The material had been "tricked" by the heat; it had relaxed too much during the hot phase, and the simplified model, which failed to see this happening, could not correct its prediction later. The history of the heat had permanently altered the internal state of the material, and a model that ignored the speed of that change could not recover.

Beyond just measuring the stress, the study introduced a new layer of verification called an entropy audit. In physics, the second law of thermodynamics dictates that energy in a system must always be dissipated or lost as heat during irreversible processes like relaxation; it can never be created out of nothing. The researchers built their simulation so that it could calculate exactly how much energy was being dissipated at every single moment. They found that the accurate model consistently showed a positive amount of energy being lost, which is physically correct. The simplified model, while it might look okay on the surface, failed to capture the true energy flow. By tracking this energy, the researchers could prove that their complex model was not just guessing, but was mathematically consistent with the laws of physics. Over the entire 800-second cycle, the accurate simulation showed that the material accumulated 57.01 kilojoules of dissipated energy per cubic meter, a precise number that serves as a fingerprint of the material's behavior.

The study also tested how well the computer code handled the math, specifically looking at whether the results changed if the simulation took smaller or larger steps in time. They found that their method was incredibly stable. Even when the simulation took time steps that were larger than the fastest moment of relaxation occurring in the hot material, the results remained accurate. This is because the code used a special mathematical trick to solve the equations exactly for each step, rather than approximating them in a way that could lead to errors or instability. This finding is crucial for engineers, as it means they can run these complex simulations efficiently without needing to break the time down into impossibly tiny fractions, provided they use the right mathematical approach.

The implications of these findings are clear for anyone designing with polymers. The study demonstrates that you cannot simply take the properties of a plastic measured at room temperature and use them to predict how it will behave in a hot environment without adjusting the timing of its internal changes. If you ignore the fact that heat speeds up the molecular clock, you will overestimate how much force the material will hold, potentially leading to over-engineered, heavy, and expensive designs. Conversely, if you are trying to predict when a seal might fail or a damper might lose its effectiveness, ignoring this acceleration could lead to a catastrophic underestimation of the risk. The error is not just a temporary glitch during the heating phase; it leaves a permanent mark on the material's memory, affecting how it behaves long after the temperature has returned to normal.

This research serves as a blueprint for better modeling. It shows that the best way to verify a computer model of a polymer is not just to see if it matches a single curve of data, but to check if it obeys the laws of energy and if it correctly handles the interaction between heat and time. By providing a clear, reproducible set of numbers and code, the study offers a standard against which future models can be tested. It proves that when dealing with materials that remember their past and react to their environment, the only way to get the answer right is to respect the complex dance of time, temperature, and energy that defines them. The study does not claim to have solved every problem in polymer mechanics, but it has firmly established that ignoring the temperature dependence of time is a mistake that cannot be ignored, and that checking the energy balance is a powerful tool to ensure the math is telling the truth.