Non-local strain gradient isogeometric analysis of thermally loaded FG-TPMS sandwich nanoplates on elastic foundations

At the very smallest scales, where materials are measured in billionths of a meter, the rules of how things bend and vibrate change. In our everyday world, a steel beam vibrates in a predictable way based on its weight and stiffness. But shrink that beam down to the size of a virus, and the atoms themselves begin to whisper to one another across distances, softening the material's response. At the same time, the very act of bending creates internal gradients that make the material harder to deform. These competing forces, known as non-local effects and strain gradients, mean that the tiny structures of the future cannot be designed using the same old blueprints. Add to this the fact that these structures often operate in hot environments where heat changes their properties, and the challenge of predicting how they will behave becomes a complex puzzle of physics and geometry.

Researchers at Van Lang University and Hanoi University of Science and Technology have tackled this puzzle by creating a new way to simulate the vibrations of a specific type of advanced sandwich panel. These panels are not made of simple layers of metal and plastic. Instead, they are constructed with a core that looks like a complex, repeating sponge made of a triply periodic minimal surface, a mathematical shape that fills space with the least amount of surface area possible. This core is sandwiched between two outer sheets that are functionally graded, meaning their composition changes gradually from one material to another, like a gradient from ceramic to metal. The team used a sophisticated computer framework called isogeometric analysis, which allows them to model these intricate shapes with extreme precision, to see how these nanoplates would vibrate when heated and supported by an elastic foundation.

The study focused on a specific design where the core is made of titanium alloy and the outer sheets blend titanium alloy with zirconia ceramic. The researchers simulated the plate under a range of conditions, testing how different shapes of the internal sponge, different ways of distributing the material density inside that sponge, and different temperatures would affect the plate's fundamental vibration frequency. They found that the two competing size effects act like a tug-of-war. The non-local interactions, which represent the long-range atomic whispers, consistently lower the vibration frequency, making the structure effectively softer. In contrast, the strain-gradient effects, which account for the internal resistance to bending, raise the frequency, making the structure harder. The final behavior of the plate depends on which of these two forces wins the battle.

One of the most surprising discoveries was that simply adding more material to the core does not always make the plate vibrate better. The researchers tested various average densities for the sponge-like core and found that increasing the density often lowered the vibration frequency. This happens because adding more material increases the weight, or inertia, of the plate faster than it increases the stiffness. In this specific configuration, the extra weight slows the vibration down more than the extra stiffness speeds it up. This suggests that for these tiny structures, a lighter core might actually perform better than a heavier one, a counterintuitive result that challenges the usual engineering instinct to add mass for strength.

The shape of the internal sponge also mattered, though the differences were subtle. The team tested three distinct geometric patterns for the core: a primitive cubic shape, a gyroid shape that twists like a spiral, and an IWP shape that looks like a wrapped package. The primitive cubic pattern consistently produced the highest vibration frequencies, making it the most rigid of the three options. However, the researchers noted that the specific arrangement of the material density within the core had a more significant impact than the geometric shape itself. They found that a steady, one-way increase in density from the bottom to the top of the core performed slightly better than a symmetric distribution where the density was highest in the middle.

Heat proved to be a relentless force that lowered the vibration frequency in every scenario. As the temperature rose, the materials softened, and the heat created internal stresses that further reduced the plate's ability to vibrate quickly. The researchers also looked at how the thickness of the core relative to the total plate thickness affected performance. They found that the relationship was not a straight line. A core that was too thin offered little benefit, while a core that was too thick made the plate too heavy and flexible. The sweet spot was an intermediate thickness, where the core was thick enough to separate the stiff outer sheets effectively but not so thick that it dominated the weight of the structure. This optimal ratio remained consistent even as the temperature changed.

Finally, the study examined how the plate would behave if it were resting on a supportive surface, like a mattress that pushes back when pressed. The researchers modeled two types of support: one that simply pushes up vertically and another that also resists the shearing motion between adjacent points. The addition of this supportive foundation significantly increased the vibration frequency, acting as a stabilizer. More importantly, a foundation with shear resistance made the plate much less sensitive to the damaging effects of heat. While the heat still lowered the frequency, the drop was much smaller when the plate was well-supported. This suggests that for these tiny, heat-sensitive structures, the environment in which they are placed is just as critical as the materials they are made from.

The work provides a clear, high-fidelity map for engineers who wish to design these next-generation nanostructures. By combining the competing effects of atomic-scale physics with the macroscopic geometry of the sandwich panel, the researchers have shown that there is no single "best" design. Instead, the optimal performance comes from balancing the weight of the core against its stiffness, choosing the right geometric pattern, and accounting for the thermal environment. The findings confirm that while these materials are incredibly complex, their behavior can be predicted and tuned, opening the door to lighter, more efficient components for future technologies that operate at the very edge of the physical world.