Nonlocality and Rotational Effects on Microconcentrated Diffusive Thermoelastic Solids within the Three-Phase-Lag Heat Conduction Theory

Imagine a solid object, like a block of metal or a piece of ceramic, that is not just sitting still but is spinning rapidly while also being heated and having different chemicals moving through it. In the world of physics, these three things—heat, mechanical movement, and the flow of matter—are usually treated as separate stories. However, in reality, they are deeply intertwined. When a material heats up, it expands; when it expands, it creates stress; and when chemicals move through it, they can change how it reacts to heat and force. For decades, scientists have used classical rules to predict how these materials behave, but those rules assume that every point in the material only feels the influence of its immediate neighbors. This works well for large objects, like bridges or engines, but it breaks down when we look at materials at the microscopic or nanoscopic level, where atoms can "feel" the influence of other atoms far away. Furthermore, when these materials are part of a spinning system, like a turbine or a satellite component, the rotation adds invisible forces that twist and pull the material in ways standard models often miss. Understanding how heat, stress, and chemical flow interact in these tiny, spinning, and complex materials is crucial for designing the next generation of advanced engineering components, from micro-sensors to high-speed machinery.

A team of researchers from Zagazig University in Egypt has taken a closer look at this complex interaction. They focused on a specific type of material called a "diffusive thermoelastic solid with microconcentration." In plain terms, this is a solid material where heat travels, mechanical waves move, and tiny particles diffuse through the structure, all while the concentration of these microscopic particles varies from place to place. To make their study realistic, they added two critical factors that are often ignored in simpler models: the "nonlocal" effect, which accounts for those long-range atomic influences, and the effects of rotation. They also used a sophisticated theory of heat conduction known as the "three-phase-lag" model. Unlike older theories that assume heat moves instantly or with a single delay, this model recognizes that heat flow has a more complex timing, involving delays in the temperature gradient, the heat flux, and the thermal displacement. By combining these advanced concepts, the researchers built a mathematical framework to simulate how such a material would behave under specific conditions.

The researchers did not perform physical experiments with spinning blocks of metal in a lab. Instead, they developed a detailed set of equations to describe the physics of the situation and then solved them using a powerful mathematical technique called the normal mode method. They simulated a half-space, which is essentially a very large block of material that extends infinitely in one direction but has a flat surface on the other. They applied specific conditions to this surface, such as a sudden thermal load, and then watched how the disturbance traveled through the material. They ran their simulations with different settings to see how changing the "nonlocal" parameter (which controls the strength of the long-range atomic interactions) and the speed of rotation would alter the results. They tracked five key physical quantities: how much the material moved (displacement), how hot it got (temperature), how the chemical concentration changed, the internal stress building up inside the material, and the chemical potential driving the diffusion.

The results of these simulations revealed that both nonlocality and rotation have a profound impact on the material's behavior. When the researchers turned on the nonlocal effect, they found that the material responded more strongly to the applied loads. The displacement of the material, the temperature rise, and the concentration of diffusing particles were all higher in the nonlocal model compared to a traditional local model. This suggests that at small scales, the material is more sensitive to external forces because its internal structure allows influences to travel further than previously thought. The rotation also played a significant role. When the material was set to spin, the mechanical and thermal responses changed noticeably. The spinning motion amplified the displacement and the temperature changes, particularly near the surface where the load was applied. The rotation essentially added a layer of complexity that made the waves of heat and stress travel differently, altering how quickly they faded out as they moved deeper into the material.

One of the most interesting findings was how the "three-phase-lag" theory compared to older models. The researchers found that this more advanced theory generally predicted a stronger and more pronounced response than the older Green-Naghdi Type II theory. This indicates that the delays in heat transfer, which the three-phase-lag model accounts for, are not just minor details but are essential for accurately describing how these materials behave, especially when combined with nonlocal effects and rotation. The study showed that the three-phase-lag model captures the nuances of the interaction between the thermal, mechanical, and diffusion fields better than its predecessors. For instance, the stress within the material, which is the internal force resisting deformation, was calculated to be higher in the three-phase-lag simulations, suggesting that engineers using older models might underestimate the stress in high-speed, micro-scale applications.

The researchers also observed that the various physical fields—temperature, stress, concentration, and displacement—did not just jump around randomly. Instead, they followed smooth, predictable patterns. Near the surface where the load was applied, the effects were strongest, with high temperatures and significant movement. As the researchers looked deeper into the material, these effects gradually faded away, eventually returning to zero. This smooth decay confirmed that their mathematical model was stable and physically realistic. The simulations showed that the nonlocal parameter and the speed of rotation acted as tuning knobs. By adjusting them, the researchers could see how the material's response changed. For example, increasing the nonlocal parameter made the material's reaction more intense, while increasing the rotation speed also boosted the magnitude of the physical fields. The study concluded that ignoring either of these factors would lead to an incomplete and potentially inaccurate picture of how these materials behave.

Ultimately, this work provides a deeper theoretical understanding of how microstructured materials behave when subjected to the combined challenges of heat, chemical diffusion, and rapid rotation. The findings suggest that for engineers designing components that operate at the micro or nano scale, or in high-speed rotating environments, it is no longer safe to rely on classical models that ignore long-range atomic interactions or the complex timing of heat flow. The study offers a more robust framework for predicting how these materials will react, which could help in the design of more reliable and efficient engineering systems. By showing that nonlocality and rotation significantly alter the distribution of temperature, stress, and concentration, the research highlights the need for these advanced models in the future of materials science. The work stands as a simulation-based investigation that clarifies the complex interplay of forces in diffusive thermoelastic solids, offering a clearer path forward for analyzing and designing advanced materials that operate under extreme and coupled conditions.