Micro-metal products have recently enjoyed high demand. In addition, metal microforming has drawn increasing attention due to its net-forming capability, batch manufacturing potential, high product quality, and relatively low equipment cost. Micro-hydromechanical deep drawing (MHDD), a typical microforming method, has been developed to take advantage of hydraulic force. With reduced dimensions, the hydraulic pressure development changes; accordingly, the lubrication condition changes from the macroscale to the microscale. A Voronoi-based finite element model is proposed in this paper to consider the change in lubrication in MHDD according to open and closed lubricant pocket theory. Simulation results agree with experimental results concerning drawing force. Changes in friction significantly affect the drawing process and the drawn cups. Moreover, defined wrinkle indexes have been shown to have a complex relationship with hydraulic pressure. High hydraulic pressure can increase the maximum drawing ratio (drawn cup height), whereas the surface finish represented by the wear is not linearly dependent on the hydraulic pressure due to the wrinkles.

The tribological behavior of traditional oil-in-water (O/W) lubricants (1.0 wt.%) and nano-TiO₂ additive lubricants (1.0–9.0 wt.%) during micro rolling of titanium foils were analyzed. In this study, the surface morphologies of titanium foils under various lubrication conditions were assessed, and the corresponding lubrication mechanisms were revealed. The tribological behavior of nano-TiO₂ additive lubricants during micro rolling of titanium foils was also explored through a series of characterization methods. The utilization of nano-TiO₂ additive lubricants in micro rolling reduces the surface roughness of titanium foils. Moreover, it effectively inhibits the generation of indentations and cracks during rolling processes, enhancing the surface quality of rolled specimens. Additionally, owing to the synergism of rolling, tribo-film, mending and polishing effects of the nanoparticles, both the rolling force and surface roughness were minimized by using lubricants containing 3.0 wt.% TiO₂ nanoparticles. Overall, an optimal concentration (3.0 wt.%) of TiO₂ nanoparticles in water-based nanolubricants was obtained with enhanced tribological properties and lubrication performance during micro rolling of titanium foils.

Water-based lubricants with different fractions of TiO₂ nanoparticles ranging from 1.0 to 9.0 wt.% were utilized to study the lubrication mechanisms during micro rolling tests and the tribological behaviour of nanolubricants during the micro rolling of copper foils. The results indicate that the application of TiO₂ nanolubricants remarkably improves the surface quality of rolled copper foils during rolling processes. For lubricants with inadequate TiO₂ nanoparticles, it is found that few TiO₂ nanoparticles enter the contact regions between the rolls and foils, causing insufficient lubrication during rolling processes. Instead, for lubricants with excessive TiO₂ nanoparticles, obvious agglomeration occurs at the contact regions and promotes the generation of voids on the surface of the rolled foils, thereby deteriorating the surface quality of the rolled copper foils. In addition, it is found that the surface quality of rolled foils is improved by utilizing a large reduction ratio. Overall, the fraction of 3.0 wt.% TiO₂ nanolubricants is optimal to improve the lubrication conditions at the contact regions, thereby improving the surface quality of the rolled copper foils.

With the help of Maple, the precise traveling wave solutions of three fractal-order model equations related to water waves, including hyperbolic solutions, trigonometric solutions, and rational solutions, are obtained by using function expansion method. An isolated wave solution is selected from the solution of each nonlinear dispersive wave model equation, and the influence of fractional order change on these isolated wave solutions is discussed. The results show that the fractional derivatives can modulate the waveform, local periodicity, and structure of the isolated solutions of the three model equations. We also point out the construction rules of the auxiliary equations of the extended (G′/G)-expansion method. In the “The Explanation and Discussion” section, a more generalized auxiliary equation is used to further emphasize the rules, which has certain reference value for the construction of the new auxiliary equations. The solutions of fractional-order nonlinear partial differential equations can be enriched by selecting other solvable equations as auxiliary equations.