In this paper, a novel size-dependent functionally graded (FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton’s principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.

In this study, scanning electron microscopy and electrochemical techniques were used to investigate the effect of the Cu content on the microstructure and corrosion resistance of plasma electrolytic oxidation (PEO) coatings on Al-xCu alloys. The results showed that Cu in the substrate alloy was used to promote the intensity and the transformation process of the microdischarges, which led to the undulating-ribbon feature of cross-sectional morphologies. The formation of pores and cracks in the coatings was facilitated by the oxygen release due to the oxidation of O 2− and the dissolution of copper oxide. Meanwhile, the average porosity of PEO coatings distinctly increased with the increase in the Cu content, which is the key factor causing the decreased corrosion resistance of PEO coatings. Due to their high strength to weight ratio, Cu-containing aluminum alloys are one of the most promising materials for use in aerospace and automobile applications. However, their application is limited by low corrosion resistance. Therefore, plasma electrolytic oxidation (PEO) is considered to be a well-known process for the in situ production of oxide coatings on Al and its alloys, which can offer better corrosion protection as compared to the other surface modification processes for substrate alloys. [1][2][3][4] In general, many processing factors, such as the composition of electrolyte, 5-7 additives, 8-10 applied power modes, 11-13 electrical parameters, 14,15 temperature, 16 and oxidation time, 17 are used to control the PEO process. These parameters not only affect the PEO process but also the performance of the prepared coatings. The characteristics of the formed PEO coatings could also be affected by the alloying elements in substrate alloys. However, this has not been studied in much detail. To the best of our knowledge, the literature contains several papers that describe the effect of multiple elements (such as Si, Mg, Cu, Li, and Fe) on the properties of the formed PEO coatings for commercial aluminum alloys.18-20 While, only a few studies focused on the effect of single alloying element in substrate. Among others, Ali et al. studied the PEO treatment of simple binary Al-Si alloys and reported that the thickness of PEO coatings significantly decreased, whereas their porosity increased with the increase in the Si content. The decomposition of volatile silicon oxides, such as SiO in the formed coatings during the PEO process, contributed to the high porosity. Furthermore, Si exhibits low electrical conductivity, which adversely affects the formation of microdischarges and the oxidation kinetics of the alloy. 21 Meanwhile, Oh et al. studied the effect of Mg on the properties of PEO coatings on Al-Mg alloys and demonstrated that when the amount of Mg was higher than 3 wt% the formation of α-Al 2 O 3 was inhibited.22 It is also reported that Zn is the primary alloying element for the 7××× series Al alloys, and the peculiar function of Zn during the formation of PEO coatings on binary Al-Zn alloys has also been investig...

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