Bending Analysis of Isotropic Rectangular Kirchhoff Plates Subjected to Non-Uniform Heating Using the Fourier Transform Method

The object of this paper is the bending analysis of isotropic rectangular Kirchhoff plates subjected to non-uniform heating (NUH) using the Fourier transform method. The bottom and top surfaces of the plate are assumed to have different changes in temperature, whereas the change in temperature of the mid-surface is zero. According to classical plate theory, the governing equation of the plate contains second derivatives of the NUH; these derivatives are zero by constant value of the NUH, which leads to its absence in the governing equation. This paper presented an approach by which Fourier sine transform was utilized to describe the NUH, while the double trigonometric series of Navier and the simple trigonometric series of Lévy were utilized to describe the deflection curve. Thus, the NUH appeared in the governing equation, which simplified the analysis. Rectangular plates simply supported along all edges were analyzed, bending moments, twisting moments, and deflections being determined. In addition, rectangular plates simply supported along two opposite edges were analyzed; the other edges having various support conditions (free, simply supported, and fixed).

The Fourier Transform Method for Second-Order Integro-Dynamic Equations on Time Scales

In this chapter we introduce the Fourier transform on arbitrary time scales and deduct some of its properties. In the chapter are given some applications for second-order integro-dynamic equations on time scales.

Bending Analysis of Isotropic Rectangular Kirchhoff Plates Subjected to a Thermal Gradient Using the Fourier Transform Method

The object of this paper is the bending analysis of isotropic rectangular Kirchhoff plates subjected to a thermal gradient (TG) using the Fourier transform method. The bottom and top surfaces of the plate are assumed to have different changes in temperature, whereas the change in temperature of the mid-surface is zero. According to classical plate theory, the governing equation of the plate contains second derivatives of the TG; these derivatives are zero by constant value of the TG, which leads to the absence of the TG in the governing equation. This paper presented an approach by which Fourier sine transform was utilized to describe the TG, while the double trigonometric series of Navier and the simple trigonometric series of Lévy were utilized to describe the deflection. Thus, the TG appeared in the governing equation, which simplified the analysis. Rectangular plates simply supported along all edges were analyzed, bending moments, twisting moments, and deflections being determined. In addition, rectangular plates simply supported along two opposite edges were analyzed, the other edges having various support conditions (free, simply supported, and fixed).

Detection and characterization of folded-chain clusters in the structured melt of isotactic polypropylene

Despite ceaseless efforts in past decades, the memory effect of semi-crystalline polymers has not been elucidated completely yet. An important reason why is that residual lamellar crystals in the structured melt are difficult to characterize. Recently, we developed a new small-angle X-ray scattering (SAXS) theory [Li et al. (2019). IUCrJ, 6, 968–983] and Fourier transform method [Li et al. (2020). CrystEngComm, 22, 3042–3058] for lamellar crystals that could derive structural information from SAXS readily. In this study, we tried to employ the new theory and method to characterize residual lamellar crystals in the structured melt. It was found that although scattering peaks cannot be observed in raw scattering profiles, they actually exist. Subtracting free-melt scattering and multiplying by q 4 benefit the observation of these weak scattering peaks. With the new Fourier transform method, it was found that indeed as proposed previously, thicker lamellar crystals exist in the structured melt. To determine the lateral size of residual lamellar crystal especially, a new method was developed under the guidance of the new theory. With the new method, it was found that although the crystallinity is very low (∼1% at 174°C), the lateral sizes in the structured melts are still large, e.g. 45.3 nm at 174°C, much greater than the critical nucleation size. This implies that these residual lamellar crystals can act as athermal nuclei after quenching to a lower temperature, as proposed by Ziabicki & Alfonso [(1994). Colloid Polym. Sci. 272, 1027–1042; (2002). Macromol. Symp. 185, 211–231] more than 20 years ago. The methodologies proposed here could also be applied to other polymer lamellar systems.