Though widely used in modelling nano- and micro- structures, Eringen’s differential model shows some inconsistencies and recent study has demonstrated its differences between the integral model, which then implies the necessity of using the latter model. In this paper, an analytical study is taken to analyze static bending of nonlocal Euler-Bernoulli beams using Eringen’s two-phase local/nonlocal model. Firstly, a reduction method is proved rigorously, with which the integral equation in consideration can be reduced to a differential equation with mixed boundary value conditions. Then, the static bending problem is formulated and four types of boundary conditions with various loadings are considered. By solving the corresponding differential equations, exact solutions are obtained explicitly in all of the cases, especially for the paradoxical cantilever beam problem. Finally, asymptotic analysis of the exact solutions reveals clearly that, unlike the differential model, the integral model adopted herein has a consistent softening effect. Comparisons are also made with existing analytical and numerical results, which further shows the advantages of the analytical results obtained. Additionally, it seems that the once controversial nonlocal bar problem in the literature is well resolved by the reduction method.

BackgroundEsophageal squamous cell carcinoma (ESCC) is one of the most common malignancies. Early diagnosis is critical for guiding the therapeutic management of ESCC. It is imperative to find more effective biomarkers of ESCC.MethodsTo identify novel biomarkers for esophageal squamous cell carcinoma (ESCC), specimens from 10 patients with ESCC were subjected to a comparative proteomic analysis. The proteomic patterns of ESCC samples and normal esophageal epithelial tissues (NEETs) were compared using two-dimensional gel electrophoresis. And differentially expressed proteins were identified using MALDI-TOF-MS/MS. For further identification of protein in selected spot, western blotting and immunohistochemistry were employed.ResultsTwelve proteins were up-regulated and fifteen proteins were down-regulated in the ESCC samples compared with the NEET samples. Up-regulation of galectin-7 was further confirmed by western blotting and immunohistochemistry. Furthermore, immunohistochemical staining of galectin-7 was performed on a tissue microarray containing ESCC samples (n = 50) and NEET samples (n = 10). The expression levels of galectin-7 were markedly higher in the ESCC samples than in the NEET samples (P = 0.012). In addition, tissue microarray analysis also showed that the expression level of galectin-7 was related to the differentiation of ESCC.ConclusionsThe present proteomics analysis revealed that galectin-7 was highly expressed in ESCC tissues. The alteration in the expression of galectin-7 was confirmed using a tissue microarray. These findings suggest that galectin-7 could be used as a potential biomarker for ESCC.

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