Acta Mech DOI 10.1007/s00707-017-1872-x ORIGINAL PAPER Shahriar Dastjerdi · Mohammad Abbasi · Leila Yazdanparast A new modified higher-order shear deformation theory for nonlinear analysis of macro- and nano-annular sector plates using the extended Kantorovich method in conjunction with SAPM Received: 21 October 2016 / Revised: 17 April 2017 © Springer-Verlag Wien 2017 Abstract In this research, the nonlinear local and nonlocal analysis of an annular sector plate is studied and solved based on a new modified higher-order shear deformation theory. Due to the shortcomings of HSDT in the two-dimensional nonlinear analysis, it is modified by eliminating the defects, and a comprehensive theory is presented for analyzing the mechanical behavior of an annular sector sheet in general form. The strain field is developed by considering the von Karman assumptions and also the nonlocal theory of Eringen from which the classical local analysis can be deduced conveniently by neglecting the small-scale effects. Whereas the annular sector plate is assumed, the sector, annular/circular, rectangular and solid circular plates can be simulated. Afterward, the nonlocal constitutive equations are derived and solved by using the two- dimensional SAPM (Dastjerdi et al. in Compos B Eng 98, 78–87, 2016). Moreover, a combination of the extended Kantorovich method and one-dimensional SAPM is applied. Since the presented theory is relatively new and similar studying was not available in order to compare the results, a comparison is done with the results of lower-order theories. Finally, the effect of various parameters, such as boundary conditions, different theories, nonlocal and local analyses, loading and the size of the plate, on the mechanical behavior of an annular sector plate are investigated. 1 Introduction Plates are one of the most applicable parts which are used extremely often in many structures such as buildings, airplanes, and industrial machines. Moreover, it is difficult to find a structure without at least a plate in it. Consequently, predicting the mechanical behavior of plates has been examined by many researchers. A plate can be studied for vibration, stability and bending behaviors. Plates usually are considered as two-dimensional structures where the thickness dimension is insignificant in comparison with the other dimensions. According to their thickness, plates can be categorized into three parts: (1) thin plates, (2) moderately thick plates and (3) thick plates. If the width to the thickness ratio is equal or greater than 10, then the plate will be considered a thin plate. The bending deformation can be supposed small when the ratio of thickness to maximum deflection is more than 5. The thin plates are usually analyzed with the classical plate theory (CLPT) which subjected to small deformation assumptions which states that after bending, the perpendicular section of the intermediate- level remains perpendicular to the surface and the normal stresses on the surface are negligible. If the deflection S. Dastjerdi · M. Abbasi Engineering Department, School of Mechanical Engineering, Shahrood Branch, Islamic Azad University, Shahrood, Iran L. Yazdanparast Department of Mechanical Engineering, Shahrood University of Technology, Shahrood, Iran S. Dastjerdi (B ) Engineering Department, School of Mechanical Engineering, Shahrood Branch, Islamic Azad University, Shahrood, Iran E-mail: dastjerdi_shahriar@yahoo.com