Singularity analysis near the vertex of the V‐notch in Reissner's plate by coupling the asymptotic expansion technique with the interpolating matrix method

Abstract

A new numerical method for calculating the singularity orders of V‐notches in Reissner's plate is proposed in this paper. By introducing the asymptotic expansion of the generalised displacement field at the notch tip into the equilibrium equations of a plate, a set of characteristic ordinary differential equations with respect to the singularity order are established. In addition, by adopting the variable substitution technique, the obtained non‐linear characteristic equations are transformed into linear ones, which are solved by the interpolating matrix method. The singularity orders of moments and shear forces can be obtained simultaneously and can be distinguished from the corresponding characteristic angular functions conveniently. Four types of boundary conditions are proposed to investigate the influence of boundary conditions on the singularity order values. The effect of the Poisson's ratio on the singularity orders of the V‐notch in Reissner's plate is discussed. The present method is versatile for the singularity analysis of single material V‐notches and bi‐material V‐notches, and can be easily extended to multi‐material V‐notches.

On ensuring structural integrity for configurations with stress singularities: a review

“…Recent reviews of such asymptotic identifications are available . Additional recent asymptotic identifications are also available …”

“…Often these additional identifications are for more complex local configurations than those in the references in the cited reviews: sometimes they simply fill in a gap in the literature. For classical elasticity with local configurations comprised of isotropic and piecewise homogeneous, linear elastic materials, all the stress singularities asymptotically identified that have been found to date since 2004 are discussed in the studies . Examples of stress singularities identified since 2004 for anisotropic elastic materials are given in the previous studies ; for functionally graded elastic materials; for finite strain in elastic materials; and for varying size scales …”

Investigation of thickness effects on the bending singularities of a notched plate

“…14 This theory has been applied to solve bending singularity in a notched plate. 19 Chen investigated the singular stress fields of a notched plate by using the integral transform method through FSDPT. 17 Both the moment and transverse shear force singularities were considered.…”

“…Cheng et al proposed an interpolating matrix method based on FSDPT to investigate the singularities of the notched plate. 19 Chen investigated the singular stress fields of a notched plate by using the integral transform method through FSDPT. 20 In Chen's results, the stress singularity orders and the stress intensities are presented in exact forms.…”

“…They also proved that the imaginary part of the singularity order caused the oscillation of the singular stress fields near the wedge apex. In the above results, [15][16][17][18][19][20][21] the singular parameters were determined in exact forms by theoretical approaches. For complicated geometric structure, it is difficult to solve the stress intensities near the notch tip, and the singular finite elements were developed according to the asymptotic displacement fields near the notch tip.…”