Vibrations of a strip with delamination in the framework of the one-parameter Aifantis model of gradient elasticity theory

Abstract

The problems on in-plane and anti-planar steady-state vibrations of an isotropic elastic strip with delamination at the lower boundary has been investigated. The goal of the study is to analyze the stress-strain state in the crack tips areas and to construct a crack opening function being the main mechanical characteristics in the crack theory problems. The problems under study have been solved in the framework of the nonclassical gradient elasticity theory (GET) on the basis of the one-parameter model proposed by Aifantis. The boundary integral equations (BIE) are obtained with respect to crack opening functions or their derivatives. The analysis of BIEs is carried out, regular and irregular parts are distinguished, the obtained BIEs with singular (e.g., with hypersingular, with cubic singularity) integrals are solved via collocation methods, approximating Chebyshev polynomials, quadrature formulas for singular integrals. For the in-plane problem solution, the simplified Ru-Aifantis method has been applied. The Ru-Aifantis method allows to divide the initial boundary value problem into two sub-problems - the classical linear elasticity theory (LTE) problem and the simplified boundary value problem for finding the gradient solution which includes the solution found via the classical theory. For each of the problems, semi-analytical expressions for the functions of crack opening have been constructed, and the analysis of the stress-strain state in the area of crack tips has been carried out. The problems have also been solved in the case of a crack with small relative length, the analysis of BIE depending on small parameters ratio has been carried out, and explicit expressions for the crack opening functions have been obtained. Numerical calculations have been performed; the applicability conditions for the asymptotic method are determined, and a comparative analysis of the results obtained on the basis of GET and LTE models, depending on the values of the gradient parameter and the delamination length, is realized.