Geometrically Nonlinear Free Vibration of Composite Materials: Clamped-Clamped Functionally Graded Beam with an Edge Crack Using Homogenisation Method

Abstract

The problem of geometrically nonlinear free vibration of a clamped-clamped functionally graded beam containing an open edge crack in its center is studied in this paper. The study is based on Euler-Bernoulli beam theory and Von Karman geometric nonlinearity assumptions. The cracked section is modeled by an elastic spring connecting two intact segments of the beam. It is assumed that material properties of the functionally graded composites are graded in the thickness direction and estimated through the rule of mixture. The homogenisation method is used to reduce the problem to that of isotropic homogeneous cracked beam. Direct iterative method is employed for solving the eigenvalue equation for governing the frequency nonlinear vibration, in order to show the effect of the crack depth and the influences of the volume fraction on the dynamic response.

Geometrically nonlinear free and forced vibrations analysis of clamped-clamped functionally graded beams with multicracks

“…Where K * ir , B * i jkr and M * ir stand for the dimensionless classical rigidity tensor, the nonlinear rigidity tensor and the mass tensor, respectively defined in [1].…”

“…The present work aims to investigate the geometrically nonlinear free and forced vibrations of MATEC Web of Conferences 211, 02002 (2018) https://doi.org/10.1051/matecconf/201821102002 VETOMAC XIV clamped-clamped Functionally Graded beams with multi-cracks, located at different positions, based on the Euler-Bernoulli beam theory and the Von Karman geometrical nonlinearity assumptions. A homogenization procedure was used in [1,2] to reduce the problem under consideration to that of an equivalent isotropic homogeneous multi-cracked beam. The closed-form solutions and transfer matrix method used previously in [3] are employed and the resulting equation is solved iteratively by the Newton Raphson method.…”

Analysis of the associated stress distributions to the nonlinear forced vibrations of functionally graded multi-cracked beams

“…The applicability of this approach shows that the nonlinear effect does not appear only via the amplitude frequency dependence but also via the amplitude dependence of the cracked FGB deflection shapes. The purpose of the present contribution is the application of the non-linear forced vibration model previously reported in [25] using a multimode response approach and combined with the homogenization procedure based on the neutral surface approach previously adopted in [26,27]. This allowed an explicit calculation of multidimensional non-linear frequency response curves, the corresponding amplitude dependence deflection shapes and the associated curvatures distributions of a multi-cracked FGB subjected to a uniformly distributed harmonic force applied over the beam length.…”