Internal resonance and bending analysis of thick visco-hyper-elastic arches

Abstract

Abstract In this study, a comprehensive analysis of visco-hyper-elastic thick soft arches under an external time-independent as well as time-dependent loads is presented from bending and internal resonance phenomenon perspectives. Axial, transverse and rotation motions are considered for modelling the thick and soft arch in the framework of the Mooney–Rivlin and Kelvin–Voigt visco-hyper-elastic schemes and third-order shear deformable models. The arch is assumed to be incompressible and is modelled using von Kármán geometric nonlinearity in the strain–displacement relationship. Using a virtual work method, the bending equations are derived. For the vibration analysis, three, coupled, highly nonlinear equations of motions are obtained using force-moment balance method. The Newton–Raphson method together with the dynamic equilibrium technique is used for the bending and vibration analyses. A detailed study on the influence of having visco-hyper-elasticity and arch curvature in the frequency response of the system is given in detail, and the bending deformation due to the applied static load is presented. The influence of having thick, soft arches with different slenderness ratios is shown, and the forced vibration response is discussed. Moreover, internal resonance in the system is studied showing that the curvature term in the structure can lead to three-to-one internal resonances, showing a rich nonlinear frequency response. The results of this study are a step forward in studying the visco-hyper-elastic behaviour of biological structures and soft tissues. Graphic abstract