RETRACTED ARTICLE: Chaotic simulation of the multi-phase reinforced thermo-elastic disk using GDQM

In this research, a mathematical derivation is made to develop a nonlinear dynamic model for the nonlinear frequency and chaotic responses of the multi-scale hybrid nano-composite reinforced disk in the thermal environment and subject to a harmonic external load. Using Hamilton’s principle and the von Karman nonlinear theory, the nonlinear governing equation is derived. For developing an accurate solution approach, generalized differential quadrature method (GDQM) and perturbation approach (PA) are finally employed. Various geometrically parameters are taken into account to investigate the chaotic motion of the viscoelastic disk subject to harmonic excitation. The fundamental and golden results of this paper could be that in the lower value of the external harmonic force, different FG patterns do not have any effects on the motion response of the structure. But, for the higher value of external harmonic force and all FG patterns, the chaos motion could be seen and for the FG-X pattern, the chaosity is more significant than other patterns of the FG. As a practical designing tip, it is recommended to choose plates with lower thickness relative to the outer radius to achieve better vibration performance.

     

Wave propagation simulation in an electrically open shell reinforced with multi-phase nanocomposites

Wave propagation simulation in a multi-hybrid nanocomposite (MHC)-reinforced doubly curved open shell covered with piezoelectric actuator is examined for the first time. The third-order shear deformation theory (third-order SDT) is applied to formulate the stress–strain relations. Rule of the mixture and modified Halpin–Tsai model are engaged to provide the effective material constants of the MHC-reinforced open shell. By employing Hamilton’s principle, the governing equations of the structure are derived. Via the compatibility rule, the bonding between the smart layer and sandwich open shell is modeled. Also, with the aid of Maxwell's equation, the mechanics of the piezoelectric layer are formulated. Afterward, a parametric study is carried out to investigate the effects of the CNTs’ weight fraction, various FG face sheet patterns, small radius to total thickness ratio, the thickness of the smart layer, externally applied voltage, and carbon fiber angle on the phase velocity of the MHC-reinforced open shell. Another necessary consequence is that as the externally applied voltage to the piezoelectric layer of the smart open shell increases, there will be seen an enhancement on the phase velocity or wave response of the system and without a doubt this issue is much more substantial at the lower wave number. It is also observed that when the applied voltage is more than zero, we can find a range for the fiber angle that these values are the critical fiber angle and this critical range will expand by increasing the external electrical load. The useful suggestion of this study is that for designing the structure, we should attention to the FG pattern and higher value of the wavenumber, simultaneously. The presented study outputs can be used in ultrasonic inspection techniques and structural health monitoring.

     

On the phase velocity simulation of the multi curved viscoelastic system via an exact solution framework

The analysis of the wave propagation behavior of a sandwich structure with a soft core and multi-hybrid nanocomposite (MHC) face sheets is carried out in the framework of the higher-order shear deformation theory (HSDT). In order to take into account the viscoelastic influence, the Kelvin-Voight model is presented. In this paper, the constituent material of the core is made of an epoxy matrix which is reinforced by both macro- and nano-size reinforcements, namely carbon fiber (CF) and carbon nanotube (CNT). The effective material properties like Young's modulus or density are derived utilizing a micromechanical scheme incorporated with the Halpin–Tsai model. Then, on the basis of an energy-based Hamiltonian approach, the equations of motion are derived. The detailed parametric study is conducted, focusing on the combined effects of the viscoelastic foundation, CNT' weight fraction, core to total thickness ratio, small radius to total thickness ratio, and carbon fiber angle on the wave propagation behavior of sandwich structure. The results show that as well as increasing the phase velocity of the sandwich structure by increasing the wave number, this influence will be much more effective by increasing the damping factor. It is also observed that there is a critical value for the viscoelastic foundation that the relation between wave number and phase velocity will change from direct to indirect. The presented study outputs can be used in ultrasonic inspection techniques and structural health monitoring.