three-dimensional theory of elasticity

Double-walled structures are widely used in various industries such as aerospace, marine, and automotive. Therefore, in this paper, the sound transmission in these structures is investigated. Due to the influence of rotation and shear parameters by increasing the thickness of the cylindrical shell, the Newton-based three-dimensional theory of elasticity is used. In order to solve the governing equation of motion for a cylinder, it is assumed that the displacement field is the sum of the gradients of a scalar potential field and the curl of a vector potential field. As a result, the shell motion equation becomes two separate wave equations, which solve the displacement field. To confirm the obtained equations, the present results are compared with the results of other researchers in this field that have been obtained with other theories such as classical shell theory. Finally, the effect of various parameters such as properties and thickness of the fluid layer, Mach number, and material of the cylinders are investigated. The results show that in double-walled structures with the air gap, the acoustic impedance (the speed of sound multiplied by density) of the fluid is the main and effective factor in sound control. Any fluid with more Acoustic impedance behaves better in sound control and improves sound transmission loss.

In this paper, the Three-dimensional (3-D) theory of elasticity has been used for sound transmission loss (TL) of a thin-walled cylindrical shell with infinite length. This structure is excited by an obliquely plane wave. Governing equations of the thin shell have been derived in radial, axial, and circumferential directions. Then, Helmholtz decomposition is applied to solve the equations. Therefore, the displacement field is considered in terms of Lame potential functions This method describes the vibrational behavior of the shells more accurately than other theories, including the classical theory and the first and third order of shear, due to the consideration of the effects of rotation and shear. A comparison of the present method results with those obtained from classical shell theory (CST) indicates an excellent agreement. However, at the high frequencies the classical encounter insufficient accuracies as a result of increasing the rotational terms as well as shear wave effects. The results show that with increasing shell thickness, due to increased flexural stiffness of the shell, the sound transmission loss has increased. By increasing the Mach number, due to the negative stiffness, reduces the sound transmission loss in the stiffness-controlled region and conversely increases the sound transmission loss in the mass-controlled region due to damping in the structure. Finally, it is shown that the critical and coincidence frequencies increase with the growing up Mach number.

Researchesdone by researchers on the forced vibration analysis of rectangular plates based on the three-dimensional elasticity theory are limited to the analytical solutions for simply-supported boundary conditions or linear analyses. In this study, by eliminating the previous limitations, the forced vibration of functionally graded rectangular plates with different boundary conditions is examined based on three-dimensional theory of elasticity and taking into account the geometrically nonlinearity.The rectangular plate is made of the aluminum and alumina in which are distributed through the thickness direction according to a power law for functionally graded materials. In order to achieve the governing equations and corresponding boundary conditions in terms of displacements, Green-Lagrange, stress-strain relations and Hamilton's principle are used. The nonlinear coupled governing equations are discretized in the space domain using the generalized differential quadrature (GDQ) method.Then, utilizing the numerical-based Galerkin scheme, one can obtain a time-varying set of ordinary differential equations of Duffing type. The arc-length method is employed to solve the vectorized form of nonlinear parameterized equations. Finally, to have a comprehensive study on the effects of geometrical parameters,forcingamplitude and damping ratio on the frequency-response curve of functionally graded rectangular plateswith different boundary conditions are examined.