In this paper, the large amplitude forced vibrations of thin rectangular plates made of different types of rubbers are investigated both experimentally and theoretically. The excitation is provided by a concentrated transversal harmonic load. Clamped boundary conditions at the edges are considered, while rotary inertia, geometric imperfections and shear deformation are neglected since they are negligible for the studied cases. The von Karman nonlinear strain-displacement relationships are used in the theoretical study; the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model, which introduces nonlinear damping. An equivalent viscous damping model has also been created for comparison. In-plane pre-loads applied during the assembly of the plate to the frame are taken into account. In the experimental study, two rubber plates with different material and thicknesses have been considered; a silicone plate and a neoprene plate. The plates have been fixed to a heavy rectangular metal frame with an initial stretching. The large amplitude vibrations of the plates in the spectral neighbourhood of the first resonance have been measured at various harmonic force levels. A laser Doppler vibrometer has been used to measure the plate response. Maximum vibration amplitude larger than three times the thickness of the plate has been achieved, corresponding to a hardening type nonlinear response. Experimental frequency-response curves have been very satisfactorily compared to numerical results. Results show that the identified retardation time increases when the excitation level is increased, similar to the equivalent viscous damping but to a lesser extent due to its nonlinear nature. The nonlinearity introduced by the Kelvin-Voigt viscoelasticity model is found to be not sufficient to capture the dissipation present in the rubber plates during large amplitude vibrations.

Experimental and theoretical study on large amplitude vibrations of clamped rubber plates

In this paper, the stability analysis of moderately thick time-dependent viscoelastic plates with various shapes is studied using the Laplace–Carson transformation and simple hp cloud meshless method. The shear effect of the plate is described by the first order shear deformation theory. The mechanical properties of the materials are supposed to be linear viscoelastic based on the constant bulk modulus. The displacement field is assumed to be the product of two functions, one being a function of geometrical parameters and the other a known exponential function of time. The simple hp cloud method is used for discretization which is based on Kronecker-delta properties. Thus, the essential boundary conditions can be imposed directly. A numerical investigation is made by employing the inverse of Laplace–Carson transformation. The time history of buckling coefficients of viscoelastic plates of various shapes with different boundary conditions is considered. Moreover, a number of numerical results are presented to study the effect of thickness, aspect ratio, different boundary conditions, and various shapes on the time history of buckling coefficients of the viscoelastic plate.

Stability analysis of arbitrarily shaped moderately thick viscoelastic plates using Laplace–Carson transformation and a simple hp cloud method

Creep and recovery of viscoelastic laminated composite plates

Original Research Paper Received 03 October 2014 Accepted 06 December 2014 Available Online 03 January 2015 In this study, the non-linear bending analysis of isotropic and orthotropic rectangular plates is performed by Dynamic Relaxation (DR) method. In order to model the plate, the four-variable refined plate theory, which is new and simple higher-order shear deformation theory and has good capability in analysis of thick plates, is adopted. Despite the first-order shear deformation plate theory, this theory does not need the shear correction factor, predicts shear strains and stress parabolically across the plate thickness and satisfies the zero stress conditions on free surfaces. The governing equations are obtained using virtual work principle and the Von-Karman nonlinear terms are considered in strain-displacement equations. The non-linear coupled governing equations are solved by DR method combined with finite difference technique, and for this purpose computer code is provided in MATLAB software. In order to demonstrate the accuracy of the present method, the numerical results are compared with the existing ones and very good agreement is observed. Also, the effects of side-to-thickness ratio and boundary conditions on the results are examined. Finally, the variations of shear effects by changing the plate thickness and also changing the orthotropy ratio in orthotropic plates are investigated.

Non-linear bending analysis of thick rectangular plates by four- variable refined plate theory and Dynamic Relaxation method

In this study, an analytical procedure for the bending problem of a viscoelastic sandwich plate with a corrugated core is presented. Reissner–Mindlin plate theory and N-termed Prony series are employed to define the elastic and time-dependent contributions of the governing equations, respectively. Three different corrugation patterns, i.e., rectangular, trapezoidal, and triangular, are examined. Moreover, the structure is analyzed under both simply support and clamp boundary conditions. The calibrated material parameters of polymethyl methacrylate (PMMA) for the Generalized Maxwell rheological model are employed to show the viscoelastic response of the structure. A 3D finite element simulation of the problem is also conducted to confirm the accuracy of the analytical formulation. The two well-known creep and stress relaxation phenomena of the viscoelastic materials are examined for the mentioned corrugation cores and both boundary conditions analytically and numerically. The time-dependent dimensionless deflection and resultant von Mises stress distributions are provided. Besides, the variation of the results with various rise-times and applied load are studied in detail. The von Mises stress contours of the upper surface of the structure at the end of the creep test are also presented. The finite element method outcomes verify the analytical results with excellent compatibility. The proposed analytical procedure can be used as an efficient tool to study the effects of various parameters such as material, geometrical constants, and corrugation pattern on bending of viscoelastic sandwich plates with corrugated core problems for design and optimization, which involves a high number of simulations.

Development of an analytical framework for viscoelastic corrugated-core sandwich plates and validation against FEM

Geometrically nonlinear governing equations for a plate with linear viscoelastic material are derived. The material model is of Boltzmann superposi- tion principle type. A third-order displacement field is used to model the shear deformation effects. For the solution of the nonlinear governing equations the Dynamic Relaxation (DR) iterative method together with the finite difference discretization technique is used. Finally, the numerical results for the critical buckling load for simply supported edge constraints are reported. In order to justify the accuracy of the results, the elastic plate critical buckling loads are obtained and compared with the existing results. The correlations are very satisfactory. The numerical results are presented for Classical Plate Theory (CPT), First order- Shear Defonmation Plate Theory (FSDT) and Higher Order- Shear Deformation Plate Theory (HSDT). In the case of thick plate the differences among the three theories are highlighted, however, for thin plate the variations are very small.

http://journals.iau.ir/article_512601_2d5960240fd8c3e6224dbb165e863d30.pdf

Manuchehr Salehi

Papers

Non-Linear Analysis of Viscoelastic Rectangular Plates Subjected to In-Plane Compression