Journal of Solid Mechanics
Volume:6 Issue: 1, Winter 2014

In this paper, vibrations and bifurcation of a damped system consists of a mass grounded by linear and nonlinear springs and a nonlinear damper is studied. Nonlinear equation of motion is derived using Newton’s equations. Approximate analytical solutions are obtained using multiple time scales (MTS) method. For free vibration, the approximate analytical results are compared with the numerical integration results. Forced vibrations of the system in primary and secondary resonant cases are studied and the effects of different parameters on the frequency-responses are investigated. Moreover, bifurcation of the system is studied considering different control parameters.

There are mainly two methods of deep drawing analysis; experimental and analytical/numerical. Experimental analysis can be useful in analyzing the process to determine the process parameters that produce a defect free product, and the analytical/numerical modeling can be used to model and analyze the process through all stages of deformation. This approach is less time consuming and more economical. Sheet metal forming often includes biaxial in-plane deformation with non-proportional strain paths. In deep drawing of cylindrical cup, the deformation in the flange in dominated by pure shear deformation, while it changes to plane strain when the material is drawn into the die. This paper deals with the analysis of deep drawing of circular blanks into axi-symmetric cylindrical cup using numerical modeling. The blank drawability has been related both theoretically and experimentally with the initial diameter of the blank and deep drawing parameters. The strains in the radial and circumferential directions have been measured. A correlation on the flange thickness variation by taking into account the work hardening with the analytical and experimental values also has been searched.

In this paper, the transverse vibrations of rectangular plate with circular central hole have been investigated and the natural frequencies of the mentioned plate with point supported by Rayleigh-Ritz Method have been obtained. In this research, the effect of the hole is taken into account by subtracting the energies of the hole domain from the total energies of the whole plate. To determine the kinetic and potential energies of plate, admissible functions for rectangular plate are considered as beam functions and it has been tried that the functions of the deflection of plate, in the form of polynomial functions proportionate with finite degrees, to be replaced by Bessel function, which is used in the analysis of the vibrations of a circular plate. Consideration for a variety of edge conditions is given through a combination of simply supported, clamped and free boundary conditions. In this study, the effects of increasing the diameter of the hole and the effects of number of point supported on the natural frequencies were investigated and the optimum radius of the circular hole for different boundary conditions are obtained. The method has been verified with many known solutions. Furthermore, the convergence is very fast with any desirable accuracy to exact known natural frequencies.

In this research, the analytical and numerical investigation of a cylindrical shell made of functionally graded materials (FGMs) reinforced by laminated composite subjected to internal pressure is presented. Using the infinitesimal theory of elasticity, the analytical solution of stress and strain in vessels made of FGMs is studied first. It is assumed that the elasticity modulus follows a power law distribution in the thickness direction and Poisson''s ratio considered to be constant for simplicity. The results of the finite element method using ABAQUS software for in-homogeneity constant in the range of -2 to 2 have been compared with the analytical results. The comparison represents good coincidence between analytical and numerical results and confirms the accuracy of stress and strain solutions presented for vessel made of FGMs. The stress and strain solutions in laminated composite vessels are then investigated. Finally, modeling of FGM vessel reinforced by composite laminates with different lay-up is taken into consideration. The obtained results demonstrate that in the cylindrical shell reinforced by laminated composites, the maximum stress is considerably less than the maximum stress in the pressure vessels made of just composites or FGMs.

In this work, a new mathematical model of thermoelasticity theory has been considered in the context of a new consideration of heat conduction with fractional order theory. A functionally graded isotropic unbounded medium is considered subjected to a periodically varying heat source in the context of space-time non-local generalization of three-phase-lag thermoelastic model and Green-Naghdi models, in which the thermophysical properties are temperature dependent. The governing equations are expressed in Laplace-Fourier double transform domain and solved in that domain. Then the inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand are obtained numerically in complex domain by using Laguerre’s method and the inversion of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the thermal displacement, temperature and thermal stress are obtained for a hypothetical material. Finally, the obtained results are presented graphically to show the effect of non-local fractional parameter on thermal displacement, temperature and thermal stress. A comparison of the results for different theories (three-phase-lag model, GN model II, GN model III) is presented and the effect of non-homogeneity is also shown. The results, corresponding to the cases, when the material properties are temperature independent, agree with the results of the existing literature.

In this paper, free vibration analysis of orthotropic functionally graded material (FGM) cylinders was carried out by a Mesh-Free method. In this analysis, moving least squares shape functions are used for approximation of displacement field in the weak form of equilibrium equation. Essential boundary conditions are imposed by transformation method. In this simulation, an axisymmetric model is used. The orthotropic FGM cylinders are assumed to be a mixture of two isotropic materials as fiber and matrix. The volume fraction of the fiber is changed in the radial direction. Consequently, mechanical properties of these cylinders are changed in the radial direction. Free vibration analysis of orthotropic FGM cylinders with any arbitrary combination of boundary conditions is possible by the proposed model. Natural frequencies obtained from the presented model are in good agreement with results of finite element simulation and other results from literature. Effects of various types of boundary conditions, geometrical parameters, and mechanical properties on the natural frequencies are studied.

Flexible Polyurethane (PU) foam samples with different densities and chemical formulations were tested in quasi-static stress-strain compression tests. The compression tests were performed using the Lloyd LR5K Plus instrument at fixed compression strain rate of 0.033 s-1 and samples were compressed up to 70% compression strains. All foam samples were tested in the foam rise direction and their compression test stress results were modeled using a constitutive Polymeric or Phenomenological Foam Model (PFM). In this research, a new constitutive PFM model that consists of mechanical systems such as dashpots and springs was formulated to be used for different strain rate experiments. The experimental compression test results for different strain rates were compared to the PFM model results for all foam samples. Both modeling and experimental results showed pretty good agreement. From curve fitting of the experimental tests with the PFM model; different mechanical materials’ coefficients such as elastic and viscous parameters were computed. These mechanical parameters are indeed important characteristics for viscoelastic materials. This model can be used for constant and variable strain rates and for characterizing biomechanical material applications such as bone tissues, muscle tissues and other cellular materials.

In the present work, the free vibration behavior of rectangular graphene sheet under shear in-plane load is studied. Nonlocal elasticity theory has been implemented to study the vibration analysis of orthotropic single-layered graphene sheets (SLGSs) subjected to shear in-plane load. The SLGSs is embedded on a viscoelastic medium which is simulated as a Visco-Pasternak foundation. Using the principle of virtual work, the governing equations are derived for the rectangular nanoplates. Differential quadrature method (DQM) is employed and numerical solutions for the vibration frequency are obtained. The influence of surrounding elastic medium, material property, aspect ratio, nonlocal parameter, length of nanoplate and effect of boundary conditions on the vibration analysis of orthotropic single-layered graphene sheets (SLGSs) is studied. Six boundary conditions are investigated. Numerical results show that the vibration frequencies of SLGSs are strongly dependent on the small scale coefficient and shear in-plane load. The present analysis results can be used for the design of the next generation of nanodevices that make use of the vibration properties of the graphene.