Values Of Length Scale Parameter Research Articles

Inextensional vibrations of thin spherical shells using strain gradient elasticity theory

We study inextensional vibrations of the spherical shell using strain gradient elasticity theory under Kirchhoff-Love hypotheses. For admissibility of the inextensional deformations the shell is punctured by making a tiny hole. The shell is assumed to be made of linearly elastic, homogeneous, and isotropic material. The closed form expression for the frequencies of inextensional modes of vibration of spherical caps of various polar angles and that of nearly complete spherical shell are derived using Rayleigh's method. Towards this, the displacement and velocity fields are obtained by setting the membrane strains to zero. Further, to facilitate the existence of inextensional vibrations, boundaries are assumed to be traction free. The frequencies are computed for the positive and negative signs in the strain gradient term of the constitutive law. The computed frequencies, employing negative sign are found to be higher than those computed using classical elasticity theory. The opposite effect is seen when the frequencies are computed with the positive sign. Further, the frequencies are found to saturate when the positive sign is used in the constitutive law. Parametric studies viz. variation in frequencies with the circumferential wavenumbers for different values of length scale parameter, variation in frequencies with circumferential wavenumbers for different values of polar angles and a given length scale parameter, variation of frequency with increasing polar angle for a given circumferential wavenumber, variation of saturation wavenumber with increasing polar angle, and increasing length scale parameter are carried out. The frequencies of nearly a spherical shell with increasing circumferential wavenumber are also computed.

On the static, vibration, and transient responses of micro-plates made of materials with different microstructures

In the present paper, for the first time, theoretical analysis for static, dynamic, and transient responses of micro-plates made of different materials were comprehensively performed. The constitutive relations of analyzed materials satisfy assumptions of the modified couple stress theory. Additionally, values of length scale parameters of studied materials are taken from well-established experimental results presented in the literature. A generalized model of micro-plates made from Cu, Ni, Ti, and Epoxy satisfies assumptions of Reissner–Mindlin plate theory. The formulated generalized boundary value problem is solved by analytical and the radial basis function finite difference (RBF-FD) methods. A convergence study is performed to show the advantages and correctness of the obtained numerical solution. The meshless numerical method is easy to implement and it provides accurate results that are in excellent agreement with analytical solutions. Effects of material parameters, boundary conditions, and assumptions of classical and the modified couple stress theory on the static bending, free vibration and transient responses of micro-plates are presented and discussed.

Novel nickle foil micro-bend tests and the need for a relook at length scale parameter’s numerical value

The current study provides a discussion on two aspects of size-effects which is known to be a characteristic behavior of micro-scaled structures. First it provides the experimentalists with a novel micro-bend technique wherein the micro-cantilever is loaded at different points along its length. This eases the experimental process by reducing the efforts of the experimentalists in sample preparation (as the same sample can be used for obtaining results corresponding to different L/H ratio). Also, precision cutting and mounting of the samples to obtain the micro-cantilever of desirous overhang length is not required. Further, the above described technique is used to test nickle micro-cantilevers of three thicknesses (15, 25, 50 μm) and different L/H (=200, 300, 400) ratios. They are elastically bent to undergo small deflections. Apart from reporting the obvious size-effects for smaller thicknesses and fitting the experiments with a model for determining the length scale parameter, size-effects for beams with different thicknesses and varying L/H ratios are presented. Secondly, a discussion on the widely fluctuating nature of numerical values that the length scale parameter assumes is presented. And it is postulated that the numerical value of length scale parameter is a function of material, model and geometry of beam cross-section.

Static, stability and dynamic analyses of second strain gradient elastic Euler–Bernoulli beams

A simplified second strain gradient Euler–Bernoulli beam theory with two non-classical elastic coefficients in addition to the classical constants is presented. The governing equation and the associated classical and non-classical boundary conditions are derived with the aid of variational principles. The simplified second strain gradient theory is governed by an eighth-order differential equation with displacement, slope, curvature and triple derivative of displacement as degrees of freedom. This theory can be reduced to the first strain gradient and classical Euler–Bernoulli beam theories. Analytical solutions for static behaviour, free vibration and stability analyses are presented for different boundary conditions and length scale parameters. Using the numerical Laplace transform, a spectral element is developed for dynamic analysis of a cantilever beam subjected to a Gaussian pulse. Further, spectrum and dispersion relations are derived to study wave propagation characteristics. The gradient effects on the structural response are assessed and compared with the corresponding first strain gradient and classical beam theories. Observations show that the second strain gradient theory exhibiting stiffer behaviour in comparison to the first strain gradient and classical theories. The beam deflection decreases whereas frequencies and buckling load increase for increasing values of the gradient coefficient in comparison to the first strain gradient and classical theories. The forced response for a finite beam reveals a decrease in the amplitude and a shift to smaller time values with an increase in the value of length scale parameter. Additionally, the second strain gradient beam shows a dispersive behaviour, and for a given frequency the wavenumber decreases and the phase speed increases with an increase in the length scale parameter as compared to the first strain gradient beam theory.

Dynamic response of the nonlocal strain-stress gradient in laminated polymer composites microtubes

This study presents the frequency analysis of a size-dependent laminated polymer composite microtube using a nonlocal strain-stress gradient (NSG) model. By applying energy methods (known as Hamilton’s principle), the motion equations of the laminated micro tube composites are developed. The thermodynamic equations of the laminated microtube are based on first-order shear deformation theory (FSDT), and a generalized differential quadrature method (GDQM) is employed to find the model for the natural frequencies. The results show that by considering C-F boundary conditions (BCs) and every even layers’ number in lower value of length scale parameter, the frequency of the structure drops by soaring this parameter. However, this matter is inverse in its higher value. Eventually, the ply angle’s influences, nonlocality as well as length scale element on the vibration of the laminated composite microstructure are investigated.

Application of nonlocal strain–stress gradient theory and GDQEM for thermo-vibration responses of a laminated composite nanoshell

In this article, thermal buckling and frequency analysis of a size-dependent laminated composite cylindrical nanoshell in thermal environment using nonlocal strain–stress gradient theory are presented. The thermodynamic equations of the laminated cylindrical nanoshell are based on first-order shear deformation theory, and generalized differential quadrature element method is implemented to solve these equations and obtain natural frequency and critical temperature of the presented model. The results show that by considering C–F boundary conditions and every even layers’ number, in lower value of length scale parameter, by increasing the length scale parameter, the frequency of the structure decreases but in higher value of length scale parameter this matter is inverse. Finally, influences of temperature difference, ply angle, length scale and nonlocal parameters on the critical temperature and frequency of the laminated composite nanostructure are investigated.

Thermoelastic Damping of Functionally Graded Material Micro-Beam Resonators Based on the Modified Couple Stress Theory

Thermoelastic damping (TED) is one of the main internal energy dissipation mechanisms in micro-/nano-resonators. Accurate evaluation of TED is important in the design of micro-electromechanical systems and nano-electromechanical systems. In this paper, a theoretical analysis on the TED in functionally graded material (FGM) micro-beam resonators is presented. Equations of motion and the heat conduction equation governing the thermodynamic coupling free vibration of non-homogenous micro-beams are established based on the Euler–Bernoulli beam theory associated with the modified couple stress theory. Material properties of the FGM micro-beam are assumed to change in the depth direction as power-law functions. The layer-wise homogenization method is used for solving the heat conduction equation. By using the mathematical similarity of eigenvalue problem between the FGM beam and the reference homogeneous one, the complex natural frequency including TED is expressed in terms of the natural frequency of the isothermal homogenous beam. In the presented numerical results, influences of various characteristic parameters, such as beam thickness, material gradient index, structure size, vibration mode and boundary conditions, on TED are examined in detail. It shows that TED decreases with the increases in the values of length scale parameters because the latter lead to the increase in structural stiffness.

The effect of multidimensional temperature distribution on the vibrational characteristics of a size-dependent thick bi-directional functionally graded microplate

This article develops the modified couple stress theory to study the free vibration of bi-directional functionally graded microplates subjected to multidimensional temperature distribution. Third-order shear deformation and classical theories of plates are adapted for free vibration analysis of thick and thin microplates, respectively. Employing the third-order shear deformation theory, both normal and shear deformations are considered without the need for shear correction factor. Material of the bi-directional functionally graded microplate is graded smoothly through the length and thickness of the microplate. Gradient of the material is assumed to obey from the power law in terms of the volume fraction of the constituents. Assuming the uniform and nonuniform temperature distributions, the effect of thermal environment on dynamic behavior of the microplate is discussed in detail. Applying the Ritz method, the displacement field is expanded by admissible functions which satisfy the essential boundary conditions, and Hamilton principle is employed to determine the natural frequencies of the microplate. Developed model has been applied to determine the natural frequencies in problems of thin/thick, one-directional/bi-directional functionally graded, and homogeneous/nonhomogeneous microplates. Effects of parameters such as the thermal environment, power law indexes [Formula: see text] and length scale parameter on free vibration of these problems are studied in detail. The results show that higher values of length scale parameter and temperature rise decrease the natural frequency of the bi-directional functionally graded microplate. According to results obtained by classical and third-order shear deformation theories, the third-order shear deformation theory is proposed for vibration analysis of microplates with thickness-to-length ratio less than five.

Dynamic characterisation of functionally graded imperfect Kirchhoff microplates

This paper aims at thorough investigation of the nonlinear forced vibration characteristics of functionally graded imperfect microplates. To this end, the modified couple stress (MCS) theory together with the Mori-Tanaka homogenisation technique is employed in order to take into account the size-dependent behaviour and the through-thickness variable material properties of the functionally graded imperfect microsystem, respectively. A geometric imperfection as a slight curvature is considered and all the out-of-plane and in-plane motions are retained in the nonlinear modelling and simulations. The nonlinear equations of motion are derived using the Lagrange equations; the large dimensional multi-degree-of-freedom truncated model is solved using a parameter-continuation method. Comprehensive investigations into the effect of material gradient parameter as well as the geometric imperfection on the dynamic characteristics of the functionally graded imperfect microplate are performed through extensive numerical simulations. Finally, the importance of incorporating small-size influences, via the MCS theory, is emphasized comparing the vibrations of the microsystem for different values of length-scale parameter.

A most general strain gradient plate formulation for size-dependent geometrically nonlinear free vibration analysis of functionally graded shear deformable rectangular microplates

In this work, the size-dependent geometrically nonlinear free vibration of functionally graded (FG) microplates is investigated. For this purpose, with the aid of Hamilton’s principle, a nonclassical rectangular microplate model is developed based on Mindlin’s strain gradient theory, Mindlin’s plate theory and the von Karman geometric nonlinearity. For some specific values of the length scale material parameters, the simple form of size-dependent mathematical formulation based on the modified strain gradient theory (MSGT) and modified couple stress theory is obtained. The generalized differential quadrature method, numerical Galerkin scheme, periodic time differential operators and pseudo arc-length continuation method are utilized to determine the geometrically nonlinear free vibration characteristics of FG microplates with different boundary conditions. The parametric effects of thickness-to-material length scale ratio, material gradient index, length-to-thickness ratio, length-to-width ratio and boundary conditions on the nonlinear free vibration characteristics of FG microplates are studied through various numerical examples presented. It is found that a considerable difference exists between the results of various elasticity theories at small values of length scale parameter. A more precise prediction can be provided by using the size-dependent plate model based on the MSGT.

Torsional Vibration Analysis of Carbon Nanotubes Based on the Strain Gradient Theory and Molecular Dynamic Simulations

In the current study, the torsional vibration of carbon nanotubes is examined using the strain gradient theory and molecular dynamic simulations. The model developed based on this gradient theory enables us to interpret size effect through introducing material length scale parameters. The model accommodates the modified couple stress and classical models when two or all material length scale parameters are set to zero, respectively. Using Hamilton's principle, the governing equation and higher-order boundary conditions of carbon nanotubes are obtained. The generalized differential quadrature method is utilized to discretize the governing differential equation of the present model along with two boundary conditions. Then, molecular dynamic simulations are performed for a series of carbon nanotubes with different aspect ratios and boundary conditions, the results of which are matched with those of the present strain gradient model to extract the appropriate value of the length scale parameter. It is found that the present model with properly calibrated value of length scale parameter has a good capability to predict the torsional vibration behavior of carbon nanotubes.

Size-dependent bending, buckling and free vibration of functionally graded Timoshenko microbeams based on the most general strain gradient theory

In the present investigation, the bending, buckling and free vibration responses of Timoshenko microbeams made of functionally graded materials (FGMs) are studied. To take size effect into account, the most general strain gradient elasticity theory is incorporated into the classical Timoshenko beam theory to develop a size-dependent beam model containing five additional material length scale parameters. The model accommodates the beam models based on the strain gradient theory (SGT), the modified strain gradient theory (MSGT), the modified couple stress theory (MCST) and the classical theory (CT) as special cases. By using Hamilton’s principle, the governing equations and corresponding boundary conditions are derived. Afterward, the governing equations and associated boundary conditions are discretized by employing generalized differential quadrature (GDQ) method. Selected numerical results are given to demonstrate the size-dependent mechanical characteristics of FGM microbeams. Moreover, a comparison between the various beam models on the basis of MCST, MSGT and CT are presented. It is observed that the critical buckling loads and natural frequencies predicted by the beam models based on MSGT and CT are the maximum and minimum values, respectively. By increasing the value of length scale parameter, the deflection curve of FGM microbeam tends to the curve obtained by CT.

Thermal buckling analysis of rectangular microplates using higher continuity p-version finite element method

In this article, thermal buckling characteristics of rectangular flexural microplates (FMP) subjected to uniform temperature are investigated using higher continuity p-version finite element framework. Invariant form of the governing equation for a microplate with non-local effects based on “modified couple stress theory” is extended for thermal buckling analysis of FMP by considering the strain gradient effects. In this case, the constitutive equation for strain gradient model is based on one constant. Galerkin weak form of the governing equation is derived and subsequently solved for a variety of boundary conditions using higher continuity p-version finite elements to extract critical thermal buckling loads. The computational procedure is verified by comparing its predictions to those of the classical theory and analytic microplate studies that are based on the same strain gradient model. Investigations indicate that length scale parameter affects the computed flexural stiffness of a plate, and the effect is directly proportional to the value of gradient coefficient considered for that plate. Hence, there is a strong influence of length scale parameter on value of the thermal buckling load. Depending on boundary conditions and value of length scale parameter used in numerical experiments, the classical plate model severely underestimates (up to 90%) the thermal buckling load for microplates. Therefore, it is concluded and strongly suggested that the classical plate theory should not be used to predict structural response of microplates.

Wedge indentation of thin films modelled by strain gradient plasticity

A plane strain study of wedge indentation of a thin film on a substrate is performed. The film is modelled with the strain gradient plasticity theory by Gudmundson [Gudmundson, P., 2004. A unified treatment of strain gradient plasticity. Journal of the Mechanics and Physics of Solids 52, 1379–1406] and analysed using finite element simulations. Several trends that have been experimentally observed elsewhere are captured in the predictions of the mechanical behaviour of the thin film. Such trends include increased hardness at shallow depths due to gradient effects as well as increased hardness at larger depths due to the influence of the substrate. In between, a plateau is found which is observed to scale linearly with the material length scale parameter. It is shown that the degree of hardening of the material has a strong influence on the substrate effect, where a high hardening modulus gives a larger impact on this effect. Furthermore, pile-up deformation dominated by plasticity at small values of the internal length scale parameter is turned into sink-in deformation where plasticity is suppressed for larger values of the length scale parameter. Finally, it is demonstrated that the effect of substrate compliance has a significant effect on the hardness predictions if the effective stiffness of the substrate is of the same order as the stiffness of the film.