Internal Material Length Scale Parameter Research Articles

Application of Modified Couple-Stress Theory to Nonlinear Vibration Analysis of Nanobeam with Different Boundary Conditions

PurposeIn the present study, the nonlinear vibration analysis of a nanoscale beam with different boundary conditions named as simply supported, clamped-clamped, clamped-simple and clamped-free are investigated numerically.MethodsNanoscale beam is considered as Euler-Bernoulli beam model having size-dependent. This non-classical nanobeam model has a size dependent incorporated with the material length scale parameter. The equation of motion of the system and the related boundary conditions are derived using the modified couple stress theory and employing Hamilton’s principle. Multiple scale method is used to obtain the approximate analytical solution.ResultNumerical results by considering the effect of the ratio of beam height to the internal material length scale parameter, h/l and with and without the Poisson effect, υ are graphically presented and tabulated.ConclusionWe remark that small size effect and poisson effect have a considerable effect on the linear fundamental frequency and the vibration amplitude. In order to show the accuracy of the results obtained, comparison study is also performed with existing studies in the literature.

Non-linear vibration analysis of specially orthotropic tapered micro-plates with arbitrary located crack: A non-classical analytical approach

The present work aims to study the non-linear vibrations in a cracked orthotropic tapered micro-plate. Linear and parabolic variation in the plate thickness is assumed in one as well as two directions. The partial crack is located in the centre, and it is continuous; this crack’s location is arbitrary and can be varied within the centre-line. Based on classical plate theory, the equilibrium principle is applied, and the governing equation of tapered orthotropic plate is derived. Additionally, the microstructure’s effect has been included in the governing equation using the non-classical modified couple stress theory. The simplified line spring model is used to consider the impact of partial crack on the plate dynamics and is incorporated using in-plane forces and bending moments. The introduction of Berger’s formulation brings the non-linearity in the model in terms of in-plane forces. Here, Galerkin’s method has been chosen for converting the derived governing equation into time-dependent modal coordinates, which uses an approximate solution technique to solve the non-linear Duffing equation. The crack is considered along the fibres and across the fibres to show the effect of orthotropy. Results are presented for an orthotropic cracked plate with non-uniform thickness. The effects of the variation of taper constants, crack location, crack length, internal material length scale parameter on the fundamental frequency are obtained for two different boundary conditions. The non-linear frequency response curves are plotted to show the effect of non-linearity on the system dynamics using the method of multiple scales, and the contribution of taper constants and crack parameters on non-linearity is shown with bending-hardening and bending-softening phenomenon. It has been found that vibration characteristics are affected by the taper parameters and fibre direction for a cracked orthotropic plate.

An effective analytical approach to nonlinear free vibration of elastically actuated microtubes

Based on Hamiltonian principle and a modified couple stress theory, a nonlinear differential equation of motion of electrostatically actuated microbeams is presented. The method of Galerkin–Bubnov decomposition is employed to convert the established system into reduced discrete modal equation. The analytical approximate solution to nonlinear free vibration is obtained by means of the optimal auxiliary functions method. The influence of internal material length scale parameter and outer diameter on the dynamic behavior is considered. The proposed procedure is effective, convenient and does not require linearization or small parameters assumption. The main advantage of this technique consists in that it provides a convenient way to control the convergence of the approximate solution in a very rigorous way.

Fluid-Induced Nonlinear Vibration of a Cantilevered Microtube with Symmetric Motion Constraints

This paper investigates the dynamic behavior of a cantilevered microtube conveying fluid, undergoing large motions and subjected to motion-limiting constraints. Based on the modified couple stress theory and the von Kármán relationship, the strain energy of the microtube can be deduced and then the governing equation of motion is derived by using the Hamilton principle. The Galerkin method is applied to produce a set of ordinary differential equations. The effect of the internal material length scale parameter on the critical flow velocity is investigated. By using the projection method, the Hopf bifurcation is demonstrated. The results show that size effect on the vibration properties is significant.

Analytical Approximate Solution for Nonlinear Behavior of Cantilever FGM MEMS Beam with Thermal and Size Dependency

This paper is concerned with the nonlinear static behavior of a cantilever functionally graded material (FGM) microbeam with the influence of thermal stress and the intermolecular force. A significant extension of the recent works of constructing analytical approximate solutions to a clamped-clamped FGM MEMS/NEMS beam is formed in the paper. Based on the modified couple stress theory and an internal material length-scale parameter, the governing equations account for the microbeam size dependency. Note that the thermal force and moment in boundary condition make procedure of solution more complex. The combinations of Galerkin method and the assumption of deflection function are used to establish analytical approximate solutions which have brief expressions. Good agreements of approximate results are found for large range of free-end deflection of cantilever FGM microbeam through comparing them with numerical solutions and other existing results. The influence of various physical parameters, such as the material attributes (Young modulus, Poisson ratio, etc.), the electrostatic gap, and microsize of the beam on mechanical behavior, or Pull-In voltage of electrostatically actuated microbeams, could also be investigated with these analytical expressions.

Effect of Thermal Environment on Vibration Analysis of Partially Cracked Thin Isotropic Plate Submerged in Fluid

Based on a non classical plate theory, an analytical model is proposed for the first time to analyze free vibration problem of partially cracked thin isotropic submerged plate in the presence of thermal environment. The governing equation for the cracked plate is derived using the Kirchhoff’s thin plate theory and the modified couple stress theory. The crack terms are formulated using simplified line spring model whereas the effect of thermal environment is introduced using thermal moments and in-plane forces. The influence of fluidic medium is incorporated in governing equation in form fluids forces associated with inertial effects of its surrounding fluids. Applying the Galerkin’s method, the derived governing equation of motion is reformulated into well known Duffing equation. The governing equation for cracked isotropic plate has also been solved to get central deflection which shows an important phenomenon of shift in primary resonance due to crack, temperature rise and internal material length scale parameter. To demonstrate the accuracy of the present model, few comparison studies are carried out with the published literature. The variation in natural frequency of the cracked plate with uniform rise in temperature is studied considering various parameters such as crack length, fluid level and internal material length scale parameter. Furthermore the variation of the natural frequency with plate thickness is also established.

A comprehensive study on the size-dependent hygrothermal analysis of exponentially graded microplates on elastic foundations

In this paper, the effects of hygrothermal conditions on various behaviors, such as bending, free vibration, mechanical and thermal buckling, of exponentially graded microplates lying on two-parameter elastic foundations are investigated. The trigonometric four-variable plate theory incorporated to the modified couple stress theory (MCST) is employed to derive the equations of motion. The present MCST contains an internal material length scale parameter, thus it can capture the size effect. The microplate is assumed to be subjected to a temperature rise and moisture concentration which are varied linearly through the thickness of the plate. Based on an exponential law, the material properties of the microplate are graded only in z direction. The equations of motion are solved analytically to obtain the displacements, stresses, eigenfrequencies and critical buckling load and temperature of the microplates. The present results are validated by comparing them with those previously published. The numerical examples reveal that considering the size effect and/or the elastic foundations leads to an increment in plate stiffness and thereby leads to a decrement in the deflection and an increment in eigenfrequency and buckling loads. It is also shown that the size effect is negligible for the thicker plate.

Effect of crack location on vibration analysis of partially cracked isotropic and FGM micro-plate with non-uniform thickness: An analytical approach

An analytical model is presented for nonlinear vibration analysis of variable thickness, thin isotropic and functionally graded rectangular micro-plate containing a partial crack located within the centre line of the plate. Linear and parabolic thickness variation is considered in one or both the in-plane directions of the plate. The thickness variation is such that the volume of the plate is equal to that of the uniform thickness plate. The continuous line crack is parallel to one of the edges of the plate. The equations of motion are derived using the equilibrium principle based on Classical Plate Theory while the size effect is incorporated using the modified couple stress theory. The partial crack is represented by bending moment and in-plane force according to the simplified line spring model. The effect of in-plane forces is considered by employing the Berger's formulation. The derived governing equation is converted into a cubic nonlinear Duffing equation by employing Galerkin's method. The effect of nonlinearity is established by deriving the frequency response equation for the cracked variable thickness plate using the method of multiple scales. The nonlinear frequency response curves show the phenomenon of bending hardening or softening. The influence of crack length, crack location, internal material length scale parameter, boundary conditions, gradient index, unidirectional and bidirectional taper constant on the fundamental frequency of square plate is demonstrated. Similar to uniform thickness plate, it is found that the presence of crack affects the vibration characteristics of variable thickness plate. When compared to uniform thickness plate, the effect of crack can be reduced by varying the thickness of the plate. Thus, it is concluded that in a given volume, it is better to employ variable thickness plate as far as the vibration characteristics are considered.

Elastic stability of curved nanobeam based on higher-order shear deformation theory and nonlocal analysis by finite element approach

In the present work, elastic stability analysis of curved nanobeams is investigated using the differential constitutive law consequent to Eringen's strain-driven integral model coupled with a higher-order shear deformation theory accounting for through thickness stretching effect. The formulation developed here is general in the sense that it can be deduced to realise the influence of different structural theories and analyses of nanobeams. The governing equations derived are solved employing finite element method using a 3-nodes curved beam element. The model developed here is validated considering problems for which analytical/numerical solutions are available in the literature. For comparison purpose, results are also presented for various structural theories obtained from the present formulation. The influence of structural and material parameters such as thickness ratio, beam length, rise of the curved beam, boundary conditions, and size-dependent or nonlocal parameter are brought out on the buckling behaviours of curved nanobeams. It is observed that the type of buckling mode pertaining to the lowest critical value can be different depending on geometrical and internal material length scale parameter, and boundary conditions.

Size effect on cracked functional composite micro-plates by an XIGA-based effective approach

Failure of structures and their components is one of major problems in engineering. Studies on mechanical behavior of functionally graded (FG) microplates with defects or cracks by effective numerical methods are rarely reported in literature. In this paper, an effective numerical model is derived based on extended isogeometric analysis (XIGA) for assessment of vibration and buckling of FG microplates with cracks. Based on the modified couple stress theory, the non-classical theory of Reissner–Mindlin plate is extended to capture microstructure, and thus, the size effect. In such theory, possessing C1-continuity is straightforward with the high-order continuity of non-uniform rational B-spline. Due to the use of enrichments in XIGA, crack geometry is independent of the computational mesh. Numerical examples are performed to illustrate the effects of microplate aspect ratio, crack length, internal material length scale parameter, material distribution, and boundary condition on the mechanical responses of cracked FG microplates. The obtained results are compared with reference solutions and that shows that the frequency and buckling loads increases with decreasing the size of FG microplates and crack length. The convergence of the present method is also studied.

Size-dependent pull-in analysis of a composite laminated micro-beam actuated by electrostatic and piezoelectric forces: Generalized differential quadrature method

In this paper, a size-dependent model of clamped-clamped composite laminated electrostatic Euler–Bernoulli microbeams with piezoelectric layers attached was proposed based on a new modified couple stress theory for anisotropic elasticity. With the aid of Hamilton's principle, the nonlinear differential governing equation was established and solved by utilizing the generalized differential quadrature method. The present model, which can be reduced to the isotropic beam model takes the residual stresses, fringing effects and midplane stretching effects into consideration. A good agreement exists between numerical results of the present reduced model and those of available literature. Numerical analysis reveals that the size effect is significant when geometry sizes of a beam are comparable to the internal material length scale parameter (hereafter referred to as MLSP). Furthermore, the pull-in voltage dependence of the residual stress, midplane stretching, MLSP as well as the voltage applied to the piezoelectric layers are also investigated. The present model can accurately predict the static behavior and guide the design of composite laminated micro-beam actuated by both the electrostatic and piezoelectric forces.

A size-dependent zigzag model for composite laminated micro beams based on a modified couple stress theory

This study developed a new composite laminated size-dependent beam model based on a re-modified couple stress theory and a refined zigzag theory. Unlike the classical theory, the proposed model only includes one internal material length scale parameter and can capture the scale effects. In addition, the current model satisfies a priori interlaminar continuity conditions and can accurately predict the displacements and stresses, particularly the interlaminar stresses. Reissner’s Mixed Variational Theorem is employed to derive the equilibrium equations and related boundary conditions. Numerical examples of simply supported cross-ply beams subjected to transverse sinusoidal loads are analytically solved by directly applying the derived formulas. The results prove that this model can be applied to investigate the scale effects, which increase the stiffness and decrease the displacements and stresses. Moreover, the results obtained with present model are compared with the exact solutions to validate the present formulation.

Effect of thermal environment on free vibration and buckling of partially cracked isotropic and FGM micro plates based on a non classical Kirchhoff's plate theory: An analytical approach

A non classical analytical model is presented for free vibration of the cracked rectangular isotropic and functionally graded micro plates in the presence of thermal environment. The new approach is based on the combination of classical plate theory and the modified couple stress theory. The crack terms are based on the Line Spring Model whereas the thermal environment is accommodated in the form of thermal moment and in-plane forces. The solution for fundamental frequencies of the cracked plate is obtained by discounting geometric nonlinearity. A non classical relation for geometrically linear thermal buckling phenomenon of a cracked micro-plate is also proposed. The variation of critical buckling temperature and fundamental frequency with uniform rise in temperature of the cracked plate is studied for various crack length ratios, gradient index and internal material length scale parameter. Furthermore the variation of the buckling coefficient with plate aspect ratio and thickness of the cracked micro-plate is established. A comparison is presented between the classical results and present results as affected by thermal environment and microstructure of the plate. It is concluded that the presence of cracks affects the critical buckling temperature and fundamental frequencies of micro plates. The consideration of size effect increases the buckling temperature which shows the significance of the internal material length scale parameter in the presence of thermal environment.

Experimental and analytical investigations of vibrational behavior of U-shaped atomic force microscope probe considering thermal loading and the modified couple stress theory

In this study, experimental and analytical models of a U-shaped atomic force microscopic (AFM) probe are studied based on the modified couple stress theory (MCST). The experimental setup is a commercially fabricated U-shaped probe AN2-300 mounted in the atomic force microscope afm+ system. In the analytical model, the U-shaped probe of the AFM is simulated as a three-beam model (TBM). The governing equations of motion and boundary conditions are obtained by combination of basic equations of the modified couple stress theory and Hamilton’s principle. It is found that the natural frequencies of the AFM predicted by the MCST are size-dependent. The difference between the natural frequencies predicted by the MCST and the classical beam model is very significant when the ratio of characteristic size to internal material length scale parameter is approximately equal to one, but it is diminishing with increase in the ratio. In order to validate the results, the mathematical and experimental models have been compared with each other. It is seen that the analytical model based on MCST has excellent agreement with experimental results. Also, the natural frequencies of the probe have been obtained for different temperature values, aspect ratios, and small scale factor amounts. The results show, natural frequency decrease with increase in temperature value.

Size-dependent static bending of a micro-beam with a surface-mounted 0–1 polarized PbLaZrTi actuator under various boundary conditions

This article presents modified couple stress theory–based size-dependent deflection curve equations for a micro-beam with a surface-mounted 0–1 polarized PbLaZrTi patch under various boundary conditions. In these models, only an internal material length scale parameter is used to capture the size-dependent mechanical behaviors. When this parameter is set to zero, these models are simplified to the available solution based on the classical Euler–Bernoulli theory. The effect of the actuator stiffness on the deflection of the beam is thoroughly discussed. It is observed that whether the deformation of the beam is sensitive to the added actuator or not depending on the stiffness of the host beam. The larger the stiffness of the host beam, the less the effect of the added actuator. It is also observed that the less the ratio of the beam thickness to the actuator stiffness, the larger the ratio of the size-dependent solution to the local one.

Size-dependent static bending and free vibration of 0–3 polarized PLZT microcantilevers

In this paper, analytical solutions for size-dependent static bending and free vibration of a pure 0–3 polarized PbLaZrTi (PLZT) cantilever are developed. This paper also makes the first attempt to investigate the static bending of a cantilever metal beam bonded with discretized 0–3 polarized PLZT actuator based on the modified couple stress theory and composite laminated beam theory. These models involve an internal material length scale parameter used to capture the size effect. In the limit when the internal material length scale parameter goes to zero, this model reduces to classical (local) solutions available in the literature. Exact solutions for the normalized static deflection are obtained as a function of the actuator thickness and the internal material length scale parameter. The simulations show that the size-dependent results developed by the present models have a remarkable difference with those got by the classical solutions when the ratio of the actuator thickness to the internal material length scale parameter is small. It is also observed that an increase in the stiffness parameter of the substrate beam gives rise to an increase in the effect of the material length scale parameter on tip deflections of the cantilever metal beam.

Size-dependent constituent equations of piezoelectric bimorphs

In this paper, two size-dependent constituent equations for symmetrical/heterogeneous piezoelectric bimorph actuators are developed using the modified couple stress theory and the principle of minimum total potential energy. The present model involves an internal material length scale parameter used to capture the size effect. In the limit when the internal material length scale parameter goes to zero, this model reduces to classical (local) constituent equations available in the literatures. Exact solutions for the normalized static deflection are obtained as a function of the ratio of the actuator thickness to the internal material length scale parameter. The simulations show that the static deflection developed by the present model has a remarkable difference with those got by the classical constituent equations when the ratio of the beam thickness to the internal material length scale parameter is small. It is also observed that the stiffness of elastic layers show directionally opposite effects on tip deflections of symmetrical piezoelectric bimorph actuator and heterogeneous one.

Effect of fibre orientation on non-linear vibration of partially cracked thin rectangular orthotropic micro plate: An analytical approach

A nonlinear analytical model for vibration analysis of partially cracked thin orthotropic plate as affected by the fibre orientation and arbitrary position of a partial crack is presented. The approach is based on the classical plate theory and the effect of microstructure is incorporated using the modified couple stress theory. A single partial crack located at an arbitrary position along the line of symmetry is parallel to one of the edges of the plate. The crack considered is in the form of a continuous line and its effect is incorporated in the form of moment and in-plane forces based on a simplified line spring model. Non-linearity in the model is introduced by Berger׳s formulation for the in-plane forces while the Galerkin׳s method with modal functions is used to obtaining the solution of the governing differential equation. A nonlinear frequency response relation and peak amplitude is obtained by the application of the method of multiple scales. Extensive results for the fundamental frequencies and peak amplitude as affected by the fibre orientation, plate aspect ratio, crack length, crack position and internal material length scale parameter are presented for two boundary conditions. It is found that the fibre orientation and the microstructure of the plate affect the fundamental frequencies of the partially cracked orthotropic plates. The linear and nonlinear frequency response curves based on the classical plate theory and the modified couple stress theory are plotted which shows the effect of internal material length scale parameter and crack position on the bending hardening or softening phenomenon.

Effect of microstructure on vibration characteristics of partially cracked rectangular plates based on a modified couple stress theory

An analytical model for vibration analysis of partially cracked isotropic and functionally graded rectangular plates is presented based on a modified couple stress theory. A single internal material length scale parameter capturing the microstructure is incorporated in the equation of motion of cracked plates derived using the equilibrium principle. Two crack configurations (i) a single partial continuous line crack and (ii) two perpendicular continuous line cracks located at the centre of the plate are considered for analysis. The effect of the single and the two perpendicular partial cracks is incorporated in the form of moment and in-plane forces based on a simplified line spring model. Berger׳s formulation for the in-plane forces makes the model nonlinear and Galerkin׳s method is used for its solution. Results for the fundamental frequencies as affected by plate aspect ratio, crack length, internal material length scale parameter and gradient index are presented for various boundary conditions. It is found that the microstructure of the plate affects the fundamental frequencies.

Axial Buckling and Dynamic Stability of Functionally Graded Microshells Based on the Modified Couple Stress Theory

A nonclassical first-order shear deformation shell model is developed to analyze the axial buckling and dynamic stability of microshells made of functionally graded materials (FGMs). For this purpose, the modified couple stress elasticity theory is implemented into the first-order shear deformation shell theory. Unlike the classical shell theory, the newly developed shell model contains an internal material length scale parameter to capture efficiently the size effect. By using the Hamilton's principle, the higher-order governing equations and boundary conditions are derived. Afterward, the Navier solution is utilized to predict the critical axial buckling loads of simply-supported functionally graded (FG) microshells. Moreover, the governing equations are written in the form of Mathieu–Hill equations and then Bolotin's method is employed to determine the instability regions. A parametric study is conducted to investigate the influences of static load factor, axial wave number, dimensionless length scale parameter, material property gradient index, length-to-radius and length-to-thickness aspect ratios on the axial buckling and dynamic stability responses of FGM microshells. It is revealed that size effect plays an important role in the value of critical axial buckling load and instability region of FGM microshells especially corresponding to those with lower aspect ratios.