Modified Couple Stress Theory Research Articles

Dynamic Stability Analysis of Size-Dependent Viscoelastic/Piezoelectric Nano-Beam

This paper analyzes the dynamic stability of an isotropic viscoelastic Euler–Bernoulli nano-beam using piezoelectric materials. For this purpose, the size-dependent theory was used in the framework of the modified couple stress theory (MCST) for piezoelectric materials. In order to capture the geometrical nonlinearity, the von Karman strain displacement relation was applied. Hamilton’s principle was also employed to obtain the governing equations. Furthermore, the Galerkin method was used in order to convert the governing partial differential equations (PDEs) to a nonlinear second-order ordinary differential one. Dynamic stability analysis was performed and the effects of such parameters as viscoelastic coefficients, size effect, and piezoelectric coefficient were investigated. The results showed that in this system, saddle points, central points, Hopf bifurcation points, and fork bifurcation points could be created, and the phase portraits connecting these equilibrium points exhibit periodic orbits, heteroclinic orbits, and homoclinic orbits.

Analytical investigation of the refined zigzag theory for electro-magneto vibration response of the viscoelastic FG-GPLRC sandwich microplates

Vibration behaviors of the viscoelastic sandwich microplates with graphene nanoplatelets (GPLs)-reinforced/Epoxy layer as core and sensor/actuator layers as piezoelectric face sheets are checked in this study. Small scale effects on the structure are computed by nonlocal elasticity modified couple stress theory (MCST). The piezoelectric layers and the core are exposed to the electromagnetic field. Modified Halpin-Tsai model and rule of mixtures (ROM) are assumed to obtain mechanical properties due to functionally graded graphene nanoplatelets (FG-GPLs)’s distributions as uniform, parabolic and linear. The viscoelastic sandwich piezoelectric microplates are embedded in orthotropic Visco Pasternak medium. In addition, the Kelvin-Voigt model is employed to calculate the viscoelastic properties of the structure. The equations of motion are derived using Hamilton’s principle based on refined zigzag theory (RZT). The effects of some remarkable parameters such as geometrical dimensions of the system, magnetic field, external voltage, small scale parameter and etc. are used to investigate frequency and damping of the system. Furthermore, decrement in system frequency is indicated when external voltages grow from to It is expected that this research will be able to address the challenges ahead in smart equipment and devices, and with control capability, it also can be used in the high-tech parts.

Dynamic stability analysis of porous functionally graded microplates using a refined isogeometric approach

This paper is devoted to analyzing dynamic stability behaviors of porous functionally graded (FG) microplates subjected to periodic uniaxial/biaxial in-plane compressions by utilizing a refined plate theory (RPT) in conjunction with modified couple stress theory (MCST) within isogeometric analysis (IGA) framework. In this setting, the displacement field of microplates is expressed based on four-variable RPT-based IGA, while the effects of microstructure could be thoroughly captured by employing MCST involving only one material length scale parameter. To define dynamic instability region boundaries, Bolotin’s method is adopted in the present work. Three different dispersions of internal porosities i.e. even, uneven, and logarithmic uneven through the thickness direction are investigated in detail. In addition, porosity-dependent material properties are estimated via a modified rule of mixture. A number of numerical investigations are carried out to study, for the first time, the influences of several key factors such as porosity coefficient and porosity distribution, material length scale, power law index, static and dynamic load factors on dynamic instability characteristics of FG microplates considering internal porosity.

Electro-magnetic vibration behavior of smart curved micro-/nanoshells

This paper investigates dynamic results of a small-scale-dependent panel subjected to a combination of mechanical, electrical, and magnetic loads using the shear defamation theory. Modified couple stress theory (MCST) is used for accounting small-scale effects in small sizes. Governing equations of motion are derived based on Hamilton’s principle in the cylindrical coordinate system. The micro panel is composed of an elastic core integrated with two piezoelectric/piezomagnetic layers at top and bottom. Microlength scale parameter is included in the couple stress tensor and symmetric curvature tensor to capture size dependency. The analytical solution is applied for the solution of the motion equations. The extended parametric results are provided to show natural frequencies in terms of main significant input parameters such as initial electric/magnetic loads, span angle, length to thickness, radius to thickness, and length to radius ratio. Furthermore, the effect of higher mode numbers along the axial and circumferential directions is studied on the responses. The numerical results indicate that the natural frequencies are increased with an increase in initial magnetic potential and a decrease in initial electric potential.

Size-Dependent Stability of a Cantilevered Piezoelectrically Actuated Micropipe Conveying Fluid

Size-dependent effects of a cantilevered piezoelectrically actuated micropipe conveying fluid are investigated. Based on the modified strain gradient beam theory, the model of system is obtained using Hamilton's principle. The motion equation is discretized into ordinary differential equations by Generalized Differential Quadrature Method (GDQM). A stability analysis of the system is completed through eigenvalue analysis. Numerical results show the effect of geometrical shape size, and length scale parameters on critical flow velocity, and critical voltage. Results prove that the modified strain gradient theory (MSGT) has a higher critical flow velocity and critical voltage than predicted by modified couple stress theory (MCST) and classical theory (CT).

Static stability and free vibration characteristics of a micro laminated beam under varying axial load using modified couple stress theory and Ritz method

This paper explores the buckling and free vibration behaviors of micro laminated composite beams (MLCBs) exposed to varying axial loads using the modified couple stress theory (MCST), with arbitrary boundary conditions and layups. The size effect is captured in MCST by taking into account the material length scale parameters. In the axial direction, the applied load has either constant, linear, or parabolic variations. The equilibrium equations are derived and solved using Hamilton’s principle and Ritz method respectively. The size effect is observed to be more pronounced when the thickness of the beam is similar to the material length scale parameter, and it nearly vanishes as the beam thickness increases. The small size effect variation is different for different boundary conditions. For parabolically increasing loading case, the rate of reduction of the fundamental frequency with an increase in applied load intensity is greater compared to other loadings.

Modified couple stress-based nonlinear static bending and transient responses of size-dependent sandwich microplates with graphene nanocomposite and porous layers

This paper investigates the nonlinear static bending and dynamic transient responses of the rectangular and circular sandwich microplates composed of functionally graded graphene nanocomposite (GNC) face sheets and a porous polymeric core layer based on the modified couple stress theory (MCST) within the isogeometric analysis (IGA) framework. The four-variable higher-order shear deformation refined plate theory (RPT) and the von Kármán large deflection assumption are synthetically employed to describe the kinematics of the microplate. The nonlinear governing equations of motion are derived from the Hamilton’s principle. The nonlinear bending problem is solved by the Newton–Raphson technique under a displacement criterion strategy, while the dynamic responses under transient loads are obtained by the Newmark integration scheme in conjunction with Newton–Raphson iterative procedure. Comparisons are carried out to test the accuracy of iteration procedure in present formulation and the reliability in capturing small scale effect of present model. Obtained outcomes reveal that raising the material length scale (MLS) parameter and volume fraction index of graphene nanofillers both leads to reductions in nonlinear static deflections, dynamic amplitudes of deflection and periods of motion. The examples for porosity coefficient show that embedding pores in core layer leads to slight changes in terms of static and dynamic responses and an improvement in weight reduction of the sandwich microplate.

Free vibration analysis of a cylindrical micro-shell based on MCST

Dynamic analysis of a functionally-graded (FG) cylindrical micro shell is investigated in this work. Effect of small parameter in micro scale is included in fundamental relations based on modified couple stress theory (MCST). For more precise analyzing the structure, third-order shear deformation theory (TSDT) is employed as kinematic relations. The shell is made from FG materials, using power function law. A solution procedure based on an analytical method is developed for special boundary condition. The numerical results are computed in graphical form to seek how changes of main parameters of the study such as micro parameter, in-homogeneity index and some geometric values change the free vibration responses. As an important output of this paper, it is concluded that an increase in micro parameter leads to increase of natural frequency. Furthermore, increase of in-homogeneous index yields more soft structure and then decrease of natural frequency.

Finite element formulation for nano‐scaled beam elements

AbstractIn the present study, size‐dependent buckling and free vibration behaviors of single‐walled boron nitride nanotube (SWBNNT) are performed in conjunction with various size‐dependent elasticity theories. Modified couple stress theory (MCST) and Eringen's nonlocal elasticity theory are used for size‐dependent models of SWBNNT. Also, the buckling loads and frequencies are obtained by using local theory to emphasize the effects and differences of these size‐dependent theories. Consequently, three different elasticity theories (two non‐classical and one classical) are utilized to achieve the detailed buckling and vibration analyses of SWBNNT. In this study, the buckling loads and frequencies of SWBNNTs are obtained via presented finite element formulation. In the finite element procedures based on two different size‐dependent elasticity theories, matrices containing the small size parameter are derived. With these matrices containing the small size parameters, eigenvalue problems for buckling and free vibration analyses are formed. The buckling loads and frequency values of the SWBNNTs under the size effect are obtained. The influences of the dimensionless nonlocal parameter, dimensionless material length scale parameter, length‐to‐diameter ratio and boundary conditions on nanotube's buckling and vibration characteristics are investigated. In addition to these influences, the rotary inertia effect neglected in many other studies is also examined.

Thermoelastic damping analysis in micro-beam resonators in the frame of modified couple stress and Moore–Gibson–Thompson (MGT) thermoelasticity theories

The thermoelastic coupling between strain and temperature fields arising out from temperature gradient, while bending of an oscillating structure, leads to irreversible heat conduction, which causes energy dissipation in such vibrating systems. The estimation of thermoelastic damping (TED) in micro- and nano-mechanical resonators is a significant aspect of acquiring a high quality factor (Q-factor). Yang et al. [Couple stress based strain gradient theory for elasticity. Int J Solids Struct. 2002;39(10):2731–2743.] developed the modified couple stress theory (MCST) that involves only one material length scale parameter. The present work attempts to investigate TED in micro/nano-beam resonators utilizing MCST and the recently proposed Moore–Gibson–Thompson (MGT) generalized thermoelasticity theory. Adopting the frequency approach method, we derive the size-dependent expression of the inverse quality factor for evaluating TED in rectangular micro/nano-beam resonators. Furthermore, the impact of the material length scale parameter on TED associated with MCST for silicon micro-beam resonator is investigated in detail by comparing the present results with those predicted by classical continuum theory (CCT). The obtained results are also compared with the existing results for the Lord–Shulman (LS) and Green–Naghdi (GN-III) thermoelasticity theories. Moreover, the effects of other important parameters like beam thickness, beam length, aspect ratio, phase-lag on TED in the present context are highlighted.

Static bending of functionally graded single-walled carbon nanotube conjunction with modified couple stress theory

This paper focused on studying the transeverse deflection behavior of three functionally graded microbeam models: material, porous material, and functionally graded single-walled carbon nanotube-reinforced composite (FG-SWCNTRC) microb beam. To calculate micro size effects of the non-classical beam model, the modified couple stress theory (MCST) uses just length material scale parameters. A numerical solution to the static deflection equation is the Lagrange multiplier method. The characteristics studied included length, material parameter ratio, volume fraction of material, porosity and carbon nanotube, SWCNT distribution types, boundary conditions, and aspect ratio (length/thickness). The static behavior of FG-micro beams demonstrates that the theory of modified couple stress (MCST) produces more accurate results than classical beams. This is especially true if the beam thickness is close to the length scale parameter. As the CNT volume fraction grows and the porosity volume fraction falls, the microbeam FG structure deflects less.

Coupled stress based formulation for static and dynamic analyses of a higher-order shear and normal deformable FG-GPL reinforced microplates

In this paper, size-dependent static bending and also free and forced vibration analyses of composite microplates are studied based on the modified couple stress theory (MCST) and quasi-3D sinusoidal shear deformation theory. The composite microplate is composed of epoxy reinforced with functionally graded graphene nanoplatelets (GPLs). The governing equations are derived utilizing Hamilton's principle and are solved for simply supported microplate using Navier's approach. The accuracy of the presented solution is confirmed and the effects of various parameters on the static and dynamic deflections and natural frequencies of the microplate are investigated including material length scale parameter and surface area, thickness, mass fraction, and distribution pattern of the GPLs. Numerical results confirm that subjoining the GPLs to the polymer reduces the static and dynamic deflections of the microplates and increases the natural frequencies. It is concluded that to achieve the highest reduction in the static and dynamic deflections and the highest growth in the natural frequencies, it is more helpful to use the GPLs with the larger surface areas and put them as far as away from the neutral surface of the microplate.

Computer modeling via a numerical method for the cross-section effect on the forced vibration of FG nonuniform and imperfect microtubes

This paper examines the nonlinear forced vibrational behaviors of axially functionally graded (AFG) cylindrical microscale tubes under external dynamic stress, including the imperfect porosity. Using the Hamilton principle, the first-order shear deformation beam theory is linked with the modified couple stress theory (MCST) and the nonlinear von-Kármán theory to get the nonlinear governing equations and associated boundary conditions. For cross-sections that vary throughout the tube length, both uniform and non-uniform functions involving the three kinds are assumed. Furthermore, the microtube is constructed of functionally graded materials that vary smoothly throughout the tube length, and the porosity distribution is in the radial direction, resulting in two-dimensional functionally graded (2D-FG) structures with both porosity and material distributions. The nonlinear results are obtained after solving the linear motion equations using the generalized differential quadrature technique (GDQM) and the iteration approach. Finally, the effects of several factors on the natural and resonance frequencies are investigated, including the cross-section, FG parameter, porosity parameter, amplitude, external excitation frequency, and small-scale effect.

A comprehensive vibration analysis of rotating truncated sandwich conical microshells including porous core and GPL-reinforced face-sheets

The first-order shear deformation theory (FSDT) and modified couple stress theory (MCST) are employed in this paper to investigate the free vibrational behavior of rotating truncated conical sandwich microshells. It is assumed that the sandwich microshell is made of polymer, the core is porous, and two face sheets are reinforced with graphene nanoplatelets (GPLs). To compute the effective mechanical properties of the face sheets, the rule of mixture and Halpin-Tsai model are utilized. The equations of motion are derived using Hamilton’s principle incorporating the centrifugal acceleration, the Coriolis acceleration, and the initial hoop tension. To solve the governing equations in the circumferential and meridional directions, a trigonimetric functions and the differential quadrature method (DQM) are utilized, respectively. Accuracy and convergence of the presented solution are verified, and the influence of various parameters on the backward and forward frequencies of the rotating truncated conical sandwich microshells are investigated, including different boundary conditions, geometrical characteristics of the microshell, rotational speed, material length scale parameter, porosity parameter, distribution pattern of the pores, and also mass fraction, dispersion pattern and geometrical characteristics of the GPLs (thickness-to-length and aspect ratios).

Thermoelastic responses in rotating nanobeams with variable physical properties due to periodic pulse heating

Recently, many researchers have been interested in the so-called theories of nonlocal elasticity (NET) and modified couple stress (MCS). The effect of size and dependence on the micron scale was predicted using this theory. A new set of equations was created that governs this theory including the equation for a couple of moments and the use of the concept of couple stress elasticity. Also, the generalized thermoelastic model with phase lag is used. The present work concentrated on introducing thermoelastic rotating nanobeams structural analysis under periodic pulse heating by applying the non-local MCS theory and generalized thermoelastic theory. The thermal conductivity of the material is considered to be variable and linearly dependent on changes in temperature. Laplace transform and state space methods are used to obtain the physical fields of the nanobeam. The effects of size and non-local factors, variability of thermal conductivity, centrifugal force due to rotation and couple stress on different field variables are visually presented and examined in depth.

Hygrothermal environment effect on the critical buckling load of FGP microbeams with initial curvature integrated by CNT-reinforced skins considering the influence of thickness stretching

Abstract Due to the need for structures with refined properties to bear against different loading conditions, recently, carbon nanotubes (CNTs) have been used widely to reinforce them. The extremely high stiffness of CNTs makes them significant as one of the best reinforcements to improve the mechanical behaviors of structures. This work focuses on microbeam buckling response with an initial curvature that includes three layers. The mid-layer that is known as the core is constituted of functionally graded porous (FGP) materials and two CNT-reinforced composite skins are bonded to the core to integrate it. The whole structure is affected by the hygrothermal environment and springs and shear layers are put below it. For the first time, for such a structure, a refined shear deformation theory (RSDT) as a higher-order theory that considers thickness stretching effect in polar coordinates is used that presents more accurate results, especially for deeply curved beams. Modified couple stress theory (MCST) in combination with the virtual displacement principle is utilized to establish the governing equations. The obtained results demonstrate the significance of porosity percentage and CNTs’ addition to the skins on the critical nanotubes buckling load. Also, the different behaviors of the microstructure at various temperatures are analyzed and discussed in detail.

Effect of applied electric potential and micro length scale parameters on the electroelastic analysis of three-layered shear deformable micro-shell

This paper uses higher-order shear deformation theory and modified couple stress theory (MCST) to the electroelastic results of FG micro-shell integrated with piezoelectric thin sheets subjected to electrical and mechanical loads rested on Pasternak's foundation. Third-order shear deformation theory (TSDT) is used for the description of the displacement field. Effect of micro-size is applied using MCST with the introduction of one micro-length scale parameter. Governing equations are derived based on the principle of virtual work. Micro-shell is composed of a FG micro core and two piezoelectric hollow shells. The numerical results are obtained for the simply-supported boundary conditions. Longitudinal and radial displacements are presented in terms of important parameters such as applied electric potentials, micro length scale parameter, dimensionless geometric parameters and two parameters of Pasternak's foundation.

Nonlinear vibration of fractional viscoelastic micro-beams

Nonlinear vibration of a fractional viscoelastic micro-beam is investigated in this paper. The Euler–Bernoulli beam theory and the nonlinear von Kármán strain are used to model the beam. The small-scale effects are considered by employing the Modified Couple Stress Theory (MCST). The viscoelastic material of the beam is modeled via the fractional Kelvin–Voigt model. Utilizing the Hamilton’s Principle, a partial fractional differential equation is derived as the governing equation of motion. The Finite Difference Method (FDM) and the Galerkin method are used together for solving the partial fractional differential equation. The FDM is utilized to discretize the time domain, and the Galerkin method is employed to discretize the space domain. In this paper, the FDM and the Shooting method are coupled together to find the periodic solution of the fractional micro-beam and draw the corresponding amplitude–frequency curve. The effects of the order of the fractional derivative, viscoelastic model, and the micro-scale are studied numerically here in this study. Numerical simulations suggest that, the effect of the fractional derivative is very strong and must be considered for modeling the viscoelastic behavior; especially when the amplitude is high. Results also show that the effects of the nonlinear viscoelastic part are considerable when the amplitude is high; this may happen when the excitation frequency is near the natural frequency, at which the maximum amplitude occurs.

Nonlinear Static Analysis of a Bi-directional Functionally Graded Microbeam Based on a Nonlinear Elastic Foundation Using Modified Couple Stress Theory

Nonlinear Static Analysis of a Bi-directional Functionally Graded Microbeam Based on a Nonlinear Elastic Foundation Using Modified Couple Stress Theory

Mechanically induced electric and magnetic fields in the bending and symmetric-shear deformations of a microstructure-dependent FG-MEE composite beam

A new FG-MEE composite microbeam model is developed using a general higher-order deformation theory (GHDT) to account for the symmetric thickness-shear and thickness-stretch deformations of a beam and a modified couple stress theory (MCST) to describe the microstructure-dependent size effect. Based on the basic assumption of field variables and Hamilton’s principle, the one-dimensional second-order coupled equations of motion and complete boundary conditions are obtained simultaneously. The present coupled equations include the microstructure dependence and higher-order extensions, which can reduce to the size-dependent Bernoulli-Euler, Timoshenko and Mindlin-Medick theories. Static problems of a simply supported beam loaded by uniform loading and a concentrated second-order moment are analytically solved by directly applying the newly developed equations. For the bending case, the numerical results show that the magnitudes of displacement, electric, magnetic, stress, electric displacement and magnetic flux fields are smaller than those predicted by classical theory. In addition, it is found that FG parameter can be used to control the electric and magnetic fields. For the concentrated second-order moment case, a new phenomenon of local electric and magnetic fields are demonstrated in the present FG-MEE composite microbeam structure. These results are useful for theoretically/numerically estimating or designing the MEMS of FG-MEE composites.