scholarly journals A Modified Couple Stress Elasticity for Non-Uniform Composite Laminated Beams Based on the Ritz Formulation

Molecules ◽  
2020 ◽  
Vol 25 (6) ◽  
pp. 1404 ◽  
Author(s):  
Farajollah Zare Jouneghani ◽  
Hamidraza Babamoradi ◽  
Rossana Dimitri ◽  
Francesco Tornabene
Keyword(s):  
Boundary Conditions ◽  
Couple Stress ◽  
Scale Parameter ◽  
Laminated Composite ◽  
Length Scale Parameter ◽  
Smart Devices ◽  
Small Scale ◽  
Length Scale ◽  
Couple Stress Elasticity ◽  
Modified Couple Stress

Due to the large application of tapered beams in smart devices, such as scanning tunneling microscopes (STM), nano/micro electromechanical systems (NEMS/MEMS), atomic force microscopes (AFM), as well as in military aircraft applications, this study deals with the vibration behavior of laminated composite non-uniform nanobeams subjected to different boundary conditions. The micro-structural size-dependent free vibration response of composite laminated Euler–Bernoulli beams is here analyzed based on a modified couple stress elasticity, which accounts for the presence of a length scale parameter. The governing equations and boundary conditions of the problem are developed using the Hamilton’s principle, and solved by means of the Rayleigh–Ritz method. The accuracy and stability of the proposed formulation is checked through a convergence and comparative study with respect to the available literature. A large parametric study is conducted to investigate the effect of the length-scale parameter, non-uniformity parameter, size dimension and boundary conditions on the natural frequencies of laminated composite tapered beams, as useful for design and optimization purposes of small-scale devices, due to their structural tailoring capabilities, damage tolerance, and their potential for creating reduction in weight.

Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

2021 ◽  
pp. 107754632110482
Author(s):  
J Ranjan Banerjee ◽  
Stanislav O Papkov ◽  
Thuc P Vo ◽  
Isaac Elishakoff
Keyword(s):  
Free Vibration ◽  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Dynamic Stiffness ◽  
Modified Couple Stress Theory ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Stress Theory ◽  
Modified Couple Stress

Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

Flexural Vibration of an Atomic Force Microscope Cantilever Based on Modified Couple Stress Theory

2015 ◽  
Vol 15 (07) ◽  
pp. 1540025 ◽  
Author(s):  
Li-Na Liang ◽  
Liao-Liang Ke ◽  
Yue-Sheng Wang ◽  
Jie Yang ◽  
Sritawat Kitipornchai
Keyword(s):  
Size Effect ◽  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Flexural Vibration ◽  
Modified Couple Stress Theory ◽  
Length Scale ◽  
Stress Theory ◽  
Atomic Force ◽  
Modified Couple Stress

This paper is concerned with the flexural vibration of an atomic force microscope (AFM) cantilever. The cantilever problem is formulated on the basis of the modified couple stress theory and the Timoshenko beam theory. The modified couple stress theory is a nonclassical continuum theory that includes one additional material parameter to describe the size effect. By using the Hamilton's principle, the governing equation of motion and the boundary conditions are derived for the AFM cantilevers. The equation is solved using the differential quadrature method for the natural frequencies and mode shapes. The effects of the sample surface contact stiffness, length scale parameter and location of the sensor tip on the flexural vibration characteristics of AFM cantilevers are discussed. Results show that the size effect on the frequency is significant when the thickness of the microcantilever has a similar value to the material length scale parameter.

Temperature-dependent vibration analysis of a FG viscoelastic cylindrical microshell under various thermal distribution via modified length scale parameter: a numerical solution

2017 ◽  
Vol 26 (1-2) ◽  
pp. 9-24 ◽  
Author(s):  
Hamed Safarpour ◽  
Kianoosh Mohammadi ◽  
Majid Ghadiri
Keyword(s):  
Boundary Conditions ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Differential Quadrature Method ◽  
Equations Of Motion ◽  
Shear Deformation Theory ◽  
Functionally Graded ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Temperature Dependent

AbstractIn this article, the vibrational analysis of temperature-dependent cylindrical functionally graded (FG) microshells surrounded by viscoelastic a foundation is investigated by means of the modified couple stress theory (MCST). MCST is applied to this model to be productive in design and analysis of micro actuators and micro sensors. The modeled cylindrical FG microshell, its equations of motion and boundary conditions are derived by Hamilton’s principle and the first-order shear deformation theory (FSDT). For the first time, in the present study, functionally graded length scale parameter which changes along the thickness has been considered in the temperature-dependent cylindrical FG microshell. The accuracy of the present model is verified with previous studies and also with those obtained by analytical Navier method. The novelty of the current study is consideration of viscoelastic foundation, various thermal loadings and size effect as well as satisfying various boundary conditions implemented on the temperature-dependent cylindrical FG microshell using MCST. Generalized differential quadrature method (GDQM) is applied to discretize the equations of motion. Then, some factors are investigated such as the influence of length to radius ratio, damping, Winkler and Pasternak foundations, different temperature changes, circumferential wave numbers, and boundary conditions on natural frequency of the cylindrical FG microshell. The results have many applications such as modeling of microrobots and biomedical microsystems.

Effects of the Length Scale Parameter on the Thermoelastic Damping of a Microbeam Considering the Couple Stress Theory

2016 ◽  
Vol 08 (06) ◽  
pp. 1650083 ◽  
Author(s):  
Mohammad Fathalilou ◽  
Ghader Rezazadeh
Keyword(s):  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Theory Of Elasticity ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Complex Frequency ◽  
Thermoelastic Damping ◽  
Stress Theory ◽  
Ambient Temperatures

This paper studies the thermoelastic damping in microbeams considering the couple stress theory with microstructure. This theory includes the microinertia effects, coming from the kinetic energy due to the velocity gradient through the differential macroelements. A Galerkin-based reduced order model and complex frequency approach have been used to determine the quality factor. For a gold microbeam as a case study, the obtained results for different ambient temperatures, beam lengths and thicknesses are compared to those obtained using the classic theory of elasticity. The comparison has been made for different values of the length scale parameter. The effects of the microinertia term on the magnitude of the thermoelastic damping have also been investigated and shown that for which conditions these effects are significant.

Thermal Effect on Static Bending, Vibration and Buckling of Reddy Beam Based on Modified Couple Stress Theory

2013 ◽  
Vol 332 ◽  
pp. 331-338 ◽  
Author(s):  
Ali Reza Daneshmehr ◽  
Mostafa Mohammad Abadi ◽  
Amir Rajabpoor
Keyword(s):  
Thermal Effect ◽  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Length Scale ◽  
Bending Vibration ◽  
Stress Theory ◽  
Micro Scale ◽  
Modified Couple Stress

A microstructure-dependent Reddy beam theory (RBT) which contain only one material length scale parameter and can capture the size effect in micro-scale material unlike the classical theory is developed .using the variational principle energy the governing equation of motion is derived based on modified couple stress theory for the simply supported beam. the equations obtained are solved by Fourier series and the influence of the length scale parameter and thermal effect on static bending, vibration and buckling analysis of micro-scale Reddy beam is investigated.

scholarly journals Nonlinear Vibration Analysis of Axially Functionally Graded Microbeams Based on Nonlinear Elastic Foundation Using Modified Couple Stress Theory

2020 ◽  
Vol 64 (2) ◽  
pp. 97-108
Author(s):  
Mehdi Alimoradzadeh ◽  
Mehdi Salehi ◽  
Sattar Mohammadi Esfarjani
Keyword(s):  
Differential Equation ◽  
Forced Vibration ◽  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Length Scale ◽  
Stress Theory ◽  
Modified Couple Stress

In this study, a non-classical approach was developed to analyze nonlinear free and forced vibration of an Axially Functionally Graded (AFG) microbeam by means of modified couple stress theory. The beam is considered as Euler-Bernoulli type supported on a three-layered elastic foundation with Von-Karman geometric nonlinearity. Small size effects included in the analysis by considering the length scale parameter. It is assumed that the mass density and elasticity modulus varies continuously in the axial direction according to the power law form. Hamilton's principle was implemented to derive the nonlinear governing partial differential equation concerning associated boundary conditions. The nonlinear partial differential equation was reduced to some nonlinear ordinary differential equations via Galerkin's discretization technique. He's Variational iteration methods were implemented to obtain approximate analytical expressions for the frequency response as well as the forced vibration response of the microbeam with doubly-clamped end conditions. In this study, some factors influencing the forced vibration response were investigated. Specifically, the influence of the length scale parameter, the length of the microbeam, the power index, the Winkler parameter, the Pasternak parameter, and the nonlinear parameter on the nonlinear natural frequency as well as the amplitude of forced response have been investigated.

Vibration analysis of two-dimensional micromorphic structures using quadrilateral and triangular elements

2022 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Mina Kohansal Vajargah ◽  
Reza Ansari
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Scale Parameter ◽  
Side Length ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Two Dimensional ◽  
Content Type ◽  
Triangular Elements ◽  
Micro Structures

PurposeThe paper aims to presents a numerical analysis of free vibration of micromorphic structures subjected to various boundary conditions.Design/methodology/approachTo accomplish this objective, first, a two-dimensional (2D) micromorphic formulation is presented and the matrix representation of this formulation is given. Then, two size-dependent quadrilateral and triangular elements are developed within the commercial finite element software ABAQUS. User element subroutine (UEL) is used to implement the micromorphic elements. These non-classical elements are capable of capturing the micro-structure effects by considering the micro-motion of materials. The effects of the side length-to-length scale parameter ratio and boundary conditions on the vibration behavior of 2D micro-structures are discussed in detail. The reliability of the present finite element method (FEM) is confirmed by the convergence studies and the obtained results are validated with the results available in the literature. Also, the results of micromorphic theory (MMT) are compared with those of micropolar and classical elasticity theories.FindingsThe study found that the size effect becomes very significant when the side length of micro-structures is close to the length scale parameter.Originality/value The study is to analyze the free vibrations of 2D micro-structures based on MMT; to develop a 2D formulation for micromorphic continua within ABAQUS; to propose quadrilateral and triangular micromorphic elements using UEL and to investigate size effects on the vibrational behavior of micro-structures with various geometries.

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