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

2015 ◽  
Vol 15 (04) ◽  
pp. 1450070 ◽  
Author(s):  
Raheb Gholami ◽  
Reza Ansari ◽  
Abolfazl Darvizeh ◽  
Saeed Sahmani
Keyword(s):  
Dynamic Stability ◽  
Couple Stress ◽  
Shell Theory ◽  
Scale Parameter ◽  
Functionally Graded ◽  
Governing Equations ◽  
First Order ◽  
Axial Buckling ◽  
Aspect Ratios ◽  
Modified Couple Stress

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.

Magnetic field effect on free vibration of smart rotary functionally graded nano/microplates: A comparative study on modified couple stress theory and nonlocal elasticity theory

2018 ◽  
Vol 29 (11) ◽  
pp. 2492-2507 ◽  
Author(s):  
Mohammad Hassan Shojaeefard ◽  
Hamed Saeidi Googarchin ◽  
Mohammad Mahinzare ◽  
Seyed Ahmad Eftekhari
Keyword(s):  
Elastic Material ◽  
Nonlocal Elasticity ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Nonlocal Elasticity Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Governing Equations ◽  
Stress Theory ◽  
Modified Couple Stress

In this article, free vibration behavior of a rotating nano/microcircular plate constructed from functionally graded magneto-elastic material is simulated with the first-order shear deformation theory. For the sake of comparison, the nonlocal elasticity theory and the modified couple stress theory are employed to implement the small size effect in the natural frequencies behavior of the nano/microcircular plate. The governing equations of motion for functionally graded magneto-elastic material nano/microcircular plates are derived based on Hamilton’s principle; comparing the obtained results with those in the literature, they are in a good agreement. Finally, the governing equations are solved using the differential quadrature method. It is shown that the vibrational characteristics of functionally graded magneto-elastic material nano/microcircular plates are significantly affected by non-dimensional angular velocity, size dependency of the Eringen’s and the modified couple stress theories, and power law index for clamped and hinged boundary conditions. Results show that a critical point occurs by increasing the angular velocity and the effect of several parameters are changed after this point.

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.

Free Vibration of Edge Cracked Functionally Graded Microscale Beams Based on the Modified Couple Stress Theory

2017 ◽  
Vol 17 (03) ◽  
pp. 1750033 ◽  
Author(s):  
Şeref Doğuşcan Akbaş
Keyword(s):  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Beam Model ◽  
Cracked Beam ◽  
Stress Theory ◽  
Modified Couple Stress

In this study, the free vibration analysis of edge cracked cantilever microscale beams composed of functionally graded material (FGM) is investigated based on the modified couple stress theory (MCST). The material properties of the beam are assumed to change in the height direction according to the exponential distribution. The cracked beam is modeled as a modification of the classical cracked-beam theory consisting of two sub-beams connected by a massless elastic rotational spring. The inclusion of an additional material parameter enables the new beam model to capture the size effect. The new nonclassical beam model reduces to the classical one when the length scale parameter is zero. The problem considered is investigated using the Euler–Bernoulli beam theory by the finite element method. The system of equations of motion is derived by Lagrange’s equations. To verify the accuracy of the present formulation and results, the frequencies obtained are compared with the results available in the literature, for which good agreement is observed. Numerical results are presented to investigate the effect of crack position, beam length, length scale parameter, crack depth, and material distribution on the natural frequencies of the edge cracked FG microbeam. Also, the difference between the classical beam theory (CBT) and MCST is investigated for the vibration characteristics of the beam of concern. It is believed that the results obtained herein serve as a useful reference for research of similar nature.

Size and Foundation Effects on the Vibration of Buckled Functionally Graded Microplates Within the Modified Couple Stress Theory Framework

2018 ◽  
Vol 10 (06) ◽  
pp. 1850068 ◽  
Author(s):  
Jia Lou ◽  
Liwen He ◽  
Jie Yang ◽  
Sritawat Kitipornchai ◽  
Huaping Wu
Keyword(s):  
Elastic Foundation ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Linear Perturbation ◽  
Governing Equations ◽  
Stress Theory ◽  
Post Buckling ◽  
Modified Couple Stress

In the present paper, the vibration behavior of a buckled functionally graded (FG) microplate lying on a nonlinear elastic foundation is studied. The modified couple stress theory is utilized to capture the size effect of the FG microplate, and the Mindlin plate theory with the von Karman’s geometric nonlinearity is adopted to describe its deflection behavior. Based on these assumptions and Hamilton’s principle, the governing equations and associated boundary conditions are derived for the FG microplate. By linearizing the governing equations around a pre-buckling/post-buckling state, linear perturbation equations are obtained. After substituting the pre-buckling/post-buckling deformation and assumed vibration mode into the linear perturbation equations and applying the Galerkin method, an eigenvalue problem is obtained, from which the free vibration frequency of the FG microplate around its pre-buckling/post-buckling state can be determined analytically. Based on the obtained closed-form solutions, numerical examples are also presented to investigate the effects of the material length scale parameter to thickness ratio, the power law index, and the stiffness of the elastic foundation on the vibration behavior of the buckled FG microplate.

scholarly journals Theories and Analysis of Functionally Graded Beams

2021 ◽  
Vol 11 (15) ◽  
pp. 7159
Author(s):  
Junuthula N. Reddy ◽  
Eugenio Ruocco ◽  
Jose A. Loya ◽  
Ana M. A. Neves
Keyword(s):  
Shear Deformation ◽  
Analytical Solutions ◽  
Functionally Graded ◽  
Thickness Variation ◽  
Governing Equations ◽  
Third Order ◽  
First Order ◽  
Graded Material ◽  
Modified Couple Stress ◽  
Shear Deformation Theories

This is a review paper containing the governing equations and analytical solutions of the classical and shear deformation theories of functionally graded straight beams. The classical, first-order, and third-order shear deformation theories account for through-thickness variation of two-constituent functionally graded material, modified couple stress (i.e., strain gradient), and the von Kármán nonlinearity. Analytical solutions for bending of the linear theories, some of which are not readily available in the literature, are included to show the influence of the material variation, boundary conditions, and loads.

Vibration of size-dependent functionally graded sandwich microbeams with different boundary conditions based on the modified couple stress theory

2017 ◽  
Vol 22 (2) ◽  
pp. 220-247 ◽  
Author(s):  
Nuttawit Wattanasakulpong ◽  
Arisara Chaikittiratana ◽  
Sacharuck Pornpeerakeat
Keyword(s):  
Boundary Conditions ◽  
Couple Stress ◽  
Volume Fraction ◽  
Couple Stress Theory ◽  
Slenderness Ratio ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Governing Equations ◽  
Stress Theory ◽  
Modified Couple Stress

This paper investigates flexural vibration of functionally graded sandwich microbeams supported by different axially immovable boundary conditions. The governing equations of free vibration problem are based on Timoshenko beam theory and the modified couple stress theory which are taking into account the important effects of shear deformation, rotary inertia and material length scale parameter. To solve the governing equations presented in the forms of coupled differential equations for vibration analysis of the beams with various boundary conditions, an effective tool, namely Chebyshev collocation method, is employed to find out accurate solutions with many important parametric studies. The effects of material volume fraction index, layer thickness ratio, slenderness ratio, boundary condition, temperature rise, etc. on natural frequencies of the beams are taken into account and discussed in details. The numerical results of the beams in ambient temperature and high thermal environment are presented in several tables and figures that can serve as benchmarks for further investigations in the field of FG sandwich microbeam analysis.

Vibration characteristics of rotating truncated conical shells reinforced with agglomerated carbon nanotubes

2021 ◽  
pp. 107754632110004
Author(s):  
Hassan Afshari ◽  
Hossein Amirabadi
Keyword(s):  
Carbon Nanotubes ◽  
Differential Quadrature Method ◽  
Shear Deformation Theory ◽  
Free Vibration Analysis ◽  
Functionally Graded ◽  
Governing Equations ◽  
Conical Shells ◽  
First Order ◽  
Effective Mechanical Properties ◽  
Comprehensive Study

In this article, a comprehensive study is conducted on the free vibration analysis of rotating truncated conical shells reinforced with functionally graded agglomerated carbon nanotubes The shell is modeled based on the first-order shear deformation theory, and effective mechanical properties are calculated based on the Eshelby–Mori–Tanaka scheme along with the rule of mixture. By considering centrifugal and Coriolis accelerations and initial hoop tension, the set of governing equations is derived using Hamilton’s principle and is solved numerically using the differential quadrature method Convergence and accuracy of the presented model are confirmed and the effects of different parameters on the forward and backward frequencies of the rotating carbon nanotube-reinforced truncated conical shells are investigated.

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