Formulation for Static Behavior of the Viscoelastic Euler-Bernoulli Micro-Beam Based on the Modified Couple Stress Theory

2012 ◽  
Author(s):  
E. Taati ◽  
M. Nikfar ◽  
M. T. Ahmadian
Keyword(s):  
Size Effects ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Closed Form Solution ◽  
Modified Couple Stress Theory ◽  
Form Solution ◽  
Equilibrium Equations ◽  
Stress Theory ◽  
Modified Couple Stress ◽  
Micro Structures

In this work an analytical solution is presented for a viscoelastic micro-beam based on the modified couple stress theory which is a non-classical theory in continuum mechanics. The modified couple stress theory has the ability to consider small size effects in micro-structures. It is strongly emphasized that without considering these effects in such structures the solution will be wrong and not suitable for designing systems in micro-scales. In this study correspondence principle is used for deriving constitutive equations for viscoelastic material based on the modified couple stress theory. Governing equilibrium equations are obtained by considering an element of micro-beam. Closed-form solution for the static deflection of simply supported micro-beam is presented. Numerical results show that when the size of system is near the length scale parameter, the classical response will intensely be deviated from the correct solution observed in laboratories contrary to the modified couple stress which reflects the size effects.

scholarly journals Vibration Analysis of a Postbuckled Microscale FG Beam Based on Modified Couple Stress Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-20 ◽  
Author(s):  
R. Ansari ◽  
M. A. Ashrafi ◽  
T. Pourashraf ◽  
M. Hemmatnezhad
Keyword(s):  
Power Law ◽  
Couple Stress ◽  
Thermal Environment ◽  
Couple Stress Theory ◽  
Closed Form Solution ◽  
Modified Couple Stress Theory ◽  
Power Law Distribution ◽  
Postbuckling Behavior ◽  
Stress Theory ◽  
Modified Couple Stress

On the basis of modified couple stress theory, the postbuckling behavior of the Euler-Bernoulli microscale FG beams is investigated by means of an exact solution method. The modified couple stress theory as a nonclassical continuum theory is capable of interpreting the size dependencies which become more significant at micro/nanoscales. The Von-Karman type nonlinear strain-displacement relationships are employed. The thermal effects are also incorporated into formulation. The governing equation of motion and the corresponding boundary conditions are derived using Hamilton’s principle. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A closed-form solution is obtained for the postbuckling deformation which is beyond the critical buckling load. To study the vibrations taking place in the vicinity of a buckled equilibrium position, the linear vibration problem is exactly solved around the first three buckled configurations. The natural frequencies of the lowest vibration modes around each of the first three buckled configurations are obtained. The influences of power-law exponent, boundary condition, length scale parameter, and thermal environment changes on the static deflection and free vibration frequencies are studied. A comparison is also made between the present results and those obtained via the classical beam theories.

Buckling and Post-Buckling of Symmetric Functionally Graded Microplate Lying on Nonlinear Elastic Foundation Based on Modified Couple Stress Theory

2018 ◽  
Vol 18 (09) ◽  
pp. 1850110 ◽  
Author(s):  
Chengyang Wu ◽  
Jia Lou ◽  
Liwen He ◽  
Jianke Du ◽  
Huaping Wu
Keyword(s):  
Elastic Foundation ◽  
Couple Stress ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Functionally Graded ◽  
Equilibrium Equations ◽  
Stress Theory ◽  
Post Buckling ◽  
Modified Couple Stress ◽  
Nonlinear Elastic

This paper is concerned with the buckling and post-buckling behaviors of a simply supported symmetric functionally graded (FG) microplate lying on a nonlinear elastic foundation. The modified couple stress theory is used to capture the size effects of the FG microplate, and the Mindlin plate theory with von Karman’s geometric nonlinearity taken into account is adopted to describe its deflection behavior. Based on these assumptions and the principle of minimum potential energy, the equilibrium equations of the FG microplate and associated boundary conditions are derived. By applying the Galerkin method to the equilibrium equations, closed-form solutions for the critical buckling load and the load–displacement relation in the post-buckling stage are obtained. Furthermore, the effects of the power law index, the material length scale parameter to thickness ratio, the stiffness of the elastic foundation, and in-plane boundary conditions on the buckling and post-buckling behaviors of the FG microplate are discussed in detail.

Nonlinear Snap-Through Instability of FGM Shallow Micro-Arches with Integrated Surface Piezoelectric Layers Based on Modified Couple Stress Theory

2019 ◽  
Vol 19 (08) ◽  
pp. 1950088 ◽  
Author(s):  
Hadi Babaei ◽  
M. Reza Eslami
Keyword(s):  
Differential Equations ◽  
Couple Stress ◽  
Volume Fraction ◽  
Couple Stress Theory ◽  
Modified Couple Stress Theory ◽  
Equilibrium Equations ◽  
Virtual Displacement ◽  
Stress Theory ◽  
Piezoelectric Layers ◽  
Modified Couple Stress

Based on the modified couple stress theory, an attempt is made in this study to analyze the nonlinear snap-through instability of shallow sandwich arches. The microstructure-dependent functionally graded material (FGM) arch with surface bonded piezoelectric actuator layers is analyzed. The piezo-FGM sandwich arch is subjected to uniform transverse pressure load in thermo-electrical environment. All material properties of the FGM micro arch are assumed to be temperature- and position-dependent. The governing equilibrium equations of the piezo-FGM sandwich arch are established with the aid of virtual displacement principle and the uncoupled thermoelacticity theory. The obtained governing differential equations are based on the first-order shear deformation shallow arch theory of the Timoshenko and von Kármán nonlinear assumptions. These equilibrium equations contain three coupled ordinary differential equations in terms of displacements. The nondimensional governing equations are solved for the cases of piezo-FGM sandwich arches with simply supported and clamped boundary conditions by using the two-step perturbation technique. Analytical closed-form solutions are derived to give the deflected shape of the piezo-FGM sandwich arch with immovable ends. Comparison is made with the existing results for the cases of FGM arch without couple stress and piezoelectric layers, where good agreement is obtained. The nonlinear behavior of the sandwich arches is highly affected by the couple stress, piezoelectric layers, temperature change, volume fraction index, and geometrical properties of the arch.

Size-Dependent Torsional Buckling of Carbon Nano-Peapods Based on the Modified Couple Stress Theory

2017 ◽  
Vol 09 (02) ◽  
pp. 1750030 ◽  
Author(s):  
Yaghoub Tadi Beni ◽  
Fahimeh Mehralian
Keyword(s):  
Couple Stress ◽  
Couple Stress Theory ◽  
Md Simulations ◽  
Modified Couple Stress Theory ◽  
Torsional Moment ◽  
Equilibrium Equations ◽  
Stress Theory ◽  
Modified Couple Stress ◽  
Minimum Potential ◽  
Torsional Instability

Using the modified couple stress theory and the shell model, this paper investigates the torsional instability of carbon nano peapods (CNPs), which are the hybrid structures made by inserting C[Formula: see text] fullerenes into CNT. Based on the modified couple stress theory as a higher order elasticity theory and using the first-order shear deformation shell theory, the equilibrium equations are derived through the principle of the minimum potential energy. The critical torsional moment is determined for the (10,10) CNT and the C[Formula: see text]@(10,10) CNP, and compared with that of MD simulations in the literature. Then, the increase in critical torsional moment is represented with the increase in the length scale parameter in both (10,10) CNT and C[Formula: see text]@(10,10) CNP based on the modified couple stress theory. Besides, by comparing the critical torsional moment in the C[Formula: see text]@(10,10) CNP and the (10,10) CNT, 100 percent increase in buckling resistance due to the presence of C[Formula: see text] fullerene is exhibited as well. In addition, the CNP surrounding is modeled as Pasternak foundation, and the effects of Pasternak stiffness are shown on the critical torsional moment.

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