scholarly journals Static and Dynamic Solutions of Functionally Graded Micro/Nanobeams under External Loads Using Non-Local Theory

Vibration ◽  
2020 ◽  
Vol 3 (2) ◽  
pp. 51-69
Author(s):  
Reza Moheimani ◽  
Hamid Dalir
Keyword(s):  
Power Index ◽  
Linear Equations ◽  
Local Effect ◽  
Functionally Graded ◽  
Local Theory ◽  
Classical Elasticity ◽  
Graded Materials ◽  
Nano Scale ◽  
Non Local ◽  
External Loads

Functionally graded materials (FGMs) have wide applications in different branches of engineering such as aerospace, mechanics, and biomechanics. Investigation of the mechanical behaviors of structures made of these materials has been performed widely using classical elasticity theories in micro/nano scale. In this research, static, dynamic and vibrational behaviors of functional micro and nanobeams were investigated using non-local theory. Governing linear equations of the problem were driven using non-local theory and solved using an analytical method for different boundary conditions. Effects of the axial load, the non-local parameter and the power index on the natural frequency of different boundary condition are assessed. Then, the obtained results were compared with those obtained from classical theory. These results showed that a non-local effect could greatly affect the behaviors of these beams, especially at nano scale.

On Free Vibration of Functionally Graded Euler-Bernoulli Beam Models Based on the Non-Local Theory

2012 ◽  
Author(s):  
Reza Moheimani ◽  
M. T. Ahmadian
Keyword(s):  
Free Vibration ◽  
Axial Load ◽  
Power Index ◽  
Theory Of Elasticity ◽  
Functionally Graded ◽  
Local Theory ◽  
Local Parameter ◽  
Bernoulli Beam ◽  
Non Local ◽  
Euler Bernoulli Beam

In this paper, the governing equations and boundary conditions of a functionally graded Euler-Bernoulli beam are developed based on the non-local theory of elasticity. Afterward, the free vibration is investigated and the effects of the axial load, the non-local parameter and the power index on the natural frequency of a hinged-hinged beam is assessed. The results indicate that the non-local parameter has a decreasing effect on the frequency while the power index has an increasing effect. It is also noted that the effect of the axial load is increasing too.

The Non-Local Theory Solution of a Crack in the Functionally Graded Piezoelectric Materials Subjected to the Harmonic Anti-Plane Shear Stress Waves

2007 ◽  
Vol 353-358 ◽  
pp. 258-262
Author(s):  
Zhen Gong Zhou ◽  
Lin Zhi Wu
Keyword(s):  
Shear Stress ◽  
Piezoelectric Materials ◽  
Electric Displacement ◽  
Stress Waves ◽  
Functionally Graded ◽  
Local Theory ◽  
Plane Shear ◽  
Functionally Graded Piezoelectric Materials ◽  
Non Local ◽  
Functionally Graded Piezoelectric

In this paper, the non-local theory of elasticity was applied to obtain the dynamic behavior of a Griffith crack in functionally graded piezoelectric materials under the harmonic anti-plane shear stress waves. The problem can be solved with the help of a pair of dual integral equations. Unlike the classical elasticity solutions, it is found that no stress and electric displacement singularities are present at the crack tips, thus allows us to use the maximum stress as a fracture criterion.

Zum Materialgesetz eines elastischen Mediums mit Momentenspannungen

1965 ◽  
Vol 20 (3) ◽  
pp. 336-359 ◽  
Author(s):  
F. Hehl ◽  
E. Kröner
Keyword(s):  
Constitutive Relations ◽  
Short Review ◽  
Local Effect ◽  
Classical Elasticity ◽  
Couple Stresses ◽  
State Of Stress ◽  
Order Of Magnitude ◽  
Non Local ◽  
Peierls Model ◽  
Edge Dislocations

If through an element of area of a continuum there acts not only a force but also a couple, we have to introduce besides the force-stresses the so-called couple-stresses. In this article we emphasize the importance of couple-stresses in dislocated solids.—§ 2 gives a short review of the present state of the theory of couple-stresses. In classical elasticity couple-stresses are to be interpreted as a non-local effect intimately connected with the range of the atomic forces. The couplestresses are of a higher order in this range than force-stresses and can therefore usually be neglected.In the field theory of dislocations couple-stresses generally are of the same order of magnitude as force-stresses, however. Hence they cause considerable effects. In § 3 we determine the macroscopic observable couple-stresses of homogeneously distributed screw and edge dislocations through averaging over their microscopic fluctuating stress field. With the PEIERLS model we show in § 4 that the core of a dislocation produces an asymmetric state of stress and for that reason also couple-stresses, which are negligibly small under certain circumstances. Introducing a simple polycrystal model we derive in § 5 the constitutive relations for couple-stresses and dislocation density in an isotropic form. The results are discussed in § 6.

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