scholarly journals Postbuckling and Free Nonlinear Vibration of Microbeams Based on Nonlinear Elastic Foundation

2018 ◽  
Vol 2018 ◽  
pp. 1-17 ◽  
Author(s):  
Dang Van Hieu
Keyword(s):  
Nonlinear Vibration ◽  
Elastic Foundation ◽  
Scale Parameter ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Nonlinear Elastic Foundation ◽  
Material Length Scale Parameter ◽  
Material Length Scale ◽  
Nonlinear Elastic ◽  
Material Length

In this paper, post-buckling and free nonlinear vibration of microbeams resting on nonlinear elastic foundation subjected to axial force are investigated. The equations of motion of microbeams are derived by using the modified couple stress theory. Using Galerkin’s method, the equation of motion of microbeams is reduced to the nonlinear ordinary differential equation. By using the equivalent linearization in which the averaging value is calculated in a new way called the weighted averaging value, approximate analytical expressions for the nonlinear frequency of microbeams with pinned–pinned and clamped–clamped end conditions are obtained in closed-forms. Comparisons with previous solutions are showed accuracy of the present solutions. Effects of the material length scale parameter and the axial compressive force on the frequency ratios of microbeams; and effect of the material length scale parameter on the buckling load ratios of microbeams are investigated in this paper.

Nonlinear Effects on Resonance Behaviour of Beams in Micro Scale

2011 ◽  
Vol 110-116 ◽  
pp. 4178-4186
Author(s):  
H. Nourbakhsh ◽  
R. Mohammadzadeh ◽  
M. Rafiee ◽  
R. Rafiee
Keyword(s):  
Scale Parameter ◽  
Lateral Displacement ◽  
Couple Stress Theory ◽  
Primary Resonance ◽  
Modified Couple Stress Theory ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Material Length Scale Parameter ◽  
Material Length Scale ◽  
Material Length

Nonlinear free and forced oscillation of microscale simply supported beams is investigated in this paper. Introducing a material length scale parameter, the nonlinear model is conducted within the context of non-classical continuum mechanics. By using a combination of the modified couple stress theory and Hamilton’s principle the nonlinear equation of motion is derived. The nonlinear frequencies of a beam with initial lateral displacement are discussed. Equations have been solved using an exact method for free vibration and multiple times scales (MTS) method for forced vibration and some analytical relations have been obtained for natural frequency of oscillations. The results have been compared with previous work and good agreement has been obtained. Also forced vibrations of system in primary resonance have been studied and the effects of different parameters on the frequency-response have been investigated. It is shown that the size effect is significant when the ratio of characteristic thickness to internal material length scale parameter is approximately equal to one, but is diminishing with the increase of the ratio. Our results also indicate that the nonlinearity has a great effect on the vibration behavior of microscale beams.

scholarly journals A nonlocal strain gradient shell model incorporating surface effects for vibration analysis of functionally graded cylindrical nanoshells

2019 ◽  
Vol 40 (12) ◽  
pp. 1695-1722 ◽  
Author(s):  
Lu Lu ◽  
Li Zhu ◽  
Xingming Guo ◽  
Jianzhong Zhao ◽  
Guanzhong Liu
Keyword(s):  
Strain Gradient ◽  
Scale Parameter ◽  
Nonlocal Parameter ◽  
Functionally Graded ◽  
Length Scale ◽  
Proposed Model ◽  
Material Length Scale Parameter ◽  
Material Length Scale ◽  
Material Length ◽  
Nonlocal Strain Gradient

AbstractIn this paper, a novel size-dependent functionally graded (FG) cylindrical shell model is developed based on the nonlocal strain gradient theory in conjunction with the Gurtin-Murdoch surface elasticity theory. The new model containing a nonlocal parameter, a material length scale parameter, and several surface elastic constants can capture three typical types of size effects simultaneously, which are the nonlocal stress effect, the strain gradient effect, and the surface energy effects. With the help of Hamilton’s principle and first-order shear deformation theory, the non-classical governing equations and related boundary conditions are derived. By using the proposed model, the free vibration problem of FG cylindrical nanoshells with material properties varying continuously through the thickness according to a power-law distribution is analytically solved, and the closed-form solutions for natural frequencies under various boundary conditions are obtained. After verifying the reliability of the proposed model and analytical method by comparing the degenerated results with those available in the literature, the influences of nonlocal parameter, material length scale parameter, power-law index, radius-to-thickness ratio, length-to-radius ratio, and surface effects on the vibration characteristic of functionally graded cylindrical nanoshells are examined in detail.

A Modified Gradient Plasticity Theory for Micro-Bending and Micro-Torsion Size Effects

2004 ◽  
Author(s):  
George Z. Voyiadjis ◽  
Rashid K. Abu Al-Rub
Keyword(s):  
Size Effects ◽  
Scale Parameter ◽  
Plasticity Theory ◽  
Material Behavior ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Gradient Plasticity ◽  
Non Local ◽  
Material Length Scale ◽  
Material Length

The definition and magnitude of the intrinsic length scale are keys to the development of the theory of plasticity that incorporates size effects. Gradient plasticity theory with a material length scale parameter is successfully in capturing the size dependence of material behavior at the micron scale. However, a fixed value of the material length-scale is not always realistic and that different problems could require different values. Moreover, a linear coupling between the local and non-local terms in the gradient plasticity theory is not always realistic and that different problems could require different couplings. A generalized gradient plasticity model with a non-fixed length scale parameter is proposed. This model assesses the sensitivity of predictions in the way in which the local and non-local parts are coupled. The proposed model gives good predictions of the size effect in micro-bending tests of thin films and micro-torsion tests of thin wires.

Prediction of the Critical Energy Release Rate of Nanostructured Solids Using the Laplacian Version of the Strain Gradient Elasticity Theory

2018 ◽  
Vol 774 ◽  
pp. 447-452 ◽  
Author(s):  
Michal Kotoul ◽  
Petr Skalka ◽  
Tomáš Profant ◽  
Martin Friák ◽  
Petr Řehák ◽  
...  
Keyword(s):  
Dft Calculations ◽  
Strain Gradient ◽  
Scale Parameter ◽  
Gradient Elasticity ◽  
Length Scale ◽  
Strain Gradient Elasticity ◽  
Material Length Scale Parameter ◽  
Gradient Elasticity Theory ◽  
Material Length Scale ◽  
Material Length

The aim of the paper is quantify the material length scale parameter of the simplified form of the strain gradient elasticity theory (SGET) using first principles density-functional theory (DFT). The single material length scale parameterlis extracted from phonon-dispersions generated by DFT calculations and, for comparison, by adjusting the analytical SGET solution for the displacement field near the screw dislocation with the DFT calculations of this field. The obtained results are further used in the SGET modeling of cracked nanopanel formed by the single tungsten crystal where due to size effects and nonlocal material point interactions the classical fracture mechanics breaks down.

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