Calculation of the Additional Constants for fcc Materials in Second Strain Gradient Elasticity: Behavior of a Nano-Size Bernoulli-Euler Beam With Surface Effects

2012 ◽  
Vol 79 (2) ◽  
Author(s):  
H. M. Shodja ◽  
F. Ahmadpoor ◽  
A. Tehranchi
Keyword(s):  
Strain Gradient ◽  
Gradient Elasticity ◽  
Surface Effects ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Strain Gradient Elasticity ◽  
Euler Beam ◽  
Concentrated Load ◽  
Material Constants ◽  
Nano Size

In addition to enhancement of the results near the point of application of a concentrated load in the vicinity of nano-size defects, capturing surface effects in small structures, in the framework of second strain gradient elasticity is of particular interest. In this framework, sixteen additional material constants are revealed, incorporating the role of atomic structures of the elastic solid. In this work, the analytical formulations of these constants corresponding to fee metals are given in terms of the parameters of Sutton-Chen interatomic potential function. The constants for ten fcc metals are computed and tabulized. Moreover, the exact closed-form solution of the bending of a nano-size Bernoulli-Euler beam in second strain gradient elasticity is provided; the appearance of the additional constants in the corresponding formulations, through the governing equation and boundary conditions, can serve to delineate the true behavior of the material in ultra small elastic structures, having very large surface-to-volume ratio. Now that the values of the material constants are available, a nanoscopic study of the Kelvin problem in second strain gradient theory is performed, and the result is compared quantitatively with those of the first strain gradient and traditional theories.

Size Dependent Vibration Behavior of an AFM with Sidewall and Top-Surface Probes Based on the Strain Gradient Elasticity Theory

2015 ◽  
Vol 07 (03) ◽  
pp. 1550046 ◽  
Author(s):  
Mohammad Abbasi
Keyword(s):  
Elasticity Theory ◽  
Strain Gradient ◽  
Gradient Elasticity ◽  
Strain Gradient Theory ◽  
Strain Gradient Elasticity ◽  
Size Dependent ◽  
Vibration Behavior ◽  
Vertical Extension ◽  
Strain Gradient Elasticity Theory ◽  
Gradient Elasticity Theory

In this paper, the size-dependent vibration behavior of an atomic force microscope with assembled cantilever probe (ACP) is analyzed utilizing the modified strain gradient elasticity theory. The proposed ACP comprises a horizontal cantilever, a vertical extension and two tips located at the free ends of the cantilever and extension. Because the vertical extension is located between the clamped and free ends of the microcantilever, the cantilever is modeled as two beams. The results of the current model are compared to those evaluated by both modified couple stress and classical beam theories. The results indicate that the resonant frequency and sensitivity of the proposed ACP is strongly size-dependent especially when the contact stiffness is very low or it is very high. The results also declare that utilizing the strain gradient theory is essential in the analysis of the vibration behavior of the proposed ACP.

Study of the sensitivity and resonant frequency of the flexural modes of an atomic force microscopy microcantilever modeled by strain gradient elasticity theory

2013 ◽  
Vol 228 (8) ◽  
pp. 1299-1310 ◽  
Author(s):  
Mohammad Abbasi ◽  
Ardeshir Karami Mohammadi
Keyword(s):  
Atomic Force Microscopy ◽  
Resonant Frequency ◽  
Strain Gradient ◽  
Contact Stiffness ◽  
Current Model ◽  
Gradient Elasticity ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Force Microscopy ◽  
Atomic Force

In this study, the resonant frequency and sensitivity of an atomic force microscopy microcantilever are analyzed utilizing the strain gradient theory, and then the governing equation and boundary conditions are derived by a combination of the basic equations of the modified strain gradient theory and the Hamilton principle. Afterward, the resonant frequency and sensitivity of the proposed atomic force microscopy microcantilever are obtained numerically. The results of the current model are compared to those evaluated by both modified couple stress and classic beam theories. Results show that utilizing the strain gradient theory in the analysis of atomic force microscopy microcantilever dynamic behavior is necessary especially when the contact stiffness is high and the thickness of the microcantilever approaches the internal material length scale parameter.

scholarly journals Instability Characteristics of Free-Standing Nanowires Based on the Strain Gradient Theory with the Consideration of Casimir Attraction and Surface Effects

2017 ◽  
Vol 24 (3) ◽  
pp. 489-507 ◽  
Author(s):  
Hamid M. Sedighi ◽  
Hassen M. Ouakad ◽  
Moosa Khooran
Keyword(s):  
Surface Energy ◽  
Strain Gradient ◽  
Dynamic Instability ◽  
Research Work ◽  
Gradient Elasticity ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Free Standing ◽  
Gradient Elasticity Theory ◽  
Casimir Attraction

AbstractSize-dependent dynamic instability of cylindrical nanowires incorporating the effects of Casimir attraction and surface energy is presented in this research work. To develop the attractive intermolecular force between the nanowire and its substrate, theproximity force approximation(PFA) for small separations, and the Dirichlet asymptotic approximation for large separations with a cylinder-plate geometry are employed. A nonlinear governing equation of motion for free-standing nanowires – based on the Gurtin-Murdoch model – and a strain gradient elasticity theory are derived. To overcome the complexity of the nonlinear problem in hand, a Garlerkin-based projection procedure for construction of a reduced-order model is implemented as a way of discretization of the governing differential equation. The effects of length-scale parameter, surface energy and vacuum fluctuations on the dynamic instability threshold and adhesion of nanowires are examined. It is demonstrated that in the absence of any actuation, a nanowire might behave unstably, due to the Casimir induction force.

MODELING THE SIZE DEPENDENT STATIC AND DYNAMIC PULL-IN INSTABILITY OF CANTILEVER NANOACTUATOR BASED ON STRAIN GRADIENT THEORY

2014 ◽  
Vol 06 (05) ◽  
pp. 1450055 ◽  
Author(s):  
HAMID M. SEDIGHI ◽  
A. KOOCHI ◽  
M. ABADYAN
Keyword(s):  
Strain Gradient ◽  
Time History ◽  
Gradient Elasticity ◽  
Beam Theory ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nanoscale Systems ◽  
Scale Parameters ◽  
Size Dependent ◽  
Electromechanical Instability

It is well-established that mechanical behavior of nanoscale systems is size dependent. In this paper, strain gradient elasticity theory is used for mathematical modeling of size dependent electromechanical instability of cantilever nanoactuator. The nanoactuator is modeled using Euler–Bernoulli beam theory and equation of motion is derived using Hamilton's principle. In order to solve the nonlinear governing equation, reduced order method (ROM) is employed. The dynamic pull-in instability of the nanoactuator is investigated through plotting the time history and phase portrait of the system. Static and dynamic pull-in voltage of nanoactuator as a function of dimensionless length scale parameters is determined. The obtained results show that when thickness of the nanoactuator is comparable with the intrinsic material length scales, size effect can substantially influence the pull-in behavior of the system.

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