scholarly journals 1D HERMITE ELEMENTS FOR C1-CONTINUOUS SOLUTIONS IN SECOND GRADIENT ELASTICITY

2016 ◽  
Vol 7 ◽  
pp. 33-37 ◽  
Author(s):  
Christian Liebold ◽  
Wolfgang H. Müller
Keyword(s):  
Finite Element ◽  
Size Effect ◽  
Numerical Solution ◽  
Stiffness Matrix ◽  
Gradient Elasticity ◽  
Theory Of Elasticity ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Additional Material ◽  
Hermite Finite Elements

We present a modified strain gradient theory of elasticity for linear isotropic materials in order to account for the so-called size effect. Additional material length scale parameters are introduced and the problem of static beam bending is analyzed. A numerical solution is derived by means of a finite element approach. A global C1-continuous displacement field is applied in finite element solutions because the higher-order strain energy density additionally depends on second gradients of displacements. So-called Hermite finite elements are used that allow for merging gradients between elements. The element stiffness matrix as well as the global stiffness matrix of the problem is developed. Convergence, C1-continuity and the size effect in the numerical solution is shown. Experiments on bending stiffnesses of different sized micro beams made of the polymer SU-8 are performed by using an atomic force microscope and the results are compared to the numerical solution.

scholarly journals Thermo-electro-mechanical behaviour of Nano-sized structures

2020 ◽  
Vol 310 ◽  
pp. 00060
Author(s):  
Miroslav Repka ◽  
Ladislav Sator
Keyword(s):  
Size Effect ◽  
Mechanical Behaviour ◽  
Strain Gradient ◽  
Mechanical Response ◽  
Scale Parameter ◽  
Theory Of Elasticity ◽  
Strain Gradient Theory ◽  
Length Scale Parameter ◽  
Gradient Theory ◽  
Length Scale

Thermo-electro-mechanical behaviour of the nano-sized structures is analysed by the finite element method (FEM). The mechanical response of the nano-sized structures cannot be modelled with classical continuum theories due to the size effect phenomenon. The strain gradient theory with one length scale parameter has been applied to study size effect phenomenon. The coupled theory of thermo-electricity has been used together with strain gradient theory of elasticity. The governing equations have been derived and incorporated into the commercial software Comsol via weak form module. The influence of the length scale parameter on mechanical response of the structures is investigated by some numerical examples.

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.

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.

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