scholarly journals Free vibration of rectangular nanoplate strips

2019 ◽  
Author(s):  
Jaan Lellep ◽  
Mainul Hossain
Keyword(s):  
Free Vibration ◽  
Cross Sections ◽  
Vibration Analysis ◽  
Theory Of Elasticity ◽  
Local Theory ◽  
Natural Vibrations ◽  
Size Dependent ◽  
Nano Structures ◽  
Non Local ◽  
Local Compliance

Natural vibrations of nanobeams and nanosheets are investigated with the help of nonlocal theories of elasticity. The vibration analysis is based on the size-dependent non-local theory of elasticity developed by A. C. Eringen. It is assumed that the nano-structures under consideration have rectangular cross sections with piece wise constant dimensions and that the nanoplates are weakened with defects. The influence of the crack on the vibration of the nanoplate is assessed with the aid of additional local compliance developed in previous papers. Numerical results are presented for one- and two-stepped nanoplates.

scholarly journals Natural vibrations of nanostrips with cracks

2021 ◽  
Vol 25 (1) ◽  
pp. 87-105
Author(s):  
Mainul Hossain ◽  
Jaan Lellep
Keyword(s):  
Cross Sections ◽  
Nonlocal Theory ◽  
Theory Of Elasticity ◽  
Material Behavior ◽  
Rotational Inertia ◽  
Natural Vibrations ◽  
Piecewise Constant ◽  
Theory Of Plates ◽  
Local Compliance

Employing the main equations of the theory of plates accounting for the rotational inertia the transverse vibrations of nanobeams and nanostrips are investigated. The nano strips under consideration have piecewise constant dimensions of cross sections. The nanosheets are weakened by cracks at re-entrant corners of steps. While the material behavior corresponds to the Eringen’s nonlocal theory of elasticity it is assumed that the cracks produce additional local compliance, which can be evaluated with the aid of the stress intensity factor at the cracktip. A numerical algorithm for determination of natural frequencies of nanosheets is developed.

scholarly journals Natural vibrations of stepped nanobeams with defects

2019 ◽  
Vol 23 (1) ◽  
pp. 143-158 ◽  
Author(s):  
Jaan Lellep ◽  
Artur Lenbaum
Keyword(s):  
Stress Intensity Factor ◽  
Cross Section ◽  
Cross Sections ◽  
Natural Vibration ◽  
Nonlocal Theory ◽  
Theory Of Elasticity ◽  
Natural Vibrations ◽  
Piecewise Constant ◽  
Local Compliance

Exact solutions for the transverse vibration of nanobeams based on the nonlocal theory of elasticity are presented. The nanobeams under consideration have piecewise constant dimensions of cross sections and are weakened with crack-like defects. It is assumed that the stationary cracks occur at the re-entrant corners of steps and that the mechanical behaviour of the nanomaterial can be modelled with the Eringen's nonlocal theory. The influence of cracks on the natural vibration is prescribed with the aid of additional local compliance at the weakened cross section. The local compliance is coupled with the stress intensity factor at the crack tip. A general algorithm for determination of eigenfrequencies is developed. It can be used in the case of an arbitrary finite number of steps and cracks.

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.

Study of internal stresses in rock massif within the framework of elastic layered block models with inclusions of the hierarchical structure of the L-rank.

2021 ◽  
Author(s):  
Olga Hachay ◽  
Andrey Khachay
Keyword(s):  
Hierarchical Structure ◽  
Classical Theory ◽  
Acoustic Waves ◽  
Degrees Of Freedom ◽  
Heterogeneous Media ◽  
Theory Of Elasticity ◽  
Rock Massif ◽  
Local Theory ◽  
Internal Stresses ◽  
Non Local

<p>In recent years, new models of continuum mechanics, generalizing the classical theory of elasticity, have been intensively developed. These models are used to describe composite and statistically heterogeneous media, new structural materials, as well as in complex massifs in mine conditions. The paper presents an algorithm for the propagation of longitudinal acoustic waves in the framework of active well monitoring of elastic layered block media with inclusions of hierarchical type of L-th rank. Relations for internal stresses and strains for each hierarchical rank are obtained, which constitute the non local theory of elasticity. The essential differences between the non local theory of elasticity and the classical one and the connection between them are investigated. A characteristic feature of the theory of media with a hierarchical structure is the presence of scale parameters in explicit or implicit form. This work focuses on the study of the effects of non locality and internal degrees of freedom, reflected in internal stresses, which are not described by the classical theory of elasticity and which can be potential precursors of the development of a catastrophic process in a rock massif. Thanks to the use of a model of a layered block medium with hierarchical inclusions, it is possible, using borehole acoustic monitoring, to determine the position of the highest values ​​of internal stresses and, with less effort, to implement the method of unloading the rock massif. If it is necessary to conduct short-term predictive monitoring of geodynamic regions and determine a more accurate position of the source of a dynamic phenomenon using borehole active acoustic observations, it is necessary to use the values ​​of the tensor of internal hierarchical stresses as a monitored parameter.</p>

Fracture analysis of nanobeams based on the stress-driven non-local theory of elasticity

2020 ◽  
pp. 1-10
Author(s):  
Sabrina Vantadori ◽  
Raimondo Luciano ◽  
Daniela Scorza ◽  
Hossein Darban
Keyword(s):  
Fracture Analysis ◽  
Theory Of Elasticity ◽  
Local Theory ◽  
Non Local
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