On governing equations for a nanoplate derived from the 3D gradient theory of elasticity

2021 ◽  
pp. 108128652110575
Author(s):  
Gennadi Mikhasev
Keyword(s):  
Constitutive Equations ◽  
Three Dimensional ◽  
Low Frequency ◽  
Gradient Elasticity ◽  
Theory Of Elasticity ◽  
Gradient Theory ◽  
Governing Equations ◽  
Nonlocal Effects ◽  
Internal Length ◽  
Variable Surface

The paper is concerned with the asymptotically consistent theory of nanoscale plates capturing the spatial nonlocal effects. The three-dimensional (3D) elasticity equations for a thin plate are used as the governing equations. In the general case, the plate is acted upon by dynamic body forces varying in the thickness direction, and by variable surface forces. The thickness of the plate is assumed to be greater than the characteristic micro/nanoscale measure and much smaller than the in-plane characteristic dimension (e.g., the wave or deformation length). The 3D constitutive equations of gradient elasticity are used to link the fields of nonlocal stresses and strains. Using the asymptotic approach, a sequence of relations for stresses and displacements in the form of polynomials in the transverse coordinate with coefficients depending on time and in-plane coordinates was obtained. The asymptotically consistent 2D differential equation governing vibration (or static deformation) of a plate accounting for both transverse shears and the spatial nonlocal contribution of the stress and strain fields was derived. It was revealed that capturing nonlocal effects in all directions leads to an increase in the correction factor compared with the well-known 2D theories based on kinematic hypotheses and the Eringen-type gradient constitutive equations. The effect of the internal length scales parameters on free low-frequency vibrations and displacements of a plate is discoursed.

scholarly journals On the refined stress analysis in the applied elasticity problems accounting of gradient effects

2019 ◽  
Vol 489 (6) ◽  
pp. 585-591
Author(s):  
E. V. Lomakin ◽  
S. A. Lurie ◽  
L. N. Rabinskiy ◽  
Y. O. Solyaev
Keyword(s):  
Scale Parameter ◽  
Three Dimensional ◽  
Fundamental Property ◽  
Gradient Elasticity ◽  
Theory Of Elasticity ◽  
The Body ◽  
Length Scale Parameter ◽  
Gradient Theory ◽  
Length Scale ◽  
Dimensional Solution

The paper proposes an extension of the approaches of gradient elasticity of deformable media, which consists in using the fundamental property of solutions of the gradient theory - ​the smoothing of singular solutions of the classical theory of elasticity, converting them into a regular class not only for the problems of micromechanics, where the length scale parameter is of the order of the materials characteristic size, but for macromechanical problems. In these problems, the length scale parameter, as a rule, can be found from the macro-experiments or numerical experiments and does note have an extremely small values. It is shown, by attracting numerical three-dimensional modeling, that even one-dimensional gradient solutions make it possible to clarify the stress distribution in the constrained zones of the body and in the area of the loads application. It is shown that additional length scale parameters of the gradient theory are related with specific boundary effects and can be associated with structural geometric parameters and loading conditions that determine the features of the classical three-dimensional solution.

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.

Dislocations in gradient elasticity revisited

2006 ◽  
Vol 462 (2075) ◽  
pp. 3465-3480 ◽  
Author(s):  
Markus Lazar ◽  
Gérard A Maugin
Keyword(s):  
Gradient Elasticity ◽  
Theory Of Elasticity ◽  
Gradient Theory ◽  
Two Dimensional ◽  
Helmholtz Equations ◽  
Plastic Distortion ◽  
Second Gradient ◽  
The Fourier Transform ◽  
Fourier Type ◽  
Edge Dislocations

In this paper, we consider dislocations in the framework of first as well as second gradient theory of elasticity. Using the Fourier transform, rigorous analytical solutions of the two-dimensional bi-Helmholtz and Helmholtz equations are derived in closed form for the displacement, elastic distortion, plastic distortion and dislocation density of screw and edge dislocations. In our framework, it was not necessary to use boundary conditions to fix constants of the solutions. The discontinuous parts of the displacement and plastic distortion are expressed in terms of two-dimensional as well as one-dimensional Fourier-type integrals. All other fields can be written in terms of modified Bessel functions.

On the Correspondence between Two- and Three-Dimensional Elastic Solutions of Crack Problems

2016 ◽  
Vol 713 ◽  
pp. 18-21 ◽  
Author(s):  
Andrei G. Kotousov ◽  
Zhuang He ◽  
Aditya Khanna
Keyword(s):  
Scale Effects ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Two Dimensional ◽  
Governing Equations ◽  
Singular Stress ◽  
Linear Elastic ◽  
Crack Problems ◽  
Elastic Fracture ◽  
Stress States

The classical two-dimensional solutions of the theory of elasticity provide a framework of Linear Elastic Fracture Mechanics. However, these solutions, in fact, are approximations despite that the corresponding governing equations of the plane theories of elasticity are solved exactly. This paper aims to elucidate the main differences between the approximate (two-dimensional) and exact (three-dimensional) elastic solutions of crack problems. The latter demonstrates many interesting features, which cannot be analysed within the plane theories of elasticity. These features include the presence of scale effects of deterministic nature, the existence of new singular stress states and fracture modes. Furthermore, the deformation and stress fields near the tip of the crack is essentially three-dimensional and do not follow plane stress or plane strain simplifications. Moreover, in certain situations the two-dimensional solutions can provide misleading results; and several characteristic examples are outlined in this paper.

Refined hyperbolic shear deformation plate theory

2014 ◽  
Vol 229 (15) ◽  
pp. 2675-2686 ◽  
Author(s):  
K Nareen ◽  
RP Shimpi
Keyword(s):  
Shear Deformation ◽  
Plate Theory ◽  
Plate Thickness ◽  
Three Dimensional ◽  
Theory Of Elasticity ◽  
Governing Equations ◽  
Dimensional Theory ◽  
Classical Plate Theory ◽  
Shear Deformation Plate Theory ◽  
Elasticity Solutions

The paper presents a novel shear deformation plate theory involving only two variables. Taking a cue from exact three-dimensional theory of elasticity solutions for a plate, hyperbolic functions are used for describing displacement variation across plate thickness. The theory involves only two governing equations, which are uncoupled for statics and are only inertially coupled for dynamics. The shear stress free surface conditions are satisfied. No shear correction factor is required. The theory is variationally consistent, has a strong similarity with classical plate theory, and is simple, yet accurate. Illustrative examples for free vibration and for static flexure demonstrate the effectiveness of the theory.

scholarly journals ОСНОВНЫЕ УРАВНЕНИЯ, ГРАНИЧНЫЕ УСЛОВИЯ И ВАРИАЦИОННЫЙ ПРИНЦИП ПЛОСКОЙ ЗАДАЧИ ГРАДИЕНТНОЙ ТЕОРИИ УПРУГОСТИ ДЛЯ ПРЯМОУГОЛЬНОЙ ОБЛАСТИ / BASIC EQUATIONS, BOUNDARY CONDITIONS AND VARIATION PRINCIPLES OF THE PLANE PROBLEM IN THE GRADIENT THEORY OF ELASTICITY FOR A RECTANGULAR DOMAIN

2021 ◽  
pp. 20-28
Author(s):  
L. Sargsyan
Keyword(s):  
Boundary Conditions ◽  
Plane Problem ◽  
Virtual Work ◽  
Gradient Elasticity ◽  
Theory Of Elasticity ◽  
Rectangular Domain ◽  
Gradient Theory ◽  
Principle Of Virtual Work ◽  
Balance Equations ◽  
Basic Equations

В работе приведены основные уравнения плоской задачи градиентной теории упругости для прямоугольной области и устанавливается принцип возможных перемещений с соответствующем вариационном уравнением. Из вариационного уравнения теории упругости и для прямоугольной области все граничные условия. / The paper demonstrates the basic equations of the plane problem in the frames of the theory of gradient elasticity and establishes the principle of virtual work along with its variation equations. The basic balance equations of the plane problem of the theory of gradient elasticity and the boundary conditions for the rectangular plane are derived.

scholarly journals An energy residual-based approach to gradient effects within the mechanics of generalized continua

2012 ◽  
Vol 21 (3-4) ◽  
pp. 101-121
Author(s):  
Castrenze Polizzotto ◽  
Aurora A. Pisano
Keyword(s):  
Long Range ◽  
Constitutive Equations ◽  
Gradient Elasticity ◽  
Main Tool ◽  
Higher Order ◽  
Gradient Theory ◽  
Generalized Continua ◽  
Standard Format ◽  
Gradient Effects ◽  
Insulation Condition

AbstractGeneralized continua exhibiting gradient effects are addressed through a method grounded on the energy residual (ER)-based gradient theory by the first author and coworkers. A main tool of this theory is the Clausius-Duhem inequality cast in a form differing from the classical one only by a nonstandard extra term, the (nonlocality) ER, required to satisfy the insulation condition (its global value has to vanish or to take a known value). The ER carries in the nonlocality features of the mechanical problem through a strain-like rate field, being the specific nonlocality source, and a concomitant higher-order long-range stress (or microstress) field. The thermodynamic restrictions on the constitutive equations are determined by the latter inequality with no need for microstress equilibrium equations, whereas the principle of virtual power (PVP) is left in a standard format. The derived state equations include a set of partial differential equations involving the nonlocality-source strain-like quantity and the related long-range stress, as well as the associated higher-order boundary conditions determined by the insulation condition. Second-grade materials within gradient elasticity, gradient plasticity and crystal plasticity, as well as materials with microstructure (micromorphic and Cosserat materials) are considered to derive the pertinent constitutive equations. The proposed ER-based approach to gradient effects is shown to constitute a more straightforward and “economic” way to formulate the relevant constitutive equations than the PVP-based one.

Three-Dimensional Dynamic Stress and Vibration Analyses of Thick Singular-Kernel Fractional-Order Viscoelastic Annular Rotating Discs Under Nonuniform Loads

2019 ◽  
Vol 20 (01) ◽  
pp. 2050007 ◽  
Author(s):  
M. Shariyat ◽  
R. Mohammadjani
Keyword(s):  
Fractional Order ◽  
Dynamic Stress ◽  
Equations Of Motion ◽  
Viscoelastic Model ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Theory Of Elasticity ◽  
Constitutive Law ◽  
Sensitivity Analyses ◽  
Governing Equations

In this paper, the dynamic stress and radial/lateral vibration of circular/annular discs made of fractional-order viscoelastic materials under nonuniform mechanical loads are investigated for the first time, utilizing the exact 3D theory of elasticity, rather than the plate theories. The governing equations of motion of the disc are derived based on the Kelvin–Voigt fractional viscoelastic model. To solve these equations, the spatial partial and the time ordinary derivatives are replaced by adequate central, backward or forward finite difference expressions. Then the resulting Caputo-type time-dependent system of the coupled integro-differential governing equations of the fractional-order is solved by a novel numerical procedure. Namely, a time-marching procedure is employed to extract the time histories of the responses, in the space-time domain for various time and spatial distributions. Finally, comprehensive sensitivity analyses and various 3D plots are presented and discussed. In this regard, effects of the fractional-order of the constitutive law, viscoelastic parameters, material rigidity, distribution and time variation patterns of the nonuniform distributed transverse loads, and boundary conditions on the distributions of the displacement and stress components are investigated.

scholarly journals Gradient Elasticity Formulations for Micro/Nanoshells

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
Bohua Sun ◽  
E. C. Aifantis
Keyword(s):  
Constitutive Relation ◽  
Gradient Elasticity ◽  
Theory Of Elasticity ◽  
Gradient Theory

The focus of this paper is on illustrating how to extend the second author’s gradient theory of elasticity to shells. Three formulations are presented based on the implicit gradient elasticity constitutive relation1 -ld2∇2σij=Cijkl(1-ls2∇2)εkland its two approximations1+ls2∇2-ld2∇2σij=Cijklεklandσij=Cijkl(1+ld2∇2-ls2∇2)εkl.

Propagation and Reflection of Plane Waves in a Rotating Magneto-Elastic Fibre-Reinforced Semi Space with Surface Stress

2020 ◽  
Vol 22 (4) ◽  
pp. 939-958
Author(s):  
Indrajit Roy ◽  
D. P. Acharya ◽  
Sourav Acharya
Keyword(s):  
Surface Stress ◽  
Incident Angle ◽  
Plane Waves ◽  
Three Dimensional ◽  
Elastic Fibre ◽  
Amplitude Ratios ◽  
Governing Equations ◽  
Rotation Surface ◽  
Wave Velocities ◽  
Magnetic Field Rotation

AbstractThe present paper investigates the propagation of quasi longitudinal (qLD) and quasi transverse (qTD) waves in a magneto elastic fibre-reinforced rotating semi-infinite medium. Reflections of waves from the flat boundary with surface stress have been studied in details. The governing equations have been used to obtain the polynomial characteristic equation from which qLD and qTD wave velocities are found. It is observed that both the wave velocities depend upon the incident angle. After imposing the appropriate boundary conditions including surface stress the resultant amplitude ratios for the total displacements have been obtained. Numerically simulated results have been depicted graphically by displaying two and three dimensional graphs to highlight the influence of magnetic field, rotation, surface stress and fibre-reinforcing nature of the material medium on the propagation and reflection of plane waves.

Sign in / Sign up

Export Citation Format

Share Document