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.

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.

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.

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 Towards Recognition of Scale Effects in a Solid Model of Lattices with Tensegrity-Inspired Microstructure

Solids ◽  
2021 ◽  
Vol 2 (1) ◽  
pp. 50-59
Author(s):  
Wojciech Gilewski ◽  
Anna Al Sabouni-Zawadzka
Keyword(s):  
Mechanical Properties ◽  
Continuum Model ◽  
Scale Effects ◽  
Theory Of Elasticity ◽  
Solid Model ◽  
Gradient Theory ◽  
General Description ◽  
Normal Forces ◽  
Second Gradient ◽  
Proposed Model

This paper is dedicated to the extended solid (continuum) model of tensegrity structures or lattices. Tensegrity is defined as a pin-joined truss structure with an infinitesimal mechanism stabilized by a set of self-equilibrated normal forces. The proposed model is inspired by the continuum model that matches the first gradient theory of elasticity. The extension leads to the second- or higher-order gradient formulation. General description is supplemented with examples in 2D and 3D spaces. A detailed form of material coefficients related to the first and second deformation gradients is presented. Substitute mechanical properties of the lattice are dependent on the cable-to-strut stiffness ratio and self-stress. Scale effect as well as coupling of the first and second gradient terms are identified. The extended solid model can be used for the evaluation of unusual mechanical properties of tensegrity lattices.

Колебания слоя с расслоением в рамках градиентной теории упругости

2021 ◽  
pp. 3-15
Author(s):  
А.О. Ватульян ◽  
О.В. Явруян
Keyword(s):  
Boundary Integral Equations ◽  
Relative Length ◽  
Theory Of Elasticity ◽  
Boundary Integral ◽  
Stress Fields ◽  
Gradient Theory ◽  
Main Attention ◽  
Asymptotic Approach ◽  
Crack Tips ◽  
Analytical Expressions

The direct problem of antiplane oscillations of a layer with delamination in the context of the gradient theory of elasticity is considered. The gradient model proposed by Aifantis is taken as a mathematical model. The main attention has been paid to the analysis of mechanical fields at the crack bank and at its tips - stress concentrators. The study is carried out using the method of boundary integral equations (BIE). The BIE for the gradient of displacement field on the crack is constructed. The analysis of the constructed BIE is carried out and the cubic singularity is explicitly revealed. The solution of singular BIE is constructed using approximating Chebyshev polynomials. A study for cracks of small relative length - asymptotic approach is carried out, simple semi-analytical expressions for constructing the crack swap function are obtained, the range of efficiency of the asymptotic approach is obtained. The stress fields in the area of the crack tips are constructed. Numerical results of computational experiments are presented.

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.

Sign in / Sign up

Export Citation Format

Share Document