scholarly journals Asymptotic method of construction of an applied model of deformation of thin beams by the gradient theory of elasticity

2020 ◽  
Vol 73 (1) ◽  
pp. 46-62
Author(s):  
L.S. Sargsyan ◽  
S.H. Sargsyan
Keyword(s):  
Asymptotic Method ◽  
Theory Of Elasticity ◽  
Gradient Theory ◽  
Applied Model ◽  
Thin Beams

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.

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 and disclinations in the gradient theory of elasticity

1999 ◽  
Vol 41 (12) ◽  
pp. 1980-1988 ◽  
Author(s):  
M. Yu. Gutkin ◽  
E. C. Aifantis
Keyword(s):  
Theory Of Elasticity ◽  
Gradient Theory

Acoustical activity in the framework of the rotation-gradient theory of elasticity

1986 ◽  
Vol 33 (8) ◽  
pp. 5795-5800 ◽  
Author(s):  
K. V. Bhagwat ◽  
R. Subramanian
Keyword(s):  
Theory Of Elasticity ◽  
Gradient Theory

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.

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.

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