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.

Fundamental Concepts of Strength of Materials

2015 ◽  
pp. 10-30
Keyword(s):  
Constitutive Model ◽  
Constitutive Equations ◽  
Virtual Work ◽  
Elementary Theory ◽  
Elastic Beam ◽  
Theory Of Elasticity ◽  
Principle Of Virtual Work ◽  
Basic Tool ◽  
Strength Of Materials

This chapter presents the concepts of strength of materials that are relevant to the analysis of frames. These are the modified Timoshenko theory of elastic beams (Sections 2.1-2.3) and the Euler-Bernoulli one (Section 2.4). These concepts are not presented as in the conventional textbooks of strength of materials. Instead, the formulations are described using the scheme that is customary in the theory of elasticity and that was described in Chapter 1 (Section 1.1.1) (i.e. in terms of kinematics, statics, and constitutive equations). Kinematics is the branch of mechanics that studies the movement of solids and structures without considering its causes. Statics studies the equilibrium of forces; the basic tool for this analysis is the principle of virtual work. The constitutive model that describes a one-to-one relationship between stresses and deformations completes the formulation of the elastic beam problem. Finally, in Section 2.5, some concepts of the elementary theory of torsion needed for the formulation of tridimensional frames are recalled.

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 Strain Gradient Theory Based Dynamic Mindlin-Reissner and Kirchhoff Micro-Plates with Microstructural and Micro-Inertial Effects

Dynamics ◽  
2021 ◽  
Vol 1 (1) ◽  
pp. 49-94
Author(s):  
Stylianos Markolefas ◽  
Dimitrios Fafalis
Keyword(s):  
Boundary Conditions ◽  
Strain Gradient ◽  
Virtual Work ◽  
Plate Thickness ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Static Case ◽  
Cauchy Stress ◽  
Gradient Effect ◽  
Micro Plates

In this study, a dynamic Mindlin–Reissner-type plate is developed based on a simplified version of Mindlin’s form-II first-strain gradient elasticity theory. The governing equations of motion and the corresponding boundary conditions are derived using the general virtual work variational principle. The presented model contains, apart from the two classical Lame constants, one additional microstructure material parameter g for the static case and one micro-inertia parameter h for the dynamic case. The formal reduction of this model to a Kirchhoff-type plate model is also presented. Upon diminishing the microstructure parameters g and h, the classical Mindlin–Reissner and Kirchhoff plate theories are derived. Three points distinguish the present work from other similar published in the literature. First, the plane stress assumption, fundamental for the development of plate theories, is expressed by the vanishing of the z-component of the generalized true traction vector and not merely by the zz-component of the Cauchy stress tensor. Second, micro-inertia terms are included in the expression of the kinetic energy of the model. Finally, the detailed structure of classical and non-classical boundary conditions is presented for both Mindlin–Reissner and Kirchhoff micro-plates. An example of a simply supported rectangular plate is used to illustrate the proposed model and to compare it with results from the literature. The numerical results reveal the significance of the strain gradient effect on the bending and free vibration response of the micro-plate, when the plate thickness is at the micron-scale; in comparison to the classical theories for Mindlin–Reissner and Kirchhoff plates, the deflections, the rotations, and the shear-thickness frequencies are smaller, while the fundamental flexural frequency is higher. It is also observed that the micro-inertia effect should not be ignored in estimating the fundamental frequencies of micro-plates, primarily for thick plates, when plate thickness is at the micron scale (strain gradient effect).

Strain Gradient Finite Element Analysis on the Vibration of Double-Layered Graphene Sheets

2016 ◽  
Vol 13 (03) ◽  
pp. 1650011 ◽  
Author(s):  
Wei Xu ◽  
Lifeng Wang ◽  
Jingnong Jiang
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Scale Effect ◽  
Strain Gradient ◽  
Virtual Work ◽  
Gradient Elasticity ◽  
Kirchhoff Plate ◽  
Small Scale ◽  
Graphene Sheets ◽  
Simply Supported

A nonlocal Kirchhoff plate model with the van der Waals (vdW) interactions taken into consideration is developed to study the vibration of double-layered graphene sheets (DLGS). The dynamic equations of multi-layered Kirchhoff plate are derived based on strain gradient elasticity. An explicit formula is derived to predict the natural frequency of the DLGS with all edges simply supported. Then a 4-node 24-degree of freedom (DOF) Kirchhoff plate element is developed to discretize the higher order partial differential equations with the small scale effect taken into consideration by the theory of virtual work. It can be directly used to predict the scale effect on the vibrational DLGS with different boundary conditions. A good agreement between finite element method (FEM) results and theoretical natural frequencies of the vibration simply supported double-layered graphene sheet (DLGS) validates the reliability of the FEM. Finally, this new FEM is used to investigate the effect of vdW coefficients, sizes, nonlocal parameters, vibration mode and boundary conditions on the vibration behaviors of DLGS.

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.

Size effect in piezoelectric semiconductor nanostructures

2021 ◽  
pp. 1045389X2110530
Author(s):  
Jan Sladek ◽  
Vladimir Sladek ◽  
Miroslav Repka ◽  
Ernian Pan
Keyword(s):  
Continuum Model ◽  
Virtual Work ◽  
Semiconductor Nanostructures ◽  
Gradient Theory ◽  
Principle Of Virtual Work ◽  
The Finite Element Method ◽  
Strain Gradients ◽  
Numerical Examples ◽  
Piezoelectric Semiconductor ◽  
Mixed Fem

A gradient theory is applied to the mechanical constitutive equations for piezoelectric semiconductor nanostructures. This is achieved by considering the strain gradients in the constitutive equation with high-order stresses and electric displacements in advanced continuum model. The C1 continuous interpolations of displacements or a mixed formulation is required in the finite element method (FEM) due to the presence of the second-order derivative on the elastic displacements. A mixed FEM is then developed from the principle of virtual work. Numerical examples clearly show the significant effect of flexoelectricity on the induced electric potential and electric current in the piezoelectric semiconductor nanostructures.

scholarly journals Modified Nonlocal Strain Gradient Elasticity for Nano-Rods and Application to Carbon Nanotubes

2019 ◽  
Vol 9 (3) ◽  
pp. 514 ◽  
Author(s):  
Raffaele Barretta ◽  
Marko Čanadija ◽  
Francesco Marotti de Sciarra
Keyword(s):  
Carbon Nanotubes ◽  
Boundary Conditions ◽  
Strain Gradient ◽  
Scale Effects ◽  
Nonlocal Problem ◽  
Gradient Elasticity ◽  
Strain Gradient Theory ◽  
Gradient Theory ◽  
Nano Rods ◽  
Nonlocal Strain Gradient

Nowadays, the modified nonlocal strain gradient theory provides a mathematically well-posed and technically reliable methodology to assess scale effects in inflected nano-structures. Such an approach is extended in this paper to investigate the extensional behavior of nano-rods. The considered integral elasticity model, involving axial force and strain fields, is conveniently shown to be equivalent to a nonlocal differential problem equipped with constitutive boundary conditions. Unlike treatments in the literature, no higher-order boundary conditions are required to close the nonlocal problem. Closed-form solutions of elastic nano-rods under selected loadings and kinematic boundary conditions are provided. As an innovative implication, Young’s moduli of Single-Walled Carbon Nanotubes (SWCNT) weare assessed and compared with predictions of Molecular Dynamics (MD). New benchmarks for numerical analyses were also detected.

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.

Basic Equations of Dynamic Elastic Stability for Thin-Walled Structural Members Subjected to Follower Loads and Their Application to Non-Conservative Torque Problems

1985 ◽  
Vol 9 (2) ◽  
pp. 75-83 ◽  
Author(s):  
Tadayoshi Aida
Keyword(s):  
Virtual Work ◽  
Elastic Stability ◽  
Structural Members ◽  
Principle Of Virtual Work ◽  
Torsional Moment ◽  
Channel Section ◽  
Numerical Examples ◽  
Thin Walled ◽  
Basic Equations ◽  
The Stability

The basic equations and the boundary conditions, in which the effect of an initial torsional moment Mz0 is included, and needed for the analysis of the dynamic elastic stability of thin-walled structural members subjected to follower loads are derived by introducing the concept of initial stress and using the principle of virtual work. The stability problems of columns with a channel section subjected to a non-conservative torque are investigated in terms of numerical examples.

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