Pinnola Department of Constructions for Engineering and Architecture, University of Naples Federico II, Naples, Italy. Vaccaro Department of Constructions for Engineering and Architecture, University of Naples Federico II, Naples, Italy.
Structural Models
A nonlocal elastic Kirchhoff-Love plate model is then derived based on the lattice plate difference equations. The microstructure-based nonlocal model differs slightly from an Eringen Kirchhoff-Love stress gradient plate model.
1 Introduction
The chapter presents exact deflection and vibration solutions of the Hencky-Bar-Chain model (denoted as the Hencky beam model) for general boundary conditions. A non-locally elastic Kirchhoff-Love plate model is then constructed from the difference grid equations of the plate grids.
2 Discrete and Nonlocal Rods 2.1 Axial Lattices
- Lattice Formulation: Governing Equations
- Lattice Formulation: Resolution
- Nonlocal Continualized Model and Eringen’s Model
- Nonlocal Solutions
- Lattice with Direct and Indirect Interactions
Tarasov [67] also commented on the possible loss of final positivity of the macroscopic elastic energy (obtained by the continualization procedure) due to the structure of the generalized lattice-level interactions. Equation (61) is an approximation of Eq. 54) which is the exact natural frequency of the generalized lattice problem.
3 Discrete and Nonlocal Beams 3.1 Hencky-Bar-Chain Model
Continualised Nonlocal Beam Model
Regarding this dependence of the length scale on the type of analysis (bending, vibration, static bending), Wang et al. This nonlocal model can be elaborated by the discrete equilibrium equation and the discrete formulation of the constitutive buckling law
Buckling and Vibrations Analyses of Discretized Beam
The ability of the non-local beam theory to predict the natural frequencies of the Hencky-Bar-Chain model is also valid for higher frequencies. The exact formulas valid for all natural frequencies in the Hencky-Bar-Chain model include the fundamental frequency:. 108).
4 Discrete and Nonlocal Plates 4.1 Hencky-Bar-Chain Net
Eringen’s Nonlocal Plate Model
The small length scale parameter that exists in the Eringen's non-local model can be calculated based on the exact solution of the micro-structured plate model. A remarkable result is that the small length scale coefficient0 in the non-local rectangular plate varies with respect to both buckling modes and aspect ratios (see Fig.6).
Microstructure-Based Nonlocal Plate Model
This nonlocal slab model is similar to Eringen's nonlocal slab model except for the spatially correlated derivatives highlighted by the last two terms in Eq. Here it can be seen that the mesh-based nonlocal plate model differs slightly from the stress gradient Eringen plate model.
5 Conclusions
This explains the geometrical dependence of the scale factor of Eringen's non-local model, when we match the results of this model and the microstructured results. The corrected non-local model used in Eq. 152) has a length scale independent of the mode and the geometry of the plate.
Wang CM, Zhang H, Challamel N, Duan WH (2017) On boundary conditions for buckling and vibration of nonlocal beams. Challamel N, Lerbet J, Wang CM, Zhang Z (2014) Analytical length scale calibration of nonlocal continuum from a microstructured buckling model.
Mindlin [29] then assumed the cohesion within the above theories by introducing Taylor series (expansion) of the stretch of the non-local integral constitutive equation, leading to the derivation of the equations of the gradient elasticity theory. The above-mentioned inconsistencies of the non-local differential form are overcome provided that the integral voltage form can be reconsidered.
2 Eringen’s Formulation 2.1 Governing Equations
A Discussion About Eringen’s Nonlocal Stress Models
- Integral Equations, Fundamental Solutions and Green Functions The nonlocal integral constitutive equation being generally hard to handle is a first
- Two Phase Nonlocal Integral Constitutive Model
- Two Phase Constitutive Model with the Modified Kernel
The non-local stresses at a reference point x are defined by a weighted average via a non-local modulus (PDF) and the macroscopic (Cauchy) stresses of all the points of the body. The modified kernel can be found in Eq. 2), and this can then be considered as a physical extension of the non-local theory of Eq.
The Correlation Between Eringen’s Nonlocal Model and Mindlin’s Gradient Model
The violation of the latter at the boundary of a finite domain is obvious, since there are no material points outside the body that contribute to the stresses [48]. The classical types of a kernel do not satisfy the normalization condition on the boundary of the body, because they are normalized in an infinite domain.
3 Applications for One-Dimensional Problems
Beam Equilibrium Equations
- Integral Stress Models
- Differential Stress Model
The uniform cross section and the constant stiffness of the beam are denoted by S. The distributed mass, the rotational inertia, the density, the ~ moment of inertia, the cross section and the dynamic transversely distributed force of the beam are denoted respectively by m0,m2,ρ, I,Sandq(x,t).
Static Problems
- Direct Approach to the Nonlocal Integral Model
- Calculation of Static Deflection Through FEM
The transverse deflection of the TPNI stress pattern exhibits a monotonically increasing behavior as the nonlocal parameter increases. Furthermore, the transverse deviation of the nonlocal differential model is identical to that of the classical-local model.
Dynamical Problems
- Free Vibration Problem of a Beam
Moreover, all the eigenfrequencies of the TPNI stress model exhibit a monotonic behavior with respect to the non-local parameter (Fig.17b, c). The eigenfrequencies of the non-local models appear to have a softening response compared to that of the classical-local model in Fig.20.
4 Conclusions
The responses of the nonlocal integral models also differ from those of the nonlocal differential model. Eptaimeros KG, Koutsoumaris CC, Karyofyllis IG (2020) Eigenfrequencies of microtubules embedded in cytoplasm by nonlocal integral elasticity.
Nonlocal Mechanics in the Framework of the General Nonlocal Theory
One important long-standing issue is the identification of nonlocal parameters and length scales. Section 5 presents an approach to identify the nonlocal parameters of the nonlocal theory and the material coefficients of the strain and couple stress theories.
2 Eringen’s Nonlocal Theory
- Nonlocal Mechanics of Particles
- Nonlocal Continuum Mechanics
- Eringen’s Constitutive Equations
- Nonlocal Field Equation
- Dispersion Relations
- Limitations of Eringen’s Nonlocal Theory
Thus, the dynamic equilibrium of the material can be obtained based on the nonlocal theory integrating Eqs. This shows that Poisson's ratio is independent of the nonlocal field in the context of Eringen's nonlocal theory.
3 General Nonlocal Theory
- Equilibrium and Constitutive Equations
- Nonlocal Moduli
- Propagation of Dispersive and Aggregative Waves
- Comparison to Eringen’s Nonlocal Theory
In addition, the superiority of the general nonlocal theory over Eringen's nonlocal theory is demonstrated. Unlike Eringen's nonlocal theory, the general nonlocal theory gives Poisson's ratio depending on the nonlocal and dispersion characteristics of the material.
4 Relation to Strain Gradient and Couple Stress Theories
- Strain Gradient Theory
- Couple Stress Theory
- Relation to Mindlin’s Strain Gradient Theories
- Relation to Couple Stress Theory
- Wave Propagation
In the following, we show that form I of the nonlocal theory can be reduced to the deformation gradient theory. For a weak nonlocal moment field (i.e., weak nonlocality), Form II of the nonlocal theory can be reduced to the pair stress theory by expanding the rigid rotation fieldθ.
5 Infeasibility of Nonlocal Strain Gradient Theory
When more gradients were considered (up to the 4th order gradient), the deformation gradient theory gave exactly the same results as the general nonlocal theory. Nevertheless, the pair stress theory yields longitudinal acoustic waves identical to those of the classical theory.
6 Identification of Nonlocal Parameters and Length Scales of Strain Gradient and Couple Stress Theories
It is clear that strain gradient and couple stress theories can capture the reduction of wave frequency with wave number and damping behaviors like nonlocal theory. The use of the length scales of strain and couple stress theories was limited to only positive real values in the literature.
![Fig. 6 Identification of nonlocal parameters, λ and μ . The general nonlocal theory (curves) fits the experimental acoustic phonons along [100] direction (closed circles) of Ag, Au, Cu, diamond, Si, Pt, and graphite](https://thumb-ap.123doks.com/thumbv2/azpdfco/10391818.0/135.659.87.575.85.578/identification-nonlocal-parameters-nonlocal-experimental-acoustic-direction-graphite.webp)
7 Conclusions
According to Mindlin, the strain energy density function is real positive definite when the length scales are either real positive or complex [2,7,8]. Lim CW, Zhang G, Reddy J (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation.
Displacement Based Nonlocal Models for Size Effect Simulation in Nanomechanics
These other terms may account for the presence of small-scale microstructure effects. Specifically, after reviewing the main nonlocal models in Section 2, Section 3 is devoted to the displacement-based nonlocal model.
2 An Overview on the Nonlocal Models
Integral Models
Hence, the elastic stress tensor at a point x of the continuous field Bi is the output of the convolution integral between the local elastic stress field ε and the scalar kernel λ. E I(λL)2M(x) (8) with the following constitutive BCs. where the signs in Eq. 9) are due to the fact that the first derivative of the special kernel is an odd function.
Gradient Models
57], choosing the bi-exponential attenuation function, Eq. 10) is equivalent to the differential equation. λL)2∂x(kfχ)(L) (12) The non-local strain gradient model is therefore based on two parameters, λ and λl. It should be noted that forλ→0 and from Eq. 10), we recover the pure stress gradient law based on the nuclear impulsivity property.
Mixture Models
3 Displacement Based Nonlocal Model
Nonlocal Rod
- Discrete Model of the Nonlocal Rod
- Differential Formulation of the Nonlocal Problem
A corresponding discrete mechanical model of the shear-based non-local rod is introduced in this section. The effect of the non-local parameter λ on the mechanical response of the rod is investigated.
Dynamical Problem of Nonlocal Beam Model
- Finite Element Formulations
- Dynamical Analysis of Nonlocal Beams
Natural frequencies and modal shapes are evaluated for different values of the nonlocal parameter, i.e. λ. Figure 10 shows the first four natural frequencies of a simply supported beam as a function of the nonlocal parameter.
4 Concluding Remarks
Eringen AC (1983) On the differential equations of nonlocal elasticity and solutions of screw dislocations and surface waves. Challamel N, Zorica D, Atanackoviˇc TM, Spasiˇc DT (2013) On the fractional generalization of nonlocal Eringen elasticity for wave propagation.
Elasticity Models for Finite Domains
In recent years, several inconsistencies and paradoxical results have been discovered in the existing solutions of nonlocal elasticity approaches. 12] showed discrepancies between nonlocal differential models and nonlocal integral beam models for various boundary conditions.
2 One-Dimensional Well-Posed Nonlocal Elasticity Theory for Finite Domains
Nonlocal Integral Constitutive Equation
They demonstrated that the transformation of the non-local integral form to its differential counterpart is ill-posed unless the additional constitutive boundary conditions (the constitutive boundary conditions were first represented by Benvenuti and Simone [7]) are satisfied. In addition, they indicated that the inconsistency between the equilibrium conditions and the constitutive boundary conditions leads to the existing paradoxes between the integral and differential forms of the non-local theory.
Differential Constitutive Equation and Its Boundary Conditions
Therefore, the boundary conditions are rewritten as 16) It should be noted that the same equations can be derived in Ref. In general, a mathematical problem is well-posed if it satisfies two conditions: (1) the problem has a unique solution for a specific boundary condition, and (2) the solution depends continuously on its parameters, including the boundary conditions.
3 Equilibrium Equation and Boundary Conditions 3.1 Governing Equation of a Nanorod
Governing Equation of a Nanobeam
Now, by differentiating twice Eq. 42) with respect to tox and substituting the resulting equation into Eq. 35) the governing equation of the beam is obtained as. 40–42), the higher-order boundary conditions are derived as By setting the axial force parameter f =0, Eq. 51)–(57) reduces to the governing equation and boundary conditions for the bending of the well-posed nonlocal Euler beam under distributed transverse lasq¯.
4 Numerical Results
Static Deformation of Nanorods
Figure 5 shows the variations of the stress and the strain along the axis of the nanorod for various values of non-local parameter. It should be noted that the maximum displacement (1) of the nanorod does not depend on the non-local parameter.
Bending of Nanobeams
However, the deflections obtained from the current model are larger than those of the local model. Variations of the beam deflection are obtained from the local, non-local differential and current models and plotted in Figure 10.
Buckling of Nanobeams
The new formulation of nonlocal elasticity is focused on expressing the dynamic equilibrium requirements based on the nonlocal field of residual stresses. Mathematically speaking, solutions can be easily obtained using the differential form of the nonlocal elasticity (Eq.
2 Nonlocal Residual Elasticity
Nonlocal Continuum Mechanics: Balance Laws
The local stress (σ) is conjugated by direct neighbor interactions, while the nonlocal residual stress (τ) is conjugated by non-neighbor interactions. Equation (13) shows that the obliquely symmetric parts of the local stress tensor σ and the nonlocal residual stress tensor τ cancel out.
Nonlocal Continuum Mechanics: Constitutive Model
F+F+∇·σsym+∇·τsym=ρu¨ (14), where the balance equations depend on the symmetrical parts of the stress tensors,σsym andτsym. The magnitude of these Lamé moduli depends on the distance between two non-neighboring particles within the elastic domain (|x−x|).
Boundary Value Problem and Solution Procedure
In the context of this iterative procedure, a boundary value problem of the local type is corrected for the non-local field and then solved. Next, the determined non-local residual stress is substituted into the local boundary value problem (20)1 and (20)2.
3 Application to Euler-Bernoulli Beams
Thus, the nonlocal Eringen elasticity completely hides the role of the nonlocal residual field and does not provide any information about it. Figures 3,4 and 5 show the effectiveness of the proposed nonlocal iterative residual elasticity to model the nonlocal problems.
