4 Numerical Results
4.3 Buckling of Nanobeams
In this subsection, the dimensionless buckling loads of the cantilever beam will be given for different models and nonlocal parameters. Table1lists the critical buckling loads for different values of nonlocal parameters from zero to 0.25. As can be inferred from this table, the nonlocal parameter decreases the buckling loads in comparison with those of the local results. Furthermore, the buckling loads of the nonlocal dif- ferential model are smaller than ones predicted by the present model. It should be noted that the negative sign shows the dimensionless load f is compressive.
Table 1 The buckling loads of the cantilever beam
λ Local model Nonlocal differential
model
Present model
0.00 −2.4674 −2.4674 −2.4674
0.05 −2.4674 −2.4523 −2.4540
0.10 −2.4674 −2.4080 −2.4213
0.15 −2.4674 −2.3376 −2.3803
0.20 −2.4674 −2.2458 −2.3401
0.25 −2.4674 −2.1377 −2.3065
5 Conclusions
In this chapter, we developed a modified strain-based integral model for finite domains and utilize it to predict the static deformation of nanorods and the static behavior of nanobeams (bending and buckling). Numerical results for several static problems were solved and compared with the local and nonlocal differential models.
The main conclusions of this chapter are as follows:
• The developed models obey the locality recovery condition, which implies that the classical Hooke’s law is recovered in the presence of a uniform strain field, no matter the value of the length scale parameter.
• Using the well-posed nonlocal elastic models, the nonlocal paradoxical results are resolved.
• Elastic displacement solutions of the studied nanorods and nanobeams, formulated according to the well-posed models, indicate the softening behavior when the nonlocal parameter is increased.
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Mohamed Shaat
Abstract Motivated by the existing complications of finding solutions of Eringen’s nonlocal model, an alternative model is developed here. The new formulation of the nonlocal elasticity is centered upon expressing the dynamic equilibrium requirements based on a nonlocal residual stress field. This new nonlocal elasticity is explained from the lattice mechanics and continuum mechanics points of view. Boundary value problems obtained based on the new nonlocal elasticity are solved following a pro- posed iterative procedure. This iterative procedure is centered upon correcting the solution of the classical field problem for the nonlocal residual field of the elastic domain. Convergence analyses are presented to show the convergence of the iterative procedure to the solution of the nonlocal field problem. The iterative procedure is an integrated part of the proposed nonlocal elasticity. Therefore, the newly developed nonlocal elasticity is given the name “iterative nonlocal residual elasticity”.
1 Introduction
In classical mechanics, a body consists of an infinite number of particles each of which is a mass point. Each particle undergoes interactions only with the nearest particles (neighbor particles). Whereas the interactions between non-neighbor par- ticles are weaker than interactions between neighbor particles, these non-neighbor interactions are exist and may contribute to the continuum mechanics in certain occa- sions. Therefore, the nonlocal theory was developed to propose a generalized theory, which models the particle exhibits interactions with its neighbors and non-neighbors.
According to Eringen [1, 2], the dynamic equilibrium of a solid elastic body is conditional by the global balance of its body forces, external surface tractions, and inertia forces. The nonlocal theory, as early proposed by Eringen, postulates the condition of the global balance, thus [2]:
M. Shaat (
B
)Mechanical Engineering Department, Abu Dhabi University, P.O.BOX 1790, Al Ain, United Arab Emirates
e-mail:[email protected];[email protected]
© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 E. Ghavanloo et al. (eds.),Size-Dependent Continuum Mechanics Approaches, Springer Tracts in Mechanical Engineering,
https://doi.org/10.1007/978-3-030-63050-8_6
169
tji,j+Fi+Fi=ρ∂2ui
∂t2 (1)
whereρis the mass density,uiis the displacement field of a material particle, andt is the time.Fiis a local-type body force, which is conjugate to neighbor interactions.
Fiis a nonlocal body force. The nonlocal body force (Fi) is a residual field, which is generated due to non-neighbor interactions. This nonlocal residual body force yields to the condition [1,2]:
V
FidV =0 (2)
whereVis the volume of the elastic body. For a homogeneous-isotropic-linear elastic material,Fi=0 [1,2].
The balance laws of nonlocal continua (Eq. (1)) depend on a total stress field (tij), which sums neighbor interactions and non-neighbor interactions at a specific pointx belongs to the elastic body. This total stress field (tij) was expressed as a functional of the deformation gradients of all points of the elastic body [1,2]. Thus, for a homogeneous-isotropic-linear elastic material, the constitutive equations were determined based on some invariance and thermodynamics requirements with the form [1,2]:
tij(x)=
V
λx−xεrr
x
δij+2μx−xεij
x dV
x
(3)
whereεijis the infinitesimal strain.λandμare Lame moduli, which are functionals ofx−x. The integration over the elastic domain,V, was involved to collect all non-neighbor interactions.
With Eringen’s manipulation of the nonlocal theory, the balance laws are identical to those of the classical continuum mechanics [1,2]. However, the constitutive law is different where the total stress field (tij) was introduced to model neighbor and non-neighbor interactions via integral operators (Eq. (3)). This total stress field is commonly known as “nonlocal stress field”.
Complications of finding solutions of nonlocal elasticity problems have been early discussed [3]. According to Eq. (3), the field equation of the nonlocal problem is an integro-partial differential equation. Analytical solutions for integro-partial differential equations are difficult to be determined, especially for mixed boundary value problems [3]. This motivated Eringen [3] to formulate the nonlocal elasticity in terms of singular differential equations. By assuming that (i) the Lame moduli exhibit the same attenuation with the distancex−xaccording to (ii) a Green’s function of a differential operator with constant coefficients, the nonlocal field problem was formulated as the following partial differential equation [3]:
σji,j+L
Fi−ρ∂2ui
∂t2
=0 (4)
whereLis the differential operator as defined by Eringen [3], i.e.Ltij =σij.σij is the local stress of the classical elasticity.
Mathematically speaking, solutions can be easily obtained using the differential form of the nonlocal elasticity (Eq. (4)). Nonetheless, the ability of this form to secret exact solutions, which should be consistent with the solutions of the integral nonlocal model is controversial. Discrepancies were observed between solutions of the differential nonlocal field problems and integral nonlocal field problems [4–7].
Details on the paradoxes of the differential nonlocal elasticity can be found in Ref.
[4–15]. Therefore, the current focus of research on nonlocal mechanics depends on the integral nonlocal model [16–18].
Moreover, the “ill-posedness” of Eringen’s nonlocal elasticity was discussed in Ref. [15]. It was revealed that the transformation of the integral constitutive law (Eq. (3)) into a differential nonlocal constitutive law secretes inconsistencies between the natural boundary conditions of the nonlocal equilibrium and the boundary con- ditions of the constitutive model generated upon the transformation [15].
An example of the paradox and inconsistency of the differential nonlocal elasticity can be exhibited by considering a nonlocal problem of an elastic body under a body force equals its inertia force, in particularFi−ρ∂2ui/∂t2 =0. An example of the latter case is a nonlocal static problem with a zero body force. For this case, Eq. (4) reduces to:
σji,j=0 (5)
which gives the stress field identical to that of the classical elasticity with no depen- dency on the nonlocal character of the body. This is a clear violation of the concept of the nonlocal mechanics.
The illustrated example and the paradoxes and inconsistencies expressed in pre- vious studies imply that the transformation of the integral nonlocal constitutive law (Eq. (3)) into a differential nonlocal constitutive law as proposed by Eringen [3]
may be not convenient to express nonlocal boundary value problems, especially in bounded domains [15]. Thus, the transformation of the integral nonlocal constitutive law into a differential nonlocal constitutive law is an injective process in infinite domains, yet it is not in finite domains [16]. However, with the deep inspection of the situation, one can realize the fact that the transformation into the differential constitutive law proposed in Ref. [3] is mathematically consistent. So, the question is: what are the reasons behind the paradoxes of Eringen’s nonlocal elasticity?
Motivated by the aforementioned complications and debates regarding Eringen’s nonlocal elasticity presented in Eqs. (1)–(4), an alternative form of the nonlocal theory is developed in this chapter. This new form of the nonlocal elasticity depends on splitting the total nonlocal stress field (tij) into a local stress (σij) and a nonlocal residual stress (τij), i.e.tij=σij+τij. The local stress (σij) is a stress field that models the material stiffness due to neighbor (local) interactions. The nonlocal residual stress
(τij) models the material stiffness due to non-neighbor (nonlocal) interactions. In the context of the new nonlocal elasticity, the balance laws are similar to those of the classical elasticity of an elastic continuum with a residual stress field. The proposed nonlocal theory is centered upon modifying the classical elasticity for an inherited nonlocal residual field. At the boundary of the elastic body, the surface tractions depend on both the local stress and the nonlocal residual stress.
The new manipulation of the nonlocal theory proposed here permits implementing an iterative procedure to extract solutions of nonlocal field problems. This iterative procedure is centered upon correcting the solution of the classical field problem for the nonlocal residual field of the elastic domain. In the context of the proposed approach, the classical field problem of an elastic domain with a pre-formed nonlocal residual stress is solved. Convergence analyses are presented to show the convergence of the iterative procedure to the solution of the nonlocal field problem formed based on the proposed nonlocal theory. It should be mentioned that the iterative procedure is an integrated part of the theory. Therefore, the newly developed form of the nonlocal theory will be give the name “iterative nonlocal residual elasticity”.