Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams

doi:10.1016/j.ijmecsci.2016.10.036

International Journal of Mechanical Sciences

Volume 121, February 2017, Pages 151-156

https://doi.org/10.1016/j.ijmecsci.2016.10.036 Get rights and content

Highlights

•
Nonlocal Eringen elastic model is critically addressed.
•
Equivalence between integral and differential formulations is proven.
•
The solution of nonlocal elastic equilibrium problems is discussed.
•
Paradoxes in elastic nanobeams for MEMS and NEMS are explained.

Abstract

A debated issue, in applications of Eringen's nonlocal model of elasticity to nanobeams, is the paradox concerning the solution of simple beam problems, such as the cantilever under end-point loading. In the adopted nonlocal model, the bending field is expressed as convolution of elastic curvature with a smoothing kernel. The inversion of the nonlocal elastic law leads to solution of a Fredholm integral equation of the first kind. It is here shown that this problem admits a unique solution or no solution at all, depending on whether the bending field fulfils constitutive boundary conditions or not. Paradoxical results found in solving nonlocal elastostatic problems of simple beams are shown to stem from incompatibility between the constitutive boundary conditions and equilibrium conditions imposed on the bending field. The conclusion is that existence of a solution of nonlocal beam elastostatic problems is an exception, the rule being non-existence for problems of applicative interest. Numerical evaluations reported in the literature hide or shadow this conclusion since nodal forces expressing the elastic response are not checked against equilibrium under the prescribed data. The cantilever problem is investigated as case study and analytically solved to exemplify the matter.

Introduction

A challenging paradox of nonlocal mechanics is commonly considered to be faced in looking for the bending solution of elastic beams obeying the elastic integral nonlocal law Eq. (1) according to which the bending field is got by convolution of elastic curvature with the special smoothing kernel of Eq. (3), depicted in Fig. 1.

Striking examples are Bernoulli-Euler nonlocal cantilever nanobeams under end-point loading which find applications in microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS) as actuators or sensors.

The paradox, first detected in [1] and later claimed in [2], was that some bending solutions of integral-based nonlocal elastic beams are found to be identical to the classical (local) solution. This affirmation has been repeated several times in literature but a fully clarifying treatment has not yet been contributed.

Recently the issue has been newly drawn to attention by the discussion in [3] where the relation between integral and differential formulations of the nonlocal constitutive law is addressed and a treatment of paradoxical examples is performed by numerical computations based on the integral formulation.

The contradiction between equilibrium and nonlocal constitutive conditions is there considered responsible for preventing the use of the differential constitutive formulation and capable to explain differences between the results obtained by means of the integral formulation with those obtained by the differential formulation.

Our approach is more basic.

It is shown that the nonlocal integral elastic law is equivalent to a problem composed of constitutive differential and boundary conditions. These boundary conditions arise in a natural way in detecting the Green's function of differential problems defined on a bounded domain and provide an effective test to discriminate whether a bending field is obtainable by integral convolution or not.

In Proposition 3.1 it will be proved that fulfilment of constitutive boundary conditions by the bending field is necessary and sufficient condition in order to assure existence and uniqueness of the solution of the integral equation defining the corresponding elastic curvature.

At this point a general discussion of the elastostatic problem is appropriate. Firstly we observe that:

1.
The bending field solution of the elastostatic problem has to fulfil equilibrium with the imposed loading.
2.
The elastic curvature has to fulfil kinematic compatibility under the imposed boundary constraints and has to be associated with a bending field that meets the constitutive boundary conditions.

It follows that a solution of the elastostatic problem will exist only if the bending field, univocally detected among the equilibrated ones by imposing the conditions of kinematic compatibility to the corresponding elastic curvature field, will also meet the constitutive boundary conditions. This verification generally fails in cases of applicative interest.

The consequent interpretation of paradoxical examples is different from the one usually adduced in literature.

Our analysis reveals in fact that no paradox occurs since in all claimed examples the elastostatic problem does not admit solution, and it is exactly the presumed existence of a solution that lies at the root of all paradoxical results.

As a matter of fact, elastic beam problems formulated according to Eringen's nonlocal integral law, as a rule do not admit solution, existence being the exception.

Ill-posedness of nonlocal elastostatic problems is put into evidence by general considerations and by a specific discussion of the well-known paradox of nonlocal cantilevers under end-point loading.

Mixing of local and nonlocal material behaviours considered in literature are discussed in Section 6. It is shown that the local elastic fraction of the mixture has a beneficial effect and induces well-posedness. This effect is however abruptly cancelled when the local fraction vanishes so that a singular behaviour is expected in the limit of a vanishing local fraction, the inherent ill-posedness of fully nonlocal problems being not eliminated.

Other proposed remedies to overcome paradoxical results, such as numerical computations of discretised formulations, hide or shadow ill-posedness of nonlocal problems, an effect that should be checked by explicitly verifying equilibrium between nodal forces, expressing the elastic response, and the prescribed data.

Section snippets

Integral formulation

In the wake of the original formulation of nonlocal elasticity contributed by Eringen in [4], a nonlocal elastic law for the Bernoulli-Euler beam model is usually formulated by expressing the bending field $M$ in terms of the curvature field $χ_{el}$ by means of the integral convolution law $M_{λ} (x) = (ϕ_{λ} ⋆ (K \cdot χ_{el})) (x) = \int_{a}^{b} ϕ_{λ} (x - y) \cdot (K \cdot χ_{el}) (y) dy,$ with $L = b - a > 0$ beam length, $K = I_{E}$ local elastic bending stiffness, with $I_{E}$ second moment of the field of Euler elastic moduli $E$ on the beam cross section. The smoothing

Differential formulation and boundary conditions

The integral equation (1) in the unknown curvature field $χ_{el}$ and with the bending field $M$ as data, is known as Fredholm equation of the first kind. In general a solution of this kind of integral equations does not exist and, when it does, uniqueness cannot be assured [5]. However the following peculiar result is consequent to the choice of the special kernel Eq. (3).

Proposition 3.1

The constitutive integral equation (1) with the special kernel equation (3) admits, for any $λ > 0$ , either a unique solution or no

Green's function

The result in Proposition 3.1 can also be stated by saying that the special kernel Eq. (3) with $λ > 0$ is the Green's function associated with the differential problem $M_{λ} (x) - L_{c}^{2} \cdot M_{λ}^{″} (x) = δ (x),$ with the homogeneous boundary conditions (5).

In fact, by linearity, the solution of the differential expression Eq. (6) is provided by convolution of the Green's function with the datum $K \cdot χ_{el}$ , as expressed by Eq. (1).

By linearity again, the homogeneous constitutive boundary conditions (5) will be fulfilled also

Nonlocal elastostatic problem

In the geometrically linearised Bernoulli-Euler beam model, the displacement fields are required to be square integrable together with the first and second generalised derivatives, so that boundary values of the displacement fields and of their first derivatives can be properly considered. This kinematical space is a Hilbert space denoted by $V := H^{2} (a, b)$ .

Formulation of the nonlocal elastostatic problem is completed by adding to the constitutive law equation (1) the following items concerning

Local/nonlocal mixture

A two-phases constitutive mixture, defined by a convex combination of local and nonlocal phases, was introduced in [9], [10]. This mixture model has been recently resorted to in [2], [7], [11], [12], [13].

The nonlocal constitutive law is accordingly expressed by $M_{λ} (x) = ξ (K \cdot χ_{el}) (x) + (1 - ξ) \int_{a}^{b} ϕ_{λ} (x - y) \cdot (K \cdot χ_{el}) (y) dy,$ with $0 \leq ξ \leq 1$ phase parameter.

The fully nonlocal law is recovered by setting $ξ = 0$ , while the standard local law corresponds to $ξ = 1$ .

A procedure analogous to the one in the proof of Proposition 3.1

Paradox of cantilever under end-point loading

A case-study in nonlocal elastic beam theory is represented by a cantilever under end-point loading. The discussion at the end of Section 5.1 reveals that this statically determinate problem does not admit solution.

Anyway non-existence of a solution can also be verified by a direct analysis which puts into evidence singularities of the involved fields when a wrong procedure is carried out.

The equilibrium differential condition $M_{λ}^{″} (x) = q (x) = 0,$ and the boundary equilibrium conditions, require that

Concluding remarks

The constitutive law of nonlocal elasticity proposed by Eringen in [4], was adapted to unidimensional beam models in [1] and thence widely adopted, with a multitude of investigations dealing with static and dynamic behaviour of micro and nano-beams and applications to MEMS and NEMS.

A proliferation of contributions has spread out in the literature, notwithstanding various signals were indicating that something basic was not going in the right way.

To contradictory outcomes of various analyses,

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Constitutive boundary conditions and paradoxes in nonlocal elastic nanobeams

Highlights

Abstract

Introduction

Section snippets

Integral formulation

Differential formulation and boundary conditions

Green's function

Nonlocal elastostatic problem

Local/nonlocal mixture

Paradox of cantilever under end-point loading

Concluding remarks

Int J Eng Sci

Int J Eng Sci

Mech Res Comm

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