Noël Challamel
email: noel.challamel@univ-ubs.fr

Zhen Zhang

C M Wang

• J N Reddy

Q Wang
email: q.wang@umanitoba.ca

Thomas Michelitsch

Bernard Collet
email: bernard.collet@upmc.fr

J N Reddy
email: jnreddy@tamu.edu

Keywords: Nonlocal model, Elasticity, Nonconservative model, Energy functional, Self-adjoint formulation, Microstructured media, Beam mechanics

In this paper, the self-adjointness of Eringen's nonlocal elasticity is investigated based on simple one-dimensional beam models. It is shown that Eringen's model may be nonself-adjoint and that it can result in an unexpected stiffening effect for a cantilever's fundamental vibration frequency with respect to increasing Eringen's small length scale coefficient. This is clearly inconsistent with the softening results of all other boundary conditions as well as the higher vibration modes of a cantilever beam. By using a (discrete) microstructured beam model, we demonstrate that the vibration frequencies obtained decrease with respect to an increase in the small length scale parameter. Furthermore, the microstructured beam model is consistently approximated by Eringen's nonlocal model for an equivalent set of beam equations in conjunction with variationally based boundary conditions (conservative elastic model). An equivalence principle is shown between the Hamiltonian of the microstructured system and the one of the nonlocal continuous beam system. We then offer a remedy for the special case of the cantilever beam by tweaking the boundary condition for the bending moment of a free end based on the microstructured model.

Introduction

Nonlocal elasticity theories provide efficient reversible mechanisms for capturing small scale effects induced by the microstructure or the discrete nature of the material at a smaller scale. In order to allow for these small length scale effects in microstructured materials (such as nanostructures for instance), Eringen's nonlocal elasticity based on an implicit differential equation between the stress and the strain quantities may be used [START_REF] Eringen | On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves[END_REF][START_REF] Eringen | Nonlocal Continuum Field Theories[END_REF]. Eringen's nonlocal elasticity can be presented in its simplest form, expressed for uniaxial case, as

σ (x) -l 2 c σ ′′ (x) = Eε (x) (1)
where σ (x) is the uniaxial stress, ε (x) the uniaxial strain, x the spatial coordinate, l c the length scale parameter responsible for the nonlocal effect, E the modulus of elasticity, and the double prime denotes the second derivative with respect to x. The first nonlocal solutions in structural mechanics are rather recent and date from the beginning of this century, with specific applications to nanotechnology. Surprisingly, we have only one decade of active research in the field of nonlocal structural mechanics, since the appearance of two pioneering papers by Peddieson et al. [START_REF] Peddieson | Application of nonlocal continuum models to nanotechnology[END_REF] and Sudak [START_REF] Sudak | Column buckling of multiwalled carbon nanotubes using nonlocal continuum mechanics[END_REF] in 2003. New solutions have then been found and developed following these two papers, which include nonlocal Euler-Bernoulli, nonlocal Timoshenko, and nonlocal higher-order beam theories (see for instance [START_REF] Reddy | Nonlocal theories for bending, buckling and vibration of beams[END_REF][START_REF] Reddy | Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates[END_REF][START_REF] Elishakoff | Carbon Nanotubes and Nanosensors: Vibrations, Buckling and Ballistic Impact[END_REF][START_REF] Challamel | Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams[END_REF]).

Eringen's nonlocal elasticity appears to be not only an engineering phenomenological model that is able to introduce, rather arbitrarily, some scale effects, but it has also been shown to capture Born-Karman lattice dispersion curves related to the discreteness of the lattice structure (see [START_REF] Eringen | On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves[END_REF] for the calibration of the small length scale on wave properties of lattice structures). Hence, such nonlocal models have found numerous applications in the field of wave propagation in heterogeneous materials (see [START_REF] Maugin | Nonlinear Waves in Elastic Crystals[END_REF]or [START_REF] Lazar | On a theory of nonlocal elasticity of bi-Helmholtz type and some applications[END_REF]). The calibration and the micromechanics justification of Eringen's nonlocal elasticity in beam mechanics are more recent. Challamel et al. [START_REF] Challamel | Analytical length scale calibration of nonlocal continuum from a microstructured buckling model[END_REF][START_REF] Challamel | Nonlocal equivalent continuum for the buckling and the vibrations of microstructured beams[END_REF]orWangetal. [START_REF] Wang | Calibration of Eringen's small length scale coefficient for initially stressed vibrating nonlocal Euler beams based on microstructured beam model[END_REF] showed, with a so-called continualization method, that microstructured bending systems globally behave as nonlocal beams of Eringen's type. It has been shown also that the spatial nonlocality is different for static analysis (buckling problems) as compared to dynamics analysis. The same kind of results has been also obtained for discrete shear systems continualized as nonlocal Timoshenko beam elements (see [START_REF] Duan | Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams[END_REF]or [START_REF] Zhang | Eringen's small length scale coefficient for buckling of nonlocal Timoshenko beam based on a microstructured beam model[END_REF]).

Self-adjointness of Eringen's elasticity applied to beam mechanics

One question that is still debated about the applicability and the limitations of Eringen's model is whether the model is self-adjoint and if it can be justified from energy considerations. In a seminal paper dated from 2007, Reddy [START_REF] Reddy | Nonlocal theories for bending, buckling and vibration of beams[END_REF] mentioned the possible nonself-adjointness of Eringen's model and the impossibility to build the underlying quadratic functional for such nonlocal beam models (see, also [START_REF] Reddy | Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates[END_REF]). Later, following the works of Adali [START_REF] Adali | Variationalprinciples for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory[END_REF][START_REF] Adali | Variational principles for transversely vibrating multi-walled carbon nanotubes based on nonlocal Euler-Bernoulli beam models[END_REF], it has been shown that the governing equations of Eringen's type nonlocal elasticity applied to several beam mechanics models may be derived from a single-energy functional, with some variationally based boundary conditions (see also [START_REF] Elishakoff | Carbon Nanotubes and Nanosensors: Vibrations, Buckling and Ballistic Impact[END_REF]or [START_REF] Challamel | Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams[END_REF]). In the present paper, we attempt to reconcile both approaches: strictly speaking, it can be shown for specific boundary conditions that Eringen's nonlocal elasticity is not self-adjoint and one cannot derive the governing equations (including the consistent boundary conditions) of the nonlocal models from an energy functional. This confirms Reddy's anticipated result [START_REF] Reddy | Nonlocal theories for bending, buckling and vibration of beams[END_REF]. However, we also find that Eringen's nonlocal elasticity may be self-adjoint for some other boundary conditions. We also find that the nonself-adjointness of Eringen's nonlocal model is not physically justified from the microstructural point of view. In other words, a correction of the boundary conditions is needed when considering the nonlocal model of Eringen for accurately describing microstructure effects.

The focus of this paper on the possible nonself-adjointness of the differential-based Eringen's model does not cover some other integral-based nonlocal models available in the seminal book of Eringen [START_REF] Eringen | Nonlocal Continuum Field Theories[END_REF], where for instance the kernel of the nonlocal model is assumed to have symmetrical spatial properties, a property that may be lost with the differential-based nonlocal model investigated in this paper herein.

As recently shown in [START_REF] Challamel | Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams[END_REF], it is possible to build a functional for the bending, vibration, and buckling problems using nonlocal Euler-Bernoulli beams, as well as higher-order beam models, to obtain the correct differential equations of Eringen's beam models. The works of Adali [START_REF] Adali | Variationalprinciples for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory[END_REF][START_REF] Adali | Variational principles for transversely vibrating multi-walled carbon nanotubes based on nonlocal Euler-Bernoulli beam models[END_REF] and Elishakoff et al. [START_REF] Elishakoff | Carbon Nanotubes and Nanosensors: Vibrations, Buckling and Ballistic Impact[END_REF]f o r nonlocal Timoshenko or higher-order beam models contain energy functionals. Apparently, it is possible, by an inverse procedure that uses the governing equations of the nonlocal model to construct the functional, to have a self-adjoint formulation of Eringen's beam models. However, there is a need for the clarification of the meaning of variationally consistent boundary conditions (i.e., boundary conditions derived from the functional constructed using the inverse procedure) that may differ from the boundary conditions obtained by the direct approach.

Considering both buckling and vibration problems, we will show in this paper that:

the buckling problem of the Eringen nonlocal beams (Euler-Bernoulli, Timoshenko or higher-order shear beam models) is self-adjoint, i.e., it is possible to derive both the differential equations and the associated boundary conditions from the variation of a single functional; and the vibration problem is different in nature. The hinged-hinged beam problem is self-adjoint, but not selfadjoint for the clamped-free nonlocal beam (or for any boundary condition that involves the specification of a generalized force including a concentrated force and a concentrated torque).

The meaning of this possible nonself-adjointness for the free-edge condition is discussed at the end of this paper.

Dependency of Eringen's kernel on boundary conditions

Let us consider the moment-curvature relation in the nonlocal Euler-Bernoulli beam theory based on Eringen's differential model:

M (x) -l 2 c M ′′ (x) = EIχ (x) withχ (x) = w ′′ (x) (2)
where M is the bending moment, I the second moment of area, χ the curvature, and w the deflection of the beam. Eringen's nonlocal elasticity appears to be efficient in capturing the scale effects of microstructured media such as microstructured beams (see for instance [START_REF] Challamel | Analytical length scale calibration of nonlocal continuum from a microstructured buckling model[END_REF][START_REF] Challamel | Nonlocal equivalent continuum for the buckling and the vibrations of microstructured beams[END_REF]). The length scale parameter l c can then be computed from the microstructure cell size.

As pointed out by Eringen [START_REF] Eringen | On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves[END_REF], this differential equation shows that the bending moment is a spatially weighted average of the curvature variable with the weighting function being the Green's function G (x, y) of the differential system associated with the relevant boundary conditions:

M (x) = EI L 0 G (x, y)χ (y) dy ( 3
)
where L is the length of the beam. If for instance, the beam is rotationally free at the boundary (hinge-hinge supported beams for instance), the Green's function is expressed as [START_REF] Elishakoff | Carbon Nanotubes and Nanosensors: Vibrations, Buckling and Ballistic Impact[END_REF]:

M (0) = M (L) = 0 ⇒ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ G (x, y) = 1 l c sinh L-y lc sinh L lc sinh x l c if x ≤ y G (x, y) = 1 l c sinh L-x lc sinh L lc sinh y l c if x ≥ y (4)
Note that the Green's function is symmetrical in this case, i.e., G (x, y) = G (y, x). The Green's functions strongly depend on the boundary conditions and are obtained from the differential equations (see for instance [START_REF] Roach | Green's Functions[END_REF]):

G (x, y) -l 2 c ∂ 2 G (x, y) ∂ x 2 = δ (x -y) (5)
with the solution expressed by

G (x, y) = A (y) cosh x l c + B (y) sinh x l c if x < y G (x, y) = C (y) cosh x l c + D (y) sinh x l c if x > y (6)
The introduction of the four following boundary conditions leads to the Green's function of Eq. ( 4):

G (0, y) = 0, G (L , y) = 0, G y -, y -= G y + , y + and ∂G ∂ x y + , y + - ∂G ∂ x y -, y -= -1 l 2 c ( 7
)
Following the same procedure, it is also possible to express other Green's function. For instance, in the case of translationally free boundaries (see also [START_REF] Elishakoff | Carbon Nanotubes and Nanosensors: Vibrations, Buckling and Ballistic Impact[END_REF]):

V (0) = M ′ (0) = V (L) = M ′ (L) = 0 ⇒ ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ G (x, y) = 1 l c cosh L-y lc sinh L lc cosh x l c if x ≤ y G (x, y) = 1 l c cosh L-x lc sinh L lc cosh y l c if x ≥ y ( 8
)
where V is the shear force variable. For such boundary conditions, the nonlocal operator is self-adjoint and the nonlocal energy functional is given, for instance by Challamel and Wang [START_REF] Challamel | The small length scale effect for a non-local cantilever beam: a paradox solved[END_REF](seealso [START_REF] Polizzotto | Nonlocal elasticity and related variational principles[END_REF]or [START_REF] Eringen | Nonlocal Continuum Field Theories[END_REF]formore general energy functionals applied to three-dimensional nonlocal elastic media):

π = L 0 L 0 1 2 EIG(x, y) χ (y) χ (x) dxdy = L 0 1 2 M (x) χ (x) dx (9)
As detailed in [START_REF] Challamel | The small length scale effect for a non-local cantilever beam: a paradox solved[END_REF] and citing Roach [START_REF] Roach | Green's Functions[END_REF] for instance, following the Green-type identity, the nonlocal operator is self-adjoint for unmixed or periodic boundary conditions. This is typically the case of the moment-free boundary conditions (see for instance Eq. ( 4) for the hinge-hinge supported beam, where M is prescribed and equal to 0), or shear-free boundary conditions (see Eq. ( 8) for translational-free boundary conditions). This is no longer the case for the clamped-free boundary condition. Therefore, Eq. ( 9) is not valid for all the boundary conditions, and this explains a part of the apparent paradox (the self-adjointness of Eringen's nonlocal model depends on the considered boundary conditions).

The nonlocal system is self-adjoint if:

L 0 M (x) δχ (x) dx = L 0 δ M (x) χ (x) dx (10)
It can be shown by integration by parts that [START_REF] Challamel | The small length scale effect for a non-local cantilever beam: a paradox solved[END_REF]:

L 0 χ (x) δ M (x) dx = L 0 M (x) δχ (x) dx - l 2 c EI M ′ δ M L 0 + l 2 c EI Mδ M ′ L 0 (11)
Hence, the self-adjointness is controlled by the following equality:

M ′ δ M L 0 -Mδ M ′ L 0 = 0( 1 2
) As a consequence, in order to fulfil the conditions Eq. ( 12), the following boundary conditions are associated with a self-adjoint formulation of Eringen's nonlocal Euler-Bernoulli beam model:

M (0) = M (L) = 0, M ′ (0) = M ′ (L) = 0, M (0) = M ′ (L) = 0orM ′ (0) = M (L) = 0 ( 13
)
, whereas the following boundary conditions (including the clamped-free boundary conditions) lead to a nonself-adjoint formulation of Eringen's-based nonlocal Euler-Bernoulli beam model:

M (0) = M ′ (0) = 0o rM (L) = M ′ (L) = 0( 1 4 )
In the case of Eq. ( 14), the self-adjointness of the problem cannot be established (and the condition Eq. ( 12) is not necessarily valid). For instance, for the clamped-free boundary conditions

V (0) = M (0) = 0 ⇒ G (x, y) = 0i fx ≤ y G (x, y) = 1 l c sinh y-x l c if x ≥ y (15)
It appears in this case that G (x, y) is no longer symmetrical in this case, and the energy formulation Eq. ( 9) cannot be used in such a case.

Buckling problem

The buckling equations for a beam loaded by an axial load P are obtained from the equilibrium equations

M ′′ + Pw ′′ = 0 ( 16
)
and the Eringen's-based nonlocal moment-curvature law:

M -l 2 c M ′′ = EIw ′′ (17)
leading to the differential equation for the deflection:

EI -Pl 2 c w (4) + Pw ′′ = 0 (18)
In this case, it is possible to show that the problem is self-adjoint whatever the boundary conditions, and the energy functional U is shown to be expressed simply by (see [START_REF] Elishakoff | Carbon Nanotubes and Nanosensors: Vibrations, Buckling and Ballistic Impact[END_REF][START_REF] Adali | Variationalprinciples for multi-walled carbon nanotubes undergoing buckling based on nonlocal elasticity theory[END_REF]or [START_REF] Challamel | Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams[END_REF]):

U [w] = π [w] -W e [w] = L 0 1 2 Mw ′′ - 1 2 Pw ′2 dx = L 0 1 2 EI -Pl 2 c w ′′2 - 1 2 Pw ′2 dx ( 19
)
where π is the internal bending energy, whereas W e is the work done by the axial force. The differential equation for the buckling of a nonlocal Euler-Bernoulli column is obtained from the application of the variational principle δU = 0, leading to the differential equations Eq. ( 18), with the natural and essential boundary conditions:

EI -Pl 2 c w ′′ δw ′ L 0 = 0a n d -EI -Pl 2 c w ′′′ + Pw ′ δw L 0 = 0( 2 0 )
which can also be expressed with the static variables as

M δw ′ L 0 = 0a n d -M ′ + Pw ′ δw L 0 = 0 (21)
In this case, the problem is self-adjoint whatever the boundary conditions, as the bending moment is directly proportional to the curvature:

M = EI -Pl 2 c χ (22)
which is a kind of modified "locality", without any Green's functions. Therefore, the discussion on the selfadjoint property is simplified in this case, due to the fact that the nonlocal law degenerates into a kind of local law in Eq. ( 22).

Vibration problem

Now, for the vibrations analysis, we have

M -l 2 c M ′′ = EIw ′′ M ′′ =-ρ A ẅ ( 23
)
where ρ is the volumetric mass density and A is the cross-sectional area. We finally obtain the nonlocal bending wave equation:

EI ∂ 4 w ∂ x 4 -µl 2 c ∂ 4 w ∂ x 2 ∂t 2 + µ ∂ 2 w ∂t 2 = 0( 2 4 )
with µ = ρ A.Asmentionedin [START_REF] Challamel | Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams[END_REF], this wave equation corresponds to the vibration problem of a Rayleigh's beam, when the radius of gyration of the cross section r is assumed to be equal to the nonlocal length scale l c .Adali [START_REF] Adali | Variational principles for transversely vibrating multi-walled carbon nanotubes based on nonlocal Euler-Bernoulli beam models[END_REF] suggested that the nonlocal wave equation can be derived from the following strain energy and kinetic energy functionals:

U [w] = L 0 1 2 EIw ′′2 dx and T [w] = 1 2 L 0 µ ẇ2 + µl 2 c ẇ′2 dx ( 25
)
where µ is the mass per unit length of the beam material. The dynamic equations are obtained via the Hamilton principle, leading to Eq. ( 24), with the variationally based boundary conditions:

-EIw ′′′ + µl 2 c ẅ′ δw L 0 = 0a n d EIw ′′ δw ′ L 0 = 0 ( 26
)
or equivalently:

-M ′ δw L 0 = 0a n d M + µl 2 c ẅ δw ′ L 0 = 0 ( 27
)
It means that the bending-type free boundary condition M + µl 2 c ẅ = 0 is different in this case from Eringen's model which should be simply M = 0 and then the variationally based model based on Eq. ( 27) is not strictly equivalent to Eringen's model:

-M ′ δw L 0 = 0a n d Mδw ′ L 0 = 0 (28)
Note that only the boundary conditions differ from the variationally based model and Eringen's nonlocal model. As a second consequence, Eringen's model can be considered from the following variational statement:

t 2 t 1 δU -δT -µl 2 c ẅδw ′ L 0 dt = 0 ( 29
)
which is clearly nonself-adjoined, as µl 2 c ẅδw ′ L 0 cannot be derived from a potential. This additional term µl 2 c ẅδw ′ L 0 responsible of the nonconservativeness of the nonlocal vibrations problem can be attributed to a nonconservative inertia moment applied at the end of a Rayleigh's beam. Note that for the hingedhinged beam problem, this last term is vanishing as w = 0 along the boundaries, and the nonlocal elasticity problem is then self-adjoined (equivalent to the considering of the energy functional). This proof confirms the nonself-adjointness of Eringen's model without boundary corrections. This result is in agreement with Reddy's analysis [START_REF] Reddy | Nonlocal theories for bending, buckling and vibration of beams[END_REF] who has already mentioned the possible nonself-adjointness of Eringen's model, and the impossibility to build the underlying quadratic functional for such nonlocal beam models (see, also [START_REF] Reddy | Nonlocal nonlinear formulations for bending of classical and shear deformation theories of beams and plates[END_REF]). This nonself-adjointness explains why nonsymmetrical mass matrix can be derived for Eringen's nonlocal beam analyses in finite element beam analyses [START_REF] Phadikar | Variational formulation and finite element analysis for nonlocal elastic nanobeams and nanoplates[END_REF].

The particular boundary conditions for having the vibration problem nonself-adjoint is the clamped-free boundary condition (see Fig. 1). In Fig. 2, the two boundary conditions Eq. ( 27) called the proposed nonlocal model (exact self-adjoint nonlocal model) and Eq. ( 28) (exact nonself-adjoint Eringen's nonlocal model) are compared for the frequency sensitivity with respect to the small length scale. It is shown in Fig. 2 that the small length scale effect has a softening influence in case of the self-adjoint nonlocal elasticity model, whereas surprisingly, a stiffening effect is noticed with the nonself-adjoint nonlocal elasticity mode ( [START_REF] Lu | Dynamic properties of flexural beams using a nonlocal elasticity model[END_REF][START_REF] Wang | Vibration of nonlocal Timoshenko beams[END_REF]or [START_REF] Reddy | Nonlocal continuum theories of beams for the analysis of carbon nanotubes[END_REF]).

By assuming free harmonic motion, the differential equation is obtained:

EIw (4) + µl 2 c ω 2 w ′′ -µω 2 w = 0( 3 0 )
where ω is the circular frequency of vibration. The frequency equation of the variationally based corrected nonlocal Euler-Bernoulli beam model is obtained from the fourth-order differential equation: where λ 2 = µ EI ω 2 , e 0 a = l c and µ = ρ A. e 0 is a dimensionless number to be calibrated, and a is a length scale related to the size of the microstructured cell, responsible for possible nonlocal effects. The general solution for Eq. ( 31)is

d 4 w dx 4 + λ 2 (e 0 a) 2 d 2 w dx 2 -λ 2 w = 0 (31)
w(x) = C 1 cos αx + C 2 sin αx + C 3 cosh βx + C 4 sinh βx (32)
where

α = λ 4 (e 0 a) 4 + 4λ 2 + (e 0 a) 2 λ 2 2 1 2 ,β = λ 4 (e 0 a) 4 + 4λ 2 -(e 0 a) 2 λ 2 2 1 2
The variationally based boundary conditions Eq. ( 27) in the case of vibrating, clamped-free, nonlocal Euler-Bernoulli beam are written as:

w (0) = 0,w ′ (0) = 0,w ′′ (L) = 0a n d EIw ′′′ (L) + µl 2 c ω 2 w ′ (L) = 0 (33)
We note that these boundary conditions are different from the ones of the local model, especially for the last one dealing with the shear force. Considering the last boundary condition in a local format such as EIw ′′′ (L) = 0, it would lead to significantly different numerical results for this case. The variationally based boundary conditions are expressed for the nonlocal Euler-Bernoulli beam problem as:

-EIw ′′′ -µl 2 c ω 2 w ′ δw L 0 = 0a n d EIw ′′ δw ′ L 0 = 0( 3
4 ) By using the correct boundary conditions of the variationally based clamped-free problem listed in Eq. ( 33), one gets the following determinant of coefficient matrix

1010 0 α 0 β γ sin(αL) -γ cos(αL)η sinh(β L)η cosh(β L) -α 2 cos(αL) -α 2 sin(αL)β 2 cosh(β L)β 2 sinh(β L) = 0 ( 35
)
where

γ = α 3 -λ 2 l 2 c α = αβ 2 and η = β 3 + λ 2 l 2 c β = βα 2 .
Therefore, the characteristic equation is given by

(γβα 2 + αηβ 2 ) + (γβ 3 + ηα 3 ) cos(αL) cosh(β L) + (-ηβ α 2 + γαβ 2 ) sin(αL) sinh(β L) = 0 (36)
This is the frequency equation of the self-adjoint, nonlocal clamped-free Euler-Bernoulli beam. This frequency equation can be also presented in the form of

2 + β 2 α 2 + α 2 β 2 cos(αL) cosh(β L) + β α + -α β sin(αL) sinh(β L) = 0 (37)
or in a more condensed form of

2 + 2 + l 4 c α 2 β 2 cos(αL) cosh(β L) -l 2 c αβ sin(αL) sinh(β L) = 0 ( 38
)
The frequency equation of the exact nonself-adjoint Eringen's nonlocal model is calculated from the following boundary conditions

w (0) = 0,w ′ (0) = 0, EIw ′′ (L) + µl 2 c ω 2 w (L) = 0a n d EIw ′′′ (L) + µl 2 c ω 2 w ′ (L) = 0 (
39) By using these last boundary conditions of the nonself-adjoint clamped-free problem, one gets the following determinant of coefficient matrix

101 0 0 α 0 β γ sin(αL) -γ cos(αL)ηsinh(β L)ηcosh(β L) -γ α cos(αL) -γ α sin(αL) η β cosh(β L) η β sinh(β L) = 0( 4 0 )
Therefore, the characteristic equation is given by

γβ 3 + α 3 η + 2γηcos(αL) cosh(β L) + -ηβ 3 + γα 3 sin(αL) sinh(β L) = 0 ( 41
)
which can be also presented in the form:

β 2 α 2 + α 2 β 2 + 2cos(αL) cosh(β L) + -β α + α β sin(αL) sinh(β L) = 0( 4 2 )
or in a more condensed form as shown by [START_REF] Lu | Dynamic properties of flexural beams using a nonlocal elasticity model[END_REF]:

2 + l 4 c α 2 β 2 + 2cos(αL) cosh(β L) + l 2 c αβ sin(αL) sinh(β L) = 0 ( 43
)
This is the frequency equation of the nonself-adjoint, nonlocal clamped-free Euler-Bernoulli beam. These frequency sensitivity is shown in Fig. 2, based on these two frequency equations Eq. ( 38) (exact self-adjoint nonlocal model) and Eq. ( 43) (exact nonself-adjoint Eringen's nonlocal model).

6 Microstructured beam model: discrete versus nonlocal equivalent model

Microstructured model and continualization method

The self-adjointness of Eringen's nonlocal model is now investigated through a microstructured model, as the micromechanics-based Eringen's model. The microstructured cantilever that comprises n-cells is shown in Fig. 3. The buckling, or vibration of this discretized beam of length L and n rigid segments connected by elastic rotational springs of stiffness C, is studied in this part. The beam is loaded by an axial force denoted by P. The beam is composed of n repetitive cells of length denoted by a(the total length of the structure L is equal to L = n × a).

The elastic potential energy of the deformed rotational springs of this discrete clamped system comprising n-cells is given by

V = C 2 n-1 i=2 w i+1 -2w i + w i-1 a 2 + C 1 2 w 1 a 2 where w i = w (x = x i = ia) ( 44
)
and C = EI/a is the stiffness of each repetitive rotational spring. C 1 is the stiffness of the connection attached to the clamped part.

The work done W e by the axial load is given by

W e = P 2 a n i=1 w i+1 -w i a 2 (45)
The kinetic energy T of the discrete system with equal concentrated inertia masses denoted by m is given by

T = m 2 n-1 i=1 ẇ2 i + m 4 ẇ2 n ( 46
)
where the superdot denotes differentiation with respect to time t,µ = m/a for the correspondence between the discrete and the continuous system, and µ is the mass per unit length of the beam material. Only half of the characteristic mass m is attributed to the end nodes, as the end nodes have only one contributing rigid segment. The application of Hamilton's principle leads to the equations of motion for the discrete system:

P a (w i+1 -2w i + w i-1 ) + EI a 3 (w i+2 -4w i+1 + 6w i -4w i-1 + w i-2 ) + µa ẅi = 0 ( 47
)
Fig. 3 Clamped-free microstructured model

This discrete equation is also the finite difference equation for a "local" continuous Euler-Bernoulli beam problem. As already outlined by [START_REF] Silverman | Numerical computation of buckling loads by finite differences[END_REF], the discrete microstructured system is mathematically equivalent to the finite difference format of the so-called "local" continuum (see [START_REF] Silverman | Numerical computation of buckling loads by finite differences[END_REF][START_REF] Salvadori | Numerical computation of buckling loads by finite differences[END_REF], and the use of Hencky's chain as a physical support of numerical-based finite difference method). The fourth-order finite difference equation can be also introduced from the constitutive law and the governing equations as

M i = EI w i-1 -2w i + w i+1 a 2 (48)
and

M i-1 -2M i + M i+1 a 2 + P w i-1 -2w i + w i+1 a 2 + µ ẅi = 0 (49)
The discrete equations can be extended to an equivalent continuum via a continualization method. The following relation between the discrete and the equivalent continuous system w i = w (x = ia) holds for a sufficiently smooth deflection function as:

w (x + a) = ∞ k=0 a k ∂ k x k! w (x) = e a∂ x w (x) with ∂ x = ∂ ∂ x (50)
The generalized bending problem is then governed by the following pseudo-differential equation (see recently [START_REF] Challamel | Discrete systems behave as nonlocal structural elements: bending, buckling and vibration analysis[END_REF])

µ ∂ 2 w ∂t 2 + 16 EI a 4 sinh 4 a 2 ∂ x w + 4P a 2 sinh 2 a 2 ∂ x w = 0 (51)
The pseudo-differential operator can be efficiently approximated by the Padé's approximant (see for instance [START_REF] Rosenau | Dynamics of nonlinear mass-spring chains near the continuum limit[END_REF][START_REF] Wattis | Quasi-continuum approximations to lattice equations arising from the discrete non-linear telegraph equation[END_REF]or [START_REF] Andrianov | Improved continuous models for discrete media[END_REF] for axial wave applications):

4 a 2 sinh 2 a 2 ∂ x = ∂ 2 x 1 -l 2 c ∂ 2 x +••• with l 2 c = a 2 12 ( 52
)
The generalized Laplacian can be defined from this pseudo-operator as:

2 = 4 a 2 sinh 2 a 2 ∂ x = ∂ 2 x 1 -l 2 c ∂ 2 x +••• (53)
Such a generalized Laplacian has been recently used and extended by Michelitsch et al. [START_REF] Michelitsch | Nonlocal constitutive laws generated by matrix functions: lattice dynamic models and their continuum limits[END_REF] for discrete systems with long-range interactions. It is also possible to introduce the first central difference as the square root of this generalized Laplacian, 2 = δδ = δ 2 :

δ = 2 a sinh a 2 ∂ x ( 54
)
where the first central difference is introduced by Sheppard (1899) (see [START_REF] Sheppard | Central differences formulae[END_REF][START_REF] Richardson | The approximate arithmetical solution by finite differences of physical problems involving differential equations with an application to the stresses in a masonry dam[END_REF]). The bending equation can be reformulated using this generalized Laplacian:

µ ∂ 2 w ∂t 2 + EIδ 4 w + Pδ 2 w = 0 (55)
which is very analogous to the continuous Euler-Bernoulli beam model, except that the spatial differentiation is replaced by the central difference δ. By using the Padé's approximants of the pseudo-differential operators, the pseudo-differential equation can be efficiently approximated by the linear differential equation given by

µ 1 -2l 2 c ∂ 2 x ∂ 2 t w + EI∂ 4 x w + P 1 -l 2 c ∂ 2 x ∂ 2 x w = 0 (56)
which can be considered as the nonlocal equivalent medium of the microstructured model. As already discussed by Challamel et al. [START_REF] Challamel | Nonlocal equivalent continuum for the buckling and the vibrations of microstructured beams[END_REF]orW angetal. [START_REF] Wang | Calibration of Eringen's small length scale coefficient for initially stressed vibrating nonlocal Euler beams based on microstructured beam model[END_REF], this nonlocal Euler-Bernoulli equation shows that the length scale of the dynamics problem is different from the one of the buckling problem.

Energy equivalence between continuum and discrete media

It can be shown that the nonlocal equations Eq. ( 56) obtained from expanding the difference system in a partial differential system using Padé's approximant can be derived from Hamilton's principle by considering the Lagrangian L = T -U of the nonlocal continuous system:

t 2 t 1 δLdt = t 2 t 1 (δT -δU ) dt = 0 with T [w] = 1 2 L 0 µ ẇ2 + 2µl 2 c ẇ′2 dx, U [w] = L 0 1 2 EI -Pl 2 c w ′′2 - 1 2 Pw ′2 dx and l 2 c = a 2 12 ( 57
)
One question is now to obtain this Lagrangian directly from the foregoing Lagrangian of the discrete system

L = n-1 i=1 µa 2 ẇ2 i - EI 2a w i+1 -2w i + w i-1 a 2 + 1 2 Pa w i+1 -w i a 2 ( 58
)
where the specific boundary effects have been omitted. This question has been already investigated by Rosenau [START_REF] Rosenau | Hamiltonian dynamics of dense chains and lattices: or how to correct the continuum[END_REF][START_REF] Rosenau | Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit[END_REF] for axial vibrating systems, where the discrete energy of the discrete axial system is continualized for obtaining a nonlocal equivalent energy. With regard to the bending problem, one may use a change of variable. Consider the regular deflection variable w defined by

∂ 2 x w = w i+1 -2w i + w i-1 a 2 (59)
This condition can be inverted, i.e.,

w i = Q (a∂ x ) w with Q (a∂ x ) = ∂ 2 x 4 a 2 sinh 2 a 2 ∂ x (60)
Rosenau [START_REF] Rosenau | Hamiltonian dynamics of dense chains and lattices: or how to correct the continuum[END_REF][START_REF] Rosenau | Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit[END_REF] has shown the equivalence between the Hamiltonian of the discrete axial system and the one of the nonlocal continuous system. In this paper, we follow the same reasoning and show the equivalence between the Hamiltonian of the discrete bending system and the one of the nonlocal continuous system. It is possible to expand this differential operator

Q (a∂ x ) w = 1 - (a∂ x ) 2 12 + o a 4 w (61)
The kinetic energy of the discrete system can be continualized thanks to this differential operator:

L 0 ẇ2 i dx = L 0 Q (a∂ x ) ẇ Q (a∂ x ) ẇ dx = L 0 ẇ Q * (a∂ x ) Q (a∂ x ) ẇ dx ( 62
)
where Q * (a∂ x ) = Q (-a∂ x ) is the self-adjoint operator (see for instance [START_REF] Wong | An Introduction to Pseudo-Differential Operators[END_REF]). The summation is here approximated with the integral operator, omitting the quadrature approximation. In this case, for symmetry reasons, the self-adjoint operator is equal to the operator itself Q * = Q, and the pseudo-differential operator involved in the kinetic energy can expanded as:

Q * Q = Q 2 = 1 - (a∂ x ) 2 6 + o a 4 ( 63
)
The kinetic term can then be rewritten as:

L 0 ẇ2 i dx = L 0 ẇQ * Q ẇ dx = L 0 ẇ Q 2 ẇ dx = L 0 ẇ ẇ - a 2 6 ẇ′′ + o a 4 dx (64)
An integration by parts finally gives

L 0 ẇ2 i dx = L 0 ẇ2 + a 2 6 ẇ′2 + o a 4 dx - a 2 6 ẇ ẇ′ L 0 ( 65
)
By omitting the boundary terms, the kinetic energy of the nonlocal system is then corrected by:

T [w] = 1 2 L 0 µ ẇ2 + 2µl 2 c ẇ′ 2 dx and l 2 c = a 2 12 ( 66
)
which is similar to the reasoning of Rosenau [START_REF] Rosenau | Hamiltonian dynamics of dense chains and lattices: or how to correct the continuum[END_REF][START_REF] Rosenau | Compact and almost compact breathers: a bridge between an anharmonic lattice and its continuum limit[END_REF] for axial vibrating systems.

The reasoning for the expansion of the potential energy of the discrete system is similar. The potential energy depends on the difference:

w i+1 -w i a = a∂ x exp (a∂ x ) -1 4sinh 2 a∂ x 2 ∂ x w with w i = a 2 ∂ 2 x 4sinh 2 a∂ x 2 w (67)
A new differential operator can be introduced as:

w i+1 -w i a = R (a∂ x ) ∂ x w with R (a∂ x ) = a∂ x exp (a∂ x ) -1 4sinh 2 a∂ x 2 (68)
The potential energy can then be obtained from:

L 0 w i+1 -w i a 2 dx = L 0 R (a∂ x )(∂ x w) R (a∂ x )(∂ x w) dx = L 0 (∂ x w) R * (a∂ x ) R (a∂ x )(∂ x w) dx (69)
It is possible to expand this differential operator as:

R * (a∂ x ) R (a∂ x ) = (a∂ x ) 2 4sinh 2 a∂ x 2 = 1 - (a∂ x ) 2 12 + o a 4 (70)
The potential energy term can then be rewritten as:

L 0 w i+1 -w i a 2 dx = L 0 w ′ w ′ - a 2 12 w ′′′ + o a 4 dx (71)
An integration by parts finally gives

L 0 w i+1 -w i a 2 dx = L 0 w ′2 + a 2 12 w ′′2 + o a 4 dx - a 2 12 w ′ w ′′ L 0 (72)
By omitting the boundary terms, the potential energy of the nonlocal system is then corrected by:

U [w] = L 0 1 2 EI -Pl 2 c w ′′2 - 1 2 Pw ′2 dx and l 2 c = a 2 12 ( 73
)
This closes the proof of the energy equivalence between the discrete bending system and the nonlocal continuous bending system.

Clamped condition defined from continuum system

There is some debate about the calibration of the end stiffness of the discrete system for the equivalent clamped section. One can consider that the continuum kinematics at the clamped section is the reference, and one looks to the clamped equivalent behaviour of the discrete system. This approach has been followed by Challamel et al. [START_REF] Challamel | Nonlocal equivalent continuum for the buckling and the vibrations of microstructured beams[END_REF] and is presented here again for the sake of completeness. Another approach followed thereafter is based on the calibration of the stiffness equivalence with respect to the discrete medium.

In this first approach where the continuum is chosen as the reference medium, the end stiffness is calibrated from the deflection and moment equivalence principle at the end connection

w 1 = w (a) and M 0 = M (0) (74)
where w 1 is the deflection at the end of the first cell (i = 1) and M 0 is the moment at the clamped boundary.

In the first element connected to the clamped extremity, these quantities are related by

w 1 = w (a) = θ 1 × a = M 0 C 1 × a (75)
For simplicity, the calibration is proposed from using the local Euler-Bernoulli theory, even if we have in mind that using the nonlocal model would have been more accurate. For the local Euler-Bernoulli beam, the natural frequencies expression of the clamped-free cantilever is obtained from (see for instance [START_REF] Blevins | Formulas for Natural Frequency and Mode Shape[END_REF]):

cos ⎛ ⎝ 4 µω 2 EI L ⎞ ⎠ × cosh ⎛ ⎝ 4 µω 2 EI L ⎞ ⎠ =-1 (76)
and the vibrations mode is given by:

w w 0 = cos ⎛ ⎝ 4 µω 2 EI x ⎞ ⎠ -cosh ⎛ ⎝ 4 µω 2 EI x ⎞ ⎠ - cos 4 µω 2 EI L + cosh 4 µω 2 EI L sin 4 µω 2 EI L + sinh 4 µω 2 EI L ⎡ ⎣ sin ⎛ ⎝ 4 µω 2 EI x ⎞ ⎠ -sinh ⎛ ⎝ 4 µω 2 EI x ⎞ ⎠ ⎤ ⎦ (77)
Now using the correspondence principle of Eq. ( 74), the equivalent stiffness in the clamped section of the discrete system is computed from:

M (0) = EIw ′′ (0) =-2EIw 0 ω µ EI = C 1 a w (a) and a = L n (78)
One finally obtains a complex frequency-dependent stiffness law: A first-order expansion leads to: Figure 4 shows excellent agreement between the frequencies obtained from microstructured beam model and the self-adjoint nonlocal beam theory (adopting e 0 = 0.408) for a clamped-free beam. On the other hand, matching of frequencies is not possible if the nonself-adjoint nonlocal Eringen beam theory was used. This demonstrates that the microstructured beam model can point to the "correct" boundary conditions for Eringen's beam theory when confronted with a nonself-adjoint formulation such as the vibration problem of a clampedfree beam. Another point of view for the calibration of the end stiffness is to consider that the reference medium is the discrete system, and one seeks the best match for the equivalent continuum.

C 1 C = -2ω µ EI L 2 n 2
C 1 C = 2 + A n +••• with A = 2
The finite difference system is exactly solved in this part, as already investigated in [START_REF] Challamel | Discrete systems behave as nonlocal structural elements: bending, buckling and vibration analysis[END_REF][START_REF] Leckie | The effect of lumped parameters on beam frequencies[END_REF][START_REF] Santoro | Accuracy of the finite difference method in stochastic setting[END_REF] for discrete bending systems. Analytical solutions for the vibrations of pure discrete shear system are available in [START_REF] Luongo | Parametric, external and self-excitation of a tower under turbulent wind flow[END_REF], whereas discrete shear/bending systems have been recently analysed in [START_REF] Duan | Development of analytical vibration solutions for microstructured beam model to calibrate length scale coefficient in nonlocal Timoshenko beams[END_REF][START_REF] Zhang | Eringen's small length scale coefficient for buckling of nonlocal Timoshenko beam based on a microstructured beam model[END_REF] within nonlocal mechanics. The fourth-order linear finite difference Eq. (47) restricted to the vibration terms can be written as

w i+2 -4w i+1 + 6w i -4w i-1 + w i-2 - 2 n 4 w i = 0 with 2 = ω 2 µL 4 EI ( 81
)
The vibrations mode can be obtained from w i = Aλ i , which by substitution in Eq. (81) leads to the characteristic equation:

1 λ + λ 2 -4 1 λ + λ + 4 - 2 n 4 = 0( 8 2 )
Equation ( 82) admits four solutions which may be expressed as (see [START_REF] Challamel | Discrete systems behave as nonlocal structural elements: bending, buckling and vibration analysis[END_REF][START_REF] Santoro | Accuracy of the finite difference method in stochastic setting[END_REF]):

λ 1 = cos θ -j sin θ, λ 2 = cos θ + j sin θ, λ 3 = 2 -cos θ -(2 -cos θ) 2 -1and λ 4 = 2 -cos θ + (2 -cos θ ) 2 -1 with θ = arccos 1 - 2n 2 and j 2 =-1 (83)
The solution can be expressed with a real functional basis, in the following format:

w i = A 1 cos (θi) + A 2 sin (θi) + A 3 λ i 3 + A 4 λ i 4 (84)
For the clamped-free problem investigated herein, the boundary conditions are detailed below. The deflection of the discrete system vanishes at the boundary, i.e.,

w 0 = 0 ⇒ A 1 + A 3 + A 4 = 0 (85)
The second condition at the clamped section is the continuity of the deflection in both parts of the clamped section:

w -1 = w 1 ⇒ 2 A 2 sin θ + A 3 λ 3 - 1 λ 3 + A 4 λ 4 - 1 λ 4 = 0 (86)
By writing the moment equivalence between the finite difference system and the physically based microstructured system, one obtains the following equivalent clamped stiffness C 1 = 2C, as shown by:

M 0 = C 1 θ 1 = w -1 -2w 0 + w 1 a C = 2Cθ 1 if w 0 = 0a n dw -1 = w 1 (87)
The third condition is the vanishing of the bending moment at the free end, i.e., M n = 0:

w n+1 -2w n + w n-1 = 0 ⇒-A 1 cos (nθ ) -A 2 sin (nθ ) + A 3 λ n 3 + A 4 λ n 4 = 0 (88)
Finally, the last boundary condition deals with the free shear force condition which is written here for the discrete system as:

w n+1 -3w n + 3w n-1 -w n-2 = -2 2n 4 w n ⇒ A 1 [sin θ sin (nθ ) + cos (nθ)] + A 2 [-sin θ cos (nθ ) + sin (nθ )] -A 3 λ n-1 3 1 -2n 2 λ 3 -A 4 λ n-1 4 1 -2n 2 λ 4 = 0 (89)
The half factor in the last shear boundary condition is due to the half term of the inertia load in the last cell (see Fig. 3). The crucial role of the free end of a discrete system has already been pointed out by Wallis [START_REF] Wallis | Effect of free ends on the vibration frequencies of one-dimensional lattice[END_REF] or Kivshar et al [START_REF] Kivshar | Nonlinear surface modes in monoatomic and diatomic lattices[END_REF] for the axial chain, in relation with surface modes. In the present study, the free end has a lumped mass m/2(seeFig.3), as also considered for instance by Leckie and Lindberg [START_REF] Leckie | The effect of lumped parameters on beam frequencies[END_REF] for the discrete bending system. This shear force boundary condition is also equivalent to:

M n -M n-1 =-ω 2 a 2 µ 2 w n ( 90
)
As M n = 0, this last boundary condition can be also written as:

M n-1 = ω 2 a 2 µ 2 w n (91)
as detailed by Leckie and Lindberg [START_REF] Leckie | The effect of lumped parameters on beam frequencies[END_REF] for instance for free-free discrete finite difference beams. Indeed, Leckie and Lindberg [START_REF] Leckie | The effect of lumped parameters on beam frequencies[END_REF] studied the free vibrations of discrete finite difference beams with various boundary conditions including the clamped-clamped case, the simply supported case of the free-free beams, but did not report results for the clamped-free case. Interestingly, Leckie and Lindberg [START_REF] Leckie | The effect of lumped parameters on beam frequencies[END_REF] also mentioned the analogy between the microstructured model and the finite difference beam equations in the dynamics case. The natural frequencies of the discrete clamped-free system are then obtained from the determinant calculation:

10 1 1 02 s i n θλ 3 -1 λ 3 λ 4 -1 λ 4 -cos (nθ) -sin (nθ ) λ n 3 λ n 4 cos (nθ) + sin θ sin (nθ ) sin (nθ ) -sin θ cos (nθ ) -λ n-1 3 1 -2n 2 λ 3 -λ n-1 4 1 -2n 2 λ 4 = 0 (92)
where θ,λ 3 ,andλ 4 are given by Eq. ( 83). These results coincide with the numerical discrete results based on the equivalent clamped stiffness C 1 = 2C. These results can be also compared to the nonlocal Eringen's model with the modified clamped boundary conditions. Equations ( 31)and( 32) are still valid in this case (nonlocal Eringen's type Euler-Bernoulli beam), but the new boundary conditions are

w (0) = 0,w (a) = w (-a) ,w ′′ (L) = 0a n d EIw ′′′ (L) + µl 2 c ω 2 w ′ (L) = 0 with l 2 c = (e 0 a) 2 = a 2 6 ( 93
)
By using these last boundary conditions, one gets the following determinant of coefficient matrix In Fig. 5, the discrepancies are clearly highlighted for a small number of cells, between the two ways for modelling the clamped section. However, in both cases, the softening effect of the small length scale effect is noticed. Furthermore, as expected, a perfect agreement is also observed between the nonlocal Eringen's model with the clamped boundary conditions w (0) = 0andw (a) = w (-a), and the discrete model based on w -1 = w 1 .

1010 0s i n (αa) 0s i n h (βa) γ sin(αL) -γ cos(αL)η sinh(β L)η cosh(β L) -α 2 cos(αL) -α 2 sin(αL)β 2 cosh(β L)β 2 sinh(β L) = 0( 9

Conclusions

The following conclusions may be drawn from this study:

1. The buckling problem of Eringen's beam is shown to be self-adjoint for all boundary conditions. A modified equivalent energy functional of the nonlocal elasticity beam can be given in term of the local deflection. This nonlocal elasticity problem is clearly a conservative problem (in the sense of deriving from a potential). 2. Eringen's Euler-Bernoulli beam mechanics is also shown to be self-adjoint for hinged-hinged boundary conditions, whatever the analysis considered (buckling or vibration analysis). 3. For vibrations analyses, the clamped-free problem of the Eringen's-based nonlocal beam leads to a nonselfadjoint problem. In this case, it is not possible to build an associated functional energy, and the problem is clearly nonconservative. In this last case, a surprising and paradoxical result has been observed, where the small length scale effect tends to stiffen the local system, in contrast to the softening result obtained for all the other boundary conditions. 4. Eringen's nonlocal elasticity has been recently shown to be an accurate description of microstructured (discrete) media. As the discrete media is considered as a conservative system at the local scale, it is rational to have a conservative macro-scale (conservative nonlocal elasticity equivalent continuum, or self-adjoint macroscopic system). From the microstructure equivalent correspondence, Eringen's corrected nonlocal elasticity with self-adjoint boundary conditions is recommended to describe the softening phenomenon induced by the small length scale effect, a conclusion that cannot be reached within nonself-adjoint Eringen elasticity. We therefore recommend the use of the differential equations with the proposed associated boundary conditions within the micromechanics-based energy arguments. 5. It has been shown in this paper how some discrete systems may be accurately approximated by some Eringen's-based nonlocal continuum models. The discrete system considered in this study is based on some unidimensional repetitive bending cells, which can be viewed as a structural paradigm in structural mechanics. However, alternative nonlocal or gradient theories may also be supported by some other physical or theoretical supports (see the historical perspective [START_REF] Dell'isola | At the origins and in the vanguard of peri-dynamics, non-local and higher gradient continuum mechanics. An underestimated and still topical contribution of Gabrio Piola[END_REF]). For instance, higher gradient elasticity theories may be also deduced starting from discrete repetitive truss cells [START_REF] Alibert | Truss modular beams with deformation energy depending on higher displacement gradients[END_REF].

Fig. 1 Fig. 2

 12 Fig. 1 Clamped-free continuous nonlocal beam model

Fig. 4

 4 Fig. 4 Comparison between frequencies obtained from microstructured beam model (with the clamped stiffness calibration) and self-adjoint nonlocal Eringen beam theory with e 0 = 0.408 for a clamped-free beam; calibration of the clamped stiffness of the discrete model with respect to the continuous beam model

Fig. 5

 5 Fig. 5 Comparison between frequencies obtained from microstructured beam model and self-adjoint nonlocal Eringen beam theory with e 0 = 0.408 for a clamped-free beam; parameter A defined in Eq. (80); effect of the boundary conditions at the clamped section


Jojomon Joseph
email: jojomon.joseph@partner.kit.edu

Michel Delanaye

Rabia Nacereddine

Andres Giraldo

Mehdi Rouabah

Jan G Korvink

Juergen J Brandner

Numerical and experimental investigation of a

INTRODUCTION

Combined heat and power (CHP) production is a process of producing both electricity and heat with high e ciency. According to the International Energy Agency (IEA), CHP shows the potential to reduce the primary energy consumption in the residential sector. Besides, they help to reduce greenhouse emissions, promotes decentralisation of electricity production and has lower initial investment and distribution losses compared to conventional grids [START_REF] Pehnt | Micro cogeneration towards decentralized energy systems[END_REF][START_REF] Hawkes | Cost-e↵ective operating strategy for residential micro-combined heat and power[END_REF][START_REF]IEA Energy E ciency Policy Recommendations[END_REF]. However, miniaturisation and cost reduction remain a challenge for the industry to reach broad adoption [START_REF] Hawkes | Cost-e↵ective operating strategy for residential micro-combined heat and power[END_REF]. The MITIS Micro Gas Turbine (MGT) system consists of a compressor, combustor, turbine, recuperator and generator.

The compressor and turbine are radial-flow designs, and similar to conventional engine turbochargers. The combustion chamber works in flameless regime, providing high e↵ectiveness with low emissions [START_REF] Delanaye | Development of a recuperated flameless combustor for an inverted Bryton cycle microturbine used in residential micro-CHP[END_REF][START_REF] Beith | Small and micro combined heat and power (CHP) systems, advanced design, performance, materials and applications[END_REF]. The air preheating is carried out by a wire-net micro heat exchanger. Preheating typically doubles the electrical e↵ectiveness of the unit while reducing the amount of heat recoverable [START_REF] Hawkes | Cost-e↵ective operating strategy for residential micro-combined heat and power[END_REF][START_REF] Beith | Small and micro combined heat and power (CHP) systems, advanced design, performance, materials and applications[END_REF]. The objective of this paper is to investigate the performance of a wire-net micro heat exchanger which preheats the combustion air, therefore reducing the amount of injected fuel [START_REF] Beith | Small and micro combined heat and power (CHP) systems, advanced design, performance, materials and applications[END_REF] for the micro-CHP system. A compact heat exchanger (CHE) regularly consists of microchannels with collectors optimised for the distribution of the flow. Here we use a plate gas to gas (air) counterflow CHE, with wire-net microchannel configuration (patent number: WO 20161124472, Figure 1). The wire-net placed in between the partition foils shifts the critical Reynolds number to lower values and enhances heat transfer mechanism by localised laminar/ turbulent mixing. The wire-net structures influence the CHE performance. Balaras [START_REF] Balaras | A Review of Augmentation Techniques for Heat Transfer Surfaces in Single-Phase Heat Exchangers[END_REF] reviewed and classified heat transfer enhancement techniques as surface, fluid and combined and compound methods. The surface method includes extended geometries such as wire-net structures (fins/protrusions), structural vibrations etc.

High convective heat transfer is associated with larger protrusions which creates greater channel obstruction with an increased acceleration of the flow [START_REF] Roeller | Heat Transfer and Turbulent Flow Characteristics of Isolated Three-Dimensional Protrusions in Channels[END_REF]. Conversely, smaller protrusions yield more intense 3D transport with high turbulence intensity, resulting in high heat transfer. Tao et al. [START_REF] Tao | A Unified Analysis on Enhancing Single Phase Convective Heat Transfer With Field Synergy Principle[END_REF] suggested heat transfer enhancement by decreasing the thermal boundary layer, introducing protrusions inside the microchannels and by increasing the velocity gradient near the heated surface. Conjugate heat transfer (CHT) analysis carried out by Paolo [START_REF] Paolo | Conjugate heat and mass transfer in drying: A modeling review[END_REF] on a cube, pointed that the heat transfer was minimum when the boundary layer was undisturbed. Transverse vortex generators and longitudinal vortex generators play a major role in local and global heat transfer enhancements [START_REF] Fiebig | Vortex generators for compact heat transfer exchangers[END_REF]. However, longitudinal vortex generators perturb the viscous sublayer along the flow direction and enhance the heat transfer [START_REF] Incropera | Introduction to Heat Transfer[END_REF] as compared to transverse vortex generators. The study made by [START_REF] Fiebig | Vortices, Generators and Heat Transfer[END_REF] M.Fiebig revealed that counterrotating vortex generators arrangements enhanced the heat transfer by more than twice that of the corotating configuration. In addition, he inferred that longitudinal vortices are remarkably persistent in both laminar and turbulent boundary layers and enhance the heat transfer locally and globally. Johnson and Joubert in 1969 made a study on the influence of longitudinal vortex on heat transfer and found that cylindrical drag was reduced due to separation delay caused by the longitudinal vortex enhanced the heat transfer [START_REF] Johnson | The influence of vortex generators on the drag and heat transfer from a circular cylinder normal to an airstream[END_REF]. The e↵ect of flow pattern influences the global heat transfer and pressure losses.

High heat transfer and low-pressure losses are formed along the flow reattachment and separation lines of leading-edge vortices [START_REF] Hajime | Local heat transfer around a wall-mounted cube in the turbulent boundary layer[END_REF]. However, areas of minimum convective heat transfer were found in regions of maximum turbulence production areas (coincided with maximum Reynolds normal stresses) due to high-velocity shear gradients. Thus, the maximum heat transfer coe cient occurred in the proximity of the reattachment points, where the wall shear stress is small, but the velocity fluctuations normal to the wall are high [START_REF] Meinders | Experimental Study of the Local Convection Heat Transfer From a Wall-Mounted Cube in Turbulent Channel Flow[END_REF]. Valencia [START_REF] Valencia | Turbulent flow and heat transfer in a channel with a square bar detached from the wall[END_REF] showed that at Reynolds number 2000 heat transfer was enhanced by 30% due to turbulence together with a five-fold increase in friction factor. Increasing the blockage ratio and reducing the aspect ratio increased the global Nusselt number [START_REF] Meis | Heat transfer enhancement in micro-channels caused by vortex promoters[END_REF]. The heat transfer coe cient was very low on the upstream side in dimple like protrusions while downstream due to flow reattachment, heat transfer coe cient was increased [START_REF] Turnow | Flow structures and heat transfer on dimples in a staggered arrangement[END_REF]. The surface roughness also plays an essential role in the micro-channel performance [START_REF] Croce | Carlo Nonino: Three-Dimensional Roughness E↵ect On Micro-channel Heat Transfer And Pressure Drop[END_REF] since at micro level it is nearly impossible to obtain a smooth surface. Morini and Spiga [START_REF] Morini | The Role of the Viscous Dissipation in Heated Microchannels[END_REF] investigated the role of viscous dissipation in the liquid flowing through heated microchannels. Heat transfer enhancement in the microdevices cannot be solved by indefinitely reducing the microchannel dimensions due to viscous dissipation e↵ect. The partition foil that separates the hot and cold fluid plays a vital role in heat transfer enhancement. Brandner et al. [START_REF] Yang | Design and Experimental Investigation of a Gas-to-Gas Counter-Flow Micro Heat Exchanger[END_REF][START_REF] Yang | Experimental analysis of the influence of wall axial conduction on gas-to-gas micro heat exchanger e↵ectiveness[END_REF] compared the performance of heat exchange devices with various microstructures manufactured on thin foils to enhance heat transfer. The smaller the foil thickness, the lower the axial conduction thereby leading to high heat transfer. Low heat conductivity material could improve the CHE e↵ectiveness rather than very low or high heat conductivity materials. Among the flow arrangements, countercurrent flow arrangement will provide high e↵ectiveness [START_REF] Kays | Compact heat exchangers[END_REF]. However, the cross-flow configuration tends to give similar results to the counter-current arrangement for higher Reynolds number and partition foils having low thermal conduction.

WIRE-NET HEAT EXCHANGER

The counterflow gas-gas (air) wire-net compact CHE arrangement (Figure 1, top left) is similar to the conventional plate-type CHE arrangement. Smooth partition foils replace the conventional corrugated partition foils. A wire-net is inserted in between the partition foils to bring the corrugation e↵ects and thereby induce localised laminar/ turbulent mixing. In addition, the wire-net also prevent buckling of partition foils and therefore enhances the mechanical stability.

The CHE consists of collectors, two hot and two The collectors distribute the flow to the microchannels, and their inlets are placed perpendicular to the microchannel inlets. In this paper, computational fluid dynamics (CFD) computations were carried out only for microchannels while experimental tests were performed on a complete CHE model with collectors. CFD computations were adopted due to the complexity in experimentally investigating the microchannel flow physics. The primary aim of the CFD simulations is to examine the microchannel design features and use the experimental results to analyse the collector performance.

The inverted Brayton cycle (IBC) has a subatmospheric hot inlet side/ hot side, and Brayton cycle (BC) has a high-pressure cold inlet side/ cold side. The pressure at the outlet was increased and decreased (based on IBC/BC operating conditions) step by step to achieve the desired pressure inside the microchannels. The two di↵erent operating conditions, IBC and BC are mentioned in Table 1.

Numerical microchannel model

Boundary conditions

The CHE assembly is shown in Figure 1, top left. Two types of geometrical symmetry exist in a plane that is parallel to the inlet flow direction. The first symmetry is along half of the collector diameter of hot and cold collector inlets (see Figure 1, bottom), and second symmetry is at the centre of the wire-net CHE (Figure 1, top right). It is computationally expensive to carry out a CHT analysis by holding the first symmetry due to complex channels with a large computational domain. In conclusion, the rectangular first symmetry, (see Figure 1, bottom) and second symmetry (see Figure 1, top right) is shown in Figure 1.

Similarly, periodic boundary conditions were implemented in the plane which is perpendicular to the inlet flow direction. In other words, a periodicity exists in half of the microchannel foil thickness (Figure 1, top) on every hot and cold side alternatively. Thus, all hot and cold microchannels were simplified into a single hot and cold microchannels with partition foils above it, as shown in Figure 1, top right. In conclusion,

• Periodicity between the partition foils of hot and cold side • Symmetric boundary conditions on the sides.

• No-slip conditions at wall (fluid and solid) boundaries CHE performance depends on a wide range of parameters, e.g. wire-net cell size, microchannel dimensions and thickness of the partition foil. In this paper, square type wire-nets were used in which the opening size, longitudinal and diagonal size of the wire-net remains constant and all together it is mentioned as wire-net sizing. The structural radius of the wire-net is kept constant to decrease complexity.

CHT solver specifications

A total mass flow (m to ) of air passes through the counter-flow collector of CHE. A single collector inlet mass flow is calculated from the total mass flow based on the number of collector inlets. The microchannel inlet flow is calculated from the overall mass flow based on the number of collector inlets, the number of microchannels and MSR coe cient. The latter is obtained from simplifying the computational domain with the help of symmetric and periodic boundary conditions. The number of microchannels along with the microchannel size reduction coe cient is utilized to calculate the microchannel inlet mass flow from the collector inlet mass flow.

m µ = m to n pl ⇤ MSR (1)
The microchannel inlet mass flow is calculated using Eq.( 1). Moreover, several microchannel inlet mass flows were calculated for di↵erent microchannel and collector arrangements, to investigate the dependency of CHE performance on mass flow.

The temperature di↵erence between hot and cold inlets is at its minimum 650 K. Thus, compressibility and radiation e↵ects may also have a significant impact on the performance. However, the e↵ect of radiation to the outside was not taken into consideration because of the excellent insulation (5 cm thick special incombustible woollen layer) to the ambient. CHT simulations were carried not only based on the total massflow (m to ) (using Eq. ( 1)), but also for several mass flows to investigate the dependency of microchannel performance on the particular range of massflows. The Reynolds number (based on inlet hydraulic diameter) in the microchannel and collectors are in the range of 30-2000 and 2500-20000 respectively. The governing di↵erential equations were discretised using a finite volume method on a tetrahedral mesh with 12 to 14 million cells. The continuity and momentum equation is expressed as,

@⇢ @t + r • (⇢v) = S (2)
@(⇢ṽ) @t + r • (⇢ṽṽ) = r • P + ⇢g + F (3)
S-is the mass added to the continuous phase from the dis-persed second phase which is zero for the numerical model. Where ⇢g and F are the gravitational body force and external body forces which is zero for the CHT model. The energy conservation equation is expressed as,

@(⇢E) @t +r•(ṽ(⇢E + P )) = r• 0 @ X j h j J j 1 A +S h (4)
Where S h is the heat source and E = h P ⇢ +0.5v 2 . h is the enthalpy for ideal gas and J j is the di↵usion flux of di↵erent species. The energy equation for solid regions are,

@(⇢h) @t + r • (ṽ⇢h) = r • (Kr • T ) + S h ( 5
)
Where K is the conductivity and the part of equation containing this term represents convective energy transfer due to rotational or translational motion of the solids.

At low mach numbers, the density does not depend on pressure but only on temperature. For a perfect gas we have,

P = ⇢RT (6)
Such that, @⇢ = ( @⇢ @P ) T @P + ( @⇢ @T

) P @T = ⇢ T @T (7)
since, @⇢ @P = 0 and the density simplifies into,

⇢ = ⇢ 0 T 0 T ( 8
)
The flow is thus considered incompressible, but density varies as a function of the inverse of the temperature as well as the fluid is dilatable. Thus a segregated solver was introduced intead of a coupled solver. Some of the general assumptions made for the CHT analysis are:

• 3D, Steady-state, laminar for a microchannel without wire-net and turbulent model for a microchannel with wire-net. • Ideal gas, steady-state, incompressible flow (segregated solver) • Radiation losses and body forces were neglected.

• Smooth partition foils relative to microchannel hydraulic diameter to neglect surface roughness e↵ects.

The porosity of the wire-net microchannels is approximately 83%, which will induce the transition to localised turbulence at low Reynolds number [START_REF] Kanaris | Optimal design of a plate heat exchanger with un-dulated surfaces[END_REF]. The CHT simulations were performed using the CFD solver FLUENT. For the velocity-pressure coupling, the SIM-PLE algorithm has been applied. The thermal eciency, ", % is calculated using the relation

", % = C h (T h in T h out ) min(C c , C h )(T h in T c in ) ⇤ 100 (9) C c = m µc Cp c ; C h = m µh Cp h (10)
with Cp h , c being the hot and cold fluid capacity rates. T h in , h out and T c in , c out are the inlet and outlet temperatures of the counter-flow channels. The total pressure losses, P ,% is calculated using, The SST k ! model switches between wall function and Low Re-implementation. Since its computationally cheap as compared to the RSM model, all the parametric studies were carried out using the SST k ! model. Besides, a detailed investigation of the wire-net flow physics is made using a higher order Reynolds stress turbulence model to obtain the full velocity gradient tensor. This could detail the e↵ect of anisotropic flow physics in the isotropic wirenet microchannels. The mesh for the RSM model was highly refined (relative to the computational mesh of the k ! model) at proximities and curvatures to capture any sharp gradients.

P

Mesh sensitivity analysis

Mesh resolution along with the mesh quality plays a vital role in the accuracy and convergence of the CFD results. There is no exact methodology to investigate the error due to spatial discretisation. The only method is to conduct an independent mesh study. The standard way to characterise the mesh refinement is by checking the global and local convergence of flow characteristics. Here in this study, the convergence (local) of the temperature distribution along the central line is examined and plotted in Figure 2, top.

Three di↵erent mesh models (highly refined, average and coarse mesh) were selected based on the mesh sensitivity analysis.

The cell number was regulated by managing the base size. In order to decrease the solver complexity and computational size, all three mesh models were converted into polyhedral cells using the Ansys Fluent solver. CHT simulations were conducted for three di↵erent mesh models by holding the same inlet boundary conditions. The temperature profile (along with the centre of the microchannel) was plotted from inlet to the outlet of a single hot inlet side for the counterflow CHT model. From the analysis, its evident that the uncertainty in the temperature distribution is relatively low (¡1%) between the highly refined model and the average mesh model (see Figure 2, top).

EXPERIMENTAL TESTING

The primary objective of the experimental testing is to characterise the collector pressure losses and compare the CHT thermal e↵ectiveness relative to the overall CHE e↵ectiveness. Figure 2, centre shows the test bench configuration for the CHE. The temperature is measured with type K thermocouples, with a standard limit of uncertainty equal to 2. is taken from a GAS-VIEW device, which can measure volumetric flow in a range of 167 L/min to 850 L/min, with an accuracy of ±3%F S. Moreover, e↵ectiveness is calculated using Eq. ( 9). To reduce heat losses to the environment, there was a 5 cm layer of insulating material all around the CHE and the inlet and outlet collectors. Furthermore, pressure sensors were installed in the primary collectors that feed the mass flow to the secondary collectors and then to the microchannels. The temperature was analysed for 10 minutes and the uncertainty in temperature measurements for experimental hot inlet temperatures (based on CFD inlet conditions) is depicted in Figure 2, centre. The instrumental uncertainties is shown in Table 2.

Table 3 shows the thermal e↵ectiveness and pressure losses obtained at di↵erent temperatures. The thermal e↵ectivness di↵erences between the CFD and experimental result is in the range of 0.3 to 4.4 % have been obtained. The di↵erences mainly arise due to conductivity and radiation losses to the environment as well as the e↵ect of collector maldistribution which is not included in the CFD model. In conclusion, the microchannel CHT model predicts the thermal e ciency with reasonable accuracy (< 4.4%).

Apart from that, the collector pressure loss is calculated from the pressure loss di↵erence between CFD and experiments at all temperatures, which is around 40-50 % of the microchannel pressure losses. Based on the heat exchanger arrangment, the contribution of collector towards thermal e↵ectiveness is inadequate. Thus, it is not worth loosing high-pressure/ power losses in the collectors. Besides, the integrated secondary collectors will also contribute to the CHE

CHT RESULTS AND DISCUSSION

CHT Simulations were carried out for several mass flows to investigate the dependency of microchannel performance on mass flow. The temperature drop/temperature potential (at each cross-section) between the counter-flow channels is a major factor that influences the heat exchanger performance. The pressure losses and thermal e↵ectiveness were calculated using Eq. (9 & 11). From the CHT analysis, we found an optimal Reynolds number/mass flow where thermal e↵ectiveness reaches a maximum, see Figure 3, top & Figure 5. Studies made by Y. C. Fu et al. [START_REF] Fu | An experimental investigation on heat transfer enhancement of sprayed wire-mesh heat exchangers[END_REF] on wire-net structures of compact heat exchangers showed a similar trend for thermal e ciency. The Reynolds number based on microchannel inlet is calculated using Eq. ( 12),

Re = ⇢D h V in µ (12)
In general we consider three mass flow regimes from the thermal e ciency curve,

• Smallest mass flow/Reynolds number • Optimum mass flow /Reynolds number • Highest mass flow /Reynolds number

The temperature distribution along the centre of the free microchannel is depicted in Figure 4, top. The non-dimensionalised velocity distribution along the centre of a single cold-inlet channel is illustrated in Figure 4, bottom.

Along the microchannel length, the velocity increases from cold-inlet to cold-outlet due to high-temperature gain from the hot-inlet side.

In order to balance the momentum equation, the density decrease due to the temperature increase is equilibrated by the velocity increase [START_REF] Mushtaq | Influence of channel geometry on the performance of a counter flow microchannel heat exchanger[END_REF].

At lower Reynolds, the potential/ temperature di↵erence between the counter-flow channels (perpendicular to the flow direction) is nearly zero (see Figure 4, top) due to high residence time inside the counter-flow channels [START_REF] Schlichting | Boundary-Layer Theory[END_REF]. However, the slope of the temperature drop curve at smaller Reynolds number is less steep (relative to other cases, see Figure 4) due to low enthalpy input in the microchannel inlets. As the Reynolds number increases (from smaller to optimum Reynolds number), the thermal e ciency also increases, due to an increased input in enthalpy. Thereby, the slope of the temperature gets steeper at an optimum Reynolds number (Figure 3, top). The enthalpy input is higher at higher Reynolds number; still, the thermal e ciency drops as it advances from optimum to higher Reynolds number (see Figure 3). This is because, the boundary layer is thin at higher mass flows and in a relatively thin boundary layer, friction Shear Stress (viscous shearing force, Eq. 14) is very large [START_REF] Hong | Data reduction of average friction factor of gas flow through adiabatic micro-channels[END_REF]. This is evident from the shear velocity profile (Eq. 14) which is shown in Figure 3(b). The section of centrally advected non-disturbed mass flow increases and does not enhance the heat transfer.

⌧ w = µ[ @u @y ] w ( 13
)
U ⇤ = µ r ⌧ ⇢ ( 14
)
It is evident from the Figure 4, top that there is a thermal enhancement at higher Reynolds number due to the wire-net localised mixing. This could increase the working range and thereby reduces the size and cost of the CHE. The wire-net configuration shifts the critical Reynolds number to smaller values due to the wire-net structures localised turbulent mixing. From the design point of view, size and in turn costs will be reduced by,

• Shifting the optimum working range to higher mass flows • Flattening the e↵ectiveness curve at higher mass flow regimes

Parametric study

A parametric study was performed for various microchannel lengths and heights. However, the study was not carried for various microchannel widths, because an increase in width will introduce strong recirculation zones thereby increasing the pressure losses together with a reduction in thermal e ciency. The strong recirculation will change the pure counterflow to cross-flow configuration thereby decreasing the e ciency.

. Micro channel length

Increasing the microchannel length enhances the thermal e↵ectiveness for both IBC and BC operating conditions. An increase in length gives more surface area improving heat recovery. Variation of thermal e↵ectiveness is saturated for BC operating condition compared to IBC configuration (see Figure 5, top). The optimum velocity (thermal e ciency) is shifted to higher inlet velocities for IBC operating conditions (compared to BC) while the optimum mass flow (thermal e↵ectiveness) remains same for IBC and BC operating conditions. The lower pressure of IBC with respect to the BC influences the inlet velocities and thereby shifted the optimum to higher velocity (see Figure 5, top).

Micro channel wire net cell size (WNs)

Figure 5(centre) shows the thermal e↵ectiveness for varying wire-net cell size.

An increase in wire-net cell size (2.5 mm to 3 mm) did not enhance the thermal e↵ectiveness but decreased the pressure losses (see Figure 4, bottom). Increase in wire-net cell size results in less number of perturbations/ square wirenet structures (for the same microchannel length), thereby lowers pressure losses.

A further increase in wire-net cell size will induce buckling of the partition foils due to the pressure di↵erence between the counter-flow microchannels.

Micro channel foil thickness (Fth)

Figure 5(centre) and Figure 4(bottom) shows the microchannel thermal e↵ectiveness and pressure losses for two di↵erent foil thicknesses.

The pressure losses did not change remarkably based on foil thickness. However, foil thickness strongly influenced the microchannel thermal e ciency.

As the foil thickness decreases, axial conduction also decreases improving heat transfer between the channels. On the other hand, thinner foils may reduce the life of the CHE because of hot corrosion and erosion. As the thickness decreases, oxidation becomes a major limiting factor for its service life [START_REF] Lai | High-temperature corrosion and material applications[END_REF]. The operating conditions (BC and IBC) could introduce buckling of the thin partition foils.

Micro-CHP cycle e ciency on overall e↵ectiveness

We, MITIS developed a computer code to estimate the e↵ectiveness of the MGT cycle at IBC and BC conditions. This code integrates all of the MGT components for both IBC and BC.

Flow physics

RSM turbulence model was implemented to infer the physics since it solves the turbulent stress transport equation and obtains the full velocity gradient tensor rather than the highly simplified eddy viscosity models. Thus, it could capture the anisotropic wire-net flow physics. The K-omega SST model is an eddy-viscosity model, which calculates only the diagonal part of the velocity gradient tensor. RSM solves for the main physical mechanisms that drive turbulence: production, redistribution, turbulent transport, viscous di↵usion, dissipation. RSM solves for 27 terms instead of 9 which makes it computationally expensive. In order to reduce the computational complexity, 1 Bar pressure operating condition was maintained in both cold and hot microchannels simultaneously.

Surface Nusselt number and wall shear stress

In a locally turbulent flow, heat transfer outside laminar sublayer occurs through conduction and convection. But the heat transfer mechanism inside the viscous sublayer is dominated by conduction [START_REF] Qiuwang | Experimental study of heat transfer enhancement in narrow rectangular channel with longitudinal vortex generators[END_REF]. As a result, thermal resistance appears in the viscous sublayer. In the wire-net structure, minimum and maximum local Nusselt number occur at locations where the cross section is at its minimum and maximum. The local Nusselt number is higher in the converging sector of the wire-net rather than in the diverging section. This is because the converging section has a higher average velocity and velocity gradient that increases heat transfer ratio. At higher mass flow, Nusselt number is higher, and heat transfer is enhanced compared to the smaller mass flow. This substantially proves the validation of Figure 6, top. The wall shear stress is proportional to Nusselt number [START_REF] Mahmud | Numerical Prediction of Fluid Flow and Heat Transfer in a Wavy Pipe[END_REF] as seen in Figure 6 (top & bottom) and increases locally near the wirenet intersections thereby enhances the heat transfer together with higher pressure losses.

Wire-net induced vortices

Figure 7, a shows the velocity.lambda 2 plotted in the midplane of the microchannel geometry. Higher order RSM analysis was performed to have the symmetric part S ij = 1 2 ( @Ui @xj + @Uj @xi ); and the skew-symmetric part ⌦ ij = 1 2 ( @Ui @xj + @Uj @xi ); of the velocity gradient tensor. 2 criterion looks for local pressure minimum but removes the e↵ects from unsteady straining and viscosity by cancelling those terms [START_REF] Vivianne | Methods for Vortex Identification[END_REF]. A vortex is defined as a connected region with two negative eigenvalues of S 2 + ⌦ 2 . The vorticity mixing is maximum at a higher mass flow (1.57e-5 kg/s), which is expected due to the localised turbulence. The strong counter-rotating vortices near the wire-net intersections shown in Figure 7, a mix the centrally convected non-disturbed mass flow. The counter-rotating vortices that arise from the near wall increased the wall-normal velocity fluctuations and thereby enhanced the heat transfer. 4.2.3. Turbulence production on heat transfer enhancement Higher order RSM directly calculates the three normal turbulent stresses ( ⇢u 2 , ⇢v 2 , ⇢w 2 ) and three tan-gential turbulent shear stresses ( ⇢uv, ⇢vw, ⇢wu) by solving the stress transport equation [START_REF] Pope | Turbulent Flows[END_REF]. The RSM model captures the anisotropy of the flow pattern [START_REF]Comparison of turbulence models in simulating swirling pipe flows[END_REF]. Turbulent kinetic energy production (mean of turbulent normal stresses, see Figure 7, b is dominant after the cylindrical cross-section which is due to the localised mixing of wire-net cross-sectional structures. Among the three normal stresses, wall normal stress appears to be dominant (> 75%) as seen in Figure 7, c which shows an enhanced mixing of centrally convected fluids and also an increased pressure loss. As a result, the major share of the localised turbulence induced by the wire-net enhanced the heat transfer rather than just increasing the pressure losses. The strong wall-normal velocity fluctuations are produced by the wire-net crosssectional structures together with the near wall regions of partition foils, this takes energy from the streamwise velocity and damps the energy to the walls thereby enhancing heat transfer. In particular, turbulent production terms do not require modelling in RSM since they only involve moments of order less or equal to two [START_REF] Manceau | Industrial codes for CFD[END_REF]:

P ij = u i u k @U i @x k u j u k @U j @x k (15)
In the RSM turbulence production terms (see Eq. 14) are not modelled. Influence of the wire-net intersections is evident from the turbulence production term, ⇢uvdU /dx. The stream wise mean velocity gradient, dU /dx is supposed to be positive and negative before and after the circular cross-section due to strong favorable and adverse pressure gradient zones [START_REF] Farhana | Numerical analysis of the flow separation and adverse pressure gradient in laminar boundary layer over a flat plate due to a rotating cylinder in the vicinity[END_REF]. The ⇢uv, vertical flux of stream wise momentum produced by the wire-net walls suppress these strong flow detachment/ separation zones behind the cylindrical cross-sections. This separation is suppressed by the nearby wire-net cross-sectional walls as seen in Figure 7, d thereby, enhancing the mixing.

CONCLUSION

When the Reynolds number increases, the pressure loss also increases in line with a heat transfer enhancement. Thus, the best way to obtain higher thermal e↵ectiveness and heat transfer is to shifts the critical Reynolds number to lower values by using perturbators. The metallic wire-net inserted in the flow passages provided the required sti↵ness and enhanced the microchannel e↵ectiveness. Thermal e↵ectiveness reaches its maximum at a particular optimum mass flow due to the boundary layer development.

A parametric study was carried on various CHE parameters for MGT optimisation.

The strong counter-rotating vortices near the wire-net intersections mix the centrally convected non-disturbed mass flow. The flow separation is alternatively suppressed by the adverse and favourable pressure gradients (created by the wire-net intersections). This could reduce the size of flow detachment zones lowering the pressure losses with an enhanced heat transfer. Among the turbulent Reynolds stresses, wall-normal velocity stress is found to be dominant. As a result, the significant share of the localised turbulence induced by the wire-net enhanced the heat transfer rather than increasing only the pressure losses.

From the experimental testing of the CHE, it was found that the collector pressure losses is around 40-50 % of the microchannel pressure losses. Furthermore, thermal e↵ectiveness is in good agreement with the CHT model, and it validates the microchannel CHT model assumptions. Additionally, it is possible to increase the predicted e↵ectiveness by changing the insulating material or with more compact insulation, and thereby, reducing heat losses to the environment. Thus, in turn, this will increase the calculated e↵ectiveness and reduce the di↵erences between CFD and experiments in terms of microchannel thermal e↵ectiveness.

FIGURE 1 .

 1 FIGURE 1. Arrangement of wire-net brazed CHE and simplification scheme to reduce the computational domain cold collector inlets with several microchannels made from a pile of counterflow passages separated by thin foils. The flow channels are delimited by frames with integrated collectors. The geometries of both frames

FIGURE 2 .

 2 FIGURE 2. Mesh sensitivity analysis (top), CHE test bench (centre), Uncertianity analysis (bottom)

  2 C. Additionally, measurement of volumetric flow

FIGURE 3 .

 3 FIGURE 3. Thermal e↵ectivness (top) and pressure drop charecteristics (bottom) of free microchannels and wire-net microchannels.

FIGURE 4 .

 4 FIGURE 4. Temperature distribution and velocity (nondimensional) distribution along the microchannel length

  Figure 5(bottom) shows the cycle e↵ectiveness of the MGT system for various microchannel inlet mass flows. Microchannel inlet mass flow corresponds to the MGT mass flow/ total CHE mass flow based on Eq. (1). The wirenet cell size of 2.5 mm shows higher pressure losses and thermal e↵ectiveness compared to 3 mm wirenet cell size as shown in Figure 4(bottom) & Figure 5(bottom). At optimum mass flow, cycle e↵ectiveness is a↵ected by the thermal e↵ectiveness pattern of the CHE. At higher mass flows, pressure losses have a more significant influence on cycle performance irrespective of thermal e↵ectiveness.

FIGURE 5 .

 5 FIGURE 5. Thermal e↵ectiveness for di↵erent length (top), Thermal e↵ectiveness on various Fth & WNs sizing (centre), Combined e↵ect of CHE pressure & thermal e↵ectiveness on MGT

FIGURE 6 .FIGURE 7 .

 67 FIGURE 6. Surface Nusselt number (top), Wall shear stress (bottom)

  migrate2015.eu) is supported by European Community H2020 Framework under the Grant Agreement No. 643095. The research of MITIS has been funded by grants Nanocogen+ and Nanocogen+2 from DGO4 and DGO6 directorates of Wallonia. a l e n t h a l p y ( J o u l e s ) micro-CHP micro-Combined Heat & Power IEA International energy agency IBC Inverted brayton cycle J Di↵usion flux of di↵erent species K C o n d u c t i v i t y ( W / ( m . K ) ) S, S h Mass source, heat source MSR Microchannel size reduction MGT Micro gas turbine RSM Reynolds stress model P, Pin/Pout,P ab Inlet and outlet total pressure (Pa), Inlet absolute pressure (Pa) D h Hydraulic diameter R Universal gas constant (J/mol. K) Re Reynolds number U Mean velocity along flow direction(m/s) ⌧w Wall shear stress (Pa) U ⇤ Friction/shear velocity (m/s) V Mean velocity perpendicular to flow direction (m/s) Vin Inlet velocity (m/s) u Velocity fluctuations along flow direction (m/s) v Wall normal velocity fluctuations (m/s) w Velocity fluctuations in the plane parellel to partition foils (m/s) ⇢ Density (kg/m3) T, T h,c in,out

  P in , P out are the inlet and outlet total pressure losses. A parametric study was conducted using the lower order 3 equation SST k ! turbulence model. SST k ! model was implemented by Kanaris et al.[START_REF] Pelletier | CFD simulation of heat transfer in compact brazed plate heat exchangers[END_REF] andPelletier et al. 26], for Re in the range of respectively 500-6000 and 3500.

	, % =	P in P out P ab	⇤ 100	(11)

TABLE 1 .

 1 CHE operating conditions for BC and IBC.

	Instrument	Range (0-FS)	Accuracy
	Vol. Flowmeter Pressure sensor Thermocouples	167 L/min -850 L/min ± 3%FS 0-2 BAR ± 0.25%FS 200 C to 1250 C 2.2 C

TABLE 2 .

 2 Characteristics and accuracy of instrumentation

TABLE 3 .

 3 Experimental

	574	85.1	85.5	0.67	0.33
	774	83.5	86.1	0.74	0.44
	974	82.3	86.8	1.02	0.58

T h,c in (K) ⌘exp(%) ⌘CF D (%) Pexp(%) P CF D(%) and CFD results pressure losses.

  Inlet and outlet total temperature for hot and cold inlet sides (K) m, mcoCollector mass flow (kg/s) Cpc, Cp h Specific heat capacity for cold and hot (J/kg.K) mto Total inlet mass flow (kg/s) mµ, m µh , mµc Inlet and outlet mass flow for hot and cold inlet sides (kg/s)

	n pl	Number of plates/micro channels
	Fth	Partition foil thickness
	WNs	Wire-net cell sizing
	µ	Viscosity (kg/m.s)
	⌘exp	Experimental thermal e↵ectiveness (%)
	⌘CF D	CFD thermal e↵ectiveness (%)
	Pexp	Experimental pressure losses (%)
	PCF D	CFD pressure losses (%)
	"	Thermal e↵ectiveness
	2	Lambda 2 vorticity citeria
	Pij	Turbulence production term