On the solution of the purely nonlocal theory of elasticity as a limiting case of the two-phase theory

In the recent literature stance, purely nonlocal theory of elasticity is recognized to lead to ill-posed problems. Yet, we show that a meaningful energy bounded solution of the purely nonlocal theory may still be defined as the limit solution of the two-phase nonlocal theory. For this, we consider the problem of free vibrations of a flexural beam under the two-phase theory of nonlocal elasticity with an exponential kernel, in the presence of rotational inertia. After recasting the integro-differential governing equation and the boundary conditions into purely differential form, a singularly perturbed problem is met that is associated with a pair of end boundary layers. A multi-parametric asymptotic solution in terms of size-effect and local fraction is presented for the eigenfrequencies as well as for the eigenforms for a variety of boundary conditions. It is found that simply supported end conditions convey the weakest boundary layer and that, surprisingly, rotational inertia affects the eigenfrequencies only in the classical sense. Conversely, clamped and free conditions bring a strong boundary layer and eigenfrequencies are heavily affected by rotational inertia, even for the lowest mode, in a manner opposite to that brought by nonlocality. Remarkably, all asymptotic solutions admit a well defined and energy bounded limit as the local fraction vanishes and the purely nonlocal model is retrieved. ∗Corresponding author Email addresses: mikhasev@bsu.by (Gennadi Mikhasev), andrea.nobili@unimore.it (Andrea Nobili) Preprint submitted to International Journal of Solids and Structures August 8, 2019 Therefore, we may define this limiting case as the proper solution of the purely nonlocal model. Finally, numerical results support the accuracy of the proposed asymptotic approach.


Abstract
In the recent literature stance, purely nonlocal theory of elasticity is recognized to lead to ill-posed problems. Yet, we show that a meaningful energy bounded solution of the purely nonlocal theory may still be defined as the limit solution of the two-phase nonlocal theory. For this, we consider the problem of free vibrations of a flexural beam under the two-phase theory of nonlocal elasticity with an exponential kernel, in the presence of rotational inertia. After recasting the integro-differential governing equation and the boundary conditions into purely differential form, a singularly perturbed problem is met that is associated with a pair of end boundary layers. A multi-parametric asymptotic solution in terms of size-effect and local fraction is presented for the eigenfrequencies as well as for the eigenforms for a variety of boundary conditions. It is found that simply supported end conditions convey the weakest boundary layer and that, surprisingly, rotational inertia affects the eigenfrequencies only in the classical sense. Conversely, clamped and free conditions bring a strong boundary layer and eigenfrequencies are heavily affected by rotational inertia, even for the lowest mode, in a manner opposite to that brought by nonlocality.
Remarkably, all asymptotic solutions admit a well defined and energy bounded limit as the local fraction vanishes and the purely nonlocal model is retrieved.
verted to in the limit of zero length scale and that (b) normalization is satisfied TPNM is solved numerically or it is reduced, by adopting the solution presented 55 in [28], to an equivalent higher-order purely differential model with a pair of ex-56 tra boundary conditions. Despite this reduction, the differential model is still 57 difficult to analyse, especially in the neighbourhood of the PNLM, that is for ξ 1 58 small. In this respect, we believe that the asymptotic approach may be put to 59 great advantage in predicting the mechanical behaviour of nanoscale structures 60 for a vanishingly small ξ 1 [36,19]. 61 In this paper, we consider free vibrations of a flexural beam taking into ac- less, they admit a perfectly meaningful, energy bounded limit, which may be 72 taken as the solution of the PNLM. We point out that the existence of such 73 limit has been observed numerically in [10] for free-free end conditions.
while rotational equilibrium lends Here, v = v(x, t) is the vertical displacement,Q andM are the dimensional 79 shearing force and the bending moment, respectively, ρ is the mass density, 80 J = ρI is the mass second moment of inertia per unit length of the beam, that is proportional to the second moment of area I, S is the cross-sectional area 82 and q(x) the vertical applied load. As well-known, it is I = Sr 2 A , where r A is 83 the radius of gyration. Assuming that the beam is homogeneous and that its 84 cross-section is constant along the length, Eqs. (1,2) give where EI is the beam flexural rigidity, L the beam length and K(|x −x|, κ)

95
In what follows, we consider the Helmholtz kernel which is frequently used for 1D problems [30]. We note that for the Helmholtz 97 kernel the following transformations are valid and In particular, Eq.(7) corresponds to [30, Eq.(6)] and it may be rewritten as whereupon K(|s−ŝ|, ε) is the Green's function of the singularly perturbed oper- It is trivial matter to prove impulsivity, i.e. lim ε→0 K(|s − s|, ε) = δ(s −ŝ), where δ(s) is Dirac's delta function. Furthermore, we observe that Eq.
Here, use have been made of Eqs.(4,5) and we have let the dimensionless ratios together with the microstructure parameter Clearly, θ plays the role of an aspect ratio squared and ε is a scale effect. As-108 suming w ∈ C 6 [0, 1], twice differentiating Eq.(8), taking into account Eqs. (6,7) 109 and then subtracting, we get the governing equation in purely differential form where, hereinafter, we adopt the shorthand ξ = ξ 1 . Eq.(10) is a singularly 111 perturbed ODE [14], with respect to the small parameter ε √ ξ.

Boundary conditions 113
Eq.(10) is supplemented by suitable boundary conditions (BCs) at the ends.
For clamped ends (C-C conditions), we have two pairs of kinematical conditions For simply supported (S-S) ends having let For free-free (F-F) ends, one has The nonlocal end bending moments (13) may be rewritten in differential form with the help of the original integro-differential equation (8): Consequently, the BCs may be recast in differential form as where, making use of the connections (6,7), we have Besides, to rule out spurious solutions which may have appeared owing to double differentiation, we introduce a pair of additional BCs. Indeed, evaluating at the beam ends the differential with respect to s of the original governing equation (8), one arrives at Dropping rotational inertia, the additional boundary conditions (18) coincide their h and λ w , respectively. However, it should be remarked that in [9] the 118 original integro-differential problem is reduced to the equivalent differential form 119 extending to dynamics the original argument developed in [34] for statics. Such 120 argument takes advantage of a result presented in [27], which really applies to 121 inhomogeneous integral equations with a given right-hand side. In the case of 122 dynamics, however, this right-hand side is a problem unknown, for it is really 123 an acceleration term, and therefore the applicability of the reduction formula is 124 questionable. The general solution of the ODE (10) is where the constants b j are the roots of the characteristic polynomial in ζ As detailed in [31,22], this bi-cubic may be turned in canonical form by the This polynomial possesses three real roots provided that and indeed, for ε √ ξ 1, we get, to leading order, Besides, we have, at leading order, 132 q = 2 27(ξε 2 ) 3 and q > 0, whereupon out of the three real roots, two, say Z 3 < Z 2 < 0, are negative and one, say Z 1 , is positive. Upon reverting to the original variable ζ, we see that ζ 2 3 < 0 < ζ 2 2 < ζ 2 1 . Indeed, we get the expansions (the sign is immaterial) whence ζ 1,2 are convey an exponential solution, while ζ 3 is related to an oscillatory solution. It is worth noticing that ζ 1 blows up as (ε √ ξ) → 0, that is The parameter ς is named the index of variation of the edge effect integrals, whose solution is sought in the form of an asymptotic series Substitution of (23) into (22) lends a sequence of differential equations in the 169 unknowns w where a ij (i = 1, 2; j = 0, 1, 2, . . .) are constants that will be determined in the 172 following from the boundary conditions.
The leading term in the series corresponds to the solution of the classical local problem and λ 0 is the classical eigenvalue. Substituting (25) into the governing 178 Eq.(10) and equating coefficients of like powers of ε leads to the sequence of 179 differential equations: where At leading order, one finds the homogeneous forth order ODE • the k th -order approximation generates two equations coupling the con- In light of the definition (20b), we find the eigenfrequencies and, by (9), the corresponding dimensional frequencies ω 0 = EI ρS (λ 0 /L) 2 .

204
Moving to first-order terms, we again obtain a set of homogeneous boundary as well as formulas for the leading amplitude in the boundary layer (24): where C 1 is an arbitrary constant. Without loss of generality, one can assume 210 w 1 ≡ 0, for this amounts to taking C = C 0 + εC 1 + . . . .

211
In the second-order approximation, when taking into account the outcomes of the previous step, we have again a homogeneous set of boundary conditions and a 11 = a 21 = 0. The associated differential equation for w 2 reads We thus arrive at the inhomogeneous BVP on "spectrum". Upon observing that the homogeneous boundary-value problem arising at leading order is selfconjugated and therefore possesses the solution z(s) = w 0 (s), we deduce the compatibility condition for the BVP (33,34) 1 0 w 0 (s)L 2 w 0 (s)ds = 0, which readily gives the correction for the eigenvalue: .
On taking into account this result, Eq. (34) turns homogeneous and, without 215 loss of generality, we can assume w 2 ≡ 0.

216
Considering the third-order approximation, one obtains the inhomogeneous 217 boundary conditions for the inhomogeneous ODE The compatibility condition for the boundary-value problem (35,36) works whence we get the next correction term for the eigenvalue .
The eigenform correction w 3 , satisfying the boundary conditions (35) where Consequently, making use of (37), we get Breaking at this step the asymptotic procedure for seeking the eigenvalues λ k and the associated eigenfunctions w k , we obtain the asymptotic expansion Remarkably, this expression is independent of θ and this unexpected feature
In the leading approximation, one has the classical boundary conditions as well as the frequency equation In particular, if θ = 0, one arrives at the classical frequency equation, cosh λ 0 cos λ 0 = 1, valid for a Bernoulli-Euler beam that disregards the rotational inertia of the cross-section, the corresponding eigenmode being Besides, we get In the first-order approximation, one has the inhomogeneous ODE (26) and the procedure of splitting the boundary conditions gives 242 w 1 (0) = w 1 (1) = 0, The compatibility conditions for the BVP (44,45) reads Dw 0 (s)w 0 (s)ds = 0, whence, accounting for Eqs.(45), one obtains the correction where part-integration has been used at the denominator. Now, we can write and Eq. (48) gives Similarly, Eq.(47) becomes Breaking the asymptotic procedure at this step, we can write down the 247 approximate formula for the nonlocal-to-local frequency ratio that, in the absence of rotary inertia, reduces to  The asymptotic expansion for the eigenmode reads where w 0 and w 1 belong to the outer solution and they are given by (28)  To fix ideas, let the left beam end be clamped and the right simply supported.

261
The correspondent boundary conditions are given by (11a), (12b) and the pair 262 of additional conditions (18). In this case, we arrive at γ 1 = 2 and γ 2 = 3 for 263 the left and for the right boundary layer, respectively.

264
At leading order, one has the classical boundary conditions whence we get the constants in the general solution (28) The first-order approximation yields and a 10 and a 20 are defined by Eqs.(43a,32b) 269 The inhomogeneous equation (44) The solution of the BVP (44,54) has the form (47) as for the C-C case, yet with 273 different coefficients. Indeed, in the special case θ = 0, Eq.(56) simplifies to and the particular solution becomes Finally, we arrive at the following asymptotic expansion for the frequency that, in the case θ = 0, reduces to   In the first-order approximation, one arrives at the following boundary con- together with the right boundary layer amplitude