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On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory
On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory
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On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory Gennadi Mikhasev, Andrea Nobili PII: DOI: Reference:
S0020-7683(19)30455-X https://doi.org/10.1016/j.ijsolstr.2019.10.022 SAS 10522
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
8 August 2019 3 October 2019 21 October 2019
Please cite this article as: Gennadi Mikhasev, Andrea Nobili, On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory, International Journal of Solids and Structures (2019), doi: https://doi.org/10.1016/j.ijsolstr.2019.10.022
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On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory Gennadi Mikhaseva , Andrea Nobilib,∗ a Department
of Bio- and Nanomechanics, Belarusian State University, 4 Nezavisimosti Ave., 220030 Minsk, Belarus b Engineering Department ”Enzo Ferrari”, Universit` a degli Studi di Modena e Reggio Emilia, 41125 Modena, Italy
Abstract In the recent literature stance, purely nonlocal theory of elasticity is recognized to lead to ill-posed problems. Yet, we show that, for a beam, 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, for simply supported end, the weakest boundary layer is formed and, surprisingly, rotational inertia affects the eigenfrequencies only in the classical sense. Conversely, clamped and free end 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: [email protected] (Gennadi Mikhasev), [email protected] (Andrea Nobili)
Preprint submitted to International Journal of Solids and Structures
October 31, 2019
Therefore, we may define this limiting case as the proper solution of the purely nonlocal model for a beam. Finally, numerical results support the accuracy of the proposed asymptotic approach. Keywords: Two-phase nonlocal elasticity, Nonlocal theory of elasticity, Asymptotic method, Free vibrations
1
1. Introduction
2
The classical linear theory of elasticity suffers from the well known defect
3
of not encompassing an internal length scale, which feature gives rise to self-
4
similar predictions. Yet, any real material possesses an internal microstructure
5
and some characteristic length thereof. Consequently, classical elasticity may
6
be assumed as a suitable model inasmuch as the physical phenomena of interest
7
occur at a scale much greater than the internal characteristic length of the ma-
8
terial. Failure to meet this condition is effectively demonstrated by, for instance,
9
the singular stress field at the tip of a crack and by the non-dispersive nature
10
of wave propagation. Extensions of classical elasticity have been proposed, in
11
the form of generalized continuum mechanics (GCM), in an attempt to reme-
12
diate these shortfalls. An excellent historical overview of GCM, together with
13
extensive bibliographic details, may be found in [20]. Among GCM theories,
14
we mention the theory of micro-polar elasticity [3, 4, 28], the couple-stress and
15
strain-gradient elasticity theories [37, 26] and the nonlocal theory of elasticity
16
[8]. In particular, following [8], ”linear theory of nonlocal elasticity, which has
17
been proposed independently by various authors [...], incorporates important
18
features of lattice dynamics and yet it contains classical elasticity in the long
19
wave length limit”. Nonlocal elasticity is based on the idea that the stress
20
state at a point is a convolution over the whole body of an attenuation function
21
(sometimes named kernel or nonlocal modulus) with the strain field [36]. Al-
22
though several attenuation functions may be considered, they need to comply
23
with some important properties which warrant that (a) classical elasticity is re-
2
24
verted to in the limit of zero length scale and that (b) normalization is satisfied
25
[7, 17]. As an example, Helmholtz and bi-Helmoltz kernels have been widely
26
used in 1-D problems, their name stemming from the differential operators they
27
are Green’s function of [9, 16]. In [6], a variational argument is adopted to de-
28
duce the governing equations for a purely nonlocal Euler-Bernoulli beam, whose
29
eigenfrequencies are numerically investigated.
30
Since nonlocal elasticity naturally leads to integro-differential equations whose
31
solution is most often impractical, an equivalent differential nonlocal model
32
(EDNM) was developed in [7]. In such differential form, nonlocal elasticity has
33
been extensively applied to study elastodynamics of beams and shells as de-
34
scribed in the recent review [5] and with special emphasis on the application
35
to nanostructures [32]. Generally, EDNM leads to interesting mechanical ef-
36
fects, such as increased deflections and decreased buckling loads and natural
37
frequencies (softening effect), when compared to classical elasticity. However, a
38
number of pathological results have also emerged, which are often referred to as
39
paradoxes [18, 11, 16]. For instance, for a cantilever beam under point loading,
40
nonlocality brings no effect [27, 35, 2, 17]. It should be remarked that many
41
studies based on the EDNM employ boundary conditions in terms of macro-
42
scopic stresses, i.e. in classical form, and therefore they disregard the important
43
effect of the boundary through nonlocality. Although this approach may be still
44
adopted for long structures or in the case of localized deformations occurring
45
away from the boundaries [23, 24], it is generally inaccurate.
46
Very recently, Romano et al. [33] claimed that Eringen’s purely nonlocal
47
model (PNLM) leads to ill-posed problems for the differential form of the model
48
is consistent inasmuch as an extra pair of boundary conditions, termed con-
49
stitutive, is satisfied (see also [19]). In [8, Eq.(6.4)] and in [1], a two-phase
50
nonlocal model (TPNL) was introduced, within the context of 3D elasticity,
51
which combines, according to the theory of mixtures, purely nonlocal elasticity
52
with classical elasticity, by means of the volume fractions ξ1 and ξ2 = 1 − ξ1 .
53
This model is immune from the inconsistencies of the PNLM and it has been
54
adopted to solve the problem of static bending [36] and buckling [38] of Euler3
55
Bernoulli (E-B) beams. Static axial deformation of a beam is considered in
56
[29, 39], while semi-analytical solutions for the combined action of axial and
57
flexural static loadings is given in [21]. Axial and flexural free vibrations of
58
beams have also been considered in [22] and in [10]. In these works, either the
59
TPNM is solved numerically or it is reduced, by adopting the solution presented
60
in [31], to an equivalent higher-order purely differential model with a pair of ex-
61
tra boundary conditions. Despite this reduction, the differential model is still
62
difficult to analyse, especially in the neighbourhood of the PNLM, that is for ξ1
63
small. In this respect, we believe that the asymptotic approach may be put to
64
great advantage in predicting the mechanical behaviour of nanoscale structures
65
for a vanishingly small ξ1 [38, 22].
66
In this paper, we consider free vibrations of a flexural beam taking into ac-
67
count rotational inertia (Rayleigh beam), within the TPNM and having assumed
68
the Helmholtz attenuation function. The integro-differential model is reduced
69
to purely differential form with an extra pair of boundary conditions. Spotlight
70
is set on developing asymptotic solutions valid for small microstructure and/or
71
little local fraction. These solutions feature a pair of boundary layers located
72
at the beam ends, whose strength depends on the constraining conditions. Nu-
73
merical results support the accuracy of the expansions. Most remarkably, the
74
asymptotic approach allows to investigate the behaviour of the solution in the
75
neighbourhood of the PNLM, where the expansions are non-uniform. Nonethe-
76
less, they admit a perfectly meaningful, energy bounded limit, which may be
77
taken as the solution of the PNLM. We point out that the existence of such
78
limit has been observed numerically in [11] for free-free end conditions.
79
2. Problem formulation
80
2.1. Governing equations
81
For a flexural beam, vertical equilibrium gives ρS
ˆ ∂2v ∂Q = + qˆ(x) 2 ∂t ∂x
4
(1)
82
while rotational equilibrium lends J
ˆ ∂M ∂2ϕ ˆ =− + Q. 2 ∂t ∂x
(2)
83
ˆ and M ˆ are the dimensional Here, v = v(x, t) is the vertical displacement, Q
84
shearing force and the bending moment, respectively, ρ is the mass density per
85
unit volume, J = ρI is the mass second moment of inertia per unit length of
86
the beam, that is proportional to the second moment of area I, S is the cross-
87
sectional area and qˆ(x) the vertical applied load. Assuming that the beam is
88
homogeneous and prismatic, Eqs.(1,2) give ˆ ∂2M ∂2v ∂4v − ρS + J + qˆ = 0, ∂x2 ∂t2 ∂x2 ∂t2
(3)
89
that governs transverse vibrations of flexural beams accounting for rotational
90
inertia. In the following, we take qˆ ≡ 0. In the mixed nonlocal theory (MNLT)
91
of elasticity, we have [10]
2 ˆ = −EI ξ1 ∂ v + ξ2 M ∂x2
ZL 0
∂2v K(|x − x ˆ|, κ) 2 dˆ x , ∂x ˆ
(4)
92
where EI is the beam flexural rigidity, L the beam length and K(|x − x ˆ|, κ) is
93
the kernel or attenuation function. The kernel is positive, symmetric, it rapidly
94
decays away from x and it satisfies the normalization condition Z K(|x − x ˆ|, κ)dˆ x = 1. R
95
The constitutive equation (4) is also obtained from consideration of a more gen-
96
eral form of kernel discussed in [17]. The nonlocal parameter κ = e0 a depends
97
on the scale coefficient e0 as well as on the internal length scale a. ξ1 and ξ2
98
take up the role of volume fractions and they represent, respectively, the local
99
and the nonlocal phase ratios, such that ξ1 + ξ2 = 1 and ξ1 ξ2 ≥ 0. When ξ1 = 0,
100
Eq.(4) degenerates into the purely nonlocal model (PNLM), while, in contrast,
101
the case ξ1 = 1 corresponds to classical local elasticity.
102
In what follows, we consider the Helmholtz kernel 1 |x − x ˆ| K(|x − x ˆ|, κ) = exp − , 2κ κ 5
(5)
103
which is frequently used for 1D problems (indeed, it is named special kernel in
104
[33]). We note that for the Helmholtz kernel the following transformations are
105
valid d ds
Z1
e
|s−ˆ s| − ε
s
0
106
Z1 Zs s ˆ s s ˆ 1 s y(ˆ s)dˆ s= e− ε y(ˆ s)dˆ s − e− ε e ε y(ˆ s)dˆ s , eε ε
and d2 ds2
Z1
e−
|s−ˆ s| ε
y(ˆ s)dˆ s=
(6)
0
1 ε2
0
Z1
e−
0
|s−ˆ s| ε
2 y(ˆ s)dˆ s − y(s). ε
(7)
In particular, Eq.(7) corresponds to [33, Eq.(6)] and it may be rewritten as Z 1 2 ˆ|, ε) 2 d K(|s − s − K(|s − sˆ|, ε) + δ(|s − sˆ|) y(ˆ s)dˆ s = 0, ε ds2 0 whereupon K(|s− sˆ|, ε) is the Green’s function of the singularly perturbed oper2
d ator Hε = 1 − ε2 ds 2 . It is trivial matter to prove impulsivity, i.e. limε→0 K(|s −
sˆ|, ε) = δ(s − sˆ), where δ(s) is Dirac’s delta function. Furthermore, we observe that Eq.(6), evaluated at the beam ends s = 0, 1 and for ξ = 0, lends the constitutive boundary conditions [33, Eq.(5)] ˆ dM ˆ (0), (0) = ε−1 M ds
and
ˆ dM ˆ (1). (1) = −ε−1 M ds
107
Thus, the constitutive boundary conditions are really the expression, on the
108
domain boundary, of a general feature of the solution that is related to the
109
integral operator (4).
110
111
112
113
114
Introducing the dimensionless axial co-ordinate s = x/L and under the assumption of time-harmonic motion, we write n o ˆ,Q ˆ = w(s), EI M (s), EI Q(s) exp(ıωt), v, M L L2
where ı is the imaginary unit. Upon multiplying throughout by L4 /EI, Eq.(3)
may be turned in dimensionless form Z 1 d2 w ξ2 |ˆ s − s| d2 w(ˆ s) d4 w 4 −2 + exp − dˆ s − λ4 w = 0. (8) ξ1 4 + λ θ − ε ξ2 ds ds2 2ε3 0 ε dˆ s2
Here, use have been made of Eqs.(4,5) and we have let the dimensionless ratios r 2 J ρSL4 ω 2 A 4 θ= = , λ = , (9) ρSL2 L EI 6
together with the microstructure parameter ε=
κ 1. L
115
Here rA is the radius of gyration. Clearly, θ plays the role of an aspect ratio
116
squared and ε is a scale effect. Assuming w ∈ C 6 [0, 1], twice differentiating
117
Eq.(8), making use of Eq.(7) and then subtracting the original equation (8), we
118
get the governing equation in purely differential form ε2 ξ
4 d6 w d2 w 2 4 d w 4 2 − (1 − ε − λ + λ4 w = 0, θλ ) (ε + θ) ds6 ds4 ds2
(10)
120
where, hereinafter, we adopt the shorthand ξ = ξ1 . Eq.(10) is a singularly √ perturbed ODE [15], with respect to the small parameter ε ξ.
121
2.2. Boundary conditions
119
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
w(0) = w0 (0) = 0,
(11a)
w(1) = w0 (1) = 0.
(11b)
For simply supported (S-S) ends
122
w(0) = 0,
M (0) = ξw00 (0) + M0 = 0,
(12a)
w(1) = 0,
M (1) = ξw00 (1) + M1 = 0,
(12b)
having let M0 =
1−ξ 2ε
Z1
s ˆ
s)dˆ s, e− ε w00 (ˆ
M1 =
0
1 − ξ −1 e ε 2ε
Z1
s ˆ
e ε w00 (ˆ s)dˆ s.
(13)
0
For free-free (F-F) ends, one has M (0) = 0,
Q(0) = ξw000 (0) + θλ4 w0 (0) + ε−1 M0 = 0,
(14a)
M (1) = 0,
Q(1) = ξw000 (1) + θλ4 w0 (1) − ε−1 M1 = 0.
(14b)
7
The nonlocal end bending moments (13) may be rewritten in differential form with the help of the original integro-differential equation (8): M0 = −ε2 ξwiv (0) + 1 − ξ − ε2 θλ4 w00 (0) + ε2 λ4 w(0), M1 = −ε2 ξwiv (1) + 1 − ξ − ε2 θλ4 w00 (1) + ε2 λ4 w(1).
(15a) (15b)
Consequently, the BCs may be recast in differential form through M (0) = w00 (0) + ε2 N0 ,
(16a)
M (1) = w00 (1) + ε2 N1 ,
(16b)
Q(0) = ξw000 (0) + θλ4 w0 (1) + ε−1 M0 ,
(16c)
Q(1) = ξw000 (1) + θλ4 w0 (1) − ε−1 M1 ,
(16d)
where, making use of the connections (6,7), we have N0 = ε−2 (ξ2 w00 (0) − M0 ) = −ξwiv (0) − θλ4 w00 (0) + λ4 w(0),
(17a)
N1 = ε−2 (ξ2 w00 (1) − M1 ) = −ξwiv (1) − θλ4 w00 (1) + λ4 w(1).
(17b)
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 original governing equation (8), differentiated once with respect to s, one arrives at ε3 ξwv (0) − ε2 ξwiv (0) − (1 − ξ − ε2 θλ4 )[εw000 (0) − w00 (0)] −ε3 λ4 w0 (0) + ε2 λ4 w(0) = 0,
(18a)
ε3 ξwv (1) + ε2 ξwiv (1) − (1 − ξ − ε2 θλ4 )[εw000 (1) + w00 (1)] −ε3 λ4 w0 (1) − ε2 λ4 w(1) = 0.
(18b)
123
Dropping rotational inertia, the additional boundary conditions (18) coincide
124
with the constitutive boundary conditions recently obtained by Fern´ andez-S´ aez
125
and Zaera [10, Eqs.(59) and (60)], provided that we replace our ε and λ4 with
126
their h and λw , respectively. However, it should be remarked that in [10] the
127
original integro-differential problem is reduced to the equivalent differential form 8
128
extending to dynamics the original argument developed in [36] for statics. Such
129
argument takes advantage of a result presented in [30], which really applies to
130
inhomogeneous integral equations with a given right-hand side. In the case of
131
dynamics, however, this right-hand side is a problem unknown, for it is really
132
an acceleration term, and therefore the applicability of the reduction formula is
133
questionable.
134
3. Exact solution of the boundary-value problems The general solution of the ODE (10) is w(s) =
6 X
cj exp (bj s) ,
j=1
135
where the constants bj are the roots of the characteristic polynomial in ζ ε2 ξζ 6 − (1 − ε2 θλ4 )ζ 4 − (ε2 + θ)λ4 ζ 2 + λ4 = 0.
(19)
136
As detailed in [34, 25], this bi-cubic may be turned in canonical form by the
137
substitution Z = ζ 2 − Z0 , it being Z0 = (1 − ε2 θλ4 )/(3ε2 ξ). Hence, Eq.(19)
138
becomes Z 3 − pZ − q = 0, where "
# 2 λ4 θε2 − 1 4 2 p = (ξε ) +λ θ+ε > 0, 3ξε2 " 3 # λ4 θ + ε2 λ4 θε2 − 1 2 λ4 θε2 − 1 2 −1 4 q = −(ξε ) λ + + . 3ξε2 27ξ 2 ε4 2 −1
This polynomial possesses three real roots provided that ∆= 139
p3 q2 − <0 4 27
√ and indeed, for ε ξ 1, we get, to leading order, ∆ = −λ4
4 + θ2 λ4 . 108(ξε2 )4
9
140
Besides, we have, at leading order, q=
2 27(ξε2 )3
and q > 0, whereupon out of the three real roots, two, say Z3 < Z2 , are negative and one, say Z1 , is positive. Upon reverting to the original variable ζ, we see that ζ32 < 0 < ζ22 < ζ12 . Indeed, we get the leading order solutions (the sign is immaterial) 1 ζ1 = √ , ε ξ with
ζ2 = α,
ζ3 = ıβ,
s
r 1 2 θ 2 λ4 , α = λ − θλ + 1 + 2 4 s r 1 2 θ2 λ4 β=λ θλ + 1 + , 2 4 141
142
(20a) (20b)
whence ζ1,2 convey an exponential solution, while ζ3 is related to an oscillatory √ contribution. It is worth noticing that ζ1 blows up as (ε ξ) → 0, that is for a
143
vanishingly small scale effect or in the purely nonlocal situation. Indeed, this
144
very root accounts for the edge effect in this problem and it describes a boundary
145
layer.
146
147
We observe that, in general, the frequency equation for the ODE (10), subject to suitable boundary conditions, appears in transcendental form F (λ; ξ, ε) = 0,
148
wherein λ is the sought-for eigenvalue. The numerical solution of this equation is
149
not straightforward matter, especially for very small values of the local fraction
150
ξ, see e.g. [10] and [36] where plots are given for ξ > 0.1 and ξ > 0.05, respec-
151
tively. Indeed, when looking for the numerical roots of (19), we observe, after
152
[34], that the transformation to canonical form lends a considerable numerical
153
advantage over Cardano’s formulas, in that it provides purely real solutions.
154
Conversely, Cardano’s formulas are likely to introduce a very small spurious
155
imaginary component, which is most likely the cause of the numerical difficulty
156
encountered in the literature when dealing with small ξ. 10
157
To estimate the eigenvalue λ for any ξ and, in particular, in the limiting case
158
of the PNLM (that occurs as ξ → 0), we consider an asymptotic expansion in
159
the small parameter ε.
160
4. Asymptotic solution of the boundary-value problems
161
Following a standard asymptotic argument [15, 22] and similarly to the ex-
162
traction of the edge effect in shells [12, 13], we seek a solution of the eigenvalue
163
problem through superposition of a solution, w(m) , valid in the interior of the
164
beam (the so-called outer solution), with a pair of boundary layers, w1
165
(e) w2 ,
(e)
fading off away from the left and from the right beam end, respectively, (e)
(e)
w(s, ε) = w(m) (s) + εγ1 w1 (s, ε) + εγ2 w2 (s, ε), 166
and
(21)
where γ1,2 > 0 and we have the order relations ∂w(m) ∼ w(m) , ∂s
(e)
∂wi ∂s
(e)
∼ ε−ς wi
as
ε → 0.
167
The parameter ς is named the index of variation of the edge effect integrals,
168
while γ1,2 are the indices of intensity of the edge effect integrals near the left
169
and right ends, respectively. The indices γ1,2 depend on the boundary conditions
170
and should be specified for each end.
171
4.1. Boundary layer
172
To derive an equation describing the behaviour of the solution in the vicinity
173
of the ends (boundary layer), we zoom in by assuming s = ες σ and 1 − s = ες σ,
174
respectively for the left and for the right end. For either case, one obtains the
175
distinguished limit ς = 1 and Eq. (10) is rewritten as (e)
ξ
(e)
(e)
4 2 d6 wi (e) 2 4 d wi 2 4 2 d wi − 1 − ε θλ − ε λ θ + ε + ε4 λ4 wi = 0, dσ 6 dσ 4 dσ 2
(22)
176
whose solution is sought in the form of an asymptotic series in the small param-
177
eter ε 1
(e)
wi
(e)
(e)
(e)
= wi0 + εwi1 + ε2 wi2 + . . . , 11
i = 1, 2.
(23)
178
Substitution of (23) into (22) lends a sequence of differential equations in the
179
unknowns wij (σ), i = 1, 2; j = 0, 1, 2, . . .. Here, we simply give the first two
180
(e)
terms of the expansion in the original variable s h i 4 s s s (e) − √ − √ 2 − ε√ ξ √ w1 (s, ε) = a10 e ε ξ + εe ε ξ a11 + a10 θλ 2(1−ξ) s + O ε e , ξ (e)
w2 (s, ε) = a20 e
√ − ε1−s ξ
+ εe
√ − ε1−s ξ
h i 1−s 4 2 − ε√ξ √ a21 + a20 θλ 2(1−ξ) , (1 − s) + O ε e ξ
(24)
181
where aij (i = 1, 2; j = 0, 1, 2, . . .) are constants that will be determined in the
182
following from the boundary conditions.
183
4.2. The outer solution
184
185
The displacement w(m) as well as the eigenvalue λ are sought in the form of an asymptotic series w(m) = w0 + εw1 + ε2 w2 + . . . , (25) 2
λ = λ0 + ελ1 + ε λ2 + . . . . 186
The leading term in the series corresponds to the solution of the classical local
187
problem and λ0 is the classical eigenvalue. Substituting (25) into the governing
188
Eq.(10) and equating coefficients of like powers of ε leads to the sequence of
189
differential equations: k X
Lj wk−j = 0,
k = 0, 1, 2, . . . ,
(26)
j=0
where L0 z =
190
d4 z d2 z d2 z + θλ40 2 − λ40 z, L1 z = −4λ30 λ1 Dz, Dz = z − θ 2 , 4 ds ds ds 4 2 d6 z d z d z L2 z = −ξ 6 − θλ40 4 + λ40 2 − 2λ20 (3λ21 + 2λ0 λ2 )Dz, ds ds ds 4 2 d z d z L3 z = −4θλ30 λ1 4 + 4λ30 λ1 4 − 4λ0 (λ20 λ3 + λ31 + 2λ0 λ1 λ2 )Dz, . . . ds ds
At leading order, one finds the homogeneous forth order ODE L0 w0 = 0, 12
(27)
191
whose general solution w0 (s) = c01 sin(βs) + c02 cos(βs) + c03 e−αs + c04 eα(s−1) ,
(28)
192
depends on the constants, c0i , i ∈ {1, 2, 3, 4}, to be determined through the
193
boundary conditions. However, the ODE (27) is subject to six boundary con-
194
ditions and the problem is to determine which of these correspond to the outer
195
solution and which pertain to the boundary layer [15]. The procedure of split-
196
ting the boundary conditions also gives the indices of intensity of the boundary
197
layer, γ1 , γ2 , as well as the constants c0k , aij . For this, one needs to insert the
198
expansions (21,24,25) into the boundary conditions and equate coefficients of
199
like powers of ε, while imposing the following requirements:
200
201
• in the leading approximation, every end condition should be homogeneous and coincide with those of the classical local theory;
202
• the k th -order approximation generates two equations coupling the con-
203
stants ai(k−1) with the previous order approximation wk−1 (s) evaluated
204
at the boundaries.
205
4.3. Beam with simply supported ends
206
Let both beam ends be simply supported (S-S conditions), as given by the
207
boundary conditions (12) rewritten in differential form through Eqs.(16a,16b),
208
together with the additional constraints (18).
209
(21,24,25) into these conditions, we determine the strength of either boundary
210
layer: γ1 = γ2 = 3.
Substituting the expansions
At leading order, we arrive at the homogeneous classical boundary conditions w0 (0) = w0 (1) = w000 (0) = w000 (1) = 0, 211
which give c01 = C, c02 = c03 = c04 = 0 and the classical eigenforms w0 (s) = C sin(βs),
212
β = πn,
n = 1, 2, . . . .
(29)
In light of the definition (20b), we find the eigenfrequencies (n)
λ0 = λ0 ≡
πn , [1 + θ(πn)2 ]1/4 13
n = 1, 2, . . . ,
(30)
213
214
215
and, using (9), the corresponding dimensional frequencies ω0 =
q
EI ρS
(λ0 /L)2 .
Moving to first-order terms, we again obtain a set of homogeneous boundary conditions w1 (0) = w1 (1) = w100 (0) = w100 (1) = 0,
(31)
as well as formulas for the leading amplitude in the boundary layer (24): p p p p a10 = − ξ(1 − ξ)w0000 (0) = Cβ 3 ξ(1 − ξ), p p p p a20 = ξ(1 − ξ)w0000 (1) = C(−1)n+1 β 3 ξ(1 − ξ).
(32a) (32b)
216
Consideration of the inhomogeneous ODE (26) arising in this approximation,
217
alongside the associated homogeneous boundary conditions (31), yields the com-
218
patibility condition λ1 = 0, whence w1 = C1 sin(βs),
219
where C1 is an arbitrary constant. Without loss of generality, one can assume
220
w1 ≡ 0, for this amounts to taking C = C0 + εC1 + . . . .
221
222
In the second-order approximation, when taking into account the outcomes of the previous step, we have again a homogeneous set of boundary conditions w2 (0) = w2 (1) = w200 (0) = w200 (1) = 0,
223
(33)
and a11 = a21 = 0. The associated differential equation for w2 reads L0 w2 = −L2 w0 ≡ ξ
4 d2 w0 d6 w0 4 d w0 3 + θλ − λ (λ + 4θλ ) + 4λ30 λ2 w0 . (34) 0 2 0 0 ds6 ds4 ds2
We thus arrive at the inhomogeneous boundary value problem (BVP) ”on spectrum”. Upon observing that the homogeneous BVP (33,34) arising at leading order is self-conjugated and therefore possesses the solution z(s) = w0 (s), we deduce the compatibility condition Z1
w0 (s)L2 w0 (s)ds = 0,
0
which readily gives the correction for the eigenvalue: λ2 = −
β 2 [λ40 (1 + θβ 2 ) − ξβ 4 ] . 4λ30 (1 + θβ 2 ) 14
224
On taking into account this result, Eq. (34) turns homogeneous and, without
225
loss of generality, we can assume w2 ≡ 0.
226
227
Considering the third-order approximation, one obtains the inhomogeneous boundary conditions √ √ w3 (0) = −a10 = −Cβ 3 ξ(1 − ξ), √ √ w3 (1) = −a20 = C(−1)n β 3 ξ(1 − ξ), w300 (0)
=
θλ40 a10
=
Cθλ40 β 3
√
ξ(1 −
√
(35)
ξ),
√ √ w300 (1) = θλ40 a20 = (−1)n+1 Cθλ40 β 3 ξ(1 − ξ) 228
for the inhomogeneous ODE L0 w3 = −L3 w0 ≡ 4λ30 λ3 Dw0 .
(36)
The compatibility condition for the BVP (35,36) works out − w300 (1)w00 (1) + w300 (0)w00 (0) − w3 (1)w0000 (1) + w3 (0)w0000 (0) Z 1 4 0 0 3 + θλ0 [w3 (0)w0 (0) − w3 (1)w0 (1)] + 4λ0 λ3 (w0 − θw000 )w0 ds = 0, 0
229
whence we get the next correction term for the eigenvalue √ √ β 6 ξ(1 − ξ) . λ3 = λ30 (1 + θβ 2 )
(37)
The eigenform correction w3 , satisfying the boundary conditions (35), is given by the sum of a particular solution w3p of Eq.(36), with the homogeneous solution w3o . The former reads w3p (s) = C3p s cos(βs), where C3p = 2Cλ30 λ3
p β5 p 1 + θβ 2 = 2C 2 ξ(1 − ξ). 2 2 2 β(α + β ) α +β
Consequently, making use of (37), we get w3 (s) = Cβ 3
p
ξ(1 −
p ξ) {c32 cos(βs) + c33 exp(−αs)
+ c34 exp[α(s − 1)] − 2c32 s cos(βs)} , 15
Figure 1: 1st (left) and 2nd (right) eigenfrequencies ω for a S-S beam (solid, black), with ε = 0.01, 0.05 and 0.075, superposed onto the 1-term (dashed, red) and the 2-term (dotted, blue) asymptotic approximation, normalized with respect to the classical local frequency ω0 , Eq.(39)
with the constants c32 = −β 2 /(α2 + β 2 ), c33 = 21 α2 eα (1 − coth α) [eα + (−1)n ] /(α2 + β 2 ), c34 = − 21 α2 eα (1 − coth α) [(−1)n eα + 1] /(α2 + β 2 ). Breaking at this step the asymptotic procedure for seeking the eigenvalues λk and the associated eigenfunctions wk , we obtain the asymptotic expansion i h p p λ = λ0 1 − 41 ε2 β 2 (1 − ξ) + ε3 β 2 ξ (1 − ξ) + O(ε4 ) ,
where β and λ0 are determined by (29) and (30), respectively. Up to an undetermined factor, the associated eigenmode reads p p n ξ(1 − ξ) c32 cos(πns) + c33 exp(−αs) s s−1 o √ + c34 exp[α(s − 1)] − 2c32 s cos(πns) + exp − √ + (−1)n+1 exp ε ξ ε ξ 4 + O ε . (38) w(s) = sin(πns) + ε3 (πn)3
230
It is of interest to compare the dimensional natural frequency, ω, determined
231
with the TPNM, with its classical counterpart, ω0 , evaluated within the frame-
232
work of local elasticity, i.e. for ξ = 1. When taking into account the definition
16
233
(9), we arrive at the relation ω = ω0
λ λ0
2
p p 1 = 1 − ε2 (πn)2 (1 − ξ) + 2ε3 (πn)2 ξ(1 − ξ) + O ε4 . 2
(39)
234
Remarkably, this expression is independent of θ and this unexpected feature
235
is indeed confirmed by the numerical solution of the TPNM, see Fig.5. Fig.1
236
plots the approximation (39) in the range 0 < ξ < 1 against the numerical
237
solution of the TPNM (given for ξ > 0.01) for the scale parameter ε = 0.01, 0.05
238
and 0.075. It appears that the 1-term asymptotic approximation is remarkably
239
effective for small values of ε. The numerical solution of the TPNM given in
240
Fig.1 compares favourably with the corresponding solution depicted in Fig.4 of
241
[11] that, however, pertains to the range ξ1 > 0.1, presumably owing to the
242
numerical difficulties that may arise in the neighbourhood of the PNLM.
243
244
As a special case of Eq.(39), one obtains the eigenfrequency ratio corresponding to the PNLM (i.e. for ξ = 0) ω 1 = 1 − ε2 (πn)2 + O ε4 . ω0 2
245
(40)
4.4. Beam with clamped ends Consideration of a beam with clamped ends requires enforcing (11) and (18) on Eqs.(21,24,25). We thus get the strength of the boundary layer γ1 = γ2 = 2. In the leading approximation, one has the classical boundary conditions w0 (0) = w0 (1) = w00 (0) = w00 (1) = 0,
246
that give the constants c01 = 2α(cosh α − cos β) c02 = 2α sin β − 2β sinh α,
c03 = β (eα − cos β) − α sin β,
(41)
c04 = −eα α sin β + β (eα cos β − 1) ,
247
as well as the frequency equation 2 1 2 θλ0
sin β sinh α + cos β cosh α − 1 = 0. 17
(42)
In particular, if θ = 0, one arrives at the classical frequency equation, cosh λ0 cos λ0 = 1, valid for a Bernoulli-Euler beam, the corresponding eigenmode being U (λ0 ) w0 (s) = C U (λ0 s) − V (λ0 s) , V (λ0 ) where S(x), T (x), U (x), V (x) are the well-known Krylov-Duncan functions [14, §14.4.3] S(x) = 12 (cosh x + cos x),
T (x) = 21 (sinh x + sin x),
U (x) = 21 (cosh x − cos x),
V (x) = 21 (sinh x − sin x).
Besides, we get
248
p p ξ 1 − ξ w000 (0), p p = ξ 1 − ξ w000 (1).
a10 =
(43a)
a20
(43b)
In the first-order approximation, one has the inhomogeneous ODE (26) L0 w1 = 4λ30 λ1 Dw0 ,
(44)
and the procedure of splitting the boundary conditions gives w1 (0) = w1 (1) = 0, p w10 (0) = 1 − ξ w000 (0), p w10 (1) = − 1 − ξ w000 (1).
(45a) (45b) (45c)
The compatibility conditions for the BVP (44,45) reads w10 (1)w000 (1) − w10 (0)w000 (0) − w1 (1)w0000 (1) + w1 (0)w0000 (0) − 249
4λ30 λ1
Z1
Dw0 (s)w0 (s)ds = 0,
0
whence, accounting for Eqs.(45), one obtains the correction √ 1 − ξ [w000 (0)]2 + [w000 (1)]2 λ1 = −λ0 , R1 4 [w000 (s)]2 ds 0
18
(46)
250
where part-integration has been used at the denominator. Now, we can write
251
the problem solution w1 (s) = c11 sin(βs) + c12 cos(βs) + c13 e−αs + c14 eα(s−1) + w1p (s),
(47)
where w1p (s) = 2
λ30 λ1 s α2 + β 2
1 + θβ 2 [−c01 cos(βs) + c02 sin(βs)] β i 1 − θα2 h −αs α(s−1) c03 e + − c04 e α
(48)
is the particular solution of Eq.(44) with the coefficients c0j being given by Eqs.(41). In the special case of no rotational inertia, θ = 0, Eq.(46) may be reduced to the very simple expression λ1 = −2λ0 (1 − and Eq.(48) gives w1p (s) =
p
ξ),
p U (λ0 ) λ1 0 U (λ0 s) . sw0 (s) = −2C(1 − ξ)λ0 s T (λ0 s) − λ0 V (λ0 )
Similarly, Eq.(47) becomes w1 (s) = C(1 − 252
253
p
T (λ0 ) V (λ0 s) + w1p (s). ξ)λ0 T (λ0 s) − V (λ0 )
Breaking the asymptotic procedure at this step, we can write down the approximate formula for the nonlocal-to-local frequency ratio p [w000 (0)]2 + [w000 (1)]2 ω = 1 − 12 ε 1 − ξ + O ε2 , R1 00 ω0 [w0 (s)]2 ds
(49)
0
254
that, in the absence of rotary inertia, reduces to
p ω = 1 − 4ε(1 − ξ) + O ε2 . ω0
(50)
255
Fig.2 plots the approximated ratio (50) onto the numerical solution of the TPNM
256
and shows that the 1-term correction provides excellent agreement for the fun-
257
damental mode. It is also clear from Eq.(50) that, as in the S-S situation, a
258
perfectly reasonable limit is retrieved for the PNLM, i.e. for ξ → 0. 19
Figure 2: 1st (left) and 2nd (right) eigenfrequencies ω for a C-C beam (solid, black) in the absence of rotatory inertia, θ = 0, and with ε = 0.01, 0.05 and 0.075, superposed onto the 1-term (dotted, blue) asymptotic approximation, normalized with respect to the classical local frequency ω0 , Eq.(50)
259
The asymptotic expansion for the eigenmode reads w = w0 + εw1 + O ε2 ,
260
261
262
(51)
where w0 and w1 belong to the outer solution and they are given by (28), with coefficients (41), and by (47), respectively. We observe that the boundary layer terms are O ε2 and therefore they do not appear explicitly in (51). To incor-
263
porate them consistently, one needs to consider the successive approximation
264
term, ε2 w2 , for the outer solution.
265
4.5. Beam with clamped and simply supported ends
266
To fix ideas, let the left beam end be clamped and the right simply supported.
267
The correspondent boundary conditions are given by (11a), (12b) and the pair
268
of additional conditions (18). In this case, we arrive at γ1 = 2 and γ2 = 3 for
269
the left and for the right boundary layer, respectively. At leading order, one has the classical boundary conditions w0 (0) = w00 (0) = w0 (1) = w000 (1) = 0,
20
whence we get the constants in the general solution (28) c01 = −2λ20 α2 β −2 cosh α + cos β , c02 = 2 λ20 sin β + α2 sinh α ,
(52a) (52b)
c03 = −λ20 sin β − β 2 cos β − eα α2 , c04 = eα β 2 cos β − λ20 sin β + α2 , (n)
together with Eq.(43a). The eigenvalues λ0 = λ0
(52c) (52d)
are found from the transcen-
dental equation α cosh α sin β − β cos β sinh α = 0, 270
that, when θ = 0, boils down to T (λ0 )U (λ0 ) = S(λ0 )V (λ0 ).
271
272
273
The last equation amounts to the well known classical equation tanh λ0 = tan λ0 , while the correspondent eigenmodes are given by S(λ0 ) V (λ0 s) . w0 (s) = C U (λ0 s) − T (λ0 ) The first-order approximation yields w1 (0) = 0,
274
(53)
p w10 (0) = 1 − ξ w000 (0),
w1 (1) = w100 (1) = 0,
(54)
and a10 and a20 are defined by Eqs.(43a,32b)
√ √ a10 = Cλ20 ξ 1 − ξ , √ √ h a20 = Cλ30 ξ(1 − ξ) V (λ0 ) −
S 2 (λ0 ) T (λ0 )
i
(55) .
275
The inhomogeneous equation (44), subject to the boundary conditions (54),
276
possesses a solution provided that compatibility is satisfied, whereby we get the
277
first eigenfrequency correction λ1 = −λ0
√ 1 − ξ [w000 (0)]2 . R1 00 2 4 [w0 (s)] ds 0
21
(56)
Figure 3: 1st (left) and 2nd (right) eigenfrequencies ω for a C-S beam (solid, black) in the absence of rotatory inertia, θ = 0, and with ε = 0.01, 0.05 and 0.075, superposed onto the 1-term (dotted, blue) asymptotic approximation, normalized with respect to the classical local frequency ω0 , Eq.(59)
278
The solution of the BVP (44,54) has the form (47) as for the C-C case, yet with
279
different coefficients. Indeed, in the special case θ = 0, Eq.(56) simplifies to λ1 = −λ0 (1 − and the particular solution becomes w1p (s) =
p
ξ),
λ1 S(λ0 ) U (λ0 s) , s w00 (s) = Cλ1 s T (λ0 s) − λ0 T (λ0 )
whence p S(λ0 )U (λ0 ) w1 (s) = Cλ0 (1 − ξ) T (λ0 s) − V (λ0 s) + w1p (s) T (λ0 )V (λ0 ) p S(λ0 ) U (λ0 ) = Cλ0 (1 − ξ) (1 − s)T (λ0 s) + sU (λ0 s) − V (λ0 s) . T (λ0 ) V (λ0 )
(57)
280
281
Finally, we arrive at the following asymptotic expansion for the frequency ratio
p [w00 (0)]2 ω + O ε2 = 1 − 12 ε 1 − ξ 1 0 R 00 ω0 [w0 (s)]2 ds
(58)
0
282
that, in the case θ = 0, reduces to
p ω = 1 − 2ε 1 − ξ + O ε2 . ω0 22
(59)
Figure 4: 1st (left) and 2nd (right) eigenfrequencies ω for a cantilever beam (solid, black) in the absence of rotatory inertia, θ = 0, and with ε = 0.01, 0.05 and 0.075, superposed onto the 1-term (dotted, blue) asymptotic approximation, normalized with respect to the classical local model frequency ω0 , according to Eq.(59)
283
Eq.(59) is plotted in Fig.3 alongside the numerical solution of the TPNM. Al-
284
though the accuracy of the expansion is restricted to small values of ε, we still
285
appreciate a limit as the TPNM tends to the PNLM.
286
4.6. Cantilever Beam For a cantilever beam we have, at leading order, w0 (0) = w00 (0) = w000 (1) = w0000 (1) + θλ40 w00 (1) = 0, and the constants in the general solution (28) are given by Eqs.(52), i.e. they are the same as in the C-S case. The secular equation now reads 1 + 12 θ2 λ40 cosh α cos β − 21 θλ20 sinh α sin β + 1 = 0,
287
that, in the special case of vanishing rotational inertia, reduces to S 2 (λ0 ) − T (λ0 )V (λ0 ) = 0.
288
This formula coincides with the classical result cosh λ0 cos λ0 + 1 = 0 and the
289
corresponding eigenforms are still given by Eq.(53).
290
291
In the first-order approximation, one arrives at the following boundary conditions w1 (0) = 0, w100 (1) = 0,
p w10 (0) = 1 − ξ w000 (0),
w1000 (1) + λ40 θw10 (1) = −4λ30 λ1 θw00 (1). 23
(60)
together with the right boundary layer amplitude a20 =
p p ξ 1 − ξ [w100 (1) + w0000 (1)] ,
292
the left being given by Eq.(43a). The compatibility condition for the inho-
293
mogeneous BVP (44, 60) is still given by Eq.(56) and, as a consequence, the
294
eigenfrequency ratio (58) and the corresponding eigenmode correction are once
295
again retrieved. Fig.4 compares the normalized eigenfrequency ω/ω0 as nu-
296
merically evaluated for the TPNM with the 1-term expansion (59) and shows
297
good accuracy. Besides, the numerical solution curve matches the corresponding
298
result given in Fig.5 of [11].
299
5. Purely nonlocal model From the previous analysis, it clearly appears that the situation ξ → 0 lends a perfectly admissible eigenfrequency which, therefore, can be assumed as the proper solution to the PNLM. We now consider what happens to the eigenmodes and for this we need to investigate the behaviour of the boundary √ √ layer term Bξ (s) = ξ exp[−s/(ε ξ)], 0 ≤ s ≤ 1, as ξ → 0. Clearly, this is a transcendentally small term for s > 0 and Bξ (s) → 0 uniformly. Non uniformity arises when we consider s = 0 for then a boundary layer appears that may be √ studied taking the rescaled variable s∗ = s/(ε ξ), see [15]. This boundary layer
300
is vanishingly small as ξ → 0 but not so are its derivatives with respect to s 0, 0, s > 0, s 6= 0, Bξ0 (s) → and Bξ00 (s) → . −ε−1 , s = 0, +∞, s = 0,
301
static axial deformation. We may now ask whether this unboundedness in the
This result is the analogue of the steep boundary layer described in [39] under
304
second derivative leads to an unbounded bending energy. To answer this we Rη first observe that ∀η > 0, 0 Bξ00 (s)ds → ε−1 uniformly and therefore Bξ00 (s) is
305
tion Mξ of the boundary layer Bξ to the bending moment M through Eq.(4),
302
303
proportional to Dirac’s delta function. Indeed, when considering the contribu-
24
306
we find Mξ (0) → (2ε2 )−1 ,
307
at leading order. If we use this result in, say, the eigenmodes (38) for a S-S beam,
308
we easily see that the boundary condition M (0) = 0 is satisfied at leading order,
309
for the boundary layer cancels out the contribution of the outer solution. At the
310
same time, the constitutive BCs are asymptotically satisfied for a vanishingly
311
small ξ due to the asymptotic procedure applied above. We then conclude that,
312
in the limit as ξ → 0, the boundary layer warrants the fulfilment of all boundary
313
conditions and it brings a finite contribution to the bending energy. From the
314
standpoint of displacements, we get w(s) → w(m) + εγ1 −1 a10 R(−s) + εγ2 −1 a20 R(s − 1),
315
where R(s) is the ramp function. For a S-S beam, we have γ1 = γ2 = 3 and a10 = (−1)n+1 a20 = Cβ 3 .
316
Whence, a finite jump in the rotation and a concentrated couple at the beam
317
ends is produced. This is perhaps not so surprising, for solutions in the sense
318
of distributions are to be expected when an integral form of the constitutive
319
equation is adopted. Consequently, from a mathematical standpoint, an energy
320
bounded solution of the PNLM may be consistently defined as the limit of the
321
TPNM, although it is meaningful in the sense of distributions and we may want
322
to reject it on physical grounds.
323
6. Influence of rotational inertia
324
We now consider the effect of including rotational inertia when considering
325
the solution of the TPNM. For a S-S beam, Fig.5 plots the eigenfrequency ratio
326
ω/ω0 for mode numbers n = 1, 2 and 4, with θ = 0, 1/100 and 1/10. It appears
327
that, for the S-S end conditions, rotational inertia is irrelevant for the purpose
328
of determining the frequency ratio (yet it still affects ω0 ). For a C-C beam,
329
Fig.6 plots the eigenfrequency ratio ω/ω0 for mode numbers n = 1 and 4, with 25
Figure 5: Eigenfrequency ω for modes 1, 2 and 4 for a S-S beam, normalized over the classical frequency ω0 , for θ = 0, 1/100 and 1/10, as a function of the local model fraction ξ. As it occurs for the asymptotic expansion (39), the frequency ratio is unaffected by rotational inertia and curves overlap
Figure 6: Eigenfrequency ratio ω/ω0 for modes 1 (left panel) and 4 (right) for a C-C beam for θ = 0 (solid, black), θ = 1/100 (dashed, blue) and 1/10 (dotted, red), as a function of the local model fraction ξ
Figure 7: Eigenfrequency ratio ω/ω0 for modes 1 (left panel) and 4 (right) for a C-S beam for θ = 0 (solid, black), θ = 1/100 (dashed, blue) and 1/10 (dotted, red), as a function of the local model fraction ξ
26
Figure 8: Eigenfrequency ratio ω/ω0 for modes 1 (left panel) and 4 (right) for a C-F beam for θ = 0 (solid, black), θ = 1/100 (dashed, blue) and 1/10 (dotted, red), as a function of the local model fraction ξ
330
θ = 0, 1/100 and 1/10. This time, rotational inertia plays an important role in
331
the direction of contrasting the softening effect induced by the nonlocal fraction.
332
Indeed, this hardening effect is already well manifest in the fundamental mode
333
and, as expected, it becomes stronger for higher modes. Besides, encompassing
334
rotational inertia of the cross-section has a significant bearing on higher modes,
335
regardless of the actual value of θ. The same qualitative picture appears in Fig.7
336
and in Fig.8, respectively for C-S and C-F beams. It appears that the softening
337
effect is stronger moving from S-S to C-C, C-F and then to C-S.
338
7. Conclusions
339
The purely nonlocal theory of beam elasticity has recently attracted consid-
340
erable attention for the controversial results it conveys. Indeed, this model is
341
believed to lead to ill-posed problems, owing to the appearance of a pair of con-
342
stitutive boundary conditions which are generally incompatible with the natural
343
boundary conditions. In this paper, we approach the problem from a different
344
perspective and carry out an asymptotic analysis of the free vibrations of flexural
345
beams endowed with rotational inertia, within the two-phase theory of nonlocal
346
elasticity. We show that the nonlocal term contributes with a boundary layer
347
whose strength greatly varies for different end conditions. In the case of simply
348
supported beams, the boundary layer is the weakest and we provide a two-term
349
correction for the classical solution. Remarkably, this situation is affected by the 27
350
presence of rotational inertia only in the classical sense. Conversely, clamped-
351
clamped, clamped-supported and clamped-free (i.e. cantilever) conditions bring
352
a much stronger boundary layer, a for these we provide a single correction term.
353
Numerical results confirm the accuracy of the asymptotic approach and show
354
that rotational inertia is very relevant in contrasting the softening effect con-
355
nected to the nonlocal phase. Most interestingly, for any end condition, the
356
asymptotic solution still exists and its energy remains bounded in the limit of
357
the purely nonlocal theory, that is for a vanishingly small local phase. This is in
358
contrast to what is anticipated in the literature, see, for instance, [33]. We are
359
therefore in the position of attaching a meaning to the purely nonlocal theory,
360
as the limit of the two-phase theory. In so doing, we encounter a solution that
361
is defined in the sense of distributions (for the curvature) and that, although
362
questionable from a physical standpoint, is mathematically sound. This result
363
is quite general for it extends to statics and, presumably, to axial vibrations in
364
a rod.
365
Acknowledgements
366
367
368
GM gratefully acknowledges a Visiting Professor Position granted by the University of Modena and Reggio Emilia in the AY 2018/2019. AN welcomes support from FAR2019, Piano di sviluppo dipartimentale
369
DIEF, DR nr. 498/2019.
370
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Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 477
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
Journal Pre-proof
On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory Gennadi Mikhasev, Andrea Nobili PII: DOI: Reference:
S0020-7683(19)30455-X https://doi.org/10.1016/j.ijsolstr.2019.10.022 SAS 10522
To appear in:
International Journal of Solids and Structures
Received date: Revised date: Accepted date:
8 August 2019 3 October 2019 21 October 2019
Please cite this article as: Gennadi Mikhasev, Andrea Nobili, On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory, International Journal of Solids and Structures (2019), doi: https://doi.org/10.1016/j.ijsolstr.2019.10.022
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On the solution of the purely nonlocal theory of beam elasticity as a limiting case of the two-phase theory Gennadi Mikhaseva , Andrea Nobilib,∗ a Department
of Bio- and Nanomechanics, Belarusian State University, 4 Nezavisimosti Ave., 220030 Minsk, Belarus b Engineering Department ”Enzo Ferrari”, Universit` a degli Studi di Modena e Reggio Emilia, 41125 Modena, Italy
Abstract In the recent literature stance, purely nonlocal theory of elasticity is recognized to lead to ill-posed problems. Yet, we show that, for a beam, 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, for simply supported end, the weakest boundary layer is formed and, surprisingly, rotational inertia affects the eigenfrequencies only in the classical sense. Conversely, clamped and free end 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: [email protected] (Gennadi Mikhasev), [email protected] (Andrea Nobili)
Preprint submitted to International Journal of Solids and Structures
October 31, 2019
Therefore, we may define this limiting case as the proper solution of the purely nonlocal model for a beam. Finally, numerical results support the accuracy of the proposed asymptotic approach. Keywords: Two-phase nonlocal elasticity, Nonlocal theory of elasticity, Asymptotic method, Free vibrations
1
1. Introduction
2
The classical linear theory of elasticity suffers from the well known defect
3
of not encompassing an internal length scale, which feature gives rise to self-
4
similar predictions. Yet, any real material possesses an internal microstructure
5
and some characteristic length thereof. Consequently, classical elasticity may
6
be assumed as a suitable model inasmuch as the physical phenomena of interest
7
occur at a scale much greater than the internal characteristic length of the ma-
8
terial. Failure to meet this condition is effectively demonstrated by, for instance,
9
the singular stress field at the tip of a crack and by the non-dispersive nature
10
of wave propagation. Extensions of classical elasticity have been proposed, in
11
the form of generalized continuum mechanics (GCM), in an attempt to reme-
12
diate these shortfalls. An excellent historical overview of GCM, together with
13
extensive bibliographic details, may be found in [20]. Among GCM theories,
14
we mention the theory of micro-polar elasticity [3, 4, 28], the couple-stress and
15
strain-gradient elasticity theories [37, 26] and the nonlocal theory of elasticity
16
[8]. In particular, following [8], ”linear theory of nonlocal elasticity, which has
17
been proposed independently by various authors [...], incorporates important
18
features of lattice dynamics and yet it contains classical elasticity in the long
19
wave length limit”. Nonlocal elasticity is based on the idea that the stress
20
state at a point is a convolution over the whole body of an attenuation function
21
(sometimes named kernel or nonlocal modulus) with the strain field [36]. Al-
22
though several attenuation functions may be considered, they need to comply
23
with some important properties which warrant that (a) classical elasticity is re-
2
24
verted to in the limit of zero length scale and that (b) normalization is satisfied
25
[7, 17]. As an example, Helmholtz and bi-Helmoltz kernels have been widely
26
used in 1-D problems, their name stemming from the differential operators they
27
are Green’s function of [9, 16]. In [6], a variational argument is adopted to de-
28
duce the governing equations for a purely nonlocal Euler-Bernoulli beam, whose
29
eigenfrequencies are numerically investigated.
30
Since nonlocal elasticity naturally leads to integro-differential equations whose
31
solution is most often impractical, an equivalent differential nonlocal model
32
(EDNM) was developed in [7]. In such differential form, nonlocal elasticity has
33
been extensively applied to study elastodynamics of beams and shells as de-
34
scribed in the recent review [5] and with special emphasis on the application
35
to nanostructures [32]. Generally, EDNM leads to interesting mechanical ef-
36
fects, such as increased deflections and decreased buckling loads and natural
37
frequencies (softening effect), when compared to classical elasticity. However, a
38
number of pathological results have also emerged, which are often referred to as
39
paradoxes [18, 11, 16]. For instance, for a cantilever beam under point loading,
40
nonlocality brings no effect [27, 35, 2, 17]. It should be remarked that many
41
studies based on the EDNM employ boundary conditions in terms of macro-
42
scopic stresses, i.e. in classical form, and therefore they disregard the important
43
effect of the boundary through nonlocality. Although this approach may be still
44
adopted for long structures or in the case of localized deformations occurring
45
away from the boundaries [23, 24], it is generally inaccurate.
46
Very recently, Romano et al. [33] claimed that Eringen’s purely nonlocal
47
model (PNLM) leads to ill-posed problems for the differential form of the model
48
is consistent inasmuch as an extra pair of boundary conditions, termed con-
49
stitutive, is satisfied (see also [19]). In [8, Eq.(6.4)] and in [1], a two-phase
50
nonlocal model (TPNL) was introduced, within the context of 3D elasticity,
51
which combines, according to the theory of mixtures, purely nonlocal elasticity
52
with classical elasticity, by means of the volume fractions ξ1 and ξ2 = 1 − ξ1 .
53
This model is immune from the inconsistencies of the PNLM and it has been
54
adopted to solve the problem of static bending [36] and buckling [38] of Euler3
55
Bernoulli (E-B) beams. Static axial deformation of a beam is considered in
56
[29, 39], while semi-analytical solutions for the combined action of axial and
57
flexural static loadings is given in [21]. Axial and flexural free vibrations of
58
beams have also been considered in [22] and in [10]. In these works, either the
59
TPNM is solved numerically or it is reduced, by adopting the solution presented
60
in [31], to an equivalent higher-order purely differential model with a pair of ex-
61
tra boundary conditions. Despite this reduction, the differential model is still
62
difficult to analyse, especially in the neighbourhood of the PNLM, that is for ξ1
63
small. In this respect, we believe that the asymptotic approach may be put to
64
great advantage in predicting the mechanical behaviour of nanoscale structures
65
for a vanishingly small ξ1 [38, 22].
66
In this paper, we consider free vibrations of a flexural beam taking into ac-
67
count rotational inertia (Rayleigh beam), within the TPNM and having assumed
68
the Helmholtz attenuation function. The integro-differential model is reduced
69
to purely differential form with an extra pair of boundary conditions. Spotlight
70
is set on developing asymptotic solutions valid for small microstructure and/or
71
little local fraction. These solutions feature a pair of boundary layers located
72
at the beam ends, whose strength depends on the constraining conditions. Nu-
73
merical results support the accuracy of the expansions. Most remarkably, the
74
asymptotic approach allows to investigate the behaviour of the solution in the
75
neighbourhood of the PNLM, where the expansions are non-uniform. Nonethe-
76
less, they admit a perfectly meaningful, energy bounded limit, which may be
77
taken as the solution of the PNLM. We point out that the existence of such
78
limit has been observed numerically in [11] for free-free end conditions.
79
2. Problem formulation
80
2.1. Governing equations
81
For a flexural beam, vertical equilibrium gives ρS
ˆ ∂2v ∂Q = + qˆ(x) 2 ∂t ∂x
4
(1)
82
while rotational equilibrium lends J
ˆ ∂M ∂2ϕ ˆ =− + Q. 2 ∂t ∂x
(2)
83
ˆ and M ˆ are the dimensional Here, v = v(x, t) is the vertical displacement, Q
84
shearing force and the bending moment, respectively, ρ is the mass density per
85
unit volume, J = ρI is the mass second moment of inertia per unit length of
86
the beam, that is proportional to the second moment of area I, S is the cross-
87
sectional area and qˆ(x) the vertical applied load. Assuming that the beam is
88
homogeneous and prismatic, Eqs.(1,2) give ˆ ∂2M ∂2v ∂4v − ρS + J + qˆ = 0, ∂x2 ∂t2 ∂x2 ∂t2
(3)
89
that governs transverse vibrations of flexural beams accounting for rotational
90
inertia. In the following, we take qˆ ≡ 0. In the mixed nonlocal theory (MNLT)
91
of elasticity, we have [10]
2 ˆ = −EI ξ1 ∂ v + ξ2 M ∂x2
ZL 0
∂2v K(|x − x ˆ|, κ) 2 dˆ x , ∂x ˆ
(4)
92
where EI is the beam flexural rigidity, L the beam length and K(|x − x ˆ|, κ) is
93
the kernel or attenuation function. The kernel is positive, symmetric, it rapidly
94
decays away from x and it satisfies the normalization condition Z K(|x − x ˆ|, κ)dˆ x = 1. R
95
The constitutive equation (4) is also obtained from consideration of a more gen-
96
eral form of kernel discussed in [17]. The nonlocal parameter κ = e0 a depends
97
on the scale coefficient e0 as well as on the internal length scale a. ξ1 and ξ2
98
take up the role of volume fractions and they represent, respectively, the local
99
and the nonlocal phase ratios, such that ξ1 + ξ2 = 1 and ξ1 ξ2 ≥ 0. When ξ1 = 0,
100
Eq.(4) degenerates into the purely nonlocal model (PNLM), while, in contrast,
101
the case ξ1 = 1 corresponds to classical local elasticity.
102
In what follows, we consider the Helmholtz kernel 1 |x − x ˆ| K(|x − x ˆ|, κ) = exp − , 2κ κ 5
(5)
103
which is frequently used for 1D problems (indeed, it is named special kernel in
104
[33]). We note that for the Helmholtz kernel the following transformations are
105
valid d ds
Z1
e
|s−ˆ s| − ε
s
0
106
Z1 Zs s ˆ s s ˆ 1 s y(ˆ s)dˆ s= e− ε y(ˆ s)dˆ s − e− ε e ε y(ˆ s)dˆ s , eε ε
and d2 ds2
Z1
e−
|s−ˆ s| ε
y(ˆ s)dˆ s=
(6)
0
1 ε2
0
Z1
e−
0
|s−ˆ s| ε
2 y(ˆ s)dˆ s − y(s). ε
(7)
In particular, Eq.(7) corresponds to [33, Eq.(6)] and it may be rewritten as Z 1 2 ˆ|, ε) 2 d K(|s − s − K(|s − sˆ|, ε) + δ(|s − sˆ|) y(ˆ s)dˆ s = 0, ε ds2 0 whereupon K(|s− sˆ|, ε) is the Green’s function of the singularly perturbed oper2
d ator Hε = 1 − ε2 ds 2 . It is trivial matter to prove impulsivity, i.e. limε→0 K(|s −
sˆ|, ε) = δ(s − sˆ), where δ(s) is Dirac’s delta function. Furthermore, we observe that Eq.(6), evaluated at the beam ends s = 0, 1 and for ξ = 0, lends the constitutive boundary conditions [33, Eq.(5)] ˆ dM ˆ (0), (0) = ε−1 M ds
and
ˆ dM ˆ (1). (1) = −ε−1 M ds
107
Thus, the constitutive boundary conditions are really the expression, on the
108
domain boundary, of a general feature of the solution that is related to the
109
integral operator (4).
110
111
112
113
114
Introducing the dimensionless axial co-ordinate s = x/L and under the assumption of time-harmonic motion, we write n o ˆ,Q ˆ = w(s), EI M (s), EI Q(s) exp(ıωt), v, M L L2
where ı is the imaginary unit. Upon multiplying throughout by L4 /EI, Eq.(3)
may be turned in dimensionless form Z 1 d2 w ξ2 |ˆ s − s| d2 w(ˆ s) d4 w 4 −2 + exp − dˆ s − λ4 w = 0. (8) ξ1 4 + λ θ − ε ξ2 ds ds2 2ε3 0 ε dˆ s2
Here, use have been made of Eqs.(4,5) and we have let the dimensionless ratios r 2 J ρSL4 ω 2 A 4 θ= = , λ = , (9) ρSL2 L EI 6
together with the microstructure parameter ε=
κ 1. L
115
Here rA is the radius of gyration. Clearly, θ plays the role of an aspect ratio
116
squared and ε is a scale effect. Assuming w ∈ C 6 [0, 1], twice differentiating
117
Eq.(8), making use of Eq.(7) and then subtracting the original equation (8), we
118
get the governing equation in purely differential form ε2 ξ
4 d6 w d2 w 2 4 d w 4 2 − (1 − ε − λ + λ4 w = 0, θλ ) (ε + θ) ds6 ds4 ds2
(10)
120
where, hereinafter, we adopt the shorthand ξ = ξ1 . Eq.(10) is a singularly √ perturbed ODE [15], with respect to the small parameter ε ξ.
121
2.2. Boundary conditions
119
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
w(0) = w0 (0) = 0,
(11a)
w(1) = w0 (1) = 0.
(11b)
For simply supported (S-S) ends
122
w(0) = 0,
M (0) = ξw00 (0) + M0 = 0,
(12a)
w(1) = 0,
M (1) = ξw00 (1) + M1 = 0,
(12b)
having let M0 =
1−ξ 2ε
Z1
s ˆ
s)dˆ s, e− ε w00 (ˆ
M1 =
0
1 − ξ −1 e ε 2ε
Z1
s ˆ
e ε w00 (ˆ s)dˆ s.
(13)
0
For free-free (F-F) ends, one has M (0) = 0,
Q(0) = ξw000 (0) + θλ4 w0 (0) + ε−1 M0 = 0,
(14a)
M (1) = 0,
Q(1) = ξw000 (1) + θλ4 w0 (1) − ε−1 M1 = 0.
(14b)
7
The nonlocal end bending moments (13) may be rewritten in differential form with the help of the original integro-differential equation (8): M0 = −ε2 ξwiv (0) + 1 − ξ − ε2 θλ4 w00 (0) + ε2 λ4 w(0), M1 = −ε2 ξwiv (1) + 1 − ξ − ε2 θλ4 w00 (1) + ε2 λ4 w(1).
(15a) (15b)
Consequently, the BCs may be recast in differential form through M (0) = w00 (0) + ε2 N0 ,
(16a)
M (1) = w00 (1) + ε2 N1 ,
(16b)
Q(0) = ξw000 (0) + θλ4 w0 (1) + ε−1 M0 ,
(16c)
Q(1) = ξw000 (1) + θλ4 w0 (1) − ε−1 M1 ,
(16d)
where, making use of the connections (6,7), we have N0 = ε−2 (ξ2 w00 (0) − M0 ) = −ξwiv (0) − θλ4 w00 (0) + λ4 w(0),
(17a)
N1 = ε−2 (ξ2 w00 (1) − M1 ) = −ξwiv (1) − θλ4 w00 (1) + λ4 w(1).
(17b)
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 original governing equation (8), differentiated once with respect to s, one arrives at ε3 ξwv (0) − ε2 ξwiv (0) − (1 − ξ − ε2 θλ4 )[εw000 (0) − w00 (0)] −ε3 λ4 w0 (0) + ε2 λ4 w(0) = 0,
(18a)
ε3 ξwv (1) + ε2 ξwiv (1) − (1 − ξ − ε2 θλ4 )[εw000 (1) + w00 (1)] −ε3 λ4 w0 (1) − ε2 λ4 w(1) = 0.
(18b)
123
Dropping rotational inertia, the additional boundary conditions (18) coincide
124
with the constitutive boundary conditions recently obtained by Fern´ andez-S´ aez
125
and Zaera [10, Eqs.(59) and (60)], provided that we replace our ε and λ4 with
126
their h and λw , respectively. However, it should be remarked that in [10] the
127
original integro-differential problem is reduced to the equivalent differential form 8
128
extending to dynamics the original argument developed in [36] for statics. Such
129
argument takes advantage of a result presented in [30], which really applies to
130
inhomogeneous integral equations with a given right-hand side. In the case of
131
dynamics, however, this right-hand side is a problem unknown, for it is really
132
an acceleration term, and therefore the applicability of the reduction formula is
133
questionable.
134
3. Exact solution of the boundary-value problems The general solution of the ODE (10) is w(s) =
6 X
cj exp (bj s) ,
j=1
135
where the constants bj are the roots of the characteristic polynomial in ζ ε2 ξζ 6 − (1 − ε2 θλ4 )ζ 4 − (ε2 + θ)λ4 ζ 2 + λ4 = 0.
(19)
136
As detailed in [34, 25], this bi-cubic may be turned in canonical form by the
137
substitution Z = ζ 2 − Z0 , it being Z0 = (1 − ε2 θλ4 )/(3ε2 ξ). Hence, Eq.(19)
138
becomes Z 3 − pZ − q = 0, where "
# 2 λ4 θε2 − 1 4 2 p = (ξε ) +λ θ+ε > 0, 3ξε2 " 3 # λ4 θ + ε2 λ4 θε2 − 1 2 λ4 θε2 − 1 2 −1 4 q = −(ξε ) λ + + . 3ξε2 27ξ 2 ε4 2 −1
This polynomial possesses three real roots provided that ∆= 139
p3 q2 − <0 4 27
√ and indeed, for ε ξ 1, we get, to leading order, ∆ = −λ4
4 + θ2 λ4 . 108(ξε2 )4
9
140
Besides, we have, at leading order, q=
2 27(ξε2 )3
and q > 0, whereupon out of the three real roots, two, say Z3 < Z2 , are negative and one, say Z1 , is positive. Upon reverting to the original variable ζ, we see that ζ32 < 0 < ζ22 < ζ12 . Indeed, we get the leading order solutions (the sign is immaterial) 1 ζ1 = √ , ε ξ with
ζ2 = α,
ζ3 = ıβ,
s
r 1 2 θ 2 λ4 , α = λ − θλ + 1 + 2 4 s r 1 2 θ2 λ4 β=λ θλ + 1 + , 2 4 141
142
(20a) (20b)
whence ζ1,2 convey an exponential solution, while ζ3 is related to an oscillatory √ contribution. It is worth noticing that ζ1 blows up as (ε ξ) → 0, that is for a
143
vanishingly small scale effect or in the purely nonlocal situation. Indeed, this
144
very root accounts for the edge effect in this problem and it describes a boundary
145
layer.
146
147
We observe that, in general, the frequency equation for the ODE (10), subject to suitable boundary conditions, appears in transcendental form F (λ; ξ, ε) = 0,
148
wherein λ is the sought-for eigenvalue. The numerical solution of this equation is
149
not straightforward matter, especially for very small values of the local fraction
150
ξ, see e.g. [10] and [36] where plots are given for ξ > 0.1 and ξ > 0.05, respec-
151
tively. Indeed, when looking for the numerical roots of (19), we observe, after
152
[34], that the transformation to canonical form lends a considerable numerical
153
advantage over Cardano’s formulas, in that it provides purely real solutions.
154
Conversely, Cardano’s formulas are likely to introduce a very small spurious
155
imaginary component, which is most likely the cause of the numerical difficulty
156
encountered in the literature when dealing with small ξ. 10
157
To estimate the eigenvalue λ for any ξ and, in particular, in the limiting case
158
of the PNLM (that occurs as ξ → 0), we consider an asymptotic expansion in
159
the small parameter ε.
160
4. Asymptotic solution of the boundary-value problems
161
Following a standard asymptotic argument [15, 22] and similarly to the ex-
162
traction of the edge effect in shells [12, 13], we seek a solution of the eigenvalue
163
problem through superposition of a solution, w(m) , valid in the interior of the
164
beam (the so-called outer solution), with a pair of boundary layers, w1
165
(e) w2 ,
(e)
fading off away from the left and from the right beam end, respectively, (e)
(e)
w(s, ε) = w(m) (s) + εγ1 w1 (s, ε) + εγ2 w2 (s, ε), 166
and
(21)
where γ1,2 > 0 and we have the order relations ∂w(m) ∼ w(m) , ∂s
(e)
∂wi ∂s
(e)
∼ ε−ς wi
as
ε → 0.
167
The parameter ς is named the index of variation of the edge effect integrals,
168
while γ1,2 are the indices of intensity of the edge effect integrals near the left
169
and right ends, respectively. The indices γ1,2 depend on the boundary conditions
170
and should be specified for each end.
171
4.1. Boundary layer
172
To derive an equation describing the behaviour of the solution in the vicinity
173
of the ends (boundary layer), we zoom in by assuming s = ες σ and 1 − s = ες σ,
174
respectively for the left and for the right end. For either case, one obtains the
175
distinguished limit ς = 1 and Eq. (10) is rewritten as (e)
ξ
(e)
(e)
4 2 d6 wi (e) 2 4 d wi 2 4 2 d wi − 1 − ε θλ − ε λ θ + ε + ε4 λ4 wi = 0, dσ 6 dσ 4 dσ 2
(22)
176
whose solution is sought in the form of an asymptotic series in the small param-
177
eter ε 1
(e)
wi
(e)
(e)
(e)
= wi0 + εwi1 + ε2 wi2 + . . . , 11
i = 1, 2.
(23)
178
Substitution of (23) into (22) lends a sequence of differential equations in the
179
unknowns wij (σ), i = 1, 2; j = 0, 1, 2, . . .. Here, we simply give the first two
180
(e)
terms of the expansion in the original variable s h i 4 s s s (e) − √ − √ 2 − ε√ ξ √ w1 (s, ε) = a10 e ε ξ + εe ε ξ a11 + a10 θλ 2(1−ξ) s + O ε e , ξ (e)
w2 (s, ε) = a20 e
√ − ε1−s ξ
+ εe
√ − ε1−s ξ
h i 1−s 4 2 − ε√ξ √ a21 + a20 θλ 2(1−ξ) , (1 − s) + O ε e ξ
(24)
181
where aij (i = 1, 2; j = 0, 1, 2, . . .) are constants that will be determined in the
182
following from the boundary conditions.
183
4.2. The outer solution
184
185
The displacement w(m) as well as the eigenvalue λ are sought in the form of an asymptotic series w(m) = w0 + εw1 + ε2 w2 + . . . , (25) 2
λ = λ0 + ελ1 + ε λ2 + . . . . 186
The leading term in the series corresponds to the solution of the classical local
187
problem and λ0 is the classical eigenvalue. Substituting (25) into the governing
188
Eq.(10) and equating coefficients of like powers of ε leads to the sequence of
189
differential equations: k X
Lj wk−j = 0,
k = 0, 1, 2, . . . ,
(26)
j=0
where L0 z =
190
d4 z d2 z d2 z + θλ40 2 − λ40 z, L1 z = −4λ30 λ1 Dz, Dz = z − θ 2 , 4 ds ds ds 4 2 d6 z d z d z L2 z = −ξ 6 − θλ40 4 + λ40 2 − 2λ20 (3λ21 + 2λ0 λ2 )Dz, ds ds ds 4 2 d z d z L3 z = −4θλ30 λ1 4 + 4λ30 λ1 4 − 4λ0 (λ20 λ3 + λ31 + 2λ0 λ1 λ2 )Dz, . . . ds ds
At leading order, one finds the homogeneous forth order ODE L0 w0 = 0, 12
(27)
191
whose general solution w0 (s) = c01 sin(βs) + c02 cos(βs) + c03 e−αs + c04 eα(s−1) ,
(28)
192
depends on the constants, c0i , i ∈ {1, 2, 3, 4}, to be determined through the
193
boundary conditions. However, the ODE (27) is subject to six boundary con-
194
ditions and the problem is to determine which of these correspond to the outer
195
solution and which pertain to the boundary layer [15]. The procedure of split-
196
ting the boundary conditions also gives the indices of intensity of the boundary
197
layer, γ1 , γ2 , as well as the constants c0k , aij . For this, one needs to insert the
198
expansions (21,24,25) into the boundary conditions and equate coefficients of
199
like powers of ε, while imposing the following requirements:
200
201
• in the leading approximation, every end condition should be homogeneous and coincide with those of the classical local theory;
202
• the k th -order approximation generates two equations coupling the con-
203
stants ai(k−1) with the previous order approximation wk−1 (s) evaluated
204
at the boundaries.
205
4.3. Beam with simply supported ends
206
Let both beam ends be simply supported (S-S conditions), as given by the
207
boundary conditions (12) rewritten in differential form through Eqs.(16a,16b),
208
together with the additional constraints (18).
209
(21,24,25) into these conditions, we determine the strength of either boundary
210
layer: γ1 = γ2 = 3.
Substituting the expansions
At leading order, we arrive at the homogeneous classical boundary conditions w0 (0) = w0 (1) = w000 (0) = w000 (1) = 0, 211
which give c01 = C, c02 = c03 = c04 = 0 and the classical eigenforms w0 (s) = C sin(βs),
212
β = πn,
n = 1, 2, . . . .
(29)
In light of the definition (20b), we find the eigenfrequencies (n)
λ0 = λ0 ≡
πn , [1 + θ(πn)2 ]1/4 13
n = 1, 2, . . . ,
(30)
213
214
215
and, using (9), the corresponding dimensional frequencies ω0 =
q
EI ρS
(λ0 /L)2 .
Moving to first-order terms, we again obtain a set of homogeneous boundary conditions w1 (0) = w1 (1) = w100 (0) = w100 (1) = 0,
(31)
as well as formulas for the leading amplitude in the boundary layer (24): p p p p a10 = − ξ(1 − ξ)w0000 (0) = Cβ 3 ξ(1 − ξ), p p p p a20 = ξ(1 − ξ)w0000 (1) = C(−1)n+1 β 3 ξ(1 − ξ).
(32a) (32b)
216
Consideration of the inhomogeneous ODE (26) arising in this approximation,
217
alongside the associated homogeneous boundary conditions (31), yields the com-
218
patibility condition λ1 = 0, whence w1 = C1 sin(βs),
219
where C1 is an arbitrary constant. Without loss of generality, one can assume
220
w1 ≡ 0, for this amounts to taking C = C0 + εC1 + . . . .
221
222
In the second-order approximation, when taking into account the outcomes of the previous step, we have again a homogeneous set of boundary conditions w2 (0) = w2 (1) = w200 (0) = w200 (1) = 0,
223
(33)
and a11 = a21 = 0. The associated differential equation for w2 reads L0 w2 = −L2 w0 ≡ ξ
4 d2 w0 d6 w0 4 d w0 3 + θλ − λ (λ + 4θλ ) + 4λ30 λ2 w0 . (34) 0 2 0 0 ds6 ds4 ds2
We thus arrive at the inhomogeneous boundary value problem (BVP) ”on spectrum”. Upon observing that the homogeneous BVP (33,34) arising at leading order is self-conjugated and therefore possesses the solution z(s) = w0 (s), we deduce the compatibility condition Z1
w0 (s)L2 w0 (s)ds = 0,
0
which readily gives the correction for the eigenvalue: λ2 = −
β 2 [λ40 (1 + θβ 2 ) − ξβ 4 ] . 4λ30 (1 + θβ 2 ) 14
224
On taking into account this result, Eq. (34) turns homogeneous and, without
225
loss of generality, we can assume w2 ≡ 0.
226
227
Considering the third-order approximation, one obtains the inhomogeneous boundary conditions √ √ w3 (0) = −a10 = −Cβ 3 ξ(1 − ξ), √ √ w3 (1) = −a20 = C(−1)n β 3 ξ(1 − ξ), w300 (0)
=
θλ40 a10
=
Cθλ40 β 3
√
ξ(1 −
√
(35)
ξ),
√ √ w300 (1) = θλ40 a20 = (−1)n+1 Cθλ40 β 3 ξ(1 − ξ) 228
for the inhomogeneous ODE L0 w3 = −L3 w0 ≡ 4λ30 λ3 Dw0 .
(36)
The compatibility condition for the BVP (35,36) works out − w300 (1)w00 (1) + w300 (0)w00 (0) − w3 (1)w0000 (1) + w3 (0)w0000 (0) Z 1 4 0 0 3 + θλ0 [w3 (0)w0 (0) − w3 (1)w0 (1)] + 4λ0 λ3 (w0 − θw000 )w0 ds = 0, 0
229
whence we get the next correction term for the eigenvalue √ √ β 6 ξ(1 − ξ) . λ3 = λ30 (1 + θβ 2 )
(37)
The eigenform correction w3 , satisfying the boundary conditions (35), is given by the sum of a particular solution w3p of Eq.(36), with the homogeneous solution w3o . The former reads w3p (s) = C3p s cos(βs), where C3p = 2Cλ30 λ3
p β5 p 1 + θβ 2 = 2C 2 ξ(1 − ξ). 2 2 2 β(α + β ) α +β
Consequently, making use of (37), we get w3 (s) = Cβ 3
p
ξ(1 −
p ξ) {c32 cos(βs) + c33 exp(−αs)
+ c34 exp[α(s − 1)] − 2c32 s cos(βs)} , 15
Figure 1: 1st (left) and 2nd (right) eigenfrequencies ω for a S-S beam (solid, black), with ε = 0.01, 0.05 and 0.075, superposed onto the 1-term (dashed, red) and the 2-term (dotted, blue) asymptotic approximation, normalized with respect to the classical local frequency ω0 , Eq.(39)
with the constants c32 = −β 2 /(α2 + β 2 ), c33 = 21 α2 eα (1 − coth α) [eα + (−1)n ] /(α2 + β 2 ), c34 = − 21 α2 eα (1 − coth α) [(−1)n eα + 1] /(α2 + β 2 ). Breaking at this step the asymptotic procedure for seeking the eigenvalues λk and the associated eigenfunctions wk , we obtain the asymptotic expansion i h p p λ = λ0 1 − 41 ε2 β 2 (1 − ξ) + ε3 β 2 ξ (1 − ξ) + O(ε4 ) ,
where β and λ0 are determined by (29) and (30), respectively. Up to an undetermined factor, the associated eigenmode reads p p n ξ(1 − ξ) c32 cos(πns) + c33 exp(−αs) s s−1 o √ + c34 exp[α(s − 1)] − 2c32 s cos(πns) + exp − √ + (−1)n+1 exp ε ξ ε ξ 4 + O ε . (38) w(s) = sin(πns) + ε3 (πn)3
230
It is of interest to compare the dimensional natural frequency, ω, determined
231
with the TPNM, with its classical counterpart, ω0 , evaluated within the frame-
232
work of local elasticity, i.e. for ξ = 1. When taking into account the definition
16
233
(9), we arrive at the relation ω = ω0
λ λ0
2
p p 1 = 1 − ε2 (πn)2 (1 − ξ) + 2ε3 (πn)2 ξ(1 − ξ) + O ε4 . 2
(39)
234
Remarkably, this expression is independent of θ and this unexpected feature
235
is indeed confirmed by the numerical solution of the TPNM, see Fig.5. Fig.1
236
plots the approximation (39) in the range 0 < ξ < 1 against the numerical
237
solution of the TPNM (given for ξ > 0.01) for the scale parameter ε = 0.01, 0.05
238
and 0.075. It appears that the 1-term asymptotic approximation is remarkably
239
effective for small values of ε. The numerical solution of the TPNM given in
240
Fig.1 compares favourably with the corresponding solution depicted in Fig.4 of
241
[11] that, however, pertains to the range ξ1 > 0.1, presumably owing to the
242
numerical difficulties that may arise in the neighbourhood of the PNLM.
243
244
As a special case of Eq.(39), one obtains the eigenfrequency ratio corresponding to the PNLM (i.e. for ξ = 0) ω 1 = 1 − ε2 (πn)2 + O ε4 . ω0 2
245
(40)
4.4. Beam with clamped ends Consideration of a beam with clamped ends requires enforcing (11) and (18) on Eqs.(21,24,25). We thus get the strength of the boundary layer γ1 = γ2 = 2. In the leading approximation, one has the classical boundary conditions w0 (0) = w0 (1) = w00 (0) = w00 (1) = 0,
246
that give the constants c01 = 2α(cosh α − cos β) c02 = 2α sin β − 2β sinh α,
c03 = β (eα − cos β) − α sin β,
(41)
c04 = −eα α sin β + β (eα cos β − 1) ,
247
as well as the frequency equation 2 1 2 θλ0
sin β sinh α + cos β cosh α − 1 = 0. 17
(42)
In particular, if θ = 0, one arrives at the classical frequency equation, cosh λ0 cos λ0 = 1, valid for a Bernoulli-Euler beam, the corresponding eigenmode being U (λ0 ) w0 (s) = C U (λ0 s) − V (λ0 s) , V (λ0 ) where S(x), T (x), U (x), V (x) are the well-known Krylov-Duncan functions [14, §14.4.3] S(x) = 12 (cosh x + cos x),
T (x) = 21 (sinh x + sin x),
U (x) = 21 (cosh x − cos x),
V (x) = 21 (sinh x − sin x).
Besides, we get
248
p p ξ 1 − ξ w000 (0), p p = ξ 1 − ξ w000 (1).
a10 =
(43a)
a20
(43b)
In the first-order approximation, one has the inhomogeneous ODE (26) L0 w1 = 4λ30 λ1 Dw0 ,
(44)
and the procedure of splitting the boundary conditions gives w1 (0) = w1 (1) = 0, p w10 (0) = 1 − ξ w000 (0), p w10 (1) = − 1 − ξ w000 (1).
(45a) (45b) (45c)
The compatibility conditions for the BVP (44,45) reads w10 (1)w000 (1) − w10 (0)w000 (0) − w1 (1)w0000 (1) + w1 (0)w0000 (0) − 249
4λ30 λ1
Z1
Dw0 (s)w0 (s)ds = 0,
0
whence, accounting for Eqs.(45), one obtains the correction √ 1 − ξ [w000 (0)]2 + [w000 (1)]2 λ1 = −λ0 , R1 4 [w000 (s)]2 ds 0
18
(46)
250
where part-integration has been used at the denominator. Now, we can write
251
the problem solution w1 (s) = c11 sin(βs) + c12 cos(βs) + c13 e−αs + c14 eα(s−1) + w1p (s),
(47)
where w1p (s) = 2
λ30 λ1 s α2 + β 2
1 + θβ 2 [−c01 cos(βs) + c02 sin(βs)] β i 1 − θα2 h −αs α(s−1) c03 e + − c04 e α
(48)
is the particular solution of Eq.(44) with the coefficients c0j being given by Eqs.(41). In the special case of no rotational inertia, θ = 0, Eq.(46) may be reduced to the very simple expression λ1 = −2λ0 (1 − and Eq.(48) gives w1p (s) =
p
ξ),
p U (λ0 ) λ1 0 U (λ0 s) . sw0 (s) = −2C(1 − ξ)λ0 s T (λ0 s) − λ0 V (λ0 )
Similarly, Eq.(47) becomes w1 (s) = C(1 − 252
253
p
T (λ0 ) V (λ0 s) + w1p (s). ξ)λ0 T (λ0 s) − V (λ0 )
Breaking the asymptotic procedure at this step, we can write down the approximate formula for the nonlocal-to-local frequency ratio p [w000 (0)]2 + [w000 (1)]2 ω = 1 − 12 ε 1 − ξ + O ε2 , R1 00 ω0 [w0 (s)]2 ds
(49)
0
254
that, in the absence of rotary inertia, reduces to
p ω = 1 − 4ε(1 − ξ) + O ε2 . ω0
(50)
255
Fig.2 plots the approximated ratio (50) onto the numerical solution of the TPNM
256
and shows that the 1-term correction provides excellent agreement for the fun-
257
damental mode. It is also clear from Eq.(50) that, as in the S-S situation, a
258
perfectly reasonable limit is retrieved for the PNLM, i.e. for ξ → 0. 19
Figure 2: 1st (left) and 2nd (right) eigenfrequencies ω for a C-C beam (solid, black) in the absence of rotatory inertia, θ = 0, and with ε = 0.01, 0.05 and 0.075, superposed onto the 1-term (dotted, blue) asymptotic approximation, normalized with respect to the classical local frequency ω0 , Eq.(50)
259
The asymptotic expansion for the eigenmode reads w = w0 + εw1 + O ε2 ,
260
261
262
(51)
where w0 and w1 belong to the outer solution and they are given by (28), with coefficients (41), and by (47), respectively. We observe that the boundary layer terms are O ε2 and therefore they do not appear explicitly in (51). To incor-
263
porate them consistently, one needs to consider the successive approximation
264
term, ε2 w2 , for the outer solution.
265
4.5. Beam with clamped and simply supported ends
266
To fix ideas, let the left beam end be clamped and the right simply supported.
267
The correspondent boundary conditions are given by (11a), (12b) and the pair
268
of additional conditions (18). In this case, we arrive at γ1 = 2 and γ2 = 3 for
269
the left and for the right boundary layer, respectively. At leading order, one has the classical boundary conditions w0 (0) = w00 (0) = w0 (1) = w000 (1) = 0,
20
whence we get the constants in the general solution (28) c01 = −2λ20 α2 β −2 cosh α + cos β , c02 = 2 λ20 sin β + α2 sinh α ,
(52a) (52b)
c03 = −λ20 sin β − β 2 cos β − eα α2 , c04 = eα β 2 cos β − λ20 sin β + α2 , (n)
together with Eq.(43a). The eigenvalues λ0 = λ0
(52c) (52d)
are found from the transcen-
dental equation α cosh α sin β − β cos β sinh α = 0, 270
that, when θ = 0, boils down to T (λ0 )U (λ0 ) = S(λ0 )V (λ0 ).
271
272
273
The last equation amounts to the well known classical equation tanh λ0 = tan λ0 , while the correspondent eigenmodes are given by S(λ0 ) V (λ0 s) . w0 (s) = C U (λ0 s) − T (λ0 ) The first-order approximation yields w1 (0) = 0,
274
(53)
p w10 (0) = 1 − ξ w000 (0),
w1 (1) = w100 (1) = 0,
(54)
and a10 and a20 are defined by Eqs.(43a,32b)
√ √ a10 = Cλ20 ξ 1 − ξ , √ √ h a20 = Cλ30 ξ(1 − ξ) V (λ0 ) −
S 2 (λ0 ) T (λ0 )
i
(55) .
275
The inhomogeneous equation (44), subject to the boundary conditions (54),
276
possesses a solution provided that compatibility is satisfied, whereby we get the
277
first eigenfrequency correction λ1 = −λ0
√ 1 − ξ [w000 (0)]2 . R1 00 2 4 [w0 (s)] ds 0
21
(56)
Figure 3: 1st (left) and 2nd (right) eigenfrequencies ω for a C-S beam (solid, black) in the absence of rotatory inertia, θ = 0, and with ε = 0.01, 0.05 and 0.075, superposed onto the 1-term (dotted, blue) asymptotic approximation, normalized with respect to the classical local frequency ω0 , Eq.(59)
278
The solution of the BVP (44,54) has the form (47) as for the C-C case, yet with
279
different coefficients. Indeed, in the special case θ = 0, Eq.(56) simplifies to λ1 = −λ0 (1 − and the particular solution becomes w1p (s) =
p
ξ),
λ1 S(λ0 ) U (λ0 s) , s w00 (s) = Cλ1 s T (λ0 s) − λ0 T (λ0 )
whence p S(λ0 )U (λ0 ) w1 (s) = Cλ0 (1 − ξ) T (λ0 s) − V (λ0 s) + w1p (s) T (λ0 )V (λ0 ) p S(λ0 ) U (λ0 ) = Cλ0 (1 − ξ) (1 − s)T (λ0 s) + sU (λ0 s) − V (λ0 s) . T (λ0 ) V (λ0 )
(57)
280
281
Finally, we arrive at the following asymptotic expansion for the frequency ratio
p [w00 (0)]2 ω + O ε2 = 1 − 12 ε 1 − ξ 1 0 R 00 ω0 [w0 (s)]2 ds
(58)
0
282
that, in the case θ = 0, reduces to
p ω = 1 − 2ε 1 − ξ + O ε2 . ω0 22
(59)
Figure 4: 1st (left) and 2nd (right) eigenfrequencies ω for a cantilever beam (solid, black) in the absence of rotatory inertia, θ = 0, and with ε = 0.01, 0.05 and 0.075, superposed onto the 1-term (dotted, blue) asymptotic approximation, normalized with respect to the classical local model frequency ω0 , according to Eq.(59)
283
Eq.(59) is plotted in Fig.3 alongside the numerical solution of the TPNM. Al-
284
though the accuracy of the expansion is restricted to small values of ε, we still
285
appreciate a limit as the TPNM tends to the PNLM.
286
4.6. Cantilever Beam For a cantilever beam we have, at leading order, w0 (0) = w00 (0) = w000 (1) = w0000 (1) + θλ40 w00 (1) = 0, and the constants in the general solution (28) are given by Eqs.(52), i.e. they are the same as in the C-S case. The secular equation now reads 1 + 12 θ2 λ40 cosh α cos β − 21 θλ20 sinh α sin β + 1 = 0,
287
that, in the special case of vanishing rotational inertia, reduces to S 2 (λ0 ) − T (λ0 )V (λ0 ) = 0.
288
This formula coincides with the classical result cosh λ0 cos λ0 + 1 = 0 and the
289
corresponding eigenforms are still given by Eq.(53).
290
291
In the first-order approximation, one arrives at the following boundary conditions w1 (0) = 0, w100 (1) = 0,
p w10 (0) = 1 − ξ w000 (0),
w1000 (1) + λ40 θw10 (1) = −4λ30 λ1 θw00 (1). 23
(60)
together with the right boundary layer amplitude a20 =
p p ξ 1 − ξ [w100 (1) + w0000 (1)] ,
292
the left being given by Eq.(43a). The compatibility condition for the inho-
293
mogeneous BVP (44, 60) is still given by Eq.(56) and, as a consequence, the
294
eigenfrequency ratio (58) and the corresponding eigenmode correction are once
295
again retrieved. Fig.4 compares the normalized eigenfrequency ω/ω0 as nu-
296
merically evaluated for the TPNM with the 1-term expansion (59) and shows
297
good accuracy. Besides, the numerical solution curve matches the corresponding
298
result given in Fig.5 of [11].
299
5. Purely nonlocal model From the previous analysis, it clearly appears that the situation ξ → 0 lends a perfectly admissible eigenfrequency which, therefore, can be assumed as the proper solution to the PNLM. We now consider what happens to the eigenmodes and for this we need to investigate the behaviour of the boundary √ √ layer term Bξ (s) = ξ exp[−s/(ε ξ)], 0 ≤ s ≤ 1, as ξ → 0. Clearly, this is a transcendentally small term for s > 0 and Bξ (s) → 0 uniformly. Non uniformity arises when we consider s = 0 for then a boundary layer appears that may be √ studied taking the rescaled variable s∗ = s/(ε ξ), see [15]. This boundary layer
300
is vanishingly small as ξ → 0 but not so are its derivatives with respect to s 0, 0, s > 0, s 6= 0, Bξ0 (s) → and Bξ00 (s) → . −ε−1 , s = 0, +∞, s = 0,
301
static axial deformation. We may now ask whether this unboundedness in the
This result is the analogue of the steep boundary layer described in [39] under
304
second derivative leads to an unbounded bending energy. To answer this we Rη first observe that ∀η > 0, 0 Bξ00 (s)ds → ε−1 uniformly and therefore Bξ00 (s) is
305
tion Mξ of the boundary layer Bξ to the bending moment M through Eq.(4),
302
303
proportional to Dirac’s delta function. Indeed, when considering the contribu-
24
306
we find Mξ (0) → (2ε2 )−1 ,
307
at leading order. If we use this result in, say, the eigenmodes (38) for a S-S beam,
308
we easily see that the boundary condition M (0) = 0 is satisfied at leading order,
309
for the boundary layer cancels out the contribution of the outer solution. At the
310
same time, the constitutive BCs are asymptotically satisfied for a vanishingly
311
small ξ due to the asymptotic procedure applied above. We then conclude that,
312
in the limit as ξ → 0, the boundary layer warrants the fulfilment of all boundary
313
conditions and it brings a finite contribution to the bending energy. From the
314
standpoint of displacements, we get w(s) → w(m) + εγ1 −1 a10 R(−s) + εγ2 −1 a20 R(s − 1),
315
where R(s) is the ramp function. For a S-S beam, we have γ1 = γ2 = 3 and a10 = (−1)n+1 a20 = Cβ 3 .
316
Whence, a finite jump in the rotation and a concentrated couple at the beam
317
ends is produced. This is perhaps not so surprising, for solutions in the sense
318
of distributions are to be expected when an integral form of the constitutive
319
equation is adopted. Consequently, from a mathematical standpoint, an energy
320
bounded solution of the PNLM may be consistently defined as the limit of the
321
TPNM, although it is meaningful in the sense of distributions and we may want
322
to reject it on physical grounds.
323
6. Influence of rotational inertia
324
We now consider the effect of including rotational inertia when considering
325
the solution of the TPNM. For a S-S beam, Fig.5 plots the eigenfrequency ratio
326
ω/ω0 for mode numbers n = 1, 2 and 4, with θ = 0, 1/100 and 1/10. It appears
327
that, for the S-S end conditions, rotational inertia is irrelevant for the purpose
328
of determining the frequency ratio (yet it still affects ω0 ). For a C-C beam,
329
Fig.6 plots the eigenfrequency ratio ω/ω0 for mode numbers n = 1 and 4, with 25
Figure 5: Eigenfrequency ω for modes 1, 2 and 4 for a S-S beam, normalized over the classical frequency ω0 , for θ = 0, 1/100 and 1/10, as a function of the local model fraction ξ. As it occurs for the asymptotic expansion (39), the frequency ratio is unaffected by rotational inertia and curves overlap
Figure 6: Eigenfrequency ratio ω/ω0 for modes 1 (left panel) and 4 (right) for a C-C beam for θ = 0 (solid, black), θ = 1/100 (dashed, blue) and 1/10 (dotted, red), as a function of the local model fraction ξ
Figure 7: Eigenfrequency ratio ω/ω0 for modes 1 (left panel) and 4 (right) for a C-S beam for θ = 0 (solid, black), θ = 1/100 (dashed, blue) and 1/10 (dotted, red), as a function of the local model fraction ξ
26
Figure 8: Eigenfrequency ratio ω/ω0 for modes 1 (left panel) and 4 (right) for a C-F beam for θ = 0 (solid, black), θ = 1/100 (dashed, blue) and 1/10 (dotted, red), as a function of the local model fraction ξ
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θ = 0, 1/100 and 1/10. This time, rotational inertia plays an important role in
331
the direction of contrasting the softening effect induced by the nonlocal fraction.
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Indeed, this hardening effect is already well manifest in the fundamental mode
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and, as expected, it becomes stronger for higher modes. Besides, encompassing
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rotational inertia of the cross-section has a significant bearing on higher modes,
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regardless of the actual value of θ. The same qualitative picture appears in Fig.7
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and in Fig.8, respectively for C-S and C-F beams. It appears that the softening
337
effect is stronger moving from S-S to C-C, C-F and then to C-S.
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7. Conclusions
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The purely nonlocal theory of beam elasticity has recently attracted consid-
340
erable attention for the controversial results it conveys. Indeed, this model is
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believed to lead to ill-posed problems, owing to the appearance of a pair of con-
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stitutive boundary conditions which are generally incompatible with the natural
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boundary conditions. In this paper, we approach the problem from a different
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perspective and carry out an asymptotic analysis of the free vibrations of flexural
345
beams endowed with rotational inertia, within the two-phase theory of nonlocal
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elasticity. We show that the nonlocal term contributes with a boundary layer
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whose strength greatly varies for different end conditions. In the case of simply
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supported beams, the boundary layer is the weakest and we provide a two-term
349
correction for the classical solution. Remarkably, this situation is affected by the 27
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presence of rotational inertia only in the classical sense. Conversely, clamped-
351
clamped, clamped-supported and clamped-free (i.e. cantilever) conditions bring
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a much stronger boundary layer, a for these we provide a single correction term.
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Numerical results confirm the accuracy of the asymptotic approach and show
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that rotational inertia is very relevant in contrasting the softening effect con-
355
nected to the nonlocal phase. Most interestingly, for any end condition, the
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asymptotic solution still exists and its energy remains bounded in the limit of
357
the purely nonlocal theory, that is for a vanishingly small local phase. This is in
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contrast to what is anticipated in the literature, see, for instance, [33]. We are
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therefore in the position of attaching a meaning to the purely nonlocal theory,
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as the limit of the two-phase theory. In so doing, we encounter a solution that
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is defined in the sense of distributions (for the curvature) and that, although
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questionable from a physical standpoint, is mathematically sound. This result
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is quite general for it extends to statics and, presumably, to axial vibrations in
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a rod.
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Acknowledgements
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367
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GM gratefully acknowledges a Visiting Professor Position granted by the University of Modena and Reggio Emilia in the AY 2018/2019. AN welcomes support from FAR2019, Piano di sviluppo dipartimentale
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DIEF, DR nr. 498/2019.
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Declaration of interests
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. 477
☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: