3 Discrete and Nonlocal Beams 3.1 Hencky-Bar-Chain Model
3.2 Continualised Nonlocal Beam Model
As followed for the axial lattice, the discrete equations of the beam lattice (Hencky- Bar-Chain model) are continualized to derive an equivalent nonlocal beam con- tinuum. The relation between the discrete and continuous displacements wi = w(x=xi =i a)also requires a smooth function as:
w(x+a,t)= ∞ k=0
ak k!
∂kw(x,t)
∂xk = [ea∂/∂x]w(x,t) (68) The following pseudo-differential operators can be expressed as:
wi−1+wi+1−2wi =
ea∂/∂x +e−a∂/∂x −2
w(x,t)=4sinh2 a
2
∂
∂x
w(x,t) (69)
and
wi+2−4wi+1+6wi−4wi−1+wi−2= e2a∂/∂x −4ea∂/∂x +6−4e−a∂/∂x +e−2a∂/∂x
w(x,t)= 16sinh4
a 2
∂
∂x
w(x,t) (70)
The generalized governing equation for bending problem is then expressed as follows
μ ∂2w
∂t2 +16E I a4sinh4
a 2
∂
∂x
w +4P a2 sinh2
a 2
∂
∂x
w=q (71) The pseudo-differential operator can be efficiently approximated by Padé’s approximant (Rosenau [33]; Wattis [75] or Andrianov et al. [49]):
4 a2sinh2
a 2
∂
∂x
= ∂
2
∂x2
1−lc2∂∂x22
+... with l2c= a2
12 (72)
The Padé approximant is applied to the lattice beam equations. Hencky-Bar-Chain model can then be approximated by a nonlocal continuous formulation given by
μ
1−2lc2 ∂2
∂x2 ∂2w
∂t2 +E I∂4w
∂x4 +P
1−l2c ∂2
∂x2 ∂2w
∂x2
=
1−2lc2 d2 d x2
q (73)
which can be considered as a nonlocal continuous approximation of the lattice beam model. For the pure buckling problem (μ=0 andq =0), Eq. (73) simplifies in
E Id4w d x4 +P
1−l2c d2 d x2
d2w
d x2 =0 (74)
Equation (74) can be equivalently obtained from a nonlocal Euler-Bernoulli beam where the nonlocality is of Eringen’s type (in the sense that the bending moment- curvature differential law is a stress gradient model, first introduced by Eringen [3]
for one-dimensional or three-dimensional nonlocal elasticity):
M−lc2d2M
d x2 =E Id2w
d x2 and d2M
d x2 = −Pd2w
d x2 (75)
A review of various nonlocal beam formulations (based on Eringen’s nonlocal- ity) applied to Euler-Bernoulli, Bresse-Timoshenko or higher-order kinematics is available in Elishakoff et al. [5] or Challamel [76]. Equation (75) shows that the
Hencky-Bar-Chain system can be captured by an Eringen’s type nonlocal elastic model [3] applied at the beam scale (stress gradient model), with a scaling factor lc=a/
2√ 3
of the nonlocal model calibrated from the length of the rigid elements.
For the uncoupled bending problem (μ=0 andP =0), Eq. (73) reduces to E Id4w
d x4 =q−2lc2d2q
d x2 (76)
This above differential equation can be equivalently derived from the following second-order differential equations:
M−2lc2d2M
d x2 =E Id2w
d x2 and d2M
d x2 =q (77)
One recognizes an Eringen’s based nonlocal beam model, but with a scaling factor lc=a/√
6, which is different from the buckling problem. Finally, the pure vibration problem (q =0 andP =0), is governed by
μ
1−2lc2 ∂2
∂x2 ∂2w
∂t2 +E I∂4w
∂x4 =0 (78)
This Rayleigh-type equation (in the sense that the nonlocal bending wave equation is corrected by some nonlocal rotary effects similar to the rotary effect introduced by Bresse (1859) and Rayleigh [55] for correcting the Euler-Bernoulli beam model) is equivalent to that considered in an Eringen’s based nonlocal model applied at the beam scale (see Challamel [76]; Zhang et al. [77]):
M−2lc2d2M
d x2 =E Id2w
d x2 and d2M
d x2 = −μ∂2w
∂t2 (79)
The small length scale parameterlc=a/√
6 appears again in the vibration analy- sis (Challamel et al. [78]; Wang et al. [79]). A surprising finding is that the calibration of Eringen’s length scale parameter is apparently dependent on the type of analysis, namely bending, buckling or vibration (see the discussion in Challamel et al. [78];
Wang et al. [79]). This would suggest that the Hencky-Bar-Chain for statics/dynamics problems is not strictly speaking captured by a stress gradient of Eringen’s type with a constant length scale. Regarding this length-scale dependence on the type of anal- ysis (buckling, vibration, static bending), Wang et al. [79] calibrated the nonlocal length scale with respect to the axial load intensity.
It is also possible to change the point of view, and to consider local constitutive laws but additional nonlocal inertia contributions.
M =E I∂2w
∂x2 and ∂2M
∂x2 = −μ∂2w
∂t2 +2μlc2
∂4w
∂x2∂t2 (80)
This set of equation can be obtained from a “local” elastic potential energy formula and a revised form (or “nonlocal”) of kinetic energy:
U[w] = 1 2
L 0
E I(∂2w
∂x2)
2
d x
and T[w]=1 2
L 0
μ (∂w
∂t )2+2μlc2(∂2w
∂x∂t)
2
dx (81)
This modification of the kinetic energy in the vibration analysis of microstructures is followed by Mindlin [53] and by Polyzos and Fotiadis [54]. The enriched kinetic energy Eq. (81) clearly contains an additional nonlocal rotary inertia.
An alternative model is derived based on the nonlocal corrections of the moment- curvature relationship and the balance equation with a nonlocal rotary term:
M−lc2∂2M
∂x2 =E I∂2w
∂x2 and ∂2M
∂x2 = −μ∂2w
∂t2 +μlc2 ∂4w
∂x2∂t2 (82) The coupling of both equations in Eq. (82) again gives a bending wave equation very close to Eq. (78), when the higher-order term inl4cis neglected:
μ
1−2lc2 ∂2
∂x2 ∂2w
∂t2 +
E I+μlc4
∂2
∂t2 ∂4w
∂x4 =0 (83) In order to resolve the apparent paradox of the dependence of Erigen’s length scale on the type of problem considered, we develop herein another point of view which considers that the Hencky-Bar-Chain system behaves like a nonlocal beam with just one small length scale but with modified equilibrium equations.
This nonlocal model can be elaborated from the discrete equilibrium equation and the discrete formulation of the bending constitutive law Eq. (66), continualized as follows:
M(x)=4E I a2 sinh2
a 2
∂
∂x
w(x) (84)
When applying the Padé approximant to the pseudo-differential operator in Eq.
(84), Eringen’s nonlocal elastic constitutive law with only one length scale is clearly recognized from this last equation:
M−lc2∂2M
∂x2 =E I∂2w
∂x2 and lc= a 2√
3 (85)
The governing equation can also be built from the continualization of Eq. (67), which again is approximated using a Padé approximant, as:
∂2M
∂x2 +P∂2w
∂x2 +μ
1−lc2 ∂2
∂x2 ∂2w
∂t2 =
1−lc2 d2 d x2
q (86)
The governing equation is no more local, and has been enriched with some nonlo- cal rotary inertia and an additional external load term. It is then possible to solve the paradox of apparent dependence of the length scale, with a nonlocal model coupled to modified nonlocal equilibrium equation. The nonlocal generalization of Newton’s second law has already been formulated in a different context by Milton and Willis [80] or Charlotte and Truskinovsky [81]. For the beam lattice model considered in this chapter (Hencky-Bar-Chain model), the nonlocal moment-curvature relationship and the governing equation are written by
M =E I ∂2w
∂x2
and ∂2M
∂x2
+P ∂2w
∂x2
+μ∂2w
∂t2 =q (87) where the angle bracket represents the nonlocal average operator associated with the pseudo-differential operator, given by Eq. (84) for the moment-curvature law, or Eq. (85) for its Padé approximant.
This correction Eq. (86) in the equilibrium equation may resolve the paradox of the dependence of length scale on the analysis type (buckling or vibrations analyses).
When correcting both the balance equation and the constitutive law by nonlocal terms, the nonlocal length scale calibrated from the beam lattice is no more dependent on the configuration.