3 Displacement Based Nonlocal Model
3.1 Nonlocal Rod
∂xχ(0)= 1
λLχ(0)+ α kf
∂xM(0)− α kfλLM(0)
∂xχ(L)= − 1
λLχ(L)+ α
kf∂xM(L)+ α
kfλLM(L) (21)
b(x)
L u(x) A
x y
z
Nj Qj Nj+1
b(xj)
Δx Fig. 1 Nonlocal elastic rod
Considering the Fig.1, the equilibrium equation of the arbitrary volume element Vj is expressed as follows
Nj + Qj + b(xj)Ax=0 (23)
where b(xj)is the axial volume force,Nj is the difference between the contact forces and Qj is the resultant of long-range interactions applied to the centroid of volume elementVj. That is,
Qj = ∞ h=−∞,h=j
Q(h,j)= ∞ h=−∞,h=j
q(h,j)VhVj (24)
whereq(h,j)represents the specific long-range interaction between the volume ele- mentsVhandVj shown in Fig.2. This force is expressed as
q(h,j)=(uh−uj)g(xh,xj) (25) whereuis the axial displacement of the rod andg(xh,xj)is a positive and symmetric attenuation function. In virtue of Eq. (24), the equilibrium in Eq. (23) can be rewritten as
Nj+ ∞ h=−∞,h=j
q(h,j)(Ax)2+b(xj) x=0 (26)
Dividing Eq. (26) byx and taking the limitx→0 lead to the following integro-differential equation
A∂xσl(x) + A2 ∞
−∞q(x, ξ)dξ = −b(x) (27)
Fig. 2 Elastic long-range
interaction qx qx
η
whereσldenotes the local Cauchy stress while the second term at the left-hand side is related to the nonlocal interactions.
In Eq. (27),q(x, ξ)is the long-rang force exerted by the volume at abscissaξ on the element located atx. That is,
q(x, ξ)= [u(ξ)−u(x)]g(x, ξ) (28) which represents the nonlocal constitutive relation. By taking into account equa- tion (28), the classical local stress-strain law and the strain-displacement relation ε(x)= dud x, it is possible to rewrite the equilibrium equation (27) in terms of axial displacementu. That is,
E A∂x2u(x) + A2 ∞
−∞[u(ξ)−u(x)]g(x, ξ)dξ = −b(x) (29) If a rod of finite lengthL, divided intomvolume elementsVj = Ax =AL/m, is considered, the equilibrium equation (23) forx→0 takes the form
E A∂x2u(x) + A2 L
0
[u(ξ)−u(x)]g(x, ξ)dξ = −b(x) (30)
3.1.1 Discrete Model of the Nonlocal Rod
An equivalent discrete mechanical model of the displacement based nonlocal rod is introduced in this section. The idea is that contact forces can be modeled by springs of stiffnessKl=E A/x=E Am/L, while the nonlocal forces can be represented by springs with distance-decaying stiffness. That is,
Knlj h =(Ax)2g(xj,xh) (31) Hence, the equilibrium equations of the point-spring model in Fig.3are expressed as follows
(Kl+Knl)u=f (32)
whereKlis the following symmetric tridiagonal matrix.
Fig. 3 Discrete elastic scheme of the local and nonlocal interactions
Knl13
K24nl
K14nl
Kl+K12nl Kl+K23nl Kl+K34nl
F1 F4
1 2 3 4
Kl =
⎡
⎢⎢
⎢⎢
⎣
Kl −Kl 0 0 ... 0 2Kl −Kl 0 ... 0 ... ... ... ...
2Kl −Kl Kl
⎤
⎥⎥
⎥⎥
⎦ (33)
whileKnl is a symmetric and full matrix. That is,
Knl =
⎡
⎢⎢
⎢⎢
⎣
K11nl −K12nl −K13nl ... ... −K1mnl K22nl −K23nl ... ... −K2mnl ... ... ... ...
Kmnl−1m−1−Kmnl−1m Kmmnl
⎤
⎥⎥
⎥⎥
⎦ (34)
andfT =x[F1 ... Fm]is the load vector in whichFj= Ab(xj)are the external nodal forces per unit length.
Kinematic and/or static boundary conditions can be imposed by assigning the first and last components of the displacement and load vectors.
By taking the limitx→0, Eq. (30) can be obtained from Eq. (32), that repre- sents a discretized version of the previous model.
3.1.2 Differential Formulation of the Nonlocal Problem
In order to obtain an equivalent differential formulation of the integro-differential model of the nonlocal rod, Eq. (30) is firstly rewritten as
E A∂x2u(x) + A2 L
0
u(ξ)g(x, ξ)dξ−A2u(x)γ (x)= −b(x) (35) where
γ (x)= L
0
g(x, ξ)dξ (36)
Now, by selecting the bi-exponential attenuation function of the type of Eq. (3), the corresponding differential formulation of the Eq. (35) can be found in a similar way to the previous section [15]. Specifically, by assuming that
g(x, ξ)= Enl
2λL5 exp
−|x−ξ| λL
(37) the following differential equation
−E A∂x4u(x)− E A
(λL)2 +A2γ (x)
∂x2u(x)+2A2∂xγ (x)∂xu(x)+ +
1 (λL)2
E A2
L4 −A2γ (x)
+A2γ (x)
u(x)= − b(x)
(λL)2 (38) with the following BCs
E A∂x3u(0)−A2
γ (0)∂xu(0)+∂xγ (0)u(0)
= 1
λL
E A∂x2u(0)−A2γ (0)u(0)+b(0) E A∂x3u(L)−A2
γ (L)∂xu(L)+∂4xγ (L)u(L)
=
− 1 λL
E A∂x2u(L)−A2γ (L)u(L)+b(L)
(39) is equivalent to the integro-differential equation in Eq. (35).
The integro-differential equation in Eq. (35) requires some specific numerical algorithms for its solution [58]. On the contrary, Eq. (38) with the BCs in Eq. (39) provides an alternative formulation of the displacement based nonlocal problem that can be readily solved. For example, consider a micro-rod constrained atx=0 with lengthL =300μm, rectangular cross section with widthb=60μmand thickness h =25μm, elastic modulusE=2.80 GPa,Enl =28 TPa, and forced atx=Lby a point load forceF =20 N.
The effect of the nonlocal parameterλon the mechanical response of the rod is investigated. Figure4shows the normalized axial displacement for different values of the nonlocal parameter. Each displacement function is normalized with respect to the related maximum value atx=L.
Numerical results show that the increasing of nonlocal parameter yields an incre- ment in terms of rod stiffness. Specifically, Fig.5shows the maximum displacement unl(L)normalized respect to the local oneul(L)for different values of nonlocal parameter. We can observe that the displacement decreases if the nonlocal parameter grows up. This behavior may be easily explained by focusing on the attenuation function (37). The nonlocal parameterλ plays a double role, it scales the whole attenuation function and at the same time modifies the velocity of decaying with the distance. An increase in the value ofλ results in a larger scaling of the function, i.e. with lower peak, and in a slower decaying with the distance. From the numer- ical application of Figs.4 and5, it may be asserted that the influence ofλin the
Fig. 4 Normalized displacement of nonlocal rod for different values ofλ
Fig. 5 Maximum
normalized displacement for different values ofλ
velocity of the attenuation function decaying is predominant and, as expected, the rod stiffness increases with λ. Further, a strain localization effect at the rod ends occurs. This effect is more significant for larger values ofλ, and is due to the fact that volume elements in the neighborhood of the rod ends are involved in a smaller amount of nonlocal interactions compared with volume elements in the central part of the rod. The behavior of the nonlocal rod may be further manipulated by varying also the nonlocal modulus Enl and/or introducing coefficients weighting the local and nonlocal contributions (similarly to Eq. (16)), resulting in a very flexible model.