This kind of model has some advantages. First of all, the nonlocal interactions are defined from a correspondent mechanical description, involving the presence of long-range springs connecting non-adjacent volume elements. For this reason, it is also known as mechanically based nonlocal model. Moreover, in presence of bounded domains, the nonlocal contributions to boundary conditions vanishes and this allows to enforce them as in a local continuum without any mathematical inconsistency.
The displacement based nonlocal model has been successfully specialized to beam elements. In this context a finite element formulation with a closed formulation of non-local stiffness matrix elements has been obtained [44]. This approach is also suitable for modelling beams with viscoelastic long-range interactions both in quasi- static and dynamic conditions [46,47].
Moreover, an analogous approach has been successfully developed in order to simulate non-local effects in the micromechanics of fluid flow, with the specific appli- cation of blood flowing along micro arterial vessels [51]. To this aim, the non-local forces are defined as dependent on the relative velocity instead of relative displace- ment. In this chapter, the mechanically based non-local model is introduced and its particularization to rods and beams are developed. Specifically, after an overview of the main nonlocal models in the Sect.2, the Sect.3is devoted to the displacement based nonlocal model. In the latter nonlocal rod and beam models are derived and some applications in both static and dynamic conditions are provided.
M(x)= L
0
λ(x,x)k¯ f(x)χ(¯ x)d¯ x¯ (2) where the beam axis coincides with the x-axis andkf(x) is the elastic bending stiffness. The attenuation kernelλcan be selected among exponential, Gaussian or power-law type functions and must satisfy the properties of symmetry, positivity and limit impulsivity. A frequent choice is the bi-exponential function defined as follows
λ(x,x¯)= 1 2λL exp
−|x− ¯x|
λL
(3) whereλLis a characteristic length.
By choosing the kernel in Eq. (3), it can be proved [15] that the integral model expressed by Eq. (2) is equivalent to the second-order differential equation
∂x2M(x)− 1
(λL)2M(x)= − E I
(λL)2χ(x) (4)
with the following constitutive boundary conditions (BCs)
∂xM(0)= 1 λLM(0)
∂xM(L)= − 1
λLM(L) (5)
being kf(x)=E I, where E is the elastic modulus and I is the cross-sectional moment of inertia around the bending axis. Equations (5) are clearly in contrast with the static boundary conditions of most structural schemes. Therefore, the incom- patibility between the equilibrium requirements and the constitutive nonlocal law reveals that applying the strain-driven model to bounded domains leads to ill-posed mechanical problems.
An efficient strategy to overcome the limits of Eringen’s formulation has been developed in Refs. [15, 25]. It consists in a new nonlocal law, the stress-driven model, formally obtained by swapping the roles of stress and strain fields. Hence, the elastic strain tensor at a point x of the continuous bodyBis the output of the convolution integral between the local elastic strain fieldεand the scalar kernelλ. That is,
ε(x)=
Bλ(x,x)C(¯ x¯)σ(x¯)dx¯ (6) whereCis the elastic complianceC=E−1.
Applied to a one-dimensional model, such as the Bernoulli-Euler beam introduced before, the stress-driven model gives
χ(x)= L
0
λ(x,x)¯ M(¯z)
kf(¯x)dx¯ (7)
The equivalent differential problem, obtained by assuming the attenuation func- tion of Eq. (3) is expressed by a second-order differential equation
∂x2χ(x)− 1
(λL)2χ(x)= − 1
E I(λL)2M(x) (8) with the following constitutive BCs
∂xχ(0)= 1 λLχ(0)
∂xχ(L)= − 1
λLχ(L) (9)
where the signs in Eq. (9) are consequence of the fact that the first derivative of the special kernel is an odd function.
Unlike Eringen’s formulation, the stress driven model provides exact solutions to nonlocal continuous problems and it is able to describe the actual behavior of micro- and nano-structures.
2.2 Gradient Models
Another nonlocal theory is the gradient model proposed in Ref. [35]. Such model overcomes the ill-posedness of the strain-driven model and it is able to simulate dispersion and wave propagation at atomic-scale [52,53]. The approach combines the nonlocal strain-driven model by Eringen with the theory of strain gradient elasticity formulated by Mindlin [54,55]. Nonlocal strain gradient model for a slender micro- or nano-beam is formulated by expressing the bending moment in terms of elastic flexural curvatureχand of its derivative∂xχ. That is,
M(x)= L
0
α0(x,x, λ¯ 0)kf(x)χ(¯ x)d¯ x¯
−l2∂x
L 0
α1(x,x, λ¯ 1)kf(x)∂¯ x¯χ(x)d¯ x¯ (10) The characteristic lengthlis introduced to make the two terms dimensionally com- patible. The kernelsα0andα1are attenuation functions depending on two nonlocal parametersλ0,λ1.
Following the treatment in Refs. [35] and [56], the attenuation kernels can be assumed to be coincident and equal to the bi-exponential function. Implicitly, it is also assumed the coincidence of nonlocal parametersλ0=λ1=λ. A second parameter, λl =l/L, may be associated with the characteristic lengthl of the strain gradient term.
As shown in Ref. [57], with the choice of the bi-exponential attenuation function, Eq. (10) is equivalent to the differential equation
(kfχ)(x)−l2∂x2(kχf)(x)=M(x)−(λL)2∂x2M(x) (11) equipped with the constitutive boundary conditions
∂xM(0)= 1
λLM(0)+ l2
(λL)2∂x(kfχ)(0)
∂xM(L)= − 1
λLM(L)+ l2
(λL)2∂x(kfχ)(L) (12) Hence, the nonlocal strain gradient model is based on two parameters, λ and λl. As proved in Ref. [57], displacement solutions exhibit softening and stiffening responses for increasing nonlocal and gradient parameters, respectively. Thus, the nonlocal strain gradient law is able to model a wide class of small scale problems.
It has to be noted that forλ→0 and from Eq. (10), we recover the pure strain gradient law, by virtue of the kernel impulsivity property. When the characteristic lengthlapproaches to zero, from the same equation, we get the fully nonlocal strain- driven model, which is ill-posed.
A stress gradient nonlocal model can be obtained by formally swapping the roles of stress and strain in Eq. (10). That is,
χ(x)= L
0
φλ(x,x)¯ M(¯x)
kf(x¯)dx¯−l2∂x
L 0
φλ(x,x)¯ ∂x¯M(¯x)
kf(x¯) dx¯ (13) By selecting the bi-exponential attenuation function, the integral formulation Eq.
(13) reverts to the differential equation
χ(x)−(λL)2∂x2χ(x)=(k−1f M)(x)−l2∂x2(k−1f M)(x) (14) equipped with the constitutive boundary conditions
∂xχ(0)= 1
λLχ(0)+ l2
(λL)2∂x(k−1f M)(0)
∂xχ(L)= − 1
λLχ(L)+ l2
(λL)2∂x(k−1f M)(L) (15)
2.3 Mixture Models
Another strategy to formulate a well-posed nonlocal model is based on the two phases approach. It consists of a convex combination between local and nonlocal responses expressed as follows
M(x)=α (kf χ)(x)+(1−α) L
0
λ(x,x)(k¯ f χ)(x)d¯ x¯ (16) whereα∈[0, 1] and the nonlocal term is expressed by following the strain-driven approach. As shown in Ref. [30], for a fixedα, the response exhibits a softening behavior for increasing nonlocal parameter λ. When α approaches to zero, from Eq. (16), we recover the purely nonlocal strain-driven model, which is ill-posed. In Ref. [30], it has been shown that by choosing the kernel (3), it can be proved that the integral model expressed by Eq. (16) is equivalent to the second-order differential equation
∂x2M(x)− 1
(λL)2M(x)= − kf
(λL)2χ(x)+αkf ∂x2χ(x) (17) with the following constitutive boundary conditions (BCs)
∂xM(0)= 1
λLM(0)+αkf∂xχ(0)− 1
λLkf χ(0)
∂xM(L)= − 1
λLM(L)+αkf∂xχ(L)+ 1
λLkf χ(L) (18) A two phases mixture model can be also formulated by following the stress-driven approach, as shown below
χ(x)=α M
kf
(x)+(1−α) L
0
λ(x,x)¯ M
kf
(x)d¯ x¯ (19) In this case, for a fixedα, the response exhibits a stiffening behavior for increasing the nonlocal parameterλand for anyα∈[0, 1] the model is always well-posed.
The equivalent differential problem is expressed by the second-order differential equation as follow
∂x2χ(x)− 1
(λL)2χ(x)= − 1
kf(λL)2M(z)+ α kf
∂x2M(x) (20) with the following constitutive BCs
∂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)