Displacement Based Nonlocal Models for
1 Introduction
In its first formulation, continuum mechanics theory was based on the idea that the matter mechanically interacts through local force interactions [1–3]. Following this approach, under certain assumptions, the solid continuum matter can be approxima- tively modeled as a set of discrete elements connected by elastic springs in which the parameters depend on the mechanical properties of the material which constitutes the matter. This approach allows to well describe the mechanical and thermodynamical behavior of several structures and materials providing well-posed formulations and accurate results. However, the local continuum mechanics fails in describing some structures at small-scale where size-effects, long range interactions and non-elastic forces influence the mechanical phenomena. Moreover, some mechanical effects, also at macro-scale, derive from micro- and nano-scale phenomena, such as stress- tip concentrations, dispersion of elastic waves, edge effects, shear bands. In this context, the local continuum theory may be inadequate in modeling these phenom- ena. For this reason, a new model able to take into account other mechanical effects was born in the last century. Pursuing this purpose the nonlocal model was formulated recently.
From the first formulations and the pioneering works [4–8], nonlocal theory is based on the idea that the stress at a given point of the continuum matter depends of on the entire local stress field of the domain. Nonlocal elasticity models provide the definition of an enriched continuum as compared to classical theories contin- uum definition. Such models are able to capture small-scale phenomena avoiding computational expensive procedures [9,10].
Nonlocal approaches introduce a constitutive relation where long distance inter- actions exchanged within the body are described by internal parameters. Specifically, in many formulations of this kind, the long-range interaction is described by a con- volution integral in the stress-strain relation. One of the first formulation has been provided by Eringen [11,12]. In this model, the nonlocal stress is the output of a convolution integral between the entire elastic strain field and a particular attenu- ation function depending on an internal length-scale parameter. Since the input in the Eringen constitutive law is the strain field, this model is also called strain-driven approach. Substantially, this nonlocal theory provides a constitutive relation which is not pointwise but it is based on an integral average. Eringen’s approach provides an accurate tool able to provide good results for screw dislocation and surface waves in unbounded domain. However, some issues appear when this model is applied to real structural problems in which, usually, the continuum domain must be considered bounded.
These issues are related to some meaningless boundary conditions which appear in a nonlocal Eringen finite domain. Specifically, from the integral formulation, it is possible to obtain the corresponding differential stress-strain relation by selecting a proper kernel. This relation is equal to the integral one only if proper boundary conditions are selected. By using the classical Eringen’s approach, the integral for- mulation leads to an ill-posed differential problem in terms of meaningless elastic
boundary conditions [13–16]. Several ways have been developed to overcome the ill-posedness of Eringen’s nonlocal model for finite domain.
Specifically, in literature, there are some integral nonlocal formulations able to overcome this drawback, e.g. the two-phase local/nonlocal models [17–20], the strain-difference nonlocal elasticity [21–24], the stress-driven approach [15, 25].
Among these integral approaches, the latter still provides an integral formulation as in Eringen’s approach but the nonlocal problem is set in a different way. Specifically, the output in the constitutive law is the nonlocal strain which is obtained as a con- volution between an attenuation kernel and the entire stress field. In this way, the corresponding differential formulation avoids the boundary condition issues of Erin- gen’s strain-driven one. By following this stress-driven approaches, several applica- tions are performed [26–28] and other models are developed, such as local-nonlocal mixture stress-driven models [29] and two-phase models [30].
The aforementioned formulations provide an integral equation in the stress-strain relation. These approach are also known as strong nonlocality [31]. However, other nonlocal models exist and they are obtained following another approach known as weak nonlocality. In this context, we remember the gradient models proposed in Refs. [32–35]. Another class of nonlocal models which involves all the previous ones are the fractional-order models [36–38]. However, some of these formulations can not be considered a different approach but a kind of generalization. Specifically, as it has been shown in Refs. [39, 40], in all the integral nonlocal approaches if the convolution kernel is of power-law kind fractional-order operators may appear in the stress-strain relation. Moreover, such operators are characterized by a strong nonlocality due to their power-law kernels [41] and this peculiarity may be useful in several continuum mechanics applications [42,43].
The aforementioned formulations are based on the assumption that the stress- strain relation of the nonlocal continuum is enriched by the introduction of additional contributions in terms of gradients or integrals of the strain and or stress fields. These other terms are able to take into account the presence of the effects of microstructures at small-scale. Some of the cited formulations presents some drawbacks. Specifically, in the strong nonlocality the selected kernel, that is the decaying function, must respects some geometrical constraints, whereas the weak nonlocality lack of an evident mechanical description. However, another class of different approach to take into account these effects exists, which avoid the restrictions of the previous approaches. This formulation is known as mechanically based nonlocal model and introduces nonlocal interactions among different locations of the body in terms of central long-range elastic [31,36,44] and/or viscoelastic body forces [45–47]. Such forces are proportional to the interactive volumes or masses of the solid. In other words, in comparison with the local continuum this model provides an enriched domain with nonlocal forces defined as dependent from the relative displacements between the volume elements and all the other volume elements in the domain [44,48, 49]. For this reason this class of mechanically based model is known as displacement based nonlocal models. The model belongs to the class of integral or strong nonlocal models, and it can be seen as a modification of the peridynamic formulation provided by Silling [50].
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.