Review on nonlocal continuum mechanics: Physics, material applicability, and mathematics

Abstract

The classical continuum mechanics assumes that a material is a composition of an infinite number of particles each of which is a point that can only move and interact with its nearest neighbors. This classical mechanics has limited applications where it fails to describe the discrete structure of the material or to reveal many of the microscopic phenomena, e.g., micro-deformation and micro-dislocation. This observation motivated the need for a general point of view that instills the fact that the material particle is a volume element that would deform and rotate, and the material is generally a multiscale material. In addition, the particle's equilibrium should not be considered in isolation from its nonlocal interactions with other particles of the material. Material models with these features are the nonlocal microcontinuum theories.

Whereas review articles and books on microcontinuum theories and nonlocal mechanics would be found in the literature, no review that extensively deals with the fundamentals of nonlocal mechanics from the physics, material, and mathematical points of view has been presented so far. There is a current scientific debate on the benefits of applying nonlocal theories to various fields of mechanics. This is due to a lack of understanding of the physics behind these theories. In addition, questions on the applicability of nonlocal mechanics for various materials are not answered yet. Furthermore, mathematicians revealed paradoxes and complications of finding solutions of nonlocal field problems. In this review, we shed light on these folders. We give extensive interpretations on the physics of nonlocal mechanics of particles and elastic continua, and the applicability of nonlocal mechanics to multiscale materials and single-scale materials is interpreted. In addition, the existing complications of solving nonlocal field problems, and the various methods and approaches to overcome these complications are collected and discussed from the physical and material points of view. Furthermore, we define the open forums that would be considered in future studies on nonlocal mechanics.

Introduction

Continuum mechanics is the most practical approach to describe the mechanics of various types of materials. In the classical approach of continuum mechanics, a material is composed of a set of idealized-infinitesimal material particles, each of which is a mass point that only interacts with its neighbor particles. The stress field in the classical continuum mechanics is expressible based on Noll's local constitutive axioms of deformations of simple materials (Truesdell, 1977). Thus, the stress at a point depends on the motions of the points of the continuum (or the deformation field that generates due to these motions), which are located within a finite distance from that point. This finite distance depends on the local material properties of the continuum. Nevertheless, the existence of long-range interactions between non-neighbor particles was reported in various theoretical and experimental studies, and the effects of these interactions on the dispersion of elastic waves and mechanical properties of materials have been extensively documented (e.g., Bouchet et al., 2010; Di Paola et al., 2010; Zingales, 2011). In addition, both natural and man-made materials have complicated microstructures with exceptional micro-and-nano-scale phenomena. The classical continuum mechanics fails to describe physical phenomena in which the long-range interactions play a major role. In addition, it cannot observe many of the micro-/nano-scale phenomena, especially the ones that require breaking down the material into multiple scales. In other words, the properties and behaviors of materials captured by the classical continuum mechanics are invariant with respect to time and length scales, and more notably size effects cannot be captured by this classical mechanics (Hütter, 2017). It has shown a clear failure to capture microscopic phenomena such as micro-rotation, micro-deformation, micro-dislocation, or micro-twinning (Mindlin, 1964; Eringen, 2002; Neff et al., 2014; Shaat and Abdelkefi, 2016a). Because of the aforementioned limitations of the classical continuum mechanics, various types of generalized models of continuum mechanics – such as micromorphic, micropolar, nonlocal, and high-order strain/rotation gradient mechanics – have been developed. These generalized models reflected effects of long-range interactions on the material mechanics, captured many of the micro-/nano-scale phenomena and the material dispersion properties (Kunin, 1984; Shaat, 2019), and modeled the realistic microstructures of materials (Kunin, 1968, 1982; 1983; Maugin, 1993; Capriz, 1989; Gurtin and Podio-Guidugli, 1992).

Advanced theories of continuum mechanics – dated back to Voigt (1887) – in which the material is modeled as a colony of independently moving and/or deforming particles have been developed. These theories are now called microcontinuum field theories (Eringen, 1999, 2002). Voigt (1887) early studied the elasticity and piezoelectricity of crystals in consideration of the crystallographic-rotation of polar molecules, which were considered rigid objects in a crystal structure. Subsequently, the Cosserat theory (Cosserat and Cosserat, 1909) was developed assuming that the material is a composition of rigid particles, and the particles exhibit independent rotations. In the Cosserat theory, the continuum is subjected to surface and body couples, which are independent of the surface and body forces. This theory was then further extended by Gunther (1958), Grioli (1960), Aero and Kuvshinskii (1961), and Schaefer (1962). Eringen (1966, 1968a, 1969) developed the micropolar theory and the microstretch theory for linear and nonlinear elasticities. The micropolar theory is a generalized version of the Cosserat theory, which accounts for both static and dynamic effects of the micro-rotation fields. In 1964, Eringen (1964a,b) developed the micromorphic theory. In the context of the micromorphic theory, the material is a composition of a large number of material particles, and each one of these particles can move, deform, and rotate. The micromorphic theory is a generalized model of continua, which can be simplified to the micropolar/Cosserat theory and microstretch theory (Eringen, 1968b). Recently, various simplified versions of the general micromorphic theory have been developed (Neff et al., 2014; Romano et al., 2016; Barbagallo et al., 2017; Shaat, 2018a). A fundamental feature of the microcontinuum theories is the formulation of the kinematical variables considering independent microstructures. This resulted in special kinematical variables such as the gradients of the micro-deformation fields and the coupling variables that measure the difference between the micro- and macro-scale fields. Because of these special kinematical variables, the microcontinuum theories are capable of describing the physical microstructures of materials (Capriz, 1989; Gurtin and Podio-Guidugli, 1992), exploring the material dispersion characteristics (Kunin, 1968, 1982; 1983, 1984; Trovalusci and Pau, 2014; Shaat, 2019), and the multiscale modeling of materials (Trovalusci, 2014; Shaat, 2018a).

Nonlocal mechanics may be viewed as a general material model. The microcontinuum theories, e.g., micromorphic, micropolar and microstretch theories, reveal nonlocality (Kunin, 1984; Maugin, 1993). Sources of the nonlocality in these theories are the gradients of the micro-deformations and the kinematical variables of the coupling between the different degrees of freedom. This nonlocality can be understood by realizing the ability of these theories to describe the spatial dispersion of the elastic waves (Kunin, 1984; Shaat, 2019). Thus, the nonlocal fields are implicitly modeled by these theories (Kunin, 1984; Maugin, 2017; Tuna and Trovalusci, 2020). However, other theories represented the nonlocal fields using convolutions of the kinetical and kinematical variables. These convolutions model the long-range effects of the material's kinematical degrees of freedom. Despite all microcontinuum theories reveal nonlocal fields implicitly, the continuum theories that utilize convolution-type constitutive equations of the nonlocal fields are commonly named in the literature ‘nonlocal theories’. This has also led to the classification of the nonlocal theories into ‘explicit’ and ‘implicit’ nonlocal theories (Kunin, 1968, 1982; 1983, 1984; Tuna and Trovalusci, 2020; Polizzotto, 2001). Nonetheless, a classification of the nonlocal theories is not abstractive, as many factors should be considered. For example, the nonlocal field due the gradient and the coupling kinematical variables may be different than the nonlocal field that is modeled by the convolutions of the kinetical and kinematical variables.

The first nonlocal continuum model was developed by Kröner (1967) in which the long-range effect of the cohesive forces was modeled. Then, Eringen and co-workers (Eringen and Edelen, 1972; Eringen, 1972) developed the current version of nonlocal mechanics. They derived attenuation functions that describe the decay of the long-range interactions between two particles with the distance between them. These functions gave the constitutive equations and the balance equations of the continuum theory depend on integral functionals of the kinematical variables. In the early versions of the nonlocal theory, it resembled the classical mechanics theory in modeling the particles of the material as mass points that exhibit only translational motions. Subsequently, the nonlocal microcontinuum mechanics was developed as a more general version of nonlocal mechanics (Eringen, 1999, 2002). A unified approach of nonlocal field theories for elastic solids, viscous fluids, electromagnetic solids and fluids, memory-dependent elastic solids, and media with microstructure was developed and presented in a book by Eringen (2002).

Various models of nonlocal mechanics were developed. Polizzotto (2001) reformulated Eringen's nonlocal elasticity for elastic continua with discontinuities by deriving attenuation functions that depend on the geodetical path of the nonlocal field. A nonlocal elasticity model that depended on the strain-difference of two distinct points in the continuum was represented by Polizzotto et al. (2004). Eringen (2006) formulated a nonlocal model in which the balance equations are formed without nonlocal residuals, and singularities and discontinuities are smoothed out. These forms of nonlocal mechanics were effectively used to model elastic continua with non-smooth deformation phenomena (Lazopoulos, 2006, 2016) and peridynamic continua (Silling et al., 2003; Silling and Lehoucq, 2010). In 2000, the peridynamics nonlocal theory was introduced by Silling (2000) as a computational tool of nonlocal mechanics. In addition, applications of the fractional calculus into the nonlocal continuum mechanics were early introduced by Gubenko (1957), Rostovtsev (1959) and recently by Lazopoulos (2006), Di Paola et al. (2009a,b), Cottone et al. (2009a,b), Drapaca and Sivaloganathan (2012), Challamel et al. (2013), Atanackovic et al. (2014a,b), Tarasov (2014), and Sumelka and Blaszczyk (2014). Fractional calculus is an effective mathematical approach that can describe physical phenomena in continuous matter using fractional derivatives with nonlocal influence (Oldham and Spanier, 1974; Poldubny, 1999).

Although several review articles (Arash and Wang, 2012; Rafii-Tabar et al., 2016; Askari et al., 2017; Behera and Chakraverty, 2017; Srinivasa and Reddy, 2017) and books (Gopalakrishnan and Narendar, 2013; Karličić et al., 2015; Ghavanloo et al., 2019) on the theoretical foundation and applications of nonlocal mechanics would be found in the literature, reviews that extensively deal with the fundamentals of nonlocal mechanics from the physics, material, and mathematics points of view have not been presented so far. Whereas theories of nonlocal mechanics have great potential to model advanced materials, the current implementations of these theories indicate a scientific debate on their benefits to various fields of mechanics. Many of the current implementations of nonlocal mechanics lack the crucial understanding of the physics behind these theories. In addition, the phenomena that would be captured by these theories and their applicability to various materials are still open. One may observe in the literature that the mechanics of the same material has been studied using different theories. Values were, as well, assigned to various material parameters defined by the theories of nonlocal mechanics with neither clarifications nor connections to the investigated materials. In addition, an extensive debate on the mathematics of nonlocal mechanics is currently found in the literature. The mathematical considerations of nonlocal mechanics have explained the paradoxes and complications of finding solutions of nonlocal field problems. Nonetheless, in the recent years, the mathematics of nonlocal mechanics in isolation from its physics have led to many questionable interpretations of nonlocal mechanics. In this review, we discuss new insights into the aforementioned folders by giving extensive interpretations on the physics behind the various theories of nonlocal mechanics, the material applicability of each one of these theories, and the mathematics of nonlocal mechanics. Nonlocal mechanics of particles and nonlocal mechanics of elastic continua are discussed. In addition, the applicability of various theories of nonlocal mechanics to single-scale and multiscale materials is examined. Moreover, the existing complications of solving nonlocal field problems are reviewed. The various efforts to overcome these complications are also collected and discussed from the physical and material points of view. Aspects of the thermodynamics and the variational principles of theories of nonlocal mechanics are, however, beyond the scope of this review. These aspects have been fully established in early studies of nonlocal mechanics and have been well interpreted by Edelen and Laws (1971), Eringen (1999, 2002), and Polizzotto (2003a,b).

The physics of nonlocal mechanics is discussed in Sections 2 Nonlocal mechanics of particles, 3 Nonlocal mechanics of elastic continua. Then, the nonlocal mechanics of multiscale materials and single-scale materials is discussed in Sections 4 Nonlocal mechanics of multiscale materials, 5 Nonlocal mechanics of single-scale materials. The different microcontinuum theories are reviewed in Section 4. The essential equations of these theories are presented in the context of nonlocal mechanics. In addition, their applicability to multiscale materials is discussed. The nonlocal theory and its reduction to the strain gradient and rotation gradient theories are presented in Section 5. In addition, the applicability of nonlocal theories to single-scale materials is interpreted. The mathematical aspects of nonlocal mechanics are reviewed and discussed in Section 6. The solutions of nonlocal field equations for unbounded and bounded domains are reviewed and compared explaining the complications and limitations of the nonlocal models. Finally, the open forums and the future prospects of research on nonlocal mechanics are defined and presented in Sections 7 Open questions, 8 Concluding remarks and future prospects.