General Non-Local Continuum Mechanics: Derivation of Balance Equations
Abstract
:1. Introduction
- 1.
- The first approach is based on the use of various integral and integro-differential operators of general form, which are not self-consistent with each other and do not form a calculus. Eringen’s book [5] and Rogula’s work [6] can be attributed to this approach. As an example, it should also be noted the works [7,8,9] that described continuous media with non-locality in time. Note some modern reviews of various aspects of nonlocal mechanics in [10,11,12,13,14]. The models of non-local media described in these reviews do not assume that integral and integro-differential operators form a calculus.It should be emphasized that the so-called differential (gradient) models, which are based on the differential equations of integer orders, cannot be considered as a tool for describing non-local media from a mathematical point of view. This is due to the well-known fact that integer-order differential equations are defined in an infinitesimally small neighborhood of the point under consideration and, therefore, are a tool for describing only local media. However, differential (gradient) models can be used to obtain some corrections to standard local continuum models by taking into account derivatives of the next order.From a mathematical point of view, the lack of mutual consistency between the integral and integro-differential operators, which do not form a calculus, leads to the absence of a non-local analogue of vector calculus for such operators. At the same time, the importance of vector calculus as a tool for describing continuous media within the framework of standard local mechanics is obvious. As will be proved in this article, the fundamental theorems of calculus, which describe the relationship between differential and integral operators, make it possible to derive balance equations.The advantage of the first approach is the use of a wide class of operator kernels in integral and integro-differential operators. However, the price for such generality is a narrower class of problems in mechanics, for which explicit solutions can be obtained for the equation of nonlocal continuum.
- 2.
- The second approach is based on the use of integral and integro-differential operators with the power-law kernels and forming a calculus, which is called fractional calculus (FC). These operators are usually called fractional integrals and derivatives of non-integer orders (see books [15,16,17,18,19] and Volumes 1 and 2 of Handbook [20,21]). The FC has a very long history [15,22,23,24,25,26], and this calculus has a wide application in mechanics and physics (for example, see [27,28,29,30,31,32,33,34,35], Volumes 3 and 4 of Handbook [36,37] and reviews [38,39]). The theory of integral and differential equations of non-integer-orders is powerful tool to describe the media and processes with nonlocality in space and time.First time the fractional derivatives with respect to space coordinates have been applied to nonlocal elastic continuum by Gubenko [40,41] in 1957. Then, the fractional calculus (FC) began to be actively used to describe continuum with power-law type of non-locality. Note some basic models and results in application of FC to nonlocal continuum mechanics: (1) Carpinteri, Cornetti and Sapora [42,43,44] consider nonlocal elastic continua modelled by a fractional calculus approach; (2) Cottone, Di Paola, Zingales and Marino, Failla [45,46,47,48,49] proposed fractional mechanical models for the dynamics of non-local continuum; (3) Drapaca and Sivaloganathan [50] describe a fractional model of continuum mechanics; (4) Challamel, Zorica, Atanackovic and Spasic [51] proposed a fractional generalization of Eringen’s nonlocal elasticity for wave propagation; (5) connection between continuous and lattice nonlocal models is proved in [27,52,53] by continuous limit of discrete systems with long-range interaction; (6) Tarasov, Zaslavsky, Edelman, Korabel [54,55,56,57] describe fractional dynamics of media with long-range interaction; (7) nonlocal model with power-law spatial dispersion in electrodynamics [58] and continuum mechanics [59,60,61,62,63] are described; (8) Yu and Zhai [64] proposed fractional Navier-Stokes equations with power-law nonlocality.Vector calculus plays an important role in describing continuous media. For the first time, fractional vector calculus (FVC) that allows taking into account nonlocality was proposed in 2008 in the work [65] (see also Chapter 11 in book [27]). The fractional vector calculus was proposed in the form of self-consistent formulation, in which differential and integral vector operators are satisfy the vector analogs of fundamental theorems. The fractional generalizations of the Green’s, Stokes’, and Gauss’s theorems are formulated and proved in [27,65]. In this nonlocal vector calculus, the Caputo fractional derivatives and Riemann-Liouville fractional integrals are used to take into account power-law spatial non-locality. Then, after 2008, other works, which consider special aspects of self-consistent formulations of the FVC, are published by Bolster, Meerschaert and Sikorskii in 2012 [66]; D’Ovidio and Garra in 2014 [67]; Tarasov in 2014, 2015 [68,69]; Ortigueira, Rivero and Trujillo in 2015 [70]; Agrawal and Xu in 2015 [71]; Ortigueira and Machado in 2018 [72]; (g) Cheng and Dai in 2018 [73]. Unfortunately, almost all works were devoted to only one type of nonlocality and mainly to power-law nonlocality.In the framework of the second approach, there are many fractional differential and integral equations that have exact and approximate solutions that describe various problems of non-local mechanics. This fact is an important advantage of the second approach in comparison with the first approach to nonlocal mechanics. Despite this advantage, the second approach has a serious disadvantage. In the models used in the framework of the second approach, only media and materials with power-law nonlocality in space and time were considered.
- 3.
- The third approach, which is proposed in this article, is designed to combine the advantages of the first and second approaches. To describe a wide class of nonlocal continua and media, it is important to use a wide class of operator kernels for which integral and integro-differential operators would form a calculus. Self-consistency of integral and differential operators is provided by non-local analogues of the first and second fundamental theorems of some general calculus. It is also important to have a non-local analogue of the vector calculus, in which non-local analogues of the theorems of standard vector calculus would hold.The possibility of formulating the third approach to nonlocal continuum mechanics is based on the general fractional calculus. The term general fractional calculus has been suggested in article [74] by Kochubei in 2011 (see also [75,76,77]). In works [74,75], the concepts of general fractional derivative (GFD) and general fractional integral (GFI) are proposed and the fundamental theorems of the GFC were proved. The GFC is based on the concept of kernel pairs, which was proposed by Sonin in 1884 work [78,79] (note that sometimes a French transliteration of his surname as “Sonine” is used instead of English transliteration [80]).In this paper, it is proposed to use the Luchko form of the general fractional calculus (GFC) to formulate a continuum mechanics of general nonlocality in space and time. This very important form of the GFC was proposed by Luchko in 2021 [81,82,83] (see also [84,85]). In works [81,82], the GFD and GFI are suggested and the general fundamental theorems of the GFC are proved.Then, in work [86], a general fractional vector calculus (GFVC) was proposed based on the general fractional calculus in the form of Luchko as a generalization of the approach that is suggested in paper [65] (see also Chapter 11 in book [27] (pp. 241–264)). The GFVC allows us to formulate continuum mechanics with general form of spatial nonlocality. This approach is proposed to be used to derive the balance equations for general non-local media in this article.It should be noted that in 2021 a non-local vector calculus was also proposed by D’Elia, Gulian, Olson and Karniadak [87] as a generalization of the Meerschaert, Mortensen and Wheatcraft approach to FVC. However, this calculus is not related to the general fractional calculus.
- (a)
- To derive balance equations in general integral form, the second fundamental theorem of the GFC is used;
- (b)
- The fractional analogue of the Titchmarsh theorem and the first fundamental theorem of the GFC are used to derive balance equation of general nonlocal continuum in the general fractional differential form from the GF integral form;
- (c)
- Using the General FVC, the general fractional differential equations for conservation of mass, momentum, and energy are proposed for a wide class of regions and surfaces in general nonlocal media.
2. General Fractional Integrals and Derivatives
2.1. Definitions of GFI and GFD
2.2. Examples of Kernel Pairs
2.3. General Fractional Integral and Derivative for
2.4. General Fractional Analogue of Titchmarsh Theorem
2.5. Triple GFI
2.6. Surface GFI
2.7. General Fractional Divergence
2.8. General Fractional Gauss Theorem for Z-Simple Region
3. Concepts of General Nonlocal Continuum
3.1. General Density Function
3.2. General Nonlocal Continuum
3.3. Nonlocality of Mass in Continuum
- (A)
- First, the standard derivation method that applies to local media is valid for flux fields, in which flux changes are small and linear or piecewise linear within the fixed region W.
- (B)
- The size of the fixed area and the scale of the measurements must be large compared to the scale of the heterogeneity in the medium. These restrictions are necessary due to the fact that changes in the flow in the fixed region W can be approximated by the Taylor series with derivatives of first orders.
- (1)
- First, for local media, it is quite logical that the size of a fixed region tends to zero, and thus an infinitely small region of medium can be used. However, for nonlocal media, this procedure is not entirely logical.
- (2)
- Second, the coefficients of the Taylor series are described as fractional derivatives from a to some upper limit, which tends to a even for finite fixed regions. This leads to the fact that the mass balance equation for a nonlocal medium should be described only by fractional derivatives on the infinitesimal interval with . A possible approach to solve this problem is proposed in [92] for power-law nonlocality. If using the Taylor series to express in terms of fractional derivatives on a finite interval. Then, the remainder (more precisely, the difference between the two remainder terms) will not tend to zero and these terms cannot be neglected (see Remark 3).
- (3)
- Third, the equations were obtained only for the power-law type of nonlocality of the medium. In addition to this, it should also be emphasized that in article [90] other laws of conservation of nonlocal media were not derived.
4. General Nonlocal Continuity Equation
4.1. Function Spaces for Derivation of Balance Equations
4.2. Mass of Nonlocal Continuum
4.3. Mass Flow of Nonlocal Continuum
4.4. Mass Balance Equation
4.5. General FVC Form of Mass Balance Equation
5. General Fractional Equation for Momentum
- (a)
- the transfer of momentum across the boundary;
- (b)
- the momentum of surface forces;
- (c)
- the momentum of mass forces.
5.1. Momentum Transfer across Boundaries
5.2. The Momentum of the Mass Force
5.3. The Momentum of the Surface Force
5.4. General Fractional Momentum Equation
5.5. GF Divergence of a Dyadic Product
5.6. General FVC Form of Momentum Balance Equation
5.7. General Fractional Equilibrium Equations for Stresses
6. General Fractional Equation for Total Energy
- (a)
- the energy transport across the boundaries of the region ;
- (b)
- the work of the mass forces ;
- (c)
- the work of the surface forces ;
- (d)
- the heat flux through the boundary (internal heat sources are not considered).
6.1. Change of Total Energy Due to the Transfer across Boundaries
6.2. Work of Mass Forces
6.3. Work of Surface Forces
6.4. Change of Total Energy by the Heat Flow
6.5. Change of Total Energy of All Sources
6.6. General FVC Form of Energy Balance Equation
6.7. Spatial Power-Law Nonlocality without Memory
7. Conclusions
- To derive general balance equation in the GF integral form, the second fundamental theorems (Theorems 2 and 3) of the GFC is used;
- The first fundamental theorems (Theorems 1 and 4) of the GFC and the proposed fractional analogue of the Titchmarsh theorem (Theorem 6) are used to derive differential form of general balance equations from the integral form of balance equations;
- Using the general fractional vector calculus, the balance equations are suggested for a wide class of regions and surfaces of the general nonlocal continuum.
- The mass balance equation for continuum with general space and time nonlocality:
- −
- The mass balance equation in the GF integral form for general nonlocal continuum is given by Theorem 9 for regions in the form of a parallelepiped.
- −
- The mass balance equation in the GF differential form for general nonlocal continuum is given by Theorem 10 for regions in the form of a parallelepiped.
- −
- Using the general fractional vector calculus, the mass balance equation in GF differential form is described by Equation (133) for a wide class of regions and surfaces.
- The momentum balance equation for continuum with general space and time nonlocality:
- −
- The momentum balance equation in the GF integral form for general nonlocal continuum is given by Theorem 11 for regions in the form of a parallelepiped.
- −
- The momentum balance equation in the GF differential form for general nonlocal continuum is given by Theorem 12 for regions in the form of a parallelepiped.
- −
- Using the general fractional vector calculus, the momentum balance equation in GF differential form is described by Equation (195) for a wide class of regions and surfaces.
- The energy balance equation for continuum with general space and time nonlocality:
- −
- The energy balance equation in the GF integral form for general nonlocal continuum is given by Theorem 13 for regions in the form of a parallelepiped.
- −
- The energy balance equation in the GF differential form for general nonlocal continuum is given by Theorem 14 for regions in the form of a parallelepiped.
- −
- Using the general fractional vector calculus, the energy balance equation in GF differential form is described by Equation (282) for a wide class of regions and surfaces.
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
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Kernel of GFI | Kernel of GFD : |
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Tarasov, V.E. General Non-Local Continuum Mechanics: Derivation of Balance Equations. Mathematics 2022, 10, 1427. https://doi.org/10.3390/math10091427
Tarasov VE. General Non-Local Continuum Mechanics: Derivation of Balance Equations. Mathematics. 2022; 10(9):1427. https://doi.org/10.3390/math10091427
Chicago/Turabian StyleTarasov, Vasily E. 2022. "General Non-Local Continuum Mechanics: Derivation of Balance Equations" Mathematics 10, no. 9: 1427. https://doi.org/10.3390/math10091427
APA StyleTarasov, V. E. (2022). General Non-Local Continuum Mechanics: Derivation of Balance Equations. Mathematics, 10(9), 1427. https://doi.org/10.3390/math10091427