Research Article | Open Access
Fractional Gradient Elasticity from Spatial Dispersion Law
Nonlocal elasticity models in continuum mechanics can be treated with two different approaches: the gradient elasticity models (weak nonlocality) and the integral nonlocal models (strong nonlocality). This paper focuses on the fractional generalization of gradient elasticity that allows us to describe a weak nonlocality of power-law type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. We demonstrate how the continuum limit transforms the equations for lattice with this spatial dispersion into the continuum equations with fractional Laplacians in Riesz's form. A weak nonlocality of power-law type in the nonlocal elasticity theory is derived from the fractional weak spatial dispersion in the lattice model. The continuum equations with derivatives of noninteger orders, which are obtained from the lattice model, can be considered as a fractional generalization of the gradient elasticity. These equations of fractional elasticity are solved for some special cases: subgradient elasticity and supergradient elasticity.
The theory of derivatives and integrals of noninteger orders [1–3] allows us to investigate the behavior of materials and media that are characterized by nonlocality of power-law type. Fractional calculus has a wide application in mechanics and physics (e.g., see [4–14]). Nonlocal elasticity theories in continuum mechanics can be treated with two different approaches : the gradient elasticity theory (weak nonlocality) and the integral nonlocal theory (strong nonlocality). The fractional calculus allows us to formulate a fractional generalization of nonlocal elasticity models in two forms: the fractional gradient elasticity models (weak power-law nonlocality) and the fractional integral nonlocal models (strong power-law nonlocality). The idea to include some fractional integral term in the equations of the elasticity has been proposed by Lazopoulos in . Fractional models of integral nonlocal elasticity are considered in different papers; see, for example, [16–22]. The microscopic models of fractional integral elasticity are also described. For this reason, the fractional integral elasticity models are not discussed here.
This paper focuses on the fractional generalization of gradient elasticity which describes a weak nonlocality of power type. We suggest a lattice model with spatial dispersion of power-law type as a microscopic model of fractional gradient elastic continuum. Complex lattice dynamics has been the subject of continuing interest in the theory of elasticity. As it was shown in [23, 24] (see also [25–27]), the equations with fractional derivatives can be directly connected to lattice models with long-range interactions. In this paper, we consider models of lattices with spatial dispersion and its continuous limits. We define a map of lattice models into continuum models. A connection between the dynamics of lattice system of particles with long-range interactions and the fractional continuum equations is proved by using the transform operation [23, 24]. We make the transformation to the continuous limit and derive the fractional equation, which describes the dynamics of the nonlocal elastic materials. We show how the continuous limit for the lattice with fractional weak spatial dispersion gives the corresponding continuum equation of the fractional gradient elasticity. The continuum equations of fractional elasticity are solved for some special cases: subgradient elasticity and supergradient elasticity.
2. Lattice Equations
The lattice is characterized by space periodicity. In an unbounded lattice, we can define three noncoplanar vectors , , , such that displacement of the lattice by the length of any of these vectors brings it back to itself. The vectors , , are the shortest vectors by which a lattice can be displaced and be brought back into itself. As a result, all spatial lattice points can be defined by the vector , where are integer. If we choose the coordinate origin at one of the sites, then the position vector of an arbitrary lattice site with is written: In a lattice, the sites are numbered in the same way as the particles, so that the vector is at the same time “number vector” of a corresponding particle.
We assume that the equilibrium positions of particles coincide with the lattice sites . A lattice site coordinate differs from the coordinate of the corresponding particle, when particles are displaced relative to their equilibrium positions. To define the coordinates of a particle, it is necessary to indicate its displacement with respect to its equilibrium positions. We denote the displacement of a particle with vector from its equilibrium position by the vector field .
The equation of motion of lattice particle is where is the mass of particle and are components of the external on-site force. The italics are the coordinate indices. We assume the summation over doubly repeated coordinate indices from 1 to 3. The coefficients describe the interparticle interaction in the lattice. For simplicity, we assume that all particles have the same mass .
It is easy see one important property of the coefficients . Assume the lattice to be displaced as a whole: . Then the internal lattice state cannot be changed in case of absence of external forces. As a result, (2) gives These conditions should be satisfied for any particle in the lattice, that is, for any vector . Equation (3) follows from the conservation of total momentum in the lattice.
For an unbounded homogeneous lattice, due to its homogeneity the matrix has the form where elements of of (2) are satisfied by condition In a simple lattice, each particle is an inversion center, and we have Using condition (5), we can represent (2) in the form This equation of motion has the invariance with respect to its displacement of lattice as a whole in case of absence of external forces even if condition (5) is not satisfied. It should be noted that the noninvariant terms lead to the divergences in the continuous limit .
Equation of motion (7) is equation for three-dimensional displacement vectors. In this paper, we will use the simplest model to describe the lattice, where all particles are displaced in one direction; we assume that the displacement of particle from its equilibrium position is determined by a scalar rather than a vector. This model allows us to describe the main properties of the lattice using simple equations.
The equations of motion for one-dimensional lattice system of interacting particles have the form where we use the summation condition over repeated indexes. Here are displacements from the equilibrium, is the coupling constant for interparticle interactions in the lattice, and the term characterizes an interaction of the particles with the external on-site force.
3. Transform Operations for Lattice Equations
Let us define the operation that transforms the lattice equations for into the continuum equation for a scalar field . In order to obtain continuum equation from the lattice equations, we assume that are Fourier coefficients of some function . We define the field on by the equation where and is the interparticle distance. For simplicity, we assume that all particles have the same interparticle distance . Equations (9) can be used to obtain the Fourier transform in the limit (). Then change the sum to an integral, and (9) become We replace the discrete function by continuous field considering . We assume that , where denotes the passage to the limit (). Here is a Fourier transform of the field and is a Fourier series transform of , where we use . The function can be derived from in the limit .
As a result, we define the map from a lattice model into a continuum model by the transform operation , which is the combination [23, 24] of the following operations:(1)the Fourier series transform: (2)the passage to the limit : (3)the inverse Fourier transform:
The similar transformations can be performed for differential equations to map the lattice equation into an equation for the elastic continuum. Therefore, the operation allows us to realize transformation of lattice models of interacting particles into continuum models [23, 24].
Let us consider the Fourier series transform of the interaction term.
Proposition 1. Let be such that conditions hold. Then the Fourier series transform maps the term where is a position of the th particle, into the term where
Proof. To derive the Fourier series transform of the interaction term (16), we multiply (16) by and sum over from to . Then Using conditions (15), we introduce the notations Using , the function (20) can be represented by From (22), we can see that is a periodic function where is an integer. Using (21) and (20), the first term on the right-hand side of (19) gives Here we use (15) and and Using , the second term on the right-hand side of (19) has the form Equations (24) and (26) give the expression where is defined by (20).
Let us give the statement that describes the Fourier transform of the lattice equations.
Proposition 2. The Fourier series transform maps the lattice equations of motion where satisfies conditions (15), into the continuum equation where , , and is an operator notation for the Fourier series transform.
Proof. To derive the equation for the field , we multiply (28) by and sum over from to . Then Using (21), the left-hand side of (30) has the form The second term of the right-hand side of (30) is The Fourier series transform maps the interaction term (16) into expression (17). As a result, we obtain (30) in the form (29), where is an operator notation for the Fourier series transform of .
4. Fractional Weak Spatial Dispersion
4.1. Weak Spatial Dispersion
Spatial dispersion is the dependence of on the wave vector that leads to nonlocal properties of the continuum. The spatial dispersion gives nonlocal connection between the stress tensor and the strain tensor . The tensor at any point of the continuum is not uniquely defined by the values of at this point. It also depends on the values of at neighboring points , located near the point .
A nonlocal constitutive relation between the stress and the strain can be understood on the basis of analysis of a lattice model. The particles of the lattice oscillate about their equilibrium positions and interact with each other. The equations of oscillations of the lattice particles with the local (nearest-neighbor) interaction give the partial differential equation of integer orders in the continuum limit [23, 24]. Note that the lattice with nonlocal (long-range) interactions in the continuous limit can give fractional partial differential equations for nonlocal continuum [23, 24].
Qualitatively describing the process we can say that the fields of the elastic wave move particles from their equilibrium positions at a given point , which causes an additional shift of the particles in neighboring and more distant points in some neighborhood. Therefore, the properties of the medium, and hence the stress tensor field , depend on the values of strain tensor field not only in a selected point but also in its neighborhood.
The size of the area in which the kernel is significant is determined by the characteristic length of interaction . The size of the area is usually of the order of the lattice constant. Wavelength of elastic wave is several orders larger than the size of this region, so the values of the field of elasticity wave do not change for a region of size . In other words, the wavelength usually holds . In such lattice, the spatial dispersion is weak. To describe the lattice dynamics, it is enough to know the dependence of the function only for small values and we can replace this function by the Taylors polynomial series. For an isotropic linear medium, we use Here we neglect a frequency dispersion, and so , , do not depend on the frequency .
4.2. Fractional Taylor Series Approach
The weak spatial dispersion in the media with power-law type of nonlocality cannot be described by the usual Taylor approximation. The fractional Taylor series is very useful for approximating noninteger power-law functions . For example, the usual Taylor series for the nonlinear power-law function has infinitely many terms for noninteger .
If we use the fractional Taylor's formula (see Appendix A) we get a finite number of terms. For example, Taylor's series in the Odibat-Shawagfeh form that contains the Caputo fractional derivative has two terms for (34). Using for the case , we get As a result, we have and the fractional Taylor’s series approximation of function (34) is exact.
4.3. Weak Spatial Dispersion of Power-Law Types
We consider properties of the lattice with weak spatial dispersion that is described by the function of a noninteger power-law type. In the continuous limit this model gives a model of continuum with power-law nonlocality.
The weak spatial dispersion will be called -type, if the function (38) satisfies condition where and . The weak spatial dispersion (and the interparticle interaction in the lattice) will be called -type, if the function satisfies conditions (40) and where and .
Similarly we define the weak spatial dispersion and the interaction in the lattice of the -type. For the weak spatial dispersion of the -type, the function can be represented in the form where and As a result, we can use the following approximation for weak spatial dispersion:
If for all , we can use the usual Taylor’s formula. In this case, we have the usual case of the weak spatial dispersion. In general, we should use a fractional generalization of Taylor's series (see Appendix A). If the orders of the fractional Taylor series approximation will be correlated with the type of weak spatial dispersion, then the fractional Taylor series approximation of will be exact. In the general case , we can use the fractional Taylor’s formula in the Dzherbashyan-Nersesian form (see Appendix A). For the special cases , where and/or , we could use the other kind of the fractional Taylor’s formulas.
5. Fractional Gradient Elasticity Equation for Continuum
In the continuous limit the equation for lattice with the interaction of the -type gives the equation for continuum of the fractional gradient model.
Proposition 3. In the continuous limit the lattice equation of motion with the weak spatial dispersion of the -type gives the fractional continuum equation of the form where is the fractional Laplacian of order in Riesz’s form (see Appendix B), the variables and are dimensionless, , , and are finite parameters.
Proof. The Fourier series transform of (45) gives (29). After division by the cross-sectional area of the medium and the interparticle distance , the limit for (29) gives
where is the mass density, is the interparticle distance, , and
Here we use (44), and () are finite parameters that are defined by (47). Note that satisfies the condition
The expression for can be considered as a Fourier transform of the interaction term (see Proposition 1). Note that for the limit , if are finite parameters.
In the limit , (48) gives where The inverse Fourier transform of (51) has the form where Here, we use the connection between the Riesz fractional Laplacian and its Fourier transform (see Appendix B and [1–3]): in the form Substitution of (54) into (53) gives the continuum equation (46).
Equations (46) and (47) give the close relation between the discrete microstructure of lattice with weak spatial dispersion of power-law type and the fractional gradient models of weak nonlocal continuum.
Let us consider the special case for integer . If the function has the form then we get the well-known equation Here where is Young’s modulus, is the spring stiffness, and is the mass density.
If we can use the spatial dispersion law in the form then we have the equation of the gradient elasticity as where , The scale parameter of the gradient elasticity is connected with the coupling constants of the lattice by the equation The second-gradient term is preceded by the sign that is defined by .
Similarly, we can consider more general model of lattice with fractional weak spatial dispersion of -type Then the continuum equation for fractional gradient model has the form where we use new notation for the constants, . Note that and are dimensionless.
6. Solution of Fractional Gradient Elasticity Equation
6.1. Plane Wave Solution
Let us consider the plane waves . Then (65) gives
The fractional analog of the Green function (see Section 5.5.1. in ) is given by where .
The following relation holds (see Lemma 25.1 of [1, 2]) for any suitable function such that the integral in the right-hand side of (70) is convergent. Here is the Bessel function of the first kind. As a result, the Fourier transform of a radial function is also a radial function.
Using relation (70), the Green function (69) can be represented (see Theorem 5.22 in ) in the form of the integral with respect to one parameter: where and and is the Bessel function of the first kind.
For the 3-dimensional case, we use Then we have For the 1-dimensional case, we use Then we have (see Theorem 5.24 in  pages 345-346) the function
If and , then (66) (see, e.g., Section 5.5.1. pages 341–344 in ) has a particular solution . Such particular solution is represented in the form of the convolution of the functions and as follows: where the Green function is given by (71).
In 3-dimensional case, the function does not depend on the angles. Therefore, we can use the spherical coordinates and then reduce the integration in (76) to by integrating with respect to the angles where and .
6.2. Static Solution
Let us consider the statics (, i.e., ) in the suggested fractional gradient elasticity model. We can consider the fractional partial differential equation (66) with and , when , and also the case where , , , , , , which is given by Equation (78) has the following particular solution (see Theorem 5.23 in ) that is represented in the form of the convolution of the functions as with the Green function where and .
These particular solutions allow us to describe static fields in the elastic continuum with the weak spatial dispersion of -type.
7. Fractional Weak Spatial Dispersion of -Type
7.1. Fractional Gradient Elasticity Equation for Dispersion of -Type
If we have the dispersion law in the form where , , and , then we have the fractional gradient elasticity equation where If and , we have the well-known equation of the gradient elasticity : where The second-gradient term is preceded by the sign that is defined by , where .
Equation (82) is the fractional partial differential equation (78) with , and such equation has the particular solution  of the firm where the Green type function is given by Here is the Bessel function of the first kind.
7.2. Point Load Problem for Fractional Gradient Elasticity
Let us consider point load problem for an infinite elastic continuum (see pages 25-26 in ) and determine a deformation of an infinite gradient continuum, when a force is applied to a small region in it. We consider this Thomson's problem for nonlocal elastic continuum with fractional weak spatial dispersion of the form (81). If we consider the deformation at distances , which are larger than the size of the region, then we can assume that the force is applied at a point. In this case, we have Then the displacement field of fractional gradient elasticity has a simple form of the particular solution (79) that is proportional to the Greens function where is given by (80). Therefore, the displacement field (86) for the force that is applied at a point (88) has the form
From a mathematical point of view, there are two special cases: fractional weak spatial dispersion of -type with and ; fractional weak spatial dispersion of -type with , , and .
From the point of view of the nonlocal elasticity theory, it is useful to distinguish the two following particular cases:(i)subgradient elasticity ( and ),(ii)supergradient elasticity ( and ).
Note that for the first case the order of the fractional Laplacian is less than the order of the first term related to the usual Hooke’s law. In the second case the order of the fractional Laplacian is greater than the order of the first term related to the Hooke law. The names of the sub- and supergradient elasticity caused by the analogy with the names of anomalous diffusion [6–8] such as subdiffusion and superdiffusion.
7.3. Subgradient Elasticity Model
The subgradient elasticity is characterized by the fractional weak spatial dispersion of -type with and . Fractional model of nonlocal continuum with this spatial dispersion is described by (82) with and , given by The order of the fractional Laplacian is less than the order of the first term related to the usual Hooke’s law. As a simple example, consider the square of the Laplacian; that is, .
As a result, the displacement field for the force that is applied at a point in the continuum with this type of fractional weak spatial dispersion is given by on the long distance .
7.4. Supergradient Elasticity Model
The supergradient elasticity is characterized by the fractional weak spatial dispersion of -type with and . For the nonlocal continuum with the weak spatial dispersion of the -type, where , and , the displacement field for the fractional gradient model which is described by (82) includes two parameters . As an example of the nonlocal continuum with this type of spatial dispersion, we highlight the case of supergradient elasticity, where and . In this case (82) has the form The order of the fractional Laplacian is greater than the order of the first term related to the Hooke law. If , (96) becomes (84). Therefore, the case can be considered as close as possible () to the usual gradient elasticity (84).
For the displacement field that is described by (82), where , and and the force is applied at a point (88), we have the following asymptotic behavior: We note that this asymptotic behavior does not depend on the parameter . The field on the long distances is determined only by term with () that can be interpreted as a fractional nonlocal “deformation” of Hooke's law.
We note the existence of a maximum for the function in the case .
The asymptotic behavior of the displacement field for is given by where we use Euler's reflection formula for Gamma function. The asymptotic relation (98) is not directly related to the supergradient case. Note that the above asymptotic behavior does not depend on the parameter , and relations (98)-(99) do not depend on . The displacement field on the short distances is determined only by term with () that can be considered as a fractional nonlocal “deformation” of the gradient term.
A lattice model with spatial dispersion of power-law type is suggested. Gradient elasticity is considered as a phenomenological theory representing continuum limit of lattice dynamics, where the length scales are much larger than interatomic distances. In the continuum limit we derive continuum equations with spatial derivatives of noninteger order . The correspondent continuum equations describe fractional generalization of gradient elasticity (the supergradient elasticity model) for and a special form of fractional integral elasticity (the subgradient elasticity model) for . The suggested lattice model with spatial dispersion can be considered as a microscopic basis for the fractional nonlocal elastic continuum. We can note that a fractional nonlocal continuum model can be obtained from different microscopic or lattice models [32, 33]. The main advantage of the suggested approach is that we can use the Taylor series in the wave vector space instead of Taylor expansion in a coordinate space. It allows us to use these models as a microstructural basis of unified description of fractional (and integer) gradient models with positive and negative signs of the strain gradient terms. The suggested approach can be generalized for three-dimensional case of gradient elasticity. The proposed lattice model can also be easily generalized for the case of the high-order gradient elasticity and the correspondent fractional extension by using the next terms of fractional Taylor series. The suggested lattice models with long-range interactions can be important to describe the nonlocal elasticity of materials at microscale and nanoscales [34–36], where the interatomic and intermolecular interactions are prevalent in determining the properties of these materials.
A. Fractional Taylor Formula
A.1. Riemann-Liouville and Caputo Derivatives
The left-sided Riemann-Liouville derivatives of order are defined by We can rewrite this relation in the form where is a left-sided Riemann-Liouville integral of order :
The Caputo fractional derivative of order is defined by where is a left-sided Riemann-Liouville integral (A.3) of order . In (A.16) we use and . The main distinguishing feature of the Caputo fractional derivative is that, like the integer order derivative, the Caputo fractional derivative of a constant is zero.
Note also that the third term in (A.16) involves the fractional derivative of the fractional derivative, which is not the same as the fractional derivative. In general, Then the coefficients of the fractional Taylor series can be found in the usual way, by repeated differentiation. This is to ensure that the fractional derivative of order of the function is a constant. The repeated fractional derivative of order gives zero. Then the coefficients of the fractional Taylor series can be found in the usual way, by repeated differentiation.
A.2. Fractional Taylor's Series in the Riemann-Liouville Form
Let be a real-value function such that the derivative is integrable. Then the following analog of Taylor formula holds (see Chapter 1. Section 2.6 [1, 2]): where are left-sided Riemann-Liouville derivatives and
A.3. Riemann Formal Version of the Generalized Taylor’s Series
A.4. Fractional Taylor's Series in the Trujillo-Rivero-Bonilla Form
The Trujillo-Rivero-Bonilla form of generalized Taylor's formula  is where and
A.5. Fractional Taylor's Series in the Dzherbashyan-Nersesian Form
Let be increasing sequence of real numbers such that
We introduce the following notation [41, 42] (see also Section 2.8 in [1, 2]): In general, . Fractional derivative differs from the Riemann-Liouville derivative by finite sum of power functions since (see equation (2.68) in )
A.6. Fractional Taylor's Series in the Odibat-Shawagfeh Form
The fractional Taylor series is a generalization of the Taylor series for fractional derivatives, where is the fractional order of differentiation, . The fractional Taylor series with Caputo derivatives  has the form where is the Caputo fractional derivative of order .
B. Riesz Fractional Derivatives and Integrals
Fractional integration and fractional differentiation in the -dimensional Euclidean space can be defined as fractional powers of the Laplace operator. For and “sufficiently good” functions , , the fractional Laplacian in Riesz's form (the Riesz fractional derivative) is defined in terms of the Fourier transform by The Riesz fractional integration is defined by
The Riesz fractional integration can be realized in the form of the Riesz potential defined as Fourier's convolution of the form where the function is the Riesz kernel. If and , the function is defined by If , then The constant has the form
Obviously, the Fourier transform of the Riesz fractional integration is given by This formula is true for functions belonging to Lizorkin's space. The Lizorkin space of test functions on is a linear space of all complex-valued infinitely differentiable functions whose derivatives vanish at the origin: where is the Schwartz test-function space. The Lizorkin space is invariant with respect to the Riesz fractional integration. Moreover, if belongs to the Lizorkin space, then where and .
For , the fractional Laplacian in Riesz's form can be defined in the form of the hypersingular integral by where and is a finite difference of order of a function with a vector step and centered at the point :