Stresses and strains in a deformable fractal medium and in its fractal continuum model

Physics Letters A 377 (2013) 2535–2541

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Physics Letters A www.elsevier.com/locate/pla

Stresses and strains in a deformable fractal medium and in its fractal continuum model Alexander S. Balankin Grupo “Mecánica Fractal”, ESIME-Zacatenco, Instituto Politécnico Nacional, México DF 07738, Mexico

a r t i c l e

i n f o

Article history: Received 13 June 2013 Accepted 16 July 2013 Available online 22 July 2013 Communicated by A.R. Bishop Keywords: Fractal materials Fractal continuum mechanics Constitutive equations

a b s t r a c t The model of fractal continuum accounting the topological, metric, and dynamic properties of deformable physical fractal medium is suggested. The kinematics of fractal continuum deformation is developed. The corresponding geometric interpretations are provided. The concept of stresses in the fractal continuum is defined. The conservation of linear and angular momentums is established. The mapping of mechanical problems for physical fractal media into the corresponding problems for fractal continuum is discussed. © 2013 Published by Elsevier B.V.

1. Introduction Most natural and engineering materials are inherently heterogeneous due to the presence of microstructure [1]. In the past two decades it was recognized that microstructures of real heterogeneous materials frequently possess formidably complicated architecture characterized by statistical scale invariance over many length scales [1–7]. For such materials the classical approximation of homogeneous Euclidean continuum is inapplicable, because the heterogeneities play an important role on almost all scales. At the same time, the fractal geometry offers helpful scaling concepts to characterize and model the scale invariant structures of heterogeneous media [1–30]. While there is no canonical definition of fractals, mathematically a fractal is commonly viewed as an object the metric dimension of which D (e.g., Hausdorff, Minkowski, self-similarity, etc.) is larger than its topological dimension d [2], except of special cases such as the Hilbert space-filling curves with D = d [31]. It is obvious that fractals with d  D < n cannot continuously fill the embedding Euclidean space E n . Consequently, the properties of fractal structures are essentially discontinuous non-differentiable functions of the Euclidean coordinates in E n [32]. Accordingly, to deal with fractal materials, it was suggested the concept of fractal continuum [33] the overall properties of which are defined as the analytic envelopes of non-analytic functions characterizing the fractal ΦnD ⊂ E n under study [32–45]. In this way, the fracn tal continuum Φ D ⊂ E n can be defined as n-dimensional region of E n equipped with appropriate fractional metric, measure, and

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vector differential calculus, such that its properties (density, displacements, velocities, etc.) are describable by the continuous (or, at worst, piecewise continuous) differentiable functions of space and time variables [42,43]. It should be emphasized that in contrast to fractals, the topological dimension of fractal continuum Φ Dn ⊂ E n is, per definition, equal to the dimension of the embedding Euclidean space, that is dFC = n > D. This immediately implies n that the density of admissible states in Φ D ⊂ E n should be scale dependent [44,45]. Although the measure of fractal continuum can be fixed by a 3 natural requirement that the mass of any region W ⊂ Φ D of characteristic size L should display the same scaling behavior as the fractal medium under study, that is M ( W ) ∝ L D [33], there are very different ways to define the metric and fractional calculus in the fractal continuum. Accordingly, quite different models of fractal continua were suggested to aboard the problems of mechanics and electrodynamics in fractal media [33–46]. These different models lead to quite different solutions of the same problems for the modeled fractal medium. Nonetheless that the strength of fractal continuum models can be ranked by its abilities to explain and predict the results of experimental studies, the mathematical and physical self-consistency of the model should be assured before its applications to a specific problem. The mechanics of deformable medium cannot be deduced from the laws of mechanics of material points and rigid bodies. Hence, additional assumptions are needed to introduced, such that new notions of internal and external forces, stresses, and the equilibrium equation should emerge. In this context, the mechanical behavior of fractal media has topological and geometrical aspects which should be accounted within the fractal continuum framework. The fractal (mass or metric) dimension D characterizes how

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the extensive (e.g. mass) and intensive (e.g. density, ρ ∝ L n− D ) properties of heterogeneous medium scale with system size in E 3 , but it tells us nothing about the connectivity and topological properties of the fractal, such that fractals of different topology and connectivity can have the same mass (metric) dimension [11]. Therefore, to account the fractal topology of medium one needs to n endow the fractal continuum model Φ D ⊂ E n with additional appropriate dimensional numbers. In this Letter, we suggest the model of deformable fractal continuum accounting the topology and metric of fractal material. The model is used to develop the fractal continuum mechanics of heterogeneous materials with scale-invariant (micro-)structures. Fig. 1. Mapping of essentially discontinuous Menger sponge Φ3D ⊂ E 3 (d f = d =

d

3 2. Fractal continuum αs Φ D ⊂ E3

d

3 D = ln 20/ ln 3) into the fractal continuum αs Φ D ⊂ E 3 with D = ln 20/ ln 3, D S = ln 8/ ln 3, and ζ = ln 2.5/ ln 3 < α = d /3 = ln 20/ ln 27.

Although, in mathematics, fractals can be defined without any reference to the embedding space [47], in real life fractal materials reside in the three-dimensional space and occupy a well-defined volume V 3 in E 3 . Accordingly, the scaling properties of fractal pattern Φ3D ⊂ E 3 can be characterized by a set of fractional dimensionalities [16]. Most definitions of dimension numbers are based on the concept of fractal covering by balls (cubes, tubes, etc.) of some size ε , or at most ε . In mathematics these covers are considered in the limit ε → 0. At the same time, it was noted that, in many cases, the number of n-dimensional coats need to cover the mathematical fractal of linear size L in E 3 scales as N ∝ ( L /ε ) D . It is precisely this power-law behavior gives rise to use the powerful tools of fractal geometry to deal with physical patterns Φ3D ⊂ E 3 exhibiting statistical scale invariance only within a wide, but bounded interval of length scale ξ0 < ε  L < ξC , where ξ0 and ξC are the lower and upper cut-offs of the physical origin [48]. Hence, strictly speaking, physical fractals are closer to the concept of pre-fractals obtained after finite number of iterations, whereas the true fractal can be obtained in the limit of infinite number of iterations (ε → 0 while ξ0 = 0). To model the fractal medium within a continuum framework, in this work we define three-dimensional fractal contind 3 uum αs Φ D ⊂ E 3 as three-dimensional region of the embedding Euclidean space E 3 filled with continuous matter and endowed with appropriate fractional measure, metric, and norm, as well as a set of rules for integro-differential calculus and a proper Laplacian accounting the metric, connectivity and topological properties of the modeled fractal medium.

fractal medium is made, ρc is the density of fractal continuum (for example, in the case of porous fractal medium ρc = (1 − φ)ρ0 , while φ is the total porosity). Notice that ξ0 > 0 accounts the prefractal nature of physical fractal medium and so the corresponding fractal continuum should obey the scaling property (1) for L > ξ0 only. To account the topological properties of fractal medium, we noted that the connectivity and topology of the fractal Φ3D ⊂ E 3 can be specified by the chemical dimension d and the fractal (i ) dimensions D S of intersections between the fractal and the Cartesian planes in the embedding Euclidean space E 3 [42]. The chemical dimension quantifies how the “elementary” structural units of the (pre-)fractal structure are “glued” together to form the entire fractal object [16]. So, d tells us “how many directions” the observer feels in the configuration space by making static measurements. Therefore, d determines the minimal number of independent coordinates needs to define the point position in the fractal, in the same way as the topological dimension d determines the number of independent coordinates (e.g. the Cartesian coordinates) in the Euclidean manifold. Hence, the number of mutually orthogonal independent coordinates which can be defined in the fractal with d < 3 is less than 3. Although one can speak about the fractional number of coordinates [49], in a fractal with 2  d < 3 it is 3 more convenient to define two fractional coordinates χi , Ai ∈ Φ D , 3 such that the infinitesimal volume elements in Φ D ⊂ E 3 can be decomposed as

dV D = dχi (xi ) dAi (x j =i ), 2.1. The measure of fractal continuum Since the fractal continuum has topological dimension dFC = 3 > D we can use the conventional rules of Lebesgue integration in E 3 , whereas the fractal measure can be accounted via the definition of scale dependent density of admissible states c 3 (x) 3 in Φ D ⊂ E 3 . In this way, the mass of any cubic (or spherical) region 3 W ⊂ ΦD of the characteristic size L is assumed to scale as



M (W ) =





ρc c3 (xi ) dV 3 xi ∈ E 3

ρ0 dV F = W ∈Φ3D



ρc dV D = m0

=

(i )

fractal dimension D S , while dχi is the infinitesimal length element along the lines parallel to the Cartesian axis i = j , k normal to the Cartesian plane (x j , xk ). Accordingly, dV D can be decomposed (1 )

ξ0

(3 )

D ,

(1 )

(2 )

(3 )

(3 )

= c 1 dx3 c 2 d A 2 = c 3 dV 3 where d A 2

L

(1 )

(1)

3 W ∈Φ D

where dV F , dV 3 and dV D are the infinitesimal volume elements in the fractal medium Φ3D , Euclidean space E 3 , and fractal contin3 uum Φ D , respectively, xi are the Cartesian coordinates in E 3 (see Fig. 1), ξ0 is the characteristic size of elemental Euclidean components of mass m0 and mass density ρ0 from which the physical

(2 )

(2 )

dV D = dχζ dA D = c 1 dx1 c 2 d A 2 = c 1 dx2 c 2 d A 2 (i )

W ∈E3





(2)

where dAi is the infinitesimal area element on the intersection be3 tween Φ D ⊂ E 3 and the Cartesian plane (x j , xk ) ∈ E 3 having the

(3)

denote the infinitesimal area elements on two(i )

dimensional Cartesian planes and c 2 (x j =i ) is the density of ad3 and the Cartesian missible states on the intersection between Φ D (i )

plane normal to the i-axis, while c 1 (xi ) is the density of admissible states along lines parallel to the i-axis (see Fig. 1). From Eq. (3) immediately follows that (i )

(i )

c 3 (xk ) = c 1 (xi )c 2 (x j =i ),

(4)

but the functional form of c 3 (xk ) can be defined in the unique (1) (2) (3) way as c 3 = c 1 (x1 )c 1 (x2 )c 1 (x3 ), if and only if d = 3, such

A.S. Balankin / Physics Letters A 377 (2013) 2535–2541

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where 1−ζi

ζi x pi

Δi (x pi , xqi ) = ξ0

ζ − xqii ,

(11)

3 whereas the norm in α Φ D is defined as

R =

3 

1/2α x2Riα

(12)

,

i

3 Fig. 2. Definition of distance between two points P , Q ∈ Φ D ⊂ E3.

that the modeled fractal medium can be treated as the Cartesian γ γ γ product Φ3D = Φ1 1 × Φ1 2 × Φ1 3 ⊂ E 3 of three manifolds with the fractal dimensions γi  1. In this case the fractal medium is pathdisconnected (or totally discontinuous, if all γi < 1, as the Candor (i )

dusts [27]), while D S = γ j + γk and D = γ1 + γ2 + γ3 < 3. Controversially, in the case of path-connected fractals with d < 3 (k)

( j)

(i )

c 2 (xi , x j ) = c 1 (xi )c 1 (x j )

(5)

and so, (1 )

(2 )

(3 )

c 3 (x1 , x2 , x3 ) = c 1 (x1 )c 1 (x2 )c 1 (x3 ),

(6)

whereas the equalities (3) and (4) are hold. Therefore, to fulfill the “constitutive” requirement (1), the densities of admissible states should satisfy the following relations



( i )  −1 2 − D S



(i ) (i )

d A 2 c2 = D S



while

(i )

ξ0

,

1−ζi ζi

dxi c 1 = ζi−1 ξ0 (i )

L

(i )

DS

L ,

(7)

where the fractal dimension of coordinate dimension

χi is defined as the co-

ζi = D − D (S1)

(8)

3 of intersection Ai ∈ Φ D . The second relation of Eq. (7) immediately implies that in the Cartesian coordinates

(i )

1−ζi ζi −1 xi ,

c 1 = ξ0

(9) (i )

but the functional forms of c 2 and c 3 cannot be defined in the unique way. To this respect, it should be emphasized that the non(i ) uniqueness of c 2 and c 3 is an intrinsic feature of transformation 3 from two fractional coordinates (χi , Ai ) in Φ D into three Cartesian 3 coordinates in E and so, in essence, this is the price one should pay to deal with the path-connected fractal media with d < 3 (see Fig. 1).

x21 + x22 + x23 . To ac-

count the fractal topology together with the fractal metric of the fractal medium Φ3D under study, the distance between two points 3 P , Q ∈ α ΦD (see Fig. 2) can be defined as

Δ( P , Q ) =

3  i

1/2α Δ2i α (x pi , x pi )

,

Making use the fractional metric (10), (11) the local partial d derivative in αs Φ3D ⊂ E 3 can be defined as follows

∇iH

f = lim

xi →x i

f (x i ) − f (xi )

Δi (x i , xi )

(10)

 =

| xi | ξ0

1−ζi

∂ f, ∂ xi

(13)

where f (x j ) is a continuous differentiable function of x j ∈ E 3 , while ∂/∂ xi denotes the conventional partial derivative and the codimension ζi is defined by Eq. (8). It is imperative to point out that the local partial derivative (13) is governed by the fractal metric (11), but it is independent of the fractal continuum topology characterized by scaling exponent 0.5  α  1, even though the distance between two points in the fractal continuum depends on its topology (see Eq. (10)). Besides, it is pertinent to note that the local fractional derivative (10) resembles the Hausdorff derivative heuristically introduced in [53] and further modified in [41–43] to the form which can be obtained from (13) by the linear coordinate transformation |xi | → |xi | + ξ0 accounting the pre-fractal nature of Φ3D . Furthermore, it is a straightforward matter to verify that the Hausdorff derivative (13) is inverse to the following fractional integral



∇iH f dζ xi =

In Ref. [43] was defined the fractional metric in the fractal 3 continuum Φ D ⊂ E 3 which is consistent with the fractal measure 3 defined by Eqs. (3)–(8), but the norm in Φ D ⊂ E 3 was assumed to be the conventional Euclidean norm x =

2.3. Integro-differential calculus and Laplacian in the fractal continuum



2.2. The metric and norm in the fractal continuum



where the fractional dimension d f = 3α can be interpreted as the minimal dimension of fractional space F α (see Refs. [50–52]) in which the fractal α Φ3D can be embedded. It is a straightforward matter to verify that the distance defined by Eqs. (10) and (11) satisfies all conventional criteria required of metrics. The geometric interpretation of metric (11) is given in Figs. 1 and 2. Notice also that Eq. (12) satisfies the conventional requirements of the norm definition if α  0.5 and so d f > 3/2. Furthermore, the dynamical properties of fractal medium Φ3D are governed by the spectral dimension d s defined via the scaling relation Ω ∝ ωds −1 , where Ω(ω) is the density of fractal vibration modes with frequency ω [16]. The spectral dimension is therefore fundamental for any diffusive process, such as the random walk (RW). Specifically, it manages the probability P (0, s) ∝ sds /2 that random walker returns to the origin after s steeps within the fractal [16]. Hence the intrinsic time metric of diffusion in the fractal 1−β continuum can be defined as dτ = τ0 t β−1 dt, where τ0 the characteristic time associated with the pre-fractal nature of Φ3D and β = ds / D [43].

n xi ∈Φ D

xi ∈ E 3

(i )



(i )





c 1 ∇iH f dxi = xi ∈ E 3



∂f dxi , ∂ xi

(14)

where c 1 is given by Eq. (9), while x ∈ E 3 dxi denotes the coni ventional Lebesgue integral. Notice, that with this definition, the d operation of integration in αs Φ3D ⊂ E 3 differs from the known definitions of fractional integrals, although visually resembles some of them, e.g. the Riemann–Liouville fractional integral (see, for review, Refs. [29,45]). 3 The vector differential calculus in Φ D ⊂ E 3 based on the Hausdorff partial derivative was developed in [41–43]. It is a straightforward matter to understand that this calculus is also applicable

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A.S. Balankin / Physics Letters A 377 (2013) 2535–2541

d

in αs Φ3D ⊂ E 3 . Specifically, the Hausdorff del operator is defined as  H = e1 ∇ ζ +e 2 ∇ ζ +e 3 ∇ ζ , where ei ∈ E 3 are base vectors, whereas ∇ 1 2 3 the Green–Gauss divergence theorem reads as



f · n d A D =



div H f dV D ,

(15)

W

A

 = nk ek where f = f k ek is any vector field accompanied by flow, n  H · f , whereas rot H f = is a vector of normal, and div H f = ∇  H × f . ∇ The possible definitions of the Laplacian on fractals and in fractional spaces were widely discussed in literature (see Refs. [50–52, 54–58]). Although, generally, the fractional Laplace operator cannot be presented as the sum of all the unmixed second partial derivatives in the Cartesian coordinates, in the case of fractal continuum ds D 3 1 Φ3 ⊂ E with d f = 3 the Hausdorff Laplacian can defined in a  straightforward way as  H = ∇

H

H

·∇

the displacement field in the initial (reference) configuration of 3 3 α Φ D ⊂ E can be defined as

ds



f =

( i ) − 2

c1

i



∂2 f ∂ x2i



(i ) (i )

α − ζi ∂ f xi

= c 3 (x)c 3−1 ( X ) J dV D =

(i )

(i )

in the special case of ζi = D − D S ≡ 1, Eq. (16) coincide with the definition of the generalized Laplacian in an isotropic fractional space F α (see Refs. [50,51]). Consequently, in the case of α = 1 and (i ) ζi = D − D S = 1, the generalized Laplacian (16) converts into the conventional Laplacian in the Euclidean space, even when D < 3 (i ) and D S < 2. 3. Mechanics of fractal continuum The continuum mechanics comes into play when one wants to examine what is going on inside a body in a smoothed picture that does not go into detail about the forces and motions of the sub-scale constituents. To develop the fractal continuum mechanics we need first to define the kinematics of deformations and stresses and then establish the balance (conservation) and constitutive laws for fractal continua. Generally, some kind of deformations can change the metric in the deformed continuum [59–61]. Evolving metrics have been extensively studied in mathematics [62]. Furthermore, in [63] was developed a geometric theory of thermoelasticity in which thermal strains are buried in a temperature-dependent Riemannian material manifold, such that a change of temperature leads to a rescaling of the material metric with a clear physical meaning. A geometric theory of growth mechanics with evolving metrics was suggested in [64]. Although, the metric evolution can be also accounted within a fractal continuum framework, below we assume that metric of fractal continuum models studied in this work is invariant under deformations and temperature changes. 3.1. Kinematics of fractal continuum deformation d

3 The fractal continuum αs Φ D ⊂ E 3 represents a region in a threedimensional Euclidean space filled with continuous matter. Accordingly, we can assume that at time t = 0 it occupies a region W 0 ⊂ ds 3 ds 3 3 3 α Φ D ⊂ E and, at time t > 0, occupies a region W t ⊂ α Φ D ⊂ E

 ∈ E 3) (see Fig. 3). The reference W 0 (x ∈ E 3 ) and current W t ( X configurations are supposed to be bounded, open, and connected. Therefore, the motion of fractal continuum can be determined by d 3 ⊂ E 3 as a functhe position x ∈ E 3 of the material points in αs Φ D

 ∈ E 3 and the time t. Specifically, tion of the reference position X  = (υ1 , υ2 , υ3 ) ∈ Φ D3 ⊂ E 3 describing the displacement vector υ

(i )

(i )

c 1 ( X i )c 2 ( X j =i )

J dA0i dXi

( 0)

= J D dV D

(16)

accounting the fractal topology of medium α Φ3D [43,44], such that

( 0)

(i )

c 1 (xi )c 2 (x j =i )

( 0)



∂ xi

(i )

dV D = dχi dAi = c 3 (x) dV 3 = c 1 c 2 dxi d A 2 = c 3 (x) J dV 3

[41–43]. However if d f < 3



+

ζi

whereas in the current coordinates υi = − X i (xi , t )]. Notice that υi ≈ ζi (xi /ξ0 )ζi −1 u i , where u i are the components of the  ∈ E 3 (see Fig. 3). displacement vector u Furthermore, as in the case of classical continuum mechanics, here we assume that the matter is impenetrable (one portion of fractal continuum never penetrates into another) and indestructible (no region of positive, finite volume is deformed into one of zero or infinite volume). Accordingly, the volume change of d 3 W t ⊂ αs Φ D (see Fig. 3) can be presented as

d

3  

(17) 1−ζ ζ ξ0 i [xi i

the generalized Laplacian in αs Φ3D ⊂ E 3 takes the form

 HF



υi = ξ01−ζi xζi i ( X i , t ) − X iζi ,

(18)

where J = det[∇ X i x j ] is the conventional Jacobian of transforma(0)

tion in E 3 (dV 3 = J dV 3 ), whereas (i )



(i )

c (xi )c 2 (x j =i ) ∂χ j J = (1i ) J = det (i ) c3 ( Xk ) ∂ Xi c 1 ( X i )c 2 ( X j =i ) c 3 (xk )

JD =

 (19) d

3 is the Jacobian of transformation in the fractal continuum αs Φ D , where χi = Xi + υi , while the components of displacement vector υi are defined by Eq. (17). Therefore, for every t > 0, functions χ (X , t ) ∈ dαs Φ D3 and x( X , t ) ∈ E 3 are smooth one-to-one maps of every material point of W 0 onto W t , such that there exists a unique inverse of (17), at least locally, if and only if J D is not identically zero, that is 0 < J D < ∞. Notice that the term in the square brackets in (19) is nothing d 3 more than the deformation gradient in αs Φ D and so a deformation conserves the volume of a region W t if and only if J D = 1. Consequently, the Lagrangian (Green) strains in the fractal continuum ds 3 H H H H α Φ D should be defined as E i j = 0.5(∇ X i υi + ∇ X j υ j + ∇ X j υk ∇ X i υk ), whereas the Eulerian (Almansi) strain tensor is given by e i j = 0.5(∇ H υ + ∇iH υ j − ∇ Hj υk ∇iH υk ). In the limit of infinitesimally j i small deformations, both tensors are converted into the infinitesimal strain tensor

εi j =

1 2

 1 ∇ Hj υi + ∇iH υ j = 2



1 ∂ υi ( j) c1 ∂ x j

+

1 ∂υ j (i ) c 1 ∂ xi

 ,

(20) (i )

where υi are defined by Eq. (17), while the scaling functions c 1 are given by Eq. (9). 3.2. Material derivative, Reynolds transport theorem, and continuity equation

Using the conventional rule for determinant differentiating, it is straightforward to obtain the generalized Euler’s identity for fractal continua in the following form



d dt

 D

J D = J D ∇iH v i ,

(21)

where (d/dt ) D denotes the fractal material (Lagrangian) derivative d 3 and the components of velocity vector v = v i ei in αs Φ D ⊂ E 3 are H 1−β ∂ υi /∂ t. Accordingly, the material defined as v i = ∇t υi = (t /τ0 ) d

3 ⊂ E 3 should be defined as derivative in αs Φ D

A.S. Balankin / Physics Letters A 377 (2013) 2535–2541

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3 Fig. 3. Mappings of fractal medium Φ3D ⊂ E 3 and the corresponding fractal continuum Φ D ⊂ E 3 from the original (reference) to the deformed (current) configurations.



d dt

 D

ψ = ∇tH ψ + v k ∇kH ψ,

(22)

where ψ(xi , t ) is any extensive quantity accompanied by a moving region and the usual summation convention over repeated indices is assumed, such that the generalization of the Reynolds transport theorem for a fractal continuum reads as follows



d dt

 

 ψ dV D =

ds

Wt



 ∇tH ψ + ∇kH (ψ v k ) dV D

Wt



 ∇tH ψ

=

dV D +

Wt

(k)

ψ v k nk d A D ,

(23)

∂W

where the first term on the right-hand side is the time rate of change of ψ within the control volume of W t and the second term represents the net flow of ψ across the control surface ∂ W d 3 ⊂ E 3 . Furthermore, using the definition of the of region W t ∈ αs Φ D fractal material derivative (22), the equation of mass conservation d 3 for fractal continuum αs Φ D ⊂ E 3 can be presented in the form

  ∇tH ρc = − div H ρc ∇tH v .

(24)

Notice that Eqs. (21)–(24) take their conventional forms for the (i ) classical (Euclidean) continuum if D S = D − 1 and d s = D, even when D < 3 and α < 1. 3.3. Stresses in fractal continuum The stress distribution in the (pre-)fractal medium is a nonanalytic function of the Cartesian coordinates. Accordingly, the stress field in the fractal continuum is defined as an analytic envelope of stress field in Φ3D ⊂ E 3 . Following to the concepts of classical continuum mechanics, the forces that act on the fractal continuum or its part can be divided into two categories: those that act by contact with the surface, called surface tractions ( F s ), and those that act at a distance, called the volume or body forces ( F b ). If dF i  D = n d A D , where is a contact force acting on the deformed area d A  is the unit outer normal to the element of area d A D ∈ dαs Φ D3 , then n the stress (traction) vector can be defined as

  f s (x, t ) = lim d F (x, t ) = lim c −1 d F , 2 dAD

d A D →0

d A 2 →0 d A D →0

d A2

(25)

where c 2 = (d A D /d A 2 ). Assuming that this limit exists one can define the normal and shear stresses in the usual way. The geometric character of stress in continuum mechanics was outlined in [65]. In general, stress is not uniformly distributed over a fractal continuum, and may vary with time. Therefore the stress tensor must be defined for each point and each moment, by considering an infinitesimal particle of the medium surrounding that point, and taking the average stresses in that particle as being the stresses at the point. d 3 The total force acting on the fractal continuum αs Φ D ⊂ E 3 can be presented in the general form as

f =



Φ

f b dV D +



f s d A D ,

(26)

∂Φ

whereas the deformations of fractal continua should satisfy the laws of momentum conservation allied with the Newton’s first and second laws. The principle of linear momentum states that the time rate of change of the linear momentum is equal to the resultant force acting on the body. The principle of angular momentum asserts that the time rate of change of the moment of momentum of a body with respect to a given point is equal to the moment of the surface and body forces with respect to that point. Specifically, the principle of linear momentum for fractal continua impales that







 dV D = 0, div H σi j + f b − ρc ∇tH ∇tH υ

(27)

Wt d

 is the acceleration field of αs Φ D3 ⊂ E 3 . Using the where ∇tH ∇tH υ continuity equations (24), the law of linear momentum conserd 3 vation in fractal continuum αs Φ D ⊂ E 3 can be presented in the following local form

ρc ∇tH ∇tH υi = f b(i) + div H σi j .

(28)

Notice that Eq. (28) converts into the conventional equation of the density of linear momentum balance if the fractal dimension of (i ) any intersection is D S = D − 1 and d s = D, even when D < 3.

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A.S. Balankin / Physics Letters A 377 (2013) 2535–2541

Assuming that there do not exist any internal angular momentum, body couples, and couple stresses in the fractal continuum ds 3 3 α Φ D ⊂ E , the principle of angular momentum implies that



d

 

dt

D

 =

ρc e i jk ξ01−ζi xζi i v j dV D

Wt 1−ζi ζi xi

e i jk ξ0

 f j dV D +

W

1−ζi ζi xi

e i jk ξ0

σmj nm d A D ,

(29)

equations for elasto-plastic and visco-elastic fractal continua can be constructed in the same way taking into account the experimentally observed mechanical behavior of fractal medium to be modeled. More generally, the plastic deformations can lead to metric changes in the deformed fractal continuum. These, can be accounted within the framework suggested in [63,64]. Besides, the constitutive laws of visco-elastic and elasto-plastic fractal continua can require the use of not-local fractional calculus accounting the “memory effects” in the fractal material under the study.

∂W 1−ζ

ζ

where e i jk is the permutation tensor and ξ0 i xi i = χi (see Fig. 3). Consequently, on account of the fractional Green–Gauss theorem (15) and Reynolds’s transport theorem (23), the law of angular momentum conservation in the fractal continuum can be presented in the local form as

e i jk σ i j = 0.

(30)

From Eq. (30) immediately follows that in the absence of any internal angular momentum, body couples, and couple stresses in the d 3 fractal continuum αs Φ D ⊂ E 3 the stress tensor is symmetric, that is

σi j = σ ji ,

(31)

regardless of whether or not it is in equilibrium. 3.4. Constitutive equations for deformable fractal continua The constitutive laws of solid mechanics cannot be deduced from the general laws of continuum mechanics, but can be defined from physical experiments (for example, Hooke’s law of elasticity, micropolar elasticity law, visco-elastic law, strain-hardening plasticity law, etc.). In the case of solid materials with (pre-)fractal (micro-)structure the constitutive equations are dependent of the mechanical properties of matter, as well as on the fractal features of the (micro-)structure. Hence, the constitutive laws for deformable fractal continua can be either determined experimentally for modeled fractal materials, or defined by the mapping of classical constitutive relations into the fractal continuum framework. In this way, there are a number of rules that must be used to define constitutive equations that are admissible from the rational and physical standpoints. First of all, constitutive equations should be invariant under any change of reference frame. Furthermore, in the local theory, the stress tensor at a given point should be independent of movements occurring at finite distance from this point. Finally, the current rheological and thermodynamic state of the material should be completely determined by the history of the thermo-kinetic process experienced by the material [66]. Within this framework, the constitutive law for linear elastic isotropic (ζi ≡ ζ ) fractal continuum takes the form





σi j = μ ∇ Hj υi + ∇iH υ j + λ∇kH υk δi j = 2μεi j + λεkk δi j ,

(32)

where the deformation tensor εi j is defined by Eq. (20), while λ and μ are the effective Lame coefficients of the fractal continuum. Substituting Eq. (32) into Eq. (28) we obtain the wave equations in the linear elastic fractal continuum. Accordingly, it is straightforward matter to see that fractal (micro-)structure of physical medium causes the coupling of the longitudinal and transversal acoustic waves, as well as wave localization in accordance with the results of numerical simulations on fractals [67–69] and experiments with fractal materials [70,71]. In the case of anisotropic elastic fractal continuum σi j = C i jkl εkl , where C i jkl is the stiffness tensor, while deformation tensor εi j is defined by Eq. (20). The constitutive equation for the Newtonian fluids flow in fractally permeable media was constructed in [42]. The constitutive

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