Thermo-poromechanics of fractal media

Abstract

This article advances continuum-type mechanics of porous media having a generally anisotropic, product-like fractal geometry. Relying on a fractal derivative, the approach leads to global balance laws in terms of fractal integrals based on product measures and, then, converting them to integer-order integrals in conventional (Euclidean) space. Proposed is a new line transformation coefficient that is frame invariant, has no bias with respect to the coordinate origin and captures the differences between two fractal media having the same fractal dimension but different density distributions. A continuum localization procedure then allows the development of local balance laws of fractal media: conservation of mass, microinertia, linear momentum, angular momentum and energy, as well as the second law of thermodynamics. The product measure formulation, together with the angular momentum balance, directly leads to a generally asymmetric Cauchy stress and, hence, to a micropolar (rather than classical) mechanics of fractal media. The resulting micropolar model allowing for conservative and/or dissipative effects is applied to diffusion in fractal thermoelastic media. First, a mechanical formulation of Fick’s Law in fractal media is given. Then, a complete system of equations governing displacement, microrotation, temperature and concentration fields is developed. As a special case, an isothermal model is worked out.

This article is part of the theme issue ‘Advanced materials modelling via fractional calculus: challenges and perspectives’.

1. Motivation

The objective of this paper is to develop a continuum physics-type framework for diffusion of liquids and gases in elastic–inelastic deformable solid materials. Three considerations underpin the development. (1) Since such diffusion causes expansion/contraction of the solid matrix, the processes are coupled. (2) Since the diffusion generally takes place through a hierarchical network of channels, their fractal geometry has to be admitted in both the diffusion processes and the mechanics. (3) Since the network is generally anisotropic, an appropriate fractal geometry with spatial anisotropy has to be involved. The theoretical strategy in this article follows the approach to continuum-type mechanics of heterogeneous porous media of fractal type.

The basic ideas, extending the dimensional regularization concepts and proceeding in the vein of a field theory, hark back to the pioneering work by Tarasov [1,2], who developed continuum-type equations of fractal porous media, and extended them to a range of problems in electrodynamics, fluid mechanics, heat transfer, wave motion, etc. [3]. The power of this approach stems from the fact that much of the framework of continuum mechanics/physics may be generalized and partial differential equations (with derivatives of integer order) may still be employed. While the original formulation was based on the Riesz measure and restricted to isotropic and fluid-like media, the present authors introduced a framework of product measure [4–7], grasping the anisotropy of fractal geometry (i.e. different fractal dimensions in different directions) generally in solid materials. This modelling strategy has been classified as the ‘fractional-integral approach’ in the comprehensive review article [3]; note that four other approaches to fractal media—each having its pros and cons—are also possible, but the fractional-integral one has already been established in a range of mechanics and physics problems. These include wave equations in several settings (one-dimensional and three-dimensional elastodynamics and a fractal Timoshenko beam) [8,9], extremum and variational principles [10], equations of turbulence in fractal media [11], electromagnetism [7] and fractal planetary rings [12]. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers.

The paper is organized as follows: first, it gives the basic concepts of product measure and the key results of tensor calculus on anisotropic fractals. On this basis, we develop continuum mechanics involving the conservation laws (of mass, microinertia, linear momentum, angular momentum and energy) as well as the second law of thermodynamics. It is next shown how the generally anisotropic fractal structure denies the symmetry of the Cauchy stress tensor and necessitates the introduction of a micropolar continuum mechanics. The resulting thermomechanical model, based on a theory developed in [13,14], is applied to thermodiffusion in fractal poroelastic media.

2. Calculus on fractal media

This section develops the product measure formulation, relying on the line, surface and volume transformation coefficients for mapping the fractal porous medium into an approximating homogenized continuum. On this basis, the basic relations of fractal tensor calculus are developed such as the Green–Gauss theorem, divergence and curl operations, and identities of the vector calculus.

(a). Fractal derivative and integral on anisotropic fractals

The mass of a fractal body occupying a region $\mathcal{W}$ in the Euclidean space ${\mathbb{E}}^3$ obeys a power law

$$m(R)=k{R}^D.$$ 2.1

Here R is the length scale of measurement and k is a proportionality constant, and D (typically <3) is the fractal dimension. Note that (2.1) can be applied to a pre-fractal, i.e. a fractal-type, physical object with lower and upper cutoffs of R. In [1,2,15], Tarasov employed a fractional integral to represent the mass as

$$m(\mathcal{W})={\int }_{\mathcal{W}}\rho (\mathbf{\text{r}})\mathrm{d}{\mathrm{V}}_{\mathrm{D}}={\int }_{\mathcal{W}}\rho (\mathbf{\text{r}}){\mathrm{c}}_3(\mathrm{D},\mathrm{R})\mathrm{d}{\mathrm{V}}_3,$$ 2.2

where the first and second equalities, respectively, involve fractional (Riesz-type) integrals and conventional integrals, while the coefficient c₃ (D, R) = const. · R^D−3 provides a transformation between the two and indicates a density of state associated with the fractal media. This formulation (now primarily of historical interest) had several drawbacks. (1) It involves a fractional derivative that does not give zero when applied to a constant function (e.g. frame translation or rigid body motion), which is an unphysical property. (2) The mechanics-type derivation of wave equations yields a result different from the variational-type derivation. (3) The three-dimensional wave equation does not correctly reduce to the one-dimensional wave equation. (4) It works only in isotropic media. These limitations have motivated a new formulation replacing (2.2) by a more general power law relation with respect to each coordinate, thus allowing for anisotropic fractals:

$$m(R)\sim x_1^{{\alpha }_1}{x}_2^{{\alpha }_2}{x}_3^{{\alpha }_3},$$ 2.3

where each α_k ∈ (0, 1] plays the role of a fractal dimension in the direction x_k. The mass distribution is then specified via a product formula

$$m(x_1,x_2,x_3)={\iiint }_{\mathcal{W}}\rho (x_1,x_2,x_3)\mathrm{d}\mu ({\mathrm{x}}_1)\hspace{0.17em}\mathrm{d}\mu ({\mathrm{x}}_2)\hspace{0.17em}\mathrm{d}\mu ({\mathrm{x}}_3).$$ 2.4

Here the length measurement in each coordinate x_k is provided by

$$\mathrm{d}\mu ({\mathrm{x}}_{\mathrm{k}})={\mathrm{c}}_1^{(\mathrm{k})}({\alpha }_{\mathrm{k}},{\mathrm{x}}_{\mathrm{k}})\mathrm{d}{\mathrm{x}}_{\mathrm{k}},$$ 2.5

where $c_1^{(k)}$ is called a line transformation coefficient in the direction x_k. The basic idea of the product measure goes back to [5]. Relation (2.5) implies that the infinitesimal fractal volume element dV_D is the product

$$\begin{array}{rl}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}=\mathrm{d}\mu ({\mathrm{x}}_1)\hspace{0.17em}\mathrm{d}\mu ({\mathrm{x}}_2)\hspace{0.17em}\mathrm{d}\mu ({\mathrm{x}}_3)& =c_1^{(1)}{c}_1^{(2)}{c}_1^{(3)}\hspace{0.17em}\mathrm{d}{\mathrm{x}}_1\hspace{0.17em}\mathrm{d}{\mathrm{x}}_2\hspace{0.17em}\mathrm{d}{\mathrm{x}}_3={\mathrm{c}}_3\hspace{0.17em}\mathrm{d}{\mathrm{V}}_3,\\ \mathrm{with}{\mathrm{c}}_3& =c_1^{(1)}{c}_1^{(2)}{c}_1^{(3)}.\end{array}$$ 2.6

By analogy to (2.5), c₃ is called a volume transformation coefficient.

Equation (2.3) implies that each α_k plays the role of a fractal dimension in the direction x_k. While it is noted that the anisotropic fractal body’s fractal dimension is not necessarily the sum of projected fractal dimensions following (2.4), we quote from a book on mathematics of fractals [16]: ‘Many fractals encountered in practice are not actually products, but are product-like’.

Focusing on a cubic-shaped dV_D in figure 1, for each surface element $S_d^{(k)}$ defined by a direction x_k normal to it, the surface transformation coefficient $c_2^{(k)}$ is set up in terms of its two in-plane linear coefficients ($c_1^{(i)}$ and $c_1^{(j)}$) or, equivalently, in terms of the volume coefficient c₃ and the line transformation coefficient $c_1^{(k)}$ in direction normal to the surface

$$c_2^{(k)}=c_1^{(i)}{c}_1^{(j)}=\frac{c_3}{c_1^{(k)}},i\ne j\hspace{0.17em}\mathrm{and}\hspace{0.17em}\mathrm{i},\mathrm{j}\ne \mathrm{k}.$$ 2.7

Figure 1.

Roles of the transformation coefficients $c_1^{(k)}$, $c_2^{(i)}$ and c₃ in homogenizing a fractal body of volume dV_D, surface dS_d and lengths dl_α into an Euclidean continuum of volume dV₃, surface dS₂ and length dx.

To specify the $c_1^{(k)}$ coefficients, we note that a variety of forms may exist to represent the fractional power law relation (2.3). In particular, there are two forms currently in use, namely, the form used by Tarasov [1,2,15] and Balankin et al. [17] from the Riesz fractional integral (actually modified by dropping the integral limits), or the one we used in [4–6] according to the fractional integral of Jumarie [18], respectively,

$$c_1^{(k)}={\alpha }_k{x}_k{}^{{\alpha }_{\mathrm{k}}-1},\mathrm{k}=1,2,3$$ 2.8

and

$$c_1^{(k)}={\alpha }_k{(l_k-x_k)}^{{\alpha }_k-1},\mathrm{k}=1,2,3.$$ 2.9

where l_k is the total length (integral interval) along x_k. Both these forms have drawbacks.

• 1.The objectivity (frame invariance) does not hold in both forms because a translation in the coordinate origin results in the change of the coordinate x_k for a given point and, therefore, a different density of state $c_1^{(k)}({\alpha }_k,x_k)$ associated with that point.
• 2.The distribution of $c_1^{(k)}({\alpha }_k,x_k)$ (density of state) in the form (2.8) is biased towards the coordinate origin, while in (2.9) to the boundary, each of which corresponds to very specific fractal media; see the discussion in [19].
• 3.The formulations cannot differentiate between two fractal media having the same fractal dimension but different density distributions and mechanical behaviours.

To overcome the aforementioned issues, we propose a combination of both forms involving the integral limits $x_k^a,x_k^b$:

$$c_1^{(k)}=l_k^a{(x_k-x_k^a)}^{{\alpha }_k-1}+l_k^b{(x_k^b-x_k)}^{{\alpha }_k-1},x_k\in (x_k^a,x_k^b);\hspace{0.17em}k=1,2,3.$$ 2.10

More specifically, $x_k^a$ and $x_k^b$ refer to the lower and upper integral limits in the coordinate axis of k-direction, respectively; $l_k^a$ and $l_k^b$ are the length scale parameters. This formulation indeed follows the original Riesz fractional integral by incorporating both lower and upper integral limits but generalizes it through introducing $l_k^a$ and $l_k^b$ (in Riesz fractional integral $l_k^a=l_k^b$). It is expected that the above equation is frame invariant since the values of $(x_k-x_k^a)$ and $(x_k^b-x_k)$ are independent of frames. In addition, the formulation can now capture the differences between two fractal media having the same fractal dimension but different density distributions by varying $l_k^a$ and $l_k^b$.

Note that $c_1^{(k)}$ in (2.10) has a fractional length dimension, which is understandable since in mathematics a fractal set has a finite measure only with respect to its Hausdorff dimension. In practice, it is convenient to work with dimensionless variables to avoid issues of dimensional inconsistencies. Thus, we suggest replacing x_k by x_k/l₀ [5] (l₀ is a characteristic scale, e.g. the mean pore size).

(b). Fractal tensor calculus

To have a working continuum theory, a generalization of the Green–Gauss theorem is needed. Following the theory put forth in [4,5], that generalization is formulated within the framework of product formulae discussed above as

$${\int }_{\mathrm{\partial }\mathcal{W}}{f}_k{n}_k\hspace{0.17em}\mathrm{d}{\mathrm{S}}_{\mathrm{d}}={\int }_{\mathcal{W}}({\mathrm{f}}_{\mathrm{k}}{\mathrm{c}}_2^{(\mathrm{k})}),\hspace{0.17em}{}_{\mathrm{k}}\hspace{0.17em}{\mathrm{c}}_3^{-1}\hspace{0.17em}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}={\int }_{\mathcal{W}}{\mathrm{f}}_{\mathrm{k}}{,}_{\mathrm{k}}{\mathrm{c}}_2^{(\mathrm{k})}\hspace{0.17em}{\mathrm{c}}_3^{-1}\hspace{0.17em}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}={\int }_{\mathcal{W}}\frac{{\mathrm{f}}_{\mathrm{k}}{,}_{\mathrm{k}}}{{\mathrm{c}}_1^{(\mathrm{k})}}\hspace{0.17em}\mathrm{d}{\mathrm{V}}_{\mathrm{D}},$$ 2.11

where (2.7) has been used, observing that $c_2^{(k)}$ is independent of x_k. This leads to the definition of the fractal derivative (in general, fractal gradient) operator

$${\mathrm{\nabla }}^D\varphi ={\mathbf{\text{e}}}_k{\mathrm{\nabla }}_k^D\varphi \mathrm{and}{\mathrm{\nabla }}_{\mathrm{k}}^{\mathrm{D}}\varphi =\frac{1}{{\mathrm{c}}_1^{(\mathrm{k})}}\frac{\mathrm{\partial }\varphi }{\mathrm{\partial }{\mathrm{x}}_{\mathrm{k}}},$$ 2.12

where e_k are the base vectors in ${\mathbb{E}}^3$. This operator has three desirable properties.

• (1)It is the ‘inverse’ operator of a fractal integral: $\int {\mathrm{\nabla }}_k^{D}f\hspace{0.17em}\mathrm{d}{\mu }^{\mathrm{D}}({\mathrm{x}}_{\mathrm{k}})={\mathrm{\nabla }}_{\mathrm{k}}^{\mathrm{D}}\int \mathrm{f}\hspace{0.17em}\mathrm{d}{\mu }^{\mathrm{D}}({\mathrm{x}}_{\mathrm{k}})=\mathrm{f}$.
• (2)Leibniz’s rule is satisfied: ${\mathrm{\nabla }}_k^D(AB)={\mathrm{\nabla }}_k^D(A)B+A{\mathrm{\nabla }}_k^D(B)$.
• (3)Its operation on any constant is zero, which indeed is a property not possessed by the usual fractional derivative of Riemann–Liouville.

We note that the fractal derivative (2.12) is equivalent to the conformable fractional derivative in [20] and the Hausdorff derivative in [21]. It can be also introduced through a difference formula that involves fractal metric [17]. It is pertinent to point out that the fractal derivative is different from the Riesz fractional derivative or other non-local fractional derivatives. The formulation follows from the Green–Gauss theorem which is more suitable in the continuum mechanics context. This will be discussed in the next section. The spatial fractal derivative (2.12) describes size effects in fractal media. Consequently, the fractional generalization of Reynold’s transport theorem is [5]

$${\left(\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\right)}_D{\int }_{\mathcal{W}}P\hspace{0.17em}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}={\int }_{\mathcal{W}}[{\left(\frac{\mathrm{\partial }}{\mathrm{\partial }\mathrm{t}}\right)}_{\mathrm{D}}\mathrm{P}+{\mathrm{\nabla }}_{\mathrm{k}}^{\mathrm{D}}({\mathrm{v}}_{\mathrm{k}}\mathrm{P})]\mathrm{d}{\mathrm{V}}_{\mathrm{D}},$$ 2.13

implying that the fractal material time derivative follows as

$${\left(\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\right)}_{D}P={\left(\frac{\mathrm{\partial }}{\mathrm{\partial }t}\right)}_{D}P+v_k{\mathrm{\nabla }}_{\mathrm{k}}^{\mathrm{D}}\mathrm{P}$$ 2.14

where (∂/∂t)_D is the time fractional derivative. Since fractal media of interest to us exhibit spatial fractality only, a conventional time derivative can be used so that (∂/∂t)_D = (∂/∂t). When the medium also displays the fractal characteristics of temporal response—as is sometimes the case in viscoelasticity—a fractional time derivative (such as Caputo) is to be used [22,23]. Clarification of a cause–effect relationship between the fractal spatial structure and a fractional-type calculus in viscoelastic and other constitutive (e.g. thermoelastic) responses remains an open challenge [24].

Since fractional viscoelasticity has been studied extensively, here we leave the choice of time derivative to be dictated by the specific model and mainly focus on spatial fractal effects.

From a homogenization standpoint, the above developments allow interpretation of the fractal (intrinsically discontinuous and multiscale) medium transformed to an equivalent homogenized continuum model, that is

$$\mathrm{d}{\mathrm{l}}_{\alpha }={\mathrm{c}}_1\hspace{0.17em}\mathrm{d}\mathrm{x},\mathrm{d}{\mathrm{S}}_{\mathrm{d}}={\mathrm{c}}_2\hspace{0.17em}\mathrm{d}{\mathrm{S}}_2,\hspace{0.17em}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}={\mathrm{c}}_3\mathrm{d}{\mathrm{V}}_3,$$ 2.15

where dl_α, dS_d, dV_D represent the line, surface and volume elements in the fractal body, while dx, dS₂, dV₃, respectively, denote those in the homogenized model, see figure 1. The coefficients c₁, c₂, c₃ provide relations between both pictures. Standard image analysis techniques (such as the ‘box method’ or the ‘sausage method’) allow a quantitative calibration of these coefficients for every direction and every cross-sectional plane.

Another advantage of the product formulation is the direct generalization to tensor calculus. To see this clearly, on account of (2.12), the fractal divergence of a vector field is

$$\mathrm{div}\hspace{0.17em}\mathbf{\text{f}}={\mathrm{\nabla }}^{\mathrm{D}}\cdot \mathbf{\text{f}}\mathrm{or}\hspace{0.17em}{\mathrm{\nabla }}_{\mathrm{k}}^{\mathrm{D}}{\mathrm{f}}_{\mathrm{k}}=\frac{1}{{\mathrm{c}}_1^{(\mathrm{k})}}\frac{\mathrm{\partial }{\mathrm{f}}_{\mathrm{k}}}{\mathrm{\partial }{\mathrm{x}}_{\mathrm{k}}}.$$ 2.16

This leads to a fractal curl operator of a vector field

$$\mathrm{curl}\hspace{0.17em}\mathbf{\text{f}}={\mathrm{\nabla }}^{\mathrm{D}}\times \mathbf{\text{f}}\mathrm{or}\hspace{0.17em}{\mathrm{e}}_{\mathrm{jki}}{\mathrm{\nabla }}_{\mathrm{k}}^{\mathrm{D}}{\mathrm{f}}_{\mathrm{i}}={\mathrm{e}}_{\mathrm{jki}}\frac{1}{{\mathrm{c}}_1^{(\mathrm{k})}}\frac{\mathrm{\partial }{\mathrm{f}}_{\mathrm{i}}}{\mathrm{\partial }{\mathrm{x}}_{\mathrm{k}}},$$ 2.17

where e_{j k i} is the Levi–Civita permutation symbol.

The four fundamental identities of the conventional vector calculus can now be shown to hold in terms of these two new operators. Thus,

• (i)
  The fractal divergence of the fractal curl of a vector field f:

  $$\mathrm{div}\cdot \mathrm{curl}\hspace{0.17em}\mathbf{\text{f}}={\mathrm{\nabla }}_{\mathrm{j}}^{\mathrm{D}}\cdot {\mathrm{e}}_{\mathrm{jki}}{\mathrm{\nabla }}_{\mathrm{k}}^{\mathrm{D}}{\mathrm{f}}_{\mathrm{i}}=\frac{1}{{\mathrm{c}}_1^{(\mathrm{j})}}\frac{\mathrm{\partial }}{\mathrm{\partial }{\mathrm{x}}_{\mathrm{j}}}\left[{\mathrm{e}}_{\mathrm{jki}}\frac{1}{{\mathrm{c}}_1^{(\mathrm{k})}}\frac{\mathrm{\partial }{\mathrm{f}}_{\mathrm{i}}}{\mathrm{\partial }{\mathrm{x}}_{\mathrm{k}}}\right]=0.$$ 2.18
• (ii)
  The fractal curl of the fractal gradient of a scalar field f:

  $$\mathrm{curl}\times (\mathrm{grad}\hspace{0.17em}\mathrm{f})={\mathbf{\text{e}}}_{\mathrm{i}}{\mathrm{e}}_{\mathrm{ijk}}{\mathrm{\nabla }}_{\mathrm{j}}^{\mathrm{D}}({\mathrm{\nabla }}_{\mathrm{k}}^{\mathrm{D}}\mathrm{f})={\mathbf{\text{e}}}_{\mathrm{i}}{\mathrm{e}}_{\mathrm{ijk}}\frac{1}{{\mathrm{c}}_1^{(\mathrm{j})}}\frac{\mathrm{\partial }}{\mathrm{\partial }{\mathrm{x}}_{\mathrm{j}}}\left[\frac{1}{{\mathrm{c}}_1^{(\mathrm{k})}}\frac{\mathrm{\partial }\mathrm{f}}{\mathrm{\partial }{\mathrm{x}}_{\mathrm{k}}}\right]=0.$$ 2.19

  In both cases, above we can pull $1/c_1^{(k)}$ out in front of the gradient because the coefficient $c_1^{(k)}$ is independent of x_j (assuming k ≠ j, otherwise e_{j k i} = 0).
• (iii)
  The fractal divergence of the fractal gradient of a scalar field ϕ is written in terms of the fractal gradient as

  $$\mathrm{div}\cdot (\mathrm{grad}\hspace{0.17em}\mathrm{f})={\mathrm{\nabla }}_{\mathrm{j}}^{\mathrm{D}}\cdot {\mathrm{\nabla }}_{\mathrm{k}}^{\mathrm{D}}\mathrm{f}=\frac{1}{{\mathrm{c}}_1^{(\mathrm{j})}}\frac{\mathrm{\partial }}{\mathrm{\partial }{\mathrm{x}}_{\mathrm{j}}}\left[\frac{1}{{\mathrm{c}}_1^{(\mathrm{j})}}\frac{\mathrm{\partial }\mathrm{f}}{\mathrm{\partial }{\mathrm{x}}_{\mathrm{j}}}\right]=\frac{1}{{\mathrm{c}}_1^{(\mathrm{j})}}\left[\frac{\mathrm{f}{,}_{\mathrm{j}}}{{\mathrm{c}}_1^{(\mathrm{j})}}\right]{,}_{\mathrm{j}},$$ 2.20

  which gives an explicit form of the fractal Laplacian.
• (iv)
  The fractal curl of the fractal curl operating on a vector field f:

  $$\mathrm{curl}\times (\mathrm{curl}\hspace{0.17em}\mathbf{\text{f}})={\mathbf{\text{e}}}_{\mathrm{p}}{\mathrm{e}}_{\mathrm{prj}}{\mathrm{\nabla }}_{\mathrm{r}}^{\mathrm{D}}({\mathrm{e}}_{\mathrm{jki}}{\mathrm{\nabla }}_{\mathrm{k}}^{\mathrm{D}}{\mathrm{f}}_{\mathrm{i}})={\mathbf{\text{e}}}_{\mathrm{p}}{\mathrm{\nabla }}_{\mathrm{p}}^{\mathrm{D}}({\mathrm{\nabla }}_{\mathrm{k}}^{\mathrm{D}}{\mathrm{f}}_{\mathrm{r}})-{\mathrm{\nabla }}_{\mathrm{r}}^{\mathrm{D}}{\mathrm{\nabla }}_{\mathrm{r}}^{\mathrm{D}}{\mathrm{f}}_{\mathrm{p}}.$$ 2.21

With the above, the Helmholtz decomposition for fractals can be proved along the classical lines. Thus, a vector field F with known divergence and curl, none of which equal to zero, and which is finite, uniform and vanishes at infinity, may be expressed as the sum of a lamellar vector U and a solenoidal vector V:F = U + V with the operations: $\mathrm{curl}\hspace{0.17em}\mathbf{\text{U}}=\mathbf{\text{0}}$, $\mathrm{div}\hspace{0.17em}\mathbf{\text{V}}=0$ understood in the sense of (2.17) and (2.16), respectively. Similarly, the Clebsch representation can also be generalized within this fractal calculus.

3. Continuum mechanics of fractal media

The continuum mechanics of fractal media developed in earlier studies [1,5,15,25] holds with the calculus outlined above. In what follows, we outline the field equations analogously to those of classical continuum mechanics, albeit based on fractal integral and expressed in terms of the fractal derivative (2.12). The developments are made in the spatial description; see [20] for large deformations and the material description where $c_1^{(k)}$ needs to be replaced by (2.10). First, we specify the relationship between surface force, ${\mathbf{\text{F}}}^S\hspace{0.17em}(=F_k^S)$, and the Cauchy stress tensor, σ_{k l}, using fractal integrals as

$$F_k^S={\int }_S{\sigma }_{lk}{n}_l\hspace{0.17em}\mathrm{d}{\mathrm{S}}_{\mathrm{d}}={\int }_{\mathrm{S}}{\sigma }_{\mathrm{lk}}{\mathrm{n}}_{\mathrm{l}}{\mathrm{c}}_2^{(\mathrm{l})}\hspace{0.17em}\mathrm{d}{\mathrm{S}}_2,$$ 3.1

where n_l are the components of the outward normal n to S. To specify the strain, we observe, using (2.15)₁ and the definition of a fractal derivative (2.12), for small deformations the strain, ε_{i j}, is defined in terms of the displacement u_k as

$${\epsilon }_{ij}=\frac{1}{2}({\mathrm{\nabla }}_j^D{u}_i+{\mathrm{\nabla }}_i^D{u}_j)=\frac{1}{2}[\frac{1}{c_1^{(j)}}{u}_i,\hspace{0.17em}{}_j+\frac{1}{c_1^{(i)}}{u}_j{,}_i].$$ 3.2

As shown in [5], this definition of strain results in the same equations governing the wave motion in linear elastic materials when derived by a variational approach as when derived by a mechanical approach; this is the case with one-dimensional, two-dimensional and three-dimensional wave motions as well as with the elastodynamics of beams [4]. The fractal strain (3.2) and stress (3.1) form a work conjugate pair.

We next apply the balance laws for mass, linear and angular momenta, energy and entropy production to the fractal medium in order to derive the corresponding continuum equations.

(a). Fractal conservation of mass and microinertia

Begin with the conservation of mass for the entire fractal body $\mathcal{W}$,

$$\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{\int }_{\mathcal{W}}\rho \hspace{0.17em}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}=0,$$ 3.3

where ρ is the density of the medium. Since the above holds for any arbitrary subset of $\mathcal{W}$, application of the fractal Reynolds transport theorem (2.13) yields the fractal conservation of mass (i.e. fractal continuity equation)

$$\frac{\mathrm{d}\rho }{\mathrm{d}\mathrm{t}}+\rho c_1^{(k)}{\mathrm{\nabla }}_k^D{v}_k=0.$$ 3.4

The medium with a product-like fractal structure should be homogenized as a micropolar continuum [26] instead of a classical continuum. This is so because the presence of a generally anisotropic fractal structure is reflected by differences in the fractal dimensions α_i in different directions, in turn implying $c_1^{(j)}\ne c_1^{(k)},\hspace{0.17em}j\ne k$.

First, in order to determine the inertia tensor i_{k l} at any micropolar point at time t, we consider a rigid particle p having, at any time t, a volume element ${\mathcal{P}}_t$, and a helicoidal vector field (for any $\mathit{\omega }\in {\mathbb{R}}^3$) [25]

$$\mathbf{\text{v}}(\mathbf{\text{x}},t)=\mathbf{\text{v}}({\mathbf{\text{x}}}_A,t)+\mathit{\omega }\times (\mathbf{\text{x}}-{\mathbf{\text{x}}}_A).$$ 3.5

This gives all the components of i_{k l} (diagonal and off-diagonal) as

$$i_{kl}={\int }_{{\mathcal{P}}_O}[x_m{x}_m{\delta }_{kl}-x_k{x}_l]\rho (\mathbf{\text{x}})\hspace{0.17em}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}.$$ 3.6

The micropolar particle $\mathcal{P}$ is characterized by the vector ${\mathrm{\Xi }}_K$ in the material description and by the vector ξ_k in the spatial description, with the linear mapping from the first into the second: ${\xi }_k={\chi }_{kK}{\mathrm{\Xi }}_K$; Here χ_{k K} is called the microdeformation [26]. Next, consider the local equation mapping the microinertia in the reference state (I_{K L}) into that in the current state (i_{k l}): i_{k l} = I_{K L}χ_{k K}χ_{l L}. The conservation of microinertia for the entire fractal particle $\mathcal{P}$ and application of the fractal Reynolds transport theorem (2.13) lead to the fractal conservation of microinertia

$$\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{i}_{kl}=i_{kr}{v}_{lr}+i_{lr}{v}_{kr}.$$ 3.7

(b). Fractal momentum and energy balance equations

Introducing a couple-stress tensor μ_{i k} and a rotation vector φ_j augmenting, respectively, the Cauchy stress tensor τ_{i k} (thus denoted so as to distinguish it from the symmetric σ_{i k}) and the deformation vector u_i, the surface force and surface couple in the fractal setting can be specified by fractal integrals of τ and μ, respectively, as

$$T_k^S={\int }_{\mathrm{\partial }\mathcal{W}}{\tau }_{ik}{n}_i\hspace{0.17em}\mathrm{d}{\mathrm{S}}_{\mathrm{d}}\mathrm{and}{\mathrm{M}}_{\mathrm{k}}^{\mathrm{S}}={\int }_{\mathrm{\partial }\mathcal{W}}{\mu }_{\mathrm{ik}}{\mathrm{n}}_{\mathrm{i}}\hspace{0.17em}\mathrm{d}{\mathrm{S}}_{\mathrm{d}}.$$ 3.8

The above is consistent with the relation of force tractions and couple tractions to the force stresses and couple stresses on any surface element: t_k = τ_{i k} n_i, m_k = μ_{i k} n_i.

Begin with the balance law of linear momentum for $\mathcal{W}$,

$$\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}{\int }_{\mathcal{W}}\rho v_k\hspace{0.17em}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}={\int }_{\mathcal{W}}{\mathrm{X}}_{\mathrm{k}}\hspace{0.17em}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}+{\int }_{\mathrm{\partial }\mathcal{W}}{\sigma }_{\mathrm{lk}}{\mathrm{n}}_{\mathrm{l}}\hspace{0.17em}\mathrm{d}{\mathrm{S}}_{\mathrm{d}},$$ 3.9

where X_k is the body force density. Using the Reynolds transport theorem (2.13), the Green–Gauss theorem (2.11), and performing localization, we obtain the fractal linear momentum equation

$$\rho {\dot{v}}_k=X_k+{\mathrm{\nabla }}_l^D{\sigma }_{lk}.$$ 3.10

Now, proceeding in a fashion similar as before, we obtain the fractal angular momentum equation

$$\frac{e_{ijk}}{c_1^{(j)}}{\tau }_{jk}+{\mathrm{\nabla }}_j^D{\mu }_{ji}+Y_i=\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}(i_{ik}{w}_k).$$ 3.11

In the above, Y_i is the body force couple, while $v_k(={\dot{u}}_k)$ and $w_k(={\dot{\phi }}_k)$ are the deformation and rotation velocities of the micropolar particle $\mathcal{P}$, respectively.

Globally, the balance of energy in continuum (thermo)mechanics has the following form:

$$\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}(\mathcal{K}+\mathcal{U})=\mathcal{P}+\mathcal{H},$$ 3.12

where $\mathcal{K}$ is the kinetic energy, $\mathcal{U}$ is the internal energy, $\mathcal{P}$ is the power and $\mathcal{H}$ is the thermal energy supplied externally. More explicitly, we have

$$\begin{array}{rl}& \mathcal{K}={\int }_{\mathcal{W}}\frac{1}{2}\rho v_i{v}_i\hspace{0.17em}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}\mathcal{U}={\int }_{\mathcal{W}}\rho \frac{\mathrm{d}\mathrm{u}}{\mathrm{d}\mathrm{t}}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}\\ & \mathcal{P}={\int }_{\mathcal{W}}(X_i{v}_i+Y_i{w}_i)\mathrm{d}{\mathrm{V}}_{\mathrm{D}}+{\int }_{\mathrm{\partial }\mathcal{W}}({\mathrm{t}}_{\mathrm{i}}{\mathrm{v}}_{\mathrm{i}}+{\mathrm{m}}_{\mathrm{i}}{\mathrm{w}}_{\mathrm{i}})\mathrm{d}{\mathrm{S}}_{\mathrm{d}}\\ \mathrm{and}& \mathcal{H}=-{\int }_{\mathrm{\partial }\mathcal{W}}{q}_i{n}_i\hspace{0.17em}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}+{\int }_{\mathrm{\partial }\mathcal{W}}\rho \mathrm{h}\hspace{0.17em}\mathrm{d}{\mathrm{S}}_{\mathrm{d}},\end{array}\}$$ 3.13

where u is the internal energy density, q_i is the heat flux through the boundary of $\mathcal{W}$, and h is the heat generation within $\mathcal{W}$. Substituting (3.13) into (3.12), with the help of (2.11), we obtain the local rate of change of internal energy

$$\rho \dot{u}={\tau }_{ji}({\mathrm{\nabla }}_j^D{v}_i-e_{kji}{\dot{\phi }}_k)+{\mu }_{ji}{\mathrm{\nabla }}_i^D{w}_j-{\mathrm{\nabla }}_i^D{q}_i+\rho h.$$ 3.14

Since we are interested in poroelasticity, we restrict attention to small motions, that is to the infinitesimal strain tensor and the curvature–torsion tensor in fractal media

$${\gamma }_{ji}={\mathrm{\nabla }}_j^D{u}_i-e_{kji}{\phi }_k\mathrm{and}{\kappa }_{\mathrm{ji}}={\mathrm{\nabla }}_{\mathrm{j}}^{\mathrm{D}}{\phi }_{\mathrm{i}}.$$ 3.15

Then, (3.14) gives the fractal energy equation

$$\rho \dot{u}={\tau }_{ij}{\dot{\gamma }}_{ij}+{\mu }_{ij}{\dot{\kappa }}_{ij}-{\mathrm{\nabla }}_i^D{q}_i+\rho h.$$ 3.16

Assuming u to be a state function of γ_{i j} and κ_{i j} only (which is natural for an elastic solid) and assuming τ_{i j} and μ_{i j} not to be explicitly dependent on ${\dot{\gamma }}_{ij}$ and ${\dot{\kappa }}_{ij}$, we find τ_{i j} = ∂u/∂γ_{i j}, μ_{i j} = ∂u/∂κ_{i j}. That is, just like in non-fractal continuum mechanics, also in fractal media (τ_{i j}, γ_{i j}) and (μ_{i j}, κ_{i j}) are the work conjugate pairs.

(c). Fractal second law of thermodynamics

To derive the field equation of the second law of thermodynamics in a fractal medium, we begin with the global form of that law in the volume:

$$\dot{S}={\dot{S}}^{(r)}+{\dot{S}}^{(i)}\mathrm{with}\hspace{0.17em}{\dot{\mathrm{S}}}^{(\mathrm{r})}=\frac{\dot{\mathrm{Q}}}{\mathrm{T}},\hspace{0.17em}{\dot{\mathrm{S}}}^{(\mathrm{i})}\ge 0,$$ 3.17

where $\dot{S}$, ${\dot{S}}^{(r)}$ and ${\dot{S}}^{(i)}$ stand, respectively, for the total, reversible and irreversible entropy production rates. Equivalently, $\dot{S}\ge {\dot{S}}^{(r)}$. Just like in thermomechanics of non-fractal bodies [27], we now introduce the rate of irreversible entropy production $\rho {\dot{s}}^{(i)}$. With s denoting specific entropies and on account of (2.11), we arrive at a local form of the second law

$$0\le \rho {\dot{s}}^{(i)}=\rho \dot{s}+\frac{{\mathrm{\nabla }}_k^D{q}_k{,}_k}{T}-\frac{(q_k{\mathrm{\nabla }}_k^{D}T)}{T^2}+\rho h.$$ 3.18

Next, we recall the standard relation between the free energy density ψ, the internal energy density u, the entropy s and the absolute temperature T:ψ = u − T s. Thus, $\dot{\psi }=\dot{u}-s\dot{T}-T\dot{s}$.

On the other hand, with ψ being a function of γ_{j i} and κ_{j i} tensors, the internal variables α_{i j} (strain type) and ζ_{i j} (curvature–torsion type) and temperature T, we have

$$\rho \dot{\psi }=\rho \frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\gamma }_{ij}}{\dot{\gamma }}_{ij}+\rho \frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\alpha }_{ij}}{\dot{\alpha }}_{ij}+\rho \frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\kappa }_{ij}}{\dot{\kappa }}_{ij}+\rho \frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\zeta }_{ij}}{\dot{\zeta }}_{ij}+\rho \frac{\mathrm{\partial }\psi }{\mathrm{\partial }T}\dot{T}.$$ 3.19

We now adopt the conventional relations giving the (external and internal) quasi-conservative Cauchy and Cosserat (couple) stresses as well as the entropy density as gradients of ψ

$${\tau }_{ij}^{(q)}=\rho \frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\epsilon }_{ij}}{\beta }_{ij}^{(q)}=\rho \frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\alpha }_{ij}}{\mu }_{ij}^{(q)}=\rho \frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\kappa }_{ij}}{\eta }_{ij}^{(q)}=\rho \frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\zeta }_{ij}}s=-\frac{\mathrm{\partial }\psi }{\mathrm{\partial }T}.$$ 3.20

This is accompanied by a split of total Cauchy and micropolar stresses into their quasi-conservative and dissipative parts: ${\tau }_{ij}={\tau }_{ij}^{(q)}+{\tau }_{ij}^{(d)},\hspace{0.17em}{\mu }_{ij}={\mu }_{ij}^{(q)}+{\mu }_{ij}^{(d)}$, along with the relationships between the internal quasi-conservative and dissipative stresses: ${\beta }_{ij}^{(q)}=-{\beta }_{ij}^{(d)}$, ${\eta }_{ij}^{(q)}=-{\eta }_{ij}^{(d)}$. On account of the energy balance, these relations lead to

$$T\rho \dot{s}={\tau }_{ij}^{(d)}{\dot{\gamma }}_{ij}+{\beta }_{ij}^{(d)}{\dot{\alpha }}_{ij}+{\mu }_{ij}^{(d)}{\dot{\kappa }}_{ij}+{\eta }_{ij}^{(d)}{\dot{\zeta }}_{ij}-{\mathrm{\nabla }}_k^D{q}_k+\rho h.$$ 3.21

Recalling (3.18), we find the local form of the second law of thermodynamics in terms of time rates of strains and internal parameters

$$0\le T\rho {\dot{s}}^{(i)}={\tau }_{ij}^{(d)}{\dot{\gamma }}_{ij}+{\beta }_{ij}^{(d)}{\dot{\alpha }}_{ij}+{\mu }_{ij}^{(d)}{\dot{\kappa }}_{ij}+{\eta }_{ij}^{(d)}{\dot{\zeta }}_{ij}-\frac{(q_k{\mathrm{\nabla }}_k^{D}T)}{T}+\rho h,$$ 3.22

which is the fractal Clausius–Duhem inequality, i.e. a generalization of Clausius–Duhem inequality to fractal dissipative media with internal parameters. Upon dropping the internal parameters (which is the case in non-dissipative mechanical phenomena), the terms ${\beta }_{ij}^{(d)}{\dot{\alpha }}_{ij}$ and ${\eta }_{ij}^{(d)}{\dot{\zeta }}_{ij}$ vanish, implying that ${\tau }_{ij}^{(d)}$ reduces to the symmetric Cauchy stress and ${\dot{\gamma }}_{ij}$ reduces to the deformation rate d_{i j}: = v_(i,j). On the other hand, upon neglecting the micropolar effects (and thus reverting to classical continuum mechanics), the terms ${\mu }_{ij}^{(d)}{\dot{\kappa }}_{ij}$ and ${\eta }_{ij}^{(d)}{\dot{\zeta }}_{ij}$ vanish.

For non-fractal bodies, the stress tensor τ_{i j} reverts back to σ_{i j}, and (3.22) reduces to the simple well-known form [27]: $0\le T\rho {\dot{s}}^{(i)}={\sigma }_{ij}^{(d)}{\dot{\gamma }}_{ij}-(q_{k}T{,}_k)/T+\rho h$.

4. Thermodiffusion in deformable materials

The formulation developed above can now be applied to more specific thermomechanics problems. The focus here is on coupled and irreversible solid–fluid–diffusion phenomena. We begin with Fick’s Law in fractal media, and then proceed to a general formulation involving all the preceding concepts. Upon setting all the fractal dimensions to integers, one would recover the classical thermodiffusion and thermoelasticity models.

(a). Basic model

Many materials encountered in nature display fractal (complex and multiscale) systems of fissures and pores, where diffusion is present, and even coupled, with thermomechanical fields. Such phenomena motivate a generalization of continuum mechanics models [13,28] to thermodiffusion in such media. Consider a body $\mathcal{B}$, identify any subset $\mathcal{W}\subset \mathcal{B}$ with $\mathrm{\partial }\mathcal{W}$, and set up a balance of flow of matter:

$${\int }_{\mathrm{\partial }\mathcal{W}}\mathit{\eta }\cdot \mathbf{\text{n}}\hspace{0.17em}\mathrm{d}{\mathrm{S}}_{\mathrm{d}}={\int }_{\mathcal{W}}(\sigma -\frac{\mathrm{\partial }\mathrm{c}}{\mathrm{\partial }\mathrm{t}})\mathrm{d}{\mathrm{V}}_{\mathrm{D}},$$ 4.1

where η is the flux, c is the concentration of the fluid, and σ is the mass source intensity. By the Green–Gauss theorem applied to the left-hand side, we have

$${\int }_{\mathcal{W}}\frac{{\eta }_k{,}_k}{c_1^{(k)}}\mathrm{d}{\mathrm{V}}_{\mathrm{D}}={\int }_{\mathcal{W}}(\sigma -\frac{\mathrm{\partial }\mathrm{c}}{\mathrm{\partial }\mathrm{t}})\mathrm{d}{\mathrm{V}}_{\mathrm{D}},$$ 4.2

which implies the local form

$$\frac{1}{c_1^{(k)}}\frac{\mathrm{\partial }{\eta }_k}{\mathrm{\partial }{x}_k}\equiv {\mathrm{\nabla }}_k^D{\eta }_k=\sigma -\frac{\mathrm{\partial }c}{\mathrm{\partial }t}.$$ 4.3

Experiments suggest Fick’s Law ${\eta }_k=-D_{kl}{\mathrm{\nabla }}_{l}c$. Now, recall our earlier results [5] that the constitutive law should not change, whereas the field quantities should be modified as dictated by the fractal formulation. This strategy ensures that the field (e.g. wave) equation derived by the variational approach is the same as that independently derived by the Newtonian approach. We note that this justification is confirmed by experiments on the size scaling effects due to materials fractal geometry [29]. Thus, we have

$${\eta }_k=-D_{kl}{\mathrm{\nabla }}_l^{D}c.$$ 4.4

In other words, the conventional gradient should be replaced by the fractal one. When combined, the relations (4.3) and (4.4) result in

$$\frac{1}{c_1^{(k)}}\left(D_{kl}\frac{c{,}_l}{c_1^{(l)}}\right){,}_k\equiv {\mathrm{\nabla }}_k^D(D_{kl}{\mathrm{\nabla }}_l^{D}c)=\frac{\mathrm{\partial }c}{\mathrm{\partial }t}-\sigma .$$ 4.5

Now, consider the effect of diffusion on the deformability of the body which causes an irreversible deformation. With the latter represented by the strain

$${\gamma }_{ij}^d=c{\alpha }_c{\delta }_{ij},$$ 4.6

where α_c is the diffusive expansion coefficient, and the elastic strain ${\epsilon }_{ij}^e$ related to τ_{i j} by

$${\tau }_{ij}=C_{ijkl}^{(1)}{\gamma }_{kk}^e,$$ 4.7

where $C_{ijkl}^{(1)}$ is the first stiffness tensor of micropolar elasticity, we have the total strain

$${\gamma }_{ij}={\gamma }_{ij}^e+{\gamma }_{ij}^d.$$ 4.8

(If the elastic response is isotropic, (4.7) becomes ${\tau }_{ij}=\lambda {\gamma }_{kk}^e{\delta }_{ij}+(\mu +\alpha ){\gamma }_{ij}^e+(\mu -\alpha ){\gamma }_{ij}^e$, where λ and μ are the Lamé constants.) Substituting (4.6) and (4.7) into (4.8) and solving for the force stresses yields

$${\tau }_{ij}=C_{ijkl}^{(1)}{\gamma }_{ij}-{\nu }_{c}c{\delta }_{ij},$$ 4.9

where ν_c = (3λ + 2μ)α_c. (As an aside, note here the analogy of (4.9) to the Duhamel–Neumann equation of thermoelasticity.) The generalized Hooke Law is augmented by the micropolar part

$${\mu }_{ij}=C_{ijkl}^{(2)}{\kappa }_{ij},$$ 4.10

where $C_{ijkl}^{(2)}$ is the second stiffness tensor of micropolar elasticity. (If the elastic response is isotropic, (4.10) becomes μ_{i j} = βκ_{k k}δ_{i j} + (γ + ε)κ_{i j} + (γ − ε)κ_{i j}, where β, γ and ε are the micropolar elastic moduli.)

Substituting (4.9) and (4.10), respectively, into the linear momentum balance (3.10) and angular momentum balance (3.11) gives a generalization of elastodynamics accounting for the gradient of concentration

$$\begin{array}{rl}& {\mathrm{\nabla }}_j^D[C_{ijkl}^{(1)}\frac{1}{2}({\mathrm{\nabla }}_l^D{u}_k+{\mathrm{\nabla }}_k^D{u}_l)]{,}_j=\rho {\ddot{u}}_i-{\nu }_c{\mathrm{\nabla }}_i^D(c)\\ \mathrm{and}& \frac{e_{ijk}}{c_1^{(j)}}{\tau }_{jk}+{\mathrm{\nabla }}_j^D[C_{ijkl}^{(2)}{\mathrm{\nabla }}_l^D{\phi }_k]{,}_j=i_{ij}{\dot{\phi }}_j.\end{array}\}$$ 4.11

Here we have assumed the mass density and inertia tensor to be constant with time. As a special case, assuming an isotropic elastic response of a generally anisotropic fractal medium, we have a generalization of the Navier equation of elasticity accounting for the gradient of concentration

$$\begin{array}{rl}& \mu {\mathrm{\nabla }}_j^D{\mathrm{\nabla }}_j^D{u}_i+(\lambda +\mu ){\mathrm{\nabla }}_i^D{\mathrm{\nabla }}_j^D{u}_j=\rho {\ddot{u}}_i-{\nu }_c{\mathrm{\nabla }}_i^{D}c\\ \mathrm{and}& \frac{e_{ijk}}{c_1^{(j)}}{\tau }_{jk}+\mu {\mathrm{\nabla }}_j^D{\mathrm{\nabla }}_j^D{u}_i+(\lambda +\mu ){\mathrm{\nabla }}_i^D{\mathrm{\nabla }}_j^D{u}_j=i_{ij}{\dot{\phi }}_j.\end{array}\}$$ 4.12

With (4.5), we thus have a system of field equations governing diffusion coupled with mechanics.

A simplified solution method, assuming uncoupled processes, where the diffusion is slow and mechanics is fast, would first require a solution of (4.5) for the c field, and then using it to drive the mechanics equation (4.12).

(b). Thermodynamic basis of diffusion in a deformable material

Recalling the energy balance (3.16), the local entropy balance gives

$$\rho T\dot{s}=-{\mathrm{\nabla }}_i^D{q}_i+h\rho -\mathcal{M}\dot{c},$$ 4.13

where $\mathcal{M}$ is the chemical potential. Eliminating $-{\mathrm{\nabla }}_i^D{q}_i$ from the above equation and (3.16) gives

$$\rho \dot{u}={\tau }_{ij}{\dot{\gamma }}_{ij}+{\mu }_{ij}{\dot{\kappa }}_{ij}+T\dot{s}+\mathcal{M}\dot{c}.$$ 4.14

The total differential is

$$\rho \dot{u}=\frac{\mathrm{\partial }u}{\mathrm{\partial }{\gamma }_{ij}}{\dot{\gamma }}_{ij}+\frac{\mathrm{\partial }u}{\mathrm{\partial }{\kappa }_{ij}}{\dot{\kappa }}_{ij}+\frac{\mathrm{\partial }u}{\mathrm{\partial }s}\dot{s}+\frac{\mathrm{\partial }u}{\mathrm{\partial }c}\dot{c}.$$ 4.15

A comparison of both relations above implies

$${\tau }_{ij}=\frac{\mathrm{\partial }u}{\mathrm{\partial }{\gamma }_{ij}},{\mu }_{ij}=\frac{\mathrm{\partial }u}{\mathrm{\partial }{\kappa }_{ij}},T=\frac{\mathrm{\partial }u}{\mathrm{\partial }s},\mathcal{M}=\frac{\mathrm{\partial }u}{\mathrm{\partial }c}.$$ 4.16

Assuming the internal energy ψ(γ_{i j}, κ_{i j}, s, c) in the quadratic form

$$u=\frac{1}{2}{\gamma }_{ij}{C}_{ijkl}^{(1)}{\gamma }_{kl}+\frac{1}{2}{\kappa }_{ij}{C}_{ijkl}^{(2)}{\kappa }_{kl}-{\nu }_{s}s{\gamma }_{kk}-{\nu }_{c}c{\gamma }_{kk}+\frac{m}{2}{s}^2+\frac{\alpha }{2}{c}^2-\delta cs,$$ 4.17

which makes explicit the presence of the first and second stiffness tensors ($C_{ijkl}^{(1)}$, $C_{ijkl}^{(2)}$), along with the thermal expansion (ν_s s) and diffusive expansion (ν_c c) effects, their independent quadratic contributions (m s²/2, αc²/2) and their mutual coupling (δc s). The coupling between classical (strains γ_{i j}) and micropolar (curvature–torsion κ_{i j}) effects would be present in the case of a chiral-type microstructure, e.g. [8].

On account of (4.16), taking θ = T − T₀, we find

$$\begin{array}{rl}& {\tau }_{ij}=C_{ijkl}^{(1)}{\gamma }_{kl}-({\nu }_{s}s+{\nu }_{c}c){\delta }_{ij},{\mu }_{ij}=C_{ijkl}^{(2)}{\kappa }_{kl}\\ \mathrm{and}& \theta =-{\nu }_s{\gamma }_{kk}+ms-\delta c,\mathcal{M}=-{\nu }_c{\gamma }_{kk}+\alpha c-\delta s.\end{array}\}$$ 4.18

Introducing the free energy ψ(γ_{i j}, κ_{i j}, θ, c) = u − T s, (4.16) implies

$$\psi ({\gamma }_{ij},{\kappa }_{ij},\theta ,c)={\tau }_{ij}{\dot{\gamma }}_{ij}+{\mu }_{ij}{\dot{\kappa }}_{ij}-s\dot{T}+\mathcal{M}\dot{c},$$ 4.19

so that

$$\psi =\frac{1}{2}{\gamma }_{ij}{C}_{ijkl}^{(1)}{\gamma }_{kl}+\frac{1}{2}{\kappa }_{ij}{C}_{ijkl}^{(2)}{\kappa }_{ij}-{\gamma }_{\theta }s{\epsilon }_{kk}-{\nu }_{c}c{\epsilon }_{kk}-\frac{1}{2m}{\theta }^2+\frac{\alpha }{2}{c}^2-\frac{1}{\delta }c\theta .$$ 4.20

From this we find

$$\begin{array}{rl}& {\tau }_{ij}=\frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\gamma }_{ij}}=C_{ijkl}^{(1)}{\gamma }_{kl}-({\gamma }_{\theta }s+{\nu }_{c}c){\delta }_{ij},{\mu }_{ij}=\frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\kappa }_{ij}}=C_{ijkl}^{(2)}{\kappa }_{kl}\\ \mathrm{and}& s=-\frac{\mathrm{\partial }\psi }{\mathrm{\partial }\theta }={\gamma }_{\theta }{\epsilon }_{kk}+\frac{1}{m}\theta +\frac{1}{\delta }c,\mathcal{M}=\frac{\mathrm{\partial }\psi }{\mathrm{\partial }c}=-{\nu }_c{\epsilon }_{kk}+\alpha c-\frac{1}{\delta }\theta .\end{array}\}$$ 4.21

Returning to the balance of entropy, on account of the mass conservation (4.3), while for simplicity assuming σ = 0, we have

$$T\dot{s}=-{\mathrm{\nabla }}_i^D{q}_i+\mathcal{M}{\mathrm{\nabla }}_k^D{\eta }_k.$$ 4.22

This relation may be written in the following form:

$$\rho \dot{s}=-{\mathrm{\nabla }}_i^D\left(\frac{q_i}{T}\right)+\mathcal{M}{\mathrm{\nabla }}_k^D{\eta }_k+\rho {\dot{s}}^{(i)},$$ 4.23

where the first two terms represent the exchange of entropy with the environment, while the third one gives the non-negative rate of irreversible entropy production:

$$\rho {\dot{s}}^{(i)}=-[\frac{q_i{\mathrm{\nabla }}_i^{D}T}{T^2}+{\eta }_i{\mathrm{\nabla }}_i^D\left(\frac{\mathcal{M}}{T}\right)]=\frac{1}{T}[q_i{Y}_i+{\eta }_i{Z}_i].$$ 4.24

This can be interpreted as the sum of two irreversible processes:

$$\rho {\dot{s}}^{(i)}=\frac{1}{T}[q_i{Y}_i+{\eta }_i{Z}_i],$$ 4.25

in which the dissipative forces, respectively conjugate to the thermodynamic velocities q_i and η_i, are $Y_i=-{\mathrm{\nabla }}_i^{D}T/T$, $Z_i=-T{\mathrm{\nabla }}_{\mathrm{i}}^{\mathrm{D}}(\mathcal{M}/\mathrm{T})$.

As commonly done in thermomechanics, we set ${\dot{s}}^{(i)}$ equal to the dissipation function ϕ, adopted as a functional: $\rho {\dot{s}}^{(i)}=\rho \varphi (Y_i,Z_i)$. Taking the latter as a quadratic form, we obtain linear constitutive equations

$$q_i=-\frac{L_{qq}}{T}{\mathrm{\nabla }}_j^{D}T-L_{q\eta }{\mathrm{\nabla }}_{\mathrm{j}}^{\mathrm{D}}\left(\frac{\mathcal{M}}{\mathrm{T}}\right)\mathrm{and}{\eta }_{\mathrm{i}}=-\frac{{\mathrm{L}}_{\eta \mathrm{q}}}{\mathrm{T}}{\mathrm{\nabla }}_{\mathrm{j}}^{\mathrm{D}}\mathrm{T}-{\mathrm{L}}_{\eta \eta }{\mathrm{\nabla }}_{\mathrm{j}}^{\mathrm{D}}\left(\frac{\mathcal{M}}{\mathrm{T}}\right).$$ 4.26

In the above, we have assumed isotropic responses since accounting for anisotropy would only complicate the writing. The relation with the system being close to the thermodynamic equilibrium, the Onsager reciprocity L_qη = L_ηq holds. Effectively, (4.26) may be written as

$$q_i=-k_{ij}\frac{{\mathrm{\nabla }}_j^{D}T}{T}-s{\eta }_i\mathrm{and}{\eta }_{\mathrm{i}}=-\frac{{\mathrm{L}}_{\mathrm{ij}}^{(\eta \mathrm{q})}}{{\mathrm{T}}_0}{\mathrm{\nabla }}_{\mathrm{j}}^{\mathrm{D}}\mathrm{M}-\frac{{\mathrm{L}}_{\mathrm{ij}}^{(\eta \eta )}}{{\mathrm{T}}_0}(\mathrm{s}-\frac{\mathcal{M}}{{\mathrm{T}}_0}){\mathrm{\nabla }}_{\mathrm{j}}^{\mathrm{D}}\mathrm{T},$$ 4.27

where α = L_qη/L_ηη, $k=(L_{qq}{L}_{\eta \eta }-L_{q\eta }^2)/(T_0{L}_{\eta \eta })>0$. The latter inequality as well as L_{q q} > 0 and L_ηη > 0 follow from (3.22).

Substituting (4.21)₃ and (4.21)₁ into the entropy balance (4.22), upon linearization, we find

$$k^{\mathrm{\prime }}{\mathrm{\nabla }}_j^D{\mathrm{\nabla }}_j^D\theta -n\dot{\theta }-{\nu }_{\theta }{\mathrm{\nabla }}_k^D{u}_k-d\dot{c}=0,k^{\mathrm{\prime }}=\frac{k}{T_0}.$$ 4.28

On the other hand, substituting (4.27)₂ into (4.3) results in a fractal diffusion equation

$$\frac{L_{ij}^{(\eta \eta )}}{T_0}{\mathrm{\nabla }}_k^D{\mathrm{\nabla }}_k^D{\eta }_k=\frac{\mathrm{\partial }c}{\mathrm{\partial }t}.$$ 4.29

First, one can express $\mathcal{M}$ through (4.21)₄ leading to

$$(a{\mathrm{\nabla }}_k^D{\mathrm{\nabla }}_k^{D}c-m\dot{c})-{\mathrm{\nabla }}_k^D{\mathrm{\nabla }}_k^D({\nu }_c{\mathrm{\nabla }}_j^D{u}_j+\theta d)=0\mathrm{and}\mathrm{m}=\frac{{\mathrm{T}}_0}{{\mathrm{L}}^{(\eta \eta )}}.$$ 4.30

Now, substituting (4.21)_1,2 into (3.11) and (3.10), we obtain

$$\begin{array}{rl}& C_{jilk}^{(1)}{\mathrm{\nabla }}_k^D{\mathrm{\nabla }}_j^D{u}_l-e_{pkl}{C}_{jilk}^{(1)}{\mathrm{\nabla }}_j^D{\phi }_p=\rho {\ddot{u}}_i+{\nu }_c{\mathrm{\nabla }}_i^{D}c+{\nu }_{\theta }{\mathrm{\nabla }}_i^D\theta \\ \mathrm{and}& \frac{e_{ijk}}{c_1^{(j)}}{C}_{jknm}^{(1)}({\mathrm{\nabla }}_n^D{u}_m-e_{pnm}{\phi }_p)+C_{jknm}^{(2)}{\mathrm{\nabla }}_l^D{\mathrm{\nabla }}_j^D{\phi }_k=I_{ij}{\ddot{\phi }}_j.\end{array}\}$$ 4.31

The above two equations together with (4.28) and (4.29) form a set of three relations completely describing the thermoelastic diffusion in a fractal medium. Note that in the basic model there was no temperature diffusion equation and the term ${\nu }_{\theta }{\mathrm{\nabla }}_i^D\theta$ was not present in (4.11), which was arrived at without any thermodynamics. In the special case of a mechanically isotropic response, (4.31) simplify to

$$\begin{array}{rl}& (\mu +\alpha ){\mathrm{\nabla }}_j^D{\mathrm{\nabla }}_j^D{u}_i+(\lambda +\mu -\alpha ){\mathrm{\nabla }}_i^D{\mathrm{\nabla }}_j^D{u}_j+2\alpha e_{ikj}{\mathrm{\nabla }}_k^D{\phi }_j=\rho {\ddot{u}}_i+{\nu }_c{\mathrm{\nabla }}_i^{D}c+{\nu }_{\theta }{\mathrm{\nabla }}_i^D\theta \\ \mathrm{and}& (\gamma +\epsilon ){\mathrm{\nabla }}_j^D{\mathrm{\nabla }}_j^D{\phi }_i-4\alpha {\phi }_i+(\gamma +\epsilon -\epsilon ){\mathrm{\nabla }}_i^D{\mathrm{\nabla }}_j^D{\phi }_j+2\alpha e_{ikj}{\mathrm{\nabla }}_k^D{u}_j=I_{ij}{\ddot{\phi }}_j\end{array}\}$$ 4.32

or, recalling (2.16)–(2.21) and written symbolically,

$$\begin{array}{r}\begin{array}{rl}& (\mu +\alpha )\mathrm{div}(\mathrm{grad}\hspace{0.17em}\mathbf{\text{u}})+(\lambda +\mu -\alpha )\mathrm{grad}\hspace{0.17em}\mathrm{div}\hspace{0.17em}\mathrm{u}+2\alpha \hspace{0.17em}\mathrm{curl}\hspace{0.17em}\mathit{\phi }=\rho \ddot{\mathbf{\text{u}}}+{\nu }_{\mathrm{c}}\hspace{0.17em}\mathrm{grad}\hspace{0.17em}\mathrm{c}+{\nu }_{\theta }\hspace{0.17em}\mathrm{grad}\hspace{0.17em}\theta \\ \mathrm{and}& (\gamma +\epsilon )\mathrm{div}(\mathrm{grad}\hspace{0.17em}\mathit{\phi })-4\alpha \mathit{\phi }+(\gamma +\epsilon -\epsilon )\mathrm{grad}\hspace{0.17em}\mathrm{div}\hspace{0.17em}\mathit{\phi }+2\alpha \hspace{0.17em}\mathrm{curl}\hspace{0.17em}\mathbf{\text{u}}=\mathbf{\text{I}}\cdot \ddot{\mathit{\phi }}.\end{array}\}\end{array}$$ 4.33

When the fractal geometry of the medium vanishes in the sense of reduction to the Euclidean structure, the above coincide with the relations of [14]. As discussed in that work, there is a second way to proceed: to work with $\mathcal{M}$ instead of c.

(c). Special cases of thermoelastic diffusion

The isothermal diffusion represents a special case of the above studies. To this end, consider the diffusion occurring under isothermal conditions. With T = T₀ (4.19) reduces to $\psi ({\gamma }_{ij},{\kappa }_{ij},\theta ,c)={\tau }_{ij}{\dot{\gamma }}_{ij}+{\mu }_{ij}{\dot{\kappa }}_{ij}+\mathcal{M}\dot{c}$, while (4.22) simplifies to $T_0\dot{s}=h\rho -\mathcal{M}{\mathrm{\nabla }}_k^D{\eta }_k$. Additionally, the constitutive equations (4.26) reduce to

$$q_i=0,{\eta }_i=-\frac{L_{ij}^{(\eta \eta )}}{T_0}{\mathrm{\nabla }}_j^D\mathcal{M}.$$ 4.34

Now, (4.20) becomes

$$\psi =\frac{1}{2}{\gamma }_{ij}{C}_{ijkl}^{(1)}{\gamma }_{kl}+\frac{1}{2}{\kappa }_{ij}{C}_{ijkl}^{(2)}{\kappa }_{ij}-{\nu }_{c}c{\epsilon }_{kk}+\frac{\alpha }{2}{c}^2$$ 4.35

which, with reference to (4.21), leads to

$${\tau }_{ij}=\frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\gamma }_{ij}}=C_{ijkl}^{(1)}{\gamma }_{kl}-{\nu }_{c}c{\delta }_{ij},{\mu }_{ij}=\frac{\mathrm{\partial }\psi }{\mathrm{\partial }{\kappa }_{ij}}=C_{ijkl}^{(2)}{\kappa }_{kl},\mathcal{M}=\frac{\mathrm{\partial }\psi }{\mathrm{\partial }c}=-{\nu }_c{\gamma }_{kk}+\alpha c.$$ 4.36

The first of these is recognized as (4.9), which was initially formulated without any thermodynamical considerations. The second one is due to the presence of micropolar effects. Regarding the third, as we shall see below, ν_c > 0 and α > 0, signifying that the chemical potential increases with concentration going up and decreases with the dilatation γ_{k k} increasing.

Eliminating $\mathcal{M}$ from (4.34) and (4.36), results in

$${\eta }_i=-D_{ij}{\mathrm{\nabla }}_j^{D}c+{\chi }_{ij}{\mathrm{\nabla }}_j^D{\mathrm{\nabla }}_k^D{u}_k,$$ 4.37

where $D_{ij}=(L_{ij}^{(\eta \eta )}\alpha )/T_0>0$, $\chi =(L_{ij}^{(q\eta )}{\nu }_c)/T_0$. Effectively, this is a generalized Fick’s Law in fractal media. In the case of isotropic response (D_{i j} = Dδ_{i j}, χ_{i j} = χδ_{i j}), it reads ${\eta }_i=-D{\mathrm{\nabla }}_i^{D}c+\chi {\mathrm{\nabla }}_i^D{\mathrm{\nabla }}_j^D{u}_j$. Next, in the case of an isotropic and non-fractal structure it reads ${\eta }_i=-D{\mathrm{\nabla }}_{i}c+\chi {\mathrm{\nabla }}_i{\mathrm{\nabla }}_j{u}_j$ or, in symbolic notation, $\mathit{\eta }=-D\hspace{0.17em}\mathrm{grad}\hspace{0.17em}\mathrm{c}+\chi \hspace{0.17em}\mathrm{grad}\hspace{0.17em}\mathrm{div}\hspace{0.17em}\mathbf{\text{u}}$.

Upon substituting (4.37) into the balance of diffusing matter (4.3), we obtain

$${\mathrm{\nabla }}_k^D(D_{kl}{\mathrm{\nabla }}_l^{D}c)-\frac{\mathrm{\partial }c}{\mathrm{\partial }t}={\chi }_{ij}{\mathrm{\nabla }}_j^D\cdot {\mathrm{\nabla }}_k^D{\mathrm{\nabla }}_l^D{u}_l,$$ 4.38

which is coupled with (4.31). Similarly, as another special case, one can develop a model of a diffusionless thermoelasticity.

5. Conclusion

Continuum-type formulation of thermomechanics of porous media with a generally anisotropic, product-like fractal structure is made possible through a certain type of fractional calculus. Starting from calculus on product measures which has several desirable features, the article develops a fractal tensor calculus in Euclidean space. The construction of a continuum hinges on expressing the fundamental balance laws for fractal media in terms of fractal integrals and, then, converting them to integer-order integrals in conventional (Euclidean) space. An important aspect is the general lack of symmetry of the Cauchy stress tensor, necessitating the adoption of a micropolar continuum model [30]. The formulation is applied to develop a model of diffusion in fractal thermoelastic media. Initial-boundary value problems (IBVPs) can then be handled by analytical–numerical methods employed in [8,9] which (i) introduced fractal generalizations of harmonic and spherical Bessel functions, and (ii) relied on finite-element methods.

The new line transformation coefficient is novel compared to the previous works. It addresses (1) the objectivity of the formulation by introducing the integral limits, and (2) the specific density distribution and the differentiations of fractals with the same fractal dimension but different behaviours (fractal dimension alone is not enough to characterize density distributions). The formerly developed framework is still kept as simple as possible, with the local balance laws retaining the original forms, except for a modified form of the line transformation coefficient.

The proposed methodology broadens the applicability of continuum mechanics/physics to studies of highly complex and fractal-type media (multiscale polycrystals, polymer clusters, gels, rocks, micro-cracked materials, percolating networks, nervous systems, pulmonary systems, etc.). Such types of materials systems, conventionally reserved on the domain of condensed matter physics, geophysics and biophysics, will now become open to studies in the vein of continuum mechanics. More specific models can be developed within this continuum mechanics framework in order to tackle IBVPs of very complex, multiscale fractal materials. This approach has also led to a new tool in quantum mechanics [31].

The main unsolved problems and challenges in pursuing this line of research include

• —experimental validations carried out on specific fractal media. This should involve (i) measuring the fractal dimension(s) and density distributions of, say, a fractal porous material, (ii) characterizing the line transformation coefficient from experimental measurements, and (iii) solving an IVBP under the same loading conditions;
• —generalization of the currently deterministic formulation to random fractal media;
• —restriction to product-like fractals to be relaxed while keeping the theory relatively simple.

Data accessibility

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Authors' contributions

J.L. and M.O.-S. jointly conceived of and designed the study, and drafted the manuscript. J.L. has primarily contributed to §2, while M.O.-S. to §4. Both authors read and approved the manuscript.

Competing interests

We declare we have no competing interests.

Funding

This study was supported by the National Science Foundation(grant no. CMMI-1462749) and start-up funds at the University of Massachusetts Dartmouth.