Two-scale, non-local diffusion in homogenised heterogeneous media

Abstract

We study how and to what extent the existence of non-local diffusion affects the transport of chemical species in a composite medium. For our purposes, we prescribe the mass flux to obey a two-scale, non-local constitutive law featuring derivatives of fractional order, and we employ the asymptotic homogenisation technique to obtain an overall description of the species’ evolution. As a result, the non-local effects at the micro-scale are ciphered in the effective diffusivity, while at the macro-scale the homogenised problem features an integro-differential equation of fractional type. In particular, we prove that in the limit case in which the non-local interactions are neglected, classical results of asymptotic homogenisation theory are re-obtained. Finally, we perform numerical simulations to show the impact of the fractional approach on the overall diffusion of species in a composite medium. To this end, we consider two simplified benchmark problems, and report some details of the numerical schemes based on finite element methods.

Effective properties of fractional viscoelastic composites via two‐scale asymptotic homogenization

“…We start by denoting with $$$ {L}_{\mathrm{c}} $$$ and $$$ \ell $$$ the characteristic length scales of the composite medium and of its internal structure so that $$$$ 0<\varepsilon := \frac{\ell }{L_{\mathrm{c}}}\ll 1, $$$$ where $$$ \varepsilon $$$ is referred to as the smallness parameter. Moreover, following Ramírez‐Torres et al [26] and Di Stefano [31], we formally rewrite a given physical quantity $$$ \Phi :\mathcal{B}\times \left[0,{t}_{\mathrm{f}}\right[\to \mathrm{\mathbb{R}} $$$ in a two‐scale fashion, so that the dependence on the characteristic length scales is explicitly taken into account. Specifically, the multi‐scale version of a quantity $$$ \Phi \left(X,t\right) $$$ is written as [26, 31] $$$$ \Phi \left(X,t\right)=\varphi \left(x,y,t\right), $$$$ where the dimensionless variables $$$ x:= X/{L}_c $$$ and $$$ y:= X/\varepsilon $$$ are referred to as the macroscopic variable and the microscopic variable, respectively.…”

“…Moreover, following Ramírez‐Torres et al [26] and Di Stefano [31], we formally rewrite a given physical quantity $$$ \Phi :\mathcal{B}\times \left[0,{t}_{\mathrm{f}}\right[\to \mathrm{\mathbb{R}} $$$ in a two‐scale fashion, so that the dependence on the characteristic length scales is explicitly taken into account. Specifically, the multi‐scale version of a quantity $$$ \Phi \left(X,t\right) $$$ is written as [26, 31] $$$$ \Phi \left(X,t\right)=\varphi \left(x,y,t\right), $$$$ where the dimensionless variables $$$ x:= X/{L}_c $$$ and $$$ y:= X/\varepsilon $$$ are referred to as the macroscopic variable and the microscopic variable, respectively. We notice that, in this framework, the partial derivatives of $$$ \Phi $$$ with respect to the spatial coordinates $$$ {X}_i,i=1,2,3 $$$, of $$$ X $$$ can be expressed as …”

“…Moreover, following Ramírez-Torres et al [26] and Di Stefano [31], we formally rewrite a given physical quantity Φ ∶ B × [0, t f [→ R in a two-scale fashion, so that the dependence on the characteristic length scales is explicitly taken into account. Specifically, the multi-scale version of a quantity Φ(X, t) is written as [26,31] Φ(X, t) = 𝜑(x, 𝑦, t), (9) where the dimensionless variables x ∶= X∕L c and 𝑦 ∶= X∕𝜀 are referred to as the macroscopic variable and the microscopic variable, respectively. We notice that, in this framework, the partial derivatives of Φ with respect to the spatial coordinates X i , i = 1, 2, 3, of X can be expressed as…”

“…From the above expression, we can verify that 𝕖 𝜂𝑦 enjoys of the left-minor symmetry, that is, [𝕖 𝜂𝑦 ] abcd = [𝕖 𝜂𝑦 ] bacd . Now, by substituting the ansatz (26) in Equations (25a)-(25c) and taking into account the expression in (27), the second cell problem, usually referred to as the local problem, reads…”

“…Here, motivated by previous works (see, e.g., [17,20,21,[25][26][27]), we use the asymptotic homogenization method [28][29][30] to compute the effective properties of a composite with viscoelastic constitutive response. In particular, we conduct our calculations in the general setting of linear viscoelastic materials and specialize them for the case in which the local constitutive response of each constituent is described by a fractional viscoelastic model.…”