Effective Thermal Conductivity of Transversely Isotropic Materials with Concave Pores

Abstract

The aim of this paper is to extend recent elastic work to thermal problem. In the first part of the paper, approximate relations for the resistivity contribution tensor of pores of two reference shapes, supersphere and axisymmetrical superspheroid, are developed on the basis of 3D Finite Element Modelling, presented in the companion paper, and known exact solutions for the limiting cases of spherical pores. In the second part application to effective elastic coefficients of transversely isotropic materials such as clay rocks, in the frame of homogenization theory, is presented to illustrate the impact of concavity parameter on overall properties.

Igor Sevostianov is deceased.

Appendices

Appendix A Background on Property Contribution Tensors

Property contribution tensors are used in micromechanics to describe the contribution of a single inhomogeneity to the property of interest (Kachanov and Sevostianov 2018). The average strain, over representative volume $\mid \Omega \mid$, can be represented as a sum

$$\begin{aligned} \underline{\varepsilon }{} = {\textbf{r}}_{0} : \underline{\Sigma }{} + \Delta \underline{\varepsilon }{} \end{aligned}$$

(A1)

where ${\textbf{r}}_{0}$ where is the resistivity tensor of the matrix and $\underline{\Sigma }{}$ is uniform remotely applied heat flux vector. The material is assumed to be linear; hence, the extra thermal gradient $\Delta \underline{\varepsilon }{}$ due to presence of an inhomogeneity $\mathcal {E}$ is a linear function of the applied heat flux vector

$$\begin{aligned} \Delta \underline{\varepsilon }{} = f\, {\textbf{H}}_{0}^{\mathcal {E}} : \underline{\Sigma }{} , \quad \text {with } f = \frac{\mid \mathcal {E} \mid }{\mid \Omega \mid } \end{aligned}$$

(A2)

where $\mid \mathcal {E} \mid$ is the pore volume and ${\textbf{H}}_{0}^{\mathcal {E}}$ is second-rank resistivity contribution tensor of the pore. The ${\textbf{H}}_{0}^{\mathcal {E}}$ tensor is determined by the shape and size of the inhomogeneity, as well as properties of the matrix and of the inhomogeneity material. This tensor is also affected by conductive interactions. In the non-interaction approximation, it is taken by treating the inhomogeneities as isolated ones. In the case of multiple inhomogeneities, the extra strain produced by m-th inhomogeneity is $\Delta \underline{\varepsilon }{}^{(m)} = {f}^{(m)}\, {\textbf{H}}_{0}^{\mathcal {E}(m)} : \underline{\Sigma }{}$ so that the extra thermal resistivity due to all the inhomogeneities is given by

$$\begin{aligned} \Delta \underline{\varepsilon }{} = \left[ \sum {f}^{(m)}\, {\textbf{H}}_{0}^{\mathcal {E}(m)}\right] : \underline{\Sigma }{} \end{aligned}$$

(A3)

Formula (A3) highlights the fundamental importance of the resistivity contribution tensors: these tensors have to be summed up and averaged in the context of the effective conductive properties. The sum

$$\begin{aligned} \sum {f}^{(m)}\, {\textbf{H}}_{0}^{\mathcal {E}(m)} \end{aligned}$$

(A4)

properly reflects compliance contributions of individual inhomogeneities and constitutes the general microstructural parameters in whose terms the effective compliance should be expressed. The conductivity contribution tensor denoted respectively by ${\textbf{N}}_{0}^{\mathcal {E}}$ allows to calculate the extra heat flux vector induced by the presence of the inhomogeneity in a dilute situation such that

$$\begin{aligned} \Delta \underline{\sigma }= - f\,{\textbf{N}}_{0}^{\mathcal {E}} \cdot \underline{E}{} \end{aligned}$$

(A5)

where $\underline{E}{}$ is the remotely applied thermal gradient vector. We recall that the average concentration tensors ${\textbf{A}}_{0}^{\mathcal {E}}$ and ${\textbf{B}}_{0}^{\mathcal {E}}$ are defined as

$$\begin{aligned} {\left\langle {\underline{\varepsilon }{}}\right\rangle }^{\mathcal {E}} = {\textbf{A}}_{0}^{\mathcal {E}} \cdot \underline{E}{} , \quad {\left\langle {\underline{\sigma }{}}\right\rangle }^{\mathcal {E}} = - {\textbf{B}}_{0}^{\mathcal {E}} \cdot \underline{E}{} \end{aligned}$$

(A6)

with

$$\begin{aligned} {\textbf{A}}_{0}^{\mathcal {E}} = {\left( {\boldsymbol{\lambda }}_{\mathcal {E}} - {\boldsymbol{\lambda }}_{0}\right) }^{-1} \cdot {\textbf{N}}_{0}^{\mathcal {E}} , \quad {\textbf{B}}_{0}^{\mathcal {E}} = {\boldsymbol{\lambda }}_{\mathcal {E}} \cdot {\textbf{A}}_{0}^{\mathcal {E}} \end{aligned}$$

(A7)

In the general case of non-ellipsoidal shapes, contribution and concentration tensors related to an inhomogeneity need to be calculated numerically. ${\textbf{H}}_{0}^{\mathcal {E}}$ and ${\textbf{N}}_{0}^{\mathcal {E}}$ can be interrelated as

$$\begin{aligned} {\textbf{H}}_{0}^{\mathcal {E}}= - {\textbf{r}}_{0} \cdot {\textbf{N}}_{0}^{\mathcal {E}} \cdot {\textbf{r}}_{0} , \quad {\textbf{N}}_{0}^{\mathcal {E}} = - {\boldsymbol{\lambda }}_{0} \cdot {\textbf{H}}_{0}^{\mathcal {E}} \cdot {\boldsymbol{\lambda }}_{0} \end{aligned}$$

(A8)

In the case of a homogeneous inhomogeneity one has

$$\begin{aligned} {\textbf{A}}_{0}^{\mathcal {E}} = {\left( {\boldsymbol{\lambda }}_{\mathcal {E}} - {\boldsymbol{\lambda }}_{0}\right) }^{-1} \cdot {\textbf{N}}_{0}^{\mathcal {E}} , \quad {\textbf{B}}_{0}^{\mathcal {E}} = {\boldsymbol{\lambda }}_{\mathcal {E}} \cdot {\textbf{A}}_{0}^{\mathcal {E}} \end{aligned}$$

(A9)

6.1.1 Appendix A.1 Case of an Ellipsoidal Homogeneous Inhomogeneity

The ellipsoidal homogeneous inhomogeneity is of particular interest in the present since analytical expressions of contribution and concentration tensors are available and can then further be compared to the numerical ones to validate the methodology presented to calculate property contribution tensors and concentration tensors. In the particular case of an ellipsoidal inhomogeneity $\mathcal {E}$ embedded in an infinite matrix 0 of conductivity ${\boldsymbol{\lambda }}_{0}$ and resistivity ${\textbf{r}}_{0}$ tensors, resistivity ${\textbf{H}}_{0}^{\mathcal {E}}$ and conductivity ${\textbf{N}}_{0}^{\mathcal {E}}$ contribution tensors write (see Kachanov and Sevostianov 2018 for details):

$$\begin{aligned} \begin{array}{l}{{\textbf{H}}_{0}^{\mathcal {E}} ={\left[ {\left( {\textbf{r}}_{\mathcal {E}}-{\textbf{r}}_{0}\right) }^{-1}+{\textbf{Q}}_{0}^{\mathcal {E}}\right] }^{-1}}, \quad {\textbf{N}}_{0}^{\mathcal {E}}={\left[ {{\left( {\boldsymbol{\lambda }}_{\mathcal {E}}-{\boldsymbol{\lambda }}_{0}\right) }^{-1}+{\textbf{P}}_{0}^{\mathcal {E}}}\right] }^{-1}\end{array} \end{aligned}$$

(A10)

where ${\textbf{P}}_{0}^{\mathcal {E}}$ and ${\textbf{Q}}_{0}^{\mathcal {E}}$ denote the second order Hill’s tensors of the inhomogeneity. Thermal gradient concentration tensor of the ellipsoidal inhomogeneity writes

$$\begin{aligned} {\textbf{A}}_{0}^{\mathcal {E}} = \left[ \textbf{i} + {\textbf{P}}_{0}^{\mathcal {E}} : \left( {\boldsymbol{\lambda }}_{\mathcal {E}} - {\boldsymbol{\lambda }}_{0} \right) \right] ^{-1} \end{aligned}$$

(A11)

The Hill polarization tensor and resistivity contribution tensor of a spheroidal inclusion aligned in a transversely isotropic host matrix is detailed below. See Giraud et al. (2019), Barthélémy (2008) for the complete solution of arbitrarily oriented ellipsoidal inhomogeneity embedded in an orthotropic matrix. One considers a transversely isotropic matrix of conductivity tensor $\boldsymbol{{\lambda }}_{0}$ ($\underline{n}{}$ denotes unit vector on the symmetry axis, in this paper $\underline{n}{} = \underline{e}{}_{3}$)

$$\begin{aligned} \boldsymbol{{\lambda }}_{0} = {\lambda }_{0} \left( {\nu }^{2}\,{\textbf{i}}_{T} + {\textbf{i}}_{N}\right) , \quad {\textbf{i}}_{N} = \underline{n} \otimes \underline{n} , \quad {\textbf{i}}_{T} = \textbf{i} - {\textbf{i}}_{N} \end{aligned}$$

(A12)

The Hill polarization tensor ${\textbf{P}}_{0}^{\mathcal {E}}$ of a spheroidal inclusion aligned in the directions of a transversely isotropic matrix (spheroid and matrix have the same symmetry axis) writes

$$\begin{aligned} {\textbf{P}}_{0}^{\mathcal {E}}=\frac{g(\nu \gamma )}{{\nu }^{2} {\lambda }_{0}} {\textbf{i}}_{T} + \frac{1 - 2\,g(\nu \gamma )}{{\lambda }_{0}} {\textbf{i}}_{N} \end{aligned}$$

(A13)

with shape function $g(\xi )$ (see Barthélémy 2008; Giraud et al. 2019)

(A14)

Resistivity contribution tensor ${\textbf{H}}_{0}^{\mathcal {E}}$ of an insulating ${\boldsymbol{\lambda }}_{\mathcal {E}} = \textbf{0}$ aligned spheroidal pore writes

$$\begin{aligned} {\textbf{H}}_{0}^{\mathcal {E}}=\frac{1}{{\nu }^{2} {\lambda }_{0}\left( 1 - g(\nu \gamma )\right) } {\textbf{i}}_{T} + \frac{1}{2\,{\lambda }_{0}\, g(\nu \gamma )} {\textbf{i}}_{N} \end{aligned}$$

(A15)

and the limiting case of the spherical pore $\gamma = 1$ embedded in an isotropic matrix $\nu = 1$ is recovered

$$\begin{aligned} {\textbf{H}}_{0}^{\mathcal {E}}=\frac{3}{2 {\lambda }_{0}} \textbf{i} \end{aligned}$$

(A16)

Resistivity contribution tensor of an aligned penny shaped crack embedded in a TI matrix writes

$$\begin{aligned} {\textbf{H}}_{0}^{\mathcal {E}}=\frac{2}{\pi \,{\lambda }_{0}\,\nu } {\textbf{i}}_{N} \end{aligned}$$

(A17)

Appendix B Numerical Results for Approximation Formula of Resistivity Contribution Tensors

Finite element results of a superspherical pore embedded in a TI matrix are given in table 4 of paper Du et al. (2020), and recalled in Table B1.

Table B1 Coefficients of piecewise functions $f_{11}^{\text {se}}$ and $f_{33}^{\text {se}}$ of superspherical pore

Finite element results of an axisymmetrical superspheroidal pore embedded in a TI matrix are given in table 3 of paper Du et al. (2020) and recalled in Table B2.

Table B2 Coefficients of piecewise functions $f_{11}^{\text {so}}$ and $f_{33}^{\text {so}}$ of axisymmetrical superspheroidal pore