A simple model of porous media with elastic deformations and erosion or deposition

Abstract

This paper deals with a model for solids with porous channels filled by an incompressible isotropic fluid. The Darcy–Brinkman–Stokes law is derived, that represents a rate equation for the local mass flux of the fluid, presenting relaxation times in which this flux evolves towards its local-equilibrium value, viscous effects and a permeability tensor referring to a response of the system to an external agent, i.e. the fluid flow produced by a pressure gradient. The erosion/deposition phenomena in an elastic porous matrix are also studied and particular thermal porous metamaterials, that have interesting functionality, like in fluid flow cloaking, are illustrated as application of the obtained results. This derived model is completely in agreement with a theory formulated in the framework of the rational irreversible thermodynamics, where two internal variables are introduced (a symmetric structural porosity tensor and a symmetric second order tensor influencing viscous phenomena, that is interpreted as the symmetric part of the velocity gradient), when the results are considered in a first approximation and some suitable assumptions are done. The constitutive theory is worked out by using Liu’s and Wang’s theorems. The obtained theory has applications in several technological sectors, like physics of soil, pharmaceutics, physiology, etc., and contributes to the study of new porous metamaterials.

Appendices

Appendix A: Objective representation of scalar functions

Following [26,27,28] a scalar objective function f,  that depends on m scalar function, namely $a_1$, $a_2$,...,$a_m$ and l second-order symmetric tensors, namely $\textit{\textbf{A}}_1$, $\textit{\textbf{A}}_2$,...,$\textit{\textbf{A}}_l$, is represented as function of the following quantities, called invariants

$$\begin{aligned} a_s, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{A}}_i, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{A}}^2_i, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{A}}^3_i, \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}_i\textit{\textbf{A}}_j), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}^2_i\textit{\textbf{A}}_j), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}^2_i\textit{\textbf{A}}^2_j), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}_i\textit{\textbf{A}}_j\textit{\textbf{A}}_k), \end{aligned}$$

(79)

with $s=1,\ldots ,m$, $i,j,k=1,\ldots ,l$ and $i\ne j\ne k$.

Thus, we have

$$\begin{aligned} f=f\Big (a_s, {{\,\mathrm{tr}\,}}\textit{\textbf{A}}_i, {{\,\mathrm{tr}\,}}\textit{\textbf{A}}^2_i, {{\,\mathrm{tr}\,}}\textit{\textbf{A}}^3_i, {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}_i\textit{\textbf{A}}_j), {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}^2_i\textit{\textbf{A}}_j), {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}^2_i\textit{\textbf{A}}^2_j), {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}_i\textit{\textbf{A}}_j\textit{\textbf{A}}_k)\Big ), \end{aligned}$$

(80)

If we consider the scalar constitutive function $S=S(T,\varvec{\varepsilon },\textit{\textbf{m}},\textit{\textbf{r}})$ of our theoretical model we have $m=1,$ with $a_1\equiv T$, and $l=3$, with $\textit{\textbf{A}}_1\equiv \varvec{\varepsilon }$, $\textit{\textbf{A}}_2\equiv \textit{\textbf{m}}$, $\textit{\textbf{A}}_3\equiv \textit{\textbf{r}}$.

Then, we have $s=1$, $i,j,k=1,2,3$ and $i\ne j\ne k$ and the invariants for the entropy S are

$$\begin{aligned} \begin{aligned}&T, \quad {{\,\mathrm{tr}\,}}\varvec{\varepsilon }, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{m}}, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{r}}, \quad {{\,\mathrm{tr}\,}}\varvec{\varepsilon }^2, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{m}}^2, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{r}}^2, \quad {{\,\mathrm{tr}\,}}\varvec{\varepsilon }^3, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{m}}^3, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{r}}^3, \quad {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }\textit{\textbf{m}}), \\&{{\,\mathrm{tr}\,}}(\varvec{\varepsilon }\textit{\textbf{r}}), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{m}}\textit{\textbf{r}}), \quad {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{m}}), \quad {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{r}}), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{m}}^2\varvec{\varepsilon }), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{m}}^2\textit{\textbf{r}}), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{r}}^2\varvec{\varepsilon }), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{r}}^2\textit{\textbf{m}}), \\&{{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{m}}^2), \quad {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{r}}^2), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{m}}^2\textit{\textbf{r}}^2), \quad {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }\textit{\textbf{m}}\textit{\textbf{r}}), \end{aligned} \end{aligned}$$

(81)

or in Cartesian components

$$\begin{aligned} \begin{aligned}&T, \quad \varepsilon _{ii}, \quad m_{ii}, \quad r_{ii}, \quad \varepsilon _{ij}\varepsilon _{ji}, \quad m_{ij}m_{ji}, \quad r_{ij}r_{ji}, \quad \varepsilon _{ij}\varepsilon _{jk}\varepsilon _{ki}, \quad m_{ij}m_{jk}m_{ki}, \quad r_{ij}r_{jk}r_{ki}, \quad \varepsilon _{ij}m_{ji}, \\&\varepsilon _{ij}r_{ji}, \quad m_{ij}r_{ji}, \quad \varepsilon _{ij}\varepsilon _{jk}m_{ki}, \quad \varepsilon _{ij}\varepsilon _{jk}r_{ki}, \quad m_{ij}m_{jk}\varepsilon _{ki}, \quad m_{ij}m_{jk}r_{ki}, \quad r_{ij}r_{jk}\varepsilon _{ki}, \quad r_{ij}r_{jk}m_{ki}, \\&\varepsilon _{ij}\varepsilon _{jk}m_{kl}m_{li}, \quad \varepsilon _{ij}\varepsilon _{jk}r_{kl}r_{li}, \quad m_{ij}m_{jk}r_{kl}r_{li}, \quad \varepsilon _{ij}m_{jk}r_{ki}. \end{aligned} \end{aligned}$$

(82)

Being $\varvec{\varepsilon }$, $\textit{\textbf{m}}$ and $\textit{\textbf{r}}$ symmetric tensors, it is also possible to write, for instance, $\varepsilon _{ij}\varepsilon _{ji}=\varepsilon _{ij}\varepsilon _{ij}$, $m_{ij}r_{ji}=m_{ij}r_{ij}$.

Assuming for S a polynomial form, S may be expressed in the form

$$\begin{aligned} \begin{aligned} S&= S^1T+S^2{{\,\mathrm{tr}\,}}\varvec{\varepsilon }+S^3{{\,\mathrm{tr}\,}}\textit{\textbf{m}}+S^4{{\,\mathrm{tr}\,}}\textit{\textbf{r}}+S^5{{\,\mathrm{tr}\,}}\varvec{\varepsilon }^2+S^6{{\,\mathrm{tr}\,}}\textit{\textbf{m}}^2+S^7{{\,\mathrm{tr}\,}}\textit{\textbf{r}}^2+S^8 {{\,\mathrm{tr}\,}}\varvec{\varepsilon }^3+S^9{{\,\mathrm{tr}\,}}\textit{\textbf{m}}^3 \\&\quad +S^{10}{{\,\mathrm{tr}\,}}\textit{\textbf{r}}^3+S^{11} {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }\textit{\textbf{m}})+S^{12}{{\,\mathrm{tr}\,}}(\varvec{\varepsilon }\textit{\textbf{r}})+S^{13}{{\,\mathrm{tr}\,}}(\textit{\textbf{m}}\textit{\textbf{r}})+S^{14}{{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{m}})+S^{15}{{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{r}})\\&\quad +S^{16}{{\,\mathrm{tr}\,}}(\textit{\textbf{m}}^2\varvec{\varepsilon }) +S^{17}{{\,\mathrm{tr}\,}}(\textit{\textbf{m}}^2\textit{\textbf{r}})+S^{18} {{\,\mathrm{tr}\,}}(\textit{\textbf{r}}^2\varvec{\varepsilon })+S^{19}{{\,\mathrm{tr}\,}}(\textit{\textbf{r}}^2\textit{\textbf{m}})+S^{20} {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{m}}^2)\\&\quad +S^{21}{{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{r}}^2)+S^{22}{{\,\mathrm{tr}\,}}(\textit{\textbf{m}}^2\textit{\textbf{r}}^2)+S^{23}{{\,\mathrm{tr}\,}}(\varvec{\varepsilon }\textit{\textbf{m}}\textit{\textbf{r}}), \end{aligned} \end{aligned}$$

(83)

with $S^\alpha =S^\alpha (T,\varvec{\varepsilon },\textit{\textbf{m}},\textit{\textbf{r}})$, $\alpha =1,\ldots ,4$ objective scalar functions, and then depending on the invariants (81). The expression (69) is a first approximated form of (83), being $\textit{\textbf{m}}=\varvec{\dot{\varepsilon }}$.

Appendix B: Objective representation of symmetric tensor functions

In this Appendix, we consider two situations: (a) a first case where a second-order symmetric objective tensor depends on scalar functions and second-order symmetric tensors; (b) a second case where a second-order symmetric objective tensor depends on scalar functions, second-order symmetric tensors and polar vectors.

Case (a) Following [26,27,28] and [29] a second-order symmetric tensor $\textit{\textbf{H}},$ that depends on m scalar functions, namely $a_1$, $a_2$,...,$a_m$ (in the case of the pressure tensor $\textit{\textbf{P}}$, $m=1$ and $a_1\equiv T$), and l second-order symmetric tensors, namely $\textit{\textbf{A}}_1$, $\textit{\textbf{A}}_2$,..., $\textit{\textbf{A}}_l$ (in our case $l=3$ and $\textit{\textbf{A}}_1\equiv \varvec{\varepsilon }$, $\textit{\textbf{A}}_2\equiv \textit{\textbf{m}}$, $\textit{\textbf{A}}_3\equiv \textit{\textbf{r}}$), is expressed as polynomial form constructed on the following invariants

$$\begin{aligned} \textit{\textbf{U}}, \quad \textit{\textbf{A}}_i, \quad \textit{\textbf{A}}^2_i, \quad \textit{\textbf{A}}_i\textit{\textbf{A}}_j+\textit{\textbf{A}}_j\textit{\textbf{A}}_i, \quad \textit{\textbf{A}}^2_i\textit{\textbf{A}}_j+\textit{\textbf{A}}_j\textit{\textbf{A}}^2_i, \quad \textit{\textbf{A}}^2_i\textit{\textbf{A}}^2_j+\textit{\textbf{A}}^2_j\textit{\textbf{A}}^2_i, \end{aligned}$$

(84)

with $i,j=1,\ldots ,l$ and $i\ne j$, i.e.

$$\begin{aligned} \textit{\textbf{H}}=\sum _{\alpha =1}^rH^\alpha \textit{\textbf{H}}_\alpha , \end{aligned}$$

(85)

where $H^\alpha =H^\alpha (a_1,a_2,\ldots ,a_m, \textit{\textbf{A}}_1, \textit{\textbf{A}}_2,\ldots ,\textit{\textbf{A}}_l)$, $\alpha =1,\ldots ,r$, are objective scalar functions (depending on the invariants (79)) and $\textit{\textbf{H}}_\alpha$ are built on the set of invariants of the list (84).

Thus, for $\textit{\textbf{P}}=\textit{\textbf{P}}(T,\varvec{\varepsilon },\textit{\textbf{m}},\textit{\textbf{r}})$ they are

$$\begin{aligned} \begin{aligned}&\textit{\textbf{U}}, \quad \varvec{\varepsilon }, \quad \textit{\textbf{m}}, \quad \textit{\textbf{r}}, \quad \varvec{\varepsilon }^2, \quad \textit{\textbf{m}}^2, \quad \textit{\textbf{r}}^2, \quad \varvec{\varepsilon }\textit{\textbf{m}}+\textit{\textbf{m}}\varvec{\varepsilon }, \quad \varvec{\varepsilon }\textit{\textbf{r}}+\textit{\textbf{r}}\varvec{\varepsilon }, \quad \textit{\textbf{m}}\textit{\textbf{r}}+\textit{\textbf{r}}\textit{\textbf{m}}, \\&\varvec{\varepsilon }^2\textit{\textbf{m}}+\textit{\textbf{m}}\varvec{\varepsilon }^2, \quad \varvec{\varepsilon }^2\textit{\textbf{r}}+\textit{\textbf{r}}\varvec{\varepsilon }^2, \quad \textit{\textbf{m}}^2\textit{\textbf{r}}+\textit{\textbf{r}}\textit{\textbf{m}}^2, \quad \textit{\textbf{m}}^2\varvec{\varepsilon }+\varvec{\varepsilon }\textit{\textbf{m}}^2, \quad \textit{\textbf{r}}^2\textit{\textbf{m}}+\textit{\textbf{m}}\textit{\textbf{r}}^2, \\&\textit{\textbf{r}}^2\varvec{\varepsilon }+\varvec{\varepsilon }\textit{\textbf{r}}^2, \quad \varvec{\varepsilon }^2\textit{\textbf{m}}^2+\textit{\textbf{m}}^2\varvec{\varepsilon }^2, \quad \varvec{\varepsilon }^2\textit{\textbf{r}}^2+\textit{\textbf{r}}^2\varvec{\varepsilon }^2, \quad \textit{\textbf{m}}^2\textit{\textbf{r}}^2+\textit{\textbf{r}}^2\textit{\textbf{m}}^2, \end{aligned} \end{aligned}$$

(86)

where, for instance $\varvec{\varepsilon }^2\textit{\textbf{m}}+\textit{\textbf{m}}\varvec{\varepsilon }^2\equiv (\varepsilon _{ik}\varepsilon _{kl}m_{lj}+m_{ik}\varepsilon _{kl}\varepsilon _{lj})$, $\varvec{\varepsilon }^2\textit{\textbf{m}}^2+\textit{\textbf{m}}^2\varvec{\varepsilon }^2\equiv (\varepsilon _{ik}\varepsilon _{kl}m_{lp}m_{pj}+m_{ik}m_{kl}\varepsilon _{lp}\varepsilon _{pj}).$ Thus, according to (85) the general form of $\textit{\textbf{P}}$ is

$$\begin{aligned} \begin{aligned} \textit{\textbf{P}}&= P^1\textit{\textbf{U}}+P^2\varvec{\varepsilon }+P^3\textit{\textbf{m}}+P^4\textit{\textbf{r}}+P^5\varvec{\varepsilon }^2+P^6\textit{\textbf{m}}^2+P^7\textit{\textbf{r}}^2+P^8(\varvec{\varepsilon }\textit{\textbf{m}}+\textit{\textbf{m}}\varvec{\varepsilon })+P^9(\varvec{\varepsilon }\textit{\textbf{r}}+\textit{\textbf{r}}\varvec{\varepsilon }. \\&\quad +P^{10}(\textit{\textbf{m}}\textit{\textbf{r}}+\textit{\textbf{r}}\textit{\textbf{m}})+P^{11}(\varvec{\varepsilon }^2\textit{\textbf{m}}+\textit{\textbf{m}}\varvec{\varepsilon }^2)++P^{12}(\varvec{\varepsilon }^2\textit{\textbf{r}}+\textit{\textbf{r}}\varvec{\varepsilon }^2)+P^{13}(\textit{\textbf{m}}^2\textit{\textbf{r}}+\textit{\textbf{r}}\textit{\textbf{m}}^2) \\&\quad +P^{14}(\textit{\textbf{m}}^2\varvec{\varepsilon }+\varvec{\varepsilon }\textit{\textbf{m}}^2)+P^{15}(\textit{\textbf{r}}^2\textit{\textbf{m}}+\textit{\textbf{m}}\textit{\textbf{r}}^2)+P^{16}(\textit{\textbf{r}}^2\varvec{\varepsilon }+\varvec{\varepsilon }\textit{\textbf{r}}^2)+P^{17}(\varvec{\varepsilon }^2\textit{\textbf{m}}^2+\textit{\textbf{m}}^2\varvec{\varepsilon }^2) \\&\quad +P^{18}(\varvec{\varepsilon }^2\textit{\textbf{r}}^2+\textit{\textbf{r}}^2\varvec{\varepsilon }^2)+P^{19}(\textit{\textbf{m}}^2\textit{\textbf{r}}^2+\textit{\textbf{r}}^2\textit{\textbf{m}}^2), \end{aligned} \end{aligned}$$

(87)

where $P^\alpha =P^\alpha (T,\varvec{\varepsilon },\textit{\textbf{m}},\textit{\textbf{r}})$, $\alpha =1,\ldots ,19$, are objective scalar functions (and than depending on the invariants (81)). The expression (70) is a first approximated form of (87), with $\textit{\textbf{m}}=\varvec{\dot{\varepsilon }}$.

Case (b) Following [26,27,28] and [29] in the case where the second-order symmetric tensor $\textit{\textbf{H}}$ depends also on a vector $\textit{\textbf{w}}$ (as in the case of $\varvec{\mathcal {M}}$ and $\varvec{\mathcal {R}}^{(i)}$, with $\textit{\textbf{w}}\equiv \nabla T$), it is expressed as polynomial of the form (85), with $\textit{\textbf{H}}^\alpha$ the following invariants

$$\begin{aligned} \begin{aligned}&\textit{\textbf{U}}, \quad \textit{\textbf{A}}_i, \quad \textit{\textbf{A}}^2_i, \quad \textit{\textbf{A}}_i\textit{\textbf{A}}_j+\textit{\textbf{A}}_j\textit{\textbf{A}}_i, \quad \textit{\textbf{A}}^2_i\textit{\textbf{A}}_j+\textit{\textbf{A}}_j\textit{\textbf{A}}^2_i, \quad \textit{\textbf{A}}^2_i\textit{\textbf{A}}^2_j+\textit{\textbf{A}}^2_j\textit{\textbf{A}}^2_i, \\&\textit{\textbf{w}}\otimes \textit{\textbf{w}}, \quad \textit{\textbf{w}}\otimes (\textit{\textbf{A}}_i\textit{\textbf{w}})+(\textit{\textbf{A}}_i\textit{\textbf{w}})\otimes \textit{\textbf{w}}, \quad \textit{\textbf{w}}\otimes (\textit{\textbf{A}}^2_i\textit{\textbf{w}})+(\textit{\textbf{A}}^2_i\textit{\textbf{w}})\otimes \textit{\textbf{w}} \end{aligned} \end{aligned}$$

(88)

and the objective scalar functions $H^\alpha (a_1,a_2,\ldots ,a_m, \textit{\textbf{A}}_1, \textit{\textbf{A}}_2,\ldots ,\textit{\textbf{A}}_l,\textit{\textbf{w}})$ depending on the invariants

$$\begin{aligned} \begin{aligned}&a_s, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{A}}_i, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{A}}^2_i, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{A}}^3_i, \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}_i\textit{\textbf{A}}_j), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}^2_i\textit{\textbf{A}}_j), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}^2_i\textit{\textbf{A}}^2_j), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{A}}_i\textit{\textbf{A}}_j\textit{\textbf{A}}_k), \\&\textit{\textbf{w}}\cdot \textit{\textbf{w}}, \quad \textit{\textbf{w}}\cdot (\textit{\textbf{A}}_i\textit{\textbf{w}}), \quad \textit{\textbf{w}}\cdot (\textit{\textbf{A}}^2_i\textit{\textbf{w}}), \quad (\textit{\textbf{A}}_i\textit{\textbf{w}})\cdot (\textit{\textbf{A}}_j\textit{\textbf{w}}), \end{aligned} \end{aligned}$$

(89)

with $s=1,\ldots ,m$, $i,j=1,\ldots ,l$ and $i\ne j$.

In the case of $\varvec{\mathcal {M}}$ and $\varvec{\mathcal {R}}^{(i)}$ we have $m=1$ and $a_1\equiv T$, $l=3$ and $\textit{\textbf{A}}_1\equiv \varvec{\varepsilon }$, $\textit{\textbf{A}}_2\equiv \textit{\textbf{m}}$, $\textit{\textbf{A}}_3\equiv \textit{\textbf{r}}$, and $\textit{\textbf{w}}\equiv \nabla T$. Then, the second-order symmetric tensors $\varvec{\mathcal {M}}(T, \nabla T, \varvec{\varepsilon }, \textit{\textbf{m}}, \textit{\textbf{r}})$ and $\varvec{\mathcal {R}}^{(i)}(T, \nabla T, \varvec{\varepsilon }, \textit{\textbf{m}}, \textit{\textbf{r}})$ are built on the following invariants

$$\begin{aligned} \begin{aligned}&\textit{\textbf{U}}, \quad \varvec{\varepsilon }, \quad \textit{\textbf{m}}, \quad \textit{\textbf{r}}, \quad \varvec{\varepsilon }^2, \quad \textit{\textbf{m}}^2, \quad \textit{\textbf{r}}^2, \quad \varvec{\varepsilon }\textit{\textbf{m}}+\textit{\textbf{m}}\varvec{\varepsilon }, \quad \varvec{\varepsilon }\textit{\textbf{r}}+\textit{\textbf{r}}\varvec{\varepsilon }, \quad \textit{\textbf{m}}\textit{\textbf{r}}+\textit{\textbf{r}}\textit{\textbf{m}}, \\&\varvec{\varepsilon }^2\textit{\textbf{m}}+\textit{\textbf{m}}\varvec{\varepsilon }^2, \quad \varvec{\varepsilon }^2\textit{\textbf{r}}+\textit{\textbf{r}}\varvec{\varepsilon }^2, \quad \textit{\textbf{m}}^2\textit{\textbf{r}}+\textit{\textbf{r}}\textit{\textbf{m}}^2, \quad \textit{\textbf{m}}^2\varvec{\varepsilon }+\varvec{\varepsilon }\textit{\textbf{m}}^2, \quad \textit{\textbf{r}}^2\textit{\textbf{m}}+\textit{\textbf{m}}\textit{\textbf{r}}^2, \\&\textit{\textbf{r}}^2\varvec{\varepsilon }+\varvec{\varepsilon }\textit{\textbf{r}}^2, \quad \varvec{\varepsilon }^2\textit{\textbf{m}}^2+\textit{\textbf{m}}^2\varvec{\varepsilon }^2, \quad \varvec{\varepsilon }^2\textit{\textbf{r}}^2+\textit{\textbf{r}}^2\varvec{\varepsilon }^2, \quad \textit{\textbf{m}}^2\textit{\textbf{r}}^2+\textit{\textbf{r}}^2\textit{\textbf{m}}^2, \\&\nabla T\otimes \nabla T, \quad \nabla T\otimes (\varvec{\varepsilon }\nabla T)+(\varvec{\varepsilon }\nabla T)\otimes \nabla T, \quad \nabla T\otimes (\textit{\textbf{m}}\nabla T)+(\textit{\textbf{m}}\nabla T)\otimes \nabla T,\\&\nabla T\otimes (\textit{\textbf{r}}\nabla T)+(\textit{\textbf{r}}\nabla T)\otimes \nabla T, \quad \nabla T\otimes (\varvec{\varepsilon }^2\nabla T)+(\varvec{\varepsilon }^2\nabla T)\otimes \nabla T, \\&\nabla T\otimes (\textit{\textbf{m}}^2\nabla T)+(\textit{\textbf{m}}^2\nabla T)\otimes \nabla T, \quad \nabla T\otimes (\textit{\textbf{r}}^2\nabla T)+(\textit{\textbf{r}}^2\nabla T)\otimes \nabla T., \end{aligned} \end{aligned}$$

(90)

where, for instance, $\nabla T\otimes \nabla T\equiv (T_{,i}T_{,j})$, $\nabla T\otimes (\varvec{\varepsilon }\nabla T)+(\varvec{\varepsilon }\nabla T)\otimes \nabla T\equiv (T_{,i}\varepsilon _{jk}T_{,k}+\varepsilon _{ik}T_{,k}T_{,j})$ and $\nabla T\otimes (\varvec{\varepsilon }^2\nabla T)+(\varvec{\varepsilon }^2\nabla T)\otimes \nabla T\equiv (T_{,i}\varepsilon _{jl}\varepsilon _{lk}T_{,k}+\varepsilon _{il}\varepsilon _{lk}T_{,k}T_{,j})$.

Thus, we have for $\varvec{\mathcal {M}}$ and $\varvec{\mathcal {R}}^{(i)}$, according to (85), the following expressions

$$\begin{aligned} \begin{aligned} \!\!\! \varvec{\mathcal {M}}&=M^1\textit{\textbf{U}}+M^2\varvec{\varepsilon }+M^3\textit{\textbf{m}}+M^4\textit{\textbf{r}}+M^5\varvec{\varepsilon }^2+M^6\textit{\textbf{m}}^2+M^7\textit{\textbf{r}}^2+M^8(\varvec{\varepsilon }\textit{\textbf{m}}+\textit{\textbf{m}}\varvec{\varepsilon })+M^9(\varvec{\varepsilon }\textit{\textbf{r}}+\textit{\textbf{r}}\varvec{\varepsilon }) \\&\quad +M^{10}(\textit{\textbf{m}}\textit{\textbf{r}}+\textit{\textbf{r}}\textit{\textbf{m}})+M^{11}(\varvec{\varepsilon }^2\textit{\textbf{m}}+\textit{\textbf{m}}\varvec{\varepsilon }^2)++M^{12}(\varvec{\varepsilon }^2\textit{\textbf{r}}+\textit{\textbf{r}}\varvec{\varepsilon }^2)+M^{13}(\textit{\textbf{m}}^2\textit{\textbf{r}}+\textit{\textbf{r}}\textit{\textbf{m}}^2) \\&\quad +M^{14}(\textit{\textbf{m}}^2\varvec{\varepsilon }+\varvec{\varepsilon }\textit{\textbf{m}}^2)+M^{15}(\textit{\textbf{r}}^2\textit{\textbf{m}}+\textit{\textbf{m}}\textit{\textbf{r}}^2)+M^{16}(\textit{\textbf{r}}^2\varvec{\varepsilon }+\varvec{\varepsilon }\textit{\textbf{r}}^2)+M^{17}(\varvec{\varepsilon }^2\textit{\textbf{m}}^2+\textit{\textbf{m}}^2\varvec{\varepsilon }^2) \\&\quad +M^{18}(\varvec{\varepsilon }^2\textit{\textbf{r}}^2+\textit{\textbf{r}}^2\varvec{\varepsilon }^2)+M^{19}(\textit{\textbf{m}}^2\textit{\textbf{r}}^2+\textit{\textbf{r}}^2\textit{\textbf{m}}^2)+M^{20}(\nabla T\otimes \nabla T)\\&\quad +M^{21}[\nabla T\otimes (\varvec{\varepsilon }\nabla T)+(\varvec{\varepsilon }\nabla T)\otimes \nabla T]+M^{22}[\nabla T\otimes (\textit{\textbf{m}}\nabla T)+(\textit{\textbf{m}}\nabla T)\otimes \nabla T] \\&\quad +M^{23}[\nabla T\otimes (\textit{\textbf{r}}\nabla T)+(\textit{\textbf{r}}\nabla T)\otimes \nabla T]+M^{24}[\nabla T\otimes (\varvec{\varepsilon }^2\nabla T)+(\varvec{\varepsilon }^2\nabla T)\otimes \nabla T] \\&\quad +M^{25}[\nabla T\otimes (\textit{\textbf{m}}^2\nabla T)+(\textit{\textbf{m}}^2\nabla T)\otimes \nabla T]+M^{26}[\nabla T\otimes (\textit{\textbf{r}}^2\nabla T)+(\textit{\textbf{r}}^2\nabla T)\otimes \nabla T], \end{aligned} \end{aligned}$$

(91)

and

$$\begin{aligned} \begin{aligned} \varvec{\mathcal {R}}^{(i)}&=R^1\textit{\textbf{U}}+R^2\varvec{\varepsilon }+R^3\textit{\textbf{m}}+R^4\textit{\textbf{r}}+R^5\varvec{\varepsilon }^2+R^6\textit{\textbf{m}}^2+R^7\textit{\textbf{r}}^2+R^8(\varvec{\varepsilon }\textit{\textbf{m}}+\textit{\textbf{m}}\varvec{\varepsilon })+R^9(\varvec{\varepsilon }\textit{\textbf{r}}+\textit{\textbf{r}}\varvec{\varepsilon }) \\&\quad +R^{10}(\textit{\textbf{m}}\textit{\textbf{r}}+\textit{\textbf{r}}\textit{\textbf{m}})+R^{11}(\varvec{\varepsilon }^2\textit{\textbf{m}}+\textit{\textbf{m}}\varvec{\varepsilon }^2)++R^{12}(\varvec{\varepsilon }^2\textit{\textbf{r}}+\textit{\textbf{r}}\varvec{\varepsilon }^2)+R^{13}(\textit{\textbf{m}}^2\textit{\textbf{r}}+\textit{\textbf{r}}\textit{\textbf{m}}^2) \\&\quad +R^{14}(\textit{\textbf{m}}^2\varvec{\varepsilon }+\varvec{\varepsilon }\textit{\textbf{m}}^2)+R^{15}(\textit{\textbf{r}}^2\textit{\textbf{m}}+\textit{\textbf{m}}\textit{\textbf{r}}^2)+R^{16}(\textit{\textbf{r}}^2\varvec{\varepsilon }+\varvec{\varepsilon }\textit{\textbf{r}}^2)+R^{17}(\varvec{\varepsilon }^2\textit{\textbf{m}}^2+\textit{\textbf{m}}^2\varvec{\varepsilon }^2) \\&\quad +R^{18}(\varvec{\varepsilon }^2\textit{\textbf{r}}^2+\textit{\textbf{r}}^2\varvec{\varepsilon }^2)+R^{19}(\textit{\textbf{m}}^2\textit{\textbf{r}}^2+\textit{\textbf{r}}^2\textit{\textbf{m}}^2)+R^{20}(\nabla T\otimes \nabla T)\\&\quad +R^{21}[\nabla T\otimes (\varvec{\varepsilon }\nabla T)+(\varvec{\varepsilon }\nabla T)\otimes \nabla T]+R^{22}[\nabla T\otimes (\textit{\textbf{m}}\nabla T)+(\textit{\textbf{m}}\nabla T)\otimes \nabla T] \\&\quad +R^{23}[\nabla T\otimes (\textit{\textbf{r}}\nabla T)+(\textit{\textbf{r}}\nabla T)\otimes \nabla T]+R^{24}[\nabla T\otimes (\varvec{\varepsilon }^2\nabla T)+(\varvec{\varepsilon }^2\nabla T)\otimes \nabla T] \\&\quad +R^{25}[\nabla T\otimes (\textit{\textbf{m}}^2\nabla T)+(\textit{\textbf{m}}^2\nabla T)\otimes \nabla T]+R^{26}[\nabla T\otimes (\textit{\textbf{r}}^2\nabla T)+(\textit{\textbf{r}}^2\nabla T)\otimes \nabla T], \end{aligned} \end{aligned}$$

(92)

where $M^\alpha (T, \nabla T, \varvec{\varepsilon }, \textit{\textbf{m}}, \textit{\textbf{r}})$ and $R^\alpha (T, \nabla T, \varvec{\varepsilon }, \textit{\textbf{m}}, \textit{\textbf{r}})$, $\alpha =1,\ldots ,26$, are scalar objective functions, that depend on the following invariants

$$\begin{aligned} \begin{aligned}&T, \quad {{\,\mathrm{tr}\,}}\varvec{\varepsilon }, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{m}}, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{r}}, \quad {{\,\mathrm{tr}\,}}\varvec{\varepsilon }^2, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{m}}^2, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{r}}^2, \quad {{\,\mathrm{tr}\,}}\varvec{\varepsilon }^3, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{m}}^3, \quad {{\,\mathrm{tr}\,}}\textit{\textbf{r}}^3, \quad {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }\textit{\textbf{m}}), \\&{{\,\mathrm{tr}\,}}(\varvec{\varepsilon }\textit{\textbf{r}}), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{m}}\textit{\textbf{r}}), \quad {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{m}}), \quad {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{r}}), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{m}}^2\varvec{\varepsilon }), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{m}}^2\textit{\textbf{r}}), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{r}}^2\varvec{\varepsilon }), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{r}}^2\textit{\textbf{m}}), \\&{{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{m}}^2), \quad {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }^2\textit{\textbf{r}}^2), \quad {{\,\mathrm{tr}\,}}(\textit{\textbf{m}}^2\textit{\textbf{r}}^2), \quad {{\,\mathrm{tr}\,}}(\varvec{\varepsilon }\textit{\textbf{m}}\textit{\textbf{r}}), \\&\nabla T\cdot \nabla T, \quad \nabla T\cdot (\varvec{\varepsilon }\nabla T), \quad \nabla T\cdot (\textit{\textbf{m}}\nabla T), \quad \nabla T\cdot (\textit{\textbf{r}}\nabla T), \quad \nabla T\cdot (\varvec{\varepsilon }^2\nabla T), \quad \nabla T\cdot (\textit{\textbf{m}}^2\nabla T), \\&\nabla T\cdot (\textit{\textbf{r}}^2\nabla T), \quad (\varvec{\varepsilon }\nabla T)\cdot (\textit{\textbf{m}}\nabla T), \quad (\varvec{\varepsilon }\nabla T)\cdot (\textit{\textbf{r}}\nabla T), \quad (\textit{\textbf{m}}\nabla T)\cdot (\textit{\textbf{r}}\nabla T). \end{aligned} \end{aligned}$$

(93)

where, for instance, $\nabla T\cdot (\varvec{\varepsilon }^2\nabla T)\equiv (T_{,i}\varepsilon _{il}\varepsilon _{lk}T_{,k})$. Notice that the invariants of the first three lines of (93) are those of (81). Relations (72) and (73) are a first approximation of (91) and (92), respectively, being $\textit{\textbf{m}}=\varvec{\dot{\varepsilon }}$.

Appendix C: Objective representation of vector functions

Following [26,27,28] and [29] a vector function $\textit{\textbf{g}},$ that depends on m scalar function, namely $a_1$, $a_2$,...,$a_m$, (in the case of the heat flux $\textit{\textbf{q}}$, $m=1$ and $a_1\equiv T$), and l second-order symmetric tensors, namely $\textit{\textbf{A}}_1$, $\textit{\textbf{A}}_2$,...,$\textit{\textbf{A}}_l$ (in our case $l=3$ and $\textit{\textbf{A}}_1\equiv \varvec{\varepsilon }$, $\textit{\textbf{A}}_2\equiv \textit{\textbf{m}}$, $\textit{\textbf{A}}_3\equiv \textit{\textbf{r}}$) and on a vector $\textit{\textbf{w}}$ (in our case $\textit{\textbf{w}}\equiv \nabla T$), is expressed as polynomial of the invariants

$$\begin{aligned} \textit{\textbf{w}}, \quad \textit{\textbf{A}}_i\textit{\textbf{w}}, \quad \textit{\textbf{A}}^2_i\textit{\textbf{w}}, \quad \textit{\textbf{A}}_i\textit{\textbf{A}}_j\textit{\textbf{w}}, \quad \textit{\textbf{A}}_j\textit{\textbf{A}}_i\textit{\textbf{w}}, \end{aligned}$$

(94)

with $i,j=1,\ldots ,l$ and $i\ne j$, having the form

$$\begin{aligned} \textit{\textbf{g}}=\sum _{\beta =1}^n g^\beta \textit{\textbf{g}}_\beta , \end{aligned}$$

(95)

where $g^\beta =g^\beta (a_1,a_2,\ldots ,a_m, \textit{\textbf{A}}_1, \textit{\textbf{A}}_2,\ldots ,\textit{\textbf{A}}_l,\textit{\textbf{w}})$, $\beta =1,\ldots ,n$, are objective scalar functions (and than depending on the invariants (93)) and $\textit{\textbf{g}}_\beta$ are building on the set of the invariants of the list (94).

Thus, the invariants of the vector-value function $\textit{\textbf{q}}(T, \nabla T, \varvec{\varepsilon }, \textit{\textbf{m}}, \textit{\textbf{r}})$ are

$$\begin{aligned} \begin{aligned}&\nabla T, \quad \varvec{\varepsilon }\nabla T, \quad \textit{\textbf{m}}\nabla T, \quad \textit{\textbf{r}}\nabla T, \quad \varvec{\varepsilon }^2\nabla T, \quad \textit{\textbf{m}}^2\nabla T, \quad \textit{\textbf{r}}^2\nabla T, \quad \varvec{\varepsilon }\textit{\textbf{m}}\nabla T, \quad \varvec{\varepsilon }\textit{\textbf{r}}\nabla T, \quad \textit{\textbf{m}}\textit{\textbf{r}}\nabla T, \\&\textit{\textbf{m}}\varvec{\varepsilon }\nabla T, \quad \textit{\textbf{r}}\varvec{\varepsilon }\nabla T, \quad \textit{\textbf{r}}\textit{\textbf{m}}\nabla T, \end{aligned} \end{aligned}$$

(96)

and, according to (95) (with $n=13$), $\textit{\textbf{q}}$ has the form

$$\begin{aligned} \begin{aligned} \textit{\textbf{q}}&=q^1\nabla T+q^2\varvec{\varepsilon }\nabla T+q^3\textit{\textbf{m}}\nabla T+q^4\textit{\textbf{r}}\nabla T+q^5\varvec{\varepsilon }^2\nabla T+q^6\textit{\textbf{m}}^2\nabla T+q^7\textit{\textbf{r}}^2\nabla T+q^8\varvec{\varepsilon }\textit{\textbf{m}}\nabla T \\&\quad +q^9\varvec{\varepsilon }\textit{\textbf{r}}\nabla T+q^{10}\textit{\textbf{m}}\textit{\textbf{r}}\nabla T+q^{11}\textit{\textbf{m}}\varvec{\varepsilon }\nabla T+q^{12}\textit{\textbf{r}}\varvec{\varepsilon }\nabla T+q^{13}\textit{\textbf{r}}\textit{\textbf{m}}\nabla T, \end{aligned} \end{aligned}$$

(97)

where $q^\beta =q^\beta (T, \nabla T, \varvec{\varepsilon }, \textit{\textbf{m}}, \textit{\textbf{r}})$, $\beta =1,\ldots ,13$, are objective scalar functions, that depend on the invariants (93). Relation (78), $\textit{\textbf{q}}=q^1\nabla T$, is a first approximated form of (97), with $\textit{\textbf{m}}=\varvec{\dot{\varepsilon }}$.