Appendix B — Hill polarization tensors
The Hill polarization tensors defined in this section are related to an ellipsoid embedded in the whole space \(\R^3\). An ellipsoid \(\mathcal{E}_{\uu{A}}\) can be defined by an equation such as \[ \uv{x}\in\mathcal{E}_{\uu{A}} \quad\Leftrightarrow\quad \uv{x}\cdot(\trans{\uu{A}}\cdot\uu{A})^{-1}\cdot\uv{x}\leq 1 \] where \(\uu{A}\) is an invertible second-order tensor so that \(\trans{\uu{A}}\cdot\uu{A}\) is a positive definite symmetric tensor associated to 3 radii (eigenvalues \(a\geq b \geq c\) possibly written \(\rho_1 \geq \rho_2 \geq \rho_3\) for convenience) and 3 angles (orientation of the frame of orthonormal eigenvectors \(\uv{e}^{\uu{A}}_1, \uv{e}^{\uu{A}}_2, \uv{e}^{\uu{A}}_3\)) \[ \trans{\uu{A}}\cdot\uu{A}=a^2 \uv{e}^{\uu{A}}_1\otimes\uv{e}^{\uu{A}}_1 + b^2 \uv{e}^{\uu{A}}_2\otimes\uv{e}^{\uu{A}}_2 + c^2 \uv{e}^{\uu{A}}_3\otimes\uv{e}^{\uu{A}}_3 = \sum_{i=1}^3 \rho_i^2 \uv{e}^{\uu{A}}_i\otimes\uv{e}^{\uu{A}}_i \tag{B.1}\]
Useful integrals
Some integrals useful for further calculations and intimately connected to the Newtonian potential are introduced as
\[\begin{align} \uu{I}^{\uu{A}} &= \frac{\det{\uu{A}}}{4\pi} \int_{\norm{\uv{\xi}}=1} \frac{\uv{\xi}\otimes\uv{\xi}} {\norm{\uu{A}\cdot\uv{\xi}}^3}\ud S_{\xi} = \frac{1}{4\pi} \int_{\norm{\uv{\zeta}}=1} \frac{% (\uu{A}^{-1}\cdot\uv{\zeta}) \otimes (\uu{A}^{-1}\cdot\uv{\zeta}) }{\norm{\uu{A}^{-1}\cdot\uv{\zeta}}^2} \ud S_{\zeta} \\ \uuuu{U}^{\uu{A}} &= \frac{\det{\uu{A}}}{4\pi} \int_{\norm{\uv{\xi}}=1} \frac{\uv{\xi}\otimes\uv{\xi}\otimes\uv{\xi}\otimes\uv{\xi}} {\norm{\uu{A}\cdot\uv{\xi}}^3}\ud S_{\xi}\\ &= \frac{1}{4\pi} \int_{\norm{\uv{\zeta}}=1} \frac{% (\uu{A}^{-1}\cdot\uv{\zeta}) \otimes (\uu{A}^{-1}\cdot\uv{\zeta}) \otimes (\uu{A}^{-1}\cdot\uv{\zeta}) \otimes (\uu{A}^{-1}\cdot\uv{\zeta}) }{\norm{\uu{A}^{-1}\cdot\uv{\zeta}}^4} \ud S_{\zeta} \\ \uuuu{V}^{\uu{A}} &= \frac{\det{\uu{A}}}{4\pi} \int_{\norm{\uv{\xi}}=1} \frac{\uv{\xi}\sotimes\uu{1}\sotimes\uv{\xi}} {\norm{\uu{A}\cdot\uv{\xi}}^3}\ud S_{\xi} = \frac{1}{4\pi} \int_{\norm{\uv{\zeta}}=1} \frac{% (\uu{A}^{-1}\cdot\uv{\zeta}) \sotimes \uu{1} \sotimes (\uu{A}^{-1}\cdot\uv{\zeta}) }{\norm{\uu{A}^{-1}\cdot\uv{\zeta}}^2} \ud S_{\zeta}\\ &= \frac{\uu{1}\sboxtimes\uu{I}^{\uu{A}}+\uu{I}^{\uu{A}}\sboxtimes\uu{1}}{2} \end{align} \tag{B.2}\]
The proof detailed here is an intrinsic alternative (see (Barthélémy et al., 2016)) to the reasoning in components presented in (Mura, 1987).
The change of variable in (B.2) is a bijective transformation of the unit sphere into itself \[ \uv{\zeta}\mapsto\uv{\xi}=\frac{\uu{A}^{-1}⋅\uv{\zeta}}{\norm{\uu{A}^{-1}⋅\uv{\zeta}}} \;\Leftrightarrow\; \uv{\xi}\mapsto\uv{\zeta}=\frac{\uu{A}⋅\uv{\xi}}{\norm{\uu{A}⋅\uv{\xi}}} \]
The delicate point of the proof is the following result on surface elements \[ \ud S_{\zeta} =\frac{\det{\uu{A}}}{\norm{\uu{A}\cdot\uv{\xi}}^3}\, \ud S_{\xi} \;\Leftrightarrow\; \ud S_{\xi} =\frac{\det{\uu{A}^{-1}}}{\norm{\uu{A}^{-1}\cdot\uv{\zeta}}^3}\, \ud S_{\zeta} \]
It is obvious that \(\uv{\xi}\) and \(\uv{\zeta}\) play symmetrical role by inverting \(\uu{A}\).
Let’s start by differentiating the transformation relationship yielding \[ \ud \uv{\xi}=\left(\uu{1}-\uv{\xi}\otimes\uv{\xi}\right)\cdot \frac{\uu{A}^{-1}}{\norm{\uu{A}^{-1}\cdot\uv{\zeta}}}\cdot\ud\uv{\zeta} \] This transformation gradient can be decomposed as the tangent invertible application \(\uv{u}\mapsto\frac{\uu{A}^{-1}}{\norm{\uu{A}^{-1}\cdot\uv{\zeta}}}\cdot\uv{u}\) followed by an orthogonal projection over the plane of normal \(\uv{\xi}\).
The end of the proof takes advantage of the following remarks:
the transformation of a surface vector \(\ud\uv{S}\) to \(\ud\uv{s}\) by the gradient transformation \(\uv{X}\mapsto\ud\uv{x}=\uu{F}⋅\ud\uv{X}\) is \(\ud\uv{s}=(\det{\uu{F}})\,\trans{\uu{F}}^{-1}⋅\ud\uv{S}\)
as \(\uv{\xi}\) and \(\uv{\zeta}\) belong to unit spheres, \(\ud \uv{S}_{\zeta}=\ud S_{\zeta}\,\uv{\zeta}\) and \(\ud \uv{S}_{\xi}=\ud S_{\xi}\,\uv{\xi}\)
the projection of material vectors over the plane of normal \(\uv{\xi}\) becomes a projection on the line directed by \(\uv{\xi}\) for surface vectors.
It results that \[ \ud S_{\xi}=\uv{\xi}\cdot\left(\left(\frac{\det{\uu{A}^{-1}}}{\norm{\uu{A}^{-1}\cdot\uv{\zeta}}^3}\right)\, \left(\norm{\uu{A}^{-1}\cdot\uv{\zeta}}\trans{\uu{A}}\right)\right)\cdot\uv{\zeta} \ud S_{\zeta} = \frac{\det{\uu{A}^{-1}}}{\norm{\uu{A}^{-1}\cdot\uv{\zeta}}^2}\, \uv{\zeta}\cdot\uu{A}\cdot\uv{\xi} \ud S_{\zeta} \] which ends the proof by observing that \(\uv{\zeta}\cdot\uu{A}\cdot\uv{\xi}=\frac{\norm{\uv{\zeta}}^2}{\norm{\uu{A}^{-1}\cdot\uv{\zeta}}}=\frac{1}{\norm{\uu{A}^{-1}\cdot\uv{\zeta}}}\).
By symmetry, it is obvious that \(\uu{A}\) (B.1) and \(\uu{I}^{\uu{A}}\) are simultaneously diagonalizable i.e. they share the same eigenvectors: \[ \uu{I}^{\uu{A}}=I^{\uu{A}}_1\,\uv{e}^{\uu{A}}_1\otimes\uv{e}^{\uu{A}}_1 + I^{\uu{A}}_2\,\uv{e}^{\uu{A}}_2\otimes\uv{e}^{\uu{A}}_2 + I^{\uu{A}}_3\,\uv{e}^{\uu{A}}_3\otimes\uv{e}^{\uu{A}}_3 = \sum_{i=1}^3 I^{\uu{A}}_i\,\uv{e}^{\uu{A}}_i\otimes\uv{e}^{\uu{A}}_i \tag{B.3}\]
The components \(I^{\uu{A}}_i\) can be obtained by identification with expressions derived from the Newtonian potential ((Kellogg, 1929), (Eshelby, 1957), (Parnell, 2016)). They are gathered in Tab. B.1 or recalled a little later in a generic form ((Barthélémy et al., 2016), (Barthélémy, 2020)).
| ellipsoid | prolate spheroid | oblate spheroid | sphere | |
|---|---|---|---|---|
| \(a>b>c\) | \(a>b=c\) | \(a=b>c\) | \(a=b=c\) | |
| \(I^{\uu{A}}_1 =\) | \(\frac{a\,b\,c}{(a^2-b^2)\sqrt{a^2-c^2}}\left({\cal F}-{\cal E}\right)\) | \(1-2\,I^{\uu{A}}_3\) | \(c\,\frac{a^2\,\arccos{(c/a)}-c\sqrt{a^2-c^2}}{2\left(a^2-c^2\right)^{3/2}}\) | \(\frac{1}{3}\) |
| \(I^{\uu{A}}_2 =\) | \(1-I^{\uu{A}}_1-I^{\uu{A}}_3\) | \(I^{\uu{A}}_3\) | \(I^{\uu{A}}_1\) | \(\frac{1}{3}\) |
| \(I^{\uu{A}}_3 =\) | \(\frac{a\,b\,c}{(b^2-c^2)\sqrt{a^2-c^2}}\left(\frac{b\sqrt{a^2-c^2}}{a\,c}-{\cal E}\right)\) | \(a\,\frac{a\sqrt{a^2-c^2}-c^2\,\arcosh{(a/c)}}{2\left(a^2-c^2\right)^{3/2}}\) | \(1-2\,I^{\uu{A}}_1\) | \(\frac{1}{3}\) |
where \({\cal F}={\cal F}(\theta,\kappa)\) and \({\cal E}={\cal E}(\theta,\kappa)\) are respectively the elliptic integrals of the first and second kinds (see (Abramowitz and Stegun, 1972)) of amplitude and parameter1 \[ \theta=\arcsin{\sqrt{1-\frac{c^2}{a^2}}} \quad;\quad \kappa=\sqrt{\frac{a^2-b^2}{a^2-c^2}} \]
Before detailing the components of \(\uuuu{U}^{\uu{A}}\) and \(\uuuu{V}^{\uu{A}}\) it is convenient to introduce the coefficients \(I^{\uu{A}}_{ij}\) in Tab. B.2 adapted from the approach proposed in (Eshelby, 1957).
| ellipsoid | prolate spheroid | oblate spheroid | sphere | |
|---|---|---|---|---|
| \(a>b>c\) | \(a>b=c\) | \(a=b>c\) | \(a=b=c\) | |
| \(I^{\uu{A}}_{11} =\) | \(\frac{1}{3}\left(\frac{1}{a^2}-I^{\uu{A}}_{31}-I^{\uu{A}}_{12} \right)\) | \(\frac{1}{3}\left(\frac{1}{a^2}-2\,I^{\uu{A}}_{31} \right)\) | \(\frac{1}{4}\left(\frac{1}{a^2}-I^{\uu{A}}_{31} \right)\) | \(\frac{1}{5\,a^2}\) |
| \(I^{\uu{A}}_{22} =\) | \(\frac{1}{3}\left(\frac{1}{b^2}-I^{\uu{A}}_{12}-I^{\uu{A}}_{23} \right)\) | \(\frac{1}{4}\left(\frac{1}{c^2}-I^{\uu{A}}_{31} \right)\) | \(\frac{1}{4}\left(\frac{1}{a^2}-I^{\uu{A}}_{31} \right)\) | \(\frac{1}{5\,a^2}\) |
| \(I^{\uu{A}}_{33} =\) | \(\frac{1}{3}\left(\frac{1}{c^2}-I^{\uu{A}}_{23}-I^{\uu{A}}_{31} \right)\) | \(\frac{1}{4}\left(\frac{1}{c^2}-I^{\uu{A}}_{31} \right)\) | \(\frac{1}{3}\left(\frac{1}{c^2}-2\,I^{\uu{A}}_{31} \right)\) | \(\frac{1}{5\,a^2}\) |
| \(I^{\uu{A}}_{23} = I^{\uu{A}}_{32} =\) | \(\frac{I^{\uu{A}}_3-I^{\uu{A}}_2}{b^2-c^2}\) | \(\frac{1}{4}\left(\frac{1}{c^2}-I^{\uu{A}}_{31} \right)\) | \(\frac{I^{\uu{A}}_3-I^{\uu{A}}_2}{b^2-c^2}\) | \(\frac{1}{5\,a^2}\) |
| \(I^{\uu{A}}_{31} = I^{\uu{A}}_{13} =\) | \(\frac{I^{\uu{A}}_3-I^{\uu{A}}_1}{a^2-c^2}\) | \(\frac{I^{\uu{A}}_3-I^{\uu{A}}_1}{a^2-c^2}\) | \(\frac{I^{\uu{A}}_3-I^{\uu{A}}_1}{a^2-c^2}\) | \(\frac{1}{5\,a^2}\) |
| \(I^{\uu{A}}_{12} = I^{\uu{A}}_{21} =\) | \(\frac{I^{\uu{A}}_2-I^{\uu{A}}_1}{a^2-b^2}\) | \(\frac{I^{\uu{A}}_2-I^{\uu{A}}_1}{a^2-b^2}\) | \(\frac{1}{4}\left(\frac{1}{a^2}-I^{\uu{A}}_{31} \right)\) | \(\frac{1}{5\,a^2}\) |
Note that, for writing convenience, \(I^{\uu{A}}_i\) of Tab. B.1 and \(I^{\uu{A}}_{ij}\) of Tab. B.2 are here adapted from those provided in (Kellogg, 1929) and (Eshelby, 1957): they differ by a factor of \(4\pi/3\) for \(I^{\uu{A}}_{ij}\) with \(i\neq j\) and by \(4\pi\) for the others.
Components \(I^{\uu{A}}_i\) and \(I^{\uu{A}}_{ij}\) are alternatively given in a generic form in the following section to be expanded.
\[\begin{align} I^{\uu{A}}_1&=\frac{a\,b\,c}{(a^2-b^2)\sqrt{a^2-c^2}}\, \left({\cal F}-{\cal E}\right)\\ I^{\uu{A}}_3&=\frac{a\,b\,c}{(b^2-c^2)\sqrt{a^2-c^2}}\, \left(\frac{b\sqrt{a^2-c^2}}{a\,c}- {\cal E}\right)\\ I^{\uu{A}}_2&=1-I^{\uu{A}}_1-I^{\uu{A}}_3\\ I^{\uu{A}}_{ij}&=\frac{I^{\uu{A}}_j-I^{\uu{A}}_i}{\rho_i^2-\rho_j^2}\quad\forall\, i\neq j\in\{1,2,3\}\\ I^{\uu{A}}_{ii}&=\frac{1}{3}\left( \frac{1}{\rho_i^2}- \sum_{j\neq i}I^{\uu{A}}_{ij} \right) \quad\forall\, i\in\{1,2,3\} \end{align}\]
\[\begin{align} I^{\uu{A}}_2=I^{\uu{A}}_3&=a\, \frac{a\sqrt{a^2-c^2}-c^2\,\arcosh{(a/c)}} {2\left(a^2-c^2\right)^{3/2}}\\ I^{\uu{A}}_1&=1-2\,I^{\uu{A}}_3\\ I^{\uu{A}}_{1i}=I^{\uu{A}}_{i1}&=\frac{I^{\uu{A}}_i-I^{\uu{A}}_1}{a^2-c^2}\quad \forall\, i\in\{2,3\}\\ I^{\uu{A}}_{ij}&=\frac{1}{4} \left(\frac{1}{c^2}-I^{\uu{A}}_{31} \right) \quad\forall\, i,j\in\{2,3\}\\ I^{\uu{A}}_{11}&=\frac{1}{3} \left(\frac{1}{a^2}-2\,I^{\uu{A}}_{31} \right) \end{align}\]
\[\begin{align} I^{\uu{A}}_1=I^{\uu{A}}_2&=c\, \frac{a^2\,\arccos{(c/a)}-c\sqrt{a^2-c^2}} {2\left(a^2-c^2\right)^{3/2}}\\ I^{\uu{A}}_3&=1-2\,I^{\uu{A}}_1\\ I^{\uu{A}}_{3i}=I^{\uu{A}}_{i3}&=\frac{I^{\uu{A}}_3-I^{\uu{A}}_i}{a^2-c^2}\quad \forall\, i\in\{1,2\}\\ I^{\uu{A}}_{ij}&=\frac{1}{4} \left(\frac{1}{a^2}-I^{\uu{A}}_{31} \right) \quad\forall\, i,j\in\{1,2\}\\ I^{\uu{A}}_{33}&=\frac{1}{3} \left(\frac{1}{c^2}-2\,I^{\uu{A}}_{31} \right) \end{align}\]
\[\begin{align} I^{\uu{A}}_1=I^{\uu{A}}_2=I^{\uu{A}}_3&=\frac{1}{3}\\ I^{\uu{A}}_{ij}&=\frac{1}{5\,a^2}\quad\forall\, i,j\in\{1,2,3\} \end{align}\]
Note that the following identities are always satisfied (Eshelby, 1957):
\[\begin{align} \sum_{i}I^{\uu{A}}_{i} &=1\\ 3\,I^{\uu{A}}_{ii}+\sum_{j\neq i}I^{\uu{A}}_{ij} &=\frac{1}{\rho_i^2}\\ 3\,\rho_i^2\,I^{\uu{A}}_{ii}+\sum_{j\neq i}\rho_j^2\,I^{\uu{A}}_{ij} &=3\,I^{\uu{A}}_{i} \end{align}\]
Again, for symmetry reasons, \(\uuuu{U}^{\uu{A}}\) and \(\uuuu{V}^{\uu{A}}\) are expected orthotropic of axes \(\left(\ve{i}^{\uu{A}}\right)_{i=1,2,3}\). They are indeed written in Kelvin-Mandel form in the frame \(\left(\ve{i}^{\uu{A}}\right)_{i=1,2,3}\) ((Barthélémy et al., 2016), (Barthélémy, 2020))
\[ \Mat\left(\uuuu{U}^{\uu{A}}, \ve{i}^{\uu{A}}\right)= \left( \begin{array}{cccccc} \frac{3(I^{\uu{A}}_1-a^2I^{\uu{A}}_{11})}{2} & \frac{I^{\uu{A}}_2-a^2I^{\uu{A}}_{12}}{2} & \frac{I^{\uu{A}}_1-c^2I^{\uu{A}}_{31}}{2} & 0 & 0 & 0 \\ \frac{I^{\uu{A}}_2-a^2I^{\uu{A}}_{12}}{2} & \frac{3(I^{\uu{A}}_2-b^2I^{\uu{A}}_{22})}{2} & \frac{I^{\uu{A}}_3-b^2I^{\uu{A}}_{23}}{2} & 0 & 0 & 0 \\ \frac{I^{\uu{A}}_1-c^2I^{\uu{A}}_{31}}{2} & \frac{I^{\uu{A}}_3-b^2I^{\uu{A}}_{23}}{2} & \frac{3(I^{\uu{A}}_3-c^2I^{\uu{A}}_{33})}{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & I^{\uu{A}}_3-b^2I^{\uu{A}}_{23} & 0 & 0 \\ 0 & 0 & 0 & 0 & I^{\uu{A}}_1-c^2I^{\uu{A}}_{31} & 0 \\ 0 & 0 & 0 & 0 & 0 & I^{\uu{A}}_2-a^2I^{\uu{A}}_{12} \\ \end{array} \right) \]
\[ \Mat\left(\uuuu{V}^{\uu{A}}, \ve{i}^{\uu{A}}\right)= \left( \begin{array}{cccccc} I^{\uu{A}}_1 & 0 & 0 & 0 & 0 & 0 \\ 0 & I^{\uu{A}}_2 & 0 & 0 & 0 & 0 \\ 0 & 0 & I^{\uu{A}}_3 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{I^{\uu{A}}_2+I^{\uu{A}}_3}{2} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{I^{\uu{A}}_3+I^{\uu{A}}_1}{2} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{I^{\uu{A}}_1+I^{\uu{A}}_2}{2} \\ \end{array} \right) \]
In the case of a spherical inclusion (\(\uu{A}=\uu{1}\)), note that \(\uuuu{U}^{\uu{A}}\) and \(\uuuu{V}^{\uu{A}}\) are simply decomposed as \[ \uuuu{U}^{\uu{A}}=\frac{1}{3}\uuuu{J}+\frac{2}{15}\uuuu{K} \quad\textrm{ and }\quad \uuuu{V}^{\uu{A}}=\frac{1}{3}\uuuu{I} \] where \(\uuuu{I}\), \(\uuuu{J}\) and \(\uuuu{K}\) are defined in Section A.4.
Alternatively generic expressions of the components are given herebelow.
\[\begin{align} U^{\uu{A}}_{iiii}&=\frac{3(I^{\uu{A}}_i-\rho_i^2I^{\uu{A}}_{ii})}{2} \quad\forall\, i\in\{1,2,3\}\\ U^{\uu{A}}_{iijj}=U^{\uu{A}}_{ijij}=U^{\uu{A}}_{ijji}&=\frac{I^{\uu{A}}_j-\rho_i^2I^{\uu{A}}_{ij}}{2} =\frac{I^{\uu{A}}_i-\rho_j^2I^{\uu{A}}_{ij}}{2} \quad\forall\, i\neq j\in\{1,2,3\} \end{align} \tag{B.4}\] and \[\begin{align} V^{\uu{A}}_{iiii}&=I^{\uu{A}}_i\quad\forall\, i\in\{1,2,3\}\\ V^{\uu{A}}_{ijij}=V^{\uu{A}}_{ijji}&=\frac{I^{\uu{A}}_i+I^{\uu{A}}_j}{4} \quad\forall\, i\neq j\in\{1,2,3\} \end{align} \tag{B.5}\]
Hill polarization tensor in elasticity
General expression
A general expression of the elastic polarization tensor is derived in (Willis, 1977) (see also (Mura, 1987)) \[\begin{align} \uuuu{P}(\uu{A},\uuuu{C})&=\frac{1}{4\pi} \int_{\norm{\uv{\zeta}}=1} (\uu{A}^{-1}\cdot\uv{\zeta})\sotimes \Big((\uu{A}^{-1}\cdot\uv{\zeta})\cdot\uuuu{C} \cdot(\uu{A}^{-1}\cdot\uv{\zeta})\Big)^{-1} \sotimes(\uu{A}^{-1}\cdot\uv{\zeta}) \ud S_\zeta\\ &= \frac{\det{\uu{A}}}{4\pi} \int_{\norm{\uv{\xi}}=1} \frac{\uv{\xi}\sotimes (\uv{\xi}\cdot\uuuu{C} \cdot\uv{\xi})^{-1} \sotimes\uv{\xi}}{\norm{\uu{A}\cdot\uv{\xi}}^3} \ud S_{\xi} \end{align} \tag{B.6}\] where \(\uuuu{C}\) is the stiffness tensor of the reference medium.
When \(\uuuu{C}\) is arbitrarily anisotropic, it is necessary to resort to numerical cubature to estimate \(\uuuu{P}\) as proposed in (Ghahremani, 1977), (Gavazzi and Lagoudas, 1990) or (Masson, 2008). However in some cases of anisotropy, analytical solutions are available ((Withers, 1989), (Suvorov and Dvorak, 2002)). Note that the transformation technique ((Pouya, 2000), (Pouya and Zaoui, 2006)) allows to extend the set of available anisotropies leading to analytical Hill tensors (Barthélémy, 2020). The case of isotropic matrix is particularly developed in the next section.
Isotropic matrix
In this section, the matrix is assumed isotropic so that its stiffness tensor writes by means of a bulk \(k\) and shear \(\mu\) or Lamé \(\lambda\) and \(\mu\) moduli or even Young modulus \(E\) and Poisson ratio \(\nu\) with \(k=\frac{E}{3(1-2\nu)}\) and \(\mu=\frac{E}{2(1+\nu)}\). \[ \uuuu{C} = 3k\uuuu{J}+2\mu\uuuu{K} = 3\lambda\uuuu{I}+2\mu\uuuu{K} \tag{B.7}\]
Introducing (B.7) in (B.6) leads to after some algebra \[ \uuuu{P}(\uu{A},3\lambda\uuuu{I}+2\mu\uuuu{K})= \frac{1}{\lambda+2\,\mu} \uuuu{U}^{\uu{A}} +\frac{1}{\mu}(\uuuu{V}^{\uu{A}}-\uuuu{U}^{\uu{A}}) \tag{B.8}\] where the tensors \(\uuuu{U}^{\uu{A}}\) and \(\uuuu{V}^{\uu{A}}\), depending only on the ellipsoidal tensor \(\uu{A}\) of (B.1), are given by (B.4) and (B.5). Note that the writing (B.8) puts well in evidence the contributions of the shape and orientation of the ellipsoid on the one hand and the moduli of the reference medium on the other hand.
Hill polarization tensor in conductivity
General expression
The conductivity polarization tensor related to a conductivity tensor \(\uu{K}\) and an ellipsoid defined by \(\uu{A}\) is also expressed in (Willis, 1977) under the form \[\begin{align} \uu{P}(\uu{A},\uu{K})&=\frac{1}{4\pi} \int_{\norm{\uv{\zeta}}=1} \frac{(\uu{A}^{-1}\cdot\uv{\zeta})\sotimes(\uu{A}^{-1}\cdot\uv{\zeta})} {(\uu{A}^{-1}\cdot\uv{\zeta})\cdot\uu{K} \cdot(\uu{A}^{-1}\cdot\uv{\zeta})} \ud S_\zeta\\ &= \frac{\det{\uu{A}}}{4\pi} \int_{\norm{\uv{\xi}}=1} \frac{\uv{\xi}\otimes\uv{\xi}}{(\uv{\xi}\cdot\uu{K}\cdot\uv{\xi})\,\norm{\uu{A}\cdot\uv{\xi}}^3} \ud S_{\xi} \end{align} \tag{B.9}\]
Isotropic matrix
The case of an isotropic matrix \(\uu{K}=K\,\uu{1}\) is straightforward: \[ \uu{P}(\uu{A},K\,\uu{1})= \frac{1}{K}\, \left( \frac{\det{\uu{A}}}{4\pi} \int_{\norm{\uv{\xi}}=1} \frac{\uv{\xi}\otimes\uv{\xi}}{\norm{\uu{A}\cdot\uv{\xi}}^3} \ud S_{\xi}\right) = \frac{\uu{I}^{\uu{A}}}{K} \tag{B.10}\] with \(\uu{I}^{\uu{A}}\) defined in (B.2) and derived in (B.3) and Tab. B.1.
Arbitrarily anisotropic matrix
The complete solution of the second-order Hill polarization tensor in a matrix of arbitrary has been derived in (Barthélémy, 2009) from a reasoning based on Green solution. However a more direct solution, based on the idea of transformation, has been detailed in (Giraud et al., 2019) and is recalled here. It first relies on the idea of square-root of the conductivity tensor. As a positive definite symmetric second-order tensor, \(\uu{K}\) can be diagonalized in an orthonormal frame \(\left(\uu{e}^K_i\right)_{i=1,2,3}\) with strictly positive eigenvalues \(\left(K_i\right)_{i=1,2,3}\) such that \[ \uu{K}=K_1 \uv{e}^{\uu{K}}_1\otimes\uv{e}^{\uu{K}}_1 + K_2 \uv{e}^{\uu{K}}_2\otimes\uv{e}^{\uu{K}}_2 + K_3 \uv{e}^{\uu{K}}_3\otimes\uv{e}^{\uu{K}}_3 \tag{B.11}\] Its square-root is then defined by \[ \uu{K}^{½}=\sqrt{K_1} \uv{e}^{\uu{K}}_1\otimes\uv{e}^{\uu{K}}_1 + \sqrt{K_2} \uv{e}^{\uu{K}}_2\otimes\uv{e}^{\uu{K}}_2 + \sqrt{K_3} \uv{e}^{\uu{K}}_3\otimes\uv{e}^{\uu{K}}_3 \tag{B.12}\] of inverse \[ \uu{K}^{-½}=\left(\uu{K}^{½}\right)^{-1}=\frac{1}{\sqrt{K_1}} \uv{e}^{\uu{K}}_1\otimes\uv{e}^{\uu{K}}_1 + \frac{1}{\sqrt{K_2}} \uv{e}^{\uu{K}}_2\otimes\uv{e}^{\uu{K}}_2 + \frac{1}{\sqrt{K_3}} \uv{e}^{\uu{K}}_3\otimes\uv{e}^{\uu{K}}_3 \tag{B.13}\]
It follows that (B.9) rewrites \[\begin{align} \uu{P}(\uu{A},\uu{K})&=\frac{1}{4\pi} \int_{\norm{\uv{\zeta}}=1} \frac{(\uu{A}^{-1}\cdot\uv{\zeta})\sotimes(\uu{A}^{-1}\cdot\uv{\zeta})} {(\uu{A}^{-1}\cdot\uv{\zeta})\cdot\uu{K} \cdot(\uu{A}^{-1}\cdot\uv{\zeta})} \ud S_\zeta\\ &= \frac{1}{4\pi} \int_{\norm{\uv{\zeta}}=1} \frac{(\uu{A}^{-1}\cdot\uv{\zeta})\sotimes(\uu{A}^{-1}\cdot\uv{\zeta})} {(\uu{K}^{½}\cdot\uu{A}^{-1}\cdot\uv{\zeta})\cdot\uu{1} \cdot(\uu{K}^{½}\cdot\uu{A}^{-1}\cdot\uv{\zeta})} \ud S_\zeta\\ &= \uu{K}^{-½}\cdot\left(\frac{1}{4\pi} \int_{\norm{\uv{\zeta}}=1} \frac{(\uu{K}^{½}\cdot\uu{A}^{-1}\cdot\uv{\zeta})\sotimes(\uu{K}^{½}\cdot\uu{A}^{-1}\cdot\uv{\zeta})} {(\uu{K}^{½}\cdot\uu{A}^{-1}\cdot\uv{\zeta})\cdot\uu{1} \cdot(\uu{K}^{½}\cdot\uu{A}^{-1}\cdot\uv{\zeta})} \ud S_\zeta\right)\cdot\uu{K}^{-½}\\ &= \uu{K}^{-½}\cdot\uu{P}(\uu{A}\cdot\uu{K}^{-½},\uu{1})\cdot\uu{K}^{-½}\\ &= \uu{K}^{-½}\cdot\uu{I}^{\uu{A}\cdot\uu{K}^{-½}}\cdot\uu{K}^{-½} \end{align} \tag{B.14}\] where \(\uu{I}^{\uu{A}\cdot\uu{K}^{-½}}\) is deduced from \(\uu{I}^{\uu{A}}\) defined in (B.2) and derived in (B.3) and Tab. B.1 in which the characteristic tensor of the ellipsoid \(\uu{A}\) is replaced by \(\uu{A}\cdot\uu{K}^{-½}\) defining a fictitious ellipsoid. Even if \(\uu{A}\cdot\uu{K}^{-½}\) is not necessarily symmetric, it is pointed out that the new fictitious radii and eigenvectors are defined by adaptation of (B.1) leading here to the diagonalization of \(\uu{K}^{-½}\cdot\trans{\uu{A}}\cdot\uu{A}\cdot\uu{K}^{-½}\).
\({\cal F}(\theta,\kappa)=\int_{\varphi=0}^\theta \frac{\ud\varphi}{\sqrt{1-\kappa^2\,\sin^2\varphi}}\) and \({\cal E}(\theta,\kappa)=\int_{\varphi=0}^\theta \sqrt{1-\kappa^2\,\sin^2\varphi}\,\ud\varphi\)↩︎