$$ \newcommand{\C}{{\mathbb{{C}}}} \newcommand{\R}{{\mathbb{{R}}}} \newcommand{\Q}{{\mathbb{{Q}}}} \newcommand{\Z}{{\mathbb{{Z}}}} \newcommand{\N}{{\mathbb{{N}}}} \newcommand{\uu}[1]{{\boldsymbol{{#1}}}} \newcommand{\uuuu}[1]{{\symbb{{#1}}}} \newcommand{\uv}[1]{{\underline{{#1}}}} \newcommand{\ve}[1]{{\uv{{e}}_{{#1}}}} \newcommand{\x}{{\uv{{x}}}} \newcommand{\n}{{\uv{{n}}}} \newcommand{\eps}{{\uu{{\varepsilon}}}} \newcommand{\E}{{\uu{{E}}}} \newcommand{\sig}{{\uu{{\sigma}}}} \newcommand{\Sig}{{\uu{{\Sigma}}}} \newcommand{\cod}{{\uv{{\symscr{b}}}}} \newcommand{\trans}[1]{{{}^{t}{#1}}} \newcommand{\sotimes}{{\stackrel{s}{\otimes}}} \newcommand{\sboxtimes}{\stackrel{s}{\boxtimes}} \newcommand{\norm}[1]{{\lVert{{#1}}\rVert}} \newcommand{\ud}{{\,\mathrm{d}}} \newcommand{\mat}{\mathsf} \DeclareMathOperator{\arcosh}{arcosh} \DeclareMathOperator{\divz}{div} \DeclareMathOperator{\divu}{\uv{div}} \DeclareMathOperator{\hess}{hess} \DeclareMathOperator{\gradu}{\uv{grad}} \DeclareMathOperator{\graduu}{\uu{grad}} \DeclareMathOperator{\Mat}{Mat} \DeclareMathOperator{\tr}{tr} \DeclareMathOperator{\ISO}{ISO} \newcommand{\volt}[1]{{#1}^{-1\circ}} \newcommand{\dcirc}{\overset{\circ}{:}} \newcommand{\jump}[1]{\mathopen{[\![}\,#1\,\mathclose{]\!]}} $$

Appendix C — ALV Hill (polarization) tensor kernel

The elastic Hill (polarization) tensor \(\mathbb{P}_\mathcal{E}(\mathbb{C})\) for an ellipsoidal inclusion \(\mathcal{E}\) embedded in an elastic medium with stiffness \(\mathbb{C}\) is defined in Appendix B. This appendix establishes its counterpart in ageing linear viscoelasticity (ALV): the Hill tensor kernel \(\uuuu{P}_\mathcal{E}(t,t')\), and derives its analytical expression for an isotropic matrix.

ALV Hill tensor kernel formula

Consider an ellipsoidal inclusion \(\mathcal{E}\) (characterized by shape tensor \(\uu{A}\), see B.1) embedded in an infinite ALV matrix with 4th-order relaxation kernel \(\uuuu{C}(t,t')\). The response of the medium to a uniform polarization \(\uu{p}(t)\) in \(\mathcal{E}\) yields a uniform strain field inside the inclusion, related to the polarization by a Volterra kernel. This kernel is the ALV Hill tensor (Barthélémy et al., 2016):

\[ \uuuu{P}_\mathcal{E}(t,t') = \frac{\det\uu{A}}{4\pi} \int_{\|\uv{\xi}\|=1} \frac{\uv{\xi} \sotimes \volt{(\uv{\xi}\cdot\uuuu{C}\cdot\uv{\xi})}_{(t,t')} \sotimes \uv{\xi}}{\|\uu{A}\cdot\uv{\xi}\|^3} \ud S_{\uv{\xi}} \tag{C.1}\]

where:

  • \(\volt{(\uv{\xi}\cdot\uuuu{C}\cdot\uv{\xi})}(t,t')\) is the Volterra inverse (in the sense of Section 17.1) of the acoustic tensor \(\uu{K}_{\uv{\xi}}(t,t') = (\uv{\xi}\cdot\uuuu{C}\cdot\uv{\xi})(t,t')\),
  • \(\sotimes\) denotes the symmetrized tensor product,
  • \(\uu{A}\) is the shape tensor whose eigenvalues give the semi-axes \(a\geq b\geq c\) of the ellipsoid.

This formula is the exact analogue of the elastic Hill tensor (see Appendix B) with the elastic matrix inverse \((\uv{\xi}\cdot\mathbb{C}\cdot\uv{\xi})^{-1}\) replaced by the Volterra inverse \(\volt{(\uv{\xi}\cdot\uuuu{C}\cdot\uv{\xi})}\). The proof rests on the Green’s kernel of the ALV medium and is detailed in (Barthélémy et al., 2016, sec. 3).

Note

Connection to the elastic case. If \(\uuuu{C}(t,t') = \mathbb{C}_E\,H(t-t')\) (elastic kernel, proportional to the Heaviside function), the Volterra inverse reduces to \(\volt{\uuuu{C}}(t,t') = \mathbb{C}_E^{-1}\,H(t-t')\), and C.1 recovers \(\uuuu{P}_\mathcal{E}(t,t') = \mathbb{P}_\mathcal{E}(\mathbb{C}_E)\,H(t-t')\).

Isotropic ALV matrix

When the matrix relaxation kernel is isotropic,

\[ \uuuu{C}(t,t') = 3k(t,t')\,\mathbb{J} + 2\mu(t,t')\,\mathbb{K}, \]

the acoustic tensor \(\uv{\xi}\cdot\uuuu{C}\cdot\uv{\xi}\) is isotropic in the \((\uv{\xi}\otimes\uv{\xi},\,\uu{I}-\uv{\xi}\otimes\uv{\xi})\) basis and can be inverted analytically in the Volterra sense. Substituting into C.1 yields (Barthélémy et al., 2019; Barthélémy et al., 2016, sec. 4):

\[ \uuuu{P}_\mathcal{E}(t,t') = \volt{\!\left(k + \tfrac{4}{3}\mu\right)}_{(t,t')}\,\uuuu{U}^{\uu{A}} + \volt{\mu}_{(t,t')}\,\bigl(\uuuu{V}^{\uu{A}} - \uuuu{U}^{\uu{A}}\bigr) \tag{C.2}\]

where \(\uuuu{U}^{\uu{A}}\) and \(\uuuu{V}^{\uu{A}}\) are the purely geometric tensors of Appendix B:

\[ \uuuu{U}^{\uu{A}} = \frac{\det\uu{A}}{4\pi}\int_{\|\uv{\xi}\|=1} \frac{\uv{\xi}\otimes\uv{\xi}\otimes\uv{\xi}\otimes\uv{\xi}}{\|\uu{A}\cdot\uv{\xi}\|^3}\ud S_{\uv{\xi}}, \qquad \uuuu{V}^{\uu{A}} = \frac{\det\uu{A}}{4\pi}\int_{\|\uv{\xi}\|=1} \frac{\uv{\xi}\sotimes\uuuu{I}\sotimes\uv{\xi}}{\|\uu{A}\cdot\uv{\xi}\|^3}\ud S_{\uv{\xi}} \]

Tip

Time-space decoupling. C.2 shows that the ALV Hill tensor kernel factorizes into a purely temporal part (\(\volt{(k+4\mu/3)}\) and \(\volt{\mu}\), scalar Volterra inverses) and a purely geometric part (\(\uuuu{U}^{\uu{A}}\) and \(\uuuu{V}^{\uu{A}}\), the same as in the elastic case). For a given ellipsoidal shape, the geometric tensors are computed once; only the scalar Volterra inverses carry the time dependence.

Special cases. For a sphere (\(\uu{A} = a\,\uu{I}\)):

\[ \uuuu{U}^{\uu{A}} = \tfrac{1}{3}\mathbb{J} + \tfrac{2}{15}\mathbb{K}, \qquad \uuuu{V}^{\uu{A}} = \tfrac{1}{3}\uuuu{I}, \]

so that the ALV Hill tensor kernel of a sphere in an isotropic matrix is:

\[ \uuuu{P}_{sphere}(t,t') = \volt{(3k+4\mu)} \circ \left(H\,\mathbb{J} + \tfrac{3}{5}(k+2\mu) \circ \volt{\mu}\,\mathbb{K}\right) \tag{C.3}\]

For spheroids and general ellipsoids in an isotropic matrix, explicit formulas for \(\uuuu{U}^{\uu{A}}\) and \(\uuuu{V}^{\uu{A}}\) in terms of elliptic integrals are given in Appendix B and (Barthélémy et al., 2019, Appendix A).