Appendix A — Tensor algebra
A.1 Conventions of tensor algebra
This appendix presents some conventions regarding tensor algebra in the usual three-dimensional euclidean space \(E=\R^3\). In the sequel, tensor components are associated to an orthonormal frame \((\ve{i})_{i=1,2,3}\) so that introducing the notion of tensor variance is useless here. The following presentation relies on the prior knowledge of the definition of tensors as multilinear operators and the classical isomorphism between the euclidean space and its dual through the scalar product \[\begin{aligned} \phi\,\colon E & \longrightarrow E^* \\ \uv{v} & \longmapsto \uv{v}\cdot\bullet \end{aligned} \tag{A.1}\] which allows to identify vectors and linear forms.
Consider two tensors \(\mathcal{T}\) and \(\mathcal{T}'\) of respective orders \(p\) and \(q\). The tensor product \(\mathcal{T}\otimes\mathcal{T}'\) is the \((p+q)\) order tensor decomposed as \[ \mathcal{T}\otimes\mathcal{T}'=\mathcal{T}_{i_1,\ldots,i_p} \,\mathcal{T}'_{i_{p+1},\ldots,i_{p+q}}\, \ve{i_1}\otimes\ldots\otimes\ve{i_{p+q}} \tag{A.2}\] where Einstein convention of immplicit summation over repeated indices is adopted and \(\ve{i_1}\otimes\ldots\otimes\ve{i_{p+q}}\) is the multilinear form such that1 \[ (\ve{i_1}\otimes\ldots\otimes\ve{i_{p+q}})(\ve{j_1},\ldots,\ve{j_{p+q}}) = \delta_{i_1,j_1}\ldots \delta_{i_{p+q},j_{p+q}} \tag{A.3}\] The notation \(\mathcal{T}\sotimes\mathcal{T}'\) indicates a tensor product followed by a symmetrization over the last index of \(\mathcal{T}\) and the first of \(\mathcal{T}'\), i.e. \[ \mathcal{T}\sotimes\mathcal{T}'= \frac{\mathcal{T}_{i_1,\ldots,i_p} \,\mathcal{T}'_{i_{p+1},\ldots,i_{p+q}} + \mathcal{T}_{i_1,\ldots,i_{p+1}} \,\mathcal{T}'_{i_p,\ldots,i_{p+q}} }{2} \,\ve{i_1}\otimes\ldots\otimes\ve{i_{p+q}} \tag{A.4}\] It follows that \[ \uv{u}\sotimes\uv{v}=\frac{\uv{u}\otimes\uv{v}+\uv{v}\otimes\uv{u}}{2} \tag{A.5}\] and an example of generalization involving a second-order tensor \(\uu{a}\) and vectors \(\uv{u}\) and \(\uv{v}\) \[ \uv{u}\sotimes\uu{a}\sotimes\uv{v}= \frac{u_i\,a_{jk}\,v_l+u_i\,a_{jl}\,v_k+u_j\,a_{ik}\,v_l+u_j\,a_{il}\,v_k}{4} \,\ve{i}\otimes\ve{j}\otimes\ve{k}\otimes\ve{l} \tag{A.6}\]
The simple dot product or contracted product between \(\mathcal{T}\) and \(\mathcal{T}'\) involves by convention a contraction between the last index of \(\mathcal{T}\) and the first of \(\mathcal{T}'\), which leads to the \((p+q-2)\) order tensor \[ \mathcal{T}\cdot\mathcal{T}'=\mathcal{T}_{i_1,\ldots,i_{p-1},{\color{red}k}} \,\mathcal{T}'_{{\color{red}k},i_{p},\ldots,i_{p+q-2}}\, \ve{i_1}\otimes\ldots\otimes\ve{i_{p+q-2}} \tag{A.7}\]
As regards the double dot product, the classical convention consists in consuming the indices going up from the extremities, which means that a first contraction acts as in the simple dot product then a second contraction is performed between the penultimate index of \(\mathcal{T}\) and the second one of \(\mathcal{T}'\). However and alternate convention adopted here is proposed in (Brisard, 2014)2, which somehow consists in considering that the double contraction operates over the two last indices of \(\mathcal{T}\) as a pair and the two first indices of \(\mathcal{T}'\) as the corresponding pair. In other words, this operation is such that if \(\uu{a}\) and \(\uu{b}\) are two second-order tensors and \(\uuuu{T}\) is a fourth-order tensor \[ \uu{a}:\uu{b}=a_{ij}b_{ij} \quad\textrm{and}\quad \uuuu{T}:\uu{a}=T_{ijkl}\, a_{kl}\, \ve{i}\otimes\ve{j} \tag{A.8}\] and the transpose tensor \(\trans{\uuuu{T}}\) is consistently defined by \[ \trans{\uuuu{T}}:\uu{a}=\uu{a}:\uuuu{T} \quad\Leftrightarrow\quad \left(\trans{\uuuu{T}}\right)_{ijkl}=\left(\uuuu{T}\right)_{klij} \tag{A.9}\] The quadruple dot product is introduced as a scalar product between fourth-order tensors as \[ \uuuu{T}:\uuuu{T}'=T_{ijkl} T'_{ijkl} \tag{A.10}\]
Another useful operator introduced in (Brisard, 2014)3 is the modified tensor product denoted by \(\boxtimes\). The fourth-order tensor \(\uu{a}\boxtimes\uu{b}\) (where \(\uu{a}\) and \(\uu{b}\) are two second-order tensors) is defined by its operation over any second-order tensor \(\uu{p}\) and by its components \[\begin{aligned} & (\uu{a}\boxtimes\uu{b}):\uu{p}=\uu{a}\cdot\uu{p}\cdot\trans{\uu{b}} =a_{ik}\,p_{kl}\,b_{jl}\,\ve{i}\otimes\ve{j} \\ & \left(\uu{a}\boxtimes\uu{b}\right)_{ijkl}=a_{ik}\,b_{jl} \end{aligned} \tag{A.11}\]
A symmetrized version of \(\boxtimes\) denoted by \(\sboxtimes\) can also be introduced. It operates as \[\begin{aligned} & (\uu{a}\sboxtimes\uu{b}):\uu{p}= (\uu{a}\boxtimes\uu{b}):\left(\frac{\uu{p}+\trans{\uu{p}}}{2}\right) =\uu{a}\cdot\left(\frac{\uu{p}+\trans{\uu{p}}}{2}\right)\cdot\trans{\uu{b}} \\ & (\uu{a}\sboxtimes\uu{b})_{ijkl}=\frac{a_{ik}\,b_{jl}+a_{il}\,b_{jk}}{2} \end{aligned} \tag{A.12}\]
It follows from these definitions that the fourth-order identity, as an operator over second-order tensors, writes \(\uuuu{1}=\uu{1}\boxtimes\uu{1}\) where \(\uu{1}\) is the second-order identity. The fourth-order operator allowing to extract the symmetric part of a second-order tensor writes \(\uuuu{I}=\uu{1}\sboxtimes\uu{1}\). The latter tensor, which obviously complies with the conditions of minor symmetries, is classically used to play the role of fourth-order identity operating over symmetric second-order tensors.
Some remarkable relationships result from the previous definitions \[\begin{aligned} &(\uu{a}\boxtimes\uu{b}):(\uv{u}\otimes\uv{v})= (\uu{a}\cdot\uv{u})\otimes(\uu{b}\cdot\uv{v}) &(a) \\ &(\uu{a}\boxtimes\uu{b}):(\uu{c}\boxtimes\uu{d})= (\uu{a}\cdot\uu{c})\boxtimes(\uu{b}\cdot\uu{d}) &(b) \\ &(\uu{a}\sboxtimes\uu{b}):(\uu{c}\sboxtimes\uu{d})= \frac{(\uu{a}\cdot\uu{c})\sboxtimes(\uu{b}\cdot\uu{d})+ (\uu{a}\cdot\uu{d})\sboxtimes(\uu{b}\cdot\uu{c})}{2} &(c) \\ &(\uu{a}\sboxtimes\uu{a}):(\uu{b}\sboxtimes\uu{b})= (\uu{a}\cdot\uu{b})\sboxtimes(\uu{a}\cdot\uu{b}) &(d) \\ &\trans{(\uu{a}\boxtimes\uu{b})}=\trans{\uu{a}}\boxtimes\trans{\uu{b}} &(e) \\ &\trans{(\uu{a}\sboxtimes\uu{a})}=\trans{\uu{a}}\sboxtimes\trans{\uu{a}} \;\textrm{ but }\; \trans{(\uu{a}\sboxtimes\uu{b})}\neq\trans{\uu{a}}\sboxtimes\trans{\uu{b}} &(f) \\ &(\uu{a}\boxtimes\uu{b})^{-1}=\uu{a}^{-1}\boxtimes\uu{b}^{-1} &(g) \\ &(\uu{a}\sboxtimes\uu{a})^{-1}:\uu{p}=(\uu{a}^{-1}\sboxtimes\uu{a}^{-1}):\uu{p} \;\textrm{ if }\;\trans{\uu{p}}=\uu{p} \;\textrm{ but }\; (\uu{a}\sboxtimes\uu{b})^{-1}\neq\uu{a}^{-1}\sboxtimes\uu{b}^{-1} &(h) \end{aligned} \tag{A.13}\]
A.2 Kelvin-Mandel notation
The Kelvin-Mandel notation allows to write the matrix of a symmetric second-order tensor in a given orthonormal frame under the form of a vector of \(\R^6\) \[ \Mat(\eps,(\ve{i}))= \left( \begin{array}{ccc} \varepsilon_{11} & \varepsilon_{12} & \varepsilon_{31}\\ \varepsilon_{12} & \varepsilon_{22} & \varepsilon_{23}\\ \varepsilon_{31} & \varepsilon_{23} & \varepsilon_{33} \end{array} \right) \mapsto \left( \begin{array}{c} \varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ \sqrt{2}\,\varepsilon_{23} \\ \sqrt{2}\,\varepsilon_{31} \\ \sqrt{2}\,\varepsilon_{12} \end{array} \right) \tag{A.14}\]
The vector of \(\R^6\) in (A.14) corresponds to the components of the second-order tensor \(\eps\) in the basis ordered as \[ \mathcal{B}=\left( \ve{1}\otimes\ve{1}, \ve{2}\otimes\ve{2}, \ve{3}\otimes\ve{3}, \sqrt{2}\,\ve{2}\otimes\ve{3}, \sqrt{2}\,\ve{3}\otimes\ve{1}, \sqrt{2}\,\ve{1}\otimes\ve{2} \right) \tag{A.15}\]
The tensors of the basis (A.15) form an orthonormal frame spanning the space of symmetric second-order tensors equipped with the double contraction “\(:\)” as scalar product. It follows that the double contraction between symmetric second-order tensors is no other than the classical scalar product of the corresponding vectors of \(\R^6\) written according to the convention (A.14).
Moreover a fourth-order tensor with minor symetries (\(C_{jikl}=C_{ijlk}=C_{ijkl}\)), which can be seen as a linear operator acting over symmetric second-order tensors by double contraction, writes in the same convention under the form of a \(6\times 6\) square matrix (the solid lines separate blocks affected by different factors whereas the colored components highlight a central block playing a major role in the sequel) \[ \small \Mat(\uuuu{C},\mathcal{B})= \left( \begin{array}{ccc|ccc} C_{1111} & C_{1122} & C_{1133} & \sqrt{2}\,C_{1123} & \sqrt{2}\,C_{1131} & \sqrt{2}\,C_{1112} \\ C_{2211} & C_{2222} & C_{2233} & \sqrt{2}\,C_{2223} & \sqrt{2}\,C_{2231} & \sqrt{2}\,C_{2212} \\ C_{3311} & C_{3322} & \color{red}C_{3333} & \color{red}\sqrt{2}\,C_{3323} & \color{red}\sqrt{2}\,C_{3331} & \sqrt{2}\,C_{3312} \\ \hline \sqrt{2}\,C_{2311} & \sqrt{2}\,C_{2322} & \color{red}\sqrt{2}\,C_{2333} & \color{red}2\,C_{2323} & \color{red}2\,C_{2331} & 2\,C_{2312} \\ \sqrt{2}\,C_{3111} & \sqrt{2}\,C_{3122} & \color{red}\sqrt{2}\,C_{3133} & \color{red}2\,C_{3123} & \color{red}2\,C_{3131} & 2\,C_{3112} \\ \sqrt{2}\,C_{1211} & \sqrt{2}\,C_{1222} & \sqrt{2}\,C_{1233} & 2\,C_{1223} & 2\,C_{1231} & 2\,C_{1212} \end{array} \right) \tag{A.16}\] The result of \(\uuuu{C}:\eps\) writes as a classical matrix-vector product of (A.16) by (A.14).
However another way of ordering the tensors of (A.15) which proves useful for the calculation of crack compliance is based on a gathering of one set of three in-plane and another one of three out-of-plane tensors (the latter involving \(\n=\ve{3}\) assumed to be the normal of the crack and the former not) \[ \mathcal{B}^*=\Bigg( \underbrace{ \ve{1}\otimes\ve{1}, \ve{2}\otimes\ve{2}, \sqrt{2}\,\ve{1}\sotimes\ve{2} }_{\textrm{in-plane}}, \quad \underbrace{ \ve{3}\otimes\ve{3}, \sqrt{2}\,\ve{2}\sotimes\ve{3}, \sqrt{2}\,\ve{3}\sotimes\ve{1} }_{\textrm{out-of-plane}} \Bigg) \tag{A.17}\] such that the matrix of \(\uuuu{C}\) in \(\mathcal{B}^*\) is now obtained by permutations of lines and columns of (A.16) to give \[ \small \Mat(\uuuu{C},\mathcal{B}^*)= \left( \begin{array}{ccc|ccc} C_{1111} & C_{1122} & \sqrt{2}\,C_{1112} & C_{1133} & \sqrt{2}\,C_{1123} & \sqrt{2}\,C_{1131} \\ C_{2211} & C_{2222} & \sqrt{2}\,C_{2212} & C_{2233} & \sqrt{2}\,C_{2223} & \sqrt{2}\,C_{2231} \\ \sqrt{2}\,C_{1211} & \sqrt{2}\,C_{1222} & 2\,C_{1212} & \sqrt{2}\,C_{1233} & 2\,C_{1223} & 2\,C_{1231} \\ \hline C_{3311} & C_{3322} & \sqrt{2}\,C_{3312} & C_{3333} & \sqrt{2}\,C_{3323} & \sqrt{2}\,C_{3331} \\ \sqrt{2}\,C_{2311} & \sqrt{2}\,C_{2322} & 2\,C_{2312} & \sqrt{2}\,C_{2333} & 2\,C_{2323} & 2\,C_{2331} \\ \sqrt{2}\,C_{3111} & \sqrt{2}\,C_{3122} & 2\,C_{3112} & \sqrt{2}\,C_{3133} & 2\,C_{3123} & 2\,C_{3131} \end{array} \right) \tag{A.18}\] One may notice that the bottom right \(3\times 3\) block of (A.18) exactly corresponds to the colored block in (A.16).
A.3 Rotation of tensors
Recalling that the set of second-order tensors is isomorphic to the set of endomorphism in an euclidean space, it is natural to define a rotation tensor as an element of the special orthogonal group, i.e. tensors \(\uu{R}\) such that \(\trans{\uu{R}}\cdot\uu{R}=\uu{1}\) and \(\det{\uu{R}}=1\). In \(\R^3\) equipped with an orthonormal frame \((\ve{i})_{i=1,2,3}\), these tensors can be defined by three Euler angles (see Fig. A.1) \(0\leq\theta\leq\pi\), \(0\leq\phi\leq 2\pi\) and \(0\leq\psi\leq 2\pi\) such that4 \[ \small \Mat(\uu{R},(\ve{i}))= \left( \begin{array}{ccc} c_\theta c_\psi c_\phi - s_\psi s_\phi & - c_\theta c_\phi s_\psi - c_\psi s_\phi & c_\phi s_\theta \\ c_\theta c_\psi s_\phi + c_\phi s_\psi & - c_\theta s_\psi s_\phi + c_\psi c_\phi & s_\theta s_\phi \\ - c_\psi s_\theta & s_\theta s_\psi & c_\theta \\ \end{array} \right) \tag{A.19}\] A rotated frame \((\underline{e}'_i)_{i=1,2,3}\) is obtained by application of the rotation \(\uu{R}\) on the vectors \(\ve{i}\) (see Fig. A.1) \[ \forall i \in \{1,2,3\}\quad \underline{e}'_i=\uu{R}\cdot\ve{i} \tag{A.20}\]
The rotation of Eurler angles \((\theta,\phi,\psi)\) applies on a \(p\) order tensor \(\mathcal{T}\) as \[ \mathcal{T}= \mathcal{T}_{i_1,\ldots,i_p} \,\ve{i_1}\otimes\ldots\otimes\ve{i_{p}} \overset{\uu{R}}{\longmapsto} \uu{R}(\mathcal{T})= \mathcal{T}_{i_1,\ldots,i_p} \,(\uu{R}\cdot\ve{i_1})\otimes\ldots\otimes(\uu{R}\cdot\ve{i_{p}}) \tag{A.21}\]
The application of (A.21) to a second-order tensor \(\uu{a}\) therefore gives \[ \uu{R}(\uu{a})=\uu{R}\cdot \uu{a} \cdot\trans{\uu{R}} \tag{A.22}\] and to a fourth-order tensor \(\uuuu{T}\) \[ \uu{R}(\uuuu{T})=(\uu{R}\sboxtimes\uu{R}): \uuuu{T} :\trans{(\uu{R}\sboxtimes\uu{R})} \tag{A.23}\] where the fourth-order rotation tensor \(\uuuu{R}=\uu{R}\sboxtimes\uu{R}\) can be expressed in Kelvin-Mandel notation (A.16) by means of the components of \(\uu{R}\) defined in (A.19) \[ \scriptsize \Mat(\uuuu{R},\mathcal{B})= \left( \begin{array}{cccccc} R_{1 1}^{2} & R_{1 2}^{2} & R_{1 3}^{2} & \sqrt{2} R_{1 2} R_{1 3} & \sqrt{2} R_{1 1} R_{1 3} & \sqrt{2} R_{1 1} R_{1 2} \\ R_{2 1}^{2} & R_{2 2}^{2} & R_{2 3}^{2} & \sqrt{2} R_{2 2} R_{2 3} & \sqrt{2} R_{2 1} R_{2 3} & \sqrt{2} R_{2 1} R_{2 2} \\ R_{3 1}^{2} & R_{3 2}^{2} & R_{3 3}^{2} & \sqrt{2} R_{3 2} R_{3 3} & \sqrt{2} R_{3 1} R_{3 3} & \sqrt{2} R_{3 1} R_{3 2} \\ \sqrt{2} R_{2 1} R_{3 1} & \sqrt{2} R_{2 2} R_{3 2} & \sqrt{2} R_{2 3} R_{3 3} & R_{2 2} R_{3 3} + R_{2 3} R_{3 2} & R_{2 1} R_{3 3} + R_{2 3} R_{3 1} & R_{2 1} R_{3 2} + R_{2 2} R_{3 1} \\ \sqrt{2} R_{1 1} R_{3 1} & \sqrt{2} R_{1 2} R_{3 2} & \sqrt{2} R_{1 3} R_{3 3} & R_{1 2} R_{3 3} + R_{1 3} R_{3 2} & R_{1 1} R_{3 3} + R_{1 3} R_{3 1} & R_{1 1} R_{3 2} + R_{1 2} R_{3 1} \\ \sqrt{2} R_{1 1} R_{2 1} & \sqrt{2} R_{1 2} R_{2 2} & \sqrt{2} R_{1 3} R_{2 3} & R_{1 2} R_{2 3} + R_{1 3} R_{2 2} & R_{1 1} R_{2 3} + R_{1 3} R_{2 1} & R_{1 1} R_{2 2} + R_{1 2} R_{2 1} \\ \end{array} \right) \tag{A.24}\] It results that (A.23) can be seen as a classical rotation operation involving matrix multiplications in \(\R^6\).
A.4 Fourth-order isotropic tensors
This paragraph concerns fourth-order tensors operating over symmetrical second-order tensors, which allows to impose that they satisfy the minor symmetries (\(C_{jikl}=C_{ijlk}=C_{ijkl}\)). The identity operator is given by \[ \uuuu{I}=\uu{1}\sboxtimes\uu{1}= \frac{\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}}{2}\,\ve{i}\otimes\ve{j}\otimes\ve{k}\otimes\ve{l} \tag{A.25}\] of identity matrix in Kelvin-Mandel convention \[ \small \Mat(\uuuu{I},\mathcal{B})= \left( \begin{array}{ccc|ccc} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) \tag{A.26}\]
It is classically proven that any minor-symmetrical fourth-order tensor invariant by rotation (A.23) write as a linear combination on the two projectors \(\uuuu{J}\) and \(\uuuu{K}\) which respectively extract the spherical and deviatoric part of any symmetrical second-order tensor \[ \uuuu{J}:\uu{a}=\frac{1}{3}\tr{\uu{a}}\,\uu{1} \quad \textrm{and} \quad \uuuu{K}:\uu{a}=\uu{a}-\frac{1}{3}\tr{\uu{a}}\,\uu{1} \tag{A.27}\] In other words, \(\uuuu{J}\) and \(\uuuu{K}\) are defined by \[ \uuuu{J}=\frac{1}{3}\uu{1}\otimes\uu{1} \quad \textrm{and} \quad \uuuu{K}=\uuuu{I}-\uuuu{J}=\uu{1}\sboxtimes\uu{1}-\frac{1}{3}\uu{1}\otimes\uu{1} \tag{A.28}\] their components by \[ J_{ijkl}=\frac{1}{3}\delta_{ij}\delta_{ik} \quad \textrm{and} \quad K_{ijkl}=\frac{\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}}{2}-\frac{1}{3}\delta_{ij}\delta_{ik} \tag{A.29}\] and their matrices in Kelvin-Mandel notation relatively to any orthonormal frame by \[ \small \Mat(\uuuu{J},\mathcal{B})= \left( \begin{array}{ccc|ccc} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 \\ \frac{1}{3} & \frac{1}{3} & \frac{1}{3} & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right) \tag{A.30}\]
\[ \small \Mat(\uuuu{K},\mathcal{B})= \left( \begin{array}{ccc|ccc} \frac{2}{3} & \frac{-1}{3} & \frac{-1}{3} & 0 & 0 & 0 \\ \frac{-1}{3} & \frac{2}{3} & \frac{-1}{3} & 0 & 0 & 0 \\ \frac{-1}{3} & \frac{-1}{3} & \frac{2}{3} & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \\ \end{array} \right) \tag{A.31}\]
The following relationships are easily obtained \[ \uuuu{J}:\uuuu{J}=\uuuu{J} \quad ; \quad \uuuu{K}:\uuuu{K}=\uuuu{K} \quad ; \quad \uuuu{J}:\uuuu{K}=\uuuu{0} \quad ; \quad \uuuu{J}::\uuuu{J}=1 \quad ; \quad \uuuu{K}::\uuuu{K}=5 \tag{A.32}\]
The isotropisation of any fourth-order tensor \(\uuuu{T}\) is defined by (Bornert et al., 2001) \[ \ISO(\uuuu{T})=\left(\uuuu{T}::\uuuu{J}\right)\,\uuuu{J}+\left(\frac{\uuuu{T}::\uuuu{K}}{5}\right)\,\uuuu{K} \tag{A.33}\] It is easy to show that (A.33) is no other than the closest isotropic tensor to \(\uuuu{T}\) if the distance is chosen as the euclidean one i.e. associated to the scalar product (A.10). Note however that this isotropisation relying on euclidean distance to the set of isotropic tensors does not lead to the same result if \(\uuuu{T}'\) is considered instead of \(\uuuu{T}\). Other distance definitions such as log-Euclidean distance fullfilling invariance by inversion are analyzed in (Morin et al., 2020).
A.5 Fourth-order transversely isotropic tensors and Walpole basis
The Walpole basis ((Walpole, 1984), (Brisard, 2014)5) allowing to write any fourth-order transversely isotropic relatively to a an axis oriented by the unit vector \(\n\) is composed of the six following tensors built from \(\uu{1}_n=\n\otimes\n\) and \(\uu{1}_T=\uu{1}-\uu{1}_n\) \[\begin{aligned} & \uuuu{E}_1=\uu{1}_n\otimes\uu{1}_n \quad\textrm{ ; }\quad \uuuu{E}_2=\frac{\uu{1}_T\otimes\uu{1}_T}{2} &\textrm{ ; }& \uuuu{E}_3=\frac{\uu{1}_n\otimes\uu{1}_T}{\sqrt{2}} \quad\textrm{ ; }\quad \uuuu{E}_4=\frac{\uu{1}_T\otimes\uu{1}_n}{\sqrt{2}} &(a) \\ & \uuuu{E}_5=\uu{1}_T\sboxtimes\uu{1}_T-\frac{\uu{1}_T\otimes\uu{1}_T}{2} &\textrm{ ; }& \uuuu{E}_6=\uu{1}_T\sboxtimes\uu{1}_n+\uu{1}_n\sboxtimes\uu{1}_T &(b) \end{aligned} \tag{A.34}\]
Any transversely isotropic fourth-order tensor can be decomposed as \[ {\uuuu{L}} = {\ell}_{1}\,{\uuuu{E}}_{1} + {\ell}_{2}\,{\uuuu{E}}_{2} + {\ell}_{3}\,{\uuuu{E}}_{3}+ {\ell}_{4}\,{\uuuu{E}}_{4}+ {\ell}_{5}\,{\uuuu{E}}_{5}+ {\ell}_{6}\,{\uuuu{E}}_{6} \tag{A.35}\]
The six parameters can be conveniently gathered in a triplet composed of a \(2 \times 2\) matrix containing the four first parameters \(\ell_i\) (\(1\leq i \leq 4\)) and the two last parameters \(\ell_5\) and \(\ell_6\) \[ \uuuu{L} \equiv \left( L , {\ell}_{5} , {\ell}_{6} \right) , \quad L = \left( \begin{array}{cc} {\ell}_{1} & {\ell}_{3} \\ {\ell}_{4} & {\ell}_{2} \\ \end{array} \right) \tag{A.36}\]
Such a synthetic notation allows simple calculations of products and inverses which consist in classical matrix or scalar products and inverses \[\begin{aligned} & \uuuu{L} : \uuuu{M} \equiv \left( L M , {\ell}_{5} m_5 , {\ell}_{6} m_6 \right) &(a) \\ & {\uuuu{L}}^{-1} \equiv \left( {L}^{-1} , \frac{1}{{\ell}_{5}} , \frac{1}{{\ell}_{6}} \right) &(b) \end{aligned} \tag{A.37}\]
\(\delta_{ij}=1\) if \(i=j\) and \(0\) if \(i\neq j\) (Kronecker symbol)↩︎
see https://sbrisard.github.io/posts/20140219-on_the_double_dot_product.html↩︎
see https://sbrisard.github.io/posts/20140226-decomposition_of_transverse_isotropic_fourth-rank_tensors.html↩︎
with the simplifying writing convention \(c_\theta=\cos\theta\), \(s_\theta=\sin\theta\), \(c_\phi=\cos\phi\), \(s_\phi=\sin\phi\), \(c_\psi=\cos\psi\), \(s_\psi=\sin\psi\)↩︎
see https://sbrisard.github.io/posts/20140226-decomposition_of_transverse_isotropic_fourth-rank_tensors.html↩︎