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Recovering the Normal Form and Symmetry Class of an Elasticity Tensor

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Abstract

We propose an effective geometrical approach to recover the normal form of a given Elasticity tensor. We produce a rotation which brings an Elasticity tensor onto its normal form, given its components in any orthonormal frame, and this for any tensor of any symmetry class. Our methodology relies on the use of specific covariants and on the geometric characterization of each symmetry class using these covariants. An algorithm to detect the symmetry class of an Elasticity tensor is finally formulated.

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Correspondence to M. Olive.

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R. Desmorat, B. Kolev and M. Olive were partially supported by CNRS Projet 80|Prime GAMM (Géométrie algébrique complexe/réelle et mécanique des matériaux)

Appendices

Appendix A: Harmonic Components of Considered Elasticity Tensors

In this section, all the linear covariants \(\mathbf{d}\), \(\mathbf{v}\), \(\mathbf{H}\) are given in GPa and the fourth-order harmonic part \(\mathbf{H}\) is expressed in Voigt’s representation.

Cubic approximation \(\mathbf{E}^{\gamma }_{cubic}\) (5.2) of \(\mathbf{E}^{\gamma }\)::

\(\mathbf{d}^{\prime } = \mathbf{v}^{\prime } = 0\), \({\mathbf{d}_{2}}^{\prime } = 0\), \(\operatorname{tr}\mathbf{d}= 1531\), \(\operatorname{tr}\mathbf{v}= 1479\), and

$$ [\mathbf{H}^{\gamma }_{cubic}] = \begin{pmatrix} -59.1358 & 38.9089 & 20.2269 & 6.39666 & 41.9737 & -21.1614 \\ 38.9089 & -75.3102 & 36.4013 & -27.7808 & 2.27754 & 16.6041 \\ 20.2269 & 36.4013 & -56.6282 & 21.3841 & -44.2512 & 4.55736 \\ 6.39666 & -27.7808 & 21.3841 & 36.4013 & 4.55736 & 2.27754 \\ 41.9737 & 2.27754 & -44.2512 & 4.55736 & 20.2269 & 6.39666 \\ -21.1614 & 16.6041 & 4.55736 & 2.27754 & 6.39666 & 38.9089 \end{pmatrix} . $$
(A.1)
Tetragonal approximation \(\mathbf{E}^{\gamma }_{tetra}\) (5.3) of \(\mathbf{E}^{\gamma }\)::

\(\mathbf{d}^{\prime } = \mathbf{v}^{\prime } = 0\), \(\operatorname{tr}\mathbf{d}= 1531\), \(\operatorname{tr}\mathbf{v}= 1479\), and

$$ [\mathbf{H}^{\gamma }_{tetra}] = \begin{pmatrix} -59.9342 & 35.8495 & 24.0847 & 5.8239 & 46.7414 & -21.0897 \\ 35.8495 & -69.6028 & 33.7533 & -25.7103 & 1.8896 & 15.3674 \\ 24.0847 & 33.7533 & -57.8381 & 19.8864 & -48.631 & 5.7223 \\ 5.8239 & -25.7103 & 19.8864 & 33.7533 & 5.7223 & 1.8896 \\ 46.7414 & 1.8896 & -48.631 & 5.7223 & 24.0847 & 5.8239 \\ -21.0897 & 15.3674 & 5.7223 & 1.8896 & 5.8239 & 35.8495 \end{pmatrix} . $$
(A.2)
First orthotropic approximation \(\mathbf{E}^{\gamma \, (1)}_{ortho}\) (5.4) of \(\mathbf{E}^{\gamma }\)::

\(\mathbf{d}^{\prime } = \mathbf{v}^{\prime } = 0\), \(\operatorname{tr}\mathbf{d}= 1531\), \(\operatorname{tr}\mathbf{v}= 1479\) and

$$ [\mathbf{H}^{\gamma }_{orth}] = \begin{pmatrix} -57.9586 & 33.959 & 23.9997 & 5.3342 & 46.3021 & -20.3543 \\ 33.959 & -69.5995 & 35.6405 & -26.2801 & 2.88311 & 14.4327 \\ 23.9997 & 35.6405 & -59.6402 & 20.9459 & -49.1853 & 5.92151 \\ 5.3342 & -26.2801 & 20.9459 & 35.6405 & 5.92151 & 2.88311 \\ 46.3021 & 2.88311 & -49.1853 & 5.92151 & 23.9997 & 5.3342 \\ -20.3543 & 14.4327 & 5.92151 & 2.88311 & 5.3342 & 33.959 \end{pmatrix} . $$
(A.3)
Second orthotropic approximation \(\mathbf{E}^{\gamma \, (2)}_{ortho}\) (5.5) of \(\mathbf{E}^{\gamma }\)::

\(\operatorname{tr}\mathbf{d}= 1531\), \(\operatorname{tr}\mathbf{v}= 1479\),

$$\begin{aligned} \mathbf{d}^{\prime } &= \begin{pmatrix} -3.6837 & -0.661831 & 1.37627 \\ -0.661831 & 1.96893 & 0.158989 \\ 1.37627 & 0.158989 & 1.71477 \end{pmatrix} ,\\ \mathbf{v}^{\prime }&= \begin{pmatrix} -3.31669 & -0.8154 & -0.112441 \\ -0.8154 & 6.51238 & 1.36466 \\ -0.112441 & 1.36466 & -3.19569 \end{pmatrix} , \end{aligned}$$

and \(\mathbf{H}= \mathbf{H}^{\gamma }_{cubic}\) is given by (A.1) (in particular \({\mathbf{d}_{2}}^{\prime } = 0\)).

Monoclinic approximation \(\mathbf{E}^{\gamma }_{mono}\) (5.6) of \(\mathbf{E}^{\gamma }\)::

\(\mathbf{d}^{\prime } = \mathbf{v}^{\prime } = 0\), \(\operatorname{tr}\mathbf{d}= 1531\), \(\operatorname{tr}\mathbf{v}= 1479\) and

$$ [\mathbf{H}^{\gamma }_{mono}] = \begin{pmatrix} -58.7344 & 34.9674 & 23.767 & 0.6715 & 47.7714 & -18.1515 \\ 34.9674 & -67.8968 & 32.9294 & -26.3969 & 4.4758 & 18.2628 \\ 23.767 & 32.9294 & -56.6964 & 25.7254 & -52.2472 & -0.1113 \\ 0.6715 & -26.3969 & 25.7254 & 32.9294 & -0.1113 & 4.4758 \\ 47.7714 & 4.4758 & -52.2472 & -0.1113 & 23.767 & 0.6715 \\ -18.1515 & 18.2628 & -0.1113 & 4.4758 & 0.6715 & 34.9674 \end{pmatrix} . $$
(A.4)
Trigonal approximation \(\mathbf{E}^{\alpha }_{trig}\) (5.7) of \(\alpha \)-quartz Elasticity tensor::

\(\operatorname{tr}\mathbf{d}= 34.72\), \(\operatorname{tr}\mathbf{v}= 59.24\),

$$\begin{aligned} \mathbf{d}^{\prime } &= \begin{pmatrix} -0.764933 & 0.3098 & 0.4514 \\ 0.3098 & -0.0727333 & 1.1811 \\ 0.4514 & 1.1811 & 0.837667 \end{pmatrix} ,\\ \mathbf{v}^{\prime } &= \begin{pmatrix} -1.02767 & 0.4162 & 0.6064 \\ 0.4162 & -0.0976667 & 1.5867 \\ 0.6064 & 1.5867 & 1.12533 \end{pmatrix} , \end{aligned}$$

and

$$ [\mathbf{H}^{\alpha }_{trig}] = \begin{pmatrix} -1.4953 & -0.0086 & 1.504 & -0.0148 & -0.2917 & -0.8173 \\ -0.0086 & 0.6713 & -0.6626 & -0.1899 & -0.0484 & 2.0042 \\ 1.504 & -0.6626 & -0.8413 & 0.2046 & 0.3402 & -1.187 \\ -0.0148 & -0.1899 & 0.2046 & -0.6626 & -1.187 & -0.0484 \\ -0.2917 & -0.0484 & 0.3402 & -1.187 & 1.504 & -0.0148 \\ -0.8173 & 2.0042 & -1.187 & -0.0484 & -0.0148 & -0.0086 \\ \end{pmatrix}. $$
(A.5)
Transversely isotropic approximation \(\mathbf{E}^{KS}_{TI}\) (5.8) of \(\mathbf{E}^{KS}\)::

\(\operatorname{tr}\mathbf{d}= 6.0707\), \(\operatorname{tr}\mathbf{v}= 6.4911\),

$$\begin{aligned} \mathbf{d}^{\prime } &= \begin{pmatrix} 0.221833 & -0.0745 & -0.2495 \\ -0.0745 & 0.235733 & -0.2272 \\ -0.2495 & -0.2272 & -0.457567 \end{pmatrix} ,\\ \mathbf{v}^{\prime } &= \begin{pmatrix} 0.1507 & -0.0505 & -0.1695 \\ -0.0505 & 0.1601 & -0.1543 \\ -0.1695 & -0.1543 & -0.3108 \end{pmatrix} , \end{aligned}$$

and

$$ [\mathbf{H}^{KS}_{TI}] = \begin{pmatrix} 0.0176 & 0.0123 & -0.0299 & -0.0138 & -0.0969 & -0.0289 \\ 0.0123 & 0.0287 & -0.0409 & -0.0923 & -0.0195 & -0.0302 \\ -0.0299 & -0.0409 & 0.0708 & 0.106 & 0.1165 & 0.0592 \\ -0.0138 & -0.0923 & 0.106 & -0.0409 & 0.0592 & -0.0195 \\ -0.0969 & -0.0195 & 0.1165 & 0.0592 & -0.0299 & -0.0138 \\ -0.0289 & -0.0302 & 0.0592 & -0.0195 & -0.0138 & 0.0123 \end{pmatrix} . $$

Appendix B: The Generalized Cross-Product in Components

The 10 independent components of the totally symmetric third order tensor \(\mathbf{a}\times \mathbf{b}\), where both \(\mathbf{a}\) and \(\mathbf{b}\) are symmetric second order tensors, are:

$$\begin{aligned} & (\mathbf{a}\times \mathbf{b})_{111} = a_{12} b_{13}-a_{13} b_{12}, \\ & (\mathbf{a}\times \mathbf{b})_{112} = \frac{1}{3} (-a_{11} b_{13}+a_{12} b_{23}+a_{13} b_{11}-a_{13} b_{22}+a_{22} b_{13}-a_{23} b_{12}), \\ & (\mathbf{a}\times \mathbf{b})_{113} = \frac{1}{3} (a_{11} b_{12}-a_{12} b_{11}+a_{12} b_{33}-a_{13} b_{23}+a_{23} b_{13}-a_{33} b_{12}), \\ & (\mathbf{a}\times \mathbf{b})_{122} = \frac{1}{3} (-a_{11} b_{23}-a_{12} b_{13}+a_{13} b_{12}+a_{22} b_{23}+a_{23} b_{11}-a_{23} b_{22}), \\ & (\mathbf{a}\times \mathbf{b})_{123} = \frac{1}{6} (a_{11} b_{22}- a_{11} b_{33}+a_{22} b_{33}-a_{22}b_{11}+a_{33} b_{11}-a_{33}b_{22}), \\ \\ & (\mathbf{a}\times \mathbf{b})_{133} = \frac{1}{3} (a_{11} b_{23}-a_{12} b_{13}+a_{13} b_{12}-a_{23} b_{11}+a_{23} b_{33}-a_{33} b_{23}), \\ & (\mathbf{a}\times \mathbf{b})_{222} = a_{23} b_{12}-a_{12} b_{23}, \\ & (\mathbf{a}\times \mathbf{b})_{223} = \frac{1}{3} (a_{12} b_{22}-a_{12} b_{33}-a_{13} b_{23}-a_{22} b_{12}+a_{23} b_{13}+a_{33} b_{12}), \\ & (\mathbf{a}\times \mathbf{b})_{233} = \frac{1}{3} (a_{12} b_{23}+a_{13} b_{22}- a_{13} b_{33}-a_{22} b_{13}-a_{23} b_{12}+a_{33} b_{13}), \\ & (\mathbf{a}\times \mathbf{b})_{333} = a_{13} b_{23}-a_{23} b_{13}. \end{aligned}$$
(B.1)

For the components of its trace (which is a vector) one has

$$\begin{aligned} & (\operatorname{tr}(\mathbf{a}\times \mathbf{b}))_{1}=\frac{1}{3} (a_{12} b_{13}-a_{13} b_{12}+a_{22} b_{23}-a_{23} b_{22}+a_{23} b_{33}-a_{33} b_{23}), \\ & (\operatorname{tr}(\mathbf{a}\times \mathbf{b}))_{2}=\frac{1}{3} (-a_{11} b_{13}-a_{12} b_{23}+a_{13} b_{11}-a_{13} b_{33}+a_{23} b_{12}+a_{33} b_{13}), \\ & (\operatorname{tr}(\mathbf{a}\times \mathbf{b}))_{3}= \frac{1}{3} (a_{11} b_{12}-a_{12} b_{11}+a_{12} b_{22}+a_{13} b_{23}-a_{22} b_{12}-a_{23} b_{13}). \end{aligned}$$

When \(\mathbf{S}\) is a totally symmetric fourth order tensor and \(\mathbf{a}\) is a symmetric second order tensor, the ten independent components of the totally symmetric third order tensor \(\operatorname{tr}(\mathbf{S}\times \mathbf{a})\) are

$$\begin{aligned} (\operatorname{tr}(\mathbf{S}\times \mathbf{a}))_{111} = & \frac{3}{10} \Big(-a_{12} (2 S_{1113}+S_{1223}+S_{1333})+a_{13} (2 S_{1112}+S_{1222}+S_{1233})-a_{22} S_{1123} \\ & +a_{23} (S_{1122}-S_{1133})+a_{33} S_{1123}\Big), \\ (\operatorname{tr}(\mathbf{S}\times \mathbf{a}))_{112} = & \frac{1}{10} \Big(a_{11} (2 S_{1113}+S_{1223}+S_{1333})- a_{12} (2S_{1123}+S_{2223}+ S_{2333}) \\ & +a_{13} (-2 S_{1111}+2 S_{1122}+S_{2222}+S_{2233})-a_{22} (S_{1113}+3 S_{1223}+ S_{1333}) \\ & +a_{23} (3 S_{1222}- S_{1233})+ a_{33} (2S_{1223}-S_{1113})\Big), \\ (\operatorname{tr}(\mathbf{S}\times \mathbf{a}))_{113} = & \frac{1}{10} \Big(-a_{11} (2 S_{1112}+S_{1222}+S_{1233})+a_{12} (2 S_{1111}-2 S_{1133}-S_{2233}-S_{3333}) \\ & +a_{13}(2 S_{1123}+S_{2223}+S_{2333})+a_{22} (S_{1112}-2 S_{1233})+a_{23} (S_{1223}-3 S_{1333}) \\ & +a_{33} (S_{1112}+ S_{1222}+3 S_{1233})\Big), \\ (\operatorname{tr}(\mathbf{S}\times \mathbf{a}))_{122} = & \frac{1}{10} \Big(a_{11} (3 S_{1123}+S_{2223}+ S_{2333})+a_{12} (S_{1113}+2 S_{1223}+ S_{1333}) \\ & +a_{13} (S_{1233}-3 S_{1112}) -a_{22}( S_{1123}+2 S_{2223}+ S_{2333}) \\ & -a_{23} (S_{1111}+2 S_{1122}+S_{1133}-2 S_{2222})+a_{33} (S_{2223}-2 S_{1123})\Big), \\ (\operatorname{tr}(\mathbf{S}\times \mathbf{a}))_{123} = & \frac{1}{20} \Big( a_{11} (3S_{1133}-3S_{1122}-S_{2222}+S_{3333})+2a_{12} ( S_{1112}- S_{1222}) \\ &+2 a_{13} (S_{1333}-S_{1113}) +a_{22} (S_{1111}+3 S_{1122}-3 S_{2233}-S_{3333}) \\ & +2 a_{23}( S_{2223}- S_{2333}) +a_{33} (-S_{1111}-3 S_{1133}+S_{2222}+3 S_{2233})\Big), \\ (\operatorname{tr}(\mathbf{S}\times \mathbf{a}))_{133} = & \frac{1}{10} \Big(-a_{11}(3 S_{1123}+ S_{2223}+ S_{2333})+a_{12} (3 S_{1113}- S_{1223}) \\ & -a_{13} (S_{1112}+S_{1222}+2 S_{1233}) +a_{22} (2 S_{1123}-S_{2333}) \\ & +a_{23} (S_{1111}+S_{1122}+2 S_{1133}-2 S_{3333})+a_{33} (S_{1123}+S_{2223}+2 S_{2333})\Big), \\ (\operatorname{tr}(\mathbf{S}\times \mathbf{a}))_{222} = & \frac{3}{10} \Big(a_{11} S_{1223}+a_{12} (S_{1123}+2 S_{2223}+ S_{2333})+a_{13} (S_{2233}-S_{1122}) \\ & -a_{23} (S_{1112}+2 S_{1222}+S_{1233})-a_{33} S_{1223}\Big), \\ (\operatorname{tr}(\mathbf{S}\times \mathbf{a}))_{223} = & \frac{1}{10} \Big(a_{11}(2 S_{1233}- S_{1222})+a_{12} (S_{1133}-2 S_{2222}+2 S_{2233}+S_{3333}) \\ & +a_{13}(3 S_{2333}- S_{1123}) +a_{22} (S_{1112}+2 S_{1222}+S_{1233}) \\ &-a_{23} (S_{1113}+2 S_{1223}+S_{1333})-a_{33} (S_{1112}+S_{1222}+3 S_{1233})\Big), \\ (\operatorname{tr}(\mathbf{S}\times \mathbf{a}))_{233} = & \frac{1}{10} \Big(a_{11} ( S_{1333}-2 S_{1223})+a_{12} (S_{1123}-3 S_{2223}) \\ & +a_{13} (2 S_{3333}-S_{1122}- S_{2222}-2 S_{2233}) +a_{22} (S_{1113}+3 S_{1223}+S_{1333}) \\ & +a_{23} (S_{1112}+S_{1222}+2 S_{1233})-a_{33} (S_{1113}+ S_{1223}+2 S_{1333})\Big), \\ (\operatorname{tr}(\mathbf{S}\times \mathbf{a}))_{333} = & \frac{3}{10} \Big(-a_{11} S_{1233}+a_{12} (S_{1133}-S_{2233})-a_{13} (S_{1123}+S_{2223}+2 S_{2333}) \\ & +a_{22} S_{1233} +a_{23} (S_{1113}+S_{1223}+2 S_{1333})\Big). \end{aligned}$$
(B.2)

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Abramian, S., Desmorat, B., Desmorat, R. et al. Recovering the Normal Form and Symmetry Class of an Elasticity Tensor. J Elast 142, 1–33 (2020). https://doi.org/10.1007/s10659-020-09784-7

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