Isotropic Tensors

The most general second-order isotropic tensor, , is such that

for arbitrary rotations of the coordinate axes. It follows from Equation (B.24) that, to first order in the ,

(B.66) |

However, the are arbitrary, so we can write

Let us multiply by . With the aid of Equation (B.16), we obtain

(B.68) |

which reduces to

(B.69) |

Interchanging the labels and , and then taking the difference between the two equations thus obtained, we deduce that

(B.70) |

Hence,

(B.71) |

which implies that

(B.72) |

For the case of an isotropic third-order tensor, Equation (B.67) generalizes to

Multiplying by , , and , and then setting , , and , respectively, we obtain

(B.74) | ||

(B.75) | ||

(B.76) |

respectively. However, multiplying the previous equations by , , and , and then setting , , and , respectively, we obtain

(B.77) | ||

(B.78) | ||

(B.79) |

respectively, which implies that

(B.80) |

Hence, we deduce that

(B.81) | ||

(B.82) | ||

(B.83) |

The solution to the previous equation must satisfy

(B.84) |

This implies, from Equation (B.8), that

(B.85) |

For the case of an isotropic fourth-order tensor, Equation (B.73) generalizes to

(B.86) |

Multiplying the previous by , , , , and then setting , , , and , respectively, we obtain

(B.87) | ||

(B.88) | ||

(B.89) | ||

(B.90) |

respectively. Now, if is an isotropic fourth-order tensor then is clearly an isotropic second-order tensor, which means that is a multiple of . This, and similar arguments, allows us to deduce that

(B.91) | ||

(B.92) | ||

(B.93) |

Let us assume, for the moment, that

Thus, we get

Relations of the form

can be obtained by subtracting the sum of one pair of Equations (B.97)-(B.100) from the sum of the other pair. These relations justify Equations (B.94)-(B.96). Equations (B.97) and (B.101) can be combined to give

The latter two equations are obtained from the first via cyclic permutation of , , and , with remaining unchanged. Summing Equations (B.102)-(B.104), we get

(B.105) |

It follows from symmetry that

(B.106) |

This can be seen by swapping the indices and in the previous expression. Finally, substitution into Equation (B.102) yields

(B.107) |

where

(B.108) | ||

(B.109) | ||

(B.110) |