The trace of the stiffness matrix *C* is an invariant, and equals the
sum of its eigenvalues. Similarly, the trace of the compliance matrix *S* is
also invariant, and equals the sum of its eigenvalues, which are the
inverses of the eigenvalues of *C*. These facts provide two more ways
of obtaining estimates of the *G*_{eff} we seek.

*Trace estimate from S*

After eliminating the four eigenvalues associated with simple shear,
the remainder of the trace of *S* is just the sum of the inverses of
and . To obtain an estimate of *G*_{eff}, we
again make use of the known bulk modulus *K* and set

(35) |

(36) |

*Trace estimate from C*

Again eliminating the four eigenvalues associated with simple shear,
the remainder of the trace of *C* is just the sum of and
. To obtain another estimate of *G*_{eff}, we make use of
the known bulk modulus *K* as before and set

(37) |

(38) |

7/8/2003