If rock mass is in a state of initial stresses,

where imply stress components in rectangular Cartesian coordinates

(2) |

where the first subscript stands for the stain measuring position (points) and the second subscript 1 and 2 express the components in the directions of the generator and in perpendicular to it (circumferential direction), respectively , see

With an assumption of isotropic linear elasticity, we have

(3) |

where E stands for Young's modulus of the material. The coefficient matrix [A] (strain matrix:2M6) is accurately determined by use of the FMBEM.

It is easy to construct an observation equation for
determining initial stresses from the measured strains
when the matrix [A] is known. If we choose more than 6 measured
strains and make use
of the least-squares method, for instance, we have

(4) |

Therefore, the initial stresses are obtained as

where

(6) |

which is called the strain-to-stress conversion matrix (62M).

It is easily understood that the key of the stress-relief
method is the strain-to-stress conversion matrix [*D*], and thus
evaluation of the strain matrix [*A*] is of fundamental importance.