General Formulation: Tensorial Crack Influence Function
In Eq. 13.3.9, the principal stress orientations at points x and £ are reflected in the values ol the scalar function Л(х, £). For the purpose of general analysis, however, it seems more convenient to use a tensorial crack influence function referred to common structural cartesian coordinates X = X.Y s AT, Z ss AT.
and transform all the inelastic stress tensor components to X. Y,Z. The local cartesian coordinates x es Xi, у X2, z = xз at point £ are chosen so that axis у coincides with the direction of the maximum principal value of the inelastic stress tensor S(£), and axes x and г coincide with the other two principal directions.
Equations (13.3.32) and (13.3.33) may be rewritten in common structural coordinates as follows:
A Sou <- A S’,,,., I – E E (l> – 1,2, …N) (13.3.54)
І/-Л г— 1
AS;j(x) <- AS^E) + / Et)AS^(0dV(0 (13.3.55)
in which, similar to (13.3.21), we included the influence of the dominant cracks normal to the principal stress direction at each point; Rf‘Jki(^) or = CkiQj = fourth-rank coordinate rotation tensor
(programmed as a square matrix when the stress tensors are programmed as column matrices) at point £ or £ ; cgi, cu = coefficients of rotation transformation of coordinate axes (direction cosines of new axes) from local coordinates tr, at point £ (having, in general, a different orientation at each £) to common structural coordinates Xj (cgi = cos(a’k, X]),Xj = CkiXk, ou — CkiQ. i^kl)’, subscripts I, J, or к, l refer to cartesian components in the common structural coordinates or in the local coordinates at £; and Л, Е or Л[‘^(х, £) = components of a tensorial discrete or continuous nonlocal weight function (crack influence function, replacing the scalar function Л), which are equal to £~2 times the cartesian stress components ogi forrr = 1 as defined by (13.3.41) for two dimensions, or C~3 times such cartesian components as defined by (13.3.49) for three dimensions (with r — |jx — £|j).