Weight functions and fundamental fields in layered media

The determination of the elastic T-term using higher order weight functions

“…By such a construction, the displacements ur and % generate stresses which are in equilibrium with zero body forces in B and the corresponding tractions, Xr and Y~, are zero on portions of the crack faces, 8B n C. Introduce the kinematical sets: Based on these admissible sets, a function H can be defined as [2,15] and strains ~ in A; and and strains n* in B. …”

“…This parameter is introduced for dimensional consistency and it will be dropped in the sequel for the sake of clarity. The field (15) gives Ps = 0 on x ~< 0 and hence it leaves the crack faces traction free. The tractions on a piecewise smooth curve F*, originating from a point ('on the lower crack face to any point (" on the upper crack face, are non-zero.…”

“…However, these non-zero tractions on F* lead to zero resultant force and zero resultant moment about the origin. Using (1 a) and (15), we find that the displacements of the singular field, us and v s , can be expressed as …”

“…The elastic T-term can be computed by work-conjugate integral through the higher order weight functions, in the same spirit as the determination of the stress intensity factor by the Bueckner-Rice weight functions, for any combination of prescribed displacements and tractions. In this work, we employ a previously developed finite element procedure [2,14,15] to determine these higher order weight functions and to compute the elastic T-term for different specimen geometries and loading conditions.…”

“…A minimum principle can then be stated as[2,15]: Among all kinematically admissible fields u~, vf in A and u*, v* in B, the true fields u I, v I in A and ur, Vr in B render the functional H a minimum.…”