A mixed-finite element approach for full-field residual stress estimation

In this paper, a full-field residual stress identification technique has been presented via a mixed finite element approach from limited measured detail. Being an inverse problem, complete identification of residual stresses from incomplete measurement is always suffered by inherent ill-posedness. To control this issue, the identification problem has been formulated as an optimization problem via a Tikhonov type cost functional. The first part of this cost functional is associated with an energy norm of mismatch between predicted and measured residual stress. The second part represents the regularization term. A Lagrangian is then considered to satisfy the equilibrium constraint over the region of interest. The incorporation of Lagrangian with aforementioned cost functional provides a form which is very similar to the standard Hellinger-Reissner (HR) functional except the discrepancy term between the data. The minimization of this formulation leads to the solution of two-field mixed variational form associated with residual stress and Lagrangian variable. To show the efficacy of the proposed procedure a two-dimensional welded specimen is considered where the weld bead is assumed to be spatially varying in all direction for residual stress identification. Line scan details are also presented to examine the deviation between the actual and predicted residual stress in the heat affected zone and at the region away from the weld bead. The results indicate that the mixed finite element formulation has the capability to provide the complete information of residual stress despite of having very few information of measurement.