Identify the distribution of 2D residual stresses around notches based on the Willis-form equations

Abstract

ABSTRACT This paper proposes a new non-destructive method to reconstruct the distribution of residual stresses around notches from the measured displacements under external loads. For this purpose, a linear finite element method based on the Willis-form equations is firstly established to calculate the displacements, so that they explicitly contain the impact of the gradient of residual stresses, which can be efficiently determined through the analytical solution of the stress distribution around notches due to remote virtual loads. Then, the remote virtual loads are identified by minimizing the discrepancy between the calculated and the measured displacements. In this way, the distribution of residual stresses can be naturally obtained by substituting the identified remote virtual loads into the analytical solution. Because the number of remote virtual loads is very limited, the identification is very efficient. However, since this is an ill-posed problem, regularization is discussed by comparing the performance of the optimal sensor placement method, the Tikhonov-Morozov method and the Tikhonov- L-curve method. Both numerical and experimental results of a plate with an elliptical hole under tension are used to check the validity of the proposed method. It is demonstrated that the distribution of residual stresses around the notch could be successfully identified by using all regularization methods. However, the optimal sensor placement method could give more reliable results.