On the determination of residual stress distribution in plane elasticity (Q2774047)

An attempt is made to define a residual stress field \(S^{(R)}_{\alpha\beta}\) \((\alpha,\beta= 1,2)\) of two-dimensional homogeneous isotropic elastostatics for a region \(R\subset E^2\) in the form \(S^{(R)}_{\alpha\beta}= S^{(E)}_{\alpha\beta}- S^{(1)}_{\alpha\beta}\) on \(\overline\Omega\subset R\), where \(\Omega\) is a subdomain of \(R\), \(S^{(E)}_{\alpha\beta}\) is a stress field obtained from an experiment in laboratory, and \(S^{(1)}_{\alpha\beta}\) is a solution to a traction problem of two-dimensional elastostatics for the domain \(\Omega\) when the traction vector \(t_\alpha= S^{(E)}_{\alpha\beta} n_\beta\) on \(\partial\Omega\) is prescribed. The definition is illustrated for the case when \(R\) is a circle, and the residual stresses correspond to an edge dislocation, an angle discilination, and a thin circular inclusion, that means, for the case when no laboratory meausurements are involved. A motivation why the tensor field \(S^{(1)}_{\alpha\beta}\) should be a solution to the traction problem and not to a mixed problem of two-dimensional elastostatics is not given. Also, there are misprints in the paper: e.g., there are no equations (10), (13), (15), (19) and (24) to which the author refers on pp. 373, 375 and 377, respectively.