Variational eigenstrain analysis of synchrotron diffraction measurements of residual elastic strain in a bent titanium alloy bar

Abstract

Most procedures for experimental stress evaluation rely on the measurement of elastic strain followed by point-wise calculation of stress based on continuum elasticity assumptions despite the fact that the real purpose of the investigation is to characterise the state of stress everywhere in the object to the greatest possible detail. Using the example of residual elastic strain measurements in a bent titanium alloy bar taken by means of high energy synchrotron X-ray diffraction, an interpretation technique is here introduced based on the variational eigenstrain analysis. An analytical framework is presented for the solution of the direct problem of eigenstrain, that is, the calculation of residual elastic strain distribution within an inelastically bent beam containing a known distribution of eigenstrain. An inverse problem about closest matching between the model and experiment is then cast in a form that allows determination of the underlying eigenstrain distribution from a single noniterative solution of linear system. Subsequently the complete stress state can be reconstructed everywhere within the object in the form of continuous functions. The value of the approach lies in the fact that subsequent deformation modelling can be carried out with the effects of residual stresses (and their evolution) naturally incorporated. The extension of this approach to more complex geometrics within the framework of the finite element method is briefly discussed.