Xu Song e-mail: xu.song@eng.ox.ac.uk Alexander M. Korsunsky Department of Engineering Science, University of Oxford, Pari<s Road, Oxford 0X1 3PJ, UK Fully Two-Dimensional Discrete Inverse Eigenstrain Analysis of Residual Stresses in a Railway Rail Head The aim of the present study was to introduce a new algorithm for reconstructing the eigenstrain fields in engineering components. A 2D discrete inverse eigenstrain study of residual stresses was carried out on a worn railhead sample. Its residual elastic strain distribution was obtained by neutron diffraction measurement in Stress-Spec, FRMII and u.sed as the input for eigenstrain reconstruction. A new eigenstrain base function-tent was introduced to capture the fully two-dimensional variation of eigenstrain distribution. An automated .'sequential tent generation scheme was programed in ABAQUS with its pre- processor to load the experimental data and postprocessor to carry out the optimization to obtain the eigen.strain coefficients. The reconstructed eigenstrain field incurs residual stress distribution in the railhead simulation, which showed good agreement with the experimental data. [DOI: 10.1115/1.4003364] Keywords: eigenstrain, residual stress, neutron diffraction, numerical simulation 1 Introduction Residual stresses (RSs) are the type of stresses that remain after removing the original cause (e.g., external forces or heat gradi- ent). They often lead to premature failure of critical components resulting in substantial cost and, at worst, in disaster. It is now widely accepted that the existence of residual stresses in the en- gineering component is strongly linked to—indeed, controlled by—the nonunifortnly distributed inelastic deformation [I]. This has been alternatively termed inelastic strain, inherent strain [2], and, finally, eigenstrain. The term eigenstrain and the notation e* were introduced by Mura [3]. The concept has the same physical basis as inelastic strain, although mathematically there is a subtle distinction be- tween the two quantities. The difference is associated with the fact that inelastic strain that is fully compatible everywhere does not give rise to any residual stress and can therefore be called impo- tent [4,5]. In our analysis, we prefer to use eigenstrain to describe the nontrivial part of inelastic strain that is responsible for residual stress generation. The total strain associated with the displacements of material points can be expressed in the additive form, e,„i„i = e,,-i-e*. Using a known eigenstrain field to deduce the residual elastic strain (and stress) field is termed the direct problem of eigenstrain analysis. It turns out that it is relatively straightforward to implement the solution of this direct problem within the finite element (¥E) framework, since the solution requires only the introduction of the inelastic strain distribution, and the subsequent self-balancing of the model [6]. A number of studies have been carried out using this method, and good agreement was found between simulation (residual stress reconstruction) and experiments [6,7]. One of the challenges for the development of eigenstrain-based approaches to residual stress analysis is that the information about eigenstrain distributions is not always readily accessible in prac- tice. For most cases, it is the residual elastic strains, or increments of the total strains, that can be measured directly via various de- Contributed by the Applied Mechanics of ASME for pubhcation in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 5, 2010; final manuscript received December 14. 2010; accepted manuscript posted January 5, 2011; pubhshed online February 17, 2011. Assoc. Editor; Jian Cao. structive and nondestructive experimental techniques [8]. There- fore, the inverse problem of eigenstrain analysis becomes the key step toward understanding and quantifying the RS sources, that is, using a known residual elastic strain (or total strain increment) information to retrieve the underlying eigenstrain field. A range of inverse eigenstrain studies has been carried out in recent years. They can be categorized conveniently by their di- mensionality. Note, however, that this requires careful specifica- tion of the aspect of the problem being referred to the dimension of the numerical (FE) model, the dimension of the eigenstrain (components) being reconstructed, and finally the dimension of the spatial variation of strain that is being considered. In previous studies reported in the literature, the dimension of the simulation was either 2D or 3D, with various shapes considered [9,10]. Mul- tiple components of eigenstrain were also considered [11,12]. However, their spatial variations have been mostly limited to ID (i.e., representable as a line plot). So far, no fully 2D discrete inverse eigenstrain analysis has been carried out (meaning 2D sample geometry, 2D eigenstrain components, and 2D eigenstrain distribution). As in the real engineering component the residual stress distribution variation is always multi-axial, it is necessary to develop a novel method capable of dealing with the multidimen- sionality of the strain variation. 2 Methodology: The "Tent" Base Function As described in Refs. [6,9], an arbitrary eigenstrain distribution can be represented by thermal strains caused by the difference in expansion coefficients across the solid body and subjected to a uniform temperature increment, so that the thermal strain caused is AL/ALo=a,Ar, i=l q. Now let the eigenstrain distribution e,*(x,y) be unknown, and let us assume that a finite number of residual elastic strain com- ponent values t^ is known at locations {Xg,y^. The same compo- nents T^ computed from the numerical solution depend implicitly on the unknown eigenstrain distribution e*j(x,y). In order to evaluate the quality of match to the experimental data achieved, a sum-of-squares measure J is introduced: Journal of Applied Mechanics Copyright © 2011 by ASME MAY 2011, Vol. 78 / 031019-1