92 A. Baldi computations steps Fig. 14.1 Schematic representation of the computations steps involved in the integral method 2 DX k Xl Œf .k; l/ g.k Cu; l Cv/ 2 (14.3) where the summations extend over the pixel of the local area (subset). The computation of derivatives of (14.3) with respect to the parameters of the shape functions (i.e. of the u and v functions) leads to a linear system which apparently solve the problem. Actually this is not the case, because the numerical values of the derivatives depend on the point of evaluation, i.e. on the unknowns, thus an iterative solution algorithm is required. To obtain field data, the above sketched computation has to be repeated to sample the active area on a regular grid. Digital Image Correlation is not a particularly good technique for residual stress measurement. Indeed, its sensitivity is relatively low; moreover, the measurement is particularly difficult near the hole, i.e. in the most interesting region, thus researchers had to cope with artifacts [6]. iDIC completely overcome all these problems by integrating problem-specific displacement functions inside the DIC algorithm. The advantages are significant: because the shape functions are able to describe the experiment globally, there is no need to partition the domain and a single, huge set of pixel is used. This ensures high reliability and robustness against local perturbations. Moreover, the parameters controlling the displacement functions are the stress components, thus, the result of the iDIC analysis are the residual stress components (the standard process requires a two step analysis, the measurement of the displacement/strain data and successively the computation of stress components). Even though the single-step processing of iDIC is elegant and grants a robust and accurate solution (because a single reverse process is used instead of two), it becomes a problem when the residual stress varies with depth. The standard approach to this problem is the integral method [7]: the stress distribution, a smooth function of depth, is approximated with a stepwise function (Fig. 14.1). For each step, a hole increment is performed and the strain/displacement field on the surface is acquired. To correlate the acquired displacements with stresses one has to consider that for each hole increment, the observed deformation depends on the newly released stress components and on all the previous ones. This leads to the linear system G! D ! f (14.4) where! is the vector of stress components, ! f the vector of the observed strain/displacements andGis the influence matrix. The element Gij of Gis the deformation/displacement observed on surface after hole increment i caused by a unitary load at depth j. Because no influence is assumed on stress below the bottom of the hole, Gis a lower triangular matrix. Gij is a number when strain gauge are used and a block sub-matrix when optical methods are involved, because at each step, the displacement of all points imaged on the surface are estimated. The linear system (14.4) can be easily solved either using a standard algorithm or—in the case of optical methods, when the system is overdetermined—by a least squares approach, i.e. solving GTG! DGT! f (14.5)
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