PSI - Issue 5

1262 Francisco Barros et al. / Procedia Structural Integrity 5 (2017) 1260–1266 Francisco Barros et al./ Structural Integrity Procedia 00 (2017) 000 – 000 3 where represents the imaginary unit and and the transform coordinates in frequency domain. The expression in (5) can also be interpreted as the Fourier transform Phase-shift theorem [5]. The theorem in (3) can then be used to compute de desired displacement between the functions or images: ℱ{ ⋆ } = | ( , )| 2 2 ( + ) (6) Expression (6) is related to the Fourier transform of a finite impulse response function located in (Δx, Δy) [6]. Let us consider an impulse function h(x, y) , with dimensions M × N defined as: ℎ( , ) = { 0, ≠ ˅ ≠ 1, = ˄ = (7) Its Fourier transform is given by: ℱ{ℎ( , )} = ( , ) = 1 2 ( + ) ∝ ( , ) ∙ ( , ) ̅̅̅̅̅̅̅̅̅ (8) and therefore, an inverse Fourier transform of ℱ{f ⋆ g} is expected to ideally return a finite impulse response function centred on the point with coordinates (Δx, Δy) . For real speckle images or subsets this resulting function is usually superimposed to noise related to the contribution of other existent spatial frequencies. These arise from other image transformations besides simple image translation, such as rotations, scaling factors and shear deformations, the resolution in the DFT and the existence of new speckle dots with different spatial frequencies in the deformed image. As proposed by Guizar-Sicairos et al. [7], calculating the cross-correlation between two images using DFT also allows for a high resolution subpixel level computation of the location of the correlation peak, by upsampling the region near the peak through a matrix operation as shown by Soummer et al. [8]. Apart from the advantages in terms of computational speed and high attainable precision, frequency based methods can be more robust when faced with lower quality images [3], which indicates that it should be able to correctly detect correlation peaks in the absence of controlled experimental conditions, making it more suitable for field measurements where such conditions can potentially be imposed by the environment in which the measurement is performed.

2. Methodology

2.1. Overview

In this work, a frequency domain based DIC method was used to compute a 3D displacement field of a round specimen in a tensile test. For the application of DIC methods to a round specimen, it is necessary to use three-dimensional methods, as the surface and its displacements are not restricted to a single plane: not only is the surface of the specimen not flat, but there is also displacement in the direction perpendicular to the camera sensor plane due to necking. The present work focuses only on the initial stages of the tensile test, where deformations are still relatively small, and aims mainly to compare 3D displacement results obtained using Fourier domain cross-correlation calculations against displacement fields given by commercial DIC software (namely VIC-3D by Correlated Solutions) using traditional intensity based DIC methods.

2.2. Experimental setup and image acquisition

Images were taken from a quasi-static tensile test performed on a round aluminium previously painted with a speckle pattern. The images were obtained with the VIC-3D acquisition system, in a setup shown in Fig. 1, with a resolution of 2048 × 2048 pixels.