Deformation pattern based digital image correlation method and its application to residual stress measurement

Jianxin Gao* and Haixia Shang

*Corresponding author

Paper published in Applied Optics, vol.48. issue 7, 2009. pp.1371 - 1381.

Abstract: To measure directly residual stresses by digital image correlation using hole-drilling, the deformation pattern which is governed by the residual stresses is used to affine transform the image captured after the object is deformed. If the values of trial residual stress components are properly chosen, the image after affine transformation will have a maximum similarity to the original image. This turns the residual stress measurement issue into a pure numerical computational process, which leads to the direct output of residual stresses. Validation tests have proved the viability of the approach. The proposed concept and principle could be extended to other specific measurement tasks with known deformation patterns.

Keywords: Digital Image Correlation, Deformation Pattern, Affine Transformation, Optimisation, Residual Stress, Stress Measurement

OCIS codes:
100.2000    Image processing: Digital image processing
120.4880    Instrumentation, measurement, and metrology: Optomechanics

1. Introduction

Residual stresses are an important issue for the structural integrity of welded components and structures. Currently, the strain rosette, hole drilling technique is widely used for residual stress quantification in industry.^[1-3] This is a straightforward method that measures directly the strain released by the drilled hole. Residual stress is subsequently calculated from the strain. Since this technique requires the attachment of a strain rosette onto the object surface, there are some limitations in its industrial application. For example, the hole should be drilled with good positional accuracy. The location of the hole is known to affect the accuracy of the measured stress values. Only limited strain data are available which are the average over the strain pitch. Significant efforts had been made to address these issues in 1970s and 1980s. A detailed analysis of the errors associated with these issues was given by Scaramangas. ^[2]

Apart from strain gauge based techniques, non-destructive diffraction based techniques have also been developed for residual stress measurement. They include X-ray diffraction, neutron diffraction and other diffraction methods. These techniques have the capability of measuring 3D stress components in the bulk of the material. However, since they require the use of an X-ray or neutron source, which could be either hazardous to human safety or very expensive, their application in on-site engineering environments is rather limited. Therefore, there is a need to develop a technique that can be applied to measure residual stress on site in an engineering environment.

Optical techniques have long been applied to measure mechanical displacement and deformation of various materials and components. The main advantages of optical techniques are non-contact, full field measurement with high sensitivity. Optical techniques can be categorised into two groups, fringe based interferometric techniques and non-interferometric techniques. The former includes moiré, holographic and speckle interferometry, electronic speckle pattern interferometry and shearography, while the latter simply consists of image processing based techniques. Interferometric techniques usually offer high sensitivity and accuracy in displacement and/or deformation measurement.^[4-6] Since they must use coherent light, most interferometric techniques tend to be prone to environmental disturbances during measurement. Their application to on-site measurement in an engineering environment has therefore been limited. An exception is the interferometric strain/slope rosette (ISSR) method proposed by Li et al,^[7-9] which is insensitive to in-plane rigid body motions. This is normally used with ring-core cutting for releasing residual stress, which is usually more complex than hole-drilling. When used with hole-drilling, the stress calculation is more complicated due to the non-uniform release of residual stress.

Digital image correlation (DIC) is a well known non-interferometric technique, having the capability of measuring the displacement field of an object with an ordinary light source such as white light.^{[10, 11]} Without the need to form an interferometric fringe pattern, the optical set-up of DIC is rather simple. These factors mean that DIC has the potential of being employed in an engineering environment. Considerable efforts have been devoted to using DIC in conjunction with the hole-drilling technique to measure residual stress.^[12-14] However, in previous development work, DIC was simply used as a displacement measurement technique similar to interferometric optical methods. The conventional procedure for residual stress measurement starts with displacement measurement, followed by differentiation of the displacement field or other relevant numerical manipulations to obtain strain information, and finally calculates the stress values based on theoretical or numerical FEM models. There are two main shortcomings in this conventional approach. Firstly, the process for calculating residual stress is complex, and can only be carried out by professionals with a sound knowledge of mechanics and materials. It is therefore difficult for a non-professional to use this approach to measure the residual stress. Secondly, the sensitivity of the technique is relatively low. The minimum resolution of residual strain is reported to be around 3×10^-4. ^{[6, 12, 14]}

This paper presents a novel approach to DIC for residual stress measurement to address the above shortcomings. Instead of using displacement components as the basic variables in conventional digital image correlation, residual stress components are taken as the direct variables. Each set of residual stress components corresponds to a specific deformation pattern. This deformation pattern as a whole is used to inversely affine transform the digital image captured after a hole is drilled onto an object under testing. Once the residual stress components are properly chosen, the digital image after inverse affine transformation will have the highest similarity to the digital image taken before the object is drilled, i.e. the transformation will recover the original image. This turns the stress measurement issue into a purely numerical computational process, i.e. a search for a set of residual stress components that will maximise the correlation between the original image and the inverse affine transformed image after hole drilling. This can be implemented through an optimisation procedure. Since the direct variables in such a numerical optimisation process are the residual stress components, the conventional stress measurement procedure in optomechanics, which involves the complex interpretation of a displacement field, is no longer required. This will facilitate the development of a compact system for residual stress measurement with ease of use for the operator. The proposed concept and principle shall not be limited to residual stress measurement, and could be extended to other specific measurement tasks with known deformation patterns.

2. Concept of deformation pattern based DIC

The central idea of the proposed approach is to use the intrinsic parameters of a measurement task as direct variables in the computation of the correlation coefficient between two images. A set of intrinsic parameters will define a specific type of mechanical behaviour of an object under investigation. For example, in a cantilever beam under point load and bending at the free end, the intrinsic parameters consist of two variables: the load force and the bending moment. In residual stress measurement, the intrinsic parameters are residual stress components. In both cases, the displacement field is governed by the stated intrinsic parameters.

Now let us describe the novel approach to the measurement of a specific type of mechanical behaviour. For simplicity and clarity, and without losing generality, we use the mechanical behaviour of a planar residual stress state released by a hole drilled in an infinite plate, as shown in Fig.1. In this case, the three residual stress components, σ_x, σ_y, τ_xy, are the intrinsic parameters. They are obtained by measuring the displacement field or strain when a small hole is drilled to locally release the residual stress. For the convenience of expression, both Cartesian coordinate system (x, y) and polar coordinate system (r, ε=10^-6) that might be bigger than the actual strain values. Without addressing this virtual strain field, the direct stress/strain measurement would be meaningless.

To ensure the new approach works effectively, the above in-plane and out-of-plane displacement (u₀ , v₀ , w₀ , ω) should be taken into account in the correlation search process. Here for convenience and without losing generality, w₀ is defined as the relative change in the distance between the surface of the object and the imaging camera, i.e. the relative change in magnification of the imaging system. Therefore, the affine transformation becomes:

ε to 250µε, to 500µε, ..., to 2000µε. Afterwards, the sample was unloaded gradually to 10µε. At each loading/unloading step, a pair of images was taken and saved to a computer. Table 1 shows the strain gauge readings during a uni-axial extension test on an aluminium sample. Digital images are recorded at each load/unload step and are saved in the computer. By selecting any two images corresponding to different load steps, various relative stress changes can be formulated. This allows us to simulate different residual stress levels ranging from compressive stress to tensile stress, so that comprehensive validation tests on the proposed method can be made.

Table 1. Experimental recordings during uniaxial extension test on an aluminium sample. Strains were measured by strain gauges which were bonded on the reverse of the specimen.

(B) Measurement results

The actual deformation around the hole of the test sample under axial loading is the superimposition of two deformation fields. The first is the deformation field of a plate without a hole under uni-axial extension/compression. The second is a residual stress associated with the deformation field described by Equation (1). Here the residual stress has only one component which equals the axial stress. By taking into account the superimposed deformation field in the image processing software, stress results can be calculated directly. When used for measuring actual residual stress, the first part of a far field uniform stress shall be excluded and all the other settings of the algorithms remain the same.

The measurement procedure is as follows. (1) Load a configuration file which contains the basic parameters of the measurement, namely the material properties, the diameter of the hole and thickness of the sample. (2) A pair of images is loaded into the software. The image scale factor is calculated automatically by measuring the diameter of the hole in pixels which is then converted to the actual dimension. (3) Locate the centre of the hole by simply adjusting a circle representing the hole to coincide with the boundary of the hole, as shown in Fig.5. This enables the centre to be located with high accuracy, usually within half a pixel. (4) Select a region of interest (ROI) around the hole. This is done by adjusting the radii of an outer and an inner circle such that the band in between is the ROI of selection whose pixels are used in the subsequent affine transformation to calculate the residual stress. (5) Click the button on the menu to start the correlation optimisation process to directly calculate the stress components. It takes between half a minute and 10 minutes to complete the whole computational process, depending on the total number of pixels of the ROI selected and the optimisation strategy used.