Effect of surface normal variability on local surface strain measurements in StereoDIC
Introduction
The Digital Image Correlation (DIC) non-contacting, full-field displacement and strain measurement method was initially developed in the early 1980s for two-dimensional, in-plane deformation and motion measurements. The approach was extended to surface deformation and shape measurements for general surfaces through the development and use of stereo-vision systems in the early 1990s [1], [2], [3], [4], [5], [6], [7]; this method is oftentimes known as StereoDIC. These DIC-based surface displacement and strain measurement methods have been used successfully in a remarkably broad range of applications such as material science, mechanical, aerospace, biomedical, and structural engineering. The advantages of DIC techniques are (i) simplicity of the experimental setup, (ii) measurements can be obtained for both microscale and macroscale specimens that have either planar or curved surfaces (iii) high measurement accuracy for displacements and strains, (iv) applicable for both dynamic and high temperature experiments and (v) effective in the presence of large rotations, large translations and high strains . For example, StereoDIC systems [[1], [2], [3], [4], [7], [8], [9], [10]] can obtain strain of each point on a planar or a curved surface, overcoming the inherent limitations of a 2D-DIC system.
Since StereoDIC measurement systems are used extensively to measure surface deformations on curved surfaces, the shape of the surface (e.g. curvature, surface normal) is one of the measured quantities during an experiment. Interesting, the effect of variability in the measured local shape on other measurements, such as the surface strains, has not been studied extensively. In this regard, Luo et al [1] used StereoDIC to obtain surface measurements on a nominally cylindrical specimen. In their work, the authors’ measured surface shape while implementing a nonlinear optimization technique with the equation of a cylindrical surface to obtain optimal values for selected surface parameters. Estimated parameters included (a) the direction cosine of the axis of the cylinder relative to the initial stereo-vision coordinate system with the X-Z plane assumed to be orthogonal to the cylinder axis, (b) the point of intersection of the cylinder with the X-Z-plane and (c) the radius of the cylinder. The computed radius of the cylinder was then compared to an independently measured value to evaluate the measurement error. However, the authors did not use the StereoDIC data to measure deformations or determine variability in their values for the surface normal along the circumferential direction of the cylinder.
Tang and Hung [11] proposed an automatic 3-D shape measurement method and performed experiments to assess the quality of the measurements. The technique, based on the principle of phase measurement of the deformed grating pattern which carries the 3-D information of the measured object, was stated to be capable of automatically and accurately obtaining the phase map or the height information of a measured object at every pixel point without assigning fringe orders and interpreting data. Information regarding the accuracy of the surface normal at each surface point was not provided.
Chang and Ho [12] developed two algorithms for phase acquisition of deformation grating images. The phase-acquisition algorithms are sufficiently simple that high-resolution phase maps using a high-resolution area detector array can be generated in a short time. Experiments were carried out on the surface-profile reconstruction of a free curve surface, and the detection of a portrait embossed on a Chinese coin. The research was done to test the measuring error and the limitation of the phase-measuring algorithms. Again, results regarding the effect of measurement errors or surface fitting algorithms on the surface normal direction were not presented.
The stereo vision system developed by Luo et al. [2] was used to successfully reconstruct the surface of a nominally planar object. This methodology was applied to determine the point positions on the surface of a fracture specimen [3], before and after deformation, to obtain the three-dimensional displacement fields and the in-plane strain fields around the crack tip of the fracture specimen. In this case, the surface was initially flat, though no discussion was provided as to the accuracy of the surface normal at any position on the surface, or how errors in this direction will affect the strain measurements.
Since the three-dimensional digital image correlation (3D-DIC, StereoDIC) technique allows investigators to measure not only the surface shape but also the 3-D surface displacements and strain field, a method was proposed by Machado et al. [13]to determine the membrane stress tensor fields for in-plane isotropic materials . The stress and strain states were obtained at any surface point for an axisymmetric bulge test. Experimental difficulties were noted, particularly difficulties obtaining information near the edges for the chosen material. In addition, the authors indicated that high spatial resolution is needed for accurate measurement of curvatures. The authors also remarked that spatial resolution achieved in practice depends on a number of factors, including but not limited to camera resolution, lens optical quality, and marker size and quality.
Todd et al. [14]proposed computational models for determining the three-dimensional shape from texture . To achieve accurate estimates of surface relief from an image, it was observed from the same visual angle with which it was photographed or rendered. However, these models produce conflicting predictions when an image is viewed from a different visual angle. An experiment was performed to observe the apparent depth profiles of hyperbolic cylinders under a wide variety of conditions. The results showed that the apparent patterns of relief from texture are systematically underestimated; convex surfaces appear to have greater depth than concave surfaces.
To determine the slant or curvature of a surface, an optical system must take the viewing distance into account. Domini et al.[15] investigated this property to examine whether the mechanisms underlying the stereoscopic curvature “after-effects” are tuned to the specimen's disparity patterns or to some other property, such as surface curvature . The authors’ results clearly support the hypothesis that 3D after-effects appear to be caused by adaptation among mechanisms specifying surface shape rather than among mechanisms associated with the disparity pattern. Thus, their work indicates that how the surface shape is “specified,” or modeled, will play a significant role in the quality of the surface estimation.
In this regard, oftentimes the process for defining local shape begins with estimating the surface normal, n, which is then used to determine the local tangent vector, t, by projection of a unit vector defining the global x direction onto the best fit plane. Once this is completed, the binormal vector, b, is determined using the cross product. Conceptually, projection of either the x-direction or the y-direction in the global system can be used as the second step in the process, with selection based on the most effective alternative for an application [16]. Additional background regarding the errors associated with measurements of curved surfaces can be found in [17,18] and their references.
In the enclosed work, the authors focus specifically on how local surface reconstruction and local coordinate system definitions affects the accuracy of the strains at points on the reconstructed surface. Though in principle the surface position measurements can be obtained with a variety of experimental methods, the combination of stereo-vision with digital image correlation (StereoDIC) is the application of particular interest for this study.
Section snippets
Theoretical developments
Though the results obtained in this study apply to general curved surfaces, a cylindrical specimen is employed to present the basic concepts. Consider the top view of a cylindrical specimen and a stereo-vision system, shown schematically in Fig. 1 by two cameras viewing the specimen. Since the manner in which surface reconstruction is performed will affect the measurements defined for the newly defined surface, in this work reconstruction will be performed using local planar fits to the
Simulation studies
The cylindrical specimen shown in Fig. 1 will be used in all simulations. Consider an arbitrary point, P, such as shown in Fig. 1. Considering surface positions along a circular arc of radius, R, the rotational transformation matrix can be written in terms of angle, θ, as follows;with
To use these equations, recall that we must satisfy Eqs. (1) and (5) to determine the increments (δnx,
Experimental validation
To demonstrate the potential applicability of the analysis performed in this study, uniaxial tension and pure torsion experiment were performed by the investigators using a right circular cylindrical specimen. The experimental setup is shown in Fig. 6. The StereoDIC system shown in Fig. 6 consists of a pair of stereo cameras, rigid cross-bar to mount the cameras, tripod to mount the rigid bar, high intensity, low heat white LED light source, calibration grid, dedicated computer with digital
Simulations
The theoretical work shown in Section 2 indicates clearly that directional errors during the process of estimating n, t, and b at any point on a surface will affect the accuracy of the “measured” surface strains. If one develops the theory solely in terms of the components of the normal vector, as the authors have done, the results show that the error components are not independent. Rather, the requirement that b, n, and t, as well as b’, n’ and t’, are unit vectors gives rise to constraints on
Concluding remarks
Results from the analytical results, associated simulations and experimental measurements confirm that points with normal vectors rotated away from the central position (see Fig. 1) are inherently prone to have higher variability in the strain measurements when the normal vector has slight inaccuracies. In these studies, values in the range of ±0.035 for δnz correspond to small-angle errors, on the order of ± 0.2o, and yet are sufficient to predict trends that are consistent with experimental
Author credits statement
Michael Sutton: Conceptualization, Methodology, Reviewing and Editing, Supervision.
Hubert Schreier: Software Development, Review and Editing.
Farzana Yasmeen: Data Curation, Writing- Original draft preparation, Reviewing and Editing; Data Analysis, Experimentation, Formal Analysis.
Andrew Campbell: Experimentation, Image Acquisition, Some Data Analysis.
Sreehari Rajan: Experimentation, Image Analysis, Supervision, Reviewing and Editing.
Declaration of Competing Interest
None.
Acknowledgments
The support of the Department of Mechanical Engineering at the University of South Carolina through teaching assistantships is gratefully acknowledged. In addition, the support of Prof. Dimitris Rizos, Department of Civil and Environmental Engineering, is deeply appreciated. Finally, the support of UofSC Chief Financial Officer, Mr. Edward Walton, is gratefully acknowledged.
References (19)
- et al.
Measurement of curved surface by stereo vision and error analysis
Opt Lasers Eng
(1998) - et al.
Determination of displacement using an improved digital correlation method
Image Vision Comput
(1983) - et al.
3D after-effects are due to shape and not disparity adaptation
Vision Res
(2001) 3-D computer vision in experimental mechanics
Opt Lasers Eng
(2009)- et al.
Accurate measurement of three-dimensional deformations in deformable and rigid bodies using computer vision
Exp Mech
(1993) - et al.
Application of stereo vision to three-dimensional deformation analyses in fracture experiments
Opt Eng
(1994) - et al.
Measurement of curved-surface deformation in cylindrical coordinates
Exp Mech
(2000) - et al.
Advances in two-dimensional and three-dimensional computer vision
- et al.
Image correlation for shape, motion and deformation measurements
(2009)
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