Keywords: **Digital** **image** **correlation**; displacement ﬁeld; measurement un-
certainty; resolution; spatiotemporal regularization; video.
1 Introduction
In the ﬁeld of solid mechanics, since the early 1980’s [1, 2, **3**], most of the procedures dealing with **digital** **image** **correlation** (**DIC**) of 2D pictures (or **3D** volumes) are based upon the registration of a pair of pictures, a ﬁrst one corresponding to the reference conﬁguration and the second one to the deformed conﬁguration [4]. They consist in subdividing the region of interest in the reference picture into a set of independent zones of interest (ZOIs). The latter ones are small interrogation windows that are registered, and may overlap since there is no spatial constraint on neighboring windows. The advantage is that the analysis of each ZOI can be run independently of the other ones. The drawback is that the continuity of initially contiguous ZOIs is not obtained because of measurement uncertainties. This lack of continuity is one of the criteria used to estimate the quality of the registration [6]. This type of approach is referred to as ‘local’ in the sense that the registration is performed with small ZOIs with no information exchange with neighboring ZOIs.

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2.2 **3D**-**Digital** **Image** **Correlation** (**3D**-**DIC**)
The **3D**-**DIC** method is based on both **digital** **image** **correlation** (**DIC**) and computer stereo vision (utilisation of two cameras), and was developed at the end of the last century [6]. This technique uses a **DIC** algorithm to determine point correspondences between two images of an object to be measured, acquired from two di fferent view points, by two rigidly bounded cameras. The **correlation** scores are computed by measuring the similarity of a fixed subset window in the first (left) **image** to a shifting subset window in the second (right) one. A first order subset shape function and a zero normalized sum of square di fference (ZNSSD) **correlation** criterion are used. Sub-pixel **correlation** is performed using B-spline gray level interpolation. After determining the calibration parameters for each camera as well as the **3**-D relative position /orientation of the two cameras (pinhole model and radial distortion of 1rd order), the **3**-D shape of the object can be reconstructed by triangulation from the **image**-points correspondences founded by **DIC**. To determine the **3**-D displacement field, **DIC** is also used to determine **image**-points correspondences between the stereo pairs acquired before and after deformation (temporal tracking) [7]. The Hencky logarithmic surface strain field tensor is obtained from the displacement field by numerical di fferentiation. A complete description of the **3D**- **DIC** technique can be found in [8].

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keywords parallel computing, global **DIC**, high performance computing, substructuring, finite elements, do- main decomposition
1 Introduction
The analysis of the mechanical behavior of materials and structures rely more and more on full-field mea- surements. It is mainly due the large quantity of data they provide, which are particularly interesting for parameter identification purposes for instance. Among them, **Digital** **Image** **Correlation** (**DIC**) has become one of the most popular because of a favorable ease of use to accuracy ratio [ 46 , 45 ] and its ability to deal with **3D** measurements on the surface (stereo **DIC** [ 27 ]) and also in the bulk with **Digital** Volume **Correlation** (DVC [ 1 , 37 ]). In continuum solid mechanics, finite element based **DIC** (FE-**DIC** [ 44 , **3** , 11 , 17 ]) has proved to be a relevant choice since (a) it allows for interpolation-free communications with finite element simulations, (b) it significantly reduces the measurement uncertainties with respect to classical subset based approaches [ 17 ], since prescribed continuity of the unknown displacement fields acts as a regularisation. The drawback of FE-**DIC** over subset-**DIC** [ 45 ] is the computational cost when high resolution is sought for [ 24 , 35 ]. Indeed, subset based **DIC** approaches lead to a set of small independent nonlinear systems of equations which are highly parallelisable, whereas FE-**DIC** method lead to one global non-linear system whose inversion can become prohibitive with a large number of degrees of freedom [ 35 ]. In addition, due to the constantly increasing resolution of photographic sensors (standard sensors provide nowadays 29 million pixels, which can be extended up to 260 million pixels using a piezoelectric pixel shift), the manipulation and interpolation of images (large dense matrices) become more and more an issue. This problem is even more acute with tomographic images.

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The analysis of the mechanical behaviour of materials and structures rely more and more on full-field measurements. It is mainly due to the large quantity of data they provide, which are particularly interesting for parameter identification purposes for instance. Amongst them, **digital** **image** **correlation** (**DIC**) has become one of the most popular because of a favourable ease of use to accuracy ratio [1, 2] and its ability to deal with **3D** measurements on the surface (stereo **DIC** [**3**]) and also in the bulk with **digital** volume **correlation** (DVC [4, 5]). In continuum solid mechanics, finite element based **DIC** (FE-**DIC** [6–9]) has proved to be a relevant choice because (a) it allows for interpolation-free communications with finite element simulations and (b) it signif- icantly reduces the measurement uncertainties with respect to classical subset based approaches [8] because prescribed continuity of the unknown displacement fields acts as a regularisation. The draw- back of FE-**DIC** over subset-**DIC** [1] is the computational cost when high resolution is sought for [10, 11]. Indeed, subset based **DIC** approaches lead to a set of small independent nonlinear systems of equations that are highly parallelisable, whereas the FE-**DIC** method leads to one global non- linear system whose inversion can become prohibitive with a large number of degrees of freedom [11]. In addition, because of the constantly increasing resolution of photographic sensors (standard sensors provide nowadays 29 million pixels, which can be extended up to 260 million pixels using

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Orthogonal cutting tests are conducted in a **3**-axis Computer Numerical Control (CNC) milling machine DMG DMC85V equipped with linear motors. The cutting speed is generated by the X-axis displacement and set at its maximum speed of 120 m.min -1 (Figure 3a). Since a
cutting speed lower than 120 m.min -1 would be too distinct compared to the usual ones (around 400 to 500 m.min -1 ), this parameter is considered as fixed. However, the applied cutting speed is much higher than the ones used on the previous studies. The sample and its fixation (Figure 3c and Figure **3d**) are tightened to a piezoelectric dynamometer Kistler model 9119 AA2 to record cutting forces. To perform **DIC** analysis, a high-speed CCD camera PHOTRON SA-Z records the burr formation during the end of the cutting test. The frame-rate is set at 30,000 fps, ensuring an **image** acquisition each 0.033 ms (approximately 66.7 µm travelled by the cutting tool) with an exposure time of 1/400,000 s. To get a high magnification of the burr formation zone (tool exit from the workpiece), a ×10 magnification Mitutoyo objective with extended lens tubes are assembled to the camera. A 1.84 × 1.23 mm 2

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2.1 Principle
The **3**-D **DIC** method is based on both **digital** **image** **correlation** (**DIC**) [22] and stereovision, and was developed at the end of the last century [9,23–26]. The technique uses a **DIC** algorithm to determine point correspondences between two images of a specimen acquired from two rigidly bounded cameras. The **correlation** scores are computed by measuring the similarity of a fixed subset window in the first **image** to a shifting subset window in the second one. A first- order two-dimensional shape function in the subset [27] and a zero normalized sum of square difference (ZNSSD) **correlation** criterion are used. Sub-pixel **correlation** is performed using quintic B-spline grey level interpolation [28]. After determining the calibration parameters for each camera as well as the **3**-D relative position/orientation of the two cameras (pinhole model and radial distortion of 3rd order), the **3**-D specimen shape can be reconstructed from the point correspondences using triangulation. To determine the **3**-D displacement field, **DIC** is also used to determine point correspondences between the stereo pairs acquired before and after deformation. The strain field is obtained from the displacement field by numerical differentiation. A complete description of the **3**-D **DIC** technique can be found in the literature, e.g. Luo et al. [24] or Garcia [29]. In this work, we have used the Vic-**3D**
r commercial

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In this work, the authors have focused on the effects of out of plane motion on 2D **DIC** and **3D** **DIC** deformation mea surements. After developing the theoretical 2D imaging equations using established pinhole models, the predicted strain errors due to out of plane translations and rotations are presented in Sections 2 and **3**, respectively. Section 4 presents details regarding out of plane translation and rotation experiments that were performed so that images are captured simultaneously by both (a) a single camera system (2D **DIC**) and (b) a two camera stereovision system (**3D** **DIC**) to investigate the effect of out of plane motion on 2D **DIC** and **3D** **DIC** measurements. Section 5 presents results from the experimental studies and Section 6 provides a detailed discussion of the findings. Section 7 presents concluding remarks.

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Passieux · Jean-No¨ el P´ eri´ e
Received: date / Accepted: date
Abstract The use of Finite Element meshes in **Digital** **Image** **Correlation** (FE- **DIC**) is now widespread in experimental mechanics. Up to now FE have been much less used in Stereo-**DIC**. The first goal of this paper is to explain in details how to use FE in Stereo-**DIC** using a formulation in the physical coordinate system. More precisely, it is shown how to perform the calibration of possibly nonlinear models, shape and displacement measurement based on a FE mesh. In addition it is shown that with such a framework it is possible to regularise the measurement with a FE model based on the same mesh. For instance, using this technique, it is shown that it is possible to measure the rotation field of a bending plate in addition to its displacement.

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Appl. Sci. 2020, 10, 1530 2 of **3**
and it raises a whole set of experimental problems (texture, acquisition, filtering, etc.). Regarding an atypical application, balloons, the authors of [ 6 ] recall that the field of (very) large deformation is still wide open and depending on the use case, special specific experimental configurations may help. It is also a theme addressed by [ 7 ] where calibrated targets were used to evaluate measurement uncertainties in this large deformation regime. The possibility to bridge more intimately measurements and models is highlighted in [ 8 ] where experimental measurements are combined to a model to extract mechanical fields with a certain mechanical admissibility close to a shear crack at bi-material interface. A little further on in the coupling between models and measurements, [ 9 ] proposed an interesting methodology to quantitatively characterize mechanical (interlaminar) properties reputed to be difficult to identify using finite element model updating techniques. Among current topics, the coupling of **DIC** with other types of instrumentation techniques or more generally data fusion is discussed in Article [ 10 ]. A comparative analysis based on **DIC** and Accoustic Emission techniques is helpful to comprehend the characteristics of concrete fracture process zones. In addition to the classic 2D **DIC**, several variants are also illustrated in this special issue. For example, stereo-**DIC** is an ally of choice for the validation of models on complex or large poly-instrumented structures. The issue of calibrating several independent benches using valuable CAD information is discussed in [ 11 ]. Conversely, when non-planar tests are to be instrumented at small scales or in conditions of difficult access, stereo can be used with a single camera by adapting the mounting with, for example, prisms and mirrors [ 12 ]. Another variant of **DIC**, which is still in its infancy, relies on X-ray based **digital** volume imaging. Increasingly, the

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Increases in spatial resolution in **DIC** by the use of SEM imaging has enabled the observation of discrete slip events. However the conventional **DIC** codes need to be developed and adapted to correctly handle the physics of the discontinuities present at the sharp localization of plasticity. Emerging **DIC** codes with disconti- nuity implementations have been recently applied in fracture mechanics [ 35e40 ]. These new **DIC** codes are signiﬁcantly improved in their ability to identify and characterize cracks or large shear band character. Rethore et al. [ 37 , 38 ] has fully described shear band formation by coupling an extended ﬁnite element method that can compute discontinuities with the **DIC** method. Further- more, **DIC** codes have been implemented [ 35 , 40 ] that directly capture discontinuities from **DIC** computations. Valle et al. [ 35 ] used the properties of the Heaviside functions to solve the problem of kinematical discontinuities and demonstrated the use of this **DIC** code for fracture mechanisms. A subset splitting method was used by Poissant et al. [ 40 ] to characterize crack opening. Along with the discontinuity tolerant ﬁeld measurement tool, the microgrid

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limit) because the strain discontinuity that is allowed by the discretization scheme remains constant over the entire set of elements cut by the discontinuity line. Furthermore, the error level obtained with the regularized approach is 5 times less than that obtained with the standard enriched X-Q4 scheme.
The extension of the **digital** **image** **correlation** technique to enriched functions appears as straightforward technically. However, for some cases (such as weak discontinuities) this enrichment naturally leads to a poor numerical conditioning, and hence, even though the displacement basis has been enriched to capture a feature of interest, its determination may be worthless because too uncertain. It is thus absolutely essential to rely on the type of presented analysis to get a proper evaluation of the real error, uncertainty and noise sensitivity. Let us also underline the fact that if the searched feature is now included in the displacement basis, much larger meshes may be used, hence leading to a more accurate and less uncertain determination of the displacement. Last, if the enriched basis still remains poorly conditioned, then a regularization may be used, and again the noise sensitivity analysis is a useful tool to choose the amount of regularization that is required. In the X-Q4 approach, the strain discontinuity is described by the shape functions intersected by the support line of the discontinuity, whereas for X-Q4r the discontinuity is a constant. Intermediate regularizations could have been considered with a few degrees of freedom to describe the discontinuity variation along its support. This strategy is valid irrespective of the chosen type of enrichment.

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.
1. INTRODUCTION
Composite materials are increasingly used in industry. Many experiments have been made to develop models to allow very complex structure sizing. But only few experiments have been done to characterize the behaviour of those materials under complex loading, such as biaxial planar loading, and even less dealing about fatigue. The biaxial testing machine owned by the LGP laboratory allows running experiments on biaxial cruciform specimens. The issue of those specimens is that the stress repartition is not constant over the specimen. It is why the **Digital** **Image** **Correlation** technique seems to be appropriate to monitor those tests, since it allows measuring an important area on the specimen with good accuracy. Thermography is as well appropriate to monitor fatigue testing as it permits to measure temperature elevation due to damage and hysteretic phenomenon. In the following study, a new kind of specimen is created to try to better respond to the needs of biaxial testing and to fit with the manufacturing process. Then, this specimen is tested during static and cyclic loading.

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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible.[r]

Abstract: Optical full-field measurement methods such as **Digital** **Image** **Correlation** (**DIC**) are increasingly used in the field of experimental mechanics, but they still suffer from a lack of infor- mation about their metrological performances. To assess the performance of **DIC** techniques and give some practical rules for users, a collaborative work has been carried out by the Workgroup “Metrology” of the French CNRS research network 2519 “MCIMS” 1 . A methodology is proposed to assess the metrological performances of the **image** processing algorithms that constitute their main component, the knowledge of which being required for a global assessment of the whole measurement system. The study is based on displacement error assessment from synthetic speckle images. Series of synthetic reference and deformed images with random patterns have been generated, assuming a sinusoidal displacement field with various frequencies and amplitudes. Displacements are evaluated by several **DIC** packages based on various formulations and used in the French community. Evalu- ated displacements are compared with the exact imposed values and errors are statistically analyzed. Results show general trends rather independent of the implementations but strongly correlated with the assumptions of the underlying algorithms. Various error regimes are identified, for which the dependence of the uncertainty with the parameters of the algorithms, such as subset size, gray level interpolation or shape functions, is discussed.

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Despite these early volume change observations, it became a standard practice to measure the longitudinal strain and to assume incompressibility in order to reach the stress-stretch response of rubbers submitted to uniaxial tensile tests. Such an assumption renders impossible the study of possible damage and biases the estimation of the actual true stress-stretch response. Later, the measurement of strains has been improved by monitoring them in the longitudinal and transverse directions on one face of the sample. This may be achieved by video extensometry [3,4] or by **digital** **image** **correlation** [5]. Using such techniques coupled to the transverse isotropy assumption, authors reported substantial reversible volume changes [5] in disagreement with the earlier accurate dilatometry measurements [1,6-8] evidencing negligible volume changes for unfilled rubbers and substantial volume changes for filled rubbers during the first load only. However, uniaxial tensile test specimens are usually cut from calendered sheets. Diani et al. [9] observed in-plane anisotropy in such specimens, suggesting an even stronger anisotropy in the thickness direction. Such anisotropy is a plausible explanation for these conflicting results.

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Keywords: shot peening, high speed camera, particle tracking, **DIC**, process control Introduction and objectives
Shot peening process is generally controlled by Almen intensity and coverage rate measurements. The results of the shot peening surface treatment depend on the process parameters (type of machine, nozzle, shot type, mass flow, velocity, impact angle) and on the properties of the treated parts (material behaviour, roughness). Previous studies have investigated the control of shot peening via measurement of the global flow characteristics [1, 2, **3**]. Developing experimental methods to characterize a particle flux is useful to obtain the contact conditions of the shot peening process (location, velocity vector) on any interesting geometry. Using a high speed camera, this study aims at the analysis of the flux of shots ahead of the nozzle of a shot peening machine. In this study two set of process parameters are investigated:

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