# Enhanced Digital Image Correlation Analysis of Ruptures with Enforced Traction Continuity Conditions Across Interfaces

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}) or high-resolution cameras, but with comparatively large subset sizes, situations typically arising in the study of dynamic problems [3]. To resolve the displacement discontinuities on the interface associated with the propagation of dynamic shear ruptures, the studies in [1,2,3] used the commercial DIC software Vic-2D (Correlated Solutions, Inc.) to separately correlate the domains above and below the interface. While standard local DIC methods calculate displacements up to half a subset away from the interface, the “fill boundary” algorithm of Vic-2D employed in [1,2,3] uses affine transformation functions to extrapolate the displacements from the center of the subset up to the interface. This enables the use of large subsets in each side of the interface, which are essential to overcoming the noise associated with ultra-high-speed photography and the analyzed rapid ruptures. An important limitation of this approach, however, is that the extrapolated displacements are less accurate than those obtained directly by the actual correlation, especially at pixels near the interface, which have the largest distance from the subset center. This leads to errors in the strains and stresses near the interface, which are obtained from the spatial derivatives of the displacement fields. In addition, the non-coupled correlation of the domains above and below the interface results in non-physical discontinuities in the tractions across the interface.

## 2. Monitoring Dynamic Shear Ruptures in the Laboratory

#### 2.1. The Laboratory Setup

_{0}= Psin(α)cos(α) and σ

_{0}= Pcos

^{2}(α), respectively. The rupture is nucleated by a local pressure release provided by a rapid expansion of a NiCr wire filament due to an electrical discharge of Cordin 640 high-voltage capacitor (Cordin, Salt Lake City, UT, USA). A key aspect of the setup is that the low shear modulus of Homalite enables production of well-developed dynamic ruptures in samples of tens of centimeters.

^{2}, at a temporal sampling of 2 million frames/second and an exposure time of 200 ns. Moreover, the HPV-X camera was equipped with a range of prime telephoto lenses, which allowed it to monitor fields of view of different sizes [3].

#### 2.2. Digital Image Correlation to obtain Displacement Fields

#### 2.3. Post-Processing of the Displacement Fields

_{1}-x

_{2}parallel to the interface and use the central difference scheme for pixels away from the boundaries [23]:

## 3. A Post-Processing Algorithm to Enforce Traction Continuity along the Interface

_{1}-x

_{2}, such that x

_{1}is parallel to the interface. However, the method can easily be extended to any other frame. The procedure is described here for a given column of pixels at a given frame, and was implemented for all the columns and frames.

#### 3.1. Traction Continuity Conditions

^{0+}) and (

^{0−}) denote the values immediately above and below the interface, respectively, and ${\sigma}_{22\left(0\right)}{}^{0+}={\sigma}_{22\left(0\right)}{}^{0-}$ are the initial stresses. Assuming a linear elastic response for the materials above and below the interface, the condition in Equation (6) can be written in terms of strains as

^{+}) and (

^{−}) denote the material properties above and below the interface, respectively. For small strains the condition can be expressed in terms of displacement gradients as

#### 3.2. Approximating the Displacements with Local Polynomials

#### 3.3. Inverting for the Polynomial Coefficients

_{1}and 1.5% for u

_{2}, but it was enough to enforce the continuity of traction.

_{1}(Equations (1)–(4)). When enforcing TC, these gradients are computed using the polynomial centered at column j. Once the displacement fields have been updated using the procedure described above, strains can be computed from the updated displacement fields, using finite difference schemes, as discussed in Section 2.3. However, the finite differences would be using the displacements of columns j−1 and j + 1, which are obtained from other polynomials than that used to enforce TC, those centered in columns j−1 and j + 1. As such, the spatial derivatives of the modified displacement field with respect to x

_{1}would be slightly different than those obtained during the enforcement of TC, using the polynomial centered at column j. Although the differences between polynomials centered at two neighboring columns are small, they may be enough to cause some discontinuities in tractions at the interface. To ensure the continuity of tractions, we computed strains and stresses within the $2\times {n}_{\mathrm{x}2}$ band, where displacements are updated directly as the algorithm goes over each column of pixels (i.e., we calculated the displacement derivatives for column j with respect to x

_{1}using the displacements in columns j−1 and j + 1 obtained from the polynomial centered at column j. Full-field maps of stresses modified by the algorithm using n

_{x}

_{1}= 7, n

_{x}

_{2,inter}= 12, and n

_{x}

_{2,cent}= 11 (Figure 2c,d) demonstrated the capability of this approach to generate more realistic and continuous stress fields near the interface. Interestingly, because of the symmetry of the problem, the enforcement of TC led to normal stresses that are close to the background value (∼17.6 MPa) at the interface.

#### 3.4. The Effects of the Geometrical Parameters of the Polynomial

_{x}

_{1}, n

_{x}

_{2,inter}, and n

_{x}

_{2,cent}in order not to introduce unwanted field distortions. For instance, a choice of a small value of n

_{x}

_{2,inter}together with a large value of n

_{x}

_{2,cent}or vice versa can lead to stress fields that are continuous on the interface, but have sharp discontinuities (Figure 2e,f) or spurious oscillations (Figure 2g,h) in the bulk. In this section, we study the effects of different parameter combinations in more details, aiming to find the parameter sets that avoid these distortions in the stress field. We ran the algorithm 324 times, varying the geometrical parameters in the following ranges: 3 ≤ n

_{x}

_{1}≤ 9, 4 ≤ n

_{x}

_{2,inter}≤ 20, and 5 ≤ n

_{x}

_{2,cent}≤ 21. Note that for n

_{x}

_{1}≥ 13 the system was over constrained. Overall, we found that parameters in ranges of 8 ≤ n

_{x}

_{2,inter}≤ 14 (1/5 to 1/3 of the subset size) and 7 ≤ n

_{x}

_{2,cent}≤ 15 gave similar results, independent of the value of n

_{x}

_{1}.

_{x}

_{1}. For example, there was no observable difference between full-field maps of stresses obtained with n

_{x}

_{1}= 3 and n

_{x}

_{1}= 9 (Figure 5a–d). To further examine the effect of n

_{x}

_{1}, we plot the normal and shear stresses vs position along two lines perpendicular to the interface at x

_{1}= 170 and 220 pixels (Figure 5e–h). For both lines there was a “jump” larger than 5.5 MPa in the normal stress at the interface when the calculation was done with the “fill boundary” algorithm of Vic-2D without TC enforcement. The TC algorithm enforced the normal stress to be continuous across the interface, with a normal stress at the interface that was about the average of the normal stresses at the pixels immediately above and below the interface when TC was not enforced. The results were not sensitive to n

_{x}

_{1}. A small effect of n

_{x}

_{1}on the shear stress was observed for the line located at x1 = 220 pixels (Figure 5h), with a difference of 0.12 MPa between the results for n

_{x}

_{1}= 3 and 9. The shear stress at the interface when TC was enforced were smaller than the shear stresses at the pixels immediately above and below the interface when TC was not enforced.

_{x}

_{2,inter}and n

_{x}

_{2,cent}resulted in artifacts in the stress field. On the one hand, taking n

_{x}

_{2,inter}= 4 and n

_{x}

_{2,cent}= 21 (with n

_{x}

_{1}= 7) resulted in discontinuities at the boundary between the Vic-2D solution and polynomial extrapolation (Figure 2e,f), n

_{x}

_{2,inter}+ n

_{x}

_{2,cent}pixels away from the interface due to a poor fit between the stresses obtained by the polynomial and the original stresses at this region. On the other hand, taking n

_{x}

_{2,inter}= 20 and n

_{x}

_{2,cent}= 5 (with n

_{x}

_{1}= 7) resulted in spurious oscillations (Figure 2g,h). Some effects of n

_{x}

_{2,inter}were observed also when n

_{x}

_{2,cent}was fixed at a moderate value (e.g., n

_{x}

_{2,cent}= 11), especially for the shear stress. A small value of n

_{x}

_{2,inter}(e.g., n

_{x}

_{2,inter}= 6) led to large gradients in the normal stresses near the interface, as well as to small discontinuities at the pixels located n

_{x}

_{2}pixels above and below the interface (Figure 6a,b). A large value of n

_{x}

_{2,inter}(e.g., n

_{x}

_{2,inter}= 16) resulted in an artificial increase or decrease of stresses at localized regions around the interface compared with the domains above and below (Figure 6c,d). For example, the normal stress decreased locally at the interface at x

_{1}= 370 pixels (Figure 6c), while the shear stress increased locally at x

_{1}= 170 and 370 pixels (Figure 6d). Some issues are illustrated further in plots of the normal and shear stresses vs position along two lines perpendicular to the interface at x

_{1}= 170 and 220 pixels (Figure 6e–h). The value of normal stress on the interface was similar for n

_{x}

_{2,inter}≤ 14, but diverged for higher values, with a difference of about 1 MPa for n

_{x}

_{2,inter}= 20. For n

_{x}

_{2,inter}≤ 6, the TC enforcement resulted in a poor agreement between the stresses obtained by the polynomial and the original stresses also at the pixels closer to the center of the subset. The shear stress at the interface also diverged, with an increase of about 1 MPa for n

_{x}

_{2,inter}= 16, at x

_{1}= 170 pixels (Figure 6f).

_{x}

_{2,cent}was generally smaller than that of n

_{x}

_{2,inter}. Full-field maps of stresses obtained with n

_{x}

_{2,cent}= 7 and n

_{x}

_{2,inter}= 10 (Figure 7a,b), were similar to those in Figure 2c,d. However, large value of n

_{x}

_{2,cent}= 19 resulted in small discontinuities in normal stress at the pixels located n

_{x}

_{2}pixels above and below the interface (Figure 7c,d). Both at x

_{1}= 170 and 220 pixels there was a very small effect of n

_{x}

_{2,cent}on the normal stress at the interface (Figure 7e,g). However, for n

_{x}

_{2,cent}≥ 17, the fit between the stresses obtained by the polynomial and the original stresses was poor also for the pixels closer to the center of the subset. Similarly to n

_{x}

_{2,inter}, the shear stress at the interface may significantly have differed for large values of n

_{x}

_{2,cent}(Figure 7f).

## 4. Implications for Friction Analysis

_{1}= 9 mm. This is also shown locally in Figure 5h. Because the stresses were computed from displacement gradients, small changes in the displacements were enough to enforce the traction continuity on the interface, and there were only minor differences between the interface-parallel displacements and velocities obtained with TC enforcement and those obtained without TC enforcement (Figure 8c,d). The small differences in stresses resulted in some differences in the evolution of local friction with slip and slip rate (Figure 8e,f). However, the important characteristics of the local friction, such as the peak, residual levels, and weakening distance, were similar.

## 5. Conclusions

_{x}

_{2,inter}, n

_{x}

_{2,cent}, and n

_{x}

_{1}involved in the construction of the polynomials, such that undesired artifacts in the stress fields were eliminated. We found that parameter ranges of 8 ≤ n

_{x}

_{2,inter}≤ 14 (1/5 to 1/3 of the subset size) and 7 ≤ n

_{x}

_{2,cent}≤ 15 gave similar results, independent of the value of n

_{x}

_{1}. Relatively minor changes in displacement fields around the interface can produce non-negligible gradient changes, resulting in some non-uniqueness regarding the exact stress evolution towards the interface, even if the traction continuity is enforced. Future progress can be made by combining the results inferred by the analysis presented in this work with dynamic rupture modeling that matches the observed full fields. The dynamic solutions can then be interrogated for the spatial dependencies of stress fields next to the interface.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematics of the laboratory rupture experiment. Dynamic shear ruptures evolved spontaneously along the frictional interface, inclined at an angle α, of two Homalite plates under a compressional load P. Ruptures were initiated by the small burst of a NiCr wire placed across the interface and connected to a capacitor bank. The white light produced by a flash source was reflected by the specimen’s surface and captured by a low-noise high-speed camera, typically at 1–2 million frames/s. The portion of the specimen to be imaged, the field of view, was coated by a flat white paint and decorated by a characteristic speckle pattern. Next, the textured images were processed by digital image correlation (DIC) algorithms to produce a temporal sequence of full-field displacement maps. The displacement fields were then post-processed to produce velocity, strain, strain rate, and stress maps.

**Figure 2.**Full-field maps of normal (left) and shear (right) stresses measured for a super-shear crack-like rupture during a test with α = 29°, P = 23 MPa (${\sigma}_{0}=17.6$ MPa and ${\tau}_{0}=9.75$ MPa), and a field of view of 19 x 12 mm. (

**a**,

**b**) Full-field maps obtained using Vic-2D extrapolation near the interface. (

**c**,

**d**) Fulfilled maps obtained with the stress continuity constraints, using a good set of parameters n

_{x}

_{1}, n

_{x}

_{2,inter}, and n

_{x}

_{2,cent}. (

**e**–

**h**) Enforcement of stress continuity with extreme values of n

_{x}

_{2,inter}and n

_{x}

_{2,cent}leads to discontinuities and spurious oscillations in the stress fields. Note that the full-field maps are cropped and their size is slightly smaller than the reported field of view size.

**Figure 3.**(

**a**) A speckled image with two subsets of 41 × 41 pixels above and below the interface (yellow squares). The post-processing algorithm is performed on the regions inside the subsets marked by the red rectangles. (

**b**) The geometry of the pixels over which the local polynomials approximating the displacement fields are defined. The polynomials approximating ${u}_{2}^{+}$, ${u}_{1}^{+}$ extend over a rectangle above the interface, while those approximating ${u}_{2}^{-}$, ${u}_{1}^{-}$ extend over a rectangle below the interface; both rectangles include ${n}_{x1}\times {n}_{x2}$ pixels. The pixels within each rectangle are divided into a group of ${n}_{x1}\times {n}_{x2,inter}$ pixels that are closer to the interface, and a group of ${n}_{x1}\times {n}_{x2,cent}$ pixels that are closer to the center of the subset. The observed displacements in the latter group are used together with continuity of stresses constraints on the interface to construct the polynomials. (

**c**,

**d**) Two different combinations of ${n}_{x2,inter}$ and ${n}_{x2,cent}$ and their relationship with Vic-2D extrapolation. In the first combination, the group of pixels ${n}_{x1}\times {n}_{x2,cent}$, which are used as data in the inversion, include pixels where the displacement values were obtained by the “fill boundary” algorithm of Vic-2D.

**Figure 4.**Interface parallel (

**a**) and normal (

**b**) displacements at the central columns of the subsets shown in Figure 3a. The black lines represent the displacements obtained with Vic-2D, while the blue lines are the displacements modified with the traction continuity algorithm, using ${n}_{x1}=5$, ${n}_{x2,inter}=12$, and ${n}_{x2,cent}=11$. The red circles represent the observed displacements that are used to invert for the coefficients of the displacement polynomials ${u}_{2}^{p+}$, ${u}_{2}^{p-}$, ${u}_{1}^{p+}$, and ${u}_{1}^{p-}$. The black diamonds correspond to the centers of the subsets.

**Figure 5.**The effect of parameter n

_{x}

_{1}on the stress field. (

**a**–

**d**) Full-field maps of normal (left) and shear (right) stresses for n

_{x}

_{1}= 3 and 9. (

**e**–

**h**) Normal (left) and shear (right) stresses vs position along paths perpendicular to the interface at x

_{1}= 170 and 220 pixels (grey lines in (

**c**)) for different values of n

_{x}

_{1}. The stresses calculated with the “fill boundary” algorithm of Vic-2D without the traction continuity algorithm are also shown, where the black diamonds correspond to the centers of the subsets above and below the interface. In all the cases shown in this figure, n

_{x}

_{2,inter}= 10 and n

_{x}

_{2,cent}= 11.

**Figure 6.**The effect of parameter n

_{x}

_{2,inter}on the stress field. (

**a**–

**d**) Full-field maps of normal (left) and shear (right) stresses for n

_{x}

_{2,inter}= 6 and 16. (

**e**–

**h**) Normal (left) and shear (right) stresses vs position along paths perpendicular to the interface at x

_{1}= 170 and 220 pixels (grey lines in (

**c**)) for different values of n

_{x}

_{2,inter}. The circles on the curves correspond to n

_{x}

_{2,inter}, while the filled diamonds correspond to n

_{x}

_{2}. The stresses calculated with the “fill boundary” algorithm of Vic-2D without the traction continuity algorithm are also shown, where the black diamonds correspond to the centers of the subsets above and below the interface. In all the cases shown in this figure n

_{x}

_{1}= 7 and n

_{x}

_{2,cent}= 11.

**Figure 7.**The effect of parameter n

_{x}

_{2,cent}on the stress field. (

**a**–

**d**) Full-field maps of normal (left) and shear (right) stresses for n

_{x}

_{2,cent}= 7 and 19. (

**e**–

**h**) Normal (left) and shear (right) stresses vs position along paths perpendicular to the interface at x

_{1}= 170 and 220 pixels (grey lines in (

**c**)) for different values of n

_{x}

_{2,cent}. The circles on the curves correspond to n

_{x}

_{2,inter}, while the filled diamonds correspond to n

_{x}

_{2}. The stresses calculated with the “fill boundary” algorithm of Vic-2D without the traction continuity algorithm are also shown, where the black diamonds correspond to the centers of the subsets above and below the interface. In all the cases shown in this figure n

_{x}

_{1}= 7 and n

_{x}

_{2,inter}= 10.

**Figure 8.**(

**a**,

**b**) normal and shear stresses along the interface. The black curves represent the stresses ±0.5 pixels from the interface obtained without traction continuity (TC) enforcement, while the green curve represents the average of the two. The blue curves represent the stresses obtained with TC enforcement ±0.5 pixels from the interface. (

**c**,

**d**) Interface-parallel displacements and velocities ±0.5 pixels from the interface, obtained with (blue) and without (black) TC enforcement. (

**e**,

**f**) Resolved friction vs slip and slip rate at x

_{1}= 8.5 mm, obtained with (blue) and without (black) TC enforcement. Both curves represent the average of the values immediately above and below interface. The polynomial geometrical parameters are n

_{x}

_{1}= 7, n

_{x}

_{2,inter}= 10, and n

_{x}

_{2,cent}= 11.

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## Share and Cite

**MDPI and ACS Style**

Tal, Y.; Rubino, V.; Rosakis, A.J.; Lapusta, N.
Enhanced Digital Image Correlation Analysis of Ruptures with Enforced Traction Continuity Conditions Across Interfaces. *Appl. Sci.* **2019**, *9*, 1625.
https://doi.org/10.3390/app9081625

**AMA Style**

Tal Y, Rubino V, Rosakis AJ, Lapusta N.
Enhanced Digital Image Correlation Analysis of Ruptures with Enforced Traction Continuity Conditions Across Interfaces. *Applied Sciences*. 2019; 9(8):1625.
https://doi.org/10.3390/app9081625

**Chicago/Turabian Style**

Tal, Yuval, Vito Rubino, Ares J. Rosakis, and Nadia Lapusta.
2019. "Enhanced Digital Image Correlation Analysis of Ruptures with Enforced Traction Continuity Conditions Across Interfaces" *Applied Sciences* 9, no. 8: 1625.
https://doi.org/10.3390/app9081625