# Quantification of Reinforcement Debonding in Damaged Mortar via Digital Volume Correlation

Sylvain Langlois, Farid Benboudjema, Maroua Maaroufi, Fikri Hafid,

Benjamin Smaniotto, François Hild, Amélie Fau

► To cite this version:

Sylvain Langlois, Farid Benboudjema, Maroua Maaroufi, Fikri Hafid, Benjamin Smaniotto, et al. Quantification of Reinforcement Debonding in Damaged Mortar via Digital Volume Correlation. Experimental Mechanics, 2025, 65, pp.799-817. 10.1007/s11340-025-01166-1. hal-04962046

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# Quantification of Reinforcement Debonding in Damaged Mortar via Digital Volume Correlation

S. Langlois¹*, F. Benboudjema¹, M. Maaroufi¹, F. Hafid², B. Smaniotto¹, F. Hild¹, A. Fau¹

¹*Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS LMPS – Laboratoire de Mécanique Paris-Saclay, 4 avenue des sciences, 91190 Gif-sur-Yvette, France.

²RTE, Réseau de Transport d'Électricité, Research&Development, 92060 La Défense, France.

*Corresponding author(s). E-mail(s):
sylvain.langlois@ens-paris-saclay.fr;

## Abstract

**Background** Debonding between a cementitious material and a reinforcement is a mechanical phenomenon of great interest. It cannot be quantified directly through standard tests since it occurs within the material bulk.

**Objective** The goal is to develop an experimental method for quantifying debonding during in-situ pull-out tests that also induce damage in the mortar matrix.

**Method** A 1/50 scale foundation model is subjected to a pull-out test in an X-ray tomograph. A finite-element-based Digital Volume Correlation analysis with mechanical regularization is conducted based on a three-dimensional mesh constructed to reproduce the geometry of the foundation and reinforcement.

**Results** Heterogeneous regularization with a single-node mesh has little effect on the correlation residuals. Using split nodes to describe the interface drastically reduces the correlation residuals in the reinforcement. If cracking occurs in addition to debonding, introducing a heterogeneous regularization based on damaged elements improves the quantification of debonding.
**Conclusion** By splitting the nodes at the interface and localizing regularization in damaged elements, the reinforcement and mortar kinematics is better captured and thus debonding as well.

**Keywords:** Digital Volume Correlation (DVC), Pull-out, Mortar, Mechanical regularization, Damage

# 1 Introduction

Debonding is a frequent phenomenon in reinforced concrete structures. The bond behavior is essential to ensure the performance of anchors and reinforcements through the transmission of forces with the matrix [1, 2]. The experimental characterization of reinforcement-concrete bonding mostly involves pull-out tests [3], with 89% of the studies (of which 80% exclusively) relying on this test as an experimental method [4]. The bonding behavior is generally analyzed by measuring the ultimate load with respect to the slip displacement. Measuring this macroscopic information is a reliable and repeatable method [4] for understanding the impact of various parameters such as concrete formulation [5], type of reinforcement [6], development of pathologies in the matrix [7], or environmental conditions such as temperature [8] or relative humidity [9]. However, macroscopic measurements do not give access to local interfacial behavior.

Computed Tomography (CT) gives access to phenomena occurring within the material bulk. Each X-ray CT scan provides a 3D volume of the sample, revealing its microstructure in a non-destructive way [10]. This technique was used in material science as a qualitative or quantitative diagnostic tool [11]. Mechanical loading can be carried out directly in the X-ray tomograph during so-called in-situ tests [12]. Such experiments have been used to characterize spatial damage growth (distribution of crack or crack path) in concrete under mechanical loadings [13–15]. The bond behavior between a crenelated GFRP rebar and concrete has been studied in an in-situ
test [16]. Local pore volume changes at the interface were quantified through a seg-
mentation process by comparing the deformed scan to the reference scan. The strains
in the reinforcement were derived from pore volume changes, under the assump-
tions that these changes were proportional to the height variations of air voids and
that no strain of the reinforcement or concrete occurred in the radial direction. As
a result, the study was restricted to a 1D problem, with no damage considered in
concrete. This method would not be applicable in the case of pure slip at interfaces.
Debonded yarns in fiber-reinforced concrete were analyzed from sections of CT scans
after tensile failure of the specimen [17, 18]. Beyond qualitative (visual) analyses,
the debond length between a yarn and a mortar under tensile loading was quanti-
fied [19]. The interfacial crack propagation was analyzed for various load steps. For
each step, the debond length was estimated from the position of the interfacial crack
tip. As a microscopic event, debonding is very challenging to quantify precisely with
only visual inspection of CT scans.

Digital Volume Correlation (DVC) coupled with X-ray CT scans is an appropriate
tool to meet the previous challenge. DVC allows displacement fields to be measured
within the sample along the loading history. Initially, (local) DVC was based on the
measurement of displacements of small sub-volumes [20]. A method combining X-
ray CT and local DVC was proposed to quantitatively evaluate the interfacial bond
behavior in a compressive test on a dedicated specimen consisting of a cubic cement
paste sample bonded to a cubic porous aggregate [21, 22]. The study successfully
evaluated the effective area providing mechanical interlocking in the Interfacial Tran-
sition Zone (ITZ) through the quantification of strain fields. This study was possible
only when the specimen was cracked given the reported noise floor levels. With local
DVC, the strain field around a steel hook-end reinforcement during an in-situ pull-
out test from mortar has been quantified [23]. Load transfer was localized in the hook
and resulted in high local contraction and shear strains. However, no information
was provided regarding the interfacial debonding behavior. Subsequently, finite ele-
ment strategies were incorporated, whereby registrations were performed over the
whole region of interest [24] with subvoxel resolution [25]. Finite-element-DVC (FE-
DVC) was used to analyze different types of in-situ pull-out tests. The debonding
of a Fiber Reinforced Polymer (FRP) anchor in mortar was studied with this tech-
nique [26]. The 3D crack was visually analyzed through the DVC residuals. Then, a
2D quantification of the interfacial crack opening displacement was proposed based
on digital image correlation analyses of CT sections in the assumed symmetry plane
of the specimen. An image-based mesh was constructed in another study [27] to
reproduce consistently the interface between the anchor and mortar/plaster cylin-
ders. FE-DVC was used to compute displacement jumps using a mesh with split
nodes at the interface between brittle matrices and metallic fibers.

The present study aims to quantify debonding at the interface between a rein-
forcement bar and mortar in an in-situ pull-out test, which induces a crack in the
mortar matrix. A FE-DVC method with elastic regularization may not fully capture
the very high strain gradients in the vicinity of cracks [28]. A dedicated strategy is
proposed, which is based on meshes with split nodes at the interface, and localization
of the regularization based on damage in the mortar matrix. It is shown that splitting
the nodes of the interface allows the kinematics of the reinforcement to be better cap-
tured and enables debonding to be quantified. Damage-based regularization allows
debonding to be precisely captured even in the presence of matrix cracks.

## 2 In-situ pull-out test

A dedicated in-situ pull-out test is designed to analyze the behavior of a small-
scale mortar structure, which is designated as the foundation hereafter. The materials
used and the design of the test bench enabling for in-situ pull-out tests within an X-
ray tomograph are detailed. An overall description of the in-situ test is given. DVC
is introduced in both homogeneous and heterogeneous regularized versions. The
meshes used to compute displacement fields are presented, and the method for calculating debonding is explained in the case when split nodes are introduced at the interface.

## 2.1 Pull-out specimen and set-up

The specimen consists of a reinforcement anchored in a mortar foundation. The mortar contains 0-4 mm limestone sand and CEM II/A-LL 42.5R cement. The mortar of density 2400 kg/m³ is manufactured with ratios W/C = 0.5 and S/C = 3, where W, S and C are the masses of water, sand and cement, respectively. The obtained mortar has a Young's modulus $E^{\text{mortar}}$ of 20 GPa, and a tensile strength (measured in a 3-points bending test) $f_t$ of 3.5 MPa after 28 days. The choice of the reinforcement material has to be made with care for studies conducted in an X-ray tomograph. The displacement measurement via DVC relies on the contrast in the imaged material(s). This contrast is represented by gray level gradients in images, which ideally should cover a broad dynamic range, and must be minimally affected by tomographic artifacts. For those reasons, the selected reinforcement is made of Glass Fiber Reinforced Polymer (GFRP) with a Young's modulus $E^{\text{GFRP}}$ equal to 60 GPa. The reinforcement is compared to two other materials, namely, steel and inconel 718 in Appendix A.

The mortar is manufactured and poured directly into an Acrylonitrile butadiene styrene (ABS) mold, which is specifically designed to mimic a concrete foundation used in high voltage transmission lines in France. A fixing template allows the reinforcement to be sealed, centered and vertical with respect to the foundation. The sample is unmolded after 24 h, then left to cure endogenously at 20°C for 42 days to ensure the hydration of cement. The foundation shown in Figure 1(a) consists of a reinforcement ①, which has a diameter of 8 mm and 9 mm, if the crenulations are accounted for. The foundation ② is composed of a succession of one chimney and two slabs (slab 1 and slab 2) 28 mm, 10 mm and 10 mm in height. The lower part of the mortar foundation (approximately 30 mm in height) is glued directly onto an
ABS collar. The collar is then screwed to the lower platen of the testing machine. A steel clamp is designed based on ER collet (ISO 15488-2003) to hold the GFRP reinforcement. The clamp is then fixed to the testing machine as shown in Figure 1(b) and 1(c). The domain of interest (highlighted in red in Figure 1(a)) comprises the upper part of the mortar foundation and the lower part of the reinforcement such that boundary effects are avoided.

It is worth noting that the choice of sample geometry has a significant impact on the quality of the CT scans. Increasing the sample size (if concrete were used instead of mortar, for instance) requires the source-to-detector distance to be increased to scan the entire structure, which deteriorates image quality. Additionally, a higher diameter increases X-ray absorption, thereby decreasing the signal received by the detector. These degradations affect the measurement uncertainties and may prevent the quantification of low-amplitude displacements. It is preferable to reduce the structure size to enhance the resolution and minimize X-ray absorption. In this study, the diameter of the GFRP reinforcement could not be reduced by less than 8 mm, which imposed a minimum limit on the size of the mortar foundation to keep the same aspect ratio.

<figure><img src="image_2.png" /><figcaption>Fig. 1 (a) Specimen composed of a GFRP reinforcement ① and a mortar foundation ②. (b) Design of the experimental setup. (c) In-situ testing machine and specimen inside the tomograph.</figcaption></figure>
## 2.2 In-situ mechanical test

An in-situ pull-out test is carried out using the tension-compression-torsion machine (Figure 1(c)) designed for a use in the X-ray tomograph of LMPS (North Star Imaging X50+) The mortar foundation is loaded in steps. A scan is performed at each step to visualize the microstructure during mechanical loading. The loading curve for the in-situ pull-out test is shown in Figure 2. The mechanical test is carried out with a controlled stroke history. Various aspects of the setup prevent achieving a purely pull-out loading. For instance, the reinforcement is not perfectly parallel to the foundation axis, and the foundation itself is not axially aligned with the ABS collar. Each CT scan consisted of 1,200 radiographs, with a scan duration of 30 min.

The hardware parameters of the in-situ test are listed in Table 1. The reconstructed volume covered 978 × 978 × 1069 mm³ with a 45 µm/vx resolution. The best resolution achievable by scanning a sample in the testing machine is 12 µm/vx because of the distance required between the X-ray source and the testing machine. Conversely, the testing machine cannot load specimens larger than 10 cm in diameter and taller than 10 cm in height.

For the in-situ test, a preload of 50 N is applied to ensure contact between all parts of the setup and to guarantee the stability of the specimen during the first scan. To reduce the effect of relaxation during the scans, a time dwell of 10 minutes is applied before each scan. Scan 0 is considered the reference scan, from which the displacement fields are calculated via DVC. Scan 00, referred to as repeat scan, is performed immediately after scan 0 and is used to quantify measurement uncertainties (Section 3.3). The specimen is then loaded with a stroke velocity of 5 µm/s. For scan 1 (with a force of 350 N), a crack initiates at the junction between the chimney and slab 1. This crack propagates along the section of the chimney at 430 N just before scan 2. As the reinforcement is anchored in the foundation, load transfer through the foundation is still possible. Scan 3 is acquired when the applied force is equal to
<table><caption>**Table 1** CT hardware parameters</caption><tr><td>Tomograph</td><td>North Star Imaging X50+</td></tr><tr><td>X-ray source</td><td>XRayWorX XWT-240-CT</td></tr><tr><td>Target / Anode</td><td>30 W (reflection mode)</td></tr><tr><td>Filter</td><td>none</td></tr><tr><td>Voltage</td><td>210 kV</td></tr><tr><td>Current</td><td>150 µA</td></tr><tr><td>Focal spot size</td><td>5 µm</td></tr><tr><td>Tube to detector</td><td>615 mm</td></tr><tr><td>Tube to object</td><td>105 mm</td></tr><tr><td>Detector</td><td>Dexela 2923</td></tr><tr><td>Definition</td><td>1536 × 1944 pixels (2 × 2 binning)</td></tr><tr><td>Number of projections</td><td>1200</td></tr><tr><td>Angular amplitude</td><td>360°</td></tr><tr><td>Frame average</td><td>15 per projection</td></tr><tr><td>Frame rate</td><td>15 fps</td></tr><tr><td>Acquisition duration</td><td>Around 30 min</td></tr><tr><td>Reconstruction algorithm</td><td>Filtered back-projection</td></tr><tr><td>Gray Levels amplitude</td><td>16 bits</td></tr><tr><td>Volume size</td><td>978 × 978 × 1069 vx (after crop)</td></tr><tr><td>Field of view</td><td>44.0 × 44.0 × 48.1 mm³ (after crop)</td></tr><tr><td>Resolution</td><td>45 µm/vx</td></tr></table>

780 N, for which a crack is visible only in the chimney. The sample is then loaded until it breaks (i.e., for a force of 1700 N). Failure is unstable, followed by a significant drop in force, and rapid crack propagation across the junction between slabs 1 and 2.

The scans used in DVC analyses are marked in red in Figure 2. The DVC field of interest is for scan 3 for which the mortar matrix is damaged in the chimney.

<figure><img src="image_3.png" /><figcaption>**Fig. 2** Pull-out curve with scans acquired for different mechanical states.</figcaption></figure>
## 2.3 FE-based regularized DVC

The 3D displacement fields of the sample during the in-situ test are measured via global Digital Volume Correlation (DVC) [24] between two volumes with the Correli 3.2 framework [29] using 4-noded (T4) tetrahedra. Two volumes $f(\mathbf{x})$ (in the reference configuration) and $g(\mathbf{x})$ (in the deformed configuration) are correlated by considering the gray level residuals

$$ \rho(\mathbf{x}, \mathbf{u}) = f(\mathbf{x}) - g(\mathbf{x} + \mathbf{u}(\mathbf{x})). \quad (1) $$

where the displacement field $\mathbf{u}$ is discretized using T4 elements, and expressed as $\mathbf{u}(\mathbf{x}) = \sum_{n} u_n \phi_n(\mathbf{x})$, with $\phi_n$ the shape functions. The nodal displacements $u_n$, which are collected in the column vector $\{\mathbf{u}\}$, are determined by minimizing the L2-norm of the gray level residuals $\Phi_c^2 = \sum_{\text{ROI}} \rho^2(\mathbf{x}, \{\mathbf{u}\})$ over the selected region of interest (ROI)

$$ \{\mathbf{u}_{\text{DVC}}\} = \arg \min_{\{\mathbf{u}\}} \Phi_c^2(\{\mathbf{u}\}). \quad (2) $$

It was shown that the smaller the element size, the higher the displacement uncertainty [24]. If voxel-scale registrations are desired, DVC cannot be run without an appropriate regularization. One option within a finite-element framework is to penalize the previous cost function [30] using the equilibrium gap [31]. Such mechanical penalization restricts the search space for displacement fields to be mechanically admissible over a selected length scale. For internal nodes and traction-free boundaries, the corresponding cost function reads

$$ \Phi_m^2(\{\mathbf{u}\}) = \frac{1}{2} \{\mathbf{u}\}^\top [\mathbf{K}_b]^{-1} [\mathbf{K}_b] \{\mathbf{u}\} \quad (3) $$

where $[\mathbf{K}_b]$ is the stiffness matrix restricted to these nodes.
It is also possible to regularize Dirichlet nodes by adding the following cost function [32]

$$ \Phi_b^2(\{\mathbf{u}\}) = \frac{1}{2} \{\mathbf{u}\}^\top [\mathbf{K}_D]^\top [\mathbf{L}] [\mathbf{K}_D] \{\mathbf{u}\} \quad (4) $$

where $[\mathbf{L}]$ is the Laplace-Beltrami operator, and $[\mathbf{K}_D]$ the stiffness matrix associated with the Dirichlet nodes consisting of top/bottom nodes and interfacial nodes of the mesh introduced in Section 2.4. Minimizing the sum of these three cost functions requires some care. The three cost functions are normalized to be made dimensionless

$$ \tilde{\Phi}_c^2 = \frac{\Phi_c^2}{\{\mathbf{v}\}^\top [\mathbf{H}] \{\mathbf{v}\}} , \quad \tilde{\Phi}_m^2 = \frac{\Phi_m^2(\{\mathbf{u}\})}{\Phi_m^2(\{\mathbf{v}\})} , \quad \tilde{\Phi}_b^2 = \frac{\Phi_b^2(\{\mathbf{u}\})}{\Phi_b^2(\{\mathbf{v}\})} \quad (5) $$

where $[\mathbf{H}]$ is the DVC Hessian matrix and $\{\mathbf{v}\}$ the nodal displacement vector associated with a unitary displacement field [30]. Dimensionless weights are included to define the total cost function to be minimized

$$ \{\mathbf{u}_{R-DVC}\} = \arg\min_{\{\mathbf{u}\}} (\tilde{\Phi}_c^2 + w_m \tilde{\Phi}_m^2 + w_b \tilde{\Phi}_b^2). \quad (6) $$

The weights $w_m$ and $w_b$ scale as the fourth power of regularization lengths $\ell_m$ and $\ell_b$. In this study, the two weights are kept equal $w_m = w_b$. Choosing a regularization length $\ell_{reg}$ greater than the element size $\ell$ induces mechanical admissibility to be active over a domain of size $\ell_{reg}$ [33].

The displacement field is computed using a relaxation process on the regularization lengths. The registration is initially performed with a high weight $w_m$ and $w_b$ (i.e., $\ell_{reg} = 144$ voxels). Once the DVC code has converged (i.e., when the L2-norm of the displacement corrections becomes less than $10^{-3}$ voxel), it is used to initialize the new registration step with a smaller mechanical weight, typically half the previous regularization length. This relaxation process is carried out until the DVC code fails
to converge (with a maximum of 100 iterations). This step-by-step process allows for convergence of weakly regularized computations. Additionally, each result of any relaxation step is stored and discussed later on in Section 3. Uncertainties on the kinematics fields are analyzed with respect to the weight of regularization, as well as the gray-level residuals. The latter ones provide a probe of the mechanical behavior in relation to the regularization method, which is the equilibrium gap in the present case.

Regularization restricts the displacement field to be mechanically admissible usually assuming elastic properties. It is possible to take into account the presence of various phases (e.g., mortar and GFRP) by introducing a heterogeneous regularization based on the contrast C between their Young's moduli [34, 35]

$$C = \frac{E^{\text{GFRP}}}{E^{\text{mortar}}} = \left( \frac{\ell_{\text{reg}}^{\text{GFRP}}}{\ell_{\text{mortar}}^{\text{reg}}} \right)^2 . \qquad (7)$$

In the present case, C = 3 (see Section 2.1).

## 2.4 Segmentation and meshing

The displacement field is calculated using FE-DVC. The mesh is created for scan 0 that was acquired with a 45 µm resolution (Table 1). A sagittal section of the reference scan is displayed in Figure 3(a). The Avizo® data visualization and analysis software is used to perform image segmentation and to generate the mesh. The mesh aims to describe precisely the interface to study the debonding between the reinforcement and the mortar foundation. The segmentation between the reinforcement and mortar foundation cannot be based on thresholding since the gray levels of voxels belonging to GFRP are similar to those of mortar. One alternative is to segment the volume from the image gradient (Figure 3(b)) using a Canny-Deriche filter [36]. The segmentation provides two subvolumes (Figure 3(c)).
<figure><img src="image_4.png" /><figcaption>Fig. 3 (a) Sagittal section of the specimen in the reference configuration. (b) Image gradient obtained from a Canny-Deriche filter. (c) Segmentation between the GFRP reinforcement and the mortar foundation.</figcaption></figure>

A finite element mesh is created for each subvolume using T4 shape functions. The element sizes for both the reinforcement and foundation are designed large enough to ensure low measurement uncertainties. The meshes, whose properties are detailed in Table 2, are shown in Figure 4.

<table><caption>**Table 2** Mesh properties</caption><thead><tr><td>Phase</td><td>Elements</td><td>Nodes</td><td>Element size</td><td>Aspect ratio</td></tr></thead><tbody><tr><td>Mortar foundation</td><td>45,071</td><td>9,477</td><td>18 ± 3 vx</td><td>&lt; 20</td></tr><tr><td>GFRP reinforcement</td><td>4,474</td><td>1,107</td><td>18 ± 3 vx</td><td>&lt; 20</td></tr></tbody></table>

From the meshes of the mortar foundation and the GFRP reinforcement, two discretisations of the specimen are created. The first mesh merges the mortar and GFRP meshes, resulting in a mesh with only single nodes at the interface. The second mesh is built with split nodes to measure interfacial displacement jumps [27].
<figure><img src="image_5.png" /><figcaption>Fig. 4 Meshes of the mortar foundation (a), the GFRP reinforcement (b), and the whole specimen (c).</figcaption></figure>

## 2.5 Debonding quantification with split nodes

Using a mesh with split nodes, debonding is quantified by computing the displacement jump $[\mathbf{u}]$ between mortar and GFRP reinforcement at the interface (Figure 5)

$$[\mathbf{u}] = \mathbf{u}_{\text{mortar}}^{\text{interface}} - \mathbf{u}_{\text{GFRP}}^{\text{interface}}, \qquad (8)$$

where $\mathbf{u}_{\text{GFRP}}^{\text{interface}}$ denotes the displacement in GFRP and $\mathbf{u}_{\text{mortar}}^{\text{interface}}$ in mortar for the considered split node.
<figure><img src="image_6.png" /><figcaption>Fig. 5 Debonding measurement with displacement jump $\llbracket \mathbf{u} \rrbracket$ between split nodes.</figcaption></figure>

The displacement jump is then decomposed at the interfacial nodes into opening $\llbracket u_n \rrbracket$ and tangential slip $\llbracket u_t \rrbracket$

$$ \llbracket \mathbf{u} \rrbracket = \llbracket u_n \rrbracket \mathbf{e}_n + \llbracket u_t \rrbracket \quad (9) $$

where $\mathbf{e}_n$ is the normal to the interfacial surface at the node. Alternatively, it can be analyzed from its three components in cylindrical coordinates

$$ \llbracket \mathbf{u} \rrbracket = \llbracket u_r \rrbracket \mathbf{e}_r + \llbracket u_\theta \rrbracket \mathbf{e}_\theta + \llbracket u_z \rrbracket \mathbf{e}_z. \quad (10) $$

### 3 DVC analyses

The CT scans are registered to measure displacement fields with the DVC parameters listed in Table 3. The choice of the mesh and regularization lengths will be discussed in Section 3.3.
<table><caption>**Table 3** DVC analysis parameters</caption><thead><tr><th>DVC software</th><th>Correli 3.2 [29]</th></tr></thead><tbody><tr><td>Image filtering</td><td>none</td></tr><tr><td>Element length (mean)</td><td>18 vx</td></tr><tr><td>Shape functions</td><td>linear (T4 elements)</td></tr><tr><td>Mesh</td><td>see Table 2 and Figure 4</td></tr><tr><td>Matching criterion</td><td>penalized sum of squared differences</td></tr><tr><td>Interpolant</td><td>linear</td></tr></tbody></table>

## 3.1 Mechanical displacement fields

Rigid body motions are included in the measured displacement fields. The mechanical displacement of the sample $\mathbf{u}_{\text{mech}}$ is obtained from the measured displacement $\mathbf{u}_{\text{DVC}}$ as

$$ \mathbf{u}_{\text{mech}} = \mathbf{u}_{\text{DVC}} - \mathbf{u}_{\text{RBM}}, \qquad (11) $$

where the rigid body motion $\mathbf{u}_{\text{RBM}}$ reads

$$ \mathbf{u}_{\text{RBM}} = \mathbf{t} + \omega \wedge \mathbf{x} \qquad (12) $$

with $\mathbf{t}$ the rigid body translation vector, $\omega$ the rotation vector and $\mathbf{x}$ the vector corresponding to the coordinates of the nodes. In the following, all reported displacements correspond to so-called mechanical fields.

## 3.2 Uncertainty quantification

The measurement uncertainties are quantified by correlating the two scans of the reference configuration (Figure 2) where detector noise and spurious displacements induced by the sample rotation lead to non zero displacements. Since these sources of error are not constant over time, and additional sources may arise, these measurement uncertainties only represent a noise floor level [37].

Uncertainties (i.e., standard deviations $\sigma(\bullet)$ of the quantities of interest $\bullet$) are plotted in Figure 6. In particular, the standard displacement uncertainties in the vertical (pull-out) direction are shown for the mortar in Figure 6(a) and for the GFRP in
Figure 6(b). For the mortar, the displacement uncertainties vary between $10^{-2}$ and $3 \times 10^{-2}$ voxel, which is particularly low compared to the literature [26, 27, 35, 38]. For the GFRP (Figure 6(b)) the displacement uncertainties range from $10^{-2}$ and $2 \times 10^{-1}$ voxel regardless of mesh and type of regularization.

The standard uncertainties of the maximum principal strain $\epsilon_1$ are plotted in Figure 6(c) for the mortar and in Figure 6(d) for the GFRP. For both materials, increasing the regularization length reduces the measurement uncertainties for both $u_z$ and $\epsilon_1$. The mesh used to describe the interface has a negligible influence on the uncertainties for mortar (Figures 6(a) and 6(c)). For the reinforcement, the mesh with an interface described by single nodes reduces both uncertainties (Figures 6(b)) and 6(d)) compared to a mesh with split nodes. Last, heterogeneous regularization leads to a slight decrease of the uncertainties compared to a homogeneous regularization since the regularization is stronger in the reinforcement (see Equation (7)).
<figure><img src="image_7.png" /><figcaption>**Fig. 6** Standard uncertainties of vertical displacement $u_z$ (a-b), and maximum principal strain $\epsilon_1$ (c-d) as functions of the regularization length and different meshes for mortar (a,c) and GFRP (b,d).</figcaption></figure>

When the interface nodes are split, it is possible to quantify the uncertainty on the displacement jumps $[[u_z]]$, $[[u_n]]$ and $[[u_\theta]]$. The standard uncertainties, shown in Figure 7, range between $10^{-2}$ and $10^{-1}$ voxel. This result shows that sub-voxel displacement jumps can be quantified when split nodes are considered.
<figure><img src="image_8.png" /><figcaption>Fig. 7 Standard uncertainties of displacement jump $\|u_z\|$, $\|u_n\|$ and $\|u_\theta\|$ according to the regularization length</figcaption></figure>

## 3.3 Displacement analysis

DVC computes displacement fields between deformed configurations and a reference configuration, that is scan 0 (Figure 2). Figure 8 shows the root mean square (RMS) gray level residual for scans 00 and 3 as functions of the mechanical residual $\Phi_m$ for all regularization lengths used during the relaxation process.

For scan 00 (Figures 8(a) and 8(b)), the regularization length does not impact the RMS gray level residual. Since the repeat and reference scans were acquired in the same mechanical state, only very small motions are to be expected. Therefore, the introduction of an elastic penalty has no influence, except on the mechanical residuals. The gray level residuals are slightly reduced in the reinforcement for $l_{reg} = 9$ voxels ($l_{reg} < l_{elem}$). By comparing the gray level residuals in the reinforcement computed for the repeat scan 00 with split interfacial nodes for a regularization length $l_{reg} = 18$ voxels and $l_{reg} = 9$ voxels, it is observed (Figure 9(b)) that the residuals are lowered in areas with tomographic artifacts, such as the rotation axis and beam hardening due to the change of diameter in the foundation. This result means
that for $l_{reg} = 9$ voxels ($l_{reg} < l_{elem}$) the DVC procedure minimized the residuals but computed spurious displacements.

<figure><img src="image_9.png" /><figcaption>**Fig. 8** Mechanical and gray level residuals as functions of the regularization length for different interface descriptions for scan 00 (a,b) and scan 3 (c,d), for mortar (a,c) and GFRP (b,d).</figcaption></figure>

In the case of mortar for scan 3 (Figure 8(c)), the RMS gray level residual decreases as the regularization length is lowered, meaning that the corresponding displacement field is more consistent according to the DVC cost function. Conversely, the mechanical residual $\Phi_m$ increases as the weight is reduced. The introduction of heterogeneous regularization between mortar and GFRP or the use of split nodes at
the interface has little impact on both residuals for mortar. Such trend is an indication of a model error, namely, the assumption of an elastic behavior was not satisfied everywhere in mortar (i.e., in areas where mortar is cracked as visible in Figure 9(a)).

For scan 3, describing the interface with split nodes reduces the RMS gray level residual in the reinforcement (Figure 8(d)), meaning that allowing for displacement jumps improves the faithfulness of the measured displacement field (i.e., debonding occurred). This reduction and that of $\Phi_m$, for the same regularization length, prove that the segmentation between mortar and reinforcement, as well as subsequent meshing are consistent. Introducing split nodes, the elastic regularization has a very limited impact on the gray level residual in the reinforcement, meaning that material behaviour is elastic.

<figure><img src="image_10.png" /><figcaption>Fig. 9 (a) Sagittal section of the specimen at scan 3 with the crack in the chimney. (b) Difference of gray level residuals in the reinforcement computed for repeat scan 00 with split nodes for a regularization length $l_{reg}$ = 18 and 9 voxels. For the sake of legibility, the external surface of the sample is highlighted in orange.</figcaption></figure>

For mortar, a regularization length equal to the element size ($l_{elem}$ = 18 voxels) allows for the best measurement of the displacement field by reducing the RMS gray
level residual. However, such a small regularization length exacerbates measurement uncertainties for nodal displacements and maximum principal strains, even exceeding the critical extensional strain of mortar that leads to damage. Choosing $l_{reg} = 72$ voxels appears to be a good compromise between measurement uncertainties and mechanical residuals. This choice is not optimal regarding the gray level residuals and will be revisited in Section 4.2.

Sagittal sections of the displacement field computed for scan 3 as well as the maximum principal strain $\epsilon_1$ are displayed in Figure 10. The contour of the reinforcement is indicated by green dots. A crack is observed at the chimney in the mortar (i.e., zone with very high strains levels), which is consistent with the crack observed in the sagittal section of the tomographic scan shown in Figure 9(a). The test setup was designed to avoid any confinement of mortar, which may result from boundary conditions commonly referred to as pull-push in traditional pull-out tests of reinforcements from cementitious materials [4]. It replicates conditions as close as possible to an axially anchored structure in tension. Under these conditions, mortar fails along the axis of the reinforcement, thereby resulting in radial cracks that localize at the diameter transition between the chimney and the first slab.

Due to homogeneous regularization in the case of a mesh with single nodes at the interface, the computed displacement field within the reinforcement also shows a steep gradient at the location of the crack (Figure 10(a)). The high principal strains observed in mortar also impact the strain field within the reinforcement (Figure 10(d)) due to regularization. Using heterogeneous regularization with a contrast $C = 3$ (Figure 10(b)), the weight of the regularization in the GFRP is increased, the vertical displacement between $z = 300$ and 500 voxels in the reinforcement (i.e., at the location of the mortar crack) is altered, but a significant gradient is still observed within the reinforcement. In terms of maximum principal strain (Figure 10(e)), the
heterogeneous regularization has reduced the impact of mortar on the reinforcement displacement field. However, the strain levels are still high within the reinforcement.

<figure><img src="image_11.png" /><figcaption>Fig. 10 Sagittal sections of the displacement field (in voxels) in the vertical direction $u_z$ (a-c) and the maximum principal strain $\epsilon_1$ (d-f) for scan 3 for a mesh with single interfacial nodes and homogeneous regularization (a,d), single interfacial nodes and heterogeneous regularization (b,e), split interfacial nodes (c,f).</figcaption></figure>

The same sections are reported when the nodes are split in Figures 10(c) and 10(f). Introducing split nodes at the interface allows mortar and GFRP displacement fields to be separated in the DVC procedure. Thus, displacement discontinuities appear at the interface. The mortar fields show the presence of a crack, whereas the reinforcement is not cracked. The introduction of split nodes at the interface reduces the gray
level residuals (Figure 8(d)); the displacement field is better captured by allowing for displacement jumps at the interface.

On the sagittal sections of Figures 10(d-f), it is concluded that the high levels of the maximum principal strains $\epsilon_1$ in mortar are due to the crack of the chimney. Regularization smears those strains in the neighboring elements. The measurement of debonding and the influence of regularization will be discussed in the following section.

# 4 Debonding quantification

The present work aims at quantifying debonding between the reinforcement and the cementitious matrix during the pull-out test even when mortar experiences damage. The proposed strategy is exemplified for scan 3.

## 4.1 Debonding measurement

Figure 11 shows displacement jump profiles for three different components along the reinforcement height for scan 3. These profiles are displayed for the different regularization lengths used during the relaxation process. For the sake of readability, a moving vertical average over 50-voxel windows is computed to show the major trends. The measurement uncertainties, which are reported in Section 3.2, are represented by opacity around the curves.

Figure 11(a) shows the displacement jump profiles in the pull-out direction. According to Equation (8), a positive debonding means that $u_{z\text{mortar}}^{\text{interface}}$ is greater than $u_{z\text{GFRP}}^{\text{interface}}$. When $z \in [290; 340]$ voxels, $[u_z]$ varies suddenly because of the mortar cracks visible in Figure 9(a). Below the crack (i.e., $z \in [0; 290]$ voxels), the debonding is negative, indicating that the mortar foundation has moved farther downward than the reinforcement during the pull-out test. Above the crack ($z \in [340; 950]$ voxels), the debonding is positive, meaning the reinforcement has moved further downward because it remains attached to the foundation slabs. As long as the regularization
length remains smaller than the element size, the mortar crack opening is correctly captured by the DVC procedure, otherwise the regularization filters it out.

The normal displacement jump $[u_n]$ represents an interfacial opening if it is positive, and an interface closure if it is negative. Two interface openings occur; one under the reinforcement ($z = 0$ voxel) with a magnitude of 0.8 voxel, and another one at the crack, which ranges between 0.1 and 0.4 voxel depending on the regularization length. In other parts of the interface, no significant interface opening nor closure is measured. In the areas where the normal component of the displacement jump vanishes, the other components represent slip of the reinforcement with respect to the mortar interface. The non-zero orthoradial displacement jump $[u_\theta]$ (Figure 11(c)) is caused by the rotation between the GFRP reinforcement and the foundation because a slight torsion is present during the test. In areas of the mortar crack, the interface relaxes and the jumps $[u_\theta]$ vanish.
<figure><img src="image_12.png" /><figcaption>(a)</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>(b)</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>(c)</figcaption></figure>

<caption>**Fig. 11** Displacement jump profiles along the height of the reinforcement for scan 3 and different regularization lengths. (a) $[[u_z]]$, (b) $[[u_n]]$, and (c) $[[u_\theta]]$. The matrix crack location is highlighted in gray.</caption>

Figure 12 shows the displacement jumps drawn on the reinforcement surface of the GFRP reinforcement with $l_{reg} = 72$ voxels. Debonding is now quantified at the element level over the whole 2D interfacial surface. A significant positive displacement jump $[[u_z]]$ is observed above the matrix crack and negative below the crack, which is located between 290 and 340 voxels (Figure 12(a)). The normal debonding jump $[[u_n]]$ exhibits variations due to the geometry of the crenelations (Figure 12(b)). Because of the crenelated reinforcement, there is no axisymmetry of the normal debonding displacement jump. On the upper parts of the crenelations,
$[u_n]$ approaches zero, indicating no interface opening. Below the crenelations, the pull-out process induces interface openings. Between 290 and 580 voxels, $[u_n]$ is more homogeneous, with positive values both above and below the crenelations. In the same zone, the hoop displacement jump $[u_\theta]$ tends to vanish (Figure 12(c)). In the other parts of the specimen, a hoop displacement jump is observed at the interface.

<figure><img src="image_15.png" /><figcaption>Fig. 12 Displacement jump drawn on the reinforcement surface for $[u_z]$ (a), $[u_n]$ (b) and $[u_\theta]$ (c). The matrix crack location is highlighted in gray.</figcaption></figure>

## 4.2 Effect of mortar damage on debonding measurement

The debonding quantification shown in Figure 11 is affected by the regularization that dampens the displacement gradients (Figure 10(c)) or the maximum principal strains (Figure 10(f)) in mortar. This smeared damaged zone is not representative of the localized crack visible in Figure 9(a).

Heterogeneous regularization was introduced in Section 2.3 to account for the difference in elastic properties of GFRP and mortar. Similarly, damage can be considered [35] to reduce the mechanical regularization weight in fractured zones. The effective Young's modulus $\tilde{E}^{\text{mortar}}$ of damaged elements is defined by [39]

$$ \tilde{E}^{\text{mortar}} = (1 - D)E^{\text{mortar}} \qquad (13) $$
where D denotes the damage variable ranging from 0 (i.e., undamaged state) to 1 (i.e., fully damaged state)

$$1 - D = \frac{\tilde{E}_{\text{mortar}}}{E_{\text{mortar}}} = \left( \frac{\tilde{\ell}_{\text{reg}}^{\text{mortar}}}{\ell_{\text{reg}}^{\text{mortar}}} \right)^2 . \qquad (14)$$

Damaged elements are thus associated with an effective regularization length that tends to 0 when the element is completely damaged

$$\tilde{\ell}_{\text{reg}}^{\text{mortar}} = \ell_{\text{reg}}^{\text{mortar}} \sqrt{1 - D}. \qquad (15)$$

It is worth noting that the approach does not intend to quantify precisely damage [40] as carried out for plaster [28] but to detect damaged elements (i.e., $D \approx 1$) in order to locally reduce the elastic penalization. Damaged elements are identified with two criteria, namely, the maximum principal strain $\epsilon_1$ and the corresponding gray-level residual $\phi_e$ evaluated for each mortar element.

The critical extension strain that leads to damage is based on the tensile failure strain $\epsilon_t$ of mortar

$$\epsilon_t = \frac{f_t}{E_{\text{mortar}}} = 1.75 \times 10^{-4}. \qquad (16)$$

Figure 6(c) shows that $\epsilon_t$ may be less than the strain uncertainties. To detect damaged elements, the regularization length has to be high enough to reduce the uncertainties. In the present case, $\ell_{\text{reg}} = 72$ voxels is a sufficient value to tackle this problem.

The elementary residual $\phi_e$ contains high values in areas where cracks occur. The critical elementary residual that leads to damage is based on elementary residual $\Phi_e^{scan00}$ computed for the repeat scan. Since the repeat scan only enabled for a noise floor quantification of the measurement uncertainty, the value of the selected
coefficient is based on a trial and error approach

$$ \phi_e = 1.7 \times \Phi_e^{scan00}. \qquad (17) $$

Figure 13(a) shows the principal strain $\epsilon_1$ as a function of elementary residual $\phi_e$ for each mortar element. The detected damaged elements are represented by red crosses. Once the damaged elements are detected from the DVC analysis performed with a regularization length of 72 voxels, the global stiffness matrix is recomputed by accounting for the damaged elements. Figure 13(b) shows the new values of the maximum principal strain $\epsilon_1$ and elementary residual $\phi_e$ for each mortar element. The maximum principal strains increase significantly compared to the results of Figure 13(a) because the introduction of heterogeneous regularization allows for the localization of strain in the damaged elements instead of smearing it.

<figure><img src="image_16.png" /><figcaption>Fig. 13 Maximum principal strain $\epsilon_1$ vs. elementary gray level residual $\phi_e$ at scan 3 before (a) and after (b) applying heterogeneous regularization in damaged elements. The red crosses represent the detected damaged elements.</figcaption></figure>

By coupling these two criteria, it is possible to accurately distinguish the damaged elements from the virgin material. The damaged elements shown in
Figure 14(a) correspond well to the DVC residuals of Figure 14(b), while excluding artifacts. The damaged elements are connected to each other without discontinuities and represent a flat volume, which corresponds to the crack between the chimney and the first slab. The detection thresholds associated with $\epsilon_1$ and $\phi_e$ are thus validated.

<figure><img src="image_17.png" /><figcaption>Fig. 14 (a) Damaged elements detected for scan 3 (computed with $\ell_{reg} = 72$ voxels). (b) Thresholded gray level residuals showing the crack in the specimen. The low residual levels have been made transparent to render large values visible, and hence to reveal the presence of the crack. For the sake of legibility, the external surface of the sample is highlighted in orange.</figcaption></figure>

Prescribing a value $D = 1$ would disable the regularization, thereby leading to a divergence of the DVC code. The damage variable $D$ is chosen a priori such that $\tilde{l}_{reg}^{\text{mortar}} = l_{reg}^{\text{mortar}}/4$ for the damaged elements (i.e., a value of $D = 15/16$).

Figure 15(a) shows the RMS gray level residual $\Phi_c$ as a function of the mechanical residual $\Phi_m$, for mortar and GFRP. In the case of mortar, localizing the regularization in the damaged elements reduces the RMS gray level residuals when $\ell_{reg} > \ell_{elem}$. Using a regularization length of 72 voxels coupled with damage is effective because it achieves an RMS gray level residual very close to that with a low regularization weight. Additionally, the mechanical residual is also lower for $\ell_{reg} > 72$ voxels,
indicating that the elastic hypothesis is validated in the undamaged regions of mor-
tar. Last, the residuals have changed little in the reinforcement since no element is
considered damaged.

<figure><img src="image_18.png" /><figcaption>**Fig. 15** Mechanical and RMS gray level residuals for scan 3 using split nodes at the interface as functions of the regularization length for mortar (a) and GFRP (b) with and without damaged regularization.</figcaption></figure>

Figures 16(a) and 16(b) show the vertical displacement and the maximum princi-
pal strain fields, respectively. Compared to Figures 10(c) and 10(f), the displacement
gradient and strain fields are localized in the vicinity of the matrix crack. The lev-
els reached by the maximum principal strains in mortar are significantly higher.
The crack is better captured from a kinematic stand point, thus debonding between
mortar and reinforcement is better captured.
<figure><img src="image_19.png" /><figcaption>**Fig. 16** Sagittal sections for scan 3 of the displacement field in the vertical direction $u_z$ (a) and the maximum principal strain $\epsilon_1$ (b) with heterogeneous regularization and damage in mortar.</figcaption></figure>

Figure 17 shows the displacement jump profiles obtained by modifying locally the regularization length considering damage in mortar. Applying heterogeneous regularization in the detected damaged elements for a value of $l_{\text{reg}} = 72$ voxels effectively captures debonding. The displacement jumps are better quantified than when using $l_{\text{reg}} = 72$ voxels homogeneously and are comparable to the results obtained with $l_{\text{reg}} = 18$ voxels. These last DVC calculations are significantly better considering damage than using $l_{\text{reg}} = 18$ voxels homogeneously in terms of measurement uncertainties for all components of displacement jumps.
<figure><img src="image_20.png" /><figcaption>**Fig. 17** Displacement jump profiles along the height of the reinforcement for scan 3 and homogeneous regularization with $l_{reg}$ = 18 voxels and $l_{reg}$ = 72 voxels, and for heterogeneous regularization applied on damaged mortar. (a) $\llbracket u_z \rrbracket$, (b) $\llbracket u_n \rrbracket$, and (c) $\llbracket u_\theta \rrbracket$. The matrix crack location is highlighted in gray.</figcaption></figure>

Different models were proposed to describe the interfacial behavior in pull-out tests on cementitious materials [41, 42]. One of the simplest models consists in assuming a constant shear strength of the interface [43, 44]. Such model leads to a parabolic decay of the displacement jump away from the matrix crack along the loading direction, which is not consistent with the present observations (Figure 17) when $z \ge 400$ voxels. When $z \le 350$ voxels, there is a diameter variation that explains the displacement jump variation that does not decrease as expected from classical
derivations assuming a constant external diameter [41–44]. More advanced interface
models may thus be considered, which is beyond the scope of the present paper.

Figure 18 shows the nodal displacement jumps drawn on the external surface of the GFRP reinforcement when $\ell_{\text{reg}} = 72$ voxels considering regularization with damage. The displacement jumps $[u_z]$ have high amplitudes near to the crack (Figure 18(a)). They are more localized and not smeared as in Figure 12(a). There is no homogeneous distribution of the normal jumps $[u_n]$ around the cracks ($z \in [290; 340]$), the influence of the crenelations is clearly visible over the whole reinforcement (Figure 18(b)). The quantified hoop displacement jumps also appear more physical (Figure 18(c)), with significant jumps below the crack, and an abrupt zero jump at the level of the crack.

<figure><img src="image_21.png" /><figcaption>Fig. 18 Displacement jump drawn on the reinforcement surface for $[u_z]$ (a), $[u_n]$ (b) and $[u_\theta]$ (c) with heterogeneous regularization applied in damaged mortar. The matrix crack location is highlighted in gray.</figcaption></figure>

Table 4 gathers the percentage of split nodes with significant vertical, normal
hoop displacement jumps (i.e., greater than the debonding threshold defined as three
times the standard uncertainty), the amplitude of $[u_z]$ about the matrix crack, the
maximum amplitude of $[u_z]$, and the interface opening $[u_n]$ in the matrix crack,
computed in the middle of the damaged zone shown in Figure 17(b). The values
are derived from a moving average. The maximum amplitude of $[u_z]$ is computed
by subtracting the maximum and minimum levels of $\llbracket u_z \rrbracket$ along the reinforcement (Figure 17(a)). The amplitude of $\llbracket u_z \rrbracket$ about the matrix crack corresponds to the difference in displacement jumps $\llbracket u_z \rrbracket$ above and below the cracked zone (Figure 17(a)).

<table><caption>**Table 4** Influence of the regularization strategy on debonding quantification with split-node meshes using displacement jumps assessed with a moving vertical average over 50-voxel windows.</caption><thead><tr><th rowspan="2"></th><th colspan="1">ℓ<sub>reg</sub> = 18 voxels</th><th colspan="1">ℓ<sub>reg</sub> = 72 voxels</th></tr><tr><th>no damage</th><th>no damage</th><th>with damage</th></tr></thead><tbody><tr><td>Debonded nodes [u<sub>z</sub>] (%)</td><td>68</td><td>84</td><td>85</td></tr><tr><td>Debonded nodes [u<sub>n</sub>] (%)</td><td>44</td><td>68</td><td>58</td></tr><tr><td>Debonded nodes [u<sub>θ</sub>] (%)</td><td>85</td><td>93</td><td>90</td></tr><tr><td>Maximum amplitude of [u<sub>z</sub>] (voxels)</td><td>2.26 ± 0.09</td><td>1.89 ± 0.04</td><td>2.27 ± 0.04</td></tr><tr><td>[u<sub>z</sub>] about the matrix crack (voxels)</td><td>1.28 ± 0.09</td><td>0.53 ± 0.04</td><td>1.35 ± 0.04</td></tr><tr><td>[u<sub>n</sub>] in the matrix crack (voxels)</td><td>0.31 ± 0.05</td><td>0.29 ± 0.02</td><td>0.28 ± 0.02</td></tr></tbody></table>

Introducing a heterogeneous regularization within damaged elements does not affect the quantity of nodes that underwent vertical debonding. However, it significantly decreases the number of nodes exhibiting interface opening and slightly reduces those with hoop displacements. A regularization length of $ℓ_{reg} = 18$ voxels results in fewer detected nodes, which is attributed to the higher standard uncertainty (Figure 7).

The maximum amplitude of $\llbracket u_z \rrbracket$ has very similar levels when the regularization length is set to $ℓ_{reg} = 18$ voxels and when regularization is reduced in damaged elements. However it is reduced by 16% when a homogeneous regularization with $ℓ_{reg} = 72$ voxels is used. In terms of uncertainties, as expected from Figure 7, the uncertainties are lower when the regularization length is equal to 72 voxels. For the amplitude of $\llbracket u_z \rrbracket$ about the matrix crack, similar trends are observed. Notably, the value is significantly reduced with $ℓ_{reg} = 72$ voxels, showing a 60% decrease compared to the calculation that includes damage. The interface opening (Figure 17(b) is equal to ca. 0.30 voxel. The quantification is very close for the three DVC calculations
but the corresponding uncertainties are significantly lower when $\ell_{reg} = 72$ voxels.
For the last three discussed quantities, the best results are obtained when $\ell_{reg} =
72$ voxels and when matrix damage is accounted for.

**5 Conclusion**

This work aimed to measure debonding with sub-voxel resolution at the interface
between a reinforcement and mortar with FE-based DVC using mechanical reg-
ularization. An in-situ pull-out test was carried out in an X-ray tomograph on a
GFRP reinforcement embedded in a small-scale mortar foundation. A 3D image-
based mesh was created to consistently reproduce the reinforcement and foundation
geometries. One mesh described the interface with single nodes; another mesh
accounted for the interface with split nodes. The kinematics of the mortar are not
affected by the type of mesh used. However, describing the interface with single
nodes did not properly describe the kinematics at the reinforcement-foundation
interface even introducing elastic contrast instead of homogeneous regularization.
Splitting nodes at the interface enabled the DVC procedure to reduce the gray level
residuals in the reinforcement as well as the mechanical residual introduced by
elastic regularization.

Debonding was measured by distinguishing the displacement of the GFRP inter-
face from that of the mortar interface (i.e., by a displacement jump). The jumps
were analyzed according to three components, namely, vertical, normal and orthora-
dial debonding of the considered interfacial nodes. Mechanical regularization with
a homogeneous contribution within the mortar had the effect of overestimating
displacements at the interface by diffusing large displacements of nodes belong-
ing to damaged elements in mortar. To address this issue, damaged elements were
detected from the elementary maximum principal strains and gray-level residuals.
The regularization weight was then locally reduced to measure faithful displacement
fields and debonding in the fractured zones. The number of nodes that underwent
debonding was estimated. The maximum amplitude of the vertical displacement and the interface opening in the matrix crack were quantified with an uncertainty of approximately $10^{-2}$ voxel, which was less than one micrometer (for a resolution of 45 µm).

Node splitting coupled with heterogeneous regularization in damaged elements proved effective in capturing the kinematics of both materials during a pull-out test involving matrix cracking and debonding. In further studies, the test described herein may also be used to calibrate numerical models accounting for debonding and damage in reinforced concrete.

**Funding.** This research was partly funded by the French National Research Agency (ANR) under the "HEAT COFFEE" project (grant "ANR-22-CHIN-0007-01") and the French Transmission System Operator (RTE).

**Author contribution.** **Sylvain Langlois:** Conceptualization, Data curation, Formal analysis, Investigation, Methodology, Resources, Software, Validation, Visualization, Writing - original draft, Writing - review & editing. **Farid Benboudjema:** Conceptualization, Formal analysis, Funding acquisition, Project administration, Supervision, Validation, Writing - review & editing. **Maroua Maaroufi:** Supervision, Writing - review & editing. **Fikri Hafid:** Funding acquisition, Project administration. **Benjamin Smaniotto:** Data curation, Investigation, Resources, Software. **François Hild:** Conceptualization, Formal analysis, Methodology, Resources, Software, Validation, Visualization, Writing - review & editing. **Amélie Fau:** Conceptualization, Formal analysis, Methodology, Software, Supervision, Visualization, Writing - review & editing.

**Declarations**

**Conflict of Interest.** The authors have no competing interests to declare that are relevant to the content of this article.
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# Appendix A: Contrast in reinforcement

DVC relies on contrast of different voxels. This contrast is represented by gradients of gray levels in images [25]. To enhance the sensitivity to small displacement amplitudes, high contrast is desirable. Therefore, the choice of the reinforcement material is a limiting factor in the method developed herein because of its effect on the quality of CT scans for DVC purposes.

Three types of reinforcement were investigated to study their influence in CT scans, namely, steel, inconel 718 alloy and glass fiber reinforced polymer (GFRP). Steel and GFRP were obtained from a 9 mm reinforcement bar for concrete. Inconel 718 was additively manufactured by laser powder bed fusion. The fabrication parameters were degraded on purpose by increasing the thickness of the molten layer from 40 µm to 80 µm inducing porosities in the material. Those heterogeneities created a natural contrast in the images. The inconel alloy cylinder was then machined to have an outer diameter of 9 mm, incorporating crenelations.

Sagittal sections of the specimens with the three different reinforcement are shown in Figure 19. Due to its chemical composition, steel absorbs more X-rays than mortar and results in a reduced dynamic range (Figure 19(a)). The inconel alloy also absorbs more X-rays than mortar (Figure 19(b)), unlike the GFRP reinforcement (Figure 19(c)).
<figure><img src="image_22.png" /><figcaption>Fig. 19 Sagittal sections of the specimens with steel (a), inconel 718 (b) and GFRP (c). The red line corresponds to the location where gray level profiles are analyzed.</figcaption></figure>

To further investigate this choice, gray level profiles are plotted along the red line drawn in the sagittal sections (Figure 19) and displayed in Figure 20. The profile of the sample containing the steel reinforcement (Figure 20(a)) reveals inadequate contrast. Steel exhibits gray levels ranging from 200 to 255, with variations also attributable to image noise and beam hardening, whereas mortar operates within the range of 30 to 60 gray levels. Inconel 718 displays a contrast resulting from manufacturing-induced heterogeneities, visible in Figure 20(b). Strong beam hardening is observed and the gray levels are not covering the broad dynamic range of the available 8-bit depth. With the GFRP reinforcement, the gray level dynamic range improves (Figure 20(c)). This improvement provides greater contrast in mortar, which now ranges between 80 and 255 gray levels, while GFRP levels vary between 130 and 190.
<figure><img src="image_23.png" /><figcaption>**Fig. 20** Gray level profiles quantified along the straight line through the sagittal sections (Figure 19) of steel (a), inconel 718 (b) and GFRP (c) reinforcements.</figcaption></figure>

To enhance the sensitivity to small displacement amplitudes in the DVC pro-
cedure, strong contrast is desirable. Moreover, debonding occurs at the interface
between the reinforcement and mortar. It is thus undesirable to have tomographic
artifacts in these areas. These criteria are significantly degraded when steel and
inconel 718 are used. The gray level dynamic range is poor, the contrast is nonexis-
tent in the case of steel, severe beam hardening is present, and scattering appears in
the crenelations. Consequently, displacement measurements with steel or inconel 718
reinforcements will mostly rely on mechanical regularization, which is to be avoided
if possible.