Fabien Bernachy-Barbe

Benoît Bary

Sirine Al Dandachli

Christophe Bourcier

Benjamin Smaniotto

François Hild
email: francois.hild@ens-paris-saclay.fr

A Tsitova

Damage Quantification via Digital Volume Correlation with Heterogeneous Mechanical Regularization: Application to an In Situ Meso-Flexural Test on Mortar Aliaksandra Tsitova •

Keywords: Digital Volume Correlation (DVC), X-Ray tomography, Three-point flexural test, Microscale, Mechanical regularization

constituents at the mesoscale. Objective. In this study, a finite element based Digital Volume Correlation (FE-DVC) method with mechanical regularization tailored for heterogeneous materials was applied to study crack propagation in mortar.

Method. A realistic 3D mesh was built consistently with the actual microstructure based on high resolution X-ray tomography. A notched mesobeam was subjected to in situ three-point flexure. Results. The introduction of mechanical contrast in DVC analyses allowed the strain heterogeneities related to the underlying microstructure to be described and inelastic strain localization due to cracks to be captured. Furthermore, the relaxation of mechanical regularization in damaged elements improved the trustworthiness of displacement measurements. Conclusion. Estimations of crack opening displacements and characterization of the crack morphology were carried out, providing insights into the damage processes at the mesoscale.

Introduction

X-ray Computed Tomography (XCT) has become a powerful tool for the acquisition of 3D images and is widely used in material science [START_REF] Maire | Quantitative x-ray tomography[END_REF]. Static 3D images are being utilized for morphological and quantitative microstructure characterization that can be considered for image-based numerical modeling [START_REF] Trawiński | A three-dimensional meso-scale modelling of concrete fracture, based on cohesive elements and x-ray µCT images[END_REF][START_REF] Yang | X-ray computed tomography images based phase-field modeling of mesoscopic failure in concrete[END_REF]. Tomographic studies have found numerous applications for cementitious materials due to their high heterogeneity and complex compositions. One example among various applications is the microstructure characterization based on X-ray imaging. Studies were carried out on aggregate and cement particle shapes [START_REF] Erdogan | Three-dimensional shape analysis of coarse aggregates: New techniques for and preliminary results on several different coarse aggregates and reference rocks[END_REF][START_REF] Bentz | X-ray microtomography of an astm c109 mortar exposed to sulfate attack[END_REF][START_REF] Garboczi | Shape analysis of a reference cement[END_REF][START_REF] Garboczi | Three-dimensional mathematical analysis of particle shape using x-ray tomography and spherical harmonics: Application to aggregates used in concrete[END_REF][START_REF] He | Strategy on simulation of arbitrary-shaped cement grains in concrete[END_REF][START_REF] Roubin | The colours of concrete as seen by x-rays and neutrons[END_REF], pore and crack networks [START_REF] Bentz | X-ray microtomography of an astm c109 mortar exposed to sulfate attack[END_REF][START_REF] Bentz | Microstructure and transport properties of porous building materials. II: Three-dimensional x-ray tomographic studies[END_REF][START_REF] Xu | Microstructure and water absorption of ancient concrete from pompeii: An integrated synchrotron microtomography and neutron radiography characterization[END_REF][START_REF] Gallucci | 3d experimental investigation of the microstructure of cement pastes using synchrotron x-ray microtomography (µCT)[END_REF][START_REF] Promentilla | Quantification of tortuosity in hardened cement pastes using synchrotron-based x-ray computed microtomography[END_REF][START_REF] Provis | X-ray microtomography shows pore structure and tortuosity in alkali-activated binders[END_REF][START_REF] Lu | X-ray microtomographic studies of pore structure and permeability in portland cement concrete[END_REF], and cement paste morphology [START_REF] Bentz | Three-dimensional computer simulation of portland cement hydration and microstructure development[END_REF][START_REF] Gallucci | 3d experimental investigation of the microstructure of cement pastes using synchrotron x-ray microtomography (µCT)[END_REF]. Such studies allowed for example estimations of concrete permeability and hydration processes. Moreover, X-Ray based microstructures were used for mechanical and thermal property estimations [START_REF] Bouvard | Characterization and simulation of microstructure and properties of EPS lightweight concrete[END_REF], as input to finite element (FE) models [START_REF] Stamati | Tensile failure of micro-concrete: from mechanical tests to FE meso-model with the help of x-ray tomography[END_REF][START_REF] Nguyen | Initiation and propagation of complex 3d networks of cracks in heterogeneous quasi-brittle materials: direct comparison between in situ testing-microCT experiments and phase field simulations[END_REF][START_REF] Ren | Three-dimensional in situ XCT characterisation and FE modelling of cracking in concrete[END_REF] and FFT-based simulations [START_REF] Bernachy-Barbe | Effect of aggregate shapes on local fields in 3d mesoscale simulations of the concrete creep behavior[END_REF].

Beyond static imaging, the continuation of these studies was the observation of microstructural changes due to exposure to aggressive environments or thermohydro-mechanical loadings. Taking sequences of 3D images under different environmental conditions allowed the microstructure changes to be tracked. Tomographic studies were reported on cementitious materials subjected to leaching [START_REF] Burlion | X-ray microtomography: Application to microstructure analysis of a cementitious material during leaching process[END_REF][START_REF] Rougelot | About microcracking due to leaching in cementitious composites: X-ray microtomography description and numerical approach[END_REF], sulfate attacks [START_REF] Stock | X-ray microtomography (mi-croCT) of the progression of sulfate attack of cement paste[END_REF], corrosion [START_REF] Bernachy-Barbe | Using x-ray microtomography to study the initiation of chloride-induced reinforcement corrosion in cracked concrete[END_REF], and water migration [START_REF] Powierza | Water migration in one-side heated concrete: 4d in-situ CT monitoring of the moisture-clog-effect[END_REF]. One of the extensions to this technique is in situ testing when experiments are carried out inside CT scanners [START_REF] Buffiere | In situ experiments with x ray tomography: an attractive tool for experimental mechanics[END_REF]. Registering 3D scans allows different deformation mechanisms such as damage growth, crack opening, or shear banding to be quantified as a function of time or mechanical loading. In particular, it allowed complex damage processes to be visualized in concrete. The characterization of damage growth with X-CT scanning was carried out under mechanical [START_REF] Ren | Three-dimensional in situ XCT characterisation and FE modelling of cracking in concrete[END_REF][START_REF] Landis | Microstructure and fracture in three dimensions[END_REF][START_REF] Landis | X-ray tomography as a tool for micromechanical investigations of cement and mortar[END_REF][START_REF] Elaqra | Damage evolution analysis in mortar, during compressive loading using acoustic emission and x-ray tomography: Effects of the sand/cement ratio[END_REF][START_REF] Wong | Estimation of air void and aggregate spatial distributions in concrete under uniaxial compression using computer tomography scanning[END_REF] and hydrostatic in situ loadings [START_REF] Poinard | Compression triaxial behavior of concrete: the role of the mesostructure by analysis of x-ray tomographic images[END_REF], subjected to freezing-thawing cycles [START_REF] Wakimoto | Digital laminography assessment of the damage in concrete exposed to freezing temperatures[END_REF] or autogeneous shrinkage [START_REF] Mac | Analysis of autogenous shrinkage-induced microcracks in concrete from 3d images[END_REF].

Apart from image analyses and processing, Digital Volume Correlation (DVC) became a popular technique to measure three-dimensional displacement and strain fields with subvoxel resolution [START_REF] Bay | Methods and applications of digital volume correlation[END_REF][START_REF] Buljac | Digital volume correlation: Review of progress and challenges[END_REF]. DVC appeared as an extension of Digital Image Correlation (DIC) that was developed for 2D images [START_REF] Bay | Digital volume correlation: Threedimensional strain mapping using x-ray tomography[END_REF]. The principle of these methods first consisted in measuring displacements of small subvolumes registered between reference and deformed volumes. Later on, finite element ap-proaches were introduced [START_REF] Roux | Three dimensional image correlation from x-ray computed tomography of solid foam[END_REF]; the registrations are performed at the level of the whole region of interest. DVC analyses were performed for concrete subjected to in situ mechanical loadings [START_REF] Mostafavi | Three-dimensional crack observation, quantification and simulation in a quasi-brittle material[END_REF][START_REF] Mao | 3d strain evolution in concrete using in situ x-ray computed tomography testing and digital volumetric speckle photography[END_REF][START_REF] Ducoulombier | X-ray tomographic observations of microcracking patterns in fibre-reinforced mortar during tension stiffening tests[END_REF][START_REF] Hong | Interior fracture analysis of rubber-cement composites based on x-ray computed tomography and digital volume correlation[END_REF][START_REF] Lorenzoni | Combined mechanical and 3d-microstructural analysis of strain-hardening cement-based composites (SHCC) by in-situ x-ray microtomography[END_REF], hydrothermal loadings [START_REF] Bennai | Microscopic hydric characterization of hemp concrete by x-ray microtomography and digital volume correlation[END_REF], carbonation [START_REF] Chavez Panduro | Real time 3d observations of portland cement carbonation at CO2 storage conditions[END_REF] and reinforcement corrosion [START_REF] Jiang | Simulation and experimental verification of the non-uniform corrosion cracking process of reinforced mortar specimen[END_REF]. The study of damage development was possible by analyzing displacement or strain fields as they normally indicate crack locations [START_REF] Hild | Damage measurements via DIC[END_REF] respectively as discontinuities or high values. Another way of displaying crack networks was via residual fields, which are the difference images between the reference volume and the deformed volume corrected with the measured displacement field [START_REF] Hild | Three-dimensional analysis of a tensile test on a propellant with digital volume correlation[END_REF][START_REF] Hild | On the use of 3d images and 3d displacement measurements for the analysis of damage mechanisms in concrete-like materials[END_REF][START_REF] Hild | Damage measurements via DIC[END_REF][START_REF] Chateau | DVC-based image subtraction to detect microcracking in lightweight concrete[END_REF].

In the present study, an FE-DVC method with mechanical regularization tailored for heterogeneous materials was used to quantify damage in mortar. Mechanical regularization was initially introduced in a homogeneous setting for DVC analyses [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF][START_REF] Taillandier-Thomas | Localized strain field measurement on laminography data with mechanical regularization[END_REF]. Then, it was adapted to heterogeneous materials in a DIC framework [START_REF] Naylor | Mechanically regularized FE DIC for heterogeneous materials[END_REF] by taking into account the contrast in elastic properties of different phases. Damage was also added to allow localized solutions to be retrieved [START_REF] Hild | Damage measurements via DIC[END_REF]. In the present case, a mesh was built consistently with the underlying microstructure, based on an X-ray microtomography scan acquired prior to the in situ test. Then, a notched microbeam was subjected to in situ three-point flexure. In order to image a larger part of the beam, a lower resolution was selected. The realistic volume mesh was backtracked via DVC from the high resolution scan to the in situ scan.

The influence of mechanical contrast on the trustworthiness of 3D displacement and strain fields was studied. Furthermore, the damaged elements were detected in order to take into account their reduced stiffness in mechanical regularization. This technique allowed the displacement measurements to be consistent in the damaged areas. The final DVC results were used for crack opening estimations and a study of the crack morphology at the mesoscale.

The study workflow (Figure 1) summarizes the different steps of the present study, whose aim is to demonstrate the potential of mechanically regularized DVC based on the underlying microstructure for the study of damage at low scales. This section describes the specimen fabrication process and provides a description of the microstructure characterization procedure that was used to reconstruct a realistic 3D mesh. The experimental setup and protocol are then detailed. The section ends with a description of heterogeneous regularization within an FE-DVC framework.

Material and Specimen

A mortar batch was mixed using a CEM I Portland cement with a water-to-cement ratio of 0.525. A low sand-to-binder ratio was used (0.5). Sand was sieved to obtain a granulometric distribution within the range 200 µm to 2 mm in diameter. The reduction of the sand volume fraction to 18%, and elimination of fine particles, made the resulting microstructure more suitable for efficient image processing as less geometric details were distorted or lost. A thin mortar plate with dimensions 65 × 35 × 8 mm was cast in a silicon mold and unmolded after 1 day. By means of X-ray radiography, the mortar plate was checked for the absence of mesocracking and defects. After control, the plate was selected for further fabrication.

First, the top surface was ground and polished until a 5 mm thickness was reached with simultaneous control of parallelism of the top and bottom faces.

The thickness deviation was of the order 0.1 mm, which corresponds to a ∠1.5 inclination between the top and bottom surfaces. Second, the plate width was adjusted to 20 mm with a wire saw. Third, 5 × 5 × 20 mm microbeams (Figure 2) were cut from the plate with a diamond disk saw to ensure planar parallel sides.

Fourth, the notch was cut with a saw equipped with a fine (0.1 mm) wire and low tension to avoid sample degradation. The notch width was approximately 0.15 mm and its height was 1.5 mm. The specimen were stored at 100% air humidity. During preparation, the specimen surfaces were re-wetted to prevent them from drying cracking. Before the in situ test, a High Quality (HQ) scan of the central part of the microbeam was acquired with a 5.3 µm resolution using a GE Phoenix v|tome|x m CT system. The acquisition parameters are detailed in Table 1. containing the notch with dimensions 2.6 × 5 × 5 mm was selected for further segmentation (Figure 6). First, a median filter (3 × 3 × 3 voxels) was applied to mitigate noise (Figure 3(a)). Second, the segmentation of aggregates was carried out as follows. Two types of inclusions (limestone -CaCO 3 , silica -SiO 2 ) were present in mortar with different morphology and levels of attenuation that define their representation on images. Limestone inclusions were brighter (higher X-ray absorption) than the cementitious matrix, and were separated by simple thresholding (Figure 3(b)).

Silica inclusions had similar gray levels as the cement paste matrix, but showed less internal contrast and appeared as very homogeneous regions. The calculation of the image gradient norm gave a field with low values in homogeneous zones, and high levels in heterogeneous regions (Figure 3(c)). This tool allowed the homogeneous inclusions to be distinguished from the heterogeneous matrix (Figure 3(d)). These two families of inclusions were considered as a unique "aggregate" phase with identical elastic properties. Mesoporosity was present in the specimen in a small amount, and did not seem to have a major effect on the crack path during the flexural test. For these reasons, mesopores were not segmented as a separate phase and were incorporated in the matrix. Last, the notch surfaces were segmented from the volume (Figure 4(a)).

Based on the segmentation results, the Avizo tools allowed surface triangular meshes to be generated and exported in .stl format. The size of surface triangles was controlled to be at least several voxels for consistency with DVC, and with simultaneous check of excessive distortion of triangles. The surface meshes were then processed with the tools available in the Salome platform (https://www.salomeplatform.org/) to obtain 4-noded tetrahedral elements (Figure 4(b,c)). The mesh consisted of 346,472 tetrahedra and 63,465 nodes. The distribution of mesh elements and volume between the different phases is given in Table 2. In the matrix, elements adjacent to aggregates are denoted as interphase elements, and the remainder of the matrix are bulk elements. The equivalent element size, which is defined as the cubic root of the volume of each element, was equal on average to 10 vx (or 53 µm) in the HQ configuration and 6 vx in the LQ setting (Table 2). The mesh fineness was identical in the matrix and the aggregates. The in situ test consisted of several loading steps, and CT scans were performed at sustained and constant load. The experimental protocol and load levels of CT scans are shown in Figure 6(a). The scans were acquired with the Phoenix v|tome|x m scanner (Table 1). First, two successive scans were performed on the undeformed specimen for uncertainty quantification. A small preload of 1 N was applied to keep the specimen stable. Second, axial displacements were applied with 0.5 µm/s rate until sample failure. The experimental configuration along with the specimen geometry did not allow for a fine control of crack initiation and propa-gation. Failure was brittle, followed by substantial force drop, and very fast crack propagation on the major part of the ligament (Figure 6(b)). The second scan was carried out immediately after fracture. Third, an increment of axial displacement of 20 textmum was applied to further open the crack. After each loading, a 10 min dwell was applied before the next scan (i.e., force relaxation was stabilized).

In the reported test, the loading point was not fully aligned with the notch.

This off-axis loading resulted in a bifurcation of the fractured surface into two main cracks (Figure 6(b)). The Region of Interest (ROI) was chosen to encompass all fractured surfaces while maintaining a small gap with an irregular external surface. They were observed to cross the cement paste matrix, aggregate-matrix interfaces and limestone aggregates.

f (x) = g(x + u(x)) (1)
where u(x) is the sought displacement field, and x any voxel position within the ROI. The sought displacement field has to minimize the gap to gray level conser-

vation min {u} Φ 2 c = ROI [f (x) -g(x + u(x))] 2 (2)
with

u(x) = n unφ φ φn(x) (3)
where φ φ φn(x) are trial displacement fields, and un the associated Degrees Of Freedom (DOFs) gathered in the column vector {u}. In FE-DVC, the sought DOFs are the nodal displacements of a finite element discretization, and φ φ φn(x) the corresponding shape functions.

The minimization of Equation ( 2) is equivalent to iteratively solving a sequence of linear systems [START_REF] Hild | Digital image correlation[END_REF] [M]{δu} = {b} [START_REF] Erdogan | Three-dimensional shape analysis of coarse aggregates: New techniques for and preliminary results on several different coarse aggregates and reference rocks[END_REF] where [M] is the DVC Hessian matrix, whose terms are products of the shape functions by the gradient of the reference volume, {b} the residual vector, which is a function of the difference between the reference and the corrected deformed volumes, and {δu} the corrections to the current estimate of the nodal displacement vector {ũ}.

The gray level residual field ϕc = f (x) -g(x + u(x)) represents the difference between the volume in the reference configuration and that in the deformed configuration corrected by the measured displacement field. The global residual Φc, which is to be minimized, is the root mean square (RMS) of the residual field ϕc and serves as indicator of the consistency of the solution, namely, a smaller RMS residual means a more trustworthy solution. Local high residual values may mark, for example, discontinuities in displacements, or in other words, cracks [START_REF] Hild | Three-dimensional analysis of a tensile test on a propellant with digital volume correlation[END_REF][START_REF] Hild | On the use of 3d images and 3d displacement measurements for the analysis of damage mechanisms in concrete-like materials[END_REF][START_REF] Hild | Damage measurements via DIC[END_REF]. Last, it is worth noting that the residual fields are, by definition, Lagrangian (i.e., constructed in the reference configuration).

The correlation problem is ill-posed and the introduction of mechanical regularization is one of the ways to make it well-posed [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF][START_REF] Leclerc | Digital volume correlation: What are the limits to the spatial resolution[END_REF][START_REF] Taillandier-Thomas | Localized strain field measurement on laminography data with mechanical regularization[END_REF]. In regularized DVC, elasticity is enforced at a local level by the introduction of a penalty term based on the equilibrium gap

[K 1 ]{u} = {fr} (5)
where [K 1 ] is the rectangular stiffness matrix restricted to inner nodes and tractionfree surfaces, and {fr} the nodal force vector that must vanish in the absence of body forces. The mechanical penalty term is formulated as the L2-norm of the force residuals

Φ 2 m = {u} [K 1 ] [K 1 ]{u} (6)
For other boundary (i.e., Dirichlet) nodes, the fluctuations of tractions are regularized

Φ 2 b = {u} [K 2 ] [L][K 2 ]{u} (7)
where [L] is based upon the Laplace-Beltrami operator [START_REF] Mendoza | Complete mechanical regularization applied to digital image and volume correlation[END_REF], and [K 2 ] the rectangular stiffness matrix restricted to Dirichlet nodes. These additional cost functions do not have the same physical units as Φ 2 c . Therefore, to minimize their weighted sum, they are normalized

Φ2 m = Φ 2 m {v} [K] [K]{v} , Φ2 b = Φ 2 b {v} [L] T [L]{v} , Φ2 c = Φ 2 c {v} [M]{v} (8)
where v = v 0 exp(ik•x) is the trial displacement field selected in the form of a plane wave, v 0 the amplitude, and k the wave vector [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF][START_REF] Leclerc | Digital volume correlation: What are the limits to the spatial resolution[END_REF]. The global minimization is performed on the weighted sum of the normalized cost functions

(1 + wm + w b )Φ 2 t = Φ2 c + wm Φ2 m + w b Φ2 b ( 9
)
where Φ t is the total cost function. The weights wm and w b are defined as

w = wm = w b = (2πk reg ) 4 ( 10
)
where reg is the regularization length, and k = k the wave number. Up to now, mechanical regularization was based upon homogeneous, isotropic and linear elasticity [START_REF] Leclerc | Voxel-scale digital volume correlation[END_REF][START_REF] Taillandier-Thomas | Localized strain field measurement on laminography data with mechanical regularization[END_REF]. If reg is greater than the element size, mechanically inadmissible displacement fluctuations are filtered out over a spatial domain of size proportional to reg . In the case of damage or localized plastic strains, displacement discontinuities are smeared over a domain whose size that depends on reg [START_REF] Taillandier-Thomas | Localized strain field measurement on laminography data with mechanical regularization[END_REF][START_REF] Hild | Damage measurements via DIC[END_REF].

As the microstructure-based mesh differentiates matrix and aggregate elements (Figure 4(c)), the proposed regularization can take into account differences in elastic properties, as already performed in 2D DIC applied to a composite material [START_REF] Naylor | Mechanically regularized FE DIC for heterogeneous materials[END_REF]. For a two-phase material, the contrast C is introduced to denote the ratio of Young's moduli of the matrix Em and the inclusions

E i C = E i Em = w i w m = i reg m reg 2 (11)
where w i and w m denote the weight in the inclusions and in the matrix, respectively. In such formulation, C > 1 implies that inclusions are more rigid than the matrix, and more weight is put on the inclusion phase. Conversely, C < 1 corresponds to inclusions being more compliant and a larger weight is put on the matrix phase. The case C = 1 is equivalent to homogeneous elastic regularization.

Mechanically regularized DVC was implemented within the Correli 3.0 framework developed at LMT [START_REF] Leclerc | Correli 3.0. Agence pour la Protection des Programmes[END_REF]. Table 3 gathers the DVC parameters utilized in the following analyses. For the sake of simplicity, reg also denotes the regularization length in the matrix phase (i.e., m reg ), if heterogeneous regularization is performed since in the present case, C < 1 is very unlikely.

DVC Results and Discussion

This section describes the DVC analysis workflow that begins with the mesh backtracking procedure [START_REF] Auger | Poynting effects in pantographic metamaterial captured via multiscale DVC[END_REF]. Then, a quantification of uncertainties for different values of regularization length and contrasts was performed. Last, DVC analyses for different contrasts were carried out on the deformed (and damaged) volume.

Mesh Backtracking

The microstructure-based mesh was reconstructed on the HQ scan with a 5.3 µm/vx resolution acquired prior to the in situ test (see blue mesh in Figure 7). As mentioned above, the HQ and in situ (LQ) scans were acquired with different resolutions and acquisition parameters (Table 1). Before performing DVC analyses, the mesh had to be backtracked to take into account scale differences and rigid body motions between the two scans (i.e., initially mispositioned blue mesh in the LQ configuration in Figure 7(left)). This step was important as if the mesh did not match accurately the underlying microstructure, heterogeneous regularization would be meaningless.

Backtracking consists in running a DVC analysis between the HQ scan (volume f in Equation ( 1)) and the in situ scan (volume g) of the undeformed specimen to find the displacement field u b linking the two configurations [START_REF] Auger | Poynting effects in pantographic metamaterial captured via multiscale DVC[END_REF]. In the present case, the HQ scan was binned to a resolution of 10.6 µm/vx, and both scans were converted into 8-bit volumes. An auxiliary mesh was used, consisting of a rectangular hexahedral mesh composed of large T4 elements (their size was 25 vx or 250 µm in the reference HQ configuration) that encompassed the volume of the microstructure-based mesh (i.e., 4.5 × 4.75 × 2.3 mm, see the magenta mesh depicted in Figure 7). The calculation was initialized with a displacement field containing scaling and rigid body degrees of freedom. After DVC convergence, the displacement field uaux enabled the auxiliary mesh to be positioned in the LQ configuration. The interpolation of the displacement field uaux at the nodes of the meshed microstructure yielded u b , which gave the nodal coordinate corrections to fit it back to the in situ (LQ) configuration (i.e., repositioned blue mesh in the LQ configuration shown in Figure 7(right)).

Uncertainty Quantification

Uncertainty quantifications were performed by running DVC analyses on the two scans performed on the undeformed specimen. Four different contrasts were considered to investigate the effects of this parameter on the results. First, the contrast C was set to 1, which implied that the elastic properties of the different phases were identical. This choice was equivalent to homogeneous mechanical regularization.

Second, the contrast C was set to E i /Em = 70.0/13.3 = 5.26, which corresponds to realistic properties of mortar constituents when inclusions were more rigid that the cementitious matrix [START_REF] Tsitova | In-situ x-CT test on mortar microspecimen coupled with mesoscale numerical simulations of fracture[END_REF]. Third, the case C = 0.5 was considered, which implied that inclusions were more compliant than the matrix. Last, the contrast was set to a rather high value (C = 20), which was the case of highly rigid inclusions and very compliant matrix.

DVC analyses were run with different regularization lengths. For the first step, reg was set to a high value (i.e., 150 vx). Then, the displacement field at convergence was used to initialize the next step where the regularization length was decreased. This step-by-step (relaxation) process was continued until reg reached 15 vx. In this study, too small reg values (i.e., less than twice the element size)

were observed to be ineffective for mechanical regularization. When C = 20, DVC calculations with reg < 30 vx could not converge. decrease of reg leads to an increase in displacement and strains fluctuations. This trend is consistent with the trade-off between spatial resolution (here controlled by the regularization length) and measurement uncertainty. Further, the more heterogeneous the regularization (i.e., C departs from one), the higher the fluctuations since the regularization contrast increased (Equation ( 11)). On a more quantitative level, the strain uncertainty decay with reg follows power laws that are not too far from -2.5, which are expected when the dominant source of uncer-tainty corresponds to acquisition noise on the radiographs utilized to reconstruct 3D images [START_REF] Buljac | Digital volume correlation: Review of progress and challenges[END_REF].

DVC Analysis of Crack Propagation

The scan acquired right after specimen failure revealed that the crack had propagated on most of the ligament height. Therefore, DVC was run between the undeformed scan and this first scan. For the first DVC step, the regularization length was set to 100 vx, and gradually decreased down to 15 vx. The converged solution for higher reg was used as initialization for the next step with less weight put on the mechanical part. The gradual relaxation of mechanical regularization allowed most of the long wave displacement components to be captured, and then iteratively converged to a better solution with short wave length fluctuations that were gradually restored. This iterative process was carried out for C = 5.26.

3D renderings of thresholded Lagrangian residual fields are shown in Figure 9.

The maximum RMS level for reg = 100 vx was less than 1.4 times (287 gray level) the baseline RMS residual (215 gray level). The highest absolute levels correspond to the fractured surfaces, which were continuous and well defined. The absence of high residuals in the intact zones proves a good overall registration quality and efficient mechanical regularization of DVC. It is worth noting that when using the gray level residuals, the detection of the cracks was straightforward as all other heterogeneities, some of which had similar gray levels as the cracks, were no longer visible. In terms of displacements, cracks induce discontinuities that mechanical regularization tends to spread over a zone whose size depends on reg [START_REF] Hild | Damage measurements via DIC[END_REF]. The followed relaxation procedure led to displacement jumps localized in a smaller domain.

Consequently, the corresponding strains were higher and more concentrated (Figure 10).

Contrast Influence

The relaxation process was also carried out for C = 1.0 (i.e., homogeneous regularization), and the results are compared with the previous calculations. Figure 12(a) shows the RMS residual history during the relaxation process for the two considered contrasts. For C > 1, the RMS residuals were systematically lower, and dropped to 273 gray levels, while for C = 1.0 the minimum level was 279 gray levels. These levels are rather close to the baselines (i.e., 214-5 gray levels). This observation shows that the regularization used herein was not too strong and did not induce significant deviations. For the maximum principal strains (Figure 12(b)), their standard deviation was systematically larger for higher contrast, which allowed for more fluctuations especially at the interfaces between aggregates and the matrix to be captured.

Moreover, the growth of strain fluctuations was more progressive for C = 5.26

along with the relaxation of mechanical regularization.

To further investigate the influence of contrast, the regularization length reg was set to 20 vx, and the contrast C was varied between 0.5 and 20. The maximum principal strain fields (Figure 13) demonstrate how elastic contrast contributed to crack localization. The higher C, the less regularization was put on the matrix and the crack localized in fewer elements. In heterogeneous cementitious materials, the difference in mechanical properties of phases causes matrix-aggregate interfaces to debond. This mechanism as well as damage in the matrix are better captured with heterogeneous regularization. Although it may seem that taking a high value for C is a good option, the mean of maximum principal strains grows as well with an increase of C. It is a signal that too high contrasts result in "undeformed" aggregates and unphysical large strains in the totality of the matrix. Therefore, taking a contrast value greater than that based on physical properties of the constituents may result in displacement fields that are no longer physically correct. Therefore, in further calculations the contrast was set to 5.26. After adding elastic contrast for heterogeneous mechanical regularization, the analysis was continued with the introduction of damage into mechanical regularization.

A study of crack opening displacements and crack morphology at the mesoscale is finally proposed.

Damage Introduction into Mechanical Regularization

In the first stage, the heterogeneous regularization was taking into account the difference in mechanical behavior of different phases. In the present stage, damage can be considered in a similar manner [START_REF] Hild | Damage measurements via DIC[END_REF]. The weight of mechanical regularization can be significantly reduced in the fractured zone, thereby allowing for large local strains. The damaged elements were selected based on two criteria, namely, the average residual per element and Crack Opening Displacement (COD) u I , which is estimated as

u I = 1 (12)
by neglecting the regular contribution to mean strains [START_REF] Vakulenko | Continuum theory of medium with cracks[END_REF]. To study the effect of damage in mechanical regularization, two contrasts C were considered, namely, 1.0 and 5.26. The results of previous analyses with reg = 20 vx were chosen to initialize calculations and select damaged elements. The thresholds and number of selected damaged elements for both cases are listed in Table 4. For the gray level residuals, the baseline levels were observed to be equal to 214-5 (Figure 8(a)).

Consequently, the thresholds were selected as about two times this level to avoid false cracks. For the COD, their uncertainties were observed to be less than 0.1 vx with the evaluated strain uncertainties (Figure 8(b)). Consequently, the threshold COD for detecting damaged elements was set to 0.1 vx. The number of selected damaged elements for the two contrasts were very close. It is concluded that the selected criteria allowed for a reproducible description of the damaged zones. For a higher contrast (i.e., C = 5.26), the maximum principal strains in the damaged zones were higher (Figure 13), and consequently the crack opening displacement.

The number of detected damaged elements was also slightly higher. The detected damaged elements for C = 5.26 are presented in Figure 15. The damaged zone is continuous and replicates well the residual fields corresponding to fractured surfaces (Figure 9), thereby validating the choice of residual and COD criteria. The thickness of damaged zone consists of several elements since the mesh does not describe explicitly the fracture surface and, as a consequence, high residual values corresponding to the crack span over several elements in its vicinity. Introducing a lower stiffness in damaged elements will help localize the inelastic strains due to cracks. Thus the weight of mechanical regularization in the remainder of the volume could be increased. In the following calculations, the regularization length was set to reg = 60 vx in order to increase the weight put on mechanical regularization in undamaged elements for which heterogeneous elastic-ity is a very good approximation. Three different values were considered for the Young's modulus of the damaged elements E d , namely, 0.2Em, 0.1Em and 0.05Em.

Figure 16 shows the maximum principal strain fields for different Young's moduli of damaged elements. Lower mechanical weight put in the damaged zones does allow for a better localization of strains in the cracked zones. Higher damage levels also led to lower global residuals (Figure 17), meaning convergence to a better solution, as well as higher strain fluctuations. Last, setting the damaged stiffness to values closer to zero (i.e., E d < 0.05Em in this study) resulted in the divergence of the DVC calculations. The elastic modulus of the damaged area E d should be therefore set to the lowest possible value to allow for localized strains to occur, while retaining proper convergence of the DVC computations. Based on these results, in further calculations the Young's modulus of the damaged elements was set to 5% of the matrix modulus (i.e., E d = 0.05 Em). Figure 18 shows the maximum principal strain field for different contrasts and damage configurations. The introduction of damage into mechanical regularization was more effective for crack localization than the consideration of mechanical contrast between different phases (Figure 13). With the highest contrast, damage could also be better described at interfaces between the matrix and the aggregates.

Characterization of CODs

The dominant crack was observed to cross the cement paste matrix, aggregatematrix interfaces and limestone aggregates. Once the DVC analysis was performed and the damaged elements detected, the damaged elements belonging to the matrix, interphases and aggregates were quantified (Figure 20). In cementitious materials, matrix-aggregate interfaces have different chemical compositions along with higher porosity. It is considered that due to the lower properties and mismatch of mechanical properties, damage initiates preferably in such interfacial zones, which lead to debonding [START_REF] Ollivier | Interfacial transition zone in concrete[END_REF]. Experimental evidence shows that the interfacial zone width varies in the 10-50 µm range [START_REF] Scrivener | The interfacial transition zone (ITZ) between cement paste and aggregate in concrete[END_REF][START_REF] Mondal | Nanomechanical properties of interfacial transition zone in concrete[END_REF]. Since the average element size was ≈50 µm, the matrix elements adjacent to aggregates were considered as interphase elements for further analysis (Table 2). The distribution of damaged zones between the three phases is reported in Table 5. The volume fraction of damaged elements was mostly in the matrix (>90%)

with almost equal partitioning between interphases and matrix bulk. The maximum average COD is in interphases (7.3 µm), and the minimum is in the matrix bulk (6.2 µm). The highest COD fluctuations, which were characterized by their standard deviation, was in the aggregates (5.6 µm). For the other phases, the COD fluctuations were similar. The crack opening displacement was averaged over the total width of the sample and over its height with a 0.132 mm step. The crack started below the notch root at 3.0 µm width and its opening increased significantly up to 7.1 µm when it reached the notch root. Then the COD grew slowly until it reached its maximum level of 8.5 µm at the mid-ligament height, where a large aggregate with a vertical interface was located. Beyond the mid-ligament, the COD decreased down to 1.5 µm at 3.9 mm specimen height. The COD in the matrix bulk peaked at the notch root (7.4 µm) and then gradually decreased as it propagated upward. The COD at interphases was higher than in the matrix bulk over a significant part of the ligament with a maximum opening of 9.6 µm. The COD in aggregates was fluctuating considerably along the specimen height. This effect was due to the relatively small number of cracked aggregates and their sparse distribution in the volume (Figure 20(c)). However, locally the COD could reach 9.6 µm.

Conclusion and Perspectives

In the present study, an in situ three-point flexural test was carried out in an X-ray CT scanner on a small-scale mortar microbeam. A realistic 3D mesh was built consistently with the imaged underlying microstructure (based on a 5.3 µm resolution tomographic scan acquired prior to the in situ test). Mechanically regularized FE-DVC was adapted to heterogeneous materials by taking into account the difference between the elastic properties of the different phases of mortar. The introduction of elastic contrast C allowed the strain heterogeneities related to the underlying microstructure to be better described, and the inelastic strains to be better captured in the damaged zones. The contrast study showed that prescribing contrast levels above the expected properties of phases could result in unrealistic strains in the matrix. It is worth emphasizing that realistic Young's moduli of the various phases are needed to obtain trustworthy strain fields, especially when stronger mechanical regularization is enforced.

Furthermore, the damaged elements were selected based on previous computations for the introduction of damage in the regularization scheme. The Young's modulus in damaged elements was set to 5% of that in the undamaged matrix, which allowed displacement measurements to be more localized in the fractured zone. The estimation of crack opening displacements and crack morphology was carried out and gave more insight into the damage mechanisms at the mesoscale.

The study thus demonstrated the potential of mechanically regularized FE-DVC enriched with microstructure-based features for the quantification of damage at lower scales.

In future studies, the construction of 3D meshes consistent with fractured surfaces may improve the estimation of crack opening displacements [START_REF] Réthoré | Extended three-dimensional digital image correlation (X3D-DIC)[END_REF][START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF]. The test reported herein may also be used to calibrate numerical models accounting for the various damage mechanisms discussed herein.

Fig. 1

 1 Fig. 1 Experimental and numerical workflow

Fig. 2

 2 Fig. 2 Notched mortar microbeam

Fig. 3

 3 Fig. 3 Inclusion segmentation. (a) Filtered image (light inclusions -limestone aggregates, gray inclusions -siliceous aggregates). (b) Segmentation of limestone aggregates. (c) Contrast (norm of image gradient). (d) Segmentation of siliceous particles

Fig. 4

 4 Fig. 4 Surface and volume meshes (red: siliceous aggregates, gray: limestone aggregates, yellow: matrix, green: notch). (a) Surface meshes. (b) Aggregate and (c) matrix meshes

Fig. 5

 5 Fig. 5 Schematic and actual views of the in situ three-point flexural setup

Fig. 6

 6 Fig. 6 (a) CT scans performed during in situ flexure. (b) 2D section of fractured specimen and DVC ROI

Mesh see Figure 4

 4

Fig. 7

 7 Fig. 7 Schematic view of the backtracking procedure enabling the (blue) mesh based on the HQ configuration (top) to be fitted to the in situ configuration (bottom) via DVC using an auxiliary (magenta)

Figure 8 (

 8 Figure 8(a) shows the displacement uncertainties and Figure 8(b) the maximum principal strain 1 uncertainties as functions of the regularization reg . A

Figure 8 (

 8 Figure 8(c) shows the change of RMS residuals as functions of reg . Their variation remained very limited since for any regularization length, the kinematics was very simple (essentially rigid body motions). These levels are baselines to which further analyses will be compared.

Fig. 8

 8 Fig. 8 (a) Displacement and (b) maximum principal strain uncertainties as functions of the regularization length for different contrasts. (c) Corresponding RMS residuals (note their small dynamic range)

Fig. 9

 9 Fig. 9 Thresholded Lagrangian residual field showing the fractured surfaces ( reg = 15 vx and C = 5.26). (a) Face and (b) side views

Fig. 10 Figure 11 (Fig. 11

 101111 Fig. 10 Front face views of maximum principal strain 1 fields for different regularization lengths expressed in voxels (C = 5.26)

Fig. 12

 12 Fig. 12 (a) RMS gray level residuals, and (b) maximum principal strain 1 fluctuations as functions of the regularization length reg and contrast C

Fig. 13

 13 Fig. 13 Front face views of the maximum principal strain fields 1 for different contrasts ( reg = 20 vx)

Fig. 14

 14 Fig. 14 (a) Change of RMS residual with contrast and regularization length. (b) Mean and standard deviation of the maximum principal strains 1 as functions of contrast ( reg = 20 vx)

Fig. 15

 15 Fig. 15 3D renderings of damaged elements (corresponding to cracks) in the mechanical regularization ( reg = 20 vx, C = 5.26). (a) Side and (b) face views

Fig. 16

 16 Fig. 16 Front face views of the maximum principal strain fields 1 for different elastic moduli of damaged elements ( reg = 60 vx, C = 5.26)

Fig. 17

 17 Fig. 17 Standard deviation of the maximum principal strain 1 compared to the RMS residual for different elastic moduli of damaged elements ( reg = 60 vx, C = 5.26)

Fig. 18

 18 Fig. 18 Front face views of the maximum principal strain fields 1 for different contrasts and damage ( reg = 60 vx)

Fig. 19

 19 Fig. 19 Standard deviation of the maximum principal strain 1 compared to the RMS residual for different contrasts and damage implementations ( reg = 60 vx)

Fig. 20

 20 Fig. 20 Damaged elements belonging to (a) the matrix, (b) interphases, and (c) aggregates

Figure 21 (Fig. 21 (

 2121 Fig. 21 (a) Cross-section (at x = 2.5 mm) of the COD field. (b) Average COD and (c) average COD per phase along the specimen height

Table 1

 1 DVC hardware parameters

	Parameters	High Quality Scan	In situ Scan
	CT device	GE Phoenix v|tome|x M
	Target	high-flux|target
	Filter	0.1 mm Cu
	Voltage	180 kV	200 kV
	Current	57 µA	90 µA
	Tube to detector	807.3 mm
	Tube to object	21.5 mm	35.5 mm
	Detector	GE Dynamic 41|100
	Definition	1992 × 2000 (2 × 2 binning)
	Number of projections	2500	1600
	Angular amplitude	360°F
	rame average	5 per projection	3 per projection
	Frame rate	2 fps	3 fps
	Acquisition duration	2.5 h	45 min
	Gray Level amplitude	14 bit (16 bit format)
	Pixel size	5.3 µm	8.8 µm
	Avizo data visualization and analysis software was used for further image
	processing. After mechanical testing (Section 2.3), a Region Of Interest (ROI)

Table 2

 2 Characteristics of the phases in the mesh. The physical size of one voxel is 8.8 µm

		Aggregates	Matrix	Interphase	Bulk matrix
	Elements	74,120	272,352	134,474	137,878
	Nodes	7,863	-	16,232	39,279
	Volume, vx 3	1.57 • 10 7	5.50 • 10 7	2.61 • 10 7	2.89 • 10 7
	Volume fraction	22%	78%	37%	41%
	Element size, vx	6	6	6	6
	Young's modulus, GPa	70.0	13.3	13.3	13.3
	Poisson's ratio	0.26	0.26	0.26	0.26

Table 3

 3

	DVC analysis parameters
	DVC software	Correli 3.0 [57]
	Image filtering	none
	Element length (mean)	6 vx (53 µm) in in situ configuration
	Shape functions	linear (4-noded tetrahedra)

Table 4

 4 Damaged element selection

	Thresholds	C = 1	C = 5.26
	RMS residual, gray level		400
	COD , vx		0.1
	Number of damaged elements 18,076	18,206

Table 5

 5 Damaged zone characteristics

	Parameters	Total	Bulk matrix	Interphases Aggregates
	Number of elements in damaged zone	18,206	8,340	8,329	1,537
	Percentage of elements in damage zone	100%	46%	46%	8%
	Damaged volume, mm 3	2.42	1.10	1.09	0.23
	Average COD, µm	6.8	6.2	7.3	7.2
	COD standard deviation, µm	4.6	4.2	4.7	5.6

Acknowledgements The work reported herein was carried out within the framework of the CEA-EDF-Framatome agreement.

Conflict of interest

The authors declare that they have no conflict of interest.


Malcolm Dick and Caroline Archer-Parré (eds.), James Watt, 1736-1819: Culture, Innovation, Enlightenment Liverpool: Liverpool University Press, 2020. Pp. 280. ISBN: 978-1-7896-2081-8. £80.00 (hardback). ISBN: 978-1-7896-2082-5. £24.99 (paperback).

Yohann Guffroy

Ecole polytechnique fédérale de Lausanne This volume of ten texts is the result of a symposium that took place at the University of Birmingham in 2016. Unlike other biographies and books on James Watt's life, this one does not deal with the steam engine, or his business relationship with Matthew Boulton or the Soho manufactory. The authors propose a sociological portrait of Watt, exploring different aspects of his personality and his early career, mainly through his correspondence with his wives, Margaret (née Muirhead) and Ann (née MacGregor), and his son James Watt junior. This important archival material is presented in detail in the tenth chapter, by Fiona Taite, who was appointed to the Archives of Soho Project in the year 2000 to recatalogue the papers of James Watt and his family. She explains precisely how all of these papers have been preserved and have come down to us. This material is the strength of this book because, thanks to it, the authors can explore from the inside subjects like his relation to the transatlantic trade and slavery, the role of his wives in his professional life, his political ideas or his involvement in various social networks.

In the introduction, editors Malcom Dick and Caroline Archer-Parré present a state of the art from the first biographies written after Watt's death to those published following the two-hundredth anniversary of his death. They draw the evolution of the perception of James Watt, from a 'genius' to a complex personality engaged not only with the steam engine but with plenty of other activities and networks. This book is part of a historiographical renewal spanning the last forty years.

In the first chapter, Peter M. Jones deals with the milieu in which Watt grew up and the factors that may have led to his intellectual development. Jones's hypothesis is that Calvinism played a role in the intellectual development of the Watt family, beginning with the grandfather and the uncle. As Kate Croft shows in the third chapter, an important role was also played by Watt's wives, Margaret and Ann, who helped their husband coordinate his professional network and run his workshop in Birmingham. In my opinion, this chapter is particularly interesting because it helps to highlight that wives of scientists were not confined to the domestic sphere but were often fully engaged in their husbands' business, like Marie-Anne Paulze Lavoisier, who translated the texts of her husband Antoine Lavoisier and drew chemical drawings. In the fourth chapter, Larry Stewart focuses on James Watt junior and his political travel to France during the French Revolution. The correspondence between father and son reveals Watt senior's political convictions, his views on the Great Revolution and its reverberations in England, and his devotion to the English Crown.

The fifth chapter connects the themes of family and network. Frank A.J.L. James explains how James Watt and his sons, James junior and Gregory, contributed indirectly to the rise of Humphry Davy, who became in 1798 the superintendent of the Medical Pneumatic Institution in Bristol. Among other things, James shows how local or global scientific networks are based on individual careers and personalities even when organized around an institution. This insight is further developed in the following two chapters. By presenting Watt's relationship with two young enthusiastic chemists, Joseph Black and James Kair, Kristen M. Schranz shows how Watt acted as a coordinator and a guide, thus becoming a 'central sun' in this domain, as Muirhead once called him. Through the study of correspondence, this chapter reveals the vital role played by Watt in the circulation of chemical knowledge at the end of the late eighteenth century. In Chapter 7, David Philip Miller argues that Watt's natural philosophy, especially the chemistry of heat, has been essential in his understanding of the steam engine, as well as his work with Beddoes. The most interesting part of this chapter is concerned with what the author calls 'the sites [and institutions] of natural philosophy' (p. 178-6); that is, scientific societies, laboratories and the manufactories. According to Miller, Watt's philosophy was articulated through books and conversations, but also through material practice and production.

Material production is the main theme of Chapters 2, 8 and 9. Stephen Mullen explores Watt's youth as a pretext to write on the development of the Glasgow harbour and its inscription in the transatlantic trade and on the relationship between slavery and Watt's family, especially his father. Mullen's contribution is important because it sheds light on the less known topic of Watt's commercial implication in the triangular trade. Ben Russell, in the eighth chapter, tries to show that the objects in Watt's laboratory preserved in the Science Museum are more than simple material items. Indeed, their study unveils different implicit aspects, such as the mode of production, the useful knowledge, the organization of work, the site and the advertisement. Russell shows that a material history is essential to think about the figure of a scientist/engineer. Finally, in Chapter 9, Nina Baker traces the 'James Watt organ' preserved in the People Palace Museum of Social History in Glasgow. Through the study of letters, newspapers and bibliography, she shows that even if Watt made different musical instruments and some organs, this specific organ has probably been misattributed to Watt after his death.

This book is rich in interpretations of Watt's figure, showing that a 'genius' is not a lonely individual but a part of a large network of sociability, ideas and practices. It also manages to reimagine the steam engine not as the end of a journey, but rather as one stop in a long process of maturation. doi:10.1017/S0007087421000613

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