A Bouterf

S Roux

F Hild

J Adrien

E Maire

S Meille

Digital volume correlation applied to X-ray tomography images from spherical indentation tests on lightweight gypsum

Keywords: Digital volume correlation, global approach, reduced basis, strain eld, plaster

Cylinders made of lightweight gypsum are extracted from industrial plasterboard and then indented in-situ in an X-ray tomograph. The results from the in-situ experiment show that a compacted zone develops under the indenter, which is displaying a very sharp boundary with the undamaged material. Tomographic imaging during the mechanical load associated with digital volume correlation enable the displacement elds to be measured during the test. However, because of the inhomogeneous nature of the indentation test, a high spatial resolution for the displacement is called for, and because the range of displacement amplitudes is small, uncertainties on the measured displacement and strain elds are large. In this study, a new methodology is presented to address integrated digital volume correlation based on a library of elds computed from a commercial nite element software. It allows many uctuations in the estimated displacement elds to be ltered out and the measurement to be much more robust and reliable. This opens new pathways for the identication of mechanical properties.

Introduction

Plasterboard is a low-density material, which is composed of lightweight gypsum core lined with paper coating. Two main routes are commonly followed for the production of porous gypsum. In the rst method, gas bubbles are formed in the gypsum paste through very rapid mixing and chemical additives. The second one consists of foaming by air entrainment in the wet gypsum paste [START_REF] Vimmrová | Lightweight gypsum composites: Design strategies for multi-functionality[END_REF]. The objective of introducing foam in the board slurry is to create voids to further reduce the nal weight of the board at xed plaster/water ratio. To improve the adhesion between paper and plaster, starch is added in most cases. This is benecial to the board exural strength.

The relationship between the microstructure and the mechanical properties of the core of the board [START_REF] Murat | Corrélation texture cristalline-propriétés mécaniques de plâtres durcis, étude préliminaire[END_REF][START_REF] Coquard | Résistance mécanique des plâtres secs et humides[END_REF][START_REF] Meille | Etude de comportement mécanique du plâtre pris en relation avec sa microstructure[END_REF], or the evaluation of the re resistance of a partition wall made of plasterboard [START_REF] Sakji | Modélisation probabiliste et validation expérimentale du transfert thermique et du comportement thermomécanique avec endommagement d'une plaque multicouche carton-plâtre-carton soumise au feu[END_REF] have already been studied in detail. Benouis [START_REF] Benouis | Comportement mécanique des ouvrages en plaques de plâtre sur ossature métallique[END_REF] showed the orthotropic nature of plasterboard, which is a consequence of the tensile behavior of the paper. However, lightweight gypsum as such has received only little recognition as a structural part; its mechanical behavior and fracture resistance are scarcely investigated in the scientic literature. Yet, some mechanical properties are required for boards, and hence a good knowledge of the mechanical degradation is a need. In the sequel, it is proposed to use X-ray tomography to investigate mechanical properties of plasterboards.

X-ray tomography is a powerful non destructive technique allowing the material microstructure to be characterized [START_REF] Baruchel | X-Ray Tomography in Material Sciences[END_REF]. It has been used by numerous laboratories to investigate the microarchitecture of trabecular bone [START_REF] Bay | Digital Volume Correlation: Threedimensional Strain Mapping Using X-ray Tomography[END_REF], to characterize the microstructure of cellular materials [START_REF] Maire | X Ray Tomography Study of Cellular Materials: Expriments and Modelling[END_REF], to qualitatively reveal damage mechanisms in the bulk of particulate composites [START_REF] Buère | Characterization of internal damage in a MMCp using X-ray synchrotron phase contrast mincrotomography[END_REF], to visualize propagation of cracks in fatigue [START_REF] Ferrié | Fatigue crack propagation: in situ visualization using X-ray microtomography and 3D simulation using the extended nite element method[END_REF]. To make progress in the understanding of deformation and damage mechanisms in samples, it is crucial to be able to observe the latter ones at the same time as they are loaded. The development of in-situ experiments monitored by X-ray tomography started with the observation of Guvenilir et al. [START_REF] Guvenilir | Direct observation of crack opening as a function of applied load in the interior of a notched tensile sample of Al-Li2090[END_REF]. Since then, mechanical testing coupled with in-situ tomographic observations has been developed [START_REF] Bay | Digital Volume Correlation: Threedimensional Strain Mapping Using X-ray Tomography[END_REF][START_REF] Elliot | In-situ deformation of an open-cell exible polyurethane foam characterized by 3D computed mincrotomography[END_REF][START_REF] Buère | Situ Experiments with X ray Tomography: An Attractive Tool for Experimental Mechanics[END_REF]. The actual microstructures may then be used in nite element calculations in which numerical models are produced to represent as precisely as possible the architecture of these materials [START_REF] Youssef | Finite element modelling of the actual structure of cellular materials determined by X-ray tomography[END_REF][START_REF] Maire | X Ray Tomography Study of Cellular Materials: Expriments and Modelling[END_REF].

Another way to exploit the reconstructed volumes is to use Digital Volume Correlation (DVC) [START_REF] Bay | Digital Volume Correlation: Threedimensional Strain Mapping Using X-ray Tomography[END_REF][START_REF] Smith | Digital volume correlation including rotational degrees of freedom during minimization[END_REF][START_REF] Lenoir | Volumetric digital image correlation applied to X-ray microtomography images from triaxial compression tests on argillaceous rock[END_REF] to measure displacement elds in the bulk of the sample. DVC is the 3D extension of standard digital image correlation [START_REF] Sutton | Determination of displacements using an improved digital correlation method[END_REF]. It can either be local (or subset-based) or global when global constraints such as continuity are prescribed to the measured elds [START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF].

The latter approach was used to analyze the correlation between local density and strain levels in stone wool [START_REF] Hild | Three dimensional analysis of a compression test on stone wool[END_REF], stress intensity proles in a cracked sample [START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using 3D correlation of laboratory X ray microtomography images[END_REF] or to identify and validate constitutive and numerical models [START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF].

In the following, indentation tests will be carried out to analyze the behavior of gypsum.

They are relatively easy to perform on a specimen of simple shape and small volume. They can be used to characterize the mechanical behavior of materials. The spherical indenter, compared to other geometries, has two major advantages. The rst is the ability to distinguish the transition from the elastic behavior to the elastoplastic behavior, a very interesting feature especially for materials with low yield stress. With conical or pyramidal (i.e. sharp) indenters, plasticity begins as soon as contact is established with the medium. Second, it has been shown that with a pyramidal indenter, several sets of material parameters can give rise to very similar load-displacement curves. The discriminating character is somewhat more pronounced for a spherical indenter [START_REF] Chatelain | Comportement mécanique de mousses fragiles[END_REF].

Early works on the indentation of foams have suggested that their response is controlled to a large extent by their compressive strength [START_REF] Shaw | The plastic behavior of cellular materials[END_REF][START_REF] Wilsea | Indentation of foamed plastics[END_REF]. A review of the existing literature on indentation tests shows that spherical indentation was used to study ceramics (e.g. alumina whose porosity ranges from 2.5 % to 18 % [START_REF] Lawn | Indentation of Ceramics with Spheres: A Century after Hertz[END_REF]). The results conrm that a porous material is more susceptible to damage than cracking macroscopically in a brittle manner. Several works have been published on the modeling of the indentation behavior of cellular materials [START_REF] Kumar | Eect of displacement-rate on the indentation behavior of an aluminum foam[END_REF][START_REF] Onck | Size eects in foams: Experiments and modeling[END_REF]. Some of the studies concern brittle dense rocks and much less attention has been paid to the problem of indentation of porous rocks [START_REF] Suárez-Rivera | Indentation by pore collapse in porous rocks[END_REF][START_REF] Leite | Determination of unconned compressive strength and Young's modulus of porous materials by indentation tests[END_REF]. X-ray tomography has been applied to the indentation of aluminum-based metallic foams [START_REF] Kádár | X-ray tomography and nite element simulation of the indentation behavior of metal foams[END_REF]. In the latter study X-ray tomography revealed that the deformation was found to be almost restricted to a spherically shape compacted zone under the indenter. Smith and Bay [START_REF] Smith | Experimental Measurement of Stains using Digital Volume Correlation. Nontraditional Methods of Sensing Stress, Strain, and Damage in Materials and Structures: Second[END_REF] used DVC to quantify strains in aluminum foam and to compare them with simulated elds using an isotropic nite element analysis. DVC has also been used to measure strain elds in spherical indentation of a closed-cell and open-cell aluminium foams [START_REF] Bay | Methods and applications of digital volume correlation[END_REF]. Mostava et al. [START_REF] Mostafavi | Observation and quantication of three-dimensional crack propagation in poly-granular graphite[END_REF] used DVC to study the fracture behavior of a short-bar chevron notch crack propagation specimen made of polygranular nuclear graphite (i.e. a quasi-brittle material). However, in the analyzed cases the strain levels were very small, much smaller than the strain measurement uncertainty.

In this study, cylinders made of lightweight gypsum are tested in-situ by using a spherical indenter. As for other materials (see above), the nal aim of this test is to analyze the transition from the elastic behavior to the nonlinear response in the compacted zone. The actual state of strain and stress has to be evaluated. Therefore there is a need for measuring very small strain levels in most of the material and detecting high strain gradients in the transition zone. A new method of integrated digital volume correlation adapted to the spherical indentation test is presented to address these issues. It is proposed to couple in-situ experimental tests, DVC and numerical simulations. The key is to design a reduced kinematic basis, i.e. using very few degrees of freedom, yet being faithful to the kinematics of the test. The advantage of such an approach is that it may benet from prior knowledge on the expected displacement eld.

Spherical indentation test

Specimens used in the experiment are prepared from industrial plasterboard plates. The samples of the present study are cylinders 17 mm in diameter and 10 mm in height. For the in-situ experiment a spherical indenter (6 mm in diameter) is chosen. The indentation test is considered as quasi-static since a very low velocity is selected for the indenter (i.e. 5 µm/s). The applied load and displacement are recorded and monitored during the test. The experimental setup is shown in Figure 1. The specimen is imaged at the initial state (i.e. without any applied load) and at nine levels of compressive load. After each step, the crosshead displacement is interrupted and a dwell time of 20 min is chosen to avoid blurring caused by displacement of the sample during the relaxation process. The experiment has been conducted on a Phoenix V tome x tomograph (90 KV-voltage, 200 µA intensity, voxel size of 12 µm, 1,920 × 1,536 pixel at panel). 1,000 radiographs are acquired over a 360 • rotation for each scan. The total scan time lasts approximately 60 minutes.

Figure 2 shows the load-displacement curve of the test analyzed herein. A quasi-linear dependence of the applied load with the indentation depth is observed. The oscillations in the load-displacement response are due to repeated cycles of yield, collapse and densication.

As the deformation progresses, the indenter is supported by a continuously increasing crushed zone of compacted plaster. A comparable load-penetration relationship was obtained for porous rocks [START_REF] Leite | Determination of unconned compressive strength and Young's modulus of porous materials by indentation tests[END_REF].

Figure 3 shows 3D renderings of the reconstructed volumes of the sample in its reference conguration and at the eighth loading step. The compacted zone, where the microstructure is crushed due to the compression load, is clearly visible on the top part of the 3D rendering. The densied zone is located directly below the head of the indenter and a semi-ellipsoidal shape develops. However outside of the compacted region, the microstructure remains unaltered. As the load level increases, a larger crushed zone is formed under the indenter. After a certain load level, this crushed and compacted region acts as a virtually incompressible material that transfers the stresses to the surrounding intact plaster. Once the compaction zone is fully developed, the shape of the indenter is no longer relevant to the indentation process, which is totally controlled by the size and shape of the compaction zone. A similar phenomenon with a comparable shape was previously observed for highly porous refractory bricks (i.e. 65 % porosity [START_REF] Suárez-Rivera | Indentation by pore collapse in porous rocks[END_REF]), articial porous rock [START_REF] Leite | Determination of unconned compressive strength and Young's modulus of porous materials by indentation tests[END_REF] and aluminium foam [START_REF] Smith | Experimental Measurement of Stains using Digital Volume Correlation. Nontraditional Methods of Sensing Stress, Strain, and Damage in Materials and Structures: Second[END_REF][START_REF] Kádár | X-ray tomography and nite element simulation of the indentation behavior of metal foams[END_REF].

The aim of the next section is to introduce the global DVC approach to be used to measure displacement and strain elds in this type of experiment. One of the challenges is related to the fact that very small strain levels occur outside the crushed zone, and the latter is not directly accessible to standard DVC.

Global DVC

In the present study, a global approach to DVC [START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF] is used. The displacement eld is decomposed over nite element shape functions. It allows for a direct coupling with numerical simulations [START_REF] Limodin | Crack closure and stress intensity factor measurements in nodular graphite cast iron using 3D correlation of laboratory X ray microtomography images[END_REF][START_REF] Rannou | Three dimensional experimental and numerical multiscale analysis of a fatigue crack[END_REF]. If the reference (respectively, deformed) image is represented by a three-dimensional gray level-valued eld, f (x), (respectively, g(x)), the gray level conservation requires that

f (x) = g(x + u(x)) (1)
where u denotes the unknown displacement eld. Measuring a displacement eld consists of minimizing the square of the correlation residual φc

φc = |g(x + u(x)) -f (x)| (2)
The minimization of φ 2 c is a nonlinear and ill-posed problem. In particular, if no additional information is available, it is impossible to determine the displacement for each voxel independently since there are three unknowns for a given (scalar) gray level. For these reasons, a weak formulation is preferred. After integration over the whole Region Of Interest (ROI) Ω, the global correlation residual Φ 2 c reads

Φ 2 c = 1 2 Ω (g(x + u(x)) -f (x)) 2 dx (3)
in which the displacement eld is expressed in a (chosen) basis as

u(x) = i unψ n (x) (4)
where ψ n are (chosen) vector elds, and un the associated degrees of freedom. The measurement problem then consists of minimizing Φ 2 c with respect to the unknowns un. A modied Newton iterative procedure is followed to circumvent the nonlinear character of the minimization problem. Let u i denote the displacement at iteration i, and {u} i the vector containing all the unknown degrees of freedom. By assuming small increments du = u i+1 -u i of the solution, a Taylor expansion is used to linearize g(x

+ u i+1 (x)) ≈ g(x + u i (x)) + du(x) • ∇f (x)
and ∂Φ 2 c /∂{u} i is recast in a matrix-vector product as

∂Φ 2 c ∂{u} i = [M ]{du} -{b} i = {0} (5)
with

Mmn = Ω (∇f (x) • ψ m )(∇f (x) • ψ n ) (6)
and

b i m = Ω (∇f (x) • ψ m )(f (x) -g(x + u i (x))) (7)
In the above equation the corrected deformed image, g(x + u i (x)) is computed with a trilinear gray level interpolation.

At this level of generality, many choices can be made to measure displacement elds. When dealing with volumes, eight-noded elements with classical nite element bases can be used [START_REF] Roux | Three-dimensional image correlation from X-ray computed tomography of solid foam[END_REF].

The latter is referred to as C8-DVC (i.e. global DVC with 8-noded cubes). In the sequel, C8-DVC will be used since no displacement discontinuity is sought. Consequently, the region of interest is discretized with nite elements for which the displacement eld is interpolated with trilinear shape functions. Global C8-DVC is rst used to measure 3D displacement elds in the bulk of the sample. A regular mesh with C8 elements is used. The displacement eld at the n-th load level is assessed by correlating the n-th volume with the initial volume (unloaded state considered as the reference). The region of interest has a size of 576 × 576 × 256 voxels (or 6.9 × 6.9 × 3.07 mm). The size of the C8 elements was chosen to be 8 voxels (i.e. 96 µm, which is of the order of the wall size). This choice resulted in a little more than 360,000 unknown kinematic degrees of freedom.

The measured displacement eld is shown in Figure 4. As the crushed region has lost its microstructure, this region is masked from the ROI and the displacement eld remains undetermined there. A simple interpolation is used to compute the residual eld shown in Figure 4(a).

In the latter, normalized residuals are shown, namely φc/∆f , where ∆f denotes the dynamic range of the ROI in the reference conguration. It was chosen here to compute a harmonic interpolation, i.e. solving ∇ 2 u = 0 inside the mask, and matching the computed values on the boundary (Dirichlet-type boundary conditions). Let us stress that this interpolation is not used for any measurement, but just to reveal the crushed region. From the latter 3D image the region where plaster has been crushed is easily distinguished as its boundary is sharp. The mask is adjusted as being the smallest semi-ellipsoid that contains the crushed region (Figure 4(a)).

Because of the uncertainty level of the reconstructed images themselves and the modest range of observed displacement magnitude outside the crushed zone, the measurement appears as too noisy to allow for the determination of strains. Displacement uctuations are clearly visible since the dynamic range is very small (i.e. less than 2 voxels) as outside the compacted volume plaster remains mostly elastic. Many of these uctuations are due to measurement uncertainties associated with very small element sizes.

The uncertainty on the measurement of the displacement elds is estimated in a region very remote from the compacted area where strains are very small. After removal of the rigid body motions, the root mean square uctuations of the displacement eld is of the order of 0.4 voxel or ≈ 5 µm. This very high level is due to the high volume fraction of pores (≈ 75 %) in combination with small element sizes. As a consequence, the strain uncertainty for such a small element size is quite large (i.e. 5 × 10 -2 ). This rst analysis shows that even with very small element sizes, global C8-DVC provides a good rst displacement estimate. However, it is corrupted by measurement uncertainties and strain evaluations cannot be achieved faithfully at such a small scale.

To go beyond the previous results, the crucial point is to select the appropriate discretization of the sought displacement eld. Large elements possibly allow for a low uncertainty in the measured displacement at the expense of a poor spatial resolution. Conversely, ne meshes induce better spatial resolutions, but because of the fewer voxels considered per element, displacement uncertainties are larger. In addition, reconstruction artifacts are inevitably present in the image [START_REF] Davis | Artefacts in X-ray microtomography of materials[END_REF] and thus very small element sizes can hardly be used. Hence, the analysis of the displacements results from a compromise between spatial resolution and displacement uncertainty [START_REF] Bergonnier | Digital image correlation used for mechanical tests on crimped glass wool samples[END_REF][START_REF] Leclerc | Digital Volume Correlation: What are the limits to the spatial resolution?[END_REF]. Both quantities are inuenced by the image texture and the complexity of the displacement eld [START_REF] Réthoré | Shear-band capturing using a multiscale extended digital image correlation technique[END_REF].

Complexity is to be understood here as the ability to capture the actual displacement eld with a specic discretization. Because in the elastic domain, strains are expected to decay as 1/r 2 where r is the distance to the indentor center, a signicant discretization error is expected for coarse meshes in particular close to the indentor or the boundary of the crushed zone. This very rapid decrease of strain with r constitutes an intrinsic inhomogeneity (not due to the random microstructure) that is challenging for DVC. Further, Gaussian ltering (especially at boundaries) although commonly used in Digital Image Correlation cannot be trusted in the studied case, as even in the absence of noise, it will severely aect the strain eld. What would be needed is a lter that preserves a mechanically meaningful component and only damps out spurious uctuations. Recently, Leclerc et al. [START_REF] Leclerc | Digital Volume Correlation: What are the limits to the spatial resolution?[END_REF] proposed such a lter in so-called regularized DIC or DVC, where rather than using a Gaussian lter, a mechanical modeling can be used to assist DVC. This procedure could have been used here but the following procedure proposes an alternative route to circumvent the diculty of using a ne mesh (and hence a large uncertainty) in spite of the fact that the displacement elds in indentation tests are expected to display steep gradients.

Reduced basis in DVC

The idea is to design a reduced basis, which is adapted to the indentation test and thus allows for an accurate description of the test kinematics especially at the interface of the crushed zone and the remainder of the sample that is assumed to behave elastically. The fact that some a priori information on the sought displacement elds is available allows for a signicant reduction in the number of degrees of freedom. Because the studied domain excludes the crushed zone, and that no sign of deviation from linear elasticity is observed, it is assumed that the behavior remains linearly elastic in the entire domain studied by DVC. The porous gypsum is treated as a homogeneous and isotropic solid, with Poisson's ratio ν = 0.15 (note that the Young's modulus is not needed). In the present case, the displacement elds have a low sensitivity to Poisson's ratio when it is varied between 0 and 0.2.

Moreover, the boundary conditions are known over a large part of the domain boundary, namely, the face opposite to the indentation undergoes essentially rigid body motions that can be measured, all other faces are free, and nally only the displacements or forces are unknown on the surface, S, of the masked crushed region. Would the latter be known that the computation of the displacement eld in the entire domain would be obtained from a 3D linear elastic simulation. (It is observed that the crushed zone underneath the indentor constitutes an additional diculty as compared to a rigid indentor where the displacement eld over S would be a mere translation.) The spirit of the proposed approach is to design a parameterization of the displacement eld on S based on few degrees of freedom. Each of these unknown parameters i corresponds to a 3D displacement eld ψ i (x) computed from a commercial FE software (Abaqus in the present case). This set of elds ψ i (x) is the basis of kinematic elds used in DVC, as explicitly written in Equation ( 4). The measured amplitudes, ai, are also used to compute the actual displacement at the boundary of the masked region. It is worth noting that the computation time of C8-DVC is much higher than that of the reduced basis because of the number of unknowns has been drastically reduced (from 360,000 to 9 in the present case).

The boundary conditions used hereafter consist of (i) a rigid body motion on the face opposite the indentation (adjusted from a coarse C8-DVC analysis), (ii) traction-free surfaces over all other surfaces apart from the boundary of the crushed domain, S and (iii) on the S boundary, the displacement is sought as a kind of spherical harmonics expansion (truncated at low orders) for each component. More precisely, the library of elds used to describe the displacement is rst restricted to axisymmetric modes written as    ux(ai, ϕ, θ) = v cos(ϕ) sin(θ) + w cos(ϕ) cos(θ) uy(ai, ϕ, θ) = v sin(ϕ) sin(θ) + w sin(ϕ) cos(θ) uz(ai, ϕ, θ) = v cos(θ) -w sin(θ) [START_REF] Bay | Digital Volume Correlation: Threedimensional Strain Mapping Using X-ray Tomography[END_REF] where θ and ϕ are respectively the polar and azimuthal angles. As shown in Figure 5, w is a the displacement along the radial e r direction, and v along e θ . Both elds, v and w read v(θ; ai) = a1 + a2 cos(θ) + a3 cos(2θ) + a4 cos(3θ) w(θ; ai) = a5 sin(θ) + a6 sin(2θ) + a7 sin(3θ) [START_REF] Maire | X Ray Tomography Study of Cellular Materials: Expriments and Modelling[END_REF] where ai (i = 1, ..., 7) denote the amplitudes of the elds to be identied. It revealed necessary to enrich this basis by two additional elds to account for a breakdown of axisymmetry that was not anticipated initially. It was chosen to impose over S a uniform displacement along x and y, respectively for modes 8 and 9. These two elds account for a lateral motion of the indentor in addition to its normal penetration. Let us also note that the rigid body motion measured on the face opposite the indentor, as shown in Figure 6(a), is not axisymmetric either. This spurious motion may be due to a compliant indentor axis combined with slight heterogeneities in the to-be-crushed plaster. Let us emphasize the fact that beyond the specic choice made herein, the proposed approach is extremely exible and can easily be enriched to account for whatever eect could be detected and modeled, such as the present breakdown of symmetry, or possibly the occurrence of an indentation crack (not observed hereafter). Last, mixing Dirichlet and Neumann boundary conditions for dierent elds ψi is totally transparent in the present setting.

An overview of the mesh of the FE model used to compute the displacement basis, which uses tetrahedral elements, is shown in Figure 6. Close to the boundary of the crushed zone, the element size is of the order of 100 µm (or 8 voxels). In the same gure, two examples of displacement elds generated by Abaqus are presented.

From the nite element simulations, nine displacement elds dened at the location of each voxel from the nite element shape function are obtained (so that no mesh is needed anymore for the subsequent DVC analyses). However, as the displacement of each eld spreads over the entire domain, the problem to be solved at each time step is global and involves a full correlation matrix [M ]. Its size being very modest (i.e. 9 × 9) the correlation calculation is not the limiting step. Convergence is observed to be excellent, and the level of mean correlation residuals (5.14 %) is comparable to that observed with the initial C8-DVC basis (5.06 %).

This result validates the chosen kinematic basis. Let us emphasize the fact that the number of kinematic degrees of freedom has been reduced from about 360,000 to 9. Moreover for any combination of amplitudes {ai}, the resulting displacement eld obeys the equilibrium and constitutive equations as well as the traction-free or prescribed rigid motion at the basis of the specimen.

The displacement elds measured with the reduced kinematic basis are shown in Figure 7.

Comparing the displacement elds obtained by the two approaches, it is noted that the same range of displacement is measured. However most of the uctuations are ltered out, yet the long wave features are still captured. The uncertainty on the measurement of the displacements elds is estimated in the same remote region from the compacted area. The standard displacement uncertainty is of the order of 0.03 voxel or ≈ 0.3 µm. Comparing the latter with results obtained by C8-DVC (≈ 5 µm), there is a marked decrease in the standard displacement uncertainty. Similarly, the strain uncertainty is less than 2 × 10 -3 , i.e. at least one order of magnitude lower than previously observed.

Figure 8 shows the deviatoric and volumetric strain elds in the same plane as they are estimated with both procedures, namely, reduced basis (a) and C8 approach (b). It is worth noting that the dynamic range has to be increased for the standard C8 approach when compared with the results obtained with the reduced basis approach. For C8-DVC, the uctuations are mainly associated with measurement uncertainties as opposed to the new approach that is able to capture the strain gradients. Note also that the deviatoric strain is a positive quantity and hence its average value for a noisy displacement eld will remain well above the average of a noiseless eld.

From the latter determination, the elastic strain or stress components, in particular on the boundary of the crushed domain, can be computed more reliably. Figure 9 shows the results obtained with the optimal amplitudes. The displacement eld is presented in Figure 9(a).

It is observed that the optimal displacement eld is not axisymmetric. Figure 9(b) shows the maximum principal strain elds. The proposed procedure reveals a much more complex displacement eld than what is traditionally assumed for perfect spherical indentation. The results display very large dierences between the identied strain (or stress) distribution and the expected theoretical distribution. In addition, this new strategy allows very small strains to be measured.

The choice for the reduced basis may appear as somewhat arbitrary. It has to be simple (to limit uncertainty) and yet representative (to be able to account precisely for the actual displacement eld). Representativeness is checked by considering the residual eld, ideally comparable to that achieved with the rst C8 procedure. Simplicity means that a minimum number of elds is to be used. A natural basis is provided by a multipolar expansion, akin to spherical harmonics. This expansion has been further simplied to account for axisymmetry, and truncated to low orders. As discussed above, the rst choices were not representative (i.e. led to large correlation residuals) until deviation from axisymmetry was introduced. This choice is certainly not unique, and it is possible that other designs would lead to smaller numbers of elds. What is important is that the above choice does fulll the above two criteria as shown below.

In order to quantify the sensitivity of the results to the choice of displacement elds of the reduced basis, two other choices are considered. The measured rigid body motion is still prescribed on the bottom of the sample and dierent combinations of numerical elds (see Equation ( 9)) are prescribed on the boundary of the crushed zone. For the basis that contains at least the rst eld of the series (a1) and the two shear elds (i.e. of amplitudes a8 and a9), the results are very close to those obtained with 9 elds. The mean correlation residual is equal to 5.16 %, level very close to the values found by C8-DVC, 5.06 %, and with the rst set of amplitudes (i.e. a1 to a9) 5.14 %.

Any attempt to reduce further the number of elds (less than three elds, i.e. of amplitudes a1, a8 and a9) results in a signicant degradation of the residual level, showing that the proposed restriction is inappropriate. Let us emphasize on the fact that the strong assumptions that are made to reduce the kinematic basis are not to remain unchallenged. The residual eld and mean value provide objective quality estimates with appropriate spatial resolution to judge whether the initial assumptions can be deemed acceptable, or if they are to be revisited or relaxed.

Conclusions and perspectives

In situ mechanical tests within a tomograph, and in particular indentation tests, are seen to be well suited for an exhaustive characterization of the full 3D kinematics, and hence identication from such DVC results is appealing. However, material microstructure, tomography artifacts, test inhomogeneity may all lead to large uncertainties. The proposed methodology is a way to alleviate those diculties through the exploitation of known prior information within digital volume correlation.

The tomographic observations show that indentation proceeds by the growth of the crushed domain underneath the indentor, while the remainder of the medium can be considered as elastic. This a priori information is used to design a new DVC approach based on a reduced basis exploiting the knowledge of the elastic behavior and the boundary conditions. Each eld of the basis is the result of a nite element simulation outside the crushed zone. The few remaining unknowns are determined by image registration.

A very signicant uncertainty reduction is observed although the mean displacements remain of the same order of magnitude. It is worth noting that with the proposed strategy the strain and stress elds are accessible even if their level is small. This is possible because the measured displacement elds are not only kinematically admissible but also statically admissible.

This novel DVC strategy opens new pathways for the identication of mechanical properties. This type of approach is very generic and can be extended to other experimental congurations and material behaviors. It uses the nite element code as a way of generating kinematic bases.

The amplitudes associated with each considered eld are then evaluated via a global approach to digital volume correlation. This approach is non intrusive as any nite element code can be used as is.

The next step of the present analysis will consist of the identication of the behavior in the vicinity of the boundary between the crushed plaster and the pristine material. Full benet will be made by the fact the stress and strain uctuations close to the interface are measured very accurately with very ne discretizations.

Figure 1 :

 1 Figure 1: Specimen within the loading rig

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 2: Load-displacement plot of a test with ten scans

Figure 5 :Figure 6 :Figure 7 :Figure 8 :Figure 9 :

 56789 Figure 5: Spherical coordinates system

Acknowledgements

The authors wish to thank Saint-Gobain Recherche for supporting this research project, and particularly Xavier Brajer and René Gy. AB is also nancially supported by ANRT through contract no. 2010/567.


Amine Bouterf

Stéphane Roux

François Hild

Guillaume Vivier

Xavier Brajer

Eric Maire

Sylvain Meille

Damage law identification from full field displacement measurement: Application to four-point bending test for plasterboard

Keywords: Damage identification, Digital Image Correlation (DIC), Love-Kirchhoff kinematics

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The use of plaster as a building material has a very long and rich history. In their quest for producing a robust, light-weight, insulating structure element for the building industry, the concept of plasterboard was invented as early as in 1894 by two engineers from New York, A. Sackett and F.L. Kane. Plasterboard is nowadays the most commonly used low-cost building plates within the construction industry. This success is mostly due to the fact that plasterboards can be used in many applications, and engineered with enhanced performance for acoustic and fire requirements [START_REF] Vimmrová | Lightweight gypsum composites: Design strategies for multi-functionality[END_REF]. Gypsum board producers are still working for improved properties, or similar properties but with lower densities. Weight has direct commercial consequences, as any density saving impacts directly on production, transport, environmental and economical costs. Moreover, lightweight boards facilitate handling and make installation easier and more secure.

Two main methods are commonly used for the production of porous gypsum. In the first method, gas bubbles are formed in the gypsum paste through chemical additives. The second one consists of foaming by air entrainment in the wet gypsum paste [START_REF] Vimmrová | Lightweight gypsum composites: Design strategies for multi-functionality[END_REF][START_REF]Density and strength characteristics of foamed gypsum[END_REF]. The relationship between microstructure and mechanical properties of raw plaster or core of the board [START_REF] Murat | Corrélation texture cristallinepropriétés mécaniques de plâtres durcis, étude préliminaire[END_REF][START_REF] Coquard | Résistance mécanique des plâtres secs et humides[END_REF][START_REF] Meille | Etude de comportement mécanique du plâtre pris en relation avec sa microstructure[END_REF], or the evaluation of fire resistance of a partition wall with plasterboards has been thoroughly studied [START_REF] Sakji | Modélisation probabiliste et validation expérimentale du transfert thermique et du comportement thermomécanique avec endommagement d'une plaque multicouche carton-plâtre-carton soumise au feu[END_REF][START_REF] Sakji | Probabilistic Uncertainty Modeling for Thermomechanical Analysis of Plasterboard Submitted to Fire Load[END_REF]. However, plasterboard as such has received only little recognition as a structural part; its mechanical behavior and fracture resistance are scarcely investigated in the literature, although from manufacturing to installation, mechanical properties such as the flexural strength of the boards are crucial. [START_REF] Benouis | Comportement mécanique des ouvrages en plaques de plâtre sur ossature métallique[END_REF] showed the orthotropic nature of the plasterboard which is a consequence of tensile behavior of the paper. Some studies on the mechanical properties of syntactic foam core sandwich composites are also available. In these studies it is observed that the skin (paper) contributes to an increase of up to 40 % in estimated flexural strength, depending on starch content in adhesive between core and skin [START_REF] Gupta | Characterization of Flexural Properties of Syntactic Foam Core Sandwich Composites and Effect of Density Variation[END_REF]. Other studies on mechanical properties of gypsum sheathing found that the bending strength of gypsum sheathing depends on the strength of the adhesive bond between the facings and the gypsum core [START_REF] Mcgowan | Relationship between Moisture Content and Mechanical Properties of Gypsum Sheathing -Phase 2 Research[END_REF].

Therefore, it is critical for the producers to strive to create a board of even lighter weight that still passes the standard tests. To achieve this goal, it is important to make progress on the characterization of the failure mechanism in bending tests and to identify the different stages of damage and failure. Progress along these lines opens new pathways for optimized light wallboards, and hence efficient light frame construction.

In the present paper, Digital Image Correlation (DIC) is used to measure the kinematics of the plasterboards during four point bending tests up to failure. From this measurement, the ambition is to provide an effective constitutive law within the framework of plate theory (i.e., without distinguishing core and facings) to be used for describing wallboard within arbitrary flexural loadings up to ultimate failure.

Since the pioneering works of [START_REF] Sutton | Determination of displacements using an improved digital correlation method[END_REF]; [START_REF] Chu | Applications of digital image-correlation techniques to experimental mechanics[END_REF], DIC has been widely used not only in the academic world but also in industry. Recent advances enable the sought displacement field to be decomposed onto a suited library [START_REF] Hild | Digital image correlation: from measurement to identification of elastic properties, a review[END_REF] such as finite element shape functions [START_REF] Besnard | Finite-element" displacement fields analysis from digital images: application to Portevin-Le Chatelier bands[END_REF]. This type of technique has been used based on beam/plate kinematics [START_REF] Hild | Displacement measurement technique for Euler-Bernoulli kinematics[END_REF][START_REF] Réthoré | On the use of NURBS functions for displacement derivatives measurement by digital image correlation[END_REF][START_REF] Leplay | Damage law identification of a quasi-brittle ceramic from a bending test using digital image correlation[END_REF]. The main interest of this approach is that it provides full kinematic fields expressed in standard form for beam/plate theories. Moreover, because of the reduced number of degrees of freedom, measurement uncertainties are significantly reduced. Because the kinematic description has a direct mechanical meaning the interface with a further modeling for identification purposes is direct and does not involve any projection error. However, it may be noted that if the displacement field significantly deviates from that of elastic plate theory (say because of pronounced damage or cracking), then an integrated approach may reveal inaccurate. For this reason, the motion has been analyzed without specific assumption, and then the two-dimensional kinematic field has been reduced to 1D plate kinematics.

Examples of identification of elastic (or more complex constitutive laws) based on DIC measurement are numerous [START_REF] Forquin | A method to determine the toughness scatter of brittle materials[END_REF][START_REF] Hild | Digital image correlation: from measurement to identification of elastic properties, a review[END_REF][START_REF] Périé | Digital Image Correlation and biaxial test on composite material for anisotropic damage law identification[END_REF][START_REF] Eberl | Mechanical Characterization of Coatings Using Microbeam Bending and Digital Image Correlation Techniques[END_REF][START_REF] Leplay | Damage law identification of a quasi-brittle ceramic from a bending test using digital image correlation[END_REF]Grédiac and Hild, 2012). However, damage models have been less studied. [START_REF] Chalal | Direct identification of the damage behaviour of composite materials using the virtual fields method[END_REF] resorted to the virtual fields method to determine the parameters of a linear relationship between the damage parameter and the local shear strain. An alternative route is given by the equilibrium gap method that aims at identifying fields of elastic contrasts [START_REF] Claire | Identification of damage fields using kinematic measurements[END_REF][START_REF] Claire | A finite element formulation to identify damage fields: The equilibrium gap method[END_REF], which may be reinterpreted a posteriori as damage fields [START_REF] Claire | Identification of a Damage Law by Using Full-Field Displacement Measurements[END_REF][START_REF] Crouzeix | An orthotropic variant of the Equilibrium Gap Method applied to the analysis of a biaxial test on a composite material[END_REF]. More recently, it has been shown that the same type of formalism can be used to directly identify the parameters of the damage law in the case of isotropic [START_REF] Roux | Digital Image Mechanical Identification (DIMI)[END_REF] and anisotropic [START_REF] Périé | Digital Image Correlation and biaxial test on composite material for anisotropic damage law identification[END_REF][START_REF] Ben Azzouna | On the identification and validation of an anisotropic damage model by using full-field measurements[END_REF]

descriptions of damage.

Dedicated examples suited to beam-like geometries are less common [START_REF] Forquin | A method to determine the toughness scatter of brittle materials[END_REF][START_REF] Leplay | Damage law identification of a quasi-brittle ceramic from a bending test using digital image correlation[END_REF][START_REF] Hild | Calibration of constitutive models of steel beams subject to local buckling by using digital image correlation[END_REF]. In particular, [START_REF] Leplay | Damage law identification of a quasi-brittle ceramic from a bending test using digital image correlation[END_REF] used the central section of a four point bending test to evaluate a damage law of a ceramic material. In that case, the kinematics (i.e., uniform curvature) decouples from the constitutive law, and hence identification is direct. This procedure was shown to be accurate, but it does not tolerate any deviation from the basic hypothesis of constant curvature. Further, to account for local buckling in steel beams [START_REF] Hild | Calibration of constitutive models of steel beams subject to local buckling by using digital image correlation[END_REF], damage variables have been introduced either in the buckled zone or lumped in a hinge as described in the framework of Lumped Damage Mechanics [START_REF] Cipollina | A simplified damage mechanics approach to nonlinear analysis of frames[END_REF].

A comparable goal is aimed at in the present study, but with arbitrary test geometry of the plate, namely four-point but also three-point bending, or any arbitrary departure from a perfect test. The inhomogeneous kinematics makes the problem more involved as it calls for a dedicated procedure. Let us stress that the information that can be extracted from DIC is too detailed and spatially resolved for the ultimate goal as the details of core crack inception, their shape size, density, or opening are easily accessed but would drive naturally to a probabilistic approach describing crack initiation in the core, and interface debonding. Although it may appear as paradoxical, the too fine description of the kinematics is challenging for the identification of a homogeneous constitutive law [START_REF] Gras | Identification of the out-of-plane shear modulus of a 3D woven composite[END_REF].

In the following, the methodology is illustrated by analyzing a bending test on plasterboard plate. In Section 2, the experimental conditions are described, and first analyses of kinematic data extracted from displacement measurements thanks to DIC are presented. The identification procedure is summarized in Section 3. First, the identification of the elastic modulus of the plasterboard is proposed. Then, the identification of a damage law is detailed and applied to the experiments on plasterboard plate.

Experimental procedure

Plasterboard

Plasterboard is a material composed of lightweight gypsum core lined with paper coatings. Adhesion between paper and gypsum core is promoted by the addition of starch. The latter has definite advantages over other binders in applications for building interior sandwich panels because it is readily available, environmentally friendly, inexpensive, and renewable.

Material and sample geometries

Specimens used in the experiment are prepared from industrial plasterboard plates. The tested samples are cut out at a size 160 × 50 × 13 mm 3 . They are tested on a four-point bending rig with the following characteristics: -outer support span, D lower = 150 mm; -inner yoke span, D upper = 40 mm; -prescribed displacement speed 5 × 10 -2 mm/min until failure; Load-deflection curves are monitored all along the test.

Full field measurements

Global DIC is used to measure the displacements of the lateral surface with Correli-Q4 [START_REF] Besnard | Finite-element" displacement fields analysis from digital images: application to Portevin-Le Chatelier bands[END_REF]. We refer the reader to the latter publication for further details on the specific finite-element DIC code used herein. Let us however stress that although the displacement field is decomposed over finite element shape functions, there is no mechanical modeling involved at this stage. The reference and the last deformed image before failure are shown in Figure 1. A random speckle pattern has been applied on the sample surface to enhance the image contrast. Digital images are recorded during the deformation process using a digital single lens reflex camera (Canon 40D). The image definition is 3888 × 2592 pixels and stored with 8-bit digitization. The displacement at image n is assessed by correlating the n-th image with the initial image (unloaded state considered as the reference). The acquisition frequency is one image per five seconds. The physical size of one pixel is 37 µm. The size of the Q4 elements has been chosen to be 12 pixels or about 450 µm. A priori estimates of the displacement resolution are 0.04 pixel or 1.5 µm.

After the DIC evaluation of the displacement, the measurement data are projected onto plate theory kinematics by computing the deflection and rotation along any cross section. The ROI is chosen such that its horizontal axis coincides with the neutral axis of the sample. The vertical displacement field is then averaged in each section to determine the displacement field for the neutral axis. The latter is then compared with the displacement field calculated numerically via the plate model developed hereafter. The curvature fields are calculated by centered finite differences of the slope field, itself the result of the derivation of the measured displacement field.

Note that the Love-Kirchhoff kinematics (reducing to Euler-Bernoulli kinematics in the present case) could have been imposed directly in the DIC analysis [START_REF] Hild | Displacement measurement technique for Euler-Bernoulli kinematics[END_REF]. A gain in uncertainty is expected from such a procedure when it can be assumed that the elastic properties of the sample remain constant along the plate axis for extended intervals [START_REF] Hild | Calibration of constitutive models of steel beams subject to local buckling by using digital image correlation[END_REF]. For damage, this would imply that a large density of small microcracks is present. As will be seen in the sequel, this assumption is not fulfilled, and therefore it was chosen not to resort to such a potentially limiting assumption and the projection procedure was chosen in this study.

Experimental results

A typical load-deflection curve for plasterboard plate is presented in Figure 2. It exhibits two parts separated by a kink; the linear part corresponds to the elastic regime, while the second nonlinear part is the manifestation of the damage regime. Experimentally obtained load-deflection curves obtained for ten different samples under the same four point bending test fixture are shown in Figure 3. Some of the general observations from these curves of the samples during tests are listed below:

1. within the elastic region of the load-displacement curves, where no significant damage is induced, the responses of the specimens to the applied loads are similar. This is due to a comparable stiffness. However, an initial nonlinearity resulting from initial imperfect contact shows a much stronger variability; 2. the load that corresponds to the inception of the first crack in the plaster core shows a rather narrow scatter; 3. because of the flat shape of the load displacement response, the peak load is rather well defined as 271 ± 19 N, but the deflection at peak load exhibits a much broader scatter 3.0 ± 0.67 mm.

Figure 4 shows maps of the longitudinal (i.e., horizontal) displacement u DIC (x, y) and longitudinal strain xx (x, y) on the observed specimen side as obtained from DIC at the three points A, B, C indicated in Figure 2. At point A, (Figure 4(a)), the elastic regime still prevails; the displacement field is as expected from plate theory, and no discernible crack is observed in the strain field. At the kink of the load displacement curve, point B (Figure 4(b)), the inception of the first macroscopic crack occurs in the plaster core near the tensile face of the specimen. The strain map shows a very pronounced maximum on top of the crack as the discontinuity contributes to the (apparent) strain. Such observations allow for an easy determination of the presence and position of a crack in the specimen. The crack does not propagate through the thickness, nor does it lead to the sudden failure of the specimen for the paper facing is able to bear the load transfer from the plaster core. This is a manifestation of the "ductility" and enhanced strength provided by the paper facing. At point C, (Figure 4(c)) the specimen just before failure shows multiple transverse cracks. Final failure of the sample occurs when the paper bridging one of the cracks cannot withstand the tensile load it is subjected to. The crack opens up and propagates to the opposite face. Such results have already been reported in the literature [START_REF] Benouis | Comportement mécanique des ouvrages en plaques de plâtre sur ossature métallique[END_REF].

A description based on the presence of multiple cracks can be easily designed for such specimen following similar developments proposed for ceramic matrix composites. The inception of transverse matrix cracks does not directly lead to failure, but rather allows for a progressive degradation of the fibre/matrix interface giving rise to an apparent ductility that can be described at different scales [START_REF] Aveston | Single and Multiple Fracture[END_REF][START_REF] Evans | Methodology for Relating the Tensile Constitutive Behavior of Ceramic-matrix Composites to Constituent Properties[END_REF][START_REF] Hild | Matrix Cracking and Debonding in Ceramic-Matrix Composites[END_REF]. The matrix crack density tends to be almost periodic for a high value of the matrix Weibull modulus, and crack separation corresponds to a shielding mechanism (described by, say, a shear lag model) and analytical models have been very successful at describing either probabilistically the formation of successive cracks [START_REF] Curtin | Exact Theory of Fiber Fragmentation in Single-Filament Composite[END_REF][START_REF] Hui | An Exact Closed Form Solution for Fragmentation of Weibull Fibers in a Single Filament Composite with Applications to Fiber-Reinforced Ceramics[END_REF], or deterministically as a homogenized damage [START_REF] Hild | Tensile and Flexural Ultimate Strength of Fiber-Reinforced Ceramic-Matrix Composites[END_REF][START_REF] Hild | Ultimate Strength Properties of Fiber-Reinforced Composites[END_REF]. The second route is followed herein, but rather than reconstructing the constitutive law from individual constituent behavior, the effective behavior of the board as a whole is sought.

Identification scheme

The identification process consists of exploiting the measured dsiplacement fields obtained from DIC. First, the identification of the elastic modulus of the plasterboard is proposed. Then, the identification of a damage law is detailed.

Modeling

We resort to a Love-Kirchhoff description and hence its applicability relies on the satisfaction of the following assumptions: -straight lines normal to the mid-plane remain straight and normal to the mid-plane, -the plate deflection due to shear stresses is neglected, -the slope of the plate is assumed to be negligible compared to unity. In addition, invariance in the z-direction is assumed (cylindrical flexion).

Considering a simply supported plate subjected to lateral in-plane loading, the bending moment M (x) is determined by solving the equilibrium equation ( 1), where the applied transverse loads are denoted as q(x)

d 2 M (x) dx 2 = q(x), 0 ≤ x ≤ L (1)
For the four point bending test, these forces at the four points of contact are denoted q i at positions x i for 1 ≤ i ≤ 4, as shown in Figure 5). In the elastic regime, the local bending stiffness, κ(x) is constant and equal to κ 0

κ 0 = EI (1 -ν 2 ) (2)
as a function of Young's modulus E, Poisson's ratio ν, and inertia of the cross section, I = bh 3 /12 with b the specimen width, and h its thickness. The deflection v(x) obeys

κ(x) d 2 v(x) dx 2 = M (x), 0 ≤ x ≤ L (3)
where L is the outer fixture support span (the sample is supported at x 1 = 0 and x 4 = L). The direct problem consists of solving the differential equation for v(x) using as input the applied load q(x), the stiffness distribution κ(x), and the boundary conditions v(0) = 0, M (0) = 0, v(L) = 0 and M (L) = 0.

In the four-point bending test, it is usually assumed that two equal forces are applied symmetrically with respect to the specimen center (Figure 5), which leads to a constant plate curvature, strain, and stress in the central part of the specimen x 2 ≤ x ≤ x 3 . However in practice, the load is applied through cylinders in contact with the plaster board to limit indentation effects, and hence it is difficult to ascertain the exact location of the equivalent point load. Moreover, a slight dissymmetry is commonly observed, so that the ideal four-point bending description is only a crude approximation of actual tests. Further, damage will very severely amplify those imperfections.

Elastic identification

For the case of statically determinate simply supported plates and uniform bending stiffness, it is simple to solve for the deflection through quadratures.

To take into account the real conditions of the test, the model based on plate theory has been considered without postulating a priori symmetry, and exact positions of the load application points. Although the individual loads q 2 and q 3 are not assumed to be known individually, the total load F = q 2 + q 3 is measured. This can be taken into account by introducing individual loads and positions as unknowns, subjected to known constraints.

The identification is performed through the minimization of the quadratic differences between the measured and modeled deflection of the plate. The objective function to be minimized is written as

T (x i , q i , κ 0 , U, Ω) = L 0 (v DIC (x) -v mod (x, x i , q i , κ 0 )) -U -Ωx) 2 dx +λ(q 2 + q 3 -F ) (4
) where x i is the location of the applied load q i , U is a translation, and Ω a rotation as a rigid body motions inevitably occur in the experiment. The total load, F , which is measured in the experiment is accounted for as an additional constraint, q 2 + q 3 = F , through a Lagrange multiplier, λ. Note that q 1 and q 4 are naturally determined by the balance equations. T is minimized by a progressive adjustment of its parameters through a Newton-Raphson scheme. This allows for an excellent match between the actual test including its imperfection and the modeling.

In Figure 6, the identified positions of the applied load point and their theoretical positions (broken line) are plotted for ten different tests in the elastic regime. The identification procedure leads to average positions x 2 = 55.5 ± 2.7 mm and x 3 = 94.3 ± 3.2 mm. The average values are quite close to the nominal values x nom 2 = 55 mm and x nom 3 = 95 mm. However, the fluctuations around these values are estimated to be significant. In the test, the load is applied through contacting cylinders of radius 25 mm, and hence a small rotation, or indentation of the paper facing may result in a large shift in the effective load application point.

The effective elastic modulus of the specimen (computed from the bending stiffness as if the plate were made of a homogeneous material, i.e., ignoring the partition between plaster core and facings) is another output of the identification procedure. The elastic identification procedure has led to a Young's modulus of 2.52 ± 0.18 GPa. Let us underline that in spite of the observed scatter in the geometrical parameters, the standard deviation of the elastic property is quite small (i.e., 7 %). It is of interest to compare these estimates with those obtained from the classical test equation (assuming nominal values and perfect symmetry). This estimate amounts to 2.40±0.28 GPa. Although both estimates are consistent, it is observed that the latter evaluations have a much broader coefficient of variation (i.e., 12 %) than that obtained with the proposed procedure. This feature already shows one benefit of having a method that is tolerant to unavoidable imperfections of the test set-up. As will be shown below, this advantage is even more important for (and even key to) the evaluation of damage.

Damage identification

When the applied load exceeds the cracking load of the plaster core, cracks develop and propagate inside the tensile part of the bent plate, thereby reducing the bending stiffness, κ(x), of the cross section. In the present approach, it is assumed that degradation of the flexural stiffness κ(x) is accounted for by the classical damage parameter, d(x), such that

κ(x) = κ 0 (1 -d(x)) (5)
Interpretation of Equation ( 5) as a constitutive law means that d(x) is not an arbitrary function of the position x, but rather, d is a deterministic function of the maximum curvature χ(x) experienced in the past by the current cross section, x.

Accounting for Equation (5) in Equation (3) yields

κ 0 (1 -d(χ))χ(x) = M (x), 0 ≤ x ≤ L (6)
with

χ(x) = d 2 v(x) dx 2 (7)
The parameter of the constitutive law relating damage to the curvature, d(χ), is collectively denoted α. The chosen algebraic expression is not specified at this stage, and will be discussed below. The deflection is computed as a solution to this nonlinear differential equation by numerical means. The curvature is calculated through finite differences of the deflection using the mid-point rule. Finally, the minimization of the quadratic differences of experimental and modeled deflections (see Equation ( 4)) is proposed in the same spirit as for the elastic regime. Because of the stability observed in the previous determination of the geometry of the test, and to limit the number of unknowns, the positions of the applied load points have been considered as equal to their nominal values. Thus the minimization is performed over the parameter of damage law, α, the applied loads q i (subjected to known total load), and rigid body motions U and Ω.

It is to be stressed that no specific constraints on the expression of d(χ) are limiting (apart from trivial expectations such as a monotonic increase or d(χ = 0) = 0. Several algebraic forms have been tested, leading to comparable evaluations of the damage law. A simple expression with threshold is chosen to illustrate the results of the identification. Damage is here postulated to obey the following form

d(χ) = max 1 - α |χ| , 0 (8)
where α is the constitutive parameter to be identified. From this equation it is apparent that no damage occurs for |χ| < α, and hence α also plays the role of a threshold. Note that only positive curvatures are considered here, and hence even if upside down symmetry is broken (say because of different paper facings) this will not be probed in the present experimental test.

Figure 7 shows the obtained fit of the plate deflection just prior to failure, together with the measured deflection. A very good agreement is obtained. It should be emphasized that the proposed description is a homogeneous damage law, whereas the actual degradation mechanism consists of discrete cracks. Thus a perfect match between measurement and modeling can no longer be expected. However, for practical purposes, the homogeneous damage representation is much more satisfactory than a discrete crack picture that would call for a probabilistic framework. Among the different strategies that would allow for the identification of the "equivalent" damage law, the present one has the advantage to match at best the measured behavior in an inhomogeneous test, including significant damage gradient, either because the test would be a three point bending one, or because of the natural breakdown of symmetry occurring in four point bending. The excellent quality of the fit to the deflection profile up to failure is a further evidence that the homogeneous damage picture does not betray the actual behavior in a very significant fashion.

Figure 8 shows the damage variable as the function of the curvature for one sample for all load levels. Note that each load level is analyzed independently, and a new set of parameter α is measured. However, when all identified d(χ) laws are plotted on the same graph, a low scatter is observed. It is important to note the existence of a clear threshold curvature (∼ 2×10 -4 mm -1 ) for damage initiation. When the curvature exceeds this threshold curvature, a sudden increase of damage is observed followed by a slower one. The onset of damage is easily interpreted as the inception of cracks in the plaster core. A weibull distribution for the plaster strength with a Weibull modulus typically in the range from 5 to 10 would reproduce such a behavior. After crack initiation, the tensile load in plaster is reduced and transferred to the paper facing, and this in a region whose extension is of the order of the plasterboard thickness. Thus the damage mechanism can be compared to the derivation of damage unidirectional ceramic matrix composites. Crack initiation finally saturates due to screening effects with a well defined distance between cracks (as can be seen in Figure 4(c)). At this stage, the progressive softening of the paper coating due to the increased stress in front of cracks contributes to global damage, yet with a slower rate of increase.

Figure 9 shows the change during a test of the two determined forces applied at positions x 2 and x 3 (i.e., including the entire damage regime). It is clearly observed that after the first macroscopic crack occurs in the plaster core, the test becomes dissymmetric and the load is mostly concentrated on q 3 . The proposed identification procedure reveals a much more complex load distribution than what is traditionally assumed for the 4-point bending. This result displays very large differences between the identified load distribution and the expected theoretical value (i.e., half of the measured load). In particular, the specimen central zone is not subjected to a uniform bending moment.

Summary

Four-point bending tests have performed on plasterboards specimen until failure. DIC allows the mechanisms of mechanical degradation to be investigated, namely, multiple core cracking, paper facing debonding, load redistribution. However, the resulting portray would naturally be a probabilistic framework, involving too sophisticated a modeling for most practical purposes. The choice has been made to select a deterministic damage law to account for the bending strength of plasterboard, and hence a nonlinear identification procedure has been proposed. The methodology based on Digital Image Correlation allowed us to identify simultaneously: -the boundary conditions of the test; -the bending stiffness; -the full damage behavior law.

The results show that the proposed procedure is very effective as it captures the imperfections of the test, and in particular the breakdown of symmetry that inevitably occurs in the damage regime. It is shown that this dissymmetry can be extremely large in the present case, and hence a classical analysis assuming an ideal test would lead to erroneous results. Note that even in the elastic regime, such a procedure allowed for reducing the scatter in the bending rigidity to half its value when a standard identification is carried out.

The methodology proposed in this study can be applied straightforwardly to other quasi-brittle materials or structures such as reinforced concrete or ceramic matrix composites. 9: Change of the identified loads q 2 and q 3 all along the test in the damage regime. In an ideal test, the ratio q 2 /(q 2 + q 3 ) is constant and equal to 0.5

Figure 1 :

 1 Figure 1: Simply supported plate in its reference (a) and deformed (b) states for the last load level before failure. The black speckle paint applied onto the surface is used for DIC analyses. The analyzed ROI (white box) and the neutral axis (dashed line) are shown for the reference configuration

Figure 2 :Figure 4 :

 24 Figure 2: Typical experimental load-deflection curve of the four-point bending test. Point A shows the elastic response, point B the inception of cracks in the plaster core, and C final failure

Figure 6 :

 6 Figure 6: Identified positions of the applied load point and their theoretical positions (dotted lines) for ten tests

Figure 8 :

 8 Figure 8: Damage parameter versus curvature for one sample for all load levels. Note that each load level is analyzed independently, and yet the scatter in the twenty seven different determinations of this damage growth law is small

Figure

  Figure9: Change of the identified loads q 2 and q 3 all along the test in the damage regime. In an ideal test, the ratio q 2 /(q 2 + q 3 ) is constant and equal to 0.5

Acknowledgements

This work has been financially supported by Saint-Gobain Recherche and the ANRT through grant no. 2010/567. We wish to thank particularly René Gy for useful and stimulating discussions.

Change of the identified loads q 2 and q 3 all along the test in the damage regime. In an ideal test, the ratio q 2 /(q 2 + q 3 ) is constant and equal to 0.5 . . . . . . . . . . . . . . . . . . . . . 24