Morgan Bertin

François Hild
email: hild@lmt.ens-cachan.fr

Stéphane Roux

On the Identifiability of Hill-1948 Model with One Uniaxial Tensile Test

Keywords: Digital Image Correlation (DIC), Full field measurements, Identification, Plastic anisotropy Mots-clés : Corrélation d'images numériques (CIN), Mesures de champ, Identification, Anisotropie plastique

A uniaxial experiment is performed on ultra-thin specimen made of 17-7 precipitation hardened stainless steel.

An anti-wrinkling setup allows for the characterization of the mechanical behavior with Integrated Digital Image Correlation (IDIC). The result shows that a single uniaxial experiment investigated via IDIC possesses enough data (and even more) to characterize a complete anisotropic elastoplastic model.

Résumé

De l'identifiabilité du modèle de Hill-1948 à l'aide d'un seul essai de traction unixiale. Un essai de traction uniaxiale est conduit sur un échantillon en acier inoxydable 17-7 à durcissement structural. Un montage anti-plissement permet la caractérisation du comportement mécanique par corrélation d'images numériques intégrée (CINI). Il est montré que ce seul essai analysé par CINI possède assez d'information (voire plus) pour une identification complète du modèle de plasticité anisotrope.

Introduction

For engineering design purposes, von Mises' [START_REF] Roux | Mechanik der festen Körper im plastisch-deformablen Zustand[END_REF] criterion is commonly used and gives satisfactory results.

However, industrially-manufactured materials require plastic anisotropy to be accounted for. For instance, sheet metal forming calls for anisotropic plasticity models. Numerous authors have actively discussed their characterization. Whiteley [START_REF] Whiteley | The importance of directionality in deep drawing quality sheet steel[END_REF] and Lloyd [START_REF] Lloyd | Metallurgical Engineering in the Pressed-Metal Industry[END_REF] investigated the importance of directionality in deep drawing of sheet steel and so-called pressed-metals. Lankford et al. [START_REF] Lankford | New criteria for predicting the press performance of deep drawing sheets[END_REF] proposed an experimental ratio to characterize plastic anisotropy. Hill [START_REF] Hill | A theory of the yielding and plastic flow of anisotropic metals[END_REF] and Dorn [START_REF] Dorn | Stress-Strain Rate Relations for Anisotropic Plastic Flow[END_REF] introduced the first models to describe anisotropic plastic flow. These models have been extended to describe more complex constitutive laws [START_REF] Hill | Continuum micro-mechanics of elastoplastic polycrystals[END_REF]. Many developments have followed since these early propositions. However, one of the standard and most widely used model in nowadays commercial finite element codes [START_REF] Soare | Applications of a Recently Proposed Anisotropic Yield Surface to Sheet Metal Forming[END_REF] still is Hill's original proposition (i.e., the so-called Hill-1948 model [START_REF]User's Manual, Version[END_REF]).

The calibration of such models calls for several tensile tests, at least in three directions (i.e., generally 0 • , 90 • , and 45 • with respect to the rolling direction [START_REF]Standard Test Method for Plastic Strain Ratio r for Sheet Metal[END_REF]). It is worth noting that more complex test configurations are also used (e.g., plain strain tensile test, shear test, experiment on cross-shaped samples, bulge test) even though very few of them are standardized. Most of these tests consider zones where the strain and stress states are uniform so that simple extraction techniques are considered to determine the sought quantities. With the development of full-field measurement techniques [START_REF]Full-Field Measurements and Identification in Solid Mechanics[END_REF], another route consists of analyzing and designing the test as being fully heterogeneous [START_REF] Bertin | Optimization of a biaxial tensile specimen geometry for the identification of constitutive parameters based upon full field measurements[END_REF].

The present study aims at identifying an anisotropic plasticity law, investigating Hill-1948 quadratic yield function via Integrated-DIC [START_REF] Hild | Measuring stress intensity factors with a camera : integrated digital image correlation (I-DIC)[END_REF][START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF][START_REF] Réthoré | A fully integrated noise robust strategy for the identification of constitutive laws from digital images[END_REF][START_REF] Réthoré | Robust identification of elasto-plastic constitutive law parameters from digital images using 3D kinematics[END_REF][START_REF] Mathieu | Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated DIC[END_REF][START_REF] Bertin | Integrated digital image correlation applied to elastoplastic identification in a biaxial experiment[END_REF][START_REF] Hild | Toward 4D mechanical correlation[END_REF]. The key feature consists in analyzing a single sample made of ultra-thin precipitation hardened steel and tested in uniaxial tension. A sensitivity analysis is performed to assess the identifiability of an anisotropic yield surface coupled with linear kinematic hardening and isotropic elasticity. It will be shown that seven material parameters can be evaluated even though the sample geometry remains very simple.

Methodological framework

Investigated constitutive law

In the following, linear and isotropic elasticity (with Young's modulus E and Poisson's ratio ν), and linear kinematic hardening with a quadratic yield criterion [START_REF] Hill | A theory of the yielding and plastic flow of anisotropic metals[END_REF] are coupled. The total strain rate ˙ is written in terms of elastic and plastic strain rates ˙ = ˙ el + ˙ pl where ˙ el is the elastic strain rate tensor and ˙ pl the plastic strain rate tensor. As a first approximation, a linear kinematic hardening model is chosen for the back-stress rate [START_REF] Prager | A New Method of Analyzing Stresses and Strains in Work-hardening Plastic Solids[END_REF] in which C is the hardening modulus. Last, the quadratic [START_REF] Hill | A theory of the yielding and plastic flow of anisotropic metals[END_REF] anisotropic yield criterion is expressed in terms of rectangular Cartesian stress components as

f (σ) = H 1 (σ 22 -σ 33 ) 2 + H 2 (σ 33 -σ 11 ) 2 + H 3 (σ 11 -σ 22 ) 2 + 2H 4 σ 2 23 + 2H 5 σ 2 31 + 2H 6 σ 2 12 ( 1
)
where H i are constants expressed with the normal yield stress ratios (R

H 1 = 1 2 1 R 2 22 + 1 R 2 33 - 1 R 2 11 , H 4 = 3 2R 2 23 H 2 = 1 2 1 R 2 33 + 1 R 2 11 - 1 R 2 22 , H 5 = 3 2R 2 31 , H 3 = 1 2 1 R 2 11 + 1 R 2 22 - 1 R 2 33 , H 6 = 3 2R 2 12
The computations being two dimensional, R 31 and R 23 are insensitive and set to 1 hereafter (and hence H 4 = H 5 = 3/2). One of the in-plane parameters must be chosen to decorrelate its value from the yield stress σ y (i.e., H 1 is set to 1/2 in the sequel). Therefore, only three additional material parameters, i.e., H 2 , H 3 , and H 6 , remain to be determined when plastic anisotropy is accounted for. To conclude, 7 parameters (i.e., E, ν, σ y , C, H 2 , H 3 , and H 6 ) need to be calibrated with the chosen model.

Integrated-DIC

The calibration of the material parameters {p} is performed via Integrated Digital Image Correlation where the unknowns are no longer kinematic degrees of freedom but the sought constitutive parameters themselves. Integrated-DIC is a global technique that registers an image f in the reference configuration and a series of N t pictures g in the deformed configurations. Assuming gray level conservation, the inverse problem is solved by minimizing the sum of squared differences between the deformed image corrected by the trial displacement u(x, t, {p}) and the reference image

χ 2 f ({p}) = 1 2γ 2 f N Ω N t t Ω ((g(x + u(x, t, {p}), t) -f (x)) 2 (2)
with respect to {p}. In this expression Ω denotes the Region of Interest (ROI), N Ω its area in terms of the number of pixels it contains, x any considered pixel, and γ f = 323 gray levels is the standard deviation (i.e., 2% of the dynamic range of f ) of the white noise assumed to affect each image independently (including the reference one, which is responsible for the factor of 1/2 coming as a multiplicative term in this functional). The load is also of importance for the identification because it increases the number of the measured quantities and as a consequence diminishes the relative uncertainty by enhancing the material parameter sensitivities [START_REF] Bertin | Optimization of a biaxial tensile specimen geometry for the identification of constitutive parameters based upon full field measurements[END_REF]. Thus, a second objective function, χ 2 F , is introduced

χ 2 F = 1 N F N t γ 2 F {F m -F c ({p})} t {F m -F c ({p})} (3)
where {F m } are the measured reaction forces and {F c ({p})} the computed levels that depend on the chosen material parameter set, N F the number of load cells, and γ 2 F the load variance (i.e., 100 N in the present case). The identification, which is based upon both observables (i.e., gray levels and reaction forces), is achieved by minimizing the global functional χ 2 ({p})

χ 2 ({p}) = N Ω N Ω + N F χ 2 f ({p}) + N F N Ω + N F χ 2 F ({p}) (4)
where the DIC and reaction force functionals have been introduced in Equations ( 2) and (3) respectively.

The choice for the specific weight is issued from a Bayesian approach whereby the noise characteristics imposes the appropriate weights to be given to quadratic differences [START_REF] Bertin | Optimization of a biaxial tensile specimen geometry for the identification of constitutive parameters based upon full field measurements[END_REF]. In particular, quadratic differences are to be evaluated with a metric equal to the inverse noise covariance.

When nonlinear behaviors are investigated, the dependence of the displacement field with the material parameters is nonlinear as well. Consequently, sensitivity fields (i.e., displacement and reaction force derivatives with respect to the sought parameters) are computed via finite element simulations. In the present case the commercial (implicit) finite element code used is Abaqus standard [START_REF]User's Manual, Version[END_REF]. It is driven by the measured displacements (via FE DIC) on the upper and lower edges of the considered region of interest to compute the displacement fields and reaction forces. If the FE mesh is finer in Integrated DIC in comparison to DIC, the measured displacements on the coarser mesh are interpolated on the finer mesh according to the measurement discretization. A Gauss-Newton procedure is implemented to minimize the global functional by iteratively updating the material parameters [START_REF] Bertin | Integrated digital image correlation applied to elastoplastic identification in a biaxial experiment[END_REF].

Sensitivity analysis

To probe the identifiability of the 7 parameters, a sensitivity analysis is carried out. Consequently, the Hessian of the Integrated DIC code is analyzed [START_REF] Bertin | Optimization of a biaxial tensile specimen geometry for the identification of constitutive parameters based upon full field measurements[END_REF]. Figure 1(a) shows the global Hessian (H IF DIC ) of the IDIC procedure. The material parameters having the largest influence are H 1 , H 2 , and H 3 and account for the anisotropic plasticity behavior (H 1 is analyzed to show its correlations with other parameters but for identification purposes its value is kept equal to one half). This result shows that a uniaxial experiment enables for the characterization of an anisotropic behavior. It also indicates that this behavior must be accounted for when plastic anisotropy is suspected to occur. Figure 1(b) shows that the Poisson's ratio is correlated to H 1 , H 2 , and H 3 , and explains why it is necessary to have a model that separates both contributions from the Poisson's ratio and the anisotropic coefficients. Once the lowest eigen value of the complete Hessian has been taken out, its condition number is of the order of 10 5 , which is not too high given the number of remaining parameters. From this analysis, it is concluded that the 7 parameters are identifiable. The main experimental challenge concerns the specimen thickness that prevents grooved grips to be used. The solution consists in bonding the specimen ends to flat-surface aluminum alloy tabs with a 3M Scotch-Weld Structural Adhesive Film AF 126 [22]. Residual wrinkles are generated during this process near the edges. To maintain the surface flat in the center region (crucial to perform 2D-DIC analyses) an anti-wrinkling device is designed (see Figure 2). It consists in holding the specimen between two Plexiglas sheets fastened with four screws. A very small gap is left between the Plexiglas sheets to minimize as much as possible the effects of friction on the tested sample. To perform DIC analyses a random speckle pattern is applied on the monitored surface and no paint loss was observed during the test, thereby validating the hypothesis of negligible friction induced by the anti-wrinkling device. wrinkles are generated during this process near the boundaries. To maintain the surface flat in the center region (crucial to perform 2D-DIC analyses) an anti-buckling device is designed (see Figure 5.1). It consists of holding the specimen between two plexiglas sheets maintained with four screws. To perform DIC analyses a random pattern is applied on the monitored surface.

The study aims to assess the mechanical properties, and validates, the investigated constitutive laws. Section 5.2 introduces the chosen constitutive models and recalls the integrated-DIC identification methodology. Section 5.3 shows the identification results. The DIC residual remains relatively constant over the experiment. Conversely, the IDIC residuals are on average higher than DIC residuals. Let us stress that DIC is based on a number of degrees of freedom equal to 600 per image, or 114,600 for the entire experiment, whereas the IDIC approach has no more than 7 degrees of freedom. This very drastic reduction incorporates many constraints due to a priori assumption on the material behavior. The difference in residuals from DIC to IDIC is an indication of the model error that is due to the specific chosen constitutive law. Once the elastic regime ends (t ≥ 20 s), the IDIC residuals increase and exceed the DIC residuals. This result shows that the assumption of a constant anisotropic behavior, i.e., constant anisotropic coefficients as a function of the plastic strain, is not totally consistent with the experimental data. Similarly, Figure 3(b) shows that the load levels are consistent in the elastic regime and gradually degrade in anisotropic plasticity. Table 1 gathers the assessed material parameters for the studied constitutive law. Their initial values ({p 0 }) are chosen according to literature data [START_REF] Ak-Steel | [END_REF][START_REF] Leybold-Herbert | Axial-load fatigue tests on 17-7 PH stainless steel under constant-amplitude loading[END_REF]. The initial and last values of the residuals are also shown (1 st -last ) for the investigated law plus one identification with a coarser mesh (referred to as IDIC * ). All the parameters converge to a stabilized level while the residuals χ for each laws are minimized.

Numerical tools

First, the IDIC residual is greater than the DIC residual obtained with a coarse mesh (i.e., the one used to measure the boundary conditions) but lower than the IDIC * residual obtained on the same mesh. This is an indication that model errors occur. It proves that refining the mesh (Figure 2) can enhance the identification quality by decreasing the gray level residuals. This mesh difference also explains why the DIC residual is larger than the IDIC residuals during the first time steps (t ∈ [10; 20] s). The load residuals also decrease.

Table 1 Identified parameters and identification residuals via IDIC over the entire loading history cycle (IDIC * corresponds to the DIC mesh and "1 st -last" corresponds to the first and last value of the residuals) The lowest global residual corresponds to 7 times the noise level, which indicates that the model does not fully capture all the experimental behavior. Figure 4 shows the displacement field residual for the transverse component (i.e., difference between the IDIC and DIC displacement fields). The displacement difference is not noise (i.e., measurement uncertainty) but rather a smooth component that is not totally accounted for by the model. This is a clear manifestation of model error. The short wavelength component close to the upper edge is believed to be due to small wrinkles. The Lankford coefficient is an indicator used to describe the anisotropic plasticity behavior of sheet metals [START_REF]Standard Test Method for Plastic Strain Ratio r for Sheet Metal[END_REF]. It corresponds to r = p 33 / p 11 where p 11 is the transverse plastic strain, p 33 the out-of-plane plastic strain. This analysis is performed in the central area (see Figure 2(c)) of the specimen and the reported strains are the corresponding mean values. Figure 5 shows the Lankford coefficients both for DIC and IDIC results. For DIC analyses, the elastic strains are evaluated from the applied load and the knowledge of the elastic properties of the alloy. Incompressibility is assumed for the plastic strains to evaluate p 33 . The coefficient converges for both cases to a constant value as the level of plastic strain increases. The asymptotic value (i.e., about 2) is consistent with literature data of cold-rolled stainless steel [START_REF] Charca Ramos | Study of a drawing-quality sheet steel. I: Stress/strain behaviors and Lankford coefficients by experiments and micromechanical simulations[END_REF]. Second, the model does not fully capture the experimentally observed anisotropy in the early stages of plasticity. This result shows that even if the identification of the anisotropic coefficients has been achieved, the assumption of a constant anisotropic behavior is an approximation. This is consistent with the fact that the IDIC residuals are higher at the end of cycle 1 and the beginning of cycle 2 (see Figure 3).

χ χ f χ F E ν σ 0 C H 2 H 3 H 6 1 st -last 1 st -last 1 st -last GPa MPa GPa
It is to be emphasized that using full field measurements allows for the calibration of the entire set of parameters using a single experiment. Such a conclusions was also drawn for a thicker sample made of aluminum alloy [START_REF] Neggers | Improving full-field identification using progressive model enrichments International Journal of Solids and Structures[END_REF]. It also gives access to an evaluation of a significant model error (based on the value of the final residuals being well above 1, see Table 1), and it provides in addition an assessment of the deviation in the observed and fitted strains through the Lankford ratio, which may help enriching the constitutive law (say with an evolving anisotropy) in order to reduce the model error.

Conclusion

The identifiability of the Hill-1948 model has been investigated with Integrated Digital Image Correlation for a uniaxial test on ultra-thin sheet made of precipitation hardened stainless steel. A dedicated anti-wrinkling system was designed and employed to prevent non uniform out-of-plane displacements while loading and unloading the specimen. The identification is performed over the entire history accounting for three loading/unloading cycles. The sensitivity analysis proves that the anisotropic yield criterion can a priori be quantitatively evaluated. It is shown a posteriori that it is possible to determine all material parameters for the in-plane anisotropic plastic model in a single uniaxial experiment thanks

log 10 (Figure 1 .

 101 Figure 1. Global Hessian (a) and its corresponding correlation matrix (b) for the investigated model (i.e., {p} = {E, ν, σy, C, H 1 , H 2 , H 3 , H 6 } t ). The absolute value of the Hessian is used to allow for a logarithmic scale

  Figure 2 shows at the last experimental time step the measured (b) and identified (c) displacement fields associated with three-noded triangular meshes with linear interpolation (T3). For DIC, the mesh size is of the order of 60 pixels, and 40 pixels for Integrated DIC.

Figure 5 . 2 :

 52 Figure 5.2: (a,b) DIC and (c,d) IDIC (law D) displacement field at the last experimental time step. (c) also shows the area that w Lankford coefficient. The displacements are express

Figure 5 .

 5 Figure 5.4(a) shows the global Hessian (H IFDIC ) of the IDIC

Figure 5 . 2 :

 52 Figure 5.2: (a,b) DIC and (c,d) IDIC (law D) displacement fi at the last experimental time step. (c) also shows the area tha Lankford coefficient. The displacements are expr

Figure 5 .Figure 5 . 1 :

 551 Figure 5.4(a) shows the global Hessian (H IFDIC ) of the ID

2 (Figure 5 . 2 :

 252 Figure 5.2: (a,b) DIC and (c,d) IDIC (law D) displacement field at the last experimental time step. (c) also shows the area that wi Lankford coefficient. The displacements are expresse

Figure 5 .Figure 5 . 2 :

 552 Figure 5.4(a) shows the global Hessian (H IFDIC ) of the IDIC p

Figure 5 .Figure 5 . 2 :

 552 Figure 5.4(a) shows the global Hessian (H IFDIC ) of the ID

Figure 5 .Figure 5 . 1 :

 551 Figure 5.4(a) shows the global Hessian (H IFDIC ) of the IDIC procedure. The material

5. 2

 2 Numerical tools 5.2.1 Constitutive laws Four constitutive laws are investigated, namely, (A) linear and isotropic elasticity, (B) linear kinematic hardening with von Mises flow rule, (C) exponential kinematic hardening

Figure 5 . 2 :

 52 Figure 5.2: (a,b) DIC and (c,d) IDIC (law D) displacement fields U X (a,c) and U Y (b,d) at the last experimental time step. (c) also shows the area that will be used to assess the Lankford coefficient. The displacements are expressed in pixels

Figure 5 . 2 ( 5 . 2 :

 5252 Figure 5.4(a) shows the global Hessian (H IFDIC ) of the IDIC procedure. The material

Figure 5 .

 5 Figure 5.4(a) shows the global Hessian (H IFDIC ) of the IDIC p

Figure 5 . 2 :

 52 Figure 5.2: (a,b) DIC and (c,d) IDIC (law D) displacement fi at the last experimental time step. (c) also shows the area tha Lankford coefficient. The displacements are expr

Figure 5 .Figure 5 . 2 :

 552 Figure 5.4(a) shows the global Hessian (H IFDIC ) of the ID

Figure 5 .Figure 5 . 1 :

 551 Figure 5.4(a) shows the global Hessian (H IFDIC ) of the IDIC procedure. The material

5. 2 . 1

 21 Constitutive lawsFour constitutive laws are investigated, namely, (A) linear and isotropic elasticity, (B) linear kinematic hardening with von Mises flow rule, (C) exponential kinematic hardening with von Mises flow rule, and (D) linear kinematic hardening with quadraticHill [31]

Figure 5 . 2 :

 52 Figure 5.2: (a,b) DIC and (c,d) IDIC (law D) displacement fiel at the last experimental time step. (c) also shows the area that w Lankford coefficient. The displacements are express

Figure 5 .

 5 Figure 5.4(a) shows the global Hessian (H IFDIC ) of the IDIC

Figure 2 .Figure 3

 23 Figure 2. (a) Uniaxial specimen with dedicated anti-wrinkling device. (b) DIC and (c) IDIC displacement fields ux at the last experimental time step with two mesh sizes; (c) also shows the area that will be used to assess the Lankford coefficient. The displacements are expressed in pixels (1 pixel ↔ 70 µm)

Figure 3 .

 3 Figure 3. (a) Gray level residual history χ f (t) for DIC and IDIC. (b) Applied and computed reaction force history

Figure 4 .

 4 Figure 4. Transverse displacement field residual (IDIC-DIC) expressed in pixels (1 pixel ↔ 70 µm). Its corresponding root mean square level is equal to 0.19 pixel

Figure 5 .

 5 Figure 5. Lankford ratio evaluated from DIC and from the identified anisotropic elastoplastic anisotropic

to full-field measurements. Moreover, the same experimental procedure is rich enough to indicate a model error and provides some insight into directions for enrichment of the constitutive law to better describe the experiment.

Acknowledgments

The authors acknowledge the help of Dr. Hugo Leclerc in the implementation of the DIC codes. This work was supported by the Région Ile de France (THERMOFLUIDE-RT project) and BPI France (DIC-CIT project).


Jan Neggers
email: jan.neggers@ens-paris-saclay.fr

Florent Mathieu

François Hild

Stéphane Roux

Nicolas Swiergiel

Improving full-field identification using progressive model enrichments

Keywords: Anisotropic plasticity, Damage, Digital image correlation, Full-Field measurements, Identification linear elastic regime [2]

come

Introduction

There is a constant need for calibrating the parameters of material models. Most modern engineering materials are created from mixtures of multiple materials using highly specific micro-architectures [START_REF] Ashby | Engineering materials 2: an introduction to microstructures, processing and design[END_REF]. This allows them to be optimized to a high extent. The interest in these materials is often beyond their The mechanics of materials community has been prolific in providing an abundance of interesting nonlinear material models, each describing the material behavior with large sets of parameters [START_REF] Lemaitre | Handbook of Materials Behavior Models[END_REF]. At this moment, the experimental methods have reached a data density level rich enough to identify these multiparameter nonlinear models. The main source for the increase in data density comes from the maturity of full-field measurement methods such as Digital Image Correlation (DIC [START_REF] Sutton | Image Correlation for Shape, Motion and Deformation Measurements[END_REF][START_REF] Sutton | Computer Vision-Based, Noncontacting Deformation Measurements in Mechanics: A Generational Transformation[END_REF]).

The classical approach to parameter calibration is to optimize the experiment such that it is only sensitive to a limited number of material parameters [START_REF]E8 / E8M-15a[END_REF].

For instance, uniaxial experiments gained popularity due to their near homogeneous stress state over the entire sample, which allow for estimations of stress and strain from the displacements and forces. Full-field methods enable one to deviate from this path since they capture the heterogeneous kinematics of the experiment [START_REF] Bonnet | Inverse problems in elasticity[END_REF][START_REF] Baldi | Full field methods and residual stress analysis in orthotropic material. II: Nonlinear approach[END_REF][START_REF]Full-Field Measurements and Identification in Solid Mechanics[END_REF]. In the latter each material point experiences a different stress/strain history and hence it may provide a different clue about the material model during a single experiment.

The most common identification method is referred to as Finite Element Model Updating (FEMU [START_REF] Kavanagh | Finite element applications in the characterization of elastic solids[END_REF][START_REF] Meuwissen | Determination of the elasto-plastic properties of aluminium using a mixed numerical-experimental method[END_REF][START_REF] Ghouati | Identification of material parameters directly from metal forming processes[END_REF][START_REF] Avril | Overview of Identification Methods of Mechanical Parameters Based on Full-field Measurements[END_REF][START_REF] Pottier | Contribution of heterogeneous strain field measurements and boundary conditions modelling in inverse identification of material parameters[END_REF][START_REF] Wang | Shape features and finite element model updating from full-field strain data[END_REF]). In FEMU the gap between the experiment and simulations of the same experiment is minimized by optimizing (i.e. updating) the unknown model parameters. Within this paper a similar method is applied, which is referred to as Integrated-DIC [START_REF] Roux | Stress intensity factor measurements from digital image correlation: Post-processing and integrated approaches[END_REF][START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF][START_REF] Réthoré | A fully integrated noise robust strategy for the identification of constitutive laws from digital images[END_REF][START_REF] Réthoré | Robust identification of elasto-plastic constitutive law parameters from digital images using 3D kinematics[END_REF][START_REF] Mathieu | Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC[END_REF][START_REF] Neggers | Time-Resolved Integrated Digital Image Correlation[END_REF].

The latter optimizes the gap between simulation and experiment directly on the captured images by integrating the identification step in the DIC algorithm.

However, the differences between the two methods are not essential for the discussion in this paper. The interested reader is referred to Ref. [START_REF] Mathieu | Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC[END_REF] for more details on this last point.

Full-field identification methods such as FEMU and Integrated-DIC use the gap between the experiment and simulations with different metrics, which can be made totally consistent [START_REF] Mathieu | Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC[END_REF]. This gap, or residual, is not only required inside the respective algorithms, it also provides means for visualizing where it is the largest in space and time. In this paper the residuals will be used to progressively enrich the material model, thereby reducing the gap and thus improve the identification quality.

The full-field identification methods typically rely on sensitivity fields as part of the tangent operator in the iterative optimization algorithm [START_REF] Tarantola | Inverse Problems = Quest for Information[END_REF][START_REF] Cooreman | Elastoplastic material parameter identification by inverse methods: Calculation of the sensitivity matrix[END_REF][START_REF] Pagnacco | Parameter Identification from Mechanical Field Measurements using Finite Element Model Updating Strategies[END_REF]. As the name suggests they indicate the sensitivity of a certain parameter on space and time data of interest. Conversely, they show how the simulation will change with a small variation in a given parameter. In this paper it will be shown that these fields are invaluable to diagnose which part of the model is missing or wrong. The comparison between the sensitivity fields and the residual fields provides guidance on how to enrich the model and reduce their level.

Besides FEMU and Integrated-DIC there are other inverse identification methods (see e.g. [START_REF] Avril | Overview of Identification Methods of Mechanical Parameters Based on Full-field Measurements[END_REF] for an overview). This article focuses on two of these methods because both utilize the sensitivity fields in the same way. However, the concepts discussed herein may be equally valid for other identification methods such as the equilibrium gap method [START_REF] Claire | A finite element formulation to identify damage fields: the equilibrium gap method[END_REF] or the virtual fields method [START_REF] Grédiac | Applying the Virtual Fields Method to the identification of elasto-plastic constitutive parameters[END_REF][START_REF] Avril | Identification of elasto-viscoplastic parameters and characterization of Lüders behavior using digital image correlation and the virtual fields method[END_REF]. It is always possible to perform a simulation of the experiment with the obtained parameters of any identification method and compute the residual between the experiment and the simulation. These residuals are an accessible (though often omitted) tool for analyzing the quality of identification.

In Section 2, Integrated-DIC is briefly detailed in order to introduce concepts such as sensitivity fields and residual fields. Additionally, the tensile experiment on a dog-bone sample made of aluminum alloy 2219 is introduced and the finite element simulations are described. Section 3 discusses five identification cases using various constitutive models to analyze how successive enrichments can be used to assess identification quality. The five cases are discussed in parallel such that the differences between each can be discussed side-by-side. This parallel structure is beneficial for the discussion of the results. However, it is less optimal when introducing the five cases, which are inspired from each other based on results that will only be presented later on.

Identification Framework

The chosen identification method is Integrated-DIC (I-DIC). It integrates identification in digital image correlation by choosing the static and kinematic basis functions such that the degrees of freedom are directly the to-be-identified parameters. The interested reader is referred to the literature for additional details [START_REF] Hild | Digital Image Correlation: from Displacement Measurement to Identification of Elastic Properties -a Review[END_REF][START_REF] Réthoré | A fully integrated noise robust strategy for the identification of constitutive laws from digital images[END_REF][START_REF] Mathieu | Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC[END_REF][START_REF] Neggers | Time-Resolved Integrated Digital Image Correlation[END_REF]. However, the method is summarized to define certain aspects used to analyze the identification results. A related method is FEMU.

In this method the distance between simulated and measured quantities, such as displacement [START_REF] Pagnacco | Parameter Identification from Mechanical Field Measurements using Finite Element Model Updating Strategies[END_REF] and/or force [START_REF] Kavanagh | Finite element applications in the characterization of elastic solids[END_REF][START_REF] Pagnacco | Parameter Identification from Mechanical Field Measurements using Finite Element Model Updating Strategies[END_REF], are minimized by iteratively updating the finite element model parameters. The key difference between FEMU and I-DIC is that FEMU minimizes the gap based on a measured displacement field and I-DIC minimizes the image residual directly using the model to drive the kinematics. This integration has advantages for cases where a fine mesh is required to accurately capture for instance complex sample geometry [START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF] or strain concentrations [START_REF] Lindner | On the evaluation of stress triaxiality fields in a notched titanium alloy sample via integrated {DIC}[END_REF]. The displacement uncertainty of non-integrated DIC is inversely related to the element (or subset) size [START_REF] Hild | Comparison of Local and Global Approaches to Digital Image Correlation[END_REF], where for I-DIC this is not the case since the degrees of freedom are not the nodal or subset displacements but the unknown material parameters. For cases where the smallest element size is not critical for DIC it can be shown that the two methods are equivalent, provided that the noise level is small and an appropriate metric is chosen in FEMU [START_REF] Mathieu | Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC[END_REF][START_REF] Ruybalid | Comparison of the identification performance of conventional FEM updating and integrated DIC Andre[END_REF].

Integrated DIC

In Integrated-DIC the objective is to seek the optimum set of parameters {p} = {p 1 , . . . , p n } that minimize the distance between a reference image f and a series of deformed images g τ for a given region of interest and a series of time steps τ . The deformed images are back transformed using a displacement field u τ that depends on the defined parameter set {p}. The objective is to minimize the cost function,

η 2 I ({p}) = 1 2N τ N k γ 2 I Nτ τ N k k f k -gkτ ({p}) 2 , (1)
where, f k denotes the gray value of a pixel at location x k in the image of f and For the experiments discussed within this paper, the force on the grips of the tensile machine was also captured during the experiments. To include these data in the identification method a second cost function is defined,

gkτ = g τ x k + u k ({p},
η 2 F ({p}) = 1 N τ γ 2 F Nτ τ F exp τ -F τ ({p}) 2 , (2)
where, F exp τ is the measured force for time step τ and F τ the corresponding simulated force. Again, this cost function is scaled with the standard uncertainty of the force sensor γ F such that its expectation value approaches unity when converged in the presence of only acquisition noise.

Since the identified parameters must hold for both cost functions, they are combined to a single cost function,

η 2 = N k N k + 1 η 2 I + 1 N k + 1 η 2 F . (3)
This extensive addition of the two cost functions is interesting as it provides the optimal cost function for the parameters in the sense that the resulting estimate will have the smallest variance provided the solution has converged and that the only uncertainty is acquisition noise.

The cost function (3) is minimized using Gauss-Newton's iterative routine that starts with an initial guess for {p 0 } and computes the iterative updates at iteration l to the degrees of freedom {p} (l+1) = {p} l + {δp},

[M ]{δp} = {b}, (4)
[M I ] + [M F ] {δp} = {b I } + {b F }, (5)
where, [M ] is the Hessian matrix and {b} the right hand member. Typically, they are decomposed their respective counter parts for each individual cost function, namely, [M I ], [M F ], {b I } and {b F }. They are defined as (for details see [START_REF] Mathieu | Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC[END_REF][START_REF] Neggers | Time-Resolved Integrated Digital Image Correlation[END_REF]),

M Iij = 1 2N τ N k γ 2 I Nτ τ N k k ϕ Ikτ i • ∇f k ∇f k • ϕ Ikτ j , (6)
b Ii = 1 2N τ N k γ 2 I Nτ τ N k k ϕ Ikτ i • ∇f k f k -gkτ ({p}) , (7)
M F ij = 1 N τ γ 2 F Nτ τ S F τ i S F τ j , (8) b
F i = 1 N τ γ 2 F Nτ τ S F τ i F exp τ -F τ ({p}) , (9)
where, ∇f is the image gradient and [ϕ I ] and [S F ] are projection matrices that project the data space onto the parameter space. The image projection matrix is decomposed again using the FE shape-functions,

ϕ Ikτ i = ∂u kτ ∂p i ≈ Na j ∂u kτ ∂a jτ ∂a jτ ∂p i = Na j ψ kjτ S Ijτ i , (10)
where {a} are the nodal degrees of freedom of the FE mesh, N a the number of degrees of freedom and [ψ] the corresponding FE shape-functions. The decomposition of [ϕ I ] into FE shape-functions and the image sensitivity matrix [S I ] is a common choice [START_REF] Réthoré | Robust identification of elasto-plastic constitutive law parameters from digital images using 3D kinematics[END_REF][START_REF] Mathieu | Estimation of Elastoplastic Parameters via Weighted FEMU and Integrated-DIC[END_REF]. It allows the shape-functions to be reused, which are also applied in a non-integrated DIC procedure and in the FE simulations that are run during the identification process. The image and force sensitivity matrices are then computed using finite differences,

S Iijτ ≈ âijτ -a jτ p i , (11)
S F iτ ≈ Fiτ -F τ p i , (12)
where {a} and {F } are the respective nodal displacements and boundary forces for the current set of parameters while {â} i and { F } i are the corresponding values for a calculation with one of the parameters p i perturbed with a small factor, pi = p i + p i . For all the results reported in this paper the perturbation factor is set to = 0.01.

The sensitivity matrices presented in Equations ( 11) and ( 12) naturally follow from the derivation of Newton methods, and are a requirement for the optimization algorithm. However, they are also invaluable for analyzing sensitivities. They visually indicate where in space and time the experiment is sensitive to certain parameters, and will be shown and discussed later on. Because they can be computed before performing an experiment, they can be used to optimize the experiment [START_REF] Rossi | On the use of simulated experiments in designing tests for material characterization from full-field measurements[END_REF][START_REF] Bertin | Optimization of a cruciform specimen geometry for the identification of constitutive parameters based upon fullfield measurements[END_REF].

In the case of insignificant or limited sensitivity, Equation ( 4) is ill-conditioned.

This difficulty is circumvented by using a Tikhonov-type regularization [START_REF] Tikhonov | Solutions of ill-posed problems[END_REF][START_REF] Gras | Identification of a set of macroscopic elastic parameters in a 3D woven composite: Uncertainty analysis and regularization[END_REF],

where the linear system of equations is modified to, The remaining ingredients in this identification algorithm are the boundary conditions that are applied to the simulations, both when computing the image residual in Equation ( 7) and when computing the sensitivity matrices (Equations ( 11)-( 12)). There are different options each with their own merits. For the cases discussed in this paper a method was adopted that applies the dis-placement boundary conditions measured by a non-integrated FE-based DIC methods that uses the same mesh. This has the advantage of instantaneously aligning and synchronizing the measured and simulated data sets. Additionally, this method limits the simulation domain to the part that is visible within the field of view thereby reducing some computational costs. It has the disadvan-

[M ] + α[I] {δp} = {b} + α {p ref -{p}} , (13)
135
tage that these measured boundary conditions contain measurement uncertainty that is directly transferred to the identification algorithm. 4) where a single system of equations is formulated. The solution to this small system of equations is trivial and provides the update to the unknowns, which are the material parameters. The process is repeated until convergence which for the discussed cases is reached when the relative update norm is less than 10 -3 .

Experiment

The example experiment chosen as the integral part of the discussion is a relatively simple dog-bone sample made of AA2219 as shown in Figure 2(a).

The presented Integrated-DIC method can easily handle more complex geometries and inhomogeneous stress states [START_REF] Leclerc | Integrated digital image correlation for the identification of mechanical properties[END_REF][START_REF] Lindner | On the evaluation of stress triaxiality fields in a notched titanium alloy sample via integrated {DIC}[END_REF]. However, discussion of the gap between identification and measurement is equally important for classical identification methods. For that reason, a case was chosen that may seem trivial but, as will be discussed throughout the paper, contains nontrivial aspects.

The sample is 2 mm thick and 100 mm long, the width at the narrowest section is 10 mm, with the dog-bone radius of each side of the sample equal to 120 mm. The sample is loaded in a servo-hydraulic tension/compression testing machine equipped with a 50 kN load-cell. The top grip of the tensile machine is stationary while the bottom grip is driven at constant velocity of 0.01 mm/s to specific load levels (i.e. [1.8, 6.0, 8.5, 9.1, 9.3, 9.5, 9.6] kN). When a specific load level is reached the sample is unloaded to 0.1 kN upon which the next load cycle starts. At the end of the 7 th cycle the sample has not failed yet.

Failure will occur in the 10 th cycle, at a failure strain of approximately 8% and a maximum load of 9.7 kN. In the last three cycles, localization starts to set in, resulting in a behavior that is too challenging for the applied constitutive models. The application of more sophisticated models is possible and interesting but considered beyond the scope of this paper.

In the adopted Integrated-DIC algorithm only γ I and γ F are used to scale the cost functions. The application of γ I assumes that the displacement uncertainty can be propagated using the DIC Hessian [START_REF] Hild | Comparison of Local and Global Approaches to Digital Image Correlation[END_REF]. If this assumption is true, the following relation should hold,

γ ai = γ I √ 6 GL i (15) γU = γ a ≈ 0.0058 px ≈ 1 3 γ U , ( 16
)
where G is the mean field average of image gradient and L i is the element length for each node i, which is estimated as the square root of the area of the con-190 nected elements. There is a factor of three between the measured displacement uncertainty and the theoretical displacement uncertainty. There can be a number of reasons for this, not limited to, cross-pixel correlation, speckle quality and degradation, and sub-pixel interpolation errors. To remedy this gap, an effective gray level uncertainty is computed γI ≈ 3γ I = 1911 gray values, which is that ultimately used in Equation (3).

Finite Element Simulations

The Integrated-DIC routine is part of the Correli 3.0 framework, which is in continuous development at LMT [36]. The Integrated-DIC implementation configures the FE simulations with the correct parameters and subsequently calls the commercial code Abaqus implicit 1 to perform the simulations. The latter ones return the displacement field and reaction forces to the I-DIC implementation, which uses these data to compute the residuals and sensitivity fields, and then prepares for the next identification iteration (see Figure 1). The FE simulations are performed in 3D, using an extruded version of the DIC mesh as shown in Figure 3. The in-plane boundary conditions are applied identically to the front and the back sample surfaces. The out-of-plane boundary conditions are set to zero only for the front surface. Note that in the experiment the sample is clamped roughly 3 cm farther outward. The sample has an acceptable thickness to length ratio to be close to plane-stress. Therefore, it could be modeled in 2D instead of 3D. This hypothesis was tested and significant differences in the identified parameters between the 2D plane-stress and a 3D case were detected, especially for the Poisson's ratio. Modeling the sample with multiple elements over the thickness was also tested but proved not to significantly change the identification result while significantly increasing the computation time.

Identification Results

At the core of this paper is the discussion about the differences between identification cases and how the residuals of one case can guide choices made in the following one. In this section five cases are discussed, each extending on the previous one. Although the cases depend chronologically upon each other, they will be discussed in parallel such that the results can be compared side by side.

Consequently, the five cases will be introduced in parallel without the support of their data, causing some choices to seem unsupported at first sight.

Identification Cases

The five identification cases are defined as in Table 1. The first two cases (i.e. C1 and C2) allow for a direct comparison between classical (e.g. a stressstrain fit) and full-field (e.g. I-DIC) identification approaches. The Cases C2 to C4 progressively include more material parameters, and thus more freedom for the optimization method to reduce the residuals. Case C5 reduces the number of degrees of freedom by removing the Voce part of the hardening model and identifying only the Ludwik hardening model with Hill anisotropy. The following sections will discuss the details of the specific models.

Hardening Models

245

To showcase the method of successively enriching the identification algorithm a selection of hardening models is considered. There exist many interesting modern hardening models for sheet metals [START_REF] Banabic | Formability of metallic materials[END_REF], most of which include anisotropy. However, for the present paper, basic models suffice with the added benefit of not complicating the discussion. As a preliminary study, the following five hardening models are discussed in Case C1,

Elastic: σ = E , σ ≤ σ 0 , (17)
Swift:

σ = d( 0 + p ) b , σ > σ 0 , (18)
Ludwik:

σ = σ 0 + h m p , σ > σ 0 , (19)
Voce:

σ = σ ∞ -(σ ∞ -σ 0 ) exp(-n p ), σ > σ 0 , (20)
Ludwik-Voce:

σ = σ 0 + h m p + (σ ∞ -σ 0 ) exp(-n p ), σ > σ 0 , (21)
where p is the plastic strain in the load direction, E the Young's modulus, σ 0 the yield stress and 0 , d, b, h, m, σ ∞ and n are the hardening parameters of the various models [START_REF] Kleemola | On the strain-hardening parameters of metals[END_REF]. Not shown in the above equations is the Poisson's ratio ν adding another degree of freedom for identification.

Anisotropic Yielding 250

To enrich the material model with anisotropic plasticity, Hill's model is selected [START_REF] Hill | A theory of the yielding and plastic flow of anisotropic metals[END_REF] since it is a readily available option in Abaqus [START_REF]Abaqus Analysis User's Manual[END_REF]. The anisotropic plasticity criterion used in Cases C4 and C5 is defined as

H 1 (σ 22 -σ 33 ) 2 + H 2 (σ 33 -σ 11 ) 2 + H 3 (σ 11 -σ 22 ) 2 + 2H 4 σ 2 23 + 2H 5 σ 2 31 + 2H 6 σ 2 12 = 1, (22)
with

H 1 = 1 2 1 R 2 22 + 1 R 2 33 - 1 R 2 11 , H 4 = 3 2 1 R 2 23 , (23)
H 2 = 1 2 1 R 2 33 + 1 R 2 11 - 1 R 2 22 , H 5 = 3 2 1 R 2 13 , (24)
H 3 = 1 2 1 R 2 11 + 1 R 2 22 - 1 R 2 33 , H 6 = 3 2 1 R 2 12 , (25)
where R 11 , R 22 , R 33 , R 12 , R 13 and R 23 are yield stress ratios. For a planestress case, these ratios are interrelated such that four parameters suffice to fully describe the yield surface [START_REF] Banabic | Formability of metallic materials[END_REF]. Therefore, for Cases C4 and C5 the parameters R 22 , R 33 , R 12 and σ 0 are used adding 3 parameters to the identification routine.

Classical Identification

The classical way of identifying the material parameters is by fitting the hardening laws defined by Equations ( 18)-( 21) on experimental stress/strain data. The stress and strain need to be derived from the measured data, which requires some assumptions. In this case, the application of DIC allows for a more local measurement of the strain, which is provided by averaging the strain fields shown in Figure 2(c) within a virtual strain gauge that encompasses the high strain area in the center of the sample (indicated by the dashed box in Figure 2(a)). If no local measurement is used, the strain in the sample has to be estimated from the elongation considering the geometry of the specimen. Knowing the strains more locally is an advantage when computing the stresses since it can be used to correct the cross-sectional area by assuming incompressibility

σ = F A ≈ F exp( ) A 0 , ( 26
)
where σ is the yy-component of the true stress, A 0 and A the cross-sectional areas in the reference and deformed states, respectively. This estimation of the stress in a uniaxial experiment is perhaps not the most sophisticated and may introduce some identification errors that may be prevented. However, for the studied experiment the maximum strain level was of the order of 7 % and thus 260 Equation ( 26) is assumed to be reasonably accurate. with the hardening models defined in Equations ( 18)-( 21) and the difference between the fitted and measured stress ∆σyy. (b) The strain ratio obtained from the average strain inside the virtual strain gauge, the average ratio over the time interval marked by the vertical lines, is used to identify Poisson's ratio. The force/time plot shows that the chosen data points are those with significant force within the elastic regime of the experiment where the strain level is too small the ratio becomes highly sensitive to noise.

Therefore, only the time steps for the second cycle are used to calibrate ν. The second cycle is assumed to be fully elastic, which is reinforced by the third cycle that shows yielding at a much higher force level. Additionally, it is possible to measure the Lankford ratio from the ratio of the two strain components in the plane orthogonal to the loading direction,

r = xx zz ≈ 0.55, ( 27
)
where xx is measured directly while zz can be estimated supposing incompressibility. Assuming simple anisotropy, i.e. R 22 = 1, allows for the determination of R 33

R 33 = r(r + 1) 2r ≈ 0.89, (28)
which shows that the material is plastically anisotropic as expected for rolled aluminum sheets. Obtaining R 22 from single point measurements like strain gauges would require a second experiment where the material is tested at 90 • with respect to the rolling direction. Similarly identifying R 12 would require an experiment at 45 • . Full-field data may allow for identification of all three inplane anisotropy parameters from a single experiment, which will be discussed next.

Sensitivity Fields

The sensitivity matrices as defined in Equations ( 11)-( 12) can be visualized in the same space-time representation as used throughout the paper. In this visual form they are referred to as sensitivity fields, since they have a similar form as displacement fields. The five sensitivity fields shown in Figure 5 are used for identification purposes in Case C2. Considering first the displacement parts, they are normalized to bring the fields in a single plottable color range. Due to this, the fields represent the sensitivity shape, while their amplitudes are written above the respective field (p i ). These amplitudes underline the displacement sensitivity for each parameter. For example, for a 1 % change in Young's modulus, the change in displacement would be 0.013 px for the dark red areas and -0.013 px for the dark blue areas (i.e.

at the extremes of the color bar). Assuming a 1 % change in each parameter, these values show that the first two parameters (i.e. E and ν) have a sensitivity below the displacement uncertainty (γ U = 0.017 px), while the others (i.e. σ 0 , h and m) are more sensitive than the uncertainty. The same analysis can be performed on the force side of the sensitivity analysis. All force sensitivity signals are higher than the expected force uncertainty, γ F = 5 N, except for the Poisson's ratio, which is of the same order of magnitude.

For Case C3, two additional sensitivity fields are added to the identification routine, namely, those describing the exponential hardening (i.e. σ ∞ and n). Cases C4 and C5 consider anisotropic yielding, which adds another three parameters (i.e. R 22 , R 33 and R 12 ). The sensitivity fields corresponding to these five parameters are shown in Figure 6. Analyzing these sensitivity fields reveals that σ ∞ and R 22 are very sensitive in force, but also in displacements, R 33 has low force but significant displacement sensitivity, while n and R 12 are neither sensitive in displacement nor in force. The low sensitivity of n and R 12

(and also ν) will cause them to remain closer to their initial values as compared to the unregularized situation. This is a necessary drawback of using Tikhonov regularization without which the method is unstable and no solution is obtained. the elastic parameters (i.e. E and ν) would be identified using the entire data set, they would be sensitive to this stiffness reduction. Within the adopted Integrated-DIC method it is possible to counter this sensitivity by setting the sensitivity matrices [S I ] and [S F ] to zero for all time steps that occur after the onset of plasticity. In general this will result in a less optimal identification with higher residuals. However, the resulting identified elastic parameters would adhere more to how they are typically defined. Not limiting the elastic sensitivity to the initial regime would result in elastic parameters that are effective, or average, parameters. Ideally, the constitutive model should be enriched to include a mechanism such as damage that can account for this reduction in stiffness, but that is considered outside of the scope of this paper. For all full-field identification cases (i.e. C2-C5) the sensitivity fields for E and ν are set to zero after τ = 30 as indicated with the dashed purple boxes in Figure 5.

Identified Parameters

The results from the five identification cases are summarized in Table 2.

All four integrated methods converged in 10 iterations or less, which is mostly a consequence of the quality of the initial guess for which the obtained parameters from the previous case were used. All five methods returned comparable parameters, which gives confidence that their values are trustworthy. The identified elastic and plastic parameters are in line with typical values found in literature.

Without further data, it is not possible to decide which identified parameters adhere more closely to the reality. However, the residuals, which will be discussed later on, decrease from Cases C1 to C5 showing that the last case more closely describes the experiment.

ponential hardening law that significantly influences the early plasticity regime.

It is important to note that, although the plastic parameters change, the corresponding stress/strain curves are nearly indistinguishable (see Figure 7). All five parameter sets given in Table 2 are valid representations of the material behavior and will provide predictive capabilities. Since the parameters are to a change in the previously identified parameters. Consider for instance the R 22 parameter, which is strongly affected by the presence of the Voce model.

Without further experiments it is difficult to definitely conclude that this stiffness reduction is due to damage or anisotropy. However, the lower residuals obtained for Case C4 (discussed in the next section) indicate that C4 is a more likely solution.

Last, it is emphasized that full-field identification methods are able to identify complex material models from such a simple experiment. The identification capacity would only increase for inhomogeneous experiments, enabling the identification of more complex and realistic models.

Residuals

This section discusses three types of residuals, namely, displacement, image and force residuals. It should be noted that only the last two are considered in I-DIC. The displacement residual is available because a non-integrated DIC analysis is also performed. Additionally, the analysis of these full-field residuals for classical identification methods is possible due to the availability of a simulation of the experiment using the obtained parameters.

Table 3 shows the global residuals for each identification case. The most significant residual is the 317 N level in force for Case C1. A second observation is that the total residuals η consistently decrease from one Case C1 to C4. This trend proves that each of these cases describes the experiment better than its predecessor. between the experiment and its simulation. Full-field identification methods typically give access to these residuals. Classical identification cases rarely have access to these residuals, since they require simulating the experiment with the calibrated parameters, something that is typically not required for the identification [START_REF]E8 / E8M-15a[END_REF]. However, it is in all cases possible to generate these residuals and use them to analyze where in space and time the identification was within the expected accuracy. Case C1 shows the residual fields as obtained by using the parameters from classical identification. Although, this identification method has the lowest residuals in terms of stress-strain (Figure 7), it has the highest image and force 375 residuals. Whether these residuals are acceptable depends on the application of the identified model. In any way, they are much larger than the expected measurement uncertainty. Moreover, they are non-white, they have a structure or signature indicating that the gap between the model and the experiment is due to the limitations of the applied model.

residual (see the ellipse marked A in Figure 8). The absence of significant reductions in the image and displacement residuals indicates these residuals are not due to identification errors but are the consequences of limitations of the chosen constitutive model.

Case C3 is the enriched version of Case C2 by adding an exponential part to the hardening law. The additional three degrees of freedom did not significantly improve the residuals. There is a minor signature in the early plasticity regime (see the ellipse marked B in Figure 8). From these residuals it is clear that both versions of the hardening law perform comparably but do not resolve the most significant remaining residual.

Case C4 adds anisotropy to the identification. It significantly improves the full-field residuals. A signature in the x-component of displacement R ux is strongly reduced while also reducing a large portion of the image residual (see the ellipse marked C in Figure 8). It is interesting to note that the shape of the removed residual resembles the sensitivity field of R 22 and R 33 shown in Figure 6.

Case C5 removes the exponential hardening degrees of freedom that were added in Case C3 while retaining the anisotropy degrees of freedom. Similarly to the comparison of C2 and C3, in comparison between C4 and C4 the differences are limited. There is a zone in the R uy residual that is similar in signature but less pronounced in Case C4 and the same force residual signature has returned in the early plasticity regime (see the ellipses marked D and B in Figure 8 respectively).

Ultimately, the case with the lowest residuals is Case 4. This model most accurately describes the experiment at hand. However it can be argued that there are still significant residuals remaining, especially in the force signal. The force residuals are highest at the dashed lines, which are exactly at the bottom of the unloading cycles. The followed method of successively enriching the model can be continued to also isolate the constitutive behavior that is causing this residual. However, this is considered beyond the scope of this paper. An obvious candidate for future enrichments would be models that can account for the observed reduction in elastic modulus like, for instance, damage.

Conclusions

The experiment discussed herein is a uniaxial tensile test. The sample geometry was not optimized for identifying anisotropic plasticity. However through the use of full-field identification methods, it was possible to identify 10 parameters from a single experiment to varying accuracy. This analysis shows that the data density, which is currently attainable, is very rich and full-field identification methods can benefit from all of them.

Identification is a process to minimize the experiment/simulation gap. In this process, having more degrees of freedom in the material model typically allows for reducing the model error. For cases where the residuals are much greater than the expected uncertainty it is important to evaluate the shape or signature of the residual. Comparing the residual fields to sensitivity fields of perhaps previously unused parameters will highlight if the corresponding model enrichments will have an impact on the identification quality.

For full-field identification methods, tools like residual and sensitivity fields are part of the procedure. Therefore, they are accessible and valuable for gradually improving the identification results. Even for other methods, where these fields are not readily available, it is always possible to obtain them by simulating the experiment and comparing the measured quantities with their simulated counterparts. Similarly, sensitivity fields can be constructed from perturbations of the same simulation. In most cases, the goal of obtaining the material parameters is to use them in FE simulations. Consequently, this proposition is only a small and highly advisable extra step.

Last, enriching the material models gradually reduced the residuals. In particular it was shown that a significant reduction in residuals occurs when fullfield data are used instead of the classical way of identifying models with such a simple geometry. Further, as long as the parameters are independent enough, enriching the model not only identifies a more complex material behavior but also reduces the residual. The previous parameters will be identified to greater accuracy as well.

  where, {p ref } is a set of reference parameters obtained from other sources such as other experiments and or expert knowledge. [I] is the unity matrix and α the regularization strength, which is set to 10 -5 λ, where λ is the largest eigenvalue of [M ]. The consequence of this regularization is that insensitive parameters will tend to the reference levels instead of the otherwise erratic identification behavior. The chosen α parameter is set sufficiently small such that parameters with reasonable sensitivity are determined by the Integrated-DIC procedure based on the experimental data. It is always possible to set α = 0 and return to the original un-regularized system to analyze the influence of this regularization.

Figure 1 :Figure 1

 11 Figure 1: Integrated-DIC represented as a flow-chart. All blue items (with the rounded corners) are constants and are only computed before identification. The red items constitute the Integrated-DIC routine

AFigure 2 :

 2 Figure 2: (a) Reference image f and mesh applied for both DIC and I-DIC purposes. (b) Measured displacement field obtained by DIC per component (i.e. u = uxe x + uye y ) is shown for the last time step (right) and for one cross-section along time (left), the cross-section location is indicated with the dashed line (x = 165 px). The measured force is also shown below the displacement field. (c) Corresponding strain field using the same space and time visualization layout

  Part of the simulations are the displacement boundary conditions on each end of the sample. The displacements measured by DIC of the 3 rd row of nodes from each end inward are used, which are indicated with circular markers in Figure 2(a). The far edge nodes are not used because the images have a significant reduction in intensity due to vignetting at the top and bottom image edges. Moreover, the edge nodes of FE-based DIC routines are always more sensitive to noise due to their reduced connectivity [30]. To further reduce the impact of measurement noise in the simulations, the measured boundary conditions are smoothened using a cubic polynomial fit along the line of nodes for each time step.

Figure 3 :

 3 Figure 3: 2D mesh as used in the DIC analyses and 3D mesh as used in FE simulations

Figure 4 :

 4 Figure 4: (a) Stress/strain response as computed from the measured force/displacement, fitted

Figure 4 (

 4 Figure 4(a) shows that all 4 proposed hardening models can adequately describe the hardening behavior. The first three hardening models (i.e. Swift, Ludwik and Voce) each use three parameters, while the last (i.e. Ludwik-Voce) uses 5 parameters. Of the three parameter models, Ludwik's model has the 265

Figure 4 (

 4 Figure 4(b) shows the strain ratio for the first few cycles. For time steps

Figure 5 :

 5 Figure 5: Sensitivity fields in terms of displacement S I and force S F for the five material parameters used in all five cases. Note that S Ix and S Iy fields are normalized to equalize their color ranges. Their respective amplitudes are given above the figure (p i ). The purple dashed boxes in the sensitivity fields for E and ν indicate the zone where sensitivity is set to zero

Figure 6 :

 6 Figure 6: Sensitivity fields in terms of displacement S and force P for the five extra material parameters used in Case C4. The first three are also used in Case C3. Note that the Sx and Sy components are scaled with a normalization value shown above the figure (p i ) to equalize the color ranges

Figure 7 :

 7 Figure 7: Stress/strain response for each case using the parameters listed in Table 2

Figure 8 :

 8 Figure 8: Full-field residuals for all five cases (Table 1). The top two rows show the displacement residuals as compared to DIC, the third row shows the image residuals and the bottom row shows the force residuals. The ellipses marked A to D are discussed in the text

  τ ) denotes the corresponding interpolated gray values at the deformed locations in the image g for each time step τ . The number of pixels within the region of interest is N k and the number of time steps is N τ .

	The cost function is scaled with the gray level standard uncertainty of the image
	sensor γ I and the number of measurements such that the expectation value of
	this cost function approaches unity when converged if only random acquisition
	noise were present.

Table 1 :

 1 Definition of the five identification cases

		Identification method Hardening model Anisotropy Number of parameters
	C1)	Classical	Ludwik	-	5
	C2)	I-DIC	Ludwik	-	5
	C3)	I-DIC	Ludwik-Voce	-	7
	C4)	I-DIC	Ludwik-Voce	Hill 48	10
	C5)	I-DIC	Ludwik	Hill 48	8

Table 3 :

 3 Residuals remaining after convergence for each of the five identification cases, η is Figure8shows the full residuals for each case in space and time for the displacement, image, and force. They are a detailed representation of the distance

	the total residual of the cost function (Equation (3))
		R U [px] R I [%] R F [N] η [-]
	C1) Classical	0.28	4.94	317	1.66
	C2) Ludwik	0.26	4.71	197	1.58
	C3) Ludwik-Voce	0.26	4.69	217	1.57
	C4) Ludwik-Voce-Hill	0.13	3.3	214	1.11
	C5) Ludwik-Hill	0.14	3.37	197	1.13
	DIC	-	2.83	-	-
	365

Case C2 is the full-field version of Case C1, calibrating the same model parameters. The minimization on the image and force residuals will naturally reduce them. For this case the most significant reduction comes from the force

Acknowledgement

It is a pleasure to acknowledge the support of BPI France within the "DIC-CIT" project.

The hardening parameters changed more significantly between the identification cases. The low σ 0 values for C3 and C4 are the consequence of the exnot strictly uncoupled, enriching the material description will inevitably lead