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Quasi-static ultrasound elastography is a medical imaging technique dedicated to the mechanical characterization of biological tissues. This technique uses, for diagnosis, criteria mainly derived from the axial strain, which is the most accurately estimated component. However, imaging the strain tensor would be of interest, not only for a better evaluation of the tissue response to compression, but also for the implementation of reconstruction methods to access to the mechanical properties of the examined tissues.In this PhD work, two research axes were developed to analyze the elastic properties of biological tissues. The first axis focused on the development of a regularization method in order to smooth the displacement fields and to obtain less noisy, and thus usable, strain images. The results on numerical simulations, phantoms and breast tissues showed the strong contribution of this method, when facing complex and noisy data, especially for the lateral components of displacement and strain. Moreover, the proposed approach allowed to obtain shear strain images and rotation elastograms on which shear/rotation patterns were clearly detectable around the inclusions/lesions. The second research axis focused on the development of a method using the virtual work principle to reconstruct the spatial distribution of the Young's modulus within linear, elastic and isotropic media. This method requires the knowledge of the displacement/strain fields and the applied force. To overcome the lack of 3D ultrasound data, the reconstruction of the elastic modulus has been studied under the assumptions of plane stresses and plane strains. The results on numerical simulations and phantoms showed an accurate reconstruction of the Young's modulus mapping, in terms of region localization, shape and size. The modulus values are correctly estimated when the assumptions introduced in the method are respected. On the other hand, when the assumptions are no longer satisfied, a decrease in elastic contrast is observed. Nevertheless, the regions with distinct elastic properties in a medium remain clearly visible, both in simulations and in phantom data. Finally, the results on breast tissues indicate a stiffer region in agreement with the position of the lesion observable on the ultrasound imaging. These different results are promising and particularly interesting for a more complete analysis of biological tissues in quasi-static ultrasound elastography., L'élastographie ultrasonore quasi-statique est une technique d'imagerie médicale dédiée à la caractérisation mécanique des tissus biologiques. Cette technique utilise pour le diagnostic des critères issus principalement des images de la déformation axiale, qui est la composante estimée avec le plus de précision. Néanmoins, une imagerie complète du tenseur de déformation serait intéressante, non seulement pour une meilleure évaluation de la réponse du tissu à la compression, mais aussi et surtout pour la mise en œuvre de méthodes de reconstruction permettant d'accéder aux propriétés mécaniques des tissus imagés.Dans ces travaux de thèse, deux axes de recherche ont été développés afin de permettre une analyse des propriétés élastiques des tissus biologiques. Le premier axe a porté sur le développement d'une méthode de régularisation afin de lisser les champs de déplacement et d'obtenir des images de déformation moins bruitées, et donc exploitables. Les résultats sur des simulations numériques, des fantômes et des tissus mammaires in vivo, ont montré la forte contribution de cette méthode face à des données complexes et bruitées, notamment pour les composantes latérales du déplacement et de la déformation. De plus, l'approche proposée a permis d'obtenir des images de déformation en cisaillement et des élastogrammes de rotation sur lesquels des motifs de cisaillement/rotation étaient clairement détectables autour des inclusions/lésions. Le deuxième axe de recherche a été centré sur le développement d'une méthode utilisant le principe des travaux virtuels pour reconstruire la distribution spatiale du module d'Young au sein de milieux élastiques, linéaires et isotropes. Cette méthode nécessite de connaitre au préalable les champs de déplacement/déformation ainsi que la force appliquée. Pour pallier le manque de données 3D en échographie, la reconstruction du module élastique a été étudiée sous les hypothèses des contraintes et déformations planes. Les résultats sur simulations numériques et fantômes ont montré une reconstruction précise de la cartographie du module d'Young, en termes de localisation des régions, de leur forme et de leur taille. Les valeurs de module sont correctement estimées lorsque les hypothèses introduites dans la méthode sont respectées. En revanche, lorsque les hypothèses ne sont plus satisfaites, une diminution du contraste élastique est observée. Néanmoins les régions aux propriétés élastiques distinctes dans un milieu restent clairement visibles, que ce soit en simulations ou pour les données de fantômes. Enfin, les résultats sur tissus mammaires indiquent une région plus rigide en accord avec la position de la lésion observable à l'échographie. Ces différents résultats sont prometteurs et particulièrement intéressants pour une analyse plus complète des tissus biologiques en élastographie ultrasonore quasi-statique.

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A procedure for identifying plastic constitutive parameters from measured displacement fields based on virtual fields is proposed in this paper. The displacement fields of the specimen surface are obtained through measurements using global digital image correlation. Then, material property identification is implemented by utilizing piecewise virtual fields. In order to identify the material properties from the displacement fields obtained at an instant, the total strain theory is used as the constitutive equation. The relations between the material properties, measured displacement fields, measured tractions, and virtual displacements are obtained from the principle of virtual work, and they are solved numerically to obtain material properties. The effectiveness of the proposed method is demonstrated by applying it to displacement fields obtained by an elasto-plastic finite element method and experimentally obtained displacement fields of a pure aluminum specimen. The results of the numerical validation show that the elasto-plastic material properties can be identified from a single set of displacement fields under proportional loading. Meanwhile, the results of the experimental validation show that the elasto-plastic material properties can be evaluated but the discrepancy between the inversely identified stress-strain curve and the directly measured one is observed.

The present study aims at characterizing the post-necking strain hardening behavior of three sheet metals having different hardening behavior. Standard tensile tests were performed on sheet metal specimens up to fracture and heterogeneous logarithmic strain fields were obtained from a digital image correlation technique. Then, an appropriate elasto-plastic constitutive model was chosen. Von Mises yield criterion under plane stress and isotropic hardening law were considered to retrieve the relationship between stress and strain. The virtual fields method (VFM) was adopted as an inverse method to determine the constitutive parameters by calculating the stress fields from the heterogeneous strain fields. The results show that the choice of a hardening law which can describe the hardening behavior accurately is important to derive the true stress–strain curve. Finally, post-necking hardening behavior was successfully characterized up to the initial stage of localized necking using the VFM with Swift and modified Voce laws.

http://www3.interscience.wiley.com/journal/121488316/abstract?CRETRY=1&SRETRY=0 (DOI indiqué sur le site ne fonctionne pas.); International audience; The virtual fields method has been developed for extracting constitutive parameters from full-field measurements provided by optical non-contact techniques for instance. It is based on the principle of virtual work written with some particular virtual fields. This paper can be regarded as a general review summarising some 15 years of developments of this method. The main aspects of the method are first recalled in the case of both linear and non-linear constitutive equations. They are then illustrated by some recent relevant examples. Some studies underway as well as relevant issues to be addressed in the near future are eventually discussed.

International audience; This paper deals with the direct identification of bending stiffnesses of thin homogeneous anisotropic plates. The idea is to determine the unknown parameters from a single test giving a unique heterogeneous strain field. For the purpose of identification, the virtual fields method (VFM) is used. Here, the VFM is used taking into account two very recent developments: the piecewise construction of virtual fields and the noise minimization effect. Using these new aspects, improvements in terms of result accuracy are shown in comparison to previous studies. These new aspects are used as a tool to grade and to optimize different testing configurations with respect to the location of the applied loading. Some preliminary experimental results are then presented. They are in good agreement with theoretical expectations found in the first part of the study.

The Virtual Fields Method (VFM) and the Eigenfunction Virtual Fields Method (EVFM) are inverse techniques for estimating constitutive properties from full-field experimental data. In these, a set of virtual fields is used in the Principle of Virtual Work (PVW) to yield a system of algebraic equations for the unknown material parameters. In a typical experiment, one does not know the distribution of tractions over the external surface of the specimen, but the total force is generally measured. In order to still enable evaluation of the external virtual work integral that appears in PVW, in all the work to date on Virtual Fields methods, the virtual displacements are restricted to be uniform over the portion of the exterior surface where tractions are prescribed so that the external virtual work is simply the inner product of the known total force vector and the uniform value of the chosen virtual displacement vector. In this work, we show that this constraint can be relaxed to obtain a more flexible version of EVFM. The proposed modification is used to obtain orthotropic elastic constants from a simulated unnotched Iosipescu test, and is shown to yield tighter estimates than previously obtained wherein the boundary virtual displacements were constrained to be uniform. This approach, which is novel to Virtual Fields methods, allows us to include domains in the interior of the specimen and therefore, results in an EVFM formulation capable of dealing with material heterogeneity, missing data and discontinuities in specimen geometry. � 2014 Elsevier Ltd. All rights reserved.

The Virtual Fields Method (VFM - Pierron and Grediac, 2012), an inverse method based on the principle of virtual work (PVW), is being increasingly used to estimate mechanical properties of materials from full-field deformations obtained from techniques such as Digital Image Correlation, moir� and speckle interferometry and grid methods. By making specific choices for virtual fields (VFs) in PVW, one obtains a system of algebraic equations, which is then solved for the unknown material constants. Recently, a new variant of VFM, known as the Eigenfunction Virtual Fields Method (EVFM) has been proposed (Subramanian, 2013). In EVFM, principal components of the measured (i.e. true) strain fields are used to systematically generate VFs. We extend EVFM to orthotropic elastic materials in this work, and estimate the relevant material parameters from full-field strain data generated from a finite-element model of an unnotched Iosipescu test. Varying levels of Gaussian white noise are added to the synthetic strain data to evaluate the sensitivity of EVFM to input noise. It is observed that for low to moderate noise, the material properties estimated by the proposed method are relatively insensitive to noise. However, when noise levels are high, the proposed method yields large variance in some of the computed properties when compared to the state-of-the-art optimized piecewise continuous VFM (Toussaint et al., 2006; Pierron and Grediac, 2012). Some of the large variance in properties estimated from noisy data using EVFM is traced to the sensitivity of the third dominant eigenfunction and modifications to the proposed method to address this issue are suggested. � 2013 Elsevier Ltd. All rights reserved.

This work deals with the identification of constitutive parameters of elastomeric materials using the Virtual Fields Method. This method is suitable for identifying constitutive parameters from heterogeneous strain fields, but it had been used only in the case of small deformations until now. So the main originality of this work is to extend the method to the case of hyperelasticity, within the framework of large deformations. The identification method is presented after some reminders on large deformations and hyperelasticity. Then, numerical simulations enables us to illustrate the feasibility of the procedure with a special emphasis on the influence of noisy data on identified parameters. Experiments are then carried out in two cases of loading conditions. Displacement fields are measured with a digital correlation programme. Results found in terms of constitutive parameters are in good agreement with usual values of the tested materials., Ce travail est consacré à l'identification de paramètres constitutifs de matériaux élastomères avec la Méthode des Champs Virtuels. Cette méthode est adaptée à l'identification de paramètres constitutifs à partir de mesures de champs, mais elle n'avait été utilisée jusqu'à présent que dans le cas de petites déformations. La principale originalité de ce travail est de l'étendre au cas de l'hyperélasticité, dans le cadre des grandes déformations. La méthode d'identification est présentée après quelques rappels sur l'hyperélasticité et les grandes déformations. Des simulations numériques permettent d'illustrer la faisabilité de la procédure avec une attention particulière sur l'influence de données bruitées sur les paramètres identifiés. Des essais sont ensuite réalisés pour deux cas de chargement différents. Les champs de déplacements sont mesurés par corrélation d'images numériques. Les résultats trouvés en termes de paramètres constitutifs sont en bon accord avec les valeurs classiques obtenues généralement pour les matériaux testés.

This work deals with the identification of constitutive parameters of elastomeric materials using the Virtual Fields Method. This method is suitable for identifying constitutive parameters from heterogeneous strain fields, but it had been used only in the case of small deformations until now. So the main originality of this work is to extend the method to the case of hyperelasticity, within the framework of large deformations. The identification method is presented after some reminders on large deformations and hyperelasticity. Then, numerical simulations enables us to illustrate the feasibility of the procedure with a special emphasis on the influence of noisy data on identified parameters. Experiments are then carried out in two cases of loading conditions. Displacement fields are measured with a digital correlation programme. Results found in terms of constitutive parameters are in good agreement with usual values of the tested materials., Ce travail est consacré à l'identification de paramètres constitutifs de matériaux élastomères avec la Méthode des Champs Virtuels. Cette méthode est adaptée à l'identification de paramètres constitutifs à partir de mesures de champs, mais elle n'avait été utilisée jusqu'à présent que dans le cas de petites déformations. La principale originalité de ce travail est de l'étendre au cas de l'hyperélasticité, dans le cadre des grandes déformations. La méthode d'identification est présentée après quelques rappels sur l'hyperélasticité et les grandes déformations. Des simulations numériques permettent d'illustrer la faisabilité de la procédure avec une attention particulière sur l'influence de données bruitées sur les paramètres identifiés. Des essais sont ensuite réalisés pour deux cas de chargement différents. Les champs de déplacements sont mesurés par corrélation d'images numériques. Les résultats trouvés en termes de paramètres constitutifs sont en bon accord avec les valeurs classiques obtenues généralement pour les matériaux testés.

In this study, tensile loading experiments are performed on notched steel bars at an average applied strain rate of 1 s(-1). Displacement fields are measured across the specimen by coupling digital image correlation (DIC) with imaging using high-speed CCD cameras (4796 fps). Results from the experiments indicate the presence of local strain rates ranging from 0.1 to 10 s(-1) in the notched specimens. The heterogeneity of the strain rate fields provides suitable conditions for determining simultaneously all the elasto-visco-plastic Constitutive parameters governing the material behavior. For that, the whole stress fields are reconstructed in the specimen using the full-field deformation measurements. This reconstruction is repeated with different constitutive parameters until the average stress in the specimen matches the one measured with the load cell response. Perzyna's model is firstly considered for the reconstruction of stresses but it is shown to be unsuited for providing the drop in the average stress that is systematically detected at the onset of plasticity by the load cell. This drop is attributed to the sudden occurrence of plasticity in the material due to Luders effect. A modified model for elasto-visco-plasticity taking account of Luders behavior in the material is considered afterwards. It yields a better agreement between the reconstructed stresses and the load cell response, and a more accurate identification of the parameters driving the visco-plastic model. Eventually, it is shown how to use DIC measurements for replacing the load cell measurements when the transient effects in the test reach the resonance frequency of the load cell.

International audience; The virtual fields method (VFM) has been specifically developed for solving inverse problems from dense full-field data. This paper explores recent improvements regarding the identification of elasto-plastic models. The procedure has been extended to cyclic loads and combined kinematic/isotropic hardening. A specific attention has also been given to the effect of noise in the data. Indeed, noise in experimental data may significantly affect the robustness of the VFM for solving such inverse problems. The concept of optimized virtual fields that minimize the noise effects, previously developed for linear elasticity, is extended to plasticity in this study. Numerical examples with models combining isotropic and kinematic hardening have been considered for the validation. Different load paths (tension, compression, notched specimen) have shown that this new procedure is robust when applied to elasto-plastic material identification. Finally, the procedure is validated on experimental data.