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In this paper, we present simulation results for the electrostatic force between two conducting parts placed at different voltages: an atomic force microscope (AFM) sensor and a metallic sample. The sensor is composed of a cantilever supporting a conical tip terminated by a spherical apex. The simulations are based on the finite element method. For tip-sample distances (5-50 nm) and for an electrically homogeneous plane, the electrostatic force can be compared to the results obtained with the equivalent charge model and experiment. By scanning a plane with a potential step, the variation of the electrostatic force near the discontinuity gives the spatial resolution in electrostatic force microscopy (EFM). We establish then the relationships between the resolution, tipsample distance, and tip apex radius.The electrostatic force microscope results from one of many specializations of tip sensor in near-field microscopy [1, 2]. More precisely, this type of microscope is realized by applying a voltage on a conducting AFM tip. It is a good tool for imaging samples that present a gradient of electrical properties [3][4][5]. Variations of flexion of the cantilever holding the tip during a scan allow us to construct an electrical image [6] on inhomogeneous materials as well as on nanostructures (superlattices, nanoelectronics, etc.) [7][8][9]. In the simple case where the tip is in front of a conductive plane sample, we can deduce the force applied on the sensor by means of analytical expressions [10][11][12] or an equivalent charge model [13]. As soon as the geometry of the sample becomes complex (integrated circuits, dielectrics), the theoretical behavior of the system can be obtained by numerical methods such as the surface charge method [14], finite difference method [15], or finite element method [16].To determine the properties of the electrostatic force microscope in front of a sample with areas at different potentials, we propose to use the finite element method. In Sect. 1 we verify the results obtained by this numerical method in the simple case of a tip in front of a plane sample at constant potential [13]. In Sect. 2 we consider the response of the microscope near a potential step [17]. For this, we study the 3-dimensional tip-object system and determine the force applied on the tip by the finite element method and then we deduce the resolution for a potential step. Mathematical model The electrostatic problemThe problem consists in determining the interaction between an AFM sensor (tip + cantilever) and an infinite plane (both conducting). If the tip is long enough or the distance d between the tip and the sample is small, we can neglect the effect of the cantilever [18]. Then, the study is reduced to the calculation of the force exerted on a conical tip in front of a metallic plane. We treat the problem in 3-dimensional space because heterogeneities, as introduced in Sect. 2, cause the revolution symmetry to disappear. First, we must obtain the potential distribution in the space between the tip and the plan...

Abstract. In this Note we study a parameter identification problem associated with a two-dimensional mechanical problem. In a first part, the experimental technique of determining the displacement field is presented. The variational method proposed herein is based on the minimization of a separately convex functional which leads to the reconstruction of the elastic tensor and the stress field. These two reconstructed fields are continuous and piecewise linear on a triangulation of the two-dimensional problem. Some numerical and experimental examples are presented to test the performance of the algorithm. c Identification de paramètres mécaniques par mesure de champ de déplacement Résumé. Onétudie un problème d'identification de paramètres associésà deséquations de la mécani

Image processing techniques provide access to full field measurements of different thermomechanical data (strain; strain-rate, Wattrisse et al., J Exp Mech, 41:29-38, 2001; temperature, Chrysochoos and Louche, Int J Eng Sci 38:1759-1788, 2000. These techniques have become increasingly useful for obtaining fine and local descriptions of material properties. As they can measure complete thermal and mechanical fields, they can be used to identify several parameters of constitutive equations during a single deformation process using specifically designed heterogeneous tests (Grédiac, Composites: Part A 35:751-761, 2004). In Geymonat and Pagano (Meccanica 38:535-545, 2003), surface strain fields obtained by digital image correlation were used to identify the distribution of elastic parameters and stress fields by minimizing a given energy functional. In this paper, the previous method is improved through a relevant choice for stress approximation, and then extended to a wider class of elastoplastic materials. Its reliability is then checked through applications on simulated data obtained under small perturbation and plane stress assumptions. In particular, the robustness of the method with respect to measurement noise is studied on the basis of numerical data. An experimental application to heterogeneous material identification is finally proposed.