Alice Nasto

Amin Ajdari

Arnaud Lazarus

Ashkan Vaziri

Pedro M Reis
email: preis@mit.edy

Localization of deformation in thin shells under indentation

We perform a hybrid experimental and numerical study of the localization of deformation in thin spherical elastic shells under indentation. Past a critical indentation, the deformation of the shell ceases to be axisymmetric and sharp points of localized curvature form. In plates, these sharp points are known as d-cones. By way of analogy, we refer to regions of localization in shells as s-cones, for 'shell-cones'. We quantify how the formation and evolution of s-cones is affected by the indenter's curvature. Juxtaposing results from precision model experiments and Finite Element simulations enables us to explore the frictional nature of the shell-indenter contact and characterize the relative properties of strain energy focusing, at different loci of localization. Our combined experimental and computational approach allows us to gain invaluable physical insight towards rationalizing this geometrically nonlinear process.

Introduction:

Localization in thin shells is observed over a large range of length scales; from colloidal capsules at the microscale 1 to aircraft 2,[START_REF] Johnson | Crashworthiness of vehicles[END_REF] and architectural domes [START_REF] Mercatoris | [END_REF]5 . When localization is induced by indentation, the shape of the indenter affects the nature of the indenter-shell contact. This is an important consideration in applications such as Atomic Force Microscopy (AFM) 6 , which has been used to measure the mechanical response of shell structures such as microcapsules 7 and bacteria 8 . The curvature of the AFM tip relative to the curvature of the object that it indents is often non-negligible. As such, the relationship between the mechanical response and the nature of the indenter-object contact is critical for understanding the process.

The mechanics of shells is inextricably connected to its geometry [START_REF] Timoshenko | Theory of plates and shells[END_REF] and, for a shell with positive Gaussian curvature, bending and stretching are coupled. Stretching is energetically costly and the deformation of a shell is governed in large part by a need to minimize stretching, in favor of bending, to preserve the isometry of the surface (i.e. preserving the distance along the surface between any two points). TEXT ADDED HERE TO CLARIFY POINT ABOUT ISOMETRY A shell is curved in its undeformed shape, yet when it undergoes large deformations, the indented surface can develop sharp angular shapes. Energy localizes at these vertices and along straight edges, which, although locally costly, minimize stretching globally [START_REF] Vaziri | Proc. Natl. Acad. Sci[END_REF] . In the linear regime, there have been a number of attempts to rationalize the interplay between geometry and mechanics, ranging from the seminal work of Reissner [START_REF] Reissner | [END_REF] on spherical shells to the more recent study of geometry-induced rigidity in non-spherical pressurized shells 12 . There has been much less attention, however, on localization in shells under large deformations for which a general predictive framework is still aloof.

The scenario for large deformations in doubly-curved shells is not unlike what is observed in plates and other developable surfaces (e.g. cylindrical shells) under large deformations, where energy localizes along ridges and vertices, known as dcones for 'developable cones' [13][14][15][16][17] . These localized structures have been well studied and characterized through experimental, numerical, and theoretical work. Drawing an analogy with d-cones in plates and cylindrical shells 17 , we denote the localized objects in doubly-curved shells by s-cones, for 'shell cones', highlighting that they are not developable. Previous studies on large deformations and localization in shells have considered a variety of loading mechanisms. Some studies induce large deformations in shells through depressurization and characterize a variety of buckled shapes [18][19][20] . In other studies, shells are deformed by indentation, with the most common scenarios involving indentation by either a point load [START_REF] Vaziri | Proc. Natl. Acad. Sci[END_REF][START_REF] Vaziri | Thin Wall[END_REF][22][23] or a plate load [START_REF] Vaziri | Thin Wall[END_REF][23][24][25][26][27][28][29] .

Here, we fill the gap between point and plate indentation by studying indenters with intermediate curvatures. We are motivated by the fact that natural and engineering instances of indentation often occur in this intermediate range. Our goal is to develop insight and gain a better understanding on the relationship between the geometry of the shell, the shape of the indenter, and the mechanics of the localization process in the nonlinear regime of shells under indentation. Towards this end, we perform precision model experiments using hemispherical elastomeric shells and indenters with custom controlled geometries that we custom fabricate using rapid prototyping. The shells are made using a coating technique (Fig. 1a) whereby the surface of a mold (Fig. 1b) is coated by a liquid polymer, which eventually cures, resulting in a thin shell with uniform thickness (Fig. 1c). Our shells have radius R 2 = 25 mm and thickness t = 0.28 ± 0.027 mm. More details on the shell and indenter fabrication are given in the Materials and Methods section.

Our indenters have a range of radii of curvature, such that the ratio between the radius of the indenter, R 1 , and that of the shell, R 2 , defined as Γ = R 1 /R 2 (shown schematically in Fig. 1d), lies in the full range 0 < Γ < ∞, from point load to plate load, respectively (examples given in Fig. 1e). The mechanical response is quantified through indentation tests and the evolution of the deformation of the shell is captured with digital images using a camera located underneath the shell. In parallel, we perform Finite Element Analysis (FEA) of the same scenario explored experimentally, finding excellent quantitative agreement between the two. Given the predictive power of our numerics, we then exploit the simulations to gain further insight into quantities that cannot be readily accessed experimentally, such as the role of friction in the mechanical response and characterization of localization of the strain en-ergy. We find that patterns of localization are reproducible with all values of Γ, in both experiments and simulations, and show that the indenter's shape affects both the number of scones that form as well as the indentation onset for localization. The mechanical response is also significantly affected by the nature of the shell-indenter contact and important qualitative differences are observed between sharp indenters (Γ < 1) and blunt indenters (Γ > 1).

Point Indentation:

We start by considering the case of point indentation. A thin hemispherical shell is clamped at its equator. A point load is then applied to the shell's pole along its axis and the indentation is performed quasi-statically at 5 mm/min (see Materials and Methods for more details). This scenario is implemented and studied experimentally and through FEA simulations. In the experiments, the evolution of the deformation upon indentation is captured through digital imaging, as shown in the sequence of photographs in Fig. 2a) The white lines in the photographs, which aid identifying the loci of localization, correspond to light reflections from the ridge-like regions, where the shell inverts. The FEA simulations provides additional means to quantify the localization process, for example by having access to the energy density as indicated by the color map in Fig. 2b). Localized structures are associated with sharp increases in energy density. In Fig. 2a,b), we present a sequence of representative snapshots of the shell at different stages of indentation, for the experiments and the FEA simulations, respectively, at the same values of the control parameters. Excellent qualitative and quantitative agreement is found between the two. We define the dimensionless indentation as ε = δ /R 2 , where δ is the indentation displacement and R 2 is the radius of the shell. At small values ε, the cap of the shell inverts inwards, forming an axisymmetric circular ridge, known as Pogorelov ridge [START_REF] Pogorelov | Bendings of surfaces and stability of shells[END_REF] , along which the shell stretches (Fig. 2a1,b1). Due to the high energetic cost associated with stretching, past a critical value of indentation, the round ridge loses its axisymmetry and strain focuses at three conical-like vertices, the s-cones, which help reducing the stretching globally (Fig. 2a2,b2) [START_REF] Vaziri | Thin Wall[END_REF] . Once these three s-cones form, the inverted portion of the shell develops a tetrahedral shape. One vertex of the tetrahedron is located at the point where the indenter makes contact with the shell. The outer three vertices are located along a path on the surface where the shell inverts. These three s-cones are connected by straight segments, which we denote by ridges, which act as folds between the mostly undeformed outer shell and the inner inverted region. Inside the inverted tetrahedral region, additional folds form, which we refer to as gullies, that connect each of the s-cones to the point of indentation. As the shell is indented further, the s-cones travel along the shell's surface and eventually new s-cones can form. It is interesting to note that the birth of additional s-cones appears by division of an existing localized structure (Fig. 2a3,b3), rather than nucleation at a previously smooth region of the shell. Further indentation past the splitting event results in four well defined s-cones (Fig. 2 a4,b4).

ε = δ /R 2
To further quantify the process of birth and growth of scones, we perform digital image processing on the experimental frames to track the angular position of each of the s-cones (projected on a plane perpendicular to the vertical axis of indentation). The schematic definition of the s-cone's angular position, θ , is presented in Fig. 2a2). As the shell is indented, the evolution of the s-cones' angular position illustrates the splitting mechanism by which new s-cones form; at ε ∼ 0.7, the forth new s-cone emerges through the branching of one of the older three.

Localization of Energy for Point Indentation:

We proceed by further characterizing the various localized structures identified above for point load, using quantities extracted from the FEA that are not available experimentally. We shall later demonstrate the power of our FEA simulations in predictively reproducing the experimental results. In particular, our focus now goes to a quantitative comparison of the strain energy density, the energy stored in a body due to deformation per unit volume, for the s-cones, ridges, and gullies. We analyze the specific example of indentation to δ = 15mm (ε=0.6). The paths along the shell's surface that we shall use in our discussion are schematically drawn on top of the FEA snapshots shown in Fig. 3a,b).

In Fig. 3c, we trace the energy density along Path #1, which is a closed contour that traverses over the range of the three scones and the ridges that connect them. The energy density is highest at the s-cones and is minimum at the ridge's midpoint, half way between adjacent s-cones. The energy density level in the neighborhood of the s-cones is 25 times higher than along the ridges, which is significant of localization. Paths #2, #3, and #4 define meridians; they start from the pole of the shell, pass along a gully, over one of the s-cones, and end at the shell's equator, with each of the paths passing over each of the three s-cones. In Fig. 3d), we plot the energy density along these paths. The maximum of energy occurs at the pole where the indenter contacts the shell. Moving away from the pole, the energy density drops along the gullies, then sharply increases at the s-cones, before rapidly decaying by four orders of magnitude towards zero at the clamped equator of the shell since it is not deformed. The energy density curves corresponding to these three paths are superposed, highlighting the symmetry of the process. In Fig. 3e), we plot the energy density along Path #5, which starts at the pole, traces along one of the faces of the tetrahedron, crosses over the ridge's mid point, and ends at the shell's equator. As stated before, energy is localized most intensely at the pole where the shell is indented. The strain energy along the face of the tetrahedron is small (1% compared to the energy localized at the s-cones). While climbing and traversing over the ridge, there is a small increase in strain energy, indicating that there is some focusing of strain energy at the ridge (4% compared to the energy localized at the s-cones). The energy then decays towards zero at the shell's equator, consistently with the clamping boundary conditions there. In Fig. 3f), we trace the energy density along Path #6, which is a circular contour near the shell's pole and crosses over the gullies and faces of the tetrahedron. Along this path, the energy is maximum at the gullies. In summary for point indentation we find that energy is most strongly localized at the s-cones. Energy is localized to less of an extent along gullies and ridges, with energy being focused ten times more along gullies in comparison to ridges.

Varying the Shape of the Indenter:

Having described the behavior observed for point indentation, we now consider the effect of changing the shape of the indenter on the localization process, in both experiments and FEA simulations. The ratio of shell and indenter radii is varied from Γ = 0 (for point load) to Γ = ∞ (for plate load), and we consider twelve indenters between these extreme values. In our experiments, we employ rapid prototyping to design and custom fabricate indenters (rigid with respect to the shells) with the range of radii of curvature shown in Fig. 1e). More information on how the indenters are fabricated is given in the Materials and Methods section.

In Fig. 4, we present a series of representative experimental and FEA snapshots of the shells, for increasing values of the ratio of shell-to-indenter radii, Γ. In experimental frames, s-cones are identified by sharp corners in the ridge, as imaged by the white reflection line, and in the FEA simulations they are identified by small regions with a sharp increase in energy density. Localization occurs and s-cones form for all of the indenters, except for Γ = 1, 1.5 and 2. In the neighborhood of Γ = 1, the indenter has a curvature close to that of the shell. As a result, upon indentation their two surfaces remain in complete contact, preventing the formation of s-cones, which would otherwise require delamination from the indenter. We denote this neighborhood of 1 Γ 2 as the "localization band gap", since the formation of s-cones is forbidden for these geometries. The lower bound for the localization band gap is 0.8725 < Γ < 1 (uncertainty of ∼ 3%) and the upper bound lies in 2 < Γ < 2.5 (uncertainty of ∼ 22%) TEXT ABOUT BAND GAP BOUNDARIES ADDED HERE This band sets two regions with qualitatively different responses; sharp indenter for Γ 1 and blunt indenters for Γ 2. Concentrating on the experiments for sharp indenters, the number of s-cones at onset is a minimum (n = 3) for point load (Γ = 0) and it increases with increasing Γ, with the maximum number at onset forming just outside the localization band gap region. For blunt indenters, n = 5 s-cones form at onset for plate load (Γ = ∞), and increasingly more s-cones are observed as Γ is decreased towards Γ ∼ 2. Gullies that connect the s-cones to the pole of the shell form for sharp indenters, but not for blunt indenters [START_REF] Vaziri | Thin Wall[END_REF] . Above, for the case of point load, we saw that past the initial formation of 3 s-cones, a forth can emerge by splitting of a previous one. This mode of transition in the number of s-cones is also observed for other indenter geometries: n = 4 → 5 for Γ = 0.75, n = 6 → 7 for Γ = 10 and n = 5 → 6 → 7 for Γ = ∞ (plate load). The FEA simulations show qualitative good agreement with this experimental scenario with differences in the number of s-cones being at most by n = 1, possibly due to differences in the details of imperfections between experiments and FEA, or a consequence of the fact that multiple states may coexist (as is common in nonlinear systems). TEXT ABOUT CAUSES OF DIFFERENCES IN EXPERIMENT VS FEA ADDED HERE Moreover, FEA simulations also exhibit a localization band gap region for 1 Γ 2 and the same qualitatively different responses for sharp indenters versus blunt indenters. In both experiments and FEM (assuming a hyperelastic constitutive model as described in the Materials and Methods section), we see that the number of s-cones evolves for Γ > 2. Previous studies [START_REF] Vaziri | Thin Wall[END_REF] , however, have not seen such an evolution for plate load for the specific case of using a linear elastic model. Whether the discrepancy is due to the material model used or due to the frictional nature of contact that we have considered (both ingredients were used in the FEM to closely reproduce the experiments) is beyond the scope of this paper and will be addressed in a future study.

We have just seen that the geometry of the indenter affects the morphology and number of s-cones at onset. We now pro- ceed by quantifying how the indenter's geometry also sets the critical indentation, δ c , at the onset of localization, i.e. the δ at which the first set of s-cones forms. In Fig. 5, we plot the critical indentation as a function of Γ. As Γ approaches the band gap region in the neighborhood of Γ = 1, the onset of localization is significantly delayed. The critical indentation exhibits a divergence-like behavior around the band gap region, and the critical indentation is maximum on either side. The critical indentation decreases as Γ → 0 or Γ → ∞ (with the exception of the decrease in δ c between point load and Γ = 0.5, which disrupts the otherwise monotonic behavior to the left of the localization band gap). TEXT ABOUT NONMONOTONIC CRITICAL INDENTATION ADDED HERE Excellent quantitative agreement is found between experiments and simulations.

The indenter shape also effects the ridge-height, h, which we define as the vertical distance between the equator of the hemispherical shell, where it is clamped, and the point on the shell at which its surface inverts due to the indentation, thereby forming a ridge. In Fig. 6a), we plot this ridge-height measured experimentally (open symbols), as a function of indentation for point load, Γ = 1, Γ = 3, and plate load. For comparison, we superpose the corresponding data from the FEA (solid lines) for the extreme cases of point and plate load. Even for large values of indentation, the ridge-height h decays approximately linearly with indentation, ε. We therefore calculate the slope m = dh/dδ , which we plot as a function of the indenter's radii ratio, Γ, in Fig 6b). The ridge-height decreases the slowest for point indentation, with a rate of m ∼ -0.5, and the fastest for plate load, with m ∼ -1 (the ridge moves at the same rate as the indenter). For intermediate indenters (0 < Γ < ∞) the slope decreases monotonically with Γ and -0.5 < m < -1.

We now present a geometric argument that rationalizes these results. The schematic diagram in the top right corner of Fig. 6a illustrates the deformed configuration of the shell for Γ < 1. Here, the radius of the shell is R 2 = h + δ /2, which in return gives h/R 2 = 1 -0.5(δ /R 2 ) and yields the slope m = dh/dδ = -0.5, for Γ < 1. For Γ > 1, the nature of contact between the shell and the indenter is qualitatively different, as shown in the schematic diagram in the bottom left corner of Fig. 6a. Taking the radius of the indenter, R 1 , into account and assuming small deformations, we obtain R 2 = h+(δ -∆), where ∆ is the vertical distance between the pole of the indenter and the ridge-height, h (see schematic in bottom left of Fig. 6b). Using a trigonometric relationship involving R 1 , R 2 , and a horizontal line constructed between the ridges, we obtain ∆ = δ /(1 + Γ). Then, R 2 = h + δ [1 -1/(1 + Γ)] from the geometric construction, which yields the following slope for the h-δ curves,

m = dh dδ = -1 - 1 1 + Γ . ( 1
)
The dependence of m on Γ predicted by this geometric argument underlying Eq. ( 1), which is plotted as the solid line in Fig. 6b, is in excellent agreement with both the experimental and FEA data. This confirms that the variation in m is due to the different geometric nature of the indenter-shell contact proposed above, and which we explore in more detail next.

In order to achieve the excellent agreement between experiments and simulations highlighted above, we had to treat the shell-indenter interaction with care, and assume a frictional contact while neglecting adhesion. TEXT ABOUT NO AD-HESION ASSUMPTION ADDED HERE This is even more important since, as we saw, the nature and morphology of the shell-indenter contact varies for different values of Γ; from sharp indenters to blunt indenters. To stress the important of friction, we now focus on the mechanical response during the indentation process, as quantified by the indentation load, P, as a function of indentation, ε. A series of P -ε curves for increasing values of Γ is presented in Fig. 7a. For small indentation, there is a linear regime significant of the shell's stiffness, as previously described by 12,[START_REF] Vella | [END_REF] . For large indentations, all curves with Γ < 1 eventually asymptote to an approximately constant value. In contrast, the various curves with Γ > 1 exhibit an inflection point and fan out; higher values of Γ result in higher loads. Again, we observe a quantitative difference between sharp and blunt indenters. For Γ > 1, a significantly higher load is required to indent a shell when compared with sharp indenters (Γ < 1) due to the different nature of the indenter-shell contact. For sharp indenters, the poles of the indenters and the shells remain in contact throughout indentation. On the other hand, for blunt indenters, the shell delaminates from the pole of the indenter, and pushes onto the shell at its ridges. This requires a substantially higher force level (as high as a factor of 10 for ε = 0.4, when comparing Γ = 0 and Γ = 10) and also involves sliding and rolling of the two surfaces. These two scenarios are illustrated in the inset of Fig. 6b.

From the detailed comparison between the experimental and FEA force-indentation curves, we have learned that friction plays an essential role on the mechanical response. Forceindentation curves from simulations without friction showed a much lower mechanical response than what was measured experimentally. As presented in Fig. 7b), when the coefficient of friction µ is increased, the load required to indent the shell is also significantly increased. For example, at ε = 0.45, the load required to indent a shell for µ = 1.5 is increased by a factor as large as 5, when compared to doing so for a frictionless indenter-shell contact. Using an independent friction sliding test (details in Materials and Methods), the friction coefficient was experimentally measured to be µ = 1.46 ± 0.53. The large variation in this measured value can be attributed to the fact that a dry friction description is oversimplistic for polymer-polymer surface contacts. Still, in the FEA, we have treated µ as a fitting parameter, bound within the measured experimental range. This is appropriate since the experimental friction coefficients for each of the individual indenters are not precisely known due to the varying levels of roughness for indenters of different curvature, imparted by the fabrication process and set by the resolution of the CNC milling process.

Discussion and Conclusion:

Without a general predictive analytical model of the localization process at hand, we are lead to speculate that geometric frustration underpins the buckling transition from the axisymmetric state into the onset of s-cones. We believe that the mechanism is related to that recently reported by Dias et al. 32 , who studied the mechanics of folding of an annular flat plate containing a concentric circular crease. Folding the crease induces out-of-plane buckling of the plate. Further increasing the dihedral angle of the fold, results in increasingly more non-planar configurations and an increasing storage of stretching energy at the crease. Geometric and topological constrains, coupled with the mechanics of plates, dictate the permissible configurations. Similarly in our system, indentation introduces geometric frustration that disrupts the isometry of the shell's inverted cap. Once the circular ridge forms, the angle of the folded region where curvature of the shell inverts, increases as the shell is indented further which leads to a growing storage of strain energy at the ridge (the angle can be derived using an argument similar to that presented for the slope of the ridge-height vs. indentation curves). Past a critical indentation (quantified in Fig. 5) the circular ridge looses symmetry and stretching, due to its high energetic cost, is fo-cused onto localized structures, the s-cones.

We now comment on the observation that the number of s-cones at onset increases, when the curvature of the indenter approaches the shell's curvature. For this, we establish an analogy with buckling of confined elastica. When a slender beam is compressed axially, but its transverse displacement is constrained by two confining walls 33,34 , high-order buckling modes can be excited (in contrast to the classic mode-one Euler buckling when the beam is unconstrained). In this case, the buckling wavelength increases with the lateral confinement and the geometric constraint of the two confining walls is the driving mechanism for exciting high-order buckling modes. Following this analogy, point load indentation in our problem can be considered as being unconstrained. At the other extreme, when Γ ∼ 1, the surfaces of the shell and indenter are in close contact with each other, preventing the shell from delaminating from the indenter, thereby precluding local buckling (no s-cones form). In the constrained Euler buckling analogy, this corresponds to the case where the two constraining plates are separated by a distance equal to the thickness of the strip such that buckling cannot occur. For Γ > 0, and outside the localization band gap in the neighborhood of Γ ∼ 1, however, the shell-indenter contact constrains the buckling morphology, which excites higher order modes (increasing the number of scones) and affecting the onset of localization.

In conclusion, we have presented a rich scenario for the large deformation of a thin shell under indentation, as the curvature of the indenter is systematically varied, which has a strong effect on the onset and evolution of localized structures. The nature of the contact between the shell and the indenter was found to be a crucial ingredient to the process; friction between the indenter and the shell significantly affects the shells's load-bearing capacity, especially for blunt indenters (Γ > 1), which require significantly higher loads to indent shells compared to sharp indenters (Γ < 1). The robustness of the observed localization behavior and the excellent agreement found between experiments and numerics suggest that there is an underlying mechanism at play that arises from the strong interplay between geometry and mechanics. We hope that our exploratory study will help catalyze further theoretical efforts in this direction. Moreover, given the scale-invariance of the scenario presented here, our results should find uses at the microscale for AFM, where it is crucial to understand how the curvature of the tip, relative to the object being indented, affects the mechanical response.

Materials and Methods:

Experiments: The shells and the indenters are fabricated using rapid prototyping. Starting from a computer assisted design, a model of a mold was created with the targeted shell or indenter geometry. Molds were then machined from polyacetal with a CNC milling ma-chine. Shells were fabricated with a coating technique (Fig. 1a). Vinylpolysiloxane (a silicone based elastomer) was used to make the shells. VPS has a shear modulus G 0 = 454.6 kPa, a poisson ratio of ν = 0.5, and the Neohookean strain energy potential coefficients of C 10 = 227300 Pa, and D 1 = 2.36 × 10 -8 Pa -1 . The polymer was poured into a mold, which was then rotated so that the polymer wetted the entire surface of the mold. Then, the excess was poured out and the mold was left upside-down to allow the excess to drain out while the polymer cures. The resulting shells had a radius of R 2 = 25 mm and a thickness of t = 0.28 ± 0.027 mm, which is set by a combination of the viscosity of the polymer and the length of the curing time. In a scenario similar to Landau-Levich dip coating for fibers and plates 35 , this fabrication method creates shells with small variations on their thickness (∼ 9.7%).

Indenter Fabrication: Twelve indenters were designed with a variety of radii of curvature so that the ratio of the radius of the indenter and the radius of the shell, Γ = R 1 /R 2 had the following values: Γ = 0 (point load), 0.5, 0.75, 0.8725, 1, 1.5, 2, 3, 5, 7, 10, and ∞ (plate load) (Fig. 1e). Indenters with Γ ranging from 0.5 to 10 were cast with a hard polyurethane. The indenter for point load was a steel screw with a hemispherical cap (1.5 mm radius). The indenter for plate load was cut from a sheet of acrylic. All indenters can be considered rigid relative to the elastomeric shells. The polyurethane used to cast the indenters has a Young's modulus E = 147 MPa and the vinylpolysiloxane used to cast the shells has a Young's modulus E = 1.36 MPa.

Mechanical Testing: Shells are indented at the pole using the fabricated indenters at a constant speed of 5 mm/min. The compressive force, F, from the indentation by the imposed displacement, δ , is recorded using the load cell of an Instron machine with a resolution of ±100µN. The evolution of the deformation of the shell is recorded with digital images that are captured from underneath the shells.

Friction Tests: The friction from the contact between the shell and the indenter was measured through friction tests. A block of vinylpolysiloxane (the shell material) with a force sensor attached was pulled horizontally along a surface made of polyeurethane (the indenter material). The coefficient of friction, measured as the average from ten experiments of the ratio of the force required to pull the block to the weight of the block, was measured to be µ = 1.46±0.53.

Numerical Simulations: Numerical simulations were performed using the commercial finite element package ABAQUS/CAE (SIMU-LIA, Providence, RI). One half of the 3D spherical shell was modeled with a clamped boundary condition on the free edge, indenters were modeled as analytical rigid shells with a displacement-control boundary condition. For the material properties, we used an isotropic hyperelastic model using the material parameters measured independently in the experiments. Four-node thin shell elements with reduced integration (element type S4R) were used in all simulations and a mesh sensitivity study was carried out to ensure that the results are minimally sensitive to the element size. We used the free mesh scheme available in Abaqus and no initial geometric or material imperfection was included in the computational model. To capture the local instabilities in the structure, we used a stabilizing mechanism (available in Abaqus/Standard solver) based on automatic addition of volume-proportional damping, which was decreased systematically to ensure that the response was insensitive to this change. To model friction, a penalty formulation was used with the appropriate coefficient. Aknowledgements: A. A. and A. V. are thankful for the support of NSF (CMMI-1149750). A.N. acknowledges the support of NSF through a Graduate Research Fellowship (1122374) and MIT through a Ascher H. Shapiro Graduate Fellowship.

Fig. 1

 1 Fig. 1 Experimental setup a) Coating technique used to fabricate thin and uniform hemispherical shells. b) CAD model of indenter and shell molds. c) Hemispherical elastomeric shell created by the coating technique. d) Schematic of indenter, with radius R 1 and shell, with radius R 2 . e) Series of indenters ranging from Γ = R 1 /R 2 = 0 (point load) to Γ = ∞ (plate load).

Fig. 2

 2 Fig. 2 Point indentation a) Experimental snapshots (captured from underneath the shell) of the evolution of the pattern of localization for an elastomeric shell under point indentation at the pole. The white reflection corresponds to the location at which the shell inverts. b) Snapshots from FEA simulations of an elastomeric shell under point indentation at the pole. c) Angular position of s-cones vs. indentation obtained from experimental images. θ is defined as in the red schematic drawn in a2.

Fig. 3 Energy

 3 Fig. 3 Energy Localization a,b) Snapshot from a finite element simulation of a shell indented with a point load. b). Top view of a shell indented with a point load. c) Energy density along Path #1, which traces over the three s-cones and the three ridges that connect them. d) Energy density along Paths #2, #3, and #4 start from the pole of the shell, trace along one of the gullies and one of the s-cones, and end at the base of the shell. e) Energy density along Path #5, which starts from the pole of the shell, climbs between the s-cones and , traces over the ridge that connects adjacent s-cones, and ends at the base of the shell. f) Energy density along Path #6, which intersects the three gullies.

Fig. 4

 4 Fig. 4 Varying the shape of the indenters a) Snapshots from experiments with elastomeric shells indented at the pole for a variety of indenters, ranging from point load (Γ = 0) to plate load (Γ = ∞). b) The same scenario with the same snapshots in FEA simulations. The color map corresponds to strain energy density. The color and energy scale correspondence is adjusted for each of the images to aid in highlighting localization. Red circles indicate location of scones. Dotted red lines are drawn over ridges that connect s-cones along where shell is inverted. Solid red lines are drawn over gullies that connect s-cones to the pole of the shell (only present for Γ < 1). No localization (indicated by grey area) occurs for indenters with Γ = 1, 1.5, or 2.

Fig. 5

 5 Fig. 5 Critical indentation for the onset of localization versus the indenter-shell radii ratio, Γ. No localization occurs between Γ = 1 and Γ = 2 (indicated by the grey region).

1 Fig. 6

 16 Fig. 6 Ridge-Height a) Height of the ridge versus indentation for indenters with different geometries; experiments (data points) and simulations (solid lines). The schematics on the top right and bottom left corners correspond to the cases of sharp indenters, Γ < 1, and blunt indenters (indenter in red), Γ > 1, respectively. b) Slopes of ridgeheight, m vs. indentation curves from experiments (red squares) and simulations (blue circles) indenters ranging from Γ = 0 to Γ = ∞. The theory line corresponds to the prediction from Eq. (1). Inset illustrates sharp and blunt indenters.

Fig. 7

 7 Fig.7Mechanical response and the effect of friction. a) Forceindentation curves from experiments (solid lines) and finite element simulations (dashed lines) for a variety of indenters. Coefficients of friction for Γ = 0, 0.75, 1.5, 3, 5, 10 are, respectively, µ = 0, 1.5, 1.2, 1.7, 1.5 b) Force-indentation curves from simulations for Γ = 10 for no friction and a variety of friction coefficients.

to form a 4th

			1 1 1 1 1 1	2 2 2 2 2 2			4 4 4 4 4 4
					θ
	m 3	8000 7333 6667	1 1 1 1 1 1 1 1 1	2 2 2 2 2 2 2 2 2			4 4 4 4 4 4 4 4 4
		6000
		5333
		4667
		3333
		4000
		2667
		2000
		0 1333 667	0.1	m	0.6	0.72	0.98
		300			4 -
				-
		200
		100
			0	0.4 m	0.6	0.8	1 , ε = δ /R 2


come

Modeling non linear components in the frequency domain

Answer for low level. . . . . . . . . . . . . . . . . . . . 12 Figure 6 Answer for high level. . . . . . . . . . . . . . . . . . . 13

The frequency domain is the major domain used in electromagnetic compatibility (EMC). But it has an important default: it classically cannot be used when non linear components are involved. The purpose of this paper is to submit a technique, under the formalism of the tensorial analysis of networks (TAN), that may allow to model non linear components remaining in the frequency domain. The approach is based on current sources that creates the added harmonics coming from the non linear behavior of the components. An example is shown with a diode. The principle can be generalized to any circuits. We conclude giving the tracks of this generalization.

Electromagnetic compatibility works on the electromagnetic interactions that occur between electronics. Fields propagate in free space or using waveguides like lines or cavities. But all these interactions end on active devices including non linear components like diodes, transistors, etc. The response of these components depends on the level applied across them. While skin effect or radiations are easily modeled in the frequency domain, this kind of behavior is more difficult to model in the frequency domain. Previous works have shown how to define impedance on more than one law, depending on the values of some parameters like voltage, current, temperature, etc. By a similar technique we can define an impedance that changes with the amplitude of a voltage for example. This impedance can drive a current source which takes in charge the creation of harmonics coming from the non linear behavior of the component. Previous works were done on the same subject, using filters to reproduce the creation of harmonics from pure sinusoidal waveforms coming from non linear components. We try here to submit a technique as easier as possible under the TAN formalism. The way to reach our objective is different from the previous work, perhaps less general, but we hope simpler to use.

Basic idea consists in creating a generator, seted with a given parameter. This generator will take in charge the creation of the harmonics coming from the non linear behavior. If we make a simple test, for example exciting a diode with a square, symmetric signal, when the amplitude goes over the diode threshold, i.e. around 0,6 V, the signal should rapidly becomes non symmetric and only following the diode polarization. Figure 1 illustrates this mechanism.

.

Spectrum transformation

When the voltage across the diode is under the threshold, it behaves like a resistance. We know the spectrum Ŝ of a rectangular, symmetric signal, it is given under Laplace's formalism (p being Laplace's operator) by:

Ŝ(p) = A 1 p - 2 p e -τp + 1 p e -2τp (1)
τ being the half period duration and A the signal peak amplitude in the frequency domain. If the diode works as a perfect diode, the spectrum Ŝ should become the one of a single rectangular signal:

Ŝ (p) = A 1 p - 1 p e -τp (2)
So some operations should transform the spectrum of current across the diode from the function Ŝ to the function Ŝ . How to do that?

.

Impedance operator and current source

When the level of voltage (the voltage being here our control parameter) across the diode is too low, the diode must be a simple resistance of high value. If the voltage is negative, this resistance must remain very high (we don't consider the avalanche process for the moment). Now if the voltage goes over the threshold, the resistance must become low (that's the dynamic resistance of the diode) and the spectrum must become the one of a single positive rectangular waveform. It means that the negative part must be suppressed by some added source. If the dynamic resistance of the diode is R dyn , the nominal and linear current in the diode may be (the generator E having the expression of Ŝ):

i(p) = Ŝ(p) R 0 + R dyn (3)
R 0 being the generator self resistance. To suppress the negative part of this spectrum in current, we must add an external current source that creates an opposite spectrum for the negative part. If J is this current source, it is defined by:

J(p) = A R 0 + R dyn 1 p e -τp - 1 p e -2τp (4)
We can verify that:

i(p) + J(p) = Ŝ (p) R 0 + R dyn (5)
But how can we change the impedance dynamically and add the current source?

.

Impedance operators on domains

Impedances can be defined, under the tensorial analysis of networks, on domains [MAURICE(2017)]. If R i is the resistance of the diode when it is polarized in inverse (voltage across the diode is negative following the description of 1), R d when it is polarized in direct but with a level of voltage v under the threshold value and R dyn when the diode becomes active, its global impedance whatever the voltage value can be described by the function:

Z d = v D 1 R 1 i + v D 2 R 2 d + v D 3 R 3 dyn (6)
With this function, the diode behavior in impedance is now completely defined once we know the voltage developed across it. The domain functions v D x are equal to 1 when the parameter (here v) belongs to the interval x. Their profiles can be more or less abrupt (logistic functions as arctangent ones or gaussian ones can be used) and it exists some intermediate states, when the parameter value belongs to both intervals where the diode can have both impedances and behavior. The speed of change depends on the time rise of the domain functions.

.

Dynamic connectivity

It remains to add the current source which depends also of the parameter v value. Two techniques can be imagined. The first one uses domains to make this current source available when the threshold is covered. The second one acts on the connectivity between the branch currents and the meshes one.

Figure 2 shows the circuit completed with the current source as a added mesh, having its own voltage value unknown U while the current value J is known.

We can construct the connectivity C between the branch currents and the mesh ones (K 1 , K 2 ):

C = 1 0 1 1 (7)
If i q are the branch currents (one flows into R 0 and the other into Z d ), we have i q = C q m K m . The impedance operator in the branch space being defined by:

Z = R 0 0 0 Z d (8)
Changing from branch to the nodes-pair space, we apply ζ = C T ZC in order to obtain:

ζ = Z d + R 0 Z d Z d Z d (9)
This impedance operator leads us to the system of equations:

   E = (Z d + R 0 ) K 1 + Z d K 2 U = Z d K 1 + Z d K 2 (10)
The first equation should be sufficient to solve the unknown current K 1 with the known source E and current K 2 :

E -Z d K 2 = (Z d + R 0 ) K 1 (11)
But Z d is never equal to zero. it means that K 2 intervenes even when the level of v is lower than the threshold. We understand that another function acting on K 2 must be added. The simplest way to modify the influence of K 2 is to make it existing only when the threshold is exceeded. It means that the connectivity can be defined using a domain like:

C = 1 0 1 v D x (12)
With this connectivity, which depends on v, the relation between i 2 and K 2 exists only when the threshold is exceeded. For all the other values of v, the residual equation is:

E = (Z d + R 0 ) K 1 (13)
More, the fact to use a dynamic connectivity is an accurate description of the physical process involved, because the non linear current source exists effectively only when the threshold is exceeded. For all the other values of v, this source doesn't exist.

Next step is to see if our model can be computed using classical scientific programing under Python for example. First step is to choose a function for the domains. Various choices are available, more or less easy to use. Another challenge is to estimate the amplitude of the parameter v.

. Generator

We can start by making the generator. Any numeric application being a compromise between performances and accuracy, a first work consists in programming the spectrum of a known waveform and to verify that the inverse Fourier's transformation of this spectrum leads to the good time waveform. Starting from the spectrum definition given in 1, we can compute this spectrum choosing the best frequency sampling as possible to limit the number of points and to represent with a good pertinence the spectrum. The time waveform considered is 200 ns width, symmetric with a half period of 100 ns. 1000 points and a window of study of 10 µs allow to obtain a good representation of the signal. Figure 3 shows the spectrum and its correspondence in time computed under Python language. The listing of this first step is given in annex 1.

Once the generator is modeled, we can connect it through an internal resistance to a diode, changing the amplitude of the signal that we can verify in the time domain, using the previous result. To do that, we need to realize the functions of domains.

.

Functions of domains

We can use logistic functions D which is defined in general by:

D(a, c, v) = 1 1 + ae -c(v-p 0 ) - 1 1 + ae -c(v-(p 0 +ω i )) (14)
for values of v that belong to [0, 100]V. p 0 is the starting value of the window (for example p 0 = 20) and ω i the window width (for example ω i = 30). If the parameter value v is inferior to 100 V, we can choose a = 1 and c = 10. With this original values we understand that to make v around 1 V for example, we may increase c, p 0 , ω i of a scale factor f e with:

D(a, c, v) = 1 1 + ae -c fe (v-p 0 f e ) - 1 1 + ae -c fe (v-(p 0 +ω i )f e ) (15)
The program to trace the logistic curve is given in annex 2, an example of logistic curve is shown figure 4 for f e = 10 -3 .

.

Loop including the generator and the diode

We have at this step all the elements to compute the whole circuit with the generator in series with its own impedance and the diode. The various impedances controlled by the functions of domains and the functions of domains themselves must be defined before to compute the current in the diode. For the example we use simple function (it is another solution) that set the values of the variables depending on the parameter values given in arguments. Equation 11is computed. Listing is given in annex, and the result for a low parameter value is shown figure 5 while the result obtained for a high value of the parameter is shown figure 6.

In a general case, a circuit can be separated in two parts: linear parts that do not change the spectrums and non linear parts that change them. The propagation of the new spectrum created by the non linear behavior is unchanged by the passive parts. So we need only a technique to model non linear component in general, making some exception with magnetic materials. The technique submitted in this article uses the assumption that a previous characterization of the component has been made, giving the frequency working of the component. We have shown its mechanism in the case of a simple diode, but the method can be applied on any non linear source. The added harmonics are taken in charge through a current source in the nodes-pair space. In the example given, we clearly see the change in the spectrum of the signal for the current in the diode. At low level, the diode behaves like a high resistor and the current waveform follows the electromotive force waveform. But when the level becomes high enough, a current source suppresses the negative part of the source waveform, giving the result obtained for a single positive pulse. In that case, an average value appears in the spectrum that doesn't exist in the previous case where the waveform was symmetric, and so without any average value. Kron's method in the nodes-pair space is available to take into account non linear devices in the harmonic domain, coupled with the domain functions previously presented by the author[MAURICE( 2017

( E [ f ] ) ax [ f ]= s # a t [ f ]= f /(T ) ## iLT computation #--------------------- f o

Figure 1

 1 Figure 1 Case of a simple diode. . . . . . . . . . . . . . . . . . . 11 Figure 2 Added current source. . . . . . . . . . . . . . . . . . . 11 Figure 3 Generator modeling. . . . . . . . . . . . . . . . . . . . 11 Figure 4 Function of domains. . . . . . . . . . . . . . . . . . . . 12 Figure 5 Answer for low level. . . . . . . . . . . . . . . . . . . . 12 Figure 6 Answer for high level. . . . . . . . . . . . . . . . . . . 13

  )]. [MAURICE(2017)] Maurice, O. Impedances defined on domains. In Elements of theory for electromagnetic compatibility and systems; Bookelis; Publishing House: Aix en Provence, France, 2017; pp. 32-58, ISBN. e r o s (N, dtype=complex ) ax=np . z e r o s (N, dtype= f l o a t ) a t =np . z e r o s (N, dtype= f l o a t ) ut=np . z e r o s (N, dtype= f l o a t ) r e s =np . z e r o s (N, dtype= f l o a t ) T=10000E-9 f o =1./T#100 E6/N A= f o * N dt=T/N f o r f i n range ( 1 ,N) : s= f * f o w= 2 . * np . p i * s p=1 J * w # E [ f ]=A * ( 1 . / p-2./p * np . exp ( -100E-9 * p ) + 1 . / p * \ np . exp ( -200E-9 * p ) ) * np . exp ( -100E-9 * p ) r e s [ f ]= abs

  x s c a l e ( ' log ' ) p l t . x l a b e l ( ' frequency [Hz ] ' ) p l t . y l a b e l ( ' l e v e l [V ] ' ) p l t . s u b p l o t ( 1 , 2 , 2 ) # t i m e r =abs ( np . f f t . i f f t ( E ) ) p l t . p l o t ( a t [ 0 : 1 0 0 ] , ut [ 0 : 1 0 0 ] ) p l t . g r i d ( True , which = ' both ' ) p l t . x l a b e l ( ' time [ ns ] ' ) p l t . y l a b e l ( ' l e v e l [V ] . z e r o s (N, dtype= f l o a t ) av=np . z e r o s (N, dtype= f l o a t ) # f e =1E-2 # f a c t e u r d ' e c h e l l e a = 1 . c =10./ f e # r a i d e u r de pente po = 4 0 . * f e # p o i n t i n i t i a l wi = 4 0 . * f e # duree f o r i v i n range (N) : v= i v * 1E-1 * f e # v v a r i e de ( 0 a 1 0 0 ) * f e l o [ i v ] = 1 . / ( 1 . + a * np . exp(-c * ( v-po ) ) ) -1 . / ( 1 . + a * np . exp(-c * ( v-(po+wi ) ) ) ) av [ i v ]= v p l t . p l o t ( av , l o ) p l t . g r i d ( True , which = ' both ' ) p l t . x l a b e l ( ' v ' ) p l t . y l a b e l ( ' l o g i s t i c ( v ) ' ) p l t . show ( ) Annex 3 # - *coding : u t f -8 - * -" " " Created on Mon Nov 6 1 7 : 3 7 : v < 0 . 6 ) : r e t u r n 1 e l s e : r e t u r n 0 def Drdyn ( v ) : i f ( v > = 0 . 6 ) : r e t u r n 1 e l s e : r e t u r n 0 # N=1000 r e s =np . z e r o s (N, dtype= f l o a t ) c =10./ f e p o r i = -40. * f e w i r i = 8 0 . * f e pordyn = 4 0 . * f e wirdyn = 4 0 . . * f e # -0.1 p r i n t " v : "+ s t r ( v ) E=np . z e r o s (N, dtype=complex ) K1=np . z e r o s (N, dtype=complex ) K2=np . z e r o s (N, dtype=complex ) ax=np . z e r o s (N, dtype= f l o a t ) #v=np . z e r o s (N, dtype= f l o a t ) a t =np . z e r o s (N, dtype= f l o a t ) ut=np . z e r o s (N, dtype= f l o a t ) v t =np . z e r o s (N, dtype= f l o a t ) e t =np . z e r o s (N, dtype= f l o a t ) f o r f i n range ( 1 ,N) :s= f * f o w= 2 . * np . p i * s p=1 J * w # E [ f ]=Amp * ( 1 . / p-2./p * np . exp ( -100E-9 * p ) + 1 . / p * \ np . exp ( -200E-9 * p ) ) * np . exp ( -100E-9 * p ) K2 [ f ]=Amp/(Ro+Rdyn ) * ( 1 . / p * np . exp ( -100E-9 * p) -1./p * \ np . exp ( -200E-9 * p ) ) * np . exp ( -100E-9 * p ) Zd=Dri ( v ) * Ri+Drdyn ( v ) * Rdyn K1 [ f ] = ( E [ f ]+ Drdyn ( v ) * K2 [ f ] * 1 1 . ) / ( Ro+Zd ) ## #--------------------f or o t i n range (N) : # ut [ o t ]= abs ( spec [ o t ] ) a t [ o t ]= o t * dt * 1E9 f o r i i n i n range ( 1 ,N) : p=1 J * 2 . * np . p i * i i n /T ut [ o t ]= ut [ o t ]+np . r e a l ( K1 [ i i n ] * np . exp ( p * o t * dt )) #--------------------f o r o t i n range (N) : # ut [ o t ]= abs ( spec [ o t ] ) a t [ o t ]= o t * dt * 1E9 f o r i i n i n range ( 1 ,N) : p=1 J * 2 . * np . p i * i i n /T v t [ o t ]= v t [ o t ]+np . r e a l ( K2 [ i i n ] * np . exp ( p * o t * dt ) ) #--------------------f or o t i n range (N) : # ut [ o t ]= abs ( spec [ o t ] ) a t [ o t ]= o t * dt * 1E9 f o r i i n i n range ( 1 ,N) : p=1 J * 2 . * np . p i * i i n /T e t [ o t ]= e t [ o t ]+np . r e a l ( E [ i i n ] * np . exp ( p * o t * dt ) )

Figure 1 :

 1 Figure 1: Case of a simple diode.

Figure 2 :

 2 Figure 2: Added current source.

Figure 3 :

 3 Figure 3: Generator modeling.

Figure 4 :

 4 Figure 4: Function of domains.

Figure 5 :

 5 Figure 5: Answer for low level.

Figure 6 :

 6 Figure 6: Answer for high level.

Department EMC and Laser, ARIANE Group, route de Verneuil, 78 Les Mureaux, France

p l t . s u b p l o t ( 1 , 2 , 1 ) p l t . p l o t ( abs ( K1 ) ) p l t . g r i d ( True , which = ' both ' ) p l t . x s c a l e ( ' log ' ) p l t . t i t l e ( ' K1 ( f ) ' ) p l t . x l a b e l ( ' sample ' ) p l t . y l a b e l ( ' l e v e l ' ) p l t . s u b p l o t ( 1 , 2 , 2 ) p l t . p l o t ( a t [ 0 : 1 0 0 ] , ut [ 0 : 1 0 0 ] ) p l t . g r i d ( True , which = ' both ' ) p l t . x l a b e l ( ' time [ ns ] ' ) p l t . y l a b e l ( ' l e v e l [V ] ' ) p l t . t i t l e ( ' K1 ( t ) ' ) p l t . show ( )


Jérôme Baray

Albert Da Silva

Jean-Marc Leblanc

L'

L'analyse lexicale au service de la cliodynamique : traitement par intelligence artificielle de la base Google Ngram

teaching and research institutions in France or abroad, or from public or private research centers.

Qu'est ce que la cliodynamique ?

La cliodynamique est un domaine de recherche assez récent qui considère l'histoire comme un objet d'étude scientifique. δ'étymologie du terme est grecque composé de Clio (Κ / Kleiố, de έω / kleô, « célébrer, chanter »), la muse de l'histoire qui chante le passé des hommes et des cités, et de dynamique, dynamikos (« puissant », « efficace »), dérivé de α ς, dynamis (« puissance ») qui sousentend de façon générale l'analyse des changements. De nature transdisciplinaire, la cliodynamique tente ainsi d'expliquer les processus dynamiques historiques comme la montée ou l'effondrement des empires ou civilisations, les cycles économiques, les booms de population, les modes grâce à la modélisation mathématique, le datamining, l'économétrie ou encore la sociologie culturelle. Les big data agrégeant des données historiques, archéologiques ou économiques sont la matière permettant d'alimenter ces modèles quantitatifs. La cliodynamique comporte également un volet empirique examinant l'adéquation des hypothèses et des prévisions de modèles dynamiques avec les données historiques. Elle s'inscrit dans la démarche de la cliométrie ou « new economic history » qui étudie l'histoire grâce à des méthodes tirées de l'économétrie.

Enjeux de recherche

Il s'agissait d'une part de concevoir une méthode robuste d'analyse lexicale capable de traiter une série de très gros corpus datés dont le contenu évolue à travers le temps (big data) avec l'enjeu de cerner les évolutions sociétales et les grandes périodes historiques clés dans le cadre de la cliodynamique. L'analyse lexicale s'est d'autre part penchée sur les enseignements à tirer de la base de données Google books Ngram (https://books.google.com/ngrams) qui détaille le nombre d'occurrences des mots utilisés année après année dans les publications scannées et intégrées au moteur de recherche Google Books. On considère que cette base a compilé environ 20% des livres publiés dans les langues majeures. Nous nous sommes focalisés sur les ouvrages en langue anglaise publiés aux Etats-Unis et en Grande Bretagne. L'objectif a été de cerner l'utilisation plus ou moins forte de certains mots selon les époques, la période d'étude ayant été fixée de 1860 à 2008. Les ouvrages avant 1860 paraissent être en bien moins grand nombre et la base de données Google books Ngram s'arrête à l'année 2008.

Principe de la méthode

La méthode a été de constituer dans un premier temps un dictionnaire des mots anglais les plus usités en faisant abstraction des termes à double sens, préposition, articles, pronoms. Ce dictionnaire a dans cette version initiale rassemblé 1592 mots couvrant de nombreux aspects de la vie sociale et culturelle avec des termes liés à la politique, à la religion, aux arts et aux sciences, à l'industrie, aux objets, à la famille et aux sentiments.

Objectifs & Méthode Résultats

Les étapes de la recherche (big data & textmining)

• 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990

  Dans un second temps, il a été déterminé la représentation en % de chacun de ses mots au sein du dictionnaire année après année après avoir mis l'imposante base Ngram Google Books (1-gram) sur Postgresql. Certains mots comme "king" ou "queen" sont très bien représentés dans le dictionnaire au 19ème siècle avec le règne et la puissance des pouvoirs royaux en Europe, mais l'usage de ces locutions décline au 20ème siècle. On constate donc une évolution constante de la fréquence des mots dans les ouvrages au fur et à mesure des époques. La troisième étape a été d'effectuer une analyse factorielle en composantes principales centrée et normée sur le tableau décrivant la représentation des mots en % selon les années de 1860 à 2008 (1592 colonnes où les mots représentent les variables et 141 lignes représentant les individus statistiques c'est-à-dire les ouvrages publiés année après année et recensé dans Google Books). Une classification des années en groupes est réalisée grâce à un réseau de neurones (carte auto-adaptative de Kohonen en IA).

L

  'analyse lexicale au service de la cliodynamique : traitement par intelligence artificielle de la base Google Ngram Jérôme Baray -Albert Da Silva -Jean-Marc Leblanc Projet PEPS CNRS/UPE Eclavit δ'analyse des résultats indique une certaine stabilité dans l'utilisation des mots du vocabulaire avec néanmoins des périodes de rupture. δ'appréciation qualitative des nuées de mots permet d'interpréter la signification des axes et l'on s'aperçoit que l'axe 1 oppose des notions d'humanisme et de grands sentiments à celles de la rationalité froide. δ'axe 2 sépare les années fastes à celles de l'insécurité et du chaos alors que l'axe 3 oppose les périodes plus libérales & individualistes à celles plus révolutionnaires et collectivistes. Par ailleurs, certaines zones géographiques stratégiques sont bien plus évoquées à la veille de la majorité des conflits majeurs : Afghanistan, Irak, Biélorussie,… Conclusion Hormis ces enseignements, la méthode permet de : ✓ Dater et catégoriser les discours: commerciaux et marketing, communication institutionnelle, discours politiques,… ✓ Analyser des tendances sociales et de consommation, enquêtes marketing en créant un nouveau dictionnaire adapté à certaines disciples : ressources humaines, finance, marketing ✓ Réaliser des prévisions de tendances (périodes de crises, de conflits ou d'embellie économique à venir), ✓ Analyser des tendances socioéconomiques. Bibliographie • Turchin, P. (2003) Historical dynamics: why states rise and fall, Princeton University Press, Princeton, NJ. • Diebolt Claude, Haupert Michael (2015) Handbook of Cliometrics, Springer-Verlag Berlin and Heidelberg GmbH & Co.

  1860-1890 -Religion, Justice & Grandeur Groupe 1-2 : 1891-1910 -Prospérité, Joie de Vivre & Arts Groupe 1-3 : 1911-1934 -Industrie, Commerce & Transports Groupe 2-3 : 1935-1954 -Dictatures, Conflit Mondial & Sources d'Approvisionnement Groupe 3-3 : 1953-1972 -Décolonisation, Révoltes & Sciences Groupe 3-2 : 1973-1986 -Menaces, Santé et Innovations Techniques Groupe 3-1 : 1987-2006 -Communications, Informatique, Services Groupe 2-1 : 2008 -Amusement, Relations & Risques Les variations des variables factorielles 2 et 3 montrent une relation surprenante mais logique avec les périodes de prospérité et de conflit :

  Cette étude a également débouché sur la réalisation d'un site internet et d'un outil d'analyse capable de classifier et « dater » de nouveaux corpus : http://eclavit.univ-mlv.fr/DaText/index.php Pays jouant par une évocation plus fréquente de leurs noms un rôle important à la veille de crises ou de conflits (Corrélation / axe factoriel 3)


Jalil Elhassouni
email: jalil.elhassouni@gmail.com

Mehdi Bazzi

Abderrahim El Qadi

Mohamed El Haziti

Applying ontologies to data integration systems for bank credit risk management

Keywords: credit risk management, data integration, ontologies alignment

This paper proposes an ontological integration model for credit risk management. It is based on three ontologies: one is global and two are local. The global ontology describes credit risk management process and the local ontologies describe credit granting process and present the concepts necessary for the monitoring of credit system. This paper also presents the technique used for matching global ontology with local ontologies.

INTRODUCTION

The integration of data has become increasingly important in many areas. It provides decision makers and analysts with uniform access to distributed and heterogeneous sources of data (relational databases, XML documents, semantic databases...). However, the data integration processes across multiple sources constitute a complicated task because of the semantic heterogeneity encountered at the different levels: metadata, data schemas and data values. This issue is at the heart of many research areas, and this work falls within mapping schemas for bank credit risk management. It takes into consideration the requests of decision makers and the indicators used in the banking industry to measure credit risks. The main step in the risk management process is shown in Figure 1. Based on this architecture, we propose a general ontology that models the credit risk management process and two specific ontologies. The first models the process of credit allocation to clients and the second displays the concepts necessary for the monitoring of a credit system. Subsequently, we arranged the alignment of specific ontologies with the general ontology to facilitate semantic interoperation between applications based on these ontologies. Figure 1 displays the different steps of risk management process:

• Identifying and quantifying risks in order to measure their impact.

• Treating risks (according to the impact and the strategy of the bank) by transferring, reducing or accepting them. • Once the risk management strategy is defined, it is ready for implemented. Banker, statistical models and IT infrastructure assess the risk of existing and new investments. The whole cycle is continuously evaluated to detect new risks and improve the existing systems. The remainder of this paper is organized as follows: section one provides a literature review and section two explains the process of creating ontologies in the field of bank credit risk. The last section concludes with discussing future work.

I RELATED WORK

In recent decades, several studies have been conducted in the field of data integration. Two major approaches have been set up to design data integration systems [START_REF] Hacid | L'intégration de sources de données[END_REF]]:

• The mediator approach where the exploitation of the data is based on the virtual views.

• The Data Warehouse approach (DW) where the data are duplicated in the same global schema. In the mediator approach, the integration of data follows two steps:

• The integration system generates from a user request as many sub-queries as data sources request. • The integration system constructs the final response from the result of each sub-query posed to each data source and transmits it to the user. This approach is well suited where information changes quickly; i.e., when the user requests are not predictable and when the numbers of data and data sources are high.

In the Data Warehouse approach, data integration is also carried out in two stages:

• The integration system merges and stores different data sources into a single data warehouse.

• The user's request searches for the data in DW without having to access the original data sources. This approach is suitable when the information is summarized and annotated or when the user requests can be predefined.

Variuos ways of applying the above-stated approaches are used different researches. [START_REF] Calvanese | Data integration in data warehousing[END_REF] proposes a method based on a global ontology as an intermediate model between DW and source schemas. [START_REF] Lamolle | Intégration de Bases de données hétérogènes par une modélisation conceptuelle XML[END_REF]] provide a mediation model based on XML. [START_REF] Bakhtouchi | Materializing attributes annotation: A hybrid approach for databases integration[END_REF] utilize a hybrid framework to improve the performance of queries in heterogeneous and autonomous data integration systems. [START_REF] Dean | MapReduce: Simplied Data Processing on Large Clusters[END_REF] and [START_REF] Yuan | VDB-MR: MapReduce-based distributed data integration using virtual database[END_REF] develop an approach that is based on MapReduce. [START_REF] Mate | Ontology-Based Data Integration between Clinical and Research Systems[END_REF] use an ontological approach to describe and organize the medical concepts of both data and target sources. [START_REF] Krisnadhi | The geolink framework for pattern-based linked data integration[END_REF] put forword an approach which combines linked data publishing and modular ontology engineering based on ontology design patterns.

However, some of these applications are limited to integrating only one source such as [START_REF] Mate | Ontology-Based Data Integration between Clinical and Research Systems[END_REF], while others such as [START_REF] Krisnadhi | The geolink framework for pattern-based linked data integration[END_REF] and [START_REF] Berkani | Generic methodology for semantic data warehouse design: From schema definition to etl[END_REF]] integrate multiple sources. Several types of data thus are integrated in different ways. The approaches proposed in in [START_REF] Abadie | Formalisation, acquisition et mise en oeuvre des connaissances pour l'intégration virtuelle des bases de données géographiques[END_REF], [START_REF] Zhao | Ontology-based geospatial data query and integration[END_REF] and [START_REF] Aji | Hadoop GIS: a high performance spatial data warehousing system over mapreduce[END_REF] manipulate spatial data, whereas other propositions manipulate alphanumeric data as presented in [START_REF] Yuan | VDB-MR: MapReduce-based distributed data integration using virtual database[END_REF], linked data in [START_REF] Krisnadhi | The geolink framework for pattern-based linked data integration[END_REF] or ontologies-based data as presented in [START_REF] Berkani | Generic methodology for semantic data warehouse design: From schema definition to etl[END_REF].

All the approaches presented above use manual or semi-automatic solutions requiring more time and domain experts' intervention. The approach provided by [START_REF] Mate | Ontology-Based Data Integration between Clinical and Research Systems[END_REF] does not support mapping at hierarchical levels between the target and source ontologies. The target ontology is too cumbersome to use and intricate. Hence, when a new data repository is added, a complicated adjustment of the ontology may become necessary. In [START_REF] Krisnadhi | The geolink framework for pattern-based linked data integration[END_REF] the absence of data reasoning and alignment between data repositories may provoke the incoherence and reducndacy of the target data.

II ONTOLOGIES BASED DATA INTEGRATION APPROACH

Ontology is a formal, explicit specification of a shared conceptualization in terms of concepts (i.e., classes), properties and relations [START_REF] Gruber | Toward principles for the design of ontologies used for knowledge sharing[END_REF]. The term "formal" means the ontologies should be expressed in a formal and semantic language (RDF, OWL, PLIB...) allowing automatic reasoning. "Explicit" signifies ontology's acceptance by all expert community members and "Conceptualization" refers to an abstract, simplified view of the world that we wish to represent for some purposes.

According to [START_REF] Gagnon | Ontology-based integration of data sources[END_REF] the "ontologies' integration" carries at least three different meanings (and uses) of ontology integration depending on the situation:

• Integration: building a new ontology from other existing (sub) ontologies.

• Merging: building a new ontology by merging existing ontologies i.e.: building a single ontology that unifies them all. • Usage in applications: building an application ontology using one or more ontologies.

Existing ontologies are adapted if necessary.

Credit allocating to customers (Credit Scoring)

Figure 2 displas the architecture of the credit scoring system. This process can divided into two phases. The first phase shows the building of the statistical model, which is characterized by a set of steps like data gathering activities, initial analysis, model development and implementation. The second phase demonstrates the use of the statistical model to make creditrelated decision such as fixing the cut-off, setting credit limits, pricing and storing the information. Based on the risk management process presented in Figure 1, we propose a conceptual model Figure 3, where we widely use a set of variables to build credit scoring system. Some variables are related to the credit itself (Credit Requirements), while others are related to the financial profile (Financial Data) and to the personal profile (Socio Demographic) of the borrower. Other variables, however, are related to the final decision and to theborrower's rating (Decision Data). We modeled the credit allocation process on the form of ontology given in Figure 4.

Ontology for credit monitoring

Credit monitoring is related to 'post-credit decision' collection, analysis of information, and most importantly the identification of early warning signs. Its also entails ensuring the appropriate decisions are made with regard to changes in customers' credit risk.

During the credit period, economic variations may cause some changes that have an impact on credit risk. Banks have to monitor credit exposures periodically and continuously so that changes are detected on time. The Figure 5 illustrates the main steps in credit monitoring.

Figure 5.Main steps in credit monitoring.

Global process of risk management

Having thus defined credit allocation ontology and the credit monitoring ontology, we will build the global ontology, which will provide the user with a unified view, global schema, of data sources. In the modeling part of the global ontology, we organized several meetings and brainstorming sessions with the banking experts resulting in the development of the version described in Figure 6.

Ontologies alignment

In the last decade, several studies were released in the schema matching area, including ontology alignment. This ontology alignment is an important functionality in many applications such as data exchange, schema evolution, data integration etc. It is the basis of linking information, e.g., from heterogeneous sources into a common model that can be queried and reasoned upon. Subsequently, we have aligned the ontology of credit allocation with the global ontology in Figure 7 Figures 7 and8 show the mapping generated by LogMap algorithm. They illustrate the alignment of classes and data properties, which are semantically equivalent in both onthologies, and the degree of similarity between them.

Ontologies are created to follow up credits of various customers of the bank and allow for several analyses including:

• Analysis of the stability of elaborated models which requires reviewing the scoring models developed by banks each year. • Roll rate analysis: The percentage of credit customers who become increasingly delinquent on their accounts. Roll rate refers to the percentage of customers who "roll" from 30-day-late to 60-day-late categories, or from 60-day-late to 90-day-late categories, and so on.

Conclusion

In this paper, we presented our model of creating ontologies to facilitate the integration of Credit Scoring data. We proposed a general ontology that models the credit risk management process, and two specific ontologies that model the allocation and monitoring of credit. The result of the alignment shows the entities (classes and data properties) that are semantically equivalent. In our future work we plan to model the tracking credits taking into account the requests of decision makers in the process of integration. We also plan to develop a hybrid model for credit risk management based on ontologies and MapReduce approach.

Figure 1 .

 1 Figure 1. Different steps of a risk management process.

Figure 2 .

 2 Figure 2. Process of allocating credit to customers (Credit Scoring).

Figure 3 .

 3 Figure 3.Conceptual model of credit risk management.

Figure 4 .

 4 Figure 4. The ontology of the credit allocation process -partial view.

  using LogMap Ontology Matching Tool [Jiménez-Ruiz and Grau, 2011].

Figure 7 .

 7 Figure 7.The alignment file of credit allocation ontology with the global ontology in text format -partial view.

Figure 8 .

 8 Figure 8.The alignment file of credit monitoring ontology with the global ontology in text format -partial view.

Journal of Data Mining and Digital Humanities http://jdmdh.episciences.org

http://jdmdh.episciences.org

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