Berkovich Geometry Research Articles

Reconstruction anabélienne du squelette des courbes analytiques

In this work we bring to light some anabelian behaviours of analytic curves in the setting of Berkovich geometry. We show more precisely that the knowledge of the tempered fundamental group of some curves that we call analytically anabelian determines their analytic skeletons as graphs. The tempered fundamental group of a Berkovich space, introduced by Andr\'e, enabled Mochizuki to prove the first result of anabelian geometry in Berkovich geometry concerning analytifications of algebraic hyperbolic curves over $\overline{\mathbb{Q}}_p$. To that end, Mochizuki developed the categorical language of semi-graphs of anabelio\ids and tempero\ids. Our work consists in associating a graph of anabelio\ids to a Berkovich curve equipped with a minimal triangulation and in adapting the results of Mochizuki in order to recover the analytic skeleton of the curve. The novelty of this anabelian result in Berkovich geometry is that the curves we are interested in are not supposed anymore to be of algebraic nature. We show for example that the famous Drinfeld half-plane is an analytically anabelian curve.

Investigation and Measurement of Nanomechanical Properties of the HSS Powder Metallurgy ASP2017 and ASP2055 Steels

The intention of the team of authors in the presented scientific article was experimental measurement with subsequent evaluation of nanomechanical properties of selected microstructural components of the investigated high-speed steels type ASP2017 and ASP2055 produced by powder metallurgy. The whole experiment was performed under laboratory conditions on an experimental apparatus called Hysitron TI 950 Triboindeneter. The used device is a part of the laboratory equipment of the Center for Diagnostics and Quality Testing of Materials. The proposed process of experimental research of the mentioned properties of PM steels was carried out precisely because of the relatively high require-ments for the use of this type of steel for the needs of special applications in the engineering industry. Therefore, the research of nanomechanical properties such as nanohardness H and the reduced and real Young's modulus Er and the Young's modulus of elasticity Es were therefore the subject of the authors' research. The authors proposed the use of a Berkovich geometry tip as a test or indentation body, respectively. In terms of the proposed structure of the article, the authors describe the current state of research from the assigned problem using database sources. The experimental part of their research already solves the obtained nanomechanical properties of individual components of the struc-ture of the tested PM steels, as well as the distribution of individual positions of indents. These were designed using so-called SPM scans (Scanning Probe Microscopy). Nanoindentation curves were generated from them using the Triboscan evaluation software. Based on the performed measurements and the Er obtained from them, and the Es was also calculated using mathematical relations. The authors graphically expressed the final values of nanomechanical quantities and described them in a discussion of the experiments.

Mechanical characterization by multiscale indentation of particle reinforced Nickel-Alumina metal matrix nanocomposites obtained by high-kinetic processing of ball milling and spark plasma sintering

We report the mechanical characterization by multiscale indentation of particle reinforced dense Ni/Al2O3 metal matrix nanocomposites (MMNC) obtained by high-kinetic processing (HKP) of ball milling of the powders with a systematic variation of alumina nanoparticles fractions (from 1 to 20 vol. %) and spark plasma sintering (SPS). The morphology and particle size distribution of powder were evaluated as a function of milling time up to 10 h. Samples from the 10 h milled powders were densified by SPS. The mechanical properties of the sintered samples were obtained by micro and nanoindentation using diamond tips of Vickers and Berkovich geometry, respectively. The combination of HKP and SPS allowed a homogeneous dispersion of Al2O3 nanoparticles in the nickel matrix and effective reinforcing effects, which is the case of Ni/10 vol. % Al2O3 and Ni/15 vol. % Al2O3 samples. The highest hardness (4.68 ± 0.37 GPa) was obtained for Ni/15 vol. % Al2O3 MMNC, which is almost twice that of pure nickel (2.45 ± 0.22 GPa) processed at the same conditions. The highest elastic modulus (346 ± 30 GPa) was obtained for the Ni/10 vol. % Al2O3 sample. The analysis of the load-depth curves confirmed the reinforcing of the MMNCs as a function of the alumina particle content. Discussion of the possible reinforcing mechanisms is also included. The Ni/Al2O3 MMNC sintered specimens exhibit outstanding mechanical property results, which make them candidates for various high-temperature applications.

Global pluripotential theory over a trivially valued field

We develop global pluripotential theory in the setting of Berkovich geometry over a trivially valued field. Specifically, we define and study functions and measures of finite energy and the non-Archimedean Monge-Ampere operator on any (possibly reducible) projective variety. We also investigate the topology of the space of valuations of linear growth, and the behavior of psh functions thereon.

Enumeration of holomorphic cylinders in log Calabi–Yau surfaces, II : Positivity, integrality and the gluing formula

We prove three fundamental properties of counting holomorphic cylinders in log Calabi-Yau surfaces: positivity, integrality and the gluing formula. Positivity and integrality assert that the numbers of cylinders, defined via virtual techniques, are in fact nonnegative integers. The gluing formula roughly says that cylinders can be glued together to form longer cylinders, and the number of longer cylinders equals the product of the numbers of shorter cylinders. Our approach uses Berkovich geometry, tropical geometry, deformation theory and the ideas in the proof of associativity relations of Gromov-Witten invariants by Maxim Kontsevich. These three properties provide an evidence for a conjectural relation between counting cylinders and the broken lines of Gross-Hacking-Keel.

Assessment of Mechanical and Tribological Properties of Diamond-Like Carbon Coatings on the Ti13Nb13Zr Alloy

Abstract The paper presents the results of tests on the geometric structure of the surface, hardness and tribological tests of coatings produced by the method of physical vapor deposition (PVD) on the Ti13Nb13Zr alloy. Hardness was determined using MCT3 from Anton Paar, by instrumental indentation. A diamond indenter with Berkovich geometry was used for the measurement.Model tribological tests were carried out in reciprocating motion under conditions of technically dry friction, friction with lubrication of synovial fluid and Ringer’s solution. The counter-sample in the tested friction nodes was a ball of Al2O3 with a diameter of 6 mm. Nanometer hardness measurements showed that as a result of the diamond-like carbon coating (DLC), the hardness increased by an 7-fold. The lowest friction coefficients among all tested friction nodes were obtained for the material combination Ti13Nb13Zr a-C:H - Al2O3 regardless of the conditions of testing. In the case of technically dry friction, the coefficient of friction decreased by 80%, synovial fluid by 70%, and in the case of Ringer’s solution by 88% compared to the results obtained for Ti13Nb13Zr.

Tropical dynamics of area-preserving maps

We consider a class of area-preserving, piecewise affine maps on the 2-sphere. These maps encode degenerating families of K3 surface automorphisms and are profitably studied using techniques from tropical and Berkovich geometries.

Comment on van Casteren et al. (2018): softer metallic spheres do abrade harder enamel.

Comment on van Casteren et al. (2018): softer metallic spheres do abrade harder enamel.

Numerical Simulation of the Depth-Sensing Indentation Test with Knoop Indenter

Depth-sensing indentation (DSI) technique allows easy and reliable determination of two mechanical properties of materials: hardness and Young’s modulus. Most of the studies are focusing on the Vickers, Berkovich, and conical indenter geometries. In case of Knoop indenter, the existing experimental and numerical studies are scarce. The goal of the current study is to contribute for the understanding of the mechanical phenomena that occur in the material under Knoop indention, enhancing and facilitating the analysis of its results obtained in DSI tests. For this purpose, a finite element code, DD3IMP, was used to numerically simulate the Knoop indentation test. A finite element mesh was developed and optimized in order to attain accurate values of the mechanical properties. Also, a careful modeling of the Knoop indenter was performed to take into account the geometry and size of the imperfection (offset) of the indenter tip, as in real cases.

Topology of spaces of valuations and geometry of singularities

Given an algebraic variety X X defined over an algebraically closed field, we study the space R Z ( X , x ) \mathrm {RZ}{(X,x)} consisting of all the valuations of the function field of X X which are centered in a closed point x x of X X . We concentrate on its homeomorphism type. We prove that, when x x is a regular point, this homeomorphism type only depends on the dimension of X X . If x x is a singular point of a normal surface, we show that it only depends on the dual graph of a good resolution of ( X , x ) (X,x) up to some precise equivalence. This is done by studying the relation between R Z ( X , x ) \mathrm {RZ}{(X,x)} and the normalized non-Archimedean link of x x in X X coming from the point of view of Berkovich geometry. We prove that their behavior is the same.

Enumerative geometry of elliptic curves on toric surfaces

We establish the equality of classical and tropical curve counts for elliptic curves on toric surfaces with fixed $j$-invariant, refining results of Mikhalkin and Nishinou--Siebert. As an application, we determine a formula for such counts on $\mathbb P^2$ and all Hirzebruch surfaces. This formula relates the count of elliptic curves with the number of rational curves on the surface satisfying a small number of tangency conditions with the toric boundary. Furthermore, the combinatorial tropical multiplicities of Kerber and Markwig for counts in $\mathbb P^2$ are derived and explained algebro-geometrically, using Berkovich geometry and logarithmic Gromov--Witten theory. As a consequence, a new proof of Pandharipande's formula for counts of elliptic curves in $\mathbb P^2$ with fixed $j$-invariant is obtained.

Predictions of frame compliance and apex morphology in sharp nanoindentations

If all components in a nanoindentation system are well calibrated and a reference material has unique hardness, H and reduced modulus, Er independent of the indentation depths, the load, L and the penetration depth, h in the indentation loading curve of the reference material can be correlated by L=Kh2. Here the constant K is expressed by H, Er and indenter geometry constants. By using H and Er of a fused silica and the Berkovich geometry, an analytical expression for the indentation loading curve could be derived. To compare with this analytical loading curve, experimental indentation data were measured with two commercial nanoindenters. The experimental loading curves shifted leftward or rightward from the analytical loading curve and this depth deviation was attributed to improper calibration of the nanoindenters. Quantitative calibrations of frame compliance and indenter bluntness were tried for the raw nanoindentation data and this resulted in consistent nanoindentation data regardless of the used nanoindenters.

Modification of some Optical and Mechanical Properties of Amorphous As-S-Se Thin Films by Copper Introduction

Abstract. In this research experimental investigation of the influence of copper introduction on some relevant parameters in As-S-Se amorphous thin films is performed. Copper is introduced into As2(S0.5Se0.5)3 amorphous thin film in concentration of 3 at.%. Samples of As2(S0.5Se0.5)3 and Cu3(As2(S0.5Se0.5)3)97 amorphous thin films are prepared by the vacuum thermal evaporation technique from previously synthesized bulk samples. Envelope method is applied for the determination of the optical constants, using the transmission and reflection spectra. The dispersion of the refractive index is discussed in terms of the single oscillator model proposed by Wemple–DiDomenico. Values of absorption coefficients in the high absorption region are discussed according to Tauc's law.Instrumented indentation testing is performed, using the Berkovich geometry indenter, for obtaining the value of nano-hardness.All the determined parameters have shown the increase with introduction of copper into amorphous thin film.

Bruhat–Tits theory from Berkovich's point of view. II Satake compactifications of buildings

AbstractIn the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building$\mathcal{B}(\mathrm{G},k)$of a connected and reductive linear algebraic group G over a non-Archimedean fieldkin the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding$\mathcal{B}(\mathrm{G},k)$in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

Nanoindentation in elastoplastic materials: insights from numerical simulations

Finite element simulations of nanoindentation were performed on an elastoplastic material using Berkovich and conical indenters to investigate the effects of geometry on the load–displacement response of the material. The Berkovich indenter, widely used in nanoindentation experiments, is typically simplified to a theoretically equivalent 70.3° conical indenter for numerical simulations, which allows for a less computationally intensive two-dimensional (2D) axisymmetric analysis. Previous studies into the validity of this equivalence assumption for indentations in elastoplastic materials have varying conclusions. Using 2D and 3D finite element simulations, the present study investigates the response of elastoplastic materials, obeying a combined isotropic and kinematic hardening, to indentation with conical and Berkovich indenters. Simulations show that there is a clear difference in the load–displacement response of the selected material to the two indenters. The Berkovich geometry is found to produce a more localized pattern of contact stresses and plastic strains, leading to a smaller mobilized force for the same magnitude of displacement. To further validate the numerical simulations, experimental results of nanoindentation into an aluminium specimen were compared to elastoplastic finite element simulation results. Comparisons suggest that machining-induced residual stresses have likely affected the experimental results.

An improved energy method for determining Young’s modulus by instrumented indentation using a Berkovich tip

We previously proposed a method for estimating Young’s modulus from instrumented nanoindentation data based on a model assuming that the indenter had a spherical-capped Berkovich geometry to take account of the bluntness effect. The method is now further improved by releasing the constraint on the tip shape, allowing it to have a much broader arbitrariness to range from a conical-tipped shape to a flat-ended shape, whereas the spherical-capped shape is just a special case in between. This method requires two parameters to specify a tip geometry, namely, a volume bluntness ratio Vr and a height bluntness ratio hr. A set of functional relationships correlating nominal hardness/reduced elastic modulus ratio (Hn/Er) and elastic work/total work ratio (We/W) were established based on dimensional analysis and finite element simulations, with each relationship specified by a set of Vr and hr. Young’s modulus of an indented material can be estimated from these relationships. The method was shown to be valid when applied to S45C carbon steel and 6061 aluminum alloy.

Indentation size effect in spherical and pyramidal indentations

The indentation size effect (ISE) is studied for spherical and pyramidal indentations on a Ni poly-crystal. The indentation experiments were conducted using a Berkovich geometry as well as different spherical indenters with radii of 0.38, 3.8 and 51.0 µm. A strong ISE is observed for the material yielding a higher hardness at smaller depths or smaller sphere radii. The transition from elastic to plastic behaviour is associated with a pop-in in the load–displacement curve, in contrast to the conventional elastic–plastic transition as discussed by Tabor. The indentation response is modelled using Tabor's approach in conjunction with the uniaxial macroscopic stress–strain behaviour for calculating the statistically stored dislocation density for a given indenter geometry. The geometrically necessary dislocation (GND) density is calculated using a modified Nix/Gao approach, whereas the storage volume for GNDs is used as a parameter for the measured depth dependence of hardness. It will be shown that the ISE for both pyramidal and spherical indentations is related and can be understood within the same given framework. The indentation response of metallic materials can thus be modelled from pop-in to macroscopic hardness.

Identifying the limitation of Oliver and Pharr method in characterizing the viscoelastic-plastic materials with respect to indenter geometry

AbstractNanoindentation tests are widely used in recent years to characterize the mechanical properties of viscoelastic-plastic materials like polymers and biomaterials at the micro or nano-scale using the analysis method proposed by Oliver & Pharr (OP). However, recent studies revealed that the mechanical properties of viscoelastic-plastic (polymeric) materials determined using the OP method does not lead to a correct evaluation of Young's modulus. A systematic experimental study is performed with different indenter geometries like spherical and Berkovich geometries using various polymers in order to identify the limitations of the OP method.

Alignment uncertainties in ideal indentation styli

In this work, the first and second order uncertainty contributors in depth-sensing hardness measurements are investigated. Each term in the hardness expanded uncertainty is examined to determine the typical sources and the magnitude errors contributed to the measurement. In particular, using simulations based on ideal indenter geometries the depth uncertainty with area function uncertainty due to rotational misalignments between the stylus and the specimen is evaluated. Three typical styli are simulated; standard Berkovich, modified Berkovich, and Vickers geometries. Considering the maximum allowable tip and tilt from ISO 14577, the area function uncertainty was less than 1%. Also considered is the effect of the twist about the central axis which shows that, because of the asymmetric shape, the Berkovich geometries have larger errors as a function of rotational deviations than the Vickers geometry from a perfect alignment.

Determination of Plastic Properties of Polycrystalline Metallic Materials by Nanoindentation – Experiments and Finite Element Simulations

ABSTRACTThe determination of plastic properties of metallic materials by nanoindentation requires the analysis of the indentation process and the evaluation methods. Particular effects on the nanoscale, like the indentation size effect or piling up of the material around the indentation, need to be considered. Nanoindentation experiments were performed on conventional grain sized (CG) as well as on ultrafine-grained (UFG) copper and brass. The indentation experiments were complemented with finite element simulations using the monotonic stress-strain curve as input data. All indentation tests were carried out using cube-corner and Berkovich geometry and thus different amount of plastic strain was applied to the material, according to Tabors theory. We find an excellent agreement between simulations and experiments for the UFG materials from which a representative strain of εB ≈ 0.1 and εcc ≈ 0.2 is determined. With these data, the slope of the stress-strain curve is predicted for all materials down to an indentation depth of 800 nm.