Instytut Podstawowych Problemów Techniki

Streszczenie:

This paper focuses on the development of the atomistic framework for determining the lower scale mechanical parameters of single components of a metal matrix composite for final application to a micromechanical damage model. Here, the deformation and failure behavior of NiAl–Al2O3 interfaces and their components, metal and ceramic, are analyzed in depth using molecular statics calculations. A number of atomistic simulations of strength tests, uniaxial tensile, uniaxial compressive and simple shear, have been performed in order to obtain a set of stiffness tensors and strain–stress characteristics up to failure for 30 different crystalline and amorphous systems. Characteristic points on the strain–stress curves in the vicinity of failure are further analyzed at the atomistic level, using local measures of lattice disorder. Numerical results are discussed in the context of composite damage at upper microscopic scale based on images of the fracture surface of NiAl–Al2O3 composites.

Słowa kluczowe:

Metal-matrix composites (MMCs), Fracture, Computational modeling, Mechanical testing, Molecular statics

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The demand for pseudopotentials constructed for a given exchange-correlation (XC) functional far exceeds the supply, necessitating the use of those commonly available. The number of XC functionals currently available is in the hundreds, if not thousands, and the majority of pseudopotentials have been generated for LDA and PBE. The objective of this study is to identify the error in the determination of the mechanical and structural properties (lattice constant, cohesive energy, surface energy, elastic constants, and bulk modulus) of crystals calculated by DFT with such inconsistency. Additionally, this study aims to estimate the performance of popular XC functionals (LDA, PBE, PBEsol, and SCAN) for these calculations in a consistent manner.

Słowa kluczowe:

DFT, pseudopotentials, exchange–correlation functionals

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Recently, Li et al. [C. Li et al., Phys. Rev. B 107, 224106 (2023)] investigated the mechanical behavior of cubic HfC and TaC under simple shear and pure shear using first-principles calculations. Unfortunately, the paper contains some serious and fundamental flaws in the field of continuum mechanics and nanomechanics.
The results presented appear to be qualitatively and quantitatively incorrect; they would be correct if we were in the small/linear deformation/strain regime, which we are not. A correct description therefore requires a finite/nonlinear deformation/strain apparatus. The solution for simple shear, even from density functional theory calculations, must follow Frenkel’s sinusoidal solution.

Słowa kluczowe:

Simple shear, Pure shear, First principle calculation

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The ability of various interatomic potentials to reproduce the properties of silicene, that is, 2D single-layer silicon, polymorphs was examined. Structural and mechanical properties of flat, low-buckled, trigonal dumbbell, honeycomb dumbbell, and large honeycomb dumbbell silicene phases, were obtained using density functional theory and molecular statics calculations with Tersoff, MEAM, Stillinger–Weber, EDIP, ReaxFF, COMB, and machine-learning-based interatomic potentials. A quantitative systematic comparison and a discussion of the results obtained are reported.

Słowa kluczowe:

2D materials, DFT, silicene, interatomic potentials, mechanical properties, force fields

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The capacities of various interatomic potentials available for elemental germanium, with the scope to choose the potential suitable for the modeling of germanene (2D germanium) allotropes,f were investigated. Structural and mechanical properties of the flat, low-buckled, trigonal dumbbell, and large honeycomb dumbbell single-layer germanium (germanene) phases, were obtained using the density functional theory and molecular statics computations with Tersoff, modified embedded atom method, Stillinger–Weber, environment-dependent interatomic potential, ReaxFF, and machine-learning-based interatomic potentials. A systematic quantitative comparative study and discussion of the findings are given.

Słowa kluczowe:

Germanene, 2D materials, Interatomic potentials, Force fields, DFT, Mechanical properties

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Potentially superhard W1−xZrxB2 polymorphs, hP6-P63/mmc-WB2 and hP3-P6/mmm-WB2 , were thoroughly analyzed with zirconium doping in the range of x=0-25%, within the framework of the first-principles density functional theory, from both a structural and a mechanical point of view. The obtained results were subsequently compared with the properties of material deposited by the magnetron sputtering method. All predicted structures are mechanically and thermodynamically stable. Theoretical calculations suggest a decrease in hardness Hv and fracture toughness KIC of the hP6 phase with zirconium doping but no such effect on the hP3 phase. It was observed that an additional defect in the analyzed structure significantly weakens the hP6 phase but strengthens the hP3 phase. The deposited films are characterized by greater hardness but lower fracture toughness. The results of experiments show that not only is solid solution hardening responsible for strengthening the predicted new material but also the change in microstructure, the Hall–Petch effect and vacancies.

Słowa kluczowe:

Ab initio, Transition metal borides, Mechanical properties, Magnetron sputtered coatings, Hardness

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An ability of different molecular potentials to reproduce the properties of 2D molybdenum disulphide polymorphs is examined. Structural and mechanical properties, as well as phonon dispersion of the 1H, 1T and 1T' single-layer MoS2 (SL MoS2) phases, were obtained using density functional theory (DFT) and molecular statics calculations (MS) with Stillinger-Weber, REBO, SNAP and ReaxFF interatomic potentials. Quantitative systematic comparison and discussion of the results obtained are reported.

Słowa kluczowe:

2D materials, MoS2, molecular potentials, DFT, elastic constants, phonons

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Anisotropic core-shell model of a nano-grained polycrystal is extended to estimate the effective elastic stiffness of several metals of hexagonal crystal lattice symmetry. In the approach the bulk nanocrystalline material is described as a two-phase medium with different properties for a grain boundary zone and a grain core. While the grain core is anisotropic, the boundary zone is isotropic and has a thickness defined by the cutoff radius of a corresponding atomistic potential for the considered metal. The predictions of the proposed mean-field model are verified with respect to simulations performed with the use of the Large-scale Atomic/Molecular Massively Parallel Simulator, the Embedded Atom Model, and the molecular statics method. The effect of the grain size on the overall elastic moduli of nanocrystalline material with random distribution of orientations is analyzed.

Słowa kluczowe:

molecular statics, elasticity, polycrystal, effective medium, hexagonal symmetry

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Two new hypothetical zirconium diboride (ZrB 2) polymorphs: (hP6-P6 3 /mmc-space group, no. 194) and (oP6-Pmmn-space group, no. 59), were thoroughly studied under the first-principles density functional theory calculations from the structural, mechanical and thermodynamic properties point of view. The proposed phases are thermodynamically stable (negative formation enthalpy). Studies of mechanical properties indicate that new polymorphs are less hard than the known phase (hP3-P6/mmm-space group, no. 191) and are not brittle. Analysis of phonon band structure and density of states (DOS) also show that the phonon modes have positive frequencies everywhere and the new ZrB 2 phases are not only mechanically but also dynamically stable. The estimated acoustic Debye temperature, ΘD, for the two new proposed ZrB 2 phases is about 760 K. The thermodynamic properties such as internal energy, free energy, entropy and constant-volume specific heat are also presented.

Słowa kluczowe:

zirconium diboride, ab initio calculations, mechanical properties, elastic properties, phonons

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Anisotropic core-shell model of a nano-grained polycrystal, proposed recently for nanocrystalline copper, is applied to estimate elastic effective properties for a set of crystals of cubic symmetry. Materials selected for analysis differ in the lattice geometry (face-centered cubic vs. body-centered cubic) as well as the value of a Zener factor: a ratio of two shear moduli defining elastic anisotropy of a cubic crystal. The predictions are verified by means of the atomistic simulations. The dependence of the overall bulk and shear moduli on the average grain diameter is analysed. In the mean-field approach the thickness of the shell is specified by the cutoff radius of a corresponding atomistic potential, while the grain shell is isotropic and its properties are identified by molecular simulations performed for very small grains with approximately all atoms belonging to the grain boundary zone. It is shown that the core-shell model provides predictions of satisfactory qualitative and quantitative agreement with atomistic simulations. Performed study indicates that the variation of the bulk and shear moduli with the grain size changes qualitatively when the Zener anisotropy factor is smaller or greater than one.

Słowa kluczowe:

molecular statics, elasticity, polycrystal, effective medium, cubic symmetry

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Słowa kluczowe:

ab initio calculations, elastic moduli, pressure effects in solids and liquids, finite deformations, solid mechanics, deformation gradient

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Recently Haastrup et al 2018 (2D Mater. 5 042002) introduced the Computational 2D Materials Database (C2DB), which organises a variety of structural, thermodynamic, elastic, electronic, magnetic, and optical properties of around 1500 two-dimensional materials distributed over more than 30 different crystal structures. Unfortunately, the work contains serious and fundamental flaws in the field of elasticity and mechanical stability tests that make it unreliable.

Słowa kluczowe:

ab initio calculations, elastic stability, 2D materials, materials discovery

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This report presents the modeling of pressure-assisted sintering within the framework of a multiscale approach. Three individual numerical methods have been collectively applied to predict the behavior of a sintering body at three different scales. The appropriate solutions to connect each model/scale have been proposed. Molecular dynamics have been employed to evaluate the grain boundary diffusion coefficient at the atomistic scale. The obtained results of diffusive parameters have been transferred to the micromechanical model of sintering. Here, the discrete element method was used to represent the sintered material properties at the microscopic scale. Micromechanical based results have been validated by own experimental data of material density evolution, indicating the required coincidence. The transfer from micro- to the macroscopic model has been realized by determining the macroscopic viscous moduli from discrete element simulations and subsequently applying them to the continuum model of sintering. The numerical results from finite element simulations at the macroscopic scale have been compared with discrete element ones.

Słowa kluczowe:

sintering, multiscale modeling, discrete element method, molecular dynamics, finite element method

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Streszczenie:

The full elasticity tensor for nano-crystalline copper is derived in molecular simulations by performing numerical tests for a set of generated samples of the polycrystalline material. The results are analysed with respect to the anisotropy degree of the overall stiffness tensor resulting from the limited number of grain orientations and their spatial distribution. The dependence of the overall bulk and shear moduli of an isotropized polycrystal on the average grain diameter is analysed. It is found that while the shear modulus decreases with grain size, the bulk modulus shows negligible dependence on the grain diameter and is close to the bulk modulus of a single crystal. A closed-form mean-field model of effective elastic properties for a bulk nano-grained polycrystal with cubic grains, i.e. made of a material with cubic symmetry, is formulated. In the model all parameters are based on the data for a single crystal and on the averaged grain size without any need for additional fitting. It is shown that the proposed model provides predictions of satisfactory qualitative and quantitative agreement with atomistic simulations.

Słowa kluczowe:

Molecular statics, Elasticity, Polycrystal, Effective medium, Nano-crystalline copper

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Streszczenie:

A potentially new, single-atom thick semiconducting 2D-graphene-like material, called Anisotropic-cyclicgraphene, has been generated by the two stage searching strategy linking molecular and ab initio approach. The candidate was derived from the evolutionary-based algorithm and molecular simulations was then profoundly analysed using first-principles density functional theory from the structural, mechanical, phonon, and electronic properties point of view. The proposed polymorph of graphene (rP16-P1m1) is mechanically, dynamically, and thermally stable and can achieve semiconducting with a direct band gap of 0.829 eV.

Słowa kluczowe:

carbon, graphene, graphyne, ab initio calculations, Semiconductors

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Self-diffusion parameters in stoichiometric B2-NiAl solid state crystals were estimated by molecular statics/dynamics simulations with the study of required simulation time to stabilise diffusivity results. An extrapolation procedure to improve the diffusion simulation results was proposed. Calculations of volume diffusivity for the B2 type NiAl in the 1224–1699 K temperature range were performed using the embedded-atom-model potential. The results obtained here are in much better agreement with the experimental results than the theoretical estimates obtained with other methods.

Słowa kluczowe:

NiAl nickel–aluminium, diffusivity, molecular dynamics, molecular statics, embedded-atom method, sintering

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Streszczenie:

Two potentially new, 2D-graphene-like materials have been generated by the two stage searching strategy combining molecular and ab initio approach. The two candidates obtained from the evolutionary based algorithm and molecular calculations were then in depth analysed using first-principles Density Functional Theory from the mechanical, structural, phonon and electronic properties point of view. Both proposed polymorphs of graphene (oP8-P2mm) are mechanically and dynamically stable and can be metallic-like.

Słowa kluczowe:

Carbon, Graphene, Ab initio calculations, Mechanical properties, Elastic properties

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Molecular statics/dynamics estimation of constitutive parameters for a micromechanical NiAl sintering model is reported in this paper. The parameters include temperature-dependent diffusion coefficients, surface energy, and linear thermal expansion. These parameters define material behavior during sintering and are used in the sintering particle model implemented in the discrete element model. The investigated material, the NiAl intermetallic, belongs to novel materials characterized by advantageous mechanical properties. Various machine elements are manufactured from a pure NiAl powder or from powder mixtures containing the NiAl using the sintering technology. It is well known that sintering is governed by diffusion. Therefore diffusive properties are important parameters of the micromechanical model of sintering. Numerical estimation of the model parameters by simulations at the lower scale is a powerful tool alternative to experimental methods. Molecular statics and dynamics models for NiAl have been created using the embedded atom model potential. Numerical simulations have allowed us to estimate the volume, surface, and grain-boundary diffusivity for the B2-type NiAl in the 1573 to 1673 K temperature range. Dependence of the diffusion coefficients on temperature has been determined and validity of the Arrhenius-type temperature dependency has been assessed. The parameters evaluated numerically have been compared with available experimental data as well as with theoretical predictions obtained with other methods. Many of the results presented in this paper have a pioneer character and are not known in the literature.

Słowa kluczowe:

NiAl, sintering, diffusivity, molecular dynamics, molecular statics, nanoparticles

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Epitaxy of semiconductors is a process of tremendous importance in applied science and in the optoelectronics industry. The control of defects introduced during epitaxial growth is a key point in manufacturing devices of high efficiency and durability. In this work, it is demonstrated how useful hybrid reflections are for the study of epitaxial structures with anisotropic strain gradients due to patterned substrates. High accuracy in detecting and distinguishing elastic and plastic relaxations is one of the greatest advantages of measuring this type of reflection, as well as the fact that the method can be exploited in a symmetric reflection geometry on a commercial high-resolution diffractometer.

Słowa kluczowe:

optoelectronics, Group III-nitride semiconductors, epitaxial growth, X-ray multiple diffraction, interface defects

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Three mechanically and dynamically stable polymorphs of rhenium diboride (ReB2) (space group: P63/mmc, No: 194), (space group: R-3m, No: 166) and (space group: Pmmn, No: 59) were thoroughly analysed within the framework of Density Functional Theory from the structural, mechanical, optical, thermodynamical and phonon properties point of view. The calculated hardness of rhombohedral structure suggests that it can be even harder than well known hexagonal ReB2.

Słowa kluczowe:

Rhenium diboride, Phase stability, Density Functional Theory (DFT), Physical properties, Phonons

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Streszczenie:

Five potentially superhard WBx polymorphs: hP10-P63/mmc-WB4, hP16-P63/mmc-WB3, View the MathML sourcehR24-R3¯ m-WB3, hP6-P63/mmc-WB2 and oP6-Pmmn-WB2 were thoroughly analyzed within the framework of First Principles Density Functional Theory from the structural, mechanical and optical properties point of view. None of the analyzed polymorphs have a hardness greater than 40 GPa, for the hardest one hP6-P63/mmc-WB2, Hv = 39 GPa. The most stable WBx polymorph is oP6-Pmmn-WB2 with the lowest cohesive energy Ecoh = −8.299 eV/Atom. Due to our knowledge, the optical properties of WB2 and cohesive energy of tungsten borides were presented for the first time. The best optical properties for Pulsed Laser Ablation possess hP6-P63/mmc-WB2 with the lowest reflectivity 0.343 for 355 nm laser radiation.

Słowa kluczowe:

Ab initio calculations, Mechanical properties, Hardness, Optical properties

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In this paper, the choice and parametrisation of finite deformation polyconvex isotropic hyperelastic models to describe the behaviour of a class of defect-free monocrystalline metal materials at the molecular level is examined. The article discusses some physical, mathematical and numerical demands which in our opinion should be fulfilled by elasticity models to be useful. A set of molecular numerical tests for aluminium and tungsten providing data for the fitting of a hypere-lastic model was performed, and an algorithm for parametrisation is discussed. The proposed models with optimised parameters are superior to those used in non-linear mechanics of crystals.

Słowa kluczowe:

Multiscale modeling, Molecular statics, Polyconvexity, Finite elas- ticity, Finite deformations, Hyperelasticity, Monocrystalline metal, Crystal elasticity

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A proper reconstruction of discrete crystal structure with defects is an important problem in dislocation theory. Currently, procedures for dislocation core reconstruction presented in the literature usually neglect configuration changes. The present paper discusses a new approach, which uses an iterative algorithm to determine an atomistic configuration of the dislocation core. The mathematical background is based on finite deformation theory, in which an iterative algorithm searches for the new atomic configuration corresponding to the actual atomic configuration of the deformed crystal. Its application to the reconstruction of 4H-SiC crystal affected by the system of four threading dislocations is presented as an example. Molecular statics calculations suggest a lower potential energy, as well as dislocation core energy, per-atom energy, and per-atom stresses for the structure reconstructed by use of the iterative algorithm against the classical solution based on the Love's equations.

Słowa kluczowe:

dislocation, dislocation core energy, finite deformation, molecular statics

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This paper examines how prerelaxation effects the development of the mechanics of a nanoindentation simulation. In particular, the force-depth relation, indentation stress-strain curves, hardness and elastic modulus, are investigated through molecular statics simulations of a nanoindentation process, starting from initial relaxation by: (i) molecular dynamics and (ii) molecular statics. It is found that initial relaxation conditions change the quantitative response of the system, but not the qualitative response of the system. This has a significant impact on the computational time and quality of the residual mechanical behaviour of the system. Additionally, the method of determining of the elastic modulus is examined for the spherical and planar indenter; and the numerical results are compared. An overview of the relationship between the grain size and hardness of polycrystalline copper is examined and conclusions are drawn.

Słowa kluczowe:

molecular statics, molecular dynamics, nanoindentation, copper

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Streszczenie:

Our numerical approach to modeling elastic-plastic deformation comes back to the idea of the time-independent plasticity developed here at the molecular-statics level. We use a constitutive atomic model based on the second-moment approximation of the tight-binding potential coupled to a linear theory of elasticity solved simultaneously within the finite element method. Our model is applied to the nanoindentation problem for copper in which the indenter is represented by the equations of a sphere. For convenience the time-dependency of the nanoindentation problem is reduced to a quasi-static adiabatic scheme. A recurring theme in this paper is to determine the response of the proposed model for two differing systems: mono and polycrystalline copper. This paper discusses the force-depth response in terms of atomic bond-lengths, elastic-plastic deformations, and the instantaneous stiffness of the material. We report on an increased instantaneous stiffness of polycrystalline copper compared to that of its monocrystalline counterpart. From both a distinct and a comparative analysis of both systems, based on the relaxed positions of the atoms in the structure during the simulation, we deduce that plastic deformations at grain-boundaries are responsible for this change in the overall instantaneous stiffness of the material.

Słowa kluczowe:

linear elasticity, material science, molecular statics, nanoindentation, quasicontinuum methods

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Streszczenie:

Problem of locally disordered atomic structure is solved by using a hybrid formulation in which nonlinear elastic finite elements are linked with discrete atomic interaction elements. The continuum approach uses nonlinear hyperelasticity based upon the generalized strain while the atomistic approach employs the Tight-Binding Second-Moment Approximation potential to create new type of elements. The molecular interactions yielding from constitutive models of TB-SMA were turned into interactions between nodes to solve a boundary value problem by means of finite element solver.
In this paper we present a novel way of modelling materials behaviour where both discrete (molecular dynamics) and continuum (nonlinear finite element) methods are used. As an example, the nanoindentation of a copper sample is modelled numerically by applying a hybrid formulation. Here, the central area of the sample subject to a nanoindentation operation is discretised by an atomic net where the remaining area of the sample far from indenters tip is discretised by the use of a nonlinear finite element mesh.

Słowa kluczowe:

Nanostructure, Nanoindentation, Molecular statics, Finite element modelling

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Streszczenie:

Elastic model of continuum material is often used to simulate the relaxation of crystalline heterostructures. There are many reports on the successful application of the theory of elasticity to nano-sized crystalline heterostructures, even if the continuum condition for them is hardly fulfilled. On the other hand, progress in epitaxial growth allows for the preparation of stable ultra-thin layers with thickness of few monolayers. For such ultra-thin layers, results provided by continuum model and molecular statics/dynamics calculations become diverging. The key problem seems to be located at the modelling of the interface between layers, which is problematic in the continuum approach. By applying a step-wise substitutive compositional interfacial function, it is possible to obtain good agreement with molecular dynamics calculations, even for a single monolayer heterostructure. We propose another approach that uses composition as an extra parameter during finite element calculations, along with classical nodal displacements. Such an approach creates a chemo-elastic coupling that allows to interpolate the composition much like in the case of atomistic calculations.

Słowa kluczowe:

ultra-thin layers, elastic relaxation, molecular statics, finite elemenet modelling

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Streszczenie:

This paper presents modeling of a sintering process at various scales. Sintering is a powder metallurgy process consisting in consolidation of powder materials at elevated temperature but below the melting point. Sintering models at the atomistic, microscopic and macroscopic scales have been presented. Sintering is a process governed by diffusion therefore the atomistic modeling using the molecular dynamics has been focused on investigation of the diffusion process. The micromechanical model has been developed within the framework of the discrete element method. It allows us to consider microstructure and its changes during sintering. The macroscopic model is based on the continuum phenomenological approach. It combines elastic, thermal and viscous creep deformation. The methodology to determine macroscopic quantities: stress, strains and constitutive viscous properties from the discrete element simulations has been presented. Possibilities of the developed models have been demonstrated by applying them to simulation of sintering of the intermetallic NiAl powder. Own experimental results have been used to calibrate and validate numerical models.

Słowa kluczowe:

sintering, modeling, discrete element method, diffusion, molecular dynamics, macroscopic model

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Streszczenie:

Using three different methods namely, CDT (Continuous Dislocation Theory), molecular TB - SMA (Tight Binding Second Moment Approximation) type many - body potential, and MEM (Molecular Effective Medium) theory, we are looking for the best possible reconstruction of dislocations in Cu - Al 2 O 3 heterostructure.

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Streszczenie:

Most interatomic potentials, both classical and machine learning-based (MLPs), are parameterized for 3D structures. The question naturally arises whether they are suitable for modeling their 2D allotropes. In the present study, using ab initio calculations, we determined the structural and mechanical properties of 2D phases of materials such as MoS2, Si, Ge and Sn and then investigated whether the available potentials are able to reproduce these properties.

Słowa kluczowe:

2D materials; interatomic potentials

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Streszczenie:

Theory of elasticity, a continuum model of a macroscopic material is commonly used to model a relaxation of a crystalline heterostructures. There are many reports on the successful application of theory of elasticity to nanometer crystalline heterostructures, even if the continuum condition for these structures is hardly fulfilled. On the other hand progress in epitaxial growth techniques allows to prepare the stable ultra thin layers with the thickness about a single monolayer. For such extremely thin layers the theory of elasticity seems to fail in describing the relaxation process. The results provided by theory of elasticity and experimental measurements or molecular statics/dynamics become diverging. The key problem in that case seems to be located at the interface between layers and related to composition change, which is problematic in classic, elastic approach. By applying a "substitutive" composition of the interface layers which is just an interpolation, it is possible to obtain a good agreement with molecular statics, even for 1 monolayer heterostructure. Instead of classic approach to the composition within the theory of elasticity, we propose another approach which takes into account the composition as an extra degree of freedom along with classical displacement. Such approach creates a chemo-elastic coupling with composition interpolated by use of the Vegard's law. This allows to take into account a composition changes at the interface and avoid mesh refining necessary at the classic approach.

Słowa kluczowe:

theory of elasticity, semiconductor, monolayer, relaxation

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Słowa kluczowe:

Quantum dot, Threading dislocation, Piezoelectricity, Finite element modelling

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Słowa kluczowe:

Sintering, Powder Material, Ni-Al, Molecular Dynamics, Molecular Statics

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Słowa kluczowe:

sintering, multiscale modelling

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Streszczenie:

The interdiffusion of chemical components coupled with vacancy movement can cause void formation and/or spinodal decomposition in crystal growth. In the case of SiC growth on Si, the higher mobility of Si atoms compared to C results in the migration of SiC/Si interface and formation of voids in the substrate in some thermodynamic conditions. In the case of In -rich InGaN layers deposited on GaN a precipitation of metallic indium bordering with voids is observed. In the current approach we consider interdiffusion, lattice distortion and chemical maps extracted from HRTEM images of SiC/Si and InGaN/GaN. Dislocations and void surface are treated as local regions of nucleation and annihilation of the vacancies transporting the mass in FE mesh. In result, the interface and FE mesh are convected with the crystal lattice drift. In the constitutive modeling applied [1] the lattice strain and the atom fraction of chemical component are used as two independent thermodynamic variables. Due to climbing of misfit dislocations the plastic distortion tensor field is taken into account in the form of additional nodal variables. This tensor field is spanned on corner nodes of Lagrangian finite elements (FE) which gives the possibility for reconstruction of the atomistic model of dislocation network interpenetrating the considered FE mesh [2,3]. The chemo-mechanical coupling is based on the use of Vegard's law formulated in terms of Biot strain. Due to the logarithmic strain applied in hyperelastic modeling, some transformation rule is considered for Vegard's law. This rule allowed us to eliminate artificial residual stresses yielding from incompatibe fields of the atom fraction and plastic distortions spanned on nodes by means of shape functions [2]. In the case of single finite elements, the mentioned approach allowed us to reduce spurious stresses in integration points from the level 100 MPa to 10^ -5 MPa, while at the same time holding the stress components yielding from Vegard's law at the level of 1 GPa (relaxed by plastic distortions).

Słowa kluczowe:

Constitutive modelling, Finite Element Method

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Streszczenie:

Contrary to higher order elastic constants for momentum stresses the second (classical) and third-order elastic coefficients (TOEC) for symmetric elasticity are measured and tabulated successfully with good accuracy for tens of years. In the classical experimental measurements of TOEC, the correct recalculation of instantaneous stiffness changes onto TOEC has an important role. A similar problem arises in the constitutive and finite element (FE) modelling. Namely, because of a very strong dependency of TOEC on the strain measure choice, the constitutive and FE modelling of elastic materials is considered here in terms of different finite strain measures. To this aim, the known analytical formulae for calculation of two first derivatives of the isotropic tensor function of tensor variable are verified by means of the finite difference method. In result, the revised formulae are used for calculation of the tangent stiffness matrix. This paper closes with some remarks on the use of TOEC in finite element modelling.

Słowa kluczowe:

nonlinear elasticity, third-order elastic coefficients, logarithmic strain

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Słowa kluczowe:

Semiconductor, Piezoelectricity, Dislocation

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Streszczenie:

A method for reconstruction of atomistic models of dislocations and stacking faults in the interfacial region of heterostructures is presented. Its mathematical foundations come back to the algebra of the finite deformation fields related to introducing of discrete dislocations into an initially coherent interface. From the practical point of view the method concerns generation of interfacial regions with misfit/treading partial dislocations and stacking faults being formed in the interfacial region between crystal structures of different crystallographic type.

Słowa kluczowe:

atomistic models, dislocations, stacking faults, lattice distortion

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Słowa kluczowe:

molecular statics, molecular dynamics, nanoindentation, copper

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