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The main focus of this thesis is the discussion of stability of an objective (atomic) structure consisting of single atoms which interact via a potential. We define atomistic stability using a second derivative test. More precisely, atomistic stability is equivalent to a vanishing first derivative of the configurational energy (at the corresponding point) and the coerciveness of the second derivative of the configurational energy with respect to an appropriate semi-norm. Atomistic stability of a lattice is well understood, see, e.,g., [40]. The aim of this thesis is to generalize the theory to objective structures. In particular, we first investigate discrete subgroups of the Euclidean group, then define an appropriate seminorm and the atomistic stability for a given objective structure, and finally provide an efficient algorithm to check its atomistic stability. The algorithm particularly checks the validity of the Cauchy-Born rule for objective structures. To illustrate our results, we prove numerically the stability of a carbon nanotube by applying the algorithm.

Mathematics --- Mathematical model --- Elasticity theory --- Stability theory --- Objective structure --- Discrete subgroup of the Euclidean group

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The main focus of this thesis is the discussion of stability of an objective (atomic) structure consisting of single atoms which interact via a potential. We define atomistic stability using a second derivative test. More precisely, atomistic stability is equivalent to a vanishing first derivative of the configurational energy (at the corresponding point) and the coerciveness of the second derivative of the configurational energy with respect to an appropriate semi-norm. Atomistic stability of a lattice is well understood, see, e.,g., [40]. The aim of this thesis is to generalize the theory to objective structures. In particular, we first investigate discrete subgroups of the Euclidean group, then define an appropriate seminorm and the atomistic stability for a given objective structure, and finally provide an efficient algorithm to check its atomistic stability. The algorithm particularly checks the validity of the Cauchy-Born rule for objective structures. To illustrate our results, we prove numerically the stability of a carbon nanotube by applying the algorithm.

Science / Physics --- Mathematics --- Mathematical model --- Elasticity theory --- Stability theory --- Objective structure --- Discrete subgroup of the Euclidean group

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This book is a printed edition of the Special Issue of Crystals entitled Pressure-Induced Phase Transformations. It includes selected articles on the behavior of matter under high-pressure and high-temperature conditions, describing and discussing contemporary achievements, which were selected based on their relevance and scientific quality.

Research & information: general --- vanadate --- zircon --- high pressure --- band gap --- phase transition --- optical absorption --- benzene phase I --- homogeneous melting --- Ostwald’s step rule --- molecular dynamics simulation --- metastable phase --- melting transition --- Fe --- electrical resistivity --- thermal conductivity --- heat flow --- thermal and chemical convection --- sesquioxides --- phase transitions --- Laue diffraction --- mechanisms of phase transitions --- reactivity --- tungsten --- rhenium --- carbon dioxide --- carbonates --- high-pressure high-temperature experiments --- quantum spin liquids --- frustrated magnets --- quantum phase transitions --- high-pressure measurements --- phase diagram --- quantum molecular dynamics --- melting curve --- Z methodology --- multi-phase materials --- epsomite --- dehydration reaction --- Raman spectra --- electrical conductivity --- high-pressure phase transitions --- molecular crystals --- computational methods --- DFT and Force Field methods --- energy calculations --- intermolecular interactions --- Landau theory --- nonlinear elasticity theory --- perovskites --- fullerenes --- polymerization --- pressure-induced --- Raman --- infrared laser --- laser-heated diamond anvil cell --- synchrotron radiation --- extreme conditions --- vanadate --- zircon --- high pressure --- band gap --- phase transition --- optical absorption --- benzene phase I --- homogeneous melting --- Ostwald’s step rule --- molecular dynamics simulation --- metastable phase --- melting transition --- Fe --- electrical resistivity --- thermal conductivity --- heat flow --- thermal and chemical convection --- sesquioxides --- phase transitions --- Laue diffraction --- mechanisms of phase transitions --- reactivity --- tungsten --- rhenium --- carbon dioxide --- carbonates --- high-pressure high-temperature experiments --- quantum spin liquids --- frustrated magnets --- quantum phase transitions --- high-pressure measurements --- phase diagram --- quantum molecular dynamics --- melting curve --- Z methodology --- multi-phase materials --- epsomite --- dehydration reaction --- Raman spectra --- electrical conductivity --- high-pressure phase transitions --- molecular crystals --- computational methods --- DFT and Force Field methods --- energy calculations --- intermolecular interactions --- Landau theory --- nonlinear elasticity theory --- perovskites --- fullerenes --- polymerization --- pressure-induced --- Raman --- infrared laser --- laser-heated diamond anvil cell --- synchrotron radiation --- extreme conditions

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The problem of solving complex engineering problems has always been a major topic in all industrial fields, such as aerospace, civil and mechanical engineering. The use of numerical methods has increased exponentially in the last few years, due to modern computers in the field of structural mechanics. Moreover, a wide range of numerical methods have been presented in the literature for solving such problems. Structural mechanics problems are dealt with using partial differential systems of equations that might be solved by following the two main classes of methods: Domain-decomposition methods or the so-called finite element methods and mesh-free methods where no decomposition is carried out. Both methodologies discretize a partial differential system into a set of algebraic equations that can be easily solved by computer implementation. The aim of the present Special Issue is to present a collection of recent works on these themes and a comparison of the novel advancements of both worlds in structural mechanics applications.

History of engineering & technology --- direction field --- tensor line --- principal stress --- tailored fiber placement --- heat conduction --- finite elements --- space-time --- elastodynamics --- mesh adaptation --- non-circular deep tunnel --- complex variables --- conformal mapping --- elasticity --- numerical simulation --- numerical modeling --- joint static strength --- finite element method --- parametric investigation --- reinforced joint (collar and doubler plate) --- nonlocal elasticity theory --- Galerkin weighted residual FEM --- silicon carbide nanowire --- silver nanowire --- gold nanowire --- biostructure --- rostrum --- paddlefish --- Polyodon spathula --- maximum-flow/minimum-cut --- stress patterns --- finite element modelling --- laminated composite plates --- non-uniform mechanical properties --- panel method --- marine propeller --- noise --- FW-H equations --- experimental test --- continuation methods --- bifurcations --- limit points --- cohesive elements --- functionally graded materials --- porosity distributions --- first-order shear deformation theory --- shear correction factor --- higher-order shear deformation theory --- equivalent single-layer approach --- direction field --- tensor line --- principal stress --- tailored fiber placement --- heat conduction --- finite elements --- space-time --- elastodynamics --- mesh adaptation --- non-circular deep tunnel --- complex variables --- conformal mapping --- elasticity --- numerical simulation --- numerical modeling --- joint static strength --- finite element method --- parametric investigation --- reinforced joint (collar and doubler plate) --- nonlocal elasticity theory --- Galerkin weighted residual FEM --- silicon carbide nanowire --- silver nanowire --- gold nanowire --- biostructure --- rostrum --- paddlefish --- Polyodon spathula --- maximum-flow/minimum-cut --- stress patterns --- finite element modelling --- laminated composite plates --- non-uniform mechanical properties --- panel method --- marine propeller --- noise --- FW-H equations --- experimental test --- continuation methods --- bifurcations --- limit points --- cohesive elements --- functionally graded materials --- porosity distributions --- first-order shear deformation theory --- shear correction factor --- higher-order shear deformation theory --- equivalent single-layer approach

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This reprint presents a collection of contributions on the application of high-performing computational strategies and enhanced theoretical formulations to solve a wide variety of linear or nonlinear problems in a multiphysical sense, together with different experimental studies.

Technology: general issues --- History of engineering & technology --- nanotwin --- detwinning --- extreme hardness --- excellent stability --- electrospinning --- nanofibrous membrane --- geometric modeling --- uniaxial tensile --- buckling --- electromagnetic field --- nanobeam --- shifted chebyshev polynomial --- rayleigh-ritz method --- nanocomposites --- FG-CNTRC --- truncated cone --- critical combined loads --- multi-scale mechanics --- finite element analysis --- material testing --- cellulose nanofiber --- polymer composites --- tensile modulus --- cove-edges --- defects --- fracture --- graphene --- molecular dynamics --- strength --- recycling --- circular economy --- nanometric carbon-based ashes --- AJ®P --- non-piezoelectric polymers --- tactile sensors --- robotic gripper --- fluorinated epoxy resin --- fluorinated graphene oxide --- ordered filling --- elastic modulus --- glass transition temperature --- microscopic parameters --- surface bonding --- nanocone arrays --- molecular dynamics simulation --- axially functionally graded materials --- differential quadrature method --- flexural–torsional buckling --- nonlocal elasticity theory --- tapered I-beam --- n/a --- flexural-torsional buckling