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The structure of a crystalline solid, whether a metal or not, is best described by considering its simplest repeating unit, which is referred to as its unit cell. The unit cell consists of lattice points that represent the locations of atoms or ions. The entire structure then consists of this unit cell repeating in three dimensions. The three different types of unit cells present in the cubic lattice are illustrated in Figure 1.
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Solids in which the atoms, ions, or molecules are arranged in a definite repeating pattern are known as crystalline solids. Metals and ionic compounds typically form ordered, crystalline solids. A crystalline solid has a precise melting temperature because each atom or molecule of the same type is held in place with the same forces or energy. Amorphous solids or non-crystalline solids (or, sometimes, glasses) which lack an ordered internal structure and are randomly arranged. Substances that...
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A planar symmetry of charge density is obtained when charges are uniformly spread over a large flat surface. In planar symmetry, all points in a plane parallel to the plane of charge are identical with respect to the charges. Suppose the plane of the charge distribution is the xy-plane, and the electric field at a space point P with coordinates (x, y, z) is to be determined. Since the charge density is the same at all (x, y) - coordinates in the z = 0 plane, by symmetry, the electric field at P...
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Deformation occurs in axial and transverse directions when an axial load is applied to a slender bar. This deformation impacts the cubic element within the bar, transforming it into either a rectangular parallelepiped or a rhombus, contingent on its orientation. This transformation process induces shearing strain. Axial loading elicits both shearing and normal strains. Applying an axial load instigates equal normal and shearing stresses on elements oriented at a 45° angle to the load axis.
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Tetrahedral Complexes
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Metallic solids such as crystals of copper, aluminum, and iron are formed by metal atoms. The structure of metallic crystals is often described as a uniform distribution of atomic nuclei within a “sea” of delocalized electrons. The atoms within such a metallic solid are held together by a unique force known as metallic bonding that gives rise to many useful and varied bulk properties.
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Using Microwave and Macroscopic Samples of Dielectric Solids to Study the Photonic Properties of Disordered Photonic Bandgap Materials
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Geometrically nonlinear dynamic model for a hexagonal lattice.

A V Porubov1,2, A M Krivtsov1,2, I D Antonov1,2

  • 1Institute for Problems in Mechanical Engineering, Bolshoy 61, V. O., Saint-Petersburg, Russia.

Physical Review. E
|September 18, 2020
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Summary
This summary is machine-generated.

Angular stiffness significantly influences nonlinear terms in hexagonal lattice continuum models. This study develops a discrete model to analyze wave interactions, revealing key geometric nonlinear effects.

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Area of Science:

  • Solid Mechanics
  • Materials Science
  • Continuum Mechanics

Background:

  • Hexagonal lattice structures are fundamental in various materials.
  • Geometric nonlinearity is crucial for understanding material behavior under large deformations.
  • Previous models often simplified or neglected angular stiffness effects in continuum limits.

Purpose of the Study:

  • To formulate a geometrically nonlinear discrete model for hexagonal lattices.
  • To investigate the role of angular stiffness in continuum limit equations.
  • To analyze the interaction of longitudinal and shear plane strain waves.

Main Methods:

  • Formulation of a discrete hexagonal lattice model incorporating two sublattices.
  • Application of an asymptotic procedure to derive continuum limit equations.
  • Solving the nonlinear coupled equations of motion to study wave interactions.

Main Results:

  • Angular stiffness is identified as a significant contributor to geometrical nonlinear terms.
  • Nonlinear coupled equations of motion in the continuum limit were successfully obtained.
  • The interaction between longitudinal and shear plane strain waves was analyzed.

Conclusions:

  • The discrete hexagonal lattice model accurately captures geometric nonlinearities.
  • Angular stiffness is essential for precise continuum modeling of hexagonal lattices.
  • The study provides insights into wave propagation phenomena in nonlinear lattice structures.