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Related Concept Videos

Elastic Strain Energy for Shearing Stresses01:20

Elastic Strain Energy for Shearing Stresses

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As discussed in previous lessons, strain energy in a material is the energy stored when it is elastically deformed, a concept crucial in materials science and mechanical engineering. This energy results from the internal work done against the cohesive forces within the material. When a material undergoes shearing stress and corresponding shearing strain, the strain energy density, which is the energy stored per unit volume, is calculated. Within the elastic limit, where the stress is...
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Elastic Strain Energy for Normal Stresses01:22

Elastic Strain Energy for Normal Stresses

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Strain energy quantifies the energy stored within a material due to deformation under loading conditions, a fundamental concept in materials science and engineering. The strain energy can be modeled when a material is subjected to axial loading with uniformly distributed stress. In this scenario, the stress experienced by the material is the internal force divided by the cross-sectional area, and the strain induced is directly proportional to this stress through the modulus of elasticity.
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Relation between Poisson's ratio, Modulus of Elasticity and Modulus of Rigidity01:15

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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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The quantity that describes the deformation of a body under stress is known as strain. Strain is given as a fractional change in either length, volume, or geometry under tensile, volume (also known as bulk), or shear stress, respectively, and is a dimensionless quantity. The strain experienced by a body under tensile or compressive stress is called tensile or compressive strain, respectively. In contrast, the strain experienced under bulk stress and shear stress is known as volume and shear...
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Hooke's Law

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Hooke's law, a pivotal principle in material science, establishes that the strain a material undergoes is directly proportional to the applied stress, defined by a factor called the modulus of elasticity or Young's modulus.
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The generalized Hooke's Law is a broadened version of Hooke's Law, which extends to all types of stress and in every direction. Consider an isotropic material shaped into a cube subjected to multiaxial loading. In this scenario, normal stresses are exerted along the three coordinate axes. As a result of these stresses, the cubic shape deforms into a rectangular parallelepiped. Despite this deformation, the new shape maintains equal sides, and there is a normal strain in the direction of the...
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Electron Channeling Contrast Imaging for Rapid III-V Heteroepitaxial Characterization
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Modeling elastic anisotropy in strained heteroepitaxy.

Gopal Krishna Dixit1, Madhav Ranganathan1

  • 1Department of Chemistry, Indian Institute of Technology Kanpur, Kanpur 208016, India.

Journal of Physics. Condensed Matter : an Institute of Physics Journal
|June 3, 2017
PubMed
Summary

We model quantum dot growth in germanium on silicon using continuum equations. Elastic and surface energy anisotropies influence quantum dot shape and alignment during molecular beam epitaxy.

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Area of Science:

  • Materials Science
  • Surface Science
  • Condensed Matter Physics

Background:

  • Quantum dots (QDs) are crucial in semiconductor technology.
  • Heteroepitaxial growth of Ge on Si(001) is a key system for QD formation.
  • Understanding QD evolution requires modeling surface diffusion and elastic effects.

Purpose of the Study:

  • To model the growth and evolution of quantum dots in the heteroepitaxial Ge on Si(001) system.
  • To investigate the roles of surface energy and elastic anisotropy in QD morphology and alignment.
  • To analyze the impact of deposition and surface diffusion on QD formation.

Main Methods:

  • Continuum evolution equation modeling.
  • Free energy formulation including surface energy, curvature, wetting, and elastic energy.
  • Perturbation analysis with small slope approximation for elastic problem.
  • Linear stability analysis and numerical simulations of nonlinear equations.

Main Results:

  • Early instability evolution favors dot formation.
  • Elastic anisotropy dictates QD alignment in the linear regime.
  • Surface energy anisotropy governs QD shapes in the nonlinear regime.
  • For Ge on Si(001), surface energy dominates shape, while elastic energy influences alignment and elongation.

Conclusions:

  • The model accurately captures QD evolution influenced by deposition and surface diffusion.
  • Anisotropic surface and elastic energies play critical, distinct roles in QD morphology and arrangement.
  • Elastic anisotropy leads to island elongation, with coarsening limited by {113} facets.