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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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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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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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Elastodynamic image forces on dislocations.

Beñat Gurrutxaga-Lerma1, Daniel S Balint1, Daniele Dini1

  • 1Department of Mechanical Engineering , Imperial College London , London SW7 2AZ, UK.

Proceedings. Mathematical, Physical, and Engineering Sciences
|November 4, 2015
PubMed
Summary
This summary is machine-generated.

Elastodynamic image forces on dislocations near a free surface are derived, revealing retardation effects and inertial magnification for moving dislocations. Stationary dislocations approach elastostatic predictions, while moving ones show divergent forces, with edge dislocations at Rayleigh speed exhibiting repulsion.

Keywords:
dislocationedge dislocationelastodynamicimage forcescrew dislocation

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Area of Science:

  • Solid Mechanics
  • Materials Science
  • Continuum Mechanics

Background:

  • Dislocations are fundamental defects in crystalline materials.
  • Understanding forces on dislocations near surfaces is crucial for material deformation and fracture.
  • Previous studies primarily used elastostatic approximations.

Purpose of the Study:

  • Derive elastodynamic image forces on edge and screw dislocations near a planar free surface.
  • Investigate the influence of retardation and inertial effects.
  • Compare elastodynamic results with elastostatic predictions.

Main Methods:

  • Derivation of elastodynamic fields for injected dislocations.
  • Analysis of image forces considering wave propagation and reflection.
  • Asymptotic analysis for stationary and moving dislocations.

Main Results:

  • Image forces are affected by retardation; no force is experienced until dislocation fields reflect off the surface.
  • For stationary dislocations, elastodynamic forces asymptotically match elastostatic predictions.
  • For moving dislocations, inertial effects magnify forces, leading to divergence, a significant effect even at low speeds.
  • An edge dislocation moving at Rayleigh wave speed experiences a repulsive force, indicating potential core instabilities.

Conclusions:

  • Elastodynamic effects, including retardation and inertia, significantly alter image forces on dislocations compared to elastostatic models.
  • Inertial effects are substantial for moving dislocations, leading to divergent forces.
  • Repulsive forces for edge dislocations at Rayleigh speed suggest novel surface interaction phenomena and potential instabilities.