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Non-Conventional Thermodynamics and Models of Gradient Elasticity.

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Dynamics in Explicit Gradient Elasticity: Material Frame-Indifference, Boundary Conditions and Consistent

Charalampos Tsakmakis1, Carsten Broese1, Stergios Alexandros Sideris2

  • 1Institute for Mechanics, Civil Engineering, Technical University of Darmstadt, Franziska-Braun-Str. 7, 64287 Darmstadt, Germany.

Materials (Basel, Switzerland)
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PubMed
Summary

This study examines boundary conditions for explicit gradient elasticity, arguing against acceleration terms in boundary tractions based on objectivity and material frame indifference principles. New examples show distinct dynamic responses with and without these terms.

Keywords:
Mindlin’s gradient elasticityacceleration termsboundary conditionsconsistent Euler–Bernoulli beam theoryextensions of Hamilton’s principlematerial frame-indifference

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Area of Science:

  • Solid Mechanics
  • Continuum Mechanics
  • Elasticity Theory

Background:

  • Gradient elasticity theories extend classical elasticity by incorporating material length scale parameters.
  • Mindlin's theory of gradient elasticity is a prominent framework.
  • The role and formulation of boundary conditions in dynamic gradient elasticity remain areas of active research.

Purpose of the Study:

  • To rigorously analyze the boundary conditions for explicit gradient elasticity of Mindlin's type in dynamic scenarios.
  • To investigate the necessity and implications of acceleration terms in boundary tractions.
  • To provide a detailed discussion and new illustrative examples regarding the impact of acceleration terms on dynamic responses.

Main Methods:

  • Theoretical analysis of boundary conditions in explicit gradient elasticity.
  • Application of objectivity arguments to boundary tractions.
  • Incorporation of the principle of material frame indifference.
  • Development and analysis of new dynamic examples.

Main Results:

  • Arguments against the presence of acceleration terms in boundary tractions are elaborated based on objectivity.
  • The principle of material frame indifference is applied to further support the analysis.
  • Demonstration through new examples that boundary tractions with and without acceleration terms yield significantly different dynamic responses.

Conclusions:

  • The inclusion or exclusion of acceleration terms in boundary tractions has a substantial impact on the predicted dynamic behavior of gradient elastic materials.
  • Objectivity and material frame indifference provide a strong theoretical basis for refining dynamic boundary conditions in gradient elasticity.
  • Further research into precise dynamic boundary condition formulations is warranted for accurate modeling of gradient materials.