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The concept of curvature in plane curves, crucial in structural engineering, defines how sharply a beam bends under load. This curvature is determined using the curve's first and second derivatives.
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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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Curvature-dependent Eulerian interfaces in elastic solids.

Katharina Brazda1, Martin Kružík2, Fabian Rupp1

  • 1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|November 5, 2023
PubMed
Summary

We developed a sharp-interface model for two-phase hyperelastic materials, using an Eulerian approach for interfacial energy and curvature penalization. This model ensures stable phase interface behavior in topology optimization problems.

Keywords:
curvature varifoldselasticityinterfacial energymulti-phase materialsvarifolds

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Area of Science:

  • Continuum Mechanics
  • Materials Science
  • Computational Mechanics

Background:

  • Modeling multi-phase materials requires accurate representation of interfaces.
  • Existing models may face challenges with interfacial energy in deformed configurations.
  • Topology optimization demands robust methods for handling evolving geometries.

Purpose of the Study:

  • To introduce a novel sharp-interface model for two-phase hyperelastic materials.
  • To develop a fully Eulerian framework for interfacial energy.
  • To incorporate curvature penalization for stable interface evolution in topology optimization.

Main Methods:

  • Formulation of a sharp-interface model in the deformed configuration.
  • Inclusion of a geometric term with a curvature varifold for interfacial energy.
  • Proof of existence of equilibrium solutions via minimization.
  • Application of the model in an Eulerian topology optimization framework.

Main Results:

  • A fully Eulerian interfacial energy formulation for hyperelastic materials.
  • Demonstration of existence of equilibrium solutions.
  • Successful integration of curvature penalization into Eulerian topology optimization.
  • A stable and robust model for simulating two-phase materials with evolving interfaces.

Conclusions:

  • The proposed sharp-interface model provides a robust framework for analyzing two-phase hyperelastic materials.
  • The Eulerian approach simplifies interface tracking in large deformations.
  • Curvature penalization enhances the stability and geometric control of phase interfaces in optimization problems.