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An Energy Minimization Approach to Twinning with Variable Volume Fraction.

Sergio Conti1, Robert V Kohn2, Oleksandr Misiats3

  • 1Institut für Angewandte Mathematik, Universität Bonn, 53115 Bonn, Germany.

Journal of Elasticity
|July 22, 2024
PubMed
Summary
This summary is machine-generated.

This study models "twinning with variable volume fraction" in martensitic materials under bending. The energy minimization model predicts how microstructures scale with surface energy density, validated by upper and lower bounds.

Keywords:
BendingSolid-solid phase transitionsTwinning

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Area of Science:

  • Materials Science
  • Solid Mechanics
  • Crystallography

Background:

  • Macroscopic loading of materials undergoing martensitic phase transformation induces elastic domain rearrangement.
  • The specific microstructure termed "twinning with variable volume fraction" arises in single-crystal slabs with two martensite variants under bending.
  • Previous experimental work by Chopra et al. (1996) detailed this phenomenon in Indium-Thallium (InTl) alloys.

Purpose of the Study:

  • To develop an energy-minimization-based model for the "twinning with variable volume fraction" microstructure.
  • To analyze the scaling behavior of the minimum energy with respect to surface energy density.
  • To provide theoretical bounds explaining the optimality of observed microstructures.

Main Methods:

  • Utilized geometrically linear elasticity with sharp interfaces for phase boundaries.
  • Formulated nonlinear, nonconvex variational problems based on Dirichlet or Neumann boundary conditions to simulate bending.
  • Established theoretical upper and lower bounds to determine energy scaling.

Main Results:

  • The model successfully predicts the scaling of the minimum energy with surface energy density.
  • Ansatz-based upper bounds provided detailed insights into near-optimal microstructures.
  • Ansatz-free lower bounds confirmed the efficiency of these microstructures, demonstrating no significantly better arrangements exist.

Conclusions:

  • The developed energy minimization model accurately captures the behavior of "twinning with variable volume fraction" under bending.
  • The scaling laws derived are crucial for understanding microstructure evolution in martensitic phase transformations.
  • Theoretical bounds confirm the energetic favorability of the modeled microstructures.