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<p>Microstructure evolution of materials over space and time is typically studied through the solution of Cahn–Hilliard (CH)-type diffusion and deformation equations. In this work, a numerical framework implemented in commercial software, Abaqus as user element (UEL) and user material (UMAT), is presented to model the complex material behavior during microstructure evolution. This is done by formulating the system’s free energy that comprises chemical, interfacial, and elastic strain energy and phase separation dynamics. The elastic strain energy is expressed as a function of species concentration based on Khachaturyan’s elasticity theory. The species evolution is governed by the fourth-order CH equation and is subsequently transformed into two second-order equations amenable to traditional <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="707_2025_4423_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathcal {C}}^0\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">C</mi> </mrow> <mn>0</mn> </msup> </math></EquationSource> </InlineEquation> finite elements. A detailed Abaqus implementation is provided, enabling researchers and scientists to solve problems involving similar physics. The numerical examples demonstrate the effectiveness of the proposed algorithm in addressing both phase separation with and without coupled elastic deformation. The source code is made available in the Appendix.</p>

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Mixed finite element approach for Cahn–Hilliard-type diffusion coupled with elasticity

  • 期刊论文

  • 发表日期：

    2025年7月9日

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  • Kireeti Thatipalli,
  • Shiva Reddy Kondakindi,
  • Rajagopal Amirtham,
  • Sundararajan Natarajan

摘要

Microstructure evolution of materials over space and time is typically studied through the solution of Cahn–Hilliard (CH)-type diffusion and deformation equations. In this work, a numerical framework implemented in commercial software, Abaqus as user element (UEL) and user material (UMAT), is presented to model the complex material behavior during microstructure evolution. This is done by formulating the system’s free energy that comprises chemical, interfacial, and elastic strain energy and phase separation dynamics. The elastic strain energy is expressed as a function of species concentration based on Khachaturyan’s elasticity theory. The species evolution is governed by the fourth-order CH equation and is subsequently transformed into two second-order equations amenable to traditional \({\mathcal {C}}^0\) C 0 finite elements. A detailed Abaqus implementation is provided, enabling researchers and scientists to solve problems involving similar physics. The numerical examples demonstrate the effectiveness of the proposed algorithm in addressing both phase separation with and without coupled elastic deformation. The source code is made available in the Appendix.