## Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition

## Abstract

Here, we consider solid state phase transformations that are caused by free energy densities with domains of non-convexity in strain-composition space; we refer to the non-convex domains as mechano-chemical spinodals. The non-convexity with respect to composition and strain causes segregation into phases with different crystal structures. We work on an existing model that couples the classical Cahn-Hilliard model with Toupin’s theory of gradient elasticity at finite strains. Both systems are represented by fourth-order, nonlinear, partial differential equations. The goal of this work is to develop unconditionally stable, second-order accurate time-integration schemes, motivated by the need to carry out large scale computations of dynamically evolving microstructures in three dimensions. We also introduce reduced formulations naturally derived from these proposed schemes for faster computations that are still second-order accurate. Although our method is developed and analyzed here for a specific class of mechano-chemical problems, one can readily apply the same method to develop unconditionally stable, second-order accurate schemes for any problems for which free energy density functions are multivariate polynomials of solution components and component gradients. Apart from an analysis and construction of methods, we present a suite of numerical results that demonstrate the schemes in action.

- Authors:

- Univ. of Michigan, Ann Arbor, MI (United States)

- Publication Date:

- Research Org.:
- Univ. of Michigan, Ann Arbor, MI (United States). PRedictive Integrated Structural Materials Science (PRISMS) Center

- Sponsoring Org.:
- USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)

- OSTI Identifier:
- 1324034

- Alternate Identifier(s):
- OSTI ID: 1399034

- Grant/Contract Number:
- SC0008637

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Computer Methods in Applied Mechanics and Engineering

- Additional Journal Information:
- Journal Name: Computer Methods in Applied Mechanics and Engineering; Journal ID: ISSN 0045-7825

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; 36 MATERIALS SCIENCE; unconditional stability; mechano-chemistry; Cahn-Hilliard; strain gradient elasticity; spinodal decomposition; phase-field

### Citation Formats

```
Sagiyama, Koki, Rudraraju, Shiva, and Garikipati, Krishna. Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition. United States: N. p., 2016.
Web. doi:10.1016/j.cma.2016.09.003.
```

```
Sagiyama, Koki, Rudraraju, Shiva, & Garikipati, Krishna. Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition. United States. doi:10.1016/j.cma.2016.09.003.
```

```
Sagiyama, Koki, Rudraraju, Shiva, and Garikipati, Krishna. Tue .
"Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition". United States. doi:10.1016/j.cma.2016.09.003. https://www.osti.gov/servlets/purl/1324034.
```

```
@article{osti_1324034,
```

title = {Unconditionally stable, second-order accurate schemes for solid state phase transformations driven by mechano-chemical spinodal decomposition},

author = {Sagiyama, Koki and Rudraraju, Shiva and Garikipati, Krishna},

abstractNote = {Here, we consider solid state phase transformations that are caused by free energy densities with domains of non-convexity in strain-composition space; we refer to the non-convex domains as mechano-chemical spinodals. The non-convexity with respect to composition and strain causes segregation into phases with different crystal structures. We work on an existing model that couples the classical Cahn-Hilliard model with Toupin’s theory of gradient elasticity at finite strains. Both systems are represented by fourth-order, nonlinear, partial differential equations. The goal of this work is to develop unconditionally stable, second-order accurate time-integration schemes, motivated by the need to carry out large scale computations of dynamically evolving microstructures in three dimensions. We also introduce reduced formulations naturally derived from these proposed schemes for faster computations that are still second-order accurate. Although our method is developed and analyzed here for a specific class of mechano-chemical problems, one can readily apply the same method to develop unconditionally stable, second-order accurate schemes for any problems for which free energy density functions are multivariate polynomials of solution components and component gradients. Apart from an analysis and construction of methods, we present a suite of numerical results that demonstrate the schemes in action.},

doi = {10.1016/j.cma.2016.09.003},

journal = {Computer Methods in Applied Mechanics and Engineering},

number = ,

volume = ,

place = {United States},

year = {2016},

month = {9}

}