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# Effect of Modulus Heterogeneity on the Equilibrium Shape and Stress Field of *α* Precipitate in Ti-6Al-4V

1 Materials Genome Institute, Shanghai University, Shanghai, 200444, China

2 School of Materials Science and Engineering, Harbin Institute of Technology, Shenzhen, 518055, China

3 Shanghai Frontier Science Center of Mechanoinformatics, Shanghai University, Shanghai, 200444, China

4 Zhejiang Laboratory, Hangzhou, 311100, China

* Corresponding Author: Rongpei Shi. Email:

(This article belongs to the Special Issue: Computational Design and Modeling of Advanced Composites and Structures)

*Computer Modeling in Engineering & Sciences* **2024**, *140*(1), 1017-1028. https://doi.org/10.32604/cmes.2024.048797

**Received** 19 December 2023; **Accepted** 07 February 2024; **Issue published** 16 April 2024

## Abstract

For media with inclusions (e.g., precipitates, voids, reinforcements, and others), the difference in lattice parameter and the elastic modulus between the matrix and inclusions cause stress concentration at the interfaces. These stress fields depend on the inclusions’ size, shape, and distribution and will respond instantly to the evolving microstructure. This study develops a phase-field model concerning modulus heterogeneity. The effect of modulus heterogeneity on the growth process and equilibrium state of the*α*plate in Ti-6Al-4V during precipitation is evaluated. The

*α*precipitate exhibits strong anisotropy in shape upon cooling due to the interplay of the elastic strain and interfacial energy. The calculated orientation of the habit plane using the homogeneous modulus of

*α*phase shows the smallest deviation from that of the habit plane observed in the experiment, compared to the case where the homogeneous modulus of

*β*phase is adopted. In addition, the equilibrium volume of

*α*phase within the system using homogeneous

*β*modulus exhibits the largest dependency on the applied stresses. The stress fields across the

*α*/

*β*interface are further calculated under the assumption of modulus heterogeneity and compared to those using homogeneous modulus of either

*α*or

*β*phase. This study provides an essential theoretical basis for developing mechanics models concerning systems with heterogeneous structures.

## Keywords

The close correlation of microstructure with material properties makes it crucial in research and development [1]. A microstructure is usually composed of precipitates distributed in a solid matrix. The difference between the precipitates and matrix in lattice parameters and crystal structure generates elastic strain in both and, in turn, influences the morphology of the microstructure (e.g., shape and spatial distribution of the precipitates) and thermodynamic driving forces and kinetics of the precipitation process [2]. In principle, the contributions of the elastic interactions can be calculated by formulating a total elastic energy functional. A variational derivative of the total energy with respect to the microstructural variables (degrees of freedom) gives rise to the local elastic driving force for the evolution of the microstructure [3].

The elasticity solutions for a given microstructure are primarily deduced in the framework established by Eshelby for coherent precipitates [4]. A precipitate is considered coherent if the crystal lattice planes extend continuously from precipitate to matrix. Mathematically, the condition is specified as a continuation of the displacements across the precipitate-matrix boundaries. Eshelby’s approach was generalized and extended to treat multi-particle problems with realistic features in various microstructural studies [5–7]. Simplifications in numerical microstructure simulations are often made under an approximation of homogeneous elasticity, ignoring the variation in the elastic modulus among phases. The choice of the now uniform elastic modulus is taken on the major phase. Elastic assumptions using homogeneous modulus of the matrix phase (e.g., the parent phase for precipitation, the base alloys for composites, and others) have been used in systems for slip transmission across the

Since the elastic moduli generally differ between precipitates and matrix and among the precipitates (the difference also includes a rotation of the elastic modulus tensors, such as in a polycrystal), the solution for such problems can be much more complex in anisotropic solids and becomes very costly in microstructure simulations. Moulinec et al. proposed an iterative numerical method based on Fast Fourier Transforms and the Green function to investigate the effective properties of periodic composites [13], and an augmented Lagrangian method was further employed to treat elastically inhomogeneous solids, including voided materials and power-law materials [14]. The generalization of the phase-field micro elasticity theory [15] to elastically inhomogeneous systems [16–18] enables a general treatment of the elasticity problem in an elastically anisotropic and inhomogeneous solid, where coherent precipitates can take arbitrary shapes, populations, and spatial distributions. Ultimately, one can want to know how the homogeneous elasticity approximation can affect the elasticity solution (e.g., energy and stress) and the microstructure.

With the formulation, numerical calculations are performed for coherent inclusion under various approximations of elastic modulus, and the effects of the simplification on the elastic energy and stress distribution are investigated. The Ti-6Al-4V (wt.%), one of the earliest commercial titanium alloys, exhibits excellent and balanced mechanical and chemical performance [19–21] and is chosen as the working system. Ti-6Al-4V is a typical two-phase (

2.1 Phase-Field Model with Inhomogeneous Elastic Modulus for Precipitate and Matrix Phases

The current work is based on the three-dimensional multi-phase-field model for an elastically and structurally inhomogeneous system [24] of Ti-6Al-4V alloy. Within the framework of the multi-phase-field model, 12 order parameters are employed to distinguish

This study explores the equilibrium shape of

where

For a system with the assumption that the precipitate and matrix have identical elastic constants

where

where

The concurrent evolution of composition and structure follow the general form of Cahn-Hilliard diffusion equation and the Allen-Cahn equation in the multi-phase-field model by Steinbach et al. [27]. When orientations of precipitates are considered, the governing equations in this work is derived as:

where

Using

Case-I: homogeneous modulus of the matrix phase:

Case-II: homogeneous modulus of the precipitate phase:

Case-III: inhomogeneous modulus that takes the corresponding values of modulus according to the order parameter:

During the numerical process of finding the equilibrium elastic state, the elastic moduli concerning the two phases should not be set strictly equal due to the iteration of Eq. (5), where

3.1 Equilibrium Shape of

During the

External stresses can also alter the equilibrium shape or size of

3.2 Stress Fields of

Due to the mismatch of lattice parameters of

For titanium alloy with

However, when the homogeneous modulus assumption is applied, the stress field can be distinct from that with the inhomogeneous modulus shown above due to the variation in

The assumption of the elastic modulus for elastically and structurally inhomogeneous solids is critical to the phase transformation process and the corresponding elastic state. The effect of modulus heterogeneity on the equilibrium shape and stress field of grown

• The equilibrium shape (e.g., habit plane and size) of

• The local stresses across the

• In general, for transformation from the phase of high symmetry structure to the phase of low symmetry, using elastic modulus of the low symmetry phase would give more accurate calculation results on the equilibrium morphology of the precipitate.

Acknowledgement: The authors gratefully thank Prof. Yunzhi Wang at the Ohio State University for many useful discussions on the phase transformation model for titanium alloys.

Funding Statement: DQ would like to thank the financial support from the National Key Research and Development Program of China under Grant No. 2022YFB3707803, the Key Research Project of Zhejiang Laboratory under Grant No. 2021PE0AC02, and the National Natural Science Foundation of China under Grant No. U2230102. RS acknowledges the open research fund of Songshan Lake Materials Laboratory (2021SLABFK06) and Guangdong Basic and Applied Basic Research Foundation (2024A1515011873).

Author Contributions: The authors confirm their contribution to the paper as follows: study conception and design: R. Shi; analysis and interpretation of results: D. Qiu; draft manuscript preparation: D. Qiu. All authors reviewed the results and approved the final version of the manuscript.

Availability of Data and Materials: The raw/processed data and materials required to reproduce these findings cannot be shared at this time as the data and materials also form part of an ongoing study.

Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.

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## Cite This Article

**APA Style**

*α*precipitate in ti-6al-4v.

*Computer Modeling in Engineering & Sciences*,

*140*

*(1)*, 1017-1028. https://doi.org/10.32604/cmes.2024.048797

**Vancouver Style**

*α*precipitate in ti-6al-4v. Comput Model Eng Sci. 2024;140(1):1017-1028 https://doi.org/10.32604/cmes.2024.048797

**IEEE Style**

*α*Precipitate in Ti-6Al-4V,”

*Comput. Model. Eng. Sci.*, vol. 140, no. 1, pp. 1017-1028, 2024. https://doi.org/10.32604/cmes.2024.048797

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