# In Situ Stress Tensor Determination during Phase Transformation of a Metal Matrix Composite by High-Energy X-ray Diffraction

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

^{3}, compared to 7.8 g/cm

^{3}for steel alone (i.e., a decrease of 11.4% in mass). The final properties of an MMC depend on the chemical composition, on the nature of the interfaces, on the microstructure of the matrix, and on the stresses in the reinforcements and in the matrix. Many studies deal with load transfer between the phases in composite materials induced by external loading [1,2]. However, large stresses can be generated during the heat treatment, resulting from the differences in the coefficients of thermal expansion between matrix and reinforcements [3,4,5], and also from the phase transformations [6,7]. The residual stress levels and distributions are a key factor for the final properties of the MMC [8]. In a previous study [9], the evolution of the matrix and the reinforcements were analysed during the heat treatment using in situ high-energy X-ray diffraction focusing on the structural aspects. In this paper, we focus on the internal stress analysis. First, we describe the dedicated original device and the methodology that was developed to allow an in situ determination of the evolutions of internal stresses in a steel matrix composite during cooling. Then, internal stresses in the phases are analysed, emphasising the role of martensitic transformation. Finally, a 3D finite element micromechanical model is used to better understand this role, and a comparison with experimental results is discussed.

## 2. Material and Thermal Cycle

^{−1}. The temperature was held for 5 min at 900 °C, and then the sample was cooled down. To achieve cooling, the lamps of the furnace were switched off at the end of the dwell, and a controlled gas flow (argon) allowed the cooling rate to be controlled. The cooling rate was fast enough to avoid ferrite or bainite formation before the martensitic transformation at M

_{S}temperature (measured at 180 °C). The microstructure after the thermal treatment is presented in Figure 1b: the unreinforced areas composed of martensite and retained austenite (as we show later in the paper) and the reinforced areas where the matrix was mixed with TiC particles.

## 3. High-Energy X-ray Diffraction

#### 3.1. Experimental Setup

^{2}.

#### 3.2. Phase Analysis

## 4. Internal Stress Determination

## 5. Results

#### 5.1. Phase Transformation Kinetics

#### 5.2. Evolution of Mean Cell Parameters

#### 5.3. Full Width at Half Maximum

#### 5.4. Stress Evolutions

#### 5.5. Discussion

#### 5.5.1. Coefficient of Thermal Expansion

^{−6}K

^{−1}). In Figure 13, the envelopes present the variations in the stress levels for austenite and TiC, including these experimental uncertainties (for martensite, the stress variations are not shown as they would overlap the stress scattering). We can see that the variation in stress level can be very large. There were even variations large enough to find values of CTE for which the macroscopic stresses were zero.

#### 5.5.2. Stress-Free Parameters

#### 5.5.3. Macroscopic Elastic Constants

_{S}temperature could vary by about 250 MPa.

#### 5.6. Micromechanical Modelling

#### 5.6.1. Description of the Model

^{−1}. For the actual model, only two phases were taken into account: the matrix and the particles.

- d${\u03f5}_{ij}^{e}$: incremental elastic strain related to stress increment by Hooke’s law with temperature-dependent Young’s modulus and Poisson’s ratio.
- d${\u03f5}_{ij}^{p}$: incremental visco-plastic strain at high temperature and plastic strain at lower temperatures.

_{k}, n

_{k}, K

_{k}, and m

_{k}are temperature-dependent coefficients determined experimentally for each phase.

_{k}is the volume fraction of phase k.

- d${\u03f5}_{ij}^{tr}$: incremental strain due to volume change$${\u03f5}_{ij}^{tr}={y}_{k}\sum {\u03f5}_{k,0\phantom{\rule{3.33333pt}{0ex}}{}^{\circ}\mathrm{C}}^{tr},$$
- d${\u03f5}_{ij}^{pt}$: incremental strain due to transformation plasticity$$d{\u03f5}_{ij}^{pt}=\frac{3}{2}{K}_{k}{f}^{\prime}({y}_{k})d{y}_{k}{s}_{ij},$$

#### 5.6.2. Calculated Results

#### 5.6.3. Comparison with Experimental Results and Discussion

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Metal matrix composite (MMC) microstructure (

**a**) at initial state and (

**b**) after the thermal treatment.

**Figure 3.**2D patterns showing austenite and TiC diffraction signals at high temperature (HT) and showing austenite, martensite and TiC diffraction signals at room temperature (RT). Diffractograms from integrated images for the different phases (austenite, martensite ($\alpha $’), TiC).

**Figure 4.**(

**a**) Angles definition from setup configuration: (S1, S2, S3) is the sample reference system and (

**b**) evolution of ($\Phi ,\Psi $) defining the direction normal to the diffracting plane {hkl}.

**Figure 5.**Evolution of peak position ($\gamma $ (200)) over image number, and a zoomed view of image numbers 6400–6550.

**Figure 8.**Cell parameter evolutions during cooling (stress-free thermal expansion coefficients of TiC and austenite are also shown).

**Figure 9.**Evolution during cooling of the full width at half maximum (FWHM) for the different phases (steel-$\gamma $ phase (200), TiC (220), and steel-${\alpha}^{\prime}$ phase (211)) versus temperature.

**Figure 10.**Evolution of the ${sin}^{2}\Psi $ curves at different temperatures for steel-$\gamma $ phase, TiC, and steel-${\alpha}^{\prime}$ phase.

**Figure 11.**Evolution of the slopes of the ${sin}^{2}\Psi $ curve versus temperature for steel-$\gamma $ phase, TiC, and steel-${\alpha}^{\prime}$ phase.

**Figure 12.**Evolution of the intercepts of the ${sin}^{2}\Psi $ curves versus temperature for different $\chi $ angle for steel-$\gamma $ phase, TiC, and steel-${\alpha}^{\prime}$ phase.

**Figure 13.**Internal stress evolutions during cooling (because the stress states are hydrostatic, only one component of the stress tensor is presented). Envelopes show the stress variations introduced by uncertainties in thermal expansion coefficients (see Section 5.5).

**Figure 15.**Stresses and cumulated equivalent plastic strain profiles along axis 1 for different temperatures.

**Figure 16.**Evolution of the mean calculated stresses in the reinforcements and in the matrix during cooling.

C | Cr | Mo | Mn | Si | V | Ni | N |
---|---|---|---|---|---|---|---|

0.312 | 3.831 | 0.721 | 0.434 | 0.583 | 0.136 | 0.067 | 152 ppm |

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**MDPI and ACS Style**

Geandier, G.; Vautrot, L.; Denand, B.; Denis, S.
In Situ Stress Tensor Determination during Phase Transformation of a Metal Matrix Composite by High-Energy X-ray Diffraction. *Materials* **2018**, *11*, 1415.
https://doi.org/10.3390/ma11081415

**AMA Style**

Geandier G, Vautrot L, Denand B, Denis S.
In Situ Stress Tensor Determination during Phase Transformation of a Metal Matrix Composite by High-Energy X-ray Diffraction. *Materials*. 2018; 11(8):1415.
https://doi.org/10.3390/ma11081415

**Chicago/Turabian Style**

Geandier, Guillaume, Lilian Vautrot, Benoît Denand, and Sabine Denis.
2018. "In Situ Stress Tensor Determination during Phase Transformation of a Metal Matrix Composite by High-Energy X-ray Diffraction" *Materials* 11, no. 8: 1415.
https://doi.org/10.3390/ma11081415