Evolution of the Stored Energy Density in a X2CrNiMo17-12-2 Austenitic Steel Under Cyclic Loading Conditions

Abstract

In the present work, the impact of cyclic loading conditions on the accumulation of stored energy in an austenitic steel is investigated. For this purpose, in situ X-ray diffraction analyses were carried out to estimate the stored energy density from integral breadths. Specifically, uniaxial cyclic tests were performed for different stress amplitudes, loading ratios and numbers of loading cycles. The Williamson and Hall method was used to evaluate the microstructural changes resulting from a loading history. The comparison with experimental data available in the literature suggests that the number of cycles to failure is not solely related to the stored energy density. Indeed, while the loading ratio impacts the number of cycles to failure, it has no visible influence on the stored energy density. Also, whatever the loading ratio is, energy is mostly stored during the first loading cycle. No significant evolution of the stored energy density is observed during the following loading cycles. Finally, experimental results indicate that the fraction of the work that is stored within the material is small, about a few percent for the investigated loading conditions.

Appendix A Relation Between the Integral Breadth of Diffraction Peaks and the Stored Energy Density

Equation [2] exploits the relation between the width of diffraction peaks and the stored energy. Indeed, according to the Bragg equation, the lattice spacing $d_{hkl}$ associated with a $\{hkl \}$ set of equivalent planes is related to the Bragg angle $\theta _{hkl}$ with:

$$\begin{aligned} 2 d_{hkl} \sin \theta _{hkl} = \lambda \end{aligned}$$

(A1)

The differentiation of the above equation allows relating the variation of lattice spacing $\Delta d_{hkl}$ to the width of diffraction peaks $\Delta 2 \theta _{hkl}$ according to:

$$\begin{aligned} \frac{\Delta d_{hkl}}{d_{hkl}}= \frac{ \Delta 2 \theta _{hkl}}{2 \tan \theta _{hkl}} \end{aligned}$$

(A2)

In the above equation, $\Delta d_{hkl}/d_{hkl}$ can be interpreted as the root mean square strain resulting from the internal stress field produced by lattice defects. Since energy storage in crystalline materials is mostly controlled by lattice defects,^[25] proposed to estimate the stored energy from $\Delta d_{hkl}/d_{hkl}$ with:

$$\begin{aligned} u_{hkl} = \frac{3}{2} \frac{E_{hkl}}{1+2 \nu ^2_{hkl}} \left( \frac{\Delta d_{hkl}}{d_{hkl}} \right) ^2 \end{aligned}$$

(A3)

However, in a typical diffraction experiment, the width of diffraction peaks includes an instrumental contribution, which does not provide any information on the microstructure. To exclude such a contribution,^[22] used a modified version of Eq. [A2]:

$$\begin{aligned} \frac{\Delta d_{hkl}}{d_{hkl}}= \frac{ \sqrt{(\Delta 2 \theta _{hkl})^2-(\Delta 2 \theta _{0,hkl})^2} }{2 \tan \theta _{hkl}} \end{aligned},$$

(A4)

where $\Delta 2 \theta _{0,hkl}$ is the peak width measured for an annealed specimen with negligible stored energy. In the present work, an equivalent form of the above equation, which is based on reciprocal quantities, is used. The reciprocal lattice spacing $d^\star _{hkl}$ and the reciprocal integral breadth $\beta ^\star _{hkl}$ are given by:

$$\begin{aligned}&d^\star _{hkl}=\frac{2 \sin \theta _{hkl}}{\lambda } \end{aligned}$$

(A5)

$$\begin{aligned}&\beta ^\star _{hkl}=\frac{\Delta 2 \theta _{hkl}\cos \theta _{hkl}}{\lambda } \end{aligned}$$

(A6)

Combining the above relations leads to the following equation:

$$\begin{aligned} \left( \frac{\Delta d_{hkl}}{d_{hkl}} \right) ^2= \frac{\beta ^{\star ^2}_{hkl}-\beta ^{\star ^2}_{0,hkl}}{d^{\star ^2}_{hkl}} \end{aligned}$$

(A7)

Equation [2], which relates the stored energy density to the reciprocal integral breadth, is obtained by introducing the above expression in Eq. [A3].