# Investigation of Deformation Inhomogeneity and Low-Cycle Fatigue of a Polycrystalline Material
## 3 Measurements of Deformation Inhomogeneity and Predictions of Fatigue-Life Curve
### Different Measurement of Statistics of Strain Inhomogeneity  
The results above imply that throughout the cycle process, the new non-uniform plastic deformation continuously arises, and the material structure changes geometrically at the grain level (i.e., the microstructure evolution), which can be regarded as the factor leading to material degradation of the ability to resist damage. This urges us to find a method to measure the variation of deformation nonuniformity with cycle number. Since it is able to describe the extent of the inhomogeneity of a physical quantity distribution, the statistical standard deviation (SD) is applied as a fatigue indicator parameter (FIP) to describe fatigue damage. For any strain quantity, \(x\), it is calculated by:  
\[\hat{x}=\sqrt{\sum_{k=1}^{n_{\text{RVE}}}x_{k}^{2}v_{k}-\overline{x}^{2}} \tag{13}\]  
where \(n_{\text{RVE}}\) is the total number of finite elements in the RVE, \(v_{k}=\Delta V_{k}/V\), with \(\Delta V_{k}\) being the volume of k-th element and \(V\) the total volume of the RVE. This parameter is applied as the characteristics of the inhomogeneous strain components distribution of the RVE. For tension compression fatigue, SD of the longitudinal strain component can be used as FIP.  
Additionally, the inhomogeneity of the deformation can be quantified by using Shannon entropy. The entropy is initially referred to as a probability function characterizing the state of a thermodynamic system and is a measure of system's disorder. [PERSON] and [PERSON] [PERSON] and [PERSON] (1958) transferred the concept to quantitative measurement of information. For a model such as the RVE, which has the continuous variables, people can adopt the concept of entropy for continuous distribution, which is also referred to as the differential entropy (or continuous entropy). It can calculate as  
\[H=-\int_{-\infty}^{\infty}f_{x}\log f_{x}\text{d}x \tag{14}\]  
where \(f_{x}\) denotes the probability density function for a random variable. The differential entropy retains many of the properties of its discrete counterpart, but with some important differences. In an RVE, the interval [\(x^{\text{min}}\), \(x^{\text{max}}\)] of the variable \(x\) is equally divided into \(n\) subintervals [\(x_{i},x_{i+1}\)], \(i=1,2\,\ldots,n\). That is, the interval [\(x_{i},x_{i+1}\)] in calculation is taken as a constant and \(\Delta x_{i}=x_{i+1}-x_{i}=\left(x^{\text{max}}-x^{\text{min}}\right)/n\). Then, the integral of the entropy described by Equation (14) can be numerically calculated according to the following formula for the strain quantity:  
\[H_{x}=-\frac{n}{i=1}\bigg{[}\frac{p_{xi}}{\Delta x_{i}}\,\log\left(\frac{p_{xi }}{\Delta x_{i}}\right)\,\Delta x_{i}\bigg{]};\,p_{xi}\geq 0. \tag{15}\]  
where \(p_{xi}\) is the relative volume fraction, \(\Delta V_{i}/V_{\text{RVE}}\), of the region where [\(x_{i},x_{i+1}\)].  
When the RVE and crystal plasticity calculation are used to simulate the fatigue cycle process with different strain amplitude, the values of the above parameters (standard deviation and differential entropy) at the corresponding cycle peak can be calculated. Both the statistical standard deviation and differential entropy reflect the disorder degree of a distribution variable. Additionally, they increase numerically with the disorder. It implies the standard deviation and entropy are consistent in irreversibility.