Lognormal Distribution of Local Strain: A Universal Law of Plastic Deformation in Material

Given the infinite diversity of microstructural inhomogeneity, the variation in spatial distribution of local strain could be infinite. However, this study finds that the statistical distribution of local strain universally follows a lognormal distribution irrespective of phase content and deformation mechanism. Moreover, this universal law is proved conditional upon the macroscopic homogeneity of deformation on the statistical window scale, equivalent to the equality between the macrostrain calculated from the displacements at the window corners and the average of the local strain. The discovery of a lognormal distribution law suggests the existence of a minimum statistical representative window (MSRW) size that is characteristic for each material. Explorations on the dependence of MSRW size on the microstructure, deformation mechanism, and strain magnitude are expected to add new dimensions to understanding of the relationship between microstructure and mechanical properties.

Figure 1

(a)–(e) Strain maps of (a) ferritic steel, (b) martensitic steel, (c) dual phase steel, (d) stainless steel, and (e) TWIP steel. (f) Strain distribution of the steel samples fitted by a lognormal distribution (solid lines). Grain boundaries in (a), (d), and (e) are marked by black lines, while those in (b) and (c) are unable to be shown owing to the extremely fine-scaled substructures of martensite.

Figure 2

(a),(d) Strain map; (b),(e) strain distribution fitted by a lognormal distribution; (c),(f) illustration of the boundaries before and after deformation of the regions of window $A$ and $B$, respectively. The boundary before (dashed lines) and after (solid lines) deformation are indicated in (c) and (f).

Figure 3

(a) Polycrystalline geometric model used in the simulation. (b)–(d) Distribution of the logarithmic strain along the tension direction exhibited by the deformed model at the applied strain ${ϵ}_{xx}=0.01$, and the strain distributions fitted by lognormal distributions under (b) rigid constrained boundary condition, (c) periodic boundary condition, and (d) free boundary condition.