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The Hall–Petch relationship is a fundamental principle in materials science and metallurgy that describes an inverse correlation between grain size and mechanical strength. This relationship states that as the average crystal grain size of a material decreases, its yield strength (or hardness) increases. This phenomenon was independently discovered in the early 1950s by E. O. Hall and N. J. Petch through separate investigations and is named after these two scientists. Hall (1951) established the