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Text: In general, the relationship between yield strength σ y and grain size d of polycrystalline materials can be expressed by Hall-Petch formula as [1,2] σ y = σ 0 + kd -1/2 (2) The effect of rare earth elements Y and Sm on the grain-refinement strengthening of magnesium alloy AZ81 can be calculated by Hall-Petch formula. Assuming that the grain sizes of AZ81 and AZ81+1.8 (Y+Sm) alloys are d 1 and d 2 , and the yield strengths are σ y1 and σ y2 , respectively, it can be obtained from the formula (2) that σ y1 = σ 0 + kd 1
 -1/2 (3) σ y2 = σ 0 + kd 2 -1/2
 (4) According to the above two formulas, it can be obtained as follows σ y2 -σ y1 = k (d 2 -1/2 -d 1 -1/2 ) (5) Combined with the above results of microstructure analysis (see Figure 1), the grain sizes d 1 =120 µm and d 2 =40 µm can be obtained by the linear intercept method. The k value of commonly used magnesium alloys is 280-320 MPa·µm 1/2 [1,2] , and here k = 300 MPa·µm 1/2 . By substituting these **data** into formula (5), the contribution of grain-refinement strengthening by Y and Sm to the yield strength of magnesium alloy AZ81 can be calculated as σ y2 -σ y1 = 300×(40 -1/2 -120 -1/2 ) = 20 MPa Combined with the above tensile test results (see Table 1), after the addition of 1.8%Y and Sm, the yield strength of AZ81 alloy is increased by 186 -144 = 42 MPa Among which, the contribution of grain-refinement strengthening is 20 MPa, accounting for 47.6% of the total increase of yield strength. Therefore, grain-refinement strengthening is a very important strengthening mechanism for magnesium alloys, and its contribution to the strength of magnesium alloys should be paid enough attention.<|eot_id|><|start_header_id|>assistant<|end_header_id|>

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