*4.4. The E*ff*ect of Precipitates on Strength*

In principle, the possible increase of yield strength during heat treatment consists of grain boundary strengthening, vacancy hardening and precipitation hardening, and can be written in a first approximation as

$$
\Delta\sigma\_{\text{total}} = \Delta\sigma\_{\text{grains}} + \Delta\sigma\_{\text{vacancies}} + \Delta\sigma\_{\text{prericantes}} \tag{5}
$$

As the grain size was observed to stay constant during the heat treatments considered in this work (maximum 125 ◦C), changes of Δσ*total* could only arise from Δσ*precipitates* and/or from Δσ*vacancies*. For the estimation of contribution from precipitation hardening Δσ*precipitates* in Equation (5), we assume that all precipitates are non-shearable and are spherical. By use of a modified Orowan Equation (6) [63], i.e.,

$$
\Delta\sigma\_{\text{precipitates}} = M \frac{0.84Gb}{2\pi \left(1 - \nu\right)^{1/2} l} \ln \frac{r}{2b} \tag{6}
$$

the increase in yield strength Δσ*precipitates* can be estimated. In Equation (6), *M* means the Taylor factor (*M* = 4.2 for random texture Mg [64]), *G* the shear modulus (*G* = 17 GPa), *b* the Burgers vector (*b* = 0.32 nm), ν the Poisson's ratio (ν = 0.33), *l* the average interparticle spacing and *r* the average radius; for the following estimation, the values for *l* and *r* were taken from STEM images shown in Figure 16b (status after HPT + heat treatment) and Figure 16a (status after HPT only). The estimation leads to a difference of the yield strength due to precipitation hardening of Δσ*HPT* <sup>+</sup> *HT* − Δσ*HPT* = 115 − 92 = 23 MPa between the two conditions investigated for the furnace-cooled Mg5Zn0.3Ca samples. The measured difference, however, was 140 MPa; therefore, the result of these calculations is that the difference in the precipitation states cannot explain the extensive increase of the yield strength measured. The precipitates only contribute about 16% to the total increase in yield strength.
