The analytic mean-field potential sAMFPd method proposed by Wang et al. sWAMFPd is verified to be equivalent to the free-volume theory. A generalized analytic mean-field potential sGAMFPd is established in terms of the free-volume theory and the nearest-neighbor pairwise interaction assumption. The GAMFP contains an integral. By using numerical integrated formula to approximately evaluate the integral, the GAMFP is transformed to the WAMFP or other forms of the AMFP, and the WAMFP can be seen as an approximate analytic version of the GAMFP. The GAMFP is exact for nearest-neighbor Lennard-Jones sNN-LJd model solid. The numerical results for thermodynamic quantities of NN-LJ solid from the GAMFP is compared with the WAMFP and other AMFP’s with slightly different forms. The comparison shows that the numerical results from the WAMFP are almost completely in agreement with the GAMFP and are better than several other approximate AMFP’s. The GAMFP and WAMFP with quantum modification have been applied to Vinet-type solids. The numerical results show that the theoretical values of Grüneisen γG and Debye temperature ΘD for three type solids are qualitatively in agreement with experiments, but the agreement is not satisfactory quantitatively. The predicted values of bulk thermal expansivity are too large for rare-gas solids, too small for alkali halides, and for metallic solids the agreement is slightly better but also is not satisfactory. Especially the predicted variations of bulk thermal expansivity and compressibility versus temperature are fairly bad; except for copper, the prediction is fortunately acceptable. It is shown that the fundamental spirit of the GAMFP and WAMFP to use all cold energy to evaluate thermal properties is in contradiction with embedded-atom model sEAMd. It is necessary to improve the GAMFP and WAMFP in terms of the EAM by the replacement of all cold energy with only cold energy from interaction between metallic atoms.