A controversy exists among literature reports of constraints on elastic constants. In particular, it has been reported that embedded atom method (EAM) potentials generally impose three constraints on elastic constants of crystals that are inconsistent with experiments. However, it can be shown that some EAM potentials do not impose such constraints at all. This paper first resolves this controversy by identifying the necessary condition when the constraints exist and demonstrating the condition is physically necessary. Furthermore, this paper reports that these three constraints are eliminated under all conditions, by using response EAM (R-EAM) potentials.