This paper investigates several granular interaction laws used in the modeling of discrete granular media. In the considered model, each grain interacts with its neighbors with a coupled shear-normal interaction law. The analysis is performed in a geometrically exact framework allowing large rotation and displacement evolutions, without any geometrical approximations. It is shown that most of the granular interaction laws available in the literature are classified as hypoelastic interaction laws, and we precise the requirements to build some hyperelastic interaction laws that avoid artificial dissipation. We also show that the uncoupled granular interaction law is hyperelastic for all the studied models. The analysis is applied to a paradigmatic elementary system of a granular loop with a diamond pattern (a four-grain cyclic granular chain) loaded by concentrated forces. Instabilities are observed for large displacement of the diamond chain for all the classified models. It is observed that the discrepancies between each model may grow during the deformation process. The instability phenomenon is associated with the appearance of a limit load for this granular structural problem due to large nonlinear geometrical effects. Blocking phenomena may also appear for such granular structural systems due to secondary granular contacts.