One often investigates electrorheological (ER) solids by using the point-dipole (PD) approximation, which, however, is known to err considerably for touching particles due to the existence of many-body (local-field) effects and multipolar interactions. Beyond the PD model, previous attempts have been restricted to either local-field effects only or multipolar interactions only, but not both. In the present work, we have developed a many-body dipole-induced dipole model for an ER solid the lattice structure of which can be changed due to the application of external fields, in an attempt to take into account both local-field effects and multipolar interactions. The results show that the multipolar interaction can indeed be dominant over the dipolar interaction, while the local-field effect may yield an important correction. Also, the results are well understood with the aid of spectral representation theory.