Measurements of the effective electric dipole moments of conducting particle chains suspended in insulating dielectric liquid reveal the influence of dielectric breakdown between particles. This breakdown can lead to reversal of the sign of the interparticle force from attractive to repulsive. Other measurements with a vibrating sample magnetometer of the effective magnetic dipole moments of ferromagnetic particle chains provide clear evidence that chaining enhances nonlinear magnetization at a field intensity about a factor of 10 below the value at which saturation is observed for single particles. Both of these physical situations are examples of how nonlinear effects can influence the electromechanical interactions of closely spaced particles. The nonlinearity is caused by the very strong field intensification that occurs in the gaps between particles for chains parallel to an applied electric or magnetic field. A correspondingly strong nonlinear effect is predicted for the forces between particles in chains.