Nonlinear rheology was examined for concentrated suspensions of spherical silica particles (with radius of 40 nm) in viscous media, 2.27/1 (wt/wt) ethylene glycol/glycerol mixture and pure ethylene glycol. The particles were randomly and isotropically dispersed in the media in the quiescent state, and their effective volume fraction phi(eff) ranged from 0.36 to 0.59. For small strains, the particles exhibited linear relaxation of the Brownian stress sigma(B) due to their diffusion. For large step strains gamma, the nonlinear relaxation modulus G(t, gamma) exhibited strong damping and obeyed the time-strain separability. This damping was related to gamma-insensitivity of strain-induced anisotropy in the particle distribution that resulted in decreases of sigma(B)/gamma. The damping became stronger for larger phi(eff). This phi(eff) dependence was related to a hard-core volume effect, i.e., strain-induced collision of the particles that is enhanced for larger phi(eff). Under steady/transient shear flow, the particles exhibited thinning and thickening at low and high (gamma) over dot, respectively. The thinning behavior was well described by a BKZ constitutive equation using the G(t, gamma) data and attributable to decreases of a Brownian contribution, sigma B/(gamma) over dot. The thickening behavior, not described by this equation, was related to dynamic clustering of the particles and corresponding enhancement of the hydrodynamic stress at high (gamma) over dot. In this thickening regime, the viscosity growth g, after start-up of flow was scaled with a strain (gamma) over dot t. Specifically, critical strains gamma(d) and gamma(s) for the onset of thickening and achievement of the Steadily thickened state were independent of (gamma) over dot but decreased with increasing phi(eff). This phi(eff) dependence was again related to the hard-core volume effect, flow-induced collision of the particles enhanced for larger phi(eff).