We extracted stress-strain field (SSF) gradient and divergence representations from nanoindentation data sets of bulk solids often used as thin film substrates: bearing and tooling steels, silicon, glasses, and fused silica. Oscillations of the stress-strain field gradient and divergence induced in the subsurface layer by the nanoindentation were revealed. The oscillations were especially prominent in single indentation tests at shallow penetration depths, h<100 nm, whereas they were concealed in the averaged datasets of 10 and more single tests. The amplitude of the SSF divergence oscillations decayed as a sublinear power-law when the indenter approached deeper atomic layers, with an exponent -0.9 for the steel and -0.8 for the fused silica. The oscillations were interpreted as alternating strain hardening-softening cycles induced in the subsurface layer under the indenter load.