Abstract An inverse problem of identification of the elastoplastic properties of power hardening engineering materials from limited spherical indentation measurements is studied. A fast algorithm for reconstruction of the Ramberg–Osgood curve σ i = σ 0 ( e i / e 0 ) κ , with the strain hardening exponent κ ∈ ( 0,1 ) , is proposed. The main distinguished feature of this algorithm is that the only two output measured data 〈 α i , P i 〉 , i=0,1, i.e. discrete values of the penetration depth ( α i ) and the loading force ( P i ), are required for the reconstruction of the unknown Ramberg–Osgood curve. The first measured data 〈 α 0 , P 0 〉 corresponds to pure elastic deformations, and the second one to one of the plastic deformations. The second advantage of the proposed algorithm is its well-conditionedness, different from parametrization algorithms proposed in previous studies. Numerical examples related to applicability and enough accuracy of the proposed approach are presented for the noise free and noisy data.