On the derivation and analysis of the "machine equation" in finite deformations

The machine equation plays an important role in the analysis of experimental data on mechanical testing and in modeling work. Its careful derivation has been provided both for the case of infinitesimally small deformations (brief recapitulation of the results known before) and finite deformations and supplemented by the detailed analysis of several practically important cases (rigid and soft machines and intermediate situations). It has been demonstrated that in certain situations the value of generalized stiffness of the "sample+machine" system can even change the qualitative nature of plastic instabilities, and the importance of a careful assessment of experimental setup as related to the results of mechanical testing, has also been discussed [8]. The authors hope that their work, which is aimed mostly at experimenters, could serve as a useful "reference tool" for the mechanical metallurgy and mechanical engineering communities.