Closed-form functions of cross-scale indentation scaling relationships based on a strain gradient plasticity theory

Indentation scaling relationships intend to guide indentation experiments and help understand the indentation hardness. In the present study, dimensionless functions were obtained by fitting a series of numerical results of the cross-scale indentation tests based on a strain gradient plasticity theory. The closed-form indentation scaling relationships were found between dimensionless material properties and indentation responses, which were non-dimensionalised by a representative stress. The closed-form expressions were of high accuracy with both the numerical results and the experimental data. Forward and reverse analysis were then established and discussed, providing the indentation scaling relationships a quantitative result. The explicit definition of the representative stresses made it possible to obtain the specific relationship between indentation responses and each material parameter. Together with some pre-determined material parameters, the material intrinsic length was able to be derived from the indentation tests. The present work significantly enhanced the value of the cross-scale indentation scaling relationships, providing great convenience in guiding instrumented tests.