Crocker, L; Geca, P; Zhang, H (2024) Extract strain stress parameter via spherical indentation, FEA and inverse modelling. NPL Report. MAT 141

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Abstract

A computational study was undertaken to correlate elasto-plastic properties of ductile materials with instrumented indentation test with spherical indenters with radius of 5, 22.5 and 1005 um. Large deformation finite element computations were carried out for different combinations of elasto-plastic properties that covers a wide range of parameters of common pure and alloyed metals: Young’s modulus, E was varied from 60 to 210 GPa, yield strength, from 100 to 3000 MPa, with a choice of including strain hardening exponent, n, from 0 to 1. Different kinds of constitutional laws were employed to construct instrumented indentation tests, of which data generated was used for training machine learning algorithm to relate indentation data to elasto-plastic properties of materials.

For the focus of the current report, with restrictions of limited uniaxial experimental data, the results of the simulation of the indentation data carried out with 22.5 µm radius indenter was tested. Nanoindentation results show P91 (a 9%Cr steel) gives a reasonable fit between nanoindentation strain-stress data with tensile results (with the former about 8% lower of yield stress than the tensile test data), while the strain-stress data of Waspaloy (a Ni-based superalloy) is difficult to be extracted from indentation test data. Despite being able to separate initial elastic properties from plastic deformation, the yield stress was much higher (100%) than that measured from the tensile test. This is due to the size effect, which is beyond the scope of the current investigation.

Forward and reverse analysis algorithms were thus established: the forward algorithms allow for the calculation of a unique indentation response for a given set of elasto-plastic properties, whereas inverse modelling algorithms enable the extraction of elasto-plastic properties from a given set of indentation load-displacement data. A representative strain hardening exponent n, and indentation stress factor were identified which allows for the construction of an indentation loading response with experimental tensile strain-stress as input, under Johnson Cook theory. The proposed reverse analysis aims to provide a unique solution of Young’s modulus, a representative stress, and the hardening exponent. These values are very sensitive to the experimental scatter, such as materials, indenter radius, contact stiffness and indentation load used.

Given the Johnson Cook law adequately represents the full uniaxial stress-strain response, values of yield stress and hardening factor can be determined for the example cases here on the Waspaloy and P91. However, the plastic properties are very strongly influenced by even small variations in the parameters extracted from instrumented indentation experiments, such as at different loads. Due to the complex strain-stress field under indentation and representativeness of the indentation curve in relation to bulk elasto-plastic properties, it’s understood that it’s challenging to correlate all the conditions under one case study.

The current study sheds light on the method to extract high resolution strain-stress data via nanoindentation test. For some type of materials that fit a simple work hardening law, experimental methods are capable of extracting yielding stress from a multiple load indentation test, while for some materials such as Waspaloy, it’s challenging to correlate nanoindentation strain-stress data with tensile test data. With simulation, it’s possible to correlate different models with materials. It must be material-based analysis, there is no single rule that works for all. Therefore, it’s important to build up a database, with a range of materials to build a robust simulation model.

Item Type:	Report/Guide (NPL Report)
NPL Report No.:	MAT 141
Keywords:	Indentation strain-stress, inverse FEA modelling, Machine learning, Nanoindentation, spherical indentation
Subjects:	Advanced Materials > Mechanical Measurement
Divisions:	Materials and Mechanical Metrology
Identification number/DOI:	10.47120/npl.MAT141
Last Modified:	02 Sep 2026 07:48
URI:	https://eprintspublications.npl.co.uk/id/eprint/10498