SIMULATION OF THE INDENTATION OF EUROFER97 STEEL AFTER ION RADIATION BY THE FINITE ELEMENT METHOD
The combination of ion irradiation and nanoindentation presents a vast field of research which includes not only experiments but also modeling that can reveal the features of the deformation behavior of materials at the micro- and nanoscale. The modeling carried out in the work made it possible to evaluate the correspondence between the strength parameters measured in the course of tensile tests of macrosamples irradiated with neutrons and the results of dynamic instrumental indentation of the samples that were used in the experiment on ion irradiation. The increase in hardness calculated during the simulation turned out to be comparable with that obtained experimentally, which indicates efficiency of the technique. The plastic behavior inherent in the samples subject to the experiment which manifests itself in a decrease in the height of the rolls with an increase in the irradiation dose was also confirmed in the simulation.
The validation of materials planned for use as structural members in the new generation of nuclear and fusion reactors involves a study of their mechanical properties after being exposed to the conditions close to the operational conditions . Neutron irradiation experiments take a long time and, therefore, the ion beam irradiation is applied much more frequently. The materials subjected to ion irradiation have a hardened layer of few micrometres thick due to shallow penetration of heavy particles .
For specimens with 1–2 µm thick near-surface layer, that are of interest for our study, in order to obtain the hardness-depth relation either a series of low-load tests  or the dynamic tool indentation method  is required. The first method involves a large area of a sample while the second method allows of obtaining dependence of hardness versus the indentation depth in a single area.
One of the methods for analysing the stress state arising from local pressure is a finite element modelling. Geometric non-deformable objects similar in shape to real indenters, can induce elastoplastic deformations of the experimentally observed sample when embedded into the material under test. An important question is whether the properties of the test specimen correspond to characteristics of the model objects. If the aim is to obtain hardness values, tensile data of experimental specimens can be taken as the basic characteristics of the simulated materials . In this way the dependence between strength properties and tool hardness can be found.
In this paper a finite element modelling of the Berkovich type tip indentation test with the hardness values calculation was carried out samples of Eurofer97 steel irradiated to a dose of 10 displacements per atom. In describing the constructed model, the elastoplastic properties typical of neutron irradiated steel were specified. In such a way the available experimental data  and the simulation results were compared.
RESEARCH METHODS AND MATERIALS
The finite element modelling was carried out with the Abaqus software package . A standard module was used.
Since a Berkovich-type indenter has third-order axis symmetry, our simulation was performed with cyclic boundary conditions to reduce computational costs. In other words, the problem was set by constructing two geometric objects: 1/3 part of a trihedral pyramid and 1/3 part of a cylindrical sample. The values of elastic-plastic properties established for the specimens were taken from the literature . The modulus of elasticity of Eurofer97 steel was taken as 225 GPa and Poisson’s ratio: 0.3. The yield strength was different for the different layers. The following layers were defined: 1 – irradiated to a dose of 2 displacement per atom (dpa), 2 – irradiated to a dose of 8 dpa, 3 – to 11 dpa. Table 1 shows the values of yield strength, tensile strength and maximum deformation of the samples used in the simulation, obtained from the original and irradiated samples. The dose and temperature of irradiated samples are indicated. The temperature at which the tensile test was carried out was 25 °C.
Figure 1 shows a schematic representation of the sample + indentation system used as a model in this work. For Eurofer97 steel, the sufficient data on mechanical properties of irradiated samples are available in the literature (see Table 1) to simulate the irradiated layer as a set of three sub-layers (Fig.1a), close to the damage dose profile (Fig.1b) calculated in the SRIM package, for iron ion introduction with energy 5.6 MeV and flux of 1 · 1016 ions/cm2.
The sample was divided into two main sectors: the central part in which a fine grid of finite elements was set to increase modelling accuracy of the indentor-sample contact zone (Fig.2), and the peripheral part with a larger size of finite elements wherein the residual mechanical stress was propagated and damped. This important preparatory step made it possible, on the one hand, not to reduce essentially the obtained results accuracy and, on the other hand, to reduce calculation time. The dense grid must be present both on the samples and on the indenter. Otherwise, artifacts in the force diagram appear in the form of random small deviations from the monotone dependence, which leads to a large scatter in the calculated hardness values at small depths.
The hardness values were calculated using the standard formula valid for an ideal pyramid of the Berkowitz type :
RESULTS AND DISCUSSIONS
Figure 3a shows the von Mises elastoplastic stress fields (a) arising in the Eurofer97 sample with irradiated layers under the indenter as a result of the maximum force application (258 mN) and the corresponding deformation (b).
The highest stresses under the indenter occur in the area corresponding to the irradiated material. With the same indenter depth of 1 µm, the stress area is more than 5 times the deformation propagation radius.
Fig.4 shows the dependences of hardness on indenter depth obtained in the experiment and simulation for Eurofer97 samples: initial and irradiated to dose 10 dpa (in experiment) and 10.9 dpa (in simulation).
The experimental dependencies were published in .
In Fig.4a, the dependences of hardness versus indentation depth, obtained in the experiment, have the form of falling curves, while the plots for the simulation results are practically constant. This can be explained by the fact that the dimensional indentation effect stably observed in the experiment does not appear in the course of simulation. The dimensional effect is expressed in growth of hardness with decreasing indentation depth, i.e., it is related to the area size of the sample that comes into contact with the indenter and does not appear at depths greater than a certain characteristic depth.
Nevertheless, the graphs in Fig.4b show satisfactory agreement between a difference in hardness values between irradiated and unirradiated samples obtained in the experiment and in the simulation, which suggests possibility of using simulation as an initial estimate of hardness of the newly developed alloys.
In addition to obtaining the dependencies of hardness versus indentation depth, the values of plastic rolls-on occurring around the perimeter of indentation were measured. The rolls-on heights were measured using the built-in "Get coordinates" function as the distance between the grid node corresponding to the maximum indentation depth and the node corresponding to the maximum raised relief around the indentation.
Fig.6 shows the calculation results of ratio of plastic rolls-on height to the depth of the residual imprints, obtained in the experiment and in simulation.
It can be seen from Fig.6 that the values of plastic rolls-on formed at the edges of indentations during indentation correlate with the irradiation modes of samples in both simulation and experiment, which indicates possibility of using simulation as a tool for preliminary assessment of material hardness.
The deformation behaviour of heterogeneous specimens under local pressure can be studied by means of finite element modelling.
Finite element modelling of the tool indentation process has shown satisfactory agreement between the theoretically predicted hardness values and the experimental results.
Both in the experiment and in the simulation the dependences of hardness versus the penetration depth of the indentor were also obtained. Both the experiment and the simulation show differences in the samples properties before and after irradiation. Since in simulation the sample is specified as homogeneous within sublays and with sharp boundaries, while in the experiment the defects, non-homogeneities and a property gradient at the interface occur in the structure, the hardness values differ from each other. Nevertheless, if we compare the difference in hardness values between the irradiated and the original Eurofer97 samples obtained in the simulation and in the experiment, the data are in good agreement with each other and the values obtained can be considered as statistically significant.
Both in the experiment and in simulation, plastic rolls-on appeared as a result of indenter penetration. The decrease in plasticity observed with increasing irradiation dose, manifested by a decrease in the size of the rolls-on, is evident both in the experiment and in the simulation.
In the long term, numerical simulation can be used to elucidate the macro properties of the material, which is a separate task requiring at this stage a large amount of experimental data on reactor irradiation. The present work, however, shows that hardness measurements can be considered as a verified technique for the express analysis of the irradiation effect on mechanical properties, which is confirmed by the simulation described above.
PEER REVIEW INFO
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Declaration of Competing Interest. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.