Analytical determination of the constitutive behavior from micro-pillar testing: Application to a tempered martensitic steel

Micro-compression experiments were carried out on micro-pillars, with nominal diameter of 10 μ m and height 20 μ m, and fabricated from a high-chromium tempered martensitic steels developed for nuclear applications. An analytical method was proposed to calculate the imposed compressive displacement on the pillar that allows determining the constitutive behavior up to about 5% of plastic strain. The experimental micro-compression curves were showed to be adequately modeled by the ﬁnite element simulations, which were used to validate the main assumption considered for the analytical model development. C (cid:2) 2014 micro-pillars in compression investigate mechanical of

follows. Before machining by focused-ion beam (FIB), the specimen surface was mirror-polished with diamond suspension as fine as 0.25 μm. Zeiss NVision 40 dual-beam (FIB-SEM) system equipped with a FIB column with gallium liquid metal ion source and a field-emission Scanning Electron Microscope (SEM) column was used for the micro-pillar machining. First, a pre-mill of a2 0μm wide ring with an external diameter of 100 μm using 27 nA FIB milling current and "50 μm depth" in the "feature-mill" regime results in a ∼18 μm deep crater. The pre-milling step with the high beam current is required in order to reduce the overall time of the micro-pillar fabrication process. The second step is the "annular" milling of a second 35 μm wide ring with a diameter of 80 μm. The annular milling setting are: milling current is 1.5 nA with the 2048 resolution, the dwell time is the 155 ms for the outer ring to the 150 ms for the inner ring. The final shaping fine cut using the 700 pA milling current was done to minimize the tapering effect of the annular milling. This machining process was developed specifically for Eurofer97 tempered martensitic steel. The micro-compressions were carried out with a MTS Nanoindenter G200 instrument equipped with a flat diamond tip. The load-displacement data from the micro-compression tests were recorded and further used for the analysis presented hereafter. Finite element (FE) calculations were performed using the commercial code ABAQUS/Standard 6.12. The flat diamond tip was modeled as an axisymmetric deformable part with isotropic elastic properties, Young's modulus (E = 1141 GPa) and Poisson's ratio (ν = 0.07). Due to the symmetry of the micro-pillar geometry only the half of the specimen was modeled with the following dimensions: 40 × 45 μm 2 (width × height) for the substrate part, with the pillar having the typical height of ∼20 μm and a radius of ∼5 μm (the exact micro-pillar specimen geometry (radius and height) was taken into account for each simulation). The specimen was meshed with 8604 four-node bilinear axisymmetric quadrilateral elements with reduced integration. In the unirradiated case, the constitutive behavior of the unirradiated Eurofer97 steel was deduced from the corresponding tensile test and used as an input to the FE micro-pillar model. However, we have shown in Ref. 8 that the tensile test after irradiation at 150 • C presents a very small uniform elongation. The constitutive behavior of irradiated Eurofer97 used as the input for the FE models was then obtained with an inverse method as described in Ref. 8. In both cases, the plastic behavior was given in the form of tabulated true stress -true plastic strain. The Young's modulus and Poisson's ratio of the steel were respectively 210 GPa and 0.3. The contact between the diamond tip and the micro-pillar was assumed frictionless. Figure 2 shows a very good match between the experimental load-displacement curves of two micro-pillars in the unirradiated and irradiated Eurofer97 with their corresponding FE models. In Figure 2, it can be distinguished that the yielding for the irradiated Eurofer97 occurs as expected at a larger load in comparison with the unirradiated material, consistently with the tensile results in Ref. 8, and reflecting the well-known irradiation-hardening. While the micro-pillar testing consists in deforming small compression specimens, the conversion of the measured load-displacement curve into the true stress-strain curve is not straightforward due to the geometry of the micro-pillar. As a matter of fact, the final geometry of the micro-pillars is never a perfect cylinder. They are actually tapered, which has to be taken into account in the stress/strain analysis.
To extract the true stress-strain curve, we did the assumption that, despite the relatively small size of the specimen, the deformation behavior remains representative of the bulk. The justification is essentially based upon the fact that the basic microstructural unit of the tempered martensitic structure is the tempered lath whose dimensions are of the order of 0.5-1 μm in width and several μm in length. Thus, owing to the size of the micro-pillars used in this work, a larger number of these microstructural units are contained in the specimens (see Figure 1). In addition, the finite element simulations performed to reconstruct the load-displacement curve showed a very good agreement with experimental results. Since the finite element simulations are based upon a continuum approach of plasticity, the good match between the simulation and experiment is an additional indication that the mechanical response of the micro-pillars remains representative of the bulk. Furthermore, it is fully consistent with the work of Shin et al. 9 who showed that there is no significant size effect on the measured strengths of an oxide dispersion strengthened steels down to a pillar size of 1 μm. This was attributed to the fact that the studied alloy presented possesses a fine grain structure of about 450 nm.
The model developed in this study to do the conversion from the load-displacement into the true stress-strain is presented hereafter. We followed the work of Dimiduk et al. 4 who developed a formula to calculate the elastic displacement of micro-pillar. However, our model is extended in the plastic range. A schematic drawing of the pillar geometry and parameters considered is shown in Figure 3. The model is based upon the fact that the main component of the stress tensor is σ yy ,at least in the elastic regime and for moderate plastic deformation. This point was actually verified by the finite element simulations. In Figure 4, as an example we show a micro-pillar of unirradiated Eurofer97 loaded just above the load P y necessary to produce a general yielding on the top end of the pillar. The active plastic zone is indicated in red. The non-zero components of the stress tensor, calculated by finite element simulations for the series of element just below the indenter (Position 1 in Figure 4), are plotted against the radial distance in Figure 5. It is clear that σ yy dominates all the other components and that each component is only weakly dependent on the radial position. We also emphasize that the calculated yield load P y , at which the plastic zone has spread through the whole diameter on the top of the pillar, is consistent with the yield stress used as input for the simulations. Indeed, P y allows to estimate the yield stress through the equation σ y = 4P y /π D 2 0 , which was equal to −495 MPa, to compare with the input yield stress set at 470 MPa.
In Table I, we give the average values over the cross-section of the stress components at three different positions: 1) on the top of pillar just below the flat indenter, 2) at one third of the pillar height and 3) at two thirds of the pillar height (see Figure 4). Again, σ yy calculated by FE simulations appears as the dominating component, which in addition is very well approximated by the so-called experimental normal component obtained by dividing the applied load by the actual cross-section  Table I, were averaged over the micro-pillar cross-section. of the layer. A similar conclusion is drawn from the irradiated micro-pillar simulations, as well as for deformation level for which the plastic zone does not extend into the substrate. The derivation of the true stress-strain curve is based upon a fitting procedure where the parameters of the constitutive law are adjusted to reconstruct the measured displacement of the flat tip z meas as a function of the load. Actually, z meas represents the sum of the displacements of the "substrate" z s and of the pillar z p .  The displacement of the substrate is estimated from Sneddon's formula as: 10 where j represents the j th load increment, H j the actual height of the pillar at the j th load increment and ν the Poisson's ratio. The basic idea is to divide the micro-pillars in thin horizontal layers, for which the j th incremental displacement can be deduced from the j th incremental load. As justified above, the displacement of the pillar is obtained by assuming that the stress is uniformly distributed over the horizontal layers of the pillar. In other words, only the normal stress and strain components, σ yy and ε yy , are considered, which, for the sake of simplicity are simply refer as to σ and ε in the following equations. For the i th layer, we note: δP ji , δz ji,p , δσ ji , δε ji ,t h ej th incrementals of load, displacement, stress and strain respectively. δε ji is is replaced by dividing the incremental stress δσ ji by the strain-hardening coefficient˜ ji (δε ji = δσ ji /˜ ji ). Thus, the strain increment δε ji depends on δσ ji and˜ ji that are respectively an experimentally measurable quantity and an analytical quantity given by Eq. (8). The basic idea is then to fit the parameters of the strain-hardening law to reconstruct the measured displacement as a function of the load. The incremental stress is deduced from the incremental load δP ji . So the incremental pillar displacement δz ji,p of the i th layer for the j th increment of load is given by: where t ji is the actual thickness of the i th layer at the j th increment of load, and the normal stress is calculated as: The compressive incremental displacement of the entire micro-pillar corresponding to the j th incremental applied load δP j is obtained by adding all the incremental displacements of i layers as: Finally, the total compressive displacement of the entire micro-pillar at a given load is obtained by summation on all the j increments of load considered: where j represents the j th load increment. After each load increment, the geometry of the pillar is updated by assuming that: 1) the deformation is done at a constant volume, 2) D 1 , the bottom diameter of the micro-pillar (see Figure 3) is constant. Thus D 0´a nd θ are recalculated for each load increment.
To extract the constitutive behavior from the experimental data, the last step that remains to be done is to fit a function for the strain-hardening˜ to insert into Eq. (6), which yields the measured total displacement after summation. An expression of the Ludwik's equation type was used for the true stress-strain curves: which yields:˜ The previous equation shows that one can express˜ in function of the stress, which in turn is a function of the load. Thus for each layer and load increment, one has to replace˜ ji in Eq. (6) by Eq. (8). In principle, the Young's modulus, the yield stress σ y and the strain-hardening coefficients A and n have to be fitted in order to adjust the calculated displacement to the experimental one using Eq. (6). It has to be emphasized here that the elastic modulus obtained by this procedure does not yield the actual Young's modulus E. Indeed, the detailed finite element analysis of Yang et al., 11 performed for elastic deformation demonstrated that one measures a relaxed modulus E 0 instead of E, with a ratio E/E 0 strongly dependent on the geometry of the micro-pillar and that can be as large as 1.8. We found a similar result, namely the elastic part of the micro-compression tests could be best fitted in a first step with a relax modulus E 0 that was such that E/E 0 = 1.7-1.8. For the plastic part of the deformation, it is easy to get a good estimate of the yield stress σ y . It simply consists in determining the yield load and dividing it by the upper cross-section of the micro-pillar, which is the smallest and which yields at the first place (see Figure 4 and Table I). The plastic zone propagates through the specimens upon additional loading. In the third step, the strain-hardening coefficient A and n, as well as σ y , are adjusted to fit the displacement by using the least squared residuals method. In Figure 6 below, we show two typical examples of analysis of micro-pillars: one for unirradiated and one for irradiated Eurofer97. The displacements were calculated by fitting E 0 , σ y , A and n described above. The fitting parameters are indicated in Table II. As can be seen, the one-to-one correspondence between the measured displacement and the calculated ones obtained for the sets of E 0 , σ y , A and n parameters reproduce well the true stressstrain curves σ (ε) up to strains of about 5%. For the unirradiated Eurofer97, there is a direct comparison between the adjusted and the experimental tensile σ (ε) curves. Again such a direct comparison could not be done for the irradiated Eurofer97 since the tensile curve exhibit less than 2% of uniform elongation. However, the true stress-strain curves σ (ε) deduced from the micro-pillar analysis compared very well with those used to reconstruct the necking behavior of the tensile curves present in Ref. 8. While the proposed analysis of the micro-pillar is simple and powerful, it is limited to about 5% of strain. Such a limitation is related to the fact that after a certain amount of compression, the deformation does not occur any more as described. As soon as the plastic zone has propagated through the entire micro-pillar, the previous analysis cannot be regarded as a reasonable approximation: the substrate begins to deform plastically and the stress state in the micro-pillar is not dominated by the normal component any more. However, at larger deformation, the true stress-strain curves σ (ε) can be fitted with finite element simulation for strain much larger than 5%, using an inverse numerical method to reconstruct the load-displacement curve of the micro-pillar. However, it is worth outlining that the analytical method proposed in this paper allows determining the yield stress and the initial part of the strain-hardening very precisely.