Parametric Study

A full-factorial experimental matrix was designed to establish the effects of the Young’s modulus, E, the strength coefficient, K, and the hardening exponent, n on the resulting force-displacement curves of otherwise identical experiments [292]. This will aid future curve fitting

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efforts for approximating simulation results with experimental results, minimizing error between them. The factors K and n come from the relationship for true stress and true strain.

𝜎 = 𝐾𝜀^𝑛 (30)

This relationship, Hollomon’s law, helps to describe what happens in the plastic region when a material deforms and undergoes strain or work hardening. The strain hardening exponent, n, is used to describe the plasticity of a material, ranging from 0 to 1, with 0 being described as perfectly plastic and 1 as perfectly elastic. Combined with Hooke’s law, which describes the elastic portion of a material’s behavior, the formulation of the Ramberg-Osgood relationship is established, which can describe the entirety of the stress-strain curve for a strain-hardening material. compared with experimental results, in order to determine material properties of materials tested with SPT.

The median values for these factors (Level 3 in Table 4.1) were chosen as the approximate mean mechanical properties of IN939 from various sources at room temperature, being the material basis for future studies [55, 56, 296, 297]. Each factor was then varied by ± 10

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and 20% from the median value, a common practice when utilizing DoE and ANOVA, to establish a total of 5 levels for each factor, as shown in Table 4.1, for a complete test matrix of 125 experiments. The factors and levels were input into the SigmaZone QuantumXL statistical software to design the full-factorial study, which provided the factor levels necessary for each run. As the tests conducted were simulations, multiple replicates were not necessary.

Table 4.1 – Factor levels for full factorial parametric design study.

Level

Factor 1 2 3 4 5

E (MPa) 160000 180000 200000 220000 240000

K 880 990 1100 1210 1320

n 0.08 0.1 0.12 0.14 0.16

The experiments were programmed into the ANSYS model for evaluation and the effects of each case were determined by analyzing the effects on several points throughout the load-displacement curves, which correspond to either specific inflection points or were chosen as being representative of each region of the curve as denoted earlier in the experimental section, and are named after those regions as designated in [15, 273]. The design was then analyzed using ANOVA, and a regression was run to analyze the effects of each factor. The contributions of each of the independent factors are shown in the Pareto plot in Figure 4.9 in terms of percent contribution. It is important to note that since this graph only accounts for the main factors, and does not show the interactions of these factors, each set may not equal unity. The points are

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ordered in terms of their appearance on a P-δ curve in terms of stages from left to right, showing the trends of factor influence as deformation increases.

Figure 4.9 - Percent contribution of main factors to the various sections of the SPT force-displacement curve.

The normalized percentage values show that for the majority of the force-displacement curve the strength coefficient, K, is the dominant factor, and nearly always by a wide margin.

The hardening coefficient, n, is always second most influential, though the effect varies depending on the section of the curve in question, while Young’s modulus, E, is largely ineffectual. The exception to this, of course, is in the primary, elastic portion of the curve, where although E affects the outcome the most, K and n each have an almost equal effect as E. These trends are to be expected, as the power law in which K and n are used to describe material behavior pertains solely to the region of plastic deformation, while the elastic region is primarily

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described by Hooke’s law, in which only E is present. However, this contribution by K and n to the elastic portion of the P-δ curve indicates that this region is not wholly elastic, and so as was suggested in [221] a universal constant using the slope of this region to determine a correlation may not be viable. There is also a small notable effect from the Young’s modulus to the plastification region, and a nearly negligible contribution to the hardening region, which sequentially follow the elastic portion of the force-displacement curve. Logically, a trend can be seen in the diminishing influence of E as displacement progresses and damage transitions from elastic to fully plastic, with a combination of the two occurring somewhere in the plastification region. It is also worth mentioning that for all but the elastic and plastification zones, the interaction between K and n has a larger effect than E. As mentioned, this is not the case for the elastic zone since E is the dominating factor there and the interactions of E-K and E-n are much more relevant than K-n. However, for the plastification zone, even though it is the zone with the second highest effect from E, contribution from the interaction between K and n is nearly equal to that from E alone. Select responses of this study as detailed in Table 4.2 to aid in the visualization of the variations of the responses caused from altering the parameters and the different combination of such. The runs selected represent cases with varying combinations, from all high values to all low values, with varying combinations in between, to give a full spectrum of possible responses by which to gauge the effects of each variable.

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Table 4.2 - Factor levels of select runs for comparison of effects.

Run

Factor 5 13 21 75 76

E (MPa) 200000 200000 200000 240000 160000

K 880 1100 1320 1320 880

n 0.16 0.12 0.08 0.16 0.08

The responses of these runs are visualized in Figure 4.6, which the complete curves and a detailed view of the initial linear portions of such. The responses vary widely from the linear portions to failure, regardless of the values of the factors used. Specifically, the variation in the initial liner response shows its dependence on all three factors, even though the Young’s modulus has been shown to be the dominant factor, as runs 5, 13, and 21 all share the same value for E but display highly varied responses based on the combinations with the designated values of K and n.

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Figure 4.10 - Select curves from the parametric study, and the linear portions of such [292].

The curves in Figure 4.10 represent several varied cases within the study: run 13 correlates with the Level 3 settings in Table 4.1, with all of the median values for IN939 at room temperature, run 75 correlates with the Level 5 settings with the high end values for all factors, and run 76 correlates with the Level 1 settings where all factors are set to the low values.

Comparing these three sets the combined effects of altering the factors can easily be seen, beginning almost in the elastic range. The slope of the elastic region increases with increasing factor values, and a continuous divergence of the three curves occurs as displacement increases.

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Runs 5 and 21 keep the Young’s modulus, E, at the median value combined with a low K and high n or high K and low n, respectively. These two cases represent the extremes in this plot in terms of response. The initial, elastic portions of the curves match very well with the cases correlating with the Level 1 and Level 5 settings. Run 76, in which all factors are set to low, and run 5, which has a low K and high n match in the elastic zone and quickly diverge, with the gap between them staying nearly equal. Note that the combination of low K and high n lowers the force levels of this response throughout, a logical response to a material with a lower strength coefficient and higher hardening constant. On the other end of the spectrum are Run 75, which has all of the factors set to the high values, and Run 21, which combines the median value of E with a high K and a low n. In this case the two elastic portions again match but for a slightly larger displacement level, with a more pronounced difference occurring throughout the lengths of the curves, with the difference escalating slightly with increasing displacement. This more pronounced difference, the highest of which occurs at the peak load, occurs due to the low hardening coefficient allowing for more of a divergence between the two. This small selection of curves shows how prominent the effect of changing these factors and their interactions are on material response. Interesting to note is that the most extreme responses here are elicited by the combinations of the opposing extreme values of K and n, rather than by those on the same end of the tested range.

As the microstructures and material properties of AM materials, such as those from SLM, have been shown to be highly sensitive to both processing parameters and post-processing conditions, the advantages of utilizing the small punch test to evaluate them become immediately evident [128, 142]. As the changes in SLM material microstructures can range from minor to

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major depending on the variations employed in processing or post-processing, tracking them with a very sensitive method is of high importance. As SPT has been shown to be sensitive to minor changes in small volumes, this then becomes a highly attractive option for optimizing processing parameter settings and post-processing routines. The Pareto plot and the resultant curves from the parametric curves show that the model implemented here is in fact sensitive enough to track these changes as well, which is necessary in implementing the model for its intended use.