An approach to interpreting metastable austenitic material sensors for fatigue analysis

The transformation of metastable austenite to martensite under mechanical loading can be harnessed to create a material sensor which records a measure of the load history without the need for electrical energy and can be read out at arbitrary intervals via eddy current probing, thus leading to an ultra-low-power sensing solution. This paper presents possibilities of processing this load amplitude-dependent evolution of martensite content loading for component fatigue analysis. The general method is based on using a theoretical material model typically used in finite element analyses which includes hardening plasticity and phase transformation to precompute tables of stress amplitude or cumulative damage corresponding to different sensor readings which can be stored on a low power processing system onboard the component for energy-efficient lookup. At nominal single amplitude loading, the sensor can be used as a load cycle counter for known loads or as an overload detection device upon divergent martensite content rise. Interpretation of block program loading is less practical due to resolution issues. Under random loading, sequence effects get averaged out; interpretation is easiest with narrow load spectra, but information can be gained from very wide spectra as well. Multiple sensors at different locations can aid interpretation. Uncertainty due to necessary assumptions and untreated influences of temperature and loading rate is discussed.


Introduction
Structural health monitoring techniques facilitate more lightweight structures due to reducible safety margins or cost savings when performing predictive maintenance [1].The goal of the current research project is the development of a sensor-integrating machine element [2], which does not differ in shape and load capacity from a conventional sensor-less machine element.This applicability as a drop-in replacement is intended to provide an easy and cost-effective path to structural health monitoring.In order to create a sensor-integrating machine element in an unchanged shape, the sensor system needs to be wireless.This creates a challenge to design an exceptionally energy-efficient system.
Material sensors based on the transformation of metastable austenite to martensite under mechanical loading provide an auspicious solution to ultra-low-power measuring tasks, as they-unlike e. g. strain gauges-do not require a constant supply of electrical power and can be read out via eddy current probes at arbitrary intervals.As a tradeoff, interpreting the evolution of martensite content as a measure of component fatigue is considerably harder and less precise.
Primarily stainless steels like 1.4301 (AISI 304/X5CrNi18-10) or 1.4305 (AISI 303/X8CrNiS18-9) exhibit good metastable behaviour, where loads above a threshold lead to microstructural changes.At extreme overloads, this sensory utilizable property is used up in one single load cycle, when transformation to full martensite saturation takes place.At lower loads, the martensite content rises gradually over many load cycles.This transformation is only reversible through heat treatment.Martensite and austenite differ in their electromagnetic properties; martensite content can thus be recorded via eddy current probes.
The utilization of these material sensors in sensorintegrating machine elements such as the splined shaft shown in figure 1 is envisioned, where material sensors are prepared via laser heat treatment as spots on a component with defined thresholds.A microsection of such a sensor is shown in figure 2.
Eddy current probes are applied on top of these spots, and an evaluation unit for processing the eddy current measurements is integrated into the component (figure 1, [3]).Electrical energy is provided via energy harvesting, leading to the evaluation unit being mostly shut off to conserve energy, and only periodically waking up to capture the difference in martensite content between many load cycles.The decision when to wake up can be made on the criterion of available energy, the detectability of martensite content changes based on expected load spectrum continuation and eddy current sensor resolution, the severity of possible damage between wake ups when overloaded, or at important events signaled by an external trigger.
This paper discusses approaches to interpreting these leaping measurements for fatigue analysis under different conditions.

Prior methods
Previous applications of metastable material sensors are known from a wide field of applications such as civil, mining or aeronautical engineering [4,[6][7][8][9].They are summarized in table 1.They primarily used the metastable material as a peak load detection device, i. e. having it not transform during regular operation, and interpreting transformation as a sign of overloading beyond the design load or incumbent failure signalized by escalating strain due to growing fractures.Bemont [9] also experimentally derived a calibration curve for a smart rock anchor's assembly preload over the sensing coil's inductance change for ensuring correct fastening.De Backer et al [5] estimated remaining component life of plant equipment by creating calibration curves of magnetic content over load cycles for single-level design load amplitudes from cyclic experiments with material specimens.They also experimentally investigated block program loading and noticed strong load history dependence, but concluded their calibration curve to be a usable 'first approximation' and hypothesized that completely random loads would work better and with a smoother martensite evolution.

Material model
A material model of 1.4301 by Gallée et al originally developed for metal forming applications is employed, which includes the evolution of martensite content [10].The use of a theoretical model allows for the efficient analysis of different sensor evaluation concepts under a wide variety of load spectra.Overall behavior is determined via Pilvin's phase mixture rule [11] with the two phases being austenite and martensite.Austenite and martensite have different yield strengths.When austenite yields, martensite forms according to a modified relation derived by Shin et al [12].
Compared to [10] some changes were necessary, which are described in detail in [13] and briefly summarized here.The primary change is the re-fitting of model constants to lowcycle fatigue experiments by [14], as the original constants cause excessive isotropic hardening of the austenite phase.This hardening would lead to the simulated sensor not transforming any more after a small number of load cycles, as the austenite phase no longer yields.Furthermore, the hydrostatic volume differential of the two phases is implemented, and instead of Hill's yield criterion for anisotropic materials (adequate for sheet metal), von Mises yield criterion is used as a simplification for the heat treated sensor spot with equal yield strengths in both directions parallel to the component surface and no stresses perpendicular and thus no need to treat effects in this direction.Martensite growth mainly depends on specimen temperature and strain rate.These dependencies are left out of this initial analysis for simplicity as many applications work at constant temperature and strain rate.They could be modeled and measured as well but would considerably increase complexity.
Likewise, eddy current probing is not modeled, but instead it is assumed that the output of the material model is in accordance with the reading of the evaluation unit.A separate model could be added for this aspect as well.A calibration curve for correlating the eddy current probing signal and martensite content currently needs to be derived experimentally, as is e. g. done in [15] via x-ray diffraction.

Limitations
Martensite formation depends on hydrostatic stress, as is modeled in Gallée's modification [16] of Shin's model [12] as following: where z M is the martensite volume fraction, z S , β M , N, ϵ 0 and ϵ 1 are material parameters (z S being martensite saturation), ε vp A is the equivalent viscoplastic strain of the austenite phase, ṗA is the derivative of accumulated plastic strain in the austenite phase, and χ(s A ) the stress triaxiality: with a third of the trace of stresses in the austenite phase s A leading to the hydrostatic stress s H which is then related to equivalent stress s eqv .At certain χ, g is negative and thus żM becomes zero.For a constant load component ratio (i.e. outside of e. g. applications in hydraulic systems or general non-synchronous multiaxial loading), hydrostatic stress is uniquely related to the load scaling factor.The dependence of martensite evolution on equivalent stress has substantial consequences though, as it forms a sign-free amount.Equal amounts upper and lower alternating stress respectively therefore lead to different martensite evolution rates.This means that an equal difference in martensite content can be caused by a tensile load of one magnitude, or by a compressive load of another, higher magnitude.The relationship between load amplitude and martensite evolution is thus not an injective one; there is no one-to-one mapping.One thus has to impose an assumed stress ratio R on load amplitude interpretation.
Two approaches are therefore taken: For constant amplitude and block program loading, the load amplitude is interpreted with assumed constant R from martensite content rise, which gives detailed time-resolved information on the component as well as machine operation.For random loads, cumulative damage is statistically related to martensite content as a timeintegrating structural health monitoring concept.
It must be noted that attempting to time-resolvedly record completely random loads by constantly evaluating martensite content would be a far-fetched application, as strain gauges would be much better suited for this task where the material sensor cannot make use of its advantage of being energyefficiently read out in large intervals.
In this paper, alternating stress with R = −1 is exclusively used for load amplitude interpretation.

Evaluation of the model sensor
Whereas in [13] the material model was run on a volume model in a FEM program in the scope of the same research project, it is operated as an isolated element here in order to manage computational cost.
For research purposes, the model is externally loaded with pure tension.Alternatively, any arbitrary application dependent load tensor could be applied.Tensors for complex loading situations could be calculated via FEM.Fringe effects at the boundary of the sensor spot are not treated; homogenous behaviour is assumed in the sensor spot's center.
As it would be energy-inefficient to evaluate the theoretical material model onboard the evaluation unit, material sensor transformation behavior is precomputed for a range of multipliers to the load tensor or a range of random load spectra respectively, and stored in a lookup table for efficient interpolation.
In the case study of a splined shaft, a supplier of general sensing solutions would need to consider different root geometries (within the framework of the applicable geometry standards) as well as a range of ratios of torque and radial load.With the calculations performed in this paper taking days, a comprehensive precomputation to create a full catalogue of drop-in solutions is untenably computationally expensive.At the current state of computational tools, the more realistic approach would be to perform these calculations on a caseby-case basis.
The low yield strength sensor spot is strain controlled by the embedding high yield strength surface layer, as the miniature sensor has only a negligible retroaction on macroscopic component deformation.As the sensor lies on the components' surface and has a very small depth, it experiences no stress orthogonal to the surface.The material model is thus solved for zero stress in the orthogonal component and for prescribed strains derived from elastic loading of the base material in the remaining components.
As in [13], before applying operating loads, the initial state of the sensor is set up by modeling the reduction in volume due to the phase change during heat treatment leading to residual stress.

Constant load amplitude
3.1.1.Counting load cycles.Assuming a nominally constant load amplitude, one could use the material sensor for counting load cycles, e. g. the number of pressurizations of a vessel.For this purpose, a function of martensite content over load cycles needs to be precomputed and stored onboard the evaluation unit.The sensor can then be read out at any time and the number of load cycles can be deduced.
Sensitivity can be tuned by controlling the ratio of the sensors' nominal loading to its initial yield strength R e,s,0 .No amplitudes below approximately R e,s,0 can be recorded.Higher nominal loading leads to higher martensite content evolution per step and thus a clearer signal, but a lower number of recordable load cycles before the sensor is 'used up', i. e. fully transformed back to martensite.
The curves exhibit three distinct sections which are marked with red squares for the 1.30-line in figure 3: Initially, cyclic softening takes place.Then, an approximately linear period of low martensite content rise per load cycle takes place, which is followed by a period of high martensite content rise per load cycle.The latter starts after ca.30% martensite saturation, when the difference in secant stiffness of the sensor spot and the base material has decreased so much that the prescribed strains lead to higher stresses in the sensor spot which in turn lead to a higher martensite evolution rate, accelerating the process.

Overload detection device.
When the load cycle number is known from a different sensor like the motor speed tachometer, the material sensor can be used as an overload detection device (e. g. to issue a warning in case of an overfilled stirring apparatus).
Two modes of operation are possible: Firstly, the sensor can be nominally run at stresses below its triggering threshold R e,s,0 .In this case, any change in martensite content is a sign of overloading, as martensite transformation should not take place under nominal loads.In an alternative mode of operation, the sensor is nominally run at stresses just above its triggering threshold.Martensite content rise between two readings can then be compared to the expected transformation rate, stored as a curve as described in the previous section.These two modes of operation are comparable to open-circuit and closed-circuit principle safety switches.Only the latter ensures the detectability of errors in the measuring system.Whereas the former's lower computational cost is not determinative in the vast majority of cases, it working over a very high number of cycles without using up the sensor is a non-negligible advantage.
Figure 4 shows the expected rise in martensite content over 1 × 10 5 load cycles at which the sensor is read out; starting at 5 × 10 4 overloading takes place.
In order to assess the magnitude of overloading more precisely than by following the basic relation that more martensite equals higher loads, a lookup table for different scenarios can be created as is described in the following section.In the same manner, elevated damage accumulation can be rated in order to judge whether the part needs to be replaced or if usage can be continued after remedying the cause of the period of overloading.

Block program
Block program loading is e. g. relevant for production plants fed in batches.Interpretation is easiest when a wake-up signal is sent to the evaluation unit towards the end of each block, ensuring that the martensite differential was caused by only one single load level.
Sensor response is precomputed for a range of load amplitudes, numbers of load cycles, and starting martensite contents.Cumulative damage is precomputed for different load amplitudes.These data are then used for interpolation onboard the evaluation unit.
3.2.1.Omitted aspects.This choice of variables simplifies two aspects of material behavior: Initial cyclic softening could be monitored by adding a lookup dimension for accumulated plastic strain.As this quantity cannot be measured, it would need to be kept track of internally by using another lookup table for accumulated plastic strain generated from the material model.Initial cyclic softening is therefore neglected in the proposed evaluation scheme, as it only affects the very beginning of component life, whereas the primary interest for fatigue design lies in the subsequent much longer part.The lookup tables are generated for fully softened material.Due to the independence of current load amplitude evaluation from previous sensor interpretation, the initially poor predictions do not lead to error propagation.
A further small error arises from incorrectly modeling the transition between blocks; instead of tracking the kinematic hardening and plastic strain tensor, they are assumed to be zero at the beginning of a block.When the block has a sufficiently high number of load cycles, error in the first load cycle can be neglected.As a single-digit number of load cycles would also be problematic to evaluate due to undetectably small martensite content rise, this aspect is not considered important for practical use.

table resolution convergence.
The intended ultralow power SoC for the demo application sketched in figure 1 has a non-volatile flash storage size of 256 kB.Neglecting space requirements for the main program etc this would hold 65 536 4 byte-sized floats.Alternatively, interpretation of the eddy current signal could happen on a larger offboard main system controller, although this would complicate the plug and play aspect when the machine element is sensor integrated by a different party than the constructor of the overall machine.
At equal discretization of the lookup tables' dimensions with one lookup table with three dimensions (load amplitude looked up for number of load cycles, initial martensite content and martensite content rise) and one one-dimensional lookup table for damage per load cycle looked up for load amplitude, the SoC storage would hold a maximum of 40 table columns per dimension.Adding two further dimensions for temperature and speed would reduce the maximum number of columns to 9.
Table resolution convergence for linear and spline interpolation is plotted for the load amplitude lookup table in figure 5. Interpreted load amplitudes at points that are not sampling points of the underlying dataset are compared to explicitly calculated results.Convergence orders p are about 1.The computationally more expensive spline interpolation method leads to approximately one order of magnitude better results.
An acceptable error can be drawn from commonly used numbers of ranges in cycle counting methods, e. g. 64-100 [17], which would be matched at 5 columns per dimension.When additional sources of error apply, this resolution needs to be higher.
The relationship between the error in inferred cumulative damage and stress amplitude can be established by rearranging the equations used for damage calculation from [18].For alternating stress amplitudes which can be sustained for more than 103 load cycles 3 , relative error ρ caused by a ratio s of stresses s = (Young's modulus E, cyclic hardening exponent n ′ and cyclic hardening coefficient K ′ according to [18]).As c is very small, this relation can be simplified to 1% error in interpreted stress would lead to −4.9% error in calculated cumulative damage.

Eddy current sensor resolution.
When the evaluation depends on the difference between two readings, and not an absolute reading, eddy current sensor resolution is especially significant, as two sensor level readings separated by less than measurement uncertainty do not provide precise information but only imply loads below a certain threshold.With standard lab equipment, i. e. with a better measuring chain than currently available for machine element integration, martensite contents can reliably be absolutely measured via eddy current testing with percentage-point resolution [15].Percentage-point resolution determines that at best, 100 blocks can be resolved until the sensor is used up.In a real-world application, the only practical scenario would be to reset the sensor at machine maintenance intervals and thus adjust sensor sensitivity accordingly.
It is conceivable to reset 'used up' sensors in the same way they were initially manufactured.Research would be necessary to determine whether the laser heat treatment could also act as a recovery heat treatment [19], or whether the repeated large amount of plastic yielding in the sensor would lead to it being the weak spot of the component.Alternatively, different spots could be used sequentially for sensors, e. g. each tooth space.Glued on eddy current probes would be lost in a successive laser heat treatment and need to be reapplied afterwards; for such applications a force-fit sensor application would be preferable to a material-fitting method.In the case of block program loading, it might be preferable to use easily replaceable bolted-on metal strips as demonstrated in [4] instead of a sensor-integrating approach.Outside of this section on block program loading, it is highly preferable to carefully tune the sensor sensitivity to match expected component life and sensor life.
Figure 6 charts the necessary number of load cycles to induce a martensite level change of 1%.This number grows with increasing starting martensite content.At high martensite content, low load amplitudes are not resolvable any more.For a worst-case fatigue analysis, the minimum resolvable load amplitude at this martensite content therefore needs to be employed for interpretation.This equally applies to blocks that are too short to reliably analyze.Block 1 overestimates, whereas block 2 underestimates the load a bit.The load amplitude of block 3 is too low to be correctly resolved by an actual eddy current sensor; thus it is set to the lowest resolvable amplitude.When treating the simulation results directly with no regard to eddy current sensor resolution, the direct interpolation from the lookup table actually presents no issue-the result is 1.08 R e,s,0 at an amplitude of 1.10 R e,s,0 in block 3.

Random load
In order to asses random loads, a statistical relationship between martensite content and cumulative damage needs to be established.This relationship is found via the Monte Carlo method, i. e. by executing the material model simulation many times with different random load sequences and surveying the output.This was chosen as the most straightforward method.Gains in accuracy of the resulting probability distribution function at fewer load sequence evaluations can be expected from variance reduction methods like stratified or Latin Hypercube sampling [20].
For generating random load sequences, maximum lower S l and upper nominal stresses S u and load spectrum shape exponents ν u,l are chosen at random from a prescribed range with uniform probability.200 load sequences are used per sub-analysis.Maximum load sequence length is 1 × 10 6 , but sequences are not evaluated further when reaching sensor saturation or cumulative damage amounting to 1.
Cumulative damage is calculated via the 'FKM non-linear' [18] algorithm with damage parameter P RAM .Each load cycle leads to an increase in damage level by the reciprocal of the number of cycles the component could withstand at the parameters of this specific load cycle.At a cumulative damage level of 1, component failure is expected.
Fatigue material data are taken from [14] for 1.4301 sheet metal specimens which were pre-strained before cyclic testing in order to achieve martensite saturation.This is then taken as a stand-in for the component surface layer.
The same strains are initially used for the sensor material model as well as fatigue assessment, which is equivalent to positioning the material sensor near the critical spot of the component.In practice, the sensor should be placed away from the critical spot, as otherwise the component would need to be derated by the notch and mean stress effect of the sensor [13].A conversion factor can then be established between the critical spot and the sensor strain.Fatigue assessment is accordingly carried out using this multiplier for input strain and leads to higher cumulative damage of the critical spot.Maximum cumulative damage in figure 8(a) does not reach 1; this would be the case for a sufficiently high distance between the sensor and the critical spot.Subsequently, different sensor placements are investigated.
The probability of encountering maximum cumulative damage below certain values is calculated over the martensite content range.For two-variable analyses, cumulative damages at discrete martensite levels are used to build distribution functions.For higher dimensional analyses, the plot space is divided into bins and occurrences of curves crossing them are counted.
Limit curves of e. g. 90% probability are derived from these distributions for onboard evaluation of the damage state of the component.The different martensite evolution and damage accumulation at tensile and compressive loads as well as possible sequence effects get averaged out, as can be gathered from the proximity of the curves.
The curve shape with a high gradient at the beginning followed by a levelling off starting from a sensor martensite saturation level of approx.0.3 mirrors figure 3 and the accompanying explanation in section 3.1.1,where there is a rapid increase in martensite content over a lower number of damaging load cycles when the sensor spot has hardened so much that the prescribed strains lead to higher stresses and thus a higher transformation rate in the sensor spot.
Figure 8(b) shows the probability distribution derived from the curve array in figure 8(a).At a low sensor reading, cumulative damage is low with high probability.At full sensor saturation, cumulative damage is higher than approximately 0.42 with 90% probability (red colored).The probability of cumulative damage being lower than approx.0.39 is very low (blue colored), and the remaining colors mark the probability of cumulative damage lying in a slim band between those two levels.

Wide load spectrum.
When assumptions on the load spectrum are very relaxed, the array of curves fans out over much of the plot space (figure 9).At high sensor levels, possible cumulative damage values are almost equally probable, providing little information benefit to the machine operator (figure 9(b)).At low sensor levels, the limit curve is usable for narrowing possible cumulative damage, although it massively overestimates damage for some load spectra.

Sensor placement.
A reduction in sensor response to low load amplitudes can be achieved by sensor placement.In figure 10 the same load spectrum as in the previous paragraph   is used for a sensor placement which leads to a reduction in sensor strain to a factor of 75% of the load used for fatigue assessment.The horizontal part of the martensite curve does not occur, as full component damage is reached before sensor saturation.Uncertainty evidenced by the spread of the curve array narrows.Bins with too few crossings to be statistically meaningful are ignored (i.e. shown as white).As discussed in section 3.1.2,the load cycle number is readily available in power transmission applications from the Non-rotating components, e. g. chassis parts or building structures do not provide information as easily.Whereas the absolute martensite level could also be read out with a handset device at planned service intervals, an onboard evaluation unit as described in the introduction is crucial for monitoring the sensor evolution rate.

Multiple
The use of multiple sensors is another approach to improve differentiability between many low to medium and few high amplitude load cycles, which can lead to the same sensor reading, but have significantly different damaging effects.The second and subsequent sensors are placed at spots of lower nominal stress than the first one and thus capture fewer low loads or inversely only trigger at high loads.
Figure 12(a) shows cumulative damage by line color over two sensor readings with sensor 1 being the analysis from figure 9 and sensor 2 the one from figure 10.The number of load sequences is doubled to 400 in order to provide enough curves crossing individual bins to be statistically meaningful.
Following the increasing number of dimensions, a surface can be found that delimits cumulative damage at sensor martensite levels with e. g. 90% probability (figure 12(b)).Spectra with few high and highly damaging load cycles cluster at the left edge of the probability surface, where the more sensitive sensor 1 has low saturation whereas the less sensitive sensor 2 has high saturation.This information gain facilitates the discriminability of spectra leading to maximum damage levels differing by 0.05-0.1 whilst featuring a narrower confidence interval.

Uncertainty
Measurement uncertainty primarily stems from the insufficiently quantified accuracy of the material model.
Metastable martensite transformation is batch-dependent due its high sensitivity to chemical composition [9].Previous research does not contain a sufficiently large number of batches in order to be statistically analyzable.
When batch spread is too high for a certain application, two approaches are possible.Firstly, a standardized specimen could be extracted from each batch and thoroughly characterized in benchmark transformation experiments.Another approach would be to apply a few test loads to each component and try to correct martensite formation behaviour from this calibration experiment.One could then select from a database of precomputed results for different material compositions which table to store onboard the component.
Further research on this topic is necessary.Likewise, the reliability of the employed theoretical material model has not yet been experimentally proven for a high number of load cycles.Uncertainty due to this fact cannot be delimited at the current state of research.
Martensite transformation kinetics depend on temperature and loading rate.Temperature sensitivity in the Tomita and Iwamoto model [21] can be computed to be a linear factor in the martensite evolution equation of −1.45 • C −1 .Whereas many applications reach an equilibrium temperature at steady state operation and some applications in process industry even are temperature controlled, double-or triple-digit temperature swings can occur in equipment operated outdoors or not at a steady state.The error in martensite evolution would be intolerable in this case.Error during machine start up as a short part of component life can potentially be neglected.
Strain rate sensitivity cannot be given as a simple factor, as it depends on the plastic strain rate in the austenite phase (see equation ( 2)).Temperature and strain rate can be recorded by a thermometer and a tachometer (in power transmission applications) respectively.
The evaluation of block program loading highly depends on eddy current sensing resolution, as its input is the difference in martensite content between two blocks.At low eddy current sensor resolution, only blocks with a high number of load cycles can be evaluated.The absence of critical or unexpected loads and thus unthreatened structural integrity can be inferred with low resolution.As the single amplitude and random load evaluation techniques do not depend on the difference, but on total martensite content, they are much less sensitive to eddy current sensor resolution.

Conclusion
This study indicates a high of material for fatigue assessment.Due to uncertainty from not yet quantified effects, these theoretical findings possibly are best case results.Some assumptions concerning the occurring load need to be made in order overcome ambiguity.While this presents a hindrance to applications where high fidelity sensing solutions like strain gauges would be more appropriate, a chance of misinterpretation can be accepted in the case of merely economic consequences when the alternative would be to have no sensor data at all.
Using bad assumptions, it is possible to underestimate load amplitudes or cumulative damage.One therefore has to estimate possible discrepancies and investigate worst-case scenarios.For random load evaluation, the Monte Carlo load spectra can be biased towards the design assumptions but also include the less probable cases.
Single amplitude tracking-as is already practiced in some applications as cited in section 1.1-is easiest.Block program loading evaluation has a similarly high theoretical accuracy.Statistics based evaluation of random loads is possible and works better with tighter assumptions on the load spectrum and multiple sensors.Multiple sensors can be differentiated by positioning them at locations of different nominal stresses.Considering the sensor evolution over load cycles yields appreciable information gain.
While the theoretical model works for a multitude of comparatively controlled environments like production plants, it would need to be extended to include further effects (temperature, loading rate) for a broader range of conditions e. g. found in vehicular applications.Further sensor data (temperature, acceleration etc), possibly transmitted from additional networked sensor-integrated machine elements, as well as the supplementary utilisation of the Villari effect (inverse magnetostrictive effect, change of magnetic susceptibility with mechanical stress) [22], which can be evaluated from the established measurement setup [23], can complement interpretation.
Interpolating precomputed data is a suitably low-power evaluation method for sensor-integrating machine elements with energy harvesters.Instead of pure simulation, a hybrid dataset enriched with experimental or even an entirely experimental dataset is imaginable, although the required large number of curves to cover the spread for completely random loads possibly has prohibitive cost.
The next step to improve the technological readiness of the proposed methods would be experimental validation.Prior to this, research on material sensor manufacturing as well as the design of the evaluation unit and further material characterizations are planned.

Data availability statement
The data cannot be made publicly available upon publication because the cost of preparing, depositing and hosting the data would be prohibitive within the terms of this research project.The data that support the findings of this study are available upon reasonable request from the authors.

Figure 1 .
Figure 1.Concept of a component-integrated material sensor on a splined shaft.Reproduced with permission from [3].

Figure 2 .
Figure 2. Beraha II-etched microsection of a lasered lenticular sensor spot in the shot peened martensitic surface layer of a material sample.Deformation structure, including martensite, is black, austenite gold-colored.The mounting compound for metallography is dark greyish blue colored.

Figure 3 .
Figure 3. Martensite content increase over load cycles for different multiples of R e,s,0 , printed on the corresponding curve.The transition between three distinct sections is marked by red squares for the 1.30 line.

Figure 4 .
Figure 4. Overload after 5 × 10 4 load cycles.The horizontal curve shape at less than 1 × 10 4 load cycles is due to initial cyclic softening.The regular load is 1.1 R e,s,0 .

Figure 5 .
Figure 5. Lookup table resolution convergence for linear and spline interpolation.

Figure 6 .
Figure 6.Number of load cycles to cause 1% difference in martensite level at different load amplitudes.Starting martensite content printed on the corresponding curve.

Figure 7
presents a brief example for this evaluation technique.Interpreted values are marked with grey dots.

Figure 7 .
Figure 7. Interpretation of block program loading.
Figure 8(a)  shows sensor readings over cumulative damage for one single load spectrum (parameters given in figure caption) with randomized order.

Figure 10 .
Figure10.Curve array and probability distribution for the same load spectrum as in figure9with a different sensor placement leading to a reduction in the load amplitude experienced by the sensor to 75%.

Figure 11 .
Figure 11.Curve array and probability distribution for the same load spectrum as in figure 9 considering the load cycle number.

Figure 12 .
Figure 12.Cumulative damage over two sensor martensite saturation levels.Sensor 1 experiences 100%, sensor 2 75% of the load amplitude used for damage calculation.Load spectrum data are given in the caption of figure 9.
Figure 11(a) plots sensor level (sensor strain again at 100% of the load used for fatigue assessment) over load cycle numbers and indicates cumulative damage by line color.High sensor levels at low load cycles, i. e. a high transformation rate correlate with higher cumulative damage levels.A surface of maximum cumulative damage with 90% generated via binning is plotted in figure 11(b).