Introduction

Polycarbonate (PC) is a polymer having high temperature resistance, high ductility and high mechanical and impact strengths when compared to other engineering plastics. These properties make polycarbonate the normal choice for engineering applications in the civil, automotive and aeronautical industries, as well as the frequent choice for relatively high stress applications. Considering that cyclic loading is bound to happen in most of these applications, studying the fatigue properties of such a material is of utmost importance. Although it presents high transparency in the visible spectrum, polycarbonate is virtually opaque to infrared radiation, absorbing and emitting most of it. This property makes polycarbonate a good choice for infrared applications.

The present paper reports results from a line of investigation [1, 2] that puts together the study of fatigue crack initiation and propagation in annealed polycarbonate with temperature measurements taken with a micro-bolometer infrared camera. The thermographic temperature measurements were used to directly infer the fatigue response (crack initiation and crack propagation) of specific tested specimens, as well as to determine fracture mechanics stress intensity factors (SIF) of the propagating cracks using the Thermoelasticity Stress Analysis (TSA) technique [3,4,5,6,7,8,9,10,11,12,13]. Two resultant outcomes will be presented and discussed herein: fatigue propagation of a crack with length (a) per loading cycle (da/dN), and SIF range (ΔK) determination. Combining these two outcomes enables determining the so-called Paris’ crack propagation exponential law (da/dN = C. ΔKp), C and p resulting from the exponential fitting using the least square method, in annealed polycarbonate.

Fatigue crack initiation and propagation in specimens of annealed polycarbonate were studied in [1], where a survey of the literature on both subjects was presented. References [14,15,16,17,18,19] summarize the extraction of a number of relevant findings. In what concerns crack propagation in polycarbonate, Hertzberg et al. [14] concluded that the use of classical fracture mechanics concepts, such as the stress intensity factor (SIF) and Paris’ Law, was appropriate for polycarbonate. Corroborated by Hertzberg et al. [16], Gerberich et al. [15] investigated the effects of temperature on crack growth in polycarbonate specimens. Ward et al. [17] studied the effect of specimen thickness on crack growth rates in polycarbonate specimens. Pruitt et al. [18] also explored the effects of specimen thickness, as well as the effects of stress or load ratio (R = minimum stress (σmin) /maximum stress (σmax)) in one cycle of fatigue behavior in polycarbonate. Using Crack Layer theory to successfully predict crack growth rates, Moet et al. [19] investigated the effect of yielding mechanisms on fatigue crack growth in polycarbonate. However, a search of the literature revealed that no study has yet applied infrared thermography to fatigue in polycarbonate. Therefore, this paper makes full use of thermographic techniques to explore fatigue behavior in polycarbonate.

Thermoelasticity (TSA) is a non-contact, full-field stress measurement technique based on the thermoelastic effect [3]. By using infrared thermography, it is possible to measure the temperature distribution on the surface of a cyclically deforming body. This distribution can then be correlated with the stress range acting at each point. Eq. 1 describes how temperature measurements are converted into stress data in practical TSA applications.

$$ \Delta \left({\sigma}_1+{\sigma}_2\right)=A\ S $$
(1)

where Δσ1 and Δσ2 are the principal stress ranges, S is the thermoelastic signal from the TSA software (proportional to the temperature variation caused by the loading), and A is the thermoelastic coefficient, related to the material’s properties and to the thermographic setup.

Thermography, especially TSA, has been used on occasion to analyze the stress distribution around fatigue cracks in order to determine stress intensity factors. Stanley and Chan [5] used the first two terms of Westergaard’s stress function to fit the thermoelastic data around cracks in mode I and II. Stanley and Dulieu-Smith [6] based their approach on the cardioid format of the isopachic fringes around a crack tip. Lesniak et al. [7] used Williams’ asymptotic stress field equations to fit the thermal data by using the least squares method (LSM), whereas Lin et al. [8] approached the problem using the J-integral formulation in the data fitting process. Tomlinson et al. [9] developed a technique based on Muskhelishvili’s stress field equations and a multipoint-over-deterministic method (MPODM). Díaz et al. [10, 11] proposed a methodology for finding the crack tip position by looking at the thermoelastic data, and then fitting the data using a non-linear approach in order to measure SIFs of propagating cracks. Marsavina et al. [12] used TSA to investigate crack closure during fatigue, and Tomlinson and Olden [13] presented a broad review of the use of TSA in fracture mechanics and fatigue cracks. The present work uses an algorithm based on Westergaard’s solution for the stress field near the crack tip, in combination with experimental results from TSA, in order to determine stress intensity factors. Finite element simulations were conducted to calculate theoretical SIFs as a means for comparing and verifying the results.

The study outlined in the following sections consisted of two main approaches for determining the Paris’ law crack propagation in annealed polycarbonate specimens. The first approach used the SIF algorithm to find the ΔK and da/dN values (since one of the outputs of the non-linear algorithm is the crack tip position). In this procedure, TSA is the only tool used and therefore it can be implemented to automatically determine the crack propagation law. Although polycarbonate was used, this approach can also be applied to other materials that follow Paris’ law. The second approach used the thermographic temperature measurements at the propagating crack tip to innovatively find the value of da/dN and then combine these results with SIFs values. In this approach these SIF values were determined via TSA and also via the Finite Element Method (FEM) for comparison purposes. The next section summarizes experimental details regarding specimen preparation, material calibration, and infrared camera. Then, the following section presents a thorough discussion of the SIF, crack-tip localization and da/dN determination algorithms. The section after that shows the empirical relationship between crack propagation and temperature measurements. It is based on an extension of the Thermographic or Risitano’s method [20, 21], used to measure the fatigue limit and stress vs the number of fatigue curve cycles (S-N). Then a section on Results and Discussion follows, comparing crack propagation according to Paris’ law using the two approaches with results found in the literature. The paper’s conclusions are presented in the last section.

Experimental Procedures

Thermoelastic Coefficient Calibration

In order to obtain the value of A for a specific material and setup, a calibration experiment must be conducted. There are different approaches to this process [4]. In the present work, the calibration test consisted of a dog-bone specimen, with known dimensions (machined from the same polycarbonate sheet as the cracked specimens and having the same surface texture), being cyclically loaded in uniaxial tension, as shown in Fig. 1. Then, the resultant stress ranges and measured thermoelastic signals were used to calculate the value of A. For the ambient temperature, material and setup used here, the coefficient was determined to be A = 1.12 MPa/Cam units (“Cam unit” is used as the unit of the thermoelastic signal from the TSA software).

Fig. 1
figure 1

Cyclic pneumatic machine and calibration tensile specimen (dimensions in mm); specimen thickness = 3.9 mm

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Specimen Preparation

SAE keyhole specimens were machined from a single polycarbonate sheet (thickness = 3.9 mm), all taken at the same orientation. Some residual stresses from manufacturing the sheet, as well as from the machining processes, were present [1]. In order to eliminate the influence from these residual stresses, a specific heat treatment was applied to the specimens, heating them up to 160 °C in a span of 4 h. The temperature was then kept constant for 6 h, after which the specimens were cooled down to the ambient temperature in another 4 h. A photoelastic analysis (using a transmission polariscope) confirmed that birefringence caused by the residual stresses was eliminated after the heat treatment [1].

Specimen Pre-Cracking

In order to nucleate a crack and propagate it to an initial crack length (a), the specimens were cyclically loaded using the pneumatic machine shown in Fig. 1. Loading was controlled using a load cell (5000 N) connected in series with the specimen and the pneumatic muscle, which sent the signal to the servo-valve. The frequency used for all tests was 5 Hz, and the magnitude was manually controlled using a valve that regulated the air pressure entering the muscle (load amplitude of 225 N ± 1 N and R = (Min load/Max load) = 0.3 for pre-cracking).

The specimen dimensions after the pre-cracking process are shown in Fig. 2.

Fig. 2
figure 2

Key-hole fatigue specimen (dimensions in mm)

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Infrared Imaging

The infrared camera used for the experiment was the FLIR A655sc. It has a focal plane array detector composed of 640 × 480 uncooled micro-bolometers and can record full-frame data at up to 50 Hz. It has a spatial resolution of 17 μm and a sensitivity of 30 mK. Micro-bolometer infrared cameras are a less expensive alternative to the cooled quantum detectors used by most other authors [22].

There were two software programs used in combination with the infrared camera in order to accomplish the measurements required for the study. When making direct temperature measurements, the ResearchIR software program from FLIR was used.

For the TSA tests, DeltaTherm2, a software program from Stress Photonics Inc., was used to convert the temperature data into TSA data. Typically, the TSA technique requires a lock-in process, which correlates the ΔT data with a reference signal, usually obtained from the function generator that feeds the cyclic actuator. DeltaTherm2 automates this process and takes the reference signal directly from the ΔT measurements, making the whole TSA test much simpler to set up and, therefore, making possible the use of less expensive cameras that do not have the lock-in capability.

For most thermography applications, including TSA, it is advisable to prepare the specimen in a way that maximizes infrared emissivity. This is usually accomplished by painting the surface with matt black paint. Nonetheless, polycarbonate has a naturally high infrared emissivity (ε = 0.9), which made it a very suitable material for this study, since it is transparent to visible light while being “opaque” to the infrared spectrum [23].

Stress Intensity Factor Algorithm

TSA results can be understood as a complex variable, consisting of a magnitude, which is proportional to the temperature variation caused by the thermoelastic effect, and a phase angle, which is related to the signal of the stresses acting at a particular point. Figure 3 shows the typical magnitude and phase results obtained by cyclically loading the polycarbonate specimens. Figure 3(a) shows the magnitude map, while Fig. 3(b) shows the phase angle map around a crack with length a = 10 mm. It is obvious that the stresses concentrated at the crack tip cause a much higher temperature variation there.

Fig. 3
figure 3

(a) Magnitude (cam units) and (b) Phase angle map (degrees) around a crack with length a = 10 mm

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Figure 4 shows a flowchart outlining how the algorithm for stress intensity factor calculation works, including its inputs and outputs. From the TSA magnitude and phase data, it is possible to generate corrected TSA data (with the compressive parts of the stresses converted to negative magnitude):

$$ \mathrm{S}=\left\{\begin{array}{c}\hfill Mag\ if\ \left| phase\right|<{45}^o\hfill \\ {}\hfill - Mag\ if\ \left| phase\right|>{45}^o\hfill \end{array}\right. $$
(2)

where S is the corrected TSA data, Mag is the magnitude data value in “camera units,” and |phase| is the absolute value of the phase angle, in degrees.

Fig. 4
figure 4

SIF algorithm flowchart

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One other input required for the procedure is the mathematical model that is being used. In this case, a Westergaard stress function (We) was used to describe the stress field around the crack tip

$$ We\left(x,y\right)={\sigma}_1+{\sigma}_2=2\left[\mathit{\operatorname{Re}}(Z)+\mathit{\operatorname{Re}}(Z)\right] $$
(3)

with

$$ {\displaystyle \begin{array}{c}\hfill Z={\sum}_{n=0}^N\left[{A}_n{z}^{n-0.5}\right]\hfill \\ {}\hfill Y={\sum}_{m=0}^M{B}_m{z}^m\hfill \end{array}} $$
(4)

where N and M determine the number of coefficients to be used (usually 2 to 4 in the present paper), An and Bm are the data fitting coefficients, and z is the complex variable of the point coordinates:

$$ z=\left(x-{x}_o\right)+i\left(y-{y}_o\right) $$
(5)

where x and y are the horizontal and vertical coordinates of a point (placing the origin at the estimated crack tip location with the x axis pointing in the direction of the crack growth and the y axis pointing up), and i = √-1. Parameters xo and yo represent the unknown crack tip positioning errors in the horizontal and vertical directions. Next, there are two approaches, either xo = yo = 0, (i.e. the crack tip is determined assuming no error, making the problem linear), or the error values must be solved for during the optimization process, making the problem non-linear. Both approaches are used in this paper.

The final required inputs are the spatial resolution (SR) and the TSA calibration factor (A). The spatial resolution is easily determined via a known dimension in the specimen,

$$ SR=\raisebox{1ex}{$W$}\!\left/ \!\raisebox{-1ex}{$p$}\right. $$
(6)

where SR is the spatial resolution in mm/pixel, W is a known dimension of the specimen in mm (i.e. width), and p is the number of pixels the dimension measures on the thermographic image. The calibration factor (A) is determined as described above, using a calibration experiment. Nominal pixel size for the TSA experiments was 0.2 mm.

Stress Intensity Factor Algorithm – Crack Tip Estimation

This work uses a specially improved, automated version of the methodology proposed in [10] to estimate the position of the crack tip. First, the image needs to be positioned so that the crack is horizontal. Then, using the TSA magnitude map, the vertical position of the crack tip (Yo’) is estimated by plotting y vs 1/S2 curves for each vertical line in the image (perpendicular to the crack, as shown by the vertical arrow in Fig. 3(a). The resulting graphs, for the lines in front of the crack tip, have a clear 1/S2 minimum value. The mean value of y where this minimum value occurs for all the vertical lines is equal to Yo’. Figure 5(a) shows a typical y vs 1/S2 plot.

Fig. 5
figure 5

(a) Typical y vs 1/S2 plot. (b) Typical phase vs x plot

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The horizontal position of the crack tip (Xo’) is estimated using what is called the phase shift. Due to the very short plasticity area near the crack tip and the subsequent opened crack, the adiabatic conditions required by TSA are lost. This can be seen as an abrupt change in the phase angle directly in front of the crack tip. In the horizontal phase vs x curve at Yo’ (along the direction indicated by the horizontal arrow in Fig. 3(b), the position where this abrupt change occurs is equal to Xo’. Figure 5(b) shows a typical phase vs x plot.

Having determined the values of Xo’ and Yo’, the coordinates assignment step is simple: each point of the image gets assigned to an x and a y value, which are the horizontal and vertical distances from the estimated crack tip, respectively.

Stress Intensity Factor Algorithm – Model Validity Limits Verification

The points used for the data fitting process must be in accordance with the mathematical model. In [11], the authors concluded that the model predicts a linear behavior of 1/S2 with respect to the distance from the crack tip. They proposed that only the points of Fig. 5(a) that presented a linear behavior should be used in the data fitting process.

The methodology used in the present work uses the same argument, but instead of using the vertical line plots, it uses plots of 1/S2 vs r, where r is the radial distance from the estimated crack tip. With r and θ being the polar coordinates centered at Xo’ and Yo’, and θ = 0 being the horizontal line ahead of the crack, 1/S2 vs r curves are plotted for each 1 degree increment of θ between -120o and +120o, totaling 241 curve plots. Figure 6 shows an example of one of these plots.

Fig. 6
figure 6

Typical 1/S2 vs r plot

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For each of the 1/S2 vs r plots, rmin and rmax values were determined using two margins of tolerance of non-linearity, which were manually adjusted to fit the data. The authors verified that the rmin value for each analyzed 1/S2 vs r plot was influenced by the loading level and by the shape of the non-adiabatic region, which for each r changed with the θ angle. Also, it was seen that the rmax margin was related to the overall data noise levels. No attempts were made to automate the optimization process to define these margins, although mean values of rmin and rmax were calculated and named Rmin and Rmax, respectively. Hence, data points were collected inside the region defined by 2xRmin and Rmax. This selected data point location process helped to gather only linear data regions for all 241 θ-angle 1/S2 vs r plots. Using this selection process the uncertainty related to locating the crack-tip position (X’o, Y’o) ended up being ±3 pixels or less if values of 1.8× Rmin and 0.9xRmax were used instead of 2xRmin and Rmax. It will be shown in the next sections that the major variable to influence the SIF range determination and the crack tip location is the crack length: shorter crack lengths generating larger errors, and longer crack lengths generating smaller errors.

Stress Intensity Factor Algorithm – Error Function Construction and Minimization

After determining which data points can be fitted by the model, an error function (e) is constructed:

$$ e=\sum_{Data points}{\left[S- We\left(x,y\right)\right]}^2 $$
(7)

where e is the error function to be minimized, S is the TSA signal of the point, and We(x,y) is defined by eq. 3.

For the linear approach, the minimization process uses a Least Squares method (LSM) algorithm, and for the non-linear approach, it uses a Downhill Simplex method (Nelder-Mead) [24]. The main advantage of this method is that it does not require differentiation of the data, making it more robust for high gradient problems, such as the crack-tip stress field.

Stress Intensity Factor Algorithm – SIF Calculation and Real Crack Tip Location

After having calculated all the coefficients, the SIF can be calculated as:

$$ \varDelta {K}_1={A}_0A\sqrt{2\pi SR} $$
(8)

where ΔKI is the mode I SIF range in MPa√m, A0 is the first coefficient as described in eq. 4, A is the TSA calibration factor, and SR is the spatial resolution.

In the non-linear approach, the values of xo and yo, as seen in eq. 5, are determined, as well as the real location of the crack tip:

$$ {\displaystyle \begin{array}{c}\hfill {X}_o={X}_o^{\hbox{'}}-{x}_o\hfill \\ {}\hfill {Y}_o={Y}_o^{\hbox{'}}-{y}_o\hfill \end{array}} $$
(9)

where Xo and Yo give the calculated actual position of the crack tip.

Stress Intensity Factor Algorithm – Validation

First, the algorithm is tested using an emulation of TSA data. As an example, Fig. 7(a) shows the numerically generated data, with the crack tip located at (−3,-1) pixels. Figure 7(b) shows the fitted model after a manual crack tip estimation at (−6,-1) pixels.

Fig. 7
figure 7

(a) Numerically generated TSA data. (b) Calculated model. Crack position (x,y) given in pixels

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For the linear approach, where the real crack tip location error is not determined, the result of ΔKI for short crack lengths was heavily dependent on the crack tip horizontal position estimation (Xo’). As an example, for cracks as short as 3 mm errors up to 10% in ΔKI occurred for inaccuracies of 1 to 3 pixels in the estimation. For the vertical direction, the dependence was much smaller - a 1 pixel inaccuracy in Yo’ resulted in an error of 0.4% for ΔKI.

The non-linear approach, on the other hand, was able to determine correctly the crack-tip location even for initial crack-tip estimation coordinate (Xo’ and Yo’) errors of 20+ pixels in the horizontal and vertical directions; consequently, it gave on-point results for ΔKI.

After the numerical validation, the SIF determination algorithms were put to the test on actual cracks of different lengths. Table 1 and Fig. 8 show the results obtained for ΔKI of 8 different crack lengths, using the linear and the non-linear versions of the algorithm as well as results for DIC measurements and FEM simulations [1].

Table 1 SIF results for eight different crack lengths
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Fig. 8
figure 8

SIF results for eight different crack lengths

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The DIC measurements were conducted on the same specimen used for the TSA measurements. For each of the 8 different crack lengths the DIC ΔKI finding algorithm used the maximum load applied in the TSA experiments. The VIC-Snap and VIC-3D software programs from Correlated Solutions Inc. (Columbia, SC, USA) were used. DIC data analysis was carried out with a pixel size of 12 μm, with subset, step size and strain window equal to 35, 7 and 7 pixels, respectively. Algorithms similar to those described earlier for the TSA experiments were used to determine crack tip positions and ΔKI stress intensity factor ranges.

The finite element method (FEM) employed the ANSYS Workbench 15.0 software program. First, before the introduction of a crack, stress concentration factors Kt were determined along the thickness of the keyhole specimen using a 3D model. The 3D model was constructed with tetrahedral solid element SOLID186, which performs better when used in regular meshes, most of them concentrated near the notch root. Fourteen layers of elements were used along the specimen thickness. Due to the model symmetry the linear finite element analysis was run using ¼ of the model. The stress concentration was defined by the ratio of maximum stress and nominal (net) stress. The maximum stress occurred at the keyhole notch root. The stress concentration factor varied monotonically from 3.02 (specimen surface) to 3.17 (point located at half thickness). This variation was expected due to thickness and differences caused by plane stress and plane strain constraint effects. It is important to note that this variation has to be taken into consideration when comparisons are made between results obtained from different experimental and numerical methods such as TSA, DIC, 2D–FEM and 3D–FEM.

The FEM evaluation of ΔKI for 8 different crack lengths was performed using the ANSYS Workbench fracture tool [1]. The pre-meshed cracked option was used to model the crack, and in this case full model geometry was used. A maximum load of 100 N was used in all analyses. A total of 44,864 SOLID186 type elements were used. Sixteen layers were used along the model’s thickness. The thickness effect made the SIF vary along the thickness just as it did with the Kt variation. The ratio between maximum and minimum KI values was 15% for the specimen with crack length a = 10 mm. Results for the different crack lengths presented in the last column of Table 1 to be compared with the TSA and DIC results correspond to the surface FEM 3D determinations of ΔKI/P.

The results show that the SIF measurements are very accurate for the non-linear TSA algorithm. Both DIC and TSA (non-linear) results approach the FEM result for longer cracks, because both these calculations are able to locate the real crack tip through optimization, and with longer cracks this process becomes easier and more accurate. Another possible reason why the algorithm gives better results for longer cracks has to do with the fact that the actual crack front is parabolic and the relative error in the crack position diminishes with increasing crack length. The inaccuracy found with the linear TSA algorithm can be attributed to an error in the crack-tip position estimation. Because of the better capability of locating the real crack tip position, the non-linear TSA method is best suited for investigating crack propagation.

Crack Tip Location and Crack Propagation Rate

Before using the SIF algorithm to investigate fatigue crack propagation, it is important to assess its accuracy when locating the crack tip position. This was done by comparing the crack position obtained from the algorithm with the crack size measured using an optical camera. Care was taken so that the camera would take photos simultaneously with the TSA data collection. Figure 9 and Table 2 show the photos and the lengths of a propagating crack. In Fig. 9 the visually-determined and the algorithm-calculated crack tip positions are shown in the images. Table 2 presents measured and algorithm crack lengths as well their differences in milimeters and in percentages. Values in milimeters are presented to call attention to the fact that small differences in length can generate large percentage errors when crack lengths are short. Table 2 also presents columns with crack tip positions given in pixels with the objective of showing the small error (1 to 5 pixels) of crack tip location if measured in terms of pixels.

Fig. 9
figure 9

Photos of the propagating crack

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Table 2 Measured crack lengths and algorithm error
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The results show that the algorithm is sufficiently accurate when finding the real crack tip position. As predicted from the SIF results shown in the last section, as the crack propagates the error diminishes because the stresses are higher and the signal to noise ratio in TSA increases. Some error might be attributed to the fact that the visual localization of the crack tip usually yields a result at the middle of the crack front (notice that the cracks have a parabolic front, not a straight one), whereas the TSA data measures the temperature, and hence the stresses, at the surface of the specimen.

Using the result from the crack tip location, found by the non-linear TSA SIF algorithm, it is easy to calculate the crack propagation rate by dividing the length the crack has grown by the number of cycles between TSA data points (yielding da/dN).

Thermographic Crack Propagation Rate Determination

A second method for measuring the crack propagation rate was developed by combining temperature measurements at fatigue-damaged points near the crack front with an extension of the so-called Risitano’s method (or Thermographic method). Risitano’s method [20, 21] is used to measure fatigue limit and S-N (Stress vs Number of cycles to failure) curves using an energy damage parameter Φ, which is calculated as the integral of the point-temperature vs cycles plot.

The temperature of some selected points ahead of the crack tip in a specimen with a propagating crack (load range = 170 N with R = 0.35) and a reference temperature were measured during propagation, as seen in Fig. 10.

Fig. 10
figure 10

(a) Temperature map and the 11 selected points along the crack path before the propagation. (b) Temperature map and the 11 selected points along the crack path after the propagation. (c) Zoomed in detail from a, numbering the temperature measurement points

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Figure 11(a) shows the resulting ΔT (temperature at each point minus the reference temperature, taken from an unloaded polycarbonate specimen placed beside the loaded one) vs number of cycles plotted for the eleven points shown in Fig. 10. Each plot is identified by its peak, depicted by a circle mark and by the point number. Figure 11(b) shows the integrated area, related to the parameter Φ, for points 3, 6 and 10.

Fig. 11
figure 11

(a) ΔT vs N plots for the 11 points along the crack path. Numbered circle-marks depict the peak of each ΔT-N plot. (b) Integration process for Φ determination (example for points 3, 6 and 10)

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Table 3 shows the values of Φ obtained for each point after integrating the curves, as shown in Fig. 11(b).

Table 3 Damage parameter (Φ) for the 11 points along the crack path
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Excluding the two first points, which were already too close to the pre-crack tip at the beginning of the integration process, and which had already accumulated some damage, all the other points show similar results for the damage parameter (coefficient of variation 7.6%). These results suggest a possible way to predict crack paths in real time using temperature measurements and real-time integration of the damage parameter.

The fact that as the crack propagated (for higher N values), the peak temperature increased (although this was predictable since the stresses acting at the crack tip also increased), did raise the question of a possible relationship between the temperature variation and the crack growth rate. Both values presented the same exponential growth, characteristic of a non-linear snowball effect (crack propagates faster → stress increases → temperature increases → crack propagates even faster → …). In order to verify this relationship, the same setup used previously with the regular optical camera, was used to measure the crack propagation rate of the fatigue crack shown in Fig. 10. For comparison, the ΔTmax (the maximum temperature variation at the measured surface, located at the crack tip) vs the number of cycles plot was recorded.

Figure 12 shows the comparison between these two results. The ΔTmax was scaled using the ΔP (load range). As expected, the ΔTmax/ΔP vs N plot is similar to the plot that would be obtained if the peaks of each of the eleven points shown in Fig. 8 were connected. The results show that there is a clear correlation between the temperature variation and the crack growth rate. This finding proves to be very interesting, since, by using this method, a crack can potentially be monitored in real time, without the need for stress or strain measurements.

Fig. 12
figure 12

Comparison between da/dN vs N and ΔTmax/ΔP vs N

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Results and Discussion

In this section, combinations and comparisons between the methods of measuring both da/dN and ΔK, described earlier, are used in order to obtain Paris’ law curves that can be compared to values found in the literature. Table 4 summarizes the measurement methods and their combinations that were used to determine the data points present in Figs. 13, 14, 15 and 16. It also mentions results found in the literature regarding non-annealed polycarbonate.

Table 4 Summary of experimental data
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Fig. 13
figure 13

da/dN vs ΔK curves for two da/dN measuring methods (ΔK from algorithm)

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Fig. 14
figure 14

Measured and simulated da/dN vs ΔK curves

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Fig 15
figure 15

da/dN vs ΔK curves for two da/dN measuring methods (ΔK from FEM)

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Fig. 16
figure 16

Comparison between the present paper measured da/dN vs ΔK results and values found in the literature

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The first experiment consisted of a crack being propagated with ΔP = 225 N and R = 0.3. The pre-crack had a length of 4.2 mm and ended up being 10.5 mm long. Fig. 13 shows the comparison between two methods of measuring da/dN (optical camera and non-linear TSA algorithm), while the values of ΔK were all measured using the SIF-TSA algorithm.

This result clearly shows the interchangeability between the two da/dN measuring techniques, proving that the non-linear TSA algorithm can be self-sufficient when measuring crack propagation curves.

The second experiment consisted of a crack being propagated with ΔP = 160 N and R = 0.27. The pre-crack had a length of 10.4 mm and ended up being 26.6 mm long. Figure 14 shows the comparison between results obtained from the non-linear TSA algorithm and simulations performed using three-dimensional FEM.

The TSA points in the graph were taken from the algorithm (using 2xRmin in order to leave any crack closure effects out of the problem, since the FEM analysis was not able to predict them). It can be seen that TSA points agreed well with the results obtained in FEM simulation, showing that if the points near the crack tip are ignored, the algorithm can determine the theoretical SIF value. Since the plastic zone around the crack tip is not that big, the stress distribution far from it is not affected by crack closure.

The third experiment consisted of a crack being propagated with ΔP = 225 N and R = 0.3. The pre-crack had a length of 12.5 mm and ended up being 20 mm long. Fig. 15 shows the comparison between two da/dN measuring methods (optical camera and temperature measurement). The ΔK values were determined using the FEM simulation, for this test only.

This result illustrates the possibility of measuring crack growth rate using the temperature variation ahead of a crack.

For all results shown in Figs. 13, 14 and 15, the spread for shorter cracks tends to be higher than that for longer cracks. As stated above, this is because, with a constant loading range, the longer the crack, the higher the stresses acting around its tip, and consequently, the higher the temperature variation that occurs, minimizing the noise to signal ratio of the effective measurements being made during the tests.

Table 4 summarizes the experimental results presented in Figs. 13, 14, 15 and 16. It also presents results from Gerberich et al. [15] and Pruitt et al. for non-annealed polycarbonate. Figure 16 presents a summary of the data presented in the plots of Figs. 13, 14 and 15. Since all experimental data were taken from the key-hole specimens fabricated from the same polycarbonate sheet, with similar loading conditions and at similar ambient temperatures (25 °C), they can be all put in a single graph and a single tendency line can be fitted. Thus Fig. 16 shows the tendency line encompassing all six of the plots generated from the three different data sets, each from a different experiment, as presented in Figs. 13, 14, and 15. Two parallel lines were also added to the tendency line, defining a region between two times faster or two times slower propagation rates. Added to them are the two sets of results found in the literature. In their experiments Gerberich et al. [15] used single edge notch (SEN) specimens with dimensions measuring 50.8 mm by 152.4 mm and 50.8 mm by 203.2 mm with thickness = 6.35 mm, loaded cyclically with 1 Hz at 25 °C. Pruitt et al. [18] used compact tension (CT) specimens with length L = 19 mm, width measured from the center of holes W = 15 mm, height 2H = 19 mm, notch length ao = 5.33 mm and thickness = 2.2 mm, loaded cyclically with 5 Hz and R = 0.4.

Some variability of the data presented in Figs. 13, 14, 15 and 16 could be explained by the mean load dependence of fatigue crack propagation in polycarbonate, which was reported previously by other authors [18, 19], by ambient temperature variations (temperature variation in the order of 10-20 °C have been reported to cause non negligible effects on fatigue crack propagation in polycarbonate [15]), different specimen thicknesses and also by the residual stresses present in non-annealed specimens. Nonetheless, the results show reasonable agreement, once they fit inside the region defined by twice-as-fast and half-as-slow propagation rates. Consequently, it is possible to see how powerful thermography can be in studying fatigue crack propagation, especially due to its real-time measurements potential.

Conclusions

The experiments described in this paper were easy to set up and used an inexpensive combination of hardware (micro-bolometer camera) and novel software (lock-in and crack tip location) algorithm solutions, which enabled reliable crack growth rate and stress intensity factor measurements. The results presented good agreement with one another and with data found in the literature, making it possible to conclude:

  • TSA is a very reliable technique for SIF measurements in polycarbonate.

  • The crack tip location and crack growth rate can be determined simultaneously with the SIF determination by using TSA data.

  • Raw temperature measurement near the crack tip directly relates to crack growth rate.

  • Infrared thermography can single-handedly measure all necessary parameters for fracture mechanics analysis on the macro-scale.

The two methods used to measure da/dN proved reliable and have a lot of potential for future research, especially the temperature method, which is a very straightforward way of measuring the crack growth rate by means of a very fundamental relationship between the rise in temperature, the rise in stresses, and the speed at which the crack grows. Although different results could be possible for other materials, this relationship seems to be fundamental enough to assume that it will always be present.

Future work will be done concerning the automation of the process, in order to explore the real-time capability of the method. The idea of applying the Risitano damage parameter to crack propagation and the idea of exploring the relationship between da/dN and temperature variation in other materials will also be addressed in the future.