# Time- and Phase-Domain Thermal Tomography of Composites

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. 3D Modeling

^{6}W/m

^{2}), and the thermal process is followed for 10 s with an acquisition interval of 10 ms. Calculations were performed by using the ThermoCalc-3D software from Tomsk Polytechnic University, resulting in image sequences including N = 1000 IR thermograms each. The image format was 270 × 70 pixels with the lateral spatial step being 0.5 mm. From the point of view of the classical theory of heat conduction, such a TNDT model represents a multi-layer (up to 36 layers in ThermoCalc-3D) parallelepiped-like sample containing several parallelepiped-like defects (up to 40 defects in ThermoCalc-3D). The side surface of the sample is adiabatic while the front surface is heated with a square heat pulse. Both front and rear surfaces exchange energy with the ambient by convection. On the layer/layer and layer/defect boundaries, there are conditions of continuity of temperature and heat flux. The features of such a TNDT model have been thoroughly discussed elsewhere [13,16,17,18].

_{m}(timegram), and phase $\phi $ (phasegram), over the defect D2 (depth 0.5 mm), are presented. It appears that the profiles of temperature and phase look similarly with the signal plateau over the defect projection. This means that the 10 × 10 × 0.05 mm defect at the depth of 0.5 mm in CFRP can be considered as 1D, and heat diffusion takes place only at the defect borders where the signals decrease by 70% in regard to their maximal value over the defect center. The corresponding profile of τ

_{m}is also characterized by the plateau but the behavior of this parameter is more complicated due to the fact that timegrams represent a non-linear result of processing raw thermograms. The values of τ

_{m}, first, slightly drop in the areas where lateral heat diffusion starts, then, increase as $\Delta T$ values diminish up to zero in defect-free areas. Experimentally, in non-defect areas, τ

_{m}acquire random values from 1 to N because of a noisy character of $\Delta T$ signals. The noise can be either experimental or computational depending on whether experimental or synthetic images are processed. This peculiarity of producing timegrams causes round-shaped artifacts around defects when choosing particular ${\tau}_{mi}-{\tau}_{mj}$ intervals, as seen in Figure 3. The fact that $\Delta T$ signals tend to zero far from defects is used for thresholding τ

_{m}profiles. In other words, the pixel values in thermal tomograms are set to zero, where the $\Delta T$ values in the corresponding maxigrams become lower than a chosen threshold $\Delta {T}_{thr}$, thus allowing clear “footprints” of defects. Unfortunately, there is no definite rule on how to choose the threshold, however, in many cases it is about a few percent of a maximum ΔT

_{m}value in the corresponding maxigram.

## 3. Experimental Setup and Test Samples

## 4. Discussion of Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Kline, R.A.; Winfree, W.P.; Bakirov, V.F. A new approach to thermal tomography. AIP Conf. Proc.
**2003**, 22, 682–687. [Google Scholar] - Vavilov, V.; Maldague, X. Dynamic thermal tomography: New promise in the IR thermography of solids. In Proceedings of the Thermosense XIV: An International Conference on Thermal Sensing and Imaging Diagnostic Applications, Orlando, FL, USA, 22–24 April 1992; pp. 194–206. [Google Scholar]
- Ibarra-Castanedo, C.; Maldague, X. Defect depth retrieval from pulsed phase thermographic data on Plexiglas and aluminum samples. Proc. SPIE
**2004**, 5405, 348–356. [Google Scholar] - Rosencwaig, A. Thermal-wave imaging. Science
**1982**, 218, 223–228. [Google Scholar] [CrossRef] [PubMed] - Almond, D.P.; Patel, P.M. Photothermal Science and Techniques; Chapman & Hall: London, UK, 1996; p. 241. [Google Scholar]
- Mulaveesala, R.; Tuli, S. Theory of frequency modulated thermal wave imaging for nondestructive subsurface defect detection. Appl. Phys. Lett.
**2006**, 89, 191913. [Google Scholar] [CrossRef] - Mandelis, A. Theory of photothermal wave diffraction tomography via spatial Laplace spectral decomposition. J. Phys. A Math. Gen.
**1991**, 24, 2485. [Google Scholar] [CrossRef] - Munidasa, M.; Mandelis, A.; Ferguson, C. Resolution of photothermal tomographic Imaging of sub-surface defects in metals with ray-optic reconstruction. Appl. Phys.
**1992**, 54, 244–250. [Google Scholar] [CrossRef] - Winfree, W.P.; Plotnikov, Y.A. Defect characterization in composites using a thermal tomography algorithm. In Review of Progress in Quantitative Nondestructive Evaluation; Thompson, D.O., Chimenti, D.E., Eds.; Kluwer Academic Plenum Publishers: Dordrecht, The Netherlands, 1999; Volume 18, pp. 1343–1350. [Google Scholar]
- Toivanen, J.M.; Tarvainen, T.; Huttunen, J.M.J.; Savolainen, T.; Orlande, H.R.B.; Kaipio, J.P.; Kolehmainen, V. 3D thermal tomography with experimental measurement data. Int. J. Heat Mass Transf.
**2014**, 78, 1126–1134. [Google Scholar] [CrossRef] - Sun, J.G. Method for Implementing Depth Deconvolution Algorithm for Enhanced Thermal Tomography 3D Imaging. U.S. Patent No. 8465200, 2013. [Google Scholar]
- Sun, J.G. Quantitative three-dimensional imaging of heterogeneous materials by thermal tomography. J. Heat Transf.
**2016**, 138, 112004. [Google Scholar] [CrossRef] - Vavilov, V.P. Dynamic thermal tomography: Recent improvements and applications. NDT & E Int.
**2015**, 71, 23–32. [Google Scholar] - Zhang, X. Instrumentation in diffuse optical imaging. Photonics
**2014**, 1, 9–32. [Google Scholar] [CrossRef] [PubMed] - Friederich, F.; May, K.H.; Baccouche, B.; Matheis, C.; Bauer, M.; Jonuscheit, J.; Moor, M.; Denman, D.; Bramble, J.; Savage, N. Terahertz radome inspection. Photonics
**2018**, 5, 1–10. [Google Scholar] [CrossRef] - Balageas, D.L.; Krapez, J.-C.; Cielo, P. Pulsed photo-thermal modeling of layered materials. J. Appl. Phys.
**1986**, 59, 348–357. [Google Scholar] [CrossRef] - Maillet, D.; Andre, S.; Batsale, J.-C.; Degiovanni, A.; Moyne, C. Thermal Quadrupoles: Solving the Heat Equation through Integral Transforms; John Wiley & Sons Publ.: New York, NY, USA, 2000. [Google Scholar]
- Vavilov, V.P. Thermal/infrared testing. In Nondestructive Testing Handbook; Kluev, V.V., Ed.; Spektr Publisher: Moscow, Russia, 2009; pp. 1–485. [Google Scholar]
- Noethen, M.; Jia, Y.; Meyendorf, N. Simulation of the surface crack detection using inductive heated thermography. Nondestruct. Test. Eval.
**2012**, 27, 139–149. [Google Scholar] [CrossRef] - Ibarra-Castanedo, C.; Maldague, X.P. Infrared thermography. In Handbook of Technical Diagnostics: Fundamentals and Application to Structures and Systems; Czichos, H., Ed.; Springer Publ.: Berlin, Germany, 2013; pp. 175–220. [Google Scholar]
- Gao, B.; Woo, W.L.; Tian, G.Y. Electromagnetic Thermography Nondestructive Evaluation: Physics-based Modeling and Pattern Mining. Sci. Rep.
**2016**, 6, 25480. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**3D modeling and time/phase data processing in the inspection of CFRP (L = 3 mm, defect size 10 × 10 × 0.05 mm, l = 0.1; 0.5; 1.475; 2.45 and 2.85 mm).

**Figure 2.**Dependence of optimum observation time τ

_{m}(

**a**) and thermal wave phase $\phi $ (

**b**). on defect depth l in the inspection of CFRP (CFRP thermal properties: thermal conductivity 0.64 W·m

^{−1}·K

^{−1}, diffusivity 5.2·10

^{−7}m

^{2}·s

^{−1}, L = 3 and 5 mm, defect size 10 × 10 × 0.05 mm, 1, 3—numerical modeling, 2—analytical solution).

**Figure 4.**Comparing spatial profiles of temperature T, characteristic time τ

_{m}and phase $\phi $ (10 × 10 × 0.05 mm defect at 0.5 mm depth in CFRP; T and $\phi $ values are normalized to make results comparable).

**Figure 5.**Time- and phase-domain thermal tomography of impact damage in 5 mm-thick CFRP sample (10 J impact energy).

**Figure 6.**Time- and phase-domain thermal tomography of impact damage in 5 mm-thick CFRP sample (18 J impact energy).

**Figure 7.**Time- and phase-domain thermal tomography of impact damage in 5 mm-thick CFRP sample (63 J impact energy).

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**MDPI and ACS Style**

Vavilov, V.P.; Shiryaev, V.V.; Kuimova, M.V.
Time- and Phase-Domain Thermal Tomography of Composites. *Photonics* **2018**, *5*, 31.
https://doi.org/10.3390/photonics5040031

**AMA Style**

Vavilov VP, Shiryaev VV, Kuimova MV.
Time- and Phase-Domain Thermal Tomography of Composites. *Photonics*. 2018; 5(4):31.
https://doi.org/10.3390/photonics5040031

**Chicago/Turabian Style**

Vavilov, Vladimir P., Vladimir V. Shiryaev, and Marina V. Kuimova.
2018. "Time- and Phase-Domain Thermal Tomography of Composites" *Photonics* 5, no. 4: 31.
https://doi.org/10.3390/photonics5040031