ABSTRACT. In this dissertation the finite element (FE) method is applied to simulate the results of

Transcription

1 ABSTRACT HERNANDEZ VALLE, SAUL. Finite Element Analysis of Pulse Phase Thermography for Carbon Fiber Reinforced Polymer Structures. (Under the direction of Dr. Kara J. Peters). In this dissertation the finite element (FE) method is applied to simulate the results of pulse phase thermography (PPT) experiments on laminated composite plates. Specifically, the goal is to simulate the phase component of reflected thermal waves and therefore verify the calculation of defect depth through the identification of the defect blind frequency. The calculation of phase components requires a higher spatial and temporal resolution than that of the calculation of the reflected temperature. A FE modeling strategy is presented, including the estimation of the defect thermal properties, which in this case is represented as a foam insert impregnated with epoxy resin. A comparison of meshing strategies using tetrahedral and hexahedral elements reveals that temperature errors in the tetrahedral results are amplified in the calculation of phase images and blind frequencies. The linearity of the measured diffusion length (based on the blind frequency) is investigated as a function of defect depth. The simulations demonstrate that the diffusion length versus defect depth relation is well represented by an exponential function, in contrast to the linear function commonly applied. This result is consistent with previous experimental observations. The role of the defect thickness on the results is investigated; the results show that it is related to the blind frequency in an exponential fashion. The role of defect thickness on the surface initial thermal contrast is also studied at different depths and these two parameters (blind frequency and initial thermal contrast) used to propose a two-point strategy for charactering a defect under the surface. The FE knowledge and information gathered during this research was also applied to model the role on defect detectability of the delamination

2 area-perimeter ratio at constant area; a reduction in blind frequency and surface initial thermal contrast was observed with a reduction in its area-perimeter ratio. Also the role of the proximity of a second defect to the one being investigated was simulated; however not a significant impact was detected as a result of a second defect. Finally, a pyramidal FE model representing damage induced by low velocity impacts was created and simulated; the distance between the defects comprising the model was changed while the depth of the first defect was kept constant. Blind frequency and surface initial thermal contrast results were calculated for different locations and compared to single defect model results. The results showed that single defect models cannot be used directly to make analysis about low velocity impact induced damage, but they can be linearly correlated to get the results corresponding to the low velocity impact model. In addition, the blind frequency solutions can be related to the depth of the defect below by means of an exponential curve. However, when the spacing in-between the defects was increased, the results varied noticeably from one model to the next due to the presence of non-linear factors in the solution as it was mentioned earlier.

3 Copyright 2015 by Saul Hernandez Valle All Rights Reserved

4 Finite Element Analysis of Pulse Phase Thermography for Carbon Fiber Reinforced Polymer Structures By Saul Hernandez Valle A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy Mechanical Engineering Raleigh, North Carolina 2015 APPROVED BY: Dr. K. J. Peters Committee Chair Dr. J. W. Eischen Dr. Gracious Ngaile Dr. Rudolf Seracino

5 DEDICATION This work is dedicated to my mother, my wife and my family. ii

6 BIOGRAPHY Saul Hernandez Valle obtained his Bachelor of Engineering in Nuclear Engineering from the High Institute of Nuclear Science and Technology, Havana, Cuba in 1998, during the senior year he worked on numerical analysis of TRIGA MARK II research reactor using codes WIMS-D/4 and SNAP3d. After graduation he worked as researcher for the nuclear technology center located in Havana, Cuba from 1998 to 2002, during that time he earned a Master s degree in Nuclear Facilities from the High Institute of Nuclear Science and Technology, Havana, Cuba in 2001 working on Monte Carlo Method to solve the neutron transport equation in a section of (PWR) VVER nuclear reactors. In January 2006 he started the Master s program in Mechanical Engineering at North Carolina State University graduating in 2009, concentrating on thermal science, upon graduation he started his current PhD program. In 2010 he started working for NSWC Dahlgren as a Finite Element Analyst in Multiple projects. He pursued a Doctor of Philosophy degree under the direction of Dr. Kara Peters, focusing on Finite Element Analysis applied to Pulse Phase Thermography for Carbon Fiber Reinforced Polymer Structures. iii

7 ACKNOWLEDGMENTS I would like to express my gratitude to my wife Alma Delia Garcia Hernandez for all her patient and support during all this time of study. To my advisor Dr. Kara Peters for all her invaluable guidance, intelligence and abilities, without that it would have been impossible to finish this dissertation. I would like also to thank my committee members, Dr. Jeffrey Eischen, Dr. Ngaile Gracious and Dr. Rudolf Seracino. I would like to acknowledge the financial support of the National Science Foundation and the US Navy and NSWC Dahlgren education office and my Branch G24. iv

8 TABLE OF CONTENTS LIST OF TABLES... VII LIST OF FIGURES... VIII CHAPTER 1 INTRODUCTION MOTIVATION SCOPE OF THE RESEARCH THESIS OUTLINE... 4 CHAPTER 2 BACKGROUND FUNDAMENTALS OF PULSE PHASE THERMOGRAPHY THE FINITE ELEMENT METHOD APPLIED TO INFRARED THERMOGRAPHY Literature Review Guidelines for the Research in the Dissertation CHAPTER 3 FINITE ELEMENT METHODOLOGY FINITE ELEMENT FORMULATION OF THE HEAT TRANSFER EQUATION MODELING STRATEGY Geometrical and Physical Model Creation of the Finite Element Model Solving the Finite Element Model Post-Processing Interoperability Stages HEXAHEDRAL VS TETRAHEDRAL MESHES BOUNDARY CONDITIONS ESTIMATION OF DEFECT THERMAL PROPERTIES CHAPTER 4 FINITE ELEMENT MODELING OF PULSE PHASE THERMOGRAPHY INTRODUCTION SEMI-EMPIRICAL APPROACH TO PLATE HEATING WITH DEFECT UNDER THE SURFACE DEFECT DEPTH RETRIEVAL BASED ON BLIND FREQUENCY USING FINITE ELEMENT SIMULATIONS ROLE OF THICKNESS ON BLIND FREQUENCY AND INITIAL THERMAL CONTRAST Finite Element study of the Role of Thickness on Blind Frequency and Surface Thermal Contrast Two-Point Estimation Strategy for Determining Thickness and Depth FINITE ELEMENT STUDY OF THE ROLE OF DEFECT PROXIMITY v

9 4.6 FINITE ELEMENT STUDY OF THE ROLE OF DEFECT AREA PERIMETER RATIO CHAPTER 5 FINITE ELEMENT MODELING OF LOW VELOCITY IMPACT INDUCED DAMAGE IN CFRP INTRODUCTION FINITE ELEMENT SOLUTION BLIND FREQUENCY SOLUTION PYRAMIDAL MODEL RESULTS COMPARISON WITH SINGLE DEFECT MODEL RESULTS CHAPTER 6 CONCLUSIONS AND FUTURE WORK CONCLUSIONS FUTURE WORK REFERENCES APPENDICES APPENDIX A: CUBIT JOURNAL FILE APPENDIX B: ARIA INPUT EXAMPLE. HEAT FLUX ON A PLATE vi

10 LIST OF TABLES Chapter 3 Table 3.1. CFRP and Rohacell Foam 71 thermal properties Table 3.2. Blind frequency and Initial Thermal Contrast obtained using tetrahedral and hexahedral meshes Table 3.3. Properties of air for convection heat transfer coefficient Table 3.4. Properties of materials Chapter 4 Table 4.1. Coefficient G as a function of depth and the corresponding temperature contrast on the surface compared with the FE solution Table 4.2. Blind Frequency and cooling transient initial temperature contrast for each distance Table 4.3. Blind frequency and initial temperature contrast for the isolated defect case Table 4.4. Dimensions of the defects used in the simulations. a and b are the sides of the rectangular defect Table 4.5. Results for the different area-perimeter ratio Table 4.6. Blind frequency and initial temperature contrast change with respect to the square defect Chapter 5 Table 5.1. Defect dimensions and depth in the sample Table 5.2. Low velocity impact model blind frequencies at the pre-defined gauge locations Table 5.3. Single defect models blind frequency solutions at the different gauge locations.. 86 Table 5.4. Equation system 5.4 solution Table 5.5. Solutions of the blind frequency equations systems Table 5.6. Solutions of surface thermal contrast equations systems Table 5.8. Blind frequency and initial temperature contrast results for low velocity impact model and single defect model vii

11 LIST OF FIGURES Chapter 2 Figure 2.1. Pulse Phase Thermography (PPT) experimental setup at NC State University Figure 2.2. Schematic of the Fourier Transform applied to thermograms in PPT (Galmiche and Maldague, 2002) Figure 2.3. PPT phase images for specimen impacted at 1.75 m/s with minimum a frequency of 0.17 Hz... 9 Figure 2.4. Schematic of thermal wave propagation in material with defect at depth equal to the diffusion length for a given frequency, the defect is shown as a gray rectangle Figure 2.5. Schematic of relationship of blind frequency to phase plots for sound and defect regions Figure D meshed geometry used by Lulay and Safai (1994) Figure 2.7. Stepped composite panel modeled by Krishnapillai et al. (2005) Figure 2.8. Actual and simplified defect meshed geometry used by Krishnapillai et al. (2005) Figure 2.9. Meshed geometry used by Krishnapillai et al. (2006) Figure Schematic comparing 2D and 3D simulations and experimental data by Krishnapillai et al. (2006) Figure Geometry and mesh used by Hain et al. (2009) Figure Examples of virtual infrared images by Van Leeuwen et al. (2011) Figure Defect geometry of Ishikawa et al. (2012) Figure FE geometry model, noisy and clean temperature and phase comparison of Ishikawa et al. (2012) Chapter 3 Figure 3.1. Schematic of measurement and simulation windows during PPT. ΔT is the temperature of a point on front surface of specimen. Tsat is saturation temperature of thermal camera Figure 3.2. Measured PPT phase image of CFRP specimen at with multiple polymer foam inserts, Pawar and Peters (2013). Defect depth increases from bottom to top of image Figure 3.3. Flow chart of finite element modeling process Figure 3.4. Single defect model geometry created in CUBIT Figure 3.5. Mesh testing simulation: (a) hexahedral mesh; (b) tetrahedral mesh; (c) location of surface temperature measurement points Figure 3.6. Hexahedral mesh phase plot viii

12 Figure 3.7. Schematic of polymer foam defect transformation process: (a) polymer foam before compression; (b) after compression and (c) after resin infusion Figure 3.8. Calculated cooling transient curves for a surface pixel directly above the defect (perturbed region) and far from defect (sound region) Chapter 4 Figure 4.1. Heating transient finite element solution and plot of equation (4.4) Figure 4.2. G versus depth plot and the exponential fit Figure 4.3. Plot of equation (4.8) and FE solution for a defect of area 100 mm 2 and thickness 0.1 cm at a depth of cm Figure 4.4. Simulated phasegrams for 100 mm 2 defect at 0.65 mm depth, with maximum frequency, fmax, of (a) Hz, (b) Hz, (c) Hz and (d) Hz Figure 4.5. Estimation of blind frequencies for a 225 mm2 defect with depth of (a) mm, (b) mm, (c) mm, (d) mm, (e) mm and (f) mm Figure 4.6. Depth vs. diffusion length calculated from blind frequency of simulation data. Data and linear curve fits are shown for three defect areas Figure 4.7. Fits to depth vs. diffusion length calculated from blind frequency of simulation data. Exponential fits are shown as solid lines; linear fits to first three data points are shown as dashed lines Figure 4.8. Exponential fit to depth vs. diffusion length calculated from blind frequency of simulation data Figure 4.9. Blind frequency vs thickness plot, depth equal to cm Figure Blind frequencies for defects of different thickness at different depths Figure Defective sound pixel temperature contrasts for different thickness and depths Figure Temperature after the heat pulse, distance a) 1.0, b) 0.8 c) 0.6 and d) 0.4 cm Figure Phasegrams corresponding to frequency equal to 3.46 Hz, distance a) 1.0, b) 0.8 c) 0.6 and d) 0.4 cm Figure Phase vs frequency plot and blind frequency Figure Blind frequency and Initial thermal contrast at different distance (cm) between defects Figure Temperature at the beginning of the cooling transient for A/P = (a) 2.00 and (b) Figure Phasegrams at frequency = 3.46 Hz for A/P = (a) 2.00 and (b) Figure Bar graph of blind frequency and initial thermal contrast for different A/P Figure 4.19 Temperature cooling transient for A/P = Figure Phase plot for A/P = Figure 4.21 Temperature cooling transient for A/P = ix

13 Figure 4.22 Phase plot for A/P = Figure Bar plot of the relative change with respect to the square defect Chapter 5 Figure 5.1. Post-impact images of specimen impacted at 1.75 m/s: photograph of impacted surface (Pawar and Peters, 2013) Figure 5.2. Cross sectional examination give information of through the thickness damage (Alan T Nettles, 2011) Figure 5.3. (a) Phasegram obtained for CFRP laminated plate with LVID (one pixel corresponds to approximately 0.2 x 0.2 mm); (b) 3D impact damage progression using PPT (Pawar and Peters, 2013) Figure 5.4. CUBIT 3D impact damage pyramidal representation of damage progression Figure 5.5. CUBIT representation of the defects that comprised and impact damage pyramidal damage progression and the location on the surface of the temperature gauges Figure 5.6. Hexahedral mesh of the defect block Figure 5.7. Sample thermograph at the beginning of the cooling transient Figure 5.8. Cooling transients corresponding to the five gauges and a sound pixel Figure 5.9. Sample phasegram from IR_VIEW, frequency Hz Figure Phase curves corresponding to the five gauges and a sound pixel Figure Phase curves corresponding to the central gauge and a sound pixel Figure Phase curves corresponding to the second gauge and a sound pixel Figure Phase curves corresponding to the third gauge and a sound pixel Figure Phase curves corresponding to the fourth gauge and a sound pixel Figure Phase curves corresponding to the fifth gauge and a sound pixel Figure Blind frequency vs defects depth for pyramidal model with spacing = cm Figure Blind frequency vs defects depth for pyramidal model with spacing = cm Figure Blind frequency vs defects depth for pyramidal model with spacing = cm Figure Exponential fit to surface pixel location vs. diffusion length calculated from blind frequency of simulation data x

14 Chapter 1 Introduction 1.1 Motivation Various researchers have applied the finite element method to simulate the propagation of thermal waves in materials for infrared thermography inspection (Meola and Carlomargo, 2010). The solution of the thermal wave equations has thus been applied to calculate the surface temperature of the material as a function of time at either the front surface (for thermography measurements in reflection) or the rear surface (for thermography measurements in transmission). Researchers have used these calculated surface temperatures to verify experimental thermography results (Ibarra-Castanedo et al., 2011) and to optimize experimental configurations thus limiting the number of physical tests necessary (Schemutzler et al., 2014). In this work we apply the FE method to simulate the results of pulsed phase thermography (PPT) experiments on laminated composite plates, in particular the calculation of the blind frequency for defects at different depths. PPT has recently emerged as a powerful inspection technique for thin-walled, laminated composite structures, particularly composite airframes (Van Leeuen et al., 2011). The use of a square heating pulse inputs a wide bandwidth of thermal wave frequencies into the specimen with a single heat pulse. As only a single heating pulse is applied, a large surface area of the structure can be inspected rapidly. The surface temperature is then measured as a function of time and the individual frequency information is later extracted through post-processing of the data through FFT analysis. In particular, the depth of a defect can also be identified through the calculation of the blind frequency, i.e. the highest frequency at which a defect can be imaged. 1

15 It is the goal of this work to obtain FE simulations of the full PTT process and to compare the measured parameters with the simulations, in particular the surface phase contrast distribution and the resulting calculated blind frequency for different defect configurations and sizes. In particular, the goal is to consider realistic damage modes in composite laminates, for example delamination, matrix cracking and fiber fracture, which are not well represented by inserts or flat-bottom holes (Meola and Carlomargo, 2010). A second goal is to be able to estimate the influence of different geometrical configurations on the blind frequency, which have only been evaluated experimentally (Susa M et al., 2007). In addition, the simulations could predict measurement errors due to the 3D geometries of the specimens and defects. This information could then be used for design of virtual experiments in which variables like defect thickness, depth and area can be controlled by the analyst to study different aspect of thermography, material properties and the physics of the technique and to correct experimentally collected PPT data. Without the level of temperature data noise typically found in experiment data (Kitamura et. al., 2014), the FE simulations can provide an understanding of the theoretical role of each of these features. In future work, the effects of realistic noise distributions could be studied for research purposes. Existing FE Simulations of infrared thermography experiments can be divided into three categories, depending on the number of dimensions in the model. One-dimensional models are applicable for cases where the defect size is much larger than the thermal diffusion length (Ibarra-Castanedo et al., 2011) and can predict averaged behaviors of defective and nondefective areas. The most popular method to develop two- and three- dimensional models is through the finite element method, due to the fact that complicated geometries can be relatively easily 2

16 implemented. For the simulation of surface temperatures, 2D models are sometimes sufficient, although they cannot incorporate effects due to transverse heat flow in the specimen. However, transverse thermal diffusion is generated due to reflections of the boundaries of the specimen or edges of the damage region (Meola and Carlomargo, 2010), curved surface geometries or complex structural shapes (Chu T et al., 2005; Tranta and Sorger, 2012) and non-uniform thermal heating (Ptaszek et al., 2012). In addition, non-uniformity in damage boundaries through the thickness of the specimen, which is particularly relevant to inspection of impact damage in composite laminates, also requires 3D models (Fachinotti and Bellet, 2006). Threedimensional FE modeling can provide accurate reconstruction of these effects, although at increased computational cost. Using 3D models and specialized meshes is not enough to accurately simulate the problem with FE; realistic material properties are needed in addition to a good mesh. For the specific case of a defect, effort should be put into determining thermal properties that allow the accurate simulation of its interaction with the matrix around it. The challenge to calculating blind frequencies from FE simulations is the high spatial and temporal resolution required to accurately calculate the phase of the propagating waves from the calculated temperature values. The required accuracy is at least an order of magnitude greater than that required to predict surface temperatures at the nodal locations with the same level of error. In addition, specialized meshes are required to minimize computational errors that can be propagated during the simulation, specifically the modeling of the initial thermal shock and discontinuities in the heat flux at the composite-defect interfaces (Bai and Wong, 2001). However, these specialized meshes can be difficult to implement on irregular geometries. With the use of advanced parallel computing and meshing software capabilities this task can be accomplished. 3

17 1.2 Scope of the Research The main objective of this research is to simulate, by means of the FE Method, the physical PPT process and data post-processing to improve the inspection results for carbon fiber reinforced polymer structures. In this dissertation, we demonstrate the calculation of blind frequencies via FE simulation of PPT of a composite laminate with artificial defect inserts at different depths and with different geometries and evaluate the performance of the simulation for different mesh configurations. We also will use the developed methodology to provide a FE approach to the following: FE modeling of the heating transient through the material Defect depth inversion using a blind frequency analysis The sensitivity of the blind frequency to the defect thickness and the proximity of other defects A graphical method for inverting defect depth and thickness Effects of multiple interface delaminations on blind frequency calculations, as expected for low velocity impact damage 1.3 Thesis Outline Chapter 1 presents the introduction to this work, including the motivation, context and scope of the research. Chapter 2 reviews the fundamental concepts of Pulsed Phase Thermography, as well as describing how it is applied in practice. Afterwards, a review of the literature covering the FE method applied to infrared thermography is presented. A particular focus is placed on the 4

18 specific parameters output by the models and how they can be used to verify the results of infrared inspection experiments. At the end of this chapter, the needs for further modeling efforts to better represent the output of PPT experiments are summarized. Chapter 3 describes the FE methodology used in this research. Particular attention is paid to the meshing scheme and the performance of two models are compared, one meshed with tetrahedral elements and the second model meshed with hexahedral elements. The chapter also describes the estimation of the thermal properties and boundary conditions needed to solve a FE problem for PPT applications, specifically the defect properties and the heat flux applied on the surface from known test data. Chapter 4 presents the results of the FE models of the entire PPT process. First the heating transient of the thermal curve is modeled and compared with analytical solutions for defective and non-defective pixels. The depth inversion for constant thickness defect is studied and the role of the thickness, proximity of another defect and area perimeter ratio is analyzed. A two-point inversion estimation strategy is proposed for defect depth and thickness characterization. Chapter 5 describes the application of the FE methodology to the modeling of low velocity impact induced damage. A pyramidal characterization with multiple defects of representative delaminations with a CFRP laminated material was created. FE temperature solutions and blind frequencies were calculated for different pixels on the surface above the defect and compared with solutions for single defect models on the same pixel locations. Interpretation of the results and considerations on the influence of nonlinear effects on the solution are included. 5

19 Chapter 6 summarizes the conclusions from the research results and provides guidelines for future work on the topic of FE modeling of PPT. 6

20 Chapter 2 Background This chapter reviews the fundamental concepts of PPT and presents an overview of selected papers of the literature on the FE method applied to infrared thermography in general. A particular emphasis is placed on the parameters that have been modeled for different active thermography techniques and the correlation of these simulations to experimental studies. At the end of the chapter, directions for future steps, needed to fully replicate PPT experiments with FEM, are proposed. 2.1 Fundamentals of Pulse Phase Thermography PPT is a mathematical fusion between pulsed thermography and lock-in thermography. The advantage of PPT for rapid imaging of large structures is that only a single heating pulse is required to scan through the entire material thickness as compared to lock-in thermography, significantly reducing the inspection time. This benefit is at the cost of lower image resolution than that of lock-in thermography, but is often sufficient for rapid scanning applications. In addition, the depth of a particular defect can be estimated through the calculation of the blind frequency, i.e. the maximum frequency at which a particular defect demonstrates sufficient phase contrast to be observed in the phase image. In this section, we will review the fundamentals of PPT and the calculation of the depth of a defect from the PPT data. These fundamentals are important to understand the FE modeling approach in the following chapters. In PPT the material to be inspected is heated on the front surface with a square thermal pulse and then allowed to cool down to room temperature (Pawar and Peters, 2013), the thermal wave emitted from the front surface or rear surface of the material specimen is measured with a thermal camera and saved as two dimensional pixel arrays of temperature (thermograms) at 7

21 discrete time intervals (t). Typical PPT equipment is shown in figure 2.1. Figure 2.1. Pulse Phase Thermography (PPT) experimental setup at NC State University. These thermograms are measured during the cooling transient of the material for a time called observation time (tobs). A total of N thermograms are collected over the observation period, N = (tobs/t)+1. The Discrete Fourier Transform (DFT) is then applied to extract the specific frequency thermal waves from the measured thermal wave at each pixel in the captured thermogram, as shown in figure 2.2. Figure 2.2. Schematic of the Fourier Transform applied to thermograms in PPT (Galmiche and Maldague, 2002). The amplitude, A, and phase,, of the radiated wave at each pixel is reconstructed using 8

22 A = Re 2 (n) + Im 2 (n) = arctan ( Im(n) Re(n) ) (2.1) where Re and Im are the real and imaginary parts of the DFT of pixel in in the cooling transient. In PPT we consider the phase image due to its relative insensitivity to environmental noise. Figure 2.3 shows an example of a PPT phase image for a carbon fiber reinforced polymer (CFRP) laminated plate from (Pawar and Peters, 2013). The damage shown in figure 2.3 was created in the laboratory with an impactor hitting the plate at a velocity of 1.75 m/s; corresponding to a kinetic energy of 8.4 J. Figure 2.3. PPT phase images for specimen impacted at 1.75 m/s with minimum a frequency of 0.17 Hz. The blind frequency, and therefore the depth of a particular defect, is calculated from the relative phase contrast between a pixel at the surface above the defect and a pixel in a region away from the defect. Consider a thermal wave, T(z,t), propagating through a material with thermal diffusivity, as shown in figure 2.4(a). Assuming the material is infinitely deep, 9

23 the thermal wave can be written as (2.2), Figure 2.4. Schematic of thermal wave propagation in material with defect at depth equal to the diffusion length for a given frequency, the defect is shown as a gray rectangle. z/ f / f f T ( z, t) T0e cos 2 ft z i sin 2 ft z (2.2) where z is the distance of propagation, t is time, f is the wave frequency, is the material diffusivity and T0 is the temperature of the wave at the surface (Culshaw and Jaeger, 1959). The amplitude of the wave as it propagates is T( z, t) z/ f / T0e (2.3) The diffusion length,, is then defined as the distance at which the amplitude of a thermal wave of a certain frequency reduces to T = e -1 T0 = T0 (Bai and Wong, 2001) μ = α πf b (2.4) For the same wave, T = e -2 T0 = T0 when it returns to the surface. 10

24 Figure 2.4 demonstrates also the concept of depth retrieval from PPT. The defect is located at a depth corresponding to for a given frequency. The assumption in PPT is that for waves with higher frequencies (i.e. sho`rter diffusion lengths than ), the wave would not reach the defect and therefore not contribute to the phase reconstruction, as shown in figure 2.4(b). For waves with lower frequencies (i.e. longer diffusion lengths than ), the wave would enter the defect and propagate with a different phase (proportional to f / ) due to the change in thermal diffusivity, D, while it is in the defect. Figure 2.5 shows a schematic of the phase delay at the two pixels as a function of the input thermal wave frequency. The apparent blind frequency, fb, is therefore the point on this graph at which these two curves diverge, shown in figure 2.5 (Culshaw and Jaeger, 1959). In reality, the cutoff point in figure 2.5 depends on the minimum threshold of phase contrast observable in the phase images and therefore the numerical and experimental noise. Therefore, the cutoff point in figure 2.5 is referred to as the apparent blind frequency and is typically calibrated for an experimental setup. Following equation (2.4), the diffusion length calculated from this apparent blind frequency is linearly related to the defect depth z0, z C C (2.5) where C1 and C2 are the calibration constants. In FE simulations the external measurement noise is not present; therefore, the threshold values are much smaller than those typically used for experimental data. 11

25 Figure 2.5. Schematic of relationship of blind frequency to phase plots for sound and defect regions. It is important to understand the effects of the measurement parameters and noise on the calculation of the blind frequency from experimental data. First, under sampling of the thermograms creates aliasing of the frequency information, which shifts the depth relation of equation (2.5). Second, the fb calculation of figure 2.5 is based on the difference of two curves reconstructed from the DFT, both with measurement and numerical noise. While the curve fits to the data remove some noise and ringing effects in the phase vs. frequency plots, a threshold difference between the two curves is often set to determine fb. The resulting fb calculated for a given measurement system under these conditions is called the apparent blind frequency. In FE simulations the sampling conditions and numerical error can be minimized, therefore these effects are not significant. 12

26 However, equation (2.5) assumes one-dimensional heat flow through the material specimen, which is not the case for realistic damage modes (even in FE simulations) due to non-uniform thickness through the defect, the finite thickness of the specimen, reflections off the boundaries of the specimen or edges of the damage region, or non-uniform thermal heating. 2.2 The Finite Element Method Applied to Infrared Thermography Literature Review The origins of the FE Method as we know it today can be traced back to the 1950s when several researchers extended the matrix analysis of structures to continuum bodies. The space exploration of the 1960s provided financial resources for basic research; this situation allowed the creation of a firm mathematical foundation and stimulated the development of computer programs for implementing the method. Many applications like airplanes and missiles design provided areas of application. In the area of infrared thermography, many authors have implemented FE simulations with commercial codes to supplement experimental studies. One of the earliest examples is Lulay and Safai (1994) who investigated various pulsed thermal loads to determine the parameters required for highest flaw detection sensitivity in four-sheet superplastic forming and diffusion bonding (SPF/DB) structures. The authors used a 2D finite element model to estimate optimal heating techniques. The authors also made clear that their FE models were not intended to predict exact thermal response from defects but are just useful for comparison. Figure 2.6 shows the meshed geometry and temperature results for a 0.25 inch disbond using a high intensity, short duration heating. 13

27 Figure D meshed geometry used by Lulay and Safai (1994). Here we can see the 2D elements and the relatively coarse mesh, probably sufficient for the goals of the study, but not sufficient for modeling of the PPT process. The authors found that high intensity heating for a short duration provides great thermal gradients and made detection of disbonds more reliable. It was also found that in both the FE analysis and the experiments small defects (less than 0.25 inches long) were difficult to detect. In a 3D element example, Krishnapillai et al. (2005) investigated the use of FE to simulate thermography in complex non-axisymmetric geometries. The authors used a 21 nodded hexahedral element, which is a 20 node hexahedral element with a node at the center called HEXENODE. According to the authors this extra node provides improved flexibility when analyzing thin structures with high aspect ratios. However, unless the structure will deform as a consequence of the thermal load, it is not clear what computational advantages the addition of nodes to the basic 8 node hexahedral element will yield in PPT analysis. 14

28 In this work, the authors modeled a stepped composite panel with flat bottom holes drilled at various depths, as shown in figure 2.7. They first meshed the actual geometry in figure 2.8 but, the mesh resulted to be relatively poor close to the curved region. A second trial mesh was therefore run with the defect shaped as a rectangle and the equivalent area of the circular region added to the length of the rectangle as also shown in figure 2.8. Tests showed that the errors generated by using this pseudo-defect shape were approximately %. Hence this approach was deemed more acceptable than the unquantifiable errors that may occur from using a relatively poor quality mesh. However, the low error value was the result of the improvement in the mesh and the change in the geometry of the defect, creating effects that canceled each other out. It would have been interesting to check the error in the model by just refining the mesh in the actual geometry to the same level as in the simplified geometry; however, this information is not available. Figure 2.7. Stepped composite panel modeled by Krishnapillai et al. (2005). 15

29 Figure 2.8. Actual and simplified defect meshed geometry used by Krishnapillai et al. (2005). The FE simulations demonstrated that the thermal diffusivity is directly related to the inflection time of the temperature contrast evolution curves and that the transient thermal response of flat panel composite samples with internal defects can be accurately simulated using FEA. The authors also stated that the HEXENODE element was stable during the simulations and holds great potential for the analysis of the thermography of complex structures. No comparison with other mesh types, such as hexahedral elements was performed. This paper emphasizes the importance of the mesh type in 3D FE models to simulate irregularities. In another 3D FE example, Krishnapillai et al. (2006) examined the detection of subsurface delaminations in a composite flat plate setup using pulsed thermography. They recommended the use of a 3D model by stating that 1D and 2D simulations ignore a paramount effect in the form of lateral thermal diffusion. Figure 2.9 presents details on the geometry and mesh used in the simulations. 16

30 Figure 2.9. Meshed geometry used by Krishnapillai et al. (2006). They presented a very illustrative schematic comparing 2D, 3D models and experimental data, see figure They emphasized on the importance of the difference in numerical results attributed to size variations in defect area. The 2D numerical results do not change as the defect size changes. Figure Schematic comparing 2D and 3D simulations and experimental data by Krishnapillai et al. (2006). 17

31 Another point of discussion of the authors in this study was the argument that utilizing flat bottomed holes to represent realistic delaminations is misguiding. Also, using Teflon and other artificial inserts to represent cavities in composite material, while an established procedure regarding structural analysis, is unsuitable for thermal analysis, due to the large thermal contrast between the defect and surrounding materials. The authors put emphasis on understanding the properties of the anomaly in order to make accurate predictions possible, which means obtaining accurate material and thermal properties of the delamination to be detected. This step can be particularly challenging for damage induced defects. Finally, the authors recommended that predictions based on standard one point identification methods, which have proven imprecise, should be replaced by two point identification, also using the inflection position of the contrast curve. While this last conclusion is mostly specific to pulsed thermography (which is highly sensitive to environmental conditions), this paper represented a great improvement with respect to Krishnapillai In a similar effort, Susa et al. (2007), created a model (in COMSOL 3.2) of a test sample so that the FE method could be used to solve the problem of transient heat transfer occurring in experimental conditions, specifically the simulations of infrared thermography in complex geometries. The authors adjusted unknown parameters of the numerical model such as power density of the heat source used in experiment, convective heat transfer coefficients and sample surface emissivity to obtain results of numerical simulation as close as possible to those obtained experimentally. The surface temperature decay curves were extracted from the numerical model results. An unstructured mesh was used consisting of tetrahedral elements; in contrast to the HEXANODE elements applied in Krishnapillai et al. (2005 and 2006), the reason(s) to use tetrahedral elements was not explained in the paper; however, the complexity 18

32 of the geometry probably played a role in this decision. One of the conclusions of the study was that in order to compare experimental and numerical data, the realistic, non-uniform experimental heating conditions have to be replicated. On the other hand, the FE method can provide flexible numerical experiments in which the analyst can convert uncontrollable laboratory variables (ambient and boundary conditions, defect characteristics) into controllable ones in other to study testing configurations. This will be one of the ideas applied of this dissertation. Other researchers have also used tetrahedral element meshing for the simulation of infrared thermography, for example, Hain et al. (2009) created a generic 3D FE model, with a sample with a defect under the surface. The model consisted of tetrahedral elements, as in the paper of Susa et al. (2007), it seems that the complexity of the geometry was the main factor in the selection the type of mesh for the simulation. Figure 2.11 presents the meshed geometry used in the FE models. The goal of the infrared thermography inspection was to reveal defects and inhomogeneities inside the body, and for this purpose, tetrahedral elements can be sufficient. Figure Geometry and mesh used by Hain et al. (2009). The 3D FE model was solved for various materials (concrete, metals, polymers, etc.) with subsurface defects. The simulation results were used for optimization of the active infrared thermographic method, determining the optimal setting of heat pulse parameters and 19

33 the observation time. The results of FE simulations were experimentally verified in laboratory experiments and the active pulse thermography method was also successfully used at the investigation of frescos and mural paintings. In a more recent works of Van Leeuwen et al. (2011) and Cannas (2012) FE modeling was applied specifically to simulate the PPT technique for the detection of defects in concrete structures. The model of Van Leeuwen et al. (2011) consisted of 3D cubic, hexahedral elements with 1000 cells in total. The objective was to simulate various experimental setups that included different defect depths, heating powers or heating times. The model of Cannas (2012) consisted of tetrahedral elements; the complexity of the geometry and the ease of meshing with tetrahedral elements were probably the main causes in their selection. Both studies computed the surface temperature in front of sound concrete and in front of an internal defect. These surface temperature measurements were assembled to create virtual infrared camera measurements to evaluate the efficiency of PPT. Examples of these virtual camera images are shown in figure Figure Examples of virtual infrared images by Van Leeuwen et al. (2011). Van Leeuwen et al. (2011) demonstrated an important benefit of FE modeling of the thermography process; in their case the simulation of PPT improved the detection of defects even with a non-uniform heating profile. By running many simulations of different 20

34 configurations, the authors were able to identify the best PPT parameters for the experiments. It would not have been possible to replicate the same number of tests in the laboratory. This paper showed how important FE simulations can be in saving time and optimizing parameters. Even though the mesh was not refined for PPT purposes, the use of FE along with test data shows the great impact FE simulation are capable of providing in PPT applications. However, the output parameters of the PPT (depth and phasegrams) could not be compared to the experiments due to the lack of test data. Recently, Ishikawa et al. (2012) studied the detection of deeper defects by using pulsed phase thermography. As part of this effort they conducted numerical calculations based on 2D finite element method (FEM) to elucidate the effect of defect dimensions on the results. They simulated defects with various sizes in polymethylmethacrylate (PMMA) specimen. The defect geometry consisted of flat-bottomed holes inside the samples. Figure 2.13 presents the geometry used in the investigations. The decision of using flat-bottomed holes to represent defects and the 2D FEM geometry, contrasted with the recommendation from M. Krishnapillai et al. (2006). Also no mention to the mesh characteristics was found in the paper. They also in 2013, obtained 2D FE temperature solutions, see figure 2.14, to evaluate how the noise affects the detectability of a defect, they used an asymmetrical model without mentioning any characteristic of the mesh, the temperature data was contaminated with white noise and compared to a clean FE solution before and after DFT algorithm was applied to it, how the noise transmitted to the phase calculation and affected the defect detectability was evaluated, suggesting that minimization of noise in the temperature data improve the detectability in the phase image. 21

35 Figure Defect geometry of Ishikawa et al. (2012). Figure FE geometry model, noisy and clean temperature and phase comparison of Ishikawa et al. (2012) Guidelines for the Research in the Dissertation Based on the literature review presented above, the following research areas, to be addressed in this dissertation, were identified to contribute to the further development of finite element analyzes for infrared thermography: 22

36 3D tetrahedral and hexahedral models have been implemented to calculate temperature profile on the surface of specimens, but in PPT a FFT algorithm is applied to this temperature time history. Therefore, any error in the surface temperature estimation will be transmitted (and often magnified) in the FFT and phase image calculations. The accuracy of the phase image calculations further propagates into the blind frequency calculation, and ultimately translate into the defect depth estimation accuracy. It is therefore critical to minimize source of errors, to compare FE element types and spatial and temporal resolution limits and their performance specific to PPT simulations. The use of flat-bottomed holes and Teflon to represent a defect under the surface is not an accurate description of the nature of the defect formed either in real or laboratory conditions. Accurate material properties of the defect should be obtained in order to model the PPT inspection process with FE; however, these are not necessarily known a-priori. These defect properties in particular affect the transverse heat flow and its interaction with the surrounding matrix, which can significantly affect the output data from PPT. Once the proper meshing strategy and physical properties are identified, the FE models of the PPT process can be used to study the linear depth inversion procedure based on the calculation of defect blind frequencies used currently. Discrepancies from this linear behavior have been previously observed experimentally, even for artificial defects; therefore, it is paramount to fundamentally understand the theoretical relationship between blind frequency and defect depth for realistic specimen geometries, without the influence of measurement noise. 23

37 In PPT the heating stage is only used to create sufficient thermal contrast on the surface to be later discarded, however it contains valuable information about the CFRP sample and the defect under its surface. Obtaining relations that allow the estimation of the surface thermal contrast for both defective and non-defective pixels can be helpful to NDT analysts. Current defect depth prediction methods in PPT are based on one parameter, the blind frequency. As stated in the literature review; single parameter analyses can lead to inaccurate results. It is therefore important to identify other measurement parameters that could be integrated with the blind frequency to increase the depth prediction reliability. For example, using the initial temperature contrast of the cooling transient between a defective and a sound pixel and the blind frequency might provide ample data to improve current predictions. This initial thermal contrast is created in PPT after a heat flux is applied on the front surface for a certain time, being able to model this period is imperative and providing ways to estimate or predict the initial thermal contrast accurately will help the IR thermography analysts in designing good experiments and obtaining also excellent results. The defect identification process is further complicated in models of low velocity impact damage in laminated composites. For this case the extent of damage grows from the rear surface of the specimen with decreasing surface areas and is present at the multiple interfaces between plies. Therefore, the damage state cannot be represented by a single defect with a surface area and thickness. We also would like to contribute to the modeling of PPT applied to low velocity induced damage in CFRP samples. 24

38 Improvement in the areas stated above will extend previous work and research on PPT, and will be also useful contributions to the NDT community. 25

39 Chapter 3 Finite Element Methodology This chapter reviews the finite element method and details the methodology applied in this dissertation to simulate the PPT process. This chapter also compares the performance of two models, one meshed with tetrahedral elements and a second one meshed with hexahedral elements. The mesh of choice for the rest of the research will be based on these results. Finally, the estimation of the material properties and boundary conditions needed to simulate the specific experiments used in this dissertation are also presented. Specifically, the defect material properties, the convection heat transfer coefficient and the heat flux on the surface are estimated from known test data. 3.1 Finite Element Formulation of the Heat Transfer Equation The physical heat transfer through the specimen in a PPT experiment can be described as a transient, nonlinear problem. Locally, at each point in the specimen, the heat transfer is temperature dependent. In this case, the fundamental equation has the form in Cartesian coordinates (Lewis et al., 2004). q q q T c x y z t x y z Q (3.1) where qx, qy, qz are the x-y-z components of the heat flow per unit area (the heat flux), is the material density, c is the material heat capacity, T is the temperature, t is the time and Q is the internal heat generation source. Fourier s law relates the heat flow components to temperature as 26

40 q x = k x T x, q y = k y T y, q z = k z T z (3.2) where kx, ky, kz are the material thermal conductivities in the x-y-z directions. Combining these equations yields, T k T T T x ky kz Q c x x y y z z t (3.3) These fundamental equations can then be solved for known initial and boundary conditions. The initial condition takes the form of the initial temperature distribution, whereas the boundary conditions can have the following forms: a specified temperature; specified heat flux; convection in which the heat flux is proportional to the temperature difference between the body surface and the environment; and radiation in which the heat flux is proportional to the difference in the fourth power of the body surface and environment temperatures. In our model of the PPT experiments, Q = 0 through the interior of the specimen, as no heat is generated by the material itself; heat flux is only added to the specimen at the boundaries. In the FE method, equation (3.3) is discretized at the nodal locations, resulting in matrix equations to be solved for the unknown temperatures at each time step. Full details of these discretized equations are well known and given in Lewis et al. (2004). In order to accurately model the PPT process with sufficient convergence to calculate the Fourier transform frequency information and therefore the blind frequencies, small time steps must be applied over the duration of the measurement window (i.e. the truncation window). Furthermore, as the known initial condition is that prior to heating of the specimen, both the heating and cooling period must be simulated, as shown in Figure

41 In contrast, measurements are only made in the cooling period, after the surface temperature has fallen below the saturation temperature of the thermal camera, therefore the simulation results will be truncated to replicate experimental data. Figure 3.1. Schematic of measurement and simulation windows during PPT. ΔT is the temperature of a point on front surface of specimen. Tsat is saturation temperature of thermal camera. The benchmark specimen configuration chosen for this study was based on the ASTM E standard, as experimental data was available through previous testing in our laboratory (Pawar and Peters, 2013). The specimen was a 125 x 125 mm plate of 12 layers of 2D twill, carbon-fiber reinforced epoxy (CFRP). During layup and curing of the plate in a hotpress, artificial defects were introduced by placing polymer foam (Rohacell 71) square inserts between individual laminae. The inserts ranged in size from 5 x 5 mm to 15 x 15 mm. The artificial defects were also placed at different depths within the laminate. After curing, the total thickness of the plates was 2.60 mm. Once cured, the front surface of the samples was spray-painted with black, flat enamel paint to make the surface emissivity more uniform for more accurate thermal imaging. Figure 3.2 shows a PPT phase image of one specimen. 28

42 Figure 3.2. Measured PPT phase image of CFRP specimen at with multiple polymer foam inserts, Pawar and Peters (2013). Defect depth increases from bottom to top of image. The specifics of each step in the modeling and data post-process are described in the following sections. The actual modeling parameters were iterated until the model properly converged. Only the final model specifics are given here. 3.2 Modeling Strategy The computational flow chart of the modeling process used in the dissertation is presented in figure 3.3. The process is comprised of four main stages, each of which is outlined in the following sections: 1) Geometrical and Physical Model. 2) Creation of the FE model. 3) Solution of the FE model. 4) Result post-processing. More detailed descriptions of the modeling process specific to the simulation of PPT are found in sections 3.3 through

43 Stage 1 Geometrical model And Physics Stage 2 Creation of the Finite Element Model Stage 3 Solution of the Finite Element Model 4-3 Stage 4 Results Post-Processing Figure 3.3. Flow chart of finite element modeling process Geometrical and Physical Model In the first stage of the modeling process, the analyst must study the physics of the problem to be solved and make decisions on simplifying the geometry and the physics, without compromising the accuracy of the results. At this level the analyst should decide the physics to include and not include in the model, what parameters need to be estimated or calculated, material properties and geometrical details that are not relevant to the solution, e.g. geometrical details like fillets far away from the area of interest can be removed. 30

44 The geometry model in this dissertation was created using the code package CUBIT, which is a full-featured software toolkit for robust generation of two- and three-dimensional finite element meshes (grids) and geometry preparation (CUBIT, 2015). Its main goal is to reduce the time to generate meshes, particularly large hexahedral meshes of complicated and interlocking assemblies. Mesh generation algorithms include quadrilateral and triangular paving, 2D and 3D mapping, hexahedral sweeping and multi-sweeping, tetrahedral meshing, and various special purpose primitives. Even though CUBIT s main purpose is to generate meshes, it can also be used to create geometries, and this gives the analyst the advantage of not having to import the geometry into the meshing software Creation of the Finite Element Model Once the details of the geometry and the physics are defined, the analyst can move on to meshing the geometry, i.e. discretizing a geometric object into a set of finite elements for computational analysis or modeling. The analyst at this point should also study the different element and meshing types and make decisions on which one best suits the specific problem. The software package CUBIT was used for this effort of meshing the geometric solids, it can be noticed that the same code CUBIT was utilized for both creating and meshing the geometric model. For complex geometries, a software package more equipped for geometry creation and manipulation is necessary and the geometry will need to be saved (exported) in format that the meshing code understands (import the file). For management of multiple configurations, the meshing code CUBIT offers the analyst the option to create journal files, every command entered with buttons in the CUBIT GUI version has been recorded in CUBIT s command language in command tab. In this fashion, the analyst can create a model once and save all actions into a journal file from the history. 31

45 Subsequent configurations (changes to the first created model) can be generated by just making changes into the journal file and running it, see Appendix I for an example of a journal file in CUBIT Solving the Finite Element Model Stage three involves the creation of the input file for the FE code and solution of the problem, the input file structure is code dependent, but common factors among all FE codes are the following: The FE model is called from this file; The file assigns of material, initial and boundary conditions; The file defines characteristics of the numerical method, e.g., tolerances of convergence, time step (adaptive or fixed), simulation time and frequency of output; The file defines the variables to be output and their location; The FE code utilized in the dissertation was ARIA (see appendix II for an input file example), which is a Galerkin finite element based program for solving coupled-physics problems described by systems of partial differential equations (PDEs) and is capable of solving nonlinear, implicit, transient and direct-to-steady state problems in 2D and 3D on parallel architectures (ARIA, 2007). The output of this stage represents the solution of the FE software; in this dissertation the output consisted of a matrix of 200 x 200 temperature values (gauges) on the heated surface of the specimen to simulate an infrared camera resolution of the same level of resolution. The 32

46 temperatures were obtained from these gauges, during the cooling part of the PPT simulation and the values were saved in an excel CSV file for post processing using the Matlab based software IR_VIEW (Klein et al., 2008). The data was recorded (observation time) for 13 seconds at a frequency (sampling frequency) equal to 10 Hz to replicate the experiments Post-Processing Post processing the results is the next step, is consists of analyzing the data, comparing it with experiments or numerical or experimental benchmarks, plotting, calculating different parameters and drawing conclusions. The post processing is done with data analyzing codes that might be application dependent. In this dissertation IR_VIEW, Matlab and Excel were used for this purpose Interoperability Stages Interoperability is the ability of making systems and organizations to work together (inter-operate). This ability is paramount in the process of simulating a problem with FE analysis. The ideal path, 1 -> 2, 2 -> 3 and 3 -> 4 is represented with red arrows in figure 3.3. In this path, the analyst was able to correctly capture the physics, simplify the geometry efficiently, mesh the geometry, define blocks and side sets, create the input file for the FE code without making any mistakes, output the correct variables at the right time and frequency and post process the results. This ideal path can be followed only in very simple problems (both physically and geometrically). In more realistic simulations, the task becomes more difficult when the model involves more than one physical phenomena (coupled physics) and complex geometries. Revisiting figure 3.3, a more realistic approach is represented with the blue arrows. 33

47 Arrow 2 -> 1 represents the moment when the analyst after or during the meshing process has to simplify the geometry or add geometrical details. Arrow 3 -> 2 represents the revisions based on the convergence of the solution of the FE simulation, including refining the mesh on the region of interest. Arrow 4 -> 1 represents the case when revisions to the geometry are needed after obtaining a solution. The decision to modify the geometry to add or eliminate details is represented by this arrow. Arrow 4 -> 2 represents the case when the analyst, after obtaining a solution, decides to modify element blocks, sidesets or refine the mesh more or less in specific locations. Arrow 4 -> 3 represents the case when errors in the input are caught after obtaining a solution and going back to modify the input file is required. 3.3 Hexahedral vs Tetrahedral Meshes The first results presented are for the selection of the right mesh type, in particular to show why it is important of using an appropriate element type for the simulations. From the literature review it was seen that the main type of meshes used in IR FE simulations were hexahedral and tetrahedral; and that its selection can also depend on the physics of the problem to be simulated, the desired level of accuracy, the complexity of the geometry, and the ability of the finite element analyst to mesh (Benzley et al., 1995; Wallbrink et al., 2007). It is our goal in this section to compare the performance of both, tetrahedral and hexahedral models for PPT applications. For the FE simulations in this section, a geometrical model with a single defect was constructed in the program CUBIT; we called this model single defect model. CUBIT is particularly suited for meshing of large or complex geometries with 34

48 hexahedral elements. Figure 3.4 shows an example single defect model. The effect of defect depth and size was modeled by changing this model, so that individual defects could be isolated, rather than by modeling all the defects in a single geometry. Figure 3.4. Single defect model geometry created in CUBIT. Two single defect models were created and meshed with both tetrahedral and hexahedral elements to evaluate meshing errors. The defect was inserted under the surface at a 0.65 mm depth of dimensions 10 x 10 x 1 mm. To make sure that both meshes were created comparable, i.e. the differences in simulations were due to element types, not meshing resolution, the same number of element intervals were assigned on each edge of the specimen. For both single defect models, sufficient layers of elements were created between the defect and the front surface of the sample, to properly capture the physics of the PPT experiment, including the initial thermal shock at the beginning and ending of the thermal pulse, the transverse thermal wave diffusion, and the wave reflection off the defect-cfrp boundary. The total of elements was 1,077,696 in the CFRP matrix and 2304 elements for the defect in the hexahedral model and in the tetrahedral single defect model the CFRP matrix contained 4,288,181elements and the defect elements, it should be noticed the larger number of elements in the tetrahedral model with respect to the hexahedral one for the same number of intervals on the edges, creating more computationally demanding situations. The resulting 35

49 meshes for the defects are shown in Figure 3.5(a) and (b). The initial temperature condition was set to 20 C. (a) (b) (c) Figure 3.5. Mesh testing simulation: (a) hexahedral mesh; (b) tetrahedral mesh; (c) location of surface temperature measurement points. To test the simulations, the temperatures at five different elements on the front surface of the specimen were calculated. These locations, shown in Figure 3.5(c), were chosen such that number 1 is at the center of the specimen and above the center of the defect, while numbers 2-5 are at the four corners of the specimen, mm away from each edge. Temperatures were obtained from the simulations at every 0.1 s. Afterwards, the blind frequency was calculated from the temperature histories, with location 1 representing the defect and locations 2-5 representing the sound region of the specimen. As these four locations are symmetric with respect to the specimen geometry and heating, they should ideally produce the same results. These simulations were intended for evaluating mesh performance only being accuracy of the results not too important at this point, therefore the material properties were selected as the average of CFRP and Rohacell foam 71, Table 3.1, for the defect and CFRP for the matrix, for the convection heat transfer coefficient, h, a value in the laminar region for air on a vertical 36

50 wall was selected and a heat flux q= 1.0 W/cm 2 was applied on the surface for 7 seconds, both models were allowed to cool down at room temperature for 13 seconds; the simulation results are listed in Table 3.2. The temperature difference is between each location from 2-5 and the center pixel, location 1 at the beginning of the cooling period. As expected, the hexahedral mesh produced the same temperature difference for the four locations. However, the results of the simulation with the tetrahedral mesh were not symmetric, with a maximum temperature difference of 0.05 C between the locations. Table 3.1. CFRP and Rohacell Foam 71 thermal properties. Property Symbol CFRP Rohacell Foam 71 Density (g/cm 3 ) x10-2 Thermal conductivity (W/(g C)) k 8.0x x10-4 Specific heat (J/( g C)) c The resulting average temperature was also 3.9% different from that of the hexahedral mesh. As the temperature information is used to calculate the defect blind frequency, its effects on the blind frequency was also calculated, again listed in Table 3.2. For the blind frequency calculation, the tetrahedral mesh values varied as much as 9.2% from the hexahedral mesh value, using the threshold criterion of rads. Therefore hexahedral meshing was applied for all following simulations. The phase vs. frequency output for the blind frequency calculation is shown for the hexahedral mesh in figure

51 Figure 3.6. Hexahedral mesh phase plot. Table 3.2. Blind frequency and Initial Thermal Contrast obtained using tetrahedral and hexahedral meshes. Parameter Average Tetrahedral Mesh Temperature Difference ( o C) Blind Frequency (Hz) Hexahedral Mesh Temperature Difference ( o C) Blind Frequency (Hz) Boundary Conditions For remaining hexahedral simulations in this chapter, the mesh refinement was increased with respect to the previous section, the model contained now a total of 1,620,000 elements shared between the CFRP (1,617,544) and the defect (3456), the ambient temperature was set to 20 C and perfect thermal contact was defined at the defect-cfrp boundaries. As the front surface of the specimen was painted, the surface emissivity was set to 0.95 ( 38

52 thermography.com/material-1.html). The convection heat transfer coefficient, h, was estimated using the correlations of Churchill and Chu (1975) for free convection near a vertical wall. h = k L h ( / Ra L 8/27) (1+( Pr )9/16 ) 2 (3.4.a) where Pr is the Prandtl number, k thermal conductivity of air and Lh is the characteristic length, defined as the ratio of the plate surface area/perimeter. While equation 3.4.a applies for all flow regimes, it can be improved for laminar flow (RaL < 10 9 ), h = k L h ( Ra 1/4 L (1+( Pr )9/16 ) 4/9) RaL < 10 9 (3.4.b) As h is a function of surface temperature, which changed during the experiment, an average of the value at the beginning of the cooling transient and after thermal equilibrium was reached was used in the model. The Rayleigh number, RaL, which describes the relationship between buoyancy and viscosity within a fluid, was calculated as Ra L = gβ να (T s T )L 3 (3.5) where g is the acceleration of gravity, is the kinematic viscosity of air, is the thermal diffusivity of air and is the thermal expansion coefficient of air (which can be estimated as 1/T for an absolute gas where T is the absolute temperature). In equation (3.5), L is the characteristic length of the plate which in this case is the side length, Ts is the surface temperature of the plate and T is the room temperature. Values for the heat transfer coefficient 39

53 calculation are given in Table 3.3 from the experimental data in (the engineering toolbox), immediately after the heat pulse, Ts = 44 C and at thermal equilibrium, Ts = 33 C. Table 3.3. Properties of air for convection heat transfer coefficient. Property Value Property Value g 9.81 m/s 2 T 20 C x 10-6 m 2 /s Pr x 10-5 m 2 /s k W/m C L 125 mm Lh mm For both cases, RaL < 10 9, therefore the convection heat transfer coefficient is estimated as given in equation (3.4b). This approximation yielded heat transfer coefficients of 15.9 and 13.6 W/m 2 K at the end of the heat pulse and at thermal equilibrium respectively. Therefore, an average value of 15.0 W/m 2 K was used in the simulations. Finally, to check that the assumption of only convective heat transfer was reasonable, the Nusselt number was also calculated for the heat transfer conditions. The Nusselt number is defined as Nu L = hl k (3.6) It represents the ratio between convective and conductive heat transfer for the system. For the times corresponding to the end of the heat pulse and at thermal equilibrium, NuL = 19.4 and 16.5, therefore the assumption was reasonable. The remaining parameter to be included in the model as a boundary condition is the applied heat flux. Its value was determined by iteratively fitting the simulation results to the experimentally measured surface temperatures, until a reasonable match was obtained. The temperature profile of a surface pixel far away from a defect was studied to determine the applied heat flux (which was unknown in the experiment), 40

54 in this instance, only CFRP properties were needed. An applied heat flux of 1.0 W/cm 2 best fit the maximum temperature change of the pixel away from the defect; therefore, it was used for all simulations. Table 3.4 presents the material properties used in the FE models. Table 3.4. Properties of materials. Property Symbol CFRP Rohacell Epoxy Air Polymer Foam 71 Density (g/cm 3 ) x x Thermal conductivity (W/(cm C)) Specific heat (J/( g C)) k 8.0x x x x x10-3 C Estimation of Defect Thermal Properties The next step was to fit the defect properties such that the model also reproduces the behavior of a pixel above the defect. To determine the defect properties, one defective pixel above the center of the defect was selected and its predicted temperature compared to that of the experimental data while the defect properties were varied. The properties of the original polymer foam, seen in figure 3.7 (a) were given by the manufacturer. When the foam was inserted into the laminated composite it was compressed during the cure under heat and pressure, reducing the volume to approximately 33% of the original volume, as shown in figure 3.7 (b) (Pawar and Peters, 2013). 41

55 Figure 3.7. Schematic of polymer foam defect transformation process: (a) polymer foam before compression; (b) after compression and (c) after resin infusion. Further, it is assumed that resin entered the foam during this same period, as shown in figure 3.7(c). The, k and c for the final defect were estimated using the rule of mixtures, assuming any property, P, can be written in terms of the constituent properties and their volume fractions, P = M P i V i i=1 (3.7) where M is the number of constituents, and Pi and Vi are the property and volume fraction of the i th constituent respectively. In the original Rohacell foam (uncompressed) (figure 3.7(a)); the properties are a mixture of the air and polymer, as written in equation (3.8), where Vuncomp is the volume fraction of polymer. P Rohacell = (1 V uncomp )P air + V uncomp P polymer (3.8) The properties of the Rohacell foam and air are given in table 3.2. Based on the manufacturer s reported volume fraction of Vuncomp = (Schimper et al., 2009), the 42

56 properties of polymer were calculated and are also listed in table 3.4. The polymer volume fraction after compression is approximately three times the uncompressed Rohacell foam polymer fraction assuming negligible compression of the polymer (Pawar and Peters, 2013) V comp = 3V uncomp = (3.9) During fabrication of the composite under applied heat and pressure, the defect is infiltrated with resin, removing the air. Therefore, the defect properties are again modified replacing the air properties with those of resin material, also listed in table 3.4. The final defect properties are listed in table 3.5. P defect = (1 V comp )P resin + V comp P polymer (3.10) Table 3.5. Physical Properties of the Defect. Property Symb Defect Density (g/cm 3 ) 1.20 Thermal conductivity (W/(cm C)) k x10-3 Specific heat (J/( g C)) C To verify the defect properties, the temperature profiles of a surface pixel above the defect and one far from the defect were obtained with the simulations and are plotted in Figure 3.8. The simulation was run using the previously obtained boundary condition, namely, applied heat flux of 1.0 W/cm 2 on the surface and h = 15.0 W/cm 2 K. The defect size was 10 x 10 x 1 mm, located at a depth of 0.65 mm. Both the initial temperatures and the temperature profile of the pixels matched the experimental data of Pawar and Peters (2013), meaning that the defect properties are representative of the resultant defect after the preparation process. 43

57 Figure 3.8. Calculated cooling transient curves for a surface pixel directly above the defect (perturbed region) and far from defect (sound region). 44

58 Chapter 4 Finite Element Modeling of Pulse Phase Thermography 4.1 Introduction The PPT procedure includes three basic steps, these being application of a heat flux for sample heating, collecting temperature data during the cooling period and post processing collected data for defect identification. The heating process is very important because at the end of this period sufficient thermal contrast between the defective area and the sound area needs to be created in order to make good predictions about the defect. The length of the heating period is application dependent, namely, certain applications require longer heating periods than others (Pawar and Peters, 2013; Mabry et al., 2015). But the goal is the same, to induce enough temperature contrast on the surface. It is our intention, at the beginning of this chapter to put emphasis on this first step, to use FE models to model the heating process and find ways to predict the initial thermal contrast for both defective and sound pixels. This will allow the analyst to optimize the PPT process, namely, determining the heat flux and its duration to accomplish a desired thermal contrast without the need of expensive testing procedures. The second step in the PPT procedure is collecting temperature data during the cooling period and post processing it for defect identification. Then, in this chapter the entire PPT process is simulated numerically, using the boundary conditions and material properties calculated in chapter three. 45

59 Once that the blind frequencies are determined, they can provide to the analyst, along with the initial thermal contrast values, with more information about the defect under the surface. This chapter also introduces a method to determine the defect depth and thickness using the two parameters mentioned earlier (initial thermal contrast and blind frequency), taking into consideration the role of defect thickness on initial thermal contrast and blind frequency results. At the end of this chapter two case studies are presented, one on the role of the proximity of a defect to another defect blind frequency and the other one on the role of the area-perimeter ratio on the blind frequency. 4.2 Semi-Empirical Approach to Plate Heating with Defect Under the Surface In PPT, a heat flux is applied on the surface of the sample to be inspected, here a square pulse function of amplitude q and duration t. The solution for the surface (z=0) temperature of a semi-infinite solid (z > 0) due to a supply of heat at a constant rate per unit area, q for t > 0 can be written as (Carslaw and Jaeger, 1959), (T T o ) = 2q αt πk (4.1) where To is the ambient temperature, k is the thermal conductivity of the material and α its diffusivity. Solving for q, q = (T T o) πk 2 αt (4.2) Now including a term for convection at the surface, the heat flux takes the form q = (T T o) πk 2 αt + h(t T o ) (4.3) 46

60 where h is the convection heat transfer coefficient. Solving again for (T T o ) we find 2q αt (T T o ) = πk + 2 αt h (4.4) Equation (4.4) can be used to describe the temperature of a pixel on the surface of a defect free plate with convection heat transfer on its surface, as long as the pixel is far away from the defect and far away from the plate boundary. Figure 4.1 plots the temperature history from our FE simulations of the heating along with equation (4.4) results for a CFRP sample with q=1.0 W/cm 2 thermal pulse applied for t=7 seconds and h= W/cm 2 o C. The diffusivity for CFRP was calculated from the material properties in table 3.2. An excellent match between those results can be seen from this figure Equation (4.4) solution Finite element solution Temperature (oc) Time (s) Figure 4.1. Heating transient finite element solution and plot of equation (4.4). As the heat flux is being applied on the surface and heat starts diffusing toward the back of the sample, the presence of a defect alters the energy balance corresponding to a pixel 47

61 on the surface above it; the defect actually behaves like a heat source increasing the pixel temperature with respect to a sound pixel in the following fashion, q + C(T T o ) = (T T o) πk 2 αt + h(t T o ) (4.5) where term C(T T o ) represents the contribution of the defect to the energy balance of the surface pixel. Solving for (T T o ) the following equation is obtained. 2q αt (T T o ) = πk + 2 αt(h C) (4.6) We propose to defined the coefficient C as C = G(d) k d (4.7) where Is the defect thickness; d is the defect depth and kd the defect thermal conductivity. The reasoning behind this choice of C is based on a unit balance and understanding that (T T o ) decreases with d and increases with. G is coefficient that is a function of the depth will depend on the configuration and will have units of length -2. After substituting C into equation (4.7) it takes the form 2q αt (T T o ) = πk + 2 αt[h G(d) k d ] (4.8) Equation (4.8) will allow the analyst to fit coefficients G for the solution of (4.8) to match the FE at different depths. FE results for a single defect model with a defect of area 100 mm 2 and thickness 0.1 cm embedded at six different depths were used to be compared to equation (4.8) results under the same conditions. A coefficient G that minimized the 48

62 temperature difference between equation (4.8) and the FE solution was obtained for each depth, table 4.1 presents these results and also the FE solutions. Table 4.1. Coefficient G as a function of depth and the corresponding temperature contrast on the surface compared with the FE solution. Depth (cm) G (cm -2 ) Temperature contrast ( o C) Equation (4.8) Temperature contrast ( o C) FE solution Plotting G vs depth an exponential curve fitted the data with R 2 =0.99, see figure 4.2, this equation allows predicting G values for deeper locations and getting its corresponding temperature contrast on the surface by means of equation 4.9. In this equation the coefficients outside the exponential have unit cm -2 and the ones inside the exponential cm -1. G = 2742 exp( 4.727d) 2728 exp( 4.707d) (4.9) 12 G vs Depth Exponential fit 10 8 G(cm-2) Depth (cm) Figure 4.2. G versus depth plot and the exponential fit. 49

63 Now substituting equation (4.9) into (4.7) the following relation for C is obtained, C and h (convection heat transfer coefficient) have the same units, namely, W/cm 2 o C. C = k d (2742 exp( 4.727d) 2728 exp( 4.707d)) (4.10) C, as it was stated before, represents the contribution to the energy balance of a surface pixel of a defect of thickness and conductivity k d embedded at a depth d. Substituting a depth equal to cm for a thickness equal to 0.1 cm and the defect heat conductivity (table 3.6) in expression (4.10), a C value of W/cm 2 o C can be calculated. With this C value we can estimate the temperature at the end of a heating transient for a defect located at that depth, figure 4.3 presents the equation (4.8) results for the defect located at depth equal to cm and C equal to W/cm 2 o C and the FE solution for the same problem. For a sound pixel with T= o C the contrast will be o C in good agreement with the FE prediction which is o C Finite element solution Equation (4.8) solution Temperature (oc) Time (s) Figure 4.3. Plot of equation (4.8) and FE solution for a defect of area 100 mm 2 and thickness 0.1 cm at a depth of cm. 50

64 4.3 Defect Depth Retrieval based on Blind Frequency using Finite Element Simulations As it was mentioned in chapter two, the diffusion length calculated from this apparent blind frequency is assumed to be linearly related to the defect depth z0 by means of equation (2.5). The next set of simulations tested the ability of the FE simulations to accurately determine the depth of a defect based on its blind frequency. Separate single defect model simulations were performed with defects embedded inside a 12.5 cm x 12.5 cm and 2.6 mm thick CFRP plate at seven different depths: mm, mm, mm, mm, mm, mm and mm. These depths correspond to the first seven interfaces between plies, assuming that the composite plate was constructed of 12 plies. For each depth, defect surface areas of 25, 100 and 225 mm 2 were simulated. Separate simulations were performed for each defect to eliminate transverse heat flow effects between close defects. All defects had a thickness of 1.0 mm. The depth of each defect was defined to the top surface of the defect (closest to the heated surface) and all defects were located are the center of the specimen. The models were created in CUBIT and contained 1,620,000 hexahedral elements shared by two blocks that represented the CFRP matrix and the defect when material properties were assigned, for later application of the boundary conditions previously calculated in chapter 3, surface nodes were created. The FE simulations in software ARIA were initiated with a time step of 10-9 seconds, which was then adaptively changed with each time step based on the convergence criteria. The output of the FE simulations was the temperature at each node at each time step. While this spatial and temporal detail is required for accurate representation of the physics, it is much more information than that captured experimentally. Therefore, post-processing was applied 51

65 to the output of the FE simulations to replicate the experimental data. The front surface of the specimen was pixelated to represent a thermal camera pixel resolution of 200 x 200 pixels. The FE node closest to each pixel was chosen to represent the temperature for each pixel. The pixel data was then subsampled and truncated to the sample rate, 10 Hz, and truncation window, 12.8 seconds, used in the experiments. These reduced data sets were exported to MATLAB and assembled to form a sequence of simulated thermograms. The simulated thermograms were processed with IRVIEW, an open code software (Klein et al., 2008), for the calculation of the phasegrams, to replicate the processing of the experimental data. Figure 4.4 shows phasegrams obtained for simulations of the 100 mm 2 defect at a depth of 0.65 mm, calculated at different maximum frequencies, f max. (a) (c) (b) (d) Figure 4.4. Simulated phasegrams for 100 mm 2 defect at 0.65 mm depth, with maximum frequency, fmax, of (a) Hz, (b) Hz, (c) Hz and (d) Hz. 52

66 From the phasegrams, the calculation of the phase contrast-frequency information and blind frequencies was performed using the identical process as that of the experimental data. For the numerical data, a phase contrast threshold was required to set the criteria for zero phase contrast, which in this case was assumed to be radians. Figure 4.5 shows phase vs. frequency plots for a pixel far from and above the center of the defect, obtained from the simulations for a 225 mm2 defect at different depths. The location of the calculated blind frequency is also shown on each curve. Smoothing of the calculated curves was not necessary, so the blind frequency was determined simply based on the phase threshold. For this defect, the blind frequency was calculated to be 3.00 Hz. Therefore, in figure 4.5 (a) fmax is below the blind frequency, in figure 4.5 (b) fmax is approximately at the blind frequency, and in figures 4.5 (c) and (d) fmax is above the blind frequency. The defect is clearly visible in all images because the phasegrams show the cumulative phase calculated for the range 0 Hz < f < fmax. As expected, the image artifacts increase with increasing fmax. These phasegrams are similar to those seen for large defects in experimental data, such as in Figure 3.2, although the phase contrast between the defect and sound areas is heightened due to the lack of experimental noise. The phase was calculated for 64 points over the 5 Hz frequency range, therefore each blind frequency prediction is Hz = fb. The curves are not shown for the deepest defect, as there was not a point on the increasing slope where the difference between the two curves was above the threshold, i.e. the defect depth is not detectable. 53

67 (a) (d) (b) (e) (c) (f) Figure 4.5. Estimation of blind frequencies for a 225 mm2 defect with depth of (a) mm, (b) mm, (c) mm, (d) mm, (e) mm and (f) mm. 54

68 The resulting diffusion lengths, calculated from the blind frequencies using equation (2.4), are plotted in figure 4.6 for all defect surface areas. A standard practice for the application of PPT is to fit the data to equation (5.1), a linear fit for each defect area is also plotted in figure 4.6. Previous authors have shown a size effect in equation (5.1) (Wallbrink et al., 2007), so the fits were calculated separately for each defect area. Figure 4.6. Depth vs. diffusion length calculated from blind frequency of simulation data. Data and linear curve fits are shown for three defect areas. The simulation data does not follow the linear fits well, with R 2 values ranging from to When only shallow defects are considered, linearity is a reasonable assumption, however for deeper defects the location errors grow significantly. This lack of linearity for deeper defects has been observed experimentally (Pawar and Peters, 2013) and (Mabry et al., 2015). For these previous measurements, the nonlinearity was assumed to be due to material imperfections, transverse heat propagation and the finite thickness of the sample. However, the 55

69 nonlinearity is even more pronounced in the simulations where material imperfections are not included and the defect is too far from the boundaries to be significantly affected by transverse heat propagation effects. Defects were not considered at depths further than ½ the thickness of the specimen, therefore the finite thickness of the plate is also not expected to be the source. In addition, the uncertainty in the diffusion length, can be determined from the derivative of equation for the diffusion length (2.4) δμ = 1 2 δf b α πf b 3 (4.11) Using the material properties for the CFRP in section 3.5, ranges from 2.0 x10-3 to 7.8x10-3 mm for the blind frequency range of 2 to 5 Hz and is therefore not the source of the nonlinearity seen in figure 4.6. An exponential fit to the simulation data was also performed and is plotted in figure 4.7. The exponential fit matched the data extremely well, with a R 2 value for all defect areas of Figure 4.7 also shows a linear fit to the first three data points for each defect area, to show the high quality of a linear fit for shallow defects. Finally, an exponential fit was performed for all of the data points together, without consideration of the defect area and is plotted in figure 4.8. The R 2 value for the combined exponential fit was 0.993, indicating that the defect depth can be estimated with little error due to the unknown defect area. The fact that the nonlinearity observed in experimental measurements, such as those in (Pawar and Peters, 2013) and (Mabry et al., 2015), is less pronounced than in the simulation data (although still important) is potentially due to the fact that material imperfections and 56

70 interfaces introduce additional attenuation of the thermal wave and experimental noise affects the curve fitting process. Figure 4.7. Fits to depth vs. diffusion length calculated from blind frequency of simulation data. Exponential fits are shown as solid lines; linear fits to first three data points are shown as dashed lines. Figure 4.8. Exponential fit to depth vs. diffusion length calculated from blind frequency of simulation data. 57

71 The results show that an exponentially decaying representation of the diffusion length with defect depth is more accurate than the commonly applied linear fit, particularly for deeper defects. These findings are consistent with previous experimental observations. A similar concept of the blind frequency is used in lock-in thermography, again based on the concept of the diffusion length. However, in lock-in thermography only a single frequency is excited at a time. Therefore, no non-uniform amplitude distortion in the thermal wave would be observed and apparent shifts to the blind frequency would not be observed. It is also expected that the presence of losses at defect-material boundaries would further affect the nonlinearity of blind frequency calculations. Now that this finite element simulation technique has been shown to model phase and apparent blind frequency measurements from PPT, it can be used to evaluate transverse heat flow effects on given specimens. Transverse heat flow was a significant artifact in this study, and incorporated in model, as a boundary condition, now that a FE modeling strategy has been developed. Further simulations could also be applied to understand the role of the defect thermal contrast, the phase contrast threshold value and other parameters on the experimental results. Compensation for errors due to these different factors could also be built into the processing of PPT results. 4.4 Role of Thickness on Blind Frequency and Initial Thermal Contrast We have already simulated the role of the defect depth on the blind frequency; however, the defect thickness, will also potentially affect the blind frequency through the defect thermal 58

72 contrast. It is our goal, in this section, to model with FE the role of the delamination thickness on blind frequency and initial thermal contrast as well Finite Element study of the Role of Thickness on Blind Frequency and Surface Thermal Contrast FE simulations of models in which the depth was kept constant and the defect thickness was changed were solved, a total of five single defect models were created and the same boundary conditions defined before were applied. The blind frequency increased with an increment in the defect thickness in a way that can be said to be exponential, this same result was also obtained at different depths, for this reason only the graph for depth cm is presented in figure 4.9 where an exponential fit was applied to the blind frequency vs. thickness results for a defect depth equal to cm and provided an R 2 of From figure 4.9 it can be noticed also that the growth is asymptotic toward a blind frequency value at which regardless of the thickness of the defect, no much increment in blind frequency is obtained, in this case the asymptotic value is close to 3.0 Hz, this behavior is perhaps, due to that a defect opposes greater resistance to heat diffusion as its thickness increases, this translate into higher temperature on the surface above it and blind frequency, but also with larger thickness there is an increment in wave absorption in the defect material, this effect tends to lower the blind frequency, as the defect thickness increases the second effect becomes more relevant, creating an asymptotic behavior. Remembering that a similar fit was obtained for the case of constant thickness and variable depth, this result is expected and in agreement with what has seen in experiments. 59

73 3 2.9 Blind frequency (Hz) FE blind frequency, depth = mm Exponential fit thickness (mm) Figure 4.9. Blind frequency vs thickness plot, depth equal to cm Two-Point Estimation Strategy for Determining Thickness and Depth Expanding figure 4.9 to more depths, a total of thirty single defect models for variable thickness and constant depth were created, solved and the results consolidated in figures 4.10 and 4.11, the boundary conditions and the number of elements in the meshed blocks and sidesets were in agreement with the previous sections, namely, two blocks (defect and matrix) for initial conditions and material properties. Radiation (ε), convection coefficient (h) and a heat flux applied on the surface (q) for 7 seconds were the boundary conditions; the sample was allowed to cool down at room temperature for 13 seconds, during the cooling transient, temperature data was collected every 0.1 seconds. 60

74 Thickness (mm) depth=0.022 cm depth=0.043 cm depth=0.065 cm depth=0.087 cm depth=0.108 cm depth=0.130 cm Blind frequency (Hz) Figure Blind frequencies for defects of different thickness at different depths Thickness (mm) depth=0.022 cm depth=0.043 cm depth=0.065 cm depth=0.087 cm depth=0.108 cm depth=0.130 cm Surface Temperature Contrast (oc) Figure Defective sound pixel temperature contrasts for different thickness and depths. 61

75 An interesting aspect of the solution worth noticing from figures 4.10 and 4.11 is that multiple thicknesses can produce the same blind frequency or temperature contrast on the surface depending on the depth. In this fashion, and assuming that a delamination can only appear at depth corresponding to the in-between laminas interface (Barbero, 2013), a blind frequency of 2.4 Hz can be produced by defects of thickness 0.21, 0.39 and 0.9 mm at depth 0.065, and 1.08 respectively. Also an initial temperature contrast of 2.0 o C can be created by defects of thickness 0.41 and 0.61 mm thick at depth 0.21 and mm respectively. The range of temperature at which the solution is unique is very restricted, e.g. blind frequencies higher than 3.1 Hz and below 1.5 Hz find unique solutions also temperature contrasts above 2.5 and below 0.3 o C find unique solution, to solve the problem of multiple solutions for a given blind frequency, the initial thermal contrast will be added to the predictions allowing us to propose a two-point estimation strategy in the following fashion. For this CFRP sample (and any other), there are a limited number of depths possible. This is due to the fact that PPT can detect damage up to a maximum depth limit it (Pawar and Peters, 2013), and the assumption is that delamination (the imaged damage state) only occurs at the interfaces between plies. To propose a two-point defect characterization strategy, we hypothesize that the combination blind frequency and initial thermal contrast between a sound and a defective pixel is unique for a specific defect thickness and depth. This idea follows the recommendations from Krishnapillai et al. (2006) presented in the literature review. These recommendations were that one point estimations are imprecise; therefore, here a two data point graphical method is described to retrieve defect depth and thickness. The method is comprised of the following steps: 62

76 1. Determine the blind frequency and surface initial thermal contrast for the pixel of interest. 2. From the thickness vs initial thermal contrast plot (figure 4.11), determine the possible thicknesses and depths that satisfy that initial thermal contrast. 3. From the thickness vs blind frequency plot (figure 4.10), determine the possible thicknesses and depths that satisfy the calculated blind frequency. 4. Compare results obtained in steps (3) and (4). The combination blind frequency-initial thermal contrast provides a unique combination thickness-depth. To illustrate the method, the following example is presented. For a CFRP sample similar to the ones modeled in this work, a blind frequency of 2.90 Hz and initial thermal contrast after heating of 1.9 degrees were estimated. The initial temperature contrast and the blind frequency were calculated to be 1.9 K and 2.9 Hz respectively. Using the thermal contrast, it can be seen that two solutions are possible at depths and mm. Secondly, using the blind frequency of 2.9 Hz it can be observed that only two solutions are possible; the delamination can be located at or 0.65 mm. The combination blind frequency- initial temperature contrast therefore provides a unique combination of thickness and depth being 0.6 mm and mm respectively. This demonstrates that using extra information from the test; in this case the thermal contrast at the beginning of the cooling transient, information that is used during the post-processing stage, actually does provide more tools for the PPT analyst to gather knowledge about the defect under the surface that originated those conditions. 63

77 4.5 Finite Element Study of the Role of Defect Proximity The role of the proximity (or presence) of another defect in the vicinity of the defect of interest needs to be investigated. In this section FEA was used to model the heat transfer in order to estimate the difference in blind frequency introduced by a second defect; the distance between the defects was changed to measure how the blind frequency changes with the proximity. The software package Cubit was used to create and mesh the geometry and a journal file allowed the relatively easy creation of the different meshed models. As before the epoxy matrix consisted of 12 layers of Fiber Reinforced Epoxy with following dimensions: 12.5 cm x 12.5 cm and a total thickness of 2.6 mm, mm per layer. The defect geometry consisted of 2 defects of side length 1.0 cm, width 1.0 cm and thickness 1 mm at a depth of 0.65 mm in the epoxy matrix. The distance between the two defects was decreased at 100, 80, 60, 40 and 20% of the length of the defect, corresponding to 1.0 cm, 0.8 cm, 0.6 cm, 0.4 cm and 0.2 cm for a total of 5 FE simulations. The geometry was meshed with 1,620,000 hexahedral elements. The models were solved in FE code ARIA using the material properties and boundary conditions described before. The output consisted of a matrix of 200 x 200 temperature gauges that was defined to simulate an infrared camera resolution of 200 x 200. Surface temperatures were obtained every 0.1 seconds (10 Hz) from these gauges during the cooling transient (13 seconds of cooling time) and saved in a CSV file for postprocessing using IR_VIEW. Figure 4.12 presents the upper surface thermographs at the beginning of the cooling period (end of the heating pulse). 64

78 Temperature (oc) 45.6 Temperature (oc) Number of Pixels Number of Pixels Number of Pixels Number of Pixels (a) (c) Temperature (oc) 45.6 Temperature (oc) Number of Pixels Number of Pixels Number of Pixels (b) Number of Pixels (d) Figure Temperature after the heat pulse, distance a) 1.0, b) 0.8 c) 0.6 and d) 0.4 cm A matrix of 200 x 200 temperature gauges was defined to simulate an infrared camera resolution of 200 x 200. From these thermographs it can be seen that the highest temperature (45.6 o C) does not change with a reduction in the distance between defects. The thermographs collected during the cooling (observation) period were saved in a CSV file for FFT post processing using IR_VIEW. Figure 4.13 presents phasegrams for each case at 3.46 Hz and figure 4.14 shows the phase vs. frequency for the case of figure 4.13 (a). The other cases are not presented because they look very similiar to figure

79 Phase (Rad) Phase (Rad) Number of Pixels Number of Pixels Number of Pixels (a) Number of Pixels (c) Phase (Rad) Phase (Rad) Number of Pixels Number of Pixels Number of Pixels (b) Number of Pixels (d) Figure Phasegrams corresponding to frequency equal to 3.46 Hz, distance a) 1.0, b) 0.8 c) 0.6 and d) 0.4 cm Sound Pixel Defective Pixel -0.4 Phase (Rad) Blind Frequency = 3.02 Hz Frequency (Hz) Figure Phase vs frequency plot and blind frequency 66

80 The results obtained for the different distances between defects and the isolated defect case are compared in tables 4.2 (figure 4.15 presents the data of table 4.2 for better visualization) and 4.3 respectively. A significant difference was not observed in the value of the blind frequency when a second defect is in the vicinity of the defect of interest. Interestingly, the initial thermal contrast remained unchanged at o C. From these results it can be concluded that the impact on the detection of the presence of the defect in the vicinity of another one can be neglected according to the FE model simulation. Table 4.2. Blind Frequency and cooling transient initial temperature contrast for each distance. Distance (cm) Blind Frequency (Hz) Initial Temperature Contrast ( o C) Distance between defects (cm) Blind Frequency (Hz) Initial Temperature Contrast (oc) Figure Blind frequency and Initial thermal contrast at different distance (cm) between defects. Table 4.3. Blind frequency and initial temperature contrast for the isolated defect case. Blind Frequency (Hz) Initial Temperature contrast ( o C)

81 4.6 Finite Element Study of the Role of Defect Area Perimeter Ratio The delaminations created in CFRP samples can come in various shapes and sizes. The area-perimeter ratio is important because it determines the relative amount of transverse heat flow in the defective region and therefore the spread of temperature across the defect surface. Correction factors for the blind frequency may be required if the temperature distribution is highly non-uniform within the defect region. Therefore, it is paramount to investigate the impact of the defect area-perimeter ratio on its blind frequency. In this section, FE modelling is used to simulate the heat transfer in order to estimate the difference in blind frequency introduced by a change in the defect area-perimeter ratio. Temperature and FFT phasegrams obtained from the PPT processing will also be shown. The single defect model geometry consisted of the same CFRP plate, the only difference was that the defect geometry remained rectangular; however, the side dimensions were varied to change the area to perimeter ratio. The defect depth and area was kept constant and equal to cm and 100 mm 2 respectively. The perimeter of the rectangular defect was then changed to vary the area to perimeter ratio (A/P), as listed in Table 4.4. The case a =10 mm and b =10 mm was already run for previous simulations, for this reason, only plots and phasegrams will be shown for A/P = 2.0 and 1.18 mm. Table 4.4. Dimensions of the defects used in the simulations. a and b are the sides of the rectangular defect. a (mm) b (mm) Area (mm 2 ) Perimeter (mm) A/P (mm)

82 The geometry was meshed with hexahedral elements 1,620,000 elements shared by the matrix and the defect. The output consisted of a matrix of 200 x 200 temperature surface tracers. Surface temperatures were obtained every 0.1 seconds (10 Hz) from these pixels during the cooling transient (13 seconds of cooling time) and saved in a CSV file for postprocessing. Using previously obtained material properties and boundary conditions, the models were solved in FE code ARIA. In figure 4.16 the thermographs at the beginning of the cooling transient for defects with A/P = 2.00 and 1.18 are presented. A temperature reduction at the defective area can be observed as the A/P ratio decreases. This will likely reduce the corresponding blind frequency also; a similar behavior of reducing blind frequency with lower initial thermal contrast was noticed earlier when blind frequencies were calculated for defects of multiple thickness located at different depths. Figure 4.17 presents the phasegrams from IR_VIEW at frequency = 3.46 Hz, for defects with A/P = 2.00 and As expected, a reduction in phase can be appreciated as A/P decreases. Temperature (oc) Temperature (oc) Number of Pixels Number of Pixels Number of Pixels Number of Pixels 43.8 (a) (b) Figure Temperature at the beginning of the cooling transient for A/P = (a) 2.00 and (b)

83 Phase (Hz) Phase (Hz) Number of Pixels Number of Pixels Number of Pixels Number of Pixels (a) (b) Figure Phasegrams at frequency = 3.46 Hz for A/P = (a) 2.00 and (b) Selecting a pixel at the center of the defective area for both models and a sound pixel away from it, a plot of the phase of those pixels in the frequency domain will allow the user to determine the blind frequency with the criterion of rad utilized earlier in this chapter. Graphs corresponding to both A/P of interest presenting temperature transient and phase vs frequency plots are presented in figures 4.19 and From the phase plot a reduction in blind frequency can be noticed as A/P decreases from to Hz. These results were summarized in table form along with the square model of similar area for comparison purposes as listed in table 4.5 and shown in figure Table 4.6 and figure 4.21 show the respective change in blind frequency and initial thermal contrast with respect to the square model with A/P = Table 4.5. Results for the different area-perimeter ratio. Variable\area-perimeter ratio A/P 0.25 A/P 0.20 A/P 0.18 Blind Frequency (Hz) Initial Temperature Contrast ( o C)

84 A/P 0.25 A/P 0.20 A/P 0.18 Blind Frequency (Hz) Initial Temperature Contrast (oc) Figure Bar graph of blind frequency and initial thermal contrast for different A/P Sound Pixel Defective Pixel 42 Temperature (oc) Time (s) Figure 4.19 Temperature cooling transient for A/P =

85 0-0.2 Pixel in sound Region Pixel above defect -0.4 Phase (Rad) Blind Frequency = Hz Frequency (Hz) Figure Phase plot for A/P = Sound Pixel Defective Pixel 42 Temperature (oc) Time (s) Figure 4.21 Temperature cooling transient for A/P =

86 0-0.2 Pixel in sound region Pixel above defect -0.4 Phase (Rad) Blind Frequency = Hz Frequency (Hz) Figure 4.22 Phase plot for A/P = 1.18 Table 4.6. Blind frequency and initial temperature contrast change with respect to the square defect. Change Respect to square AP 0.20 AP 0.18 Blind Frequency (Hz) Initial Temperature Contrast ( o C) AP 0.20 AP 0.18 Blind Frequency (Hz) Initial Temperature Contrast (oc) Figure Bar plot of the relative change with respect to the square defect. 73

87 The shape of the defect has a direct impact on the blind frequency and also on the initial thermal contrast. A reduction occurred in both parameters occurred as the area-perimeter ratio is reduced, as can be seen in table 4.5 and figure This reduction could lead the analyst to conclude that the defect is deeper in the structure than where it really is located using a square defect as a reference. The initial thermal contrast is more sensitive to A/P changes than the blind frequency. The relative change in blind frequency was 0.27% for AP=0.20 and 4.34 for AP=0.18 with respect to the square case. The initial thermal contrast the change was 6.57% and 25.98% respectively with respect to the same square case, as seen from both tables 4.6 and figure In short, close attention should be paid then when applying results obtained using square artificial defect to characterize non-square delaminations. 74

88 Chapter 5 Finite Element Modeling of Low Velocity Impact Induced Damage in CFRP 5.1 Introduction The focus of this chapter is on FE modeling of PPT evaluation of low velocity impact damage (LVID) in CFRP samples. Cantwell et al. (1992) reviewed the impact resistance of composite materials and defined LVID as damage created by an impact velocity of up to 10 m/s. In contrast, Abrate (1991) reported the value for LVID to be less than 100 m/s. However, the approach used for our work, modeled in this section, was suggested by Joshi and Sun (1987) and Liu and Malvern (1987). This approach classifies impact damage according to the dominant damage modes. High velocity is characterized by fiber breakage due to penetration and low velocity is characterized by matrix cracking and delamination. Figure 5.1 presents a CFRP plate with low velocity induced damage. Figure 5.1. Post-impact images of specimen impacted at 1.75 m/s: photograph of impacted surface (Pawar and Peters, 2013). 75

89 There is a major difference between imaging laminates subjected to impact induced damage and laminates with typical manufacturing damage represented as flat bottom holes or inserts. For the case of low velocity impact created damage, the damage region within a lamina (or in-between laminae) is expected to decrease in size from the rear to the front (impacted) surface (Cantwell et al., 1992). A cross sectional view of a CFRP sample subjected to low velocity impact is presented in figure 5.2, it shows the expected growing shape of the induced defect. Figure 5.2. Cross sectional examination gives information of through the thickness damage (Alan T Nettles, 2011) The presence of multiple defects at different depths creates shadowing of deeper defects by more shallow ones during thermal imaging. However, due to the growing shape of the damage region (from the front to rear surface) all defects are visible when imaged from the front surface. An idea of the conical contour shape of the damage is presented in figure 5.3 (Pawar and Peters, 2013). The challenge is therefore not the detection of defects due to LVID, but instead the assessment of the extent of damage for a case such as seen in figure

90 (a) (b) Figure 5.3. (a) Phasegram obtained for CFRP laminated plate with LVID (one pixel corresponds to approximately 0.2 x 0.2 mm); (b) 3D impact damage progression using PPT (Pawar and Peters, 2013). In this chapter we investigate the capability of PPT to accurately assess the size of the damage regions at different lamina interfaces, in the presence of shadowing. Following an empirical approach, Pawar and Peters (2013) calculated the blind frequency at different pixels in the region of impact damage and estimated the depth of the damage at each pixel, using the linear depth to diffusion length relation. A threshold depth was then set for the distance to each interfacial layer to determine whether delamination had occurred at that interface. Figure 5.3 shows a typical phasegram for low-velocity impact in a 12 layer CRFP laminated composite and the resulting delamination area estimation for each interface layer. This calculation was performed assuming that there is no interaction in the PPT process between delaminations at different depths, and a linear relation of the blind frequencies versus depth. As higher resolution imaging data was not available for comparison, the accuracy of this reconstruction was not quantified. In this chapter, we evaluate the ability of PPT imaging 77

91 to reconstruct damage due to low velocity impacts at different interfaces within a laminate, through FE simulations. 5.2 Finite Element Solution Assuming that the damage is only present at the interface in-between the sample plies, the whole damage will be ideally modeled by a sequence of very thin delaminations growing in size with increasing depth in the sample. The result is a pyramidal physical representation suitable for implementation into a FE simulation. The model is not intended to represent a real defect, but to capture the thermal interaction of imperfections placed in a growing order from front to back surface, similar to what is found in low velocity impact induced damage. A geometric file describing a representation of low velocity impact induced damage in a CFRP matrix was created in software CUBIT and is shown in figure 5.4. The dimensions of the geometry were consistent with prior simulations, the only difference being the defect dimensions. The CFRP plate was, as before, 12.5 x 12.5 cm and 0.26 cm thick. The file included five defects whose geometry consisted of thin square blocks of 0.2 mm thickness, separated by mm, the number of defects included was based on figure 5.3 (a) where blind frequencies for five locations were calculated. The shallowest defect was the smallest one and the planar area was increased with depth. The dimensions and depth of each defect is listed in Table 5.1. Thus, a defect was implemented between each of the five first layers, assuming as before, that delamination will appear in-between the laminas. Figure 5.4. CUBIT 3D impact damage pyramidal representation of damage progression. 78

92 The isolated defects (without visualization of surrounding matrix material) from figure 5.4 are presented in figure 5.5. On the surface and above these five defects, pixels were selected for temperature tracking, data post processing and blind frequency calculations with respect to a sound pixel away from the damage area. The geometry presented in figure 5.4 was meshed in CUBIT, with 3,690,000 hexahedral elements shared by 2 blocks (the defects and CFRP matrix). Table 5.1. Defect dimensions and depth in the sample. Depth (cm) Dimensions (cm x cm) x x x x x 1.50 The same initial conditions, material properties and boundary conditions were applied, as previously described in chapters 3 and 4. As a reminder they included convection and radiation applied on the surface of the sample, a heat flux applied on the upper surface for 7 seconds and perfect thermal contact between matrix and defect. Figure 5.6 presents the meshed defect block. The change in curvature induced in the geometry by the impact was considered minimum and therefore negligible. Figure 5.5. CUBIT representation of the defects that comprised and impact damage pyramidal damage progression and the location on the surface of the temperature gauges. 79

93 Figure 5.6. Hexahedral mesh of the defect block. The FE problem was solved in ARIA and post processed in IR_VIEW. Temperature data was collected from gauges shown on figure 5.5 during the 13 second cooling transient and post processed for PPT and defect characterization using a DFT algorithm. Figure 5.7 presents a thermograph at the end of the heating process (beginning of the cooling transient) and the cooling curves for each temperature gauge and also for the sound pixel (defect free) used for blind frequency estimation are presented in figure 5.8. Temperature (oc) Number of Pixels Number of Pixels Figure 5.7. Sample thermograph at the beginning of the cooling transient 5.3 Blind Frequency Solution The cooling curves presented in figure 5.8 were sampled every 0.1 seconds, corresponding to a frequency of 10 Hz; the same as was done in chapter 4 for the different 80