Photothermal NDE of UD/Epoxy-Based Carbon Fibre Reinforced Laminates for Quantitative Porosity Analysis

High strength and low density make epoxy-based CFRP a highly interesting construction material for the aerospace manufacturing industry. Porosity represents an unavoidable defect and significantly weakens strength values dominated by the matrix. To evaluate the quality of safety-relevant components, non-destructive evaluation and thus the characterization of porous structures is indispensable. Pulsed thermography represents a fast, large-area and non-contact testing method that enables efficient estimation of material parameters. In this work, the authors demonstrate the quantitative estimation of porosity by pulsed thermography on a multi-axial laminate fabricated from unidirectional Prepregs for the first time. The characteristic, extensive expansion of the pores in fiber direction, is addressed by the 3D microstructure characterization of Cone beam X-ray computed tomography data. Hence, the application of effective medium theories and thus the model based porosity estimation is enabled. After the investigation of the effect of pore expansion on the effective thermal diffusivity in 3D finite element simulations, the quantitative photothermal porosity estimation on a sample with a global volume porosity of Φ=1.51%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Phi =1.51\%$$\end{document} is demonstrated. The accuracy of this fast and non-contact method for porosity estimation with pulsed thermography (ΔΦ=0.63%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta \Phi =0.63\%$$\end{document}) is comparable to the standard ultrasonic method. Consequently, an efficient estimation of porosity for large, complex shaped UD/Epoxy composite components is enabled.


Introduction
Unidirectional (UD) carbon fibre-reinforced plastics (CFRP), fabricated using thermosetting epoxy resin and carbon fiber, is of high interest for lightweight aerospace application. Compared to woven fabric CFRP, unidirectional preimpregnated fibers (Prepregs) offer improved structural properties (higher tensile strength-to-weight and stiffness-to-weight insufficient curing cycles applied in the autoclave [5,6]. Several studies [4,6,7] have shown that porosity has greatest effect on reducing matrix-dependent mechanical properties such as compressive strength, transverse tensile strength, and interlaminar shear strength (ILSS).
In the field of aerospace, the limit of 2.5% porosity for safety-relevant components must not be exceeded [8]. For this reason, the non-destructive evaluation of components and the quantitative estimation of porosity is obvious. For such purposes, optically excited pulsed thermography (PT) [9] is a fast, non-contact testing method. Zalameda and Winfree [10] determined the thermal diffusivity of porous CFRP samples to characterize the volume fraction of voids. The experimental estimation of the thermal diffusivity allows the prediction of the porosity, using mathematical models that describe the dependence on the thermophysical properties of the internal microstructure [11]. In the case of an opaque material in pulsed thermography, the infrared camera records the temperature field on the surface of the specimen. In this macroscopic scale, the length scales of the voids tend towards zero [12]. Effective material parameters, such as effective thermal diffusivity, can be used to characterize the smallscale phenomena on a macroscopic and thus measurable scale. The use of effective medium theories (EMT) enables the estimation of effective properties of heterogeneous material [11]. The authors do not attempt to provide an accurate historical overview or provide a fully comprehensive list of references with regards to EMT. For example Ringermacher et al. [13] applied EMT to porous fibre-reinforced composites in transient heat conduction. Experimental studies on woven fabric CFRP [14,15] confirm the influence of pore geometry on heat conduction in porous structures and show results of quantitative model-based photothermal porosity estimation by approximating the averaged pore shape as oblate spheroid.
Several works have been carried out describing the specific morphology of unidirectional CFRP. Hernández et al. [16] show on quasi-isotropic laminates that the pores are predominantly oriented in the fiber direction of the adjacent plies. Stone and Clarke [17], endorsed by Stamopoulos et al. [5], showed that the shape of the pores changes from "spherical" to "ellipsoidal" to "needle-shaped" for increasing void content. In addition, Little et al. [18] show that the sphericity decreases almost linearly with increasing diameter of the projected area of the pore.
In this work, the quantitative photothermal porosity estimation on multi-axial laminates, made from unidirectional laminas, is addressed for the first time. With respect to nondestructive testing, the intention of this work is to enable the model-based porosity estimation from thermographic measurements after an one-time material characterization. The derived microstructure-dependent model, can be reused for other samples of the same material system (resin, fiber), independent of the orientation of the individual plies and thus the orientation of pores. In order to meet the requirements due to the pore morphology, the authors perform their considerations on the dethermalization tensor of the general spheroid, considering three independent semi-axes (a ≥ b ≥ c), and validate their modeling approach for the porosity-dependent effective thermal diffusivity by 3D FEM simulations. On the flash-excited pulsed thermography experiment, the pixel-wise effective thermal diffusivity of a porous specimen (porosity < 2%) is estimated using the Virtual Wave Concept (VWC). As shown in previous work, this photothermal parameter estimation method can achieve higher accuracy than state-of-the-art methods [15]. Additionally, it could be shown that this method allows the application of arbitrary heating functions, e.g., a rectangular pulse, to increase the energy input into the component and thus to further increase the measurement accuracy on thick structures [19]. For the 3D microstructure characterization from Cone beam X-ray computed tomography (XCT), independent of the ply orientation of the laminate, the authors utilize statistical moments in the present work. The currently proposed general spheroid modeling approach, in combination with the highly accurate estimated effective thermal diffusivity, enables the model-based photothermal porosity estimation on multi-axial UD/Epoxy laminates for the first time.

Effective Thermal Diffusivity
All evaluations of transient thermal phenomena occurring in pulsed thermography experiments require knowledge of the thermal diffusivity α. The thermal diffusivity is a measure of the speed of heat propagation in a medium during changes of temperature over time. Its magnitude is calculated by the ratio of the thermal conductivity k to the volumetric heat capacity, which is the product of the specific heat capacity c and the density ρ. Porous CFRP is a heterogeneous material which consists of domains of different materials called phases. The discontinuous phase of the voids is distributed over the continuous phase of the carbon fiber and epoxy matrix. Since the thermal properties of the heat conduction and the volumetric heat capacity are discontinuous functions of the location (x,y,z), the heat conduction equation cannot be applied without simplifying assumptions [20].
The homogenization approach, which is based on the volume average of the potential and gradient field of temperature, allows the prediction of the effective thermal diffusivity for an equivalent homogeneous medium. Due to the heterogeneity of the material in the range of the microscale of the pore size, the heat conduction is not defined. By assuming a quasi-homogeneous material based on the averaged material behavior of the original heterogeneous material constituents, this can be achieved at macroscopic length scale.
Ringermacher et al. [13] expressed the effective thermal diffusivity for one-dimensional transient heat conduction. In three-dimensional space, the effective thermal diffusivity tensors is (1) k eff is the effective heat conduction tensor, which consists of nine independent conductivity coefficients for anisotropic media [21] and (ρc) eff is the effective volumetric heat capacity both describing a quasi-homogeneous material. The most general model for ellipsoidal inclusions is represented by the self-consistent method according to Miloh et al. [22]. The conductivity analogue to the Mori-Tanaka [23] approximation, a derivative of EMT, is according to Torquato [11] where the subscripts m and p denote the matrix and the pore phase. The porosity = V p /V is the volume fraction of the pores V p relative to the total volume V , where the total volume V = V p +V m is composed of the volume of the pores V p and the volume of the carbon fiber/epoxy resin matrix V m . The microstructure-dependent dethermalization tensor is given by Osborn [24] and the tensor trace tr(η) = η 1 +η 2 + η 3 = 1. This approximation assumes a random distribution of unidirectional orientated, ellipsoidal and non-overlapping pores parallel to the surface. As a very low porosity content (i.e., a strong dilute medium with < 10%) in a low conductivity material is expected, it is reasonable to neglect interactions between individual pores. The accuracy of such approaches is limited due to neglecting the position and the mutual interaction of inclusions [25]. However, certain physical limitations due to entropy-based information loss apply for photothermal testing [26]. This results in a finite accuracy when determining the thermal diffusivity or conductivity. Thus, given a certain measurement error, the use of a model incorporating higher order terms to derive a porosity value does not necessarily increase the overall accuracy of the method. Mayr et al. [27] have shown an approximation of the typical shape of the averaged pore in porous woven fabric CFRP by an oblate spheroid. In this work, to describe the pore geometry of multi-axial porous CFRP, made from unidirectional laminas, with sufficient accuracy, it is necessary to employ a general spheroid approach with three different semi-axes a ≥ b ≥ c corresponding to Fig. 1. According to Osborn [24], the in-plane dethermalization (see Fig. 2) is described analytically in x-direction by and in y-direction by The out-of-plane dethermalization in z-direction (see Fig. 2), is given by with and where the angles are between 0 ≤ (α, ϑ, ϕ) ≤ π/2. The expressions F(k, ϑ) and E(k, ϑ) are full elliptic integrals of the first and second kind (Legendre): where k represents the modulus and ϑ the amplitude of the integrals. To use the graphs in Fig. 2, any value of c/a respectively b/a may be chosen within the limitations a ≥ b ≥ c, where η 3 ranges between 1/3 and 1. η 3 = 1/3 represents a sphere (marked by a red cross symbol).
Based on the independence of the effective volumetric heat capacity from the shape or distribution of the pores, a simple volume averaging of the material constituents can be used In photothermal testing of porous CFRP, the spatially homogeneous excitation of the specimens surface (z = 0) results in a strongly damped thermal wave that propagates one-dimensional in z-direction and its behaviour is therefore dominated by the porosity dependent effective thermal diffusivity in this direction. Thus, using Eqs. 2 and 11 in Eq. 1 yields the material model for the effective thermal out-ofplane diffusivity This model strongly depends on the averaged pore shape, which is described by the dethermalization factor η 3 (Eq. 6). Since the long semi-axis a of needle-shaped pores in laminates are primarily perpendicular to the z-direction, a low effect on η 3 due to the pore orientation in the xy-plane is expected. Therefore, the EMT model for the thermal conductivity is sufficient at least for η 3 , which is the most important dethermalization factor for the proposed experimental method in which 1D heat flow is assured.

Procedure
Due to the effective thermal diffusivity, the procedure shown in Fig. 3 allows the characterization of small-scale phenomena and the associated effect, such as the local porosity (x, y) on a macroscopic scale.

Determination of Pore Geometry
Consider general volumetric data, where I (x, y, z) are continuously varying scalar values within a 3D domain in the coordinate system x, y and z. In this work I (x, y, z) is the X-ray attenuation detected in XCT measurements. In principle, the scalar field can be anything that describes the material in this volume. To characterize the porous microstructure in the volume , the authors introduce homogenization due to the similarity of thermophysical parameters of fiber and resin and separate into two phases by binarization of volumetric data: Matrix (0) and Pore (1). The X-ray attenuation as well as the thermophysical parameters of these two material components differ by orders of magnitude and are taken into account by means of binarization threshold ξ , where the binarized data results in Torquato [11] proposed statistical methods, using n-point correlation functions, as an impressive approach for materials characterization. Using suggested statistical methods, the properties of heterogeneous materials can be estimated. For the binarized two-phase material (Eq. 13) with the dimension ∈ R J × K × L , utilizing the first order correlation function (n=1) the spatial average of the stochastic function I B , which corresponds to the global porosity volume fraction, follows as with the running variables j = (1 . To gain information about pore distribution and morphology, higher order correlation functions have to be applied. To extract full 3D information of the microstructure from volumetric data (see Fig. 4), the authors employ a 3D autocorrelation function (3DACF) for I B as a function of the components of the lag vector γ x , γ y , and γ z and define, according to Priestley [28] ρ γ x , γ y , γ z = I B (x, y, z) with the intregral over the volume . This formal definition is inconvenient to calculate, whereby this approach corresponds to a 2-point correlation function (n=2) in each spatial direction. The Wiener-Khinthchine theorem [29,30] states that the auto-correlation function corresponds to the inverse Fourier transform of the power spectral density of a statistically stationary process hence anisotropic effects can be taken into account despite low computational effort. For ideal synthetic 2-phase material with pores of same geometry and orientation, based on the minimum of 3D autocorrelation (Fig. 5), the size of the representative Several papers [28,31] discuss bias reduction and increasing accuracy in the calculation of autocorrelation functions, ranking the Fourier method as not optimal for estimating the true values of the autocorrelation [31]. For the calculation of the relevant dethermalization factor (Eq. 6) the true pore size is irrelevant, only the anisotropy of the pore geometry is taken into account, which can be determined with sufficient accuracy by the proposed method [32].
The aforementioned approach works well for synthetic data with an uniform pore orientation or oblate spheroidlike pores. An uniform pore orientation is however more likely to be the case in unidirectional lay-ups only, whereas practical lay-ups are usually non-unidirectional (e.g., quasiisotropic lay-ups). For needle-like or arbitrarily shaped pores the computed ratios will be inaccurate. Furthermore, due to the multidimensionality of Eq. 15 its interpretation for real data becomes difficult. Considering that each pore can be described as a realization of a discrete, three dimensional probability distribution, an approach based on centralized statistical moments is employed [33]. Prior, the binarized data (Eq. 13) is segmented into individual pores by a region growing algorithm [34]. The basic idea of region growing algorithms is that voxels which belong to the same pore must be adjacent to each other (i.e., connected by their face, edge or corner) and have the same intensity (here I B = 1). For each pore p, which consists of a set of voxel indices S p = {( j, k, l) ∈ p | I B ( j, k, l) = 1}, the statistical moments can be calculated with where the centroid (mean value) of the distribution is Using Eq. 17, second-order moments are calculated to form a normalized variance matrix for each pore. By singular value decomposition of the covariance matrix (Eq. 19), the orientation (i.e., the eigenvectors) and the semiaxes (i.e., the eigenvalues) of a three dimensional ellipsoid are computed. The extents a, b, c of the representative pore is then determined by averaging the semiaxes of all pores.

Thermal Diffusivity Estimation by the Virtual Wave Concept
The Virtual Wave Concept (VWC) [35] is applied for the estimation of the porosity-dependent effective thermal diffusivity α eff . Through the proposed VWC, ultrasonic evaluation methods can be applied to photothermal data T (r, t) acquired by the infrared (IR) camera. Based on a local transformation for different time scales (t and t ), a Virtual wave signal T virt (r, t ) is calculated for feature extraction purposes. This transformation is a linear inverse problem and can be formulated as a Fredholm integral of the first kind with the underlying mathematical model, the kernel where α is the thermal diffusivity of the examined material and v c is an arbitrary virtual speed of sound. Only if the thermal diffusivity is known, the kernel K can be calculated accurately. In the case of porosity characterization depending on spatial variations of the microstructure, the effective thermal diffusivity α eff is unknown and literature based values α init have to be assumed for the initial calculation of the kernel K . In photothermal testing, e.g., after optical radiation is absorbed at the surface of the component (z = 0), the calculated virtual wave propagates into the component with  T (r, t), b corresponding virtual wave in A-Scan representation for the estimation of the ToF related effective thermal diffusivity α eff the given velocity c and is reflected at the back wall at the position L. For a pixel-by-pixel estimation of the effective thermal diffusivity, the Time-of-Flight (ToF) measurement on the virtual wave T virt (r, t ) in A-Scan representation is applied, as proposed in previous works [15,36]. In Fig. 6 the local transformation (Eq. 20) is shown for a typical experimental temperature T (r, t) vs. dimensionless time Fo for Dirac-Delta like heating and adiabatic boundary conditions. The temperature data T (r, t) is normalized with the adiabatic temperature T a = q 0 /(ρcL), where q 0 [J m −2 ] is the absorbed heat energy density in an infinite thin surface layer δ(z), ρ [kg m −3 ] is the density, c [Jkg −1 K −1 ] the specific heat and L [m] is the sample thickness. Additionally, in this example the temperature is distorted with additive white Gaussian noise (AWGN) with a standard deviation σ of 25 mK to comply to typical Quantum detector noise [37].
By the VWC the ToF method, well known from ultrasonic testing, can be applied to thermographic measurement data, independently of the selected excitation signal [19], whereby the ToF related effective thermal diffusivity is given by z denotes the spatial resolution and N is the index of max(T virt ) that corresponds to the estimated back wall peak and L is the sample thickness.

Validation of the Porosity Estimation Approach Based on Synthetic Data
Transient 3D heat conduction simulations based on finite element methods (FEM) are carried out to compare the estimated diffusivity based on the VWC (Eq. 22) with the model-based effective thermal diffusivity using the EMT in (Eq. 12). For this validation, parameters of the representative elementary volume are defined and on the one hand employed to define the EMT model (left path of Fig. 7). And on the other hand, these parameters are used to synthesise the representative elementary volume for heat transfer simulations (right path of Fig. 7).The solution of the simulation in form of a time and spatial dependent temperature field T (x, y, t) is used to estimate the effective thermal diffusivity α eff according to Sect. 3.2. As depicted in Fig. 8, the simulation domain is a heterogeneous material, consisting of a matrix material and statistically homogeneously distributed air-filled pores.The porosity = [0, 2, 4, 6, 8, 10]% and the pore semi-axis b = [20,40,60] μm are varied for each simulation whereby the other semi-axes a = 60 μm and c = 20 μm are kept constant. Modeling individual pores in the geometry would result in high computational costs and non-converging simulations, thus pores are modeled by spatially varying the material properties. The discrete, synthetic two-phase material, which is a binary data cube with (4 × 4 × 4)¯m 3 voxel size, is incorporated as an interpolatable look-up-table. That is, for each element of the meshed geometry either the pore material properties or the matrix material properties are assigned, depending on whether the element is inside or outside of a pore, respectively. The thermophysical properties of the constituents used are listed in Tab. 1.A swept mesh from the source surface (z = L) along the z-direction to the target surface (z = 0) is used for discretization. Hexahedral elements with a maximum size of 32.5 μm are used. Adiabatic boundary conditions are assumed. Usually the thermal excitation in pulsed thermography experiments is mathematically treated as a Dirac-Delta boundary condition of the second kind or Dirac-Delta initial condition and often reveal convenient and sufficiently accurate solutions. However for a proper numerical treatment, the pulse duration must be finite, thus it is identified Fig. 7 Validation of EMT based modeling approach with the temporal resolution of the simulation ( t = 0.02 s). The pulse is prescribed as a constant heat flow at the boundary z = L for this duration, to yield a heat energy density of q 0 = 3.84 kJ m −2 which in turn results in a sufficiently large temperature increase ( T ≥ 1 K) and signal to noise ratio (S N R ≥ 40). Since only measurements in transmission mode are considered, only transient temperature data of the surface at z = 0 is relevant. In order to simulate a more realistic parameter estimation, white Gaussian noise (σ = 25 mK) is added to the temperature data. As can be seen in Fig. 9, the estimated effective thermal diffusivity α eff , normalized to the void-free matrix α 0 (Table 1), are in good agreement to the EMT model in the low-porosity regime, which is relevant for composites made from UD laminas. Theoretical considerations would lead to the assumption that the EMT model for oblate spheroids may be sufficient for cigar-shaped pores, due to the low sensitivity of η 3 to changes in b (see third graph of Fig. 2 for c/a → 0).However Fig. 9 indicates, that this assumption leads to underestimating the porosity when utilizing the high accuracy method described in Sect. 3. Furthermore it is noteworthy, that with increasing porosity , the deviation between EMT model and estimates increases. This can be attributed rather to insufficient size of the total simulation volume compared to the pore size than to EMT model errors. Thermal diffusivity α m 2 /s 3.54e-7 2.16e-5 The properties of the matrix material were determined from the material data sheet and pulsed thermography measurements on a void-free specimen corresponding to Sect. 3.2. The properties of the air-filled pore were retrieved from the literature [38]

Experiment and Results
In this section, the quantitative porosity estimation from the optically excited pulsed thermography experiment is performed according to the procedure shown in Fig. 3.

Test Specimen
For the estimation of the model-based porosity in multiaxial CFRP, a specimen made of 10 unidirectional layers of different orientations (Fig. 10) is used. UD continuous carbon fibre-reinforced Prepregs with an epoxy resin content of 35.5% were utilized (CYCOM 5276-1), and the induced porosity was generated by insufficient autoclave processing parameters. Fig. 10 10-ply UD Prepregs of different orientation, resulting in a multi-axial laminate. The ROI is used for microstructural characterization (Fig. 11) The size of the specimen in lateral direction is (40 × 20) mm 2 and the locally averaged specimen thickness is L = 1.53 mm. The diffusivity α 0 = 3.54e − 7 m 2 /s of the voidfree material was determined on a non-porous sample of the same design and material corresponding to Sect. 3.2.

X-ray Computed Tomography
In order to obtain the microstructure, Cone beam X-ray computed tomography (XCT) measurements with a spatial resolution of (11 × 11 × 11)μm 3 have been carried out. The microscopic pore distribution after binarization (ξ = 0.68) of the volume is visualized in Fig. 11a.
The global volume porosity (Eq. 14) is = 1.51%. For multi-axial laminates made from UD Prepregs the preferential orientation of the long and cigar-shaped pores in fibre direction can be clearly seen. As other work [39] has shown, smaller pores occur more frequently in resin-rich areas but have no effect on the global porosity, thus pores with a volume smaller than 750 voxels (respectively 0.001 mm 3 ) were neglected in this work. As shown in Fig. 11b, compared to the synthetic data (Fig. 4c) the varying pore orientations leads to a different 3D-autocorrelation function, which makes the estimation of the representative pore difficult and inaccurate. By employing statistical moments the representative pore geometry can be computed (b/a = 0.111, c/a = 0.039). This results in the dethermalization factor η 3 = 0.733 (Eq. 6), which is strongly influenced by the pore geometry, whereby a significant expansion of the long semi-axis a of the pores is evident.

Active Thermography Setup
The employed experimental setup for optically excited pulsed thermography is shown in Fig. 12. In transmission mode, a flash lamp with the electrical energy of 6 kJ and a pulse duration of about 2ms, controlled by a signal Fig. 12 Transmission configuration measurement setup for flash excitation Fig. 13 Thickness image of the sample, obtained by a 3D scanner. The region of interest (ROI XCT) is used for further quantitative porosity estimations generator, is used to bring the sample out of thermal equilibrium.The absorbed heat energy density is approximately q 0 = 5.9 kJ m −2 , which was determined by comparing the experimental and theoretical adiabatic temperatures. The PC controlled data acquisition is synchronized to the excitation signal and is performed with a Quantum detector infrared (IR) camera equipped with a cooled Indium Antimonide (InSb) focal plane array with a resolution of 1280 × 1024 pixels. The IR camera is sensitive in the spectral range of 1.5 μm to 5.1 μm and has a NETD of 25 mK. The spatial resolution of the measurement was 0.13 mm per pixel. The measurement frequency was 200 Hz whereby the temperature data, according to the optimum evaluation time [15], was temporally truncated to evaluate only time regions containing information about the heat diffusion in the sample. The specimen thickness was acquired with a 3D scanner with an axial resolution of 26 μm and a lateral resolution of 40 μm. The high resolution point cloud was resampled laterally to an uniform grid of the same resolution as the thermal measurement, while keeping the axial resolution, thus creating a thickness image L(x, y) represented as contour plot in Fig. 13 with a standard deviation less L RO I :XCT = ±31 μm. The small structural surface irregularities can be attributed to local matrix accumulations on the bag side, due to vacuum bagging manufacturing process [3].The image is used for a pixel-wise estimation of the effective thermal diffusivity α eff (x, y) according to Eq. 22.

Quantitative Porosity Estimation Results
In addition to the one-time characterization of the microstructure, the investigation by XCT also provides the spatially resolved porosity reference values in the xy-plane shown in Fig. 14a, which are computed by summation in z-direction. For the (19 × 12) mm 2 region of interest (ROI XCT), the global volume porosity is RO I :XCT = 1.51%.The distinctive expansion and orientation of the individual pores can be attributed to the structural design of this multi-axial laminate made from UD Prepregs of different orientations. The inhomogeneous pore distribution can be attributed to insufficient curing cycles in the autoclave process. Figure 14c shows the porosity image derived from the estimated effective thermal diffusivity α eff (x, y), determined with pulsed thermography, and the EMT model, defined by the pore geometry (Eq. 12) and the thermophysical parameters of the material constituents (Table 1). Heat diffusion is an irreversible process, thus is accompanied by entropy production which is equivalent to information loss. In the photothermal results this is represented by a blurred appearance of subsurface features like pores. Thus, the thermal porosity image cannot be compared to the high-resolution XCT porosity image. To comply with the heat diffusion process, a 2D convolution filter is applied to the high-resolution XCT data to generate Fig. 14b. The kernel size of the box (mean) filter corresponds to the size of the representative elementary volume (1.5 × 1.5 × 1.5) mm 3 . The line profiles in Fig. 14df, whose horizontal and vertical positions are represented by the dashed lines in Fig. 14b and c, show porosity values for the same area of the sample, measured by pulsed thermography and XCT respectively. Despite the low global porosity of the investigated sample, the model-based porosity values from pulsed thermography agree well with the porosity values from XCT. According to a first order correlation function (Eq. 14) to calculate the global porosity volume fraction from binarized XCT measurement data, RO I :XCT = 1.51%. Computing the spatial average of the photothermal porosity values obtained pixel by pixel in the labeled area ROI:XCT, the global photothermal porosity results in RO I :PT = 2.14%. The deviation in the comparison of the global porosity on multiaxial laminates of UD prepregs of the different technologies is = 0.63%, which makes the accuracy of the photothermal porosity estimation for this sample comparable to the accuracy of the standard method ultrasonic testing [40] and the wet chemical analysis. This destructive test method for porosity analysis is the reference method for calibration of ultrasonic and XCT systems in the manufacturing aerospace industry. According to DIN EN 2564:2019-08 [41] an absolute error of ±1% can be expected.

Conclusions
Outstanding material properties, such as high tensile strength and low density, make multi-axial laminates made of UD Prepregs highly interesting for the manufacturing lightweight industry. Through part inspection and process monitoring, non-destructive testing is essential. In this context, pulsed thermography offers the possibility of a fast, non-contact and large-area method for determining material parameters. Known effective medium theory modeling approaches, e.g., the approximation of pores in woven fabric CFRP by oblate spheroids is not suitable for multi-axial laminates made of UD Prepregs. As shown in 3D FEM simulations, this leads to a critical underestimation of porosity. In this work the authors propose an photothermal porosity estimation method using effective medium theory, based on 3D pore geometry information. Through a general spheroid modeling approach, considering three independent semi-axis (a ≥ b ≥ c), the specific pore geometry of multi-axial laminates fabricated from unidirectional laminas can be taken into account with sufficient accuracy. The pores oriented predominantly in the fiber direction of the adjacent plies cannot be characterized with 2-point correlation functions in multi-axial laminates. The authors propose an approach to determine the pore anisotropy based on central statistical moments, which allows an accurate characterization of the pore expansion independent of the ply and thus pore orientation. Combined with the highly accurate estimation of the effective thermal diffusivity by the Virtual Wave Concept, the fast and non-contact quantitative photothermal porosity estimation is performed for the first time on multi-axial UD/Epoxy laminates. The utilized effective thermal out-of-plane diffusivity depends on the 3D pore expansion but not on the orientation in the XY plane and thus not on the ply orientation. The absolute error of the photothermal porosity estimation is = 0.63% and thus this method may become competitive to the ultrasonic method. The accuracy of the proposed method for photothermal quantitative porosity determination is comparable to the accuracy of wet chemical analysis, which is the reference standard method for porosity estimation in the manufacturing aerospace industry as a destructive testing method. According to the standard DIN EN 2564:2019-08, an absolute error in porosity of ±1% can be expected. The multiple advantages of pulsed thermography make it a fast, simple and non-contact inspection method for photothermal porosity estimation, which is of great economic interest for the manufacturing industry, especially for large and complex shaped components like structural components in the aerospace industry.