Development and verification of a cure-dependent visco-thermo-elastic simulation model for predicting the process-induced surface waviness of continuous fiber reinforced thermosets

Process-induced surface waviness effects represent a major concern for series production of high-quality lightweight structures based on fiber reinforced plastics (FRP). This paper suggests a method for the numerical prediction of these effects by using the example of processing glass fiber reinforced plastics (GFRP) in a resin transfer molding (RTM) process. The influence of reaction kinetics, chemical shrinkage and cure-dependent viscoelastic properties of the resin are taken into account. Furthermore, the dependence of surface quality on curing cycle, consolidation pressure, textile architecture and thickness of neat resin layer (NRL) at the part surface are investigated. The work is based on published material data and a visco-thermo-elastic simulation approach which has been previously presented and validated. All numerical results are compared to the surfaces of FRP plates that were manufactured with the corresponding parameter variations. Based on a literature survey, different surface waviness values have been identified for comparison of experimental and numerical results. Satisfactory agreement between experiments and simulations is found. Furthermore, it is shown that the analyzed NRL thickness has no relevant influence on the surface waviness while the curing temperature significantly affects the surface waviness. The role of relaxation-induced change of the surface waviness is highlighted by performing long-term measurements and corresponding time-dependent simulations. It is concluded that relaxation plays a decisive role in the appropriate selection of the subsequent surface finishing process. The suggested simulation approach provides a basis for optimization strategies to improve surface quality and reduce post-processing effort.


Introduction
Fast-curing resin systems are commonly used for costeffective production of automotive structural parts in wet compression molding (WCM) or resin transfer molding (RTM). For cosmetic parts, in addition to structural requirements, the visual and textural appearance of the surface represents a key requirement to the customer. 1 In addition, ensuring class A surface finish and affordable manufacturing costs 2 represent prerequisites for the series production of exterior parts e.g. for automotive applications.
The fiber print through (FPT) effect represents a major surface defect of fiber reinforced plastics (FRP) and emerges from heterogeneous distribution of fiber and matrix at the laminate surface. Spatially resolved resin-rich and resinpoor regions having different thermomechanical and shrinkage properties result in non-uniform contraction during processing and cause the fiber architecture to become visible to the human eye. [3][4][5] For the evaluation of the surface quality, both the waviness formed by FPT and subordinate roughness can be utilized as measurement quantity. Standardized methods are available for the calculation of the roughness and waviness quantities from primary profiles. Likewise, established industry test standards, such as the wave-scan dual (WSD), utilize the waviness as main parameter for the description of FPT. 6,7 It is well accepted that FPT and related surface properties of FRP are dependent on textile structure, 7-10 resin material, 8,9 curing cycle, 8,9,11 consolidation pressure 8,11 conditions during cooling process 12 and boundary conditions, e.g. tool surface. 7,9 The influence of the curing cycle during RTM curing was investigated by Kunze et al. 9 The authors find a beneficial effect of low curing temperatures and thus long curing cycles on surface roughness and waviness as a reduced thermal and chemical shrinkage of the resin system is assumed at lower curing temperatures. By applying a temperature gradient between the mold plates, the surface finish is slightly improved. 8 A crucial effect of the heating rate on the surface quality was found by Herring et al. 11 According to the authors, a small process window and reduced time for pore defect removal by resin transport yields increased surface roughness and porosity. The effect of consolidation pressure was investigated in. 8,11 During RTM manufacturing of GFRP laminates changing the injection pressure from 0.2 MPa to 0.6 MPa results in higher surface roughness of the parts. 8 In, 11 laminates were manufactured by the infusion process with vacuum bag pressures of 3. 2-18.4 kPa. Although the pressure shows a slight effect on the surface roughness, the porosity of the surface is significantly increased using low infusion pressures. The influence of the textile architecture on the surface characteristics is investigated in 11 by manufacturing laminates based on carbon fiber pre-impregnated unidirectional (UD) and 2 × 2 twill fabrics in autoclave processing. No significant influence of fabric architecture on surface roughness is found for the unpainted state. In contrast to that, the surface roughness of the painted surface depends on the fiber architecture used. The underlying long-term waviness is lightly influenced by the fiber architecture. In 9 , carbon fiber reinforced polymer (CFRP) laminates with different textile architecture were fabricated in vacuumassisted RTM (VARTM). Lower characteristic surface roughness is obtained using UD fiber reinforcement instead of satin weave fabric. Moreover, no improvement in surface quality was achieved by reducing the fiber volume content (FVC) which is attributed to more pronounced total volume shrinkage caused by deeper neat resin nests.
Beside the process inherent parameters, the resulting surface quality is affected by the tool surface 7,9 and additional surface modifications, such as application of gelcoat, 13,14 in-mold coating, 15 surface films, 16 and subsequent painting process. 3 To evaluate the FPT effect, the quality control systems of original equipment manufacturer (OEMs) are either based on subjective perception of professional auditors 1 or on objective parameters resulting from optical 5,17-19 and tactile measuring 7,8,20,21 methods. In 9,17 a white light interferometer (WLI) is used to determine the regions 17 and linear 9 surface roughness. Profile filters according to ISO 16610 22 are commonly used to two dimensionally distinguish roughness and waviness profiles. 23 Stylus profilometry represents a common method for tactile measurement of the surface roughness. In 7,8,20,21 stylus profilometry is applied to measure the roughness profile of composite parts manufactured in the RTM process. The wave scan dual (WSD) measuring device is well established in the automotive industry as it allows a fast assessment of the surface quality and can be used in a portable and flexible environment. 9 In 5,7 , WSD analysis is used for the surface assessment of composites. The measurement of the surface quality is complemented by analyzing the cross-section based on micrographs. 5,7 Moreover, digital image correlation (DIC) 18,19 and wavelet texture analysis 19 can be applied to measure the surface quality of exterior components. Although many experimental studies focus on roughness, the waviness values derived from the WSD have established as industrial standard to quantify FPT effect and represents a significant processing-relevant measured variable. Consequently, the investigation of the waviness is of major importance.
Besides the experimental characterization of the surface quality, modelling approaches for the description of surface properties of thermoplastic 4,24-28 and thermoset FRP 13,29-32 have been developed. First, analytical models to calculate the surface waviness of thermoset UD-GFRP were developed by  . The model describes the displacement of the part surface by an array of parallel elastic beams and only considers shrinkage effects in thickness direction. It is shown that the coefficient of thermal expansion (CTE) of the resin material significantly controls the magnitude of FPT. 34 By substantially increasing the top NRL thickness, 33 FPT effect can be significantly decreased. However, the investigated NRL thicknesses of up to 1 mm are significantly higher than usually occurring in FRP components. Lin et al. 30 investigated the FPT of GFRP for marine applications by manufacturing parts in the Seemann composites resin infusion molding process (SCRIMP), quantifying the effect and subsequently identifying causes of FPT. Furthermore, the authors proposed a thermo-elastic simulation model with a simplified textile geometry to predict cooling-induced residual stresses. Based on the model the authors developed methods to unify the distribution of residual stresses and thus reduce FPT and improve surface quality. 13,31 A one-dimensional analysis of surface displacement of a carbon fiber-reinforced plastics (CFRP) laminate was conducted in. 21 In contrast to previous studies, chemical shrinkage was included in the simulation approach, so that measured average roughness R a values of up to 1.07 µm are in good agreement with simulated surface roughness. Mold filling and in-mold pressure development was investigated numerically and experimentally in. 32 Palardy et al. found that increased pressures in the later stage of resin cure results in enhanced surface properties and reduction of FPT effect, as the tool surface can be reproduced more accurately due to better contact. In 29 a cure-dependent elastic simulation approach is introduced to predict the surface waviness of an automotive fender produced in sheet molding compound (SMC) pressing. A good agreement between the position of the measured waviness and the calculated stress distribution is found.
Although a large number of experimental studies of the FPT and the associated surface properties (waviness, roughness) have been carried out, only few attempts were made to predict FPT and resulting surface properties of FRP. The present simulation models are commonly running simplified linear elasticity models 13,31 or geometrical simulation tools 21 while the use of viscoelastic and curedependent simulation approaches is mostly limited to the computation of residual stresses and the resulting process induced deformations (PID). [36][37][38] Viscoelastic curedependent models provide a more reliable description of the epoxy resins material behavior during cure and accompanying process induced residual stresses. 39 Indeed these models also yield good results for non-conventional cure cycles and allow the consideration of relaxation effects during and after processing. 40 However, a comprehensive and complex experimental characterization of the resin material represents a prerequisite for the use of curedependent viscoelastic models. 41 To close this gap and to numerically predict the surface properties of FRP, the cure-dependent visco-thermo-elastic model introduced in 42,43 is used to predict the surface waviness of an epoxy resin composite. The underlying material data are taken from 12,44 and have previously been determined under process-specific conditions. Particular importance is paid to the temporal change of strains during processing, post-processing relaxation effects and the precise characterization and modelling of the underlying complex fiber architecture of the fabric. In addition, extensive experimental investigations of the surface waviness are performed comparing different optical measurement methods to verify the proposed simulation model (for detailed studies of the processing conditions) on surface waviness.

Methodology
In the present paper, the interaction between surface waviness and processing conditions is investigated in numerical and experimental studies. Therefore, both numerically and experimentally determined primary surface profiles are processed with the same filter methods to extract the waviness profiles from the primary surface profiles. Afterwards, the results are evaluated and experimental as well as numerical studies are directly compared. The overall methodology is schematically shown in Figure 1. First, the experimental investigation, the evaluation of the measured and numerically predicted primary surface profiles including applied filter methods are described in detail. Then the numerical methods are presented.

Experimental investigation
A fast-curing resin-hardener combination consisting of a bisphenol-A based resin system (EPIKOTEÔ Resin TRAC 06150 from Hexion Inc.) and an amine-based curing agent (EPIKUREÔ Curing Agent TRAC 06150 from Hexion Inc.) is investigated in the present work. Extensive characterization of the resin material data was carried out in preliminary work. 12,42,44 UD non crimp fabrics (NCF) from HP-Textiles GmbH, Germany with a grammage of 608 g=m 2 and tricot binding (15 g=m 2 Þ with polyethersulfone (PES) stitching is used for the manufacturing of [0/90] s laminates. The UD-NCF consists of 2200 tex glass fiber E6 from Jushi Group Co., Ltd., China with silane sizing and a filament diameter of 22 µm. Laminates (300 × 150 mm 2 ) with a thickness of 1.8 ± 0.06 mm for FVC of 52% are manufactured using VARTM process and applying consolidation pressures of 0.5 and 2.5 MPa, respectively. The average roughness of the mold surface is 0.1 µm. The mold surface is treated with solvent-based external release agent (LOCTITE FRECO-TENC 700-NC, Henkel AG & Co., KGaA, Germany). An overview of the parameters for the manufactured laminates is given in Table 1. The pressure and temperature vary in a fully factorial 2 2 experimental design. A total of 16 FRP plates is produced.
To experimentally characterize the primary surface profiles of the manufactured GFRP plates, three different optical measuring methods are employed. A laser confocal microscope (LCM) VK-X 200 of Keyence AG, Japan, is used for surface analysis. Three equidistant line measurements (0.2 mm width) with a length of 40 mm and spacing of 10 mm are carried out in the center of the component. Measurements are performed in both in-plane coordinate directions with data points being recorded every 2.8 µm in high-precision mode. The LCM measurements are carried out 1 month after the production of the FRP plates. A WLI by Fries Research & Technology, Germany is being used to determine the vertical profile of a of 40 × 40 mm 2 surface area which corresponds to a data matrix of 640 × 640 measurement points with equidistant spacing of 0.06 mm. The z-resolution of 20 nm allows reliable determination of standardized surface values. In contrast to the LCM measurements, the WLI measurements are carried out 1 year after FRP plate manufacture. After 2 years the WLI measurements are repeated. During this period, the FRP plates are stored hermetically isolated and in the absence of light to minimize aging effects. The WSD measuring device from BYK Additives & Instruments GmbH, Germany, is being used to measure the surface properties (LW , SW ) based on reflectivity in the time period 1 month after plate production. The mean value of three measurements with a scan length of 100 mm and data points being recorded every 0.027 mm is evaluated in both in-plane coordinate directions. 45 A spot size of 10 mm width is evaluated at the part surface.
Additionally, micrographs of the cross-section area are analyzed using a digital microscope VHX5000 of Keyence AG, Japan with a 100 and 200 magnification factor to both map the geometry of the NCF and to identify neat resin pocket depth and thickness of NRL at the surface ( Figure 2(a)). By WLI scans of the NCF, additional measurement data for the yarn cross section and stitching pattern are obtained (Figure 2(b)).
After determining the primary surface profile by means of the described measuring method, the measured primary surface profile is subsequently filtered for the calculation of waviness characteristics. First, form deviation is eliminated by describing it using a 2 nd order polynomial to approximate the primary profile. According to DIN EN ISO 4288, an expected average groove width of 3 mm and underlying measuring length of 40 × 40 mm 2 (WLI) and 40 mm (LCM) cut-off wavelengths of λ c ¼ 2:5 mm and λ f ¼ 8 mm are selected for removing non-relevant wavelengths by applying low-and high-pass Gaussian filters. The processing of the experimental data is illustrated in Figure 3(a).
To minimize evaluation method-dependent errors, the post-processing of the numerical data is conducted using the same approach as for the experimental data. However, the numerically predicted node displacements in z-direction is used as input data set. After removing the form deviation from the primary profile using 2 nd order polynomial approximate no further Gaussian filtering is applied as it is found that subsequent filtering strongly effects and artificially smoothens the waviness profile ( Figure 3(b)). Since the simulation data is based on an idealized geometry, no further filtering to remove nonrelevant surface structures is required. Based on the numerically and experimentally determined waviness profiles, two surface parameters W a and W t according to DIN EN ISO 4287 are selected to evaluate the surface waviness. The numerical model considers six nodes in width-direction ( Figure 5) so that the surface parameters of the six nodes rows are calculated individually based on the node displacement.  In addition to the experimental method of the WSD, a method to determine the LW for numerical models is employed. In accordance with the experimental profile filter process, at first the form deviation is eliminated. Afterwards the standard deviation, noted σ, of the surface waviness profile is calculated. A correlation equation including σ gives the LW parameter for the numerically predicted surface displacement: 28

Numerical methods
For the numerical modelling the textile architecture the open-source tool of TexGen v3.12.2 is used. 46 The modelling parameters of the UD-NCF are based on the microscopic and computed -tomography analysis of the final specimen plates (Figure 2(a)). The simulation of the curing process is carried out using abaqus © 2022 combined with a subroutine of thermo-viscoelastic material model of the matrix developed in 42,43 and extensive material characterization previously published in. 12,44 It has to be mentioned that the numerical model of the textile architecture is based on the consolidated specimens which means that the effects of impregnation, compaction and cure on yarn shape are included. Due to this, overlapping of the yarn's sections and thus reduction of the total laminate thickness reduction arise allowed. A representative volume cell is generated using TexGen. For reduction of computational cost, only the half thickness of the laminate is modelled, and two regions of interests (ROI) are simulated and evaluated in detail (Figure 4). The voxel method is used for mesh generation which excels at generating accurate meshes for a wide range of textile structures 47. Hence, the knots of the stitch yarns are not modelled in detail. Characteristic element lengths of 0.1 mm in in-plane(x-y) and  The other region (ROI 2) runs parallel between two 0°direction yarns. The 90°-direction yarns at the bottom and the stich yarns in the top of the region embedded in the resin can be seen in Figure 5.
Although the yarns can be considered as quasiinextensible 48 the material behavior of the yarns is assumed to be isotropic linear elastic. The engineering constants of stiffness and thermal expansion are calculated by rule of mixture as given in Table 2. The determination of the inner FVC of the yarn is performed with the open-source image program Fiji IMAGE J. To distinguish between glass fibers and matrix material a trainable Weka Segmentation is applied. 49 Therefore, a section of a yarn is used to train the segmentation algorithm by dividing the section into matrix and glass filament classes. This approach allows the determination of the FVC for different sections in the same picture. The experimentally determined FVC in the yarns with micrographs and the machine-learning approach of the WKEA segmentation are up to 77.8 vol.-% ( Figure 6). Therefore, the resin rich zones between the single fibers inside a yarn are assumed to have no influence on the surface profile development during process. Also, for the PES-stitch yarn an isotropic linear elastic material behavior is used. The resultant engineering constants are given in Table 1. The description of the resin material is characterized and modelled in 12,42,44 .
The initial and boundary conditions for the two regions are adapted to the RTM process. The numerical process includes four steps. A heating step from initial room temperature to process temperature by a rate of 20 K/min. A holding step with constant temperature and a cooling step by a rate of 20 K/min to room temperature. After reaching 23°C, the final relaxation step begins. The consolidation pressure is defined over all steps within the material model of the resin. No mechanical pressure is applied during curing and cooling process as an additional boundary condition. The boundary conditions of the regions are in accordance with the process and continuum mechanics. Therefore, the symmetry plane of the laminate and the cutting planes are defined with symmetric boundary conditions for the whole process except the surface plane at the top. Until the final relaxation step is reached the surface plane is fixed in normal direction as the part surface tightly sticks to the mold 30 and stays in contact with the mold surface throughout the process. 21 During the additional relaxation step representing a demolded part, deformations in all directions are allowed. Since the relative displacement of the component to the tool is of subordinate importance in the RTM process, a modelling of the adhesion to the tool surface 39 e.g. by Coulomb's friction law 50,51 is not included. Furthermore, based on the assumption that the exothermic effect of the curing reaction is less pronounced for thinwalled parts (1.8 mm thickness) with high content of reinforcement fibers, no modeling of heat conduction and exothermal effects is considered. No plasticity and damage of resin and fiber-matrix-interface is considered in the present model.

Results and discussion
The influence of the top NRL thickness on the surface waviness is analyzed using numerical investigations as the experimentally determined thickness is scattered and its influence on the formation of the surface waviness is to be evaluated first. The aim is to prove that NRL thickness, which are smaller than those in, 33 are of subordinate importance for the formation of surface waviness. Since the NRL thickness cannot be controlled experimentally, a further impact of layer thickness on the experimental results needs to be excluded. Subsequently, numerical and experimental investigations on the influence of the processspecific parameters on the developing surface waviness are presented. Particular attention is paid to the accurate numerical modelling of the process-specific conditions such as applied curing cycle and consolidation pressure. During the experimental determination of the surface parameters, it is ensured that the investigated ROI is accurately represented by the measurement data. The surface scan method and evaluation approach for WLI and LCM measurements is depicted comparatively in Figure 7. Finally, the relaxation behavior after completion of the cooling process is presented so that the advantage of the cure-dependent viscothermo-elastic approach becomes apparent.
First, the top NRL thickness is evaluated as previous microscopic investigations of cured part cross section area showed significant variations and cannot be adjusted experimentally. For this reason, three NRL values (3, 50 and 100 µm) covering the entire span of measured thickness values ( Figure 8) were selected for the first numerical consideration. ROI 1 for setup 1 is chosen as a representative cross-section, as the neat resin top layer proportion in the thickness direction is comparatively smaller than in ROI 2 ( Figure 5) and thus a greater influence is to be expected. Figure 9 depicts W t and W a values based on numerical and experimental data for different NRL. It can be stated that the NRL thickness slightly influences the numerically predicted W t and W a values. The deviation of the numerically predicted values is relatively small so that the respective values in the boxplot cannot be clearly identified. Based on a linear elastic model a contrary and more pronounced influence on the displacement of the surface has been identified by Kia. 33 However, a top coat layer thicknesses of 0.1-1.0 mm with differing stiffness values were considered, which are significantly higher than the NRL thicknesses investigated in the present investigation. As the NRL thickness shows subordinate effect on the W t and W a values, for further evaluation of processing conditions, a representative NRL thickness of 50 µm is selected.
In general, the experimental data of the LCM in Figure 9 show slightly higher W t (3.45 µm) and W a (0.91 µm) values compared to the numerical predicted values of 2.41 µm and 0.51 µm at top NRL thickness of 50 µm. The experimental values are also more widely scattered. A possible explanation for the higher scatter might relate to not equally displaying the ideally defined ROI cross section from the measurement data. Indeed the LCM data consist of equidistant line measurements so that a precise correlation with the defined ROI section in x-and y-direction is restricted by the underlying data acquisition method. Manufacturing defects such as micro dry spots or non-impregnated regions within a roving have a strong impact on the W a value and therefore tend to result in higher W a values for the experimental determination.
Since these defects are not considered in the numerical simulation model, a discrepancy can be expected between the experimental data and the simulation. Still, for the underlying processing conditions especially the WLI data show good agreement with the numerically determined W t values.
It can be stated that the influence of the NRL thickness is of minor importance for the underlying range of 3-100 µm. No significant influence of NRL thickness on the waviness due to varying NRL thicknesses is to be expected for the manufactured FRP.   Experimental and numerical analysis on the impact of the processing conditions on W t and W a The influence of cure cycle and applied consolidation pressure on the experimentally and numerically determined surface waviness is depicted in Figures 10 and 11. Changing the applied consolidation pressure from 0.5 to 2.5 MPa in the underlying material model shows no measurable influence on the surface waviness. According to 44,58 the pressure slightly influences the chemical shrinkage so that a slightly increased waviness was expected. Thus, on meso-scale the pressure-dependent effect of chemical shrinkage that is included in the present simulation approach is stated to be neglectable for the formation of surface waviness. Within ROI 2 predicted W t slightly decreases from 2.41 to 2.31 µm (Figure 9) whereas the W a increase slightly by 0.05 µm (Figure 11). Considering the changed cross section area and the shorter grove width (Figure 6), a higher W a is reasonable as this profile is mainly considered in the waviness fraction for λ c < 2:5 mm. The decreased W t can be attributed to more homogeneous in-thickness-distribution of resin poor and fiber dominated cross section area. The stitch yarn exhibits a coefficient of thermal expansion similar to that of the matrix material that has been determined to 6 × 10 À5 1/K 44 which results in a uniform thermal shrinkage of the surface.  An elevated processing temperature of 120°C leads to a significant increase in predicted W t and W a values which agrees with findings for surface roughness in literature. 9 At reduced curing temperature of 80°C, the curing process decelerates. The temperature increase caused by the exothermic nature of the reaction is reduced and the temperature difference between the curing temperature and the operating temperature of the part decreases. Assuming a linear relationship between degree of cure ξ and chemical shrinkage β, 44,59 β is reduced by decreasing final ξ values. It is noticeable that the model seems to consistently underestimate the W a values may be that the geometric model of the textile does not include deviations from the idealized elliptical geometry. As can be seen in Figure 1, deviations from the elliptical yarn cross-sectional area are present in the part. These have an influence on the average waviness so that the W a value is consistently underestimated.
Numerical and experimental values at the elevated curing temperature show significant differences which can be attributed to following aspects: The material data show particularly good fitting at 80°C. 42 Furthermore, fiber-matrixinteraction 60,61 is intensified through faster injection process which non-reproducibly influences fiber architecture and favors voids at elevated tooling temperatures. 62 Since impregnation defects during the impregnation process and subsequent formation of voids 63 are not taken into account in the numerical model, the surface waviness is underestimated in the numerical prediction. Figure 12(b) and (c) shows a laminate manufactured with a 120°C tooling temperature exhibiting impregnation defects and voids, while the laminate manufactured with a 80°C tooling temperature (Figure 12(a)) shows no apparent defects. Assuming a generic temperature cycle that is imposed on all nodes, the thermal shrinkage may be underestimated by the model due to the exclusion of temporary temperature increases by the heat generated as a consequence of the exothermal curing reaction. Since this most likely occurs at higher curing temperatures (and thus higher reaction rates δξ δt , the generated specific heat _ q ∼ H R δξ δt , Figure 9. Surface waviness W t and W a predicted by numerical model and experimental data (WLI, LCM) for ROI 1 at temperature of 80°C and consolidation pressure of 0.5 MPa as well as different NRL thickness. where H R represents the total reaction enthalpy 64 ), this may cause deviations at a temperature of 120°C. Also, the underlying numerical model considers no plasticity and damage of resin and fiber-matrix-interface which may result in a divergence between the numerical and the experimental data.
In particular, the processing temperature is of decisive importance for the level of W a and W t . However, the influence is more pronounced in the experimental investigations than in the numerical investigation. Pressure and ROI are of secondary order on the formation of W a and W t .
Experimental and numerical analysis on the impact of the processing conditions on LW In the following, the numerical and experimentally determined LW values are compared. By using equation (1) numerically determined σ values are transformed to the often applied LW scale and can directly be compared with WSD measurements. 65 As shown in Figure 13(a), the NRL thickness has no significant influence on the LW values. The LW value is underestimated by the numerical simulation when compared to the experimental investigations. However, a noticeable trend for the LW values to decrease with increasing NRL thickness can be identified.
The reason for the deviations might be that a surface area of 100 × 10 mm 2 is measured with the WSD, while only a surface area of 7 × 0.5 mm 2 is numerically calculated. Since no separation into the ROIs is possible for the experimental measurements, different sized areas and thus the mean value of a large width is being compared with a smaller and defined ROI. Furthermore, it should be noted that the correlation equation (1) was mainly validated for non-translucent carbon fiber fabric components. Since the studied GFRP have a translucent surface and therefore exhibit different reflection and transmission behavior of the surface. That circumstance can lead to measurement errors and deviations based on the reduced Figure 11. Surface waviness W a predicted by numerical model and experimental data (WLI, LCM) for ROI 1 and ROI 2 for the investigated curing temperatures and consolidation pressures. reflectivity of the components surface and resulting low image forming qualities. 66 The influence of the process parameters is shown in Figure 13(b) and is also underestimated by the simulation. Qualitatively, the results for the analysis of the process influence are comparable to those for the W a and W t values. However, the influence of temperature on the LW value is less pronounced. For the investigated crosssection ROI 2, the measured LW values with a mean value of 22.3 are below but still closest to the numerically predicted LW values with 13.8. Compared to the literature data, the experimental data are in the range of 10-40 for unpainted glass fiber reinforced epoxy composite parts. 67 Although there is no generally applicable standard for defining a Class-A surface, in the literature several values are stated for the required LW values to attain Class A surface: According to Grünberg 68 LW values up to 15 are acceptable, whereas Schubel 7 proposes more strict values of LW < 8 and Volkswagen AG a tolerable LW range of 1-5. 9 This circumstance suggests that a post-painting process is inevitable to ensure the required surface properties and that the post processing material behavior and associated subsequent relaxation processes play an important role for the prediction of final part surface waviness.

Numerical analysis -Formation of surface waviness caused by long term relaxation
To consider the relaxation effects after accomplished the cooling process, the time-dependent surface waviness profile after demolding is proposed in Figure 14. The corresponding W a and W t values are given in Table 3. At a curing temperature of 80°C, W t changes from 1.79 µm immediately after demoulding to 1.92 µm after 1 × 10 6 s. This relationship illustrates that viscoelastic material modelling of the matrix material is essential to describe the time-dependent relaxation behavior and understand the formation of the process-induced surface waviness even after completed cooling process. At a curing temperature of 120°C, W t changes from 2.88 µm immediately after demoulding to 3.08 µm after 1 × 10 6 s, which corresponds to a relaxation time of 11.5 days. After curing at different cure cycles and completed cooling process, the final glass transition temperature (T g ) values are given with 90.5°C for cure at 80°C and 108.1°C for a curing at 120°C respectively.  According to relaxation experiments conducted in previous studies 42 a decreased relaxation modulus and thus an decreased surface displacement is to be expected with elevated T g values. This correlation is confirmed based on the results of the numerical simulation. The surface displacement after curing at 120°C amounts to 0.04 µm, while curing at 80°C results in a displacement of 0.05 µm for a relaxation time of 1 × 10 4 s ( Figure 15).
A direct comparison of time-dependent development of W t is given in Figure 15. At times of 2 × 10 4 s the absolute surface displacement of the curves with T g of 108.1°C and 90.5°C intersect. From this point on, the surface displacement is significantly more pronounced with a T g of 108.1°C. Therefore, at high curing temperatures resulting in a higher T g , in addition to enhanced thermal and chemical shrinkage, more pronounced relaxation behavior is to be expected which leads to increased W t values.
In addition to the numerical analysis, the time-dependent influence on W t is analyzed experimentally. Figure 16    shows surface profiles measured by WLI 3 and 24 months after manufacturing of FRP plates. Mean W t value of the measured profiles increases by 0.09 µm from 3.94 ± 0.61 to 4.03 ± 0.82 µm in the examined time interval. A time-dependent increase in waviness is measured experimentally and thus confirms the numerical investigations. Compared to the numerical data, the experimental data show a significantly higher scatter of the measured W t values. Considering mean W t values, a more pronounced time-dependent change of W t compared to numerical analysis is found (Figure 15). The measured change in W t is small compared to the incertitude of measured W t , so that relaxation-induced change of W t occurring after long time intervals are less pronounced. However, a large number of measurements were carried out, so that this trend can clearly be identified. Although the changes in surface waviness below 2 µm caused by relaxation processes can no longer be directly resolved by the human eye, color changes induced by the displacement are perceived by the human eye as hills and valleys 65 so that waviness might reappear on the surface.
For a post-painting process of parts cured at elevated temperatures, the coating film thickness should be increased to compensate for the surface waviness that develops during the relaxation process. The present findings represent an important advice for subsequent coating processes as timedelayed surface displacement is to be considered for predicting final part surface properties.

Conclusions
In this paper, the surface waviness of GFRP with fast-curing epoxy resin matrix was investigated and analyzed numerically and experimentally. A thermo-viscoelastic simulation approach was applied to predict the formation of the surface waviness during the manufacturing process. The numerical results are compared with experimentally obtained results. To accurately represent the processing conditions and the material behavior, a cure-dependent thermo-viscoelastic simulation approach has been used and extensive characterization and modelling of the textile geometry has been carried out.
· Based on the present numerical investigation, a subordinate effect of the NRL on the resulting surface quality has been found. · Further, a good agreement between numerically predicted and experimentally determined surface waviness W t and W a has been demonstrated for the low temperature curing cycle at 80°C. WLI based experimental data and numerical simulation showed very good agreement for ROI 1 in 90°yarn direction. For higher temperatures, a deviation between numerical prediction and experimental W a and W t is observed. This may be due to a more pronounced fiber misalignment that changes the yarn geometry. Another possibility concerns the numerical assumptions, which neglect exothermal effects, plasticity and damage of resin and fiber-matrix-interface. · Based on the deviations between predicted and simulated LW values, it remains to be verified whether the correlation given in equation (1) is generally valid or must individually be determined for each investigated component, depending on the optical properties. · By considering the relaxation behavior after the cooling process, the surface displacement was simulated for a time up to 10 × 10 8 s which corresponds to 3.2 years. This resulted in an additional displacement up to 0.32 µm. The relaxation-induced change of W t has been quantified by long-term measurements, which were further verified by the numerical model prediction.
The present simulation approach provides a basis for optimization strategies to improve surface quality of fastcured GFRP and reduce post-processing effort by advanced process design. Further work will focus on generalizing equation (1) for translucent parts.