Modeling of Falling Ball Impact Test Response on Solid, Veneer, and Traditional Engineered Wood Floorings of Several Hardwoods

: Hardness is a key mechanical property of ﬂooring materials. In this study, the performance of veneer ﬂoorings (with a top layer thickness of 0.6 mm) was investigated by dynamic hardness tests, comparing it with those of traditional engineered wood ﬂoorings (with a top layer thickness of 3 mm) and solid wood ﬂoorings. Two hardwoods commonly used on wood ﬂooring, viz. Quercus robur L. and Hymenaea courbaril L., and two fast-growing hardwoods, Eucalyptus globulus Labill. and Eucalyptus grandis W. Hill ex Maiden, were tested as top layers. To compare their usage properties, a dynamic impact hardness test involving steel balls with three diameters and ﬁve different drop heights was carried out, measuring the footprint diameter (FD) and the indentation depth (ID). The data from 4800 impacts, corresponding to 180 different individual groups (4 hardwood species × 3 ball diameters × 5 drop heights × 3 ﬂoor types) were analyzed. The results showed that the general response in terms of both FD and ID was better in the engineered wood ﬂoorings than in solid wood ﬂoorings, and that the veneer ﬂoorings (0.6 mm) showed better behavior than traditional engineered wood ﬂoorings (3.0 mm). Furthermore, for the veneer ﬂoorings, the two fast-growing hardwood species tested, which have signiﬁcantly different densities, showed similar behavior to traditional hardwoods, suggesting that they would be suitable for valorization in the wood ﬂooring industry.


Introduction
The use of wood floorings in building construction is widespread, to the extent that wood has become the most used floor material [1]. At present, four main types of wood floorings are available: solid wood floorings, parquet wood floorings, engineered wood floorings, and veneer floorings. Nonetheless, as discussed below, no real differences between the latter two are defined in regions outside Europe.
Solid wood flooring is made with a single piece of hardwood (i.e., a solid piece of noble wood), while engineered wood floorings consist of an inner core with a thin noble wood layer on top (i.e., a multilayer heart with a final noble wood top layer). The modernization of the industrial flooring sector and the development of new products have resulted both in quality improvement and cost reductions, making engineered wood floorings more competitive than solid ones [2][3][4]. The current trend in the industry is towards the development of new multilayer products with lower top layer thicknesses, Table 1. Wood flooring types tested in the study, with basic information on sample sizes and properties for each of the hardwood species investigated.

Eucalyptus grandis W. Hill ex Maiden
Solid wood flooring * Length, width, and thickness values are expressed in mm; ** Mean density is expressed in kg m −3 ; *** Coefficient of variation is expressed in %.

Wood Flooring Specimens
Three types of wood floorings were assayed (Table 1), namely: • Solid wood flooring (SW), consisting of a single piece of wood (sawn, brushed, and sanded). All wood layers were glued with polyvinyl acetate (PVAc). The tested surfaces were varnished with the Bona UV system with a matte finish, consisting of 5 coats and about 80 g·m −2 [30].
All samples were manufactured ad hoc at Intasa S.A. wood flooring factory located in As Pontes de García Rodríguez (Galicia, Spain).

Dynamic Hardness Testing
The dynamic hardness of the different samples was evaluated through the ball impact test by measuring the footprint diameter (FD) and indentation depth (ID). According to ASTM D1037-99 [31] and ASTM D2394-83(1999) [32], with the modifications of some parameters of the tests indicated in AITIM recommendations [33], the FD and ID of the impact footprint left by different diameter steel balls upon free fall from a given height were measured. While it does not require sophisticated equipment, it may cause perimeter breakage of the wood fibers as a result of excessive impact energy, hampering an accurate measurement of the footprint. Hence, a modification of the procedure proposed by Acuña et al. [23] was followed instead: the wood impact test was performed with three steel balls of three different sizes (ø = 30, 40, 50 mm), and at five different reference heights (0.60, 0.75, 0.90, 1.05, and 1.20 m). The impact energies for each diameter and height combination are summarized in Table 2. In the tests, chromed steel balls, of 100Cr6 steel type (SKF, Gothenburg, Sweden), with the indicated hardness according to ISO 3290-1:2014 [34] standard, were used. Table 3 shows the characteristics of the steel balls used in the tests.  The sample from each group (4 hardwood species × 3 types of wood flooring) consisted of 50 pieces. The dimensions of the specimens, detailed in Table 1, were chosen because they are the most common in the wood flooring industry. Each specimen was subjected to at least 5 impacts with a minimum spacing of 50 mm (the location of the impacts is shown in Figure 1), resulting in a total of 4800 impacts. Following the methodology proposed by Acuña et al. [23], the specimens were placed on a completely rigid surface and were anchored to it using clamps located on the perimeter, thus avoiding any influence of the support on the deformation behavior. Subsequently, each of the reference drop heights was tested on each of the specimens using an auxiliary metallic instrument to guarantee the test conditions (precise reference height, and initial velocity u0 = 0 m·s −1 ), as shown in Figure 2. Following the methodology proposed by Acuña et al. [23], the specimens were placed on a completely rigid surface and were anchored to it using clamps located on the perimeter, thus avoiding any influence of the support on the deformation behavior. Subsequently, each of the reference drop heights was tested on each of the specimens using an auxiliary metallic instrument to guarantee the test conditions (precise reference height, and initial velocity u 0 = 0 m·s −1 ), as shown in Figure 2.
Following the methodology proposed by Acuña et al. [23], the specimens were placed on a completely rigid surface and were anchored to it using clamps located on the perimeter, thus avoiding any influence of the support on the deformation behavior. Subsequently, each of the reference drop heights was tested on each of the specimens using an auxiliary metallic instrument to guarantee the test conditions (precise reference height, and initial velocity u0 = 0 m·s −1 ), as shown in Figure 2. For each impact, the FD and ID were measured using a digital micrometer (with a ± 0.001 mm error) and a back plunger dial indicator (with a ± 0.01 mm error), both from Mitutoyo (Takatsu-ku, Kawasaki, Kanagawa, Japan). To improve the impact readings, carbon paper was placed on the specimen surface. It should be clarified that the ball impact produces an elliptical deformation due to the different compressive strengths in the parallel-and perpendicular-to-the-fibers direction, so the mean diameter of the footprint was calculated as the mean value between the D1 (direction of the fiber) and D2 (transverse to the fiber) values. For each impact, the FD and ID were measured using a digital micrometer (with a ±0.001 mm error) and a back plunger dial indicator (with a ±0.01 mm error), both from Mitutoyo (Takatsu-ku, Kawasaki, Kanagawa, Japan). To improve the impact readings, carbon paper was placed on the specimen surface. It should be clarified that the ball impact produces an elliptical deformation due to the different compressive strengths in the parallel-and perpendicular-to-the-fibers direction, so the mean diameter of the footprint was calculated as the mean value between the D1 (direction of the fiber) and D2 (transverse to the fiber) values.

Statistical Analyses
All statistical analyses were performed using R software (v. 4.1.1:2021). Steel ball diameter and drop height were selected to establish the statistical design according to ASTM D1037-99 [31] and AITIM [33] recommendations. For each group (hardwood species × ball diameter × drop height × wood flooring type), the number of tests was established by an initial and traditional variability analysis, performed with 800 tests. Accordingly, the number of replicates (n) obtained with a 95% confidence level was 26 trials. However, to ensure the adequate performance of the design, between 30 and 35 trials per group were carried out, resulting in a total of 4800 impacts, corresponding to 180 different individual groups.
The assumptions of independence, normality, and homoscedasticity were checked for the two variables under study (FD and ID) in each of the 180 sample groups. The normality of the data was tested for all populations using the Shapiro-Wilk normality test and Q-Q normal probability plots. The homoscedasticity requirement was contrasted by Bartlett's test. Since it was not met on numerous occasions, the usual comparative analysis by ANOVA could not be used. Robust comparison methods were used instead: Welch's heteroscedastic F-test with trimmed means and Winsorized variances. This robust procedure tests for equality of means by replacing the usual means and variances with trimmed means and Winsorized variances [35,36], together with bootstrapping and comparison using robust homogeneous groups.
To select the response models, generalized linear models (GLM) from different families (Gaussian and gamma families) and different link functions (identity, inverse, log) were used, selecting for each target variable (FD, ID) the one that offered the best explanation. This performance was evaluated employing residual analysis, the residual deviance [37], the Akaike information criterion (AIC) [38], the Bayesian information criterion (BIC) [39,40], and Efron's R 2 value [41]. These values, together with the likelihood-ratio test and Wald's test, make it possible to establish the model's goodness-of-fit.

Descriptive Statistics
Box plots of the two variables under study, viz. average footprint diameter and indentation depth, as a function of the wood flooring type, hardwood species used in the top layer, steel ball diameter, and drop height are presented in Figures 3 and 4, respectively. The main descriptive statistics by groups are summarized in Table S1 for SW and in Table S2 for EWF.

Model Selection
According to the box plots presented in Figures 3 and 4, several general trends in the response of the target variables, FP and ID, could be observed. These trends were influenced by: (i) the ball diameter (the larger the diameter, the larger FD and ID were); (ii) the ball drop height (the higher the drop height, the higher the values of FD and ID were); (iii) the type of wood flooring and thickness of the noble layer, with higher values in SW than in EWF, and, in the latter case, higher values of FD and ID for a larger thickness of the top layer; and (iv) the hardwood species used, with higher FD and ID values for the hardwood species with lower densities. The significance and importance of these influences were corroborated by an analysis of variance of the GLMs established for the two target variables, FD and ID, which may be expressed by Equations (1) and (2):

Model Selection
According to the box plots presented in Figures 3 and 4, several general trends in the response of the target variables, FP and ID, could be observed. These trends were influenced by: (i) the ball diameter (the larger the diameter, the larger FD and ID were); (ii) the ball drop height (the higher the drop height, the higher the values of FD and ID were); (iii) the type of wood flooring and thickness of the noble layer, with higher values in SW than in EWF, and, in the latter case, higher values of FD and ID for a larger thickness of the top layer; and (iv) the hardwood species used, with higher FD and ID values for the hardwood species with lower densities. The significance and importance of these influences were corroborated by an analysis of variance of the GLMs established for the two target variables, FD and ID, which may be expressed by Equations (1) and (2): where V1 is the drop height (in m), considered a continuous numerical variable; V2 is the diameter of the steel ball (in mm), taken as a 3-level factor; V3 is the hardwood species, taken as a 4-level factor; V4 is the wood flooring type, taken as a 2-level factor; and V5 is the top layer thickness (mm), considered a continuous numerical variable.
The reference values of the models are listed in Table 4. For the footprint diameter, a gamma GLM with inverse link function was selected, which proved to be the most explanatory and which was significantly better than the other models tested. For the selection of the model for the ID variable, two models from the gamma family were shortlisted as the best candidates, one with a logarithmic link function and the other with an inverse link function. As may be observed in Table 4, the main reference values of these two models turned out to be appreciably similar, with a very slight advantage for the model with the logarithmic link function. Nonetheless, the better distribution of errors of the gamma model with inverse link function ( Figure 5) finally led to its selection.
where V1 is the drop height (in m), considered a continuous numerical variable; V2 is the diameter of the steel ball (in mm), taken as a 3-level factor; V3 is the hardwood species, taken as a 4-level factor; V4 is the wood flooring type, taken as a 2-level factor; and V5 is the top layer thickness (mm), considered a continuous numerical variable.
The reference values of the models are listed in Table 4. For the footprint diameter, a gamma GLM with inverse link function was selected, which proved to be the most explanatory and which was significantly better than the other models tested. For the selection of the model for the ID variable, two models from the gamma family were shortlisted as the best candidates, one with a logarithmic link function and the other with an inverse link function. As may be observed in Table 4, the main reference values of these two models turned out to be appreciably similar, with a very slight advantage for the model with the logarithmic link function. Nonetheless, the better distribution of errors of the gamma model with inverse link function ( Figure 5) finally led to its selection.  The results of the analysis of the selected models allow establishing that all the variables used were statistically significant. The individual explanatory capacity of each independent variable in terms of the explained deviance is presented in Tables 5 and 6 for FD and ID, respectively. The results of the analysis of the selected models allow establishing that all the variables used were statistically significant. The individual explanatory capacity of each independent variable in terms of the explained deviance is presented in Tables 5 and 6 for FD and ID, respectively.  Based on the values presented in Table 4 and on the values obtained from the likelihood-ratio test and Wald's test for both models, presented in Tables 5 and 6, the selected models may be regarded as strongly significant and highly explanatory. The equations of the final models are presented in Equations (3) and (4):

Model Diagnostics and Performance
Cook and Weisberg marginal model plots [42] were used to reproduce nonlinear marginal relationships for the predictors. In Figure 6, it may be observed that the models, represented by the dashed red lines, fitted quite well to the marginal relationships, represented by the solid blue lines.   Based on the values presented in Table 4 and on the values obtained from the likelihood-ratio test and Wald's test for both models, presented in Tables 5 and 6, the selected models may be regarded as strongly significant and highly explanatory. The equations of the final models are presented in Equations (3)

Model Diagnostics and Performance
Cook and Weisberg marginal model plots [42] were used to reproduce nonlinear marginal relationships for the predictors. In Figure 6, it may be observed that the models, represented by the dashed red lines, fitted quite well to the marginal relationships, represented by the solid blue lines.   The responses of the selected models for FD and ID have been plotted in Figures 7 and 8. For clarity, the responses have been divided into different windows according to ball diameter and wood flooring type, representing in each of them the values of FD and ID for the different drop heights and keeping the same scale on the Y-axis to enable visual comparisons of the evolution of the response.        The differences observed in the FD among the four hardwood species in the SW group decreased with ball diameter. These differences were always statistically significant, except for Q. robur and E. globulus, which presented a similar response in all cases. In the case of EWF with a 3.0 mm top layer, the values of FD response by species for the same ball size were-in all cases-significantly lower than those found in SW and significantly higher than those found in EWF with a 0.6 mm top layer. Within each of the three wood flooring types, for the different ball sizes, the existing differences between species were smaller as the ball diameter decreased, although they were still statistically significant with the exception, as in the previous case, of Q. robur and E. globulus, which did not show differences between them. It may be observed that, for EWF with a 0.6 mm top layer, the values of the four species were not clearly differentiated, becoming significantly equal as the ball size decreased.
The general trend of the response of the ID (Figure 8) was analogous to that of the FD, although there were some notable differences. Firstly, the ID of Q. robur and E. globulus differed in all cases for SW and EWF 3.0 mm, whereas they were similar in terms of FD. Secondly, for EWF 6.0 mm, differentiation between the species was gradually lost and, finally (ball diameter = 30 mm), no difference was found between the ID values of the wood flooring types of the four species, i.e., the indentation depth for EWF 0.6 mm showed a considerably complacent behavior, in which practically no differences between species were observed or, when they did exist, they were extremely small. This could also be observed when the relationship between the target variables (FD and ID) and the impact energy was represented (Figure 9). The differences observed in the FD among the four hardwood species in the SW group decreased with ball diameter. These differences were always statistically significant, except for Q. robur and E. globulus, which presented a similar response in all cases. In the case of EWF with a 3.0 mm top layer, the values of FD response by species for the same ball size were-in all cases-significantly lower than those found in SW and significantly higher than those found in EWF with a 0.6 mm top layer. Within each of the three wood flooring types, for the different ball sizes, the existing differences between species were smaller as the ball diameter decreased, although they were still statistically significant with the exception, as in the previous case, of Q. robur and E. globulus, which did not show differences between them. It may be observed that, for EWF with a 0.6 mm top layer, the values of the four species were not clearly differentiated, becoming significantly equal as the ball size decreased.
The general trend of the response of the ID (Figure 8) was analogous to that of the FD, although there were some notable differences. Firstly, the ID of Q. robur and E. globulus differed in all cases for SW and EWF 3.0 mm, whereas they were similar in terms of FD. Secondly, for EWF 6.0 mm, differentiation between the species was gradually lost and, finally (ball diameter = 30 mm), no difference was found between the ID values of the wood flooring types of the four species, i.e., the indentation depth for EWF 0.6 mm showed a considerably complacent behavior, in which practically no differences between species were observed or, when they did exist, they were extremely small. This could also be observed when the relationship between the target variables (FD and ID) and the impact energy was represented (Figure 9).  Table 2.
As is shown in Figures 7 and 8, due to the size of the footprint and wood anisotropy, the results showed that measuring the FD is an easier and more precise method than measuring the ID.
The construction morphology of the EWF has a structural performance analogous to that of a collaborating slab. In this structure, the HDF layer is responsible for absorbing much of the impact energy, being greater the thinner the hardwood layer is, and actively forcing the outer layer to recover from the deformation caused by the impact (Figure 10a). In the absence of this HDF layer, permanent deformation would occur, essentially perpendicular to the ground, with no significant recovery in the indentation depth. This can be verified by plotting the FD:ID ratio for each wood flooring type as a function of the drop height, regardless of steel ball diameter and hardwood species. In Figure 10b, it may  Table 2.
As is shown in Figures 7 and 8, due to the size of the footprint and wood anisotropy, the results showed that measuring the FD is an easier and more precise method than measuring the ID.
The construction morphology of the EWF has a structural performance analogous to that of a collaborating slab. In this structure, the HDF layer is responsible for absorbing much of the impact energy, being greater the thinner the hardwood layer is, and actively forcing the outer layer to recover from the deformation caused by the impact (Figure 10a). In the absence of this HDF layer, permanent deformation would occur, essentially perpendicular to the ground, with no significant recovery in the indentation depth. This can be verified by plotting the FD:ID ratio for each wood flooring type as a function of the drop height, regardless of steel ball diameter and hardwood species. In Figure 10b, it may be

Final Remarks
Based on the test results obtained, two key outcomes should be highlighted: (i) E. globulus showed very similar behavior to that of Q. robur in the three wood flooring typologies in terms of the footprint diameter and the indentation depth. Hence, E. globulus, a fast-growing species, may hold promise to replace (or complement) Q. robur, a much slower-growing species, in the production of wooden floorings. As for the values of FD and ID in E. grandis, due to its lower density, significant differences versus the rest of the hardwood species were observed in SW and EWF 3.0 mm floorings, but, in the EWF with a 0.6 mm top layer, the ID variable did not show significant differences with the rest of the species. Similar conclusions were reached by other researchers [43,44], although from the comparison of different hardwood species with similar density ratios. This fast-growing species, currently destined for uses with little added value, may thus be valorized as an oak replacement in veneer floorings. (ii) Regarding the constructive typology of the floors, the replacement of part of the solid wood by a 9 mm HDF board significantly improved the behavior of the floor in terms of its performance in hardness tests, regardless of the hardwood species used. These results provide evidence that hardness is more closely related to density than to other wood properties [45], which explains why HDF board properties are more representative than those of the hardwood layer. An EWF flooring with a hardwood layer thickness of only 0.6 mm resulted in significantly lower FD and ID values than an EWF typology with a 3 mm hardwood layer thickness for three of the hardwood species (E. globulus, E. grandis, and Q. robur), while, in the case of the densest wood, from H. courbaril, the difference in FD was not as evident between the two EWF floorings. This implies that using a 9 mm HDF board and a 0.6 mm thickness of the solid wood top layer may save a significant amount of high-quality wood and lower the cost of the final product while offering better performance in terms of hardness than solid wood flooring.

Conclusions
The dynamic hardness test results demonstrated that veneer floorings (with a top layer thickness of only 0.6 mm) exhibited a performance comparable to (or better than) those of solid wood floorings and traditional engineered wood floorings (with a top layer thickness of 3 mm), with lower footprint diameter and indentation depth values. Concerning the influence of the hardwood used in the top layer, one of the two fast-

Final Remarks
Based on the test results obtained, two key outcomes should be highlighted: (i) E. globulus showed very similar behavior to that of Q. robur in the three wood flooring typologies in terms of the footprint diameter and the indentation depth. Hence, E. globulus, a fast-growing species, may hold promise to replace (or complement) Q. robur, a much slower-growing species, in the production of wooden floorings. As for the values of FD and ID in E. grandis, due to its lower density, significant differences versus the rest of the hardwood species were observed in SW and EWF 3.0 mm floorings, but, in the EWF with a 0.6 mm top layer, the ID variable did not show significant differences with the rest of the species. Similar conclusions were reached by other researchers [43,44], although from the comparison of different hardwood species with similar density ratios. This fast-growing species, currently destined for uses with little added value, may thus be valorized as an oak replacement in veneer floorings. (ii) Regarding the constructive typology of the floors, the replacement of part of the solid wood by a 9 mm HDF board significantly improved the behavior of the floor in terms of its performance in hardness tests, regardless of the hardwood species used. These results provide evidence that hardness is more closely related to density than to other wood properties [45], which explains why HDF board properties are more representative than those of the hardwood layer. An EWF flooring with a hardwood layer thickness of only 0.6 mm resulted in significantly lower FD and ID values than an EWF typology with a 3 mm hardwood layer thickness for three of the hardwood species (E. globulus, E. grandis, and Q. robur), while, in the case of the densest wood, from H. courbaril, the difference in FD was not as evident between the two EWF floorings. This implies that using a 9 mm HDF board and a 0.6 mm thickness of the solid wood top layer may save a significant amount of high-quality wood and lower the cost of the final product while offering better performance in terms of hardness than solid wood flooring.

Conclusions
The dynamic hardness test results demonstrated that veneer floorings (with a top layer thickness of only 0.6 mm) exhibited a performance comparable to (or better than) those of solid wood floorings and traditional engineered wood floorings (with a top layer thickness of 3 mm), with lower footprint diameter and indentation depth values. Concerning the influence of the hardwood used in the top layer, one of the two fast-growing hardwoods (E. globulus) showed a behavior similar to that of oak, suggesting that it may be a suitable replacement for the latter in the three types of wood floorings tested. The other fast-growing species (E. grandis), with a lower density, would only be a suitable alternative to oak for veneer floorings (0.6 mm top layer). These experimental results have been completed with two dynamic response models, one for predicting the footprint diameter and the other for the indentation depth of the footprint, which include the ball diameter, drop height, hardwood species, flooring type, and thickness of the top layer variables. These models can be used by wood flooring factories to choose the combination of wood flooring typology and thickness of the top layer, with the species analyzed in the study, to meet customers' requirements in terms of impact hardness.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to their relevance to an ongoing Ph.D. thesis.