Multi-scale mechanical characterization of the lining canvas of The Night Watch by Rembrandt

Fibre scale

The experimental methodology presented in Section “In-situ tensile tests on fibre specimens” was applied to fibre specimens extracted from the lining canvas − 10 in the warp and 11 in the weft direction − and from the reference canvas − 4 in the warp and 6 in the weft direction. Figure 9a illustrates the stress-strain response (σ_xx − ε_xx) of one representative fibre from the weft direction of the lining canvas, based on data collected from 18 ROIs. Each curve reflects a local mechanical response of the fibre, measured at a specific ROI. During the initial loading steps, the fibres undergo uncurling and straightening. As this stage primarily reflects geometric realignment rather than elastic deformation, these early data were excluded from the analysis. After removing these points, the initial portion of the stress-strain curve − depicted by a dashed line − was reconstructed by extrapolating the measurement data back to the origin using a first-order polynomial fit. The resulting curves exhibit an approximately linear trend up to abrupt fracture. This brittle failure behaviour may be attributed to fibre embrittlement caused by ageing processes, consistent with observations for other cellulose-based fibres^24,42,43,44. Since fracture occurs abruptly after the acquisition of the final height profile, GDHC cannot capture the actual strain at fracture. Therefore, for each ROI, the strain at fracture is estimated by extrapolating a first-order polynomial fit through the last three data points of the stress-strain curve up to the final stress at fracture, calculated as the last recorded force divided by the fibre’s minimum cross-sectional area. This estimation is illustrated in Fig. 9a by the dashed line at the end of each curve, with an asterisk symbol marking the local strain at fracture. The variability in the local responses of the reference fibre is considerable, reflecting its heterogeneous nature. It is worth noting that local stresses and strains measured on a single fibre allow the deduction of statistically representative material parameters (elastic modulus, ultimate tensile strength, strain at fracture) from a relatively small number of tensile tests. Similar stress-strain curves were obtained for the other fibres tested, from both the lining and reference canvases, in warp and weft directions, which are omitted here for brevity.

Based on the stress-strain responses of all tested fibre samples, the average mechanical properties are determined. For each fibre, the strain at fracture is first calculated as the mean of the local strain at fracture values across all ROIs. These values are then averaged over all specimens within each category to obtain the final average strain at fracture, ε_f. The corresponding standard deviation, computed across the fibre samples, is reported alongside ε_f in Fig. 9b. In this figure, the dark blue and red bars represent the lining canvas in the warp and weft directions, respectively, while the light blue and orange bars denote the reference canvas in the warp and weft directions, respectively. This colour scheme will be applied consistently throughout the manuscript to distinguish sample types.

The strain at fracture is broadly consistent across three of the four sample types, with mean values clustering around 0.015. A notable exception is observed for the warp-direction fibres of the reference canvas, which exhibit a significantly lower average strain at fracture of 0.011. In addition, the standard deviations for all specimen types range from 0.004 to 0.010. The average strains at fracture largely fall within the experimental range of 0.012 to 0.033 reported in the review paper ref. ⁴⁵, which summarises data on linen (flax) fibres. However, the present measurements lean toward the lower end of this range. This difference may be attributed to the prevalent use of global strain assessment methods in previous studies (e.g.,^28,29,46), where the strain is inferred from the relative displacement of the tensile stage grips. This approach may considerably overestimate strain values, as it also includes the effects of fibre slippage at the grips. In contrast, the present study employs a local strain measurement at the fibre level, providing a more accurate and representative evaluation of the strain at fracture.

The Young’s modulus for each ROI was calculated from the initial slope of the stress-strain curve, and then averaged across the ROIs to obtain fibre-level values. Subsequently, these were averaged per fibre type. Figure 9c shows that the average Young’s modulus E ranges from 42.3 to 83.6 GPa across the four fibre types. The measured values fall within the broader range of 27.6 to 103 GPa reported for flax fibres in ref. ⁴⁵. However, it should be noted that ref. ⁴⁵ does not distinguish between weft and warp fibre orientations. Moreover, the standard deviation of E ranges from 25.9 to 48.4 GPa, indicating considerable variability in Young’s modulus due to the intrinsic heterogeneity of linen fibres. Fibres oriented in the warp direction exhibit lower Young’s moduli compared to those in the weft direction. This can be explained by the fact that warp threads are typically subjected to higher tension during weaving and must remain taut on the loom afterwards. This sustained tension restricts their ability to shrink and relax when the fabric is unloaded, whereas the weft threads are free to deform more easily. The constrained shrinkage and residual stresses in the warp threads can induce micro-damage within the fibres, such as micro-cracking or fibre misalignment, ultimately leading to a reduction in their stiffness. Note further that fibres extracted from the lining canvas exhibit lower Young’s moduli compared to the corresponding fibre types from the reference canvas. This trend, observed consistently across different scales and further examined in Sections “Thread scale” and “Canvas scale”, is provisionally attributed to the more extensive degradation experienced by the lining canvas. This degradation may be hypothesised to be the result of the presence of wax-resin, the environmental display conditions of the lining canvas, and the constant stress exerted by the stretcher, since the reference canvas lacks wax-resin, was stored under controlled environmental conditions, and remained stress-free. However, the large standard deviations observed introduce significant uncertainty in this interpretation, suggesting that the mechanical properties of the lining and reference canvases may be more similar than the trend implies. The hypothesis on the influence of wax resin and display conditions on degradation will be statistically evaluated in Section “Assessing degradation: statistical comparison between the lining canvas and the reference canvas”.

The tensile strength of each fibre is determined as the ratio of the maximum applied tensile force to the smallest cross-sectional area across the fibre length. The fibre tensile strengths were averaged for each fibre type to compute the average fibre tensile strength. Figure 9d presents both the average fibre tensile strength, σ_f, and the corresponding standard deviation for the four different fibre types. The average tensile strengths range from 689 to 1738 MPa, with standard deviations between 266 and 1199 MPa. These values are consistent with previous studies on linen fibres, as reported in ref. ⁴⁵. Similar to the trend observed for the Young’s modulus, the fibre tensile strength is generally higher in the weft direction than in the warp direction. This anisotropy may again be attributed to a greater constrained shrinkage and residual stresses of fibres in the warp direction. When comparing the lining canvas to the reference canvas, fibres from the lining canvas exhibit lower strength in the warp direction but similar strength in the weft direction. A more detailed statistical analysis of fibre strength for both canvases is provided in Section “Assessing degradation: statistical comparison between the lining canvas and the reference canvas”.

Thread scale

The experimental procedure described in Section “In-situ tensile tests on thread specimens” was applied to 5 warp and 5 weft thread samples from the lining canvas, and to 4 warp and 3 weft thread samples from the reference canvas. Hence, 17 thread specimens were tested in total, and their corresponding stress-strain (σ_xx–ε_xx) responses are presented in Fig. 10a. Each curve in Fig. 10a exhibits three distinct regions. In the initial region, the slope gradually increases, corresponding to the straightening of the threads under the applied load. This straightening extends over a larger deformation range in the warp threads compared to the weft threads. This behaviour is attributed to larger constrained shrinkage and residual stresses in the warp threads, ultimately leading to micro-damage within their fibres and consequently reducing their stiffness, see Fig. 9c. The reduced fibre stiffness causes the warp threads to undergo greater initial deformation than the weft threads in order to develop a given stress during straightening. Furthermore, the small initial stress measured at the onset of this region is likely caused by straightening of some of the thread’s waviness and shrinkage during the clamping process; hence, the initial part of the curve is extended to zero stress by applying a linear extrapolation (shown as a coloured dashed line) through the first two data points. The second region of the curves is characterised by an approximately constant slope, corresponding to the linear elastic response of the threads under tension. In this region, the warp threads exhibit lower elastic stiffness than the weft threads, as shown by the slope. This discrepancy reflects the effect of a higher fibre micro-damage previously described for the initial region. The third region, leading up to the maximum load, is characterised by a progressively decreasing slope, indicating the initiation of inelastic deformation due to micro-damage and/or micro-plasticity under the applied load. The extent of this region, which thus reflects progressive failure of fibres within the thread, differs among specimen types. It is noted that fibre fracture continues well beyond the point of maximum load, persisting until all fibres have failed and the tensile force in the thread reaches zero. However, the post-peak softening regime associated with progressive fibre failure is not included in the stress-strain curves depicted in Fig. 10a, as the analysis of the mechanical response beyond the peak load lies outside the scope of the present study. Furthermore, the reference for the individual stress-strain curves in Fig. 10a is established as follows: a first-order polynomial (indicated by a black dashed line) is fitted to the linear elastic portion of the curve (i.e., the second region). The entire curve is then horizontally shifted such that the fitted polynomial intersects the origin, ensuring that zero axial strain corresponds to zero axial stress. It can be observed that the stress-strain curves display substantial variability, primarily due to heterogeneities in the flax fibres and variations introduced during the flax yarn manufacturing process. Similar variability has been reported in other experimental studies^19,47.

The strain at fracture for each thread sample is obtained from the stress-strain curves, corresponding to the (adjusted) strain at maximum load. This value is then averaged across each sample category to determine the average strain at fracture, denoted as ε_f. Figure 10b presents the average strain at fracture along with its standard deviation, indicated by the error bars. Despite frequent imaging during testing, an image corresponding exactly to the maximum load is typically unavailable. Therefore, to estimate the strain at fracture, the same approach used for the fibre specimens in Section “Fibre scale” is applied: a first-order polynomial is fitted to the final three data points of the stress-strain curve and extrapolated to the stress value at fracture, as indicated by the dashed line at the end of each curve in Fig. 10a. However, due to the higher imaging frequency employed for thread specimens (optical images) compared to fibre specimens (surface height profiles), the extrapolated segment is very short and barely visible. No consistent trend can be identified between the strain at fracture of threads from the lining canvas and the reference canvas. Nonetheless, the strain at fracture in the warp direction seems to be larger than that in the weft direction − a characteristic not fully evident at the fibre scale. Across all specimen types, the average strain at fracture ranges between 0.018 and 0.022, with standard deviations varying from 0.004 to 0.006. The average strain values align with literature data reporting values ranging from 0.017 to 0.037 for different flax threads^18,47.

Two separate sets of results are presented for the Young’s modulus and tensile strength, distinguishing between the composite and intrinsic properties of the thread. The strain at fracture, however, is unaffected by this distinction, as it is independent of the cross-sectional area. Figure 10c presents the mean value and standard deviation of the composite Young’s modulus, whereas Fig. 10d shows these values for the intrinsic Young’s modulus. The average composite Young’s modulus measured across all tested specimens falls within the range of 10.0−21.6 GPa. These values align with the range of 6.4−16.7 GPa reported in previous studies on linen (flax) yarns¹⁹, which do not differentiate between the warp and weft directions. The standard deviation of the composite Young’s modulus across all specimen types ranges from 1.6 to 6.5 GPa. This relatively large variability reflects the inherent heterogeneity at the thread scale, comparable to what is observed at the fibre level. In contrast, the average intrinsic Young’s modulus measured for the tested specimens ranges from 25.6 to 38.1 GPa, with standard deviations spanning from 5.7 to 15.2 GPa. For both the composite and intrinsic Young’s moduli, it is observed that threads extracted from the lining canvas tend to exhibit lower Young’s moduli compared to those from the reference canvas. While this trend could hypothetically be attributed to a higher degree of degradation in the lining canvas − possibly linked to the presence of wax resin and prolonged exposure to gallery conditions − the large standard deviation in the data, as discussed at the fibre scale in Section “Fibre scale”, indicates that the mechanical properties of the two canvases may be more similar than the apparent trend suggests. Statistical analyses presented in Section “Assessing degradation: statistical comparison between the lining canvas and the reference canvas” will further evaluate the significance of the differences observed between the lining and reference canvases. Furthermore, the properties measured in the warp direction are generally lower than those in the weft direction. This difference may again be attributed to the higher constrained shrinkage and residual stresses in the warp threads, which promote micro-damage formation. It is also important to note that the intrinsic Young’s moduli exhibit significantly higher values compared to the composite moduli. The intrinsic Young’s moduli represent only the load-bearing capacity of the fibre material − the stiffest constituent − while excluding the influence of the porous structure and the wax-resin. Additionally, the composite Young’s moduli display considerable variation among the sample groups, reflecting the heterogeneous distribution of wax resin and pores. When the cross-sectional area correction factors are applied, the resulting intrinsic Young’s moduli reveal a more consistent trend across the samples, with the exception of the reference canvas in the weft direction.

The strength of each thread is determined by dividing the peak load obtained from the stress-strain curves by the smallest measured cross-sectional area. The individual thread strength values are then averaged within each sample category. Figure 10e presents the average composite tensile strength σ_f along with its standard deviation, while Fig. 10f illustrates the average intrinsic strength values. Overall, the average composite strength ranges from 197 to 320 MPa, with standard deviations between 12 and 63 MPa. These average strength values are consistent with the reported range of 115–339 MPa for flax threads, presented in refs. ^18,19. The average composite strength measured in the warp direction is generally lower than that in the weft direction. This difference may be attributed to the greater micro-damage in the warp threads, which reduces their effective load-bearing capacity. Furthermore, the strength values measured from the lining canvas samples are lower than those obtained from the reference canvas samples. These observations broadly align with trends identified at the fibre scale and will be examined in detail through statistical analyses in Section “Assessing degradation: statistical comparison between the lining canvas and the reference canvas”. For all tested specimens, the average intrinsic strength ranges from 462 to 562 MPa, with standard deviations spanning from 21 to 179 MPa. After applying cross-sectional correction factors, the intrinsic strength values are significantly higher than the corresponding composite strength values. Nonetheless, no consistent pattern can be identified between the warp/weft characteristics and the properties of the lining and reference canvases.

The Young’s moduli presented in Fig. 10 are derived from the linear branch of the stress-strain curves shown in Fig. 10a, corresponding to the second phase following the straightening of the thread samples. However, the initial part of the stress-strain curves, where the slope gradually increases, is also of interest in this study. This region offers insight into the thread response under relatively low stress levels, which more closely represents the typical loading conditions experienced by the lining canvas in museum environments. The initial Young’s modulus E₀ is determined as the slope of the first-order polynomial fitted to the first two data points of each stress-strain curve, as described previously. Figure 11 illustrates the initial Young’s modulus E₀ for thread specimens from both the lining and reference canvases, in the warp and weft directions, considering composite and intrinsic properties. The average initial composite Young’s modulus measured across all tested specimens ranges from 1.2 to 5.5 GPa, with standard deviations between 0.9 and 3.5 GPa. In contrast, the average initial intrinsic Young’s modulus ranges from 1.9 to 9.6 GPa, with standard deviations between 1.4 and 8.3 GPa. For both types of initial Young’s moduli, the values in the warp direction are significantly lower than those in the weft direction, which can again be ascribed to the larger micro-damage in the warp threads. Furthermore, comparing the lining and reference canvases reveals no obvious distinction. The substantial standard deviations observed in the initial Young’s moduli of the thread specimens can likely be attributed to the inherent heterogeneity in local fibre alignment and fibre properties, see also Fig. 9.

Canvas scale

The experimental procedure described in Section “In-situ tensile tests on canvas specimens” was applied to 4 specimens each from the warp and weft directions of the lining canvas, and 6 specimens each from the warp and weft directions of the reference canvas. Figure 12a shows the resulting stress-strain curves (σ_xx–ε_xx) for the 20 canvas samples. Similarly to the thread response in Fig. 10a, these curves display an initial nonlinear region, an approximately linear elastic region, and a post-yield region leading to catastrophic failure. Consistent with the thread scale, the curves have been horizontally shifted so that the extrapolation of the linear elastic branch (black dashed line) runs through the origin of the diagram. Further, the initial part of each stress-strain curve is linearly extrapolated to zero stress using the first two data points of the curve (coloured dashed line). For reasons similar to those discussed in Section “Thread scale” with respect to the thread responses presented in Fig. 10a, canvas specimens loaded in the warp direction exhibit greater deformation during the initial straightening phase (first region) and display lower elastic stiffness in the subsequent linear elastic regime (second region), compared to those loaded in the weft direction.

The strain at fracture for each canvas specimen is determined from the stress-strain curves at the point of maximum load. The average strain at fracture, ε_f, together with its standard deviation, is presented in Fig. 12b. These results indicate that specimens loaded in the warp direction exhibit a higher strain at fracture than those loaded in the weft direction. This observation is consistent with the thread-scale behaviour reported in Section “Thread scale”. For all specimen types, the average fracture strain ranges from 0.011 to 0.035 with standard deviations between 0.002 and 0.013, with the lining canvas specimens showing higher fracture strain values in both the warp and weft directions compared to the reference canvas. It is worth noting that the strain at fracture values reported in previous studies on linen-based canvas paintings (e.g., ref. ⁸) are significantly higher − approximately 0.25 for the warp direction and 0.13 for the weft direction. However, in ref. ⁸, the strain at fracture was calculated by including the initial strain associated with the straightening phase of the threads, and should therefore be compared to the maximum, total strain values measured in the current study. As can be observed from Fig. 12a, for the lining canvas in the warp direction the total strain reaches a maximum of approximately 0.15. Although this total strain value is closer to that reported in ref. ⁸, it remains lower, likely due to differences in the materials, strain measurement methods, and evaluation criteria employed in each study. In particular, the disparity in strain measurement approaches plays a significant role: the current study uses local strain evaluation at the canvas scale, whereas ref. ⁸ relies on global strain measurements that tend to overestimate strain, see also the discussion in Section “Fibre scale”.

Following the same approach applied at the thread scale, the composite and intrinsic Young’s moduli and tensile strengths are evaluated. Figure 12c shows the mean value and standard deviation of the composite Young’s modulus E for the four canvas types, whereas Fig. 12d displays the corresponding values for the intrinsic Young’s modulus. The mean values of the composite Young’s modulus measured across all specimens range from 1.43 to 4.37 GPa, with standard deviations between 0.40 and 1.43 GPa. These values are higher than those reported in previous studies investigating the mechanical properties of linen canvases ^1,2,8. Note, however, that in these earlier works the reported elastic modulus corresponds to the initial modulus derived from the initial slope of the stress-strain curve, whereas in the present analysis the modulus corresponds to the subsequent, approximately linear region. This difference in definition accounts for the higher values obtained here. The mean values of the intrinsic Young’s modulus range between 4.86 and 17.91 GPa, with standard deviations from 1.35 to 6.26 GPa. For both the composite and intrinsic Young’s moduli, specimens loaded in the warp direction consistently show lower mean values than those loaded in the weft direction. This trend again reflects the effect of larger micro-damage in the warp fibres. Overall, the lining canvas specimens generally show Young’s moduli that are similar to or slightly lower than those of the reference canvas; a statistical analysis quantifying these variations is presented in Section “Assessing degradation: statistical comparison between the lining canvas and the reference canvas”. As expected, the intrinsic Young’s moduli are considerably larger than the composite values, and the relative differences between the warp and weft directions are more pronounced at the canvas scale than at the thread scale (Fig. 10d).

Figure 12e presents the mean and standard deviation of the composite tensile strength σ_f for the four canvas types, while Fig. 12f shows the corresponding intrinsic tensile strength values. Overall, the mean composite strength ranges from 33 to 63 MPa, with standard deviations between 4 and 24 MPa, which aligns with the measurements reported in ref. ², when differences in specimen width are taken into account. The mean value of the intrinsic ultimate tensile strength across all canvas types ranges from 101 to 193 MPa, with standard deviations between 15 and 103 MPa. Both the composite and intrinsic tensile strengths are generally lower in the warp direction compared to the weft direction. Interestingly, unlike the trends observed on the fibre and thread scales, the composite tensile strengths of the lining canvas in warp and weft directions exceed those of the reference canvas, whereas the intrinsic tensile strength does not exhibit any clear or consistent difference between the two.

Figure 13 shows the initial Young’s modulus E₀ measured at the onset of loading for the lining and reference canvases, in both warp and weft directions, with results reported for composite and intrinsic properties. As part of the recent retensioning procedure carried out during Operation Night Watch, the newly installed spring tension system applied carefully controlled loads per unit length of 2.7 N/cm and 2.0 N/cm in the warp and weft directions, respectively⁴¹. These loads correspond to stresses below 1 MPa on the lining canvas, calculated based on its thickness and composite properties. Determining the initial Young’s modulus E₀ is therefore particularly relevant, as it characterizes the canvas stiffness under these low in-situ stress levels applied by the tensioning system. Observe from Fig. 13 that, consistent with observations at the thread scale, the initial Young’s modulus E₀ in the warp direction is lower than in the weft direction. In ref. ⁸, the reported (initial) Young’s moduli range from 0.05 to 0.50 GPa for the warp direction and from 0.2 to 0.9 GPa for the weft direction. By comparison, the composite initial Young’s modulus determined in this study varies between 0.06 and 0.22 GPa for the warp direction and between 0.79 and 2.91 GPa for the weft direction. These values fall within the same order of magnitude but are modestly higher than those reported by ref. ⁸. This difference may be due to the use of local strain measurements in the present study, as opposed to the global strain evaluation applied in ref. ⁸, which tends to overestimate strain and thus underestimate the (initial) Young’s modulus. The discrepancy may also simply reflect variations in sample type, including microstructure, conditions, past conservation treatments, or manufacturing.

Scaling laws

To compare the mechanical response of the lining canvas across different structural levels, the material properties reported in Sections “Fibre scale”, “Thread scale”, and “Canvas scale” − corresponding to fibre, thread, and canvas measurements − are examined as a function of their characteristic length scales. For fibres and threads, the characteristic length scale l_c is defined as the square root of the average cross-sectional area, ranging from 5.6–18.2 μm to 285.7–549.7 μm, respectively. Canvas specimens, which have a basket weave formed by interwoven thread pairs, are idealised as a periodic structure. Hence, the characteristic length scale l_c is defined as the distance between the centres of adjacent thread pairs, calculated by dividing the specimen width (20 mm) by the number of thread pairs. Based on measured thread densities of 19 threads/cm in the warp direction and 17 threads/cm in the weft direction (see also Section “Characteristics of the canvas support of The Night Watch”), this yields characteristic length scales of approximately 1 mm for the warp direction and 1.33 mm for the weft direction.

Figure 14 summarises the measured mechanical properties of the lining canvas as a function of the characteristic length scale, l_c [μm]. Blue and red markers correspond to the warp and weft directions, respectively. A logarithmic horizontal axis is used to capture the wide range of length scales. Figure 14a and b show the composite and intrinsic Young’s modulus E, respectively, while Fig. 14c and d present the corresponding composite and intrinsic ultimate tensile strength σ_f. The strain at fracture does not exhibit a clear trend with length scale and is therefore omitted. Scaling laws relating the mechanical properties to the characteristic length scale l_c are derived from the data in Fig. 14, with the fitted relationships shown as continuous lines in each figure. The general form of the scaling law of the lining canvas is given by

where M ∈ {E, σ_f} denotes the evaluated material parameter (Young’s modulus or tensile strength), and a, b and c are calibration parameters. The values of the calibration parameters, corresponding to the Young’s modulus and tensile strength in both the warp and weft directions, along with the associated R² values, are listed in Table 3.

Figure 14 shows that the Young’s modulus and tensile strength decrease with increasing length scale in both warp and weft directions. This reduction at larger scales is attributed to an increased probability of defects within the fibres, as well as microstructural mechanisms − such as frictional interactions and slippage between fibres − that have a more prominent cumulative effect at larger scales, collectively reducing the average stiffness and tensile strength. For the composite properties, the fitted scaling laws result in R² values ranging from 0.62 to 0.90, indicating moderate to strong agreement with the experimental data. In contrast, for the intrinsic properties, the R² values fall between 0.54 and 0.75, suggesting that the scaling laws capture the overall trends but with more scatter and lower predictive accuracy.

The presented scaling laws may enable, in the future, informed predictions of the mechanical properties of the lining canvas at the canvas scale, based on non-invasive fibre-level analyses rather than destructive sampling at thread or canvas scale, which is an important advantage given the limited availability of lining canvas specimens. The scaling laws for composite properties are directly relevant to conservation efforts for The Night Watch, as they reflect its specific porosity and wax-resin impregnation. In contrast, scaling laws for intrinsic properties are expected to show greater generality, as they relate to the ideally uniform linen fibre phase. Beyond this case study, these scaling laws may also support the analysis of other historic (lining) canvases made of linen or comparable materials, broadening their applicability to conservation strategies in cultural heritage. Finally, it is noted that similar scaling laws were derived for the reference canvas, but these are not included here for brevity.

Assessing degradation: statistical comparison between the lining canvas and the reference canvas

The state of degradation of the lining canvas of The Night Watch is assessed by statistically comparing its mechanical properties with those of a reference canvas. The reference canvas, preserved in a controlled environment and free from wax-resin impregnation, serves as a benchmark to evaluate the potential effects of the Rijksmuseum gallery environment, the presence of wax-resin, and the continuous tensile stress applied to the lining canvas on its mechanical degradation. The comparison is performed across different length scales (fibres, threads, and canvas) and focuses on the intrinsic Young’s modulus, intrinsic tensile strength, and strain at fracture. Using intrinsic properties ensures a consistent basis for comparison by reducing the influence of confounding factors such as wax resin content, porosity, and cutting spread of threads.

To assess differences between the two data groups, an independent two-sample Student’s t-test is performed⁴⁸. This test evaluates whether the means of the two groups differ significantly, under the null hypothesis that the group means are equal, within a certain statistical significance. The t-statistic is calculated as

where \({\overline{X}}_{1},{\overline{X}}_{2}\) are the means on the specific sample quantities analysed, s₁, s₂ are the sample variances, and n₁, n₂ are the sample sizes. The t-value obtained through Eq. (2) for the different properties analysed is compared with a critical threshold, t_critical, which depends on the selected significance level α and the degrees of freedom of the problem (n₁ + n₂ − 2). If ∣t∣ < t_critical, the null hypothesis of equal mean values cannot be rejected. The results of the t-test for α = 0.05 are reported in Table 4.

The validity of this test relies on three key assumptions: (i) the samples are independent; (ii) the data approximately follow a normal distribution; and (iii) the two groups have similar standard deviations. In the present analysis, data independence is satisfied. Due to the small sample sizes, the normality assumption cannot be formally tested and is assumed a priori⁴⁹. The assumption of comparable standard deviations − often considered reasonable if their ratio does not exceed approximately two⁴⁹ − is met for most of the properties tested. Nevertheless, given the limited sample size, the robustness of this comparison may be limited.

To address this limitation, a Mann−Whitney U-test is additionally performed as an alternative statistical analysis⁵⁰. Unlike the t-test, the Mann−Whitney U-test is non-parametric and therefore does not assume a normal data distribution or require comparable standard deviations. The test ranks all data points, where the rank of a data point corresponds to its position in the combined data set sorted in ascending order. The test calculates the U-statistics for the two sample groups as

where R₁ and R₂ are the rank sums of the respective groups. The smaller value between U₁ and U₂ is compared to the critical value Z_critical, derived from the Mann−Whitney U distribution tables for the given sample sizes (n₁, n₂) and significance level (α). If ∣U ∣ < Z_critical, the null hypothesis that the distributions are identical cannot be rejected. The results of the U-test for a significance level α = 0.05 are also reported in Table 4.

The results of both tests are consistent, with all cases − except one − supporting the null hypothesis. The exception is the tensile strength of the canvas in the warp direction, where a significant difference is observed between the means of the two sample groups. For all other cases, acceptance of the null hypothesis suggests that environmental conditions and the presence of wax resin may have had a negligible effect on the mechanical degradation of the lining canvas. The agreement between the t– and U-tests further strengthens the reliability of this conclusion. Although definitive statements are limited by the small sample size, this statistical analysis provides valuable insight into identifying or excluding factors that potentially contribute to canvas degradation.