Deformation measurement accuracy
To evaluate the accuracy of ICT, synthetic deformation fields were added to the R-specimen datasets (Fig. 3a). For this purpose, constant strain was simultaneously introduced in both R and L directions ((varepsilon_{RR}),({ }varepsilon_{LL})). Absolute accuracy (hat{varepsilon }) was measured by adding synthetic deformation to the reference state #1 (#1 + synthetic), and then measuring with ICT strain of #1 + synthetic with respect to #1. Differential accuracy ({Delta }hat{varepsilon }) was measured by adding synthetic deformation to the deformed state #2 (#2 + synthetic), measuring with ICT strain of #2 + synthetic with respect to the reference state #1, and finally subtracting the ICT measured strain between #2 and #1.
Figure 3 shows ICT strain estimates for tracheids and wood rays. The accuracy is highest along the cell-cross section (RT for tracheids, LT for wood rays) and lowest in the cell longitudinal direction (L for tracheids, R for wood rays). Due to the tubular cell geometry (Fig. 2b,c) of both cell types, tracking is more challenging in the longitudinal cell axis, for which symmetry reduces the available landmarks (for instance, wood pits in Fig. 2) and deformation tracking accuracy. Absolute accuracy is limited by both cell segmentation and deformation parametrization. Differential accuracy is further influenced by experimental uncertainties between #1 and #2 acquisitions, such as vibration artifacts and sample relaxation strains. While absolute and differential accuracy are similar in cell-cross sections, differential accuracy is reduced with respect to absolute accuracy along the longitudinal cell axis. The sensitivity limit for strain measurements in tracheids is (varepsilon_{RR}) < 5 × 10–5 and (varepsilon_{LL}) < 2 × 10–3, whereas for wood rays (varepsilon_{LL}) < 2 × 10–4 and (varepsilon_{RR}) < 2 × 10–3. By evaluating deformations at individual cell level, ICT optimizes strain tracking accuracy, since (varepsilon_{LL}) estimates from wood rays are not influenced by the lower (varepsilon_{LL}) accuracy of adjacent wood tracheids and vice versa for (varepsilon_{RR}). Since the average tracheid lumen diameter is 35 µm, (varepsilon_{RR}) < 5 × 10–5 corresponds to a deformation of 1.8 nm, i.e. three orders of magnitude below the voxel size. A similar precision is observed for tracking changes in the tracheid lumen area (varepsilon_{Sigma }) < 5 × 10–5. Tracheid cell wall thickness deformation down to (varepsilon_{t}) < 5 × 10–3 also follows accurately the synthetic strain trends. This high sensitivity to small strains is explained by the fact that ICT computes strains using global metrics (lumen centroid, area and inclination), which average the contributions of all lumen voxels, thereby reducing the effect of discretization noise. Due to the smaller lumen diameters, wood ray lumen area deformation can presently only be tracked down to (varepsilon_{Sigma }) < 5 × 10–3. Similarly, for wood rays, due to the small lumen size and the simultaneous presence of multiple cell lumens per cluster, cell wall deformation analysis is more involved, and was left out of the scope of this work.
The tracking accuracy obtained with ICT is comparable or superior to values reported for DVC, for which typical displacement uncertainties are 0.1–0.01 voxels 33. Gillard et al.52 measured strains in bone with DVC down to 2 × 10–4–8 × 10–4. As for lateral resolution, minimum DVC correlation subsets to track displacements in wood have been reported in the range 110–330 µm, i.e. 3–9 tracheid diameters or 16–47 wood ray diameters34. ICT, on the other hand, operates at single cellular level.
Multi-scale deformation analysis of growth ring loaded radially in tension until fracture
Macroscopic deformation with independent analysis of tracheids and wood rays
Figure 4a shows a 3D segmentation of the R-specimen, corresponding to the region of interest of (R, T, L) = (2.6, 1.5, 1.8) mm marked in Fig. 1e. Out of the segmented 1.67 × 109 voxels, 3700 tracheids were automatically detected in first iteration (67.9% of void voxels), with 232 additional ones (all LW cells) in second iteration. 312 wood rays were detected, which account for ~ 3% void voxels. Figure 4b shows a density profile along R, which was calculated based on the cell wall amount of tracheid cells. Discrepancies with the specimen’s gravimetric density (400 kg m−3 ± 60 kg m−3) are explained by the natural density variability of wood and the small sample dimensions.
A growth ring (thickness 1500 µm) is defined by two sharp LW–EW discontinuities at R = 850 µm and R = 2350 µm, corresponding to the start and end of the growth season. The density increases gradually between EW and LW cells. Deformation state #2 (elastic) shows a barely visible 20 µm displacement of the density profile with respect to #1 (preload). The fracture line (density drop in state #3) occurs at 20% of the growth ring width within the EW region (R = 1150 µm). For comparison, the fracture line is also marked with an dashed red line and an arrow in Fig. 4a. Figure 4c shows the wood ray density (percentage of wood ray voxels) across the growth ring (0.5–2%). As it is also visible in Fig. 4a, wood rays extend in R direction over several growth rings. Wood ray density increases smoothly from EW and LW cells. Opposite to tracheids, it does not show a sharp LW–EW discontinuity, but reduces gradually until R = 1200 µm, where the fracture line (#3) occurs. Deformation #2 is visible as a displacement of the density profile.
Figure 4d shows macroscopic strains calculated with ICT by analyzing tracheids. The elastic deformation #2 induces heterogeneous strains along the growth ring. Radial strain (varepsilon_{RR}) shows a peak of 0.8% at mid-width of the growth ring (R = 1600 µm) with respect to a minimum (varepsilon_{RR}) of 0.4% at the LW region. (varepsilon_{RR} ) is accompanied by an approximately proportional compressive tangential strain (varepsilon_{TT}), with average Poisson ratio (v_{TR} = – frac{{varepsilon_{TT} }}{{varepsilon_{RR} }} = 0.94). ({ }varepsilon_{TT}) shows a smoother pattern than (varepsilon_{RR}), and (v_{TR}) varies between 0.69 (LW) to 1.25 (R = 1300 µm). Significant shear strains (varepsilon_{RT}) up to 0.5% are observed in the RT plane, with a similar trend to (varepsilon_{RR}). Figure 4e shows strains calculated with ICT by analyzing wood rays. The strains are consistent with Fig. 4d, suggesting that wood tracheids and wood rays deform together as a single composite unit. (varepsilon_{RR} ) and (varepsilon_{TT} ) are slightly lower (0.2%) than for tracheids, which indicates a deformation restraining effect in R direction by wood rays. (varepsilon_{LL}) is well-resolved based on wood rays, with (v_{LR} = – frac{{varepsilon_{LL} }}{{varepsilon_{RR} }} = 0.07). Fracture state #3 leads to deformation release for both tracheids and wood rays, with most of strain of state #2 vanishing. Strains (varepsilon_{RR}) and (varepsilon_{TT}) are reduced below 0.1%, and (varepsilon_{LL}) below 0.02%. Both (varepsilon_{LL}) calculated based on tracheids and (varepsilon_{RR}) calculated based on wood rays show noisy patterns, which are explained by the lower accuracy of the ICT method along the longitudinal cell axes (see Sect. Deformation measurement accuracy, Fig. 3).
Farruggia et al.53 found (v_{TR} > 1) in microtensile tests of EW, with smaller values in LW, in agreement with our values. The heterogeneous (varepsilon_{RR} ,{ }varepsilon_{RT} ) pattern across the growth ring observed in this work resemble moisture-induced free swelling in softwood, with anisotropy ratio (varepsilon_{RR} {/}varepsilon_{TT} ) increasing at mid-growth ring54,55. Yet, for moisture-induced strain, (varepsilon_{TT}) shows a constant pattern and (varepsilon_{RR}) peaks closer to the LW region. On a constrained unit cell, Rafsanjani et al.56 found highest hygroscopic R displacements at mid-width of the growth ring, with the stiffer LW pushing radially the softer EW. Jernkvist et al.57 optically measured deformation in softwood growth rings loaded in tension and found (varepsilon_{RR}) and (varepsilon_{TT}) peaking at mid-growth ring width and with a 50% decrease at LW–EW transition, together with significant (varepsilon_{RT}), in close agreement with our observations. These were explained by the apparent stiffening of the softer EW by the sharp interface to the significantly stiffer LW region. Similar trends were observed by Moden et al.58, who found less pronounced strain variations than in the growth ring density profiles. This is associated to the constraining between adjacent cells. (varepsilon_{RR}) is more heterogeneous than (varepsilon_{TT}). A possible explanation is that EW and LW cells act as a parallel system of strings in T direction, deforming as a single composite unit, while in R direction they act as a series system of elastic strings, thus allowing for heterogeneous strain behavior across cells.
We observed deformation release after fracture and no significant strain close to fracture surface, indicating fast failure and small plastic deformation. Fracture occurs in the EW region, which is structurally weaker than LW59. Wood rays act as a reinforcement preventing crack propagation60. A previous fractographic analysis of radially tensioned wood samples36 associated radial failure to the weaker ray section. Accordingly, we observed a smooth decrease of wood ray lumen diameter within the EW region, which roughly corresponded to the position of the fracture plane.
Microstructure deformations at subcellular scale
Figure 5a,b quantify the geometric deformation of wood tracheids in function of the growth ring position. The tracheid lumen size ranges between 22 and 48 µm, with smallest values at LW. The lumen geometry is anisotropic in EW (aspect ratio R/T = 1.6) and more isotropic in LW (aspect ratio R/T = 1.3). The cell wall thickness ranges between 3.4 and 4.2 µm, with thinnest values in EW (Fig. 5a). The lumen deformation (Fig. 5b) follows the macroscopic strain patterns of Fig. 4d, with the major axis following (varepsilon_{RR}) and the minor axis following (varepsilon_{TT}). In the LW region, (v_{TR} < 1 ) and the cell lumen accordingly shows a surface increase (varepsilon_{sum } =) 1.1%. In EW at peak (varepsilon_{RR}) strain, the cell lumen surface does not change ((varepsilon_{sum } approx) 0%) confirming the macroscopic observation (v_{TR} approx 1). The tension load stretches the cell wall, reducing its thickness in a range (varepsilon_{CW} =)[− 0.3%, − 1.4%]. Even at EW positions where (varepsilon_{sum } approx) 0%, due to the cell stretch along R the lumen perimeter increases and consequently the cell wall thickness is reduced. Figure 5d,e quantify the deformation of wood rays. The lumen size ranges between 5 and 13 µm (Fig. 5d). Opposite to wood tracheids, for wood rays the largest lumen sizes are observed in LW. The radial load induces an overall compression of the lumen size ((varepsilon_{sum }) = 2%) for both major and minor cell axes (Fig. 5d). The compression is largest in the EW region, where the R strain of the wood rays is highest, thereby showing a Poisson effect in both T and L directions.
The ICT method allows direct visualization of the subtle geometric contributions of different cell types to the global deformation state. Upon tensile deformation, we quantified a thinning of the cell wall thickness of around 0.8% accompanied by a similar increase of the lumen section, which accounts for a detection of 28 nm deformation in a 3.5 µm thick cell wall. Cell wall stretching upon tensile deformation has been previously hypothesized based on the observed linear dependence of elastic modulus with density over a broad range of wood species58. The lumen area remained unchanged at the center of the growth ring, with approximately equal opposed deformation of the major and minor lumen axes, allowing geometric observation of Poisson ratio (v_{TR} approx 1).
Correlations between microscopic and macroscopic deformation properties
Voxel-wise correlation coefficients r between deformation parameters for elastic deformation state #2 are provided in Appendix B as Supplementary Tables S1–S3. All r > 0.001 are significant (p < 0.05). Due to Poisson effect, (varepsilon_{RR}) shows a negative correlation with (varepsilon_{TT}) (r = − 0.639), and (varepsilon_{TT}) positively correlates with (varepsilon_{LL}) (0.372). From all calculated strain fields, (varepsilon_{TT}) calculated with both wood rays and tracheids show the strongest agreement (0.835). Shear strain (varepsilon_{RT}) increases at larger deformations (varepsilon_{RR}) (0.225).
As for geometric wood tracheid deformation, (varepsilon_{RT}) is correlated with shifts in lumen orientation (Delta e_{{Psi }}) (− 0.312). Strain (varepsilon_{RR}) increases for larger tracheid cell lumens ({Sigma }) (0.166) and thinner cell walls (t) (− 0.229), both characteristic of EW cells. The correlation with cell size is strongest for (varepsilon_{TT}), with r = − 0.262 for ({Sigma }) and r = 0.457 for t. The cell wall t is also negatively correlated with tracheid inclination (theta) (− 0.459), while (varepsilon_{RR}) increases with (theta) (0.179). The cell wall swelling (varepsilon_{t}) is negatively correlated with increase of lumen size (varepsilon_{{Sigma }}) (r = − 0.344).
As for geometric wood ray deformation, (varepsilon_{RT}) is correlated with shifts in ray cell inclination (Delta theta) (r = − 0.512). Tracheid lumen areas ({Sigma }) are correlated with wood ray inclination (theta). Poisson compression strain (varepsilon_{LL}) results in ray cluster compression (varepsilon_{ray}) (r = 0.259), which reduces the inner distances between ray lumens, and a reduction of ray lumen size (varepsilon_{{Sigma }}) (0.237). Larger ray lumen areas ({Sigma }) are correlated with larger adjacent tracheid cell walls t (r = 0.383), as found in LW regions, and show smaller deformations, with r = − 0.355 for (varepsilon_{RR}).
As the distance to wood rays (d_{ray}) increases, the strains (varepsilon_{RR}) (0.052) and (varepsilon_{TT}) (− 0.108) significantly increase in absolute terms, and the tracheid inclination (theta) is reduced (− 0.060).
Correlation analysis reveals larger deformations for thinner cell walls t and larger lumen areas, which are characteristic of EW cells. It is known from previous work that LW and EW differentiate in their chemical and nanoscopic building. For instance, the microfibril angle varies from LW and EW, and the S2-layer reduces in thickness from LW and EW, which leads to different amount of lignin/cellulose over the complete cell wall and stiffer LW than EW cells61.
The role of wood rays in wood composite structure: comparison with previous studies
Wood rays have been associated to both nutrient transport and storage and to adaptive tree reinforcement in radial direction. Burgert and Eckstein isolated single wood ray cells and subjected them to microtensile tests62. They found significantly higher strength in isolated wood ray cells than in the macroscopic wood material, which verified the radial reinforcement of wood rays63. Reiterer et al.60 and Burgert et al.64 analyzed the macroscopic effect of wood rays by identifying improved radial strength properties in wood species with similar wood structure except for the distribution of wood ray cells. Similarly, Elaieb et al.65 recently linked wood ray presence with hygroscopic stability, by showing significant correlations across species between microscopic wood ray proportion and reduced drying-induced macroscopic shrinkage. The functional adaption of wood rays has been investigated by subjecting growing trees to radial loads12, and analyzing wood microstructure and macroscopic mechanical properties at different stem locations. Wood rays act as stiff radial pins to prevent shear slipping of EW and LW layers during stem bending due to wind loads12.
Our work for the first time directly measures the mechanical deformation of wood rays embedded within a tissue composite structure. In real wood composites, due to the cells not being isolated, they influence and are influenced by the surrounding cellular microstructure. In softwoods, wood rays are small and difficult to isolate; in our experiments they accounted for only 3% of total material volume. We observed in 2.2.1 that wood rays followed similar deformation trends to wood tracheids, showing that interactions between both cell types result in macroscopic composite deformation. The strain of wood rays is locally smaller than in tracheids, showing a less compliant behavior. The negative correlation between distance to wood rays and strain shows the reinforcement effect of wood rays. Accordingly, Xue et al.66 found in cryomicrotome sections of hardwood largest shrinkages for cells most separated from wood rays.
The presence of wood rays affected the inclination of wood tracheids (theta), which wrap around the wood ray cells. Similar weaving effects were observed by28 for hardwood vessels cells. Changes in (theta) are known to affect mechanical properties in grain direction67. The radial stretching of wood rays was observed to reduce their lumen sections, showing a Poisson effect at cellular scale. Ray cells within a single ray cluster also modified their relative L-distances when pulled in radial direction. Together with the heterogeneous EW/LW cell microstructure, the geometric imbalances resulting from the embedding of tracheids and wood rays in the same composite structure, introduce shifts in tracheid lumen tension and wood ray inclination, which result in shear strains during the tension experiment.
The role of wood rays in wood fracture has also been indirectly analyzed with microscopic and micro-tomography methods, which allow visualizing cell distribution in fracture surfaces36,60. Baensch et al.36 associated radial wood failure with the weakest ray sections. Similarly, we observed smallest ray lumen diameters in earlywood at the position of the fracture plane. Reiterer et al.60 found that, for similar number of wood rays in a volume, wood species with larger wood rays show higher radial stiffness properties. Accordingly, we observed that ray lumens were larger surfaces were less compliant, showing smaller radial strains. Overall, our results suggest that wood ray with larger lumens are mechanically stronger.
Multi-scale analysis of softwood compressed longitudinally until densification
Despite large non-linear deformations due to cell densification, ICT successfully co-registers individual tracheids and wood rays in specimen L between successive compression states (Fig. 6). A main densification line develops at the sample mid-point (F1), with a secondary densification line being visible at the bottom of the sample (F2). Figure 7 exemplifies the data evaluation process of the geometrical changes of a single tracheid from reference #1 to densified state #4. F1 is visible by eye comparing the renders before (state #1) and after (state #4) densification. Longitudinal compression strain ((varepsilon_{zz}) < 0) and a reduction of the tracheid section area ((varepsilon_{sum }) < 0) at the compression plane are observed. Moreover, a shear shift in the tracheid lumen centroid ((Delta c_{Y})) and an increase of the tracheid inclination (∆θ > 0) were assesed below the compression plane, which indicate cell buckling. F2 is not visible by eye in the segmented datasets but can be identified in the deformation fields with simultaneous (varepsilon_{zz}) < 0 and (varepsilon_{sum }) < 0 at the compression plane, which indicate cell densification. For further verification of F2, Contrast to Noise Ratio calculations are included in Supplementary Materials—Appendix C (Figure S2), where F1 shows a CNR of 11.5, followed by F2 with a CNR of 4.7. Since no shear effects are present in F2, the deformation is interpreted as telescopic shortening37.
Individual Cell Tracking (ICT) method for single tracheid of L-specimen subject to compression. Based on one-dimensional cell parametrization (Fig. 2), cell geometry parameters are obtained for reference (#1) and densification (#4) states and co-registered with texture correlation in Z. The deformation fields reveal two fracture plane F1 and F2. Buckling (({Delta }theta > 0)) is observed in F1, while for this cell F2 shows telescopic shortening ((varepsilon_{LL} < 0,varepsilon_{{Sigma }} < 0)) only.
Even if the deformation effects are small for single cells, the ICT method automatically analyzes thousands of cells, revealing consistent deformation patterns. Accordingly, Fig. 8 shows profiles of the average deformation fields along the grain direction L. For state #4, an increase of density (Fig. 8a) is observed at L = 940 µm and L = 1510 µm, corresponding to compression planes F1 and F2, with a third intermediate plane F3 (L = 1150 µm) connecting F1 and F2. The compression planes are characterized by peak compression strains (varepsilon_{LL}) = − 60% (Fig. 8b). Densification is also characterized by an increase of cross-grain shear strain (varepsilon_{RT}) above F1 (Fig. 8c), which indicates cell buckling.
Average deformation profiles along grain direction for L-sample of Fig. 6. (a–f) are computed with ICT of tracheids, (g–i) with ICT of wood rays. Macroscopic strains (varepsilon_{LL} ,varepsilon_{RT}) and geometric cell deformations (cell lumen area (varepsilon_{{Sigma }}), cell wall thickness (varepsilon_{{text{t}}}), cell inclination ({Delta }theta)) are shown. Strong densification effects are observed for wood tracheids concentrated in fracture lines F1 and F2, while wood rays show more distributed deformation patterns. (varepsilon_{{Sigma }} , varepsilon_{RT}) and ({Delta }theta) provide clear indications for the onset of plastic deformation (#3), while macroscopic density and compression (varepsilon_{LL}) are less apparent.
The ICT method allows visualization of the compression process at the microstructural level. A reduction of the tracheid lumen area (varepsilon_{sum }) < 0 (Fig. 8d) and an increase in cell wall thickness (varepsilon_{t} >) 0 (Fig. 8e) indicate densification, which was concentrated at F1 and distributed more homogeneously over the specimen top part. The bottom part of the specimen below F1 was slanted at a constant tracheid inclination ∆θ = 2.5° and not compressed (Fig. 8f). The densification was clearly dominant in plane F1, but buckling effects propagated to top sample planes F2 and F3 as the deformation increased.
Wood rays show more distributed deformation patterns than tracheids. A localized increase of ray density at F1 was not observed (Fig. 8g) and the cell lumen compression was more distributed over the top part of the specimen (Fig. 8h). Finally, the wood ray inclination (Fig. 8i) was not significantly affected by the densification process. These overall shows that wood rays are less subjected to densification than wood tracheids, acting as stiff pins within the composite.
The analysis of deformation states #2 and #3 shows which parameters are more sensitive for small linear deformations (#2) and the onset of plastic deformation (#3). At state #2, deformation is only clearly visualized with (varepsilon_{sum } = 0.4%) and (varepsilon_{RT} = 0.001). Small deformations were homogeneously distributed over the sample length, with a gradual increase of cell RT shearing. In deformation state #3, the development of densification line F1 becomes clearly visible through ∆θ = 1°, (varepsilon_{sum }) = 2% and (varepsilon_{RT} = 0.01). (varepsilon_{LL}) is less sensitive than the aforementioned parameters and shows a noisy pattern which does not reveal clearly the densification process even at state #3, where (varepsilon_{LL} = 5%).
In early compression stages, deformation is homogeneously distributed over the tracheid length, whereas with higher compression they become localized around fracture lines, which absorb the compressive stresses. At the latest densification stage, collapse of tracheid lumens occur, which increase the macroscopic density. Wood rays behaved as stiff pins, showing more distributed compression patterns, and their inclination was less affected by the densification process. At early deformation stages, deformation was only visible by the buildup of shear strain and small variations of the tracheid lumen section. With larger deformation, shear build up progressed towards cell buckling. In accordance with these observations, Zauner et al.37 observed the failure area of spruce wood after uniaxial compression and identified changes of fracture line progression when crossing wood rays. In final densification stages, wood ray cells buckle and are compressed.
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