The work utilized several different, but hence complementary approaches that are convenient to outline for clarity. The work focused on detailed analysis of the region around the crack using DIC, especially on the relation between strain-based criteria derived from optical data to the crack propagation length based on such a criterion. Such an optical-based value is compared to the assessed value of Gc obtained using CBBM. In the work we distinguish the Gc as the value of strain energy release rate when maximum force is applied, and Gf which is the maximal value that can be attained. The CBBM is a method based on assumption of energy dissipation during the crack propagation, it is based on eqLEFM concept and deals thus with equivalent crack length that cannot be seen directly. Nevertheless, comparison of which size of ε1 criterion in optical measurement induces which Gc from CBBM is valuable, since it compares the methods and allows to determine reasonably safe criterion which can be applied to other cases in the same specie, anatomical direction and mode I loading. Additionally, an analysis of length of nonlinear region - length of FPZ, based on the optical data is also performed. Last, correlation between the results is critically reviewed.
2.1. SEN-TPB experiments
The experimental work followed the test procedure from NT BUILD 422 standard (NT BUILD 1993) that is used for testing fracture properties in mode I. The specimen geometry and boundary conditions are shown in Fig. 1. First, the crack-free timber beams with growth ring orientated in tangential (T), radial (R), and mixed (TR) orientations were cut from Norway spruce beams of big size. The specimens were then conditioned in a climate chamber using standard conditions (20°C and 65% relative humidity (RH) to reach 12% equilibrium moisture content (EMC). Then, wooden blocks with dimensions of 12 × 12 cm (height × width) were cut from the beams to create a middle element and glued to specimen arms using PVA-based dispersions of one-component adhesive (see Fig. 1).
The orientation of grain in the middle element is rotated 90° to the arms, the specimens were held under pressure for one hour and then the middle element was cut vertically up to the half of the specimen height to introduce an initial and narrow crack, the detail of tree ring orientation is shown in Fig. 2. All specimens were made from four different trees. In each of the groups (T, R, TR) are specimens from at least two trees (mixed half way). In T, R and TR groups, we measured 10, 12, and 9 specimens, respectively. The central parts 12x12x6 cm intended to be glued in between two arms were cut from one plank, that means, close to each other and thus should not have big differences.
At the middle element with initiated crack was then applied with a stochastic pattern using Airbrush Revell Master Class together with acrylic pigments to create randomly distributed speckles (Fig. 3) to ensure high-quality computation by DIC in further work. The areas of interest (AoI) on specimens around the crack tip were filmed from both sides of the specimen using same cameras Basler acA2440-20gm with frame rate 1 Hz and resolution 2448 × 2048 px. The optical data were used to analyze crack onset and propagation - monitoring crack length. The semi-telecentric lens Computar TEC-M55MPW with focal length 55 mm was used to minimize optical distortion. The images were further processed using DICe software (Turner et al. 2015) and used parameters of DIC were as follows: step = 30 px (equals ~ 0.675 mm), subset = 27 px, Gauss kernel = 5 px, transformation full affine. For the mechanical testing, the universal testing machine Galdabini Quasar 100 was used, equipped with a load cell with a range of 10 kN and Labtest testing software. The outputs from the test machine and sensors were acquired by the unit DEWE2602. For all samples, the load, the corresponding deflection measured by two control LVDTs (see Fig. 2) and one control LVDT to identify the initiation and opening of the crack were monitored. For the purpose LVDT sensors DTA-5G-CA (Micro-epsilon, Germany) throughout the duration of the load test were used, located on both sides of the test specimen. The average value of these LVDT sensors was used to evaluate the data. The tests were displacement-driven with a speed of 0.3 mm/min.
CBBM evaluation was made according to Dourado et al. (2011). Due to the experimental variability, the interval for initial compliance was selected individually, but always was in the region of 0.2 to 0.4 of Fmax. Other parameters, although often with minor influence to the result were: k = 0.86, β = 1.07, EL=10 GPa and ET=EL/17. Elastic properties of the arms have very low influence on the final results. It is not possible to obtain the true crack length because the initial scatter of the algorithm strongly influences the value. That is why the value of Gc was used for later analysis in processing optical data instead of the absolute crack length. Typical processing of data is shown in Fig. 4 and is described below.
2.2. Optical data post-processing
It is worth noting that the experimental determination of the crack tip position in quasi-brittle materials is difficult, since the development of FPZ is complicated and makes the method really challenging. The key questions here are: where the crack starts and what length the crack has. From the theoretical point of view it is clear - nevertheless from the experimental perspective it is a real challenge to answer those questions for a quasi-brittle material. The concept of eqLEFM serves as a good etalon and that is why we compared the optical data to the CBBM ones. Strain-based criteria are of certain importance because in testing of structural-size members, the recognition of presence of a crack is crucial for assessment of bearing capacity of notched beams. In such a case, CBBM cannot be applied because a significant amount of energy is dissipated in closure of the drying cracks, material inhomogeneities (like knots), contacts, embedments etc. Even for assessment of bearing capacity of timber beams using CZM we have to address the presence of a critical length using crack shape parameters, CZM itself is not capable of predicting critical crack growth. Despite the limitations encountered with the use of DIC, it is the only reasonable and cheap experimental technique which is not dependent on assumption of many parameters. This work intends to add one stone to the mosaic of understanding wood fracture.
The DICe algorithm provided us with raw data, namely displacement, rotation and strain fields for each image in the whole image sequence representing the time domain. Such data were further post-processed using a Python-powered set of libraries SciPy and NumPy (Scipy - Jones et al. 2001). First, the equivalent crack length (ae) and G were computed using CBBM, so we obtained ae vs. time and G vs. ae and could identify Gc and Gf (Fig. 4).
The SEN-TPB induces pure mode I and thus tensile stresses/strains are of our interest. Therefore, the whole procedure used was based on an evaluation of the first principal strain (ε1) field, resp. its maxima. The ε1 was computed for each pixel using linear algebra. The strain field is very handy especially in optical measurement, where strains computed from pixel values mean the same strains as in the studied plane when undistorted images are assumed. Another reason why principal strains instead of displacements were used for the evaluation is their invariant nature and simply the fact that displacements need two points selected and size of DIC step can become another additional parameter. Last, strain analysis was employed because of the ability to find the beginning of nonlinearity in the strain field across the height of the bent glued-in specimen. This nonlinearity ahead of the crack tip can be considered the very end of FPZ.
The whole method described here follows the CBBM principle meaning there is no change in compliance linearity without crack propagation. The crack starts with the first positive change in compliance (see Fig. 5a), where the bullets show the end of the linear region). The procedure can be further split into two tasks: (i) how to assess a strain criterion and (ii) how to determine the crack length itself based on such a criterion. For clarity, we start with the second task: the algorithm first chooses the region of interest (ROI) which is the one that is a subset of the original field, where there is the biggest peak when summed along width of the glued-in part and its size is usually ~ 5 mm in width and ~ 45 mm in length. This is crucial in order to remove imperfection influences (knots, other microcracks) from the analysis. After ROI has been selected, it detects zones along the height (45 mm) where the ε1 is higher than the given strain-based criterion (in our case ε1max = 3e-3). Because the detection utilizes discrete values of matrix element values, it employs a linear interpolation to get more precise crack length. This algorithm is applied on the whole sequence and allows to apply a set of criteria that return a set of crack lengths (as curves). In Fig. 5d we can see for each of the cameras two curves representing the crack length in time if minimal and maximal strain criterion is used.
This research is unique because we track both sides of the sample and account for different crack lengths in time. This is attained by specifying the correct strain criterion on each side of the sample - the first task mentioned above. The problem is, that if the crack starts on one side, it still does not have to be apparent on the other. So in the first step we apply to both sides a relatively safe strain criterion such as ε1max = 3e-3. Using the above described algorithm, we get curves for both sides of the sample. This is illustrated in Fig. 5b, where cracks from camera 1 (blue) and camera 2 (red) are depicted. Then, we determine the delay in time between the two curves (green curve) which allows us to analyze each side separately using the strain criterion at different positions in the sequence based on the delay. The results of this are shown in Fig. 5d, colors distinguish the cameras. The first principal strains (Y axis) along the height of the scanned area (X axis) are depicted as bright solid curves. Note that both curves (red and blue) are at different time steps and determine the probable onset of crack. Further, for each of the curves, a linear part is automatically detected and fitted by a line. The difference between the curve and the fitted line is shown in the same picture as lines with semi-transparency using the second Y axis. These curves allow reading the first value representing onset of any nonlinearity, e.g. the highest possible FPZ length. The maximum represents the mouth of the crack, it is the value when the derivative of the difference line is the highest (dashed lines). Bullets in Fig. 5c represent the value of the criteria, and from the whole procedure we can obtain two extreme values of strain criterion for each side, and further interpolate between them on the strain curve to obtain additional three points. This step outputs two vectors of five values of strain criteria and for each one, the crack length at each side of the specimen is computed.
The final crack length is computed as the average of crack lengths obtained for each criterion in the time sequence. Such an average is also presented in the results as one criterion - for example, first two criteria for each side are assessed and used to produce respective crack lengths, however, in the end are together averaged for clarity of the results. Such a joined curve (a = al+ar)/2 can be used to compute the value of Gc next to the maximal force using known Irwin-Kies (Irwin et al. 1954) equation:
$$G=\frac{{F}^{2}}{2b}\frac{\partial C}{\partial a}$$
, where F is the acting force, C is compliance, b width of the sample, and a is the crack length determined using optical method. The derivation is made using the gradient method in discrete domain. In fact, we use the same compliance as is used in the CBBM method and couple it with the crack length determined from optical data. Because all variables were noisy and the noise in this equation even amplified, we used a polynomial fit in the region of Fmax to get stable and reliable results.