## 4 Discussion  
The results of the measurements made it possible to draw a number of conclusions. For a length of 0.7 mm, the paper fibers delaminated and crushed near the fixed lower handle of the machine. In this case, the ratio of the length of the sample to its thickness is L/g = 5. For lengths of 1.3 and 2.0 mm, destruction shall also occur near the lower handle with visible horizontal displacement of the samples. For these cases, we have L/g = 9.3 and L/g = 14.3, respectively. For the fastening lengths of 3.0 (L/g = 21.4) and 4.0 mm (L/g = 28.5), the destruction of the samples takes place in the central part and the sample takes the shape of the letter V. Note that paper edges enter tangentially into both handles. However, for a length of 5 mm (L/g = 35.7), the V-shape is very clearly visible, in particular the refraction itself. The sample at the places of attachment to both handles takes the shape as for joint-like support.  
In the theory of thin-walled structures, the following classification is assumed: when L/g > 25-30, where L is a smaller overall dimension, the plates are thin-walled; when 5 < L/g < 25 the plates are of medium thickness, while L/g \(\leq\) 5 the plates are thick [16; 17]. According to this classification, the clamping length L = 0.7 mm corresponds to thick plates, and for L = 1.3, 2.0, 3.0, 4.0 plates of medium length, and for L = 5.0 thin plate. For plates of medium thickness, modifications are made in relation to thin plates [18; 25; 26].  
Figure 15 shows diagrams of changes in the deflection arrow of the sample as a function of displacement of the movable handle of the testing machine with the place where the maximum force is reached.  
Figure 14: Shape of the edges of the compressed samples at the time of destruction for the connection length: (**a**) 0.7 mm, (**b**) 1.3 mm, (**c**) 2 mm, (**d**) 3 mm, (**e**) 4 mm, (**f**) 5 mm.  
The correctness of determining the deflection arrow using the developed image analysis technique was checked by comparing measured sample shortening with the analytically determined value using the formula given in [16]:  
\[\Delta l=l\left(1-\sqrt[2]{1-\left(\frac{2\delta}{l}\right)^{2}}\right) \tag{3}\]  
where:  
\(l\)--sample length  
\(\Delta l\)--sample shortening  
\(\delta\)--deflection arrow.  
Figure 16 shows a comparison of the measured and theoretically calculated shortening of the paper sample with a clamping length of 4 mm and 5 mm as a function of its deflection. A slight difference between the measured and calculated values may result from the adopted destruction model, which does not reflect all the phenomena occurring during the test, or the inaccuracy of determining the deflection arrow by imaging methods.  
Figure 16: Comparison of the measured and theoretically calculated sample shortening during the test for clamping length of (**a**) 4 mm and (**b**) 5 mm.  
Figure 15: Dependence of the value of the deflection arrow on the displacement of the movable handle at the fastening lengths: (**a**) 3 mm, (**b**) 4 mm, (**c**) 5 mm.  
Figure 17 summarizes the diagrams of changes in the period of the sinusoidal function approximating the shape of the test sample as a function of displacement of the movable handle of the testing machine. Estimating the period until buckling occurs is not justified, since the shape of the edge of the sample does not resemble a sinusoid. From the moment the maximum force is reached, the value of the period stabilizes at a level comparable to the length of the fastening.