Symmetric Nature of Stress Distribution in the Elastic-Plastic Range of Pinus L. Pine Wood Samples Determined Experimentally and Using the Finite Element Method (FEM)

Abstract

This article presents the results of experimental research on the mechanical properties of pine wood (Pinus L. Sp. Pl. 1000. 1753). In the course of the research process, stress-strain curves were determined for cases of tensile, compression and shear of standardized shapes samples. The collected data set was used to determine several material constants such as: modulus of elasticity, shear modulus or yield point. The aim of the research was to determine the material properties necessary to develop the model used in the finite element analysis (FEM), which demonstrates the symmetrical nature of the stress distribution in the sample. This model will be used to analyze the process of grinding wood base materials in terms of the peak cutting force estimation and the tool geometry influence determination. The main purpose of the developed model will be to determine the maximum stress value necessary to estimate the destructive force for the tested wood sample. The tests were carried out for timber of around 8.74% and 19.9% moisture content (MC). Significant differences were found between the mechanical properties of wood depending on moisture content and the direction of the applied force depending on the arrangement of wood fibers. Unlike other studies in the literature, this one relates to all three stress states (tensile, compression and shear) in all significant directions (anatomical). To verify the usability of the determined mechanical parameters of wood, all three strength tests (tensile, compression and shear) were mapped in the FEM analysis. The accuracy of the model in determining the maximum destructive force of the material is equal to the average 8% (for tensile testing 14%, compression 2.5%, shear 6.5%), while the average coverage of the FEM characteristic with the results of the strength test in the field of elastic-plastic deformations with the adopted ±15% error overlap on average by about 77%. The analyses were performed in the ABAQUS/Standard 2020 program in the field of elastic-plastic deformations. Research with the use of numerical models after extension with a damage model will enable the design of energy-saving and durable grinding machines.

1. Introduction

Modern science recognizes the growing correlation between the sustainable management of wood resources and human health [1,2]. This contributes to an increase in the number of trees, especially in urban areas [3]. Trees in such places contribute to the improvement of air quality and constitute a natural method of its purification [4,5]. They can also be a bioindicator demonstrating exceeding the limits of pollution [6]. However, green infrastructure areas in cities and trees along roads require pruning and cutting processes. This raises an important issue of reducing exhaust emissions coming from machinery for grinding branches [7,8]. This can be achieved through the use of innovative power units [9], systems improving the machine’s adaptation to grinding processes [10,11] or alternative fuels that are less harmful to the environment [12,13]. Regardless of the fuel used and the fuel supply system, the reduction in engine displacement, and thus the power and torque parameters, leads to a reduction in fuel consumption and reduced exhaust gas emissions [14]. More accurate selection of the power and torque of the power unit for the implemented processing procedures requires knowledge of the system load characteristics [15,16]. Determination of material properties and the development of a simulation model may contribute to a more precise selection of power units and may support the design of more effective cutting mechanisms [17,18].

One fast developing direction of wood research are computer simulations oriented toward the possibility of selected wood properties prediction. The increase in the use of computer analyses allows replacing experiments with non-destructive simulation tests. Modeling with the finite element method (FEM) can be distinguished among such studies. The literature includes analyses regarding e.g., elastic-plastic wooden screw connections [19] or other wooden connections [20], modeling of non-linear multiphase materials [21], modeling of wood-based panels properties [22], wood roasting simulations [23], interaction simulations between wood and microwaves [24], analysis of deformation and tension of upholstered furniture [25,26].

There are articles describing the FEM models along with the mechanical load tests on wood samples, whose aim is to validate the proposed calculation models. Such studies are the current research topic, because in 2019 the three-point bending test results of spruce wood were published, whose model was made in the ABAQUS program, characterized by a Pearson’s correlation coefficient at the level of r = 0.994 [27]. The study of the coniferous wood model in a similar strength test was also carried out in 2020 [28]. In the same year and software, a model of beech wood compression in three longitudinal, radial, and tangential directions was presented, with the accuracy of covering non-linear functions at the level of R² = 0.72 [29]. The modeling of beech wood compression in the ABAQUS program was also studied by the team of Prof. Malujda, who analyzed the moisture content and temperature influence on the test samples [30]. The absence of material model of scots pine wood (Pinus L. Sp. Pl. 1000. 1753) that can be used in FEM analysis is noticed. Knowledge of the mechanical properties of the tested material is essential to achieve its development.

Tests on the properties of timber are available in the literature; however, very often they relate to selected issues only, for example: the bending strength [31,32] or composite wood [33], determination of modulus of elasticity (MOE) [32], properties during compression [34], tensile [35] and surface properties [36]. It is popular to study timber depending on geographical origin (terrain conditions) due to its mechanical properties [36,37,38]. Such studies were carried out, among others, for pine wood (Pinus sylvestris L.) by Chuchala et al. in 2017 in terms of assessing the cutting force with a band saw [37]. It has been shown that the value of power necessary for the cutting process by a band saw can be twice as large depending on the region of Poland where the wood comes from [37]. Pine tree (Pinus L.) as a popular species occurring in Europe and Asia is characterized by many test results, which include, inter alia: the impact of drying methods on moisture content and weight change [38], the effect of heat treatment on impregnation [39,40], impact of the effectiveness of plant protection products [41], it is subject to genetic evaluation [42]. Physical, mechanical, and aerodynamic properties of cones [43], energy properties [44] and mechanical properties [45] of fibers [46] or composites [24,47] are also examined.

Determining the cutting force, which can be the basic data for choosing a power unit, is an important issue for designers and recognized in many scientific publications. Models are available to determine the force during cutting with such machines as: circular saw [48,49,50,51], band saw [52], chain saw [53], large crusher [54], milling machine [55,56,57,58].

However, there are no models for designers of wood chipping machines; moreover, such machines are characterized by five basic types of cutting mechanisms: cylindrical chipper [10], disk chipper [59], drum chipper [60], hammer chipper [61] and spiral chipper [62]. As indicated by Ihnat et al. in 2020, there is a reduction in the size of timber considered to be waste [63] in the carpentry industry. It follows that machines for grinding branches and remains of carpentry processes will be characterized by lower power demand, as harder pieces of timber (larger sizes of waste) will be used otherwise.

To design more efficient wood grinding machines, strength tests of pine wood were carried out (Pinus L. Sp. Pl. 1000. 1753). The aim of this research work was to determine the numerical model of the destructive force for pine wood for various stress cases based on experimental tests. Results concerning tensile, compression and shear of anisotropic material samples in all significant directions are presented. Inter alia, material constants, such as modulus of elasticity or yield stress were determined based on the research. The published results enabled the development of the FEM model, which at this stage of research maps the elastic-plastic nature of the material in all three strength tests (tensile, compression and shear), in accordance with the research aim. It is oriented at determining the maximum value of force (stress) necessary to destroy the tested material. Models of destruction and its practical applications will be presented in further publications of the authors. The model presented then will be used to determine the maximum cutting force required to the design of economic and ecological grinding machines. Accurate knowledge of the tested material will allow validation of the developed models in the future.

2. Materials and Methods

2.1. Methodology of Strength Tests

Measurements in this study were carried out for pine wood (Pinus L. Sp. Pl. 1000, 1753—Scots pine) available for commercial sale in Poland. Two types of samples were produced. Type 1 consisted of samples acclimatized for 6 months at 20 °C so that they reached a moisture content (MC) of 8.74% ± 0.1%. Type 2 was samples prepared in the same way to achieve moisture content of 19.9% ± 0.1%. For this purpose, a climatic chamber was used. The moisture content was checked with a Mettler Toledo moisture analyzer during conditioning until the desired value was obtained. The measurement consisted of precise weighing of the sample (±0.001 g) and simultaneous drying (change of MC from measured value to oven-dry). Ten specimens were prepared for each RH (relative humidity) level, loading direction and stress type. The average density of the tested samples was 750 kg m⁻³ ± 5%. The samples were cut mechanically from timber free from defects such as knots, rot, etc. They were made from selected logs in order to obtain representative and reproducible test results with respect to European standards, according to which other scientists are testing [64,65,66]. Nine samples were tested, and the five most convergent were selected for analysis.

The tests were carried out on the MTS Insight testing machine at temperature of 25 °C and air humidity of 40%, with the use of a 50 kN load cell. The applied strain rate for each of the cases was 0.02 mm/s. The static tensile test was carried out on samples with standardized dimensions in the dog-bone shape (Figure 1a) [67]. For the static compression test, cuboidal samples and dedicated equipment for the testing machine were used. The base on which the test sample was placed consisted of two parts with spherical geometry. The convex and concave parts cooperate in such a way as to enable axial transmission of the compressive force (Figure 1b). The static shear test was also performed using dedicated instrumentation (Figure 1c). Its geometry allowed the stress conditions as close to technical shearing as possible.

Due to its unique internal structure, wood follows an orthotropic pattern along the three axes of its biological directions. These directions are: L—longitudinal or fibrous rings, R—perpendicular to the rings, T—tangent to the rings [68]. Therefore, during tensile and compression test, the samples were subjected to loads in all possible directions in relation to the fibers from the timber they were made. These directions were: longitudinal (L), tangent (T) and radial (R) as shown (with geometric dimensions) in Figure 2 and Figure 3. During the shear tests, the samples were loaded transversely to the fibers in the tangential plane, along the fibers in the tangential plane, transversely to the fibers in the radial plane and along the fibers in the radial plane, as shown in Figure 4. This figure also includes the geometrical dimensions of the used samples. The results of similar studies on the mechanical properties of walnut (Juglans regia L.) and cherry (Prunus avium L.) timber can be found in [69], while those concerning yew (Taxus baccata L.) and spruce (Picea abies [L.] Karst.) in [70].

The research work consisted of determining the yield point as a result of plotting a straight line tangent to the registered stress-strain curve. This line is defined by two points: the first is the origin of the coordinate system, the second is the point where the curve deviates from the tangent by more than 1% (as schematically shown in Figure 5). The modulus of elasticity was determined based on the inclination angle of the straight line determined in this way. The values for the shear stress tests were determined analogously.

2.2. Model Finite Element Method (FEM)

The analyses were performed in the ABAQUS/Standard 2020 software. For tensile tests (Figure 6) a 3D model of a paddle sample was made, whose side surfaces (which are in fact compressed by the jaws of the gripper used in the tensile test) of the lower gripping part of the sample were fixed, and the upper ones were given kinematic extortion with a specified displacement value. The sample was divided into finite elements of the type C3D8R: an 8-node linear brick with reduced integration and hourglass control. The results were presented in the form of stress distribution in the stretched sample.

The wood material was modeled as elastic-plastic without considering the failure mechanisms. The elastic part was modeled with the engineering constants that consisted of 3 elastic moduli E₁, E₂ and E₃, 3 Poisson ratios ν₁, ν₂ and ν₃ and 3 shear moduli G₁, G₂ and G₃. The elastic moduli as well as the theoretical Yield Point (which divides the characteristic into elastic and plastic range) were determined based on the experimental test characteristics. If the stress level exceeds the Yield Point value, the behavior of the material was determined based on the tabular relation between the stress and the strain, which was determined for each of the 18 combinations of three states of stress (tension, compression and shearing), two values of moisture content (samples type 1 and 2) and three directions of wood structure (longitudinal, radial and tangential) separately. The range of the analysis was limited up to the point of the critical strain, where the failure of the material takes place (as was tested in the experimental research).

Compression test (Figure 7) was made with the use of a 3D model, which consisted of a tested sample and two jaws—the lower fixed and the upper, to which kinematic extortion with a given displacement value was applied. The C3D8R finite elements were also used in this study, similar to the tensile test. The sample was modeled as a deformable element, while the jaws were modeled as Rigid Bodies. To prevent the sample from escaping due to the lateral force generated on the sample’s contact surface with the jaws, sockets blocking the sample movement without generating artificial stress in the sample compression section were made in the jaws. The results were presented in the form of stress distribution in the compressed sample.

To decrease the computational time 2D analysis was used for performing the simulation of the shear test (Figure 8), where the cut section constituted a square with a side of 20 mm. The element was meshed using the CPS4R elements—a 4-node bilinear plane stress quadrilateral with reduced integration and hourglass control. The jaws of the testing device used in the shear test were modeled as straight lines lying on the vertical sides of the cut section. The boundary conditions were applied in such a way that one of the reference points marked in Figure 8c was established, and kinematic extortion with a given displacement value was applied to the other one (located in the opposite corner). The results are presented in the form of stress distribution in the cutting section.

In all three studies, contact with a property in the tangential direction defined by a constant value of the friction coefficient of µ = 0.25 [71,72] was used to model the contact between the test sample made of pine wood and the instrumentation made of steel. In the crosswise direction, a “hard contact” property that prevents two elements from interpenetration was defined.

Since 18 different cases were analyzed, it is possible to determine which of these cases provide the greatest challenge for the working unit of the machine. Each case will be valid for a different wood processing technology such as chopping, shredding, cutting, etc. The presented database of models (engineering constants) for the selected types of wood can be used to compare various tools used for wood processing in a chosen technology and to determine the influence of the tool geometry. This approach will be used to improve the design process of the wood processing devices. Since all of these models were obtained based on the same experimental research and using the same methodology, it is safe to state that it will be suitable for the selected purpose. Additionally, the practice of using various simplifications in wood modeling, depending on the application of the model, can be found in the literature [73,74,75,76].

3. Results

Material Strength Test Results

Results of the test are the characteristics of static tensile, compression and shear stress-strain curves. Examples of static longitudinal tensile test results are shown in Figure 9. On their basis, the average tensile characteristics in the longitudinal direction were determined along with the determined modulus of elasticity (Figure 10). The values determined on the basis of a static tensile test in the longitudinal direction (L), radial direction (R), tangential direction (T), for samples with 8.74% and 19.9% moisture content are presented in

Table 1. Average values determined on the basis of a static tensile test; L—longitudinal direction, R—radial direction, T—tangential direction.

Direction (Tensile)	L		R		T
Sample Type	1	2	1	2	1	2
Moisture content [%]	8.74	19.90	8.74	19.90	8.74	19.90
Cross-section area $A$ [mm2]	80	80	100	100	100	100
Yield point $R_e$ [MPa]	7.675	6.772	1.324	0.473	0.84	0.32
Standard deviation	0.25	0.21	0.09	0.03	0.02	0.02
Deformation ${\epsilon }_e$ [%]	0.0022	0.0018	0.0028	0.0018	0.0048	0.0030
Standard deviation	0.0002	0.0002	0.0002	0.0002	0.0002	0.0002
Tensile strength $R_m$ [MPa]	76.39	74.28	4.98	3.63	3.28	1.68
Standard deviation	18.64	19.32	0.55	0.25	0.43	0.22
Deformation ${\epsilon }_m$ [%]	0.16	0.27	0.009	0.011	0.018	0.015
Standard deviation	0.03	0.04	0.0004	0.0005	0.0018	0.0007
Modulus of elasticity $E$ [MPa]	3838	3386	662	473	212	161
Standard deviation	5.66	4.52	2.34	2.55	2.71	3.12

.

Examples of static radial compression test results are shown in Figure 11. On their basis, the average radial compression characteristics were determined along with the determined moduli of elasticity (Figure 12). The non-linearity visible in the initial measuring range results from the unevenness of the pressed surface of the sample, which only after alignment allows achieving a linear range. The values determined on the basis of a static tensile test in the longitudinal direction (L), radial direction (R), tangential direction (T), for samples with 8.74% and 19.9% moisture content are presented in

Table 2. Average values determined on the basis of a static compression test; L—longitudinal direction, R—radial direction, T—tangential direction.

Direction (Compression)	L		R		T
Sample Type	1	2	1	2	1	2
Moisture content [%]	8.74	19.90	8.74	19.90	8.74	19.90
Cross-section area $A$ [mm2]	400	400	400	400	400	400
The limit of proportionality $R_H$ [MPa]	43.29	38.16	1.65	1.75	2.09	1.77
Standard deviation	5.1	2.8	0.24	0.33	0.19	0.15
Deformation ${\epsilon }_H$ [%]	1.86	1.63	0.64	1.21	1.72	1.78
Standard deviation	5.1	2.8	0.24	0.33	0.19	0.15
Modulus of elasticity $E$ [MPa]	3218	3495	314	245	162	145
Standard deviation	5.1	2.8	0.24	0.33	0.19	0.15

.

Examples of static shear test results along the fibers in the radial plane are shown in Figure 13. On their basis, the average shear characteristics crosswise to the fibers in the radial plane (RT) were determined along with the determined shear moduli (Figure 14). Values determined on the basis of a static compression test crosswise to the fibers in the tangential plane (TR), crosswise to the fibers in the radial plane (RT), along the fibers in the radial plane (LR), along the fibers in the tangential plane (LT), for samples with a moisture content of 8.74% and 19.9% are presented in

Table 3. Average values determined on the basis of a static shear test of a pine sample; crosswise to the fibers in the tangential plane (TR), crosswise to the fibers in the radial plane (RT), along the fibers in the radial plane (LR), along the fibers in the tangential plane (LT).

Direction (Shear)	TR		RT		LR		LT
Sample Type	1	2	1	2	1	2	1	2
Moisture content [%]	8.74	19.90	8.74	19.90	8.74	19.90	8.74	19.90
Cross section area $A$ [mm2]	400	400	400	400	400	400	400	400
Yield point for shearing test $R_t^e$ [MPa]	4.74	0.972	1.60	0.61	8.46	1.93	9.03	4.51
Standard deviation	0.14	0.21	0.23	0.28	0.12	0.11	0.21	0.24
Deformation ${\gamma }_e$ [%]	9.95	1.74	2.49	4.88	3.88	1.88	4.18	4.23
Standard deviation	0.19	0.20	0.22	0.43	0.16	0.33	0.22	0.23
Shear strength $R_t$	8.07	3.53	4.15	1.28	14.93	7.01	11.65	6.77
Standard deviation	0.54	0.61	1.04	0.34	1.09	0.71	1.32	0.92
Deformation ${\gamma }_t$ [%]	18.30	12.76	12.33	17.78	9.51	10.43	5.74	11.32
Standard deviation	1.73	2.81	2.45	3.03	2.52	0.74	0.75	2.98
Shear modulus $G$ [MPa]	135	99	114	19	372	210	390	177
Standard deviation	2.14	3.21	4.67	8.9	5.13	4.48	2.12	5.61

.

4. Engineering Constants Used in FEM Modeling of the Wood Test

FEM models require the introduction of material constants, which were determined in the course of research and taken from literature [77]. Moduli of elasticity used to model all three types of tests were determined on the basis of experimental results, similarly to shear moduli used for modeling compression and shear tests. However, the tests did not allow for accurate determination of Poisson’s ratio nor for shear moduli for the tensile test. The tables contained in the works of Kretschmann, 2010 [72] were used for this purpose. In the tables the elastic ratios representing the remaining five moduli of elasticity relative to Young’s moduli in longitudinal direction were presented for various species. The missing values, which were not specified based on the experimental research, were determined on the basis of the ratio of two extreme moduli of elasticity (in longitudinal and tangential directions) comparing the obtained result in the literature [77]. For that purpose the proper ratios (ET/EL, ER/EL, GLR/EL, GLT/EL, GRT/EL) and based on that closest out of the nine species of Pine was selected (based on Table 5-1 [10]) and then the Poisson ratio were selected (based on Table 5-2 [10]). Since data were presented for only one moisture content (12%) which lie between both values analyzed (8.74% and 19.9%) authors have assumed that since the moisture content affects the mechanical properties, for both dry and wet the specimen will behave as different species of pine wood. The values of the engineering constants for the FEM model used in the tensile test are shown in

Table 4. Engineering constants used in FEM modeling of the wood tensile test.

Mechanical Parameters	Moduli of Elasticity			Poisson’s Ratios			Shear Moduli
Engineering Constants	E1 [MPa]	E2 [MPa]	E3 [MPa]	ν12 [-]	ν13 [-]	ν23 [-]	G12 [MPa]	G13 [MPa]	G23 [MPa]
Extension 8.74% (MC)	3838	662	212	0.332	0.365	0.384	272.5	230.3	46.1
Extension 19.9% (MC)	3386	473	161	0.392	0.444	0.447	186.2	179.5	33.9

, compression in

Table 5. Engineering constants used in FEM modeling of the wood compression test.

Mechanical Parameters	Moduli of Elasticity			Poisson’s Ratios			Shear Moduli
Engineering Constants	E1 [MPa]	E2 [MPa]	E3 [MPa]	ν12 [-]	ν13 [-]	ν23 [-]	G12 [MPa]	G13 [MPa]	G23 [MPa]
Compression 8.74% (MC)	3218	314	162	0.332	0.365	0.384	372	390	114
Compression 19.9% (MC)	3495	245	145	0.28	0.364	0.389	210	177	19

, and shear in

Table 6. Engineering constants used in FEM modeling of the wood shear test.

Mechanical Parameters	Moduli of Elasticity			Poisson’s Ratios			Shear Moduli
Engineering Constants	E1 [MPa]	E2 [MPa]	E3 [MPa]	ν12 [-]	ν13 [-]	ν23 [-]	G12 [MPa]	G13 [MPa]	G23 [MPa]
Shear B 8.74% (MC)	314	162	-	0.384	-	-	114	372	390
Shear B 19.9% (MC)	245	145	-	0.389	-	-	19	210	177
Shear C 8.74% (MC)	3218	314	-	0.332	-	-	372	390	114
Shear C 19.9% (MC)	3495	245	-	0.28	-	-	210	177	19
Shear D 8.74% (MC)	3218	162	-	0.365	-	-	390	372	114
Shear D 19.9% (MC)	3495	145	-	0.364	-	-	177	210	19

.

In this research, 18 FEM analyses were performed as a combination of various parameters. Each of these cases will be valid for different wood processing technologies such as chopping, shredding, cutting, etc. It is not possible to describe all these processes by using a single material model. The main goal is to determine the database of models (engineering constants) for the selected type of wood, which can be used to compare various tools used for wood in a chosen technological process and to find the influence of the tool geometry. This will enable finding effective solutions in terms of tool design for wood machining. Since all of these models were obtained based on the same experimental research and using the same methodology, it is safe to state that they will be usable for the selected purpose.

5. Discussion

5.1. Analysis of the Results of Strength Tests of Tested Samples

Mechanical properties of pine wood (Pinus L. Sp. Pl. 1000. 1753) vary depending on the direction of the wood fibers and moisture content. Tensile tests showed a decrease in mechanical properties along with an increase in material moisture (around 11%) in the range from 3% to 64% depending on the parameter, as shown in Figure 15. Wood during the tensile test shows the greatest strength in the longitudinal direction L, then in the radial direction R, and it is the least resistant to tensile forces in the tangential direction T. The nature of the determined properties is consistent with the tests of wood of other scientists [78,79,80,81].

Compression testing showed a change in mechanical properties as the material moisture increased (around 11%), but did not contribute to deterioration in all cases (as opposed to tensile or shear strength tests). Changes in material properties depending on moisture content in the conducted tests ranged from 6% to 22% (Figure 16). Wood during the compression test shows the greatest strength in the longitudinal direction L, then in the tangential direction T, and it is the least resistant to compression forces in the radial direction R. The nature of the determined properties is consistent with the tests of wood of other scientists [78,82].

Shear tests showed a decrease in mechanical properties along with an increase in material moisture content (approximately 11%) in the range of 27% to 83% depending on the parameter, as shown in Figure 17. During the shear test, wood shows the greatest strength along the fibers in the radial plane (LR) and along the fibers in the tangential plane (LT), then in the direction crosswise to the fibers in the tangential plane (TR), and it is the least resistant to shear forces in the crosswise direction for fibers in the radial plane (RT). The nature of the determined properties is consistent with the research of other scientists in relation to wood [77,82].

5.2. Analysis of FEM Model Results and Strength Test Results

The results of the FEM model showing the material exposed to various strength tests, for both dry and wet specimens (8.74% and 19.9% moisture content), are presented in Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27, Figure 28, Figure 29, Figure 30, Figure 31, Figure 32, Figure 33, Figure 34 and Figure 35. The material characteristics in the elastic-plastic range were presented in relation to the result of tests carried out on the testing machine. The stress state distributions from the FEM model were marked on them in selected fragments of the characteristic (in the case of compression and shear tests at points corresponding to 10%, 50%, and 100% deformation causing sample destruction, while during tensile—the state just before breaking only), which show the stress distribution in tested sample. Due to the purpose of the developed model (determination of the maximum destructive force), it is more important for authors to determine the value of the point corresponding to the material strength limit rather than mapping the characteristics. To evaluate the model, the authors conducted an analysis of the model results discrepancy and strength values in two configurations. The first concerned the maximum destructive force and is presented in

Table 7. Comparison of model convergence with strength tests during tensile testing at the point of damage; L—longitudinal direction, R—radial direction, T—tangential direction.

Direction	L		R		T
Moisture content [%]	8.74	19.9	8.74	19.9	8.74	19.9
FEM model [MPa]	64	41	4.38	3.48	3.27	1.68
Strength tests [MPA]	76	74	4.98	3.57	3.04	1.65
Error [%]	16	44	12	2.5	7.5	2

,

Table 8. Comparison of model convergence with strength tests during compression test at the point of damage; L—longitudinal direction, R—radial direction, T—tangential direction.

Direction	L		R		T
Moisture content [%]	8.74	19.9	8.74	19.9	8.74	19.9
FEM model [MPa]	51	44	3.4	2.4	4.6	3.6
Strength tests [MPA]	51	43.5	3.4	2.5	4.4	3.4
Error [%]	0	1	0	4	4.5	5.5

and

Table 9. Comparison of model convergence with strength tests during the shear testing at the point of damage; crosswise to the fibers in the radial plane (RT), along the fibers in the radial plane (LR), along the fibers in the tangential plane (LT).

Direction	RT		LR		LT
Moisture content [%]	8.74	19.9	8.74	19.9	8.74	19.9
FEM model [MPa]	4.3	1.5	16	7.5	11.9	7.4
Strength tests [MPA]	4.1	1.3	15	7.1	11.6	6.8
Error [%]	5	13	6	5	3	8

. The second one concerned the mapping of the characteristics and was presented in

Table 10. Comparison of the convergence characteristics of the model with the strength tests during the tensile test; L—longitudinal direction, R—radial direction, T—tangential direction.

Direction	L		R		T
Moisture content [%]	8.74	19.9	8.74	19.9	8.74	19.9
Error 15% [%]	25	31	100	100	100	100

,

Table 11. Comparison of the convergence characteristics of the model with the strength tests during the compression test; L—longitudinal direction, R—radial direction, T—tangential direction.

Direction	L		R		T
Moisture Content [%]	8.74	19.9	8.74	19.9	8.74	19.9
Error 15% [%]	95	94	100	22	96	97

and

Table 12. Comparison of the convergence characteristics of the model with the strength tests during the shear test; crosswise to the fibers in the radial plane (RT), along the fibers in the radial plane (LR), along the fibers in the tangential plane (LT).

Direction	RT		LR		LT
Moisture Content [%]	8.74	19.9	8.74	19.9	8.74	19.9
Error 15% [%]	94	5	96	99	51	85

, where the coverage of the curve from the test results and the FEM model in the range of ±15% error were analyzed. An example of the analysis of the coverage of the characteristics is shown in Figure 36, the coverage of the characteristic is expressed in accordance with relationship (1):

$$Characteristic coverage (\pm 15)= \frac{X-Y}{X}·100\%,$$

(1)

where: X—the length of the characteristics from the strength test, Y—the sum of the lengths of the sections of the FEM model characteristics ($\pm 15\%$) overlapping with the characteristics of the strength tests.

The error of the maximum value of the destructive force determined by the FEM analysis, estimated as the arithmetic mean of the errors obtained for each analyzed case, is 8%, and the median average for these results equals 5%. The error value is in the range from 0% to 44%. The average error value in selected strength tests equals 14% for tensile, 2.5% for compression, 6.5% for shear. In contrast, the average coverage of the FEM characteristics with the results of tests from the strength test in the field of elastic-plastic deformations with the assumed ±15% error overlap with the average of around 77% arithmetically, and the median for these results equals 95.5%. The average coverage of the characteristics in selected strength tests equals 76% for tensile, 84% for compressive, 72% for shear.

5.3. Aspects of Symmetry in the Tested Samples

The symmetrical nature of the stress distribution is clearly visible on simulation models. The nature of damage in tested samples presented in Figure 37 for tensile and Figure 38 for shear shows the convergent nature of the influence of the forces. In the samples subjected to compression, the influence of the forces is hardly noticeable, therefore it has not been presented. A similar nature of the stress distribution in tested materials samples is available, for example, for composite materials [83], steel [84], wood [28,85]. Additionally, the symmetrical stress distribution is noticeable in the tested structures, e.g., frames of: road bridges [86], bicycles [87], vehicles [88], bridges [89] or machine parts: gears [90,91], turbine blades [92], and furniture connections [93].

6. Conclusions

Mathematical models that enable the prediction of destructive force value of the tested samples in various stress states (compression, shear, and tension), allow for a more accurate prediction of the effects of forces acting on the tested material. In industrial applications, such models can be used to determine cutting force during chipping processes described in the modern trend of modeling cutting force using FEM analysis [94,95,96,97]. This will contribute to the development of efficient and ecological machines.

Conducting research with the use of the developed numerical model enables prediction of values with quite high accuracy. This is guaranteed by the presented comparison with the results of experimental tests. The recorded convergence in the estimation of the strength parameters of wood processing is at the level of about 8%.

Research confirms that wood with lower moisture content is characterized by greater durability in most strength tests. This indicates that it is more advantageous to carry out the chipping processes of untreated wood.

The geometry of the samples used in the tests and the simplifications in the model, mainly based on the assumption of a homogeneous material in the entire cross-section of the sample, contribute to the symmetrical nature of the stress distribution in real and numerical samples.