Differential Energy Criterion of Brittle Fracture as a Criterion for Wood’s Transition to the Plastic Deformation Stage

Abstract

An experimental study and modeling of the behavior of wood during compression along the fibers was carried out. The nonlinear analytical dependence of the load on the strain was considered. Attention was focused on the post-peak stage of deformation in order to determine the load and displacement at which the transition to the stage of plastic deformation occurs. The work was aimed at substantiating the application of the energy criterion of brittle fracture as a criterion for the transition to the stage of plastic deformation. To achieve this goal, methods of mathematical modeling and analysis of test results were used. As an upshot, a simple and practical procedure was developed to predict the transition point to the above stage of plastic deformation. The simulation results were consistent with laboratory tests of samples and fragments of structures. The practical significance of this criterion lies in its possible use as an additional tool for analyzing the condition of some wooden structures. Energy criteria, including the one mentioned above, belong to fairly universal criteria. Accordingly, the research methodology can be adapted to analyze the behavior of, for example, composites under other types of loads in further studies.

1. Introduction

Studies of the strength and criteria for the destruction of materials and structures have remained relevant throughout the history of science and technology. To date, many strength and fracture criteria have been developed for homogeneous isotropic materials and, to a lesser extent, for anisotropic materials [1,2]. Studies show that many criteria have limited application, and some of them do not fully correspond to real materials at the micro-, meso- and (or) macro levels [3,4]. At the same time, the best results in this area are based on energy concepts [3].

The energy criteria of destruction are known, which, for example, are based on the analysis of the deformation energy density [5]. One of these criteria is based on the concept of an equivalent material, i.e., on the transformation of a physically nonlinear material into an equivalent linearly elastic material [6]; however, the prediction accuracy decreases for materials with defects [6,7]. Therefore, the development of a fracture criterion that takes into account the influence of all material defects requires further research, which is especially important for wood since the physical and mechanical properties of this material are very variable [4].

The development of destruction criteria is carried out in various directions. For example, a change in the rate of release of elastic deformation energy is used as one of the criteria for destruction [8]. In this area, a generalized criterion for the maximum energy release rate is known [9]. Energy release in the process of damage and destruction of wood [10] and rocks [11] can be estimated indirectly by recording acoustic emissions.

A number of other destruction criteria are known, which are based on the analysis of the conversion of input energy into the potential energy of elastic deformation, as well as into the energy of micro-, meso- and macro-destruction, acoustic emission [12,13,14], heat transfer [15] and other physical effects [16,17,18,19,20,21,22,23]. In any case, it is important to know the load-displacement (stress-strain) relationship since energy is determined by integrating the above dependence [24], and the graphical representation of these dependencies in the form of load-displacement (stress-strain) curves improves the understanding of the behavior of materials under load [4,25].

Load-displacement curves for uniaxial compression of wood [4,26,27] and materials such as sandstone and concrete [18,28] have certain similarities. The similarity of load-displacement curves for various indicates that some methods of analyzing rock behavior under load can be adapted to analyze wood failure, subject to certain limitations associated with different material properties [4,22,29]; for example, the acoustic emission analysis mentioned above is an effective support tool for experimental studies of both rock and wood failure [10,11,29,30,31]. The generalization of experimental data has formed the basis of a number of defining models of the mechanical behavior of wood. However, creating highly accurate nonlinear models in this area is difficult because of the large number of material properties required to calibrate the models [4,25,29]. To overcome these difficulties, brittle fracture models can be used; the advantage of such models is the small amount of input data. For example, the application of the finite element method in combination with a deterministic brittle fracture law is able to adequately reflect the load-displacement relationship of wood in structures [25,29]. However, it can be assumed that the application of analytical equations modeling the quasi-brittle behavior and fracture of wood will allow a simpler model to be developed. This work is aimed at substantiating the application of the energy criterion of brittle fracture as a criterion for the transition of wood from the quasi-elastic stage of deformation to the stage of plastic deformation.

2. Methodology and Results

2.1. Description of the Physical Model of the Research Object

To obtain a sufficiently universal model of the load-displacement relation, we do not limit the size and shape (i.e., geometric characteristics) of the object of study at the stage of choosing the research methodology, staying, however, in the space of existing materials and structures, whose physical properties and mechanical characteristics are predetermined by the conditions of their origin. Concretizing the limitations from the physical point of view, we will focus on the analysis of coniferous wood samples (pine and spruce).

We consider the loading process as a step (discrete) process, in which each step represents the destruction of the next meso-element or connection of elements in the structure [32] and is accompanied by the pulse of the aforementioned acoustic emission. The adequacy of this assumption is confirmed by the experiments known in the literature [10,14,19,30,33]. The load and displacement variations at each stage of such a process are assumed to be sufficiently small and are therefore bound by Hooke’s law. However, the above-mentioned failure of the next element leads to a decrease in the cross-sectional area and hence the stiffness at the next step of the process and, consequently, to the appearance of nonlinearity of the load-displacement relation for the process. In other words, the process under consideration is locally linear (at the level of one step) but globally nonlinear.

Detailing the object of the study, we note the following. The structure and physical properties of wood are very heterogeneous and depend on many factors that are difficult to take into account in the equations that determine the mechanical state of wood under load. Recent studies [4] have shown that wood should be considered as an intermediate link between the material and the structure. The mechanical behavior of such a structure is determined by the geometric characteristics and physical properties of its tubular and fibrous elements, as well as the conditions of their mechanical interaction. It is natural to assume that the bearing capacity of these elements is not the same; therefore, in the process of force action, the weakest micro- and meso-elements from the set of those that have not yet been destroyed are damaged one after another. So, considering wood as a structure [4], let’s discuss the methodology for obtaining and applying a load-displacement model using the term “stiffness,” characteristic of structures, the analog of which at the level of classical material is the modulus of elasticity.

2.2. Load-Displacement Curve in Compression Tests of Wood along the Fibers

This paper uses the Blagojevich analytical model [34], which belongs to the class of simple but sufficiently versatile load-displacement models, a review of which can be found in [35]. The application of modifications of this model to the analysis of frozen soils and some rocks is considered in [36,37]. However, when applying this model to the analysis of wood, it is necessary to take into account features that have not been previously considered in sufficient detail in the known literature. These features manifest themselves at the post-peak stage of deformation due to the growth of plastic strains and, as a consequence, the appearance of an almost horizontal section on the load-displacement curve. For example, for cubic spruce specimens of 40 × 40 × 40 mm (humidity 18%) under compression along the fibers with a controlled displacement of 10 mm/min (Figure 1), experimental curves (Figure 2) were obtained (in the laboratory of the Institute of Forestry, Mining and Construction Sciences, Petrozavodsk State University), which are similar to other experimental curves of comparable tests known from the literature [30,38,39,40].

We will analyze the behavior of wood under uniaxial compression using the plots in Figure 2 and the results of acoustic emission measurements in comparable tests known from the literature [30,33].

At stages 0–1 (Figure 2), the load and strain rate increase smoothly, the gaps are closed, and the weakest wood particles are crushed and compacted (particles and wood structure can be seen in microphotographs [4]). At the initial stage of loading (0–1), the tangential stiffness of the sample also increases smoothly. Nevertheless, at this stage, the acoustic emission is very small compared to the acoustic emission at other stages of wood deformation under uniaxial compression [30]. The absence of acoustic emission corresponds to the absence of wood particle fracture.

At stages 1–2, the acoustic emission remains small, but the load-displacement relation is almost linear. Nonzero acoustic emission at this stage is an indicator of micro-damage, which could lead to a decrease in the bearing capacity of the sample and the nonlinear nature of the load-displacement relation at stages 1–2. However, the fragment of curves 1–2 is almost a straight line. This circumstance can be explained by the fact that at this stage, probably, two counteracting processes are realized, namely: (a) micro-cracks emerge and the weakest wood particles (from among the remaining undamaged ones) collapse, which is indirectly confirmed by acoustic emission [30]; (b)—compression leads to an increase in density of wood particles and redistribution of internal forces in these particles

At pre-peak stages 2–3, the tangential stiffness decreases to zero at the peak, elastic properties decrease, and the wood deforms like a plastic material.

At the post-peak stages 3–4, acoustic emission increases significantly [30], which indicates an increase in the intensity of damage and destruction of particles and wood structure. The load is redistributed to the still undamaged particles. As the number of undamaged particles decreases, the average stress in the particle material increases under the same load, and the most stressed particles are turned off. Since the number of functioning particles is no longer sufficient to resist the peak load, the static equilibrium conditions correspond to a load lower than the peak value. This avalanche-like process at stages 3–4 is indirectly confirmed by the increase in acoustic emission intensity [30]. At this stage, in the fracture process zone (FPZ) [24,40,41], the volume of damaged particles increases, due to which the wood structure (as a structure [4]) degrades and at the finish of stages 3–4 is transformed into a mixture of individual particles weakly connected, which is already typical for loose materials.

At stages 4–5, due to the above-mentioned destruction, plastic deformation of the sample occurs, which is experimentally confirmed by an almost horizontal line for each of the three samples (Figure 2). At this stage, compressive loading leads to the growth of force interaction of particles and an increase in the conglomerate density of these particles. At the same time, the high intensity of acoustic emission at stages 4–5 [30] indirectly confirms that the process of fracture and compaction of these particles′ mass continues.

2.3. Load-Displacement Model

Testing of materials is necessary to obtain reliable data on their physical and mechanical properties, as well as on the features of mechanical behavior under load. However, in order to cover the entire range of possible parameter values, mathematical models are also needed, which can be divided into two large classes: numerical and analytical. Numerical models and computer technologies allow for obtaining results identical to experimental data [42]. However, like experiments, numerical methods are capable of describing a discrete state of the object under specific values of the input data. A more general view of the regularities of the mechanical behavior of the object under study is realized in analytical models using equations that predict the behavior of the object at all admissible input data. From the practical point of view, numerical and analytical models allow us to investigate different sides of the object, and experiments are necessary to check the adequacy of the models.

In this paper, we use the Blagojevich analytical model [34], which belongs to the class of simple but rather universal load-displacement models, a review of which can be found in [35]. Using [36,37], we write the basic equation of the model in terms of load-displacement ($F–u$):

$$F=F_{peak}{\left(\frac{u}{u_{peak}}{e}^{\left(1-\frac{u}{u_{peak}}\right)}\right)}^{\left(a-b\right)\stackrel{=}{H}\left(1-\frac{u}{u_{peak}}\right)+b}.$$

(1)

Heaviside function $\stackrel{=}{H}\left(1-\frac{u}{u_{peak}}\right)=1,$ if $u\le u_{peak},$ which corresponds to the pre-peak branch of the load-displacement plot; $\stackrel{=}{H}\left(1-\frac{u}{u_{peak}}\right)=0,$ if $u\ge u_{peak},$ which corresponds to the post-peak branch of the same plot. Equation (1) uses the same Blagojevich equation with separate parameters ($a$ and $b$) for the pre-peak and post-peak branches of the load-strain curve. It is important to note that using this equation provides a smooth junction of the branches at the peak point. In the special case, if $a=b=1$, we obtain the Furamura model [35,43]. Parameters $a$ and $b$ are determined using experimental data, as shown in [35]. Another way of determining these parameters is discussed below in a separate section.

Figure 3 shows experimental curves according to Figure 2 and curves according to Equation (1). The values of $F_{peak}$, $u_{peak}$, $a$ and $b$ are given in

Table 1. Values $F_{peak}$, $u_{peak}$, $a$ and $b$ for plots in Figure 3.

Curve	${\mathit{F}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}\mathbf{\left(}\mathbf{kN}\mathbf{\right)}$	${\mathit{u}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}\mathbf{\left(}\mathbf{mm}\mathbf{\right)}$	$\mathit{a}$	$\mathit{b}$
1	73.54	1.29	4.0	7.0
2	90.31	1.44	3.5	5.0
3	88.67	1.42	3.5	2.0

.

Figure 3 shows that the curves in Equation (1) model well the pre-peak condition of the samples and a small portion of the post-peak condition. That is, under post-peak loading, Equation (1) does not predict the appearance of an almost horizontal section on the load-displacement curve. From a physical point of view, the nearly horizontal section 4–5 in Figure 2, as noted above, corresponds to plastic deformation and compaction in the fracture process zone (Figure 1a). The compaction of the particles at stages 4–5 can lead to an increase in stiffness, i.e., an increase in the load with an increase in strain, as shown by one of the diagrams in [4].

The above-mentioned transition to the plastic stage of wood deformation is accompanied by significant deformations (4% or more) and, in terms of physics, means almost complete loss of elastic properties of wood. From a practical point of view, such a state can be regarded as unacceptable. It is therefore important to answer the question: how to determine the transition point to the plastic stage of deformation on the load-displacement curve? To answer this question, it is necessary to carry out a preliminary analysis of the changes in the stiffness of the sample during its loading.

2.4. The Highest Value of Tangential Stiffness and Tangential Modulus of Elasticity

The tangential stiffness can be determined at any point of the load-displacement curve (1). Consider the ascending branch of this curve. Using (1), we write the equation of the rising curve:

$$F=F_{peak}{\left(\frac{u}{u_{peak}}{e}^{(1-\frac{u}{u_{peak}})}\right)}^a;u\le u_{peak}.$$

(2)

The tangential stiffness $S_{tang}$ can be determined for any point on the load-displacement curve using relation (3):

$$S_{tang}=\frac{dF}{du}.$$

(3)

The greatest value of tangential stiffness ${\widehat{S}}_{tang}$ is of practical interest. The displacement $\widehat{u}$, which corresponds to ${\widehat{S}}_{tang}$, we find from condition (4):

$$\frac{d{\widehat{S}}_{tang}}{du}=0.$$

(4)

We get after transformations:

$$\widehat{u}=(1-\frac{1}{\sqrt{a}})u_{peak}.$$

(5)

Using (2), (3), and (5), determine the largest tangential stiffness $S=\widehat{S}$ depending on the parameter $a$:

$${\widehat{S}}_{tang}=\frac{F_{peak}}{u_{peak}}{(1-\frac{1}{\sqrt{a}})}^{a-1}{e}^{\sqrt{a}}\sqrt{a}.$$

(6)

Equation (6) contains an important characteristic (secant stiffness $S_{secant}$) similar (but not identical) to secant modulus of elasticity:

$$S_{secant}=\frac{F_{peak}}{u_{peak}}.$$

(7)

Given equality (7), rewrite Equation (6) in the form (8):

$${\widehat{S}}_{tang}=S_{secant}{(1-\frac{1}{\sqrt{a}})}^{a-1}{e}^{\sqrt{a}}\sqrt{a}.$$

(8)

Practical interest for the analysis of wood behavior under loading can be represented by the plot of the dependence of the highest value of tangential stiffness ${\widehat{S}}_{tang}$ on the parameter $a$ (Figure 4).

Commenting on the plot of dependence (8) on the parameter $a$ in Equations (1) and (2), we should pay attention to the following. If $a=1$, then $\widehat{S}=e\approx 2.718S_{secant}$. The same multiplier (2.718) for the ratio of the tangent and secant modulus of elasticity was obtained in [44] using the Furamura model, which is a particular case of model (1) at $a=b=1$. Model (1) is more universal because it predicts the ratio of secant and tangent stiffnesses $\widehat{S}/S_{secant}$ for all physically admissible values. The minimum value of this ratio is 1.701 (Figure 4).

Regarding the validity of the simulation results, we note the following. Since we consider timber as a structure [4], it is permissible to compare the dependence according to Figure 4 with the results of tests of wooden structures known from the literature [38], according to which $S_{secant}=\frac{123}{2.73}=45.1\frac{\mathrm{kN}}{\mathrm{mm}};{\widehat{S}}_{tang}=\frac{29.6}{0.38}=77.9\frac{\mathrm{kN}}{\mathrm{mm}}$ does not contradict Equation (8) and its graphical interpretation (Figure 4), according to which the condition $\frac{{\widehat{S}}_{tang}}{S_{secant}}=\frac{77.9}{45.1}=1.73$ must hold. In addition to the above relationship, it is necessary, using experimental data, to determine the value of the parameter a; this question will be answered in general form after $a$ short note.

Note that the ratios of the tangential and secant stiffness are equivalent to the ratios of the tangential and secant modulus of elasticity. For example, in the case of uniaxial compression of a specimen of length $L_0$ with cross-section $A_0$, the secant stiffness $S_{secant}$ (7) is related to the secant modulus of elasticity by relations (9):

$$S_{secant}=\frac{F_{peak}}{u_{peak}}=\frac{F_{peak}{A}_0}{{\epsilon }_{secant}{L}_0{A}_0}=\frac{{\sigma }_{peak}{A}_0}{{\epsilon }_{secant}{L}_0}=E_{secant}\frac{A_0}{L_0};E_{secant}=\frac{L_0}{A_0}{S}_{secant}.$$

(9)

2.5. Parameter a for the Load-Displacement Curve Equation

If the offsets $\widehat{u}$ and $u_{peak}$, respectively, for the inflection point and the peak point are determined experimentally, then the parameter $a$ can be found from Equation (5):

$$a=\frac{u_{peak}^2}{{(u_{peak}-\widehat{u})}^2}.$$

(10)

Using (2) and (10), we find the load $\widehat{F}$:

$$\widehat{F}=F_{peak}{(1-\frac{1}{\sqrt{a}})}^a{e}^{\sqrt{a}}.$$

(11)

The plot of the function $\frac{\widehat{F}}{F_{peak}}$ is shown in Figure 5.

Note that the plot in Figure 5 can be used to determine the parameter $a$ if the experimental values of $F_{peak}$ and $\widehat{F}$ are known.

2.6. Example

As the initial data, we use the results of tests of wood structures known in the literature [38]: $F_{peak}=123$ kN; $u_{peak}=2.73$ mm; $\widehat{u}=1.05$ mm. Using (10) and (11), we obtain: $a=2.64$; $\widehat{F}=50.1$ kN. Using $F_{peak},$ $u_{peak}$, $a$ and Equation (1), we plot the load-displacement plot (Figure 6, red line). The tangent (dashed line) passes through the point ($\widehat{u},\widehat{F}$) where the tangential stiffness is greatest; equation of the tangent:

$$F={\widehat{S}}_{tang}(u-\widehat{u})+\widehat{F}.$$

(12)

In this example $F=77.9\left(u-1.05\right)+50.1$.

Commenting on Figure 6, note the same pattern seen in Figure 2 and Figure 3, namely, good agreement with the experiment on the pre-peak branch and signs of plastic deformation on the post-peak branch of the load-displacement curve. The transition of wood to an almost plastic stage of deformation means an almost complete loss of elastic properties, so it is reasonable to consider the transformation of input energy in the process of deforming, damaging and destroying wood in view of the test results presented above.

2.7. Application of the Differential Energy Criterion for Brittle Fracture

The goal of this part of the work can be formulated as a search and substantiation of the criterion for the transition of wood from the elastic stage of deformation to the plastic behavior, i.e., from stages 3–4 to stages 4–5, according to Figure 2. Taking into account the external similarity of the ascending and (partially) descending branches of the complete load-displacement curve in tests of wood and brittle materials, consider the application of the recently proposed differential energy criterion of brittle fracture [45]. According to this criterion, in the context of this work, the point of brittle failure is located at the intersection of the post-peak branch of curve (1) and line (13):

$$F=\frac{1}{2}({\widehat{S}}_{tang}(u-\widehat{u})+\widehat{F}).$$

(13)

Equation (13) is obtained using Equation (12).

The tangential stiffness ${\widehat{S}}_{tang}$ can be replaced by the secant stiffness $S_{secant}$ (7). In this case, the point of brittle failure is at the intersection of the post-peak branch of curve (1) and line (14):

$$F=\frac{1}{2}{S}_{secant}u.$$

(14)

Since ${\widehat{S}}_{tang}>S_{secant}$, which follows from Equation (8), the use of Equations (13) and (14) leads, respectively, to upper and lower estimates of brittle fracture. These estimates predict the point at which the transition from stage 3–4 to plastic deformation stage 4–5 is realized, according to Figure 2.

Figure 7 explains the application of the differential brittle fracture energy criterion [45] and the tangential stiffness (8) to the analysis of the aforementioned wooden structure (Figure 6) [38].

Figure 8 and Figure 9 explain the application of the differential brittle fracture energy criterion [45] to the analysis of the above-mentioned wooden structure (Figure 6) [38]. Figure 9 shows that the transition to the plastic stage is realized at a load of 100 kN, which agrees with the experimental data from [38].

Commenting on Figure 9, it should be noted that the considered method does not guarantee accurate results since it is approximate and does not take into account the heterogeneity of wood structure and other factors, the exact accounting of which is extremely difficult even at the theoretical level [4]. Therefore, it is reasonable to apply upper and lower estimates of the criterion for the transition of wood to the plastic stage of deformation, i.e., this criterion should be considered as an interval variable. Application of interval analysis and fuzzy sets theory seems promising, although this way is also not simple [46,47].

2.8. Comparison of Simulation Results with Experimental Data

Using the methodology discussed above, we determine upper and lower estimates for the load at which the Figure 1 specimens enter the plastic stage of deformation. Experimental data and simulation results are shown in

Table 2. Values $F_{peak}$, $u_{peak}$, $a$, $b$, $\widehat{F}$ and $\widehat{u}$ for plots in Figure 3.

Curve by Figure 2	${\mathit{F}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}\left(\mathbf{kN}\right)$	${\mathit{u}}_{\mathit{p}\mathit{e}\mathit{a}\mathit{k}}\left(\mathbf{mm}\right)$	$\mathit{a}$	$\mathit{b}$	$\widehat{\mathit{F}}$	$\widehat{\mathit{u}}$
1	73.54	1.29	4.0	7.0	33.9	0.65
2	90.31	1.44	3.5	5.0	40.3	0.67
3	88.67	1.42	3.5	2.0	39.6	0.66

and Figure 10.

Figure 10 shows that the simulation results are consistent with experimental data and reliable enough to predict the transition of wood to the plastic stage of deformation. Each of points 4, 5 and 6 is at the intersection of the descending branch of curve (1) and line (14); this issue is discussed in more detail in the section below using Figure 10. An important conclusion for practice is that the use of secant stiffness, i.e., Equation (14), leads to more accurate estimates of the load that corresponds to the transition to the plastic stage of deformation; this stage is shown in Figure 9 by horizontal lines starting at points 4, 5 and 6, for which the load is 50.0, 63.5 and 69.0 kN, respectively.

2.9. To the Determination of Parameter $a$ and $b$

The peak load, the peak displacement, the load and displacement corresponding to the greatest tangential stiffness can be obtained from the experimental load-displacement curve, and the parameter $a$ can also be given by Equation (11) and Figure 5. However, the question remains unsolved: how can the parameter $b$ in Equations (1) and (11) and Table 1 and Table 2 be determined?

It should be noted that parameter $b$ affects only the post-peak branch of the load-displacement curve. Using the Blagojevich model [34,35], by analogy with the equation of the pre-peak branch (2), we write the equation of the post-peak curve (15):

$$F=F_{peak}{\left(\frac{u}{u_{peak}}{e}^{(1-\frac{u}{u_{peak}})}\right)}^b;u\ge u_{peak}.$$

(15)

Variants of load-displacement curves for sample 2 (Table 2) at the same value a = 3.5, but at different values $b=$ 0, 1/4, 1, 2, 3, …, 9, 10, 100 and 1000 are shown in Figure 11.

The trajectory of the states of sample 2 (from Table 2) is shown in Figure 10 as a solid thickened line and corresponds to the values $a=3.5$ and $b=5$. In accordance with the above-mentioned energy criterion of brittle fracture [45], the quasi-brittle fracture of sample 2 ends at the intersection of the descending branch (15) and the straight line (14) (thin red line in Figure 10). These points, if $b=4,5,6\mathsf{и}7,$ are located at a small distance from each other, so the approximate values of parameter b can be selected from the range from 4 to 7. This circumstance can be considered as a sign of the stability of the criterion (and the corresponding model) to small deviations of parameter $b$. This is a useful property if experimental and numerical methods are used to determine the parameter.

The parameter $b$ can be determined using experimental data and their approximation, e.g., by the least-squares method [35].

Various algorithms for determining parameter $b$ are possible; one of them is as follows: determine $u_{peak}$ and $F_{peak}$; using Equation (1) or Equations (2) and (15), construct load-displacement plots for a large enough set of theoretically possible values of parameters $a$ and $b$ (Figure 10); from this set, select the curve corresponding to the experiment.

It is important to pay attention to the property of the parameter $b$: the values of the parameter $b$ positively correlate with the stiffness of the material. Therefore, this parameter can be considered a characteristic of the material in the post-peak state, but further studies using a sufficiently large amount of experimental data for various materials are needed to confirm this assumption.

Since a feature of the post-peak stage of deformation of materials is increased acoustic emission compared to the pre-peak stage [11,30], it is logical to assume that the physical characteristics of acoustic emission are not the same for different materials and deformation stages, and therefore analysis of these differences will allow predicting the post-peak behavior of the material. Research in this direction is ongoing [31]. In addition, the physical characteristics of acoustic emission may correlate with the value of the $b$ parameter, which may be the subject of further studies.

3. Discussion and Conclusions

In the review part of this paper, it was noted that in some models, the accuracy of fracture prediction decreases for materials with defects [6,7]. In this paper, we propose a methodology that is indifferent to defects and material heterogeneity as well as to the shape of specimens, which is demonstrated by simulating a wooden structure fragment (Figure 6) and cubic shape specimens (Figure 1 and Figure 10). It was possible to increase the versatility of the approach to modeling the behavior of wood and the structural fragment by considering wood as a structure in accordance with the study [4]. Considering only the external aspects of the behavior of the material under load, we used the concept of structural stiffness (the stiffness of materials is usually characterized by the modulus of elasticity). Analysis of the behavior of heterogeneous materials, such as frozen ground, rocks, and concrete, also confirmed the effectiveness of this approach [36,45]. The main issue of this work was to determine the criterion for transition from the quasi-elastic behavior of wood to the quasi-plastic stage. This question was solved by adapting the energy criterion in differential form, which was proposed in [45]. As it was noted in the review part of this paper, the energy criteria of strength and fracture are the most universal, which makes it possible to predict the development of the presented approach and its use in future studies of the load-displacement relation for different materials and structures.

The use of acoustic emission methods [48,49] and other modern methods [11,50,51] contributes to a better understanding of the physical aspects of brittle fracture and plasticity of wood, as well as other materials at various stages of their deformation [52,53]. The input data for calculations according to the proposed model uses the concept of stiffness but not the modulus of elasticity and other physical constants of the material, so the modeling technique (under some constraints) is indifferent to the defects and features of wood, which, considered as a structure in accordance with [4]. Consideration of this feature of the calculation methodology can be useful when analyzing the load-displacement relationship for structures made of wood [54,55,56], which is a renewable resource and whose rational use is becoming increasingly important for sustainable development in modern conditions [57,58,59].

In conclusion, summing up, it is necessary to pay attention to the practical significance and prospects for the development of the presented research. Since the energy criteria, including the differential energy criterion of destruction [45], used as the basis of this work, belong to fairly universal criteria, the methodology of the presented study can be adapted to the analysis of the mechanical behavior of not only wooden structures [45], but also composite materials [60,61,62], structures [63,64] and bone [65] under other types of loads in future research.