Uniaxial compression of dry wood: experiment and modeling

. The paper considers the dependence of the load on displacement during uniaxial compression of spruce wood samples with an adjustable deformation rate (1 mm/min); the water content in the samples was equal to 6.4%. The study focuses on the analysis of criteria for brittle destruction of wood and its transition to a plastic state. The concept of virtual material transformation (CoViMT) forms the basis of the research. At the idea level, this concept is a variant of the well-known equivalent material concept (EMC). An integral criterion for brittle fracture was developed. Comparison with experimental data showed that the integral criterion and the known differential criterion lead to mismatched points of brittle fracture and transition of wood to the plastic state. Therefore, it is possible to determine the boundaries of the region of plastic states with the help of these criteria. Experiments confirmed the adequacy of the criteria when compressing wood with 6.4% and 18.0% moisture content. However, the volume of studies of the problem touched upon is small, so further research is necessary to better understand the stages of deformation of wood of different species.


Introduction
One of the main uses of wood is construction, where the problems of strength of materials and structures are key. Despite the ongoing research in this area and the progress made, the problem of wood strength remains relevant, since it is difficult to fully reflect and predict the mechanical behavior of wood under the influence of forces due to the peculiarities of its structure and variability of mechanical and physical characteristics of a meso-and microscale wood elements [1,2].
The response of wood to load is nonlinear and depends on the magnitude and direction of the load. Experimental load-displacement curves have shown that wood deforms as an elastic-plastic material that gradually transitions from nearly linear-elastic stage to nearly plastic with monotonic increases in compressive load; however, wood typically becomes brittle in tension [1]. Plastic deformations, unlike elastic deformations, do not disappear after the cessation of force impact.
The term "plastic strain" is acceptable from the point of view of material mechanics, but from the point of view of the user of wooden construction, the term "residual strain" is more acceptable. In this context, it is clear that the shape of a wooden structure (for example, a beam) should not change significantly in its life cycle, so the stage of plastic deformation of elements of real structures is undesirable. Such (residual) deformations appear under a sufficiently large load, so in order to exclude their appearance, it is necessary to know the criterion (i.e. load) of wood transition into plastic state, which can be determined experimentally or theoretically, using numerical or analytical modeling methods [2].
To ensure the adequacy of the results and to minimize the cost of the indicated predictions, simulations, experiments on specimens and adaptation of the results to real structures are usually used. The purpose of this work is to develop an integral criterion for the transition of wood under compression along the fibers into a plastic state using the full load-displacement curve. This work focuses on conceptual and methodological aspects.
To achieve the above goal, various approaches can be used, implemented in many models of behavior and failure of isotropic and, to a lesser extent, anisotropic materials [1,2]. Nevertheless, in spite of the progress in this field, many criteria have limited application because they do not fully correspond to real materials [3][4][5]. At the same time, the best results have been obtained using energy concepts [3,4].
For example, strain energy density analysis is the basis of some energy criteria for fracture [5]. One of these criteria is based on the concept of an equivalent material, namely, the conversion of a physically nonlinear material into an equivalent linearly elastic material [6], however, in some cases the prediction accuracy is reduced 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 [1].
Note that the transition of wood during compression along the fibers into the plastic state was briefly considered in [8], but using a different (differential) criterion. Given that, generally speaking, all the models are to some extent approximate, and the areas of their effective application may not coincide, it is advisable to complement the known models and approaches in order to try to close the gaps in this area and improve the understanding of wood behavior in the elastic-plastic stage.

Conceptual aspects
The considering concept of modeling the mechanical behavior of wood under compression along the fibers can be called the concept of virtual material transformation (CoViMT). At the idea level, this concept is a variant of the well-known equivalent material concept (EMC), on the basis of which approaches and models have been developed to analyze the mechanical state of metal and polymer specimens with notches [5,6,7,9,10]. In our case, samples without notches are used, but studies [8,11] have shown that the EMC idea, after adaptation, can be used in other instances, one of which is considered in this paper.
In CoViMT, as in EMC, a real elastic-plastic material with a nonlinear load-displacement (or stress-strain) relationship is equivalent to a virtual material with linear elastic behavior up to the point of brittle failure. In CoViMT, equivalence means that the elastic modulus, Poisson's ratio, absorbed energy and displacement (or strain) at the fracture point for the virtual material have, respectively, the same values as for the real material.
CoViMT assumes that the load-displacement (or stress-strain) equation for a real material is known [8], allowing the tangential stiffness (or tangential modulus of elasticity) to be determined; values of the tangential modulus of elasticity, as noted above, are the same for real and virtual material. Tangential stiffness (modulus of elasticity) use to express the potential energy of deformation of a virtual linearly elastic material. The energy absorbed by the real material is determined by integrating the load-displacement (or stress-strain) equation mentioned above. In the two relations thus obtained, the load (stress) and displacement (strain) are unknown. According to the concept, the real and the virtual material collapse under the same displacements (deformations), but under different loads (stresses). Using the condition of equality of absorbed energy at the point of brittle fracture, we can determine the displacement (strain) at the point of brittle fracture. Then, using the displacements (stresses) found, the fracture load (stresses) values of the real and virtual material can be calculated using the corresponding load-displacement (stress-strain) equations, for example from [8,11,12].
The main problem in CoViMT is the partitioning of the input energy for a virtual linearelastic material into two parts; one part of the energy is stored in the dummy material and consumed for cracking, while the rest of the energy is dissipated. In [11] shows that no more than 50% is dissipated; accordingly, at least 50% of the input energy is stored. These estimates do not contradict the known studies of brittle fracture [13], according to which, during quasi-static crack growth in a linearly elastic material, only part (0.41) of the macrolevel energy release spent on the fracture itself, while the remaining energy is dissipated. In this paper, we assume that 50% of the input energy spent on deformation and brittle fracture of the aforementioned virtual linearly elastic material.

Terminology
Regarding terminology, we note that the mechanical behavior of wood can be described in terms of "load," "displacement," and "stiffness," or in terms of "stress," "strain," and "modulus of elasticity. Both sets of terms are acceptable in CoViMT, so we have written the analogues of the terms in parentheses.
The concept presented involves a transition from a real material to a virtual material, the formulation of equivalence conditions, and the reverse transition from a virtual material to a real material. In other words, it implies a transformation of virtual material, which reflects the name of the concept (CoViMT), which, as noted above, is a variant of the well-known concept of equivalent material (EMM) [9].

Introductory remarks
This section presents the results of CoViMT simulations and a comparison with experimental data. The aforementioned load-displacement equation for a real material is an adaptation of the well-known Blagojevich model [14,15]: The input data for the calculations are model parameter ( ), peak load values ( )) and displacement ( ), which can be determined experimentally or predicted [16]. The load-displacement equation for the virtual material is a straight-line equation in which the angular coefficient is equal to the tangential stiffness (tangential modulus of elasticity) of the real material.

Experiment
The compression test specimens were made of spruce wood; their characteristics and model parameter (1) shows Table 1. The device for measuring the moisture content of the specimens, the testing machine, and the specimens before and after test shows Figure 1; compression at controlled displacement (1 mm/min).   Table 1.  Figure 2 shows that the ascending branch of curve 4 corresponds to the experimental data, but on the descending branch the adequacy of the simulation is lost if the displacement exceeds 1.5 mm (which for the test specimen with a height of 30 mm (Table 1) corresponds to a relative deformation of 5%).

Modeling
We will determine the work of force in the compression of a sample of real material by integrating equation (1); we consider this work equal to the energy spent on deformation and destruction. Numerically, this energy is equal to the area under the load-displacement graph (Figure 3).   Fig. 3. Curves 1 and 2 simulate, respectively, the load ( ) and energy ( ) dependencies on displacement (u) for a real material.
As noted above, during quasi-static crack growth in a linearly elastic material, only part of the injected energy spent on the fracture itself, while the rest is dissipated. In this paper, we assume that 50% of the input energy spent on deformation and brittle fracture of the virtual linearly elastic material. Therefore, the energy expended on deformation and fracture of the virtual material is equal to half of the area under the load-displacement diagram. The tangential stiffness (tangential modulus of elasticity) of the virtual material is equal to the highest value of the tangential stiffness (tangential modulus of elasticity) of the real material on the ascending branch of the load-displacement curve (Figure 4).

Discussion
In this section, we compare the above simulation results with experimental data. In addition, we touch on the application of the well-known differential criterion of brittle fracture [8].
Experimental data and simulation results shows in Figure 6.  (1). Straight line 5 simulates the plastic stage of deformation. Brittle fracture and transition to the plastic state corresponds to point 6, so part of curve 7 not realized. Brittle fracture and transition to plastic state in accordance with the differential criterion [8] corresponds to point 8; in this case, the dashed curve 9 models the plastic deformation stage. Figure 6 shows that the integral criterion considered above and the differential criterion [8] lead to different points of brittle fracture and transition of wood into a plastic state. Therefore, it is possible to simulate the area of possible plastic states. Lines 5 and 9 in Figure  6 are, respectively, the lower and upper boundaries of this area. Note that if the deformation exceeds 2.6 mm (8.7%), the adequacy of the model decreases.
In this work, we consider the behavior of spruce wood with a moisture content of 6.4% (Table 1); at the same time, a similar problem for wood with a moisture content of 18% was considered in [8], but using the differential brittle fracture criterion. The simulation results in these two cases differ, but do not contradict each other. However, confident conclusions are untimely, since the amount of research on the raised problem is small, so to better understand the stages of deformation of wood of different species it is necessary to continue experimental studies and develop more accurate models, taking into account the conditions on which the state and the mechanical behavior of wood depends.

Conclusion
The paper considers the dependence of the load on displacement during uniaxial compression of spruce wood samples with an adjustable deformation rate (1 mm/min); the water content in the samples was equal to 6.4%. The study focuses on the analysis of criteria for brittle destruction of wood and its transition to a plastic state.
The concept of virtual material transformation (CoViMT) is used. At the idea level, this concept is a variant of the well-known equivalent material concept (EMC). An integral criterion for brittle fracture was developed.
Comparison with experimental data showed that the integral criterion and the known differential criterion lead to mismatched points of brittle fracture and transition of wood to the plastic state. Therefore, it is possible to determine the boundaries of the region of plastic states with the help of these criteria.
However, the volume of studies of the problem touched upon is small, so further research is necessary to better understand the stages of deformation of wood of different species.