Monotonic and cyclic load of pine wood under uniaxial compression: experiments and modeling

. To analyze the behavior of wood under uniaxial monotonic deformation, we applied an analytical model to separately control of the pre-peak and post-peak branches of the load-displacement curve. Two criteria for the transition of wood into the plastic stage of deformation are considered. To analyze the behavior of wood under cyclic uniaxial deformation, we used the load-displacement curve obtained for monotonic deformation as an envelope. Equations for the plotted of load-displacement curves under cyclic loading with incomplete unloading are proposed. As an experimental and model illustration, the behavior of a pine specimen under cyclic uniaxial compression with incomplete unloading in the range from 8 to 15 kN is considered. Because the highest cyclic load exceeds the load at the point of transition of wood into the plastic stage, the deformation process has already after 54 cycles moved to the downward branch of the load-displacement curve for monotonic deformation. The simulation results are in agreement with experimental and literature data, which suggests the suitability of this method for the study of some wood-based composites.


Introduction
The value of wood as a renewable resource is only increasing over time.Large volumes of wood are used in construction, which corresponds to the criteria of sustainable development [1] and implies the continuation of research on this natural composite using modern experimental methods, computer technologies and mathematical modelling [2].Wood, as a material of building structures, is subjected to constant loading from its own weight as well as repetitive and cyclic loads [3,4,5].In this paper, uniaxial cyclic compression of pine wood specimens along the fibres is considered.This type of loading can occur in various situations such as transport and functioning of wooden structures [3,4].Under load, wood deforms and internal forces appear in its volume [2].The deformations of wood include elastic and plastic components.At low loads, elastic deformation dominates and disappears after unloading.With increasing load, the proportion of plastic deformations increases, which is manifested by the non-linear nature of the load-strain relationship.Plastic strains do not disappear after unloading, so they are also called residual strains [6].Wood exhibits complex viscous, elastic and plastic anisotropic behaviour.The complex nonlinear behavior of wood and the variability of its properties, which are explained by the conditions of origin and functioning of this natural composite [2,6,7,8,9], predetermine the need to continue research in the designated area.
This paper focuses on the behavior of wood under cyclic compression with incomplete unloading.Of practical interest are the largest values of safe load on the structure.The area of the largest theoretically possible load values is bounded by a curve called the envelope [10].The construction of the shell and the analysis of the cycle performance with respect to the degradation of the mechanical properties of wood related to key issues in the analysis of the cyclic deformation behavior of this composite [6,10,11].
The envelope for cyclic uniaxial loading parallel to the grain is almost the same as the curve of the load dependence on displacement during monotonic deformation [10].Hence, the failure point is located at the intersection of two experimental (or predicted) loaddisplacement curves plotted for monotonic and cyclic loading (Figure 1).Petrozavodsk State University provided a testing machine, specimens and a moisturemeasuring device [12].In addition to Figure 1, Figures 3 and 4 show, respectively, the experimental displacement vs. time and load vs. time dependences for the same specimen.As noted above, experimental data (Figure 1) and studies of wood behavior under cyclic and monotonic deformations known from the literature show that, the load-displacement curve can model the envelope curve [10].A similar dependence is found in studies of the behavior of rocks [13].However, unlike rocks, when wood is deformed monotonically, the displacements in the post-peak stage are significantly larger compared to those in the prepeak stage (Figure 1).For brittle materials, the post-peak strains are usually smaller than the pre-peak strains [13,14]; while there are some similarities in the load-displacement plots, there are significant differences that need to be analyses separately.
The aim of this study is to simulate the behavior of pinewood under cyclic compression with incomplete unloading.

Preliminary remarks
As noted above, the failure point is located at the intersection of two load-displacement curves plotted for monotonic and cyclic loading in uniaxial compression of wood, respectively (Figure 1).However, it is not possible to obtain both curves for a single specimen.To obtain both curves, it is necessary to use two identical specimens, while it should be borne in mind that it is impossible to achieve complete identity of wood specimens due to the natural variability of its properties [2].To obtain a sufficiently accurate picture of the cyclic loading behavior of wood, two series of similar specimens should be tested under identical conditions and the test results processed using appropriate algorithms.
Useful tools for further analysis of wood loading behavior are numerical and analytical models, the visualized results of which are load-displacement curves under monotonic and cyclic deformation [6,7,10,11,15].In this paper, we used an analytical model belonging to a certain class of models [16].A specific example is discussed in section 2.2.

Physical aspects of modeling
Modeling includes a description of the physical model of the object of study, a mathematical description of the physical model, as well as experimental verification of the simulation results and model correction (if necessary).Consider the physical aspects.As noted above, the complex structure of wood and the variability of mechanical properties make it difficult to model the behavior of wood under load [2].In addition, from a practical point of view, the model should be simple and reasonably accurate.Figure 3 shows the variability of wood properties of three specimen under uniaxial compression.Figure 5 shows that elastic deformations predominate in the pre-peak stage and plastic deformations predominate in the post-peak stage.This significant difference makes it difficult to construct an analytical dependence that models the behavior of the specimen at both the pre-peak and post-peak stages.To overcome this difficulty, we used separate modelling of the pre-peak and post-peak branches.In addition, on the post-peak branch we modelled the transition point of the specimen into the plastic deformation stage, as shown below.
Modelling the transition point of wood (as an elastic-plastic composite consisting, respectively, of elastic and plastic elements [2]) into the stage of plastic deformation can be considered as a separate problem, the exact solution of which is unknown.In order to obtain an approximate solution, we assumed (at the level of hypothesis) that the transition to the plastic state means almost complete exhaustion of elastic properties and, accordingly, brittle failure of elastic elements, after which deformation of plastic elements of the damaged composite (if such elements exist) continues.Consequently, from the physical point of view, some criterion of brittle fracture can be regarded as a criterion for the transition of an elasticplastic material from the elastic stage to the plastic deformation stage [17,18].Undoubtedly, this brief description corresponds to an idealized simplified model, since in real natural composites (which includes wood) elastic and plastic elements interact and deform together [2], with an increase in the impact on the composite accompanied by a decrease in the share of elastic deformations and, accordingly, an increase in the share of plastic deformations (Figures 1 and 5).
The selection of model parameters   ,   ,   and   can be carried out both visually, by studying and correcting the graphs of load-displacement dependence, and automatically by the method of least squares by analogy with [16].An example of the application of equation ( 1) for the specimen in Figure 2 is discussed below (Figure 6),   =17 kN;   =0.9 mm;   =1.4;   =1.9;   =0.95;   =1.0.The pre-peak and post-peak branches of the load-displacement curve have inflection points where the  2   2 = 0.This means that at each of these points the tangent stiffness  =   reaches an extreme (Figure 6).The characteristic points and lines on the load-displacement curve are: peak point (1); inflection points (2 and 3); tangents 4 and 5 are drawn through points 2 and 3, respectively; a line 6 is drawn through the point of intersection of line 4 with the abscissa axis, the tangent of the angle of inclination of which is twice less than the tangent of the angle of inclination of tangent 4 [18]; the point of intersection of line 6 with the postpeak branch of the load-displacement curve (7) and point 3 determine, respectively, the upper and lower estimates of the load, at which the transition of the specimen into the plastic stage of deformation is possible; the fragment of the post-peak branch (dotted line) does not exist in practice (Figure 5), because at point 7 (or 3) the specimen transitions to the plastic deformation stage [17], which is modelled by line 8 or 9, respectively.After passing point 3, the tangential stiffness decreases in absolute value and tends to zero, so point 3 can be considered as an indicator of the composite's transition to the plastic deformation stage.

Load-displacement model for cyclic deformation
Experiments show (Figure 1) that in uniaxial cyclic deformation of wood with incomplete unloading, line segments (Figure 7) can model the shape of the cycle, for which the tangents of the angles of inclination determine the stiffness of the specimen.Let us consider cycles 1-2-3 and 3-4-5.Based on the experiments, we assume that the slope angle of the unloading lines (1-2, 3-4, etc.) is the same for all cycles.At the same time, the angle of inclination of loading lines (2-3, 4-5, etc.) is smaller than the angle of inclination of unloading lines (1-2, E3S Web of Conferences 458, 07021 (2023) EMMFT-2023 https://doi.org/10.1051/e3sconf/2023458070213-4, etc.), and this angle decreases with increasing strain, which can be explained by energy dissipation during loading.Considering the dissipation of energy in each cycle, we can obtain equations (2) for the coordinates of points 1-5 and, by analogy, for all cycles (Figure 7).We assume that the displacement  1 and force  1 are known at point 1, and the force  2 is known at point 2. In addition, the stiffness  at unloading and the coefficient  of the above mentioned energy loss (dissipation) at loading, 0 <  < 1, are known for section 1-2.The results of calculations using equations ( 1) and ( 2) are shown in Figure 8. Calculations were carried out at   =17 kN;   =0.9 mm;   =1.4;   =1.9;   =0.8;   =0.6.The values of the coefficient  were adjusted depending on the deformation stage (Figure 8).After 54 cycles, the cyclic deformation process moved to the downward branch of the loaddisplacement curve of monotonic deformation. .

Discussion
Predicting the moment of transition to the post-peak stage of plastic deformation is necessary for efficient utilization of wood (Figures 5 and 6).To predict the transition to this stage, the differential criterion of brittle fracture, discussed earlier [17,18], can be used.However, this criterion is an approximation; therefore, given the high variability of wood properties, multiple criteria should be used to reduce errors.In this paper, we propose an alternative criterion that assumes the elastic elements (i.e., components and particles) of the natural composite may fail prior to the plastic elements.The presence of an inflection point along the post-peak branch of the load-strain curve (Fig. 6) may support this assumption.Beyond this point, the composite stiffness's absolute value approaches zero, indicating that it behaves like a plastic material.The inflection point can be located either from the load-displacement curve obtained experimentally or via equation (1) analysis.Furthermore, the practical significance of equation ( 1) and the load-displacement curve is expressed by the possibility of using this curve as an envelope [10,13] when analyzing cyclic loads on wood in both modern [4,5,20] and historic buildings [21,22].
In this paper, we considered cyclic deformation in which the highest cycle load was above the transition point to the plastic stage.In this case, the number of cycles to the limit state of the specimen was relatively small (54 cycles, Figure 8).It can be assumed that the number of cycles to unacceptable wood degradation would be much greater if the highest cycle load was less than the plastic transition point.In such cycling cases [6], the cost of experimentally testing this assumption increases, as the number of cycles can be very large for a small maximum cycle load.Nevertheless, such testing, modeling, and analysis are of practical and scientific interest.

Conclusion
The paper presents the results of experimental and theoretical studies of the behavior of pinewood under uniaxial monotonic and cyclic loading.A model with independent control of the pre-peak and post-peak branches of the load-strain curve was used to analyze the loaddisplacement relationship under monotonic deformation, which improved the accuracy of the analysis results.Taking into account previous studies known from the literature, the loadstrain curve was used as an envelope for cyclic loading.The main result is a methodology for modelling the cyclic deformation of wood under the condition of incomplete unloading (2).The modelling results presented in this paper for monotonic and cyclic loading are adequate from a physical point of view.However, making confident generalizations and drawing conclusions is premature since the number of studies on cyclic loading of wood remains small.Further experimental studies and model development are necessary to gain better insight into the deformation stages of wood of varying species.

Fig. 1 .
Fig. 1.The red solid line shows the behavior of a pine wood specimen in the form of a 20×20×30 mm prism in the experiment under cyclic compressive deformation with incomplete unloading in the range from 8 to 15 kN.The moisture content of the wood specimens is 7.4%.The displacement rate is 5 mm/min.The expected conditions of the specimen during monotonic deformation, as indicated by the dotted line, are not achieved during cyclic loading because the upper limit of the cycle (15 kN) is lower than the peak monotonic load.

Fig. 4 .
Fig. 4. Cyclic dependence of external force (load) on time.On the right -the cycle form (enlarged).

Fig. 5 .
Fig. 5. Load-displacement relationships for three specimens according to Figure 2. The circles indicate the assumed points of transition of the specimens to the plastic stage of deformation.

Fig. 6 .
Fig. 6.Characteristic points and lines on the load-displacement curve (explanations in the text).

Fig. 8 .
Fig. 8. Stages of monotonic and cyclic deformation (left) and dependence of the energy dissipation coefficient on the cycles number (right).