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In this work, we study the relative capacities of two mathematical models for the description of the stressstrain relationship of wood material subjected to damaging cyclic compressive stresses. The models studied are the MazarsPijeaudier Cabot model and the FozaoFoudjet model (FozaoFoudjet 2). The envelope curves obtained experimentally during the cyclic compression tests were approximated using the two mathematical models and the coefficient determination R^{2 }was calculated for each of the models. It appears from the observations that the two models present a good fit to the experimental results. In addition, the FozaoFoudjet 2 model used with the parameters and chosen such as 0.00≤λ_{0}≤0.015 and 0.99≤λ_{1}≤1.05, shows a better fit to the experimental results compared to the MazarPijeaudier Cabot model.
wood, cyclic compression, envelope curves, damage, mathematical models
Wood is a biological material with impressive mechanical properties whose availability and ease of processing have made it the most used material for centuries. It operates at several levels in civil engineering, shipbuilding and many other fields. Its use has seen renewed interest in recent years due to the need to promote sustainable development while respecting the environment. The use of a material in the field of mechanical construction, whether in civil engineering, shipbuilding or simply carpentry, requires a prior knowledge of the mechanical characteristics of said material. These characteristics include the physicochemical properties of the material, the mechanical properties (Young's modulus, failure load, etc.) and the likely behaviors that these materials may exhibit when subjected to certain types of common stresses. The mastery of said characteristics not only guarantees better safety conditions, but also makes optimum use of the material. The study of the material in order to guarantee better safety conditions goes through the description of the various deterioration and rupture mechanisms of this material.
A material will be well known and therefore well valued if its different plasticity and damage mechanisms are well described. This problem is the subject of several studies carried out all over the world on the material wood [110]. The purpose of this study is to study the stressstrain relationship of some tropical species when they are stressed beyond their elasticity limits by cyclic compressive stresses.
It is shown that beyond the yield strength, a material exhibits nonlinear behavior. This nonlinearity is attributed mainly to the phenomena of plasticity and damage. These two phenomena are sufficiently well described by damage theory [11, 12].
Damage is the process of gradual deterioration leading to failure of the material [11], it begins with birth followed by the growth and coalescence of micro defects. The coalescence of the micro defects is the final step in the damage process and results in the appearance within the material of macroscopic cracks, thus marking the rupture of the material [12]. The first models for describing damage appeared in the 1960s when Kachanov [13] defined a mathematical variable for describing damage. Much work has therefore emerged on the modeling of damage to materials such as metals and concrete [1116]. The modeling of the damage is generally done within the framework of the mechanics of the continuous damage developed by the authors such as Lemaitre and Chaboche [13] whose essential points are: (1) the choice of an admissible thermodynamic potential for the establishment of the constitutive laws of the observable variables (strain, and temperature) and (2) the appropriate choice of a potential of dissipation for the evolution of the internal variables (plasticity, damage). The difference between the damage models thus proposed can therefore mainly reside in the choice of the dissipation potentials and therefore in the choice of the form of evolution of the damage variable. This choice is, however, guided by experimental observations and must be made in ways that best reproduce these experimental observations [1]. There are thus many damage models in the literature which all differ from each other in the form of the law of evolution chosen for the damage [12, 14, 17]. However, the choice of a law of evolution being conditioned by experimental observation, it depends not only on the type of material studied, but also and above all on the type of stresses. The damage models are therefore developed according to the types of stress [1, 14].
One of the stress regimes responsible for material damage is the cyclic stress regime. These cyclic stresses can be low in amplitude in which case the damage is due to fatigue of the material after a high number of cycles; just as they can be of increasing amplitude and relatively high in which case damage occurs even after a low number of cycles [12]. This second type of solicitation is the subject of this study.
The modeling of the damage under cyclic load with low number of cycles was the subject of numerous works on materials such as steels and concretes. [1117] Despite the large number of models existing for the description of the behavior cyclic of materials such as concrete and steels, there is to our knowledge very little model describing the behavior of wood material under cyclic stress at low number of cycles [1821]. The objective of this study is to find, among two existing damage models for modeling the behavior of materials other than wood, the one capable of best describing the cyclic behavior at low number of cycles of wood.
Many damage models have been proposed within the framework of the modeling of cyclic behaviors, a large part of these models, based on the theories of plasticity and damage; although efficient remain difficult to apply to materials, this mainly due to the large number of parameters required for their use. A model widely used for its relative simplicity is that proposed by Mazars and pijaudiercabot [22]. It was developed by the authors for the modeling of concrete damage. This model was subsequently adopted in the literature by many authors such as: Faria; Sima; Brecolleti [2325], among many others for modeling concrete damage under cyclic stress. Recently it has been adopted in the framework of wood damage modeling by authors such as Wang [18]. The initial model as proposed by the authors is sometimes modified in order to bring improvements in the description of the damage Sima [2328], but these modifications all the time bring an additional number of parameters and make the model more complex and therefore more difficult to use. One of the recent modifications made to the model is that carried out by FozaoFoudjet [26] for the description of behavior of bamboo under cyclic stress. The model named FosaoFoudjet 2 thus proposed by the authors has been shown to be more suitable for modeling the stressdeformation curves of bamboo under cyclic stress. The adoption of one or the other of these models within the framework of the modeling of the cyclic behavior of wood should be done by taking into account the capacity of the model to accurately describe the behavior studied, in this case the stressstrain relation. In this study, we are therefore interested in a comparison of the two damage models in their capacity to describe the stressstrain relationship through the envelope curves of the wood material subjected to cyclical stress. In our work, we plot the envelope curves proposed by the MazarPijodiercabot and FozaoFoudjet models, which we compare to the envelope curves obtained experimentally.
It is shown that the envelope curve of the material subjected to cyclic compression can be approximated by a stressstrain curve of monotonic compression. This monotonic compression curve includes: a linear part (zone with elastic behavior) and a nonlinear part describing the nonlinear behavior. The envelope curve is therefore generally defined by a set of parameters obtained experimentally including (1) the initial modulus of elasticity, (2) the strain at the elastic limit, (3) the coordinates at the top of the stressstrain curve, as can be seen in Figure 1.
Figure 1. Typical monotonic uniaxial stressstrain curve [23, 26]
2.1 Presentation of models
The models object of our study are those of Mazarpijaudier –Cabot [22] and FozaoFoudjet [26]. The MazarPijaudiercabot model was proposed for the description of the behavior of concrete under cyclic stress.The FozaoFoudjet 2 model is obtained from the Mazarpijaudiercabot model by the introduction of two parameters λ_{0} and λ_{1} with 1.00≤λ_{0}≤1.38 and 0.0≤λ_{1}≤0.13. For the description of the stressstrain relationship of bamboo. The mathematical expressions of the mazarspijaudiercabot and FozaoFoudjet 2 models are presented in Table 1. The original parameters were rewritten in terms of parameters obtained in the monotonic 1 D compression tests [23, 24].
Table 1. Mathematical expressions from MazarPijeaudierCabot and from FozaoFoudjet2 models

Equations of Mazar  Pijaudier cabot’s models 
Equation of FozaoFoudjet’s models 
1 
$\sigma =(1D){{E}_{0}}\varepsilon $ 
$\sigma ={{\left( \frac{{{\varepsilon }_{c}}}{{{\varepsilon }_{0}}} \right)}^{{{\lambda }_{1}}}}(1D){{E}_{0}}\varepsilon $ with 0.0≤λ_{1}≤0.13 
2 
$D=1\frac{{{\varepsilon }_{0}}}{\varepsilon }(1B)B\exp (\frac{{{\varepsilon }_{0}}\varepsilon }{{{\varepsilon }_{c}}})$ 
$D=1\frac{{{\varepsilon }_{0}}}{\varepsilon }(1B)B\exp (\frac{{{\varepsilon }_{0}}\varepsilon }{{{\varepsilon }_{c}}})$ 
3 
$B=\frac{\left[ {{\sigma }_{c\max }} \right.\left. {{E}_{0}}{{\varepsilon }_{0}} \right]}{{{E}_{0}}\left[ {{\varepsilon }_{c}}\exp (\frac{{{\varepsilon }_{0}}{{\varepsilon }_{c}}}{{{\varepsilon }_{c}}}){{\varepsilon }_{0}} \right]}$ 
$B=\frac{\left[ {{\sigma }_{c\max }} \right.\left. {{E}_{0}}{{\varepsilon }_{0}} \right]}{{{E}_{0}}{{\varepsilon }_{0}}\left[ {{\left( {{k}_{0}} \right)}^{{{\lambda }_{0}}}}\exp (\frac{{{\varepsilon }_{0}}{{\varepsilon }_{c}}}{{{\varepsilon }_{c}}})1 \right]}$ With ${{k}_{0}}=\frac{{{\varepsilon }_{c}}}{{{\varepsilon }_{0}}}$ and 1.00≤λ_{0}≤1.38 
In these formulas,
•σ is the applied stress;
•${{\sigma }_{c\max }}$ is the maximum applied stress;
•${{E}_{0}}$ is the modulus of elasticity;
•D is the Damage of the material;
•$\varepsilon $ is the total strain;
•${{\varepsilon }_{0}}$ is the elastic strain;
•${{\varepsilon }_{c}}$ is the strain at the maximum applied stress;
λ_{0 }and λ_{1} are parameters of FozaoFoudjet 2 model.
Figure 2. Zone of extraction of specimen (a) and dimensions of specimen (b)
Wood is a biological, anisotropic, heterogeneous and hygroscopic material. Each species of wood has its own physical properties. In addition, theses physicals and mechanicals properties are all dependent on its moisture content. The knowledge of a mechanical behavior of wood begins with the knowledge of its moisture content and its intrinsic mechanical caracteistics (modulus of elasticity and the failure load ect). The species studied are sapelli and iroko. The 30mmx30mmx90mm compression specimens were taken from the heartwood of each species according to standard NFB 51003, Figure 2(a) shows the zone of extraction, while Figure 2(b) shows the dimensions of the specimen. The determination of the moisture content was made following the standard NFB 51004. The moisture content is given by the relation:
$H=\frac{{{m}_{H}}{{m}_{0}}}{{{m}_{0}}}x100$ (1)
where:
·H is the moisture content of the specimen;
·m_{0} is the mass of wood in an anhydrous state;
·m_{H} is the mass of wood with moisture H.
The average humidity rate for our essences obtained during the tests was 19% for the sapelli and 15.82% for the iroko.
The determination of maximum stress followed the standard NFB 51007. The test consisted of monotonic compressive loading at low velocity until failure of the specimen. The average breaking stress of the species determined by standard NFB 51007 were 47.98 Mpa for sapelli and 45.80 Mpa for iroko for the obtained moisture contents. For the determination of the Damage cyclic compression loading and unloading tests have been made on longitudinal and radial directions of each species. Each cycle consisted of a monotonic loading phase, followed by an elastic unloading. The first load cycle was made at a maximum stress less than one third of the average breaking stress of each species (16Mpa for the sapelli end 15.3Mpa for the iroko). The value of the maximum stress for each cycle was incremented in the next cycle and the cycles were repeated until the specimen failed. The maximum cycle number was 06 for the longitudinal direction and 05 for the radial direction. The machine used for the tests was a motorized hydraulic compression machine of the DMY brand. The temperature of the experimental environment was stabilized at 23℃.
The envelope curves of the cyclic compression tests obtained by Fothe et al. [29] were plotted and approximated by the models of MazarPijaudiercabot and FosaoFoudjet. It is observed that both models give a good approximation of the stressstrain curve for stress levels below the yield stress where the material exhibits linear behavior. For the zone with nonlinear behavior, we observe divergences between the approximations of the two models. These divergences depend on the parameters λ_{0 }and λ_{1} chosen for the FozaoFoudjet 2 model; For certain values of these parameters, 1 and 0 respectively, one obtains a superposition of the FozaoFoudjet curves on the Mazarpijaudiercabot curves. This is easily explained by the fact that the FozaoFoudjet model derives from the Mazars model by the introduction of the parametersλ_{0 }and λ_{1}. For other values of these parameters, the divergence between these two models is such that the curves of the FozaoFoudjet model are closer to the experimental curves than that of Mazarpijaudiercabot; thus making the FozaoFoudjet model more apt to describe the stressstrain relation than the Mazarpijaudiercabot model. The following Figures 3 ad represent the experimental envelope curves and those proposed by the two models for certain values of λ_{0 }and λ_{1} defined such as 1.00≤λ_{0}≤1.38 and 0.0≤λ_{1}≤0.13. For specimens in different directions.
(a) Measured and calculated data compared Sapelli in longitudinal Direction
(b) Measured and calculated data compared Iroko in longitudinal Direction
(c) Measured and calculated data compared Sapelli in Radial Direction
(d) Measured and calculated data compared Iroko in Radial Direction
Figure 3. Measured and calculated data compared
The following Tables 25 present the values of the parameters used in the plotting of the curves of the preceding figures. The coefficient of determination R^{2} [8] for the two models was calculated for each sample by the relation:
${{R}^{2}}=1\frac{\sum\limits_{1}^{n}{(\sigma _{\exp }^{i}\sigma _{cal}^{i})}}{\sum\limits_{1}^{n}{\left( \sigma _{\exp }^{i}{{{\bar{\sigma }}}_{\exp }} \right)}}$ (2)
where:
•$\sigma _{\exp }^{i}$ is the experimental value of the stress;
• $\sigma _{cal}^{i}$is the value of the stress calculated by the model;
• ${{\bar{\sigma }}_{\exp }}$is the mean value of the experimental stress
The results are given in the following Tables 25.
It comes from the preceding Tables 25 that the values of the parameters and for each of the wood species and in each of the two directions are defined such that: 0.00≤λ_{0}≤0.015 and 0.99≤λ_{1}≤1.05. These values are lower than those proposed by FozaoFoudjet in their study on bamboo. This difference can be explained by the fact that although they are both organic materials, wood and bamboo do not have the same cellular structure. The values of the coefficients of determination for each model and for all the test pieces in longitudinal directions of sapelli and iroko are such that R^{2}≥98.86%. For the radial directions, this coefficient of determination for the great majority of specimens is such that: R^{2}≥95.76%. These observations allow to conclude that the models of MazarPijaudierCabot and FozaoFoudjet give good descriptions of the stress relation of formation of the wood material subjected to cyclic compressions.
However, for the entire test specimen, the coefficient of determination of the FozaoFoudjet model is higher than that of theMazarPijaudiercabot model. This shows quantitatively that for the parameters and chosen such as 0.00≤λ_{0}≤0.015 and 0.99≤λ_{1}≤1.05. The FozaoFoudjet 2 model gives a better description of the stressstrain relationship than the MazarPijaudiercabot mode.
Table 2. Parameters used in the models for Sapelli in the longitudinal direction

Mazar’s Equation 
FozaoFoudjet 2 

specimen 
E_{0}(Mpa) 
σ_{cmax} (Mpa) 
$\varepsilon_{c}$(x10^{3}) 
$\varepsilon_{0}$(x10^{3}) 
$\lambda_{0}$ 
$\lambda_{1}$ 
$B$ 
R^{2}(%) 
$\lambda_{0}$ 
$\lambda_{1}$ 
$B$ 
R^{2}(%) 
SAPL01 
10482.38 
46.18 
5.8 
2.5 
1.00 
0.00 
2.43 
99.24 
1.02 
0.010 
2.35 
99.37 
SAPL02 
9392.17 
46.18 
7.00 
2.70 
1.00 
0.00 
2.03 
99.86 
1.01 
0.010 
1.96 
99.86 
SAPL03 
9506.12 
46.18 
7.3 
2.73 
1.00 
0.00 
1.83 
99.46 
0.99 
0.0001 
1.89 
99.58 
SAPL04 
9506.12 
46.18 
6.9 
3.4 
1.00 
0.00 
1.95 
99.60 
0.99 
0.01 
2.03 
99.73 
SAPL05 
13584.54 
46.18 
4.9 
2.1 
1.00 
0.00 
1.90 
99.71 
0.99 
0.010 
1.90 
99.74 
SAPL06 
12919.52 
46.18 
4.8 
2.6 
1.00 
0.00 
2.21 
99.49 
0.99 
0.015 
2.21 
99.61 
Table 3. Parameters used in the models for Iroko in the longitudinal direction

Mazar’s Equation 
FozaoFoudjet 2 

specimen 
${{E}_{0}}$(Mpa) 
${{\sigma }_{c\max }}$(Mpa) 
${{\varepsilon }_{c}}$(x10^{3}) 
${{\varepsilon }_{0}}$(x10^{3}) 
${{\lambda }_{0}}$ 
${{\lambda }_{1}}$ 
$B$ 
R^{2}(%) 
${{\lambda }_{0}}$ 
${{\lambda }_{1}}$ 
$B$ 
R^{2}(%) 
IROKL01 
6042.88 
46.18 
9.5 
6.00 
1.00 
0.00 
2.96 
99.94 
1.01 
0.01 
2.78 
99.94 
IROKL02 
8068.25 
42.76 
6.4 
4.4 
1.00 
0.00 
3.17 
99.79 
1.00 
0.01 
3.17 
99.77 
IROKL03 
9784.42 
42.76 
5.2 
3.7 
1.00 
0.00 
3.47 
99.68 
1.00 
0.01 
3.47 
99.71 
IROKL04 
8002.97 
42.76 
6.6 
3.1 
1.00 
0.00 
2.89 
99.69 
1.00 
0.01 
2.89 
99.75 
IROKL05 
9725.26 
46.18 
6.8 
3.1 
1.00 
0.00 
1.92 
98.88 
1.00 
0.01 
1.92 
99.06 
IROKL06 
10862.97 
42.76 
5.8 
2.6 
1.00 
0.00 
1.76 
98.86 
0.99 
0.01 
1.83 
98.99 
Table 4. Parameters used in the models for Sapelli in the radial direction

Mazar’s Equation 
FozaoFoudjet 2 

specimen 
${{E}_{0}}$ (Mpa) 
${{\sigma }_{c\max }}$(Mpa) 
${{\varepsilon }_{c}}$(x10^{3}) 
${{\varepsilon }_{0}}$(x10^{3}) 
${{\lambda }_{0}}$ 
${{\lambda }_{1}}$ 
$B$ 
R^{2}(%) 
${{\lambda }_{0}}$ 
${{\lambda }_{1}}$ 
$B$ 
R^{2}(%) 
SAPR01 
1424.60

13.68 
15.9 
4.8 
1.00 
0.00 
1.53 
94.99 
1.02 
0.01 
1.44 
95.76 
SAPR02 
1485.28 
13.68 
10.00 
4.5 
1.00 
0.00 
3.73 
90.72 
0.99 
0.01 
3.78 
91.77 
SAPR03 
1592.15 
13.68 
15.87 
4.4 
1.00 
0.00 
1.26 
97.10 
1.02 
0.01 
1.26 
97.51 
SAPR04 
1339.33 
11.11 
12.00 
4.9 
1.00 
0.00 
1.95 
99.31 
1.02 
0.01 
1.82 
99.59 
SAPR05 
1321.76 
11.11 
11.3 
5.3 
1.00 
0.00 
2.32 
97.21 
1.02 
0.01 
2.15 
97.80 
Table 5. Parameters used in the models for Iroko in the radial direction

Mazars Equation 
FozaoFoudjet 2 

specimen 
${{E}_{0}}$(Mpa) 
${{\sigma }_{c\max }}$(Mpa) 
${{\varepsilon }_{c}}$(x10^{3}) 
${{\varepsilon }_{0}}$(x10^{3}) 
${{\lambda }_{0}}$ 
${{\lambda }_{1}}$ 
$B$ 
R^{2}(%) 
${{\lambda }_{0}}$ 
${{\lambda }_{1}}$ 
$B$ 
R^{2}(%) 
IROK R01 
1227.35 
8.55 
13.6 
4.9 
1.00 
0.00 
0.91 
99.28 
1.01 
0.00 
0.86 
99.29 
IROK R02 
1531.48 
8.55 
9.20 
4.00 
1.00 
0.00 
1.29 
99.46 
1.02 
0.01 
1.20 
99.51 
IROK R03 
739.67 
8.55 
19.2 
8.4 
1.00 
0.00 
1.24 
99.38 
1.02 
0.01 
1.16 
99.36 
IROK R04 
1474.80 
8.55 
10.8 
4.4 
1.00 
0.00 
0.88 
98.19 
1.05 
0.01 
0.75 
98.49 
IROK R05 
1058.14 
8.55 
16.4 
5.8 
1.00 
0.00 
0.81 
99.69 
1.00 
0.00 
0.76 
99.73 
This work investigated the relative capacities of two damage models to describe the stressstrain relationship of wood materials subjected to cyclic compressive stresses. The models object of this study are the models of MazarPijaudier Cabot and the model of FozaoFoudjet 2. The experimental envelopes curves obtained from cyclic compression tests on the longitudinal and radial directions of the sapelli and the iroko were drawn and compared to the envelope curves proposed by each of the two models. The calculation of the coefficient of determination R^{2}for each of the models was also carried out.
The results show a good agreement between the experimental curves and the two mathematical models. For the FozaoFoudjet 2 models, this good agreement is obtained from the parameters λ_{0} and λ_{1} chosen as 0.00≤λ_{0}≤0.015 and 0.99≤λ_{1}≤1.05. The parameters of the model of FozaoFoudjet 2, retained for the modeling of the cyclic compression of wood are thus defined such as 0.00≤λ_{0}≤0.015 and 0.99≤λ_{1}≤1.05.
Moreover, for the parameters and chosen such as 0.00≤λ_{0}≤0.015 and 0.99≤λ_{1}≤1.05. The coefficient of determination for the FozaoFoudjet 2 model is greater than that of the MazarPijaudiercabot model for all of the two directions. This makes the FozaoFoudjet 2 model a better candidate for describing the behavior of wood material subjected to cyclic compressions.
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