The model is developed with the aim of describing as faithfully as possible the experimental observations made on the behavior of the wood material under creep stress. These observations are [1,2,3, 24,25,26]:

Instant elasticity,

Viscoelasticity,

Damage and rupture.
As the wood material is a biological material, its mechanical behavior is strongly influenced by its humidity level. This humidity level is also responsible for another phenomenon observed experimentally, namely mechanosorption. This phenomenon will not be taken into account in the present model. In addition, the humidity level will be assumed to be constant for our material.
Instantaneous elasticity
This is the instantaneous deformation of the material upon application of the stress. This deformation disappears entirely as soon as the stress is canceled. It is modeled by a purely elastic spring of stiffness E. The stress–strain relation is given by Hooke's law:
$$\sigma \left( t \right) = E\varepsilon \left( t \right).$$
(1)
Viscoelasticity
It is the growth of strain under constant stress. This deformation does not cancel instantly upon removal of the stress; it gradually disappears during the socalled recovery phase. The evolution of viscoelastic deformation is a function of the level of stress. [1, 3] In the literature, it is frequently modeled by simple or generalized fields of Kelvin–Voigt or Maxwell [14, 24]. One of our objectives being to take into account the existence of a finite limit in creep for low stress levels, and this with a reduced number of parameters, we carry our choice on the model of Kelvin–Voigt; because, Maxwell's body reflects the behavior of a fluid without the possibility of equilibrium for constant and nonzero stress [27]. The stress and strain relation for this model is given by:
$$\sigma \left( t \right) = E_{1} \varepsilon_{{\text{ve}}} \left( t \right) + \eta_{\text{1}} \dot{\varepsilon }_{{\text{ve}}} \left( t \right),$$
(2)
where the elastic modulus of the spring \(E_{1}\) is, \(\eta_{1}\) is the viscosity coefficient of the viscous pot, and \(\varepsilon_{{\text{ve}}} \left( t \right)\) is the viscoelastic strain.
Viscoplasticity
The works of Van der put, Gressel and Hasanni [5, 13, 14] show that wood, for certain stress thresholds has a viscoplastic behavior where the stress is a function of the strain rate, which strain does not cancel out when the stress disappears even after a long enough time, thus leaving a residual strain or even plastic strain. Plastic behavior is frequently modeled using a pad, and viscoplasticity by a series and/or parallel association of a pad and a damper.
One of our objectives being to take into account the existence of a threshold conditioning the birth of nonlinearity phenomena, we adopt a simple Bingham element, which is a parallel association of a damper and a pad. The stress strain relation for this model is:
$$\sigma = \eta^{{\text{vp}}} \dot{\varepsilon }_{{\text{vp}}} + \sigma_{\text{y}} ;{\text{ with }}\sigma \ge \sigma_{\text{y}} ,$$
(3)
where \(\eta^{{\text{vp}}}\) is the viscosity coefficient of the damper, \(\sigma_{\text{y}}\) is the threshold plastic stress, and \(\varepsilon_{{\text{vp}}}\) is the viscoplastic strain.
We have so far individually modeled each of the phenomena (elasticity, viscoelasticity, viscoplasticity) observed experimentally by single and/or combined elements. It is now a question of combining all these individual elements for the modeling of the general behavior of the material. The combination must, however, have a reduced number of parameters and be thermodynamically valid (possibility of partitioning the total deformation among others) in addition to having physically interpretable parameters and to describe fairly faithfully the behavior of the material. A series association of the spring, the Kelvin–Voigt body and the Bingham body leads to an extended standard solid body model (Fig. 1), [21].
This model admits the possibility of the partition of the deformation and presents a reduced number of parameters which are all physically interpretable. Note that if we take \(\sigma_{\text{y}} = 0\), we obtain the Burgers model, which is known [8, 13] for its ability to best describe primary creep and the onset of secondary creep. We will therefore adopt the model for the rest of our modeling.
The partition of the total deformation makes it possible to write:
$$\varepsilon = \varepsilon_{\text{e}} + \varepsilon_{{\text{ve}}} + \varepsilon_{{\text{vp}}} .$$
(4)
And the serial association makes it possible to write:
$$\sigma = \sigma_{\text{e}} = \sigma_{{\text{ve}}} = \sigma_{{\text{vp}}} .$$
(5)
Damage and failure
For fairly long durations, and stresses greater than a certain threshold, the creep deformations of the material experience an exponential growth which leads to the rupture of the material. This is the last phase of creep or tertiary creep. The literature offers several approaches for the description of this phenomenon. Authors such as Wang, Cai, Pierce et al. [13, 15, 16] have modified the components of the model by introducing a damage variable to explain tertiary creep.
This approach, although yielding significant results, encounters a certain number of difficulties, namely firstly the too large number of parameters and the difficulty in finding a physical interpretation of each parameter. The other difficulty remains the basic rheological model chosen for this modeling, namely the Burgers model. If the Burgers model is known for its ability to properly describe primary creep and the onset of secondary creep, it remains insufficient to describe the existence of a finite limit of creep strains under low stress levels.
However, any model which must serve as a basis for modeling damage and therefore tertiary creep must first be able to model as faithfully as possible the entire behavior of the material in the absence of damage. The model proposed by Reichel and Kaliske and adopted as a basic model within the framework of the present work captures very well the primary and secondary creep, and materializes well the finite limit for low levels of stresses.
However, for the description of tertiary creep, they used an approach based on strain energy density. This approach measures the strain energy \(e\left( t \right)\) defined by:
$$e\left( t \right) = \int\limits_{0}^{t} {\sigma \dot{\varepsilon }dt} .$$
(6)
This strain energy measured at each increment of the strain is compared with a threshold value characteristic of each material, threshold value beyond which the material breaks. This approach has been shown to be effective for the description of creep failure in that it allows a failure criterion to be defined. However, according to Becker [18], it remains insufficient to describe the rapid growth of strain during tertiary creep and is therefore unable to capture tertiary creep. To solve this problem, Reichel and Kaliske modified the viscoplastic viscosity coefficient by adding to it a factor governed by an arctangent function which decreases this coefficient when the strain increases. This approach, although having led to the capture of tertiary creep, does not take into account the phenomenon of damage, damage which however has a fully established responsibility in the occurrence of tertiary creep [2, 25, 26]. Indeed, the rapid growth of deformation during tertiary creep is due to the progressive decrease in rigidity, itself caused by the multiplication of gaps within the material, and therefore by damage. It therefore appears more reasonable and more realistic to take into account a damage variable in the modeling of tertiary creep.
Moreover, this approach used by the authors introduces two parameters in the tangent arc function, parameters which are not easily accessible.
The originality of our approach in our modeling consists in modifying the component of the rheological model responsible for Viscoplasticity, by introducing a damage variable to better describe the rapid growth of the strain during tertiary creep, and thus model the tertiary creep. Our approach will make it possible to take into account the damage phenomenon so observed in the tertiary phase of creep, and thus lead to a model that is more faithful to reality and at the same time simpler. This modification will be made based on a set of assumptions all based on experimental observations.
Hypothesis 1
The damage begins after a certain threshold plastic deformation. It has been observed that the initiation of the damage is consecutive to the large plastic deformations [25, 26]. There is therefore a threshold of plastic deformation conditioning the onset of the damage phenomenon. The damage variable therefore acts on the component of the solid responsible for viscoplasticity. We therefore assume that there is a defined threshold value \(\varepsilon_{D}\) such as:
$$\left\{ {\begin{array}{*{20}c} {\dot{D} \ge 0,\forall \varepsilon \ge \varepsilon_{D} } \\ {\dot{D} = 0,\forall \varepsilon \prec \varepsilon_{D} } \\ \end{array} } \right..$$
(7)
Hypothesis 2
The damage affects the coefficient of viscosity. The rapid growth of strain during tertiary creep is due to the gradual decrease in the coefficient of viscosity; the decrease is caused by the loss of rigidity, itself caused by the multiplication of gaps in the material, and therefore by damage. We therefore consider that the coefficient of viscosity of our material experiencing damage \(D\) is given by:
$$\tilde{\eta }^{{\text{vp}}} = \eta_{0}^{{\text{vp}}} \left( {1  D} \right).$$
(8)
Hypothesis 3
The damage, once initiated knows an exponential growth, according to the stress level. We therefore admit that the damage is given as a function of time and of the stress by the relation:
$$D\left( t \right) \, = \, 1 \,  \, \exp \left( {  \beta \sigma \left( {t  t_{0} } \right)} \right),$$
(9)
where \(t_{0}\) is the instant at which the damage begins, and \(\beta\) a parameter conditioning the effect of the stress on the damage. The form of Eq. (9) is chosen to ensure that the damage evolves in the interval \(\left[ {0,1} \right],\) [27]. The instant \(t_{0}\) is defined as:
$$\varepsilon \left( {t_{0} } \right) = \varepsilon_{D} .$$
(10)
By putting Eqs. (8) and (9) in Eq. (3), we obtain the following modified rheological model (Fig. 2), with \(D\) given by Eq. (9).
Expression of the total strain
The total strain \(\varepsilon \left( t \right)\) is given by relation (4) respecting the hypothesis of the partition of the total strain, in elastic, viscoelastic, and viscoplastic strain.
The elastic deformation is obtained from Eq. (1) taking into account relation (5) and is given by:
$$\varepsilon_{\text{e}} = \frac{\sigma }{{E_{\text{e}} }}.$$
(11)
The viscoelastic strain is obtained by solving Eq. (2) and is given by:
$$\varepsilon_{{\text{ve}}} = \frac{\sigma }{{E_{1} }}\left(1  \exp \left(  \frac{{E_{1} }}{{\eta_{1} }}t\right)\right).$$
(12)
The development of the viscoplastic strain being dependent on the stress level, to express them, we consider the situation where the stress is greater than the plasticity threshold. So \(\sigma \ge \sigma_{\text{y}}\); thus, we can distinguish two situations. The first situation is where the plastic deformations evolve, but have not yet reached the threshold causing the damage to start. In this case, and in accordance with relation (7) obtained from hypothesis 1, Eq. (3) allows us to write:
$$\dot{\varepsilon }_{{\text{vp}}} = \frac{{\sigma  \sigma_{\text{y}} }}{{\eta^{{\text{vp}}} }}.$$
(13)
An integration of (14) makes it possible to write:
$$\varepsilon_{{\text{vp}}} = \left( {\frac{{\sigma  \sigma_{\text{y}} }}{{\eta^{{\text{vp}}} }}} \right)t + c.$$
(14)
In Eq. (14), the integration constant \(c\) is determined by considering the condition:
$$t = 0,\varepsilon_{{\text{vp}}} = 0.$$
(15)
This gives:
The viscoplastic deformation before the onset of damage therefore has the following expression:
$$\varepsilon_{{\text{vp}}} = \left( {\frac{{\sigma  \sigma_{\text{y}} }}{{\eta^{{\text{vp}}} }}} \right)t.$$
(17)
The second situation is when the plastic deformations have evolved to the point of causing the onset of damage. Thus, Eqs. (3), (8) and (9) allow us to write:
$$\sigma = \eta^{{\text{vp}}} (1  D)\dot{\varepsilon }_{{\text{vp}}} + \sigma_{\text{y}} .$$
(18)
From Eq. (18), we get the expression for \(\dot{\varepsilon }_{{\text{vp}}}\), which gives us:
$$\dot{\varepsilon }_{{\text{vp}}} = \frac{{\sigma  \sigma_{\text{y}} }}{{\eta^{{\text{vp}}} (1  D)}}.$$
(19)
By replacing the damage \(D\) in Eq. (19) by its value given by Eq. (9), we get:
$$\dot{\varepsilon }_{{\text{vp}}} = \frac{{\sigma  \sigma_{\text{y}} }}{{\eta^{{\text{vp}}} (\exp (  \beta \sigma (t  t_{0} ))}}.$$
(20)
A rewrite of (20) gives us:
$$\dot{\varepsilon }_{{\text{vp}}} = \frac{{(\sigma  \sigma_{\text{y}} )}}{{\eta^{{\text{vp}}} }}\left[ {\exp (\beta \sigma (t  t_{0} ))} \right].$$
(21)
An integration of (21) gives us:
$$\varepsilon_{{\text{vp}}} = \frac{{\sigma  \sigma_{\text{y}} }}{{\beta \sigma \eta^{{\text{vp}}} }}\left[ {\exp (\beta \sigma (t  t_{0} ))} \right] + \varepsilon_{{\text{vp0}}} .$$
(22)
The integration constant \(\varepsilon_{{\text{vp0}}}\) is determined by looking at the instant (t_{0}) when the damage begins. We have from Eq. (17):
$$\varepsilon_{{\text{vp}}} \left( {t = t_{0} } \right) = \left( {\frac{{\sigma  \sigma_{\text{y}} }}{{\eta^{{\text{vp}}} }}} \right)t_{0} .$$
(23)
To ensure the continuity of the deformation to the left and to the right of the instant when the damage is initiated, we have, starting from Eqs. (22) and (23):
$$\left( {\frac{{\sigma  \sigma_{\text{y}} }}{{\eta^{{\text{vp}}} }}} \right)t_{0} = \frac{{\sigma  \sigma_{\text{y}} }}{{\beta \sigma \eta_{{\text{vp}}} }} + \varepsilon_{{\text{vp0}}} .$$
(24)
From relation (24), we obtain the integration constant \(\varepsilon_{{\text{vp0}}}\), which gives:
$$\varepsilon_{{\text{vp0}}} = \frac{{\sigma  \sigma_{\text{y}} }}{{\beta \sigma \eta_{{\text{vp}}} }}\left[ {\beta \sigma t_{0}  1} \right].$$
(25)
By inserting the constancy \(\varepsilon_{{\text{vp0}}}\) obtained in (24) into Eq. (22), we finally have the expression of the viscoplastic strain taking into account the damage phenomenon as follows:
$$\varepsilon_{{\text{vp}}} (t) = \frac{{\sigma  \sigma_{\text{y}} }}{{\beta \sigma \eta_{{\text{vp}}} }}\left[ {\exp (\beta \sigma (t  t_{0} ) + \beta \sigma t_{0}  1} \right].$$
(26)
Thus, the total deformation is given from Eq. (4) by:
\(\varepsilon = \varepsilon_{\text{e}} + \varepsilon_{{\text{ve}}} + \varepsilon_{{\text{vp}}} .\) This gives, by grouping the results (11), (12), (17) and (26):
$$\varepsilon (t) = \left\{ {\begin{array}{*{20}l} {\frac{\sigma }{{E_{\text{e}} }} + \frac{\sigma }{{E_{1} }}(1  \exp (  \frac{{E_{1} }}{{\eta_{1} }}t)),\forall \sigma \le \sigma_{\text{y}} ;(a)} \\ {\frac{\sigma }{{E_{\text{e}} }} + \frac{\sigma }{{E_{1} }}(1  \exp (  \frac{{E_{1} }}{{\eta_{1} }}t)) + (\frac{{\sigma  \sigma_{\text{y}} }}{{\eta^{{\text{vp}}} }})t,\forall \sigma \succ \sigma_{\text{y}} ,et,\forall t \le t_{0} ;(b)} \\ {\frac{\sigma }{{E_{\text{e}} }} + \frac{\sigma }{{E_{1} }}(1  \exp (  \frac{{E_{1} }}{{\eta_{1} }}t)) + (\frac{{\sigma  \sigma_{\text{y}} }}{{\eta^{{\text{vp}}} }})t + \frac{{\sigma  \sigma_{\text{y}} }}{{\beta \sigma \eta_{{\text{vp}}} }}\left[ {\exp (\beta \sigma (t  t_{0} )) + \beta \sigma t_{0}  1} \right],\forall \sigma \succ \sigma_{\text{y}} ,et,\forall t \ge t_{0} .(c)} \\ \end{array} } \right..$$
(27)
The system of Eq. (27) describes the dynamics of the rheological model that we propose for modeling the damage of wood material under creep stress.
This model that we propose requires a total of five independent parameters all physically interpretable and easily accessible (\(E_{\text{e}}\),\(E_{1}\),\(\eta_{1}\),\(\eta_{{\text{vp}}}\) and \(\beta\)), which is low, compared to the model proposed by Kaliske et al. [21] which has a total of six independent parameters. The model captures the rapid change in strain during tertiary creep while accounting for damage.
Determination of the model parameters
The parameters,\(E_{\text{e}}\), \(E_{1}\), \(\eta_{1}\) and \(\eta_{{\text{vp}}}\) can be determined through low stress creep tests. The parameter \(\sigma_{\text{y}}\) representing the plasticity threshold is determined experimentally through creeprecovery tests under high stresses. Its empirical value proposed in the literature is 35 to 50% of the breaking stress of the material [1, 3, 5]. The parameter \(\beta\) conditioning the evolution of the damage and the plastic strain’s rate can be determined from the strain rate of tertiary creep. It is given by the slope of the logarithm of the tertiary creep strain rate. Indeed, starting from Eq. (21) we have:
$$\begin{aligned} \ln (\dot{\varepsilon }_{{\text{vp}}} ) = & \ln \left( {\frac{{(\sigma  \sigma_{\text{y}} )}}{{\eta^{{\text{vp}}} }}\left[ {\exp (\beta \sigma (t  t_{0} ))} \right]} \right) \\ =& \beta \sigma (t  t_{0} ) + \ln \left( {\frac{{\sigma  \sigma_{\text{y}} }}{{\eta^{{\text{vp}}} }}} \right). \\ \end{aligned}$$
(28)
Relation (28) gives the logarithm of the tertiary creep strain rate. As shown by this relation (28), the logarithm of the tertiary creep rate is a line whose slope is the product of the stress by the parameter \(\beta\). From this slope, we therefore deduce the parameter \(\beta\) by:
$$\beta = \frac{\tan \alpha }{\sigma },$$
(29)
where \(\tan \alpha\) is the slope of the line representing the logarithm of the tertiary creep rate.
The parameter \(t_{0}\) depends on the stress \(\sigma\) and the plastic deformation \(\varepsilon_{D}\) damage threshold and can be obtained through relations (10) and (27b), which allow us to write:
$$\varepsilon_{D} = \varepsilon (t_{0} ) = \frac{\sigma }{{E_{\text{e}} }} + \frac{\sigma }{{E_{1} }}(1  \exp \left( {  \frac{{E_{1} }}{{\eta_{1} }}t_{0} } \right) + \left( {\frac{{\sigma  \sigma_{\text{y}} }}{{\eta^{{\text{vp}}} }}} \right)t_{0} .$$
(30)
The numerical resolution of Eq. (30) provides the parameter \(t_{0}\).
One of the essential points for determining the parameters of the model is the determination of the instant \(t_{0}\) from which the damage begins. In general, in the models proposed for the description of the failure by creep of materials, the moment \(t_{0}\) from which the damage begins is very often ignored, in favor of the moment \(t_{\text{r}}\) of failure. Indeed, authors are generally more interested in the moment \(t_{\text{r}}\) of failure than the moment of onset of damage. It seems important to us to be equally interested in the moment at which the damage begins in the modeling of failure by creep, because for certain materials, like brittle materials, the failure is so fast or sudden, that one can easily think that the moment of the beginning of the damage is almost the same one as that of rupture. This moment being linked to the threshold deformation \(\varepsilon_{D}\) of damage by relation (30), it is necessary and sufficient to determine the threshold deformation of damage \(\varepsilon_{D}\).
When the damaging stresses (\(\sigma \ge \sigma_{\text{y}}\)) are kept constant, the creep strain can only increase. This growth necessarily begins with the value: \(\varepsilon_{0} = \frac{\sigma }{{{\rm E}_{\text{e}} }}\) which is the instantaneous elastic strain. The strain field of creep failure is therefore:
$$\varepsilon \in \left[ {\varepsilon_{0} , + \infty } \right].$$
(31)
The onset of damage being consecutive to the large plastic deformations, it is evident that the threshold damage deformation is included in the universe of deformations defined by relation (31). It is therefore logical to think that any increment \(\Delta \varepsilon\) of strain starting from the value \(\varepsilon_{\text{0}} = \frac{\sigma }{{{\rm E}_{\text{e}} }}\) results in a total strain value which may be a good candidate to represent the threshold strain for damage. One of the possible increments of strain is:
$$\Delta \varepsilon = \frac{{\sigma_{\text{y}} }}{{{\rm E}_{\text{e}} }}.$$
(32)
The choice of this increment is motivated by the need to propose a model whose parameters depend on the easily accessible mechanical characteristics of the materials (modulus of elasticity, breaking load). This increment leads to a total strain value given by:
$$\varepsilon = \varepsilon_{0} + \Delta \varepsilon .$$
(33)
Relation (34) allows us to write:
$$\varepsilon = \frac{\sigma }{{{\rm E}_{\text{e}} }} + \frac{{\sigma_{\text{y}} }}{{{\rm E}_{\text{e}} }}.$$
(34)
The total strain given by relation (34) is therefore a good candidate to represent the value of the threshold damage strain. For the validation of our model, we adopt the value of the deformation of the relation (34) as the value of the threshold deformation of damage. Thus, we have:
$$\varepsilon_{D} = \frac{{\sigma + \sigma_{\text{y}} }}{{E_{\text{e}} }}.$$
(35)
The choice of this value is justified by the need to take into account the fact that the origin and development of damage strongly depend on the stress level and the mechanical characteristics of the material (elastic modulus, breaking load).