Investigation of Nonlinear Creep Behaviour of Millettia Laurentii Wood Through Zener Fractional Rheological model

Nowadays one of the principal difficulties that wood structural development and construction have to face is wood creep. Nevertheless, the secret to master and solve the creep deformation of wood relies on a sensible and exact rheological model for numerical analysis. In this research work our goal is to study the nonlinear creep behaviour of the Cameroonian wood species Millettia Laurentii known as Wengé wood through fractional calculus approach. So, we have conducted a nonlinear creep constitutive model of Millettia Laurentii wood, that is the Zener fractional rheological model, and the parameters of this model have been determined. We have studied the influence of stress level σ and fractional order n on the Millettia Laurentii wood creep process by a sensitivity analysis of the model parameters. The outcomes of this sensitivity analysis are of paramount importance because they can be used in reality to inspect the creep process and deformation amount of Millettia Laurentii wood in practical engineering. Moreover, guidance for the secure construction of Millettia Laurentii wood engineering can be given by the means of the findings of this research.


Introduction
The phenomenon in which wood deformation increases with time under long-term external load is known as wood creep. The development of creep increases the loss of stress in the structure of wood.
It also redistributes the internal force of static and statically indeterminate structures, resulting in excessive structural deformation, a significant reduction in overall strength, and even a loss of bearing capacity [1]. The internal energy of wood structures is also redistributed by this phenomenon, as a result: the deformation of the structure is increased, the strength is significantly reduced, and the bearing capacity of the structure is negatively impacted [1]. With the increasingly use of wood materials in civil engineering structures, wood creep appears as one of the principal issues that affect the development of wood structures, the security of wood constructions and their long-term stability [2][3][4][5][6]. In the particular case of wood material, the nonlinearity in wood behaviour can be observed at both lower and higher stress levels [7]. A lot of linear models have been proposed in order to simulate creep behaviour of wood [8][9][10][11][12][13]. Meanwhile, the study of nonlinear creep behaviour of wood is very complex because this material is assimilated to a certain extend as a composite material. The basic Maxwell model is referenced to propose the linearized mathematical modelling of the geometric nonlinearity theory [14 -16]. But wood creep cannot be described accurately with the mentioned model. Some research works have been devoted to the nonlinear creep behaviour of wood under high stresses [17][18][19][20][21]. Fractional derivative is a temporal differentiation operator that can allow following the evolution of a function varying with time [22]. The theory of fractional calculus is widely used in the domain of material sciences [23][24], and fractional derivative has been introduced to establish a viscoelastic rheological model that simulates the viscoelastic behaviour and the mechanical response of a material [25][26][27][28]. Millettia Laurentii (Wengé) wood is one of the oldest building materials used by humans in the south region of Cameroon and in many sub-Saharan African countries. Some research works have been devoted to the study of mechanical and physical properties of this wood [29,30], but this remains insufficient to well characterise this wood material. Therefore, further studies are necessary to promote this material in the modern world. So, it is urgent to study its creep behaviour to have an idea about the load bearing capacity of Millettia Laurentii (Wengé) wood. To achieve this goal in the current research work, the fractional rheological model of Zener is proposed.

The spring-pot
Let's use the fractional calculus operator, then equation (1) below is the constitutive equation of the Spring-pot ( fig. 1b).
where η n is the viscosity coefficient, n the order of the fractional derivative, and indicates fractional differentiation.
where D indicates differentiation and the operation −1 can be expressed as a Riemann-Liouville fractional integral, i.e., with Γ the gamma function. where εe and εve are the strains of Hooke body and viscoelastic body, respectively.
For the Hooke body, the constitutive relation is given by where E0 is the elastic modulus as shown in fig. 2 and σ the applied stress.
For the viscoelastic body, the constitutive relation is given by: where E stands for the elastic modulus, and η n is the viscosity coefficient of the spring-pot.
By substituting Eqs. (6) and (7) into Eq. (5), we obtain the total creep strain of the time-based fractional derivative model shown in fig. 2 as below:

Experimental setup
The efficacy of the fractional derivative model is dependent on its ability to adequately fit experimental data. The current experiments were carried out at Dschang University (Cameroon) using a four points flexural test machine (Fig. 4) coupled with a strain-bridge possessing a high accuracy.
The indoor temperature was 23 o C and the relative humidity was 65% during all the process. All the wood samples were extracted from the same billet of Millettia Laurentii wood, originating from Kyé-Ossi natural forest in Cameroon south region. The specimens were prepared with a required dimension of 20mm×20mm×360mm ( fig. 3).
During the test the sample (Fig. 3) is laid on the test machine in such a way that one gauge is on the top measuring the traction of the wood fibers and another one symmetrically on the opposite face of the sample (Fig. 4) evaluating the compression of the wood fibers. The wood specimens were tested under four points flexural loading following the French Norm NF B 51-003 that labels general requirements for physical and mechanical tests.

Determination of parameters of fractional derivative model
The parameter E0 is the Young modulus, it is calculated by the means of initial instantaneous elastic strain at the loading time of the wood sample. The expression of E0 is as follows: where σ is the initial stress, and ε0 stands for the initial instantaneous elastic strain. .

Figure 4 : Creep test machine
In this work the model parameters E, η and n were determined according to the Levenberg-Marquardt algorithm; which is an optimization method whose the coming down direction is a combination between the directions of the gradient and Newton-Gauss algorithms. The main advantage of this optimization method being that the out coming optimized model parameters is of high accuracy.  (2) the isovelocity creep stage with constant strain rate.

Verification of
The experimental curve characteristics can be described by using the elastomer and viscoelastic body in the model. Now, the parameters E0, E, η and n need to be solved. The Levenberg-Marquardt algorithm; which is an optimization method is carried out using the conducted model. The calculated model parameters are shown in Table 1, and the fitting results are shown in Figure 6. Accelerate creep has not occurred when the axial pressure is 38,06 MPa and 43,03 MPa, the wood creep process consists of deceleration and isometric creep.

Influence of the fractional derivative order n
To study the influence of the fractional order, the variable control method is made use to ensure that other parameters remain fixed. The fractional order is progressively incremented from 0,357 to 0,369 by 0,003 step resulting in a series of creep curves with different orders, as figure 10 shows below. From figure 10, it is straightforward that increasing the fractional order increases the duration of primary and secondary creeps. The deformation of Millettia Laurentii decreases in proportion as the fractional order is incremented. In addition, creep rate is influenced by the fractional order; that is, the greater the fractional order, the smaller the creep rate at each level. This observation is the main finding of this experimental work, since it is not consistent with the outcomes of Zhou et al. [31,32] concerning the effect of fractional order. As a matter of fact, Zhou et al. [31,32] discovered that increasing fractional order resulted in a growth of the material deformation.

Conclusion
In this research work, fractional calculus theory has been applied to better study the nonlinear creep of Millettia Laurentii wood while the fractional order is rather negatively correlated with the deformation and the creep rate of this wood species. The deformation and the creep rate will grow with the increase of σ whereas they will have rather a decreasing shape with the increase of n. In future prospects, the stress levels employed in this work were not up to the third of ultimate breaking stress of Millettia Laurentii wood, the study of nonlinear behavior of this wood species under higher stress levels will be full filled in the upcoming research works. The proposed model in this work will undergo a modification with an element that will take into account the evolution of the material towards failure point.  Sample carrying two gauges.

Figure 4
Creep test machine Figure 5 The curve of the wood creep tests under different stresses    Effect of stress level on creep process of Millettia Laurentii Figure 10 In uence of the fractional order on creep process of Millettia Laurentii