# Modeling of Hygro-Mechanical Behaviour through Raffia vinifera Fiber during the Water Diffusion Phenomenon

Authors: Harrond Nimjieu Takoudjou; Nicodème R. Sikame Tagne; Peguy R. Nwagoum Tuwa; Médard Fogue; Ebenezer Njeugna
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In an industrial context where the use of friendly materials is encouraged, natural fibers of vegetable origin become more solicited for the reinforcement of composite materials. This work deals with the modeling of the hygro-mechanical behavior through raffiavinifera fiber during the diffusion phenomenon. The modeling of water diffusion through the raffia vinifera fiber is described by Fick’s second law and taking into account the swelling phenomenon of the fiber. The equation obtained is solved numerically by the finite difference method, and the evolution of the fiber radius as a function of time is obtained. By applying the Leibniz integration rule, a mathematical expression to predict the evolution of this radius as a function of time is proposed. It is observed numerically and analytically an increase of the dimensionless fiber radius with time up to a critical value after what one observes the saturation. This model allowed us to propose a mathematical model describing the absorption kinetics of the raffia vinifera fiber through its absorption ratio. By comparing the results of this model with the experimental results from the literature, one observes a good agreement. Moreover, the induced stresses in the fiber during the water diffusion can also be estimated with the proposed mathematical model expression of fiber. These stresses increase with time and can reach between 5 and 7 GPa. The results of this work can be used to predict the behavior of the raffia vinifera fiber inside a composite material during its development.


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## Body

## 1. Introduction

Raffiavinifera is a nontimber forest product found in the Americas, Asia, and Africa [1]. This plant is fast-growing and found in swampy zones and along rivers. The stem of this palm (petiole) consists of a cork or pith surrounded by a hard and smooth shell that plays a protective role. These stems have a green coloration when fresh and grey when dry [2, 3]. They have a length of 5 to 10 m and a diameter between 2 and 10 cm [2, 3]. Cork consists of about 35.8% to 54.5% fibers [4] aligned lengthwise in a natural matrix. The mechanical properties of this natural matrix were determined by [4]. Studies on the anatomy and physical properties of raffia vinifera bamboo have shown that the physical properties of this material approach those of lightweight wood [5]. The low value of the diffusion coefficient of the fibers from the leaves [6] justifies the use of raffia leaves as a roofing material by poor populations. The thermal properties of raffia bamboo make it a good insulator [2]. It also has the ability to store heat and release it at the appropriate time. It is surely for this reason that raffia bamboo is used to make granaries for the storage of agricultural products by the populations of West Cameroon. With this property, the raffia bamboo can be used as a temperature regulator in habitats.Over the past decade, researchers have studied different components of raffia plants for various applications. For instance, Talla et al. [7, 8] presented, respectively, in 2004 a static model of the breaking strength of raffia vinifera under compression and in 2005 a static model of the breaking strength of raffia vinifera under bending. In 2007, they underline that raffia vinifera behaves like a composite [9]. The mechanical properties in flexure, and at constant moisture content, of dry raffia cork [3, 4], dry shell, and dry bamboo [3] were investigated. The water diffusion phenomenon was studied through a fiber [10, 11] and through cork [12] of raffia vinifera. The mechanical [13] and viscoelastic [14] properties of raffia fibers from the stem show that these fibers can be well used in the field of composite materials; in addition, they have an added advantage due to their high elongation [15] compared to flax and sisal fibers already used in the automotive industry. The realization of such a composite uses polymeric matrices in the form of paste whose water can be absorbed by the fiber during the process of shaping the composite. It is therefore necessary to control the physicochemical behavior of the fiber in order to better predict the mechanical behavior of the final composite. In the previous work carried out by Sikame [10, 11], the radius of the raffia fiber is assumed to be constant throughout the experiment. However, it is known that cellulosic materials, in general, are the excellent seats of shrinkage and swelling phenomena (increase in volume when they absorb water and decrease in volume when they dry). Thus, during the molding of the composite, the fiber absorbs water, and its volume increases (Figure 1). During the drying process, it loses water, shrinks, and can become detached from the matrix (delamination of the composite [16]). Therefore, the composite is defective without even being used. The study of the radius variation of the raffia vinifera fiber becomes crucial.Figure 1
A schematic of the physical model of the swelling of raffia fibers.Problems, where the domain boundary varies with time (moving surface), are encountered in several industries. According to the literature [17], this type of problem was first studied by J. Stefan in 1891. Since the boundary of the domain varies over time, its location at any date is a part of the solution of the study problem. Typically, the problem is solved numerically because of the nonlinearity introduced by the boundary shift [17–19]. The complexity of the problem is also related to the geometry of the moving surface, which is why most authors only study simple unidirectional cases in plane geometry [17, 20].To the best of our knowledge, almost all the works done on raffia fiber are experimental. There are almost no works oriented on the modeling. Moreover, in all the work done on raffia fiber so far, there are no works done on the study of the swelling or the hygro-mechanical behavior of the raffia fiber during the diffusion phenomenon. The objective of this work is to predict the hygro-mechanical behavior of raffia fibers during the water diffusion phenomenon. The objective of this work is to predict the hygro-mechanical behavior of raffia fibers during the water diffusion phenomenon. The fiber surface being in motion, the problem is nonlinear and is solved numerically by the finite difference method and implemented in the MATLAB software. The following simplifying assumptions are assumed:The physical and chemical properties of the fiber are constantAny variation of the quantity of water in the fiber induces a variation of the volume of the fiberThe diffusion of water in the fiber is done at constant temperature and the displacement of the surface is free (not constrained)This work is organized as follows: In Section2, the water diffusion through the raffia vinifera fiber is first modeled by taking into account the swelling of the fiber. Secondly, a mathematical model giving the evolution of the fiber radius as a function of time is proposed. Then, a mathematical model describing the absorption kinetics of the fiber is also proposed. Finally, the stresses induced in the fiber are determined. Section 3 presents the results obtained, and Section 4 presents the summary and conclusion.


## 2. Modeling

The mechanical and physicochemical properties of cellulosic materials vary depending on the sampling zone. For the study of raffiavinifera, there are generally 12 sampling zones as described by [10–12]: four zones in the longitudinal direction and three zones in the radial direction. Figure 2 illustrates the situation. These different zones will be considered in this work.Figure 2
Location of sampling zones along the stem: (a) longitudinal position and (b) cross-section relative to a given longitudinal position [10, 11].
(a)(b)


### 2.1. Modeling of Water Transfer through the Raffiavinifera Fiber

Mass transfer through a solid is described according to Fick’s second law [21, 22] by the following equation, where C (in mol/m3) is the concentration of the diffusing molecule and D (in m2/s) is the diffusion coefficient:(1)∂C∂t=div−Dgrad⟶C.Despite the fact that the raffia fiber has an elliptical section [13], it is assimilated in this work to a cylinder of initial radiusR0. Depending on whether the fiber loses or absorbs water, it becomes either shrunken or swollen, and its radius varies. Let R=Rt, therefore, be the radius of the fiber at timet>0. The water diffusion through the raffia fiber is described by equation (1) in cylindrical coordinates r,θ,z as follows:(2)∂Cr,t∂t=1r∂∂rrD∂Cr,t∂r+∂∂θDr∂Cr,t∂θ+∂∂zrD∂Cr,t∂z,0≤r≤R.Since the problem is cylindrical symmetry, the diffusion along the angleθ is neglected. On the other hand, the length of the fiber being very large compared to the diameter of the fiber, the fiber can be considered as an infinite cylinder, and the diffusion along the z-axis is assumed to be constant. Equation (2) is written in cylindrical coordinates following the radial coordinate r alone. A similar assumption was done by Dejam [23].(3)∂Cr,t∂t=Dr∂∂rr∂Cr,t∂r,0≤r≤Rt,t>0,with the initial conditions and the following boundary conditions:(4)Cr,t=C0fort=0and0<r≤RtCr,t=Cstfort>0andr=Rt,whereC0 is the initial concentration and Cst the surface concentration.


### 2.2. Modeling of the External Surface of Raffiavinifera Fiber

The observation of a raffia fiber by the scanning electron microscope (Figure3(a)) [15] shows that it is surrounded by an envelope of small thickness e (Figure 3(b)), which protects it against attack and convection of fluids. Diffusion through this envelope is modeled by applying two boundary conditions: the continuity of the water concentration and flux at the interface between the fiber and its envelope on the one hand and at the interface between the envelope and the external environment on the other hand. Such boundary conditions were used by Dejam to model the Shear dispersion in a capillary tube with a porous wall [23].Figure 3
(a) Cross-section of a raffia fiber given by the scanning electron microscope [15]. (b) Sketch of the external surface of the fiber.
(a)(b)At the interface, there is a coefficient determining the distribution of quantities [22, 24, 25]. Let σf and σenv be the distribution coefficients of the quantities at the fiber/envelope and envelope/external environment interface, respectively. We have(5)σf=CRt,tCenv−=CstCenv−,σenv=Cenv+Cet,where Cenv− and Cenv+ are the concentration of the envelope at the fiber/envelope and envelope/external environment interface, respectively, and Cet is the concentration of the external environment. Given that, relation (5) can still be written as follows:(6)Jenv⟶.n⟶=−DenvCenv+−Cenv−e=Denve.σfCst−σfσenvCet.At the interface, the continuity of the flow allows to write(7)−D∂Cr,t∂r|Rt=−Denv∂Cenvr,t∂r|Rt=pCst−σCet,where p=Denv/e.σf and σ=σfσenv.Relation 8 is analogous to the convective flow through a surface in the thermal context [21, 22]. Thus, the factor p (in m/s) introduced translates the permeability of the fiber, and we distinguish two limit cases according to the values ofp:Ifp=0 (impermeable surface), no flow through the fiber, and relation (7) is written as: −D∂Cr,t/∂r|Rt=0Ifp=∞, relation (7) is written as: CRt,t=Cst=σCetThe absorption of water molecules by the surface of fibers can be a kinetics reaction. Assuming that this reaction is a first-order reaction [18, 24, 26], mass conservation is written as follows:(8)∂Cst∂t=kCmax−Cst,whereCmaxis the maximum concentration of water in the fiber at saturation and k has the dimension of the inverse of time and represents a kinetic constant.


### 2.3. Modeling of RaffiaVinifera Fiber Swelling

Consider an elementary surfacedS of the fiber that moves with a speed v⟶, from a point at date t to another point at date t+dt. In the case of absorption, this displacement results from an excess of water in the fiber of volume Ve which diffuses through the fiber. The variation of the volume of the fiber corresponds to the volume of water absorbed by the fiber. Thus, (9)VT=V0+Ve,where VT,V0, and Ve denote, respectively, the total volume of the fiber, the initial volume of the fiber, and the volume of water absorbed by the fiber; it follows that(10)πR2t=πR02+∫0RtCr,t.2πrdr.Deriving this relation with respect to time and by applying the Leibniz integral rule [27, 28], it comes that(11)D∂Cr,t∂r|Rt=1−CstdRtdt.The details of this calculation are given in the appendix.The mathematical expression of the radius of the fiber as a function of time is obtained by placing equation (7) in equation (11), and it follows that(12)dRtdt=pCet−Cst1−Cst.Considering the case whereCet=cte=Cmax, the integration of the previous relation leads to(13)Rt=pkln1−C01−Cst+R0.Equation (13) is one of the main results of this work.


### 2.4. Determination of the Water Concentration

In order to facilitate the manipulation of the equations, it is preferable to make the physical quantities involved dimensionless. For this purpose, as in [23, 29], let us say(14):τ=DtR02,ϒ∗=ϒR0,R∗τ=RtR0<9.Using the chain rule, equations (3) and (11) lead to the following equations, respectively:(15)∂Cr∗,τ∂τ=1r∗∂∂r∗r∗∂Cr∗,τ∂r∗,0≤r∗≤R∗τ,τ>0,(16)∂Cr∗,τ∂r∗|R∗=1−CR∗,τdR∗τdτ,τ>0.


#### 2.4.1. Immobilization of the Domain Boundary

The fiber boundary varies with time. To simplify the numerical solution of equations (14), we transform them into an equivalent problem on a fixed domain. A new spatial variable is defined as follows [17, 18]:(17)X=r∗R∗τ=rRt,0≤X≤1.In writingφX,τ=φr∗/R∗τ,τ≡Cr∗,τ and using the chain rule, equations (15) and (16) lead to the following equations, respectively:(18)∂φ∂τ−r∗R∗τ2∂R∗τ∂τ∂φ∂X=1XR∗τ2×∂∂XXR∗τ.1R∗τ∂φ∂X,(19)1R∗τ∂φ∂X|X=1=1−φX,τ|X=1dR∗τdτ.Multiplying relation (18) by R∗τ2and introducing a new variable Zτ=1/2R∗τ2 as in [18], we obtain the following relation:(20)2Zτ∂φ∂τ=1X∂∂XX∂φ∂X+XdZτdτ∂φ∂X,0≤X≤1,τ>0,where(21)dZτdτ=11−φX,τ|X=1∂φ∂X|X=1,τ>0.


#### 2.4.2. Discretization

To solve equation (20), we will use the finite difference method. For this, equation (20) is written as follows:(22)2Zτ∂φ∂τ=∂²φ∂X²+1X∂φ∂X+XdZτdτ∂φ∂X.The spaceX is divided into n intervals of length ΔX, and a time step Δτ is chosen. The approximation of φX,τ at a point Xiand at time τk is denoted φikwith i=1,2,…,n and k=1,2,…;i=0 corresponds to the center of the cylinder, and i=n corresponds to the surface. The implicit Euler scheme is used to obtain a matrix equation of the form AX=F. The choice of this scheme is motivated by the fact that this scheme is unconditionally stable, contrary to the explicit scheme for which ∆X and ∆τ satisfy certain conditions [30]. In the unidirectional case in Cartesian coordinates, for example, ΔX and Δτ must verify τ<1/2ΔX2. In general, the terms of the diagonal of the matrix A must be positive. Under these conditions, equations (21) and (22) are written as follows:(23)2Zkφik+1−φikτ=φi+1k+1−2φik+1+φi−1k+1X²+φi+1k+1−φik+1XiX+XidZdτ|kφi+1k+1−φik+1ΔX,(24)dZdτ|k=11−φi=nkφi=nk−φi=n−1kΔX,(25)Zk+1=Zk+dZdτ|k×Δτ.


### 2.5. Determination of Absorption Kinetics

#### 2.5.1. Numerical Approach

The water absorption ratio notedgt is defined as follows [11, 12]:(26)gt=Mt−M0M∞−M0,whereM0and M∞ are the masses at the initial and saturation times of the fiber, respectively. Mt is the mass of the fiber at time t and is defined as follows:(27)Mt=∫0Rt2πrCr,tdr.Using the previous change of variable, it comes that(28)Mτ=∫012πR2τXφX,τdX.This equation is solved using the Gaussian quadrature method [31].


#### 2.5.2. Proposal of a Model

Relation (26) can still be written as follows:(29)gt=Mt−M0M∞−M0=ρeVt−V0ρeV∞−V0=R2t−R02R∞2−R02,whereV0, V∞, and Vt represent the initial fiber volume, the volume at saturation, and the fiber volume at a time t, respectively. Similarly, R0, R∞, and Rt represent the initial radius of the fiber, the radius at saturation, and the radius of the fiber at a time t, respectively. Rt is given by relation (13).(30)R∞=limt⟶∞Rt=pk.ln1−C01−Cmax+R0.


### 2.6. Diffusion-Induced Stresses

The volume change of an isotropic material is related to the deformation tensor by the following relation [32, 33]:(31)ΔVV=traceε,where εis the strain tensor defined in polar coordinate r,θ as follows:(32)ε=εrrεrθεθrεθθ=∂ur∂r00urr,where ur is the displacement along the radius of the fiber.Relations (31) and (32) lead to the following differential equation:(33)∂ur∂r+urr=ΔVV.We deduce that(34)εrr=εθθ=12ΔVV.The stresses and strains are linked by Hooke’s law [32, 33]:(35)σij=λεkkδij+2μεijwhere λandμ are the Lamé coefficients and defined as follows:(36)λ=νE1+ν1−2ν,μ=E21+ν,where E and ν denote, respectively, Young’s modulus and Poisson’s ratio. With this definition, relation (35) leads to(37)σrr=σθθ=12E1+ν1−2νΔVV.


## 2.1. Modeling of Water Transfer through the Raffiavinifera Fiber

Mass transfer through a solid is described according to Fick’s second law [21, 22] by the following equation, where C (in mol/m3) is the concentration of the diffusing molecule and D (in m2/s) is the diffusion coefficient:(1)∂C∂t=div−Dgrad⟶C.Despite the fact that the raffia fiber has an elliptical section [13], it is assimilated in this work to a cylinder of initial radiusR0. Depending on whether the fiber loses or absorbs water, it becomes either shrunken or swollen, and its radius varies. Let R=Rt, therefore, be the radius of the fiber at timet>0. The water diffusion through the raffia fiber is described by equation (1) in cylindrical coordinates r,θ,z as follows:(2)∂Cr,t∂t=1r∂∂rrD∂Cr,t∂r+∂∂θDr∂Cr,t∂θ+∂∂zrD∂Cr,t∂z,0≤r≤R.Since the problem is cylindrical symmetry, the diffusion along the angleθ is neglected. On the other hand, the length of the fiber being very large compared to the diameter of the fiber, the fiber can be considered as an infinite cylinder, and the diffusion along the z-axis is assumed to be constant. Equation (2) is written in cylindrical coordinates following the radial coordinate r alone. A similar assumption was done by Dejam [23].(3)∂Cr,t∂t=Dr∂∂rr∂Cr,t∂r,0≤r≤Rt,t>0,with the initial conditions and the following boundary conditions:(4)Cr,t=C0fort=0and0<r≤RtCr,t=Cstfort>0andr=Rt,whereC0 is the initial concentration and Cst the surface concentration.


## 2.2. Modeling of the External Surface of Raffiavinifera Fiber

The observation of a raffia fiber by the scanning electron microscope (Figure3(a)) [15] shows that it is surrounded by an envelope of small thickness e (Figure 3(b)), which protects it against attack and convection of fluids. Diffusion through this envelope is modeled by applying two boundary conditions: the continuity of the water concentration and flux at the interface between the fiber and its envelope on the one hand and at the interface between the envelope and the external environment on the other hand. Such boundary conditions were used by Dejam to model the Shear dispersion in a capillary tube with a porous wall [23].Figure 3
(a) Cross-section of a raffia fiber given by the scanning electron microscope [15]. (b) Sketch of the external surface of the fiber.
(a)(b)At the interface, there is a coefficient determining the distribution of quantities [22, 24, 25]. Let σf and σenv be the distribution coefficients of the quantities at the fiber/envelope and envelope/external environment interface, respectively. We have(5)σf=CRt,tCenv−=CstCenv−,σenv=Cenv+Cet,where Cenv− and Cenv+ are the concentration of the envelope at the fiber/envelope and envelope/external environment interface, respectively, and Cet is the concentration of the external environment. Given that, relation (5) can still be written as follows:(6)Jenv⟶.n⟶=−DenvCenv+−Cenv−e=Denve.σfCst−σfσenvCet.At the interface, the continuity of the flow allows to write(7)−D∂Cr,t∂r|Rt=−Denv∂Cenvr,t∂r|Rt=pCst−σCet,where p=Denv/e.σf and σ=σfσenv.Relation 8 is analogous to the convective flow through a surface in the thermal context [21, 22]. Thus, the factor p (in m/s) introduced translates the permeability of the fiber, and we distinguish two limit cases according to the values ofp:Ifp=0 (impermeable surface), no flow through the fiber, and relation (7) is written as: −D∂Cr,t/∂r|Rt=0Ifp=∞, relation (7) is written as: CRt,t=Cst=σCetThe absorption of water molecules by the surface of fibers can be a kinetics reaction. Assuming that this reaction is a first-order reaction [18, 24, 26], mass conservation is written as follows:(8)∂Cst∂t=kCmax−Cst,whereCmaxis the maximum concentration of water in the fiber at saturation and k has the dimension of the inverse of time and represents a kinetic constant.


## 2.3. Modeling of RaffiaVinifera Fiber Swelling

Consider an elementary surfacedS of the fiber that moves with a speed v⟶, from a point at date t to another point at date t+dt. In the case of absorption, this displacement results from an excess of water in the fiber of volume Ve which diffuses through the fiber. The variation of the volume of the fiber corresponds to the volume of water absorbed by the fiber. Thus, (9)VT=V0+Ve,where VT,V0, and Ve denote, respectively, the total volume of the fiber, the initial volume of the fiber, and the volume of water absorbed by the fiber; it follows that(10)πR2t=πR02+∫0RtCr,t.2πrdr.Deriving this relation with respect to time and by applying the Leibniz integral rule [27, 28], it comes that(11)D∂Cr,t∂r|Rt=1−CstdRtdt.The details of this calculation are given in the appendix.The mathematical expression of the radius of the fiber as a function of time is obtained by placing equation (7) in equation (11), and it follows that(12)dRtdt=pCet−Cst1−Cst.Considering the case whereCet=cte=Cmax, the integration of the previous relation leads to(13)Rt=pkln1−C01−Cst+R0.Equation (13) is one of the main results of this work.


## 2.4. Determination of the Water Concentration

In order to facilitate the manipulation of the equations, it is preferable to make the physical quantities involved dimensionless. For this purpose, as in [23, 29], let us say(14):τ=DtR02,ϒ∗=ϒR0,R∗τ=RtR0<9.Using the chain rule, equations (3) and (11) lead to the following equations, respectively:(15)∂Cr∗,τ∂τ=1r∗∂∂r∗r∗∂Cr∗,τ∂r∗,0≤r∗≤R∗τ,τ>0,(16)∂Cr∗,τ∂r∗|R∗=1−CR∗,τdR∗τdτ,τ>0.


### 2.4.1. Immobilization of the Domain Boundary

The fiber boundary varies with time. To simplify the numerical solution of equations (14), we transform them into an equivalent problem on a fixed domain. A new spatial variable is defined as follows [17, 18]:(17)X=r∗R∗τ=rRt,0≤X≤1.In writingφX,τ=φr∗/R∗τ,τ≡Cr∗,τ and using the chain rule, equations (15) and (16) lead to the following equations, respectively:(18)∂φ∂τ−r∗R∗τ2∂R∗τ∂τ∂φ∂X=1XR∗τ2×∂∂XXR∗τ.1R∗τ∂φ∂X,(19)1R∗τ∂φ∂X|X=1=1−φX,τ|X=1dR∗τdτ.Multiplying relation (18) by R∗τ2and introducing a new variable Zτ=1/2R∗τ2 as in [18], we obtain the following relation:(20)2Zτ∂φ∂τ=1X∂∂XX∂φ∂X+XdZτdτ∂φ∂X,0≤X≤1,τ>0,where(21)dZτdτ=11−φX,τ|X=1∂φ∂X|X=1,τ>0.


### 2.4.2. Discretization

To solve equation (20), we will use the finite difference method. For this, equation (20) is written as follows:(22)2Zτ∂φ∂τ=∂²φ∂X²+1X∂φ∂X+XdZτdτ∂φ∂X.The spaceX is divided into n intervals of length ΔX, and a time step Δτ is chosen. The approximation of φX,τ at a point Xiand at time τk is denoted φikwith i=1,2,…,n and k=1,2,…;i=0 corresponds to the center of the cylinder, and i=n corresponds to the surface. The implicit Euler scheme is used to obtain a matrix equation of the form AX=F. The choice of this scheme is motivated by the fact that this scheme is unconditionally stable, contrary to the explicit scheme for which ∆X and ∆τ satisfy certain conditions [30]. In the unidirectional case in Cartesian coordinates, for example, ΔX and Δτ must verify τ<1/2ΔX2. In general, the terms of the diagonal of the matrix A must be positive. Under these conditions, equations (21) and (22) are written as follows:(23)2Zkφik+1−φikτ=φi+1k+1−2φik+1+φi−1k+1X²+φi+1k+1−φik+1XiX+XidZdτ|kφi+1k+1−φik+1ΔX,(24)dZdτ|k=11−φi=nkφi=nk−φi=n−1kΔX,(25)Zk+1=Zk+dZdτ|k×Δτ.


## 2.4.1. Immobilization of the Domain Boundary

The fiber boundary varies with time. To simplify the numerical solution of equations (14), we transform them into an equivalent problem on a fixed domain. A new spatial variable is defined as follows [17, 18]:(17)X=r∗R∗τ=rRt,0≤X≤1.In writingφX,τ=φr∗/R∗τ,τ≡Cr∗,τ and using the chain rule, equations (15) and (16) lead to the following equations, respectively:(18)∂φ∂τ−r∗R∗τ2∂R∗τ∂τ∂φ∂X=1XR∗τ2×∂∂XXR∗τ.1R∗τ∂φ∂X,(19)1R∗τ∂φ∂X|X=1=1−φX,τ|X=1dR∗τdτ.Multiplying relation (18) by R∗τ2and introducing a new variable Zτ=1/2R∗τ2 as in [18], we obtain the following relation:(20)2Zτ∂φ∂τ=1X∂∂XX∂φ∂X+XdZτdτ∂φ∂X,0≤X≤1,τ>0,where(21)dZτdτ=11−φX,τ|X=1∂φ∂X|X=1,τ>0.


## 2.4.2. Discretization

To solve equation (20), we will use the finite difference method. For this, equation (20) is written as follows:(22)2Zτ∂φ∂τ=∂²φ∂X²+1X∂φ∂X+XdZτdτ∂φ∂X.The spaceX is divided into n intervals of length ΔX, and a time step Δτ is chosen. The approximation of φX,τ at a point Xiand at time τk is denoted φikwith i=1,2,…,n and k=1,2,…;i=0 corresponds to the center of the cylinder, and i=n corresponds to the surface. The implicit Euler scheme is used to obtain a matrix equation of the form AX=F. The choice of this scheme is motivated by the fact that this scheme is unconditionally stable, contrary to the explicit scheme for which ∆X and ∆τ satisfy certain conditions [30]. In the unidirectional case in Cartesian coordinates, for example, ΔX and Δτ must verify τ<1/2ΔX2. In general, the terms of the diagonal of the matrix A must be positive. Under these conditions, equations (21) and (22) are written as follows:(23)2Zkφik+1−φikτ=φi+1k+1−2φik+1+φi−1k+1X²+φi+1k+1−φik+1XiX+XidZdτ|kφi+1k+1−φik+1ΔX,(24)dZdτ|k=11−φi=nkφi=nk−φi=n−1kΔX,(25)Zk+1=Zk+dZdτ|k×Δτ.


## 2.5. Determination of Absorption Kinetics

### 2.5.1. Numerical Approach

The water absorption ratio notedgt is defined as follows [11, 12]:(26)gt=Mt−M0M∞−M0,whereM0and M∞ are the masses at the initial and saturation times of the fiber, respectively. Mt is the mass of the fiber at time t and is defined as follows:(27)Mt=∫0Rt2πrCr,tdr.Using the previous change of variable, it comes that(28)Mτ=∫012πR2τXφX,τdX.This equation is solved using the Gaussian quadrature method [31].


### 2.5.2. Proposal of a Model

Relation (26) can still be written as follows:(29)gt=Mt−M0M∞−M0=ρeVt−V0ρeV∞−V0=R2t−R02R∞2−R02,whereV0, V∞, and Vt represent the initial fiber volume, the volume at saturation, and the fiber volume at a time t, respectively. Similarly, R0, R∞, and Rt represent the initial radius of the fiber, the radius at saturation, and the radius of the fiber at a time t, respectively. Rt is given by relation (13).(30)R∞=limt⟶∞Rt=pk.ln1−C01−Cmax+R0.


## 2.5.1. Numerical Approach

The water absorption ratio notedgt is defined as follows [11, 12]:(26)gt=Mt−M0M∞−M0,whereM0and M∞ are the masses at the initial and saturation times of the fiber, respectively. Mt is the mass of the fiber at time t and is defined as follows:(27)Mt=∫0Rt2πrCr,tdr.Using the previous change of variable, it comes that(28)Mτ=∫012πR2τXφX,τdX.This equation is solved using the Gaussian quadrature method [31].


## 2.5.2. Proposal of a Model

Relation (26) can still be written as follows:(29)gt=Mt−M0M∞−M0=ρeVt−V0ρeV∞−V0=R2t−R02R∞2−R02,whereV0, V∞, and Vt represent the initial fiber volume, the volume at saturation, and the fiber volume at a time t, respectively. Similarly, R0, R∞, and Rt represent the initial radius of the fiber, the radius at saturation, and the radius of the fiber at a time t, respectively. Rt is given by relation (13).(30)R∞=limt⟶∞Rt=pk.ln1−C01−Cmax+R0.


## 2.6. Diffusion-Induced Stresses

The volume change of an isotropic material is related to the deformation tensor by the following relation [32, 33]:(31)ΔVV=traceε,where εis the strain tensor defined in polar coordinate r,θ as follows:(32)ε=εrrεrθεθrεθθ=∂ur∂r00urr,where ur is the displacement along the radius of the fiber.Relations (31) and (32) lead to the following differential equation:(33)∂ur∂r+urr=ΔVV.We deduce that(34)εrr=εθθ=12ΔVV.The stresses and strains are linked by Hooke’s law [32, 33]:(35)σij=λεkkδij+2μεijwhere λandμ are the Lamé coefficients and defined as follows:(36)λ=νE1+ν1−2ν,μ=E21+ν,where E and ν denote, respectively, Young’s modulus and Poisson’s ratio. With this definition, relation (35) leads to(37)σrr=σθθ=12E1+ν1−2νΔVV.


## 3. Results

### 3.1. Fiber Absorption Rate

The modeling of the water diffusion phenomenon through raffia fiber depends on several input parameters such as the physical properties of the fiber, initial and boundary conditions, and even geometric sizes. These parameters are taken from the literature and are recorded in Table1.Table 1
Physical properties of raffia fiber for different sampling areas along the stem [11].
Longitudinal positionRadiale positionDeffmm2/skh−1Section moyennemm2PL-1/4Périphérique1.39×10−43.3522.775Mi rayon1.44×10−42.6342.853Center1.94×10−41.9693.007PL-2/4Périphérique8.34×10−42.3512.219Mi- rayon1.50×10−42.6752.563Center2.28×10−43.4522.75PL-3/4Périphérique1.31×10−43.9752.020Mi- rayon2.00×10−43.9732.205Center2.36×10−42.082.35PL-4/4Périphérique7.12×10−52.9961.617Mi- rayon1.86×10−43.4231.828Center2.11×10−44.8111.929The values ofC0 and Cmax are calculated, respectively, from the following relationships: M0=∫0R2πrC0dr and M∞=∫0R2πrCmaxdr. With these values, Figure 4 shows the evolution of the local water concentration φ as a function of the time of a sample located after the base of the stem and at the half radius in the radial direction (PL-2/4-R2).Figure 4
Evolution of the absorption rate as a function of time of a raffia fiber located after the base of the stem and at the half radius in the radial direction (PL-2/4-R2).It can be seen from Figure4 that the swelling of the raffia vinifera fiber is felt from a dimensionless timeτ=2.5. This observation allows us to conclude that when manufacturing composites reinforced with raffia vinifera fibers from the stem, the shaping time should be less than this time to avoid swelling of the fiber. It also appears from this figure that from this time, the local concentration of water in the fiber becomes high in the case where its volume does not change (absence of swelling) compared to the case where there is variation in volume. This result is physically acceptable. Indeed, if the same quantity of water diffuses in two materials having the same physicochemical properties, then the local water concentration will be low in the material that has the greater volume.


### 3.2. Fiber Radius

The mathematical expression of the radiusRt of the fiber is given by equation (13). The numerical determination of this radius of the fiber after absorbing water is done from equation (25) and from the following relation:(38)Zτ=12R∗τ2.We deduce that(39)R∗τ=2Zτ1/2.Figure5 shows the variations of the radius of the fiber as a function of the time of a sample located after the base of the stem and at midradius in the radial direction (PL-2/4-R2). The parabolic direction of concavity turned towards the x-axis is explained by the fact that after a certain time, the fiber begins to be saturated; the local concentration of water in the fiber becomes almost constant, and therefore, the radius no longer varies significantly since it is only induced by the variation of the moisture content in the fiber. The analytical model approaches fairly well the numerical model. The shape of this curve is similar to that obtained by Kutlay et al. [17] and Swejen et al. [18]. The same observations are made for the other samples.Figure 5
Variation of the radius as a function of time of a raffia fiber located after the base of the stem and at the half radius in the radial direction (PL-2/4-R2).Figure6 shows the evolution over time of the radius of the fibers located on the cross-section of the part located at the PL-3/4 position of the stem. It shows that the fiber swelling rate increases from the center towards the peripheral, while the relative fiber swelling (ratio of the final radius of a fiber to its initial radius) increases from the peripheral towards the center.Figure 6
Evolution as a function of time of the radius of the fibers located at the PL-3/4 position of the stem during the water diffusion phenomenon.This observation is related to the fact that the diffusion coefficient increases from the peripheral towards the center, while the kinetic constantk decreases from the peripheral towards the center [11]. Indeed, when the absorption kinetics is high, the fiber absorbs water quickly, and consequently, its volume varies also quickly. Similarly, if the diffusion coefficient is high, the amount of water absorbed will be important, and consequently, the volume variation will be maximal. The same observations are made in the other sampling zones.Figure7 shows the variations over time of the radius of the fibers located at the half radius of the cross-section along the stem during the diffusion phenomenon. It follows that the relative variation in the radius of the fibers situated at the same radial position is almost identical. This is due to the fact that for a given radial position, the ratio R02k/D is almost constant whatever the longitudinal position, and this is the reason why at a given radial position, the relative variation of the radius of the fiber during the diffusion phenomenon along the stem is less significant. The same observations are made for other radial positions.Figure 7
Evolution as a function of time of the radius of the fibers located at midradius of the cross-section along the stem during the water diffusion phenomenon.


### 3.3. Water Absorption Ratio

Figure8 shows the water absorption ratio of a raffia vinifera fiber taken after the base of the stem and at the half radius in the radial direction (PL-1/4-R2) when its volume changes over time and when it does not change. Unlike Figure 4, where the local water concentration was different for the case where the volume of the fiber varied and for the case where it did not change, the absorption ratio is identical in both cases. This result is quite logical because as the name suggests, it is a ratio. This result, therefore, highlights the intrinsic character of the water absorption phenomenon. We can therefore conclude that the diffusion of water through a material does not depend on the geometry of the material but its physicochemical properties. We also observe on the same figure that the curves obtained by modeling agree well the large number of the experimental points obtained by [11]. The same observations are made for other samplings zones.Figure 8
Absorption kinetics curve of a raffia fiber located after the base of the stem and at the half radius in the radial direction (PL-2/4-R2).By exploiting the volume change of the raffiavinifera fiber during the water diffusion phenomenon, we proposed a mathematical model to predict the water absorption ratio of raffia vinifera fibers. This model is given by equation (13) and plotted in Figure 9 below. It appears from this figure that this proposal model fits more the experimental points obtained by [11]. The same observations are made for other samplings zones.Figure 9
Comparison between the proposed model and the experimental result.


### 3.4. Diffusion-Induced Stresses

The expression giving the stresses evolution in the fiber involves Young’s modulus and Poisson’s ratio. The value of Young’s modulus of raffiavinifera fiber is taken from the work of [13]. For Poisson’s ratio, we have no reliable information on its value. This coefficient will simply be taken from the interval [–1; 0.5]. Figures 10 and 11 show the time evolution of strains and stresses in the raffia fiber following a change in its hygroscopic state.Figure 10
Evolution of the deformations as a function of time of a raffia fiber located after the base of the rod and at the half radius in the radial direction (PL-2/4-R2).Figure 11
Variation of induced stresses as a function of time in a raffia fiber located after the base of the rod and at midradius in the radial direction (PL-2/4-R2).From these figures, it appears that the induced stresses and strains in the raffiavinifera fiber during the water diffusion phenomenon increase with time. For a relatively long time, these stresses and strains stabilize. This convergence is explained by the fact that after a certain time, the fiber becomes saturated, and the stress becomes constant. The same observations are made for the other sample zones.


## 3.1. Fiber Absorption Rate

The modeling of the water diffusion phenomenon through raffia fiber depends on several input parameters such as the physical properties of the fiber, initial and boundary conditions, and even geometric sizes. These parameters are taken from the literature and are recorded in Table1.Table 1
Physical properties of raffia fiber for different sampling areas along the stem [11].
Longitudinal positionRadiale positionDeffmm2/skh−1Section moyennemm2PL-1/4Périphérique1.39×10−43.3522.775Mi rayon1.44×10−42.6342.853Center1.94×10−41.9693.007PL-2/4Périphérique8.34×10−42.3512.219Mi- rayon1.50×10−42.6752.563Center2.28×10−43.4522.75PL-3/4Périphérique1.31×10−43.9752.020Mi- rayon2.00×10−43.9732.205Center2.36×10−42.082.35PL-4/4Périphérique7.12×10−52.9961.617Mi- rayon1.86×10−43.4231.828Center2.11×10−44.8111.929The values ofC0 and Cmax are calculated, respectively, from the following relationships: M0=∫0R2πrC0dr and M∞=∫0R2πrCmaxdr. With these values, Figure 4 shows the evolution of the local water concentration φ as a function of the time of a sample located after the base of the stem and at the half radius in the radial direction (PL-2/4-R2).Figure 4
Evolution of the absorption rate as a function of time of a raffia fiber located after the base of the stem and at the half radius in the radial direction (PL-2/4-R2).It can be seen from Figure4 that the swelling of the raffia vinifera fiber is felt from a dimensionless timeτ=2.5. This observation allows us to conclude that when manufacturing composites reinforced with raffia vinifera fibers from the stem, the shaping time should be less than this time to avoid swelling of the fiber. It also appears from this figure that from this time, the local concentration of water in the fiber becomes high in the case where its volume does not change (absence of swelling) compared to the case where there is variation in volume. This result is physically acceptable. Indeed, if the same quantity of water diffuses in two materials having the same physicochemical properties, then the local water concentration will be low in the material that has the greater volume.


## 3.2. Fiber Radius

The mathematical expression of the radiusRt of the fiber is given by equation (13). The numerical determination of this radius of the fiber after absorbing water is done from equation (25) and from the following relation:(38)Zτ=12R∗τ2.We deduce that(39)R∗τ=2Zτ1/2.Figure5 shows the variations of the radius of the fiber as a function of the time of a sample located after the base of the stem and at midradius in the radial direction (PL-2/4-R2). The parabolic direction of concavity turned towards the x-axis is explained by the fact that after a certain time, the fiber begins to be saturated; the local concentration of water in the fiber becomes almost constant, and therefore, the radius no longer varies significantly since it is only induced by the variation of the moisture content in the fiber. The analytical model approaches fairly well the numerical model. The shape of this curve is similar to that obtained by Kutlay et al. [17] and Swejen et al. [18]. The same observations are made for the other samples.Figure 5
Variation of the radius as a function of time of a raffia fiber located after the base of the stem and at the half radius in the radial direction (PL-2/4-R2).Figure6 shows the evolution over time of the radius of the fibers located on the cross-section of the part located at the PL-3/4 position of the stem. It shows that the fiber swelling rate increases from the center towards the peripheral, while the relative fiber swelling (ratio of the final radius of a fiber to its initial radius) increases from the peripheral towards the center.Figure 6
Evolution as a function of time of the radius of the fibers located at the PL-3/4 position of the stem during the water diffusion phenomenon.This observation is related to the fact that the diffusion coefficient increases from the peripheral towards the center, while the kinetic constantk decreases from the peripheral towards the center [11]. Indeed, when the absorption kinetics is high, the fiber absorbs water quickly, and consequently, its volume varies also quickly. Similarly, if the diffusion coefficient is high, the amount of water absorbed will be important, and consequently, the volume variation will be maximal. The same observations are made in the other sampling zones.Figure7 shows the variations over time of the radius of the fibers located at the half radius of the cross-section along the stem during the diffusion phenomenon. It follows that the relative variation in the radius of the fibers situated at the same radial position is almost identical. This is due to the fact that for a given radial position, the ratio R02k/D is almost constant whatever the longitudinal position, and this is the reason why at a given radial position, the relative variation of the radius of the fiber during the diffusion phenomenon along the stem is less significant. The same observations are made for other radial positions.Figure 7
Evolution as a function of time of the radius of the fibers located at midradius of the cross-section along the stem during the water diffusion phenomenon.


## 3.3. Water Absorption Ratio

Figure8 shows the water absorption ratio of a raffia vinifera fiber taken after the base of the stem and at the half radius in the radial direction (PL-1/4-R2) when its volume changes over time and when it does not change. Unlike Figure 4, where the local water concentration was different for the case where the volume of the fiber varied and for the case where it did not change, the absorption ratio is identical in both cases. This result is quite logical because as the name suggests, it is a ratio. This result, therefore, highlights the intrinsic character of the water absorption phenomenon. We can therefore conclude that the diffusion of water through a material does not depend on the geometry of the material but its physicochemical properties. We also observe on the same figure that the curves obtained by modeling agree well the large number of the experimental points obtained by [11]. The same observations are made for other samplings zones.Figure 8
Absorption kinetics curve of a raffia fiber located after the base of the stem and at the half radius in the radial direction (PL-2/4-R2).By exploiting the volume change of the raffiavinifera fiber during the water diffusion phenomenon, we proposed a mathematical model to predict the water absorption ratio of raffia vinifera fibers. This model is given by equation (13) and plotted in Figure 9 below. It appears from this figure that this proposal model fits more the experimental points obtained by [11]. The same observations are made for other samplings zones.Figure 9
Comparison between the proposed model and the experimental result.


## 3.4. Diffusion-Induced Stresses

The expression giving the stresses evolution in the fiber involves Young’s modulus and Poisson’s ratio. The value of Young’s modulus of raffiavinifera fiber is taken from the work of [13]. For Poisson’s ratio, we have no reliable information on its value. This coefficient will simply be taken from the interval [–1; 0.5]. Figures 10 and 11 show the time evolution of strains and stresses in the raffia fiber following a change in its hygroscopic state.Figure 10
Evolution of the deformations as a function of time of a raffia fiber located after the base of the rod and at the half radius in the radial direction (PL-2/4-R2).Figure 11
Variation of induced stresses as a function of time in a raffia fiber located after the base of the rod and at midradius in the radial direction (PL-2/4-R2).From these figures, it appears that the induced stresses and strains in the raffiavinifera fiber during the water diffusion phenomenon increase with time. For a relatively long time, these stresses and strains stabilize. This convergence is explained by the fact that after a certain time, the fiber becomes saturated, and the stress becomes constant. The same observations are made for the other sample zones.


## 4. Summary and Conclusion

The objective of this work was to model the hygro-mechanical behavior of a raffiavinifera fiber from the stem. The study of the variation of the diameter of the fiber in the function of time was done by admitting that this variation was free; the physicochemical properties of the fiber, as well as the temperature, were constant. Moreover, any variation of the hygroscopic state of the fiber induced a variation of its diameter. This study shows the following:The local concentration of water in the fiber is high when the fiber volume is assumed to be constant as opposed to the case where the fiber volume variesThe absorption ratio of the fiber is the same whether the swelling of the fiber is considered or notThe mathematical model describing the variations of the fiber radius as a function of time agrees with the numerical resultsThis radius, scaled, stabilizes around a value of 3.5The mathematical model describing the absorption kinetics of the fiber agrees with the experimental results from the literatureThe stresses induced in the fiber increase with time and can reach 5 to 7 GPaIt is obvious that any temperature variation in a material induces a variation of its moisture content (Soret effect [34]). Moreover, whatever the field of use (in the reinforcement of composites, for example), the raffia fiber will be associated with other materials, and in this case, its swelling will be prevented or constrained. The above simplifying assumptions considerably reduce the area of validity of the results of this work because they are not applicable in a highly variable environment; such is the limitation of this work. The parameters neglected in this work will be taken into account in future works in order to get closer to reality.



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*Source: 9354878-2021-11-22.xml*