1. Introduction

The fatigue behavior of vegetal-fiber composites has been investigated rarely, and even less attention has been given to composite materials reinforced with Alfa fibers. Towo and Ansell [1, 2] studied the fatigue of sisal/epoxy and sisal/polyester composites. The S–N curves were presented for reinforced composites with untreated sisal fibers and sisal fibers treated with 0.06 NaOH. The sisal/epoxy composites showed a longer fatigue life than the sisal/polyester ones. In addition, the effect of the chemical treatment on the fatigue life was more significant for the sisal/polyester reinforced composites. Silva et al. [3] identified the tensile fatigue behavior of cement composites reinforced with long aligned sisal fibers. The fatigue behavior was examined in terms of stress versus the number of cycles and the stress–strain hysteresis of the composites. Elouaer et al. [4] studied the fatigue response of different varieties of composites with an injected hemp reinforcement. The authors found that the fatigue strength of hemp/PP was higher than that of hemp/CP. Similarly, Reis et al. [5] studied the flexural behavior of manually manufactured hybrid laminated composites with a natural hemp fiber/polypropylene core and two glass fiber/polypropylene surface layers, one on each side of specimen. Thwe and Liao [6] studied the resistance of a bamboo-fiber-reinforced polypropylene composite (BFRP) and a hybrid bamboo- and glass-fiber-reinforced polypropylene composite (BGRP) to hygrothermal aging and fatigue under cyclic tensile loads. Ferreira et al. [7] studied the fatigue of composite adhesive lap joints. The adhesive used was a Bostik 7452 (Rubber and Plastics Grade) ethyl cyanoacrylate. Gassan [8] studied the fatigue of flax/epoxy and flax/polyester composites in comparison with that of composites reinforced with jute fibers. The results showed that the damping properties of flax/epoxy were better than those of jute/epoxy ones. Miraoui and Hassis [9] developed a mechanical model for vegetal-fiber-reinforced composite materials to determine the optimization parameters necessary for maximizing the stress in vegetal fibers. Ferreira et al. [10] studied short-vegetal-fiber-reinforced HDPE treated with electron beam radiation. Fotouh et al. [11] studied the fatigue behavior of hemp-fiber–reinforced HDPE composites. A model was developed to predict the fatigue behavior of the natural-fiber composites at different fiber fractions and fatigue stress ratios. Moraes et al. [12] studied the effect of loading frequency on the fatigue behavior of a coir-fiber-reinforced PP composite.

Regarding homogenization, Ghossein and Lévesque [13] studied the evaluation of analytical homogenization models for the case of randomly distributed and oriented ellipsoidal-fiber-reinforced composites. D. Notta-Cuvier et al. [14] focused on modeling the behavior of short-fiber-reinforced composites. Willoughby et al. [15] modeled a composite material with a fiber reinforcement. An asymptotic homogenization method and an approach intended to represent a macroscopically random material were used. Ma and Hu [16] developed a general micromechanical method for a micropolar composite with ellipsoidal fibers. The influence of both the shape and size of the fibers were analyzed by the method proposed. In their research work, Sobhani Aragh et al. [17] used an equivalent continuum model based on the Eshelby–Mori–Tanaka approach to estimate a constitutive law for an elastic isotropic medium (matrix) with oriented straight carbon nanotubes (CNTs).

John and Anandjiwala [18] used a natural-fiber surface treatment method to reinforce composite materials by improving their interfacial bonding and increasing their strength. A. Bessadok et al. [19] studied the effects of chemical treatments with maleic anhydride, styrene, acrylic acid, and acetic anhydride on Alfa-fiber/unsaturated polyester composites. Furthermore, Alix et al. [20] studied the behavior of flax fibers chemically modified by silane and styrene. Lastly, Arrakhiz et al. [21, 22] determined the mechanical properties of different composite materials with chemically treated Alfa fibers.

The present study focuses on the fatigue behavior of a composite material with Alfa fibers and optimization of their chemical treatment to enhance its static mechanical characteristics (Mechakra et al. [23]); an improved Mori–Tanaka model based on the Eshelby stress tensor and an inverse method were used to determine the mechanical characteristics of the composite.

2. Fatigue test

The tests were conducted on a Zwick fatigue testing machine with a maximum load of 250 kN. All experimental data obtained were stored on an acquisition card. The tests were conducted at a frequency of 1 Hz with amplitude ΔF/2 and load ratio R = σmin/σmax = 0.1. The choice of such a low frequency allowed us to avoid any side effect mainly due to self-heating of the material. All tests were conducted at room temperature [4]. The specimens were subjected to the load ratios of 65, 75, and 85% of the maximum load Fmax.

The aim of the fatigue tests was to found the S–N curves, defining the lifetime of the material at a given load. On the semi-logarithmic scale, the S–N curve can be given by a linear regression of the form [4]

$$ r=A-B\log \left({N}_f\right), $$
(1)

where is the load level ratio applied, A is material constant, B is slope of the endurance curve, and Nf is the number of cycles at break.

The form of Eq. (1) allowed us to better visualize the fatigue of the composites (polypropylene with treated and untreated short Alfa fibers) studied according to the slope (Table 1).

Table 1. Values of A and B used
Full size table

2.1. Fatigue mechanism

Three different load ratios (r = 65, 75, and 85%) were used to investigate the evolution of rigidity of the materials, defined as the current rigidity E divided by the initial rigidity at the first loading cycle, E0, with respect to the number of cycles at break [24].

An intermediate applied load r = 65% UTS was used (Fig. 1) to validate the experimental mechanical characteristics and the effect of chemical treatment in static tests [23].

Fig. 1.
figure 1

Rigidity loss of pollypropylene composites with untreated fibers (1) and Alfa fibers treated for 24 (2) and 48 (h).

Full size image

The polypropylene with Alfa fibers (treated for 24 and 48 h) exhibited a better fatigue resistance than the polypropylene with untreated ones. The residual rigidity of the composites with fibers chemically treated for 24 and 28 h was higher than of the composite with untreated fibers: by 56 and 52%, respectively, at the load r = 65% UTS (Fig, 1a); by 44.2 and 36.5%, respectively, at r = 75% UTS (Fig. 1b); by 57 and 48%, respectively, at r = 85% UTS (Fig. 1c). In addition, at r = 85% UTS, thr residual strength of the composite with fibers treated fot 24 h was higher than that with fibers treated for 48 h.

The fatigue strength of the composite with untreated fibers was 65% of that in the case of theated fibers. This can be explained by a better adhesion bond between the polypropylene matrix and theated fibers.

2.3. Process and fatigue mechanisms

During the cyclic fatigue loadings, three phases (I, II, and III) could be observed during the evolution of Young’s modulus E as a function of the number of cycles. They are illustrated in Fig. 2 for two composites, PP/Alfa-T24h and PP/Alfa-T48h, and two loading ratios, r = 65 and 85% of UTS.

  • Phase I. A considerable and rapid decrease in E, when the first cracks appeared in the matrix. This degradation is described by the initiation of damage caused by microcracks in different parts of the matrix and fiber/matrix interfaces.

  • Phase II. A moderate decrease in E (the longest phase), followed by a steady state with a slow evolution of damage, corresponding to the multiplication of transverse cracks. This phase is characterized by the propagation and increase of microcracks.

  • Phase III. An accelerated decrease in E and a rapid development of cracks and macrocracks, corresponding to the final damage and breaking.

Fig. 2.
figure 2

Development of fatigue process in composites with fibers treated for 24 (a, c) and 48 h (b, d) at 65 (a, b) and 85% UTS (c, d).

Full size image

These phases confirm the same fatigue behavior of all thermoplastic/thermoset materials reinforced with vegetal fibers, such as, e.g., the flax/epoxy and flax/polyester studied by Elouaer et al. [4].

3. Application of the Mori–Tanaka Model

A micromechanical calculation was performed with Matlab to obtain the numerical rigidity tensor of the composite by the Mori–Tanaka model. To ensure that the calculation run smoothly in the case of PP/Alfa-48h and PP/Alfa-24h composites, the model required the following information:

  • the volume fraction of reinforcement and the elastic properties of each constituent (E and ν).

  • The rigidity matrices of each constituent, with simplifying assumptions. The polypropylene resin and Alfa fibers were assumed to be isotropic.

  • The geometry and orientation of fibers. Random orientation of the fibers in the model was assumed.

In the Mori–Tanaka model, two homogenization tensors that take into account the inclusion of the Eshelby tensor are considered:

1. The pseudotensor of localization [15]

$$ {\mathbf{L}}^i=-{\left[\left({\mathbf{C}}^i-{\mathbf{C}}^0\right){\mathbf{S}}_{\mathrm{Esh}}^i+{\mathbf{C}}^0\right]}^{-1}\left({\mathbf{C}}^i-{\mathbf{C}}^0\right), $$
(2)

where Ci is the rigidity tensor of inclusion, C0 is the rigidity tensor of matrix, and \( {S}_{Esh}^i \) is the Eshelby tensor.

2. The stiffness tensor of the equivalent homogeneous material

$$ {\mathbf{C}}^c={\mathbf{C}}^0{\left\{\mathbf{I}+{\sum}_{i=1}^n{f}^i{\mathbf{L}}^i{\left[I+{\sum}_{i=1}^n{f}^i\left({\mathbf{S}}_{\mathrm{Esh}}^i-\mathbf{I}\right){\mathbf{L}}^i\right]}^{-1}\right\}}^{-1}, $$
(3)

where I is the 4th-order identity tensor, fi is the volume fraction of inclusion, and Li is the localization tensor.

The rigidity tensor of the equivalent homogeneous material is a function of material microstructure parameters, the volume fraction of inclusions, the mechanical properties of matrix and trinclusions, and the geometry of fibers. For the case of a composite material reinforced with randomly oriented fibers, they are distributed in N families. Each of them has a particular orientation, and therefore they are considered as various phases [14, 25].

3.1. Components of Eshelby tensor

The nonzero components of the Eshelby tensor for an isotropic ellipsoidal inclusion with aspect ratio α = a3/a2 are given in [15, 26]. The case of cylindrical inclusions with elliptical cross sections is considered in [16, 27].

3.1.1. Eshelby tensor. The Eshelby tensor depends on fiber geometry and the Poisson ratio of polypropylene matrix.

The fiber diameter ranged from 7.41 to 360 μm. Estimating the particular random distribution of short Alfa fibers in the composite, we determined the Representative Elementary Volume (REV) of the entire structure, which will be used in the rest of the study, calculated the Eshelby tensor, and determined fiber diameters a1 and a2. These diameters were average of the areas of different random fiber cross sections. Finally, the values of the Alfa fiber were obtained from scanning electronical microscope (SEM) observations (Fig. 3) to calculate the average diameters (Fig. 4).

Fig. 3.
figure 3

Representative elementary volume (REV): SEM pictures of Alfa fibers.

Full size image
Fig. 4.
figure 4

Different cross sections of Alfa fibers with diameters varying between 7.41 and 360 μm.

Full size image

The Eshelby tensor, which considers that the fiber is elliptical, was modified by the shape of actual fractured specimens according to SEM observations (Fig. 3). After calculation, we obtain the Eshelby tensor

$$ {\mathbf{S}}_{\mathrm{Esh}}=\left[\begin{array}{cccccc}0.7050& 0.1257& 0.3323& & & \\ {}0.1278& 0.7082& 0.3344& & \left[0\right]& \\ {}0& 0& 0& & & \\ {}& & & 0.2508& 0& 0\\ {}& \left[0\right]& & 0& 0.2492& 0\\ {}& & & 0& 0& 0.2952\end{array}\right]. $$
(4)

3.2. Polypropylene rigidity tensor

For the sake of simplicity, the polypropylene matrix was assumed isotropic, meaning that the amorphous phase and the crystalline phase had the same properties as the entire matrix. The rigidity tensor of resin has the form

$$ {\mathbf{C}}^m=\left[\begin{array}{cccccc}{\lambda}^m+2{\mu}^m& {\lambda}^m& {\lambda}^m& & & \\ {}{\lambda}^m& {\lambda}^m+2{\mu}^m& {\lambda}^m& & \left[0\right]& \\ {}{\lambda}^m& {\lambda}^m& {\lambda}^m+2{\mu}^m& & & \\ {}& & & {\mu}^m& 0& 0\\ {}& \left[0\right]& & 0& {\mu}^m& 0\\ {}& & & 0& 0& {\mu}^m\end{array}\right], $$
(5)

Here, λm and μm are the Lamé coefficients of the PP matrix. They are determined by the relations

$$ {\lambda}^m=\frac{v_m{E}_m}{\left(1+{v}_m\right)\left(1-2{v}_m\right)}\;\mathrm{and}\;{\mu}^m=\frac{E_m}{2\left(1+{v}_m\right)}, $$

where νm and Em are Young’s modulus and the Poisson ratio of the PP matrix. After calculations, we obtain the following rigidity tensor of PP:

$$ {\mathbf{C}}^m=\left[\begin{array}{cccccc}3289.29& 2192.86& 2192.86& & & \\ {}2192.86& 3289.29& 2192.86& & \left[0\right]& \\ {}2192.86& 2192.86& 3289.29& & & \\ {}& & & 548.21& 0& 0\\ {}& \left[0\right]& & 0& 548.21& 0\\ {}& & & 0& 0& 548.21\end{array}\right]. $$
(6)

Note that, in the analytical equations of the Mori–Tanaka model, the rigidity tensor of matrix is denoted by C0; in our case, it is denoted by Cm.

3.3. Rigidity tensor of Alfa fibers

The properties of the fibers were difficult to assess experimentally, and we employed an inverse method. To this end, we selected a range of fiber Young’s moduli from 18.7 to 24.9 GPa [12, 28] and calculated the macroscopic Young’s modulus. This approach was used for the Alfa fibers treated for 24 and 48h.

The rigidity tensor of Alfa fiber is given by the expression

$$ {\mathbf{C}}^f=\left[\begin{array}{cccccc}{\lambda}^f(i)+2{\mu}^f(i)& {\lambda}^f(i)& {\lambda}^f(i)& & & \\ {}{\lambda}^f(i)& {\lambda}^f(i)+2{\mu}^f(i)& {\lambda}^f(i)& & \left[0\right]& \\ {}{\lambda}^f(i)& {\lambda}^f(i)& {\lambda}^f(i)+2{\mu}^f(i)& & & \\ {}& & & {\mu}^f(i)& 0& 0\\ {}& & & 0& {\mu}^f(i)& 0\\ {}& & & 0& 0& {\mu}^f(i)\end{array}\right], $$
(7)

Here, λf and μf are the Lamé coefficients of the fibers. They are determined by the relations

$$ {\lambda}^f=\frac{v_f{E}_f}{\left(1+{v}_f\right)\left(1-2{v}_f\right)}\;\mathrm{and}\;{\mu}^f=\frac{E_f}{2\left(1+{v}_f\right)}, $$

where νf and Ef are Young’s modulus and the Poisson ratio of the fiber. The Lamé constants were variable, because they depend on the value of Ef varying in the range mentioned previously, Ef∈ [18.7, 24.9] GPa [28]. Since we did not know the exact value of the longitudinal rigidity of Alfa fiber, we preferred to scan all values from the range.

3.4. Determination of the homogenized rigidity tensor

The ultimate aim of the calculation was to estimate the mechanical characteristics of the equivalent homogeneous material (EHM) for the PP/Alfa materials containing fibers treated for 24 and 48 h. Thus, the Eshelby tensor was introduced. The rigidity tensor of PP/Alfa-T24h with a volume fraction of fibers νf = 0.266 [28] was found to be

$$ {\mathbf{C}}^c=\left[\begin{array}{cccccc}6447.75& 3620.14& 2622.41& & & \\ {}3620.14& 6440.67& 2621.75& & \left[0\right]& \\ {}2622.41& 2621.75& 14648.79& & & \\ {}& & & 2177.95& 0& 0\\ {}& \left[0\right]& & 0& 2185.18& 0\\ {}& & & 0& 0& 2005.81\end{array}\right], $$
(8)

which is transversely isotropic with five independent components.

The properties of the fibers and matrix were as follows: Ef = 18.7-24.9 GPa, νf = 0.15 [4], Vf = 6.6-26.6 vol,%, Lf = 0.659, a1 and a2 will be determined; Em = 1.535 GPa, νm = 0.4.

3.5. Estimation of fiber Young’s modulus

Using the Mori–Tanaka model and varying fiber properties, we obtained Young’s modulus of fibers by the inverse process, starting from the macroscopic elastic modulus Ec of the composite (Fig. 5).

Fig. 5.
figure 5

Determination of Young’s modulus of the Alfa fibers treated for 24 and 48h.

Full size image

For the modulus Ec = 3558.35 MPa of the PP/Alfa-T24h composite, the corresponding analytical Young’s modulus of the fiber was Ef = 24,000 MPa and its Poisson ratio was ν12 = 0.2718. The analytical Young’s modulus Ef = 19,000 MPa and Poisson ratio ν12 = 0.3030 were deduced from the same curve for the longitudinal elastic modulus Ec = 3053.38 MPa of the PP/Alfa composite reinforced with fibers treated for 48 h.

Thus, the elastic modulus of the Alfa fiber treated for 24h (Ef = 24,000 MPa) was greater than that of the Alfa fiber treated for 48 h (Ef =19,000 MPa). This result seems logical and predictable, because the rigidity of the Alfa fiber treated for 24 h was higher than that of the fiber treated for 48 h [23].

4. Parametric Study

The previous numerical study allowed us to determining Young’s moduli of fibers treated for 24 and 48 h. In its turn, a parametric study provided a better understanding of the effect of various fiber volume fractions on the mechanical properties of the two composite materials investigated experimentally.

4.1. Effect of fiber volume fraction V f on Young’s modulus E 1

Figure 6a presents Young’s modulus vs. fiber volume fraction Vf for the PP/Alfa-T48h and PP/Alfa-T24h composite materials. A linear variation in Young’s modulus is seen. The Young’s moduli obtained by the numerical simulation were similar to those obtained experimentally for PP/Alfa-T48h and PP/Alfa-T24h in the previous experimental static study (see Table 2 in [23]).

Fig. 6.
figure 6

E1 (a), G12 (b) and ν12 (c) of PP/Alfa-T24h (1) and PP/Alfa t48h (2) vs. fiber volume fraction Vf.

Full size image

4.2. Effect of fiber volume fraction V f on the shear modulus G 12

The shear modulus G12 vs. Vf in the 2D cnfiguration (determined numerically by the Mori–Tanaka model) given in Fig. 6b shows the same effect. Indeed, a quasi-isotropic character is observed for both materials in Figs. 6a, b.

4.3. Effect of fiber volume fraction V f on the Poisson ratio ν 12

Knowing that the Poisson ratio of the composite cannot be determined experimentally, the Mori–Tanaka micromechanical model was used to evaluate it for the two composite materials in relation to the fiber volume fraction Vf. Results for both the materials are illustrated in Fig. 6c.

As is seen, the Poisson ratio decreases with growing Vf for both the PP/Alfa-T48h and PP/Alfa-T24h composites in contrast to the elastic moduli E1 and G12, which both increase.

Conclusion

The fatigue tests on composite materials made by reinforcing polypropylene with 30% of untreated and chemically treated Alfa fibers were performed to their static behavior by evaluating the mechanical characteristics experimentally. Based on the results obtained, the following conclusions were drawn.

  • The PP/Alfa composite with fibers treated for 24h was the strongest material owing to a better distribution of fibers in the matrix, which agreed well with data obtained in static tests.

  • A micromechanical modeling was carried out to validate the experimental results. The Mori–Tanaka model was used due to experimental difficulties in determining the mechanical characteristics of fibers parameters. Their Young’s modulus was obtained using an inverse method, by varying fiber properties found from the curve describing the macroscopic elastic modulus of the composites.

  • The Mori–Tanaka model, based on the Eshelby tensor, was modified to take into account the cylindrical cross section of fibers and the Poisson ratio of polypropylene matrix. An elliptical cross section of the fibers was introduced and examined using SEM observations and the fracture surfaces of composite samples.