**3.3 Density**

Different methods can be used including solid pycnometers or gas pycnometers [45–47]. The choice of gases (helium for example) or immersion liquids such as toluene, ethanol and xylene is decisive for quality results [46, 47]. Fibers must be dried for at least 72 h in a desiccator containing silica (previously regenerated). Fibers are then cut into lengths of 5–15 mm and then introduced into the pycnometer which is eventually placed in the desiccator for at least 24 h. Before carrying out the hydrostatic weighing with the immersion liquids, the vortex agitation of fibers to evacuate the microbubbles between needs to be done. Significant degassing could occur at this stage and provides information on the porosity rate of the fibers [28].

#### **3.4 Mechanical characterization**

#### *3.4.1 General considerations*

In general, PFs are suitable for reinforcing plastics (thermosets and thermoplastics) and textiles manufacturing thanks to their relatively high strength and low density. The tensile strength and the modulus of elasticity of PFs are very important characteristics for the use of fibers as reinforcements in composite and textile materials. However, the tensile test data for most fibers in service have yet to be studied, as the data found in the literature are scattered and often unreliable. In fact, methods used for the characterization are not identical. **Table 4** shows the tensile mechanical properties of some plant fibers compared to synthetic fibers [48]. The properties of the fibers and their structure depend on several factors such as the origin, variety, conditions of growth and harvesting of fibers associated with the treatments, the location in the stem, the presence or absence of a lumen, measurement techniques that vary greatly from one research team to another. These factors can make a difference for the same type of fiber and influence test results.

Selection of plant fiber implies a prior study of its mechanical properties, chemical resistance, dimensional stability, separation process, etc. It is worth recalling that linear cellulosic macromolecules are linked by hydrogen bonds and are closely associated with hemicelluloses and lignin, which confer stiffness to fiber. One of the issues of natural fibers is the scattered information and the differences in mechanical properties reported. Likewise, the lack of standards for both producers and users of these materials regarding methods to collect, process, post-process and characterize plant fibers underlines the complexity in the selection.

#### *3.4.2 Quasi-static tensile test*

Quasi-static tensile test is the method commonly used in the literature for the characterization of the mechanical properties of plant fibers in the longitudinal direction. This type of characterization presents challenges linked to the assembly and to the single nature of the fiber. In addition, the geometry of the plant fiber makes it often difficult to conduct the tests. Therefore, evaluation of the mean diameter along the fiber using a microscope is necessary for the performance of the test. The single fiber is mounted on a paper frame and a drop of glue is used to stick the fibers. The role of this paper frame is to facilitate the handling and alignment of the fiber on the jaws of the experimental device as shown in **Figure 8** [32].

*Extraction, Applications and Characterization of Plant Fibers DOI: http://dx.doi.org/10.5772/intechopen.103093*


#### **Table 4.**

*Mechanical properties of some selected plant fibers versus synthetic fibers [48].*

**Figure 8.**

*Tensile test and gripping tab specimens for plant fibers.*

The large dispersion of the mechanical properties of the plant fibers observed (**Figure 9**) is mostly related to the test conditions. The research work by Ntenga et al. [14] focused on the choice of the stress speed and the gage length, in order to keep the deformation in the elastic domain and reduce this dispersion during the tests. The machine cross-head speed of 1 mm/min and the gage length of 10 mm were found to cause less dispersion of the mechanical properties in a tensile test.

### *3.4.3 Nano-indentation test*

Nanoindentation is a technique used to characterize the longitudinal and transverse mechanical properties of fibers at the cell wall scale. Commonly measured properties are Young's modulus and material hardness. In the literature, nanoindentation tests have been carried out to access both transverse and longitudinal mechanical properties on wood fibers [31, 49] and recently on flax fibers [50]. According to Cisse [51], nanoindentation only gives access to local behavior of the fiber, and the identification of mechanical properties requires knowledge and use of a behavior model. The testing technique consists of applying a force to the indenter and taking the area of the indentation, in order to determine the Young's modulus and the hardness of the material (**Figure 10(a)** and **(b)**).

A typical set of nanoindentation tests results [53] is shown in **Figure 11**.

Differences in transverse and longitudinal modulus noted between the fibers can be explained not only by the differences in micro-fibrillary angles but also by the rate of cellulose that varies between fibers. Hemp and sisal in particular have a cellulose content of around 60%, while that of flax is over 75%; however, the mechanical properties of cellulose are much superior to those of lignin, hemicelluloses and pectins, other constituents of natural fibers [50].

#### *3.4.4 Dynamic mechanical analysis*

A large amount of work exists in the field of vibration-based non-destructive testing (NDT) including an extensive survey of over 300 papers by Kong et al. [54]. Indeed, the vibration-based technique has been a very active area of research for many years, however, has always dealt with rigid bodies. As an extension of the use of

**Figure 9.** *Tensile stress/strain curves for the four cross-head speeds of gage length 10 mm [14].*

*Extraction, Applications and Characterization of Plant Fibers DOI: http://dx.doi.org/10.5772/intechopen.103093*

*(a) Nano indentation experimental device and (b) indentor impression Berkovich [52].*

**Figure 11.** *Transverse modulus of plant fibers obtained in nano indentation.*

this technique, the purpose of this section is to present the applicability of the lowfrequency vibration-based technique towards estimation of dynamic Young's modulus of natural fiber-based materials, initially having no bending stiffness. This technique enhances the applicability of non-contact acoustic non-destructive testing to the estimation of dynamic characteristics of thin materials, where the current standard method [55] is not applicable.

Let us consider a thin rectangular specimen having a length *b*, a width *a*, a thickness *h* and a density *ρ*. **Figure 12** shows the specimen configuration in a Cartesian coordinate system at equilibrium (i) and vibrating in flexural mode (ii) and (iii).

The specimen, considered as a membrane, initially has no bending stiffness. It is then slightly stretched in the y-direction, in order to make it possible to vibrate transversally (i.e. in the *z*-direction). The tensile force *F* is assumed to remain constant during small vibrations in the *y-z* plane.

In general, for a specimen having intrinsic elasticity, the equation of motion is expressed as follows:

$$\frac{\partial^2 \xi}{\partial t^2} - c^2 \left( \frac{\partial^2 \xi}{\partial \mathbf{x}^2} + \frac{\partial^2 \xi}{\partial t \mathbf{y}^2} \right) + d^2 \left( \frac{\partial^2 \xi}{\partial \mathbf{x}^2} + \frac{\partial^2 \xi}{\partial t \mathbf{y}^2} \right)^2 \xi = \frac{p(\mathbf{x}, \mathbf{y}, t)}{\rho h} \tag{1}$$

where *ξ* is the displacement normal to the plane (*x,y*) which coincides with the equilibrium position of the membrane, *c* is the velocity of propagation of bending waves, *p* is the external pressure on the surface of the membrane. *c* and *d* are defined as follow:

$$\sigma = \sqrt{\frac{T}{\rho h}} \text{ and } d^2 = \frac{Eh^2}{12\rho(1-\nu^2)}\tag{2}$$

where *E*,*T* and *ν* are the elastic modulus, tensile force per unit length of the edge, and Poisson's ratio.

The frequency equation with the fixed-fixed boundary condition shown in **Figure 12** above was derived in Mfoumou et al. [56] to obtain the frequency of vibration *ωmn* of each bending mode as follows:

$$\rho\_{\rm min} = c \sqrt{\left(\frac{\pi m}{a}\right)^2 + \left(\frac{\pi n}{b}\right)^2} \left\{ 1 + \frac{d^2}{2c^2} \left[ \left(\frac{\pi m}{a}\right)^2 + \left(\frac{\pi n}{b}\right)^2 \right] \right\}, m, n = 0, 1, 2, \dots \tag{3}$$

where *m* and *n* are the mode numbers.

**Figure 12.**

*Specimen configuration (i): undeformed, and vibrating at (ii): fundamental frequency, (iii): second frequency in flexural mode.*

For a plant fiber-based material considered as a membrane; therefore, no account of intrinsic elasticity is taken so that Eq. (3) is simplified, and the normal frequencies equation is expressed as:

$$\alpha\_{mn} = c \sqrt{\left(\frac{\pi m}{a}\right)^2 + \left(\frac{\pi n}{b}\right)^2}, m, n = 0, 1, 2, \dots \tag{4}$$

The Young's modulus can therefore be determined using the flexural resonance method by monitoring normal modes of vibration. These modes for an oscillating system are special solutions where all the parts of the system are oscillating with the same frequency. At these modes, considering only bending modes in the length direction (*m* = 0), the relationship between the frequency in hertz and the state of strain was established as follows [56]:

$$f\_{0n}^2 = \frac{E.n^2}{4.\rho.b^2}.e\tag{5}$$

thus, enabling extraction of the constant *E* from experimentally measured normal mode frequencies and corresponding strains.

#### *3.4.5 Creep test*

Both creep experiment and relaxation experiment are two techniques commonly used to characterize the delayed behavior of 'conventional' materials. A creep test consists of imposing an almost instantaneous stress load on the plant fiber and maintaining it constantly over time and then proceeding to a discharge. The resulting deformation under the action of the load is creep, and that under the action of discharge is recovery. In general, the creep responses can be broken down into three stages depending on the strain rate as shown in the following **Figure 13**. The first stage in which creep occurs at a decreasing rate is called primary creep; the second step, commonly called secondary creep, is carried out at a relatively constant speed; and the third stage, tertiary creep, occurs at an increasing rate and terminates with material fracture.

The creep test was successfully carried out on an elementary hemp fiber and the results allowed it possible to highlight the viscoelastic nature of the plant fiber [51]. **Figure 13** shows the creep test results obtained.

#### *3.4.6 Relaxation test*

#### *3.4.6.1 The context*

When a constant strain is applied to a material for a long period, cross-links or the primary bonds that form between molecules start breaking with time and spontaneously lose their bonding capability. High level of strain or long period is the main reason for intermolecular bond breakage, thus creating stress decay over time, called stress relaxation. The rate of bond breakage influences the rate of stress relaxation. Other factors control the rate of bond breakdown, such as stress on the bond, chemical interference, molecular chain mobility which allows molecular chains to move out from their position. The behavior of stress relaxation in plant fibers is also influenced by temperature, humidity, and strain levels. The stress relaxation tests are therefore mainly performed with different ranges of temperature, humidity and strain levels. The time taken to reach the end of relaxation is called relaxation time. From other studies, it is reported that at higher temperature relaxation time becomes shorter, while at lower temperature it becomes longer but the shape of relaxation does not change with temperature [57]; moreover, the variation of strain level affects the stress relaxation [58]. The literature also reports the sensitivity of this class of material to loading-directionality, and ductile and brittle phenomena [59].

#### *3.4.6.2 Stress relaxation measurement*

During structural design, the properties of the material must be considered. Elastic Modulus is one of the most important material properties describing the stiffness of the material. When a force is applied to an object, modulus of elasticity or elastic modulus gives the mathematical description of the object's tendency to be deformed elastically.

In orthotropic materials such as wood-based natural fibers, the strain quickly increases linearly with the stress, then exhibit a nonlinear behavior when the strain exceeds the proportional limits. When the stress relaxation tests are conducted for a very small deformation, the viscoelasticity of the material can be considered linear. During stress relaxation test, the material relieves stress over time as well as the elastic modulus of material *E t*ð Þ also decreases with time at a constant temperature. According to linear viscoelastic material [60], the elastic modulus relaxation can then be defined as:

$$E(t) = \frac{a(t)}{s\_o} \tag{6}$$

where, *so* is the constant strain and *a t*ð Þ is stress of material as a function of time. Indeed, elastic modulus relaxation is the relaxation of modulus of elasticity of material.

#### *3.4.6.3 Sample, experiments, and results*

A rectangular strip of specimen is placed between the clamps of the tensile test machine (see **Figure 8**), and it is slightly loaded within its elastic region. The specimen is tested in uniaxial stress-state at a strain rate of 1 mm/mm with 0.4% strain changes. The elongation is kept constant at 0.4% strain level (1 mm extension) for 5400 s and time, stress, and strain are recorded.

Experiments were carried out for paperboard (PPR) without crack and PPR with crack. Five specimens were tested for each case and each experiment continued for 5400 s (1.5 h) with 1 mm extension. The reason for taking 1 mm extension was to keep the deformation within the elastic region.

The stress relaxation of each specimen was monitored and analyzed at constant. elongation. The load, stress and time data for constant strain were obtained from the experiments. From the testing of five specimens in each case, we have plotted stress versus time curves. The plotted stress relaxation of PPR without and with the presence of a side crack is presented in **Figure 14**.

**Figure 15** show the stress relaxation behavior of PPR at different strain levels (two different extension levels, 1 mm and 0.5 mm).

#### *3.4.6.4 Formulation of relaxation*

The data obtained from the stress relaxation experiments are decreasing type of data with function of time and this type of data can be fitted to the poly-exponential function of the following form:

$$y(t) = \sum\_{i=1}^{N} a\_i e^{-b\_i t} \tag{7}$$

where, *ai* and *bi* are the unknown parameters. Several methods have been developed to estimate these parameters. The most available methods are graphical method, regression-difference equation method, method of partial sums, Fourier Transform method, Foss's method [61] for a sum of two exponentials. However, their uses are limited. For example, the graphical method is not suited where there are consistent fluctuations, regression-difference method and method of partial sums are only appropriate for equally spaced data and Fourier Transform method is suitable for

**Figure 14.** *Stress relaxation of paperboard with and without crack.*

**Figure 15.** *Stress relaxation of paperboard for 1 mm and 0.5 mm extension.*

exponentially spaced observation. On the other hand, the use of Foss method is broader than any other method, and even not equally and exponentially spaced data can be treated using this method [61].

#### *3.4.6.5 Model parameters extraction using experimental data*

The parameters of a set of mechanical models can be calculated from experimental data. MATLAB, for example, can be used to extract the parameters from the data. To analyze the suitability of the mechanical model with the experimental stress relaxation, Maxwell Model, Two-unit Maxwell Model, Modified Two-unit Maxwell Model, Standard linear solid model are constructed and then compared with the experimental relaxation. Analytical description of these models is given in [62].

In Ref. [56] we have chosen Foss method to develop curve fitting for all models and then compared with the experimental relaxation. Whereas in Ref. [15] we used the Zapas-Phillips method. The best-fitted model with the experimental data was then selected to analysis all experimental data and mathematically stress relaxation equations were derived.

To predict the stress relaxation behavior of natural fibers, we derived the mathematical equations for PPR with and without presence of crack. These equations were derived by the Modified Two-unit Maxwell model which suits best with the experimental result. Though we carried out our experimental tests with five specimens for each kind of test and among them three specimen-data were taken into consideration, but here we will construct the stress relaxation equation for only one specimen for each case.

Below the comparison, diagrams between experimental relaxation data and the Modified Two-unit Maxwell are shown in **Figures 16** and **17**. The stress relaxation equation for each case is derived using Modified Two-unit Maxwell model.

#### *3.4.7 Inverse characterization*

Suitability of materials inverse characterization, destructive or non-destructive, is widely investigated [52, 63, 64]. Furtado et al. [65] used an ultrasound shear wave

**Figure 16.** *Stress relaxation of paperboard—curve fitting.*

**Figure 17.** *Stress relaxation of paperboard with crack–curve fitting.*

viscoelastography method to determine the viscoelastic complex shear modulus of macroscopically homogeneous tissues. Ilczyszyn et al. [66] performed the mechanical characterization of flax fibers using an inverse optimization simplex method.

The aim here is to use macro-micro approaches to achieve an efficient estimation of the fiber properties. In fact, homogenization laws of the micromechanics of the elastic/viscoelastic behavior of composite materials provide relationships of the properties of these materials in terms of their constituents' properties. For an orthotropic material, the knowledge of its off-axes elastic modules in a set of *θ* directions leads to the calculation of the fibers' elastic constants for instance. Analytical relationships of elastic constants that account for the orientation of the fibers can be found in the literature. A presentation of an inverse method based on the composite cylinder assembly proposed by Hashin [67] to characterize the anisotropy of plant fibers was discussed in Ref. [64] for a transversely isotropic 2-phase composite. In this case, the

matrix and fiber phases are assumed isotropic and transversely isotropic respectively. Using a stress field in cylindrical coordinates and applying Hooke's law gives the following results:

ð Þ *x*, *y*, *z* stand for off-axis Cartesian coordinates (reference coordinate system).

ð Þ 1, 2, 3 represent material axes of a unidirectional composite.

For a tensile test in the *x*-direction, the Young modulus is expressed as:

$$\frac{1}{E\_{\text{xx}}} = \frac{1}{E\_{11}}\cos^4\theta + \left(\frac{1}{4G\_{23}} + \frac{1}{4K\_{23}} + \frac{\nu\_{12}^2}{E\_{11}}\right)\sin^4\theta + \left(\frac{1}{G\_{12}} - 2\frac{\nu\_{12}}{E\_{11}}\right)\sin^2\theta\cos^2\theta \tag{8}$$

There are five independent properties to be determined ð Þ *E*11, *G*23, *G*12, *K*23, *ν*12*:*

Analytical expressions of the five properties in terms of fiber and matrix phase properties and the volume fractions are given by:

$$E\_{11} = E\_x^f \phi^f + E\_x^m \phi^m - \frac{\nu\_{x0}^f}{\Delta} \left[ E\_x^f \left( \frac{\nu\_{rx}^f}{E\_r^f} + \frac{\nu\_{x0}^f}{E\_x^f} \right) - 2\nu^m \right] \phi^m \tag{9}$$

$$
\omega\_{12} = -2\frac{1 - \left(\nu^m\right)^2}{\mathcal{E}^m} \frac{\nu\_{x0}^f - \nu^m}{\Delta} + \nu^m \tag{10}
$$

$$k\_{23} = \frac{E^{\mathfrak{m}} \phi^{f} \left(I\_{1}I\_{4} - I\_{2}I\_{3}\right)}{2\left[-(1-\nu^{\mathfrak{m}})\phi^{f}\left(I\_{4}L\_{1} - I\_{2}L\_{2}\right) + (1+\nu^{\mathfrak{m}})\phi^{f}\left(I\_{4}L\_{1} - I\_{2}L\_{2} - I\_{4}I\_{1} + I\_{2}I\_{3}\right) - \nu^{\mathfrak{m}}\phi^{\mathfrak{m}}\left(I\_{1}L\_{2} - I\_{3}L\_{1}\right)\right]} \tag{11}$$

$$\frac{\bar{G}\_{12}}{G^m} = \frac{G^m \phi^m + G\_{rx}^f (\phi^f + \mathbf{1})}{G^m \left(\phi^f + \mathbf{1}\right) + G\_{rx}^f \phi^m} \tag{12}$$

$$\frac{\bar{G}\_{23}}{G^m} = 1 + \frac{\phi^f}{G^m \left(G^f\_{r\theta} - G^m\right) + (k^m + \frac{7}{3}G^m)\frac{f}{\left(2k^m + \frac{8}{3}G^m\right)G\_{rx}}} \tag{13}$$

with

$$\mathbf{G}\_{r\theta}^{f} = \frac{E\_r^f}{2\left(\mathbf{1} + \nu\_{r\theta}^f\right)} \mathbf{G}^m = \frac{E^m}{2(\mathbf{1} + \nu^m)} k^m = \frac{E^m}{3(\mathbf{1} - 2\nu^m)}\tag{14}$$

$$\Delta = \left[ \frac{\nu\_{r\theta}^f}{E\_r^f} - \frac{1}{E\_\theta^f} + \nu\_{x\theta}^f \left( \frac{\nu\_{rx}^f}{E\_r^f} + \frac{\nu\_{x\theta}^f}{E\_x^f} \right) \right] \frac{\phi^m}{\phi^f} - \frac{\nu^m}{E^m} \frac{\phi^m}{\phi^f} - \frac{1}{E^m} \frac{\phi^f + 1}{\phi^f} + 2 \frac{(\nu^m)^2}{E^m}; \tag{15}$$

$$I\_1 = \phi^m \left(\frac{\nu\_{rz}^f}{E\_r^f} + \frac{\nu\_{x\theta}^f}{E\_x^f}\right) + \frac{2\nu^m}{E\_m}\phi^f; I\_2 = -\left(\frac{\phi^m}{E\_x^f} + \frac{\nu\_{x\theta}^f}{E\_m}\right) \\ L\_1 = -\left(\frac{\nu\_{rz}^f}{E\_r^f} + \frac{\nu\_{x\theta}^f}{E\_x^f}\right); \tag{16}$$

$$I\_3 = \left(\frac{\nu\_{r\theta}^f}{E\_r^f} - \frac{\nu\_{x\theta}^f}{E\_\theta^f}\right) \phi^m - \frac{1}{E\_m} \left(\nu^m - \phi^f - 1\right);\tag{17}$$

$$I\_4 = \frac{\nu\_{x\theta}^f}{E\_x^f} \phi^m + \frac{\nu^m}{E\_m} \phi^f L\_2 = -\left[\frac{\nu\_{r\theta}^f}{E\_r^f} - \frac{1}{E\_\theta^f} + \frac{1}{E\_m}(\nu^m - 1)\right] \tag{18}$$

Eqs. (8)–(13) are then solved for *E <sup>f</sup> <sup>r</sup>* , *E <sup>f</sup> <sup>z</sup>* , *<sup>ν</sup> <sup>f</sup> rz*, *ν <sup>f</sup> <sup>z</sup><sup>θ</sup>*, *<sup>G</sup> <sup>f</sup> <sup>r</sup><sup>θ</sup> and G <sup>f</sup> rz* using a multiparameter optimisation algorithm.
