Open access peer-reviewed chapter

Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites for Structural Applications

By Christian Emeka Okafor and Christopher Chukwutoo Ihueze

Submitted: November 22nd 2019Reviewed: December 18th 2019Published: July 15th 2020

DOI: 10.5772/intechopen.90890

Downloaded: 53

Abstract

Plantain fiber-reinforced composite materials have demonstrated significant properties that are applicable in structural design and development. However, two major concerns arise in relation to the obvious material anisotropy and challenges imposed by structural discontinuity encountered as need for use of fasteners arises. The study assesses the extent of variation of elastic properties ( E x , E y , G xy , v xy , v yx , m x , m y ) with fiber orientation using MATLAB functions while considering the extent of variation of the tangential stresses around an idealized functional hole edge. The tensile strength of 410.15 and 288.1 MPa was recorded at 0° fiber orientation angle, while 37.3397 and 33.133 MPa were obtained at fiber orientation angle of 90° for Plantain Empty Fruit Bunch Fiber Composite (PEFBFC) and Plantain Pseudo Stem Fiber Composite (PPSFC), respectively. The tangential stress distribution at hole edge indicated maximum stress value of 119.15 and 100.587 MPa at angular position ø = 90° for PEFBFC and PPSFC, respectively. Judging from various failure indices considered, failure will be initiated at ø = 70° for PEFBFC with stress concentration factor of 2.53 and ø = 65° for PPSFC with stress concentration factor of 2.13, which are less than the stress concentration around the peak stress when angular position is 90°. Both PEFBFC and PPSFC showed similar trends in response to the design scenario considered.

Keywords

  • elastic properties
  • structural application
  • plantain fiber
  • composites
  • matrix

1. Introduction

These days several shortcomings have been observed with respect to the utilization of synthetic fiber-reinforced polymers, hence empowering the drive for more utilization of plant fiber composite in structural designs. The primary weaknesses of synthetic fiber-reinforced polymers which includes issue in afterlife disposal and non-biodegradability are completely settled by utilizing plant fibers in polymer reinforcements. Reinforced composites are prime choice for light weight structural designs and automotive body parts assembly.

Extensive literature on recent advancements in reinforced composites and its reliability are reported in classical reports of Dehmous et al. [1], Okafor et al. [2], Xie and Wang [3], Beaumont et al. [4], Pei et al. [5], Bittrich et al. [6], Prasad et al. [7], Wang et al. [8]. The hygrothermal efficiencies has been reported by Foulc et al. [9], Shettar et al. [10]. In addition, the utilization of reinforced composites predicates the reuse of domestic and agricultural residues. For example, vehicles made with fiber-reinforced composites are lighter and run on smaller engines which produce fewer emissions to the environment. Most items produced using natural fiber composites is a win-win for manufacturers. Most composite material ventures can utilize their genius green item data to build deals because customers comprehend the ecological dangers of synthetic assembling.

Due to scarcely available information regarding some new material response to structural discontinuity, superior properties of those composites are seriously compromised by the utilization of bizarrely enormous factor of safety in design. Accordingly, the quick fate of composite materials as a class of innovative materials may depend more on clear assessment of its performance in various structural design scenario. All inclusive acknowledgment of composites as eco-friendly materials will therefore depend especially on the certainty of the designer and client about the variation of its elastic properties. In a typical fiber-reinforced composites, the polymer matrix serves as a binder and deforming most times for stress distribution purposes. There are different options in the choice of matrix/fibers and the general composition of reinforced composites is shown in Figure 1. The figure identified the three major categories of polymers to include biopolymers, thermoplastics and thermosets. Biopolymers are chain like atoms created by organic biomass. Exceptional nontoxicity and biodegradable properties of biopolymers boosts their applications in composites formulation, hardware and restorative gadgets. Fuse of nano-sized support in the biopolymers to improve the properties contributes to the upgrade functional applications of the matrix.

Figure 1.

Composition of reinforced composites.

Though thermosets and thermoplastics sound similar, they have very different properties and applications [11]. Thermosets typically changes from fluid to solid state after curing chemical reaction initiated by addition of a catalyst, cross-linker, and curing agent. In the course of the chemical reaction, the material solidifies as a result of cross-linking and formation of longer molecular networks. Subsequently, any further exposure to high heat will cause the material to degrade unlike thermoplastic parts that melts and softens whenever exposed to elevated temperature, thermoset simply become set in their physical and mechanical properties after an initial treatment and therefore are no longer affected by additional heat exposure.

Again, thermoplastics are dissolvable plastics. At temperature above liquefying point, the thermoplastic condenses. The thermoplastic sets once again into solid state when the temperature is reduced and the handling temperature dips under its melting point. This inherent characteristic of thermoplastics enables its softening when heated above its melting point and re-forming as the temperature decreases below the melting point. Most of the times, the expenses of materials for creating thermoset are lower when contrasted with thermoplastic. Also thermoset is regularly simple for wetting the reinforcements and shaping last composites items. Thermoplastics will in general be harder than thermosets and require no refrigeration as uncured thermosets as often as possible do, and can be more effectively be reused and fixed. Elastomers are typically thermosets (requiring vulcanization).

Obviously literature has indicated several approximate relationships between some reinforced composites elastic constants and the homogenized modules of elasticity [12, 13, 14]. Also recent research have reported the possibility of measuring variation elastic constants of materials using ultrasonic methods [15, 16, 17, 18]. However, scanty research is available on strength analysis and variation of elastic properties in plantain fiber/polyester composites, a gap that the present study seeks to fill.

2. Background to plantain cultivation and utilization as reinforcement in polymer composites

An expanded enthusiasm for the utilization of agricultural wastes in development of reinforced composites has been on the increase. Natural fibers extracted from bio wastes offer a few points of interest over woody biomass, since they are accessible in huge amounts as leftovers and agricultural wastes [19, 20]. The plantain pseudo stem (PPS) and empty fruit bunch (EFB) strands presented in this chapter are agricultural by-products that are biodegradable and locally available from renewable agricultural sources with potentials to contribute to reduction in environmental pollution when utilized in large scale as polymeric reinforcements.

Plantain fruit is one of the staples in Nigeria and it is mainly cultivated in the tropics and ethnic enclaves [21]. It is evaluated that 70 million individuals in West and Central Africa derive most of their nourishment and vitality requirements from plantain fruit plant [22, 23]. Plantain fruit has a fare potential in the light of its huge cultivation and consumption in Nigeria and many other African countries. Akinyemi et al. [24] reported that plantain plant is the third most important plant grown after cassava and yam in Nigeria; collaborating, Kaine and Okoje [25] showed that plantain production is a very profitable enterprise as every ₦1 naira invested in plantain production yields a return on investment of about ₦12.60 kobo. In a study about economics of plantain production Kainga et al. [26] found that the associated high return on investment and short maturity period for plantain contributes to its massive cultivation in Nigeria.

Africa cultivates over 50% of worldwide production of plantain and Nigeria is one of the biggest plantain producing nations in the planet. Therefore the interest in plantain plant fiber for polymer reinforcement was as a result of its abundance and accessibility as it is evaluated that over 15.07 million tons of plantain fruit is produced each year in Nigeria with about 2.4 million metric tons produced from southern Nigeria [27, 28]. Plantain fiber also satisfied over 50% conditions for eco-friendly materials as shown in Figure 2 and can make strong reinforcement in composites. A composite which can be characterized as a physical blend of at least two unique materials, has properties that are commonly superior to those of any of the establishing materials. It is important to utilize blends of materials to tackle issues in light of the fact that any one material alone cannot suffix effectively in eco-friendly materials technology at an acceptable performance [29].

Figure 2.

Properties of eco-friendly building materials.

Cadena Ch et al. [30] and Adeniyi et al. [31] orchestrated the potentials of natural fibers from plantain pseudo stem for use in fiber-reinforced composites. It is therefore important to assess the extent of variation of elastic properties in plantain fiber-reinforced polyester composites to guard against out of plane failure during structural applications. Unfortunately most studies involving plantain fiber-reinforced composites has dwelt on assessment of tensile, flexural and hardness properties [32], optimization of hardness strengths [33], effect of water and organic extractives removal [34], effects of fiber extraction techniques [35], optimization of flexural strength [36], compressive and impact strength evaluation [37, 38, 39], effect of high-frequency microwave radiation [40], effect of chemical treatment on the morphology [41], implications of interfacial energetics on mechanical strength [42]. Although Ihueze, Okafor and Okoye [43] has reported the longitudinal (1) and transverse (2) properties of plantain fiber-reinforced composites in Figure 1, there is still need to establish the essential elastic constants at directions other than the material axis directions 1–2.

The present research efforts will further drive the interests of structural designers in the use of plantain fiber-reinforced composites because the superior strength of materials are rarely utilized to full as a result of incomplete knowledge of elastic properties which are related to various fundamental solid-state characteristics of the composites. In essence, the elastic constants of plantain fiber-reinforced composites is expected to describe the material response to external stressor and provide useful information about bonding characteristics and structural stability. Kenedi et al. [44] assessed the orthotropic elastic properties in a sandwiched composites laminates and proposed models for estimating the orthotropic elastic properties of composite materials. Hwang and Liu [45] reported that elastic modulus and Poisson’s ratio vary significantly with different braid angles in carbon fabric/polyurethane composites. Ren et al. [46] reported that elastic modulus and tensile strengths are overly dependent on the angles of fiber orientation. Kumar et al. [47] studied the influence of ±0°, ±10°, ±30°, ±40°, ±45°, ±55°, ±65°, ± 75°, and ± 90° angle ply on mechanical properties of glass-polyester composite laminate and found that that glass/polyester with 0° fiber orientation angle yields’ high strength. Cordin et al. [48] experimentally examined the effect of 0°, ±22.5°, ±45°, ±67.5° and 90° fiber orientation angles on the mechanical properties of polypropylene-lyocell composites. Ihueze et al. [49] optimally determined the tensile strengths of plantain fiber-reinforced composites considering 30°, 45° and 90° fiber orientation angles. The application of these previous studies are limited to fiber orientation angles studied, however failure may be initiated from angles other than those considered hence the need to verify the variation of important elastic constants within a wide range of fiber orientation coverage are necessary.

Additionally, researchers have provided various theoretical strategies for determination of elastic constants in reinforced composites using software codes to cover the wide range of fabricating conditions. Jules et al. [50] ascertained the effect of fibers orientation on the predicted elastic properties of long fiber composites using Monte-Carlo simulation to assign the in plane and out of plane orientation values. Venetis and Sideridis [51] developed a model to find the approximate elastic constants in unidirectional fiber-reinforced composite materials in terms of the constituent material properties. Cuartas [52] theoretically determined the elastic properties in CFRP composites which compared favorably with other methods based on tensile tests and ultrasonic characterization. Rahmani et al. [12] found that MATLAB codes are capable of predicting the elastic constants of composites with reasonable confidence.

3. Mathematical framework for assessment of extent of variation of elastic properties in plantain fiber-reinforced polyester composites

One significant property of composite materials is their plainly visible macroscopic anisotropy, which means that the properties estimated in the longitudinal direction are by far not the same as those measured in transverse direction. There are no material planes of symmetry, and normal loads create both normal strains and shear strains. This anisotropic characteristic of reinforced composites results in low mechanical properties in the out-of-plane orientation where the matrix carries the primary load. Consequently the application of reinforced composites is limited in scenarios prone to complex load paths such as lugs and fittings [53].

By implication any endeavor to comprehend the structural application of plantain fiber-reinforced polyester composite must assess the inborn anisotropy. Composites are a subclass of anisotropic materials that are delegated orthotropic. Orthotropic materials have properties that are unique in three directions with perpendicular axes of symmetry. In this way, orthotropic mechanical properties depend heavily on fiber orientation. An orthotropic ply is thus defined as that having two different material properties in two mutually perpendicular directions at a point and the two mutually perpendicular directions also form the planes of material properties symmetry at the point.

3.1 Determination of reduced stiffness matrix and compliance matrix

Considering two possible loading conditions of longitudinal (direction 1) and transverse (direction 2) in the matrix as shown in Figure 3, the resulting direct strains from Hooks law are respectively e1=v21σ1E1and e2=v12σ1E1where v12= major Poisson’s ratio and v21= minor Poisson’s ratio.

Figure 3.

Stressed single thin composite lamina. From Ref. [43].

Hence the application of both direct stresses σ1and σ2will yield corresponding strains as follows:

e1=σ1E1v21σ2E2E1
e2=σ2E2v12σ1E1E2

Putting Eqs. (1) and (2) in a matrix form, yields

e1e2=1E1v21E2v12E11E2σ1σ2E3
σ1σ2=E11v12v21v21E11v12v21v12E21v12v21E21v12v21e1e2E4

Eq. (3) is symetric about the loading diagonal such that

v21E2=v12E1E5

A combined effect of shear and direct stresses gives the reduced stiffness matrix as in Eq. (6) and reduced compliance matrix as in Eq. (7)

σ1σ2τ12=E11v12v21v21E11v12v210v12E21v12v21E21v12v21000G12e1e2e12E6
e1e2e12=1E1v21E20v21E11E20001G12σ1σ2τ12E7

Minor Poisson’s ratio is the strain resulting from a stress in the axial direction, Ihueze et al. (2013) calculated the major Poisson’s ratio v12for plantain fiber/polyester composites. However, there is need to further assess the minor Poisson’s ratio v21using Eq. (5) and Table 1 as follows

CompositesProperties
Su1(MPa)Su2(MPa)Sy(MPa)E1(MPa)E2(MPa)E(MPa)v12τmax(MPa)G12(MPa)
PEFBFC410.1537.339733.6914,9227030.9629990.100.3819.31003622.99
PPSFC288.1033.133029.2413027.56817.1759146.3050.2915.57003332.835

Table 1.

Evaluated mechanical properties of plantain fibers and plantain fibers reinforced polyester composites. From Ref. [43].

Su1,Su2are tensile strengths in the longitudinal and transverse directions respectively.

v21, PEFBFRP=E2v12E1=7030.9620.3814,922=0.179
1v12v21=10.380.179=10.068=0.93
v21, PPSFC=E2v12E1=6817.1750.2913027.5=0.152
1v12v21=10.290.152=10.068=0.956

The reduced stiffness matrix () for PEFBFC and PPSFC is obtained from Eq. (6).

For PEFBFC

E11v12v21=1492210.380.179=16011.07
v21E11v12v21=0.17916011.07=2865.98
E21v12v21=7030.96210.380.179=7544.113
PEFBFC=16011.072865.9802865.987544.1130003622.99MPa

For PPSFC

E11v12v21=13027.510.290.152=13628.23
v21E11v12v21=0.15213628.23=2071.49
E21v12v21=6817.17510.290.152=7131.53
PPSFC=13628.232071.4902071.497131.530003332.835MPa

The reduced compliance matrix (β) is obtained from Eq. (7).

For PEFBFC

1E1=114922=6.7×1051/MPa
v21E2=0.1797030.962=2.5×1051/MPa
1E2=17030.962=1.4×1041/MPa
1G12=13622.99=2.8×1041/MPa
βPEFBFC=6.7×1052.5×10502.5×1051.4×1040002.8×1041/MPa

For PPSFC

1E1=113027.5=7.7×1051/MPa
v21E2=0.1526817.175=2.2×1051/MPa
1E2=16817.175=1.5×1041/MPa
1G12=13332.835=3.0×1041/MPa
βPPSFC=7.7×1052.2×10502.2×1051.5×1040003.0×1041/MPa

3.2 Transformation of elastic constants

The know of the stress-strain relationship in the plantain/polyester composite is completely comprehended by knowing the associated independent engineering elastic constants (E 1, E 2, G 12 and v 12) as previously determined by Ihueze et al. (2013) as in Tables 1 and 2. However, there is need to further establish these elastic properties at different directions of fibers other than directions 1 and 2. Datoo [54] derived various expressions for determination of the elastic properties in the reference axes x-y for any fiber orientation as expressed in Eqs. (8)(14) where c=cosθ, s=sinθ, Ex, Ey, Gxy, vxy, mxand myare the elastic properties at any fiber orientation θrelative to a reference direction x-y.

PropertyPolyester resin
Density (g/cm3)1.2–1.5 (1400 kg/m3)
Young modulus (MPa)2000–4500
Tensile strength (MPa)40–90
Compressive strength (MPa)90–250
Tensile elongation at break (%)2
Water absorption 24 h at 20°C0.1–0.3
Flexural modulus (GPa)11.0
Poisson’s ratio0.37–0.38
Plantain pseudo stem fibers
Young modulus (MPa)23,555
UTS (MPa)536.2
Strain (%)2.37
Density (kg/m3)381.966
Plantain empty fruit bunch fibers
Young modulus (MPa)27,344
UTS (MPa)780.3
Strain (%)2.68
Density (kg/m3)354.151

Table 2.

Mechanical properties of plantain fibers and polyester resin. From Ref. [43].

1Ex=c4E1+s4E2+c2s21G122v12E1E8
1Ey=s4E1+c4E2+c2s21G122v12E1E9
1Gxy=c2s24E1+4E2+8v12E1+c2s21G12E10
vxy=Exc4+s4v12E1c2s21E1+1E21G12E11

The minor Poissons ratio with respect to the material reference axes is obtained from Eq. (5) such that

vxyEx=vyxEyE12
mx=Exc3s1G122v12E12E1cs31G122v12E12E2E13
my=Eycs31G122v12E12E1c3s1G122v12E12E2E14

3.3 Variation of engineering elastic constants with fiber orientation θ

Considering fiber orientation θ° ranging from 0° to 90° in increments of 5° the variation of Ex, Ey, Gxy, vxy, mxand myhas been assessed Plantain Empty Fruit Bunch Fiber Composite (PEFBFC) and Plantain Pseudo Stem Fiber Composite (PPSFC) using Eqs. (8)(14) and presented in Tables 3 and 4, respectively.

S/Nθ°ExEyGxyvxyvyxmxmyExE2GxyE2
1014922.0007030.9623622.9900.3800.1790.0000.0002.1220.515
2514769.7707053.3733629.2700.3780.1800.1160.0372.1010.516
31014337.6707121.3233647.4770.3720.1850.2210.0732.0390.519
41513690.4807236.9503675.7270.3620.1910.3040.1111.9470.523
52012911.6207403.7863710.9850.3500.2010.3620.1501.8360.528
62512081.3007626.6923749.2560.3360.2120.3950.1901.7180.533
73011262.7307911.7123785.9460.3200.2250.4050.2311.6020.538
83510497.9908265.7873816.3910.3040.2390.3980.2711.4930.543
9409810.4628696.2133836.5280.2880.2550.3770.3101.3950.546
10459209.6559209.6553843.5710.2710.2710.3460.3461.3100.547
11508696.2139810.4623836.5280.2550.2880.3100.3771.2370.546
12558265.78710497.9903816.3910.2390.3040.2710.3981.1760.543
13607911.71211262.7303785.9460.2250.3200.2310.4051.1250.538
14657626.69212081.3003749.2560.2120.3360.1900.3951.0850.533
15707403.78612911.6203710.9850.2010.3500.1500.3621.0530.528
16757236.95013690.4803675.7270.1910.3620.1110.3041.0290.523
17807121.32314337.6703647.4770.1850.3720.0730.2211.0130.519
18857053.37314769.7703629.2700.1800.3780.0370.1161.0030.516
19907030.96214922.0003622.9900.1790.3800.0000.0001.0000.515

Table 3.

Variation of engineering elastic constants with fiber orientation θ in PEFBFC.

S/Nθ°ExEyGxyvxyvyxmxmyExE2GxyE2
1013027.5006817.1753332.8350.2900.1520.0000.0001.9110.489
2512897.6206830.6513343.6130.2900.1540.1140.0231.8920.490
31012530.1906872.0143375.0390.2910.1590.2140.0471.8380.495
41511983.2806944.0243424.3500.2910.1680.2930.0731.7580.502
52011330.9607051.1193486.8430.2900.1800.3440.1031.6620.511
62510643.3107199.1983555.9010.2870.1940.3690.1361.5610.522
7309974.4067395.3333623.3340.2820.2090.3710.1731.4630.532
8359359.1927647.3943680.2290.2750.2240.3560.2121.3730.540
9408816.0127963.5093718.3370.2650.2390.3280.2521.2930.545
10458351.2078351.2073731.7580.2530.2530.2920.2921.2250.547
11507963.5098816.0123718.3370.2390.2650.2520.3281.1680.545
12557647.3949359.1923680.2290.2240.2750.2120.3561.1220.540
13607395.3339974.4063623.3340.2090.2820.1730.3711.0850.532
14657199.19810643.3103555.9010.1940.2870.1360.3691.0560.522
15707051.11911330.9603486.8430.1800.2900.1030.3441.0340.511
16756944.02411983.2803424.3500.1680.2910.0730.2931.0190.502
17806872.01412530.1903375.0390.1590.2910.0470.2141.0080.495
18856830.65112897.6203343.6130.1540.2900.0230.1141.0020.490
19906817.17513027.5003332.8350.1520.2900.0000.0001.0000.489

Table 4.

Variation of engineering elastic constants with fiber orientation θ in PPSFC.

Figure 4 shows the variation of elastic modulus with fiber orientation, it can be seen that the highest value of 14,922 and 13027.5 MPa in the reference x-direction (Ex)is attained in the fiber orientation angle 0° for PEFBFC and PPSFC respectively. However as fiber orientation angle changes, there is a sharp drop in the value of elastic modulus in the reference x-direction to a respective lowest value of 7030.96 and 6817.18 MPa as the fiber orientation angle increased to 90°. On the contrary, the lowest value of 7030.96 and 6817.18 MPa were recorded for elastic modulus in the reference y-direction (Ey)as the fiber orientation angle increased from 0° to 90° reaching a peak value of 14,922 and 13027.5 MPa for PEFBFC and PPSFC respectively. The implication is that reinforcements are required to be aligned in the direction of applied load [55]. Although Jones [56] intuitively suggested that highest value of material properties may not necessarily occur along the principal material directions, rather it is essential that transverse reinforcement is needed in unidirectional fiber composites which are subjected to multi axial loading [57].

Figure 4.

Variation of elastic modulus with fiber orientation in PEFBFC and PPSFC.

As can be seen in Figure 5 that shear modulus peaked at 45° fiber orientation and shear modulus was symmetric at about 45° fiber orientation angle for both PEFBFC and PPSFC considered. This implies that the higher in-plane shear resistance is achievable when fiber orientation is 45°. Also the respective minimum value of 3622.99 and 3332.83 MPa at fiber orientation 0° for PEFBFC and PPSFC can be seen to gradually increase to maximum values of 3843.57 and 3731.758 MPa at fiber orientation 45° and then reversed parabolically at 90° where it again reaches to 3622.99 and 3332.84 MPa. Similar trend was obtained by Farooq and Myler [58] who developed efficient procedures for determination of mechanical properties of carbon fiber-reinforced laminated composite panels. This trend in which the value of Gxypeaks at 45° fiber orientation angle and lowers at 0° and 90° fiber orientation angle indicates that off-axis reinforcement is very necessary for robust shear stiffness in unidirectional composites [57].

Figure 5.

Variation of shear modulus with fiber orientation in PEFBFC and PPSFC.

Figure 6 shows variation of Poisson’s ratio with fiber orientation, the graph depicts a gradual drop of major Poisson’s ratio (vxy)for PEFBFC and PPSFC respectively from 0.38 and 0.29 when fibers are aligned at 0° orientation angle to a lowest value of 0.18 and 0.15 value when fibers were aligned at 90° orientation angle. Additionally, the minor Poissons ratio (vyx)for PEFBFC and PPSFC increased respectively from 0.18 and 0.15 when fibers are aligned at 0° orientation angle to a highest value of 0.38 and 0.29 value when fibers were aligned at 90° orientation angle.

Figure 6.

Variation of Poisson’s ratio with fiber orientation in PEFBFC and PPSFC.

Figure 7 depicts the variation of shear coupling coefficient with fiber orientation, equal magnitude of shear coupling effect was obtained at 45° fiber orientation angle for both PEFBFC and PPSFC considered. Gibson [57] reported that shear coupling coefficient is a measure of the amount of shear strain developed in the xyplane per unit normal strain along the direction of the applied normal stress σx. Figure 7 clearly indicate that the maximum value of the shear coupling coefficient in the reference x-direction for PEFBFC and PPSFC was attained at 30° fiber orientation angle while the coefficient in the reference y-direction for PEFBFC and PPSFC was attained at 60° fiber orientation angle. This is an indication that as the shear-coupling ratio increases, the amount of shear coupling increases.

Figure 7.

Variation of shear coupling coefficient with fiber orientation in PEFBFC and PPSFC.

4. Tsai-Hill failure criteria assessment of longitudinal tensile strength

Failure theory is essential in determining whether the composite has failed. Literature review has shown that results of failure prediction depend on failure criterion applied and one major failure criteria used in the industries is Tsai-Hill and failure criteria. Additionally, since composites ultimate tensile strength and strain depend on the fiber orientation, a failure criterion must be used in which the applied stress system is also in material axis [54]. Tsai-Hill theory considers an interaction of the stresses in the fiber direction. It postulates that failure can only occur in reinforced composites when the failure index exceeds 1, hence Eq. (19) must be satisfied to avoid failure.

By considering an arbitrary positive angle θ with reference to the x-axis in Figure 3, Ihueze et al. [43] transformed the stresses within the global axes (x-y) into material axes 1–2 as given in Eq. (15)

σ1σ2τ12=c2s22scs2c22scscscc2s2σxσyτxyE15

where c = cosθand s = sinθ. Taking longitudinal direction stresses as σy=τxy=0and thus

σ1=σxcos2θE16
σ2=σxsin2θE17
τ12=σxcosθsinθE18

Considering Tsai-Hill failure criterion and setting the failure index as 1 for the composite failure to occur:

σ1Su12+σ2Su22+τ12τmax2σ1Su1σ2Su1=1E19

Substituting the appropriate value in Eq. (19) we have for PEFBFC

σxcos2θ410.152+σxsin2θ37.33972+σxcosθsinθ19.31002σxcos2θ410.15σxsin2θ410.15=1
σx, PEFBFC=1cos4θ410.152+sin4θ37.33972+cos2θsin2θ19.31002cos2θsin2θ410.152

And for PPSFC

σxcos2θ288.102+σxsin2θ33.13302+σxcosθsinθ15.57002σxcos2θ288.10σxsin2θ288.10=1
σx, PPSFC=1cos4θ288.102+sin4θ33.13302+cos2θsin2θ15.57002cos2θsin2θ288.102

Hence the value for σxis then calculated for orientation ranging from 0° to 90° as shown in Table 5.

S/NOrientation θPEFBFCPPSFC
10410.15288.1
25195.86152.635
310108.78686.8937
41575.455660.6747
52058.635647.3142
62548.826239.5106
73042.643934.6088
83538.603831.4352
94035.960729.4005
104534.304328.1833
115033.392627.5977
125533.071627.5303
136033.231127.9038
146533.77428.6492
157034.591329.679
167535.541830.8595
178036.444731.9912
188537.099532.8242
199037.339733.133

Table 5.

Longitudinal tensile strength variation with fiber orientation angle.

The variation of longitudinal tensile strength with fiber orientation for PEFBFC and PPSFC has been presented in Figure 8, it can be seen that the tensile strength equals 410.15 and 288.1 MPa which are the longitudinal tensile strength for PEFBFC and PPSFC respectively when fiber orientation angle is 0°; on the other hand, the tensile strength equals 37.3397 and 33.133 MPa which are the transverse tensile strength for PEFBFC and PPSFC respectively when fiber orientation angle is 90°.

Figure 8.

Variation of longitudinal tensile strength with fiber orientation for PEFBFC and PPSFC.

5. Variation of tangential stress and modulus around a structural discontinuity

Structural discontinuity arising from holes in reinforced composites created for joining or access purposes causes stress concentration at the point of discontinuity [59]. Adequate comprehension of stress redistribution pattern and concentrations is helpful for proficient and safe structural designs. Unlike in ductile materials where stress concentration is of no much ado, plantain fiber-reinforced composites may be sufficiently brittle, hence every form of stress concentration and structural discontinuity has to be properly designed. In a typical scenario where a circular hole is created in the composite as shown in Figure 9, assuming no interlaminar stresses exist around the free edge of the hole, the ply is nominally stressed by σ1, σ2, σ12some distance away from the hole as indicated. Lekhnitskii [60] derived various useful expressions for stress distribution around holes in a composite plate, the tangential elastic modulus Eat an angular position is determined using Eq. (20).

Figure 9.

Depiction of hole in the plantain fiber-reinforced composites sample.

E=1sin4E1+1G122v12E1sin2cos2+cos4E2E20

Hence the tangential stress σat the perifery of the hole with an angle is found from Eq. (21).

σ=EE1Aσ1+Bσ2+Cσ12E21

where

A=cos2+1+psin2
B=qq+pcos2sin2
C=1+q+ppsin2
p=2qv12+E1G12
q=E1E2

Using the stress transformation matrix and replacing the axes system 1–2 by radial (r)-tangential (∅), we can resolve the tangential stress σback into the material axesin Eq. (22) for proper strength evaluation

σrστr=c2s22scs2c22scscscc2s2σxσyτxyE22

At the edge of the hole, only the tangential stress σ> 0, thus σr=σr=0in Eq. (22), therefore

σ1=σx=σsin2E23
σ2=σy=σcos2E24
σ12=σxy=σcossinE25

Using the maximum stress criterion, the material will fail when any stress value in the material axes exceeds their respective ultimate strenght. Such that

σ1Su1<1E26
σ2Su2<1E27
σ12τmax<1E28

The left hand side of Eqs. (26)(28) represents the failure indices (FI). The maximum failure index (FI) for the applied stress is factored in to obtain the load factor. Due to the inherent material orthotropy, the failure zone of the plantain fiber-reinforced composite as a result of structural discontinuity may not necessarily occur at the point of maximum stress concentration, therefore it is important to assess the extent of variation of the tangential stress around the hole edge and the failure index using maximum stress theory at other points aside the point of maximum stress concentration. Also in the present consideration we take a simplified scenario where σ2=σ12=0such that the ply of dimensions 150 × 19.05 × 3.2 mm with a circular hole at the center is subjected to only nominal axial stress σPEFBFC=34MPaand σPPSFC=29MPa. Tables 6 and 7 depict the values of tangential stress, material axis stress and tangential modulus as computed using Eq. (20)(25).

S/NEøσøσ1σ2σ12F.I.1F.I.2F.I.12Stress conc.
07030.962−23.3380.000−23.3380.0000.0000.6250.000−0.69
57053.373−22.807−0.173−22.6341.9800.0000.6060.103−0.67
107121.323−21.211−0.640−20.5713.6270.0020.5510.188−0.62
157236.950−18.542−1.242−17.3004.6360.0030.4630.240−0.55
207403.786−14.786−1.730−13.0564.7520.0040.3500.246−0.43
257626.692−9.917−1.771−8.1463.7990.0040.2180.197−0.29
307911.712−3.903−0.976−2.9271.6900.0020.0780.088−0.11
358265.7873.3031.0872.217−1.5520.0030.0590.0800.10
408696.21311.7514.8556.896−5.7860.0120.1850.3000.35
459209.65521.48410.74210.742−10.7420.0260.2880.5560.63
509810.46232.51519.08013.434−16.0100.0470.3600.8290.96
5510497.99444.78430.05014.733−21.0420.0730.3951.0901.32
6011262.72658.10343.57714.526−25.1590.1060.3891.3031.71
6512081.30472.07659.20312.873−27.6070.1440.3451.4302.12
7012911.62286.02575.96210.063−27.6480.1850.2691.4322.53
7513690.47698.95192.3236.628−24.7380.2250.1781.2812.91
8014337.671109.599106.2943.305−18.7420.2590.0890.9713.22
8514769.765116.668115.7820.886−10.1300.2820.0240.5253.43
9014922.000119.152119.1520.0000.0000.2910.0000.0003.50
9514769.765116.668115.7820.88610.1300.2820.0240.5253.43
10014337.671109.599106.2943.30518.7420.2590.0890.9713.22
10513690.47698.95192.3236.62824.7380.2250.1781.2812.91

Table 6.

Variation of tangential stress, material axis stress and tangential modulus at the edge of material discontinuity in PEFBFC.

S/NEøσøσ1σ2σ12F.I.1F.I.2F.I.12Stress conc.
06817.175−20.9780.000−20.9780.0000.0000.6330.000−0.72
56830.651−20.459−0.155−20.3041.7760.0010.6130.114−0.71
106872.014−18.909−0.570−18.3393.2340.0020.5540.208−0.65
156944.024−16.346−1.095−15.2514.0860.0040.4600.262−0.56
207051.119−12.791−1.496−11.2954.1110.0050.3410.264−0.44
257199.198−8.269−1.477−6.7923.1670.0050.2050.203−0.29
307395.333−2.793−0.698−2.0951.2090.0020.0630.078−0.10
357647.3943.6351.1962.439−1.7080.0040.0740.1100.13
407963.50911.0244.5556.469−5.4280.0160.1950.3490.38
458351.20719.3919.6959.695−9.6950.0340.2930.6230.67
508816.01228.73616.86311.873−14.1500.0590.3580.9090.99
559359.19239.01426.17912.835−18.3310.0910.3871.1771.35
609974.40650.08737.56512.522−21.6880.1300.3781.3931.73
6510643.30861.65150.64011.011−23.6140.1760.3321.5172.13
7011330.95873.17564.6158.560−23.5180.2240.2581.5102.52
7511983.27783.85678.2395.617−20.9640.2720.1701.3462.89
8012530.18592.66789.8732.794−15.8470.3120.0841.0183.20
8512897.62298.52697.7780.748−8.5540.3390.0230.5493.40
9013027.500100.587100.5870.0000.0000.3490.0000.0003.47

Table 7.

Variation of tangential stress, material axis stress and tangential modulus at the edge of material discontinuity in PPSFC.

Tangential stress distribution at hole edge for PEFBFC and PPSFC are shown in Figure 10, the maximum stress value of 119.15 and 100.587 MPa was attained at angular position ø = 90° for PEFBFC and PPSFC respectively. However, considering various failure indices in Tables 5 and 6, failure will be initiated at ø = 70° for PEFBFC with stress concentration factor of 2.53 and ø = 65° for PPSFC with stress concentration factor of 2.13 which are less than the stress concentration around the peak stress when angular position is 90°.

Figure 10.

Tangential stress distribution at a hole edge for PEFBFC and PPSFC.

6. Conclusions

The utilization of plantain fiber-reinforced composites in structural applications empowers architects to acquire huge accomplishments in the usefulness, security and economy of development. These materials have high proportion of strength-to-density ratio, can be tailored to posses certain mechanical properties. The elastic constants of plantain fiber-reinforced composites depend greatly on fiber orientation with notable anisotropic characteristics which makes it less attractive for applications involving lugs and fittings. The present report amplified some notable design procedures in handling such limitations in plantain fiber-reinforced composites using relevant failure theories. Both plantain EFBFRC and PSFRC showed similar trends in response to the design scenario considered. Be that as it may be, a proficient utilization of plantain fiber-reinforced composites in structural applications requires a cautious assessment of all influential factors.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Christian Emeka Okafor and Christopher Chukwutoo Ihueze (July 15th 2020). Strength Analysis and Variation of Elastic Properties in Plantain Fiber/Polyester Composites for Structural Applications, Composite and Nanocomposite Materials - From Knowledge to Industrial Applications, Tri-Dung Ngo, IntechOpen, DOI: 10.5772/intechopen.90890. Available from:

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