The Graft of ANN-FEM Technique in Macro-mechanics of Multi-oriented Natural Fiber/Polyester Laminates

C. Emeka Okafor ^a*, C. Chukwutoo Ihueze ^b

^aDepartment of Mechanical Engineering, Nnamdi Azikiwe University, Awka, Nigeria

^bDepartment of Industrial and Production Engineering, Nnamdi Azikiwe University, Awka, Nigeria

1.     Introduction

Natural fibers have been traditionally utilized over the years to address human needs such as in production of sacks, ropes, bags, rugs, floor covers, carpet backs, binder twines, hats, mats, baskets, stuffing material for upholstery and mattresses due to their low cost, low density, fair specific modulus, high toughness, non-toxic, recyclability and availability compared with synthetic fibers [1; 2]. Additionally, these properties have become the center of attraction for the surge in utilization of natural fibers in reinforcing polymer matrix for application in automotive [3], civil works and transportation sector [4], oil and gas sector [5], wind energy industry [6], housing sector [7], maritime sector [8], power and telecommunication industries [9]. Although, the greater part of the natural fibers such as those extracted from plantain plant are hydrophilic naturally leading to poor interfacial bond with hydrophobic polymer matrix [10], it can still be used successfully as a reinforcement after effective chemical treatment [11].

Continued exploration of post-harvest plantain plant residues in the development of engineering materials such as reinforced composites has been on the increase. Natural fibers extracted from bio wastes offer significant advantage over woody biomass, since they have low micro-fibril angle, small lumen mean, high cellulose content, high crystallinity and relatively longer fibers which gives better control of fiber lay-up and orientation [12; 13; 14]. The use of biodegradable and locally available Plantain Pseudo Stem (PPS) and Empty Fruit Bunch (EFB) fibers from renewable agricultural sources in fabrication of Plantain Empty Fruit Bunch Fiber Reinforced Composite and Plantain Pseudo Stem Fiber Reinforced Composite have been reported and assessed [15; 16].

Extensive research evidence has thereafter emerged regarding various physical and mechanical properties of plantain fiber reinforced composites [15; 17; 16; 18; 11; 19; 20; 21; 22]; nonetheless, majority of these studies has concentrated on the properties of single plies which consists of unidirectional fiber reinforced lamina. Generally, reinforced composites in which fibers are aligned in one direction would be very weak in the transverse direction; consequently, any small transverse load arising from uneven lateral contraction can initiation crack. There is therefore need to combine some of the best aspects of unidirectional plantain fiber reinforced laminas in order to achieve a more robust material, this is because incorporation of variously oriented plies in one laminate will definitely lead to significant advantages when compared with conventional monolithic materials [23].

The boundless coalescence of ply orientations and stacking sequence obtainable at the design stage of laminates provides the necessary design pliancy to achieve high modulus industrial materials [24].  In this regard, laminated composites can be tailored for directional stiffness properties in longitudinal and transverse directions via combination of axis and off axis laminates. This is essential to guard against catastrophic failure in a way that peradventure any of the layers fail first, the composite can still endure more loads prior to the eventual failure of last ply [25]. Obviously, the failure stress and stiffness properties of plantain fiber reinforced laminates can be acquired through numerical techniques [26; 27]. Basically, the principles of laminated composite material mechanics can be learned at a macro-scale where mechanical behaviour of the laminate and the structure are the primary focal point of investigations [28]. This provides superior initial guess for the designer in terms of average properties of the composite material [29; 30; 31]. Advanced numerical and computational tools have evolved over the years for application in solving diverse composite design problems and skilful utilization of these tools would further diminish the design and improvement time alongside the expense of experimentation expected in the design development phase of laminates.

Rahimi, Hussain, Meon and Mahmud [32] reported that ANSYS software is capable of predicting the failure of composite laminates based on Maximum Stress and Tsai-Wu theories. Saxena and Kirtania [33] used finite element analysis and analytical methods in evaluation of symmetric cross-ply laminated composite plates, authors computational results compared closely with the theoretical results. Other researchers has reported comparable results between finite element and analytical methods for inter-laminar stresses in laminated composite beam [34], first ply failure of laminated composite plates [35], cross-ply laminate of boron/epoxy composite failure under uniformly distributed load [36], failure of inter-ply hybrid laminated composite [37], failure of carbon fiber-reinforced polymer laminated composite thin-walled I-beams [38] and failure of fiber/epoxy composite laminate interface using cohesive multi-scale model [39]. It is necessary to integrate finite element analysis and artificial neural network in order to boost accuracy and speed in design process.

Various macro-mechanical models such as elasticity models [40], plasticity models [41] and nonlinear constitutive model [42] have also been formulated to embody the constitutive affiliations of laminates. Hsuan-Teh and Lung-Shen [42] carried out constitutive modelling of composite laminates under uniaxial compression, authors detected damage onset using Tsai-Wu and the maximum stress criterion. Dimitrienko [43] developed universal models of laminated composites constitutive relations with discrete information on main conjugated stress-strain tensors pairs. Development of a constitutive equation for a novel plantain fiber reinforced laminate is required for establishment of the laminate load deformation affiliation and equivalent elastic constants.

There is also need to relate the externally applied force on plantain fiber reinforced laminate to the resulting deformation of strain stresses at ply level, this will provide the best standing to determine whether any ply has failed using appropriate failure theory. Different failure theories have been used in the literature to evaluate failure initiation in fiber reinforced laminates; Rajanish, Nanjundaradhya, Ramesh and Bhaskar [44] classified them into limit/non-interactive (maximum stress and maximum strain); interactive (TsaiHill and Tsai-Wu) and partially-interactive (Hashin-Rotem and Puck) theories. The validity and criteria for selecting any of these theories is well documented [45; 46; 47]. The non-interactive maximum principal stress criterion was adopted in the present study to account for various ply failure modes inherent in plantain fiber reinforced laminate.

2.     Methodology

2.1. Determination of effective ply thickness

The fiber mass fraction  is determined from eq. (1) and the ply thickness in terms of mass fraction of fibers is presented in eq. (2).

where  is the density of matrix while and  are the density and optimal volume fraction (50%) of fiber respectively. Properties and characteristics of the plantain fiber reinforcement–polyester matrix with detailed implementation of Archimedes principle has been reported elsewhere [15; 18] and by knowing the density of polyester resin as 1200kg/m³, the ply thickness from eq. (2) and specific mass of fiberis presented in Table 1.

The individual layers of laminate composites are held together by matrix to form a plate. The layers of the laminate composites referred in the present study are of thesame constituent materials having different reinforcing pattern. The matrix and fiber part of the laminate composite are respectively polyester and plantain fiber with properties described in Tables 2 and 3. Obviously, the analysis of laminate depends on the properties of individual layers and the detailed mechanical properties of single layer of Plantain Empty Fruit Bunch Fiber Reinforced Composite (PEFBFC) and Plantain Pseudo Stem Fiber Reinforced Composite (PPSFC) have been reported by [16].

2.2. Design of experiment and Finite Element modeling

The classical Central Composite Design (CCD) method was used to determine the number of numerical experiments to be evaluated. CCD is a useful design of experiment (DOE) template which minimizes the required number of iterations for full material assessment. It  involves varying the stacking sequence from (30/0/90)s to (30/45/90)s, laminate aspect ratio from 1.31 to 0.79 and fiber orientation from 30⁰ to 90⁰ as shown in Fig. 1. This control factor arrengement gave a total of 26 experimental runs.

The Finite Element Analysis was conducted in ANSYS Parametric Design Language (APDL). A 3D 4-node shell 181 element with six degrees of freedom was used in creating the model following the symmetry of layer code shown in Fig. 1a. According to Abhishek [35], SHELL181 is useful in evaluation of thin to moderately-thick shell structures. The material model was assigned to structural linear elastic orthotropic behaviour and then the shell section was modified to conform to fiber angle orientations and lay-up.

Table 1: Plantain fiber ply parameters determined by application of Archimedes principle

Table 2: Mechanical properties of single layer plantain fibers reinforced polyester composites [16]

Note:  are tensile strengths in the longitudinal and transverse directions respectively.

Table 3: Mechanical properties of plantain fibers and polyester resin [16]

Fig. 1: (a) Stacking sequence and (b) Boundary Condition for PEFBFL (Nodes = 825,  Elements = 768)

Each layer thickness corresponds to the ply thickness as specified in Table 1, finite element model was created using rectangular cross section and the meshing was done for the entire laminate at a time. In the post-processing phase, results of laminate stresses in x direction and deformation are viewed and plotted at nodes as shown in Fig. 2.

The data acquired were fitted to the empirical polynomial regression model of eq. (3).

where Y is responses (stress and displacement); β₀,β_i (i = 1, 2,…..k) and β_ij (i = 1, 2,…k; j = 1, 2,…k) are the model coefficients; x_i and x_j are the coded independent variables. Also, the adequacy of the model was checked using the coefficient of determination (R²).

2.3. Artificial Neural Network training scheme

The Neural Network model was trained using the raw numerical dataset obtained from the finite element simulation. The weighted sum of inputs arriving at each neuron was passed through an activation function to generate an output signal. A total of 17 (65.4%) of experimental results was used to train the network, 5 (19.2%) of the experimental result was used to validate the training while the remaining 4 (15.4%) was used for testing. In order to reduce the deviations of response predictions, a trial and error method was utilised to establish the suitable number of neurons required in the hidden layer, this was achieved by repeating the training cycle at varying number of neurons in the hidden layer. The Multi-layer perceptron which was trained with back-propagation algorithm has three input neurons representing the design independent variables (stacking sequence, laminate aspect ratio and fiber orientation), a single hidden layer of ten neurons and an output layer consisting of two neurons representing the stress and displacement as shown in Fig. 4. The network performance was assessed using Mean Square Error (MSE) and coefficient of determination (R²). Error minimization was achieved by Levenberge–Marquardt (LM) method. The Levenberg-Marquardt algorithm uses an approximation to the Hessian Matrix in a Newton-like update shown in Eq. (4).

where J = Jacobian matrix containing first derivatives of the network errors, e = vector of network errors. At , eq. (4) becomes the regular Newton’s method using an approximate Hessian matrix, however at , eq. (4) becomes gradient descent with a small step size.  The mean sum of squares of the network errors was captured using Eq. (5).

Fig. 2: (a)Laminate Stresses and (b) Displacement in x direction for PEFBFL

Fig. 3: Flowchart showing the graft of ANN-FEM technique

Fig. 4: Artificial neural network with 3 input variables, 1 hidden layer of 10 neurons and 2 output variables.

2.4. Formulation of laminate stiffness framework

Following classical laminate theory [48; 49; 29], the variation of direct strain in x and y is given in eq. (6). Classical Laminate Theory (CLT) is a predictive means used in the analysis of laminate behaviour based on its stacking sequence and material properties. The theory assumed that each layer of the laminate is both quasihomogeneous and orthotropic; the thickness of the plate is much smaller than any of its characteristic dimension; the displacements arising from mechanical loading is small compared with thickness of the plate; the in-plane strains are small compared with unity; transverse shear and normal strain  are negligible; no warping exist; each ply obeys Hooke’s Law and the plate thickness is constant throughout the laminate. Eqs. 6-20 provides the mathematical background for application of CLT.

and shear strain in x-y plane reads:

Combining eqs. 6 -7 in a matrix form gives:

Integrating in-plane force intensities and moment over the thickness of ply p in x,y and x-y-directions reads:

The knowledge of ,  and  on the plies is necessary for complete evaluation of eqs. (9 – 11). Considering two possible loading conditions of longitudinal (direction 1) and transverse (direction 2) in the matrix, the resulting direct strains from Hooks law are respectively   and  where  = major Poisson’s ratio and  = minor Poisson’s ratio. Hence the application of both direct stresses   and  will yield corresponding strains as follows:

And the stress-strain relationship for an orthotropic ply can be expressed as:

where

Combining eq. (13) with eq. (8) we have:

Eq. (14) describes the association between the applied force (N) and the resulting midplane laminate deformations. While the extensional and coupling stiffnessis presented in eqs. (15) and (16) respectively.

In similar consideration for moment about specific axis for ply 1to n  in Fig. 1 gives:

Eq. (17) gives the bending stiffness term ( which depends on transformed reduced stiffness term  and the ply cordinate terms.

Inversion of the laminate stiffness terms in eq. (13) yields the required transformed reduced compliance terms , so considering a fully populated extensional stiffness matrices (strain component of eq.  (14) 

Where

Considering a fully populated bending stiffness matrices (k component of eq. 14)

Where

3.     Results and discussions

3.1. CCD optimum conditions

The optimization exercise for stress and displacement was conducted utilizing the flexibility of the Design Expert optimization tool function. Eq. (3) was solved for the best solutions. A usual approach which involves selecting the best solution was adopted and the chosen optimal solutions gave stacking sequence = (30/45/90)s, laminate aspect ratio = 1.31, fiber orientation = 90⁰ and desirability = 1. The laminate with optimum characteristics was further analyzed by applying the Classical Laminate Theory along the thickness of specified fiber source.

3.2. ANN training evaluation and performance

The training record was used to plot the performance progress. The Mean Square Error of the trained network for PEFBFL strength prediction was 0.46, having a regression coefficient of 0.99 while the network for PEFBFL displacement prediction has Mean Square Error of 4.55442E-09 and regression coefficient of 0.93 (Figs. 5 & 6). Similarly, The Mean Square Error of the trained network for PPSFL strength prediction is 0.6557, having a regression coefficient of 0.96 while the network for PEFBFL displacement prediction has Mean Square Error of 1.25768E-09 and regression coefficient of 0.94 (Figs. 7 & 8). In all, the training data indicates a good fit and the validation results also show reasonably high R² values that are close to 1. The regression coefficient measures the correlation between the predicted responses (outputs) and the experimental responses (targets).

Fig.  5: (a) Actual value of PEFBFL stress versus predicted (b) Regression plot for validation

Fig. 6: (a) Actual value of PEFBFL displacement versus predicted (b) Regression plot forvalidation

Fig. 7: (a) Actual value of PPSFL stress versus predicted (b) Regression plot for validation

Fig. 8: (a) Actual PPSFL value of displacement versus predicted (b) Regression plot for validation

3.3. Overall reduced stiffness terms

The optimal set of plantain fiber reinforced laminate with thickness (t) which is composed of six stacked plies in the order   is represented in Fig. 9.

Minor Poisson’s Ratio is the strain resulting from stress in the axial direction. The minor Poisson’s ratio  for single layer plantain fibers reinforced polyester composites using eq. (12) and relevant data in Table 1 is found as:

Similarly, the ply reduced stiffness matrix () for a single layer of PEFBFC (N/mm²) and single layer PPSFC (N/mm²) is also evaluated and presented in eq. (21) For PEFBFC and For PPSFC in eq. (22).

Hence the transformed reduced stiffness term for the multi-oriented plantain fiber/polyester laminate described in Fig. 10 is obtained from eq. (13) and presented in Table 4 for PEFBFL and Table 5 for PPSFL.

Fig.  9: Elements of the laminate showing point force and moment strengths

Fig. 10: Six layer PEFB fiber symmetric laminate code (mm)

Table 4: values (N/mm^2) for PEFBFL (30/45/90)s

Table 5: values (N/mm^2) for PPSFL (30/45/90)_s

Since the laminate in this study is symmetric, the coupling  described in eq. (16) tends to zero while the extensional  and bending  stiffness terms are computed using the ply thickness and centroidal values described in Fig. 10 and tabulations in Table 6 for PEFBFL and in Table 7 for PPSFL.

The extensional stiffness  and bending stiffness as defined in eqs.  (15) and (18) are computed using plies 1, 2 and 3 in turn doubling the result in consideration of ply symmetry to get total laminate extensional and bending stiffness written in matrix form in  eq. (23) to (26) for PEFBFL and PPSFL.

Evaluation of eqs. (19) and (20) gives the transformed reduced compliance  and bending stiffness  matrices for PEFBFL and PPSFL as shown in eq. (27) to (30).

The laminate equivalent elastic constants are therefore computed for membrane and bending modes using appropriate expressions in Table 8.

3.4. Macroscopic assessment of ply failure based on Maximum Principal Stress Theory

Ply strength analysis was performed using Maximum Principal Stress Theory, the use of this noninteractive failure criterion was justified because of the need to ascertain probable laminar mode of failure. The load required to cause the first ply failure was obtained from the strength ratio of the plies. The maximum strength ratio was subsequently used to calculate the load factor and the load intensity required to cause the first ply failure.

Table 6: Ordinate values for PEFBFL (30/45/90)_s

Table 7: Ordinate values for PPSFL (30/45/90)_s

Table 8: Equivalent elastic constants of laminates

Notably, the load factor is inversely proportional to the strength ratio only for Maximum Principal Stress Theory. However, for Tsai-Hill failure criterion, the load factor is inversely proportional to the square root of strength ratio. For Hoffman criterion and Tsai-Wu criterion, the relationship between load factor and strength ratio has combination of quadratic and linear functions. Hence for initial design development phase of laminates, Maximum Principal Stress Theory is adjudged valid.

3.4. 1. First-ply-failure analysis:

Considering only membrane load, a partially populated compliance matrices is obtained from eq. (19) and by applying theordinate values for PEFBFL ply laminate (30/45/90)_s in Fig. 6:

6859.86

7774.292

2662.503

2791.012

And a membrane equivalent Young’s modulus (E_x) in the direction of the applied load is

where  is the total thickness of the laminate (t_p× number of layers) and   . An axial load of 476N is applied at first instance to the 25×19.05 (LW) rectangular section of Fig. 1, the force intensity  is the axial force per unit width of the section ( as  and ). Then the laminate mid-plane deformations strains  at the laminate reference axes are calculated for each ply using eq. (31).

Or

3.3.1.1.Computation of PEFBFL ply strain in material axes:

Eq. (32) is the constant strain distribution through the laminate thickness in all the three symmetric plies. However, the ply strain in the material reference axes is related to the material axes 1-2 in eq. (33). Symmetric plies are combined because only a membrane load is considered and the strain in the material axes are constant through the thickness, so once the computation is done for a definite ply configuration, the rest of the plies with the same configuration will have identical value, hence

For plies 1 and 6 at 30^o

For plies 2 and 5 at 45^o

For plies 3 and 4 at 90^o

3.3.1.2. Computation of PEFBFL ply stresses in material axes:

The ply stresses in the material axis is related to the ply strains in the material axes as shown in eq. (34).

Therefore the ply strains in material axes 1-2 for each ply are used to establish the stress-strain relationship as follows: 

For plies 1 and 6 at 30^o

For plies 2 and 5 at 45^o

For plies 3 and 4 at 90^o

Using the Maximum Principal Stress Theory which suggests that the ply will fail when any stress value in the material axes exceeds their respective ultimate strength. Such that

The left hand side of eqs. (35) represents the strength ratio (SR). The maximum Strength ratio (SR) for the applied stress is factored in to obtain the load factor. Table 9 showed the ply stresses associated strength ratios at  for PEFBFL, it can be seen that maximum strength ratio of 0.9596 occured in ply 2 at 45 degrees in shear mode. Therefore at  no ply failure has yet occurred, hence the load intensity that can cause first ply to fail according to Maximum Principal Stress Theory in the 45 degree plies in shear mode is , furthermore, since the laminate is symmetric with membrane load only, the failure of ply two and ply five are predicted simultaneously. Similar computations were carried out for PPSFL and result presented in Table 10.

Table 9: Ply stresses and associated strength ratios in PEFBFL @  and

Table 10: Ply stresses and associated strength ratios in PPSFL @  and

Table 11: values (N/mm^2) for PEFBFL when ply 2 and 5 has completely failed

3.3.2         Second-ply-failure analysis

For second-ply-failure analysis, it is assumed that all the elastic values E₁, E₂ and G₁₂ for the two already failed plies 2 and 5 are zero; hence the reduced and transformed reduced stiffness for plies 2 and 5 are zero. Therefore the new assumed transformed reduced stiffness term for PEFBFL is presented in Table 11.

The altered laminate configuration shown in Table 11 is still symmetric, so considering only membrane load, a partially populated compliance matrices is obtained from eq. 24. Computed ply strain and stresses in material axes is shown in Table 12, the ply stresses in the material axis is related to the ply strains in the material axes as shown in eq. 34. The ply strain in material axes 1-2 for each ply is hence used to establish the stress-strain relationship. Table 12 showed the ply stresses and associated strength ratios in PEFBFL at , it can be seen that maximum strength ratio of 1.27715 occured in ply 1 at 6 at 30 degrees in shear mode. Therefore at   ply 1 at 6 failure has occured, furthermore, since the laminate is symmetric with membrane load only, the failure of ply one and ply six are predicted simultaneously.

Table 13 showed the ply stresses and associated strength ratios in PPSFL at , it can be seen that maximum strength ratio of 0.59418 occurred in ply 1 at 6 at 30 degrees in shear mode. Therefore at   no subsequent ply failure has occured in this reduced stiffness laminate, therefore the load to cause the ply failure according to Maximum Principal Stress Theory in 30 degree plies in shear mode is .

3.3.3         Third-ply-failure analysis

For Third-ply-failure analysis, it is assumed that all the elastic values E₁, E₂ and G₁₂ for the four failed plies 1, 2, 5 and 6 are zero, hence the reduced and transformed reduced stiffness for plies 1, 2, 5 and 6 are zero. Thus the new transformed reduced stiffness terms for the composite are presented in Table 14.

Table 12: Ply stresses and associated strength ratios in PEFBFL @  and

Table 13: Ply stresses and associated strength ratios in PPSFL @  and

Table 14: values (N/mm^2) for PEFBFL when ply 1, 2, 5 and 6 has completely failed

The altered laminate configuration shown in Table 14 is still symmetric, so considering only membrane load, a partially populated compliance matrices is obtained from eq. (19). Following the method used in first and second ply failure analysis, the ply stresses and associated strength ratios at , is shown in Table 15. It can be seen that maximum strength ratio of 3.218395 occured in ply 3 and 4  at 90 degrees in tensile mode of transverse direction. Therefore at  ply failure has occured in PEFBFL, furthermore, since the laminate is symmetric with membrane load only, the failure of ply one and six are predicted simultaneously.

Similarly for PPSFL, it can be seen in Table 16 that maximum strength ratio of 2.690382 occurred in ply 3 and 4  at 90 degrees in tensile mode of transverse direction. Therefore at  ply failure has occured in PPSFL, furthermore, since the laminate is symmetric with membrane load only, the failure of ply one and six are predicted simultaneously.

The average stress which is the overall laminate strength (is therefore  failure load divided by laminate thickness which is  for PEFBFL. While PPSFL is . Eq. (31) gives the corresponding strain at each ply failure as   because .

4.     Conclusions

Based on the macro-mechanical analysis of (30/45/90)_s laminate under tensile loading, the following conclusions are drawn:

1. Higher number of layer contributes to increased strength and stiffness in multi-oriented natural fiber/polyester laminate.

Table 15: Ply stresses and associated strength ratios in PEFBFL @  and

Table 16: Ply stresses and associated strength ratios in PPSFL @  and

                                                          Table 17: The average values of stress (and strain ( for PEFBFL and PPSFL

Table 18: The values of stress (and deformation for PEFBFL and PPSFL

1. The laminate equivalent elastic constants (  for symmetric plantain fiber/polyester laminate have been determined in membrane and bending modes.
2. The maximum principal stress theory provided the necessary sets of physics-based failure criteria for predicting the plantain fiber/polyester laminate failure using the lamina-based longitudinal, transverse and shear strengths.
3. A steady reduction in the laminate elastic modulus was observed as a result of compromised stiffness in principal directions arising from gradual failure of the plies until the last ply failure occurred in ply 3 and 4 at 90 degrees in tensile mode of transverse direction.
4. Stresses and displacements observed in the symmetric plantain fiber/polyester laminate with specified mechanical loading condition using Classical Laminate Theory agree very closely with predictions of ANN.
5. The urgency of a fast technique that simulates macro-mechanical characteristics of laminate can be fulfilled by using an artificial neural network trained with raw experimental data obtained from finite element analysis in the range of loads and material control factors studied.
6. Considering the rich micro-structural information gathered from this stiffness and macroscopic analysis, it is believed that this technique deserves further study as an initial screening mechanism in advanced composite development programmes.

Acknowledgements

The supportive role played by Nnamdi Azikiwe University Awka in providing enabling environment for this research is highly acknowledged.

Appendix

Table 19: The ANN training data set and response for PEFBFL

Code value: (30/0/90)s =>1, (30/45/90)s=>2

Table 20: The ANN training data set and response for PPSFL

Code value: (30/0/90)s =>1, (30/45/90)s=>2

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