1 Introduction

Fiber-reinforced polymer (FRP) composites have their uses in applications of engineering due to their directional stiffness and properties of high strength-to-weight ratio. The FRP nanocomposite material laminate properties can be transformed by varying the thickness by adding or removing plies, by altering the materials for matrix and fibers, and by changing the volume fraction of nanoparticles and the fiber-angle orientation. The optimization design process has been developed to find the best presentation of composite structures in a wide range of engineering applications. The design optimization can be implemented in the FRP nanocomposites [1] for the blast resistance. The combined and individual use of a genetic algorithm, finite element methods, and discrete lay-up method are effectively used by the authors in solving the optimization problems of FRP [2,3,4,5]. The design optimization of Carbon fiber-reinforced polymer and other FRP’s are effectively carried out for various parts and the structures for different failure analyses and considering different parameters by Almeida, Do-Hyoung Kim, and Stacy Nelson [6,7,8,9]. G. Narayananaik et al. and Sohouli et al. have also applied the genetic algorithm for the failure creation mechanism and design optimization of the composite laminates [10, 11]. Furthermore, Yasser Rostamiyan et al. have successfully carried out experimental optimization of damping properties of nano epoxy-based composites, and high-impact polystyrene by mixture design approach by using RSM and the mechanical properties of epoxy-high-impact polystyrene/carbon multi-walled nanotubes ternary nanocomposites using an artificial genetic algorithm and neural network [12,13,14,15]. Boon et al. have also studied the comparison of wear performance of FRP nanocomposites using RSM [16]. Boroujeni et al., highlighted the improvement of mechanical properties by the implementation of carbon nanotubes in CFRP composites [17]. Abdolreza Mirmohseni et al. also successfully used RSM for modeling and optimization of nano CFRP composites [18]. Multi-objective optimization of the composite FRP laminates was studied by various researchers for different automobile components and structures [19,20,21]. Pelletier et al. have explored a study on the maximization of the stiffness, strength, and for minimal mass [22]. Many researchers have focused on the optimization of flexural strength, stacking sequence, and cost optimization of FRP composites [23,24,25]. Narayana Naik et al. have used the optimization technique by nature-inspired in designing composite structures [26]. Rahul et al. have optimized the impact strength of FRP laminates by using the island model parallel genetic algorithm [27]. Some of the researchers have continued their study on optimizing by considering different failure criteria, flexural properties of nanocomposites by using RSM, and optimization of variable stiffness composites [28,29,30]. Jie et al. optimized the tribological properties of fiber and matrix interface [31]. Meera Balachandran et al. have implemented the CCD method for the optimization of nanocomposites [32]. Optimization of the delamination failure, buckling load capacity, and minimal thickness is carried out in the studies of many researchers [33,34,35]. Srinivas et al. have studied the effect of carbon nanotubes on various properties of nanocomposites [36]. Chung Park et al. have done their study on optimization of composite structures by considering manufacturing cost and mechanical performance [37]. Tripathi et al. have developed a comprehensive optimization procedure named as OptiComp. This method can be effectively used for multi-objective optimization of FRP laminate design by considering the VTA and uniform thickness approach as well. They have considered ply orientation angle, thickness, the density of the material, etc. as the variables/parameters and the Tsai-Wu, Tsai–Hill, and Maximum stress theories of failure as constrains. The OptiComp seems to be a very useful tool in implementing the design optimization of plain FRP/FRP hybrid nanocomposites [38,39,40]. Leila Fakhri et al. have explored their study on polystyrene nanoclay and nanoZnO composite by RSM [41]. Balaji has studied the morphological and mechanical properties of banana FRP composites.

In the current study, the multi-objective design optimization of the CFRP hybrid nanocomposites containing nanoclay and nanoZnO particles was done by using OptiComp. The use of nanoparticles not only improves the strength of the interface but also significantly alters stress and strain propagation in composite laminates which ultimately improves in the stiffness of CFRP composite laminates.

2 Material and Methods

2.1 Materials

In the current study, the Control sample was carbon fiber-reinforced polymer (CFRP) composites. The carbon fiber of 200 GSM, standard modulus of 3K, and HCU200 code is used for the main reinforcement. Hinpoxy C epoxy resin is Bisphenol‐A-based liquid-state material. HARDENER-B is a modified amine hardener, has low viscosity, and is a colorless liquid, with a stoichiometric ratio of A:B = 10:3; the fiber and the resin both were supplied by the Hindoostan Mills Ltd. Mumbai, India. NanoZnO particles and montmorillonite nanoclay were used, supplied by Nano Research Lab. Jamshedpur, (Jharkhand), India. All the details of the properties are given in Table 1.

Table 1 Material properties
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2.2 Sample Fabrication

The carbon fiber-reinforced polymer nanocomposite laminates were manufactured by the process of hand lay-up method, followed by the process of vacuum bagging. Here, the main part of the reinforcement was unidirectional carbon fiber in the epoxy resin. Bisphenol A (DGEBA) resin was mixed up with hardener in the ratio of 100:30 of weight. The clay and Zinc Oxide nanoparticles were used as a nanofillers. For the suitable dispersion in the epoxy resin clay and Zinc Oxide Nanoparticles, first of all, nanoparticles were dried before use overnight at 750C in an oven. For the attainment of a uniform dispersion of the nanoclay and ZnO(Zinc Oxide) nanoparticles in the epoxy resin, the magnetic stirring and ultrasonication process of the mixture were carried. The % weight of nanoparticle content was used in the range of 1% to 5% in the mixture of the epoxy resin and hardener as per the results from OVAT analysis and previous research carried out. Ultra-sonication bath method was then carried out for 40 min at 40° C for uniform mixing of the nanoparticles in the epoxy resin. After ultrasonication, the prepared mixture was then magnetically stirred for around one and a half hours. The prepared mixture was used for the CFRP nanocomposite preparation. The hand lay-up-prepared sample is then followed by the process of vacuum bagging which was carried out for 4.5–5 h with a constant vacuum pr. of 720 mmHg. Fourteen plies were used, and according to the coded levels, the combination of three factors was varied. The specimen was kept in the bag for 15–18 h at room temperature for curing. The post-curing of the specimen was then done in an oven for five hours at 80 °C.

2.2.1 OptiComp

OptiComp is the comprehensive generic procedure for the design optimization of composites. With the implementation of the OptiComp, the discrete form Variable thickness approach (VTA) used in this study can find the minimum laminate thickness, and for maximum stiffness in one step instead of two-step methodology. The required laminate minimum thickness can be used by the designer in several ways. The present study found that the effect of the Variable thickness approach (VTA) in this regard is dependent on the ply thickness increment value and the ply number. Tsai-Wu theory and Maximum stress theory were used as constraints for a given loading condition. Ply angle, ply thickness, and ply number are used as a design variable in the current simulation study in discrete form. Optimization is performed using a direct value coding genetic algorithm. You can choose two approaches for designing composite laminates, i.e., Variable Thickness Approach (VTA) and Uniform Thickness Approach (UTA). In UTA, all plies are of uniform thickness, and ply angles and multiple plies are treated as design variables. In VTA, ply angle, ply count, and ply thickness are treated as design variables. In the new process, OptiComp, the following change has been made in conventional GA to costume the problem of composite laminate design optimization.

(i) Instead of binary-coded representation. Direct value-coded representation is used for chromosomes.

This type of representation gives ease in handling of multiple design variables of different nature in discrete form.

(ii) The direct value-coded chromosome demonstration can accurately catch increment value for the design variables given by the user within the limit bounds, which is hard in binary representation.

(iii) The single-point crossover mutation/function is defined in such a way to suit the chromosome representation. The chromosome representation and cross-over mutation for the new procedure OptiComp are shown in Fig. 1.

Fig. 1
figure 1

Chromosome representation, crossover, and mutation in OptiComp

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2.2.2 Maximum Stress Theory

According to the given theory of failure, a composite lamina will fail, when the value of any one of the principal stresses created in the lamina equalizes its limiting value. The limiting values are defined by their respective strengths. The lamina will fail, if

$${\sigma }_{11} = {\text{S}}_{{{\text{LC}}}} {\text{or}} {{ \sigma }}_{11} = {\text{S}}_{{{\text{Lt}}}}$$
(1)
$${\sigma }_{22} = {\text{S}}_{{{\text{TC}}}} {\text{or }}\sigma _{22} = {\text{S}}_{{{\text{Tt}}}}$$
(2)
$${\uptau }_{12} = {\text{S}}_{{{\text{Lts}}}} ,$$

where σ11 and σ22 are normal stresses created in directions 1 and 2, respectively, while τ12 is the shear stress developed in plane 1–2 for individual lamina.

2.2.3 Tsai-Wu Theory (TW)

Based on the von Mises yield criterion, according to this theory, the lamina under the consideration will fail, when the below-mentioned condition is satisfied.

$$F_{1} \sigma_{11} + F_{2} \sigma_{22} + F_{6} \tau_{12} + F_{11} \sigma_{11}^{2} + F_{22} \sigma_{22}^{2} + F_{66} \tau_{12}^{2} + 2F_{12} \sigma_{11} \sigma_{22} = 1,$$

where \(F_{1}\),\(F_{2}\),\(F_{6}\),\(F_{11}\), \(F_{22}\), \(F_{66}\), and \(F_{12}\) are the coefficients which can be calculated using strengths of the lamina in different directions as given below,

$$F_{1} = \frac{1}{{S_{{{\text{Lt}}}} }} - \frac{1}{{S_{{{\text{LC}}}} }},F_{2} = \frac{1}{{S_{{{\text{Tt}}}} }} - \frac{1}{{S_{{{\text{TC}}}} }},F_{6} = 0$$
$$F_{11} = \frac{1}{{S_{{{\text{Lt}}}} S_{{{\text{LC}}}} }},F_{22} = \frac{1}{{S_{{{\text{Tt}}}} S_{{{\text{TC}}}} }},F_{66} = \frac{1}{{S_{{{\text{Lts}}}}^{2} }},F_{12} = - \frac{1}{2}\left( {F_{11} F_{12} } \right)^{0.5}$$

A suitable biaxial test is required for determining \(F_{12}\) is

$$- \frac{1}{2}\left( {F_{11} F_{12} } \right)^{0.5} \le F_{12} \le \, 0$$

In the absence of any experimental data, the lower limit of the above equation is frequently used as F12.

3 Generalized Problem Domain

Using the classical lamination plate theory, the strains and stresses developed in each lamina are calculated. The geometry of the laminate coordinate systems and loading conditions considered in the analysis are shown in Fig. 2. The geometric parameters necessary for the laminate analysis are shown in Fig. 3.

Fig. 2
figure 2

Global and local coordinate systems for composite laminate

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Fig. 3
figure 3

Laminate geometric properties

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4 Inputs:

  • Plate Dimensions (Length and Width)

  • Loading Conditions (Nxx, Nyy, Nxy)

  • Material Properties (Elastic Properties, Poisson's Ratio, Strengths)

  • Maximum number of Plies

The elements of \(\left[ {\overline{Q}} \right]\) the stiffness matrix can be developed from the material properties

$$\left[ {\overline{Q}_{mn} } \right] = \left[ {\begin{array}{*{20}c} {\overline{Q}_{11 } } & {\overline{Q}_{12 } } & {\overline{Q}_{16 } } \\ {\overline{Q}_{12 } } & {\overline{Q}_{22 } } & {\overline{Q}_{26 } } \\ {\overline{Q}_{16 } } & {\overline{Q}_{26 } } & {\overline{Q}_{66} } \\ \end{array} } \right]$$

The elements of \(\left[ {\overline{Q}} \right]\) the stiffness matrix can be calculated as

$$\overline{Q}_{11 } = Q_{11} \cos^{4} \theta + 2\left( {Q_{12} + 2Q_{66} } \right)\sin^{2} \theta \cos^{2} \theta + Q_{22}$$
(3)
$$\overline{Q}_{12 } = Q_{12} \left( {\sin^{4} \theta + \cos^{4} \theta } \right) + \left( {Q_{11} + Q_{22} - 4Q_{66} } \right)\sin^{2} \theta \cos^{2} \theta$$
(4)
$$\overline{Q}_{22 } = Q_{11} \sin^{4} \theta + 2\left( {Q_{12} + 2Q_{66} } \right)\sin^{2} \theta \cos^{2} \theta + Q_{22} \cos^{4} \theta$$
(5)
$$\overline{Q}_{16 } = \left( {Q_{11} - Q_{12} - 2Q_{66} } \right)\sin \theta \cos^{3} \theta + \left( {Q_{12} - Q_{22} + 2Q_{66} } \right)\cos \theta \sin^{3} \theta$$
(6)
$$\overline{Q}_{26 } = \left( {Q_{11} - Q_{12} - 2Q_{66} } \right)\cos \theta \sin^{3} \theta + \left( {Q_{12} - Q_{22} + 2Q_{66} } \right)\sin \theta \cos^{3} \theta$$
(7)
$$\overline{Q}_{66} = \left( {Q_{11} + Q_{22} - 2Q_{12} - 2Q_{66} } \right)\sin^{2} \theta \cos^{2} \theta + Q_{66} \left( {\sin^{4} \theta + \cos^{4} \theta } \right),$$
(8)

where \(Q_{11} = \frac{{E_{11} }}{{1 - \gamma_{12} \gamma_{21} }}\), \(Q_{22} = \frac{{E_{22} }}{{1 - \gamma_{12} \gamma_{21} }}\), \(\gamma_{21} = \gamma_{12} \frac{{E_{22} }}{{E_{11} }}\).

\(Q_{12} = Q_{21} = \frac{{\gamma_{21} E_{11} }}{{1 - \gamma_{12} \gamma_{21} }} = \frac{{\gamma_{12} E_{22} }}{{1 - \gamma_{12} \gamma_{21} }}\), \(Q_{66} = G_{12}\).

The additionally required geometric parameters for the laminate analysis are shown in Fig. 3, t1, tj, and tn indicate 1st, jth, and nth thicknesses of laminates; h0 gives space from the mid-plane laminate to the top of the 1st lamina, while h1 indicates the space/distance from the mid-plane laminate to the bottom of the 1st lamina. hj1 specifies the distance between the top to laminate mid-plane of the jth lamina and hj indicates the distance from the bottom to the mid-plane of laminate the jth lamina. The entire thickness of the laminate is specified by with letter TZj gives the distance from the mid-plane laminate to the mid-plane of the jth lamina

[\(A_{1}\)] = [\(A^{ - 1}\)] + [\(A^{ - 1}\)] [B] [(D*)−1] [B] [\(A^{ - 1}\)].

[\(B_{1}\)] = \(-\)[\(A^{ - 1}\)] [B] [(D*)−1], [\(C_{1}\)] = [\(B_{1}\)]T.

[D*] = [D] – [B] [A−1] [B], [\(D_{1}\)] = [(D*)−1].

The curvatures [K] and mid-plane strains [\(\varepsilon^{0}\)] can be calculated as

$$[\varepsilon^{0} ] = [{\text{A}}_{1} ][{\text{N}}] + [{\text{B}}_{{1}} ][{\text{M]}}\;{\text{where}}$$
(9)
$$\left[ {\varepsilon^{0} } \right] = \left[ {\begin{array}{*{20}c} {\varepsilon_{xx}^{0} } \\ {\varepsilon_{yy}^{0} } \\ {\varepsilon_{xy}^{0} } \\ \end{array} } \right],\,\;\left[ {\text{N}} \right] = \left[ {\begin{array}{*{20}c} {N_{xx} } \\ {N_{yy} } \\ {N_{xy} } \\ \end{array} } \right],\;[M] = \left[ {\begin{array}{*{20}c} {M_{xx} } \\ {M_{yy} } \\ {M_{xy} } \\ \end{array} } \right]$$
$$[{\text{K}}] = [{\text{C}}_{1} ][{\text{N}}] + [{\text{D}}_{{1}} ][{\text{M}}]$$
(10)

The curvatures and strains at the mid-plane are used to calculate actual stresses and strains for individual lamina in global coordinate system.

$$\left[ \varepsilon \right]_{j} = \left[ {\varepsilon^{0} } \right] + Z_{j} [{\text{K}}]$$
(11)

The converted local coordinate system from the global one, for individual lamina.

Can be given by using the relations given below.

$$\varepsilon_{11} = \varepsilon_{xx} \cos^{2} \theta + \varepsilon_{yy} \sin^{2} \theta + \varepsilon_{xy} \sin \theta \cos \theta$$
(12)
$$\varepsilon_{22} = \varepsilon_{xx} \sin^{2} \theta + \varepsilon_{yy} \cos^{2} \theta - \varepsilon_{xy} \sin \theta \cos \theta$$
(13)
$$\varepsilon_{12 } = 2( - \varepsilon_{xx} + \varepsilon_{yy} )\sin \theta \cos \theta + \varepsilon_{xy} \cos^{2} \theta - \sin^{2} \theta$$
$$\left[ \sigma \right]_{j} = \left[ {\overline{Q}} \right] _{j} \left[ \varepsilon \right]_{j} \;{\text{where}}\;[\sigma ] = \left[ {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\tau_{xy} } \\ \end{array} } \right]$$
(14)

where\(Z_{j}\) is the space/distance from the mid-plane laminate to the mid-plane of the jth lamina.

The local converted coordinate system from global stresses, for individual lamina, can be given.

By using the relations below.

$$\sigma_{11} = \sigma_{xx} \cos^{2} \theta + \sigma_{yy} \sin^{2} \theta + 2\tau_{xy} \sin \theta \cos \theta$$
(15)
$$\sigma_{22} = \sigma_{xx} \sin^{2} \theta + \sigma_{yy} \cos^{2} \theta - 2\tau_{xy} \sin \theta \cos \theta$$
(16)
$$\tau_{12 } = ( - \sigma_{xx} + \sigma_{yy} )\sin \theta \cos \theta + \tau_{xy} \cos^{2} \theta - \sin^{2} \theta$$
(17)

The second approach ‘Ideal Procedure’ for solving MOOP is developed on the basis of Non-Dominated Sorting G.A. (NSGA II) in the generic procedure (Refer Fig. 4). In NSGAII, the \({\text{P}}_{{\text{t}}}\) population of the parent is used to create the offspring population \({\text{Q}}_{{\text{t}}}\). During this process, the genetic operators like crossover, selection, and mutation are applied to the population of the parent \({\text{P}}_{{\text{t}}}\). Then two populations are combined to create the \({\text{R}}_{{\text{t}}}\) population. If ‘N’ is the size of the parent population, then the size of \({\text{R}}_{{\text{t}}}\) will be '2 N.' Then by using a non-dominated sorting procedure, the entire population \({\text{R}}_{{\text{t}}}\) is classified. This allows a non-domination global check among the off spring and parent solutions. After this step, solutions from different non-dominated fronts are used one by one to fill \(P_{t + 1}\) the new population. The filling starts through the finest non-dominated front and continues by way of solutions of the second non-dominated front, succeeded by the third one and so on. As the size of the new population is 'N,' it cannot occupy all the solutions from the overall population \(R_{t}\) of size ‘2 N.’ The fronts going beyond size ‘N’ could not be accommodated and so are deleted. The allowed last front may include extra solutions than the existing slots/space in the population new one. In such a situation, crowd sorting of the solutions explained below is used to choose the member of the last front, which is placed in the least crowded region of that front.

Fig. 4
figure 4

Flowchart of Non-Dominated Sorting G.A. (NSGA II) in the generic procedure

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Crowded tournament selection: A solution ‘i’ is declared as tournament winner with another solution ‘j’ if any of the following conditions are satisfied.

  1. 1.

    If the solution 'i' has better rank, i.e., \(r_{i} < r_{j}\)

  2. 2.

    For the same ranks, solution 'i' must have a better crowding distance than solution 'j.'

The solutions from the same front will carry the same rank. At this stage, the second criterion is more important. The newly formed population \(P_{t + 1}\) can be used for developing new offspring population \(Q_{t + 1} .\), this will complete one generation of NSGAII. This process is explained in Fig. 4. The generic procedure is developed using Matlab, which has its multi-objective optimization toolbox. NSGAII under the multi-objective optimization toolbox of the Matlab considers all the design variables in continuous form. The practically available values of ply angles and ply thicknesses are in discrete form. In a multi-objective optimization module based on NSGAII, the parent population \(P_{t}\) is generated considering the user-required ply thickness and ply angle values. The nature of the chromosomes developed is similar to as mentioned in Fig. 1 for VTA and UTA. The selection operator is adopted from the Matlab toolbox, which is based on the selection method of the tournament. The single-point crossover and mutation operators as defined in Fig. 1 are applied to this population to the general population \(Q_{t}\).

By default, the Matlab MOOP toolbox is not meant to accept non-linear constraints for the 'custom' population used in the process. The non-linear constraints in the case of composite laminate design are the different theories of failure. The non-linear constraints are invoked in the NSGAII-based module developed under OptiComp using the penalty approach.

4.1 Problem Variables Within the Project Domain

Objectives:

  1. (i)

    Maximize stiffness

  2. (ii)

    Minimize weight

Design variables:

  1. (i)

    Type of fiber

  2. (ii)

    Fiber orientation angle

  3. (iii)

    Thickness of lamina

  4. (iv)

    Number of laminae

  5. (v)

    wt% of nanoparticles

Constraints:

  1. (i)

    Maximum stress theory

  2. (ii)

    Tsai-Wu theory

4.2 Formulation of Problem

Find {θn,tn},

To minimize the weight,

$${\text{W}} = \rho .a.b.T \;\;(T = \mathop \sum \limits_{i = 1}^{n} t_{i} )$$
(18)

To maximize the stiffness or to minimize the strain,

$$K_{xx} \propto \frac{1}{{\varepsilon_{xx} }}$$
$$K_{yy} \propto \frac{1}{{\varepsilon_{yy} }}$$
$$K_{xy} \propto \frac{1}{{\varepsilon_{xy} }}$$
(19)

Subjected to satisfying

  • Maximum stress theory and Tsai-Wu theory

  •  − 45° ≥ θn ≥ 90° (Ply angle incremental value 15°),

  • 0.2 ≤ tn ≤ 0.4 (Ply thickness incremental value 0.01 mm),

  • n = 1,..., Nmax (Nmax as obtained in an earlier stage).

The values used in the OptiComp, calculated by mixture rule and measured by the testing

The rule of the mixture is as given below

$$E_{L} = E_{f} V_{f} + E_{m} V_{m} = E_{f} V_{f} + E_{m} \left( {1 - V_{f} } \right) = E_{m} + V_{f} \left( {E_{f} - E_{m} } \right)$$
(20)

The values represented in Table 2 are calculated by using the law of mixture as shown in the equation above the Table. The values of the tensile strength in the longitudinal and transverse direction given in Table 1 are obtained from the experimental results.

Table 2 Calculated and experimental values of the parameters used in OptiComp
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5 Result and Conclusions

As per the results obtained from the OptiComp as shown in Tables 2 and 3, we can get the different minimum and maximum values of the weights and the stains of the laminates at the different combinations of the nanoparticles and CFRP. The comparison of the different level weights and strains at both the theories of failure as a constraint can be easily obtained from the given tables.

Table 3 Showing the result of Minimum and Maximum weight and strain range for Max stress Theory as a constraint
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As per the values given in Tables 3 and 4, the graphs are shown in Figs. 5, 6, 7, and 8. It displays the different levels and behavior of the material system of different combinations of the nanoparticles combination, i.e., nanoclay and nanoZnO percentage mixture with the CFRP.

Table 4 Showing the result of Minimum and Maximum weight and strain range for Tsai-Wu Theory as a constraint
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Fig. 5
figure 5

Bar chart showing the maximum and minimum weight for various levels of CFRP hybrid nanocomposites

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Fig. 6
figure 6

Line chart showing the maximum and minimum weight for various levels of CFRP hybrid nanocomposites

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Fig. 7
figure 7

Bar chart showing the maximum and minimum strain for various levels of CFRP hybrid nanocomposites

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Fig. 8
figure 8

Line chart showing the maximum and minimum strain for various levels of CFRP hybrid nanocomposites

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From the Pareto, optimal fronts are the multi-objective OptiComp results for the pure CFRP and new CFRP material systems with the nanoparticles shown in Fig. 9a–g for the Maximum stress theory as a constraint and Fig. 10a–g for the Tsai-Wu theory as the constraint. The simultaneous sample readings of stacking sequence angles and corresponding thickness of the laminas are as shown in Tables 5 and 6, respectively.

Fig. 9
figure 9figure 9

a–g Pareto optimal front for various levels of CFRP hybrid nanocomposite for Max. Stress theory as a constraint. a Pareto optimal front for Pure CFRP composite laminate for Max stress Th. b Pareto optimal front for Level-I CFRP nanocomposite laminate for Max stress Th. c Pareto optimal front for Level-II CFRP nanocomposite laminate for Max stress Th. d Pareto optimal front for Level-III CFRP nanocomposite laminate for Max stress Th.e Pareto optimal front for Level-IV CFRP nancomposite laminate for Max stress Th. f Pareto optimal front for Level-V CFRP nanocomposite laminate for Max stress Th. g Pareto optimal front for Level-VI CFRP nanocomposite laminate for Max stress Th

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Fig. 10
figure 10figure 10

a–g Pareto optimal front for various levels of CFRP hybrid nanocomposite for Tsai-Wu theory as a constraint. a Pareto optimal front for Pure CFRP composite laminate for Tsai-Wu Th. b Pareto optimal front for Level-I CFRP nanocomposite laminate for Tsai-Wu Th. c Pareto optimal front for Level-II CFR nanocomposite laminate for Tsai-Wu Th. d Pareto optimal front for Level-III CFRP nanocomposite laminate for Tsai-Wu Th. e Pareto optimal front for Level-IV CFRP nanocomposite laminate for Tsai-Wu Th. f Pareto optimal front for Level-V CFRP nanocomposite laminate for Tsai-Wu Th. g Pareto optimal front for Level-VI CFRP nanocomposite for Tsai-Wu Th

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Table 5 Showing samples of stacking sequence angles and thickness of laminates for Max. Stress theory as a constraint
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Table 6 Showing samples of stacking sequence angles and thickness of laminates for Tsai-Wu Theory as a constraint
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It can be observed from the Pareto charts and the readings of the Table 3, for the Maximum stress theory as constraint, the minimum weight of the laminate is going to be obtained at Level II, i.e., CFRP with 2 wt% of nanoZnO and 3 wt% of nanoclay. Simultaneously at Level –IV, we can observe the minimum strain, i.e., maximum stiffness of the material system with 3% each wt of Nanoclay and nanoZnO with CFRP. We can get the optimum results between 2 and 3% of both nanoclay and nanoZnO. We can also observe from Table 4 and corresponding Pareto optimal fronts, with Tsai-Wu theory as a constraint. The minimum weight for the CFRP hybrid nanocomposite can be obtained at Level I with 2 wt% of nanoparticles of each of clay and ZnO. The maximum stiffness can be obtained with Level II with 2 wt% of nanoclay and 3 wt% of ZnO nanoparticles. Tables 5 and 6 show the sample of the stacking sequence and thickness of the respective layer of the laminates for the respective theories as constraints. From the foresaid graphs and the results, it can be proved that the OptiComp can be effectively and efficiently used for solving the multi-objective design optimization problems related to FRP hybrid nanocomposites.