# Optimal Design of CNT-Nanocomposite Nonlinear Shells

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Solid-Shell Model

#### Stored and Complementary Energy

## 3. Isogeometric Solid-Shell Model

**NURBS basics**. A B-Spline curve is represented as

**Isogeometric interpolation**. In this subsection, the discrete isogeometric model used within the optimization strategy is summarized. The geometry is described by NURBS interpolation functions as

**Stored energy and equilibrium path**. The stored energy of the shell can be evaluated using a numerical integration as

## 4. Constitutive Formulation for CNT Nancomposite Shells

- (i)
- randomly orientated CNTs$$\langle \mathit{B}\rangle =\frac{1}{8{\pi}^{2}}{\int}_{0}^{2\pi}{\int}_{0}^{\pi}{\int}_{0}^{2\pi}\overline{\mathit{B}}sin\vartheta d\phi d\vartheta d\beta $$
- (ii)
- CNTs alignment along ${\overline{e}}_{1}$$$\langle \mathit{B}\rangle =\frac{{\int}_{0}^{2\pi}{\int}_{0}^{\pi}{\int}_{0}^{2\pi}\overline{\mathit{B}}f(\phi ,\beta )sin\vartheta d\phi d\vartheta d\beta}{{\int}_{0}^{2\pi}{\int}_{0}^{\pi}{\int}_{0}^{2\pi}f(\phi ,\beta )sin\vartheta d\phi d\vartheta d\beta},$$

## 5. Postbuckling Optimization of CNT Nanocomposite Shells

#### 5.1. Through-the-Thickness Optimization of the Aligned CNTs Volume Fraction

#### 5.2. Optimization of Randomly Orientated CNTs Volume Fraction

#### 5.3. Optimization of the in-Plane CNTs Orientation

## 6. Objective Function and Optimization Algorithm

**The objective function**. The optimization process is aimed at maximizing the collapse load of the nanocomposite shells. In buckling problems, the collapse load can be defined as the lower bound between the critical limit load ${\lambda}_{lim}$ and the load associated with a deformation limit ${\lambda}_{def}$. We denote with $\mathit{\alpha}$ the vector collecting the generic design optimization parameters. In this work, $\mathit{\alpha}$ coincides with the material variables: either $\mathbf{a}$, $\mathit{\varphi}$ or $\mathit{\vartheta}$ introduced in the previous section for the three stated optimization problems, respectively. The objective function can thus be written as

**Koiter’s method**. The structure is first discretized using the isogeometric environment described in Section 3. Then, the stored energy of each element is rewritten in a mixed form using the stresses at each integration point ${\mathit{\sigma}}_{g}$ as independent variables [17]

**The Global Convergent Method of Moving Asymptotes**. The optimal design problem is solved using a gradient-based optimizer, i.e., the Global Convergent Method of Moving Asymptotes (GCMMA) [33,38,39,40]. This algorithm devised for the optimization of objective functions requires a relatively high computational cost to evaluate the gradient and is characterized by many optimization variables. It is based on convex subsequent approximations of the objective function.

## 7. Numerical Results

- -
- OPT1: through-the-thickness optimization, i.e., variable through-the-thickness distribution of aligned CNTs across the shell mid-surface with assigned mid-surface value;
- -
- OPT2: optimization across the mid-surface, i.e., the volume fraction of randomly oriented CNTs is kept constant through the thickness but can vary within the shell mid-surface with a constraint on the overall CNTs volume;
- -
- OPT3: optimization of the orientation, i.e., the CNTs volume is assigned, the orientation of the CNTs can vary across the shell mid-surface but is constant through the thickness.

#### 7.1. Nanocomposite Beam Under Compression

**Optimization of the CNT volume fraction for random orientations**. The optimization problem OPT2 only is considered because OPT1 and OPT3 do not yield any improvement for this specific problem. The results of OPT2 are compared with the performance of a uniform distribution of ${\varphi}_{C}$ (hereafter referred to as UD). Two polynomial orders are used to describe the distribution of ${\varphi}_{C}$ across the surface: order 4 and order 9. First, we compare the linearized buckling loads reported in Table 2 for order 4. We note an increase of the first load for all average volume fractions ${\varphi}_{C}^{*}$ and in particular for the case ${\varphi}_{C}^{*}=5\%$ where the improvement reaches $20\%$. Similar considerations hold for the description with order 9 as shown in Table 3. A similar improvement is highlighted by a full nonlinear analysis in Figure 3. Finally, the optimal distribution of CNTs volume fraction is depicted in Figure 4 and Figure 5 for the two orders used to describe ${\varphi}_{C}$ over the surface. As expected, it is possible to observe that, by comparison with the initial uniform distribution, the CNTs volume fraction is greater at the midspan of the beam and lower at the end sections in order to maximize the flexural stiffness. Moreover, it is worth noting that the effectiveness ratio of the CNTs decreases by increasing ${\varphi}_{C}$ as shown in Table 1. This is the reason why the optimal distribution tends to be more uniform and equal to the maximum admissible fraction near the midspan as shown in Figure 4 and Figure 5 for high values of ${\varphi}_{C}^{*}$.

#### 7.2. Nanocomposite Plate under Compression

**Optimization of the CNTs volume fraction for random orientations**. The optimization problem OPT2 is discussed here, making use of a comparison with the solution obtained for the uniform distribution of ${\varphi}_{C}$ (UD). The linearized buckling loads are reported in Table 4 for the polynomial description of ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ of order 4. We note a general but very slight increase of the first load for all average volume fractions ${\varphi}_{C}^{*}$. Similar considerations hold for the parametrization of order 9 where the optimized solution is slightly better. The same considerations hold if we consider the full nonlinear paths shown in Figure 7. Finally, the optimal distribution of CNTs volume fraction is depicted in Figure 8 for 9th-order polynomials employed to describe ${\varphi}_{C}$ across the surface. For the simply supported square plate, the use of a nonuniform distribution of randomly CNTs within the mid-surface does not yield significant improvements.

**Optimization of the CNTs orientation within the mid-surface**. The optimization problem OPT3 is investigated next. It consists in optimizing the orientation of the CNTs in the plate plane, while keeping the volume fraction constant. The results of OPT3 are compared with those corresponding to all CNTs uniformly aligned with the load direction (UD). The linearized buckling loads are reported in Table 5 for the polynomial parametrization of the orientation of order 4. It is possible to observe that the first buckling load is increased notably by the optimization and the improvement gets better with the volume fractions. Figure 9 with the full nonlinear path shows that a higher post-buckling stiffness can be obtained for the optimized solutions compared to the uniform orientation and an even better behavior is obtained with the parametrization of order 9. The results are completed with Figure 10 depicting the optimal orientation distribution of order 9.

#### 7.3. Cylindrical Nanocomposite Panel under Compression

**Through-the-thickness optimization of aligned CNTs volume fraction**. We start with the through-the-thickness optimization problem referred to as OPT1. The CNTs are aligned along the load direction. The results are compared with those obtained for a uniform through-the-thickness distribution (UD). First, Table 6 shows that the first linearized buckling load turns out to be almost unaffected by the optimization process. On the contrary, the full nonlinear analysis reported in Figure 12 shows how the optimization globally turns the behavior from unstable into stable using the same CNTs volume. The snap-through behavior is completely suppressed at the cost of a very slight stiffness reduction in the pre-critical range. Similar results in terms of equilibrium paths are obtained with a polynomial description of the function $a({\zeta}_{1},{\zeta}_{2})$ of order 9. The shape of $a({\zeta}_{1},{\zeta}_{2})$ is very similar for the two orders and is reported in Figure 13 for order 4.

**Optimization of CNTs volume fraction for random orientations**. The optimization problem OPT2 is considered next. The results are compared with the performance of a uniform distribution of randomly oriented CNTs (UD). First, we compare the linearized buckling loads reported in Table 7 for order 4. We note a general increase of the first load for all average volume fractions ${\varphi}_{C}^{*}$ and, in particular, for the intermediate ${\varphi}_{C}^{*}$ whose improvement is about $10\%$. Similar considerations hold for the parametrization of order 9 as shown in Table 8 where the optimized solution is also slightly improved. However, the great benefit of the variable volume fraction distribution is highlighted by a full nonlinear analysis in Figure 14. The unstable behavior of the uniform distribution is made stable by the optimal CNTs distribution using the same overall amount of CNTs. The slight stiffness reduction in the pre-critical range is compensated by the complete elimination of snap-through, at least in the range of interest. Similar results in terms of equilibrium paths are obtained with a polynomial description of ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ of order 9. Indeed, the analysis leads to a very similar optimal distribution for the two orders, reported in Figure 15 for order 9.

**Optimization of CNTs orientation**. The optimization problem OPT3 is finally discussed here. It consists of optimizing the CNTs orientation within the shell mid-surface, while keeping constant the volume fraction at each point of the structure. The results of OPT3 are compared with those obtained for uniformly aligned CNTs collinear with the load direction (UD). The linearized buckling loads are reported in Table 9 for the parametrized orientation distribution of order 4. It is possible to observe that the first buckling load becomes notably larger for high volume fractions, while it remains almost the same for low CNT contents. However, looking at Figure 16, the radical change of mechanical behavior is appreciable also for low volume fractions. The optimal CNTs distribution leads to the suppression of the snap-through instability, at the cost of initial stiffness reduction for low ${\varphi}_{C}$, reduction that becomes negligible for higher volume fractions. Similar results are obtained with the parametrization of order 9. The results are completed with Figure 17 depicting the optimal orientation distribution of order 9.

## 8. Conclusions

- A numerical strategy for the optimization of the buckling and postbuckling response of nanocomposite shells with variable CNTs distribution was proposed and investigated. The method is based on an integrated isogeometric framework that employs NURBS functions to describe the geometry and displacements while the optimization variables deal with the CNT distributions within the polymer hosting matrix.
- Various CNTs distributions were investigated either through the thickness or within the mid-surface for both aligned CNTs, aligned but varying within the surface or randomly oriented. The obtained through-the thickness distributions can be practically realized in multilayer nanocomposite structures since a continuous law can be reasonably approximated by piece-wise functions when the multilayers are considered sufficiently thin.
- The outcomes of extensive numerical tests have proved that the limit load can be largely improved for optimal CNTs distributions in the sense of strategically deploying the nanofibers where the maximum elastic stiffness fighting against the negative stiffness can be attained.
- Most importantly, it has been shown that shallow shells, which are dangerously prone to snap-through, can become globally stable if the CNTs are optimally distributed. This is a remarkable result on the global stability of nanocomposite shallow shells which, when properly designed, do not show any snap-through and thus can be safely employed in engineering applications. Mention must be made of the fact that these nanocomposite panels also show the additional advantage of exhibiting enhanced damping capability thanks to the CNT/polymer interfacial dissipation which makes these structures generally more stable against dynamic loads.
- This work has shown the potential of optimizing nonlinear structural behaviors using the unprecedented flexibility afforded by the CNTs nanoreinforcement which not only acts to shift the elastic loss of stability towards higher stresses, but can also either suppress snap-through or make the response less compliant in the postbuckling range. The next step of the research will be the optimal design of high-performance and lightweight vehicles, aerostructures and devices.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Wang, B.; Tian, K.; Zhou, C.; Hao, P.; Zheng, Y.; Ma, Y.; Wang, J. Grid-pattern optimization framework of novel hierarchical stiffened shells allowing for imperfection sensitivity. Aerosp. Sci. Technol.
**2017**, 62, 114–121. [Google Scholar] [CrossRef] - Wu, Z.; Raju, G.; Weaver, P.M. Optimization of Postbuckling Behaviour of Variable Thickness Composite Panels with Variable Angle Tows: Towards “Buckle-Free” Design Concept. Int. J. Solids Struct.
**2017**, 132, 66–79. [Google Scholar] [CrossRef] - Zucco, G.; Groh, R.M.J.; Madeo, A.; Weaver, P.M. Mixed shell element for static and buckling analysis of variable angle tow composite plates. Compos. Struct.
**2016**, 152, 324–338. [Google Scholar] [CrossRef] [Green Version] - Ajayan, P.M.; Suhr, J.; Koratkar, N. Utilizing interfaces in carbon nanotube reinforced polymer composites for structural damping. J. Mater. Sci.
**2006**, 41, 7824–7829. [Google Scholar] [CrossRef] - Basiricò, L.; Lanzara, G. Moving towards high-power, high-frequency and low- resistance CNT supercapacitors by tuning the CNT length, axial deformation and contact resistance. Nanotechnology
**2012**, 23, 305–401. [Google Scholar] [CrossRef] - Formica, G.; Lacarbonara, W. Debonding model of carbon nanotubes in a nanostructured composite. Compos. Struct.
**2012**, 96, 514–525. [Google Scholar] [CrossRef] - Basiricò, L.; Lanzara, G. A monolithic functional film of nanotubes/cellulose/ionic liquid for high performance supercapacitors. J. Power Sources
**2014**, 271, 589–596. [Google Scholar] [CrossRef] - Formica, G.; Milicchio, F.; Lacarbonara, W. Hysteretic damping optimization in carbon nanotube nanocomposites. Compos. Struct.
**2018**, 194, 633–642. [Google Scholar] [CrossRef] - Taló, M.; Lanzara, G.; Krause, B.; Janke, A.; Lacarbonara, W. “Sliding Crystals” on low-dimensional carbonaceous nanofillers as distributed nanopistons for highly damping materials. ACS Appl. Mater. Interfaces
**2019**, 11, 38147–38159. [Google Scholar] [CrossRef] - Formica, G.; Milicchio, F.; Lacarbonara, W. Computational efficiency and accuracy of sequential nonlinear cyclic analysis of carbon nanotube nanocomposites. Adv. Eng. Softw.
**2018**, 125, 126–135. [Google Scholar] [CrossRef] - Formica, G.; Taló, M.; Lanzara, G.; Lacarbonara, W. Parametric identification of carbon nanotube nanocomposites constitutive response. J. Appl. Mech.
**2019**, 86, 041007. [Google Scholar] [CrossRef] - Formica, G.; Milicchio, F.; Lacarbonara, W. Storage and damping optimization in hysteretic multilayer nanocomposites. Int. J. Multiscale Comput. Eng.
**2020**, 18, 141–157. [Google Scholar] [CrossRef] - Odegard, G.M.; Gates, T.S.; Wise, K.E.; Park, C.; Siochi, E.J. Constitutive modeling of nanotube-reinforced polymer composites. Compos. Sci. Technol.
**2003**, 63, 1671–1687. [Google Scholar] [CrossRef] - Talò, M.; Krause, B.; Pionteck, J.; Lanzara, G.; Lacarbonara, W. An updated micromechanical model based on morphological characterization of carbon nanotube nanocomposites. Compos. Part B Eng.
**2017**, 115, 70–78. [Google Scholar] [CrossRef] - Thompson, J.M.T. Optimization as a generator of structural instability. Int. J. Mech. Sci.
**1972**, 14, 627–629. [Google Scholar] [CrossRef] - Riks, E. An incremental approach to the solution of snapping and buckling problems. Int. J. Solids Struct.
**1979**, 15, 529–551. [Google Scholar] [CrossRef] - Magisano, D.; Leonetti, L.; Garcea, G. How to improve efficiency and robustness of the Newton method in geometrically non-linear structural problem discretized via displacement-based finite elements. Comput. Methods Appl. Mech. Eng.
**2017**, 313, 986–1005. [Google Scholar] [CrossRef] - Eriksson, A.; Pacoste, C.; Zdunek, A. Numerical analysis of complex instability behaviour using incremental-iterative strategies. Comput. Methods Appl. Mech. Eng.
**1999**, 179, 265–305. [Google Scholar] [CrossRef] - Deml, M.; Wunderlich, W. Direct evaluation of the “worst” imperfection shape in shell buckling. Comput. Methods Appl. Mech. Eng.
**1997**, 149, 201–222. [Google Scholar] [CrossRef] - Groh, R.M.J.; Avitabile, D.; Pirrera, A. Generalised path-following for well-behaved nonlinear structures. Comput. Methods Appl. Mech. Eng.
**2018**, 331, 394–426. [Google Scholar] [CrossRef] - Lindgaard, E.; Lund, E. Nonlinear buckling optimization of composite structures. Comput. Methods Appl. Mech. Eng.
**2010**, 199, 2319–2330. [Google Scholar] [CrossRef] [Green Version] - Lindgaard, E.; Lund, E.; Rasmussen, K. Nonlinear buckling optimization of composite structures considering “worst” shape imperfections. Int. J. Solids Struct.
**2010**, 47, 3186–3202. [Google Scholar] [CrossRef] [Green Version] - Henrichsen, S.R.; Lindgaard, E.; Lund, E. Robust buckling optimization of laminated composite structures using discrete material optimization considering “worst” shape imperfections. Thin-Walled Struct.
**2015**, 94, 624–635. [Google Scholar] [CrossRef] - Koiter, W.T. On the Stability of Elastic Equilibrium. Techische Hooge School at Delft, English Transl. NASA TT-F10, 883 (1967) and AFFDL-TR70-25 (1970). Available online: https://www.worldcat.org/title/on-the-stability-of-elastic-equilibrium/oclc/13765167 (accessed on 9 December 2020).
- Magisano, D.; Leonetti, L.; Garcea, G. Koiter asymptotic analysis of multilayered composite structures using mixed solid-shell finite elements. Compos. Struct.
**2016**, 154, 296–308. [Google Scholar] [CrossRef] - Liguori, F.S.; Madeo, A.; Magisano, D.; Leonetti, L.; Garcea, G. Post-buckling optimization strategy of imperfection sensitive composite shells using Koiter method and Monte-Carlo simulation. Compos. Struct.
**2018**, 192, 654–670. [Google Scholar] [CrossRef] - Liguori, F.S.; Zucco, G.; Madeo, A.; Magisano, D.; Leonetti, L.; Garcea, G.; Weaver, P.M. Postbuckling optimization of a variable angle tow composite wingbox using a multi-modal Koiter approach. Thin-Walled Struct.
**2019**, 138, 183–198. [Google Scholar] [CrossRef] - Garcea, G.; Liguori, F.S.; Leonetti, L.; Magisano, D.; Madeo, A. Accurate and efficient a posteriori account of geometrical imperfections in Koiter finite element analysis. Int. J. Numer. Methods Eng.
**2017**, 112, 1154–1174. [Google Scholar] [CrossRef] - Liang, K.; Ruess, M.; Abdalla, M. The Koiter-Newton approach using von Karman kinematics for buckling analyses of imperfection sensitive structures. Comput. Methods Appl. Mech. Eng.
**2014**, 279, 440–468. [Google Scholar] [CrossRef] - Magisano, D.; Liang, K.; Garcea, G.; Leonetti, L.; Ruess, M. An efficient mixed variational reduced-order model formulation for nonlinear analyses of elastic shells. Int. J. Numer. Methods Eng.
**2018**, 113, 634–655. [Google Scholar] [CrossRef] - Barkanov, E.; Ozolins, O.; Eglitis, E.; Almeida, F.; Bowering, M.C.; Watson, G. Optimal design of composite lateral wing upper covers. Part I: Linear buckling analysis. Aerosp. Sci. Technol.
**2014**, 38, 1–8. [Google Scholar] [CrossRef] - Le Riche, R.; Haftka, R.T. Optimization of laminate stacking sequence for buckling load maximization by genetic algorithm. AIAA J.
**1993**, 31, 951–956. [Google Scholar] [CrossRef] - Svanberg, K. The method of moving asymptotes—A new method for structural optimization. Int. J. Numer. Methods Eng.
**1987**, 24, 359–373. [Google Scholar] [CrossRef] - Johansen, L.S.; Lund, E.; Kleist, J. Failure optimization of geometrically linear/nonlinear laminated composite structures using a two-step hierarchical model adaptivity. Comput. Methods Appl. Mech. Eng.
**2009**, 198, 2421–2438. [Google Scholar] [CrossRef] - Leonetti, L.; Liguori, F.S.; Magisano, D.; Garcea, G. An efficient isogeometric solid-shell formulation for geometrically nonlinear analysis of elastic shells. Comput. Methods Appl. Mech. Eng.
**2018**, 331, 159–183. [Google Scholar] [CrossRef] - Leonetti, L.; Magisano, D.; Liguori, F.S.; Garcea, G. An isogeometric formulation of the Koiter’s theory for buckling and initial post-buckling analysis of composite shells. Comput. Methods Appl. Mech. Eng.
**2018**, 337, 387–410. [Google Scholar] [CrossRef] - Piegl, L.; Tiller, W. The NURBS Book; Springer: Berlin/Heidelberg, Germany, 1997. [Google Scholar]
- Zillober, C. A globally convergent version of the method of moving asymptotes. Struct. Optim.
**1993**, 6, 166–174. [Google Scholar] [CrossRef] [Green Version] - Bruyneel, M.; Duysinx, P.; Fleury, C. A family of MMA approximations for structural optimization. Struct. Multidiscip. Optim.
**2002**, 24, 263–276. [Google Scholar] [CrossRef] [Green Version] - Bruyneel, M.; Fleury, C. Composite structures optimization using sequential convex programming. Adv. Eng. Softw.
**2002**, 33, 697–711. [Google Scholar] [CrossRef]

**Figure 3.**Euler nanocomposite beam: equilibrium paths for the optimal volume fraction distribution ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ with assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 4 and 9 (${\lambda}_{r}=0.0005313$ KN/mm).

**Figure 4.**Euler nanocomposite beam: optimized volume fraction ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ for assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 4.

**Figure 5.**Euler nanocomposite beam: optimized volume fraction ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ for assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 9.

**Figure 7.**Square nanocomposite plate: equilibrium paths for the optimal volume fraction distribution ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ with assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 4 and 9.

**Figure 8.**Square nanocomposite plate: optimized volume fraction distribution ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ with assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 9.

**Figure 9.**Square nanocomposite plate: equilibrium paths for the optimal CNTs orientation $\theta ({\zeta}_{1},{\zeta}_{2})$ described by Bernstein polynomials of order 4 and 9 for assigned volume fraction ${\varphi}_{C}$.

**Figure 10.**Square nanocomposite plate: optimal orientation $\theta ({\zeta}_{1},{\zeta}_{2})$ described by Bernstein polynomials of order 9 for assigned volume fraction ${\varphi}_{C}$.

**Figure 11.**Cylindrical nanocomposite: geometry (lengths in mm), NURBS control grid, loading and boundary conditions.

**Figure 12.**Cylindrical nanocomposite panel: equilibrium paths for the optimal through-the-thickness CNTs distribution with function $a({\zeta}_{1},{\zeta}_{2})$ in Equation (20) described by Bernstein polynomials of order 4 and 9 for assigned volume fraction ${\varphi}_{C}^{*}$ (${\lambda}_{r}=0.006466$ KN/mm).

**Figure 13.**Cylindrical panel: optimized variability function $a[{\zeta}_{1},{\zeta}_{2}]$ in Equation (20) described by Bernstein polynomials of order 4 for assigned ${\varphi}_{C}^{*}$.

**Figure 14.**Cylindrical nanocomposite panel: equilibrium paths for the optimal volume fraction distribution ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ with assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 4 and 9 (${\lambda}_{r}=0.006466$ KN/mm).

**Figure 15.**Cylindrical nanocomposite panel: optimized volume fraction distribution ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ with assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 9.

**Figure 16.**Cylindrical nanocomposite panel: equilibrium paths for the optimal CNTs orientation $\theta ({\zeta}_{1},{\zeta}_{2})$ described by Bernstein polynomials of order 4 and 9 for assigned volume fractions ${\varphi}_{C}$ (${\lambda}_{r}=0.006466$ KN/mm).

**Figure 17.**Cylindrical nanocomposite panel: optimized orientation $\theta ({\zeta}_{1},{\zeta}_{2})$ described by Bernstein polynomials of order 9 for assigned volume fraction ${\varphi}_{C}$.

nominal | 0.50 | 0.75 | 1.00 | 2.00 | 5.00 | 10.0 | 15.0 |

effective | 0.48 | 0.72 | 0.96 | 1.89 | 4.45 | 7.570 | 8.50 |

**Table 2.**Euler beam: linearized buckling loads for the optimal volume fraction distribution ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ with assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 4 normalized with respect to ${\lambda}_{r}=0.0005313$ KN/mm.

Mode | $0.50\%$ | $0.75\%$ | $1.00\%$ | $2.00\%$ | $5.00\%$ | $10.0\%$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | |

1 | 0.5899 | 0.5386 | 0.6951 | 0.6163 | 0.7977 | 0.6936 | 1.1867 | 1.0000 | 2.2341 | 1.8662 | 3.1037 | 2.9634 |

2 | 2.1437 | 2.1621 | 2.4200 | 2.4737 | 2.6829 | 2.7843 | 3.7129 | 4.0141 | 7.3660 | 7.4911 | 12.027 | 11.894 |

3 | 4.8159 | 4.8906 | 5.4150 | 5.5955 | 5.9816 | 6.2982 | 8.1653 | 9.0798 | 15.715 | 16.944 | 26.985 | 26.904 |

4 | 8.6074 | 8.7516 | 9.6712 | 10.013 | 10.675 | 11.270 | 14.512 | 16.248 | 27.450 | 30.322 | 48.143 | 48.142 |

**Table 3.**Euler beam: linearized buckling loads for the optimal volume fraction distribution ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ with assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 9 normalized with respect to ${\lambda}_{r}=0.0005313$ KN/mm.

Mode | $0.50\%$ | $0.75\%$ | $1.00\%$ | $2.00\%$ | $5.00\%$ | $10.0\%$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | |

1 | 0.5988 | 0.5386 | 0.7020 | 0.6163 | 0.8026 | 0.6936 | 1.1897 | 1.0000 | 2.2694 | 1.8662 | 3.2106 | 2.9634 |

2 | 2.0477 | 2.1621 | 2.3072 | 2.4737 | 2.5889 | 2.7843 | 3.6998 | 4.0141 | 6.8262 | 7.4911 | 12.350 | 11.894 |

3 | 4.7410 | 4.8906 | 5.2998 | 5.5955 | 5.8482 | 6.2982 | 8.0605 | 9.0798 | 14.120 | 16.944 | 27.260 | 26.904 |

4 | 8.4905 | 8.7516 | 9.4941 | 10.013 | 10.486 | 11.270 | 14.363 | 16.248 | 24.777 | 30.322 | 48.199 | 48.142 |

**Table 4.**Square plate: linearized buckling loads for the optimal volume fraction distribution ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ with assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 4.

Mode | $0.50\%$ | $0.75\%$ | $1.00\%$ | $2.00\%$ | $5.00\%$ | $10.0\%$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | |

1 | 0.6736 | 0.6358 | 0.7835 | 0.7276 | 0.8892 | 0.8191 | 1.2947 | 1.1810 | 2.4128 | 2.2030 | 3.5968 | 3.4943 |

**Table 5.**Square nanocomposite plate: linearized buckling loads for the optimal CNTs orientation $\theta ({\zeta}_{1},{\zeta}_{2})$ described by Bernstein polynomials of order 4 for assigned volume fraction ${\varphi}_{C}$.

Mode | $0.50\%$ | $0.75\%$ | $1.00\%$ | $2.00\%$ | $5.00\%$ | $10.0\%$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | |

1 | 0.8611 | 0.6907 | 1.0369 | 0.8038 | 1.2020 | 0.9131 | 1.7696 | 1.3199 | 3.1007 | 2.3369 | 4.5583 | 3.4909 |

**Table 6.**Cylindrical nanocomposite panel: linearized buckling loads for the optimal through-the-thickness CNTs distribution with function $a({\zeta}_{1},{\zeta}_{2})$ in Equation (20) described by Bernstein polynomials of order 4 and assigned volume fraction ${\varphi}_{C}^{*}$ normalized with respect to ${\lambda}_{r}=0.006466$ KN/mm.

Mode | $0.50\%$ | $0.75\%$ | $1.00\%$ | $2.00\%$ | $5.00\%$ | $10.0\%$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | |

1 | 0.6325 | 0.6065 | 0.7230 | 0.6878 | 0.7920 | 0.7605 | 1.0336 | 1.0000 | 1.4586 | 1.4978 | 1.9950 | 1.9873 |

2 | 0.7947 | 0.7861 | 0.9166 | 0.9108 | 1.0286 | 1.0283 | 1.4351 | 1.4507 | 2.1035 | 2.4574 | 3.3456 | 3.5521 |

3 | 1.1102 | 1.0811 | 1.2726 | 1.2461 | 1.4225 | 1.4029 | 1.9248 | 1.7974 | 2.8299 | 2.6611 | 3.7066 | 3.5924 |

4 | 1.3092 | 1.2268 | 1.4444 | 1.3339 | 1.5427 | 1.4354 | 1.9806 | 1.9787 | 2.9550 | 3.3660 | 4.6859 | 4.8864 |

5 | 1.3720 | 1.3669 | 1.6309 | 1.6260 | 1.8700 | 1.8737 | 2.7485 | 2.7820 | 4.2409 | 5.0083 | 6.9759 | 7.4824 |

6 | 1.5385 | 1.5380 | 1.8357 | 1.8323 | 2.1058 | 2.1160 | 3.1257 | 3.1646 | 4.8939 | 5.5607 | 7.6127 | 7.7939 |

7 | 2.2954 | 2.2719 | 2.5844 | 2.5079 | 2.8048 | 2.7305 | 3.6036 | 3.5502 | 5.1469 | 5.7560 | 8.2031 | 8.6383 |

8 | 2.3122 | 2.3043 | 2.6621 | 2.6583 | 2.9917 | 3.0021 | 4.2227 | 4.2762 | 6.3765 | 7.3119 | 10.0125 | 9.1880 |

**Table 7.**Cylindrical nanocomposite panel: linearized buckling loads for the optimal volume fraction distribution ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ with assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 4 normalized with respect to ${\lambda}_{r}=0.006466$ KN/mm.

Mode | $0.50\%$ | $0.75\%$ | $1.00\%$ | $2.00\%$ | $5.00\%$ | $10.0\%$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | |

1 | 0.6075 | 0.5552 | 0.6942 | 0.6353 | 0.7832 | 0.7151 | 1.1173 | 1.0310 | 1.9557 | 1.9237 | 3.1329 | 3.0530 |

2 | 0.7684 | 0.7010 | 0.8776 | 0.8022 | 0.9892 | 0.9030 | 1.4251 | 1.3019 | 2.4889 | 2.4290 | 3.9793 | 3.8548 |

3 | 1.0120 | 1.0087 | 1.1678 | 1.1543 | 1.3151 | 1.2994 | 1.8771 | 1.8736 | 3.3529 | 3.4952 | 5.3862 | 5.5450 |

4 | 1.1833 | 1.1107 | 1.3590 | 1.2708 | 1.5277 | 1.4305 | 2.2466 | 2.0625 | 3.9364 | 3.8483 | 6.2432 | 6.1075 |

5 | 1.2680 | 1.2401 | 1.4611 | 1.4190 | 1.6476 | 1.5974 | 2.3504 | 2.3032 | 4.1880 | 4.2968 | 6.7559 | 6.8168 |

6 | 1.3703 | 1.4200 | 1.5833 | 1.6249 | 1.7772 | 1.8292 | 2.5711 | 2.6374 | 4.6157 | 4.9201 | 7.3073 | 7.8052 |

7 | 1.9120 | 1.7370 | 2.1885 | 1.9875 | 2.4664 | 2.2372 | 3.5623 | 3.2256 | 6.2360 | 6.0183 | 9.9263 | 9.5510 |

8 | 2.0283 | 1.9699 | 2.3364 | 2.2541 | 2.6371 | 2.5374 | 3.7589 | 3.6584 | 6.6961 | 6.8254 | 10.807 | 10.83039 |

**Table 8.**Cylindrical nanocomposite panel: linearized buckling loads for the optimal volume fraction distribution ${\varphi}_{C}({\zeta}_{1},{\zeta}_{2})$ with assigned average value ${\varphi}_{C}^{*}$ described by Bernstein polynomials of order 9 normalized with respect to ${\lambda}_{r}=0.006466$ KN/mm.

Mode | $0.50\%$ | $0.75\%$ | $1.00\%$ | $2.00\%$ | $5.00\%$ | $10.0\%$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | |

1 | 0.6237 | 0.5552 | 0.7167 | 0.6353 | 0.8029 | 0.7151 | 1.1353 | 1.0310 | 2.0376 | 1.9237 | 3.1647 | 3.0530 |

2 | 0.7799 | 0.7010 | 0.8936 | 0.8022 | 1.0059 | 0.9030 | 1.4267 | 1.3019 | 2.5885 | 2.4290 | 4.0217 | 3.8548 |

3 | 1.0178 | 1.0087 | 1.1699 | 1.1543 | 1.3147 | 1.2994 | 1.8990 | 1.8736 | 3.4311 | 3.4952 | 5.4352 | 5.5450 |

4 | 1.1623 | 1.1107 | 1.3317 | 1.2708 | 1.5204 | 1.4305 | 2.1848 | 2.0625 | 4.0413 | 3.8483 | 6.3539 | 6.1075 |

5 | 1.2778 | 1.2401 | 1.4679 | 1.4190 | 1.6554 | 1.5974 | 2.3892 | 2.3032 | 4.3172 | 4.2968 | 6.8085 | 6.8168 |

6 | 1.3526 | 1.4200 | 1.5470 | 1.6249 | 1.7422 | 1.8292 | 2.5344 | 2.6374 | 4.6328 | 4.9201 | 7.4096 | 7.8052 |

7 | 1.9405 | 1.7370 | 2.2259 | 1.9875 | 2.5083 | 2.2372 | 3.5554 | 3.2256 | 6.4522 | 6.0183 | 10.0560 | 9.5510 |

8 | 2.0844 | 1.9699 | 2.3961 | 2.2541 | 2.6806 | 2.5374 | 3.8509 | 3.6584 | 6.9260 | 6.8254 | 10.8735 | 10.8303 |

**Table 9.**Cylindrical nanocomposite panel: linearized buckling loads for the optimal CNTs orientation $\theta ({\zeta}_{1},{\zeta}_{2})$ described by Bernstein polynomials of order 4 for assigned volume fraction ${\varphi}_{C}$ normalized with respect to ${\lambda}_{r}=0.006466$ KN/mm.

Mode | $0.50\%$ | $0.75\%$ | $1.00\%$ | $2.00\%$ | $5.00\%$ | $10.0\%$ | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | OPT | UD | |

1 | 0.5956 | 0.6065 | 0.6710 | 0.6878 | 0.7368 | 0.7605 | 0.9519 | 1.0000 | 1.6582 | 1.4978 | 2.2547 | 1.9873 |

2 | 0.7753 | 0.7861 | 0.8947 | 0.9108 | 1.0060 | 1.0283 | 1.4116 | 1.4507 | 2.5809 | 2.4574 | 3.4228 | 3.5521 |

3 | 1.0966 | 1.0811 | 1.2574 | 1.2461 | 1.4100 | 1.4029 | 1.7751 | 1.7974 | 2.8041 | 2.6611 | 3.6373 | 3.5924 |

4 | 1.2540 | 1.2268 | 1.3588 | 1.3339 | 1.4587 | 1.4354 | 1.9597 | 1.9787 | 3.5398 | 3.3660 | 4.7805 | 4.8864 |

5 | 1.3938 | 1.3669 | 1.6453 | 1.6260 | 1.8821 | 1.8737 | 2.7433 | 2.7820 | 4.8703 | 5.0083 | 6.7610 | 7.4824 |

6 | 1.5363 | 1.5380 | 1.8221 | 1.8323 | 2.0941 | 2.1160 | 3.0839 | 3.1646 | 5.5986 | 5.5607 | 7.5653 | 7.7939 |

7 | 2.2102 | 2.2719 | 2.5331 | 2.5079 | 2.7802 | 2.7305 | 3.6117 | 3.5502 | 5.8414 | 5.7560 | 7.8166 | 8.6383 |

8 | 2.3373 | 2.3043 | 2.6906 | 2.6583 | 3.0311 | 3.0021 | 4.2693 | 4.2762 | 7.3583 | 7.3119 | 9.7666 | 9.1880 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Leonetti, L.; Garcea, G.; Magisano, D.; Liguori, F.; Formica, G.; Lacarbonara, W.
Optimal Design of CNT-Nanocomposite Nonlinear Shells. *Nanomaterials* **2020**, *10*, 2484.
https://doi.org/10.3390/nano10122484

**AMA Style**

Leonetti L, Garcea G, Magisano D, Liguori F, Formica G, Lacarbonara W.
Optimal Design of CNT-Nanocomposite Nonlinear Shells. *Nanomaterials*. 2020; 10(12):2484.
https://doi.org/10.3390/nano10122484

**Chicago/Turabian Style**

Leonetti, Leonardo, Giovanni Garcea, Domenico Magisano, Francesco Liguori, Giovanni Formica, and Walter Lacarbonara.
2020. "Optimal Design of CNT-Nanocomposite Nonlinear Shells" *Nanomaterials* 10, no. 12: 2484.
https://doi.org/10.3390/nano10122484