# A Study on the Effect of Piezoelectric Nonlinearity on the Bending Behaviour of Smart Laminated Composite Beam

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

#### 2.1. Theoretical Formulation Using HSDT

#### 2.2. Nonlinear Constitutive Relation

#### 2.3. Energy Formulation

#### 2.4. Finite Element Formulation

#### 2.5. Principle of Virtual Work

## 3. Results and Discussion

#### 3.1. Validation of Results with Unimorph and Bimorph with Linear Piezoelectric Coefficients

#### 3.2. Validation of Nonlinear Analysis of Piezoelectric Cantilever Bimorph and Unimorph

#### 3.3. Nonlinear Analysis of Piezo-Actuated Laminated Composite Beam

#### 3.4. Analysis of Piezo-Actuated Laminated Composite Beam with Different End Conditions

#### 3.5. Nonlinear Analysis for Deflection and Stress Distribution of Composite Laminates

## 4. Conclusions

## Author Contributions

## Funding

## Informed Consent Statement

## Conflicts of Interest

## References

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**Figure 5.**Transverse deflection of the cantilever beam for 100 and 200 volts (Reference results are taken from Liew et al. [25]).

**Figure 6.**The maximum deflection of the beam for different end conditions against actuator voltage (Reference results are taken from Liew et al. [25]).

**Figure 8.**Variation of tip deflection with the increasing electric field for cantilever bimorph with opposite polarity (Reference results are taken from Chattaraj & Ganguli [14]).

**Figure 9.**Variation of tip deflection with the increasing electric potential for cantilever unimorph (Reference results are taken from Sumit et al. [15].

**Figure 10.**Composite laminated beam: (

**a**) symmetric cross-ply laminate; (

**b**) anti-symmetric angle-ply laminate configuration (a/h = 10).

**Figure 11.**Variation of (

**a**) deflection and (

**b**) normal stress for a thick beam with the applied voltage for symmetric cross-ply laminate.

**Figure 12.**Variation of (

**a**) deflection and (

**b**) normal stress for a moderately thick beam with the applied voltage for symmetric cross-ply laminate.

**Figure 13.**Variation of (

**a**) deflection and (

**b**) normal stress for a thin beam with the electric potential for symmetric cross-ply laminate.

**Figure 14.**Variation of (

**a**) effective piezoelectric strain coefficient and (

**b**) effective elastic property with applied electric potential.

**Figure 15.**Variation of (

**a**) deflection and (

**b**) normal stress with electric field for thick anti-symmetric angle-ply laminate.

**Figure 16.**Variation of (

**a**) deflection and (

**b**) normal stress with electric field for moderately thick anti-symmetric angle-ply laminate.

**Figure 17.**Variation of (

**a**) deflection and (

**b**) normal stress with electric field for thin anti-symmetric angle-ply laminate.

**Figure 18.**Variation of (

**a**) maximum deflection and (

**b**) normal stress with electric potential for cross-ply laminate with SS end condition.

**Figure 19.**Variation of (

**a**) maximum deflection and (

**b**) normal stress with electric potential for cross-ply laminate with CS end condition.

**Figure 20.**Variation of (

**a**) maximum deflection and (

**b**) normal stress with the electric potential for anti-symmetric angle-ply laminate for SS end condition.

**Figure 21.**Variation of (

**a**) maximum deflection and (

**b**) normal stress with the electric potential for anti-symmetric angle-ply laminate for CS end condition.

**Figure 22.**In-plane deflection, normal stress distribution for (

**a**) symmetric cross-ply laminate, (

**b**) anti-symmetric angle ply laminate and transverse shear stress distribution for (

**c**) symmetric cross-ply laminate and (

**d**) anti-symmetric angle ply laminate.

$\mathbf{E}\left(\mathbf{G}\mathbf{P}\mathbf{a}\right)$ | $\mathbf{G}\left(\mathbf{M}\mathbf{P}\mathbf{a}\right)$ | $\mathit{\upsilon}$ | ${\mathit{e}}_{31}$ | ${\mathit{e}}_{32}$ |
---|---|---|---|---|

$2.0$ | 775 | 0.29 | 0.046 cm^{−2} | 0.046 cm^{−2} |

$\mathit{x}\left(\mathbf{m}\right)$ | FE Tzou/Ye [24] | FE Jiang & Li [4] | FE Present | Theoretical Jiang & Li [4] |
---|---|---|---|---|

0.02 | 0.132 | 0.136 | 0.138 | 0.138 |

0.04 | 0.528 | 0.545 | 0.552 | 0.552 |

0.06 | 1.19 | 1.226 | 1.242 | 1.242 |

0.08 | 2.11 | 2.18 | 2.208 | 2.208 |

0.1 | 3.30 | 3.41 | 3.45 | 3.45 |

**Table 3.**Material properties (${E}_{i},{G}_{ij}in\mathrm{G}\mathrm{P}\mathrm{a}{d}_{ij}in{10}^{-12}\mathrm{m}{\mathrm{V}}^{-1})$.

Material | ${\mathit{E}}_{1}$ | ${\mathit{E}}_{2}$ | ${\mathit{E}}_{3}$ | ${\mathit{G}}_{23}$ | ${\mathit{G}}_{13}$ | ${\mathit{G}}_{12}$ | ${\mathit{\upsilon}}_{23}$ | ${\mathit{\upsilon}}_{13}$ | ${\mathit{\upsilon}}_{12}$ | ${\mathit{d}}_{31}$ | ${\mathit{d}}_{32}$ |
---|---|---|---|---|---|---|---|---|---|---|---|

PZT [25] | 63 | 63 | 63 | 24.8 | 24.8 | 24.8 | 0.28 | 0.28 | 0.28 | −166 | −166 |

PZT-4 [26] | 81.3 | - | 64.5 | - | 25.6 | 25.6 | - | 0.43 | 0.43 | −122 | −122 |

Al [26] | 68.9 | 68.9 | 68.9 | 27.6 | 27.6 | 27.6 | 0.25 | 0.25 | 0.25 | - | - |

Adhesive [26] | 6.9 | 6.9 | 6.9 | 2.46 | 2.46 | 2.46 | 0.4 | 0.4 | 0.4 | - | - |

T300/934 [26] | 132.38 | - | 107.6 | - | 56.5 | 56.5 | - | 0.24 | 0.43 | - | - |

**Table 4.**Material properties (${E}_{i},{G}_{ij}in\mathrm{G}\mathrm{P}\mathrm{a}{d}_{ij}in{10}^{-12}\mathrm{m}{\mathrm{V}}^{-1})$.

Material | ${\mathit{E}}_{1}$ | ${\mathit{E}}_{2}$ | ${\mathit{E}}_{3}$ | ${\mathit{G}}_{23}$ | ${\mathit{G}}_{13}$ | ${\mathit{G}}_{12}$ | ${\mathit{\upsilon}}_{23}$ | ${\mathit{\upsilon}}_{13}$ | ${\mathit{\upsilon}}_{12}$ |
---|---|---|---|---|---|---|---|---|---|

PZT 3203 HD [11] | 60.24 | 60.24 | 47.62 | 19.084 | 19.084 | 24.04 | 0.494 | 0.494 | 0.253 |

AS/3501 Gr/Ep [25] | 144.8 | 9.65 | - | 5.92 | 7.1 | 7.1 | - | - | 0.3 |

PZT APC 850 [15] | 63 | 63 | 63 | 24.05 | 24.05 | 24.05 | 0.31 | 0.31 | 0.31 |

Silicon [15] | 166 | 166 | 166 | 65.9 | 65.9 | 65.9 | 0.26 | 0.26 | 0.26 |

${\mathit{d}}_{31}$ | ${\mathit{d}}_{32}$ | ${\mathit{d}}_{33}$ | ${\mathit{d}}_{331}\left({\mathrm{m}}^{2}{\mathrm{V}}^{-2}\right)$ | ${\mathit{\kappa}}_{331}\left({\mathrm{m}}^{3}{\mathrm{N}}^{-1}{\mathrm{V}}^{-1}\right)$ | |||||

PZT 3203 HD [11] | −320 | −320 | 650 | −520 $\times {10}^{-18}$ | - | ||||

PZT APC 850 [15] | −175 | −175 | - | −1210 $\times {10}^{-18}$ | −6.3 $\times {10}^{-17}$ |

Elastostriction (%) | Electrostriction (%) | Both (%) | |
---|---|---|---|

Deflection | −86.16 | 691.41 | 9.49 |

Normal Stress | −98.45 | 691.42 | −87.74 |

Shear Stress | −85.81 | 691.32 | 12.23 |

Elastostriction (%) | Electrostriction (%) | Both (%) | |
---|---|---|---|

Deflection | −81.97 | 691.42 | 42.65 |

Normal Stress | −97.98 | 691.42 | −84.038 |

Shear Stress | −82.22 | 691.36 | 43.73 |

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**MDPI and ACS Style**

Akhlaq, A.; Shaik Dawood, M.S.I.; Jaffar Syed, M.A.; Sulaeman, E.
A Study on the Effect of Piezoelectric Nonlinearity on the Bending Behaviour of Smart Laminated Composite Beam. *Materials* **2023**, *16*, 2839.
https://doi.org/10.3390/ma16072839

**AMA Style**

Akhlaq A, Shaik Dawood MSI, Jaffar Syed MA, Sulaeman E.
A Study on the Effect of Piezoelectric Nonlinearity on the Bending Behaviour of Smart Laminated Composite Beam. *Materials*. 2023; 16(7):2839.
https://doi.org/10.3390/ma16072839

**Chicago/Turabian Style**

Akhlaq, Adnan, Mohd Sultan Ibrahim Shaik Dawood, Mohamed Ali Jaffar Syed, and Erwin Sulaeman.
2023. "A Study on the Effect of Piezoelectric Nonlinearity on the Bending Behaviour of Smart Laminated Composite Beam" *Materials* 16, no. 7: 2839.
https://doi.org/10.3390/ma16072839