# Series Solution-Based Approach for the Interlaminar Stress Analysis of Smart Composites under Thermo-Electro-Mechanical Loading

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Formulation

_{4}and σ

_{6}are employed, which leads to:

## 3. Computational Results

#### 3.1. Uniaxial Extension and Thermal Loading

_{3}) distribution under tensile and thermal loadings. Stress concentrations occurred at the interfaces of the layers. The result of the present method correlates well with the 3D finite element result, but the FE-based solution underestimates the stress concentrations at the free edge. All the graphs show that the present method’s solutions are greater than the 3D finite element solution, because the 3D solid elements popularly used in FE analysis are based on the displacement field; the solution cannot guarantee the statically admissible stress field, even though the displacement field is exact.

_{3}) and shear stresses (σ

_{4}) at the PZT/0 and PZT/45 interfaces of the laminates under 0.1% uniaxial extension and unit thermal loading (ΔT = 1). The maximum normal stress (σ

_{3}) concentration in both cross-ply and angle-ply laminates occurs at the free edge that can cause the PZT layer’s delamination failure. The present analysis results of free-edge of cross-ply and angle-ply indicate that the large shear stress (σ

_{4}) represents the satisfaction of the traction-free boundary condition and becomes null at the interior region. The results of the 3D-FEM solution show good agreement with the present analysis results.

#### 3.2. Reduction of Interlaminar Stress by Combined Thermo-Mechanical Loading

_{3}of cross-ply laminate under the combined 0.1% uniaxial extension loading, and the variable rise in the thermal loading throughout the laminate. The interlaminar stress distribution’s magnitude is reduced at the PZT/0 layer and 0/90 layer interface by increasing the laminate’s temperature. Without the thermal loading, the magnitude of peeling stress is 8.3 MPa; however, as the temperature loading of ΔT = 30 is applied, peeling stress is reduced to 2.9 MPa.

_{3}) and (σ

_{4}) distribution through the in-plane direction at the PZT/0 interface. The magnitude of the peeling stresses decreased significantly with the increase in thermal loading:

#### 3.3. Piezoelectric Excitation

^{5}V/m of electric excitation is applied on both surfaces of the piezoelectric actuators. Figure 7 shows the free edge σ

_{3}distribution. The maximum stress concentration occurs at the PZT/0 and PZT/45 layer interfaces. The FEM results show a good correlation with the present analysis results.

_{3}) and the shear stresses (σ

_{4}) at the PZT/0 and PZT/45 interfaces of the studied laminates. For all laminates the maximum normal stress (σ

_{3}) concentration appears at the free edge that can cause the PZT layer’s delamination failure. On the other hand, FEM results indicate that the traction-free boundary conditions are not satisfied as the FEM follows the displacement-based approach. Therefore, the present analysis results provide more accurate results as compared to the FEM-based results.

#### 3.4. Reduction in Interlaminar Stress by Combined Thermo-Electro-Mechanical Loading

_{3}of the free edge of the cross-ply laminate under the combined thermo-electro-mechanical loading with 0.1% uniaxial tension, temperature loading of ΔT = 10, and the variable electric field of (E3 = 0 V/m, E3 = 1 × 10

^{5}V/m, and E3 = 2 × 10

^{5}V/m). It can be observed that, without applying the electric field, the concentration of peeling stress is 6.3 MPa, and upon applying the excitation of 2 × 10

^{5}V/m, it is reduced to 4.7 MPa (up to 25.3% reduction). Similarly, at the PZT/0 layer in the in-plane direction, the peak value of the concentrated peeling stress drops with increased electric field loading.

_{3}at the free edge under the combined thermo-electro-mechanical loading with 0.1% uniaxial tension, temperature loading of ΔT = 10, and the variable electric field of (E3 = 0 V/m, E3 = 1 × 10

^{5}V/m, and E3 = 2 × 10

^{5}V/m). Large stress concentrations can be seen to occur on the free edge of the 45/−45 layer interface. When an electric excitation at the piezoelectric layer is applied, the peeling stress in angle-ply laminate could be significantly reduced.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Mahapatra, S.D.; Mohapatra, P.C.; Aria, A.I.; Christie, G.; Mishra, Y.K.; Hofmann, S.; Thakur, V.K. Piezoelectric Materials for Energy Harvesting and Sensing Applications: Roadmap for Future Smart Materials. Adv. Sci.
**2021**, 8, 2100864. [Google Scholar] [CrossRef] [PubMed] - Lee, S.-L. Active Vibration Suppression of Wind Turbine Blades Integrated with Piezoelectric Sensors. Sci. Eng. Compos. Mater.
**2021**, 28, 402–414. [Google Scholar] [CrossRef] - Zhang, Y.; Xue, Y.; Yuan, W.; Ma, W.; Li, J.; Li, F. Active Control of Thermo-Mechanical Buckling of Composite Laminated Plates Using Piezoelectric Actuators. Acta Mech. Solida Sin.
**2021**, 34, 369–380. [Google Scholar] [CrossRef] - Zalhaf, A.S.; Mansour, D.-E.A.; Han, Y.; Yang, P.; Yang, P.; Darwish, M.M.F. Numerical and Experimental Analysis of the Transient Behavior of Wind Turbines When Two Blades Are Simultaneously Struck by Lightning. IEEE Trans. Instrum. Meas.
**2021**, 1. [Google Scholar] [CrossRef] - Kumar Saini, M.; Kumar Bagha, A.; Kumar, S.; Bahl, S. Finite Element Analysis for Predicting the Vibration Characteristics of Natural Fiber Reinforced Epoxy Composites. Mater. Today Proc.
**2021**, 41, 223–227. [Google Scholar] [CrossRef] - Sayed, A.M.; Abouelatta, M.A.; Badawi, M.; Mahmoud, K.; Lehtonen, M.; Darwish, M.M.F. Novel Accurate Modeling of Dust Loaded Wire-Duct Precipitators Using FDM-FMG Method on One Fine Computational Domains. Electr. Power Syst. Res.
**2022**, 203, 107634. [Google Scholar] [CrossRef] - Kapuria, S.; Ahmed, A. A Coupled Efficient Layerwise Finite Element Model for Free Vibration Analysis of Smart Piezo-Bonded Laminated Shells Featuring Delaminations and Transducer Debonding. Int. J. Mech. Sci.
**2021**, 194, 106195. [Google Scholar] [CrossRef] - Mittelstedt, C.; Becker, W. Free-Edge Effects in Composite Laminates. Appl. Mech. Rev.
**2007**, 60, 217–245. [Google Scholar] [CrossRef] - Karp, B.; Durban, D. Saint-Venant’s Principle in Dynamics of Structures. Appl. Mech. Rev.
**2011**, 64, 020801. [Google Scholar] [CrossRef] - Pipes, R.B.; Pagano, N.J. Interlaminar Stresses in Composite Laminates Under Uniform Axial Extension. In Mechanics of Composite Materials; Reddy, J.N., Ed.; Solid Mechanics and Its Applications; Springer: Dordrecht, The Netherlands, 1994; Volume 34, pp. 234–245. ISBN 9789048144518. [Google Scholar]
- Kant, T.; Swaminathan, K. Estimation of Transverse/Interlaminar Stresses in Laminated Composites—A Selective Review and Survey of Current Developments. Compos. Struct.
**2000**, 49, 65–75. [Google Scholar] [CrossRef] - D’Ottavio, M.; Vidal, P.; Valot, E.; Polit, O. Assessment of Plate Theories for Free-Edge Effects. Compos. Part B Eng.
**2013**, 48, 111–121. [Google Scholar] [CrossRef] [Green Version] - Pagano, N.J.; Pipes, R.B. Some Observations on the Interlaminar Strength of Composite Laminates. Int. J. Mech. Sci.
**1973**, 15, 679–688. [Google Scholar] [CrossRef] - Tahani, M.; Nosier, A. Free Edge Stress Analysis of General Cross-Ply Composite Laminates under Extension and Thermal Loading. Compos. Struct.
**2003**, 60, 91–103. [Google Scholar] [CrossRef] - Nosier, A.; Maleki, M. Free-Edge Stresses in General Composite Laminates. Int. J. Mech. Sci.
**2008**, 50, 1435–1447. [Google Scholar] [CrossRef] - Pagano, N.J. On the Calculation of Interlaminar Normal Stress in Composite Laminate. J. Compos. Mater.
**1974**, 8, 65–81. [Google Scholar] [CrossRef] - Becker, W. Closed-Form Analysis of the Free Edge Effect in Angle-Ply Laminates. J. Appl. Mech.
**1994**, 61, 209–211. [Google Scholar] [CrossRef] - Kassapoglou, C.; Lagace, P.A. An Efficient Method for the Calculation of Interlaminar Stresses in Composite Materials. J. Appl. Mech.
**1986**, 53, 744–750. [Google Scholar] [CrossRef] - Yin, W.-L. Free-Edge Effects in Anisotropic Laminates Under Extension, Bending and Twisting, Part I: A Stress-Function-Based Variational Approach. J. Appl. Mech.
**1994**, 61, 410–415. [Google Scholar] [CrossRef] - Yin, W.-L. Free-Edge Effects in Anisotropic Laminates Under Extension, Bending, and Twisting, Part II: Eigenfunction Analysis and the Results for Symmetric Laminates. J. Appl. Mech.
**1994**, 61, 416–421. [Google Scholar] [CrossRef] - Flanagan, G. An Efficient Stress Function Approximation for the Free-Edge Stresses in Laminates. Int. J. Solids Struct.
**1994**, 31, 941–952. [Google Scholar] [CrossRef] - Cho, M.; Yoon, J.-Y. Free-Edge Interlaminar Stress Analysis of Composite Laminates by Extended Kantorovich Method. AIAA J.
**1999**, 37, 656–660. [Google Scholar] [CrossRef] - Cho, M.; Kim, H.S. Iterative Free-Edge Stress Analysis of Composite Laminates under Extension, Bending, Twisting and Thermal Loadings. Int. J. Solids Struct.
**2000**, 37, 435–459. [Google Scholar] [CrossRef] - Kim, H.S.; Cho, M.; Kim, G.-I. Free-Edge Strength Analysis in Composite Laminates by the Extended Kantorovich Method. Compos. Struct.
**2000**, 49, 229–235. [Google Scholar] [CrossRef] - Kim, H.S.; Rhee, S.Y.; Cho, M. Simple and Efficient Interlaminar Stress Analysis of Composite Laminates with Internal Ply-Drop. Compos. Struct.
**2008**, 84, 73–86. [Google Scholar] [CrossRef] - Kim, H.S.; Cho, M.; Lee, J.; Deheeger, A.; Grédiac, M.; Mathias, J.-D. Three Dimensional Stress Analysis of a Composite Patch Using Stress Functions. Int. J. Mech. Sci.
**2010**, 52, 1646–1659. [Google Scholar] [CrossRef] - Cho, M.; Rhee, S.Y. Layup Optimization Considering Free-Edge Strength and Bounded Uncertainty of Material Properties. AIAA J.
**2003**, 41, 2274–2282. [Google Scholar] [CrossRef] - Cho, M.; Rhee, S.Y. Optimization of Laminates with Free Edges under Bounded Uncertainty Subject to Extension, Bending and Twisting. Int. J. Solids Struct.
**2004**, 41, 227–245. [Google Scholar] [CrossRef] - Rhee, S.Y.; Cho, M.; Kim, H.S. Layup Optimization with GA for Tapered Laminates with Internal Plydrops. Int. J. Solids Struct.
**2006**, 43, 4757–4776. [Google Scholar] [CrossRef] [Green Version] - Nosier, A.; Bahrami, A. Interlaminar Stresses in Antisymmetric Angle-Ply Laminates. Compos. Struct.
**2007**, 78, 18–33. [Google Scholar] [CrossRef] - Lo, S.H.; Zhen, W.; Cheung, Y.K.; Wanji, C. An Enhanced Global–Local Higher-Order Theory for the Free Edge Effect in Laminates. Compos. Struct.
**2007**, 81, 499–510. [Google Scholar] [CrossRef] - Kim, T.; Atluri, S.N. Optimal Through-Thickness Temperature Gradients for Control of Interlaminar Stresses in Composites. AIAA J.
**1995**, 33, 730–738. [Google Scholar] [CrossRef] [Green Version] - Kim, T.; Steadman, D.; Hanagud, S.V.; Atluri, S.N. On the Feasibility of Using Thermal Gradients for Active Control of Interlaminar Stresses in Laminated Composites. J. Compos. Mater.
**1997**, 31, 1556–1573. [Google Scholar] [CrossRef] - Huang, B.; Kim, H.S. Reduction of Free Edge Peeling Stress of Laminated Composites Using Active Piezoelectric Layers. Sci. World J.
**2014**, 2014, 1–13. [Google Scholar] [CrossRef] [PubMed] - Khan, A.; Ko, D.-K.; Lim, S.C.; Kim, H.S. Structural Vibration-Based Classification and Prediction of Delamination in Smart Composite Laminates Using Deep Learning Neural Network. Compos. Part B Eng.
**2019**, 161, 586–594. [Google Scholar] [CrossRef] - Jiang, G.; Dong, T.; Guo, Z. Nonlinear Dynamics of an Unsymmetric Cross-Ply Square Composite Laminated Plate for Vibration Energy Harvesting. Symmetry
**2021**, 13, 1261. [Google Scholar] [CrossRef] - Huang, B.; Kim, H.S. Free-Edge Interlaminar Stress Analysis of Piezo-Bonded Composite Laminates under Symmetric Electric Excitation. Int. J. Solids Struct.
**2014**, 51, 1246–1252. [Google Scholar] [CrossRef] [Green Version] - Chopra, I. Review of State-of-Art of Smart Structures and Integrated Systems. In Proceedings of the 19th AIAA Applied Aerodynamics Conference, Anaheim, CA, USA, 11–14 June 2001; American Institute of Aeronautics and Astronautics: Anaheim, CA, USA, 2001. [Google Scholar]
- Lekhnitskii, S.G.; Fern, P.; Brandstatter, J.J.; Dill, E.H. Theory of Elasticity of an Anisotropic Elastic Body. Phys. Today
**1964**, 17, 84. [Google Scholar] [CrossRef] - Wang, S.S.; Choi, I. Boundary-Layer Effects in Composite Laminates: Part 1—Free-Edge Stress Singularities. J. Appl. Mech.
**1982**, 49, 541–548. [Google Scholar] [CrossRef] - Abaqus Analysis User’s Manual-Abaqus Version 6.14; Simulia: Johnston, RI, USA.

**Figure 1.**Configuration of the composite laminate under the combined extension, and the thermal (ΔT) and electric load.

**Figure 2.**Finite element configuration of the model exhibiting the applied axis-symmetric boundary conditions and the mesh configuration.

**Figure 3.**Free-edge interlaminar stress distribution in cross-ply, angle-ply, and quasi-isotropic layups (

**a**) under 0.1% uniaxial extension loading and (

**b**) under unit thermal loading (ΔT = 1).

**Figure 4.**Normal stress σ

_{3}and shear stress σ

_{4}at the PZT/0 and PZT/45 interface of [PZT/0/90] s and [PZT/45/−45] s composite laminates (

**a**) under 0.1% uniaxial extension and (

**b**) laminates under unit thermal loading (ΔT = 1).

**Figure 5.**Free-edge σ

_{3}in [PZT/0/90] s laminate under the combined 0.1% uniaxial extension thermal loading and variable temperature loadings (ΔT = (0, 1, 10, 20, and 30)).

**Figure 6.**(

**a**) σ

_{3}at the PZT/0 interface of cross-ply laminate under the combined 0.1% uniaxial extension thermal loading and variable temperature loadings (ΔT = (0, 1, 10, 20, and 30); (

**b**) σ

_{4}at the 0/90 interface under the combined 0.1% uniaxial extension thermal loading and variable temperature loadings (ΔT = (0, 1, 10, 20, and 30)).

**Figure 7.**Interlaminar stress distribution in cross-ply, angle-ply, and quasi-isotropic layup under the electrical excitation.

**Figure 8.**σ

_{3}and σ

_{4}at the PZT/0 and PZT/45 interface of cross-ply and angle-ply laminates under the electrical excitation.

**Figure 9.**Interlaminar stress σ

_{3}distribution (

**a**) at the free edge and (

**b**) at the PZT/0 interface.

**Figure 10.**Interlaminar stress σ

_{3}distribution in angle-ply laminate (

**a**) at the free edge and (

**b**) at the PZT/45 interface.

PZT-5H Layer [40] | Composite Lamina [14] |
---|---|

E_{1} = E_{2} = 62 × 10^{3} MPa | E_{1} = 138 × 10^{3} MPa |

E_{3} = 48 × 10^{3} MPa | E_{2} = E_{3} = 14.5 × 10^{3} MPa |

G_{12} = 23.5 × 10^{3} MPa | G_{12} = G_{23} = G_{13} = 5.9 × 10^{3} MPa |

G_{13} = G_{23} = 5.9 × 10^{3} MPa | ν_{12} = ν_{13} = ν_{23} = 0.21 |

ν_{12} = ν_{13} = ν_{23} = 0.49 | α_{1} = 1.2 × 10^{−7} (^{0}C^{−1}) |

d_{31} = d_{32} = −2.74 × 10^{−10} m/N | α_{2} = α_{3} = 8.4 × 10^{−6} (^{0}C^{−1}) |

d_{33} = 5.93 × 10^{−10} m/N | |

d_{42} = d_{51} = 7.41 × 10^{−10} m/N | |

α_{1} = α_{2} = α_{3} = 3.5 × 10^{−6} (^{0}C^{−1}) |

Approach | Contribution | Limitation |
---|---|---|

Layerwise theory [19] | Prediction of interlaminar stress in a multilayered strip of a laminate subjected to extension, bending, and twisting loading | Computationally inefficient, many degrees of freedom |

Equivalent single-layer theory [21] | Accurate prediction of inplane stresses | Out-of-plane stresses showed oscillation behavior |

Series solution based approach [37] | Minimized interlaminar stresses by electro-mechanical coupling phenomenon | Missing thermo-electro-mechanical coupling effect |

Proposed approach | Accurate prediction of interlaminar stresses under thermo-electro-mechanical loading | Only applicable for the simple geometry |

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**MDPI and ACS Style**

Khalid, S.; Lee, J.; Kim, H.S.
Series Solution-Based Approach for the Interlaminar Stress Analysis of Smart Composites under Thermo-Electro-Mechanical Loading. *Mathematics* **2022**, *10*, 268.
https://doi.org/10.3390/math10020268

**AMA Style**

Khalid S, Lee J, Kim HS.
Series Solution-Based Approach for the Interlaminar Stress Analysis of Smart Composites under Thermo-Electro-Mechanical Loading. *Mathematics*. 2022; 10(2):268.
https://doi.org/10.3390/math10020268

**Chicago/Turabian Style**

Khalid, Salman, Jaehun Lee, and Heung Soo Kim.
2022. "Series Solution-Based Approach for the Interlaminar Stress Analysis of Smart Composites under Thermo-Electro-Mechanical Loading" *Mathematics* 10, no. 2: 268.
https://doi.org/10.3390/math10020268