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Impact of in-plane follower force on the frequency response of the hybrid angle-ply laminated system via dynamic simulation and generalized differential quadrature framework

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Abstract

Central concern of this article is presenting the high-exactitude analysis on hygro-thermo-mechanical vibration of three-phase multi-scale hybrid composite angle-ply laminated rectangular plate (MHCALRP) for different couples of boundary conditions within the context of twelve-variable refined higher order shear deformation theory (RHOSDT12). Defining the kinematics of the system according to RHOSDT12 is the essential step toward achieving sufficient accuracy in elucidation of vibrational behavior of moderately thick plates. Mechanical properties of MHCALRP are calculated through two-step Halpin–Tsai homogenization process. Numerical solution to the governing differential motion equations is acquired by implementing generalized Differential quadrature method (GDQM). Accuracy of the employed approach is evaluated by a comparative study with the published studies. As a practical conclusion it is revealed that the orientation angle of the macrofibres can effectively compensate the lack of nano reinforcement as well as the absence of extensional load to overcome negative impacts of hygrothermal environment in an affordable cost. The influence of extensional load along two directions, nano/macro scale reinforcements and hygro-thermal environment on the vibrational response of the MHCALRP is completely elucidated through the parametric investigation.

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Abbreviations

h, a, and b :

Thickness, the width, and length of the rectangular plate, respectively

F and NCM:

Indicate fiber and nanocomposite matrix, respectively

RHOSDT12:

Refined high order shear deformation theory with 12 variables

\(\rho ,\,\,E,\,\nu \,\,\,and\,\,\,G\) :

Illustrate the density, Young’s module, Poisson’s ratio, and shear parameters, respectively

V NCM, V F :

Volume fractions of nanocomposite matrix and fiber, respectively

l CNT, t CNT, d CNT, E CNT and VCNT :

Indicate the length, thickness, diameter, Younge’s module, and volume fraction of carbon nanotubes, respectively

\({V}_{\mathrm{CNT}}^{*}\), W CNT :

Effective volume fraction and weight fraction of the CNTs, respectively

N t, V CNT :

Layer number and volume fraction of CNTs

\(\alpha_{11} ,\,\,\,\alpha_{22} \,\,{\text{and}}\,\,\alpha_{33}\) :

Thermal expansion coefficients of the multi-scale hybrid nanocomposite

\(\alpha_{NCM}\) :

Thermal expansion coefficient of the nanocomposite matrix

\(\beta_{11} ,\,\,\,\beta_{22} \,\,{\text{and}}\,\,\beta_{33}\) :

Moisture coefficients of the multi-scale hybrid nanocomposite

β M :

Indicates moisture coefficient of the matrix

\({Q}_{ij}\), \({\stackrel{-}{Q}}_{ij} \mathrm{and} \theta\) :

Stiffness elements, stiffness elements relates to orientation angle and the orientation angle, respectively

U, V, W :

Displacement field of a general point (x, y, z) in the plate domain in the x, y, and z-direction, respectively

U 0, V 0, W 0 :

Indicates the components of the displacement of mid-plane. θx, θy and θz represent rotations of the normal about the y-axis, x-axis, and z-axis, respectively

\({\delta u}_{{0}}^{*}\),\({\delta v}_{{0}}^{*}\),\({\delta w}_{{0}}^{*}\),\({{\delta \theta }}_{{\text{x}}}^{*}\),\({{\delta \theta }}_{{\text{y}}}^{*}\) and \({{\delta \theta }}_{{\text{z}}}^{*}\) :

The higher-order terms in Taylor’s series expansion

\({\upvarepsilon }_{{{\text{xx}}}}\), \({\upvarepsilon }_{{{\text{yy}}}}\) and \({\upvarepsilon }_{{{\text{zz}}}}\) :

Indicate normal strains in X, Y and Z directions, respectively. Moreover, illustrate the shear strain in \(X - Y\), \(Y - Z\) and \(X - Z\) planes

\({\upgamma }_{{{\text{xy}}}}\), \({\upgamma }_{{{\text{yz}}}}\) and \({\upgamma }_{{{\text{xz}}}}\) :

Illustrate the shear strain in \(X - Y\), \(Y - Z\) and \(X - Z\) planes, respectively

I i :

Represents the mass inertias

T and U :

Represent corresponding kinetic energy and strain energy of the system, respectively

W 1, W 2, and W 3 :

Indicate the work done by thermal load, hygro loading and the external work done by external in-plane forces, respectively

N T and N H :

Indicate applied forces due to variation of temperature and moisture, respectively

ΔT, ΔH:

Indicate the temperature change and moisture change, respectively

\(N_{xx}^{0} \,{\text{and}}\,N_{yy}^{0}\) :

Indicate the applied load in X and Y direction, respectively

d, b, and \(\delta\) :

Indicates d as a subscript stand for the domain grid-points, b as subscript stands for boundary grid-points and the displacement vector, respectively

\(\omega ,\,\,\overline{\omega }\) :

Linear frequency and non-dimensional natural frequency, respectively

M ij and K ij :

Components of mass, damping and stiffness matrices

M ij * and K ij * :

Components of mass, damping and stiffness matrices in the GDQ method

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Acknowledgements

Project funded by National Natural Science Foundation of China (No. 0066/2019/AFJ): Research on knowledge-oriented probabilistic graphical model theory based on multi-source data. Project funded by China Postdoctoral Science Foundation (No. 2019T120787; No. 2018M640878). Project funded by National Key R&D Program of China (No. 2017YFB0503604; No. 2017YFB0503801). Project funded by Cross and Multi Dimension Electronic Fence System Project.

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    Authors and Affiliations

    1. The Post-Doctoral Joint Research Center of Hengqin New Area, City University of Macau, Macau, 999078, China

      Wenjian Liu, Lianbing Deng & Zhiming Cai

    2. Institute of Data Science, City University of Macau, Macau, 999078, China

      Wenjian Liu & Zhiming Cai

    3. Zhuhai Da Hengqin Science and Technology Development Co., Ltd, Guangdong Hengqin New Area, 519031, China

      Lianbing Deng & Daming Li

    4. Guangdong Qinzhi Science and Technology Research Institute, Guangdong Hengqin New Area, 519031, China

      Lianbing Deng & Daming Li

    5. Department of Mechanical Engineering, Faculty of Engineering, Tarbiat Modares University, Tehran, Iran

      Alireza Rahimi

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    1. Wenjian Liu
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    Correspondence to Daming Li or Alireza Rahimi.

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    Wenjian Liu and Lianbing Deng contributed equally to this work.

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    Liu, W., Deng, L., Cai, Z. et al. Impact of in-plane follower force on the frequency response of the hybrid angle-ply laminated system via dynamic simulation and generalized differential quadrature framework. Engineering with Computers 38 (Suppl 5), 3743–3760 (2022). https://doi.org/10.1007/s00366-020-01215-4

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