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Influence of initial geometric imperfection on static and free vibration analyses of porous FG nanoplate using an isogeometric approach

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Abstract

In this paper, the size-dependent analysis of geometrically imperfect porous functionally graded (FG) nanoplates is studied by an isogeometric approach based on a new refined plate theory. Simultaneous effects of initial geometric imperfection and porosity on the natural frequency and deflection of the FG nanoplates are investigated. The initial geometric imperfection is considered as an initial curvature and modeled by an analytical function in the governing equations of the nanoplate. Material properties of porous FG nanoplate are defined by a modified power-law function, and two types of distribution for porosity are used. A four-variable refined plate theory with a new polynomial shape function is proposed. Based on Hamilton’s principle, a weak form of static and free vibration problem for nonlocal plate is derived. The discrete system of equations is solved by an isogeometric approach based on NURBS basis functions, and the accuracy of the present study is verified by comparing the results with solutions given in published papers. Present results indicate the importance of porosity parameter, porosity distributions, nonlocal parameter, material index, plate geometrical parameters, and specially imperfection amplitude on the static and free vibration behavior of FG nanoplates.

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Abbreviations

a, b, h :

Length, width, and thickness of the plate

\({C}_{ij}\) :

Elastic coefficients

\({D}^{b}\) , \({D}^{s}\) :

Material matrices

E :

Young’s modulus

F :

Global load vector

G e :

Effective shear modulus

I i :

Mass moments of inertia

K :

Global stiffness matrix

K e :

Effective Bulk modulus

M :

Global mass matrix

m :

Mass matrix

n :

Material index

p, q :

Order of NURBS function in two directions

\({P}_{I}\) :

Control point

\({R}_{I}\) :

NURBS basis function

\({u}_{I}\) :

Degrees of freedom vector of the control point I

u 0, v 0 :

Mid-plane displacements in x-, y-direction

\({V}_{c}, {V}_{m}\) :

Volume fraction of ceramic and metal

\({w}^{b}, {w}^{s}\) :

Transverse bending and shear displacements

\({w}^{i}\) :

Initial geometric imperfection

\(\stackrel{-}{w}\) :

Non-dimensional deflection

\(\alpha \) :

Porosity volume fraction

\(\gamma :\) :

Out of plane shear strain

\(\delta U, \delta V, \delta T\) :

The variation of strain energy, work done by external forces, and kinetic energy

\({\varepsilon }^{b}\) :

In-plane strain

\({\xi }^{*}\) :

Imperfection amplitude

\(\xi ,\eta \) :

Parametric spaces

\(\mu \) :

Nonlocal parameter

\(\upsilon \) :

Poisson’s ratio

\(\stackrel{-}{\omega }\) :

Non-dimensional frequency

\(\rho \) :

Density

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Authors and Affiliations

  1. Department of Mechanical Engineering, Central Tehran Branch, Islamic Azad University, Tehran, Iran

    Heidar Fazeli, Armen Adamian & Ahmad Hosseini-Sianaki

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  1. Heidar Fazeli
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  2. Armen Adamian
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  3. Ahmad Hosseini-Sianaki
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Correspondence to Armen Adamian.

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Appendices

Appendix A

The governing equations for the local plate

$$ \begin{aligned} &\delta u_{0} : N_{xx,x} + N_{xy,y} = I_{1} \ddot{u}_{0} - I_{2} \ddot{w}^{{b}}{}_{,x} + I_{4} \ddot{w}^{s}{}_{,x}\\ &\delta v_{0} : N_{yy,y} + N_{xy,x} = I_{1} \ddot{v}_{0} - I_{2} \ddot{w}^{b}{}_{,y} + I_{4} \ddot{w}^{s}{}_{,y} \\ &\delta w^{b} : (N_{xx} w^{i}{}_{,x} ){}_{,x} + (N_{yy} w^{i}{}_{,y} ){}_{,y} + (N_{xy} w^{i}{}_{,y} ){}_{,x} + (N_{xy} w^{i}{}_{,x} ){}_{,y} + M^{b}{}_{xx,xx} + M^{b}{}_{yy,yy} - 2M^{b}{}_{xy,xy} \\&\qquad\qquad\qquad =- q + I_{2} \left( { \ddot{u}_{0,x} + \ddot{v}_{0,y} } \right) - I_{3} \left( {\ddot{w}^{b}{}_{,xx} + \ddot{w}^{b}{}{}_{,yy} } \right) + I_{5} \left( {\ddot{w}^{s}{}{}_{,xx} + \ddot{w}^{s}{}_{,yy} } \right) + I_{1} \left( {\ddot{w}^{b} + \ddot{w}^{s} } \right)\\ &\delta w^{s} : -\,M^{s}{}_{xx,xx} - M^{s}{}_{yy,yy} - 2M^{s}{}_{xy,xy} + Q_{xz,x} + Q_{yz,y} + (M^{i}_{xx} w^{i}{}_{,x} ){}_{,x} + (M^{i}{}_{yy} w^{i}{}_{,y} ){}_{,y} + (M^{i}{}_{xy} w^{i}_{,x} ){}_{,y}+ (M^{i}{}_{xy} w^{i}{}_{,y} ){}_{,x}\\ &\qquad\qquad\qquad = - q - I_{4} \left( {\ddot{u}_{0,x} + \ddot{v}_{0,y} } \right) + I_{5} \left( {\ddot{w}^{b}{}_{,xx} + \ddot{w}^{b}{}_{,yy} } \right) - I_{6} \left( {\ddot{w}^{s}{}_{,xx} + \ddot{w}^{s}{}_{,yy} } \right) + I_{1} \left( {\ddot{w}^{b} + \ddot{w}^{s} } \right) \end{aligned} $$

The nonlocal governing equations in terms of displacement

$${\delta u}_{0}: {A}_{11}{({u}_{0,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,x})}_{,x}+{A}_{12} {({v}_{0,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,y})}_{,x}+{A}_{66}{({u}_{0,y}+{v}_{0,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,x})}_{,y}-{B}_{11}{{w}^{b}}_{,xxx}-{B}_{12}{{w}^{b}}_{,xyy}-2{B}_{66}{{w}^{b}}_{,xyy}+{E}_{11}{{w}^{s}}_{,xxx}+{E}_{12}{{w}^{s}}_{,xyy}+2{E}_{66}{{w}^{s}}_{,xyy}={\left(1-\mu {\nabla }^{2}\right)(I}_{1}{\ddot{u}}_{0}-{I}_{2}{{\ddot{w}}^{b}}_{,x}+{I}_{4}{{\ddot{w}}^{s}}_{,x})$$
$$\delta {v}_{0}: {A}_{21}{({u}_{0,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,x})}_{,y}+{A}_{22}{({v}_{0,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,y})}_{,y}+{A}_{66}{({u}_{0,y}+{v}_{0,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,x})}_{,x}-{B}_{21}{{w}^{b}}_{,xxy}-{B}_{22}{{w}^{b}}_{,yyy}-2{B}_{66}{{w}^{b}}_{,xxy}+{E}_{21}{{w}^{s}}_{,xxy}+{E}_{22}{{w}^{s}}_{,yyy}+2{E}_{66}{{w}^{s}}_{,xxy}={\left(1-\mu {\nabla }^{2}\right)(I}_{1}{\ddot{v}}_{0}-{I}_{2}{{\ddot{w}}^{b}}_{,y}+{I}_{4}{{\ddot{w}}^{s}}_{,y})$$
$${\delta {w}^{b}: A}_{11}{\left(({u}_{0,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,x}\right){{w}^{i}}_{,x})}_{,x}+{A}_{12}{\left(({v}_{0,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,y}\right){{w}^{i}}_{,x})}_{,x}+{A}_{21}{\left(({u}_{0,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,x}\right){{w}^{i}}_{,y})}_{,y}+{A}_{22}{\left(({v}_{0,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,y}\right){{w}^{i}}_{,y})}_{,y}+{A}_{66}\left[{\left(({u}_{0,y}+{v}_{0,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,x}\right){{w}^{i}}_{,y})}_{,x}+{\left(({u}_{0,y}+{v}_{0,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,x}\right){{w}^{i}}_{,x})}_{,y}\right]+{B}_{11}{({u}_{0,xx}+{{w}^{b}}_{,x}{{w}^{i}}_{,xx})}_{,x}+{B}_{12}{({v}_{0,xy}+{{w}^{b}}_{,xy}{{w}^{i}}_{,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,xy}-{{w}^{b}}_{,yy}{{w}^{i}}_{,x})}_{,x}+{B}_{21}{({u}_{0,xy}+{{w}^{b}}_{,xy}{{w}^{i}}_{,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,yy}-{{w}^{b}}_{,xx}{{w}^{i}}_{,y})}_{,y}+{B}_{22}{({v}_{0,yy}+{{w}^{b}}_{,y}{{w}^{i}}_{,yy})}_{,y}-2{B}_{66}\left[{u}_{0,xyy}+{v}_{0,xxy}+2{\left({{w}^{b}}_{,xy}{{w}^{i}}_{,y}\right)}_{,x}+2{\left({{w}^{b}}_{,xy}{{w}^{i}}_{,x}\right)}_{,y}+{\left({{w}^{b}}_{,x}{{w}^{i}}_{,yy}\right)}_{,x}{+\left({{w}^{b}}_{,y}{{w}^{i}}_{,xx}\right)}_{,y}\right]-{D}_{11}{{w}^{b}}_{,xxxx}-{D}_{12}{{w}^{b}}_{,xxyy}-{D}_{21}{{w}^{b}}_{,yyxx}-{D}_{22}{{w}^{b}}_{,yyyy}+4{D}_{66}{{w}^{b}}_{,xyxy}+{E}_{11}{\left({{w}^{s}}_{,xx}{{w}^{i}}_{,x}\right)}_{,x}+{E}_{12}{\left({{w}^{s}}_{,yy}{{w}^{i}}_{,x}\right)}_{,x}+{E}_{21}{\left({{w}^{s}}_{,xx}{{w}^{i}}_{,y}\right)}_{,y}+{E}_{22}{\left({{w}^{s}}_{,yy}{{w}^{i}}_{,y}\right)}_{,y}+2{E}_{66}\left[{\left({{w}^{s}}_{,xy}{{w}^{i}}_{,y}\right)}_{,x}+{\left({{w}^{s}}_{,xy}{{w}^{i}}_{,x}\right)}_{,y}\right]+{F}_{11}{{w}^{s}}_{,xxxx}+{F}_{12}{{w}^{s}}_{,xxyy}+{F}_{21}{{w}^{s}}_{,yyxx}+{F}_{22}{{w}^{s}}_{,yyyy}-4{F}_{66}{{w}^{s}}_{,xyxy}=\left(1-\mu {\nabla }^{2}\right)[-q+{I}_{2}\left({\ddot{u}}_{0,x}+{\ddot{v}}_{0,y}\right){-{I}_{3}\left({{\ddot{w}}^{b}}_{,xx}+{{\ddot{w}}^{b}}_{,yy}\right)+I}_{5}\left({{\ddot{w}}^{s}}_{,xx}+{{ \ddot{w}}^{s}}_{,yy}\right)+{I}_{1}\left({\ddot{w}}^{b}+{\ddot{w}}^{s}\right)]$$
$$\delta {w}^{s}: {-E}_{11}{({u}_{0,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,x})}_{,xx}{-E}_{12}{({v}_{0,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,y})}_{,xx}{-E}_{21}{({u}_{0,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,x})}_{,yy}{-E}_{22}{({v}_{0,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,y})}_{,yy}-2{E}_{66}{\left({u}_{0,y}+{v}_{0,x}+{{w}^{b}}_{,x}{{w}^{i}}_{,y}+{{w}^{b}}_{,y}{{w}^{i}}_{,x}\right)}_{,xy}+{F}_{11}{{w}^{b}}_{,xxxx}+{F}_{12}{{w}^{b}}_{,xxyy}+{F}_{21}{{w}^{b}}_{,yyxx}+{F}_{22}{{w}^{b}}_{,yyyy}+4{F}_{66}{{w}^{b}}_{,xyxy}-{H}_{11}{{w}^{s}}_{,xxxx}-{H}_{12}{{w}^{s}}_{,xxyy}-{H}_{21}{{w}^{s}}_{,yyxx}-{H}_{22}{{w}^{s}}_{,yyyy}-4{H}_{66}{{w}^{s}}_{,xyxy}-{Z}_{33}{w}^{s}+{{D}^{s}}_{44}{{w}^{s}}_{,yy}+{{D}^{s}}_{55}{{w}^{s}}_{,xx}=\left(1-\mu {\nabla }^{2}\right)[{-I}_{4}\left({\ddot{u}}_{0,x}+{\ddot{v}}_{0,y}\right){+{I}_{5}\left({{\ddot{w}}^{b}}_{,xx}+{{\ddot{w}}^{b}}_{,yy}\right)-I}_{6}\left({{\ddot{w}}^{s}}_{,xx}+{{ \ddot{w}}^{s}}_{,yy}\right)+{I}_{1}\left({\ddot{w}}^{b}+{\ddot{w}}^{s}\right)]$$

Appendix B

See (Fig. 

Fig. 12
figure 12

Flowchart of various steps involved in the solution process

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12).

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Fazeli, H., Adamian, A. & Hosseini-Sianaki, A. Influence of initial geometric imperfection on static and free vibration analyses of porous FG nanoplate using an isogeometric approach. J Braz. Soc. Mech. Sci. Eng. 43, 200 (2021). https://doi.org/10.1007/s40430-021-02847-3

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