Size-dependent nonlinear bending behavior of porous FGM quasi-3D microplates with a central cutout based on nonlocal strain gradient isogeometric finite element modelling

Abstract

With the aid of the non-uniform rational B-spline (NURBS)-based isogeometric technique, for the first time, the size-dependent geometrically nonlinear bending characteristics of microplates made of porous functionally graded materials (FGMs) having a central cutout with different shapes are studied. The nonlocal strain gradient continuum elasticity within the framework a hybrid higher-order quasi-3D plate theory is adopted to describe the kinematic relations via only four unknowns. To capture the effective material properties, a porosity-dependent rule of mixture is employed. The nonlocal strain gradient nonlinear load–deflection responses are obtained corresponding to various geometrical and material parameters as well as different boundary conditions. It is revealed that the significance of both the nonlocal and strain gradient reduces. This prediction is the same for all values of the material property gradient index as well as the porosity index. Also, it is demonstrated that a central cutout leads to change the trend of load–deflection response, and this change occurs at a specific value for the applied distributed load which depends on several parameters such as the cutout geometry and boundary conditions. In addition, it is displayed that corresponding to different maximum deflections, the significance of the strain gradient size effect in the absence of nonlocality on the nonlinear flexural stiffness of a porous FGM microplate is more than that of the nonlocal size effect in the absence of the strain gradient size dependency.

1 Introduction

Recently, through advancement of materials science and technology, porous structures have been manufactured to develop lightweight as well as controlled pore systems with desired mechanical properties and functionality. Accordingly, several investigations have been carried out. Cheng et al. [1] presented a study on the multifaceted applications of cellulosic porous materials in environment, health and energy. Wang et al. [2] presented a review study on the photocatalytic and electrocatalytic applications of two-dimensional porous materials. Guo et al. [3] employed the microwave cavity interference enhancement technique to image nano-defects in porous materials putting the metal waveguides to use. Zhang et al. [4] prepared nitrogen-doped hierarchical porous carbon materials with the aid of a template free method for application in CO₂ capture. Yu et al. [5] fabricated porous carbon materials using corn straw as anode materials for lithium ion batteries. Safaei [6] explored the effect of embedding a porous core on the free oscillation response of a sandwich composite plate. Gao et al. [7] and Moradi-Dastjerdi et al. [8] analyzed, respectively, the wave propagation and static performance of porous plates reinforced with carbon nanotubes. Lin et al. [9] introduced an antibacterial, thermo and light-responsive porous composite material having smart titanium particles.

To capture the influences of different small scale effects, it is necessary to implement them into the classical continuum elasticity. Accordingly, various non-classical continuum elasticity theories have been introduced to accomplish this purpose. In the last two decades, several investigations have been carried out to analyze size-dependent mechanical behaviors of structures having small scaled dimensions [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]. To mention some recent studies in this field, Li et al. [29] developed a size-dependent inhomogeneous beam model accounting the through-length variation of material properties for nonlocal strain gradient Euler–Bernoulli beams made of axially functionally graded material (FGM). Nguyen et al. [30] introduced a refined quasi-3D isogeometric analysis for FGM microplates having seventh-order shear deformation including couple stress size effect. Joshi et al. [16] considered the temperature effect on vibration response of cracked Kirchhoff FGM microplates on the basis of the strain gradient elasticity. Radic and Jeremic [31] explored the vibration and buckling behaviors of orthotropic double-layered graphene sheets subjected to hygrothermal loading on the basis of the differential form of nonlocal theory of elasticity. Sahmani and Aghdam [32,33,34] constructed nonlocal hybrid FGM shell models to predict nonlinear static instability of cylindrical nanoshells under various loading conditions. Al-Shujairi and Mollamahmutoglu [35] constructed nonlocal strain gradient beam model for buckling and free vibrations of FGM sandwich microbeams in thermal environments. Jia et al. [36] investigated the thermos-electro-mechanical buckling behavior of FGM composite microbeams based upon the modified couple stress theory of elasticity. Thanh et al. [37] predicted static and free vibrations of couple stress-based FGM carbon nanotube reinforced composite nanoplates. Taati [38] examined buckling and postbuckling responses of FGM composite modified couple stress-based microbeams. Hajmohammad et al. [39] studied bending and buckling characteristics of FGM composite annular microplates with piezoelectric facesheet within the framework of the nonlocal continuum elasticity. Soleimani and Tadi Beni [40] reported an axisymmetric shell element formulation with the aid of a two node shell element incorporating couple stress type of size dependency.

To mention some more recent studies, Ghorbani Shenas et al. [41] analyzed thermal prebuckling and postbuckling of pre-twisted rotating FGM composite microbeams subjected to a thermal environment on the basis of the modified strain gradient continuum elasticity. Sahmani et al. [42,43,44,45] anticipated the nonlinear free and forced vibrations of FGM nanoshells incorporating modal interactions in the presence of surface stress size effect. Sobhy and Zenkour [46] explored the influences of porosity and inhomogeneity on the size-dependent buckling and oscillations of FGM composite quasi-3D nanoplates. Phung-Van et al. [47] studied the porosity-dependent nonlinear transient responses of FGM nanoplates in the presence of nonlocal type of size effect. Aria and Friswell [48] investigated the hygro-thermal vibration and buckling responses of FGM sandwich temperature-dependent microbeams. Jun et al. [49] proposed a modified nonlocal elasticity theory incorporating much more general constitutive equations containing three characteristics lengths to analyze buckling behavior of nanobeams. Sahmani and Safaei [50,51,52] studied size-dependent nonlinear mechanical responses of bi-directional FGM nanobeams. Thai et al. [53] introduced a modified strain gradient-based computational model for free vibration behavior of FGM composite multilayer microplates. Thanh et al. [54] utilized the modified couple stress elasticity for thermal bending and buckling of composite laminate microplates. Fang et al. [55] constructed a new nonlocal Euler–Bernoulli beam model for vibrations and thermal buckling of FGM composite nanobeams in thermal environment. Yuan et al. [56,57,58] employed different size-dependent continuum theories to investigate nonlinear behaviors of FGM truncated conical microshells. Sarthak et al. [59] studied dynamic buckling of curved nanobeams with the aid of nonlinear nonlocal finite element method together with a higher-order shear flexible plate model. Thai et al. [60] developed a size-dependent Kriging meshfree model to analyze deformation as well as free vibrations of FGM carbon nanotube reinforced nanobeams. Zhang et al. [61] employed a two-node strain gradient Reddy beam element for static and dynamic analysis of microbeams. Sahmani and Safaei [62] presented a surface elastic-based conical shell model for nonlinear vibration characteristics of FGM conical nanoshells. The size-dependent shear buckling characteristics of FGM skew nanoplates are analyzed by Fam et al. [63], and Yuan et al. [64]. Karamanli and Vo [65] carried out a study on bending, buckling and free vibrations of FGM sandwich microbeams based on the modified strain gradient elasticity theory. Fan et al. [66] investigated the dynamic stability of conical microshells surrounded by a viscoelastic medium based on the couple stress elasticity. Guo et al. [67] reported the three-dimensional nonlocal buckling loads of composite nanoplates with coated one-dimensional quasicrystal. Ghane et al. [68] conducted a flutter instability analysis of fluid-conveying nanotubes under an external magnetic field based upon the nonlocal strain gradient Timoshenko beam model. Mao et al. [69] explored the free vibration response of FGM piezoelectric composite microplates within the framework of the nonlocal continuum elasticity. Thanh et al. [70] introduced a geometrically nonlinear size-dependent plate model for porous FGM microplates based on the modified couple stress theory. Fan et al. [71,72,73] employed the isogeometric method for size-dependent nonlinear responses of porous FGM microplates.

In the current study, through combination of the nonlocal strain gradient continuum elasticity and a hybrid-type quasi-3D plate theory, the size-dependent geometrically nonlinear flexural behavior of porous FGM microplates having a central cutout with different shapes is investigated. The material properties of microplates are approximated via a porosity-dependent rule of mixture. With the aid of the NURBS-based isogeometric approach, the possibility of flexibly meeting higher-order derivatives is achieved. Several case studies including various porosity dispersion patterns, different material gradient indexes, boundary conditions, and shapes of the central cutout are presented.

2 Quasi-3D nonlocal strain gradient porous FGM plate model

In the current investigation, typical rectangular microplates having a central cutout made of a porous functionally graded material (FGM) are taken into consideration. To this purpose, three different kinds of porous distribution scheme are supposed as shown schematically in Fig. 1. Accordingly, a porosity-dependent rule of mixture is employed to estimate the material fulfilling the partition of unity in the following form [74]

$${\mathcal{P}}\left( z \right) = {\mathcal{P}}_{c} \left[ {\left( {\frac{1}{2} + \frac{z}{h}} \right)^{k} - \frac{\Gamma }{2}} \right] + {\mathcal{P}}_{m} \left[ {1 - \left( {\frac{1}{2} + \frac{z}{h}} \right)^{k} - \frac{\Gamma }{2}} \right],$$

(1)

in which $\Gamma$ and k are the porosity index and the material property gradient index, respectively.

Fig. 1

Illustration schematically a porous FGM microplate having a central cutout

Consequently, the effective Young’s modulus and Poisson’s ratio of porous FGM microplates relevant to each kind of the porosity dispersion scheme can be extracted based on the porosity-dependent rule of mixture as

$$E\left( z \right) = \left( {E_{{\text{c}}} - E_{{\text{m}}} } \right)\varphi_{1} \left( z \right) + E_{{\text{m}}} - \left( {E_{{\text{c}}} + E_{{\text{m}}} } \right)\Gamma \varphi_{2} \left( z \right),$$

(2a)

$$\nu \left( z \right) = \left( {\nu_{{\text{c}}} - \nu_{{\text{m}}} } \right)\varphi_{1} \left( z \right) + E_{{\text{m}}} - \left( {\nu_{{\text{c}}} + \nu_{{\text{m}}} } \right)\Gamma \varphi_{2} \left( z \right),$$

(2b)

where,

$$\varphi_{1} \left( z \right) = \left( {\frac{1}{2} + \frac{z}{h}} \right)^{k} ,\quad \varphi_{2} \left( z \right) = \left\{ {\begin{array}{*{20}l} \frac{1}{2} \hfill & {\text{U-PFGM}} \hfill \\ {\frac{1}{2} - \frac{\left| z \right|}{h}} \hfill & {\text{O-PFGM}} \hfill \\ { - \frac{\left| z \right|}{h}} \hfill & {\text{X-PFGM}} \hfill \\ \end{array} } \right.$$

(3)

In Figs. 2, 3 and 4, the variation of the dimensionless effective Young’s modulus ($E\left( z \right)/E_{c}$) through the plate thickness and porosity index of porous FGM microplates are plotted corresponding to different values of the material property gradient index.

Fig. 2

Variation of Young’s modulus of a U-PFGM microplate with porosity index and through thickness corresponding to different material property gradient indexes

Fig. 3

Variation of Young’s modulus of an O-PFGM microplate with porosity index and through thickness corresponding to different material property gradient indexes

Fig. 4

Variation of Young’s modulus of a X-PFGM microplate with porosity index and through thickness corresponding to different material property gradient indexes

Within the framework of a higher-order shear deformation plate theory, the displacement field can be expressed as

$${\mathcal{U}}_{x} \left( {x,y,z} \right) = u\left( {x,y} \right) - z\frac{{\partial w\left( {x,y} \right)}}{\partial x} + {\mathbb{f}}\left( z \right)\left( {\psi_{x} \left( {x,y} \right) + \frac{{\partial w\left( {x,y} \right)}}{\partial x}} \right),$$

(4a)

$${\mathcal{U}}_{y} \left( {x,y,z} \right) = v\left( {x,y} \right) - z\frac{{\partial w\left( {x,y} \right)}}{\partial y} + {\mathbb{f}}\left( z \right)\left( {\psi_{y} \left( {x,y} \right) + \frac{{\partial w\left( {x,y} \right)}}{\partial y}} \right),$$

(4b)

$${\mathcal{U}}_{z} \left( {x,y,z} \right) = w\left( {x,y} \right),$$

(4c)

in which $u, v, w$ are the mid-plane displacement variables along $x, y,$ and $z$ axes, respectively. Also, $\psi_{x} , \psi_{y}$ are the rotations about $y$-axis and $x$-axis, respectively. ${\mathbb{f}}\left( z \right)$ represents the transverse shear shape function to take shear deformation into account.

By separating the transverse displacement variable into the bending and shear components, and implementing the transverse normal shape function ${\mathbb{g}}\left( z \right)$ to take the normal strains through the thickness into consideration, one will have

$${\mathcal{U}}_{x} \left( {x,y,z} \right) = u\left( {x,y} \right) - z\frac{{\partial w_{b} \left( {x,y} \right)}}{\partial x} + \left({\mathbb{f}}{\left( z \right) - z} \right)\frac{{\partial w_{s} \left( {x,y} \right)}}{\partial x},$$

(5a)

$${\mathcal{U}}_{y} \left( {x,y,z} \right) = v\left( {x,y} \right) - z\frac{{\partial w_{b} \left( {x,y} \right)}}{\partial y} + \left({\mathbb{f}}{\left( z \right) - z} \right)\frac{{\partial w_{s} \left( {x,y} \right)}}{\partial y},$$

(5c)

$${\mathcal{U}}_{z} \left( {x,y,z} \right) = w_{b} \left( {x,y} \right) + \left( {1 + {\mathbb{g}}\left( z \right)} \right)w_{s} \left( {x,y} \right),$$

(5c)

where $w_{b} \left( {x,y} \right)$ and $w_{s} \left( {x,y} \right)$ denote, respectively, the bending and shear displacement variables. By assuming a sinusoidal shear function for ${\mathbb{f}}\left( z \right)$, and a trigonometric shape function for ${\mathbb{g}}\left( z \right)$, the hybrid-type quasi-3D higher-order shear deformation theory can be achieved. So, it is supposed that

$${\mathbb{F}}\left( z \right) = \sin \left( {\frac{\pi z}{h}} \right) - z,$$

(6a)

$${\mathbb{G}}\left( z \right) = 1 + \frac{5}{12\pi }\cos \left( {\frac{\pi z}{h}} \right),$$

(6b)

Figure 5 demonstrates the through-thickness profiles of the introduced shape functions and their derivatives.

Fig. 5

Variation of shape functions and their derivatives through plate thickness considered for the developed hybrid higher-order quasi-3D plate model

Now, the strain–displacement equations including the von-Karman geometric nonlinearity can be written within the developed hybrid-type quasi-3D higher-order shear deformation theory as below

$$\begin{aligned} \varepsilon_{xx} & = \frac{\partial u}{{\partial x}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)^{2} - z\frac{{\partial^{2} w_{b} }}{{\partial x^{2} }} + {\mathbb{F}}\left( z \right)\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }} \\ \varepsilon_{yy} & = \frac{\partial v}{{\partial y}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right)^{2} - z\frac{{\partial^{2} w_{b} }}{{\partial y^{2} }} + {\mathbb{F}}\left( z \right)\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }} \\ \varepsilon_{zz} & = \frac{{{\text{d}}{\mathbb{G}}\left( z \right)}}{{{\text{d}}z}}w_{s} \\ \gamma_{xy} & = \frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}} + \left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)\left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right) - 2z\frac{{\partial^{2} w_{b} }}{\partial x\partial y} + 2{\mathbb{F}}\left( z \right)\frac{{\partial^{2} w_{s} }}{\partial x\partial y} \\ \gamma_{xz} & = \left( {\frac{{{\text{d}}{\mathbb{F}}\left( z \right)}}{{{\text{d}}z}} + {\mathbb{G}}\left( z \right)} \right)\frac{{\partial w_{s} }}{\partial x} \\ \gamma_{yz} & = \left( {\frac{{{\text{d}}{\mathbb{F}}\left( z \right)}}{{{\text{d}}z}} + {\mathbb{G}}\left( z \right)} \right)\frac{{\partial w_{s} }}{\partial y}. \\ \end{aligned}$$

(7)

Accordingly, the stress–strain constitutive equations can be expressed in the following form

$$\left\{ {\begin{array}{*{20}c} {\sigma_{xx} } \\ {\sigma_{yy} } \\ {\sigma_{zz} } \\ {\tau_{xy} } \\ {\tau_{yz} } \\ {\tau_{xz} } \\ \end{array} } \right\} = \left[ {\begin{array}{*{20}c} {Q_{11} \left( z \right)} & {Q_{12} \left( z \right)} & {Q_{13} \left( z \right)} & 0 & 0 & 0 \\ {Q_{12} \left( z \right)} & {Q_{22} \left( z \right)} & {Q_{23} \left( z \right)} & 0 & 0 & 0 \\ {Q_{13} \left( z \right)} & {Q_{23} \left( z \right)} & {Q_{33} \left( z \right)} & 0 & 0 & 0 \\ 0 & 0 & 0 & {Q_{44} \left( z \right)} & 0 & 0 \\ 0 & 0 & 0 & 0 & {Q_{55} \left( z \right)} & 0 \\ 0 & 0 & 0 & 0 & 0 & {Q_{66} \left( z \right)} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\varepsilon_{xx} } \\ {\varepsilon_{yy} } \\ {\varepsilon_{zz} } \\ {\gamma_{xy} } \\ {\gamma_{yz} } \\ {\gamma_{xz} } \\ \end{array} } \right\},$$

(8)

where,

$$\begin{aligned} Q_{11} \left( z \right) & = Q_{22} \left( z \right) = Q_{33} \left( z \right) = \frac{{\left( {1 - \nu \left( z \right)} \right)E\left( z \right)}}{{\left( {1 - 2\nu \left( z \right)} \right)\left( {1 + \nu \left( z \right)} \right)}}, \\ Q_{12} \left( z \right) & = Q_{13} \left( z \right) = Q_{23} \left( z \right) = \frac{\nu \left( z \right)E\left( z \right)}{{\left( {1 - 2\nu \left( z \right)} \right)\left( {1 + \nu \left( z \right)} \right)}}, \\ Q_{44} \left( z \right) & = Q_{55} \left( z \right) = Q_{66} \left( z \right) = \frac{E\left( z \right)}{{2\left( {1 + \nu \left( z \right)} \right)}}. \\ \end{aligned}$$

(9)

On the basis of the nonlocal strain gradient continuum elasticity, the total stress tensor can be expressed as follows [75]

$$\Phi_{ij} = \sigma_{ij} - \nabla \sigma_{ijm}^{*} ,$$

(10)

where the classical and higher-order stresses can be defined, respectively, as below,

$$\sigma_{ij} = \mathop \int \limits_{V} \chi_{1} \left( {x^{\prime},x,e_{1} } \right)C_{ijkl} \varepsilon_{kl} {\text{d}}V,$$

(11a)

$$\sigma_{ijm}^{*} = l^{2} \mathop \int \limits_{V} \chi_{2} \left( {x^{\prime},x,e_{2} } \right)C_{ijkl} \varepsilon_{kl,m} {\text{d}}V,$$

(11b)

in which $e_{1}$ and $e_{2}$ represent the nonlocal parameters regarding the size effect of the nonlocal stress. Also l denotes the material length scale parameter to take the strain gradient size dependency into account. $\varepsilon_{kl}$, $\varepsilon_{kl,m}$ and $C_{ijkl}$ are the strain components, strain gradient components and elastic coefficients. In accordance with nonlocal strain gradient theory, it is assumed that the two kernel functions of $\chi_{1} \left( {x^{\prime},x,e_{1} } \right)$ and $\chi_{2} \left( {x^{\prime},x,e_{2} } \right)$ should satisfy the associated conditions introduced by Eringen [76] as follow

$$\sigma_{ij} - e_{1}^{2} \left( {\frac{{\partial^{2} \sigma_{ij} }}{{\partial x^{2} }} + \frac{{\partial^{2} \sigma_{ij} }}{{\partial y^{2} }}} \right) = C_{ijkl} \varepsilon_{kl} ,$$

(12a)

$$\sigma_{ijm}^{*} - e_{2}^{2} \left( {\frac{{\partial^{2} \sigma_{ijm}^{*} }}{{\partial x^{2} }} + \frac{{\partial^{2} \sigma_{ijm}^{*} }}{{\partial y^{2} }}} \right) = l^{2} C_{ijkl} \varepsilon_{kl,m} .$$

(12b)

As a result, the generalized constitutive equation on the basis of the nonlocal strain gradient elasticity can be written as

$$\begin{aligned} & \left[ {1 - e_{1}^{2} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} }}{{\partial y^{2} }}} \right)} \right]\left[ {1 - e_{2}^{2} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} }}{{\partial y^{2} }}} \right)} \right]\Phi_{ij} \\ & \quad = \left[ {1 - e_{1}^{2} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} }}{{\partial y^{2} }}} \right)} \right]C_{ijkl} \varepsilon_{kl} - l^{2} \left[ {1 - e_{2}^{2} \left( {\frac{{\partial^{2} }}{{\partial x^{2} }} + \frac{{\partial^{2} }}{{\partial y^{2} }}} \right)} \right]C_{ijkl} \frac{{\partial^{2} \varepsilon_{kl} }}{{\partial x^{2} }} \\ \end{aligned}$$

(13)

With assumption of $e_{1} = e_{2} = e$, one will have

$$\Phi_{ij} - e^{2} \left( {\frac{{\partial^{2} \Phi_{ij} }}{{\partial x^{2} }} + \frac{{\partial^{2} \Phi_{ij} }}{{\partial y^{2} }}} \right) = C_{ijkl} \varepsilon_{kl} - l^{2} C_{ijkl} \left( {\frac{{\partial^{2} \varepsilon_{kl} }}{{\partial x^{2} }} + \frac{{\partial^{2} \varepsilon_{kl} }}{{\partial y^{2} }}} \right).$$

Accordingly, the variation of the strain energy for a porous FGM microplate modeled via the nonlocal strain gradient hybrid-type quasi-3D higher-order shear deformation theory can be expressed as

$$\delta \Pi_{S} = \mathop \int \limits_{S} \mathop \int \limits_{{ - \frac{h}{2}}}^{\frac{h}{2}} \Phi_{ij} \delta \varepsilon_{ij} {\text{d}}z{\text{d}}S,$$

(15)

Also, the virtual work caused by the external distributed load ${{q}}$ can be written as

$$\delta \Pi_{W} = \mathop \int\limits_{S} {\mathscr{q}}\delta w{\text{d}}S,$$

(16)

On the basis of the virtual work principle, and through substituting Eqs. (7) and (8) into Eq. (15), one will have

$$\mathop \int \limits_{S} \left\{ {\delta \left( {{{\mathfrak{P}}}_{{\varvec{b}}}^{T} } \right){\varvec{\xi}}_{{\varvec{b}}} {{\mathfrak{P}}}_{{\varvec{b}}} - l^{2} \delta \left( {\nabla^{2} {{\mathfrak{P}}}_{{\varvec{b}}}^{T} } \right){\varvec{\xi}}_{{\varvec{b}}} {{\mathfrak{P}}}_{{\varvec{b}}} + \delta \left( {{{\mathfrak{P}}}_{{\varvec{s}}}^{T} } \right){\varvec{\xi}}_{{\varvec{s}}} {{\mathfrak{P}}}_{{\varvec{s}}} - l^{2} \delta \left( {\nabla^{2} {{\mathfrak{P}}}_{{\varvec{s}}}^{T} } \right){\varvec{\xi}}_{{\varvec{s}}} {{\mathfrak{P}}}_{{\varvec{s}}} } \right\}{\text{d}}S = \mathop \int \limits_{S} \left( {1 - e^{2} \nabla^{2} } \right)\delta w{\text{d}}S,$$

(17)

in which

$$\begin{aligned} {{\mathfrak{P}}}_{{\varvec{b}}} & = \left[ {\begin{array}{*{20}l} {\frac{\partial u}{{\partial x}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)^{2} } \hfill & { - \frac{{\partial^{2} w_{b} }}{{\partial x^{2} }}} \hfill & {\frac{{\partial^{2} w_{s} }}{{\partial x^{2} }}} \hfill & 0 \hfill \\ {\frac{\partial v}{{\partial y}} + \frac{1}{2}\left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right)^{2} } \hfill & { - \frac{{\partial^{2} w_{b} }}{{\partial y^{2} }}} \hfill & {\frac{{\partial^{2} w_{s} }}{{\partial y^{2} }}} \hfill & 0 \hfill \\ {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}} + \left( {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \right)\left( {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \right)} \hfill & { - 2\frac{{\partial^{2} w_{b} }}{\partial x\partial y}} \hfill & {2\frac{{\partial^{2} w_{s} }}{\partial x\partial y}} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & {w_{s} } \hfill \\ \end{array} } \right]^{T} \\ {\varvec{\xi}}_{{\varvec{b}}} & = \left[ {\begin{array}{*{20}c} {{{A}}_{{{b}}} } & {{{B}}_{{{b}}} } & {{{C}}_{{{b}}} } & {{{E}}_{{{b}}} } \\ {{{B}}_{{{b}}} } & {{{D}}_{{{b}}} } & {{{F}}_{{{b}}} } & {{{G}}_{{{b}}} } \\ {{{C}}_{{{b}}} } & {{{F}}_{{{b}}} } & {{{H}}_{{{b}}} } & {{{K}}_{{{b}}} } \\ {{{E}}_{{{b}}} } & {{{G}}_{{{b}}} } & {{{K}}_{{{b}}} } & {{{J}}_{{{b}}} } \\ \end{array} } \right],\;{{\mathfrak{P}}}_{{\varvec{s}}} = \left[ {\begin{array}{*{20}c} {\frac{{\partial w_{s} }}{\partial x}} \\ {\frac{{\partial w_{s} }}{\partial y}} \\ \end{array} } \right],\;\;{\varvec{\xi}}_{{\varvec{s}}} = \mathop \int \limits_{{ - \frac{h}{2}}}^{\frac{h}{2}} \left( {\frac{{{\rm d}{\mathbb{F}}\left( z \right)}}{{\rm d}z} + {\mathbb{G}}\left( z \right)} \right)^{2} \left[ {\begin{array}{*{20}c} {Q_{44} \left( z \right)} & 0 \\ 0 & {Q_{55} \left( z \right)} \\ \end{array} } \right]{\text{d}}z \\ \end{aligned}$$

(18)

where,

$$\begin{aligned} \left\{ {{{A}}_{{{b}}} ,{{B}}_{{{b}}} ,{{C}}_{{{b}}} } \right\} & = \mathop \int \limits_{{ - \frac{h}{2}}}^{\frac{h}{2}} \left\{ {1,z,{\mathbb{F}}\left( z \right)} \right\}\left[ {\begin{array}{*{20}c} {Q_{11} \left( z \right)} & {Q_{12} \left( z \right)} & 0 & {Q_{13} \left( z \right)} \\ {Q_{12} \left( z \right)} & {Q_{22} \left( z \right)} & 0 & {Q_{23} \left( z \right)} \\ 0 & 0 & {Q_{66} \left( z \right)} & 0 \\ {Q_{31} \left( z \right)} & {Q_{32} \left( z \right)} & 0 & {Q_{33} \left( z \right)} \\ \end{array} } \right]{\text{d}}z \\ \left\{ {{{D}}_{{{b}}} ,{{E}}_{{{b}}} ,{{F}}_{{{b}}} ,{{G}}_{{{b}}} } \right\} & = \mathop \int \limits_{{ - \frac{h}{2}}}^{\frac{h}{2}} \left\{ {z^{2} ,\frac{{{\text{d}}{\mathbb{G}}\left( z \right)}}{{{\text{d}}z}},z{\mathbb{F}}\left( z \right),z\frac{{{\text{d}}{\mathbb{G}}\left( z \right)}}{{{\text{d}}z}}} \right\}\left[ {\begin{array}{*{20}c} {Q_{11} \left( z \right)} & {Q_{12} \left( z \right)} & 0 & {Q_{13} \left( z \right)} \\ {Q_{12} \left( z \right)} & {Q_{22} \left( z \right)} & 0 & {Q_{23} \left( z \right)} \\ 0 & 0 & {Q_{66} \left( z \right)} & 0 \\ {Q_{31} \left( z \right)} & {Q_{32} \left( z \right)} & 0 & {Q_{33} \left( z \right)} \\ \end{array} } \right]{\text{d}}z \\ \left\{ {{{H}}_{{{b}}} ,{{K}}_{{{b}}} ,{{J}}_{{\varvec{b}}} } \right\} & = \mathop \int \limits_{{ - \frac{h}{2}}}^{\frac{h}{2}} \left\{ {{\mathbb{F}}^{2} \left( z \right),{\mathbb{F}}\left( z \right)\frac{{{\text{d}}{\mathbb{G}}\left( z \right)}}{{{\text{d}}z}},\left( {\frac{{{\text{d}}{\mathbb{G}}\left( z \right)}}{{{\text{d}}z}}} \right)^{2} } \right\}\left[ {\begin{array}{*{20}c} {Q_{11} \left( z \right)} & {Q_{12} \left( z \right)} & 0 & {Q_{13} \left( z \right)} \\ {Q_{12} \left( z \right)} & {Q_{22} \left( z \right)} & 0 & {Q_{23} \left( z \right)} \\ 0 & 0 & {Q_{66} \left( z \right)} & 0 \\ {Q_{31} \left( z \right)} & {Q_{32} \left( z \right)} & 0 & {Q_{33} \left( z \right)} \\ \end{array} } \right]{\text{d}}z \\ \end{aligned}$$

(19)

3 Isogeometric solution methodology

The isogeometric numerical solving process has been widely use in recent years [77,78,79,80,81,82,83,84,85,86,87,88]. Within a one-dimensional domain, the associated knot vector can be expressed in a non-decreasing form as below:

$${\mathbb{K}}\left( \xi \right) = \left\{ {\xi_{1} ,\xi_{2} ,\xi_{3} , \ldots ,\xi_{m + n + 1} } \right\},$$

(20)

where m and n stand for the number of basis function and the order of the B-spline basis function. In addition, it is necessary that each ith knot satisfies the condition of $0 \le \xi_{i} \le 1$. As a consequence, the B-spline basis function is written based on the recursive Cox–de Boor formula as below

$${\mathcal{X}}_{i,0} \left( \xi \right) = \left\{ {\begin{array}{*{20}l} 1 \hfill &\quad {\xi_{i} \le \xi < \xi_{i + 1} } \hfill \\ 0 \hfill &\quad {\text{else,}} \hfill \\ \end{array} } \right.$$

(21a)

$${\mathcal{X}}_{i,n} \left( \xi \right) = \frac{{\xi - \xi_{i} }}{{\xi_{i + n} - \xi_{i} }}{\mathcal{X}}_{i,n - 1} \left( \xi \right) + \frac{{\xi_{i + n + 1} - \xi }}{{\xi_{i + n + 1} - \xi_{i + 1} }}{\mathcal{X}}_{i + 1,n - 1} \left( \xi \right),$$

(21b)

Within a two-dimensional domain, the tensor product of two basic functions can be utilized to achieve the associated B-spline basis function as follows

$${\mathcal{F}}_{i,j}^{p,q} \left( {\xi ,\eta } \right) = \mathop \sum \limits_{i = 1}^{m} {\mathfrak{J}}_{i} \left( {x,y} \right){\mathcal{P}}_{i} ,$$

(22)

where ${\mathcal{P}}_{i}$ denote the control points within the bi-directional control net, and

$${\mathfrak{J}}_{i} \left( {\xi ,\eta } \right) = \frac{{{\mathcal{X}}_{i,p} \left( \xi \right){\mathcal{X}}_{j,q} \left( \eta \right){\mathfrak{W}}_{i,j} }}{{\mathop \sum \nolimits_{i = 1}^{m} \mathop \sum \nolimits_{j = 1}^{n} {\mathcal{X}}_{i,p} \left( \xi \right){\mathcal{X}}_{j,q} \left( \eta \right){\mathfrak{W}}_{i,j} }},$$

(23)

in which ${\mathcal{X}}_{i,p} \left( \xi \right)$ and ${\mathcal{X}}_{j,q} \left( \eta \right)$ represent, respectively, the shape functions of orders $p$ and $q$ along $\xi$ and $\eta$ directions. Also, ${\mathfrak{W}}_{i,j}$ is the relevant weight coefficient. Thereafter, the knot vector of ${\mathbb{K}}\left( \eta \right)$ is employed to extract the derivation of the shape function ${\mathcal{X}}_{j,q} \left( \eta \right)$. In Fig. 6, the considered cubic elements for square microplates with and without a central cutout are illustrated.

Fig. 6

Illustration of cubic elements for square microplates with geometrical parameters: a microplate without a central cutout, b microplate with a square cutout, c microplate with a circular cutout

By taking the non-uniform rational B-spline (NURBS)-based isogeometric analysis into consideration, an approximation for the associated displacement field within a plate-type domain can be given in the following form.

$$\left\{ {\tilde{u}^{i} ,\tilde{v}^{i} ,\tilde{w}_{b}^{i} ,\tilde{w}_{s}^{i} } \right\}^{T} = \mathop \sum \limits_{i = 1}^{m \times n} T_{i} \left( {x,y} \right)\left\{ {\begin{array}{*{20}c} {u^{i} } \\ {v^{i} } \\ {w_{b}^{i} } \\ {w_{s}^{i} } \\ \end{array} } \right\},$$

(24)

where,

$$T_{i} \left( {x,y} \right) = \left[ {\begin{array}{*{20}c} {{\mathfrak{J}}_{i} \left( {x,y} \right)} & 0 & 0 & 0 \\ 0 & {{\mathfrak{J}}_{i} \left( {x,y} \right)} & 0 & 0 \\ 0 & 0 & {{\mathfrak{J}}_{i} \left( {x,y} \right)} & 0 \\ 0 & 0 & 0 & {{\mathfrak{J}}_{i} \left( {x,y} \right)} \\ \end{array} } \right].$$

(25)

In accordance with Eq. (7), and considering Eq. (24), the strain components can be rewritten as below

$${{\mathfrak{P}}}_{{\varvec{b}}} = {{\mathfrak{P}}}_{{\varvec{b}}}^{{\varvec{L}}} + {{\mathfrak{P}}}_{{\varvec{b}}}^{{{\varvec{NL}}}} = \mathop \sum \limits_{i = 1}^{m \times n} {{\mathcal{T}}}_{{{\varvec{Lb}}}}^{{\varvec{i}}} {\mathbb{X}} + \mathop \sum \limits_{i = 1}^{m \times n} \frac{1}{2}{{\mathcal{T}}}_{{{\varvec{NLb}}}}^{{\varvec{i}}} {\mathbb{X}},\quad {{\mathfrak{P}}}_{{\varvec{s}}} = \mathop \sum \limits_{{{\varvec{i}} = 1}}^{{{\varvec{m}} \times {\varvec{n}}}} {{\mathcal{T}}}_{{\varvec{s}}}^{{\varvec{i}}} {\mathbb{X}},$$

(26)

where

$$\begin{aligned} {{\mathcal{T}}}_{{{\varvec{Lb}}}}^{{\varvec{i}}} & = \left\{ {\begin{array}{*{20}c} {{{\mathcal{T}}}_{{{\varvec{b}}1}}^{{\varvec{i}}} } & {{{\mathcal{T}}}_{{{\varvec{b}}2}}^{{\varvec{i}}} } & {{{\mathcal{T}}}_{{{\varvec{b}}3}}^{{\varvec{i}}} } & {{{\mathcal{T}}}_{{{\varvec{b}}4}}^{{\varvec{i}}} } \\ \end{array} } \right\}^{{\varvec{T}}} ,\quad {{\mathcal{T}}}_{{{\varvec{NLb}}}}^{{\varvec{i}}} = \left\{ {\begin{array}{*{20}c} {{{\mathcal{T}}}_{{{\varvec{b}}5}}^{{\varvec{i}}} } & 0 & 0 & 0 \\ \end{array} } \right\}^{{\varvec{T}}} {{\mathcal{T}}}_{{\varvec{G}}}^{{\varvec{i}}} ,\;\;{\mathbb{X}} = \left\{ {\begin{array}{*{20}c} {u^{i} } \\ {v^{i} } \\ {w_{b}^{i} } \\ {w_{s}^{i} } \\ \end{array} } \right\}, \\ {{\mathcal{T}}}_{{{\varvec{b}}1}}^{{\varvec{i}}} & = \left[ {\begin{array}{*{20}c} {{\mathfrak{J}}_{i,x} \left( {x,y} \right)} & 0 & 0 & 0 \\ 0 & {{\mathfrak{J}}_{i,y} \left( {x,y} \right)} & 0 & 0 \\ {{\mathfrak{J}}_{i,y} \left( {x,y} \right)} & {{\mathfrak{J}}_{i,x} \left( {x,y} \right)} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right],\quad {{\mathcal{T}}}_{{{\varvec{b}}2}}^{{\varvec{i}}} = - \left[ {\begin{array}{*{20}c} 0 & 0 & {{\mathfrak{J}}_{i,xx} \left( {x,y} \right)} & 0 \\ 0 & 0 & {{\mathfrak{J}}_{i,yy} \left( {x,y} \right)} & 0 \\ 0 & 0 & {2{\mathfrak{J}}_{i,xy} \left( {x,y} \right)} & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} } \right], \\ {{\mathcal{T}}}_{{{\varvec{b}}3}}^{{\varvec{i}}} & = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & {{\mathfrak{J}}_{i,xx} \left( {x,y} \right)} \\ 0 & 0 & 0 & {{\mathfrak{J}}_{i,yy} \left( {x,y} \right)} \\ 0 & 0 & 0 & {2{\mathfrak{J}}_{i,xy} \left( {x,y} \right)} \\ 0 & 0 & 0 & 0 \\ \end{array} } \right],\quad {{\mathcal{T}}}_{{{\varvec{b}}4}}^{{\varvec{i}}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{\mathfrak{J}}_{i} \left( {x,y} \right)} \\ \end{array} } \right], \\ {{\mathcal{T}}}_{{{\varvec{b}}5}}^{{\varvec{i}}} & = \left[ {\begin{array}{*{20}c} {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} & 0 \\ 0 & {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} \\ {\frac{{\partial w_{b} }}{\partial y} + \frac{{\partial w_{s} }}{\partial y}} & {\frac{{\partial w_{b} }}{\partial x} + \frac{{\partial w_{s} }}{\partial x}} \\ \end{array} } \right],\quad {{\mathcal{T}}}_{{\varvec{G}}}^{{\varvec{i}}} = \left[ {\begin{array}{*{20}c} 0 & 0 & {{\mathfrak{J}}_{i,x} \left( {x,y} \right)} & {{\mathfrak{J}}_{i,x} \left( {x,y} \right)} \\ 0 & 0 & {{\mathfrak{J}}_{i,y} \left( {x,y} \right)} & {{\mathfrak{J}}_{i,y} \left( {x,y} \right)} \\ \end{array} } \right] \\ \end{aligned}$$

(27)

As a result, the variation of the strain tensor can be expressed as

$$\delta \left( {{{\mathfrak{P}}}_{{\varvec{b}}} } \right) = \delta \left( {{{\mathfrak{P}}}_{{\varvec{b}}}^{{\varvec{L}}} } \right) + \delta \left( {{{\mathfrak{P}}}_{{\varvec{b}}}^{{{\varvec{NL}}}} } \right) = \mathop \sum \limits_{i = 1}^{m \times n} \left( {{{\mathcal{T}}}_{{{\varvec{Lb}}}}^{{\varvec{i}}} + {{\mathcal{T}}}_{{{\varvec{NLb}}}}^{{\varvec{i}}} } \right)\left\{ {\begin{array}{*{20}c} {\delta u^{i} } \\ {\delta v^{i} } \\ {\delta w_{b}^{i} } \\ {\delta w_{s}^{i} } \\ \end{array} } \right\},\quad \delta \left( {{{\mathfrak{P}}}_{{\varvec{s}}} } \right) = \mathop \sum \limits_{i = 1}^{m \times n} {{\mathcal{T}}}_{{\varvec{s}}}^{{\varvec{i}}} \left\{ {\begin{array}{*{20}c} {\delta u^{i} } \\ {\delta v^{i} } \\ {\delta w_{b}^{i} } \\ {\delta w_{s}^{i} } \\ \end{array} } \right\}.$$

(28)

After that, the discretized form of the nonlinear differential equations of the system can be presented as follows

$${{\mathfrak{T}}}\left( {\mathbb{X}} \right){\mathbb{X}} = {\mathbb{p}},$$

(29)

where ${{\mathfrak{I}}}\left( {\mathbb{X}} \right)$ stands for the global stiffness matrix including linear and nonlinear parts as below

$${{\mathfrak{I}}}_{L} = \mathop \int \limits_{S} \left\{ {\left( {{{\mathcal{T}}}_{{{\varvec{Lb}}}}^{{\varvec{i}}} } \right)^{{\varvec{T}}} {\varvec{\xi}}_{{\varvec{b}}} {{\mathcal{T}}}_{{{\varvec{Lb}}}}^{{\varvec{i}}} - l^{2} \nabla^{2} \left( {{{\mathcal{T}}}_{{{\varvec{Lb}}}}^{{\varvec{i}}} } \right)^{{\varvec{T}}} {\varvec{\xi}}_{{\varvec{b}}} {{\mathcal{T}}}_{{{\varvec{Lb}}}}^{{\varvec{i}}} + \left( {{{\mathcal{T}}}_{{\varvec{s}}}^{{\varvec{i}}} } \right)^{{\varvec{T}}} {\varvec{\xi}}_{{\varvec{s}}} {{\mathcal{T}}}_{{\varvec{s}}}^{{\varvec{i}}} - l^{2} \nabla^{2} \left( {{{\mathcal{T}}}_{{\varvec{s}}}^{{\varvec{i}}} } \right)^{{\varvec{T}}} {\varvec{\xi}}_{{\varvec{s}}} {{\mathcal{T}}}_{{\varvec{s}}}^{{\varvec{i}}} } \right\}{\text{d}}S,$$

(30a)

$$\begin{aligned} {{\mathfrak{I}}}_{NL} \left( {\mathbb{X}} \right) & = \mathop \int \limits_{S} \left\{ {\frac{1}{2}\left( {{{\mathcal{T}}}_{{{\varvec{Lb}}}}^{{\varvec{i}}} } \right)^{{\varvec{T}}} {\varvec{\xi}}_{{\varvec{b}}} {{\mathcal{T}}}_{{{\varvec{NLb}}}}^{{\varvec{i}}} - \frac{{l^{2} }}{2}\nabla^{2} \left( {{{\mathcal{T}}}_{{{\varvec{Lb}}}}^{{\varvec{i}}} } \right)^{{\varvec{T}}} {\varvec{\xi}}_{{\varvec{b}}} {{\mathcal{T}}}_{{{\varvec{NLb}}}}^{{\varvec{i}}} + \left( {{{\mathcal{T}}}_{{{\varvec{NLb}}}}^{{\varvec{i}}} } \right)^{{\varvec{T}}} {\varvec{\xi}}_{{\varvec{b}}} {{\mathcal{T}}}_{{{\varvec{Lb}}}}^{{\varvec{i}}} } \right. \\ & \quad \left. { - l^{2} \nabla^{2} \left( {{{\mathcal{T}}}_{{{\varvec{NLb}}}}^{{\varvec{i}}} } \right)^{{\varvec{T}}} {\varvec{\xi}}_{{\varvec{b}}} {{\mathcal{T}}}_{{{\varvec{Lb}}}}^{{\varvec{i}}} + \frac{1}{2}\left( {{{\mathcal{T}}}_{{{\varvec{NLb}}}}^{{\varvec{i}}} } \right)^{{\varvec{T}}} {\varvec{\xi}}_{{\varvec{b}}} {{\mathcal{T}}}_{{{\varvec{NLb}}}}^{{\varvec{i}}} - \frac{{l^{2} }}{2}\nabla^{2} \left( {{{\mathcal{T}}}_{{{\varvec{NLb}}}}^{{\varvec{i}}} } \right)^{{\varvec{T}}} {\varvec{\xi}}_{{\varvec{b}}} {{\mathcal{T}}}_{{{\varvec{NLb}}}}^{{\varvec{i}}} } \right\}{\text{d}}S \\ \end{aligned}$$

(30b)

Also, the load vector can be given as

$${\mathbb{p}} = \mathop \int \limits_{S} \left( {1 - e^{2} \nabla^{2} } \right){\mathscr{q}}\left\{ {\begin{array}{*{20}c} 0 \\ 0 \\ {{\mathfrak{J}}_{i} \left( {x,y} \right)} \\ {{\mathfrak{J}}_{i} \left( {x,y} \right)} \\ \end{array} } \right\}{\text{d}}S.$$

(31)

Thereafter, an iterative solution methodology based on the Newton–Raphson technique is employed to solve Eq. (29). To this purpose, the residual force vector is introduced as follows

$${{\mathcal{R}}}\left( {\mathbb{X}} \right) = {{\mathfrak{I}}}\left( {\mathbb{X}} \right){\mathbb{X}} - {\mathbb{p}} = \left[ {{{\mathfrak{I}}}_{L} + {{\mathfrak{I}}}_{NL} \left( {\mathbb{X}} \right)} \right]{\mathbb{X}} - {\mathbb{p}}.$$

(32)

Accordingly, the considered increment in the value of displacement vector is as below

$${\mathbb{X}} = - \frac{{{{\mathcal{R}}}\left( {{\mathbb{X}}^{i} } \right)}}{{{{\mathfrak{I}}}_{L} + {{\mathfrak{I}}}_{NL} + {{\mathfrak{I}}}_{G} }},$$

(33)

where

$${\mathbb{X}}^{i + 1} = {\mathbb{X}}^{i} + {\mathbb{X}},$$

(34)

and ${{\mathfrak{I}}}_{G}$ is the geometric stiffness matrix which can be achieved in the following form

$${{\mathfrak{I}}}_{G} = \mathop \int \limits_{S} \left\{ {\left( {{{\mathcal{T}}}_{{{\varvec{G}}1}} } \right)^{{\varvec{T}}} {\varvec{N}}_{{\varvec{x}}} {{\mathcal{T}}}_{{{\varvec{G}}1}} - l^{2} \nabla^{2} \left( {{{\mathcal{T}}}_{{{\varvec{G}}1}} } \right)^{{\varvec{T}}} {\varvec{N}}_{{\varvec{x}}} {{\mathcal{T}}}_{{{\varvec{G}}1}} + \left( {{{\mathcal{T}}}_{{{\varvec{G}}2}} } \right)^{{\varvec{T}}} {\varvec{N}}_{{\varvec{y}}} {{\mathcal{T}}}_{{{\varvec{G}}2}} - l^{2} \nabla^{2} \left( {{{\mathcal{T}}}_{{{\varvec{G}}2}} } \right)^{{\varvec{T}}} {\varvec{N}}_{{\varvec{y}}} {{\mathcal{T}}}_{{{\varvec{G}}2}} } \right\}{\text{d}}S,$$

(35)

in which

$${{\mathcal{T}}}_{{{\varvec{G}}1}} = \left[ {\begin{array}{*{20}c} 0 & 0 & {{\mathfrak{J}}_{i,x} \left( {x,y} \right)} & 0 \\ 0 & 0 & {{\mathfrak{J}}_{i,y} \left( {x,y} \right)} & 0 \\ \end{array} } \right],\quad {{\mathcal{T}}}_{{{\varvec{G}}2}} = \left[ {\begin{array}{*{20}c} 0 & 0 & 0 & {{\mathfrak{J}}_{i,x} \left( {x,y} \right)} \\ 0 & 0 & 0 & {{\mathfrak{J}}_{i,y} \left( {x,y} \right)} \\ \end{array} } \right],$$

(36)

and ${\varvec{N}}_{{\varvec{x}}}$, ${\varvec{N}}_{{\varvec{y}}}$ are the load-type stress resultants along x axis and y axis, respectively.

4 Numerical results and discussion

The dimensionless nonlocal strain gradient load–deflection curves associated with the porosity-dependent nonlinear flexural response of the uniformly distributed loaded porous FGM microplates with and without a central cutout having different shapes are presented. It is assumed that the top and bottom surfaces of porous FGM microplates are ceramic-rich and metal-rich, respectively. The material properties are: E_c = 210 GPa, v = 0.24 for the ceramic phase, E_m = 70 GPa, v = 0.35 for the metal phase [89]. Also, the dimensionless load and deflection are calculated as $\overline{P} = PL_{1}^{2} /E_{m} h^{3}$, $W = w/h$. Moreover, the geometric parameters are assumed as h = 20 µm, L₁ = 50 h, L₁/L₂ = 1.

At first, the validity of the present solution process is checked. To this end, by neglecting the size dependent terms, the nonlinear load–deflection response associated with the geometrically nonlinear flexural response of a square composite plate is obtained and compared with that presented by Singh et al. [90] as shown in Fig. 7. A very good can be observed which confirms the introduced numerical solution methodology.

Fig. 7

Comparison of load deflection curves associated with the nonlinear bending of composite square plate under inform distributed load

In Figs. 8 and 9, the classical and nonlocal strain gradient load–deflection curves associated with the nonlinear bending response of U-PFGM microplates without any central cutout are displayed for various values of the nonlocal parameter and strain gradient parameter, respectively. By comparing the nonlocal strain gradient curves with the classical counterparts, it is deduced that by taking the strain gradient type of size dependency into consideration, the maximum deflection associated with a given applied distributed load gets smaller which indicated the stiffening influence of the couple stress size effect. However, in the presence of the nonlocal size effect, an opposite pattern is found which represents the softening influence of it. This observation has a similar pattern for the both fully simply supported (SSSS) and fully clamped (CCCC) boundary conditions.

Fig. 8

Dimensionless classical and nonlocal strain gradient load–deflection responses associated with the nonlinear bending of U-PFGM microplates corresponding to various values of the nonlocal parameter in the absence of strain gradient size effect (l = 0 µm, $\Gamma = 0.4$, $k = 0.5$, $a/L = d/L = 0$)

Fig. 9

Dimensionless classical and nonlocal strain gradient load–deflection responses associated with the nonlinear bending of U-PFGM microplates corresponding to various values of the strain gradient parameter in the absence of nonlocal size effect (e = 0 µm, $\Gamma = 0.4$, $k = 0.5$, $a/L = d/L = 0$)

Figures 10 and 11 illustrate the classical and nonlocal strain gradient load–deflection curves relevant to the nonlinear bending behavior of U-PFGM microplates without any central cutout corresponding to different material property gradient indexes, and respectively, for various nonlocal and strain gradient parameters. It is demonstrated that by moving from the fully ceramic-rich microplate to the fully metal-rich one, the rate of load–deflection variation decreases significantly, so a specific maximum deflection occurs at a lower applied distributed load. Also, the gap between the nonlocal strain gradient curve and its classical counterpart becomes larger by moving from the fully metal-rich microplate to the fully ceramic-rich one.

Fig. 10

Dimensionless classical and nonlocal strain gradient load–deflection responses associated with the nonlinear bending of U-PFGM microplates corresponding to various values of the material gradient index in the absence of strain gradient size dependency (l = 0 µm, $\Gamma = 0.4$, $a/L = d/L = 0$)

Fig. 11

Dimensionless classical and nonlocal strain gradient load–deflection responses associated with the nonlinear bending of U-PFGM microplates corresponding to various values of the material gradient index in the absence of nonlocal size dependency (e = 0 µm, $\Gamma = 0.4$, $a/L = d/L = 0$)

The classical and nonlocal strain gradient load–deflection curves associated with the nonlinear bending characteristics of porous FGM microplates without a central cutout are depicted in Figs. 12 and 13 corresponding to different porosity dispersion patterns. It is seen that by taking only the nonlocality into consideration, the gap between nonlinear bending curves relevant to various patterns of porosity dispersion is a bit lower for the nonlocal strain gradient case than the classical one. But, in the presence strain gradient size dependency and in the absence of nonlocality, the observation becomes reverse. In addition, a same trend can be observed for different patterns of porosity dispersion, and this issue is for the both types of SSSS and CCCC boundary conditions.

Fig. 12

Dimensionless classical and nonlocal strain gradient load–deflection responses associated with the nonlinear bending of porous FGM microplates corresponding to various porosity dispersion patterns in the absence of strain gradient size dependency (l = 0 µm, $\Gamma = 0.4$, $k = 0.5$, $a/L = d/L = 0$)

Fig. 13

Dimensionless classical and nonlocal strain gradient load–deflection responses associated with the nonlinear bending of porous FGM microplates corresponding to various porosity dispersion patterns in the absence of nonlocal size dependency (e = 0 µm, $\Gamma = 0.4$, $k = 0.5$, $a/L = d/L = 0$)

In Tables 1 and 2, the classical and nonlocal strain gradient dimensionless distributed loads associated with given maximum deflections resulted from the nonlinear bending response of porous FGM microplates without any central cutout are tabulated corresponding to different material property gradient indexes and in the presence and absence of one of the nonlocal and strain gradient size dependencies. The percentages presented in parentheses indicate the difference between the nonlocal strain gradient distributed load and its classical counterpart. It is revealed that for a larger maximum deflection, the significance of the both nonlocal and strain gradient reduces. This prediction is the same for all values of the material property gradient index. Furthermore, it can be seen that corresponding to different maximum deflections, the both nonlocality and strain gradient size effects on the distributed load is a bit lower for a fully clamped porous FGM microplate than that a fully simply supported one. It is revealed that among various patterns of porosity dispersion, the geometrically nonlinear bending stiffness associated with the O-PFGM and X-PFGM microplates is minimum and maximum, respectively.

Table 1 Classical and nonlocal strain gradient dimensionless distributed loads of porous FG composite microplates corresponding to different nonlocal parameters, porosity dispersion patterns, maximum deflections, and various material property gradient indexes ($\Gamma = 0.4$, l = 0 µm)

Table 2 Classical and nonlocal strain gradient dimensionless distributed loads of porous FG composite microplates corresponding to different strain gradient parameters, porosity dispersion patterns, maximum deflections, and various material property gradient indexes ($\Gamma = 0.4$, e = 0 µm)

In Fig. 14, the nonlocal strain gradient nonlinear bending responses of O-PFGM and X-PFGM microplates without any central cutout are shown corresponding to various values of the porosity index. It is obvious that for a porous FGM microplate with a higher value of the porosity index, the gap between nonlocal strain gradient nonlinear bending curves associated with the O-PFGM and X-PFGM dispersion patterns enhances. This anticipation may be related to this fact that a higher porosity index results in bigger internal pores that makes increase the importance of the role of porosity dispersion pattern. This conclusion can be deduced for the both types of SSSS and CCCC edge supports.

Fig. 14

Dimensionless nonlocal strain gradient load–deflection responses associated with the nonlinear bending of porous FGM microplates corresponding to various values of the porosity index (e = l = 120 µm, $k = 0.5$, $a/L = d/L = 0$)

Tables 3 and 4 give the classical and nonlocal strain gradient dimensionless distributed loads associated with given maximum deflections resulted from the nonlinear bending response of porous FGM microplates without any central cutout corresponding to different porosity indexes and in the presence and absence of one of the nonlocal and strain gradient size dependencies. The percentages presented in parentheses indicate the difference between the nonlocal strain gradient distributed load and its classical counterpart. It is observed that corresponding to different maximum deflections, the significance of the strain gradient size effect in the absence of nonlocality on the nonlinear flexural stiffness of a porous FGM microplate is more than that of the nonlocal size effect in the absence of the strain gradient size dependency.

Table 3 Classical and nonlocal strain gradient dimensionless distributed loads of porous FG composite microplates corresponding to different nonlocal parameters, porosity dispersion patterns, maximum deflections, and various porosity indexes ($k = 0.5$, l = 0 µm)

Table 4 Classical and nonlocal strain gradient dimensionless distributed loads of porous FG composite microplates corresponding to different strain gradient parameters, porosity dispersion patterns, maximum deflections, and various porosity indexes ($k = 0.5$, e = 0 µm)

To indicate the influence of a central cutout with different shapes on the nonlocal strain gradient nonlinear bending characteristics of porous FGM microplates, the nonlinear load–deflection curves associated with U-PFGM microplates with square and circular shapes having various side lengths and diameters are plotted in Figs. 15 and 16, respectively. It can be found that a central cutout leads to change the trend of load–deflection response. As a consequence, for smaller value of the applied distributed load, the induced maximum deflection for a microplate without any central cutout is higher than that induced in microplates with a central cutout. However, by increasing the distributed load, this pattern becomes vice versa. Accordingly, there is a specific value for the applied distributed load that this shift of trend occurs, and it depends on several parameters such as the cutout geometry and boundary conditions.

Fig. 15

Influence of a square central cutout on dimensionless nonlocal strain gradient load–deflection responses associated with the nonlinear bending of U-PFGM microplates (e = l = 120 µm, $\Gamma = 0.4$, $k = 0.5$)

Fig. 16

Influence of a circular central cutout on dimensionless nonlocal strain gradient load–deflection responses associated with the nonlinear bending of U-PFGM microplates (e = l = 120 µm, $\Gamma = 0.4$, $k = 0.5$)

5 Conclusion

In this work, the nonlocal strain gradient geometrically nonlinear flexural response of porous FGM microplates having a central cutout with different shapes was predicted. To accomplish this issue, a hybrid-type quasi-3D higher-order shear deformation plate theory was formulated within the framework of the nonlocal strain gradient continuum elasticity. Afterwards, using the NURBS-based isogeometric approach, the possibility of flexibly meeting higher-order derivatives was achieved.

It was indicated that by taking the strain gradient type of size dependency into consideration, the maximum deflection associated with a given applied distributed load gets smaller which indicated the stiffening influence of the couple stress size effect. However, in the presence of the nonlocal size effect, an opposite pattern is found which represents the softening influence of it. Also, the gap between the nonlocal strain gradient curve and its classical counterpart becomes larger by moving from the fully metal-rich microplate to the fully ceramic-rich one. Additionally, it was found that for a larger maximum deflection, the significance of the both nonlocal and strain gradient reduces. This prediction was the same for all values of the material property gradient index as well as porosity index.

Furthermore, it was seen that corresponding to different maximum deflections, the significance of the strain gradient size effect in the absence of nonlocality on the nonlinear flexural stiffness of a porous FGM microplate is more than that of the nonlocal size effect in the absence of the strain gradient size dependency. In addition, it was revealed that a central cutout leads to change the trend of load–deflection response. As a consequence, for smaller value of the applied distributed load, the induced maximum deflection for a microplate without any central cutout is higher than that induced in microplates with a central cutout. However, by increasing the distributed load, this pattern becomes vice versa.