An isogeometric approach to free vibration analysis of bi-directional functionally graded porous doubly-curved shallow microshells with variable length-scale parameters

Abstract

This study uses iso-geometric investigation, which is based on the non-uniform rational B-splines (NURBS) basis function, to investigate natural oscillation of bi-directional functionally graded porous (BFGP) doubly-curved shallow microshells placed on Pasternak foundations with any boundary conditions. The characteristics of the present material vary in both thickness and axial directions along the x-axis. To be more specific, a material length-scale coefficient of the microshell varies in both thickness and length directions as the material’s mechanical properties. One is able to develop a differential equation system with varying coefficients that regulate the motion of BFGP double-curved shallow microshells by using Hamilton principle, Kirchhoff–Love hypothesis, and modified couple stress theory. The numerical findings are reported for thin microshells that are spherical, cylindrical, and hyperbolic paraboloidal, with a variety of planforms, including rectangles and circles. The validity and effectiveness of the established model are shown by comparing the numerical results given by the proposed formulations with previously published findings in many specific circumstances. In addition, influences of length scale parameters, power-law indexes, thickness-to-side ratio, and radius ratio on natural oscillation responses of BFGP microshells are investigated in detail.

Graphical abstract

Keywords

Kirchhoff–Love’s shell theory / isogeometric analysis / bi-directional functionally graded / free vibration / variable length-scale parameter

1 Introduction

Shells are a specific kind of structural element that can be identified by their geometry, which consists of a three-dimensional solid with a very thin layer when measured in comparison to the other dimensions, and by the stress resultants that are calculated in the middle plane of the structure, which show components that are both coplanar and normal to the surface. Shells are also unique in that their internal stresses have parts that are both parallel to the surface and normal to it. From a plate, a shell can be created in one of two fundamental ways: either by first constructing the center surface of the structure as a single or doubly-curved surface, or by applying loads that are coplanar to the plane of a plate and generating large stresses. Both of these methods involve the generation of large stresses [1,2]. In a technical sense, the majority of the surfaces of mechanical structures have some kind of shell texture, ranging in scale from macroscopic to micro- and nano-scale. In particular, as a result of significant advances in material and computer sciences, micro- and nano-scale components are increasingly being incorporated into a wide variety of electronic devices, including semiconductor components, thermal batteries, medical sensors, and medical robotic systems, where numerous instances of microshell designs may be found. As a result, the calculation and design of this sort of structure in engineering practice rely heavily on the findings of research into the mechanical behavior of shell structures.

In recent years, a computational approach known as isogeometric analysis (IGA) has emerged. This method makes it possible to include finite element investigation into typical designing implements that are based on non-uniform rational B-splines (NURBS). As a consequence of this, it is now feasible to make use of data set during the progression of developing models, verifying those models, and improving those models [3–5]. Much work has gone into the Finite Element Method (FEM) framework, and B-splines have been widely used as benchmark functions. Hughes and his collaborators [6,7] have developed a framework for studying isometries, and they have unified and presented their work under the label IGA. Since then, isogeometric approaches have found widespread use in computer approaches, especially for simulating mechanical responses of advanced structures. Cottrell et al. [8] carried out an IGA of structural vibrations. By applying IGA to structural excitations, the sequel reawakens Argyris’ geometrical creativity. After its introduction, IGA has been applied to thin-walled structures. This method employs three-dimensional solid modeling, beam models without revolution, and plate designs. Hughes et al. [9] discussed isogeometric quadrature rules based on NURBS. Dörfel et al. [10] demonstrated the potential use of the T-spline. This permits T-junctions, which in FEM are hanging nodes. Buffa et al. [11] discretize Maxwell equations in two dimensions. Inspired by the unique paradigm of IGA of Hughes et al. [6], they proposed a technique based on bi-variable B-spline expansions. They constructed parametric B-spline spaces with varied regularity between elements. They exploited these spaces, together with physical push-forward, to handle the Maxwell’s eigenvalue problem. Bazilevs and Akkerman [12] demonstrated that the residual-based variational multiscale formulations universally conserve rotational motion. Zhuang and team [13–16] presented a comprehensive energy approach for the static bending, oscillation, and buckle investigation of classical plate structures. For two-dimensional acoustic issues, Shaaban et al. [17] suggested an incidental boundary element approach based on isogeometric study. Cohen et al. [18] indicated that an analogous conception of modeling performance exists in isogeometric evaluation. Valizadeh et al. [19] used a NURBS-based isogeometrical finite element modeling to examine the mechanical response of Functionally Graded (FG) plates. To study the random mechanical performance of FG material plates with inherent material unpredictability, Dsouza et al. [20] developed a non-intrusive approach that was coupled with an NURBS-based isogeometrical FEM. Within the framework of an IGA, Hu et al. [21] created a formulation that overcomes the problem of deteriorating Reissner–Mindlin shell formulation. IGA is used in many different fields, and may be investigated using the supplemental sources [22–29].

The field of computational mechanics has made significant strides in the open application of IGA approaches to evaluation of mechanical reactions of structures made of complex and advanced materials. The development of computational mechanics has made these advances possible. Since the suggestion that this strategy be used, there has been significative development in the total number of researchers engaged in research and development, where micro/nanostructures are the subject of extensive investigation and are put to use in a variety of different aspects of day-to-day life. The following are some remarkable accomplishments. Ansari and Norouzzadeh [30] examined buckling behaviors of different shape nanoplates made of FGM based on an isogeometric model. Fan et al. [31] predicted shear buckling responses of FG nanoplates subjected to surface reaction. Norouzzadeh and Ansari [32] carried out an isogeometric oscillation investigation of smoothly graded nanoplates taking nonlocal and surface effects into calculations. Fan et al. [33] examined oscillations of microstructures with a center square abatement. The mechanical characteristics of microplates were determined using a simple function which integrates material gradient and porosity dependence. C¹ continuity was satisfied using NURBS-based isogeometrics. Luat et al. [34] analyzed the bending, free oscillation, and buckling of an unique bi-functionally graded sandwich nanobeams. Nguyen [35] and Dung et al. [36] conducted nonlinear bending investigations of microplates placed on defective elastic media, and they used FEM based on the improved couple stress hypothesis to establish the nonlinear finite element equations. Qiu et al. [37] explored porosity-dependent nonlinear post-buckling on microplates (modified couple stress theory), showing that third-order shear deformation plate hypothesis generated stress-coupling nonlinear differential equations (third-order shear deformation plate theory). The first IGA of BDFG rectangular plates in a fluid environment was presented by Pham et al. [28]. IGA, which is based on NURBS technology, was used by Rahmouni et al. [38] in order to evaluate the stress dispensation and first ply failure strength of hybrid multilayer composites that had been exposed to uniaxial loads. Studies on the mechanical responses of structures with various numerical methods can be found in Refs. [39–50].

On the basis of the above analysis, the IGA technique is one of the most dependable methods that can be used when estimating the mechanical responses of a structure. Over a long period of time, it has become clear that this method works, and scholars are still interested in improving, expanding, and perfecting it. Computer science made progress at the same time that new materials technology was developing. Commercial organisations are devoting more and more of their investigations and development endeavors to making structures on the microscale and nanoscale levels. In present work, by using the IGA, based on the NURBS basis function, the free oscillation of bi-directional functionally graded porous (BFGP) doubly-curved microshells resting on two-parameter substrates with arbitrary boundary circumstances is explored. The material’s thickness and axial direction undergo modifications along the x-axis. More precisely, the length-scale parameter of microshells is seen as a function of the spatial coordinates, and its value fluctuates according to the material gradient parameters. The motion of BFGP doubly-curved shallow microshells may be represented by a set of variable coefficient differential equations derived from Hamilton’s principle, the improved couple stress, and classical shell theories. The findings of this investigation on the mechanical response of this doubly-curved microshell structure have extremely significant implications for the design calculations of medical microelectromechanical devices.

The remainder of this paper is organized in the following manner. The conceptual formulations are presented in Section 2, where the description of BFGP doubly-curved shallow microshells, modified couple stress hypothesis, and motion equations in general are fully described. Section 3 introduces the isogeometric formulation for free oscillation analysis of shell structures. The numerical results are presented in Section 4 along with the verification and a comprehensive parameter study. Section 5 outlines some novel findings and the contributions of present investigation.

2 Theoretical formulations

2.1 Description of bi-directional functionally graded porous doubly-curved shallow microshells

A doubly-curved shallow microshell, which is fabricated from BFGP mixed with ceramic and metal phases, is shown in Fig.1. The principal radius of the microshell with respect to the x-axis is R_1 and the principal radius with respect to the y-axis is R_2. The geometrical parameters of the microshell are shown in Fig.1, where a and b are the curve’s lengths, and h is the thickness. The structure is resting on Pasternak elastic medium defined by two coefficients (k_{\mathrm{w}},k_{\mathrm{s}}), k_{\mathrm{w}} denotes the Winkler stiffness and k_{\mathrm{s}} represents the Pasternak stiffness. The porosity laws of microshell materials use the following mathematical models [51].

Type E: even porosity distribution

\begin{array}{}\text{(1a)}& \vartheta =\frac{\xi }{2},\end{array}

Type U: uneven porosity distribution

\begin{array}{}\text{(1b)}& \vartheta =\frac{\xi }{2}\left(1-\frac{2\left|\mathit{\text{z}}\right|}{h}\right),\end{array}

which \vartheta is the parameter of the porous dismemberment and \xi is the porous coefficient.

The rules of composition of the mechanical characteristics of materials with porosities include: Young’s modulus E\left(\mathrm{G}\mathrm{P}\mathrm{a}\right), mass density \rho \left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^3\right), length-scale parameter l\left(\text{μ}\mathrm{m}\right), and Poisson ratio υ represented by the following equation [52].

For a rectangular microshell:

\begin{array}{}\text{(2a)}& E\left(x,\mathit{\text{z}}\right)=E_{\mathrm{m}}+\left(E_{\mathrm{c}}-E_{\mathrm{m}}\right){\left(\frac{1}{2}+\frac{\mathit{\text{z}}}{h}\right)}^{{\alpha }_{\mathit{\text{z}}}}{\left(\frac{x}{a}\right)}^{{\alpha }_x}-\vartheta \left(E_{\mathrm{c}}+E_{\mathrm{m}}\right),\end{array}

\begin{array}{}\text{(2b)}& \rho \left(x,\mathit{\text{z}}\right)={\rho }_{\mathrm{m}}+\left({\rho }_{\mathrm{c}}-{\rho }_{\mathrm{m}}\right){\left(\frac{1}{2}+\frac{\mathit{\text{z}}}{h}\right)}^{{\alpha }_{\mathit{\text{z}}}}{\left(\frac{x}{a}\right)}^{{\alpha }_x}-\vartheta \left({\rho }_{\mathrm{c}}+{\rho }_{\mathrm{m}}\right),\end{array}

\begin{array}{}\text{(2c)}& l\left(x,\mathit{\text{z}}\right)=l_{\mathrm{m}}+\left(l_{\mathrm{c}}-l_{\mathrm{m}}\right){\left(\frac{1}{2}+\frac{\mathit{\text{z}}}{h}\right)}^{{\alpha }_{\mathit{\text{z}}}}{\left(\frac{x}{a}\right)}^{{\alpha }_x}-\vartheta \left(l_{\mathrm{c}}+l_{\mathrm{m}}\right),\end{array}

\begin{array}{}\text{(2d)}& v\left(x,\mathit{\text{z}}\right)=v_{\mathrm{m}}+\left(v_{\mathrm{c}}-v_{\mathrm{m}}\right){\left(\frac{1}{2}+\frac{\mathit{\text{z}}}{h}\right)}^{{\alpha }_{\mathit{\text{z}}}}{\left(\frac{x}{a}\right)}^{{\alpha }_x}-\vartheta \left(v_{\mathrm{c}}+v_{\mathrm{m}}\right).\end{array}

For a circular microshell with the radius R in the x-direction.

\begin{array}{}\text{(3a)}& E\left(x,\mathit{\text{z}}\right)=E_{\mathrm{m}}+\left(E_{\mathrm{c}}-E_{\mathrm{m}}\right){\left(\frac{1}{2}+\frac{\mathit{\text{z}}}{h}\right)}^{{\alpha }_{\mathit{\text{z}}}}{\left(\frac{\left|x\right|}{R}\right)}^{{\alpha }_x}-\vartheta \left(E_{\mathrm{c}}+E_{\mathrm{m}}\right),\end{array}

\begin{array}{}\text{(3b)}& \rho \left(x,\mathit{\text{z}}\right)={\rho }_{\mathrm{m}}+\left({\rho }_{\mathrm{c}}-{\rho }_{\mathrm{m}}\right){\left(\frac{1}{2}+\frac{\mathit{\text{z}}}{h}\right)}^{{\alpha }_{\mathit{\text{z}}}}{\left(\frac{\left|x\right|}{R}\right)}^{{\alpha }_x}-\vartheta \left({\rho }_{\mathrm{c}}+{\rho }_{\mathrm{m}}\right),\end{array}

\begin{array}{}\text{(3c)}& l\left(x,\mathit{\text{z}}\right)=l_{\mathrm{m}}+\left(l_{\mathrm{c}}-l_{\mathrm{m}}\right){\left(\frac{1}{2}+\frac{\mathit{\text{z}}}{h}\right)}^{{\alpha }_{\mathit{\text{z}}}}{\left(\frac{\left|x\right|}{R}\right)}^{{\alpha }_x}-\vartheta \left(l_{\mathrm{c}}+l_{\mathrm{m}}\right),\end{array}

\begin{array}{}\text{(3d)}& v\left(x,\mathit{\text{z}}\right)=v_{\mathrm{m}}+\left(v_{\mathrm{c}}-v_{\mathrm{m}}\right){\left(\frac{1}{2}+\frac{\mathit{\text{z}}}{h}\right)}^{{\alpha }_{\mathit{\text{z}}}}{\left(\frac{\left|x\right|}{R}\right)}^{{\alpha }_x}-\vartheta \left(v_{\mathrm{c}}+v_{\mathrm{m}}\right),\end{array}

where subscripts ‘m’ and ‘c’ denote metallic and ceramic components, respectively; {\alpha }_{\mathit{\text{z}}} and {\alpha }_x are the power-law indexes of the material through the thickness and x-direction. In the case of {\alpha }_{\mathit{\text{z}}}={\alpha }_x=0, the BFGP doubly-curved shallow microshell becomes a homogeneous isotropic metal porous doubly-curved microshell; and in the case of {\alpha }_{\mathit{\text{z}}}=0,{\alpha }_x\ne 0, the BFGP doubly-curved shallow microshell becomes uniaxial FGP doubly-curved shallow microshell; when {\alpha }_x=0,{\alpha }_{\mathit{\text{z}}}\ne 0, the BFGP doubly-curved shallow microshell becomes a usual FGP doubly-curved shallow microshell.

By setting l_{\mathrm{c}}=l_{\mathrm{m}}=l=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}, the consequently modified couple stress hypothesis is reduced to the classic version. Compared to other available works investigating BFGP microstructures, this is the originality of the present paper. In addition, the effective length-scale parameter of BFGP microstructures is dependent on power-law indexes. Using theoretical methodologies is challenging to determine the values of these coefficients. Instead, the value of this parameter must be identified through experimental investigations or simulations of atomistic mechanics.

Based on classical shell deformation theory, the displacement field at any point that is located at a distance z from the central surface can be obtained [53]. This is done by representing respectively the central surface displacement of a doubly-curved microshell via u_0, v_0, and w_0 in the x-, y-, and z-direction.

(4)\{\begin{array}{rl}& u_x=\left(1+\frac{\mathit{\text{z}}}{R_1}\right)u_0\left(x,y\right)-\mathit{\text{z}}{w}_{0,x}\left(x,y\right),\\ & u_y=\left(1+\frac{\mathit{\text{z}}}{R_2}\right)v_0\left(x,y\right)-\mathit{\text{z}}{w}_{0,y}\left(x,y\right),\\ & u_{\mathit{\text{z}}}=w_0\left(x,y\right).\end{array}

By modifying the microshells’ curvature components in different ways, it is possible to create four distinct kinds of shallow microshell structures, which are described below.

1) For R_1\to \mathrm{\infty },R_2\to \mathrm{\infty }: the microshell becomes a flat (FL) microplate.

2) For R_1\to \mathrm{\infty }: the microshell becomes a cylindrical (CY) microshell.

3) For R_1=R_2: the microshell becomes a spherical (SP) microshell.

4) For R_1={-R}_2: the microshell becomes a hyperbolic paraboloidal (HP) microshell.

Below are illustrations of shallow microshells having a rectangular planform (Fig.2).

2.2 Modified couple stress theory

According to modified couple stress hypothesis proposed by Yang et al. [54], curvature and strain components are presented as

\begin{array}{}\text{(5a)}& {\epsilon }_{ii}=u_{i,i};i=x,y,\mathit{\text{z}};\end{array}

\begin{array}{}\text{(5b)}& {\epsilon }_{ij}=u_{i,j}+u_{j,i};j=x,y,\mathit{\text{z}};j\ne i,\end{array}

(6){\chi }_{ij}=\frac{1}{2}\left({\mathrm{\Phi }}_{i,j}+{\mathrm{\Phi }}_{j,i}\right);i=x,y,\mathit{\text{z}};j=x,y,\mathit{\text{z}};

in which {\epsilon }_{ij} are the symmetric curvature tensor, {\chi }_{ij} are parts of the local strain tensor. x, y, and z components of the rotation vector {\mathrm{\Phi }}_i are described as follows:

(7){\mathrm{\Phi }}_x=\frac{1}{2}\left(u_{\mathit{\text{z}},y}-u_{y,\mathit{\text{z}}}\right);{\mathrm{\Phi }}_y=\frac{1}{2}\left(u_{x,\mathit{\text{z}}}-u_{\mathit{\text{z}},x}\right);{\mathrm{\Phi }}_{\mathit{\text{z}}}=\frac{1}{2}\left(u_{y,x}-u_{x,y}\right).

Given this, we can derive the constitutive equation.

(8){\sigma }_{ij}=\lambda {\epsilon }_{kk}{\delta }_{ij}+2\mu {\epsilon }_{ij};{\theta }_{ij}=2\mu l^2{\chi }_{ij},

where {\sigma }_{ij} are parts of the local stress tensor, {\theta }_{ij} are parts of the deviatoric portion of the symmetric couple stress tensor; \lambda and \mu are Lamé’s constants, and using Eqs. (4)–(7), the nonzero strain parts, curvature and rotation, may be expressed as approximating that (1+\mathit{\text{z}}/R_1\approx 1;1+\mathit{\text{z}}/R_2\approx 1), the nonzero strain components can be written as:

(9){\epsilon }_{xx}=u_{x,x}+\frac{u_{\mathit{\text{z}}}}{R_1}=u_{0,x}+\frac{w_0}{R_1}-\mathit{\text{z}}\left(w_{0,xx}-\frac{u_{0,x}}{R_1}\right),

(10){\epsilon }_{yy}=u_{y,y}+\frac{u_{\mathit{\text{z}}}}{R_2}=v_{0,y}+\frac{w_0}{R_2}-\mathit{\text{z}}\left(w_{0,yy}-\frac{v_{0,y}}{R_2}\right),

(11){\epsilon }_{xy}=u_{x,y}+u_{y,x}=u_{0,y}+v_{0,x}-\mathit{\text{z}}\left(2w_{0,xy}-\frac{u_{0,y}}{R_1}-\frac{v_{0,x}}{R_2}\right).

The following steps are required to derive the components of the curvature tensor:

\begin{array}{}\text{(12a)}& {\chi }_{xx}=w_{0,xy}-\frac{v_{0,x}}{2R_2};{\chi }_{yy}=-w_{0,xy}+\frac{u_{0,y}}{2R_1};{\chi }_{\mathit{\text{zz}}}=\frac{u_{0,y}}{2R_1}-\frac{v_{0,x}}{2R_2};\end{array}

\begin{array}{}\text{(12b)}& {\chi }_{xy}=\frac{1}{2}\left(\frac{u_{0,x}}{2R_1}-w_{0,xx}\right)-\frac{1}{2}\left(\frac{v_{0,y}}{2R_2}-w_{0,yy}\right);\end{array}

\begin{array}{}\text{(12c)}& {\chi }_{x\mathit{\text{z}}}=-\frac{1}{4}\left(u_{0,xy}-v_{0,xx}\right)-\frac{1}{4}\mathit{\text{z}}\left(\frac{u_{0,xy}}{R_1}-\frac{v_{0,xx}}{R_2}\right);\end{array}

\begin{array}{}\text{(12d)}& {\chi }_{y\mathit{\text{z}}}=\frac{1}{4}\left(u_{0,yy}-v_{0,xy}\right)-\frac{1}{4}\mathit{\text{z}}\left(\frac{u_{0,yy}}{R_1}-\frac{v_{0,xy}}{R_2}\right).\end{array}

After that, the constitutive relations for linear elastic are expressed as follows:

\begin{array}{}\text{(13a)}& {\sigma }_{xx}=\frac{E(x,\mathit{\text{z}})}{1-v{(x,\mathit{\text{z}})}^2}\left({\epsilon }_{xx}+v(x,\mathit{\text{z}}){\epsilon }_{yy}\right);\end{array}

\begin{array}{}\text{(13b)}& {\sigma }_{yy}=\frac{E(x,\mathit{\text{z}})}{1-v{(x,\mathit{\text{z}})}^2}\left({\epsilon }_{yy}+v(x,\mathit{\text{z}}){\epsilon }_{xx}\right);\end{array}

\begin{array}{}\text{(13c)}& {\sigma }_{xy}=\frac{E(x,\mathit{\text{z}}){\epsilon }_{xy}}{2(1+v(x,\mathit{\text{z}}))}.\end{array}

(14)\begin{array}{rl}& \left\{{\theta }_{xx},{\theta }_{yy},{\theta }_{\mathit{\text{zz}}},{\theta }_{xy},{\theta }_{x\mathit{\text{z}}},{\theta }_{y\mathit{\text{z}}}\right\}\\ =& \frac{E(x,\mathit{\text{z}})l^2(x,\mathit{\text{z}})}{1+v(x,\mathit{\text{z}})}\left\{{\chi }_{xx},{\chi }_{yy},{\chi }_{\mathit{\text{zz}}},{\chi }_{xy},{\chi }_{x\mathit{\text{z}}},{\chi }_{y\mathit{\text{z}}}\right\}.\end{array}

2.3 Motion equations general

The Kirchhoff–Love shell hypothesis equations of motion, which were determined using Hamilton’s principle, are then provided for the BFGP doubly-curved shallow microshell resting on a two-parameter elastic medium.

(15){\int }_{t_1}^{t_2}\left(\delta U+\delta U_{\mathrm{f}}+\delta V-\delta K\right)\mathrm{d}t=0,

where \delta U,\delta U_{\mathrm{f}},\delta V, and \delta K refer to the variation of the strain energy, the change of the Pasternak elastic foundation strain energy, the variation of the potential energy, and the change of the kinetic energy, respectively.

It is possible to present the variation in strain power of the doubly-curved microshell using the formula.

(16)\begin{array}{rl}\delta U=& {\int }_{S_{\mathrm{e}}}{\int }_{-h/2}^{h/2}({\sigma }_{xx}\delta {\epsilon }_{xx}+{\sigma }_{yy}\delta {\epsilon }_{yy}+{\sigma }_{xy}\delta {\epsilon }_{xy}+{\theta }_{xx}\delta {\chi }_{xx}+{\theta }_{yy}\delta {\chi }_{yy}\\ & +{\theta }_{\mathit{\text{zz}}}\delta {\chi }_{\mathit{\text{zz}}}+2{\theta }_{xy}{\delta \chi }_{xy}+2{\theta }_{x\mathit{\text{z}}}\delta {\chi }_{x\mathit{\text{z}}}+2{\theta }_{y\mathit{\text{z}}}\delta {\chi }_{y\mathit{\text{z}}})\mathrm{d}\mathit{\text{z}}\mathrm{d}x\mathrm{d}y\\ =& {\int }_{S_{\mathrm{e}}}({\mathit{\epsilon }}^{\mathrm{T}}{\mathit{D}}_{\mathrm{b}}\delta \mathit{\epsilon }+{\mathit{\psi }}^{\mathrm{T}}{\mathit{D}}_{\mathrm{m}\mathrm{b}}\delta \mathit{\psi }+{\mathit{\chi }}^{\mathrm{T}}{\mathit{D}}_{\mathrm{m}\mathrm{s}}\delta \mathit{\chi })\mathrm{d}x\mathrm{d}y,\end{array}

where S_{\mathrm{e}} is the area of the shell element and the components \mathit{\epsilon },\mathit{\psi },\mathit{\chi },{\mathit{D}}_{\mathrm{b}},{\mathit{D}}_{\mathrm{m}\mathrm{b}}, and {\mathit{D}}_{\mathrm{m}\mathrm{s}} are determined from the known elements as follows:

(17)\mathit{\epsilon }=\left\{\begin{array}{c}{\mathit{\epsilon }}^0\\ {\mathit{\epsilon }}^1\end{array}\right\};{\mathit{\epsilon }}^0=\left\{\begin{array}{c}{u}_{0,x}+{\displaystyle \frac{w_0}{R_1}}\\ v_{0,y}+{\displaystyle \frac{w_0}{R_2}}\\ u_{0,y}+v_{0,x}\end{array}\right\};{\mathit{\epsilon }}^1=\left\{\begin{array}{c}{w}_{0,xx}+{\displaystyle \frac{u_{0,x}}{R_1}}\\ w_{0,yy}+{\displaystyle \frac{v_{0,y}}{R_2}}\\ 2w_{0,xy}-{\displaystyle \frac{u_{0,y}}{R_1}}-{\displaystyle \frac{v_{0,x}}{R_2}}\end{array}\right\};

(18)\mathit{\psi }=\left\{\begin{array}{c}{w}_{0,xy}-{\displaystyle \frac{v_{0,x}}{2R_2}}\\ -w_{0,xy}+{\displaystyle \frac{u_{0,y}}{2R_1}}\\ {\displaystyle \frac{u_{0,y}}{2R_1}}-{\displaystyle \frac{v_{0,x}}{2R_2}}\\ {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{u_{0,x}}{2R_1}}-w_{0,xx}\right)-{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{v_{0,y}}{2R_2}}-w_{0,yy}\right)\end{array}\right\};

(19){\mathit{D}}_{\mathrm{b}}=\left[\begin{array}{cc}\mathit{A}& \mathit{B}\\ \mathit{B}& \mathit{F}\end{array}\right];{\mathit{D}}_{\mathrm{m}\mathrm{b}}={\mathit{A}}_{\mathrm{m}\mathrm{b}};{\mathit{D}}_{\mathrm{m}\mathrm{s}}=\left[\begin{array}{cc}{\mathit{A}}_{\mathrm{m}\mathrm{s}}& {\mathit{B}}_{\mathrm{m}\mathrm{s}}\\ {\mathit{B}}_{\mathrm{m}\mathrm{s}}& {\mathit{F}}_{\mathrm{m}\mathrm{s}}\end{array}\right];

where \mathit{A},\mathit{B},\mathit{F},{\mathit{A}}_{\mathrm{m}\mathrm{b}},{\mathit{A}}_{\mathrm{m}\mathrm{s}},{\mathit{B}}_{\mathrm{m}\mathrm{s}}, and {\mathit{F}}_{\mathrm{m}\mathrm{s}} are the material’s stiffness; this is a function that is dependent on x and y, and it is determined using the following formula:

(20)\left(\mathit{A},\mathit{B},\mathit{F}\right)={\int }_{-h/2}^{h/2}{Q}_{\mathrm{b}}\left(1,\overline{\mathit{\text{z}}},{\overline{\mathit{\text{z}}}}^2\right)\text{d}\mathit{\text{z}};

(21){\mathit{A}}_{\mathrm{m}\mathrm{b}}={\int }_{-h/2}^{h/2}{Q}_{\mathrm{m}\mathrm{b}}\text{d}\mathit{\text{z}};\left({\mathit{A}}_{\mathrm{m}\mathrm{s}},{\mathit{B}}_{\mathrm{m}\mathrm{s}},{\mathit{F}}_{\mathrm{m}\mathrm{s}}\right)={\int }_{-h/2}^{h/2}{Q}_{\mathrm{m}\mathrm{s}}\left(1,\overline{\mathit{\text{z}}},{\overline{\mathit{\text{z}}}}^2\right)\text{d}\mathit{\text{z}};

where Q_{\mathrm{b}},Q_{\mathrm{m}\mathrm{b}}, and Q_{\mathrm{m}\mathrm{s}} are the material’s tensors and \overline{\mathit{\text{z}}}=\mathit{\text{z}}-t_0;t_0 is the distance between the microshell’s neutral surface and its mean surface. These factors are computed as follows:

(22)\begin{array}{rl}& Q_{\mathrm{b}}=\frac{E\left(x,\mathit{\text{z}}\right)}{1-v{\left(x,\mathit{\text{z}}\right)}^2}\left[\begin{array}{ccc}1& v\left(x,\mathit{\text{z}}\right)& 0\\ v\left(x,\mathit{\text{z}}\right)& 1& 0\\ 0& 0& {\displaystyle \frac{1-v\left(x,\mathit{\text{z}}\right)}{2}}\end{array}\right];\\ & Q_{\mathrm{m}\mathrm{b}}={\displaystyle \frac{E\left(x,\mathit{\text{z}}\right)l^2\left(x,\mathit{\text{z}}\right)}{1+v\left(x,\mathit{\text{z}}\right)}}\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 2\end{array}\right];\\ & Q_{\mathrm{m}\mathrm{s}}=\frac{E\left(x,\mathit{\text{z}}\right)l^2\left(x,\mathit{\text{z}}\right)}{1+v\left(x,\mathit{\text{z}}\right)}\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right];t_0={\displaystyle \frac{{\displaystyle {\int }_{-h/2}^{h/2}E\left(x,\mathit{\text{z}}\right)\mathit{\text{z}}\mathrm{d}\mathit{\text{z}}}}{{\displaystyle {\int }_{-h/2}^{h/2}E\left(x,\mathit{\text{z}}\right)\mathrm{d}\mathit{\text{z}}}}}.\end{array}

The following factors determine the extent to which the Pasternak elastic foundation strain energy varies:

(23)\delta U_{\mathrm{f}}=\underset{S_{\mathrm{e}}}{\overset{}{\int }}\left(k_{\mathrm{w}}{w}_0\delta w_0-k_{\mathrm{s}}\left(w_{0,xx}+w_{0,yy}\right)\delta w_0\right)\mathrm{d}x\mathrm{d}y.

The calculation for the variation of potential energy \delta V goes as follows:

(24)\delta V=\underset{S_{\mathrm{e}}}{\overset{}{\int }}\left(q_0-N_x^0{w}_{0,xx}-N_y^0{w}_{0,yy}\right)\delta w_0\mathrm{d}x\mathrm{d}y.

The mechanical compressive forces N_x^0 and N_y^0 that are operating in the direction tangentially to the radius of curvature of the microshell and, N_x^0=N_y^0=N^0. If nothing changes, force N^0 is uniformly applied to the edges of the shell structure.

The difference of the kinetic energy \delta K of the doubly-curved microshell can be determined using the formula:

(25)\begin{array}{rl}\delta K=& {\int }_{S_{\mathrm{e}}}{\int }_{-h/2}^{h/2}\rho \left(x,\mathit{\text{z}}\right)(\left(\left(1+\frac{\overline{\mathit{\text{z}}}}{R_1}\right){\dot{u}}_0-\overline{\mathit{\text{z}}}{\dot{w}}_{0,x}\right)\\ & \cdot \delta \left(\left(1+\frac{\overline{\mathit{\text{z}}}}{R_1}\right){\dot{u}}_0-\overline{\mathit{\text{z}}}{\dot{w}}_{0,x}\right)+\left(\left(1+\frac{\overline{\mathit{\text{z}}}}{R_2}\right){\dot{v}}_0-\overline{\mathit{\text{z}}}{\dot{w}}_{0,y}\right)\\ & \cdot \delta \left(\left(1+\frac{\overline{\mathit{\text{z}}}}{R_2}\right){\dot{v}}_0-\overline{\mathit{\text{z}}}{\dot{w}}_{0,y}\right)+{\dot{w}}_0\delta {\dot{w}}_0)\mathrm{d}\mathit{\text{z}}\mathrm{d}x\mathrm{d}y\\ =& {\int }_{S_{\mathrm{e}}}\left({\dot{\mathit{u}}}^{\mathrm{T}}{\mathit{G}}_{\mathrm{m}}\delta \dot{\mathit{u}}\right)\mathrm{d}x\mathrm{d}y,\end{array}

where

(26)\mathit{u}=\left\{\begin{array}{ccccc}{u}_0& v_0& w_0& w_{0,x}& w_{0,y}\end{array}\right\},

(27)\begin{array}{rl}& {\mathit{G}}_{\mathrm{m}}={\int }_{-h/2}^{h/2}\rho \left(x,\mathit{\text{z}}\right)L_{\mathit{\text{z}}}^{\mathrm{T}}{L}_{\mathit{\text{z}}}\mathrm{d}\mathit{\text{z}};\\ & L_{\mathit{\text{z}}}=\left[\begin{array}{ccccc}1+{\displaystyle \frac{\overline{\mathit{\text{z}}}}{R_1}}& 0& 0& -\overline{\mathit{\text{z}}}& 0\\ 0& 1+{\displaystyle \frac{\overline{\mathit{\text{z}}}}{R_2}}& 0& 0& -\overline{\mathit{\text{z}}}\\ 0& 0& 1& 0& 0\end{array}\right].\end{array}

The free oscillation of the BFGP doubly-curved shallow microshell may be characterized by the equation below, which can be obtained by entering Eqs. (16) and (23)−(25) into Eq. (15).

(28)\begin{array}{rl}& \underset{S_{\text{e}}}{\int }\left({\mathit{\epsilon }}^{\text{T}}{\mathit{D}}_{\text{b}}\delta \mathit{\epsilon }+{\mathit{\Psi }}^{\text{T}}{\mathit{D}}_{\text{mb}}\delta \mathit{\Psi }+{\mathit{\text{χ}}}^{\text{T}}{\mathit{D}}_{\text{ms}}\delta \mathit{\text{χ}}\right)\text{d}x\text{d}y\\ & +\underset{S_{\text{e}}}{\int }\left(k_{\text{w}}{w}_0\delta w_0+\left(k_{\text{s}}+N^0\right)\left(w_{0,x}\delta w_{0,x}+w_{0,y}\delta w_{0,y}\right)\right)\text{d}x\text{d}y\\ =& \underset{S_{\text{e}}}{\int }\left({\ddot{\mathit{u}}}^{\text{T}}{\mathit{G}}_{\mathrm{m}}\delta \mathit{u}\right)\text{d}x\text{d}y.\end{array}

3 Isogeometric formulation for free oscillation

3.1 Non-uniform rational B-splines basis functions

In a single-dimensional space, the knot vector \mathit{k}\left(\zeta \right) is a set of non-decreasing numbers between zero and one. This set is represented by the equation \mathit{k}\left(\zeta \right)= \{{\zeta }_1=0,\dots ,{\zeta }_i,\dots ,{\zeta }_{n+p+1}=1\}, where i is the knot index, {\zeta }_i is the ith knot, n is the number of basis functions, and p is the degree of the polynomial. The notation N_{i,p}\left(\zeta \right) represents the recurrence contention of the ith B-spline basis function of degree p [54,55].

(29)

and

(30)N_{i,p}\left(\zeta \right)=\frac{\zeta -{\zeta }_i}{{\zeta }_{i+p}-{\zeta }_i}{N}_{i,p-1}\left(\zeta \right)+\frac{{\zeta }_{i+p+1}-\zeta }{{\zeta }_{i+p+1}-{\zeta }_{i+1}}{N}_{i+1,p-1}\left(\zeta \right),\mathrm{f}\mathrm{o}\mathrm{r}p⩾1.

Multiplying two univariate B-spline basis functions is one of the steps involved in the production of two-dimensional NURBS basis functions [54]:

(31)C_{i,j}^{p,q}\left(\zeta ,n\right)=\frac{N_{i,p}\left(\zeta \right)N_{j,q}\left(\eta \right)w_{i,j}}{{\displaystyle {\sum }_{i=1}^n{\displaystyle {\sum }_{j=1}^m{N}_{i,p}\left(\zeta \right)N_{j,q}\left(\eta \right)w_{i,j}}}},

where w_{i,j} represents the weight; N_{i,p}\left(\zeta \right) and N_{j,q}\left(\eta \right) indicate the B-spline basis functions of order p in the \zeta and \eta directions, respectively; and N_{j,q}\left(\eta \right) adheres to the recurrence formula provided in Eq. (30) with the knot vector \mathit{k}\left(\eta \right), which has the identical contention as \mathit{k}\left(\zeta \right).

3.2 Non-uniform rational B-splines-based finite element formulation

The following equations can be used to figure out how far apart microshells with two curves are on the middle surface:

(32){\mathit{d}}^{\mathrm{h}}={\sum }_{I=1}^{Ne}{\mathit{C}}_I{\mathit{d}}_I.

with

(33){\mathit{d}}_I=\left[u_{0\mathrm{e}},v_{0\mathrm{e}},w_{0\mathrm{e}}\right],

in which {\mathit{C}}_I and {\mathit{d}}_I represent the shape functions and the unknown displacement vector at controlling point I, respectively, where Ne=(p+1)(q+1) is the number of control points per physical element.

By substituting Eq. (32) into Eq. (4), the displacement components u_0,v_0, and w_0 can be computed as

u_0={\sum }_{I=1}^{Ne}\left[\begin{array}{ccc}{\mathit{C}}_I& 0& 0\end{array}\right]{\mathit{d}}_I={\sum }_{I=1}^{Ne}{\mathit{H}}_I^{\mathrm{u}}{\mathit{d}}_I;

(34)v_0={\sum }_{I=1}^{Ne}\left[\begin{array}{ccc}0& {\mathit{C}}_I& 0\end{array}\right]{\mathit{d}}_I={\sum }_{I=1}^{Ne}{\mathit{H}}_I^{\mathrm{v}}{\mathit{d}}_I;

w_0={\sum }_{I=1}^{Ne}\left[\begin{array}{ccc}0& 0& {\mathit{C}}_I\end{array}\right]{\mathit{d}}_I={\sum }_{I=1}^{Ne}{\mathit{H}}_I^{\mathrm{w}}{\mathit{d}}_I.

Supplanting Eq. (34) into Eq. (28) and employing following equations, one obtains the equilibrium equation for the free oscillation of the microshell as:

(35)\mathit{M}\ddot{\mathit{d}}+\left({\mathit{K}}_{\text{s}}+{\mathit{K}}_{\text{f}}\right)\mathit{d}=0,

where \mathit{M}={\displaystyle \sum _{I=1}^{Ne}{M}_{\mathrm{s}}^I,{\mathit{K}}_{\mathrm{s}}={\displaystyle \sum _{I=1}^{Ne}{K}_{\mathrm{s}}^I,{\mathit{K}}_{\mathrm{f}}={\displaystyle \sum _{I=1}^{Ne}{K}_{\mathrm{f}}^I}}} and d are the mass matrix, the stiffness matrix of the microshell, and the stiffness matrix of the outer shell components (elastic foundation and compressive force tangential to the radius of the shell) and the shell’s vector of degrees of freedom, respectively. The parts of the matrix are determined by the following equations.

The mass matrix is calculated as:

(36)M_{\mathrm{s}}^I={\int }_{S_{\text{e}}}\left({\mathit{N}}_I^{\mathrm{T}}{\mathit{G}}_{\mathrm{m}}{\mathit{N}}_I\right)\mathrm{d}x\mathrm{d}y.

The stiffness matrix is calculates as

(37)\begin{array}{rl}{K}_{\mathrm{s}}^I=& {\int }_{S_{\mathrm{e}}}({\left\{\begin{array}{c}{\mathit{B}}_I^1\\ {\mathit{B}}_I^2\end{array}\right\}}^{\mathrm{T}}{\mathit{D}}_{\mathrm{b}}\left\{\begin{array}{c}{\mathit{B}}_I^1\\ {\mathit{B}}_I^2\end{array}\right\}+{\mathit{B}}_{\mathrm{m}I}^{\mathrm{T}}{\mathit{D}}_{\mathrm{m}\mathrm{b}}{\mathit{B}}_{\mathrm{m}I}\\ & +{\left\{\begin{array}{c}{\mathit{B}}_{\mathrm{m}I}^{\mathrm{s}1}\\ {\mathit{B}}_{\mathrm{m}I}^{\mathrm{s}2}\end{array}\right\}}^{\mathrm{T}}{\mathit{D}}_{\mathrm{m}\mathrm{s}}\left\{\begin{array}{c}{\mathit{B}}_{\mathrm{m}I}^{\mathrm{s}1}\\ {\mathit{B}}_{\mathrm{m}I}^{\mathrm{s}2}\end{array}\right\})\mathrm{d}x\mathrm{d}y,\end{array}

(38)\begin{array}{rl}{K}_{\mathrm{f}}^I=& \underset{S_{\mathrm{e}}}{\int }(k_{\mathrm{w}}{({\mathit{H}}_I^{\mathrm{w}})}^{\mathrm{T}}{\mathit{H}}_I^{\mathrm{w}}+(k_{\mathrm{s}}+N^0)\\ & \cdot ({({\mathit{H}}_{I,x}^{\mathrm{w}})}^{\mathrm{T}}{\mathit{H}}_{I,x}^{\mathrm{w}}+{({\mathit{H}}_{I,y}^{\mathrm{w}})}^{\mathrm{T}}{\mathit{H}}_{I,y}^{\mathrm{w}}))\mathrm{d}x\mathrm{d}y,\end{array}

in which ‘T’ is denote the transfer matrix, and the following quantities are defined:

(39){\mathit{N}}_I={\left\{\begin{array}{c}{\mathit{H}}_I^{\mathrm{u}}\\ {\mathit{H}}_I^{\mathrm{v}}\\ {\mathit{H}}_I^{\mathrm{w}}\\ {\mathit{H}}_{I,x}^{\mathrm{w}}\\ {\mathit{H}}_{I,y}^{\mathrm{w}}\end{array}\right\}}^{\mathrm{T}};{\mathit{B}}_I^1=\left[\begin{array}{c}{\mathit{H}}_{I,x}^{\mathrm{u}}+{\displaystyle \frac{{\mathit{H}}_I^{\mathrm{w}}}{R_1}}\\ {\mathit{H}}_{I,x}^{\mathrm{v}}+{\displaystyle \frac{{\mathit{H}}_I^{\mathrm{w}}}{R_2}}\\ {\mathit{H}}_{I,y}^{\mathrm{u}}+{\mathit{H}}_{I,x}^{\mathrm{v}}\end{array}\right];{\mathit{B}}_I^2=\left[\begin{array}{c}{\displaystyle \frac{{\mathit{H}}_{I,x}^{\mathrm{u}}}{R_1}}-{\mathit{H}}_{I,xx}^{\mathrm{w}}\\ {\displaystyle \frac{{\mathit{H}}_{I,y}^{\mathrm{v}}}{R_2}}-{\mathit{H}}_{I,yy}^{\mathrm{w}}\\ {\displaystyle \frac{{\mathit{H}}_{I,y}^{\mathrm{u}}}{R_1}}+{\displaystyle \frac{{\mathit{H}}_{I,x}^{\mathrm{v}}}{R_2}}-2{\mathit{H}}_{I,xx}^{\mathrm{w}}\end{array}\right],

(40)\begin{array}{rl}{\mathit{B}}_{\mathrm{m}I}=& [{\mathit{H}}_{I,xy}^{\mathrm{w}}-\frac{{\mathit{H}}_{I,x}^{\mathrm{v}}}{2R_2},\frac{{\mathit{H}}_{I,y}^{\mathrm{u}}}{2R_1}-{\mathit{H}}_{I,xy}^{\mathrm{w}},\frac{{\mathit{H}}_{I,y}^{\mathrm{u}}}{2R_1}-\frac{{\mathit{H}}_{I,x}^{\mathrm{v}}}{2R_2},\\ & {\frac{1}{2}\left({\mathit{H}}_{I,yy}^{\mathrm{w}}-{\mathit{H}}_{I,xx}^{\mathrm{w}}\right)+\frac{1}{4}\left(\frac{{\mathit{H}}_{I,x}^{\mathrm{u}}}{R_1}-\frac{{\mathit{H}}_{I,y}^{\mathrm{v}}}{R_2}\right)]}^{\mathrm{T}},\end{array}

(41){\mathit{B}}_{\mathrm{m}I}^{\mathrm{s}1}=-\frac{1}{4}\left[\begin{array}{c}{\mathit{H}}_{I,xy}^{\mathrm{u}}-{\mathit{H}}_{I,xx}^{\mathrm{v}}\\ {\mathit{H}}_{I,yy}^{\mathrm{u}}-{\mathit{H}}_{I,xy}^{\mathrm{v}}\end{array}\right];{\mathit{B}}_{\mathrm{m}I}^{\mathrm{s}2}=-\frac{1}{4}\left[\begin{array}{c}{\displaystyle \frac{{\mathit{H}}_{I,xy}^{\mathrm{u}}}{R_1}}-{\displaystyle \frac{{\mathit{H}}_{I,xx}^{\mathrm{v}}}{R_2}}\\ {\displaystyle \frac{{\mathit{H}}_{I,yy}^{\mathrm{u}}}{R_1}}-{\displaystyle \frac{{\mathit{H}}_{I,xy}^{\mathrm{v}}}{R_2}}\end{array}\right].

Setting: \mathit{d}=\overline{\mathit{d}}\mathrm{s}\mathrm{i}\mathrm{n}\left(\omega t\right), natural vibrational frequency \omega of the BFGP doubly-curved shallow microshell is determined as follows

(42)\mathrm{d}\mathrm{e}\mathrm{t}\left|\left({\mathit{K}}_{\mathrm{s}}+{\mathit{K}}_{\mathrm{f}}\right)-{\omega }^2\mathit{M}\right|=0.

The following boundary conditions (BCs) are taken into account for each of the numerical examples that follow:

1) Simply supported (S):

u_0=w_0=w_{0,x}=0 at y=0,b and v_0=w_0=w_{0,y}=0 at x=0,a.

2) Clamped (C)

u_0=v_0=w_0=w_{0,x}=w_{0,y}=0 at all edge.

For present paper, only fully simply supported boundary (SSSS) and fully clamped supported boundary (CCCC) are used for BFGP doubly-curved shallow microshells with a circular planform. The following boundary settings are employed for BFGP doubly-curved shallow microshells that have a rectangular planform, as shown in Fig.3.

4 Numerical results

Using MATLAB’s programming language, a package of algorithms for the free vibrational investigation of BFGP doubly-curved shallow microshell resting on Pasternak elastic medium with a variable length-scale parameter is developed. This program has the benefit of being able to calculate two-curvature shells of any shape; the author uses rectangular and circular shells as examples. In this collection of programs, convergence and accuracy assessment studies are validated, and the effect of geometrical and material features on the particular vibration response is clarified. The following dimensionless formulae are used for ease in providing numerical results.

(43)\begin{array}{rl}& {\mathrm{\Omega }}_1={\omega }_{11}{a}^2\sqrt{{\rho }_{\mathrm{c}}h/D_{\mathrm{c}}};{\mathrm{\Omega }}_1^{\ast }={\omega }_{11}\frac{a^2}{h}\sqrt{{\rho }_{\mathrm{c}}/E_{\mathrm{c}}};\\ & {\overline{\mathrm{\Omega }}}_1={\omega }_{11}\frac{a^2}{{\text{π}}^2}\sqrt{{\rho }_{\mathrm{c}}h/D_{\mathrm{c}}};\\ & N^0=N^{00}\frac{E_{\mathrm{c}}{h}^3}{10a^2};K_{\mathrm{w}}=\frac{k_{\mathrm{w}}{a}^4}{D_{\mathrm{c}}};\\ & K_{\mathrm{s}}=\frac{k_{\mathrm{s}}{a}^2}{D_{\mathrm{c}}};D_{\mathrm{c}}=\frac{E_{\mathrm{c}}{h}^3}{12\left(1-{\nu }_{\mathrm{c}}^2\right)}.\end{array}

The mechanical properties of BFGP doubly-curved shallow microshells are given as follows:

Ceramic: Al₂O₃, E_c = 380 GPa, ρ_c = 3800 kg·m⁻³, υ_c = 0.3.

Metallic: Al, E_m = 70 GPa, ρ_m = 2700 kg·m⁻³, υ_m = 0.3.

4.1 Convergence rate and precision evaluation

The operation characteristics, including convergence rate evaluation and numerical verification, are explored in this part. First, the natural frequencies {\mathrm{\Omega }}_1 of simply supported shallow SP microshells are compared with those of rectangular planforms shown in Tab.1 and Fig.5. Thin shells are examined with three distinct values of material length-scale ratio {\displaystyle \frac{l}{h}} = 0, 0.5, and 1. Seven distinct finite element meshes are analyzed for each shell case to determine the convergence rate of the present IGA approach. As shown in Tab.1 and Fig.5, while a fast convergence rate of the investigation is archived for polynomial orders p=3 and p=4, solutions utilizing p=2 exhibit a relatively slow convergence rate toward analytic solutions, as reported by Dehrouyeh-Semnani and Mostafaei [56]. Based on the aforementioned results, the 11 × 11 cubic (p=4) NURBS element mesh is replete for all cases. Unless otherwise specified, the mesh depicted in Fig.4(a) is used throughout the remainder of this article.

Tab.2 shows a comparison of the non-dimensional natural frequency {\mathrm{\Omega }}_1^{\ast } of rectangular-shaped FG microplates. The plate thickness is specified as average (a/h = 20) and thin (a/h = 100) with four values of the material length-scale coefficient l/h=0.2,0.4,0.8,1.0, and 3 of the power-law index {\alpha }_{\mathit{\text{z}}}=0,1,10. The findings {\mathrm{\Omega }}_1^{\ast } provided by the paper are totally consistent with the results reported in Ref. [57] as seen in Tab.2.

The proposed findings on the dimensionless natural frequencies {\mathrm{\Omega }}_i of isotropic shallow microshells with rectangular and circular planforms are compared with the published results of Dehrouyeh-Semnani and Mostafaei [56], as presented in Tab.3 and Tab.4. Among the modified parameters are the boundary conditions CCCC and SSSS, as well as the two-curvature shell types CY, SP, and HP. According to the data table, the findings given by the research utilizing the IGA approach are totally compatible with the finite element results published by Dehrouyeh-Semnani and Mostafaei [56].

4.2 Rectangular planform

In this part, study of the effect of geometric parameters on the natural frequency of a BFGP doubly-curved shallow microshell with a rectangular planform is reported. Unless otherwise specified, the following are fixed in the input data: a/h = 50, R₁ = 5a, b = a, α_z = 0.5, α_x = 1, ξ = 0.1, l_c = h, l_m = l_c/2, k_w = 10, k_s = 1, and N^{00}=0. First, the effect of various grading-indices and the material length-scale ratio l_{\mathrm{c}}/h on the dimensionless natural frequency {\overline{\mathrm{\Omega }}}_1 of BFGP SP, HP, and CY shallow microshells with rectangular planform is presented in Tab.5–Tab.7. Here, α_x = α_z = α = 0, 0.5, 1, 2, 5, 10; coefficients l_c = 0, 0.5h, h and boundary conditions SSSS, CCCC, SFSF, and CSCS are used for each shell type. As seen, the natural frequency values for various two-curvature shells fall as the coefficient α grows. This is very comprehensible given that the fraction of metal in the material structure is continually growing, resulting in a reduction in the system’s stiffness. Unlike the rise in grading index, the increase in material length-scale parameter l_{\mathrm{c}} makes the shell structure more stiff. Consequently, the natural frequency will grow; this increase is independent of boundary circumstances or shell structure. Given the identical input value for the three types of two-curvature examples analyzed, the SP shell will provide the greatest natural frequency, while the CY shell will produce the lowest natural frequency. Before continuing with the design of genuine curved shell structures, this should be kept in mind. The strength of the IGA numerical approach is that it can handle the majority of difficult-to-implement boundary conditions with great precision. The influence of different boundary conditions and porosity coefficients on the natural frequency of various shells are shown in Tab.8. According to the boundary conditions, the shell’s hardness decreases in the following sequence: CCCC, CSCS, SSSS, CFCF, SFSS, and SFSF. As a result of its use, flexible boundary research using an elastic system to change the stiffness at the boundary point is gaining popularity. Tab.8 reveals that an increase in the porous coefficient results in a decrease in the natural frequency. Nevertheless, the presence of porosities has the benefit of making the structure lighter; hence constructions with porosities are becoming more popular and widespread.

Following this, a series of numerical data are presented in the form of graphs for the purpose of evaluating the effect of the parameters. The impact of varying the material length-scale parameter l_{\mathrm{c}},l_{\mathrm{m}} on the natural frequency of various types of curved shells is seen in Fig.6. It can be seen that the natural frequency of oscillation grows as l_{\mathrm{c}} or l_{\mathrm{m}} increases, and when l_{\mathrm{c}}=l_{\mathrm{m}}=h, the frequency achieves its maximum value Fig.7 clarifies the simultaneous impacts of ceramic and metal ratio control coefficients in {\alpha }_{\mathit{\text{z}}},{\alpha }_x materials on the unique brilliance of distinct shells. All of the data indicate that a rise in {\alpha }_{\mathit{\text{z}}} or {\alpha }_x causes a drop in the natural frequency value, and this decrease becomes more rapid when 0⩽{\alpha }_{\mathit{\text{z}}}({\alpha }_x)⩽2. However, the rate of decline moderates when the metal ratio approaches its maximum value.

A two-parameter elastic foundation is then used. As shown in Fig.8, changing the values of the foundation’s stiffness coefficient k_{\mathrm{w}},k_{\mathrm{s}} has a big effect on the stiffness of the curved shell structure. As the stiffness of the elastic substrate goes up, the natural frequency of the two-curvature shells goes up dramatically. The majority of investigations indicate that the natural frequency results vary approximately linearly with the elastic foundation stiffness.

Next, the analysis of the change of the structure’s curvature radius to the shell’s natural frequency is shown in Fig.9 with various boundary conditions. Those findings indicate that an increase in the radius of curvature of the shell decreases its natural frequency. This is fully comprehensible given that, as the radius of curvature grows, the structure eventually converts into a plate and the its stiffness diminishes.

As depicted in Fig.10, the influence of the mechanical compressive force N^{00} operating in the tangential direction of the shell on the unconstrained vibration of the shell is analyzed. As the shell’s stiffness increases when the compressive force N^{00} increases, the natural frequency {\overline{\mathrm{\Omega }}}_1 increases as well. In six investigated boundary conditions, it was found that the rigidity of the shell increased with the number of mechanical constraints. The natural frequency {\overline{\mathrm{\Omega }}}_1 decreases progressively according to the following principles, as shown in Fig.10: CCCC, CSCS, SSSS, CFCF, SFSS, and SFSF.

4.3 Circular planform

Comparable research was conducted for microshells with circular and rectangular planforms. The control grid utilized for convergence and precision is shown in Fig.4(b). Unless otherwise specified, the following are fixed in the input data: R = a, R/h = 50, R₁ = 5R, α_z = 0.5, α_x = 1, ξ = 0.1, l_c = h, l_m = l_c/2, k_w = 10, k_s = 1, and N^{00}=0. In the framework of classical theory l_{\mathrm{c}}/h=0, modified couple stress theory, isotropic materials {\alpha }_x={\alpha }_{\mathit{\text{z}}}=\alpha =0, and bidirectional BFGP materials, Tab.9–Tab.11 presents the first natural frequency {\overline{\mathrm{\Omega }}}_1 of shallow microshells with circular planform (of radius a) and various curvature radii ratios (R_1/R_2). The findings for two boundary conditions, SSSS and CCCC, and two types of porosity, E and U, are provided. As demonstrated in Tab.9 for the SP shallow microshell, the natural oscillation frequency {\overline{\mathrm{\Omega }}}_1 tends to grow gradually as l_{\mathrm{c}} increases, but the natural frequency tends to drop as the power-law index increases. In addition, comparable outcomes may be obtained from Tab.10 and Tab.11 for shallow microshells that are CY, SP, and HP. The impact of the coefficients l_{\mathrm{c}} and \alpha on the natural frequency of vibration is seen to be independent of the boundary conditions and porosity form. Tab.12 presents the results of further research on the influence of porosity coefficient \xi =0,0.1, and 0.2 on the first, second, and fourth natural frequencies of BFGP shallow microshells with circular planforms. It can be observed that the porosity has a substantial impact on the natural frequency of the shell, and it can also be seen that the frequency has a tendency to drop as the porosity coefficient increases. In any curved shell configuration, the natural frequency is determined by the pore law Type U is greater than the natural frequency determined by the pore law Type E.

The influence of the length-scale parameters l_{\mathrm{c}} and l_{\mathrm{m}}, the grading indexes {\alpha }_{\mathit{\text{z}}} and {\alpha }_x, and the stiffness elastic foundation parameters k_{\mathrm{w}},k_{\mathrm{s}} with relation to the fundamental frequency of a BFGP microshell that is shallow and has a circular planform may be seen quite clearly in Fig.11–Fig.13. Observations made for circular cases are similar to those for rectangular examples.

Like rectangular shells, the shell structure of a sphere becomes more rigid when subjected to mechanical compression, N^{00}, leading to a higher natural frequency of vibration. Four distinct shells are shown in Fig.14 with their respective first-two non-dimensional natural frequencies asfunctions of compression force N^{00}.

Last but not least, the first nine oscillating configurations of the BFGP shallow microshell with a rectangular and circular planform may be shown in Fig.15 and Fig.16. When the material is transformed in two directions, it is clear to see that, despite the fact that the CCCC and SSSS symmetry boundaries are being utilized, the shell deformation is always skewed toward the material that has a greater metallic composition.

5 Conclusions

An isogeometric approach for BFGP free oscillation analysis shallow curved microshell based on modified couple stress theory with a changeable material length-scale parameter is established. The numerical results for rectangular and circular planforms with four distinct curvature radii are provided. Studies indicate that the natural frequency tends to grow as the length-scale characteristic of the material increases; it is independent of the shell shape and boundary circumstances. In the meanwhile, a rise in the power-law index and the porosity coefficient always decreases the shell structure’s stiffness, resulting in significant reduction in the natural frequency. Adjustment of various boundary conditions will result in distinct frequency bands and vibrational shapes. In addition, the rigidity of the elastic foundation plays a vital part in the working process of the shell, as it contributes to the shell rigidity. Comparing the findings of rectangular and circular planforms leads to the conclusion that the change in planform has a major impact on the mode shape exchange of shallow microshells based on both classical and non-classical models.

The new points of this work are as follows.

1) A doubly-curved microshell with an arbitrary planform and boundary conditions is considered.

2) Bidirectional FG material combined with the modified couple stress theory is presented.

3) Free vibration of a doubly-curved microshell using Kirchhoff–Love’s shell theory and the isogeometric FEM is explored.

4) Collection of effects of shell shape, porosities, elastic foundation, grading-index, and boundary conditions on the individual vibration ability of microshells are investigated.

The results of this research will be used as a starting point to study how BFGP microshells behave mechanically. Future research might focus on static buckling, dynamic buckling, and forced vibration of the BFGP microshell based on the paper’s methodology and program.

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