Analytical Treatment of the Size-Dependent Nonlinear Postbuckling of Functionally Graded Circular Cylindrical Micro-/Nano-Shells

Abstract

Mindlin’s strain gradient theory (SGT) is the most popular size-dependent higher-order gradient elasticity theory capable of describing the mechanical behavior of structures at micro-/nanoscale, in which the strain gradient terms are included in the strain energy density. In this article, based on Mindlin’s SGT and classical Donnell’s shell theory, the size-dependent nonlinear postbuckling characteristics of circular cylindrical micro-/nanoscale shells under the action of axial compressive loads are studied. For some specific values of the gradient-based material parameters, the present general micro-/nano-shell formulation can be reduced to those based on simple forms of the strain gradient elasticity theory such as the modified strain gradient theory and the modified couple stress theory. The micro-/nano-shells are assumed to be made from functionally graded materials whose properties vary across the thickness direction based on a power-law distribution function. To consider the geometric nonlinearity, the von Kármán relations are used. After obtaining the potential energy of the system including strain gradient effects, an analytical variational approach is utilized to solve the postbuckling problem for small-scale shells with simply supported ends. Finally, selected numerical results are presented to investigate the influence of different parameters such as volume fraction index, length scale parameter and radius-to-thickness ratio on the nonlinear postbuckling behavior of micro-/nano-shells.

Appendix

Herein, the procedure of derivation of governing equations and boundary conditions associated with the small-scale shells based upon the Mindlin’s SGT is presented.

Using Eqs. (6)–(10) and (13) as well as by defining the following relations

$$T_{ijk} = \mathop \int \limits_{ - h/2}^{h/2} \tau_{ijk} {\text{d}}z,\quad M_{ijk} = \mathop \int \limits_{ - h/2}^{h/2} \tau_{ijk} \bar{z}{\text{d}}z$$

(45)

the variations of strain potential energies of small-scale shells corresponding to the classical and higher-order stresses can be stated as follows

$$\delta \varPi_{C} = \mathop \int \limits_{A}^{ } \left\{ {N_{xx} \delta \phi_{0} + M_{xx} \delta \phi_{1} + N_{yy} \delta \varphi_{0} + M_{yy} \delta \varphi_{1} + N_{xy} \delta \kappa_{0} + M_{xy} \delta \kappa_{1} } \right\}{\text{d}}A,$$

(46)

$$\delta \varPi_{NC} = \int_{A} {\left\{ {T_{xxx} \delta \left( {\frac{{\partial \phi_{0} }}{\partial x}} \right) + M_{xxx} \delta \left( {\frac{{\partial \phi_{1} }}{\partial x}} \right) + T_{yxx} \delta \left( {\frac{{\partial \phi_{0} }}{\partial y}} \right) + M_{yxx} \delta \left( {\frac{{\partial \phi_{1} }}{\partial y}} \right) + T_{zxx} \delta \phi_{1} + T_{xyy} \delta \left( {\frac{{\partial \varphi_{0} }}{\partial x}} \right) + M_{xyy} \left( {\frac{{\partial \varphi_{1} }}{\partial x}} \right) + T_{yyy} \delta \left( {\frac{{\partial \varphi_{0} }}{\partial y}} \right) + M_{yyy} \delta \left( {\frac{{\partial \varphi_{1} }}{\partial y}} \right) + T_{zyy} \delta \varphi_{1} + T_{xxy} \delta \left( {\frac{{\partial \kappa_{0} }}{\partial x}} \right) + M_{xxy} \delta \left( {\frac{{\partial \kappa_{1} }}{\partial x}} \right) + T_{yyx} \delta \left( {\frac{{\partial \kappa_{0} }}{\partial y}} \right) + M_{yyx} \delta \left( {\frac{{\partial \kappa_{1} }}{\partial y}} \right) + T_{zxy} \delta \kappa_{1} - \frac{{T_{yxz} }}{R}\delta \kappa_{0} - \frac{{M_{yxz} }}{R}\delta \kappa_{1} - \frac{{2T_{yyz} }}{R}\delta \varphi_{0} - \frac{{2M_{yyz} }}{R}\delta \varphi_{1} } \right\}}$$

(47)

Moreover, the variation of work done by the external in-plane loads $\left( {N_{x}^{0} , N_{yy}^{0} {\text{and}} N_{xy}^{0} } \right)$ and external transverse load $q$ can be expressed as

$$\delta \varPi_{P} = \mathop \int \limits_{A}^{ } \left[ {N_{xx}^{0} \frac{\partial w}{\partial x}\delta \frac{\partial w}{\partial x} + N_{xy}^{0} \frac{\partial w}{\partial x}\delta \frac{\partial w}{\partial y} + N_{xy}^{0} \frac{\partial w}{\partial y}\delta \frac{\partial w}{\partial x} + N_{yy}^{0} \frac{\partial w}{\partial y}\delta \frac{\partial w}{\partial y} + q\frac{\partial w}{\partial y}w} \right]{\text{d}}A$$

(48)

By utilizing the principle of virtual work and calculus of variations, the governing equations can be obtained from Eqs. (46)–(48) as the following form:

$$\frac{{\partial \bar{N}_{xx} }}{\partial x} + \frac{{\partial \bar{N}_{xy} }}{\partial y} = 0,$$

(49)

$$\frac{{\partial \bar{N}_{yx} }}{\partial x} + \frac{{\partial \bar{N}_{yy} }}{\partial y} = 0,$$

(50)

$$\frac{{\partial^{2} \bar{M}_{xx} }}{{\partial x^{2} }} + 2\frac{{\partial^{2} \bar{M}_{xy} }}{\partial y\partial x} + \frac{{\partial^{2} \bar{M}_{yy} }}{{\partial y^{2} }} - \frac{{\bar{N}_{yy} }}{R} + \frac{\partial }{\partial x}\left( {\bar{N}_{xx} \frac{\partial w}{\partial x}} \right) + \frac{\partial }{\partial x}\left( {\bar{N}_{xy} \frac{\partial w}{\partial y}} \right) + \frac{\partial }{\partial y}\left( {\bar{N}_{xy} \frac{\partial w}{\partial x}} \right) + \frac{\partial }{\partial y}\left( {\bar{N}_{yy} \frac{\partial w}{\partial y}} \right) + N_{xx}^{0} \frac{{\partial^{2} w}}{{\partial x^{2} }} + 2N_{xy}^{0} \frac{{\partial^{2} w}}{\partial x\partial y} + N_{yy}^{0} \frac{{\partial^{2} w}}{{\partial y^{2} }} + q = 0.$$

(51)

Moreover, the corresponding boundary conditions are obtained as

$$\delta u = 0\quad {\text{or}}\quad \tilde{N}_{xx} n_{x} + \tilde{N}_{xy} n_{y} = 0$$

(52)

$$\delta v = 0\quad {\text{or}}\quad \tilde{N}_{xy} n_{x} + \tilde{N}_{yy} n_{y} = 0$$

(53)

$$\delta w = 0 or \left( {\frac{{\partial \tilde{M}_{xx} }}{\partial x} + \frac{{\partial \tilde{M}_{xy} }}{\partial y} + \tilde{N}_{xx} \frac{\partial w}{\partial x} + \tilde{N}_{xy} \frac{\partial w}{\partial y} + \frac{{T_{xyy} }}{R}} \right)n_{x} + \left( {\frac{{\partial \tilde{M}_{yy} }}{\partial y} + \frac{{\partial \tilde{M}_{xy} }}{\partial x} + \tilde{N}_{yy} \frac{\partial w}{\partial y} + \tilde{N}_{xy} \frac{\partial w}{\partial x} + \frac{{T_{yyy} }}{R}} \right)n_{y} = 0$$

(54)

$$\delta \left( {\frac{\partial w}{\partial x}} \right) = 0\quad {\text{or}}\quad \left( {\tilde{M}_{xx} - T_{xxx} \frac{\partial w}{\partial x} - T_{xxy} \frac{\partial w}{\partial y}} \right)n_{x} + \left( {\tilde{M}_{xy} - T_{yxx} \frac{\partial w}{\partial x} - T_{yyx} \frac{\partial w}{\partial y}} \right)n_{y} = 0,$$

(55)

$$\delta \left( {\frac{\partial w}{\partial y}} \right) = 0\quad {\text{or}}\quad \left( {\tilde{M}_{xy} - T_{xxy} \frac{\partial w}{\partial x} - T_{xyy} \frac{\partial w}{\partial y}} \right)n_{x} + \left( {\tilde{M}_{yy} - T_{yyx} \frac{\partial w}{\partial x} - T_{yyy} \frac{\partial w}{\partial y}} \right)n_{y} = 0,$$

(56)

$$\delta \left( {\frac{\partial u}{\partial x}} \right) = 0\quad {\text{or}}\quad T_{xxx} n_{x} + T_{yxx} n_{y} = 0, \quad \delta \left( {\frac{\partial u}{\partial y}} \right) = 0\quad {\text{or}}\quad T_{xxy} n_{x} + T_{yyx} n_{y} = 0$$

(57)

$$\delta \left( {\frac{\partial v}{\partial x}} \right) = 0 \quad {\text{or}}\quad T_{xxy} n_{x} + T_{yyx} n_{y} = 0,\quad \delta \left( {\frac{\partial v}{\partial y}} \right) = 0\quad {\text{or}}\quad T_{xyy} n_{x} + T_{yyy} n_{y} = 0$$

(58)

$$\delta \left( {\frac{{\partial^{2} w}}{{\partial x^{2} }}} \right) = 0 \quad {\text{or}}\quad M_{xxx} n_{x} + M_{yxx} n_{y} = 0$$

(59)

$$\delta \left( {\frac{{\partial^{2} w}}{{\partial y^{2} }}} \right) = 0\quad {\text{or}}\quad M_{xyy} n_{x} + M_{yyy} n_{y} = 0$$

(60)

$$\delta \left( {\frac{{\partial^{2} w}}{\partial x\partial y}} \right) = 0\quad {\text{or}}\quad M_{xxy} n_{x} + M_{yyx} n_{y} = 0$$

(61)

where

$$\begin{aligned} & \bar{N}_{xx} = N_{xx} - \frac{{\partial T_{xxx} }}{\partial x} - \frac{{\partial T_{yxx} }}{\partial y}, \quad \bar{N}_{yy} = N_{yy} - \frac{{\partial T_{xyy} }}{\partial x} - \frac{{\partial T_{yyy} }}{\partial y} - \,\frac{{2T_{yyz} }}{R}, \\ & \bar{N}_{xy} = N_{xy} - \frac{{\partial T_{xxy} }}{\partial x} - \frac{{\partial T_{yyx} }}{\partial y} - \frac{{T_{yxz} }}{R}, \quad \bar{M}_{xx} = M_{xx} + T_{zxx} - \frac{{\partial M_{xxx} }}{\partial x} - \frac{{\partial M_{yxx} }}{\partial y}, \\ & \bar{M}_{yy} = M_{yy} + T_{zyy} - \frac{{\partial M_{xyy} }}{\partial x} - \frac{{\partial M_{yyy} }}{\partial y} - \frac{{2M_{yyz} }}{R}, \\ & \bar{M}_{xy} = M_{xy} + T_{zxy} - \frac{{\partial M_{xxy} }}{\partial x} - \frac{{\partial M_{yyx} }}{\partial y} - \frac{{M_{yxz} }}{R}. \\ \end{aligned}$$

(62)