本文發展基於Reissner混合變分原理的有限雙曲率層殼法歸一理論，將其應用於具簡支承邊界條件之多層功能性梯度材料雙曲率殼，分析該結構承受機械荷載之靜態行為。此理論可用於探求受壓力加載之功能性梯度材料薄膜-基座雙曲率殼之層間應力與變位沿厚度方向之分佈，並將所得解與基於虛位移原理的有限雙曲率層殼法之結果進行比較。多層功能性梯度材料殼之材料性質會沿厚度方向平滑漸變，本文使用兩相材料混合原理與Mori-Tanaka微觀定理估計其有效楊氏係數。三角函數與Lagrange多項式分別用以表示各層主要變數在面內與厚度方向之分佈。文中各主要變數沿厚度方向展開的冪級數階數保持相同，分別為線性、二次與三次。藉由比較文獻中之三維、二維精確解與本文推衍之Reissner混合變分原理有限雙曲率層殼法所得解，驗證得本方法的準確性高及收斂速度快。

SUMMARY

A unified formulation Reissner’s mixed variational theorem (RMVT) -based finite doubly curved layer (FDCL) methods is developed for the static analysis of simply-supported, multilayered functionally graded material (FGM) doubly curved shells under mechanical loads. The trigonometric functions and Lagrange polynomials are used to interpolate the in-surface and thickness variations of the primary variables of each individual layer, respectively. The accuracy and convergence rate of these RMVT-based FDCL methods are validated by comparing their solutions with the exact three-dimensional and accurate two-dimensional solutions available in the literature.

INTRODUCTION

Shell structures possess initial curvatures which may increase their gross stiffness and loading capacity. By changing the curvature radii, the DCS can have a variety of different types of shell structures.

In order to overcome drawbacks of fiber-reinforced composite (FRC) structures, functionally graded materials (FGMs) has been used to replace FRC materials to form FGM structures, the material properties of which vary gradually and smoothly through the thickness direction.

Comprehensive surveys about laminated FRC shells and multilayered FGM shells for such 2D and 3D approaches can be found in Carrera (2000a, 2000b, 2003a), Carrera and Ciuffreda (2005), Noor and Burton (1989, 1990a, 1990b), Soldatos (1994), Wu and Liu (2016) and Wu et al. (2008).

The formulation of RMVT-based models is extended to the static analysis of laminated FRC and multilayered FGM DCSs under mechanical loads. The accuracy and convergence rate of these RMVT-based FDCL methods are examined in the literature.

RMVT-based FDCL methods

A simply-supported, multilayered FGM DCS subjected to mechanical loads on the top and bottom surfaces, is considered, as shown in Fig. 1(a). A global doubly curved coordinate system is located at the mid-surface of the shell, and a set of local thickness coordinates are located at the mid-surface of each individual layer, as shown in Fig. 1(b) and 1(c).

The primary field variables in this formulation, namely the elastic displacement and transverse stress components of a typical layer of the shell. n denotes the related orders used for expansions of each primary variable, and when n=1, 2 and 3, the FDCL methods are called linear, quadratic and cubic ones, respectively. For a typical layer, the linear constitutive equations are given by Eq. (2.2). The strain-displacement relations for each individual layer are written as Eq. (2.3)-(2.8).

The RMVT is used to derive the Euler-Lagrange equations of the shell for RMVT-based FDCL methods, and its corresponding energy functional is written in the form of Eq. (2.9). Introducing Eqs. (2.13)-(2.15) in Eq. (2.9) and then imposing the stationary principle of the Reissner energy functional, we obtain the Euler-Lagrange equations of the shell, as Eq. (2.16).

Using Eq. (2.16) and assembling the local stiffness matrix and forcing vector of each layer, in which continuity conditions at the interfaces between adjacent layers are imposed, we may construct the global stiffness matrix and forcing vector for the pressure-loaded DCS. The primary variables and the in-surface stress components at each nodal surface can then be determined.

PVD-based FDCL methods

The principle of virtual displacements is a displacement-based energy principle, in which only the displacement components are regarded as the primary variables. After introducing Eq. (3.1) and Eqs. (2.13)-(2.15) in Eq. (3.2) and imposing the stationary principle of the potential energy functional, we obtain the Euler-Lagrange equations of the shell, as Eq. (3.4).

After using Eq. (3.4) and continuity conditions at the interfaces, we may construct the global stiffness matrix and forcing vector for the pressure-loaded DCS by following the standard process of the FEMs. The displacement components at each nodal surface can then be determined. Subsequently, the in-plane and transverse stress components at the nodal surfaces can be obtained using the primary variables and Hooke’s law, as Eq. (3.5)-(3.7).

The transverse stresses are calculated using Hooke’s law which might lead to poor predictions. As a result, a set of analytical formulae derived from the stress equilibrium equations is recommended by using the successive approximation method. According to this method, the solutions of transverse stress components obtained using Eqs. (3.8)-(3.10) will approach the exact 3D solutions.

RESULT AND DISCUSSION

Laminated FRC DCSs and plates
Table 1 shows the linear-, quadratic- and cubic-order solutions of PVD- and RMVT-based FDCL methods for the displacement and stress components induced at certain positions in simply-supported, laminated shells and plates. The convergence rates of the RMVT-based FDCL methods are faster than those of PVD-based FDCL methods.

Figure 2 shows solutions for the through-thickness distributions of various field variables induced in the simply-supported, pressure-loaded DCSs.

Bi-layered Homogeneous film-FGM substrate DCSs
Figure 3 shows the through-thickness distributions of the effective Young’s modulus for the FGM film-substrate DCS estimated by using the rule of mixtures and Mori-Tanaka scheme. It is shown the results obtained using these two models are minor.

Figure 4 shows the results obtained using the rule of mixtures and Mori-Tanaka scheme with regard to the through-thickness distributions of various field variables induced in a Bi-layered Homogeneous film-FGM substrate DCS.

The simply-supported and pressure-loaded FGM film-substrate DCS in Fig. 4 is reconsidered in Fig. 5, except for changing the values of . It can be seen that the in-surface stress components induced at the film-substrate interface of the shell are discontinuous, while they are continuous at the film-substrate interface for the FGM film-substrate DCS. The transverse shear stress components induced in the homogeneous film-substrate DCS ( ) are reduced when the homogeneous substrate is replaced with the FGM substrate ( ).

Bi-layered FGM film-homogeneous substrate DCSs
Figure 6 shows the through-thickness distributions of the effective Young’s modulus for the FGM film-substrate DCS estimated by using the rule of mixtures and Mori-Tanaka scheme.

The simply-supported and pressure-loaded FGM film-substrate DCS in Fig. 7 is reconsidered in Fig. 5, except for changing the values of and the sequence of the material layers. The in-surface stress components induced at the film-substrate interface of the shell are continuous at the film-substrate interface for the FGM film-substrate DCS. However, the slope of in-surface stress components in film layer increase rapidly because the slope of the effective Young’s modulus in film layer is higher.

CONCLUSION

In the implementation of various PVD- and RMVT-based FDCL methods, the results show that the RMVT-based FDCL methods are superior to the PVD-based ones. When comparing the results obtained using RMVT-based FDCL methods with other 2D theories, it is shown that ( , LM4)> ( , LD4)>LM1>LD1>ED4>ED2>ED1, in which the symbol “>” means more accurate and a faster convergence rate.

The through-thickness distributions of various field variables induced in the FGM film-substrate DCSs appear to be layer-wise linear and higher-order polynomial variations for the displacement and stress components, respectively, which is inconsistent with the kinematic and kinetic assumptions of most of conventional 2D ESLTs of laminated FRC shells, although this might not be suitable for the analysis of multilayered FGM shells. Some more advanced 2D theories of multilayered FGM shells thus need to be developed.

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