Modeling of cylindrical composite shell structures based on the Reissner’s Mixed Variational Theorem with a variable separation method
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Abstract
This work deals with the modeling of laminated composite and sandwich shells through a variable separation approach based on a Reissner’s Variational Mixed Theorem (RMVT). Both the displacement and transverse stress fields are approximated as a sum of products of separated functions of the inplane coordinates and the transverse coordinate. This approach yields to a nonlinear problem that is solved by an iterative process, in which 2D and 1D problems are separately considered at each iteration. In the thickness direction, a fourthorder expansion in each layer is used. For the inplane description, classical Finite Element method is used. Numerical examples involving several representative shell configurations (deep/shallow, thick/thin) are addressed to show the accuracy of the present method. It is shown that it can provide quasi3D results less costly than classical LW computations. In particular, the estimation of the transverse stresses, which are of major importance for damage analysis, is very good.
Keywords
Composite Shell Separation of variables RMVTIntroduction
Composite shells are widely used in the industrial field (aerospace, automotive, marine, medical industries...) due to their excellent mechanical properties, especially their high specific stiffness and strength. For composite design, accurate knowledge of displacements and stresses is required. One way consists in considering the threedimensional modelisation. However, due to the complexity of such numerical simulations, it is desirable to take advantage of the geometric ratios and to represent the problem as a twodimensional model by referring to shell theories. There are two ways to define the approximation of the displacement field. A “pure shell” model can be considered in which the displacement is associated with the local curvilinear vectors, and strain and stress are deduced using the differential geometry [1]. Alternatively, the shelllike solid approach [2] is widely used to formulate shell Finite Element (FE), in particular in commercial software. In this case, the displacement vector is defined in the global cartesian frame. The jacobian matrix transformation is used to express strain and stress with respect to the reference frame defined on the middle surface in order to introduce the constitutive law. With respect to the “pure shell” approach, differentiation is simplified and the curvatures need to be directly calculated [3]. The development of efficient computational models for the analysis of shells appears thus of major interest.

The Equivalent Single Layer Models (ESLM), to which belong the Classical Shell Theory (CST/Koiter) and First Order Shear Deformation Theory (FSDT/Nagdhi). The reader can refer to [5] for a detailed description of the assumptions on the strain field to derive different shell models. CST leads to inaccurate results for composites because both transverse shear and normal strains are neglected, see [6, 7, 8] for shallow laminated shells. FSDT is the most popular model due to the possibility to use a C\({}^0\) FE, but it needs shear correction factors and the transverse normal strain is still neglected (cf. [9, 10, 11, 12]). So, Higherorder Shear Deformation Theories (HSDT) have been developed to overcome these drawbacks. Different kinematics including 7 [13, 14] or 9 parameters [15] have been proposed. Reddy has derived a 5 parameter model starting from a thirdorder theory by considering the homogeneous top/bottom conditions for the transverse shear stresses [16]. Note also the variable kinematics approach based on the Carrera’s Unified Formulation developed in [17, 18].
In the ESLM context, a simple way to improve the estimation of the mechanical quantities consists in adding one zig–zag function, called Murakami’s function, in the expression of the displacement. In this way, the slope discontinuity at the interface between two adjacent layers is introduced. It allows to describe the socalled zig–zag effect. It has been carried out in conjunction with the FSDT in [19] and [20] based on dedicated mixed formulation. It has been also used with the HSDT in [21] and [22] including 9 and 13 parameters, respectively.

The LayerWise Models (LWM), in which the expansion of the mechanical quantities is defined over each layer independently. Some of these works are based on a linear distribution of the inplane displacements through each layer, without taking into account the transverse normal stress. The transverse displacement can be constant across the whole thickness, such as in [23, 24], or in each layer separately, as in [25]. But, this type of approach fails to predict accurate transverse stresses, unless using dedicated postprocessing steps [24]. Thus, higherorder approaches taking into account the transverse normal effect have been developed. The threedimensional constitutive law is used. Second, third and fourthorder expansions are discussed in [4, 26, 27]. In this framework, Kulikov [28] has developed the sampling surfaces method, see also the previously mentioned work [18]. In all the aforementioned LWM, the number of unknowns depends on the number of layers, which may thus affect the performance in terms of computational cost.
It should be noted that the mentioned works are based on the Finite Element method for linear elasticity problem in mechanics and applied to laminated composites, knowing that many other approaches (meshless, analytical, semianalytical...) are involved in open literature. Furthermore, the fundamental subject about the shear and membrane locking of shell is not addressed here. So, this above literature deals with only some aspects of the broad research activity about composite shells. An extensive assessment of different approaches for various theories and/or finite element applications can be found in [37, 38, 39, 40, 41, 42, 43, 44, 45] and more recently in [46].
Nevertheless, in the framework of the failure analysis of composite structures, involving the free edge effect for instance, the prediction of the interlaminar stresses is of major interest. In particular, the difficulty is to welldescribe the interlaminar continuous transverse stresses. Most of the ESLM fail to represent these in the most critical cases, unless using a postprocessing treatment [11, 24, 47, 48, 49]. To overcome these drawbacks, alternative formulations to the displacementbased approach have been developed. On the one hand, several techniques have been devised to correct the transverse shear locking pathology affecting FSDTbased plate/shell elements, most of which can be stated from hybridmixed approaches [50]. For composites, an assumed strain approach has been adapted in a FSDT [51] or in a LWM [52]. On the other hand, assumed partial/complete stress field over the laminate thickness independently aims at increasing the accuracy of this one. Without considering the transverse normal stress, some authors [53, 54] have adopted a partial hybrid stress approach based on the HellingerReissner Variational Principle (HRVP). Using a HSDT approach for displacements, Yong [53] has developed a generalized stress assumption for the transverse shear stresses only, whereas inplane stresses are also involved in [54]. Alternative hybrid approaches take into account the transverse normal effect. The Fraeijs de VeubekeHuWashizu multifield variational principle [55, 56] and the Reissner Mixed Variational Theorem [57] are used assuming the transverse shear stresses only. Note that Sgambitterra et al. [14] have introduced a mixedfield assumptions (\(\gamma _{\alpha 3}\) and \(\sigma _{33}\)) to derive a hybrid formulation enforcing the compatibility straindisplacement relations to be leastsquares compatible through the shell thickness. As a complement to these hybrid methods, mixed formulations are addressed in conjunction with FSDT [58] (HRVP) or including the Murakami’s zig–zag function [20] (Jing and Liao’s functional [59]). An advanced method is proposed in [60] by considering all interlaminar stresses between two adjacent layers as primary variables and also a higherorder LW displacement.
In the same way, the partially Reissner’s Mixed Variational Theorem (RMVT) assuming two independent fields for all displacement and transverse stress variables allows to ensure a priori interlaminar continuous transverse stress fields. The consideration of the transverse normal stress as an independent assumed variable seems to be important to obtain accurate distributions through the thickness of the shell (see e.g. [55]). The RMVT approach comes from the works of Reissner, see [61, 62]. It was first applied for multilayered structures in [63] and then, in [64] with higher order displacement field and [65] with a Layerwise approach for both displacements and transverse stress fields. Afterwards, the approach was widely developed with a systematic approach based on the Carrrera’s Unified Formulation to provide a large panel of 2D models for composite structures based on ESL and/or LW descriptions of the unknowns [66, 67, 68]. It has been also applied for shell structures in [69, 70, 71]. Note that the reader can refer to [27] for a survey on RMVT in this framework, and to [37, 72] for a further discussion.
In the present work, the RMVT is considered because it is a natural variational tool for composite structures as it allows to formulate independent approximations for mechanical variables that are required to be continuous across the stacking direction. It is used in conjunction with a variable separation method, namely the Proper Generalized Decomposition (PGD). The main purpose consists in taking advantage of this promising approach to decrease the high computational cost of a LW approach. In fact, interesting features have been shown in the model reduction framework [73, 74, 75]. So, the aim of the present paper is to assess this particular representation of the unknowns in the framework of a mixed formulation to model cylindrical laminated and sandwich shells. Both displacements and transverse stresses are written under the form of a sum of products of bidimensional polynomials of \((\xi ^1,\xi ^2)\) and unidimensional polynomials of z. A piecewise fourthorder Lagrange polynomial of z is chosen across the thickness of each layer. As far as the variation with respect to the inplane coordinates \((\xi ^1,\xi ^2)\) is concerned, a 2D eightnode quadrilateral FE is employed. With the PGD approach, each unknown function of \((\xi ^1,\xi ^2)\) is approximated using one degree of freedom (dof) per node of the mesh and the LW unknown functions of z are global for the whole shell. This process yields to two separate linear problems, i.e., a 2D problem in \((\xi ^1,\xi ^2)\) and a 1D problem in z, in which the number of unknowns is much smaller compared to a classical Layerwise approach. These two problems are solved successively within an iterative scheme. The interesting feature of this approach lies on the possibility to have a higherorder zexpansion and to refine the description of the mechanical quantities through the thickness without substantially increasing the computational cost. This is particularly suitable for the modeling of composite structures. Note that this method has been successfully applied to displacementbased plate/shell models in [76, 77, 78, 79] and also to RMVT plate models in [80].
We now outline the remainder of this article. First, the shell definition and the differential geometry are recalled. The RMVT formulation is described and the separation of the inplane and outofplane displacements/ transverse stresses is introduced. The principles of the PGD are precised in the framework of our study. The particular assumption on the assumed variables yields to a nonlinear problem, that is solved within an iterative process. The FE discretization is also described and finally, numerical tests are addressed. A preliminary convergence study is performed. Then, the influence of classical shell assumptions on the strains and the number of numerical layers are studied. The approach is also assessed for a wide range of applications: deep/shallow shells, crossply/angleply configurations, different slenderness ratios and different degrees of anisotropy for a sandwich are considered. The accuracy of the results is evaluated by comparison with a 2D elasticity solution from [81], or results available in the open literature [79, 82, 83].
Shell definitions and differential geometry
Reference problem description
The definition of the strain field
Constitutive relation
In the framework of the RMVT formulation, the Hooke’s law has to be rewritten under a convenient mixed form.
The weak form of the boundary value problem
The formulation of the problem is based on the Reissner’s partially Mixed Variational Theorem [61], denoted RMVT. In this formulation, the Principle of Virtual Displacement is modified by introducing the constraint equation to enforce the compatibility of the transverse strain components. This term also depends on the assumed transverse stresses, see also [17, 72]. Thus, the problem can be formulated as follows:
Application of the proper generalized decomposition to the cylindrical shell
In this section, we develop the application of the PGD for shell analysis with a mixed formulation. This work is an extension of a previous study on composite cylindrical shell structures [79].
The cylindrical geometry
The displacement and transverse stress field
In this paper, a classical eightnode FE approximation is used in \(\Omega \) and a LW description is chosen in \(\Omega _{z}\) as it is particulary suitable for the modeling of composite structures.
The strain field for the cylindrical composite structure
The problem to be solved

\({\varvec{v}}^{(k+1)}, \, \varvec{\tau }^{(k+1)}\) satisfy Eq. (29) for \({\varvec{f}}\), \( {\varvec{f}}_{\sigma }\) set to \({\varvec{f}}^{(k)}\) and \({\varvec{f}}_{\sigma }^{(k)}\)

\({\varvec{f}}^{(k+1)}\), \({\varvec{f}}_{\sigma }^{(k+1)}\) satisfy Eq. (30) for \({\varvec{v}}\), \(\varvec{\tau }\) set to \({\varvec{v}}^{(k+1)}\), \(\varvec{\tau }^{(k+1)}\)
Finite element discretization
Finite element problem to be solved on \(\Omega \)

\( {\varvec{q}}^{v\sigma }\) is the vector of the nodal displacements / transverse stresses associated with the finite element mesh in \(\Omega \),

\({\mathbf {K}}_{z}(\tilde{f},\tilde{f}_{\sigma })\) is the stiffness matrix obtained by summing the elements’ stiffness matrices \( \displaystyle {\mathbf {K}}_{z}^e(\tilde{f},\tilde{f}_{\sigma })= \int _{\Omega _{e}} \Big [ {{\varvec{B}}}_{\xi }^{T} {\varvec{k}}_{z}^v(\tilde{f}){{\varvec{B}}}_{\xi } + {{\varvec{B}}}_{\xi }^{T} {\varvec{k}}_{z}^{v\sigma }(\tilde{f},\tilde{f}_{\sigma }) {{\varvec{N}}}_{\sigma \xi } + {{\varvec{N}}}_{\sigma \xi }^T {\varvec{k}}_{z}^{\sigma v}(\tilde{f},\tilde{f}_{\sigma }) {{\varvec{B}}}_{\xi } + {{\varvec{N}}}_{\sigma \xi }^T {\varvec{k}}_{z}^{\sigma \sigma }(\tilde{f}_{\sigma }) {{\varvec{N}}}_{\sigma \xi } \Big ] \sqrt{a} d\Omega _e \)

\( {\varvec{\mathcal {R}}}_{v}(\tilde{f},\tilde{f}_{\sigma }, {\varvec{u}}^m, {\varvec{\sigma }}_{nM}^m) \) is the equilibrium residual obtained by summing the elements’ residual load vectors \( \displaystyle {\varvec{\mathcal {R}}}_{v}^e(\tilde{f},\tilde{f}_{\sigma }, {\varvec{u}}^m, {\varvec{\sigma }}_{nM}^m) = \int _{\Omega _{e}} \Big [ {{\varvec{N}}}_{\xi }^T {\varvec{t}}_z(\tilde{f})  {{\varvec{B}}}_{\xi }^{T} {\varvec{\sigma }}_z(\tilde{f}, {\varvec{u}}^m, {\varvec{\sigma }}_{nM}^m) {{\varvec{N}}}_{\sigma \xi }^T {\varvec{\varepsilon }}_z(\tilde{f}_{\sigma }, {\varvec{u}}^m, {\varvec{\sigma }}_{nM}^m) \Big ] \sqrt{a} d\Omega _e.\)
Finite element problem to be solved on \(\Omega _{z}\)

\( {\varvec{q}}^{f f_\sigma }\) is the vector of degree of freedom associated with the F.E. approximations in \(\Omega _{z}\),
 \( {\mathbf {K}}_{\xi }(\tilde{v},\tilde{\tau })\) is obtained by summing the elements’ stiffness matrices:$$\begin{aligned} {\mathbf {K}}_{\xi }^e(\tilde{v},\tilde{\tau })= & {} \int _{\Omega _{ze}} \Big [{{\varvec{B}}}_{z}^{T} {\varvec{k}}_{\xi }^f(\tilde{v}) {{\varvec{B}}}_{z} + {{\varvec{B}}}_{z}^T {\varvec{k}}_{\xi }^{f f_\sigma }(\tilde{v},\tilde{\tau }) {{\varvec{N}}}_{\sigma z} \nonumber \\&+ {{\varvec{N}}}_{\sigma z}^T {\varvec{k}}_{\xi }^{f_\sigma f}(\tilde{v},\tilde{\tau }) {{\varvec{B}}}_{z} + {{\varvec{N}}}_{\sigma z}^T {\varvec{k}}_{\xi }^{f_\sigma f_\sigma }(\tilde{\tau }) {{\varvec{N}}}_{\sigma z} \Big ] \mu dz_e \end{aligned}$$(48)
 \({\varvec{\mathcal {R}}}_{f}(\tilde{v},\tilde{\tau }, {\varvec{u}}^m, {\varvec{\sigma }}_{nM}^m)={\varvec{\mathcal {R}}}_{f}^F(\tilde{v})  {\varvec{\mathcal {R}}}_{f}^{Coup}(\tilde{v},\tilde{\tau }, {\varvec{u}}^m, {\varvec{\sigma }}_{nM}^m)\) is a equilibrium residual with \({\varvec{\mathcal {R}}}_{f}^F(\tilde{v}) = \left. {{\varvec{N}}}_{z}^T {\varvec{t}}_{\xi }(\tilde{v} ) \mu \right _{z=z_F} \) and \( {\varvec{\mathcal {R}}}_{f}^{Coup}(\tilde{v},\tilde{\tau }, {\varvec{u}}^m, {\varvec{\sigma }}_{nM}^m) \) is obtained by the summation of the elements’ residual vectors given by$$\begin{aligned} \int _{\Omega _{ze}} \Big [ {{\varvec{B}}}_{z}^{T} {\varvec{\sigma }}_{\xi }(\tilde{v}, {\varvec{u}}^m, {\varvec{\sigma }}_{nM}^m) + {{\varvec{N}}}_{\sigma z}^T {\varvec{\varepsilon }}_{\xi }(\tilde{\tau }, {\varvec{u}}^m, {\varvec{\sigma }}_{nM}^m) \Big ] \mu dz_e \end{aligned}$$(49)
Numerical results
In this section, an eightnode quadrilateral FE based on the Serendipity interpolation functions is used for the unknowns depending on the inplane coordinates. The geometry of the shell is approximated by this classical FE in the parametric space. The geometrical transformation is based on an explicit map \(\vec \Phi \). A Gaussian numerical integration with 3 \(\times \) 3 points is used to calculate the elementary matrices.
Several static tests are presented with the aim of validating our approach and evaluating its efficiency. First, a convergence study is carried out to determine the suitable mesh for the further analysis. Then, the influence of the approximation on the factor \(1/\mu \) is addressed [1, 85]. Three orders of expansion are considered. The influence of the numerical layers is also studied to illustrate the possibility to refine the transverse description of the displacements and the stresses. The present approach is assessed for deep/shallow and thick/thin crossply/angleply/sandwich shells to show the wide range of validity.

Geometry: Composite crossply cylindrical shell, \(R = 10\), \(\phi \in \{ \pi /8, \pi /3, \pi /2\}\), with the following stacking sequences \([0^{\circ }]\), \([0^{\circ } / 90^{\circ }]\), \([0^{\circ } / 90^{\circ } / 0^{\circ }]\). All layers have the same thickness. \(\displaystyle S=\frac{R}{e} \in \{2, 4, 10, 40, 100\}\). The panel is supposed infinite along the \(x_2 = \xi ^2\) direction (\(b_{\mathcal {C}} = 8 a_{\mathcal {C}})\) (Cf. Fig. 2).

Boundary conditions: Simplysupported shell along its straight edges (transverse and tangential displacements are fixed on the \((\xi ^1,\xi ^2)\) mesh), sinusoidal pressure along the curvature: \(q(\xi ^1)= q_{0} \sin {\dfrac{\pi \xi ^1}{R \phi }}\)

Material properties: \( \begin{array}{lcl} E_{L} &{} = &{} 25 \, \text{ GPa } \; , \; E_{T} = 1 \, \text{ GPa } \; , \; G_{LT} = 0.2 \, \text{ GPa } \; , \\ G_{TT} &{} = &{} 0.5 \,\text{ GPa } \; , \; \nu _{LT} = \nu _{TT} = 0.25. \end{array}\) where L refers to the fiber direction, T refers to the transverse direction.

Mesh: Only a quarter of the structure is meshed. The mesh is constituted of \(N_x \times N_y\) elements in the \(\xi ^1\) and \(\xi ^2\) directions respectively. A space ratio is considered in these two directions (ratio between the size of the larger and the smaller element).

Numerical layers: \(N_z\) is the total number of numerical layers.

Number of dofs: \(Ndof_{xy} =6 (3 N_x N_y + 2 (N_x + N_y) +1)\) and \(Ndof_z = 24 \times \alpha NC +6\) are the number of dofs of the two problems associated with \(v_j^i\) and \(f_j^i\) respectively. \(\alpha \) is the number of numerical layers per physical layer. So the total number of dofs is \(Ndof_{xy} + Ndof_{z}\).

Results: The results are made nondimensional using:
\( \displaystyle \bar{u} =u_1(0,b_{\mathcal {C}}/2,z) \frac{E_T }{e q_0 S^3},\qquad \bar{w} = u_3(a_{\mathcal {C}}/2,b_{\mathcal {C}}/2,z) \frac{100 E_T }{e q_0 S^4} \)
\( \bar{\sigma }_{\alpha \alpha } = \dfrac{ \sigma _{\alpha \alpha }(a_{\mathcal {C}}/2,b_{\mathcal {C}}/2,z)}{q_{0} S^2}, \)
\( \bar{\sigma }_{13} = \dfrac{\sigma _{13}(0,b_{\mathcal {C}}/2,z) }{q_{0} S}, \quad \bar{\sigma }_{33} = \dfrac{ \sigma _{33}(a_{\mathcal {C}}/2,b_{\mathcal {C}}/2,z)}{q_{0}} \)

Reference values: The 2D exact elasticity results are obtained as in [81].

LM4: It refers to the systematic work of Carrera and his “Carrera’s Unified Formulation” (CUF), see [37, 72, 86]. A LayerWise model based on a RMVT approach where each component is expanded until the fourthorder, is given. \(24 NC+6\) unknown functions per node are used in this kinematic.

VSLD4: It refers to the work on the proper generalized decomposition with a spatial separation between (x, y) and z. A fourthorder expansion is used for the 1D problem associated to the zdirection. The formulation is based on a displacement approach (see [79]).
Convergence study
Convergence study—one layer \([0^{\circ }]\)—S = 40—\(\phi =\pi /3\)— \(N_z = NC\)—mesh \(N_x \times 10\) with space ratio (50)
\({{\varvec{N}}}_{{\varvec{x}}}\)  \({\bar{{\varvec{u}}}}{{\varvec{(0,e/2)}}}\)  \(\bar{{{\varvec{w}}}}{{\varvec{(L,0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{11}}}}{{\varvec{(e/2)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{13}}}}{{\varvec{(0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{33}}}}\) max  

4  VSLM4  9.3287  0.0779  \(\) 0.8406  0.3181  \(\) 8.7001 
Error  0.51%  0.02%  8.79%  44.08%  22.50%  
8  VSLM4  9.3744  0.0780  \(\) 0.7908  0.5061  \(\) 7.5489 
Error  0.02%  0.10%  2.35%  11.02%  6.29%  
14  VSLM4  9.3769  0.0780  \(\) 0.7786  0.5484  \(\) 7.2645 
Error  0.00%  0.04%  0.77%  3.60%  2.29%  
20  VSLM4  9.3775  0.0780  \(\) 0.7756  0.5590  \(\) 7.1931 
Error  0.01%  0.03%  0.38%  1.73%  1.28%  
26  VSLM4  9.3775  0.0780  \(\) 0.7744  0.5633  \(\) 7.1654 
Error  0.01%  0.02%  0.22%  0.97%  0.89%  
32  VSLM4  9.3775  0.0780  \(\) 0.7738  0.5651  \(\) 7.1503 
Error  0.01%  0.02%  0.14%  0.67%  0.68%  
Navier  9.3778  0.0780  \(\) 0.7727  0.5688  \(\) 7.1074  
Exact  9.3765  0.0780  \(\) 0.7727  0.5688  \(\) 7.1021 
Influence of the expansion order for the factor \(1/\mu \)
Influence of the orderapproximation of \(1/\mu \) with respect to S—one layer \([0^{\circ } ]\)—\(\phi =\pi /3\)—mesh 26 \(\times \) 10—\(N_z = NC\)
S  Expansion  Model  \(\bar{{{\varvec{u}}}}{{\varvec{(0, e/2}}}{} \mathbf{)}\)  \(\bar{{{\varvec{w}}}}{{\varvec{(L,0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{11}}}}{{\varvec{(e/2)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{13}}}}{{\varvec{(0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{33}}}}\) max 

2  Zeroorder  VSLM4  3.9747  0.7982  \(\) 1.4093  0.4688  0.8883 
Error  16.67%  19.83%  42.59%  15.76%  11.17%  
2  Oneorder  VSLM4  4.9928  1.0179  \(\) 2.2682  0.5908  0.9324 
Error  4.68%  2.24%  7.59%  6.16%  6.76%  
2  Twoorder  VSLM4  4.7940  0.9769  \(\) 2.1818  0.5776  0.8967 
Error  0.51%  1.88%  11.12%  3.78%  10.33%  
Exact  4.7696  0.9956  \(\) 2.4546  0.5565  1.0000  
4  Zeroorder  VSLM4  2.3665  0.2792  \(\) 1.0406  0.5287  0.9225 
error  10.39%  10.50%  21.81%  7.87%  7.75%  
4  Oneorder  VSLM4  2.6745  0.3153  \(\) 1.3452  0.5958  0.9163 
error  1.28%  1.08%  1.08%  3.82%  8.37%  
4  Twoorder  VSLM4  2.6386  0.3112  \(\) 1.3261  0.5909  0.9062 
Error  0.09%  0.25%  0.36%  2.97%  9.38%  
Exact  2.6408  0.3120  \(\) 1.3309  0.5739  1.0000  
10  Zeroorder  VSLM4  2.5623  0.1094  \(\) 0.8071  0.5558  \(\) 1.3651 
Error  4.54%  4.55%  9.30%  4.13%  9.35%  
10  Oneorder  VSLM4  2.6932  0.1150  \(\) 0.8937  0.5838  \(\) 1.4207 
Error  0.33%  0.33%  0.44%  0.69%  5.66%  
10  Twoorder  VSLM4  2.6846  0.1146  \(\) 0.8908  0.5829  \(\) 1.4165 
Error  0.01%  0.01%  0.11%  0.55%  5.94%  
Exact  2.6843  0.1146  \(\) 0.8898  0.5798  \(\) 1.5059  
40  Zeroorder  VSLM4  9.2631  0.0770  \(\) 0.7553  0.5564  \(\) 7.0856 
Error  1.21%  1.20%  2.25%  2.19%  0.23%  
40  Oneorder  VSLM4  9.3800  0.0780  \(\) 0.7746  0.5629  \(\) 7.1657 
Error  0.04%  0.04%  0.25%  1.05%  0.89%  
40  Twoorder  VSLM4  9.3775  0.0780  \(\) 0.7744  0.5633  \(\) 7.1654 
Error  0.01%  0.02%  0.22%  0.97%  0.89%  
Exact  9.3765  0.0780  \(\) 0.7727  0.5688  \(\) 7.1021 
Considering different opening angles of the structure in Table 3, it can be inferred from these results that the use of the twoorder expansion allows us again to decrease significantly the error rate for both displacements and stresses. Nevertheless, the influence on the transverse normal stress does not appear for the shallow shells (\(\phi \le \pi /3\)).
Influence of the orderapproximation of \(1/\mu \) with respect to \(\phi \)—one layer \([0^{\circ } ]\)—\(S=4\)—mesh 26 \(\times \) 10—\(N_z = NC\)
\({\varvec{\phi }}\)  Expansion  Model  \(\bar{{{\varvec{u}}}}{{\varvec{(0,e/2)}}}\) max  \(\bar{{{\varvec{w}}}}{{\varvec{(L,0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{11}}}}{{\varvec{(e/2)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{13}}}}{{\varvec{(0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{33}}}}\) max 

\(\phi =\pi /8\)  Zeroorder  VSLM4  0.1758  0.0206  \(\) 0.1721  0.1465  0.9699 
Error  20.37%  14.13%  35.61%  8.30%  3.01%  
Oneorder  VSLM4  0.1943  0.0237  \(\) 0.2138  0.1676  0.9782  
Error  11.98%  1.10%  20.01%  4.94%  2.18%  
Twoorder  VSLM4  0.1925  0.0235  \(\) 0.2117  0.1670  0.9698  
Error  12.79%  1.97%  20.78%  4.57%  3.02%  
Exact  0.2207 (\(e/2\))  0.0240  \(\) 0.2673  0.1597  1.0000  
\(\phi =\pi /3\)  Zeroorder  VSLM4  2.3665  0.2792  \(\) 1.0406  0.5287  0.9225 
Error  10.39%  10.50%  21.81%  7.87%  7.75%  
Oneorder  VSLM4  2.6745  0.3153  \(\) 1.3452  0.5958  0.9163  
Error  1.28%  1.08%  1.08%  3.82%  8.37%  
Twoorder  VSLM4  2.6386  0.3112  \(\) 1.3261  0.5909  0.9062  
Error  0.09%  0.25%  0.36%  2.97%  9.38%  
Exact  2.6408  0.3120  \(\) 1.3309  0.5739  1.0000  
\(\phi =\pi /2\)  Zeroorder  VSLM4  19.7700  1.2434  \(\) 2.4271  0.9709  \(\) 1.4613 
Error  9.90%  9.85%  21.19%  8.83%  15.41%  
Oneorder  VSLM4  22.3131  1.4019  \(\) 3.1382  1.0922  \(\) 1.6628  
Error  1.69%  1.63%  1.89%  2.57%  3.75%  
Twoorder  VSLM4  21.9662  1.3801  \(\) 3.0886  1.0822  \(\) 1.6379  
Error  0.10%  0.06%  0.28%  1.62%  5.19%  
Exact  21.9433  1.3793  \(\) 3.0799  1.0649  \(\) 1.7275  
\(\phi =5\pi /6\)  Zeroorder  VSLM4  1268.7500  39.7516  \(\) 15.1031  4.0278  \(\) 13.0670 
Error  9.01%  8.99%  20.62%  10.01%  10.39%  
Oneorder  VSLM4  1436.1875  44.9875  \(\) 19.6075  4.5312  \(\) 14.8620  
Error  3.00%  3.00%  3.05%  1.24%  1.92%  
Twoorder  VSLM4  1409.4375  44.1469  \(\) 19.2512  4.4970  \(\) 14.5900  
Error  1.08%  1.08%  1.18%  0.48%  0.05%  
Exact  1394.3688  43.6759  \(\) 19.0264  4.4756  \(\) 14.5826 
Influence of the number of numerical layers
One layer \([0^{\circ }]\)—\(\phi =\pi /3\)—mesh 26 \(\times \) 10—\(N_z = 4 \times NC\)—twoorder expansion of \(1/\mu \)
S  Model  \(\bar{{{\varvec{u}}}}{{\varvec{(0,e/2)}}}\)  \(\bar{{{\varvec{w}}}}{{\varvec{(L,0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{11}}}}{{\varvec{(e/2)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{13}}}}{{\varvec{(0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{33}}}}\) max 

2  VSLM4  4.7645  0.9938  \(\) 2.4609  0.5560  1.0111 
Error  0.11%  0.18%  0.25%  0.09%  1.11%  
Exact  4.7696  0.9956  \(\) 2.4546  0.5565  1.  
4  VSLM4  2.6411  0.3119  \(\) 1.3329  0.5720  1.0110 
Error  0.01%  0.02%  0.16%  0.32%  1.10%  
Exact  2.6408  0.3120  \(\) 1.3309  0.5739  1.  
10  VSLM4  2.6847  0.1147  \(\) 0.8907  0.5784  \(\) 1.5176 
Error  0.02%  0.02%  0.09%  0.24%  0.77%  
Exact  2.6843  0.1146  \(\) 0.8898  0.5798  \(\) 1.5059 
Influence of the orderapproximation of \(1/\mu \)—one layer \([0^{\circ } ]\)  \(S=4\)—mesh 26 \(\times \) 10—numerical layers \(N_z = 4 \times NC\)
\(\varvec{\phi }\)  Expansion  Model  \(\bar{{{\varvec{u}}}}{{\varvec{(0,e/2)}}}\) max  \(\bar{{{\varvec{w}}}}{{\varvec{(L,0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{11}}}}{{\varvec{(e/2)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{13}}}}{{\varvec{(0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{33}}}}\) max 

\(\phi =\pi /8\)  Zeroorder  VSLM4  \(\) 0.0085  0.0211  \(\) 0.2094  0.1415  1.0027 
Error  9.30%  11.86%  21.63%  11.44%  0.27%  
Oneorder  VSLM4  \(\) 0.0096  0.0243  \(\) 0.2694  0.1621  1.0103  
Error  2.59%  1.30%  0.82%  1.48%  1.03%  
Twoorder  VSLM4  \(\) 0.0095  0.0241  \(\) 0.2670  0.1616  1.0023  
Error  1.72%  0.46%  0.09%  1.14%  0.23%  
Exact  \(\) 0.0094  0.0240  \(\) 0.2673  0.1597  1.0000  
\(\phi =\pi /3 \)  Zeroorder  VSLM4  2.3664  0.2797  \(\) 1.0442  0.5119  1.0101 
Error  10.39%  10.32%  21.54%  10.80%  1.01%  
Oneorder  VSLM4  2.6764  0.3160  \(\) 1.3518  0.5766  1.0221  
Error  1.35%  1.30%  1.57%  0.49%  2.21%  
Twoorder  VSLM4  2.6411  0.3119  \(\) 1.3329  0.5720  1.0110  
Error  0.01%  0.02%  0.16%  0.32%  1.10%  
Exact  2.6408  0.3120  \(\) 1.3309  0.5739  1.0000 
In Table 6, the suitable number of numerical layers is given for different slenderness ratios so as to obtain an error rate of about 1% for \(\phi =\pi /8\) (the most critical test). As expected, it can be seen that the number of numerical layers increases with the thickness of the structure. Even considering a structure with only one physical layer, the refinement of the description of mechanical quantities is required as their distributions through the thickness could be very complex for a shell structure. For illustration, the distributions of the displacement \(\bar{u}\) and the stresses \(\bar{\sigma }_{11}\), \(\bar{\sigma }_{13}\) through the thickness are provided in Fig. 4. An oscillating behavior occurs for the displacement and the inplane stress, which is quite different than the plate case. Moreover, an asymmetrical distribution can be observed. The maximum value of the transverse shear stress is obtained near the top of the homogeneous shell.
For the present approach, it should be also noted that the additional computational cost inducing by the zrefinement is negligible as only the number of dofs of the 1D problem increases (\(Ndof_z = 24 \times \alpha NC +6\), see Table 6). The size of the 2D problem remains unchanged. This is a main difference with respect to a LW approach where the total number of unknowns would be \(Ndof_{xy} \times Ndof_{z} = Ndof_{xy} \times (24 \times \alpha NC +6)\).
One layer \([0^{\circ }]\)—\(\phi =\pi /8\)—mesh 26\(\times \)10—twoorder expansion of \(1/\mu \)
S  Model  \({{\varvec{N}}}_{{\varvec{z}}}{{\varvec{/}}}{{\varvec{NC}}}\)  \({{\varvec{Ndof}}}_{{\varvec{z}}}\)  \(\bar{{{\varvec{u}}}}{{\varvec{(0,e/2)}}}\)  \(\bar{{{\varvec{w}}}}{{\varvec{(L,0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{11}}}}{{\varvec{(e/2)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{13}}}}{{\varvec{(0)}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{33}}}}\) max 

2  VSLM4  8  198  0.2156  0.0867  \(\) 0.3054  0.1502  0.9979 
Error  1.03%  1.53%  0.78%  1.85%  0.21%  
Exact  0.2179  0.0854  \(\) 0.3078  0.1475  1.0000  
4  VSLM4  4  102  0.2202  0.0241  \(\) 0.2670  0.1616  1.0023 
Error  0.22%  0.46%  0.09%  1.14%  0.23%  
Exact  0.2207  0.0240  \(\) 0.2673  0.1597  1.0000  
10  VSLM4  2  54  0.1334  0.0051  \(\) 0.1484  0.1826  1.0000 
Error  0.01%  0.04%  0.05%  1.02%  0.00%  
Exact  0.1335  0.0051  \(\) 0.1483  0.1808  1.0000 
Bending analysis of laminated shells under a sinusoidal pressure
Crossply test case
Angleply test case
Three layers \([45^{\circ }/45^{\circ }/45^{\circ }]\,\)– \(\phi =1\)—S = 4
\({{\varvec{z/e}}}\)  \(\bar{{{\varvec{u}}}}\)  \(\bar{{{\varvec{w}}}} \)  \(\bar{\varvec{\sigma }}_{{{\varvec{11}}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{13}}}}\)  \(\bar{\varvec{\sigma }}_{{{\varvec{33}}}}\)  

VSLM4  exact  VSLM4  Exact  VSLM4  Exact  VSLM4  Exact  VSLM4  Exact  
\(\) 1/2  \(\) 2.41  \(\) 2.42  7.297  \(\) 1.16  \(\) 1.16  0.00  0  0.00  0  
\(\) 1/3  \(\) 2.14  \(\) 2.14  7.329  \(\) 0.43  \(\) 0.43  0.42  0.42  \(\) 0.41  \(\) 0.41  
\(\) 1/6\(^{}\,\)  \(\) 1.68  \(\) 1.69  7.329  0.05  0.05  0.48  0.48  \(\) 0.26  \(\) 0.26  
\(\) 1/6\(^+\)  \(\) 1.68  \(\) 1.69  7.329  \(\) 0.61  \(\) 0.61  0.48  0.48  \(\) 0.26  \(\) 0.26  
0  \(\) 0.02  \(\) 0.02  7.320  7.319  \(\) 0.02  \(\) 0.02  0.59  0.60  \(\) 0.15  \(\) 0.15 
1/6\(^{}\,\)  1.51  1.51  7.318  0.51  0.51  0.43  0.43  0.27  0.27  
1/6\(^+\)  1.51  1.51  7.318  \(\) 0.04  \(\) 0.04  0.43  0.43  0.27  0.27  
1/3  1.84  1.84  7.325  0.39  0.39  0.32  0.32  0.56  0.56  
1/2  2.03  2.02  7.334  0.92  0.92  0.00  0  1.00  1. 
Bending analysis of a sandwich shell under a sinusoidal pressure

Geometry: Cylindrical shell with \(R = 10\), \(S=4\). The thickness of each face sheet is \(\frac{e}{10}\). The panel is supposed infinite along the \(x_2 = \xi ^2\) direction.

Boundary conditions: Simplysupported shell along its straight edges, subjected to a sinusoidal pressure along the curvature: \(q(\xi ^1)= q_{0} \sin {\dfrac{\pi \xi ^1}{R \phi }}.\)

Material properties: Face: \(E_{face}= 73\) GPa, \(\nu _{face} = 0.34\), \(G_{face} = 27.239\) GPa Core: \(E_{L} = E_{T} = \alpha 0.01 \) MPa, \(E_{zz} = \alpha 75.85 \) MPa, \(\nu = 0.01 \), \(G = \alpha 22.5 \) MPa, with \(\alpha = 1\) (B1) or \(\alpha = 10^{2}\) (B2)

Mesh: Mesh 26 \(\times \) 10 with a space ratio 50 is used for the quarter of the plate.

Number of dofs: \(Ndof_{xy} = 5.118\) and \(Ndof_z = 24 \times NC +6 { = 78}\)
 Results: The results are made nondimensional using:$$\begin{aligned} \displaystyle \bar{w}= & {} u_3(a_{\mathcal {C}}/2,b_{\mathcal {C}}/2,z) \frac{10 E_{face} }{S^4 e q_0} \\ \bar{\sigma }_{13}= & {} \dfrac{\sigma _{13}(0,b_{\mathcal {C}}/2,z) }{q_{0} S}, \; \; \; \bar{\sigma }_{33} = \dfrac{ \sigma _{33}(a_{\mathcal {C}}/2,b_{\mathcal {C}}/2,z)}{q_{0}} \end{aligned}$$

Reference values: the LM4 results are given in [83].
Bending analysis of laminated shells under a constant pressure
Conclusion
In this paper, a variable separation method in the framework of Reissner’s Mixed Variational Theorem is proposed for the modeling of laminated composite and sandwich shells. A 8node FE for the inplane unknown approximation and a fourthorder LW description for the thickness unknown approximation are used. In this formulation, all interface conditions are exactly satisfied. The approach has been assessed through different benchmarks proposed in the open literature. The influence of classical strain assumptions (approximation of \(1/\mu \)) is discussed. We have shown the importance of the expansion of this term for thick shells. At the same time, the accuracy of the results could be also improved by using numerical layers in each physical layer without increasing significantly the computational cost. In fact, the number of layers has no influence on this cost as only the cost of the 1D problem is affected. This is particularly interesting in the framework of a mixed approach, where the number of unknowns involving both displacements and transverse stresses becomes very important in a classical LW method. Comparisons with exact reference solutions, results available in open literature have shown the very good accuracy of the method for a wide range of applications. Deep/shallow, very thick/thin shell structures with various stacking sequences and high anisotropy configurations can be modeled with an excellent accuracy. So, the present work can provide quasi3D results avoiding expensive 3D FEM or LW computations. Therefore, this method seems to have very attractive features.
Acknowledgements
Not applicable.
Authors’ contributions
All authors have read and approved the final manuscript.
Funding
Not applicable.
Availability of data and materials
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Notes
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