Different kinematics including 7 [ 13 , 14 ] or 9 parameters [ 15 ] have been proposed. In this way, the slope discontinuity at the interface between two adjacent layers is introduced. It allows to describe the so-called 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 Layer-Wise 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 in-plane 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 post-processing steps [ 24 ]. Thus, higher-order approaches taking into account the transverse normal effect have been developed.
The three-dimensional constitutive law is used. Second-, third- and fourth-order expansions are discussed in [ 4 , 26 , 27 ].
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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. As an alternative, refined models have been developed in order to improve the accuracy of ESL models avoiding the additional computational cost of LW approach. The deduced model can be derived from the CST [ 29 ], as well as from an HSDT with a third-order theory [ 30 , 31 , 32 , 33 ] or the sinus model [ 1 ].
The number of parameters of the resulting displacement model varies from 5 to While the above references propose models still relying on the 2D constitutive law, the papers [ 35 , 36 ] extend the zig—zag approach to include the transverse normal stress.
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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, semi-analytical 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. 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 well-describe the interlaminar continuous transverse stresses. Most of the ESLM fail to represent these in the most critical cases, unless using a post-processing treatment [ 11 , 24 , 47 , 48 , 49 ]. To overcome these drawbacks, alternative formulations to the displacement-based approach have been developed. 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 in-plane 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.
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An advanced method is proposed in [ 60 ] by considering all interlaminar stresses between two adjacent layers as primary variables and also a higher-order LW displacement. 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. 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.
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. 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. A piecewise fourth-order Lagrange polynomial of z is chosen across the thickness of each layer.
This process yields to two separate linear problems, i. These two problems are solved successively within an iterative scheme. The interesting feature of this approach lies on the possibility to have a higher-order z -expansion 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.
We now outline the remainder of this article. First, the shell definition and the differential geometry are recalled. The principles of the PGD are precised in the framework of our study. The particular assumption on the assumed variables yields to a non-linear 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 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 ]. For a point on the shell middle surface, the covariant basis vectors defining the tangent plane to the middle surface are usually obtained as follows:.
Their covariant, contravariant and mixed form definitions are given by:. The geometry of a shell can also be defined using contravariant or mixed forms. Furthermore, covariant and contravariant differentiation involving Christoffel symbols are not detailed here and readers can refer to the book [ 84 ].
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The stress tensor is obtained from the strain tensor using the constitutive equations. In case of laminated shells composed of orthotropic plies, this reference frame ensures to consistently take into account the different material orientations of the layers. The tensorial relation is:. It should be noted that the stress tensor is defined in the covariant reference frame, whereas the strain components are defined in the contravariant frame. If the frame is assumed to be orthonormal then covariant and contravariant components are equal, that is super-script and sub-script are interchangeable.
The shell can be made of NC perfectly bonded orthotropic layers. Using the previous assumptions and the separation between transverse and in-plane components, the three dimensional constitutive law of the k th layer is given by:.
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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:. The body force is not considered in this expression. 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 ]. A cylindrical shell see Fig.
The strain field in Eq. Equation 18 must be introduced at this level in order to obtain the compatible strain expansion along the normal coordinate z of the shell. So, we obtain:. The resolution of Eq. The test functions defined by Eqs.
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From Eqs. These two equations are linear. The strains and the assumed transverse stresses in Eq. The introduction of the finite element approximation Eq. The strain in Eq. The introduction of the finite element discretization Eq. In this section, an eight-node quadrilateral FE based on the Serendipity interpolation functions is used for the unknowns depending on the in-plane coordinates.
The geometry of the shell is approximated by this classical FE in the parametric space. 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. 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. Unless otherwise mentioned, the numerical assessments are based on the test case proposed by Ren [ 81 ]. It concerns a semi-infinite simply-supported cylindrical shell submitted to a sinusoidal pressure, as detailed below:.
All layers have the same thickness. Mesh: Only a quarter of the structure is meshed. A space ratio is considered in these two directions ratio between the size of the larger and the smaller element. Reference values: The 2D exact elasticity results are obtained as in [ 81 ]. Few iterations are required to reach the convergence of the fixed point algorithm cf.