### Preliminaries of multi-layered shell model

Figure 1 shows a typical laminated shell structure with curvatures. This geometry can be described in the orthogonal curvilinear reference system (\(\alpha ,\beta ,z\)), in which \(\alpha \) and \(\beta \) indicate the two in-plane directions and *z* the through-thickness direction which is usually measured with reference to the middle surface. The infinitesimal in-plane area *dS* and volume *dV* can be written as:

$$\begin{aligned} d S= & {} H_\alpha \; H_\beta \; d\alpha \; d\beta \; = H_\alpha \; H_\beta \; d\Omega , \end{aligned}$$

(1)

$$\begin{aligned} dV= & {} H_\alpha H_\beta H_z \; d\alpha \; d\beta \;dz. \end{aligned}$$

(2)

where \(d\Omega \) is the infinitesimal in-plane area on the middle surface, and \(H_\alpha , H_\beta \) and \(H_z\) are:

$$\begin{aligned} \begin{aligned} H_\alpha = A (1 + z/R_\alpha ), \;\;\; H_\beta = B (1 + z/R_\beta ), \;\;\; H_z = 1. \end{aligned} \end{aligned}$$

(3)

in which \(R_\alpha \) and \(R_\beta \) are the principal radii of the middle surface, *A* and *B* the coefficients of the first fundamental form of \(\Omega \). The present work considers only shells with constant curvatures, for which \(A=B=1\). For more details about shell formulations, the reader is referred to [44, 45].

The strains and stresses defined in the curvilinear reference system are:

$$\begin{aligned} {\varvec{\epsilon }}= & {} \{\epsilon _{\alpha \alpha }, \epsilon _{\beta \beta }, \epsilon _{zz}, \epsilon _{\alpha z}, \epsilon _{\beta z}, \epsilon _{\alpha \beta } \}^T \end{aligned}$$

(4)

$$\begin{aligned} {\varvec{\sigma }}= & {} \{\sigma _{\alpha \alpha }, \sigma _{\beta \beta }, \sigma _{zz}, \sigma _{\alpha z}, \sigma _{\beta z}, \sigma _{\alpha \beta } \}^T \end{aligned}$$

(5)

The strains \({\varvec{\epsilon }}\) can be obtained through the geometrical relations:

$$\begin{aligned} {\varvec{\epsilon }} = {\varvec{b}} {\varvec{u}} \end{aligned}$$

(6)

wherein \({\varvec{u}}=\{u, v, w\}^T\) is the displacement vector, and \({\varvec{b}}\) the differential operators matrix, whose explicit expression reads:

$$\begin{aligned} {\varvec{b}} = \begin{bmatrix} \frac{\partial _\alpha }{H_\alpha }&\quad 0&\quad \frac{1}{H_\alpha R_\alpha } \\ 0&\quad \frac{\partial _\beta }{H_\beta }&\quad \frac{1}{H_\beta R_\beta } \\ 0&\quad 0&\quad \partial _z \\ \partial _z -\frac{1}{H_\alpha R_\alpha }&\quad 0&\quad \frac{\partial _\alpha }{H_\alpha } \\ 0&\quad \partial _z -\frac{1}{H_\beta R_\beta }&\quad \frac{\partial _\beta }{H_\beta } \\ \frac{\partial _\beta }{H_\beta }&\quad \frac{\partial _\alpha }{H_\alpha }&\quad 0 \end{bmatrix} \end{aligned}$$

(7)

The stresses can be attained from the constitutive equations as follows:

$$\begin{aligned} \begin{aligned} {\varvec{\sigma }} = {\varvec{C}} {\varvec{\epsilon }} \end{aligned} \end{aligned}$$

(8)

in which \({\varvec{C}}\) is the material coefficients matrix which is obtained by transforming the original form \({\varvec{C}}_0\) from the material coordinate system (1, 2, 3) to the global system (\(\alpha ,\beta ,z\)). The orthotropic material coefficients are characterized by nine independent coefficients, namely Young’s moduli, shear moduli, and Poisson ratios [7].

### Carrera Unified Formulation (CUF) for refined 2D models

Through Carrera Unified Formulation (CUF), the displacement field of a shell structure can be assumed as:

$$\begin{aligned} {\varvec{u}}(\alpha ,\beta ,z)= F_\tau (z) {\varvec{u}}_{\tau }(\alpha ,\beta ) \end{aligned}$$

(9)

where \({\varvec{u}}_{\tau }(\alpha ,\beta )\) are the in-plane displacement vectors, and functions \(F_{\tau }(z)\) are related to the theories of structures (TOS). Since \(F_{\tau }(z)\) depends only on the thickness coordinates, they are also referred to as thickness functions. By increasing the polynomial order of these thickness functions, the shell kinematic models can be refined. Both Equivalent-Single-Layer (ESL) and Layer-Wise (LW) models can be accounted for in this framework, as elaborated by Carrera et al. [46]. The FSDT [47] can be treated as a particular case of the HOT models adopting Taylor expansions (TE). In the LW model framework, Lagrangian polynomial expansions (LE) can be used to formulate kinematics with only translational degrees of freedom. More discussions about these two types of refined kinematic models can be found in the work of Carrera et al. [46]. By using LE, the interfacial continuity of transverse stresses can be approximately achieved when the thickness functions are adequately refined, as demonstrated by Carrera et al. [26].

When the FE discretization is introduced, the in-plane displacements of a shell structure are approximated through the shape functions \(N_i(\alpha , \beta )\) through:

$$\begin{aligned} {\varvec{u}}_{\tau }(\alpha ,\beta ) = N_i(\alpha , \beta ) {\varvec{u}}_{i\tau } \end{aligned}$$

(10)

in which \({\varvec{u}}_{i\tau }\) are nodal unknown variables. The above expression leads to FE formulation in the framework of CUF:

$$\begin{aligned} \begin{aligned} {\varvec{u}}(\alpha ,\beta ,z)&= F_\tau (z) N_i(\alpha , \beta ) {\varvec{u}}_{i\tau } \\ \delta {\varvec{u}}(\alpha ,\beta ,z)&= F_s(z) N_j(\alpha , \beta ) \delta {\varvec{u}}_{js} \end{aligned} \end{aligned}$$

(11)

where \(\delta \) indicates the virtual variation. The above expression is compact through the use of repeated indexes. By applying the Principle of Virtual Displacements (PVD), the general expressions of the stiffness matrix and load vector of the FE model, namely the *Fundamental Neuclei* (FNs), can be obtained. The explicit expressions of the FNs are given in [12]. As expounded by Carrera et al. [46], the CUF-type FE formulation is independent of the kinematic assumptions adopted and is a general framework for the development of refined FE models. For more details about the derivation of shell FE formulations in the framework of CUF, the reader is referred to the work of Carrera et al. [46].

### Decomposition of strain energy in refined shell models

For a general laminated shell structure, the strain energy can be decomposed into four parts as follows:

$$\begin{aligned} E_{pn}= & {} \frac{1}{2} \int _V (\varepsilon _{\alpha \alpha } \; \sigma _{\alpha \alpha } + \varepsilon _{\beta \beta } \; \sigma _{\beta \beta } ) \; dV \end{aligned}$$

(12)

$$\begin{aligned} E_{ps}= & {} \frac{1}{2} \int _V \varepsilon _{\alpha \beta } \; \sigma _{\alpha \beta } \; dV \ \end{aligned}$$

(13)

$$\begin{aligned} E_{zs}= & {} \frac{1}{2} \int _V ( \varepsilon _{\alpha z} \; \sigma _{\alpha z} + \varepsilon _{\beta z} \; \sigma _{\beta z} ) dV \end{aligned}$$

(14)

$$\begin{aligned} E_{zz}= & {} \frac{1}{2} \int _V \varepsilon _{zz} \; \sigma _{zz} \; dV \end{aligned}$$

(15)

where \(E_{pn}\) represents the in-plane normal energy, \(E_{ps}\) the in-plane shear energy, \(E_{zs}\) the transverse shear energy, and \(E_{zz}\) the thickness stretch energy. The transverse shear energy allows us to evaluate the shear locking effects in shell elements. The introduction of the thickness stretch energy makes it convenient to assess the performance of the adopted structural theory. Note that the above decomposition applies to arbitrarily laminated shell structures.

To calculate the strain energy components, their corresponding stiffness matrices are necessary. These matrices can be obtained through standard FE procedure in the framework of CUF. Taking the transverse shear energy \(E_{zs}\) as an example, by recalling the PVD, one has:

$$\begin{aligned} \delta E_{zs} = \int _V (\delta \varepsilon _{\alpha z} \sigma _{\alpha z} + \delta \varepsilon _{\beta z} \sigma _{\beta z} ) dV = \delta {\varvec{u}}_{js} \cdot {\varvec{k}}^{zs}_{ij\tau s} \cdot {\varvec{u}}_{i\tau } \end{aligned}$$

(16)

wherein \({\varvec{k}}^{zs}_{ij\tau s}\) is the FNs for the transverse shear stiffness matrix of the element. By substituting the displacement approximations (Eq. 11) into the geometrical relations (Eq. 6), and considering the constitutive equations (Eq. 8), one obtains:

$$\begin{aligned}&\begin{Bmatrix} \delta \varepsilon _{\alpha z}\\ \delta \varepsilon _{\beta z} \end{Bmatrix} = {\varvec{b}}_{zs} \cdot \delta {\varvec{u}} = \begin{bmatrix} \partial _z -\frac{1}{H_\alpha R_\alpha }&\quad 0&\quad \frac{\partial _\alpha }{H_\alpha } \\ 0&\quad \partial _z -\frac{1}{H_\beta R_\beta }&\quad \frac{\partial _\beta }{H_\beta } \end{bmatrix} \cdot N_j F_s \delta {\varvec{u}}_{js} \end{aligned}$$

(17)

$$\begin{aligned}&\begin{Bmatrix} \sigma _{\alpha z}\\ \sigma _{\beta z} \end{Bmatrix} = {\varvec{C}}_{zs} \cdot {\varvec{\varepsilon }} = \begin{bmatrix} C_{41}&\quad C_{42}&\quad C_{43}&\quad C_{44}&\quad C_{45}&\quad C_{46} \\ C_{51}&\quad C_{52}&\quad C_{53}&\quad C_{54}&\quad C_{55}&\quad C_{56} \end{bmatrix} \cdot ({\varvec{b}} \; N_i F_{\tau } {\varvec{u}}_{i\tau } ) \end{aligned}$$

(18)

thus the FNs for the transverse shear stiffness matrix \({\varvec{k}}^{zs}_{it\tau s}\) can be obtained as:

$$\begin{aligned} {\varvec{k}}^{zs}_{ij\tau s} = \int _V ({\varvec{b}}_{zs}^T N_j F_s) \; {\varvec{C}}_{zs} \; ({\varvec{b}} \; N_i F_\tau ) \; dV \end{aligned}$$

(19)

Through the assembly of \({\varvec{k}}^{zs}_{ij\tau s}\) according to the standard procedure of FE formulation in CUF framework, the transverse shear stiffness matrix \({\varvec{K}}_{zs}\) can be obtained. When the displacement solutions are obtained, the transverse shear strain energy can be calculated through:

$$\begin{aligned} E_{zs} =\frac{1}{2} \int _V ( \varepsilon _{\alpha z} \; \sigma _{\alpha z} + \varepsilon _{\beta z} \; \sigma _{\beta z} ) dV =\frac{1}{2} \; {\varvec{u}}^T \cdot {\varvec{K}}_{zs} \cdot {\varvec{u}} \end{aligned}$$

(20)

The in-plane normal stiffness matrix \({\varvec{K}}_{pn}\), in-plane shear stiffness matrix \({\varvec{K}}_{ps}\), and out-of-plane normal stiffness matrix \({\varvec{K}}_{zz}\) can be achieved accordingly by means of the following FNs:

$$\begin{aligned} {\varvec{k}}^{pn}_{ij\tau s}= & {} \int _V ({\varvec{b}}_{pn}^T N_j F_s) \; {\varvec{C}}_{pn} \; ({\varvec{b}} \; N_i F_\tau ) \; dV \end{aligned}$$

(21)

$$\begin{aligned} {\varvec{k}}^{ps}_{ij\tau s}= & {} \int _V ({\varvec{b}}_{ps}^T N_j F_s) \; {\varvec{C}}_{ps} \; ({\varvec{b}} \; N_i F_\tau ) \; dV \end{aligned}$$

(22)

$$\begin{aligned} {\varvec{k}}^{zz}_{ij\tau s}= & {} \int _V ({\varvec{b}}_{zz}^T N_j F_s) \; {\varvec{C}}_{zz} \; ({\varvec{b}} \; N_i F_\tau ) \; dV \end{aligned}$$

(23)

wherein \({\varvec{b}}_{pn}\), \({\varvec{b}}_{ps}\), and \({\varvec{b}}_{zz}\) are the sub-matrices of the differential operators matrix \({\varvec{b}}\) as in Eq. 7, and their explicit expressions are:

$$\begin{aligned} {\varvec{b}}_{pn}= & {} \begin{bmatrix} \frac{\partial _\alpha }{H_\alpha }&\quad 0&\quad \frac{1}{H_\alpha R_\alpha } \\ 0&\quad \frac{\partial _\beta }{H_\beta }&\quad \frac{1}{H_\beta R_\beta } \end{bmatrix} \end{aligned}$$

(24)

$$\begin{aligned} {\varvec{b}}_{ps}= & {} \begin{bmatrix} \frac{\partial _\beta }{H_\beta }&\quad \frac{\partial _\alpha }{H_\alpha }&\quad 0 \end{bmatrix} \end{aligned}$$

(25)

$$\begin{aligned} {\varvec{b}}_{zz}= & {} \begin{bmatrix} 0&\quad 0&\quad \partial _z \end{bmatrix} \end{aligned}$$

(26)

and their corresponding material coefficients matrices (sub-matrices of the material coefficients matrix \({\varvec{C}}\)) are as follows:

$$\begin{aligned} {\varvec{C}}_{pn}= & {} \begin{bmatrix} C_{11}&\quad C_{12}&\quad C_{13}&\quad C_{14}&\quad C_{15}&\quad C_{16} \\ C_{21}&\quad C_{22}&\quad C_{23}&\quad C_{24}&\quad C_{25}&\quad C_{26} \end{bmatrix} \end{aligned}$$

(27)

$$\begin{aligned} {\varvec{C}}_{ps}= & {} \begin{bmatrix} C_{61}&\quad C_{62}&\quad C_{63}&\quad C_{64}&\quad C_{65}&\quad C_{66} \end{bmatrix} \end{aligned}$$

(28)

$$\begin{aligned} {\varvec{C}}_{zz}= & {} \begin{bmatrix} C_{31}&\quad C_{32}&\quad C_{33}&\quad C_{34}&\quad C_{35}&\quad C_{36} \end{bmatrix} \end{aligned}$$

(29)

In the end, the complete stiffness FNs can be obtained as the summation of these terms as:

$$\begin{aligned} {\varvec{k}}_{ij\tau s} = {\varvec{k}}^{pn}_{ij\tau s} + {\varvec{k}}^{ps}_{ij\tau s} + {\varvec{k}}^{zs}_{ij\tau s} + {\varvec{k}}^{zz}_{ij\tau s} \end{aligned}$$

(30)

If the multi-layered shell has symmetric lamination properties, the neutral surface of bending will coincide with the geometrical middle surface, and the in-plane normal strain energy, as in Eq. 12, can be further decomposed into membrane energy \(E_{memb}\) and bending energy \(E_{bend}\) conveniently through:

$$\begin{aligned} E_{memb}= & {} \frac{1}{2} \int _V ( \varepsilon _{\alpha \alpha }^0 \; \sigma _{\alpha \alpha } + \varepsilon _{\beta \beta }^0 \; \sigma _{\beta \beta } ) \; dV \end{aligned}$$

(31)

$$\begin{aligned} E_{bend}= & {} \frac{1}{2} \int _V [ (\varepsilon _{\alpha \alpha }-\varepsilon _{\alpha \alpha }^0) \; \sigma _{\alpha \alpha } + (\varepsilon _{\beta \beta }-\varepsilon _{\beta \beta }^0) \; \sigma _{\beta \beta } ] \; dV \end{aligned}$$

(32)

wherein \(\varepsilon _{\alpha \alpha }^0\) and \(\varepsilon _{\beta \beta }^0\) are the normal strains due to the mid-surface straining, and they can be attained by means of:

$$\begin{aligned} \begin{Bmatrix} \varepsilon _{\alpha \alpha }^0\\ \varepsilon _{\beta \beta }^0 \end{Bmatrix} = {\varvec{b}}_{pn} \cdot {\varvec{u}} = \begin{bmatrix} \frac{\partial _\alpha }{H_\alpha }&\quad 0&\quad \frac{1}{H_\alpha R_\alpha } \\ 0&\quad \frac{\partial _\beta }{H_\beta }&\quad \frac{1}{H_\beta R_\beta } \end{bmatrix} \cdot N_j F_{s}(0) \; {\varvec{u}}_{js} \end{aligned}$$

(33)

By following the procedure described before, \({\varvec{k}}_{ij\tau s}^{memb}\), the FNs for the membrane stiffness matrix \({\varvec{K}}_{memb}\), can be derived. The bending energy can be then obtained through:

$$\begin{aligned} E_{bend} = E_{pn} - E_{memb} \end{aligned}$$

(34)

This separation of membrane and bending energy components provides the convenience to evaluate the existence of membrane locking and better understand the structural responses. It should be noted that \(E_{pn}\) and \(E_{ps}\) are both in-plane strain energy components, however since in laminated plates and shells the calculation \(E_{ps}\) does not dependent on a specific neutral surface as the membrane and bending energy components do, it is considered apart in the present article.