Composite Structures 309 (2023) 116542

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Composite Structures
journal homepage: www.elsevier.com/locate/compstruct

Dynamic analysis of anisotropic doubly-curved shells with general

boundary conditions, variable thickness and arbitrary shape
Francesco Tornabene a, *, Matteo Viscoti a, Rossana Dimitri a, Luciano Rosati b
a
Department of Innovation Engineering, School of Engineering, Università del Salento, 73100 Lecce, Italy
b
Department of Structures for Engineering and Architecture, Università degli Studi di Napoli Federico II, 80138 Napoli, Italy

A R T I C L E I N F O A B S T R A C T

Keywords: In the present contribution a general formulation is proposed to account for general boundary conditions within
Anisotropic laminated structures the dynamic analysis of anisotropic laminated doubly-curved shell having arbitrary shape and variable thickness.
Doubly-curved shells Different analytical expressions are considered for the shell thickness variation along the geometrical principal
Equivalent Single Layer
directions, and the distortion of the physical domain is described by a mapping procedure based on Non-Uniform
GDQ method
General boundary conditions
Rational Basis Spline (NURBS) curves. Mode frequencies and shapes are determined employing higher-order
theories within an Equivalent Single Layer (ESL) framework. The related fundamental relations are tackled
numerically by means of the Generalized Differential Quadrature (GDQ) method. The dynamic problem is
derived from the Hamiltonian Principle, leading to a strong formulation of the governing equations. General
external constraints are enforced along the edges of the shell employing a distribution of linear springs
distributed on the faces of the three-dimensional solid and accounting for a spatial coordinate-dependent stiffness
along both in-plane and out-of-plane directions. Moreover, a Winkler-type foundation with general distribution
of linear springs is modelled on the top and bottom surfaces of the shell. A systematic set of numerical examples
is carried out for the validation of the proposed theory by comparing mode frequencies with predictions from
refined three-dimensional finite element analyses. Finally, we perform a sensitivity analysis of the dynamic
response of mapped curved structures for different spring stiffnesses and general external constraints, according
to various kinematic assumptions.

1. Introduction specifically, moderately thick shells can be analysed according to the so-
called Equivalent Single Layer (ESL) approach, whereas a Layer-Wise
Recent advances in many engineering fields encourage the employ­ (LW) formulation is required by thick shells [5–6]. Unlike LW, in the
ment of curved lightweight structures in the design process. Most ap­ case of ESL theories, the number of field variables does not depend on
plications require an optimized static and dynamic behaviour of such the number of layers within the laminate. For very thick structures, a
structural members [1], whose design and material properties can in­ three-dimensional elasticity theory is required to provide accurate re­
fluence their mechanical response. Among innovative materials, lami­ sults [7], whereas moderately thick shells commonly rely on an ESL
nated composites are largely used in several engineering fields due to model [8–11]. Based on this approach, the three-dimensional kinematic
their lightweight nature and high stiffness and strength [2]. An accurate and mechanical quantities defined within each layer are referred to the
superposition of differently-oriented layers provides the best solution for geometric reference surface, according to some a-priori assumptions,
the specific engineering problem object of analysis. However, for com­ thus reducing the problem to two-dimensions.
plex structural shapes, several issues must be considered for a proper At the same time, classical methods set the original hypotheses
mechanical modelling, primarily an efficient geometrical description of focusing on the in-plane behaviour of plates and shells [12–15]. In this
arbitrary shapes [3–4]. way, warping and in-plane coupling effects along each direction of the
With particular reference to displacement-based elastic theories, a structure are well described, especially if refined hypotheses are
two-dimensional problem can be set for a three-dimensional structure assumed. In addition, in such theories the out-of-plane field variable
with fixed geometrical proportions in the structural thickness. More component is assumed to be constant along the entire thickness of the

* Corresponding author.
E-mail address: francesco.tornabene@unisalento.it (F. Tornabene).

https://doi.org/10.1016/j.compstruct.2022.116542
Received 27 September 2022; Accepted 26 November 2022
Available online 9 December 2022
0263-8223/© 2022 Elsevier Ltd. All rights reserved.
F. Tornabene et al. Composite Structures 309 (2023) 116542

structure. As a consequence, there are effects that are not predictable if analytical closed-form solution can be found only for some specific
the lamination scheme assumes a softcore configuration [16–18]. boundary conditions [73–74]. The Generalized Differential Quadrature
Moreover, shear stresses through-the-thickness profiles are discontin­ (GDQ) method [75–78] has been proven to be one of the most powerful
uous, so that the shear correction factor is required in the formulation algorithms belonging to this class [79]. Based on the Weierstrass inter­
[19]. In this way, the macromechanical contribution of the actual stress polation theory, it provides a quadrature weighted expression of the
distribution is considered. derivative of the unknown function with respect to the values assumed
A milestone in the analysis of shells can be traced to the introduction by the function in a discrete set of points selected from the definition
of the generalized approach for the description of the displacement field domain. The success of the method lies on the definition of the weighting
[20–23]. According to this methodology, each component of the actual coefficients and the assessment of the computational domain [80–81].
field variable is described in a unified manner by means of the so-called Very accurate results can be obtained if non-uniform grid distributions
thickness functions, embedding the shape variation for each field are employed [82–84]. Moreover, there are several studies in which
component. In this way, several theories can be included in a unified GDQ has been applied for the static [85–86] and dynamic [87–89]
model and a systematic set of analyses can be easily performed by simply analysis of shell structures made of advanced materials. Generally
setting a different analytical expression for the thickness functions. speaking, GDQ-based problems show a great accordance with respect to
Accordingly, the accuracy of an ESL theory can be easily assessed very refined solutions, even though a significantly reduced number of
starting from a proper definition of the through-the-thickness displace­ Degrees of Freedom (DOFs) is employed. The GDQ method has been,
ment field, taking into account both polynomial [24–25] and trigono­ also, successfully applied to problems involving singularities [90–91] as
metric [26–27] functions. Thus, a smooth continuous variation can be well as time integration simulations [92]. Moreover, moving from such a
obtained along the thickness of the shell. In laminated structures, procedure, it is also possible to perform integral operations [93] by
however, very complicated coupling effects can occur between two means of the Generalized Integral Quadrature (GIQ).
adjacent layers, and the displacement field can show a singularity. Thus, When the unknown field variable is expressed from the interpolation
some thickness function must be set in order to account for such phe­ of the values assumed in some specific point of a pre-determined
nomena, such as the so-called zigzag functions [28–30]. At the same computational grid, the so-called weak formulation of the differential
time, the refined theories [31–32] account for the actual shear stiffness problem comes out. In this way, the differentiation order of the solution
distribution in each layer of the laminate. In addition, the thickness decreases. The classical Finite Element Method (FEM) follows this
functions can be introduced to account for some delamination issues at approach, accounting for a decomposition of the physical domain [94].
the interlaminar stage [33–36]. Unlike FEM, when the GDQ method is applied, a higher order smooth
In the general case of doubly-curved structures, the fundamental variation of the solution is guaranteed along the entire physical domain
governing equations are written in a curvilinear principal reference [95]. On the other hand, when a weak formulation of the structural
system, following the well-known isogeometric approach [37–38]. For problem is adopted, boundary conditions are defined from the simple
shells with arbitrary geometries, a distortion algorithm is introduced to elimination of the DOFs related to the applied kinematic constraint. In
analyse them in their unmapped original shape. In the case of singu­ contrast, strong form theories require some analytical expressions of the
larities like holes, a domain decomposition is essential [39–42] since a external constraints which should be numerically implemented by
parametrization in principal coordinates is difficult to be achieved. All means of the GDQ approach [96–98]. As a consequence, non-
the above-mentioned procedures are based on an isogeometric mapping conventional external constraints can be expressed by means of a se­
of the physical domain based on Non-Uniform Rational Basis Spline ries of pre-determined distributions of linear springs [99–102], whereby
(NURBS) curves [43–46] employed to define the structural edges. It several configurations can be defined once the governing parameters are
should be remarked that classical distortion algorithms are based on a properly calibrated [103–104]. In the present work, a general formu­
polynomial description of the boundary edges in each computational lation for the enforcement of external boundary conditions is presented.
domain [47]. A distribution of linear elastic springs is applied along each edge of
Another interesting aspect regarding shell structural theories is the arbitrarily shaped anisotropic shell structures, whose vibration is here
proper description of the mechanical behaviour for each layer. Due to analysed according to a unified formulation based on the ESL strategy.
relatively recent manufacturing issues, many theories have been pro­ Unlike other ESL formulations for the dynamic analysis of shell struc­
posed in the literature, accounting for isotropic, non-homogeneous and tures, in the present work a surface distribution of linear elastic springs
orthotropic media [47–51]. Most innovative materials, however, exhibit is associated with each boundary of the three-dimensional solid and
a completely anisotropic behaviour, and require the introduction of accounts not merely for an in-plane general dispersion of the springs, as
some coupling terms within the formulation, together with a series of other formulations, but also for a generalized out-of-plane distribution
homogenization methods for a proper definition of the equivalent me­ of linear elastic springs. In this way, a three-dimensional configuration
chanical properties. The interested reader can refer to some works of the boundary conditions has been modelled. Furthermore, the cali­
regarding to the application of Functionally Graded Materials [52–58], bration of the position and shape parameters of the adopted distribu­
Carbon Nanotubes [59–62], honeycomb and grid lattices [63–65] to tions allows one to constrain the structure in a specific point of its edges,
spatial structures. even though a spectral collocation method is adopted for the numerical
Referring to the dynamic analysis of shells, the structural response implementation of the differential problem.
can be successfully oriented thanks to a redistribution of the inertial Following the ESL approach, the reference middle-surface is
masses by properly setting a thickness variation along the parametric described by means of a parametrization in curvilinear principal co­
lines [66–68]. To this purpose, very interesting results can be read for ordinates, and isogeometric NURBS-based mapping is used to account
both linear static problems and free vibration analysis [69–70]. for any possible domain distortion. A general relation is provided for the
The governing equations can be easily derived from the application thickness variation of each layer along the entire physical domain. A
of energy principles, depending on the nature of the problem. For a free two-dimensional model is assessed starting from the computation of the
vibration problem, the Hamiltonian Principle can be set in a variational elastic strain energy. The equilibrium equations are obtained from the
form, computing the elastic strain energy and inertia energy contribu­ Hamiltonian Principle. A completely anisotropic constitutive relation is
tions. Following the integration by parts rule, a strong formulation of the assigned to each layer of the stacking sequence, and the generalized
structural problem is obtained, without any reduction of the differen­ elastic coefficients are defined for the entire lamination scheme. The
tiability order. Then, a boundary value problem is obtained from the fundamental equations are derived with respect to the unknown vector
energy stationary component [71], that can be easily tackled numeri­ of the generalized displacement field. The final relations are solved
cally with a spectral collocation method as reliable tool [72], because an numerically by means of the GDQ method, providing a solution directly

2
F. Tornabene et al. Composite Structures 309 (2023) 116542

in a strong form. The proposed theory is validated by a free vibration dinate system O α 1 , α 2 , α 3 is introduced, where axes α 1 and α 2 are
′

analysis of a mapped laminated plate with different external boundary defined from the principal directions of the reference surface, and α 3 =
constraints. Mode frequencies and shapes are successfully compared to ζ is taken parallel to the outward normal direction of the shell surface
those provided by a three-dimensional finite element approach. A (Fig. 1). In other words, any arbitrary point P of the shell is univocally
parametric analysis is also performed to check for the influence of some identified from its position vector R(α 1 , α 2 , ζ), and it can be obtained as
boundary conditions key parameters on the dynamic response of those a shift of its projection P’ located on the reference surface, whose po­
structures. Finally, a comprehensive set of modal analyses is performed sition vector is denoted as r(α 1 , α 2 ), leading to the following relation
on singly-curved and doubly-curved panels with different lamination [22]:
schemes and external constraints, employing various Higher Order
h(α 1 , α 2 )
Shear Deformation Theories (HSDTs). The importance of the field var­ R(α 1 , α 2 , ζ) = r(α 1 , α 2 ) + z n(α 1 , α 2 ) (1)
iable expansion order is pointed out, as well as the introduction of a 2
zigzag assumption, for accurate predictions of complicated warping and where h(α1 , α2 ) is the shell thickness expression and z = 2ζ/h is a
stretching effects. All the analyses presented in the article, and the dimensionless through-the-thickness coordinate, such that z ∈ [ − 1, 1].
related formulation, are implemented in the DiQuMASPAB software Moreover, n(α 1 , α 2 ) denotes the midsurface outward normal unit vec­
[105]. tor. All these quantities are computed referring to each point (α 1 , α 2 ) of
the reference surface. The latter can be expressed following the pro­
2. Equivalent single layer theory for a doubly-curved shell cedure reported in Ref. [22], taking into account the first order de­
structure rivatives of r(α 1 , α 2 ) with respect to in-plane coordinates α i = α 1 , α 2 ,
marked with r,i = ∂r/∂αi :
In the present section an ESL fomulation is provided for the dynamic
analysis of laminated doubly-curved shell structures made of generally r,1 × r,2
n = ⃒⃒ ⃒ (2)
anisotropic materials. The geometry of the structure is described by r,1 × r,2 ⃒
using the main outcomes of differential geometry, whereas the equi­ For the sake of clarity, the vector product in Eq. (2) has been indi­
librium equations are derived from the Hamiltonian principle. The cated with the symbol × . In order to provide a physical meaning to the
constitutive relationship of each layer of the structure is intended to be geometric representation reported in Eq. (1), the in-plane coordinates
linear elastic, accounting for generally anisotropic materials. [ ] [
variation must be restricted to a closed interval α01 , α11 × α02 , α12 ,
]

being α i and α i , for i = 1, 2, the extremes of the variation interval of

0 1
2.1. Geometric description of a doubly curved shell each principal coordinate.
As far as the shell thickness is concerned, each layer is characterized
Doubly-curved shells are three-dimensional geometric structures due by a width h k defined from the through-the-thickness locations ζ k and
to their spatial configuration, but they can be investigated by consid­ ζ k+1 of the extreme sheets referred to each k-th layer:
ering a reference midsurface. In this way, the main outcomes of differ­
ential geometries can be applied to this geometric entity and a proper
parameterization of such structures can be derived. A curvilinear coor­

Fig. 1. Representation of a doubly-curved shell with respect to the reference middle surface according to the ESL description of the displacement field based on the
introduction of the thickness functions set along the principal directions of the structure. The physical domain mapping for the geometric description of an arbitrary-
shaped structure stems from a distortion algorithm based on NURBS curves.

3
F. Tornabene et al. Composite Structures 309 (2023) 116542

∑
l ∑
l √̅̅̅̅̅̅̅̅̅̅̅̅ √̅̅̅̅̅̅̅̅̅̅̅̅
h(α 1 , α 2 ) = h k (α 1 , α 2 ) = (ζ k+1 (α 1 , α 2 ) − ζ k (α 1 , α 2 ) ) (3) A1 = r,1 ⋅r,1 , A2 = r,2 ⋅r,2 (9)
k=1 k=1

2.2. Kinematic relations for the equivalent single layer model

being l the total number of layers within the laminate [22]. The thick­
ness variation can be accounted for the model directly from Eq. (3), by
Following the ESL methodology, a unified formulation is provided
introducing a reference k-th layer’s height h0k . In this way, the actual for the description of the three-dimensional displacement field U(α 1 , α 2 ,
layer thickness h k can be expressed for each reference surface as:
ζ, t) = [U 1 U 2 U 3 ] T , referred to the above-described orthogonal curvi­
h k (α 1 , α 2 ) = h0k h̄( α 1, α 2) linear principal reference system and using a variable order kinematical
model. Hereafter we will assume the notation α 3 = ζ. For each principal
= h0k (1 + δ 1 ϕ 1 (α 1 , α 2 ) + δ 2 ϕ 2 (α 1 , α 2 ) + δ 3 ϕ 3 (α 1 , α 2 )
direction, an axiomatic assumption can be made for the thickness vari­
+ δ 4 ϕ 4 (α 1 , α 2 ) + δ̄ ) (4) ation of the field variable quantity via the introduction of a generalized
thickness function Fατ i (ζ), assigned to each αi for i = 1, 2, 3, at each τ-th
As it can be seen, a thickness variation law h̄(α 1 , α 2 ) valid for all
order of the kinematic expansion for τ = 0, ..., N + 1, being N the
layers is introduced. As stated in Eq. (4), it consists of a constant shift δ̄
maximum order of the displacement field assumption. Employing a
and four scaling factors δ f along with their corresponding variation
matrix formulation, one gets [22]:
relations ϕ f (α1 , α 2 ) for f = 1, ..., 4. The latter are expressed in a unified
⎡ ⎤ ⎡ ⎤⎡ ⎤
manner by means of two dimensionless coordinates ᾱ i and ̃ αi , for i = 1, U1 F ατ 1 0 0 ⎢u1 ⎥
(τ)

2, defined as: ⎢ ⎥ ∑
⎢ ⎥
N +1 ⎢
⎢
⎥⎢
⎥ (τ) ⎥
⎢ U2 ⎥ = ⎢ 0 Fατ 2 0 ⎥⎢ u2 ⎥ ⎥ ⇔
⎣ ⎦ τ=0 ⎣ ⎦⎢
⎣ ⎦
α i − α0i α3 (10)
ᾱi = U3 0 0 Fτ (τ)
u3
α1i − α0i for i = 1, 2 (5)
∑
N +1
αi = 1 − ᾱi
̃ U(α 1 , α 2 , ζ, t) = F τ (ζ)u(τ) (α 1 , α 2 , t)
In the following, we describe the thickness variation expressions
τ=0

ϕ f (α1 , α 2 ) for f = 1, ..., 4, as adopted in the present work [22]: where the unknown three-dimensional field has been expressed, for each

{ {
¯αp1 1 ¯αp2 2
ϕ 1 (α1 , α2 ) = p1 , ϕ 2 (α1 , α2 ) =
(sin(π(n1 ᾱ1 + α1m ) ) ) (sin(π(n2 ᾱ2 + α2m ) ) )p 2
⎧ { (6)
⎨ αp1 3
̃ αp2 4
̃
ϕ 3 (α1 , α2 ) = , ϕ 4 (α1 , α2 ) =
α1 + α3m ) ) )p 3
⎩ (sin(π (n3 ̃ α2 + α4m ) ) )p 4
(sin(π(n4 ̃

τ = 0,...N + 1, in terms of a generalized displacement field vector u(τ) =

Being n f ∈ N, p f ∈ R and αfm ∈ [0, 1] for f = 1, ..., 4. As a conse­ u(τ) (α 1 , α 2 , t), independently from the thickness. As it will be seen, the
quence, a generalized thickness variation can be obtained from a com­ N + 1-th expansion order is used when the kinematic model accounts for
bination of a power and a sinusoidal analytical expression. Moreover, the zigzag interlaminar effects; otherwise, an expansion up to τ = N is
the approach introduced in Eqs. (4)–(6) provide a symmetric expression considered. A generalized axiomatic assumption is adopted here for the
with respect to the reference surface. description of the displacement field variable, and the formulation of the
Since an orthogonal curvilinear principal parameterization has been structural problem turns out to be independent from the actual selection
provided for the definition of r(α 1 , α 2 ), it is possible to express the of the thickness function itself. Unlike three-dimensional formulations,
principal radii of curvature R i (α1 , α2 ) for i = 1, 2 in each point of the in the present work the unknown variables of the problem are the
middle surface, referred to the α 1 and α 2 directions, as follows [22]: generalized displacement field components rather than the three-
r,i ⋅r,i dimensional ones. However, when the order of the kinematic expan­
R i (α1 , α2 ) = − for i = 1, 2 (7) sion increases, the solution converges to the real through-the-thickness
r,ii ⋅n
profile of the displacement field. Based on the ESL formulation
( )
being r,ij = ∂2 r/ ∂αi ∂αj for i, j = 1, 2, the second order derivatives of assessed in Eq. (10), a generalized structural model is thus obtained for
the dynamic analysis of shells, as it is shown in Fig. 1. Thus, the choice of
r(α 1 , α 2 ) with respect to α 1 , α 2 , and n(α 1 , α 2 ) the normal vector defined the thickness functions set Fτ can provide several structural models
in Eq. (2). Starting from Eq. (7), the geometric quantity Hi (α 1 , α 2 , ζ) is [15,17]. Moreover, classical approaches like the Classical Plate Theory
defined for an arbitrary point of the three-dimensional structure, for (CPT) [8] and First Order Shear Deformation Theory (FSDT) [12,13] are
each α i = α 1 , α 2 , accounting for the curvature of the midsurface: efficiently embedded in the formulation.
ζ Some consideration should be made on the choice of the kinematic
Hi = 1 + (8) expansion. In the present work, a power expansion has been selected in
Ri
all the examples of investigations. Although some remarks are traced in
In the following, the Lamè Parameters A 1 (α1 , α2 ) and A 2 (α1 , α2 ) of the literature concerning for the correct choice of the thickness functions
the reference surface r(α 1 , α 2 ) are presented, coming directly from the set. More specifically, there are some cases [24–25] in which a poly­
computation of the first fundamental form of r(α 1 , α 2 ), as discussed in nomial set is adopted, whereas other theories [26–27] embed a trigo­
Ref. [22]: nometric expression of the field variable. For laminated structures, we

4
F. Tornabene et al. Composite Structures 309 (2023) 116542

provide a zigzag through-the-thickness kinematic assumption. There­ ⎡ ⎤

1
fore, at the τ = (N + 1)-th order, the generalized Murakami’s formu­ ⎢H 0 0 0 0 0 0 0 0⎥
lation [22,28–29] is employed for each principal direction α i = α 1 , α 2 , ⎢ 1
⎢
⎥
⎥
α 3 , namely: ⎢
⎢ 0 1
0 0 0 0 0 0 0⎥
⎥
( ) ⎢ H2 ⎥
⎢ ⎥
2 ζ + ζk ⎢ ⎥
αi
FN+1 (ζ) = ( − 1)k z k = ( − 1)k ζ − k+1 for i = 1, 2, 3 ⎢
⎢ 0
1 1 ⎥
ζk+1 − ζk ζk+1 − ζk ⎢ 0 0 0 0 0 0⎥ ⎥
⎢ H1 H2 ⎥
(11) Dζ = ⎢ ⎥ (15)
⎢ 1 ∂ ⎥
⎢ 0 0 0 0 0 0 0⎥
where k = 1, ..., l refers to the arbitrary layer of the lamination scheme ⎢
⎢ H1 ∂ζ ⎥
⎥
⎢ ⎥
composed by l laminae. ⎢ 1 ∂ ⎥
⎢ 0 0 0 0 0 0 0⎥
For instance, the EDZ4 displacement field assumption requests a ⎢
⎢ H2 ∂ζ ⎥
⎥
fourth-order for the kinematic expansion [22], taking into account also ⎢
⎣ ∂⎦
⎥
the Murakami’s zigzag expression reported in Eq. (11), as follows: 0 0 0 0 0 0 0 0
∂ζ
k
U 1 = u(0) (1) 2 (2) 3 (3) 4 (4) (5)
1 + ζu 1 + ζ u 1 + ζ u 1 + ζ u 1 + ( − 1) z k u 1 Starting from the unified expression for the displacement field re­
U 2 = u(0) (1) 2 (2) 3 (3) 4 (4) k (5)
2 + ζu 2 + ζ u 2 + ζ u 2 + ζ u 2 + ( − 1) z k u 2
(12) ported in Eq. (10), it is possible to derive the three-dimensional kine­
U 3 = u 3 + ζu 3 + ζ2 u 3 + ζ3 u 3 + ζ4 u 3 + ( − 1)k z k u 3
(0) (1) (2) (3) (4) (5) matic relations reported in Eq. (13) so that the three-dimensional strain
T
vector ε(α1 , α2 , ζ, t) = [ ε1 ε2 γ12 γ13 γ 23 ε3 ] is expressed
where u1 ,u2 ,u3 for τ = 0, ..., 5 are the unknown variables of the
(τ ) (τ ) (τ ) following the ESL approach in terms of the generalized strain vector
[ ]T
problem. ε(τ) (α 1 , α 2 , t) = ε(1τ) ε(2τ) γ(1τ) γ(2τ) γ(13
τ)
γ(23) ω(13) ω(23) ε(3) , for
τ τ τ τ

For a steady identification of the displacement field assumption, a each τ = 0, ..., N + 1, see Ref. [22]:
structural theory nomenclature is adopted. More specifically, the symbol
N +1 ∑
∑ 3 N +1 ∑
∑ 3 N +1 ∑
∑ 3
E refers to the ESL approach, D refers to the displacement-oriented ε= D ζ DαΩi F τ u(τ) = Z (τ)α i DαΩi u(τ) = Z (τ)α i ε(τ)α i
formulation, whereas N is the maximum order of the kinematic expan­ τ= 0 i=1 τ= 0 i=1 τ= 0 i=1
sion adopted in each principal direction. When Z is introduced in the (16)
nomenclature, the τ = (N + 1)-th order is added to the model, and the
Thus, the three-dimensional strains ε(α1 , α2 , ζ, t) are related to their
thickness function defined in Eq. (11) is taken into account.
corresponding generalized expressions, based on the introduction of a
As mentioned before, in an ESL framework, the position vector for a
kinematic matrix collecting the through-the-thickness derivatives of the
doubly-curved shell can be expressed from the midsurface r(α 1 , α 2 ), as
thickness functions set Fατ i (ζ) for each τ-th expansion order and for each
already defined in Eq. (1). Starting from the three-dimensional elasticity
principal direction α i = α 1 , α 2 , α 3 :
problem, the kinematic strain vector ε(α 1 , α 2 , ζ, t) = ⎡ αi ⎤
[ ε 1 ε 2 γ 12 γ13 γ 23 ε 3 ]T can be expressed in principal curvilinear Fτ
0 0 0 0 0 0 0 0 ⎥
⎢H
coordinates [22]: ⎢ 1 ⎥
⎢ αi ⎥
( ) ⎢ F ⎥
∑
3 ⎢ 0 τ
0 0 0 0 0 0 0 ⎥
⎢ ⎥
ε = DU = Dζ (DαΩ1 + DαΩ2 + DαΩ3 )U = Dζ DαΩi U (13) ⎢ H2 ⎥
⎢ α α ⎥
i=1 ⎢ F i
F i ⎥
⎢ 0 0 τ τ
0 0 0 0 0 ⎥
⎢ H1 H2 ⎥
where D is a three-dimensional kinematic operator. This operator can be ⎢ ⎥
Z (τ)α i
=⎢ αi αi ⎥ (17)
⎢ F ∂ F ⎥
split in a through-the-thickness part denoted with Dζ , which includes the ⎢ 0 0 0 0 τ
0 τ
0 0 ⎥
⎢ H1 ∂ζ ⎥
outward normal direction derivatives, and three operators DαΩi , i = 1, 2, 3 ⎢
⎢
⎥
⎥
αi αi
accounting for the in-plane derivatives. Recalling the curvature thickness ⎢
⎢ 0 0 0 0 0
Fτ
0
∂Fτ
0
⎥
⎥
⎢ ⎥
H i = 1 +ζ/R i for i = 1, 2 defined in Eq. (8), the differential operators ⎢ H 2 ∂ ζ ⎥
⎢ ⎥
introduced in Eq. (13) can be written in an extended form as [22]: ⎣ ∂Fτα i ⎦
0 0 0 0 0 0 0 0
⎡ ⎤ ⎡ ⎤ ∂ζ
1 ∂ 1 ∂A 1 ⎡ ⎤
⎢ 0 0⎥ ⎢0 0⎥ 1 The generalized strain vector ε(τ) (α 1 , α 2 , t) introduced in Eq. (16) is
⎢ A 1 ∂α1 ⎥ ⎢ A A ∂α ⎥ ⎢ 0 0 ⎥
⎢ ⎥
expressed, for each τ = 0, ..., N + 1, as follows:
1 2 2
⎢ ⎥ ⎢ ⎥ ⎢ R1 ⎥
⎢ 1 ∂A 2 ⎥ ⎢ 1 ∂ ⎥ ⎢ ⎥
⎢ 0 0⎥ ⎢ ⎥ ⎢ 1 ⎥
⎢0 0⎥ ⎢0 0 ⎥
⎢ A 1 A 2 ∂α1
⎢
⎥
⎥ ⎢
⎢
A 2 ∂α2 ⎥
⎥
⎢
⎢ R2 ⎥
⎥ ε(τ) αi = D αΩi u(τ) for τ = 0, 1, 2, …, N, N + 1 i = 1, 2, 3 (18)
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 1 ∂A1 ⎥ ⎢ 1 ∂ ⎥ ⎢0 0
⎢− 0 0⎥ ⎢0 0⎥ 0 ⎥
⎢ A 1 A2 ∂α2
⎢
⎥
⎥ ⎢ A 1 ∂α1 ⎥
⎢
⎢
⎥
⎥ 2.3. Anisotropic elastic constitutive relation
⎢ ⎥ ⎢ ⎥ ⎢0 0 0 ⎥
⎢ 1 ∂ ⎥ ⎢ 1 ∂A ⎥ ⎢ ⎥
⎢ 2 ⎥ ⎢ ⎥
α1 ⎢
DΩ = ⎢ A 2 ∂α2
0 0⎥ α2
⎥, DΩ = ⎢
⎢
0 −
A 1 A2 ∂α1 ⎥
0 α
⎥ DΩ = ⎢
3
⎢0 0
1 ∂ ⎥
⎥ Based on the kinematic relations (16) within an ESL framework, it is
⎢ ⎥
⎢ 1 ⎥ ⎢
⎢0
⎥
⎥
⎢
⎢ A 1 ∂α1 ⎥
⎥ possible to derive the constitutive elastic equations starting from the
⎢ ⎥ 0 0
⎢ − 0 0⎥ ⎢
⎢
⎥
⎥
⎢
⎢ 1 ∂ ⎥
⎥ generalized Hooke’s law for each layer of the stacking sequence.
⎢ R1 ⎥ ⎢ 1 ⎥ ⎢0 0 ⎥
⎢
⎢
⎥ ⎢0 − 0⎥ ⎢ A 2 ∂α2 ⎥ Referring to the k-th layer, the so-called material reference system
⎢ 0 0 0⎥⎥ ⎢ R2 ⎥ ⎢ ⎥
⎢ ⎥ ⎢ 0 ⎥ O α 1 α 2 ζ(k) is rotated with respect to the geometric principal reference
′ (k) (k)
⎢ ⎥ ⎢0 ⎥ ⎢0 0 ⎥
⎢ 1 0 0⎥⎥ ⎢ 0 0 ⎥ ⎢ ⎥
⎢ system O α 1 α 2 ζ. Thus, each layer of the stacking sequence features the
′
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢0 0 0 ⎥
⎢ 0 0 0 ⎥ ⎢
⎣
0 1 0 ⎥
⎦
⎣ ⎦ following anisotropic behaviour [22]:
⎣ ⎦ 0 0 1
0 0 0 0 0 0

(14)

5
F. Tornabene et al. Composite Structures 309 (2023) 116542

⎡ ⎤ ⎡ ⎤⎡ ⎤
The stiffness matrix C̄ is now referred to the three-dimensional
(k)
(k) (k) (k) (k) (k) (k) (k) (k)
⎢̂σ ⎥ ⎢ C11 C12 C16 C14 C15 C13 ⎥⎢ ̂ε ⎥ [ ]T
⎢ 1 ⎥ ⎢ ⎥⎢ 1 ⎥ stress and strain vectors σ = σ 1 σ 2 τ 12 τ 13 τ 23 σ 3 and ε(k) =
(k) (k) (k) (k) (k)
(k) (k)
⎢̂ (k) ⎥ ⎢ (k) (k) (k) (k) (k) (k) ⎥⎢ (k) ⎥
⎢ σ2 ⎥ ⎢ C12 C22 C26 C24 C25 C23 ⎥⎢ ̂ ε ⎥ [ ]T
⎢ (k) ⎥ ⎢ (k) ⎥⎢ 2 ⎥
⎢ ̂τ
⎢ 12
⎥ ⎢C
⎥ ⎢ 16
(k)
C26 (k)
C66 (k)
C46 (k)
C56 (k) ⎥⎢ (k)
C36 ⎥⎢ ̂γ 12
⎥
⎥ (k) ε(k) (k) (k) (k) (k) (k)
1 ε 2 γ 12 γ 13 γ 23 ε 3 , respectively. Each component of Eq. (20) is
⎢ (k) ⎥ = ⎢ (k) ⎥⎢ ̂ (k) = C(k) ̂
⎥ ⇔ σ ε
⎢ ̂τ ⎥ ⎢C denoted with¯E ij in order to easily refer to the rotated three-dimensional
(k) (k) (k) (k) (k) ⎥⎢ (k) ⎥ (k)
⎢ 13 ⎥ ⎢ 14 C24 C46 C44 C45 C34 ⎥⎢ ̂γ 13 ⎥
⎢ (k) ⎥ ⎢ (k) (k) ⎥⎢ (k) ⎥
stiffness coefficients ¯C(k) ij , and to reduced terms ¯Q ij valid in the case of
(k) (k) (k) (k)
⎢ ̂τ 23 ⎥ ⎢ C15 C25 C56 C45 C55 C35 ⎥⎢ ̂γ 23 ⎥ (k)
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎣ (k) ⎦ ⎣ (k) (k) ⎦⎣ (k) ⎦
plane-stress assumption. Starting from the computation of the elastic
(k) (k) (k) (k)
σ3
̂ C13 C23 C36 C34 C35 C33 ̂ε3
strain energy, taking into account all the constitutive relations reported
(19) in Eqs. (19)–(20), the generalized ESL constitutive relation is derived in
[ ]T terms of the generalized stress resultant vector S(τ)α i (α1 , α2 , t) =
being ̂ (k) = ̂
σ σ (k) σ (k) τ (k) τ (k) τ (k) σ (k) and ̂ε (k) = [ ]T
1 ̂ 2 ̂ 12 ̂ 13 ̂ 23 ̂ 3 N(1) i N(2) i N(12) i N(21) i T(1) i T (2) i P(1) i P(2) i S(3) i , as detailed in
τα τα τα τα τα τα τα τα τα
[ ]T
ε (k) ε (k) the three-dimensional stress and strain Ref. [22]:
(k) (k) (k) (k)
̂ 1 ̂ 2 ̂γ 12 ̂γ 13 ̂γ 23 ̂ε 3
vectors, respectively, and C ij for i, j = 1, ..., 6 the elastic stiffness con­
(k) N +1 ∑
∑ 3
S(τ)αi = A(τη)αi αj ε(η)αj for τ = 0, 1, 2, ..., N, N + 1, αi = α1 , α2 , α3
stants of the k-th layer referred to O Once the elastic 1 α2 ζ .
α(k)
′ (k) (k)
η=0 j=1
constitutive relationship is expressed for the entire laminate, a homog­ (21)
enization technique should be developed in order to obtain a single
relation valid for the entire structure. In the present work, setting ζ(k) = In Eq. (21), a generalized 9 × 9 stiffness matrix A comes out for
(τη)α i α j

each τ, η-th order of the kinematic assumption (10) and for each α i α j =
ζ a rotation transformation T , characterized by an angle θ k between
(k)
α 1 , α 2 , α 3 principal direction. In an extended matrix form, one gets:
1 and α 1 , can be defined. In this way, the stiffness matrix C
α(k) in Eq.
(k)

(19) is expressed with respect to the geometric reference system

(22)

O α 1 α 2 ζ defined for the shell under consideration [22]: If we introduce the notations Fταi = ∂0 Fταi /∂ζ0 and Fη j = ∂0 Fη j /∂ζ0 that
′ α α

⎡ ⎤ allow us to indicate and identify that the 0-th order derivatives of the
thickness functions Fταi , Fτ j coincide with the thickness functions them­
α
⎢¯E(k) ¯E(k) (k) (k) (k) (k)
12 ¯E 16 ¯E 14 ¯E 15 ¯E 13 ⎥
⎢ 11 ⎥ selves, the generic component of the generalized stiffness matrix
⎢ (k) ⎥
⎢¯E12 ¯E22 ¯E26 ¯E24 ¯E25 ¯E(k)
(k) (k) (k) (k)
23 ⎥
⎢ (k)
⎢¯E
⎥
(k) ⎥
A(τη)α i α j can be expressed by the following condensed relation, accord­
⎢ ¯E(k) (k) (k) (k)
25 ¯E 66 ¯E 46 ¯E 56 ¯E 36 ⎥ ing to the nomenclature introduced in Eq. (22):
(20)
(k)
C̄ =T C T (k) (k) (k)T
= ⎢ 16 ⎥
⎢¯E(k) ¯E(k) (k) (k) (k) (k) ⎥
⎢ 14 24 ¯E 46 ¯E 44 ¯E 45 ¯E 34 ⎥
⎢ (k) (k) ⎥
⎢¯E15
⎢ ¯E(k) (k) (k) (k)
25 ¯E 56 ¯E 45 ¯E 55 ¯E 35 ⎥
⎥
⎣ (k) (k) (k) (k) (k) (k) ⎦
¯E13 ¯E23 ¯E36 ¯E34 ¯E35 ¯E33

for τ, η = 0, 1, 2, ..., N, N + 1
for n, m = 1, 2, 3, 4, 5, 6
∫
∑l ζk+1 ∂f Fηα j ∂g Fτα i H1 H2
(23)
(τη)[fg]α α
A nm (pq)i j = ¯B(k) p q dζ for p, q = 0, 1, 2
k=1
ζk
nm
∂ζf ∂ζg (H1 ) (H2 )
for αi , αj = α1 , α2 , α3
for f , g = 0, 1

6
F. Tornabene et al. Composite Structures 309 (2023) 116542

where H k for k = 1, 2 is defined in Eq. (8). In Eq. (23) one can find the f (τη)α i (− ) (− ) (− ) (+) (+) (+)
generic τ, η-th order of the field variable expansion, as well as the indexes L ii = kif f (− ) Fηαi (− ) Fταi (− ) H1 H2 + kif f (+) Fηαi (+) Fταi (+) H1 H2 for
n, m of the stiffness coefficients referred to the k-th layer. The latter has i = 1, 2, 3
been denoted with ¯B(k)
nm so that Eij =¯Cij , or Eij =¯Qij . According to (28)
(k) (k) (k) (k)

Eq. (23), also the shear correction factor κ(ζ) can be embedded in the According to Eq. (28), a variation of linear springs can be assessed on
model when lower order theories cannot predict the actual through-the- the top and bottom surfaces of the shell, from the enforcement of two
thickness shear stresses distribution [22], namely: ( )
bivariate functions f (±) ξ 1 , ξ 2 . If a constant Winkler distribution of
{
¯E(k) for n, m = 1, 2, 3, 6 springs with stiffnesses k(±)
if is defined on the top and bottom surface, the
¯B(k) = nm
(24)
nm
κ(ζ)¯E(k)
nm
for n, m = 4, 5 function at issue should be defined at each point of the physical domain,
such that:
The generalized ESL constitutive relation reported in Eq. (21) can be ( )
rearranged in such a way that each component of the stress resultant f (±) ξ 1 , ξ 2 = 1 (29)
vector S(τ)α i (α1 , α2 , t), for each τ = 0, ...N + 1, can be directly expressed Moreover, a Gaussian two-dimensional spring distribution can be
in terms of the generalized kinematic vector u(η) (α1 , α2 , t), with η = 0,..., ( )
applied to the external surfaces of the shell. In this case, f (±) ξ 1 , ξ 2
N + 1: reads as follows:
⎡ ⎤ ⎡ ⎤
(( )2 ( )2 )
(±) (±)
⎢ (τ)αi ⎥ ⎢ (τη)αi α1 ⎥ ( ) − 1
ξ1 − ξ
1m +
ξ2 − ξ
2m
⎢N
⎢ 1
⎥
⎥
⎢O
⎢ 11 O(21τη)αi α2 O(31τη)αi α3 ⎥
⎥ f (±)
ξ 1, ξ 2 =
1
e
2
Δ
(±)
1
Δ
(±)
2 (30)
⎢ (τ)αi ⎥ ⎢ (τη)αi α1 τη α α ⎥ 2πΔ(±) (±)
1 Δ2
τη αi α2
⎢ N2
⎢
⎥
⎥
⎢ O12
⎢ O(22 ) O(32 ) i 3 ⎥ ⎥
⎢ (τ)αi ⎥ ⎢ (τη)αi α1 τη αi α2 τη α α ⎥
⎢ N12 ⎥ ⎢ O13 O(23 ) O(33 ) i 3 ⎥⎡ ⎤ where Δ 1 , Δ 2 > 0 and ξ 1m , ξ 2m > 0 are the position and scaling pa­
(±) (±) (±) (±)
⎢ ⎥ ⎢ ⎥
⎢ N (τ)αi ⎥ ⎢ O(τη)αi α1 τη α α
O(24 ) i 2 O(34 ) i 3 ⎥
τη α α (η)
⎢ 21
⎢ (τ)α
⎥ ∑ ⎢
⎥ N+1 ⎢ 14
⎥⎢ u1
⎥ ⎥ rameters of the distribution, referred to the top ( + ) and bottom (− )
(τη)αi α3 ⎥⎢ (η) ⎥
⎢T i
⎢ 1
⎥=
⎥
⎢ O(τη)αi α1
⎢ 15
(τη)αi α2
O25 O35 ⎥⎢ u ⎥ (25) surface of the shell, for each parametric line of the computational
⎢ (τ)αi ⎥ η=0 ⎢ (τη)αi α1 ⎣ 2 ⎦
τη α α τη α α ⎥ (η) domain.
⎢ T2
⎢
⎥
⎥
⎢ O16
⎢ O(26 ) i 2 O(36 ) i 3 ⎥ ⎥ u3
⎢ (τ)αi
⎢ P1
⎥
⎥
⎢ (τη)αi α1
⎢ O17 (τη)α α (τη)α α ⎥ Linear elastic boundary springs can be applied on the edge faces of
O27 i 2 O37 i 3 ⎥
⎢
⎢ (τ)αi
⎥
⎥
⎢
⎢ (τη)αi α1 (τη)α α (τη)α α ⎥
⎥ the structure by means of predefined distributions of normal and shear
⎢ P2 ⎥ ⎢ O18 O28 i 2 O38 i 3 ⎥ (k)αmj
⎢ (τ)α
⎢S i
⎥
⎥
⎢
⎢ O(τη)αi α1
⎥ stresses along each boundary edge (Fig. 2). Let us denote with k if for
⎣ 3 ⎦ ⎣ 19 O(29τη)αi α2 O(39τη)αi α3 ⎥
⎦
i = 1, 2, 3, j = 1, 2 and m = 0, 1 the stiffness of the boundary linear
springs at issue in each principal direction. The following definitions can
( )
be introduced, accounting for their variation f αm1 , α 2 along the shell
2.4. Winkler-type foundation and general boundary conditions [ ]
boundaries, setting α 1 = αm1 , with m = 0, 1 and α 2 ∈ α02 , α12 :
In the present work, a general distribution of surface linear elastic ¯σ(k)
( m ) (k)αm1 ( m ) (
f α 1 , α 2 U 1 αm1 , α 2 , ζ
)
1 α 1 , α 2 , ζ = − k 1f
springs is applied on the top and the bottom surface of the shell. ( ) ( ) ( )
(31)
m
(k)α 1
Furthermore, a general distribution of springs is considered for each ¯τ(k) m
12 α 1 , α 2 , ζ = − k 2f f αm1 , α 2 U 2 αm1 , α 2 , ζ for m = 0, 1
( ) m ( ) ( )
lateral surface of the three-dimensional doubly-curved solid. In this way, ¯τ(k) m (k) α
f αm1 , α 2 U 3 αm1 , α 2 , ζ
13 α 1 , α 2 , ζ = − k 3f
1

a general configuration of boundary conditions is carried out within a

continuum model. In the same way, the generalized boundary conditions are applied to
The elastic foundation acting on the top and the bottom shell surfaces the edges of the structure identified with α 2 = αm2 for m = 0, 1, letting
[ ] ( )
is implemented following the Winkler hypothesis in each principal di­ α 1 ∈ α01 , α11 . A general distribution of spring stiffness f α 1 , αm2 is
rection. If U i for i = 1, 2, 3 are the three-dimensional displacement
(±) accounted as follows:
components acting on the top ( + ) and bottom ( − ) surface of the shell, ( ) (k)αm2 ( ) ( )
¯τ(k) m
12 α 1 , α 2 , ζ = − k 1f f α 1 , αm2 U 1 α 1 , αm2 , ζ
the elastic foundation of stiffnesses k 1f , k 2f , k 3f yield a vector of
(±) (±) (±)
( ) (k)αm2 ( ) ( )
[ ]T ¯σ(k) m
2 α 1 , α 2 , ζ = − k 2f f α 1 , αm2 U 2 α 1 , αm2 , ζ for m = 0, 1 (32)
external actions qefk = q1efk , q2efk , q3efk whose components are
(±) (±) (±) (±)
( ) (k)αm2 ( ) ( )
¯τ(k) m
23 α 1 , α 2 , ζ = − k 3f f α 1 , αm2 U 3 α 1 , αm2 , ζ
expressed as [22]:
Introducing in Eq. (31) the ESL formulation of the displacement field
q(±) (±) (±)
iefk = − k if U i for i = 1, 2, 3 (26) reported in Eq. (10), the following expressions are obtained:
To account for the virtual work of the elastic foundation, all the (k) ( ) (k)αm ( )∑N+1 α (η) ( )
¯σ 1 αm1 , α 2 , ζ = − k 1f 1 f αm1 , α 2 F 1 (ζ)u 1 αm1 , α 2
quantities introduced in Eq. (26) must be computed according to the ESL η=0 η
( m ) (k)αm1 ( m )∑N+1 α η ( )
formulation set in Eq. (10). If the generalized ESL actions set q(fkτ) is ¯τ(k)
12 α 1 , α 2 , ζ = − k 2f f α 1, α 2 F 2 (ζ)u(2 ) αm1 , α 2 for m = 0, 1
η=0 η
defined for each τ-th order of the kinematic expansion, one gets [22]: )∑N+1
(k) ( ) (k)αm1 ( η ( )
⎡ ⎤ ⎡ ⎤⎡ ⎤ ¯τ 13 αm1 , α 2 , ζ = − k 3f f αm1 , α 2 F α 3 (ζ)u(3 ) αm1 , α 2
η=0 η
(τ)
(33)
f (τη α τ
q
⎢ 1 fk ⎥ ⎢ L 11 ) 1 0 0 ⎥⎢ u(1 ) ⎥
⎢ ⎥ ∑
N +1 ⎢ ⎥⎢ (τ) ⎥
⎢ ⎥
q(fkτ) = − Lf (τη) u(τ) ⇔ ⎢ q(2τ)fk ⎥ = − ⎢ 0 0 ⎥ ⎢ ⎥ The unified formulation for the field variable can be also applied to
f (τη α
⎢ L 22 ) 2 ⎥⎢ u 2 ⎥
⎢ ⎥ ⎣ ⎦⎣ ⎦
⎣ (τ) ⎦ η=0
f (τη)α 3 (τ) Eq. (32), leading to:
q 3 fk 0 0 L 33 u3

(27)

f(τη)α i
where L ii for i = 1, 2, 3 can be expressed as:

7
F. Tornabene et al. Composite Structures 309 (2023) 116542

(k) ( ) (k)αm ( )∑N+1 α (η) ( ) dimensional ESL model. Taking into account Eqs. (33)–(34), the
¯τ 12 α 1 , αm2 , ζ = − k 1f 2 f α 1 , αm2 F 1 (ζ)u 1 α 1 , αm2
η=0 η
following compact relation is assessed, following the procedure from
( ) (k)αm2 ( )∑N+1 α η ( ) Ref. [22]. Thus, the generalized boundary conditions for α 1 = αm1 for
¯σ (k) m
2 α 1 , α 2 , ζ = − k 2f f α 1 , αm2 F 2 (ζ)u(2 ) α 1 , αm2
η=0 η
for m = 0, 1 [ ]
m = 0, 1 and α 2 ∈ α02 , α12 , are defined as:
(k) ( ) (k)αm2 ( )∑N+1 (η) ( )
¯τ 23 α 1 , αm2 , ζ = − k 3f f α 1, α 2 m
Fαη 3 (ζ)u 3 α 1 , α 2
m
⎡ ⎤
η=0 ⎡ ⎤ (τη)α ⎡ ⎤
L f 1(2)α1 m 0 0
(34) ¯ N
⎢ 1 ⎥
(τ)α1 ⎢
⎢ 1 ⎥ u(1η)
⎥ ⎢ ⎥
⎢ (τ)α2 ⎥ ∑⎢
N +1
⎥⎢ (η) ⎥
(40)
(τη)α
In the present work, the general boundary conditions along the edges ⎢¯N 12 ⎥ = −
⎣ ⎦
⎢ 0
⎢
L f 2(2)α2 m
1
0 ⎥⎢ u 2 ⎥
⎥⎣ ⎦
of the structure have been assessed in terms of two different distribu­ (τ)α3 η=0 ⎣ ⎦ (η)
¯T 1 (τη )α u3
0 0 L f 3(2)α2 m
tions of linear springs, namely a Double-Weibull and a super elliptic one. 1

The external constraints are obtained by properly setting the governing

parameters for each distribution. Thus, the following dimensionless and the generalized external constraints along α 2 = αm2 for m = 0, 1 and
[ ]
coordinates (ξ̄, ̃
ξ) ∈ [0, 1] are introduced at each side: α 1 ∈ α01 , α11 read as follows:
⎡ ⎤
α r − α0r ̃ α1r − α r ⎡ ⎤ (τη)α ⎡ ⎤
ξ̄ = , ξ= 1 = 1 − ξ̄ for r = 1, 2 (35) ¯N (τ)α1 ⎢ L f 1(1)α1 m
2
0 0 ⎥ u(1η)
α1r − α0r α r − α0r ⎢ 21 ⎥ ∑⎢
N+1 ⎢ ⎥ ⎢ ⎥
⎢ (τ)α2 ⎥ ⎥⎢ (η) ⎥
(41)
(τη)α
⎢¯N 2 ⎥ = − ⎢ 0 L f 2(1)α2 m 0 ⎥⎢ u 2 ⎥
Based on Eq. (35), a Double-Weibull (D) linear springs distribution ⎣ ⎦ ⎢
η=0 ⎣
2 ⎥⎣
(τη)α 3 ⎦ u(η)
⎦
¯T (2τ)α3
can be enforced as follows: 0 0 L f 3(1)αm 3
2
( )p ( ξ̃ )p
− ¯ξ
(36)
−
In Eqs. (40)–(41) the fundamental boundary coefficients L fi(p)αimn have
(τη)α
f (ξ) = 1 − e ξ̄m + e ξ̃m
been introduced, for each τ, η-th kinematic expansion order, and α i = α 1 ,
where ξ̄ m ∈ [0, 1] and ̃
ξ m ∈ [0, 1], whereas p denotes the order of the α 2 , α 3 , m = 0, 1, n = 1, 2 and p = 1, 2. They can be expressed in a
distribution. In the same way, a super elliptic (S) analytical expression is condensed form as:
introduced:
l ∫
∑ ζ k+1
⃒ ⃒p
(42)
(τη)α (k)αmn
− ⃒ m⃒
ξ̄− ξ̄
L fi(p)αimn = k if λ̄F αη i Fατ i H p dζ
f (ξ) = e ξ̃m (37) k=1 ζk

being ξ̄ m ∈ [0, 1] and ̃

ξ m ∈ [0, 1] the position and the scaling parameter 2.5. Basic governing equations
of the distribution, respectively, whereas p ∈ R is a power exponent.
For each component of the external spring loads (33)–(34), we The equilibrium dynamic relations are now derived from the
introduce a shape function λ̄ = λ̄(ζ) dependent on the thickness direc­ Hamiltonian Principle [22], accounting for the time integral, within the
tion, such that a constant, linear and parabolic spring dispersion can be arbitrary interval [t 1 , t 2 ] of the total elastic strain energy Φ, the inertial
defined as λ̄ = 1, λ̄ = 2ζ/h, and λ̄ = 1 − (2ζ/h)2 (Fig. 2). When boundary contribution T and the virtual work L f induced by the external loads
loads are embedded in the ESL model, some equivalent quantities should related to the elastic springs [22]:
be defined on the reference surface, accounting for the stress distribu­ ∫ t2 ( ) ∫ t2 ( )
[ ] [ ]
tion. Referring to the physical domain α01 , α11 × α02 , α12 of the struc­ δ T − Φ + L f dt = 0 → δT − δΦ + δL f dt = 0 (43)
tural model, the following quantities can be defined for each τ-th t1 t1

kinematic expansion order at the edges characterized by α 1 = αm1 , for More specifically, the kinetic term δT is assessed taking into account
[ ]
m = 0, 1 and α 2 ∈ α02 , α12 . If l is the number of laminae occurring in the the inertial contribution of each k-th layer of a l laminae stacking
stacking sequence, one has [22]: sequence to the overall structure [22]:
∫ l ∫
∑ ζk+1 ∫ ∫
( ) ∑l ζk+1
¯N (1τ)α1 αm1 , α 2 = ¯σ (k) α1
1 λ̄F τ H2 dζ
δT = ρ(k) (δU)T Ü H 1 H 2 A 1 A 2 dα 1 dα 2 dζ (44)
k=1 ζk k=1 ζk α1 α2

l ∫
∑ ζk+1
τ α2 ( )
¯N (12) αm1 , α 2 = ¯τ(k) α2
12 λ̄F τ H2 dζ (38) where ρ(k) is the density associated with each k-th layer of the stacking
k=1 ζk [ ]T
∫ sequence and Ü = Ü 1 Ü 2 Ü 3 the second order time derivative of the
( ) ∑l ζk+1
¯T (1τ)α3 αm1 , α 2 = ¯τ(k) α3
13 λ̄F τ H2 dζ
k=1 ζk three-dimensional displacement field vector. The virtual variation δΦ of
the elastic strain energy can be stored in Eq. (43) accounting for the
In the same way, the boundary stresses can be computed on the edges
[ ] generalized stress resultant S(τ)α i (α1 , α2 , t) and the higher order ESL
with α 2 = αm2 for m = 0, 1 and α 1 ∈ α01 , α11 as follows [22]:
strains ε(τ)α i (α1 , α2 , t) for each τ = 0, ..., N +1 and αi = α 1 , α 2 , α 3 :
∫
τ)α1 ( ) ∑l ζk+1
∑ 3 ∫ ∫
N +1 ∑
¯N (21 α 1 , αm2 = ¯τ(k) α1 ( )T
12 λ̄Fτ H1 dζ δΦ = δε(τ)αi S(τ)αi A1 A2 dα1 dα2 (45)
k=1 ζk
τ=0 i=1 α1 α2
l ∫ ζk+1
∑
(τ)α2 ( )
¯N 2 α 1 , αm2 = (k)
¯σ 2 λ̄Fτα2 H1 dζ (39) The complete demonstration of Eq. (45) can be found in Ref. [22].The
ζk
energetic contribution δL f of the external loads consists of a part related
k=1
∫
( ) ∑l ζk+1 [ ]T
¯T (τ2)α3 α 1 , αm2 = ¯τ(k) α3
23 λ̄Fτ H1 dζ to the top and bottom surface actions q fk = q 1 fk q 2 fk q 3 fk reported
(τ ) (τ ) (τ ) (τ )
k=1 ζk
in Eq. (27) related to the Winkler foundation, and another one ac­
The generalized stress resultants of Eqs. (38)–(39) can be expressed counting for boundary stresses (38), (39), according to Eq. (40) [22]:
in terms of the generalized displacement components u(iη) for i = 1, 2, 3
and η = 0,...,N + 1. In this way, they can be easily embedded in the two-

8
F. Tornabene et al. Composite Structures 309 (2023) 116542

N +1 ∫ ∫ (
∑ ) ⎡ ⎤
τ τ τ τ τ τ
δL f = q(1 )fk δu(1 ) + q(2fk) δu(2 ) + q(3fk) δu(3 ) A 1 A 2 dα 1 dα 2 + I 0(τη)α 1 α 1 0 0
τ=0 α1 α2 ⎢ ⎥
M(τη) =⎢ 0 I 0(τη)α 2 α 2 0 ⎥ for τ, η = 0, ..., N + 1 (49)
N +1 ∮ (
∑ ) ⎣ ⎦
(τ)α (τ) (τ)α (τ) (τ)α (τ) 0(τη)α 3 α 3
+ ¯N 21 1 δu 1 +¯N 2 2 δu 2 +¯T 2 3 δu 3 A 1 dα 1 + 0 0 I
(46)
τ=0
α1
N +1 ∮ (
∑ ) where each component I0(τη)α i α j for i, j = 1, 2, 3 can be computed as:
(τ)α (τ) (τ)α (τ) (τ)α (τ)
+ ¯N 1 1 δu 1 +¯N 12 2 δu 2 +¯T 1 3 δu 3 A 2 dα 2
l ∫
∑ ζk+1
τ=0
α2
I 0(τη)α i α j = ρ(k) Fτα i Fηα j H1 H2 dζ for α i , α j = α 1 , α 2 , α 3 ,
In the previous relation, the two last integrals account for an inte­ k=1 ζk

gration along the curved edges of the physical domain characterized by τ, η = 0, ..., N + 1
α 2 and α 1 constant, respectively, whereas the first one is a surface in­ (50)
tegral referred to the entire physical domain. Employing the virtual Introducing in Eq. (47) the generalized constitutive relationship
variation of the kinetic energy derived in Eq. (44), the elastic contri­ (21), as well as the ESL kinematic relation reported in Eq. (16), the
bution of Eq. (45), and boundary springs reported in Eq. (46) in the application of the variational theorems leads to the final form of the
Hamiltonian Principle of Eq. (43), the equilibrium relations for an fundamental equations:
anisotropic doubly-curved shell are derived in the following compact
form [22]: ∑
N +1
( ) ∑
N +1
L(τη) − Lf (τη) u(η) = M(τη) ü(η) for τ = 0, ..., N + 1 (51)
∑
3 ∑
N +1 η=0 η=0
D*Ωαi S(τ)αi + q fk = (47)
(τ)
M(τη) ü(η) for τ = 0, ..., N + 1
i=1 η=0
where Lf(τη) is the fundamental matrix related to the surface Winkler
[ ]T elastic support, accounted in Eq. (27) for any τ, η-th kinematic expansion
where ü(η) = ü 1 ü 2 ü 3
(η ) (η ) (η )
is the second order time derivative of the order. The elastic fundamental differential matrix L(τη) , for each
τ, η = 0, ..., N +1 order, assumes the following expanded form [22]:
generalized displacement field u(η) , for each η = 0, ..., N + 1, whereas
D*Ωαi for i = 1, 2, 3 stand for the following equilibrium operators of di­
mensions 3 × 9, according to Ref. [22]:

⎡ ⎤T ⎡ ⎤T
⎡ ⎤T
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 1 ∂ 1 ∂A2 ⎥ ⎢ ⎥ ⎢ ⎥
⎢ + 0 0⎥ ⎢ ⎥ ⎢ ⎥
⎢ A1 ∂α1 A1 A2 ∂α1 ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢0
⎢
⎢ 1 ∂A 2
⎥
⎥ ⎢ − 1 ∂A 1
A1 A2 ∂α2
0⎥
⎥
⎢
⎢0 0 − 1
R1
⎥
⎥
⎢ − 0 0⎥ ⎢ ⎥ ⎢ ⎥
⎢ A1 A2 ∂α1 ⎥ ⎢ 1 ∂ ∂A1 ⎥ ⎢ ⎥
⎢ ⎥ ⎢0
⎢ A2 ∂α2
+ A11A2 ∂α2 0⎥
⎥
⎢0
⎢ 0 − 1
R2
⎥
⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 1 ∂A1 ⎥ ⎢0
⎢ 0 0⎥ ⎢
1 ∂
+ A11A2 ∂A2
0⎥
⎥
⎢0
⎢ 0 0 ⎥
⎥
⎢ A1 A2 ∂α2 ⎥ ⎢ A1 ∂α1 ∂α1
⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢0 0 0 ⎥
⎢ ⎥ ⎢0 1 ∂A2
0⎥ ⎢ ⎥
⎢ 1 ∂ 1 ∂A1 ⎥ ⎢ A1 A2 ∂α1 ⎥ ⎢ ⎥
⎢ + 0 0⎥ (48)
D*Ωα 1 = ⎢ A2 ∂α2 A1 A2 ∂α2 ⎥ D*Ωα 2 =⎢
⎢0 0
⎥
0⎥ D*Ωα 3 =⎢
⎢0 0 1 ∂
A1 ∂α1
+ A11A2 ∂A2 ⎥
∂α1 ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 1 ⎥ ⎢ ⎥ ⎢0 ∂A1 ⎥
⎢ ⎥ ⎢0 1
0⎥ ⎢ 0 1 ∂
+ A11A2
⎢ 0 0⎥ A2 ∂α2 ∂α2 ⎥
⎢ R1 ⎥ ⎢ R2 ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎢0 0 0⎥ ⎢0 0 0 ⎥
⎢ 0 0 0⎥
⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎢0 − 1 0⎥ ⎢0 0 0 ⎥
⎢ − 1 0 0⎥
⎥ ⎢
⎢
⎥
⎥
⎢
⎢
⎥
⎥
⎢ ⎥ ⎢0 0 0⎥ ⎢0 0 − 1 ⎥
⎢ 0 0 0⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ 0 0 0⎥ ⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥ ⎣ ⎦
⎢ ⎥ ⎣ ⎦
⎣ ⎦

The inertial mass matrix M(τη) , defined for each τ, η = 0,...,N + 1, is 3 ∑

∑ 3
D*Ωαi A(τη)αi αj DΩj
α
now reported in an extended matrix form [22]: L(τη) =
i=1 j=1
⎡ ⎤
(τη)α α
1 1 (τη)α1 α2 (τη)α1 α3
⎢ L11 L12 L13 ⎥
⎢ (τη)α2 α1 ⎥
= ⎢ L21 L(22τη)α2 α2 L(23τη)α2 α3 ⎥ for τ, η = 0, ..., N + 1 (52)
⎣ ⎦
(τη)α3 α1
L31 L(32τη)α3 α2 L(33τη)α3 α3

Thus, the fundamental relation reported in Eq. (51) is assembled, for

any τ, η expansion order. In an extended matrix form, one gets [22]:

9
F. Tornabene et al. Composite Structures 309 (2023) 116542

⎡ ⎤⎞
⎛⎡ ⎤ ⎢ ⎥ ⎟⎡ (0) ⎤
L(00) L(01) ⋯ L(0(N) ) L(0(N+1) ) ⎢ Lf (00) Lf (01) ⋯ Lf (0(N) ) Lf (0(N+1) ) ⎥⎟ u
⎜⎢ ⎥ ⎢ ⎥ ⎟⎢ (1) ⎥
⎜⎢ L(10) L(11) ⋯ L(1(N) ) L(1(N+1) ) ⎥ ⎢ Lf (10) Lf (11) ⋯ Lf (1(N) ) Lf (1(N+1) ) ⎥ ⎟⎢ u ⎥
⎜⎢ ⎥ ⎢ ⎥ ⎟⎢ ⎥
⎜⎢ ⋮ ⋮ ⋱ ⋮ ⋮ ⎥− ⎢ ⋮ ⋮ ⋱ ⋮ ⋮
⎥ ⎟⎢
⎥ ⎟⎢ ⋮ ⎥=
⎜⎢ ⎥ ⎢ ⎥
⎜⎢ ((N)0 ) ⎥ ⎢ ⎥ ⎟⎢ ⎥
⎝⎣ L L((N)1 ) ⋯ L((N)(N) ) L((N)(N+1) ) ⎦ ⎢⎢ L
f ((N)0 )
Lf ((N)1 ) ⋯ Lf ((N)(N) ) Lf ((N)(N+1) ) ⎥ ⎟⎣ u(N)
⎥⎟ ⎦
⎢ f ((N+1)0 ) ⎥ ⎟ (N+1)
L((N+1)0 ) L((N+1)1 ) ⋯ L((N+1)(N) ) L((N+1)(N+1) ) ⎣L Lf ((N+1)1 ) ⋯ Lf ((N+1)(N) ) Lf ((N+1)(N+1) ) ⎦⎠ u
(53)
⎡ ⎤⎡ ⎤
M(00) M(01) ⋯ M(0(N) ) M(0(N+1) ) ü(0)
⎢ ⎥⎢ (1) ⎥
⎢ M(10) M (11)
⋯ M (1(N) )
M (1(N+1) )
⎥⎢ ü ⎥
⎢ ⎥⎢ ⎥
=⎢
⎢ ⋮ ⋮ ⋱ ⋮ ⋮ ⎥⎢ ⋮
⎥⎢
⎥
⎥
⎢ ⎥⎢ (N) ⎥
⎣ M((N)0 ) M((N)1 ) ⋯ M((N)(N) ) M((N)(N+1) ) ⎦⎣ ü ⎦
M((N+1)0 ) M((N+1)1 ) ⋯ M((N+1)(N) ) M((N+1)(N+1) ) ü(N+1)

3. Governing equations for distorted domains being w i (for each i = 0,..., n) a coefficient defining the influence of the
corresponding control point P i on the i-th basis function Ni,p (u), con­
In the previous sections, the fundamental governing equations have
sisting of a piecewise polynomial. The latter is a univariate B-Spline
been derived for a laminated doubly curved shell in a principal coor­
curve which has been defined setting a = 0 and b = 1. In particular, the
dinate parametrization setting. However, in cases of shells with arbi­
extended clamped knot vector Ω has been set so that Ω =
trary geometries, a key issue consists in finding a curvilinear principal [ ]
coordinate system, according to differential geometry principles. For a, …, a , up+1 , …, um− p− 1 , b, …, b , being p +1 the first and the last
⏟̅̅̅̅̅⏞⏞̅̅̅̅̅⏟ ⏟̅̅̅̅̅⏞⏞̅̅̅̅̅⏟
this reason, a mapping procedure of the physical domain is adopted p+1 p+1
(Fig. 1). Based on the domain distortion, it is possible to apply Eq. (53) knot multiplicity and m the number of the selected breakpoints. To sum
also for any arbitrary shaped structure. To this end, the blending func­ up, the well-known recursive procedure [22] has been adopted for the
tions α1 = α1 (ξ1 , ξ2 ) and α2 = α2 (ξ1 , ξ2 ) should be assessed first [22]: derivation of Ni,p (u) due to its computationally-friendly nature. More
specifically, it provides a piecewise polynomial, taking into account a

1( )
α 1 (ξ1 , ξ2 ) = (1 − ξ2 )ᾱ1(1) (ξ1 ) + (1 + ξ1 )ᾱ1(2) (ξ2 ) + (1 + ξ2 )ᾱ1(3) (ξ1 ) + (1 − ξ1 )ᾱ1(4) (ξ2 ) +
2
(54)
1( )
− (1 − ξ1 )(1 − ξ2 )α1(1) + (1 + ξ1 )(1 − ξ2 )α1(2) + (1 + ξ1 )(1 + ξ2 )α1(3) + (1 − ξ1 )(1 + ξ2 )α1(4)
4

1( )
α 2 (ξ1 , ξ2 ) = (1 − ξ2 )ᾱ2(1) (ξ1 ) + (1 + ξ1 )ᾱ2(2) (ξ2 ) + (1 + ξ2 )ᾱ2(3) (ξ1 ) + (1 − ξ1 )ᾱ2(4) (ξ2 ) +
2
(55)
1( )
− (1 − ξ1 )(1 − ξ2 )α2(1) + (1 + ξ1 )(1 − ξ2 )α2(2) + (1 + ξ1 )(1 + ξ2 )α2(3) + (1 − ξ1 )(1 + ξ2 )α2(4)
4

As it can be seen, Eqs. (54)–(55) are based on a NURBS description of step function for p = 0, and a linear combination of Ni,p− 1 (u) and
( )
the p = 1, ..., 4 external edge ᾱ1(p) , ᾱ2(p) of the structure within the Ni+1,p− 1 (u) for an arbitrary p. This means that
( )
computational domain, whereas α1(q) , α2(q) for q = 1, ..., 4 represents {
1 if ui ⩽u < ui+1
the domain corners. In this way, the mapped shell geometry is Ni,0 (u) =
completely defined in a unified manner in a rectangular parent element. 0 otherwise (57)
When blending functions are assessed, a new set of natural coordinates u − ui ui+p+1 − u
Ni,p (u) = Ni,p− 1 (u) + Ni+1,p− 1 (u)
ξ 1 ∈ [ − 1, +1] and ξ 2 ∈ [ − 1, +1] is introduced for the computational ui+p − ui ui+p+1 − ui+1
domain mapping. The parameters ξ 1 and ξ 2 are defined from the
The mapping algorithm of the physical domain reported in Eqs. (54)–
normalization of the NURBS parametrization C(u) of the structural
(55) is employed for the coordinate transformation α i (ξ1 , ξ2 ) for i = 1,2.
edges, being u ∈ [a, b] the general variable for the description of the
In this way, the fundamental relations of Eq. (53) are suitable for arbi­
curve, and a, b ∈ R. If we denote by p the degree of the curve, the
trarily shaped domains. As a consequence, the first order derivatives of
function C(u) reads as follows [22]:
the above-described blending functions are required. In a compact form,
∑n
Ni,p (u)wi Pi they are accounted as follows [22]:
C(u) = ∑i=0 n (56)
i=0 Ni,p (u)wi

10
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 2. Schematic representation of a general boundary distribution of linear elastic springs along the edges of a doubly-curved shell, for α i = α 1 , α 2 or ξ i = ξ 1 , ξ 2 .
A general surface variation is set along the three-dimensional thickness of the shell by properly setting a boundary distribution of elastic springs. A constant, linear
and parabolic distribution can be adopted along the thickness of the laminated structure.

⎡ ⎤ ⎡ ⎤⎡ ⎤
∂ ∂ξ1 ∂ξ2 ∂
⎢ ∂α ⎥ ⎢ ∂α ∂ξ1 1 ∂α2 ∂ξ1 1 ∂α1
⎢ 1⎥ ⎢ 1 ∂α1 ⎥⎢ ⎥
⎥⎢ ∂ξ1 ⎥ ξ1,α1 = = , ξ1,α2 = =−
⎢ ⎥=⎢ ⎥⎢ ⎥ (58) ∂α1 det(J) ∂ξ2 ∂α2 det(J) ∂ξ2
⎣ ∂ ⎦ ⎣ ∂ξ1 ∂ξ2 ⎦⎣ ∂ ⎦ (63)
∂ξ2 1 ∂α2 ∂ξ2 1 ∂α1
∂α2 ∂α2 ∂α2 ∂ξ2 ξ2,α1 = =− , ξ2,α2 = =
∂α1 det(J) ∂ξ1 ∂α2 det(J) ∂ξ1
By employing the inverse transformation, a unified expression of the
In this way, the multiplication by J− 1 of the first order derivatives
derivatives with respect to the natural coordinates ξ1 , ξ2 can be easily
with respect to the natural coordinates ξ1 , ξ2 leads to a direct compu­
provided:
tation, within Eq. (53), of the derivatives with respect to α 1 , α 2 for an
⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎡ ⎤
∂ ∂α1 ∂α2 ∂ ∂ arbitrarily shaped geometry.
⎢ ∂ξ ⎥ ⎢ ∂ξ
⎢ 1⎥ ⎢ 1 ∂ξ1 ⎥⎢ ⎥
⎥⎢ ∂α1 ⎥
⎢ ∂α ⎥
⎢ 1⎥ Starting from Eq. (61), the second-order derivatives with respect to
⎥=⎢ ⎥ = J⎢ (59)
⎢
⎣ ∂ ⎦ ⎣ ∂α1
⎥⎢
∂α2 ⎦⎣ ∂ ⎦ ⎣ ∂ ⎦
⎥ the principal coordinates α 1 , α 2 can also be performed in terms of ξ1 ,ξ2 ,
∂ξ2 ∂ξ2 ∂ξ2 ∂α2 ∂α2 as shown in Ref. [22]:

where J is the Jacobian matrix of the transformation, so that [22]: ∂2 ∂2 ∂2

= ξ21,α1 2 + ξ22,α1 2 + 2ξ1,α1 ξ2,α1
∂2
+ ξ1,α1 α1
∂
+ ξ2,α1 α1
∂
(64)
∂α21 ∂ξ1 ∂ξ2 ∂ξ 1 ∂ξ2 ∂ξ1 ∂ξ2
∂α1 ∂α2 ∂α2 ∂α1
det(J) = − (60)
∂ξ1 ∂ξ2 ∂ξ1 ∂ξ2
∂2 ∂2 ∂2 ∂2 ∂ ∂
= ξ21,α2 2 + ξ22,α2 2 + 2ξ1,α2 ξ2,α2 + ξ1,α2 α2 + ξ2,α2 α2 (65)
If det(J) ∕
= 0 at each point of the computational domain according to ∂α22
∂ξ1 ∂ξ2 ∂ξ1 ∂ξ2 ∂ξ1 ∂ξ2
Eq. (60), the derivative transport relation of Eq. (59) can be inverted,
and the following expression comes out: ∂2 ∂2 ∂2 (
= ξ1,α1 ξ1,α2 2 + ξ2,α1 ξ2,α2 2 + ξ1,α1 ξ2,α2
⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ ∂α1 ∂α2 ∂ξ1 ∂ξ2
∂ ∂ ∂ξ1 ∂ξ2 ∂
⎢ ∂α ⎥ ⎢ ∂ξ ⎥ ⎢ ∂α ⎥⎢ ⎥ ) ∂2 ∂ ∂
⎢ 1⎥ ⎢ 1 ⎥ ⎢ 1 ∂α1 ⎥⎢ ∂ξ1 ⎥ (66)
⎢ ⎥ = J− 1 ⎢ ⎥=⎢ ⎥⎢ ⎥ (61) + ξ1,α2 ξ2,α1
∂ξ1 ∂ξ2
+ ξ1,α1 α2
∂ξ1
+ ξ2,α1 α2
∂ξ2
⎣ ∂ ⎦ ⎣ ∂ ⎦ ⎣ ∂ξ1 ∂ξ2 ⎦⎣ ∂ ⎦
∂α2 ∂ξ2 ∂α2 ∂α2 ∂ξ2 The extended version of all the coefficients occurring in Eqs. (64)–
− 1 (66) is provided in Appendix I.
Since the inverse of the Jacobian matrix J is computed according to
In cases of shells described by arbitrarily shaped domains, some re­
the following unified relation [22]:
marks are required to assess the static and kinematic boundary condi­
⎡ ⎤
∂α2 ∂α2 tions. Actually, a local coordinate system of axes nn , ns and nζ is set for
− [ ]
1 ⎢ ⎢ ∂ξ2 ∂ξ1 ⎥
⎥ ξ ξ each structural edge, which is described by means of a NURBS according
J− 1 = ⎢ ⎥ = ξ1,α1 ξ2,α1 (62) to Eqs. (54)–(55). It is useful to define such directions with respect to the
det(J) ⎣ ∂α1 ∂α1 ⎦ 1,α2 2,α2
− principal coordinate system O α 1 , α 2 , ζ, so that n n = [n n1 n n2 n n3 ]T ,
′
∂ξ2 ∂ξ1
n s = [n s1 n s2 n s3 ]T and n ζ = [n ζ1 n ζ2 n ζ3 ]T with the components
The comparison of the J− 1 expression of Eq. (62) and the inverse
defined from the direction cosines of the vectors at issue. When the
coordinate transformation collected in Eq. (61) provides an extended
mapped edges are parallel to the principal parametric lines of the shell at
form of the coefficients introduced in Eq. (62):
issue, the local coordinate system coincides with the global one, except

11
F. Tornabene et al. Composite Structures 309 (2023) 116542

for the orientation of each axis. can be found in Ref. [22].

General boundary conditions defined in Eq. (40) can be expressed According to the coordinate blending transformation reported in
[ ]T
Eqs. (54)–(55), the computational domain is described in terms of nat­
accounting for the generalized displacement field vector u n u s u ζ
(τ ) (τ) (τ )
( )
ural coordinates, so that ξ 1 , ξ 2 ∈ [ − 1, 1] × [ − 1, 1]. Thus, a non-
for τ = 0, ..., N + 1, so that external constraints are developed for an
uniform discrete set of points is selected, based on the well-known
irregular edge of the structure. To this end, the following relations are
Chebyshev-Gauss-Lobatto (CGL) grid distribution [22]:
useful [22]: ( )
p− 1
τ
u(nτ) = nn1 u(1 ) + nn2 u(2 )
τ ξrp = − cos π , p = 1, ..., IP , r = 1, 2, p = 1, ..., IP
IP − 1 (72)
(τ) (τ)
u(sτ) = ns1 u 1 + ns2 u 2 (67) IP = IN , IM , for ξrp ∈ [ − 1, 1]
τ τ
u(ζ ) = u(3 )
More specifically, a two-dimensional discrete grid distribution of
In the same way, the generalized stress resultants N(nτ)α 1 , N(nsτ)α 2 and IN × IM points along the natural coordinates ξ 1 , ξ 2 has been defined.
Referring to the collocation points defined in Eq. (72), the arbitrary n-th
T (ζτ)α 3 , for each τ = 0, ..., N + 1, are evaluated for the mapped geometry
and m-th order derivatives of a two-dimensional function f(ξ 1 , ξ 2 ), with
according to the following relations:
n, m = 0, 1, 2 along ξ 1 and ξ 2 can be expressed as a weighted sum of the
τα τα
Nn(τ)α1 = N1( ) 1 n2n1 + N2( ) 1 n2n2 + N12
( ) 1 τα ( ) 1
nn1 nn2 + N21 nn1 nn2
τα values f(ξ 1k , ξ 2l ) assumed in the grid of Eq. (72) for k = 1, ..., I N and l =
1, ..., I M . In conclusion, the following relation is derived [22]:
(τ)α2
Nns =
τα
N1( ) 2 nn1 ns1 +
τα
N2( ) 2 nn2 ns2 (τ)α2
+ N12 nn1 ns2
τα
( ) 2
+ N21 nn2 ns1 (68)
τα τα ⃒ ( )
(τ)α3
Tζ( ) 3 = T1( ) 3 nn1 + T2 nn2 ∂n+m f (ξ 1 ,ξ 2 ) ⃒⃒ ∑ IN
ξ 1 (n)
∑IM
ξ 2 (m) i = 1, 2,..., IN
≅ ϛik ϛjl f (ξ 1k ,ξ 2l )
∂ξn1 ∂ξm2 ⃒ξ =ξ , ξ =ξ j = 1, 2,..., IM
In this way, the general boundary conditions assessed on irregular 1 k=11i 2 l=1 2j

mapping can be referred to the curvilinear principal coordinate system. (73)

Furthermore, kinematic boundary conditions can be also applied on the The weighting coefficients with r = 1, 2, p = i, j, q = n,m and v =
ξ r (q)
ϛpv
shell mapped geometry [22]. In the present article, the fully clamped (C)
k,l required in Eq. (73) are computed from the following recursive
boundary condition is accounted for as follows:

τ
u(nτ) = u(sτ) = u(ζ ) = 0 for τ = 0, 1, 2, ..., N, N + 1, at ξ 1 = − 1 or ξ 1 = 1, − 1⩽ξ 2 ⩽1
τ
(69)
u(nτ) = u(sτ) = u(ζ ) =0 for τ = 0, 1, 2, ..., N, N + 1, at ξ 2 = − 1 or ξ 2 = 1, − 1⩽ξ 1 ⩽1

In the case of free edge (F), the following relations must be assessed relationship presented in Ref. [22], in turn derived from the properties
on the interested edges: of the Lagrange polynomials L , valid for q⩾1:

τ
Nn(τ) = 0, (τ)
Nns = 0, Tζ( ) = 0 for τ = 0, 1, 2, ..., N, N + 1, at ξ 1 = − 1 or ξ 1 = 1, − 1⩽ξ 2 ⩽1
(70)
(τ)
Nns = 0, Nn(τ) = 0, Tζ(τ) =0 for τ = 0, 1, 2, ..., N, N + 1, at ξ 2 = − 1 or ξ 2 = 1, − 1⩽ξ 1 ⩽1

For an easier identification of the external edges of the structure it is

( )
useful to adopt the following nomenclature [22]: L (1)
(
ξ rp
) ξ r (q− 1)
ϛpv
ϛξpvr (1) = , ξ r (q)
ϛpv = q ϛξpvr (1) ϛpp
ξ r (q− 1)
− p∕
=v
West edge (W) → − 1⩽ξ 1 ⩽1, ξ2 = − 1 (ξ rp − ξ rv )L (1) (ξ rv ) ξ rp − ξ rv
South edge (S) → ξ 1 = − 1, − 1⩽ξ 2 ⩽1 ∑N
(71) ϛξppr (q) = − ϛξ r (q)
=p pv
v=1 v∕
p=v
East edge (E) → − 1⩽ξ 1 ⩽1, ξ2 = 1
(74)
North edge (N) → ξ 1 = 1, − 1⩽ξ 2 ⩽1
When q = 0, the GDQ interpolating coefficients ϛpvr are defined for
ξ (q)

4. Numerical implementation each r = 1, 2 so that = δ pv , being δ pv the Kronecker delta. Note

ξ (0)
ϛpvr
that Eq. (73) provides a general formulation for mixed derivatives of the
In the present section, we provide the main fundamentals of the GDQ two-dimensional function f(ξ 1 , ξ 2 ). Pure derivatives with respect to ξ 1 ,
method, as here applied to treat numerically the governing equations. ξ 2 can thus be easily performed by properly setting n, m = 0. When the
The extended numerical implementation of the present higher order governing differential equation is expanded for each τ, η = 0,...,N + 1, as
problem is here avoided for the sake of conciseness. It is based on the reported in Eq. (53), it is numerically solved by means of the quadrature
definition of a non-uniform two-dimensional grid within the computa­ procedure of Eq. (73), redefined in matrix form. To this end, we employ
tional domain and the application of the classic GDQ approach to the well-known vectorization of a generic A matrix of dimension I N ×
bivariate smooth functions for the approximation of derivatives. Basi­
I M , which provides a I N I M × 1 column vector accounted as
cally, the proposed procedure is based on the vectorization of the
discrete values assumed by the solution along the computational grid, →
A = Vec(A) (75)
together with the Kronecker product for the extension of the one-
dimensional GDQ rule to the multi-dimensional case. As far as the nu­ In this way, if the generic element of A is identified by A ij for i = 1,
→
merical assessment of boundary conditions is concerned, further details ..., I N and j = 1,..., I M , it can be associated with the component A k of A

12
F. Tornabene et al. Composite Structures 309 (2023) 116542

with k = 1, ..., I N I M according to the following relation: points if compared to classical approaches. Based on the numerical
( ) GDQ method of Eq. (73) for derivatives of a generic order, the pro­
i = 1, ..., I N j = 1, ..., I M
A k = A ij for (76) cedure allows a quadrature-based computation of integrals of a generic
k k = i + (j − 1)I N
thickness-dependent function f(ζ) according to the following expres­
Nevertheless, for the generic two-dimensional function f(ξ 1 , ξ 2 ), its sion [22]:
arbitrary (n + m)-th derivation order can be numerically assessed ac­ ∫ ζk+1 ∑
IT
cording to the following matrix expression: f (ζ)dζ =
( ) ( )
w(k+1)g − wkg f ζ g (81)
ζk g=1
→(n+m) →
f = ϛξ 1 (n)ξ 2 (m) f (77)
being I T the number of discrete points selected in the closed interval
being f and f (n+m) the matrices of the values assumed by the function f [ζ k , ζ k+1 ], identified with ζ g for g = 1, ..., I T . In the present work, each
and its (n + m)-th order derivative, respectively, at each point of the k-th layer of the stacking sequence of an l laminate has been discretized
computational domain, vectorized according to Eq. (75). Starting from by means of the CGL distribution, as follows [22]:
Eq. (73), the GDQ weighting coefficients of Eq. (74) for the derivatives ( )
g− 1
along ξ 1 and ξ 2 are collected in a I N I M × I N I M matrix ϛξ 1 (n)ξ 2 (m) , ξkg = − cos π , k = 1, ..., l, g = 1, ..., IT for ξkg ∈ [ − 1, 1]
IT − 1
developed employing the well-known Kronecker product [22], as
(82)
follows:
The discretization employed in Eq. (82), defined in the closed in­
ϛξ 1 (n)ξ 2 (m) = ϛξ 2 (m) ⊗ ϛξ 1 (n) (78)
terval [ − 1, 1], can be shifted to the physical interval [ζ k , ζ k+1 ] via the
following coordinate transformation [22]:
being ϛξ r (q) for r = 1, 2 and q = n, m a squared matrix of order I Q = I N ,
I M . As also observed in the previous sections, when n = 0 or m = 0 the ζkg =
ζk+1 − ζk ( )
ξkg + 1 + ζk , k = 1, ..., l, g = 1, ..., IT for
identity matrix should be introduced, setting ϛξ r (0) = I. 2 (83)
When the discrete form of the fundamental governing equation for a ξkg ∈ [ − 1, 1]
doubly-curved shell is considered according to Eq. (53), for each τ, η = 0, Since the GIQ method reported in Eq. (81) comes from the general
..., N + 1, partial derivatives with respect to α 1 and α 2 principal di­ properties of the fundamental theorem of integrals, the weighting co­
rections can be performed starting from the derivation with respect to ξ 1 efficients wkg and w(k+1)g are computed according to the recursive
and ξ 2 , according to Eq. (77). This means that matrices ϛα1 (1)α2 (0) and formulation set in Eq. (74). The interested reader can find more details
ϛα1 (0)α2 (1) must be introduced for n, m = 0, 1, as follows: in Ref. [22]. In the present article, a free vibration analysis of doubly-
→ ◦ → ◦ curved shells of arbitrary geometries is performed. Based on Eq. (51),
ϛα1 (1)α2 (0) = P 11 ϛξ1 (1)ξ2 (0) + P 21 ϛξ1 (0)ξ2 (1)
→ ◦ → ◦ (79) for each η-th order of the kinematic expansion, we define the generalized
ϛα1 (0)α2 (1) = P 12 ϛξ1 (1)ξ2 (0) + P 22 ϛξ1 (0)ξ2 (1)
displacement field vector u(η) (α 1 , α 2 , t) by applying a separation vari­
able method among the time-dependent and space-dependent quantities
where Pri for r = 1, 2 and i = 1, 2, collects the coefficients of the first
[22]:
order derivatives of the blending transformations of Eq. (54)–(55), ac­
cording to Eq. (63), and the symbol ◦ denotes the well-known Hadamard u(η) (α 1 , α 2 , t) = U(η) (α 1 , α 2 )eiωt (84)
product [22]. As far as the second order derivatives with respect to α 1
and α 2 are concerned, the weighting coefficient matrices are calculated being ω the circular frequency and U(η) (α 1 , α 2 ) the corresponding modal
starting from Eqs. (64)–(66), providing the following equations: shape vector, for each η = 0, ..., N + 1. Employing Eq. (84) in the

→o2 →o2 → → → →
ϛα1 (2)α2 (0) = P 11 ◦ϛξ1 (2)ξ2 (0) + P 21 ◦ϛξ1 (0)ξ2 (2) + 2 P 11 ◦ P 21 ◦ϛξ1 (1)ξ2 (1) + P 111 ◦ϛξ1 (1)ξ2 (0) + P 211 ◦ϛξ1 (0)ξ2 (1)
→o2 →o2 → → → →
ϛα1 (0)α2 (2) = P 12 ◦ϛξ1 (2)ξ2 (0) + P 22 ◦ϛξ1 (0)ξ2 (2) + 2 P 12 ◦ P 22 ◦ϛξ1 (1)ξ2 (1) + P 122 ◦ϛξ1 (1)ξ2 (0) + P 222 ◦ϛξ1 (0)ξ2 (1) (80)
→ → → → (→ → → → ) → →
ϛα1 (1)α2 (1) = P 11 ◦ P 12 ◦ϛξ1 (2)ξ2 (0) + P 21 ◦ P 22 ◦ϛξ1 (0)ξ2 (2) + P 11 ◦ P 22 + P 12 ◦ P 21 ◦ϛξ1 (1)ξ2 (1) + P 112 ◦ϛξ1 (1)ξ2 (0) + P 212 ◦ϛξ1 (0)ξ2 (1)

As it can be seen, the coordinate transformation has been performed fundamental relations (51), the free vibration governing equations for
thanks to the computation of the matrices Prij , with r = 1, 2 and i,j = 1, each τ = 0, ..., N +1 assume the following form:
2. They collect all the second order derivatives ξr,αi ,αj of the natural
∑
N +1
( ) ∑
N +1
coordinates ξr with respect to αi and αj , which have been performed L(τη) − Lf (τη) U(η) + ω2 M(τη) U(η) = 0 for
following the Eqs. (64)–(66). The numerical integration of the gener­ η=0 η=0 (85)
alized stiffness coefficients A , for τ, η = 0, ..., N +1 and α i , α j =
(τη)αi αj τ = 0, 1, 2, ..., N, N + 1
α 1 , α 2 , α 3 , follows the Generalized Integral Quadrature (GIQ) algo­ It is useful to rearrange the DOFs of the discretized system from a
rithm. The use of such a numerical technique for the computation of separation within the computational grid defined in Eq. (72), of DOFs
integrals lies in the fact that Eq. (23) accounts for an integration on a denoted with δb related to the boundaries (“b” nodes) from the other
variable interval. Furthermore, a general expression is considered here unknown variables δd referred to the inner points (“d” points) [22]:
for the thickness functions. Moreover, the GIQ method allows one to [ ][ ] [ ][ ]
compute integrals also for the case of a general through-the-thickness f f
L̄bb − L̄bb L̄bd − L̄bd δb 2 0 0 δb
variation of the material properties within each layer, as happens for = ω (86)
L̄ − L̄
f
L̄ − L̄
f δd 0 M̄dd δd
example when FGM laminae are considered in the lamination scheme.
db db dd dd

Furthermore, the GIQ has been demonstrated to converge more steadily The final form of the discrete stiffness matrix of Eq.(86), as well as
to stable and accurate solutions with a reduced number of integration the mass one, follows the DOFs rearrangement at issue. In this way, it is

13
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 3. Geometric input data for the dynamic analysis of a rectangular plate and a catenoid mapped with a domain subjected to general external boundary conditions
and characterized by variable thickness along the principal lines. The equations of each reference surface have been written in curvilinear principal coordinates [22].
The computational domain mapping has been performed starting from the NURBS description of the structural edges.

14
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 4. Geometric input data for the dynamic analysis of an elliptic cone and a revolution paraboloid mapped with a general domain subjected to general external
boundary conditions and characterized by variable thickness along the principal lines. The equations of each reference surface have been written in curvilinear
principal coordinates [22]. The computational domain mapping has been performed starting from the NURBS description of the structural edges.

15
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 5. Geometric input data for the dynamic analysis of a hyperbolic paraboloid and a helicoid mapped with a domain subjected to general external boundary
conditions and characterized by variable thickness along the principal lines. The equations of each reference surface have been written in curvilinear principal
coordinates [22]. The computational domain mapping has been performed starting from the NURBS description of the structural edges.

16
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 6. Geometric input data for the dynamic analysis of a pseudosphere and a hyperbolic hyperboloid mapped with a domain subjected to general external boundary
conditions and characterized by variable thickness along the principal lines. The equations of each reference surface have been written in curvilinear principal
coordinates [22]. The computational domain mapping has been performed starting from the NURBS description of the structural edges.

17
F. Tornabene et al. Composite Structures 309 (2023) 116542

Table 1
First ten mode frequencies of a rectangular plate [22] with variable thickness and mapped geometry. It has been externally constrained with a three-dimensional set of
linear boundary springs characterized by different distributions along the external edges. The analysis refers to different higher order theories. The results have been
compared to the outcomes of a refined three-dimensional finite element model.
( )
Mapped Rectangular Plate BKDDD FBKDDD F
Double - Weibull distribution – sensitivity analysis

Mode 3D FEM Constraint edge parameter

f [Hz] CFCF
̃ξm = 1.0 ξm = 0.9
̃ ̃ξm = 0.8 ξm = 0.7
̃ ̃ξm = 0.6 ξm = 0.5
̃ ̃ξm = 0.4 ξm = 0.3
̃ ̃ξm = 0.2 ξm = 0.1
̃ ̃ξm = 0.01

DOFs 2131302 16182 – EDZ4

1 63.640 63.863 62.862 61.827 59.155 56.164 54.082 47.983 41.955 32.996 27.918 16.261
2 75.555 76.172 72.400 69.159 61.941 56.958 55.878 52.940 49.623 44.186 40.542 33.159
3 166.031 166.836 162.489 158.280 146.968 136.448 132.961 121.351 110.666 96.884 90.669 80.633
4 194.315 195.387 189.065 183.002 166.937 155.238 152.391 141.886 133.010 122.039 116.387 103.872
5 217.638 219.076 215.663 213.381 209.005 205.298 203.245 197.591 190.531 178.896 171.842 140.108
6 309.013 310.869 299.775 289.143 260.982 241.515 236.579 221.795 211.452 200.892 196.331 160.005
7 342.572 344.788 339.216 332.549 306.842 287.641 284.135 270.152 260.762 230.826 206.117 188.428
8 384.917 385.252 369.826 359.554 338.960 322.839 317.187 296.501 273.344 253.085 248.915 230.337
9 399.353 402.587 392.647 383.841 361.980 347.923 342.152 327.145 311.575 288.672 274.983 236.531
10 451.292 455.244 453.349 450.898 411.060 387.193 381.371 367.548 358.864 348.114 319.463 251.909
11 490.249 493.880 475.257 457.631 446.555 440.898 439.072 430.757 405.346 356.747 346.905 263.261
12 517.665 521.864 513.919 505.232 477.691 464.298 461.800 444.417 428.891 407.808 375.217 337.963
13 603.235 608.425 595.627 578.489 534.359 503.077 488.862 464.249 453.505 420.779 403.776 370.087
14 627.283 629.087 611.318 592.314 542.258 517.868 507.933 483.076 461.564 437.285 420.456 399.500
15 654.545 661.282 644.724 629.918 581.716 557.838 551.083 522.744 485.943 457.292 455.744 448.887
16 696.882 699.867 682.727 655.432 619.780 582.690 563.635 539.386 529.247 515.898 507.417 492.622
17 705.620 711.873 687.665 669.943 630.041 623.970 622.586 618.150 610.531 581.213 562.873 529.271
18 725.366 732.300 723.831 715.100 696.228 685.661 678.722 655.130 625.864 605.316 595.073 571.799
19 770.096 779.986 776.124 772.448 748.286 719.991 707.982 687.333 674.726 659.422 651.716 636.849
20 806.781 816.369 810.318 799.403 765.335 754.096 748.053 732.547 718.648 703.638 696.052 679.692
(k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11
Boundary Springs: ξ̄m = 0 (W), ξ̄m = 1 (E), p = 1000, k 1f = k 1f = k 1f = k 1f = 1⋅1021 N/m3 , k 2f = k 2f = k 2f = k 2f = 1⋅1021 N/m3 , k 3f = k 3f =
(k)ξ02 (k)ξ12
k 3f = k 3f = 1⋅1021 N/m3
Lamination Scheme: (0/30/45), h01 = h03 = 0.02 m, h02 = 0.03 m, 1st graphite-epoxy, 2nd triclinic, 3rd graphite-epoxy
Thickness Variation: ϕ 1 (α 1 , α 2 ) =¯αp11 , ϕ 2 (α 1 , α 2 ) =¯αp21 , p 1 = p 2 = 1, δ 1 = δ 2 = 1.0
Isogeometric Mapping: Domain-01
Numerical Issues: CGL computational grid with IN = 31 and IM = 33 discrete points

possible to reduce the dimension of the discrete system of Eq. (86) taking characterized by different syngonies within each layer. More specif­
into account only the DOFs related to the inner part δd of the compu­ ically, anisotropic, orthotropic and isotropic continua have been
tational domain, since the remaining ones δb are already subjected to the considered. To the first class belongs the following triclinic material
( (k) )
external constraints. From the application of such DOFs condensation, ρ = 7750 kg/m3 , whose stiffness matrix has been taken from the
one gets: DiQuMASPAB database [105]. It is expressed according to the conven­
( ((
f ) ( f )( f )− 1 ( f )
) ) tion adopted in Eq. (19) with respect to the k-th layer material reference
¯M−dd1 L̄dd − L̄dd − L̄db − L̄db L̄bb − L̄bb L̄bd − L̄bd − ω2 I δd
system O α 1 α 2 ζ(k) :
′ (k) (k)

=0 ⎡ ⎤
(87) (k)
⎢ C11 C12
(k) (k)
C16
(k)
C14
(k)
C15
(k)
C13 ⎥
⎢ ⎥
⎢ (k) (k) ⎥
being I the identity matrix.
(k) (k) (k) (k)
⎢ C12 C22 C26 C24 C25 C23 ⎥
⎢ (k) ⎥
⎢C C26
(k) (k)
C66
(k)
C46
(k)
C56 C36 ⎥
(k)
⎢ ⎥
5. Applications and results C(k) = ⎢ 16 ⎥
⎢ C(k) (k)
C24
(k)
C46
(k)
C44
(k)
C45 C34 ⎥
(k)
⎢ 14 ⎥
⎢ (k) (k) (k) (k) (k) (k) ⎥
⎢ C15 C25 C56 C45 C55 C35 ⎥
In the present section we analyze the modal response of some ⎢ ⎥
⎣ (k) (k) ⎦
laminated panels, involving zero-curved, single-curved and doubly-
(k) (k) (k) (k)
C13 C23 C36 C34 C35 C33
curved structures, characterized by an arbitrary shape, completely ⎡ ⎤
anisotropic layers and a general thickness variation along the principal 98.84 53.92 0.03 1.05 − 0.1 50.78
⎢ 53.92 99.19 0.55 − 0.18 50.87 ⎥
parametric lines. The physical domain is described, for each case, ⎢ 0.03 ⎥
⎢ 0.03 0.02 ⎥
assuming a parameterization in principal coordinates. In addition, the ⎢
=⎢
0.03 22.55 − 0.04 0.25 ⎥ GPa (88)
⎢ 1.05 0.55 − 0.04 21.1 0.07 1.03 ⎥
use of NURBS-based blending functions, as discussed in Eqs. (54)–(55), ⎣ − 0.1 − 0.18 0.25
⎥
0.07 21.14 − 0.18 ⎦
allows one to solve the free vibration linear discrete system reported in
50.78 50.87 0.02 1.03 − 0.18 87.23
Eq. (87) referred to a squared computational domain (Fig. 1). For each
case study, we describe the mapped domain, together with the The three-dimensional stiffness matrix of the trigonal material [105]
( )
geometrical description of the structures and thickness variation, as ρ(k) = 2649 kg/m3 is accounted, instead, as follows:
shown in Figs. 3-6. The lamination schemes involve materials

18
F. Tornabene et al. Composite Structures 309 (2023) 116542

Table 2
Mode frequencies of a rectangular plate [22] with variable thickness and mapped geometry composed by three layers of general-oriented anisotropic and orthotropic
materials. The external West (W) and East (E) edges have been constrained with a super elliptic distribution of linear springs with a pre-determined three-dimensional
translational stiffness in each principal direction. The influence of the extension of the spring distribution on the modal response has been pointed out.
( )
Mapped Rectangular Plate BKDDD FBKDDD F
Double - Weibull distribution – sensitivity analysis

Mode 3D FEM Constraint edge parameter

f [Hz] CFCF
̃ξm = 1.0 ξm = 0.9
̃ ̃ξm = 0.8 ξm = 0.7
̃ ̃ξm = 0.6 ξm = 0.5
̃ ̃ξm = 0.4 ξm = 0.3
̃ ̃ξm = 0.2 ξm = 0.1
̃ ̃ξm = 0.01

DOFs 2131302 16182 – EDZ4

1 63.640 63.863 62.862 61.827 59.155 56.164 54.082 47.983 41.955 32.996 27.918 16.261
2 75.555 76.172 72.400 69.159 61.941 56.958 55.878 52.940 49.623 44.186 40.542 33.159
3 166.031 166.836 162.489 158.280 146.968 136.448 132.961 121.351 110.666 96.884 90.669 80.633
4 194.315 195.387 189.065 183.002 166.937 155.238 152.391 141.886 133.010 122.039 116.387 103.872
5 217.638 219.076 215.663 213.381 209.005 205.298 203.245 197.591 190.531 178.896 171.842 140.108
6 309.013 310.869 299.775 289.143 260.982 241.515 236.579 221.795 211.452 200.892 196.331 160.005
7 342.572 344.788 339.216 332.549 306.842 287.641 284.135 270.152 260.762 230.826 206.117 188.428
8 384.917 385.252 369.826 359.554 338.960 322.839 317.187 296.501 273.344 253.085 248.915 230.337
9 399.353 402.587 392.647 383.841 361.980 347.923 342.152 327.145 311.575 288.672 274.983 236.531
10 451.292 455.244 453.349 450.898 411.060 387.193 381.371 367.548 358.864 348.114 319.463 251.909
11 490.249 493.880 475.257 457.631 446.555 440.898 439.072 430.757 405.346 356.747 346.905 263.261
12 517.665 521.864 513.919 505.232 477.691 464.298 461.800 444.417 428.891 407.808 375.217 337.963
13 603.235 608.425 595.627 578.489 534.359 503.077 488.862 464.249 453.505 420.779 403.776 370.087
14 627.283 629.087 611.318 592.314 542.258 517.868 507.933 483.076 461.564 437.285 420.456 399.500
15 654.545 661.282 644.724 629.918 581.716 557.838 551.083 522.744 485.943 457.292 455.744 448.887
16 696.882 699.867 682.727 655.432 619.780 582.690 563.635 539.386 529.247 515.898 507.417 492.622
17 705.620 711.873 687.665 669.943 630.041 623.970 622.586 618.150 610.531 581.213 562.873 529.271
18 725.366 732.300 723.831 715.100 696.228 685.661 678.722 655.130 625.864 605.316 595.073 571.799
19 770.096 779.986 776.124 772.448 748.286 719.991 707.982 687.333 674.726 659.422 651.716 636.849
20 806.781 816.369 810.318 799.403 765.335 754.096 748.053 732.547 718.648 703.638 696.052 679.692
(k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11
Boundary Springs: ξ̄m = 0 (W), ξ̄m = 1 (E), p = 1000, k 1f = k 1f = k 1f = k 1f = 1⋅1021 N/m3 , k 2f = k 2f = k 2f = k 2f = 1⋅1021 N/m3 , k 3f = k 3f =
(k)ξ02 (k)ξ12
k 3f = k 3f = 1⋅1021 N/m3
Lamination Scheme: (0/30/45), h01 = h03 = 0.02 m, h02 = 0.03 m, 1st graphite-epoxy, 2nd triclinic, 3rd graphite-epoxy
Thickness Variation: ϕ 1 (α 1 , α 2 ) =¯αp11 , ϕ 2 (α 1 , α 2 ) =¯αp21 , p 1 = p 2 = 1, δ 1 = δ 2 = 1.0
Isogeometric Mapping: Domain-01
Numerical Issues: CGL computational grid with IN = 31 and IM = 33 discrete points

⎡ ⎤
(k)
(k)
⎢ C11 (k)
C12
(k)
C16
(k)
C14
(k)
C15
(k)
C13 ⎥ E 1 = 137.90 GPa G(k)
12 = 7.10 GPa ν(k)
12 = 0.30
⎢ ⎥
⎢ (k) (k) (k) (k) (k) (k) ⎥ E(k) = 8.96 GPa
(k)
G 13 = 7.10 GPa ν(k)
13 = 0.30
(91)
⎢ C12 C22 C26 C24 C25 C23 ⎥ 2
⎢ (k) ⎥
⎢C (k)
C26 (k)
C66 (k)
C46 (k)
C56 (k)
C36 ⎥ E(k)
3 = 8.96 GPa (k)
G 23 = 6.21 GPa ν(k)
23 = 0.49
⎢ 16 ⎥
C(k) = ⎢ (k) ⎥
⎢C (k)
C24 (k)
C46 (k)
C44 (k)
C45 (k)
C34 ⎥
⎢ 14
⎢ (k)
⎥
⎥ whereas ρ(k) = 1450 kg/m3 is the associated material density.
(k) (k) (k) (k) (k)
⎢ C15
⎢ C25 C56 C45 C55 C35 ⎥
⎥ In some cases, an isotropic central core has been included within the
⎣ (k) ⎦ ( (k)
lamination scheme. In particular, zirconia E = 168.00
(k) (k) (k) (k) (k)
C13 C23 C36 C34 C35 C33
) (
⎡ ⎤ GPa, ν(k) = 0.3, ρ(k) = 5700 kg/m3 , steel E(k) = 210.00 GPa,
86.74 6.99 0 0 − 17.91 11.91 ) ( (k)
⎢ 6.99 86.74 0 0 17.91 11.91 ⎥ ν(k) = 0.3, ρ(k) = 7800 kg/m3 and aluminum E =
⎢ ⎥ )
⎢ 0 0 39.88 − 17.91 0 0 ⎥ 70.00 GPa, ν(k) = 0.3, ρ(k) = 2707 kg/m3 materials [105] have been
=⎢ ⎥ GPa
⎢ 0
⎢ 0 − 17.91 57.94 0 0 ⎥ ⎥ adopted in different combinations.
⎣ − 17.91 17.91 0 0 57.94 0 ⎦ The external boundary conditions are assessed employing the
11.91 11.91 0 0 0 107.20 generalized formulation derived in Eqs. (40)–(41) for each side of the
(89) mapped shell object of analysis (Fig. 2). Besides, they are obtained from
The orthotropic materials considered in the present work refer to the a correct definition of the governing parameters of the linear springs
graphite-epoxy and glass–epoxy composites, whose linear elastic distributed along the structural edges, taking into account both in-plane
behavior is described by means of the well-known engineering con­ and out-of-plane external constraints thickness variation, as reported in
Eqs. (40)–(41). Moreover, a second set of linear springs have been set
stants, referred to the material reference system O α 1 α 2 ζ(k) , as follows
′ (k) (k)
along the shell top and bottom surfaces, following the Winkler gener­
[105]:
alized model introduced in Eq. (27).
E(k) G(k) ν(k) A validation analysis starts considering the vibration response of a
1 = 53.78 GPa 12 = 8.96 GPa 12 = 0.25
mapped rectangular plate, as provided by the present structural model
(k)
E 2 = 17.93 GPa G(k)
13 = 8.96 GPa ν(k)
13 = 0.25
(90)
and a three-dimensional FEM simulation. The natural frequencies are
E(k)
3 = 17.93 GPa G(k)
23 = 3.45 GPa ν(k)
23 = 0.34 summarized comparatively in Tables 1-2, and the related mode shapes
The density of the material of interest has been assumed equal to are represented in Figs. 7-9. Moreover, a parametric investigation is
ρ(k) = 1900 kg/m3 . The orthotropic behavior with respect to performed to check for the influence of the lateral constraints shape
parameter, as well as the stiffness of the selected boundary springs, as
O α(k) (k) (k)
of the graphite-epoxy layer [105] considers the following
′
1 α2 ζ
summarized in Table 3. Then, we investigate the influence on the dy­
elastic constants:
namic response of curved structures of the higher order assumption of
the field variable according to Eq. (10). In particular, the assumed field

19
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 7. Mode shapes of a mapped laminated rectangular plate with variable thickness and general boundary conditions. A linear thickness variation has been
selected. The external constraint has been obtained from a Double - Weibull distribution of translational springs distributed along the three principal directions so
that the four corners of the structure are bounded. Corresponding mode frequencies, as well as other analysis information, have been collected in Table 1.

variable through-the-thickness hypotheses are identified with the ESL

nomenclature introduced in the previous sections. The zigzag formula­ U 1 (α 1 , α 2 , ζ, t) = u(0) (1)
1 (α 1 , α 2 , t) + ζu 1 (α 1 , α 2 , t)

tion (Z) of Eq. (11) is adopted too. When the shear correction factor κ(ζ) U 2 (α 1 , α 2 , ζ, t) = u(0) (1)
2 (α 1 , α 2 , t) + ζu 2 (α 1 , α 2 , t)
(92)
is included in the model according to Eq. (24), it is declared as a su­ (0)
U 3 (α 1 , α 2 , ζ, t) = u 3 (α 1 , α 2 , t)
perscript. Moreover, in cases of reduced elastic coefficients Eij = Qij
(k) (k)
Besides, when the TSDT model is employed, a third order expansion
within each stiffness matrix for each k-th layer according to Eq. (20), the
of the in-plane components of the field variable is considered as follows
symbol RS is introduced as a subscript. Some simulations rely on clas­
[22]:
sical ESL theories. For the sake of completeness the FSDT displacement
field is defined as [22]:

20
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 8. Mode shapes of a mapped laminated rectangular plate with variable thickness and general boundary conditions. A linear thickness variation has been
selected. The external constraint has been obtained from a super elliptic distribution of translational springs distributed along the three principal directions so that
the midpoints belonging to the four edges of the structure are bounded. Corresponding mode frequencies, as well as other analysis information, have been collected
in Table 1.

21
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 9. Mode shapes of a mapped laminated rectangular plate with variable thickness and general boundary conditions. A linear thickness variation has been
selected. The external constraint has been obtained from a super elliptic distribution of translational springs distributed along the three principal directions applied
on the West (W) and East (E) sides of the structure. Corresponding mode frequencies, as well as other analysis information, have been collected in Table 1.

22
F. Tornabene et al. Composite Structures 309 (2023) 116542

U 1 (α 1 , α 2 , ζ, t) = u(0) (1) 2 (2) 3 (3)

1 (α 1 , α 2 , t) + ζu 1 (α 1 , α 2 , t) + ζ u 1 (α 1 , α 2 , t) + ζ u 1 (α 1 , α 2 , t)

U 2 (α 1 , α 2 , ζ, t) = u(0) (1) 2 (2) 3 (3)

2 (α 1 , α 2 , t) + ζu 2 (α 1 , α 2 , t) + ζ u 2 (α 1 , α 2 , t) + ζ u 2 (α 1 , α 2 , t)
(93)
U 3 (α 1 , α 2 , ζ, t) = u(0)
3 ( α 1 , α 2 , t)

For an easy identification of the general external constraints within properly setting the parameters of the general thickness expression re­
each structural side, we adopt the notation BKTTT , which means that the ported in Eq. (4). As it can be seen in Table 1, a lamination scheme made
side at issue is bounded (B) with a set of linear springs (K) oriented along of three generally oriented layers has been employed, characterized by a
α 1 , α 2 and ζ principal directions (Figs. 1-2). The symbol T stands for the central triclinic core covered by two external orthotropic graphite-epoxy
linear spring distribution along the edge, for each direction. More spe­ skins. Four different boundary conditions have been modelled, and the
cifically, T = D denotes the Double – Weibull distribution introduced in first ten natural frequencies have been provided for each case,
Eq. (36), whereas the super elliptic distribution of Eq. (37) is identified employing different higher order theories. The fully-clamped (CCCC)
with T = S. boundary condition has been obtained according to Eq. (69). Among all
Natural frequencies of doubly-curved shells subjected to general the ESL higher order theories considered in the numerical investigation,
boundary conditions have been reported in Tables 4-10, whereas the the most accurate results with respect to the 3D FEM outcomes are
mode shapes are collected in Figs. 10-22. The formulation presented in provided by the EDZ4 theory. This is related to the complex nature of the
this work provides very accurate results with respect to the FEM solu­ lamination scheme, with coupling issues between adjacent laminae, and
tions, despite the lower DOFs required by the GDQ approach. In addi­ complex in-plane and out-of-plane deformation effects within each
tion, the differential geometry description of shells allows one to easily layer. A fully-constrained condition at the corners of the structure in the
address very complex structures, as well as the higher order assumption mapped geometry has been modelled taking into account, for each side
( )
is capable of considering several warping and coupling effects that of the structure, a Double – Weibull distribution BKDDD of linear springs
cannot be depicted by classical ESL approaches. In this way, a complete with reference stiffness equal to 1⋅1021 N/m3 in each principal direction,
three-dimensional capability can be achieved even in the case of a two- setting also ξ̄ m = ̃
ξ m = 0.0025 and p = 20. As it can be seen in Table 1,
dimensional model. the choice of a higher order model with N = 4 is essential to capture
warping effects along the thickness, as well as the zigzag kinematic
assumption of Eq. (11) for the N + 1-th expansion order. Furthermore,
5.1. Modal analysis of a mapped plate
Fig. 7 shows that the symmetry of the boundary conditions does not
induce symmetric modes due to the anisotropy of the layers in the
We now assess the accuracy of the higher order theory presented in
stacking sequence. Moreover, such very complex external constraints
this work, based on the free vibration analysis performed on a mapped
induce, in higher modes, some localized deformations which are
rectangular plate. The main geometric features of the structure have
completely disregarded by lower order ESL theories. Finally, the GDQ
been reported in Fig. 3, referred to the Domain-01. In both α 1 and α 2
method has proved to be an accurate numerical capable to interpret such
principal directions, a linear thickness variation has been selected by

Table 3
Parametric analysis on first ten mode frequencies of a laminated rectangular plate [22] with variable thickness externally constrained by a set of boundary linear
translational springs applied on the East (E) and West (W) edges of the structure, whereas North (N) and South (S) sides of the shell are fully clamped. Reference
solutions have been calculated with both the ESL GDQ approach and the finite element method. As can be seen, natural frequencies decrease, for each mode number, as
the springs’ thickness decreases.
( )
Mapped Rectangular Plate CBKCCC CBKCCC
Linear Boundary Springs – sensitivity analysis

k f [N/m3 ] Mode f [Hz]

1 2 3 4 5 6 7 8 9 10 DOFs

3D FEM 0⋅10 0 189.811 309.828 418.143 472.516 561.695 672.275 714.288 732.047 885.974 893.054 2131302
(CCCC)
CCCC 0⋅100 191.448 312.641 424.668 477.000 570.809 679.497 729.446 743.686 904.394 905.330 16182
CBCB 1⋅1021 191.448 312.641 424.668 477.000 570.809 679.497 729.446 743.686 904.394 905.330 16182
CBCB 1⋅1020 191.448 312.641 424.668 477.000 570.809 679.497 729.446 743.686 904.394 905.330 16182
CBCB 1⋅1015 191.461 312.661 424.697 477.025 570.850 679.521 729.491 743.738 904.416 905.392 16182
CBCB 1⋅1013 194.645 317.039 431.657 482.025 580.121 684.848 737.350 759.753 908.817 918.267 16182
CBCB 1⋅1012 172.031 172.596 262.376 358.305 447.225 522.679 571.898 618.313 663.252 784.893 16182
CBCB 1⋅1011 125.175 239.683 246.290 300.771 352.251 414.991 467.260 573.874 594.966 658.730 16182
CBCB 1⋅1010 119.689 178.693 246.778 282.975 325.546 326.378 387.837 389.226 389.282 443.478 16182
CBCB 1⋅105 63.861 76.178 166.838 195.393 219.081 310.877 344.791 385.306 402.599 455.248 16182
CBCB 1⋅100 63.864 76.175 166.840 195.392 219.080 310.878 344.792 385.306 402.598 455.248 16182
CFCF 0⋅100 63.864 76.175 166.840 195.392 219.080 310.878 344.792 385.306 402.598 455.248 16182
3D FEM 0⋅100 63.640 75.555 166.031 194.315 217.638 309.013 342.572 384.917 399.353 451.292 2131302
(CFCF)
(k)ξ01 (k)ξ11 (k)ξ01 (k)ξ11 (k)ξ01 (k)ξ11
Boundary Springs: Constant distribution, k 1f = k 1f = k f , k 2f = k 2f = k f , k 3f = k 3f = kf
Lamination Scheme: (0/30/45), h01
= h03
= 0.02 m, h02
= 0.03 m
Laminae Material Sequence: 1 graphite-epoxy, 2nd triclinic, 3rd graphite-epoxy
st

ESL Analysis: CGL computational grid with IN = 31 and IM = 33 discrete points, EDZ4 displacement field
Thickness Variation: ϕ 1 (α 1 , α 2 ) =¯αp11 , ϕ 2 (α 1 , α 2 ) =¯αp21 , p 1 = p 2 = 1, δ 1 = δ 2 = 1.0
Isogeometric Mapping: Domain-01

23
F. Tornabene et al. Composite Structures 309 (2023) 116542

Table 4
Mode frequencies of a mapped catenoid [22] characterized by variable thickness and subjected to general boundary linear elastic constraints. The results have been
provided by the unified ESL approach employing different higher order theories. Elastic translational springs have been distributed along the structure in all the
principal directions in order to obtain different kinds of elastic supports. A sinusoidal thickness variation has been set along the physical domain.
Mapped Catenoid (a = 0.85 m)

Mode FSDTκ=1.2
RS
FSDTZ RS TSDT RS TSDTZ RS ED1 RS EDZ1 RS ED2κ=1.2 EDZ2 ED3 EDZ3 ED4 EDZ4
f [Hz]

DOFs 5046 7569 10092 12615 5046 7569 7569 10092 10092 12615 12615 15138
( )
Clamped Opposite Edges FBKSSS FBKSSS
1 79.650 79.280 78.155 78.530 78.682 79.518 78.912 77.713 78.732 79.054 78.531 79.135
2 127.752 127.008 125.098 124.904 127.014 128.800 126.395 125.033 126.059 125.862 125.475 125.355
3 157.365 156.683 154.962 155.160 156.450 158.050 156.876 154.514 156.428 156.594 155.846 156.055
4 263.850 262.853 260.755 260.386 259.129 265.925 263.988 261.032 263.115 262.501 262.314 261.642
5 297.436 296.274 291.856 290.822 294.327 297.688 294.612 290.528 293.975 292.703 292.989 291.523
6 339.774 338.280 332.788 331.563 337.601 340.799 335.981 331.474 335.152 333.833 333.733 332.130
7 454.421 452.713 448.363 447.777 445.382 458.147 453.663 448.677 452.069 451.044 450.512 449.519
8 532.191 529.955 522.704 522.600 526.499 530.117 525.672 517.861 525.230 524.931 523.928 523.584
9 538.690 534.236 526.735 525.436 533.769 534.916 531.932 522.755 529.638 528.354 527.963 526.251
10 638.399 635.502 629.633 628.565 628.869 639.028 636.517 627.367 633.980 632.482 631.816 630.244
Boundary Springs:
North (N) edge: super elliptic distribution (ξ̄m = 0, ̃
ξm = 0.23, p = 1000),
South (S) edge: super elliptic distribution (ξ̄m = 1, ̃
ξm = 0.23, p = 1000),
(k)ξ01 (k)ξ11 (k)ξ0 (k)ξ11 (k)ξ01 (k)ξ11
k 1f = 1⋅1021 N/m3 , k 2f 1 = k 2f
= k 1f = 1⋅1021 N/m3 , k 3f = k 3f = 1⋅1021 N/m3
( )
Clamped Half Corners BKSSS BKSSS BKSSS BKSSS
1 164.778 163.454 160.020 158.397 164.520 162.408 160.971 158.084 160.051 158.552 159.590 157.391
2 202.815 201.954 199.338 198.970 200.193 202.595 201.580 198.448 200.741 200.227 200.151 199.564
3 260.273 259.581 257.358 259.958 257.071 258.071 257.894 253.486 258.880 261.287 258.321 262.303
4 330.007 328.372 325.096 325.875 326.995 330.995 327.801 323.192 327.712 328.291 326.479 327.929
5 353.012 353.718 350.375 351.048 350.334 353.467 352.351 346.776 353.292 353.803 351.908 352.907
6 405.357 403.741 399.330 397.378 404.219 410.293 403.374 398.637 403.101 401.303 400.845 398.325
7 533.028 530.502 522.581 521.224 527.680 528.418 525.524 516.959 524.318 522.738 523.060 521.100
8 570.632 568.689 565.682 564.924 553.821 575.986 576.232 569.951 573.826 571.881 572.961 571.824
9 586.363 584.852 577.487 577.198 581.094 587.051 582.986 573.805 582.466 581.942 580.571 580.191
10 633.816 630.781 622.343 621.141 622.975 631.284 628.401 618.961 625.670 623.977 624.335 622.535
0 1 0 1 0 1 0 1
Boundary Springs: super elliptic distribution (ξ̄m = 0.5, ̃ξm = 0.01, p = 40), k 1f 1 = k 1f 1 = k 1f 2 = k 1f 2 = 1⋅1021 N/m3 , k 2f 1 = k 2f 1 = k 2f 2 = k 2f 2 = 1⋅1021 N/m3 ,
(k)ξ (k)ξ (k)ξ (k)ξ (k)ξ (k)ξ (k)ξ (k)ξ

(k)ξ0 (k)ξ1 (k)ξ0 (k)ξ1 3

k 3f 1 = k 3f 1 = k 3f 2 = k 3f 2 = 1⋅1021 N/m
( )
Clamped Edge Areas FB DDD FBKDDD
K

1 203.957 203.065 201.325 201.045 196.960 203.547 203.637 200.980 202.553 201.904 202.374 201.714
2 221.594 220.819 219.334 219.066 216.283 224.309 222.379 220.082 221.531 220.955 220.781 220.196
3 258.021 256.607 252.585 252.059 255.560 256.056 254.583 250.428 253.691 253.046 253.134 252.402
4 287.970 286.107 281.520 280.957 285.309 286.594 284.164 279.613 283.115 282.482 282.306 281.510
5 376.633 375.318 372.418 371.854 365.134 376.304 376.197 371.577 374.630 373.478 374.080 372.897
6 463.161 461.607 458.101 457.416 448.495 465.191 464.340 458.486 462.046 460.604 461.065 459.721
7 496.165 493.589 487.655 486.840 487.056 493.750 491.749 484.890 489.880 488.603 489.046 487.702
8 565.496 563.577 556.512 555.617 559.286 564.546 560.276 551.932 558.877 557.785 557.543 556.251
9 568.825 565.720 556.730 555.740 562.673 566.093 561.631 553.701 559.801 558.574 558.056 556.756
10 572.058 569.720 564.830 563.867 566.090 579.792 572.050 565.801 570.817 569.542 567.801 566.507
0 1 0 1 0 1 0 1
Boundary Springs: Double - Weibull distribution (ξ̄m = 0.05, ̃ξm = 0.05, p = 100), k 1f 1 = k 1f 1 = k 1f 2 = k 1f 2 = 1⋅1021 N/m3 , k 2f 1 = k 2f 1 = k 2f 2 = k 2f 2 = 1⋅
(k)ξ (k)ξ (k)ξ (k)ξ (k)ξ (k)ξ (k)ξ (k)ξ

(k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12

1021 N/m3 , k 3f = k 3f = k 3f = k 3f = 1⋅1021 N/m3
Lamination Scheme: (0/iso/30), h01 = h03 = 0.02 m, h02 = 0.04 m
Laminae Material Sequence: 1st trigonal material, 2nd zirconia, 3rd trigonal material
Numerical Issues: CGL computational grid with IN = IM = 31 discrete points
Thickness Variation: ϕ i (α 1 , α 2 ) = (sin(π(n i ᾱ i + α im ) ) )p i , n i = n 1 = n 2 = 1, p i = p 1 = p 2 = 2, α im = α 1m = α 2m = 0, δ 1 = δ 2 = 0.7
Isogeometric Mapping: Domain-01

phenomena with a reduced number of discrete points. dimensional model provides bending modes coupled with out-of-plane
It must be remarked that the choice of different boundary conditions stretching effects. Such a behavior can be depicted only by the EDZ4
does not affect the accuracy of the solution provided by the proposed kinematic assumption. As also shown in Fig. 9, mode shapes exhibit a
( )
model, as it can be seen from the second investigation BKSSS BKSSS BKSSS BKSSS complex warping phenomenon along the thickness of the plate in higher
in which the structure has been fixed at the middle of each edge. For modes. Lower modes show a non-standard bending direction, because of
each principal direction the external constraints consider a super elliptic the complete anisotropy of the lamination scheme.
distribution (S) of linear springs, as described in Eq. (37), with a refer­ In the last investigation carried out on the mapped rectangular plate,
ence stiffness equal to 1⋅1021 N/m3 . For this case, the governing pa­ the structure is clamped (C) in two opposite straight edges only at half of
the side. All the remaining parts are left free (F). Such a general
rameters have been set as ξ̄ m = 0.5, ̃ξ m = 0.01 and p = 40. It has been
boundary condition has been set employing a super elliptic distribution
shown that, even though classical approaches reported in Eqs. (92)–(93)
of linear springs. Also in this case, 3D FEM-based outcomes are well
are modelled together with the zigzag thickness function of Eq. (11),
predicted only for higher order assumptions of the displacement field,
lower modes are not suitably described, since the refined three-
according to Eq. (10), due to the complexity of the deformation for each

24
F. Tornabene et al. Composite Structures 309 (2023) 116542

Table 5
Modal analysis of a mapped elliptic cone [22] with variable thickness laminated with anisotropic, orthotropic and isotropic layers enforced with boundary and surface
elastic supports, employing different higher order theories. Boundary linear springs have been set so that the structure is constrained in the four outer corners. Gaussian
distribution has been employed for the description of the concentrated springs in the three principal directions located on the bottom surface. The influence of the
Winkler elastic surface support on the modal response of the shell has been shown.
Mapped Elliptic Cone (a = 0.1 m, b = 0.15 m, α = 30 deg)

Mode FSDTκ=1.2
RS
FSDTZ RS TSDT RS TSDTZ RS EDZ1 RS ED2κ=1.2 EDZ2 ED3 EDZ3 ED4 EDZ4
f [Hz]

DOFs 5394 8091 10788 13485 8091 8091 10788 10788 13485 13485 16182
( )
Winkler Foundation and Boundary Springs BKSSS FBKSSS F
1 1550.048 1545.468 1541.775 1534.978 1549.612 1543.381 1534.738 1546.916 1537.616 1545.171 1533.852
2 1627.817 1615.560 1610.704 1608.351 1632.221 1613.510 1603.508 1613.805 1610.424 1611.858 1607.209
3 1715.475 1699.338 1693.167 1690.052 1732.036 1699.060 1686.131 1700.041 1695.187 1697.357 1691.272
4 1846.731 1829.989 1822.411 1819.417 1847.999 1830.364 1818.318 1834.821 1830.254 1832.642 1827.439
5 1865.565 1856.857 1852.206 1848.151 1872.092 1847.142 1833.096 1849.995 1844.443 1846.460 1840.175
6 2080.372 2070.928 2066.398 2063.146 2099.703 2082.738 2069.970 2082.549 2077.185 2080.411 2073.921
7 2530.336 2510.172 2503.223 2497.348 2560.202 2508.428 2489.979 2508.469 2501.640 2504.137 2495.010
8 2690.442 2678.320 2673.642 2666.058 2693.575 2685.293 2671.239 2685.107 2674.717 2683.452 2670.426
9 3212.145 3189.888 3181.010 3175.909 3221.689 3183.811 3162.042 3188.761 3180.752 3184.696 3175.511
10 3536.852 3515.432 3506.816 3499.085 3544.882 3520.161 3500.669 3521.786 3512.885 3519.033 3506.613
Boundary Springs: Double - Weibull distribution (ξ̄m = ̃ξm = 0.025, p = 20),
(k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12
k 1f = k 1f = k 1f = k 1f = 1⋅1021 N/m3 , k 2f = k 2f = k 2f = k 2f = 1⋅1021 N/m3 , k 3f = k 3f = k 3f = k 3f = 1⋅1021 N/m3
21 3
Winkler Foundation: Gaussian distribution (ξ1n = 0.5, ξ2m = 0.7, Δ 1 = Δ 2 = 0.01), k 1f k 2f k 3f = 1⋅10 N/m
(− ) (− ) (− )
= =
( )
Boundary Springs BKSSS FBKSSS F
1 636.107 633.198 630.955 629.962 641.705 636.200 632.398 635.855 634.772 635.178 633.553
2 886.733 881.083 879.205 878.037 901.820 883.957 876.968 883.677 881.911 882.051 879.824
3 1553.910 1549.092 1545.770 1538.800 1552.113 1547.142 1538.698 1550.429 1540.490 1548.832 1537.011
4 1607.850 1596.657 1591.971 1589.430 1611.463 1596.161 1586.441 1596.092 1592.252 1594.452 1589.054
5 1759.077 1748.778 1744.921 1741.473 1787.387 1753.237 1739.955 1753.599 1748.714 1750.350 1744.474
6 1981.477 1973.769 1969.986 1967.395 1992.751 1985.077 1974.443 1984.744 1980.897 1983.305 1977.997
7 2524.178 2504.285 2497.607 2491.814 2553.681 2502.108 2483.982 2502.538 2495.800 2498.228 2489.272
8 2653.713 2643.590 2640.444 2631.672 2661.960 2654.944 2642.003 2654.120 2641.280 2652.496 2638.032
9 2686.892 2672.832 2664.732 2660.252 2709.553 2676.715 2659.103 2677.485 2671.440 2674.364 2665.847
10 3071.885 3060.884 3057.445 3054.049 3097.019 3084.963 3069.800 3085.868 3079.313 3083.479 3076.355
(k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12
Boundary Springs: Double - Weibull distribution (ξ̄m = ̃ξm = 0.025, p = 20), k 1f = k 1f = k 1f = k 1f = 1⋅1021 N/m3 , k 2f = k 2f = k 2f = k 2f = 1⋅1021 N/m3 ,
(k)ξ0 (k)ξ1 (k)ξ0 (k)ξ1 21 3
k 3f 1 = k 3f 1 = k 3f 2 = k 3f 2 = 1⋅10 N/m
Lamination Scheme: (30/iso/70), h01 = h03 = 0.003 m, h02 = 0.005 m
Laminae Material Sequence: 1st trigonal material, 2nd steel, 3rd glass–epoxy
Numerical Issues: CGL computational grid with IN = 33 and IM = 31 discrete points
αp24 , p 4 = 20, δ 4 = 0.7
Thickness Variation: ϕ 4 (α 1 , α 2 ) = ̃
Isogeometric Mapping: Domain-02

mode, as shown in Fig. 9. Since a general anisotropic behavior of the distribution of springs oriented along each principal direction. The
homogenized model has been adopted, lower modes are not symmetric. stiffness of the translational springs has been varied. For both CCCC and
In addition, in higher modes, very complex deformations can be seen in CFCF configurations, a reference solution has been provided by means of
the region involved by a varying structural stiffness. a refined 3D FEM model: a perfect alignment with the results provided
In Table 2, we summarize the results from a parametric analysis with by the ESL formulation is seen. In the latter, external boundary condi­
respect to a three-layer mapped rectangular plate. Both West (W) and tions have been assigned by following the kinematic approach provided
East (E) edges of the structure have been clamped with a Double – in Eq. (69). Free (F) edges have been modelled starting from Eq. (70). It
Weibull distribution of linear springs oriented along the α 1 , α 2 and ζ has been shown that the decrease of the natural frequencies does not
principal directions, characterized by a reference stiffness value equal to follow the decay of the springs stiffness, since the influence of such
1⋅1021 N/m3 , according to Eq. (36). More specifically, the influence of design parameters is evident only between 1⋅1012 N/m3 and
the constraint edge parameter ̃ ξ m has been checked on the first twenty 1⋅105 N/m3 .
mode frequencies of the plate. A reference CFCF solution has been
provided by a refined 3D FEM simulation, and it has been compared to
5.2. Free vibration analysis of doubly-curved shells
predictions from the ESL theory, setting ̃
ξ m = 1.0, with an excellent
accuracy. After this preliminary validation, different values of ̃ξ m have We now present a series of modal analyses carried out on various
been considered, defining a reduction in the clamped area of West (W) doubly-curved panels of different curvatures and mapped geometries.
and East (E) shell edges. As expected, a reduction of all mode frequencies Different thickness variations along the parametric lines have been
is pointed out. Each analysis of the considered parametric investigation selected, together with general boundary conditions. Moreover, the
has required only 16182 DOFs, embedding the EDZ4 displacement field adopted lamination schemes are always unsymmetric and they are ob­
assumption according to Eq. (10) and a CGL computational grid char­ tained by layers with different material symmetries.
acterized by IN × IM = 31 × 33 discrete points. On the other hand, the The first set of simulations have been led on a catenoid, whose
reference 3D FEM model required a total number of DOFs equal to reference surface r(α 1 , α 2 ), according to Eq. (1), has been described in
2131302. Fig. 3. The Domain-01 has been selected for the isogeometric mapping,
In Table 3, we report the first ten mode frequencies for the same along with a two-dimensional sinusoidal thickness variation character­
mapped plate, constrained in West (W) and East (E) edges by a constant ized by a single wave. Two external layers of trigonal materials are

25
F. Tornabene et al. Composite Structures 309 (2023) 116542

Table 6
Mode frequencies of a mapped revolution parabolic shell [22] characterized by variable thickness and subjected to general boundary linear elastic constraints. The
results have been provided by the unified ESL approach employing different higher order theories. Elastic translational springs have been distributed along the
structure in all the principal directions in order to obtain different kinds of elastic supports. A sinusoidal and power thickness variation has been set along the physical
domain.
Mapped Revolution Parabolic Shell (R b = 0.01 m)

Mode FSDTκ=1.2
RS
FSDTZ RS TSDT RS TSDTZ RS EDZ1 RS ED2κ=1.2 EDZ2 ED3 EDZ3 ED4 EDZ4
f [Hz]

DOFs 5046 7569 10092 12615 7569 7569 10092 10092 12615 12615 15138
( )
Clamped Portion Edge FFBKDDD F
1 254.228 253.442 251.400 250.166 255.194 253.219 251.153 254.078 252.498 253.465 253.126
2 580.178 578.198 572.224 569.314 581.174 572.583 570.042 574.145 571.356 573.640 569.342
3 847.640 848.718 836.266 834.021 854.515 833.135 830.294 838.488 838.343 840.777 839.220
4 1083.495 1079.466 1075.448 1070.430 1082.068 1074.685 1067.297 1081.045 1074.336 1082.645 1076.299
5 1452.672 1455.553 1441.937 1438.497 1458.070 1436.217 1429.822 1446.389 1442.937 1446.453 1442.648
6 1725.737 1721.237 1708.694 1706.115 1727.932 1705.015 1697.224 1712.165 1706.621 1709.146 1702.255
7 2067.198 2057.864 2041.066 2033.351 2061.869 2041.630 2029.336 2053.000 2040.636 2045.520 2035.037
8 2368.604 2368.044 2351.474 2346.518 2370.475 2349.703 2335.060 2358.254 2348.471 2354.231 2343.512
9 2540.819 2537.859 2522.309 2517.584 2571.080 2529.919 2517.674 2535.891 2528.701 2531.729 2527.158
10 2764.731 2757.516 2738.926 2730.741 2768.193 2738.190 2722.105 2752.016 2739.674 2744.294 2733.419
Boundary Springs: Double - Weibull distribution (ξ̄m = 0.05, ̃ξm = 0.05, p = 100),
(k)ξ01 (k)ξ11 (k)ξ0 (k)ξ1 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12
k 1f = k 1f 2 = k 1f 2 = 1⋅1021 N/m3 ,
= k 1f k 2f = k 2f = k 2f = k 2f = 1⋅1021 N/m3 , k 3f = k 3f = k 3f = k 3f = 1⋅1021 N/m3
( )
Clamped Corners BKDDD BKDDD BKDDD BKDDD
1 608.367 609.302 602.091 612.768 598.207
601.065 599.220 604.716 604.023 607.042 597.405
2 929.892 927.987 923.403 933.087 918.143
925.507 919.196 930.803 925.265 932.603 930.958
3 1300.096 1301.242 1287.168 1300.464 1283.497
1285.496 1279.117 1293.289 1290.338 1290.816 1275.145
4 1470.400 1472.420 1458.586 1476.755 1455.017
1457.640 1452.390 1467.566 1468.042 1470.046 1469.916
5 1567.612 1563.456 1556.351 1572.613 1551.423
1568.118 1555.110 1573.407 1562.748 1574.618 1564.626
6 1875.472 1871.351 1852.889 1888.042 1847.929
1858.623 1849.050 1864.120 1856.628 1857.742 1827.832
7 2077.297 2071.921 2061.927 2086.257 2049.940
2063.500 2049.097 2077.754 2066.284 2081.600 2059.658
8 2268.594 2264.757 2244.110 2276.776 2238.710
2252.121 2239.775 2258.516 2252.769 2252.071 2242.865
9 2446.282 2451.378 2427.879 2463.422 2423.470
2424.020 2415.430 2440.258 2437.783 2435.667 2427.462
10 2753.342 2748.848 2722.923 2768.261 2717.979
2731.747 2717.494 2744.200 2732.683 2743.236 2735.020
Boundary Springs: Double - Weibull distribution (ξ̄m = 0.0025, ̃ξm = 0.0025, p = 20),
(k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ0 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12
k 1f = k 1f = 1⋅1021 N/m3 , k 2f 1 = k 2f
= k 1f = k 1f = k 2f = k 2f = 1⋅1021 N/m3 , k 3f = k 3f = k 3f = k 3f = 1⋅1021 N/m3
( )
Winkler Foundation and Boundary Springs BKDDD FFF
1 178.382 177.464 183.731 188.083 171.496 174.672 177.501 182.866 188.902 175.605 156.040
2 473.301 472.646 470.966 472.156 475.700 467.528 466.652 474.333 476.959 464.768 465.420
3 888.584 887.151 859.486 852.495 893.447 853.970 837.898 873.406 866.914 844.397 834.123
4 944.212 941.109 930.064 927.308 939.047 923.002 918.436 931.090 929.096 919.872 918.572
5 1163.179 1161.490 1141.928 1136.981 1176.680 1147.257 1130.792 1145.224 1134.002 1126.886 1125.027
6 1287.009 1283.271 1259.685 1259.160 1287.335 1240.897 1201.309 1234.109 1208.801 1209.551 1214.152
7 1569.070 1571.806 1516.877 1505.195 1570.060 1547.380 1525.760 1539.533 1536.325 1460.209 1415.666
8 1772.136 1770.177 1669.708 1651.876 1777.856 1663.028 1603.487 1684.156 1665.109 1633.841 1610.008
9 1981.432 1979.153 1789.322 1786.559 1997.844 1812.475 1778.347 1796.458 1798.085 1783.534 1749.421
10 2274.132 2271.357 2044.332 2037.543 2279.667 2108.517 2020.118 2045.607 2032.883 1949.098 1918.887
Winkler Foundation: Gaussian distribution (ξ1n = 0.5, ξ2m = 0.75, Δ 1 = Δ 2 = 0.01), k 1f = k 2f = k 3f = 1⋅1021 N/m3
(− ) (− ) (− )

(k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11

Boundary Springs: West (W) Edge: Double - Weibull distribution (ξ̄m = 0.0025, ̃
ξm = 0.0025, p = 20), k 1f = k 1f = k 1f = k 1f = 1⋅1021 N/m3 , k 2f = k 2f =
(k)ξ02 (k)ξ12 21 3 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 21 3
k 2f = k 2f = 1⋅10 N/m , k 3f = k 3f = k 3f = k 3f = 1⋅10 N/m
Lamination Scheme: (30/0/45), h01 = h02 = h03 = 0.003 m
Laminae Material Sequence: 1st graphite-epoxy, 2nd triclinic material, 3rd trigonal material
Numerical Issues: CGL computational grid with IN = IM = 31 discrete points
Thickness Variation: ϕ 1 (α 1 , α 2 ) = (sin(π(n 1 ᾱ 1 + α 1m ) ) )p 1 , ϕ 2 (α 1 , α 2 ) =¯αp22 , n 1 = 1, p 1 = 2, p 2 = 20, α 1m = 0, δ 1 = 0.7, δ 2 = 1.0
Isogeometric Mapping: Domain-02

considered in the central core of isotropic zirconia. Three different the analysis, since the higher order expansion of the out-of-plane field
boundary conditions have been considered, namely clamped opposite variable influences the final outcomes of the simulation. In Figs. 10-11,
( ) ( )
edges FBKSSS FBKSSS , clamped half corners BKSSS BKSSS BKSSS BKSSS and clam­ we report the mode shapes for the catenoidal panel under consideration.
( K K
) More in detail, in Fig. 10 we represent the eigenvectors of the catenoidal
ped edge areas FBDDD FBDDD . The first ten mode frequencies have been
reported in Table 4. A convergence of the natural frequencies can be panel externally bounded only at a portion of the mapped edges: in this
observed for an increasing order of the kinematic expansion. Moreover, case, the asymmetric lamination generalized stiffness matrix for the
the adoption of the zigzag hypothesis according to Eq. (11) introduces entire lamination scheme induces a series of coupled bending and
some coupling issues between laminae, that yield different values of warping deformations for all mode shapes. Very complicated de­
mode frequencies. The influence of anisotropic materials is evident in formations can be seen in the case of localized pointed clamping, as
the case of clamped half corners. As it can be seen from the mode fre­ occurs in Fig. 11, in line with predictions from refined 3D FEM models.
quencies, there are no coupled modes despite the symmetric geometry of In Table 5, the proposed model has been applied to a laminated
the panel in terms of physical domain, thickness distribution, and elliptic cone, whose geometry has been mapped by means of the so-
external constraints. In the last case, the stretching effect is involved in called Domain-02. All the useful information regarding the assumed

26
F. Tornabene et al. Composite Structures 309 (2023) 116542

Table 7
Mode frequencies of a mapped hyperbolic paraboloid [22] with variable thickness subjected to general surface and boundary linear elastic constraints. The results have
been provided by the unified ESL approach employing different higher order theories. Elastic translational springs have been distributed along the structure in all the
principal directions in order to obtain different kinds of elastic supports. A sinusoidal thickness variation has been set along the physical domain.
Mapped Hyperbolic Paraboloid (k 1 = 4, k 2 = 4.5)

Mode FSDTκ=1.2
RS
FSDTZ RS TSDT RS TSDTZ RS ED1 RS EDZ1 RS ED2κ=1.2 EDZ2 ED3 EDZ3 ED4 EDZ4
f [Hz]

DOFs 5046 7569 10092 12615 5046 7569 7569 10092 10092 12615 12615 15138
( )
Winkler Foundation and Boundary Springs BKSSS BKSSS BKSSS BKSSS
1 98.424 98.084 95.460 75.797 95.304 98.040 96.706 96.624 97.672 96.737 88.879 94.774
2 99.190 98.810 99.328 87.872 95.944 98.769 97.519 97.377 123.343 99.145 103.919 96.254
3 163.913 163.601 147.329 97.896 162.390 162.343 161.251 160.847 123.343 132.487 109.651 159.424
4 169.209 169.532 166.041 150.970 167.312 168.197 167.079 166.684 155.630 149.178 109.651 169.169
5 171.001 170.751 179.514 165.638 169.436 173.057 170.376 169.958 169.720 166.962 161.667 169.169
6 191.128 191.268 185.523 180.985 182.402 184.954 183.318 182.640 180.203 174.576 170.077 183.868
7 194.041 191.946 185.523 185.205 191.252 191.650 188.887 188.389 186.654 185.512 180.958 183.868
8 205.932 206.703 209.112 186.772 204.266 204.689 202.916 202.499 199.172 190.758 187.373 212.019
9 215.314 215.157 209.112 209.919 210.140 214.636 212.645 212.142 210.623 209.243 201.355 212.019
10 228.335 226.797 259.658 209.919 225.353 225.610 222.570 222.171 254.525 211.336 215.253 230.849
( K K )
Boundary Springs B SSS B SSS BKSSS BKSSS
1 76.576 76.442 74.662 44.699 74.505 77.776 76.359 76.181 87.275 80.469 87.028 71.960
2 95.807 95.486 94.104 57.817 92.838 96.008 95.000 94.865 112.039 98.463 87.137 91.285
3 128.009 127.616 126.553 89.712 123.600 128.418 127.053 126.978 112.039 121.625 106.558 126.054
4 150.066 149.778 140.519 126.934 146.704 150.023 149.436 149.068 129.765 132.788 106.558 149.761
5 165.417 165.672 156.843 143.635 163.470 165.899 164.337 163.985 155.831 144.151 121.517 165.858
6 169.929 169.971 169.946 157.812 168.807 168.477 167.372 166.964 167.808 157.245 156.159 171.702
7 175.655 175.740 180.235 169.448 174.360 176.331 174.848 174.497 170.861 170.657 174.317 176.504
8 204.377 204.972 199.389 184.342 201.178 203.860 202.177 201.819 194.815 189.627 174.317 205.420
9 208.168 207.637 208.398 201.188 203.513 207.790 206.203 205.790 207.867 197.333 194.845 205.420
10 211.143 210.891 229.837 205.917 207.680 211.115 209.285 208.770 231.098 208.722 215.060 225.285
Winkler Foundation: Gaussian distribution (ξ1n = ξ2m = 0.5, Δ 1 = Δ 2 = 0.01), k 1f = k 2f = k 3f = 1⋅1021 N/m3
(− ) (− ) (− )

0 1 0 1 0 1 0 1
Boundary Springs: super elliptic distribution (ξ̄m = 0.5, ̃ξm = 0.01, p = 40), k 1f 1 = k 1f 1 = k 1f 2 = k 1f 2 = 1⋅1021 N/m3 , k 2f 1 = k 2f 1 = k 2f 2 = k 2f 2 = 1⋅1021 N/m3 ,
(k)ξ (k)ξ (k)ξ (k)ξ (k)ξ (k)ξ (k)ξ (k)ξ

(k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12

k 3f = k 3f
= k 3f = k 3f = 1⋅1021 N/m3
( )
Clamped Half Edges BKSSS FFBKSSS
1 13.606 13.576 12.202
3.780 13.294 13.727 13.522 13.506 10.849 16.135 12.063 13.994
2 28.638 28.573 26.393
20.413 28.464 28.393 28.163 28.133 29.200 47.534 53.921 30.229
3 60.371 60.432 59.933
60.722 59.209 61.431 60.409 60.320 62.831 47.534 53.921 59.586
4 61.909 61.757 64.396
68.937 61.198 61.666 61.341 61.251 64.433 59.897 60.496 59.586
5 120.263 119.967 120.143
120.010 118.299 119.789 119.298 119.120 120.046 109.568 108.529 118.886
6 134.525 134.713 132.774
132.528 132.233 135.284 133.917 133.753 130.219 132.829 133.489 132.909
7 179.589 179.653 180.619
176.062 176.248 179.028 177.801 177.538 180.578 157.856 146.078 174.544
8 186.199 185.795 180.619
184.828 182.076 189.502 185.617 185.427 180.578 192.005 189.181 182.832
9 209.172 208.408 204.356
198.269 203.879 210.854 207.738 207.354 203.811 192.005 199.303 204.892
10 217.943 217.961 219.002
219.606 214.084 218.046 216.228 215.891 217.569 199.005 199.303 212.268
Boundary Springs: Super elliptic distribution (ξ̄m = 0, ̃ξm = 0.47, p = 1000),
(k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12
k 1f = k 1f = k 1f = k 1f = 1⋅1021 N/m3 , k 2f = k 2f = k 2f = k 2f = 1⋅1021 N/m3 , k 3f = k 3f = k 3f = k 3f = 1⋅1021 N/m3
Lamination Scheme: ( − 45/30/70/20), h01 = h02 = h03 = h04 = 0.03 m
Laminae Material Sequence: 1st triclinic material, 2nd graphite-epoxy, 3rd glass–epoxy, 4th trigonal material
Numerical Issues: CGL computational grid with IN = IM = 31 discrete points
Thickness Variation: ϕ i (α 1 , α 2 ) = (sin(π(n i ᾱ i + α im ) ) )p i , n i = n 1 = n 2 = 1, p i = p 1 = p 2 = 2, α im = α 1m = α 2m = 0, δ 1 = δ 2 = 1.0
Isogeometric Mapping: Domain-03

geometry have been reported in Fig. 4. In this case, a power thickness not included in the model, the fourth order of the kinematic expansion
variation has been selected along the meridian direction of the shell, so allows for the convergence of results. Some mode shapes of the present
that a mass concentration is held near the West (W) side of the structure. investigations are collected in Fig. 12. Namely, the influence of the
The laminate consists of three layers of trigonal material (89), isotropic surface support is evident from the bending component of the structural
steel and orthotropic glass–epoxy (90). Two different cases for boundary deformation in all the reported modes.
conditions have been implemented. In the first one, identified by Moreover, the isogeometric mapping by means of the Domain-02 has
( K )
B SSS FBKSSS F , the structure has been fixed in the four corners by setting been applied to a revolution parabolic shell. In this case, the shell
a super elliptic linear spring distribution along West (W) and East (E) thickness has been varied following a combination of power and one-
edges. Moreover, a Gaussian distribution of Winkler elastic support, wave sinusoidal analytical expressions. The lamination scheme em­
defined according to Eq. (30), has been set, accounting for a point beds an orthotropic graphite-epoxy (91), a triclinic material (88) and a
support on the bottom surface. The second configuration of boundary trigonal continuum (89). Three different external constraints have been
conditions is exactly the same as the previous one, but does not involve considered, and the first ten mode frequencies have been computed by
any surface support. With particular reference to the latter constraint means of different displacement field assumptions in Table 6. In the first
( )
settings, lower mode frequencies are slightly influenced by the analysis, FFBKDDD F boundary conditions are considered. The structure
assumption of the zigzag thickness function defined in Eq. (11), even has been fixed on two portions of the East (E) edge, thanks to the use of
though a higher order power expansion is held within the field variable the Double – Weibull distribution of linear springs, introduced in Eq.
features of Eq. (10). Moreover, when the Winkler elastic foundation is (36). The corresponding mode shapes are reported in Fig. 13. The out-of-

27
F. Tornabene et al. Composite Structures 309 (2023) 116542

Table 8
Mode frequencies of a mapped helicoid [22] characterized by variable thickness and subjected to general boundary linear elastic constraints. The results have been
provided by the unified ESL approach employing different higher order theories. Elastic translational springs have been distributed along the structure in all the
principal directions in order to obtain different kinds of elastic supports. A sinusoidal thickness variation has been set along the physical domain.
Mapped Helicoid (a = 1 m)

Mode FSDTκ=1.2
RS
FSDTZ RS TSDT RS TSDTZ RS ED1 RS EDZ1 RS ED2κ=1.2 EDZ2 ED3 EDZ3 ED4 EDZ4
f [Hz]

DOFs 5046 7569 10092 12615 5046 7569 7569 10092 10092 12615 12615 15138
( )
In-Plane General Constraint – Spring Parabolic Thickness distribution BKDD FBKDD C
1 122.010 122.281 122.064 122.056 113.414 125.041 122.610 122.444 122.891 122.800 123.025 123.235
2 144.034 144.421 144.191 144.061 133.676 150.790 144.538 144.337 144.774 144.606 144.779 144.625
3 155.531 155.870 155.539 155.359 145.639 163.489 155.962 155.753 155.971 155.660 156.621 156.525
4 182.544 183.399 182.722 181.901 175.474 195.157 183.004 182.792 183.410 182.664 182.360 181.226
5 223.767 224.527 223.942 223.649 209.370 232.842 224.623 224.301 225.060 224.689 224.854 224.520
6 246.950 247.376 246.869 246.533 225.446 256.115 248.081 247.650 248.219 247.855 247.297 246.786
7 258.643 259.449 258.932 258.834 244.601 273.562 259.795 259.400 260.326 260.174 259.853 259.867
8 309.549 310.914 309.924 309.426 298.628 322.230 310.420 310.013 311.069 310.487 310.663 310.151
9 344.798 346.161 345.140 344.588 327.748 365.952 345.945 345.478 346.793 346.243 346.566 346.481
10 360.214 361.418 360.662 360.653 343.471 383.167 361.497 361.024 362.215 362.008 362.538 362.481
0 1 0 1
Boundary Springs: Double - Weibull distribution (ξ̄m = 1, ̃ξm = 0.27, p = 1000), k 1f 1 = k 1f 1 = 1⋅1021 N/m3 , k 2f 1 = k 2f 1 = 1⋅1021 N/m3
(k)ξ (k)ξ (k)ξ (k)ξ

( )
General Constraint – Spring Parabolic Thickness distribution BKDDD FBKDDD C
1 124.367 124.646 124.501 124.775 116.903 128.263 125.355 125.176 125.691 125.841 125.867 126.450
2 150.562 150.935 150.719 150.565 140.936 157.216 151.306 151.090 151.540 151.358 151.507 151.262
3 176.871 177.293 176.881 176.278 166.492 185.302 177.391 177.157 177.380 176.740 177.675 176.750
4 207.743 208.985 208.278 208.347 203.891 227.361 208.635 208.402 209.636 209.687 208.593 208.682
5 256.006 256.771 256.310 256.547 237.156 270.764 257.323 256.908 257.853 257.924 257.196 257.212
6 287.992 289.339 288.545 288.202 272.950 300.008 289.288 288.803 290.096 289.691 290.024 289.494
7 313.138 314.418 313.618 313.348 298.497 324.110 314.250 313.833 315.006 314.707 314.673 314.346
8 357.031 358.893 357.579 356.948 338.933 382.403 359.092 358.556 360.238 359.471 360.031 359.168
9 362.698 363.760 362.914 362.576 342.997 392.605 364.206 363.716 364.783 364.316 364.836 364.302
10 410.670 412.118 411.230 411.021 386.741 430.702 412.125 411.519 413.013 412.636 412.756 412.164
0 1 0 1
Boundary Springs: Double - Weibull distribution (ξ̄m = 1, ̃ξm = 0.27, p = 1000), k 1f 1 = k 1f 1 = 1⋅1021 N/m3 , k 2f 1 = k 2f 1 = 1⋅1021 N/m3
(k)ξ (k)ξ (k)ξ (k)ξ

Cantilever Boundary Conditions (FFFC)

1 12.209 12.212 12.148 12.017 12.157 13.872 12.325 12.316 12.237 12.051 12.040 11.715
2 49.266 49.461 49.603 50.425 48.249 53.907 49.430 49.390 49.824 50.657 51.790 53.518
3 73.150 73.261 73.153 73.042 69.417 77.339 73.362 73.281 73.386 73.303 73.189 73.120
4 92.932 93.121 92.960 92.754 87.284 100.444 93.338 93.224 93.377 93.141 93.305 93.059
5 156.966 157.380 156.935 156.216 150.420 167.572 157.411 157.244 157.570 156.938 156.441 155.562
6 187.883 188.240 187.891 187.637 175.960 198.109 188.571 188.329 188.742 188.536 188.280 187.858
7 208.626 208.881 208.619 208.358 193.368 216.635 209.377 209.099 209.412 209.145 209.044 208.722
8 229.133 229.458 229.236 229.026 212.921 241.190 230.023 229.708 230.151 229.915 229.922 229.691
9 245.708 246.400 245.751 244.745 235.774 254.707 246.478 246.181 246.565 245.428 245.984 244.710
10 293.509 294.076 293.598 293.191 269.184 303.286 294.629 294.209 294.873 294.396 294.554 294.172
Lamination Scheme: (45/iso/70), h01 = h02 = h03 = 0.02 m
Laminae Material Sequence: 1st triclinic material, 2nd aluminium, 3rd triclinic material
Numerical Issues: CGL computational grid with IN = IM = 31 discrete points
Thickness Variation: ϕ i (α 1 , α 2 ) = (sin(π(n i ᾱ i + α im ) ) )p i , n i = n 1 = n 2 = 1, p i = p 1 = p 2 = 2, α im = α 1m = α 2m = 0, δ 1 = δ 2 = 1.0
Isogeometric Mapping: Domain-03

plane power ESL formulation provides a rapid convergence of both low found in Table 8. Referring to the former, the super elliptic distribution
and high natural frequencies. In the second case study, the four corners of linear springs have been applied to the structure, together with a
have been fixed employing the above-mentioned Double – Weibull concentrated Winkler support in the middle of the structure. The
distribution. Also in this case, a rapid convergence of results is noticed as importance of such elastic foundation on lower modes can be seen from
the kinematic order increases. The influence of the zigzag function the natural frequencies values reported in Table 7. The first nine ei­
introduced in Eq. (11), together with a higher order displacement field genvectors of this last example at issue, calculated by means of the EDZ4
assumption, is more evident in the last configuration identified as theory, are reported in Figs. 15-16. In the case of clamped half edges
( K ) ( K )
B DDD FFF , where the shell is fixed at two extreme points of the West B SSS FFBKSSS , the zigzag thickness function according to Eq. (11) and
(W) edge, together with a point constraint acting at the bottom surface, the use of the higher order theory is crucial, since it allows one to
obtained from a Gaussian distribution of Winkler springs in each prin­ consider an additional vibration mode of the structure that cannot be
cipal direction. Actually, lower frequencies decrease as the displacement otherwise detected by lower order theories, as well as the coupling of
field unified formulation becomes more complicated. For this last case, some mode shapes. In Table 8, a free vibration analysis has been per­
the first nine mode shapes have been reported in Fig. 14. formed for the previously described helicoidal panel. Apart from the
A mapping by means of the Domain-03 has been introduced in Fig. 5, cantilevered boundary conditions configuration (FFFC), two arrange­
characterized by four curved edges. It has been applied to a hyperbolic ments of general external constraints have been considered, taking into
paraboloid and helicoidal panel. These two structures contain a one- account a parabolic-through-the-thickness distribution of linear elastic
( )
wave sinusoidal thickness variation, according to Eqs. (4), (6). The springs. In the first case, denoted by BKDD FBKDD C , two portions of the
first ten mode frequencies of the above panels have been computed for West (W) and East (E) side of the shell have been clamped by a proper
various external restraints taking into account different higher order setup of a Double – Weibull distribution of linear springs, according to
theories. The results of the analysis held on the hyperbolic paraboloid Eq. (36). As it can be seen, general boundary conditions have been set
are arranged in Table 7, whereas those referred to the helicoid can be

28
F. Tornabene et al. Composite Structures 309 (2023) 116542

Table 9
Mode frequencies of a mapped pseudosphere [22] of revolution with a tractrix meridian characterized by variable thickness and subjected to general surface and
boundary linear elastic constraints. The results have been provided by the unified ESL approach employing different higher order theories. Elastic translational springs
have been distributed along the structure in all the principal directions in order to obtain different kinds of elastic supports. A power thickness variation has been set
along the physical domain.
Mapped Pseudosphere (a = 1.5 m)

Mode FSDTκ=1.2
RS
FSDTZ RS TSDT RS TSDTZ RS EDZ1 RS ED2κ=1.2 EDZ2 ED3 EDZ3 ED4 EDZ4
f [Hz]

DOFs 5046 7569 10092 12615 7569 7569 10092 10092 12615 12615 15138
( )
Three-Points Constraint BKDDD FFBKDDD
1 132.566 132.770 132.200 131.874 132.902 131.802 130.959 132.356 131.615 133.053 132.249
2 238.474 236.532 234.838 233.291 236.528 233.619 231.397 235.150 232.996 234.770 233.047
3 332.070 332.512 330.315 329.418 335.442 332.325 330.500 333.388 331.697 332.671 331.470
4 399.071 402.328 399.491 398.348 405.907 399.097 396.198 406.167 403.191 404.538 402.620
5 442.837 442.728 438.437 437.777 447.405 442.220 440.167 442.492 440.587 441.789 440.638
6 570.894 578.494 572.572 569.979 582.100 573.912 570.202 581.800 577.484 580.863 578.567
7 601.917 601.763 595.587 592.836 606.063 597.130 592.946 608.541 604.379 606.347 604.568
8 712.135 712.116 706.418 705.596 719.744 714.024 711.528 713.190 711.618 712.641 711.929
9 739.361 745.584 739.310 737.535 750.355 741.149 737.377 748.287 744.376 748.139 746.395
10 780.042 789.332 781.887 781.829 800.816 790.836 788.772 800.102 799.030 795.918 795.055
Boundary Springs: North (N) and West (W) edges: Double - Weibull distribution (ξ̄m = ̃
ξm = 0.0025, p = 20),
(k)ξ01 (k)ξ02 (k)ξ0 (k)ξ0 (k)ξ0 (k)ξ0
k 1f = 1⋅1021 N/m3 , k 2f 1 = k 2f 2 = 1⋅1021 N/m3 , k 3f 1 = k 3f 2 = 1⋅1021 N/m3
= k 1f
( )
Clamped Quarter Edges FBKSSS BKSSS F
1 272.570 278.327 272.816 273.021 259.609 256.682 255.536 264.307 263.683 254.173 252.797
2 309.561 311.467 308.358 307.537 313.686 308.425 306.092 312.665 310.500 311.611 310.251
3 384.621 388.170 385.454 386.775 398.975 399.614 399.870 398.289 399.124 395.606 396.102
4 466.920 480.373 477.290 476.160 488.540 479.243 478.067 494.980 492.393 490.706 491.321
5 492.680 493.478 493.344 491.729 490.419 485.016 480.661 495.209 492.393 494.237 491.759
6 561.090 563.696 559.288 557.991 563.670 551.607 547.505 562.646 558.771 562.490 559.953
7 633.202 636.755 627.435 624.265 643.197 630.992 626.341 640.611 635.137 639.886 636.667
8 780.971 787.750 780.682 778.090 798.389 786.858 781.564 795.631 789.857 791.929 788.251
9 865.565 879.567 870.640 868.140 887.157 870.595 866.051 892.395 887.344 890.310 888.054
10 898.530 908.870 903.850 908.581 924.708 918.438 918.602 928.270 930.543 923.326 922.774
Boundary Springs: Super elliptic distribution (ξ̄m = 1.0, ̃ξm = 0.25, p = 20),
(k)ξ11 (k)ξ12 (k)ξ11 (k)ξ12 (k)ξ11 (k)ξ12
k 1f = k 1f = 1⋅1021 N/m3 , k 2f = k 2f = 1⋅1021 N/m3 , k 3f = k 3f = 1⋅1021 N/m3
Two Clamped Edges (FCFF)
1 312.388 318.739 317.535 317.052 298.024 297.305 295.887 304.864 303.565 302.887 301.493
2 501.598 516.300 512.712 512.098 526.253 518.415 516.766 533.852 532.146 531.888 531.287
3 560.495 560.956 558.135 558.001 567.142 560.430 557.773 562.957 561.101 561.930 561.106
4 657.637 657.953 650.755 649.248 666.575 654.185 649.868 661.784 657.624 659.196 657.187
5 822.181 837.681 829.859 828.466 845.241 831.458 828.114 845.600 841.889 843.346 841.677
6 895.778 897.798 890.405 888.528 907.678 894.720 888.980 903.705 898.435 899.953 897.038
7 1019.937 1024.393 1012.275 1009.327 1034.845 1014.433 1007.568 1030.200 1023.254 1026.050 1022.544
8 1051.266 1053.173 1038.904 1035.615 1064.888 1041.884 1034.070 1053.890 1045.730 1049.198 1044.863
9 1196.403 1212.604 1204.554 1201.236 1223.266 1205.197 1197.853 1228.676 1220.439 1223.483 1219.211
10 1289.146 1285.160 1269.122 1265.109 1303.028 1275.534 1265.023 1281.371 1270.670 1275.454 1269.686
Lamination Scheme: (30/iso/iso/45), h01 = h02 = h03 = h04 = 0.007 m
Laminae Material Sequence: 1st trigonal material, 2nd aluminium, 3rd steel, 4th trigonal material
Numerical Issues: CGL computational grid with IN = IM = 31 discrete points
Thickness Variation: ϕ i (α 1 , α 2 ) =¯αpi i , p i = p 1 = p 2 = 10, δ 1 = δ 2 = 1.2
Isogeometric Mapping: Domain-04

only along the in-plane principal directions α 1 , α 2 . A similar configu­ have been reported in Table 9. In particular, three-points constraints
( ) ( K ) ( )
ration BKDDD FBKDDD C has also been considered in which linear springs B DDD FFBKDDD , clamped quarter edges FBKSSS BKSSS F and (FCFF) con­
are distributed along α 1 , α 2 , ζ. As expected, mode frequencies do not figurations have been studied. As it can be seen, if in the first configu­
differ too much between these two case studies. However, as it can be ration there is a very rapid convergence of results as the higher order
( )
seen from a comparison between Figs. 17-18, the corresponding mode model gets complicated, in the FBKSSS BKSSS F stable results are reached
shapes differ due to the presence of the out-of-plane deformation in the for higher and lower modes when the ED4 theory is assumed, regardless
former configuration. of the use of the zigzag behaviour set in Eq. (11). When the structure is
The last two examples have been obtained from the physical domain clamped at two edges (FCFF), results provided by higher order theories
mapping declared in Fig. 6, characterized by two curved sides and two differ from those provided by classical ESL approaches, underlying the
straight edges. In particular, a pseudosphere and a revolution hyperbolic importance of the through-the-thickness stretching effect for all mode
hyperboloid have been blended with the considered mapped domain. A shapes, as it appears in Figs. 19-20. In Table 10 one can find the modal
power thickness variation has been selected along all parametric lines. response of a revolution hyperbolic hyperboloid mapped with the
The mapped pseudosphere has been laminated with four generally ori­ domain reported in Fig. 6, tackled by means of the ESL formulation
ented layers, accounting for two outer skins of trigonal material (89) and presented in this work, and various higher order theories. The lamina­
a non-homogeneous central core made of aluminium and steel isotropic tion scheme consists of two external layers of glass–epoxy (90) and a
sheet, whose elastic properties have been defined in the previous sec­ central trigonal layer (89). The first boundary conditions layup con­
tions. Mode frequencies for three different sets of boundary conditions siders a Double – Weibull distribution of linear elastic springs accounted

29
F. Tornabene et al. Composite Structures 309 (2023) 116542

Table 10
Mode frequencies of a mapped revolution hyperbolic hyperboloid [22] characterized by variable thickness and subjected to general boundary linear elastic constraints.
The results have been provided by the unified ESL approach employing different higher order theories. Elastic translational springs have been distributed along the
structure in all the principal directions in order to obtain different kinds of elastic supports. A sinusoidal thickness variation has been set along the physical domain.
Mapped Revolution Hyperbolic Hyperboloid (a = 1 m, c = 2 m)

Mode FSDTκ=1.2
RS
FSDTZ RS TSDT RS TSDTZ RS ED1 RS EDZ1 RS ED2κ=1.2 EDZ2 ED3 EDZ3 ED4 EDZ4
f [Hz]

DOFs 5046 7569 10092 12615 5046 7569 7569 10092 10092 12615 12615 15138
( )
Clamped Corners BKDDD BKDDD BKDDD BKDDD
1 161.440 161.651 161.160 160.968 160.750 163.155 161.606 160.641 161.820 161.642 161.215 161.010
2 202.591 202.995 202.509 202.038 201.629 203.625 202.652 201.409 203.243 202.665 202.443 201.749
3 209.595 209.913 209.088 208.584 208.767 211.645 209.533 208.230 209.915 209.300 209.070 208.453
4 259.500 259.707 258.768 257.690 258.544 262.512 259.295 257.705 259.893 258.666 258.641 257.461
5 285.687 286.041 285.011 284.187 284.654 287.798 285.135 283.474 285.814 285.026 284.572 283.624
6 316.785 317.860 316.827 315.973 315.358 319.213 316.580 314.941 318.010 317.076 316.718 315.607
7 349.227 349.622 348.556 347.374 347.749 350.813 348.536 346.455 349.609 348.232 348.271 346.752
8 380.155 380.374 378.157 376.829 379.509 384.885 378.423 376.312 379.413 378.137 377.175 375.910
9 433.661 434.217 432.253 430.056 432.630 439.350 432.307 429.723 433.987 431.874 431.552 429.230
10 482.159 483.225 482.213 480.466 479.777 487.075 482.667 479.415 484.223 482.250 482.087 480.112
(k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12 (k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12
Boundary Springs: Double - Weibull distribution (ξ̄m = 0.0025, ̃ξm = 0.0025, p = 20), k 1f = k 1f = k 1f = k 1f = 1⋅1021 N/m3 , k 2f = k 2f = k 2f = k 2f = 1⋅
(k)ξ01 (k)ξ11 (k)ξ02 (k)ξ12
1021 N/m3 , k 3f = k 3f = 1⋅1021 N/m3
= k 3f = k 3f
( )
Clamped Half Edges – Spring Linear Thickness distribution BKSSS FFBKSSS
1 50.687 51.003 50.700 50.563 50.568 51.411 50.378 50.138 50.762 50.576 50.408 50.451
2 51.793 52.186 51.820 51.830 51.736 52.307 51.528 51.464 51.866 51.855 51.647 51.654
3 112.887 113.254 112.749 112.383 112.571 113.970 112.490 111.960 112.945 112.703 112.410 112.073
4 176.204 176.822 175.631 174.943 175.858 178.193 175.138 174.557 176.096 175.364 175.105 174.346
5 261.445 261.885 260.111 258.888 261.053 264.247 259.636 258.482 260.839 259.460 259.119 257.792
6 276.455 277.448 276.563 275.915 275.668 280.739 276.439 275.232 277.742 276.975 276.456 275.568
7 362.348 362.668 361.881 360.852 360.115 364.492 362.502 359.796 363.030 361.889 361.531 360.406
8 379.344 379.263 377.793 376.865 377.848 382.324 378.193 376.235 378.737 377.713 377.033 375.951
9 403.351 403.671 403.450 402.517 401.821 404.905 403.822 401.086 404.521 403.410 403.411 402.223
10 416.406 416.147 414.658 413.510 415.780 423.136 415.703 413.749 416.201 414.933 413.981 412.810
0 0 0 0 0 0
Boundary Springs: Super elliptic distribution (ξ̄m = 0, ̃ξm = 0.47, p = 1000), k 1f 1 = k 1f 2 = 1⋅1021 N/m3 , k 2f 1 = k 2f 2 = 1⋅1021 N/m3 , k 3f 1 = k 3f 2 = 1⋅1021 N/m3
(k)ξ (k)ξ (k)ξ (k)ξ (k)ξ (k)ξ

Cantilever Boundary Conditions (FFFC)

1 26.163 26.392 26.003 25.982 26.159 26.480 25.781 25.710 25.962 25.882 25.708 25.790
2 52.520 52.716 52.449 52.182 52.504 54.066 52.493 52.442 52.751 52.522 52.509 52.314
3 121.238 121.383 120.534 119.887 121.087 122.007 120.285 119.874 120.921 120.045 119.984 119.162
4 132.048 132.409 132.190 131.725 131.666 132.798 131.025 130.049 132.204 131.822 131.315 130.972
5 168.725 168.669 168.455 167.307 168.568 173.300 168.800 168.379 169.680 168.627 169.407 167.975
6 212.060 211.893 212.235 209.807 211.529 215.076 212.182 210.820 213.628 211.060 213.115 210.525
7 254.410 254.254 252.874 252.345 253.764 257.478 253.578 252.509 253.811 253.164 252.563 251.835
8 297.558 294.923 288.057 288.968 296.542 296.351 293.722 288.128 289.109 289.888 288.188 289.001
9 366.338 366.270 363.814 362.858 365.569 372.382 364.818 362.844 365.336 364.410 362.729 361.672
10 383.861 383.745 381.302 380.337 382.981 388.319 381.848 380.160 382.436 381.447 380.225 379.225
Lamination Scheme: (30/20/70), h01 = h03 = 0.01 m, h02 = 0.02 m
Laminae Material Sequence: 1st glass–epoxy, 2nd trigonal material, 3rd glass–epoxy
Numerical Issues: CGL computational grid with IN = IM = 31 discrete points
̃ i + α im ) ) )p i , n i = n 3 = n 4 = 1, p i = p 3 = p 4 = 10, α im = α 3m = α 4m = 0, δ 1 = δ 2 = 1.0
Thickness Variation: ϕ i (α 1 , α 2 ) = (sin(π(n i α
Isogeometric Mapping: Domain-04

in Eq. (36), for all the shell edges, leading to four clamped corners. In convergence, since ED1 and EDZ1 theories do not provide results com­
this case, the warping phenomenon occurs in bending modes and parable with other higher order theories.
therefore a linear through-the-thickness variation of the displacement
field components is inadequate for a proper derivation of modal eigen­ 6. Conclusions
values. Such warping effect is not related to interlaminar issues, as it
appears in Fig. 21, since not only the ED1, but also the EDZ1 theory In the present work, an ESL formulation for the free vibration anal­
provide results not comparable with other higher order theories. The ysis of anisotropic doubly-curved shells with arbitrary geometry sub­
second simulation on the hyperbolic hyperboloid has been performed by jected to general boundary conditions has been proposed. The mapping
( )
clamping the area near a corner of the structure, setting BKSSS FFBKSSS . In of the physical domain has been obtained by means of a NURBS-based
addition, a linear through-the-thickness distribution of linear springs has set of blending functions. A general thickness variation has been set
been considered. It is worth noting that the use of a completely aniso­ along the principal parametric lines of the structure, whereas a general
tropic stacking sequence induces unsymmetric modes, as it can be seen anisotropic material has been considered in each layer of the lamination
from mode frequencies values, and mode shapes in Fig. 22. The last free scheme. Unlike previous studies regarding general boundary conditions
vibration analysis accounts for cantilever boundary conditions (FFFC). within an ESL model, non-conventional constraints have been modelled
In this case, the structure exhibits some stretching effects in all modes, along each lateral surface of the structural solid starting from an arbi­
with clear discrepancies with respect to classical ESL theories. We recall trary distribution of linear springs not only alongside the boundary of
that they account for constant through-the-thickness out-of-plane the two-dimensional physical domain, but on the entire lateral surface of
displacement components. Moreover, a parabolic (N = 2) displacement the three-dimensional shell along both in-plane and out-of-plane di­
field assumption is at least required to ensure a satisfactory rections. In this way, the calibration of the distribution governing

30
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 10. Mode shapes of a mapped catenoid constrained in two edges with general boundary conditions. A sinusoidal thickness variation has been selected. The
external constraint has been obtained from a super elliptic distribution of translational springs along the edges of the mapped geometry by properly selecting the
governing parameters. Corresponding mode frequencies have been collected in Table 4.

31
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 11. Mode shapes of a mapped catenoid constrained in the middle point of each edge of the structure. A sinusoidal thickness variation has been selected. The
external constraint has been obtained from a super elliptic distribution of translational springs along the edges of the mapped geometry by properly selecting the
governing parameters. Corresponding mode frequencies have been collected in Table 4.

32
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 12. Mode shapes of a mapped elliptic cone constrained in its four outer corners. The influence of a concentrated spring applied on the bottom surface has been
pointed out. Corresponding mode frequencies have been collected in Table 5.

33
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 13. Mode shapes of a mapped revolution parabolic shell constrained in two portions of an edge. A sinusoidal thickness variation has been selected. The external
constraint has been obtained from a Double - Weibull distribution of translational springs along the edges of the mapped geometry by properly selecting the gov­
erning parameters. Corresponding mode frequencies have been collected in Table 6.

34
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 14. Mode shapes of a mapped revolution parabolic shell constrained in two portions of an edge and with a surface spring applied on the bottom surface. A
sinusoidal thickness variation has been selected. The external constraint has been obtained from a Double - Weibull distribution of translational springs along the
edges of the mapped geometry by properly selecting the governing parameters, as well as a Gaussian distribution for the definition of the restraint on the bottom
surface. Corresponding mode frequencies have been collected in Table 6.

35
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 15. Mode shapes of a mapped laminated hyperboloid paraboloid with variable thickness and general boundary conditions. A sinusoidal thickness variation has
been selected. The external constraint has been obtained from a super elliptic distribution of translational springs distributed along the three principal directions
applied on all the sides of the structure, and from a Gaussian distribution of linear springs applied on the bottom surface. Corresponding mode frequencies, as well as
other analysis information, have been collected in Table 7.

36
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 16. Mode shapes of a mapped laminated hyperbolic paraboloid with variable thickness and general boundary conditions. A sinusoidal thickness variation has
been selected. The external constraint has been obtained from a super elliptic distribution of translational springs distributed along the three principal directions
applied on all the sides of the structure. Corresponding mode frequencies, as well as other analysis information, have been collected in Table 7.

parameters has led to the definition of point constraints without recur­ Hamiltonian Principle directly in the strong form and they have been
ring to the classical domain decomposition methodology. Furthermore, numerically tackled by means of the GDQ method. In this way, the so­
a Winkler elastic foundation with general distribution has been taken lution has been derived in an efficient way even though a reduced mass
into account on the top and the bottom surfaces of the shell. The storage has been required. The model has been validated with respect to
fundamental governing equations have been derived employing the refined 3D FEM simulations, with a perfect match between the relevant

37
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 17. Mode shapes of a mapped helicoid constrained with in-plane elastic springs on two sides of the physical domain. A sinusoidal thickness variation has been
selected. The external constraint has been obtained from a Double - Weibull distribution of translational springs along the edges of the mapped geometry by properly
selecting the governing parameters. Corresponding mode frequencies have been collected in Table 8.

solutions. Different physical domain mappings have been investigated, obtain very accurate results, despite the limited reduced number of
for structures with various curvatures, lamination schemes and discrete points implemented in each example.
displacement field assumptions. The influence of the selection of gov­
erning parameters for the assessment of surface and lateral boundary Declaration of Competing Interest
conditions has been pointed out. The proposed formulation provides an
ESL formulation with 3D capability for a structural problem character­ The authors declare that they have no known competing financial
ized by very complex geometry, stacking sequence and external con­ interests or personal relationships that could have appeared to influence
straints. To sum up, the advanced numerical algorithm allows one to the work reported in this paper.

38
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 18. Mode shapes of a mapped helicoid constrained with elastic springs on two sides of the physical domain. A sinusoidal thickness variation has been selected.
The external constraint has been obtained from a Double - Weibull distribution of translational springs along the edges of the mapped geometry by properly selecting
the governing parameters. Corresponding mode frequencies have been collected in Table 8.

39
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 19. Mode shapes of a mapped revolution pseudosphere constrained in the neighbourhood of a shell corner. A power thickness variation has been selected. The
external constraint has been obtained from a super elliptic distribution of translational springs along South (S) and East (E) edges of the mapped geometry. Cor­
responding mode frequencies have been collected in Table 9.

40
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 20. Mode shapes of a mapped revolution pseudosphere constrained in three corners. A power thickness variation has been selected. The external constraint has
been obtained from a Double - Weibull distribution of translational springs along the edges of the mapped geometry by properly selecting the governing parameters.
Corresponding mode frequencies have been collected in Table 9.

41
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 21. Mode shapes of a mapped revolution hyperbolic hyperboloid constrained in its corners. A sinusoidal thickness variation has been selected. The external
constraint has been obtained from a Double - Weibull distribution of translational springs along the edges of the mapped geometry by properly selecting the gov­
erning parameters. Corresponding mode frequencies have been collected in Table 10.

42
F. Tornabene et al. Composite Structures 309 (2023) 116542

Fig. 22. Mode shapes of a mapped revolution hyperbolic hyperboloid constrained in its corners. A sinusoidal thickness variation has been selected. The external
constraint has been obtained from a super elliptic distribution of translational springs along the edges of the mapped geometry by properly selecting the governing
parameters. Corresponding mode frequencies have been collected in Table 10.

43
F. Tornabene et al. Composite Structures 309 (2023) 116542

Data availability

No data was used for the research described in the article.

Appendix I

Second order derivatives with respect to principal coordinates α 1 , α 2 should be assessed by means of their corresponding ones with respect to
natural coordinates ξ 1 ,ξ 2 ∈ [ − 1, 1] × [ − 1, 1], employing the NURBS-based blending function presented in Eqs. (54)–(55). For the direct computation
of second-order derivatives set in Eqs. (64)–(66), the following definitions are required:
( ( )2 )
1 ∂α2 ∂2 α2 ∂α2 det(J)ξ1 ∂α2 ∂2 α2 ∂α2 ∂α2 det(J)ξ2
ξ1,α1 α1 = − − + (A.1)
det(J)2 ∂ξ2 ∂ξ1 ∂ξ2 ∂ξ2 det(J) ∂ξ1 ∂ξ22 ∂ξ1 ∂ξ2 det(J)
( ( )2 )
1 ∂α1 ∂2 α1 ∂α1 det(J)ξ1 ∂α1 ∂2 α1 ∂α1 ∂α1 det(J)ξ2
ξ1,α2 α2 = − − + (A.2)
det(J)2 ∂ξ2 ∂ξ1 ∂ξ2 ∂ξ2 det(J) ∂ξ1 ∂ξ22 ∂ξ1 ∂ξ2 det(J)
( )
1 ∂α2 ∂2 α1 ∂α2 ∂α1 det(J)ξ1 ∂α2 ∂2 α1 ∂α2 ∂α1 det(J)ξ2
ξ1,α1 α2 = 2
− + + − (A.3)
det(J) ∂ξ2 ∂ξ1 ∂ξ2 ∂ξ2 ∂ξ2 det(J) ∂ξ1 ∂ξ22 ∂ξ1 ∂ξ2 det(J)
( ( )2 )
1 ∂α2 ∂2 α2 ∂α2 ∂α2 det(J)ξ1 ∂α2 ∂2 α2 ∂α2 det(J)ξ2
ξ2,α1 α1 = − + + − (A.4)
det(J)2 ∂ξ2 ∂ξ21 ∂ξ2 ∂ξ1 det(J) ∂ξ1 ∂ξ1 ∂ξ2 ∂ξ1 det(J)
( ( )2 )
1 ∂α1 ∂2 α1 ∂α1 ∂α1 det(J)ξ1 ∂α1 ∂2 α1 ∂α1 det(J)ξ2
ξ2,α2 α2 = − + + − (A.5)
det(J)2 ∂ξ2 ∂ξ21 ∂ξ2 ∂ξ1 det(J) ∂ξ1 ∂ξ1 ∂ξ2 ∂ξ1 det(J)

( )
1 ∂α2 ∂2 α1 ∂α1 ∂α1 det(J)ξ1 ∂α2 ∂2 α1 ∂α2 ∂α1 det(J)ξ2
ξ2,α1 α2 = 2
− + + − (A.6)
det(J) ∂ξ1 ∂ξ1 ∂ξ2 ∂ξ2 ∂ξ1 det(J) ∂ξ2 ∂ξ21 ∂ξ1 ∂ξ1 det(J)

where det(J)ξ1 and det(J)ξ2 are the derivatives of the determinant of the Jacobian matrix J of the blending functions reported in Eqns. (54)–(55) with
respect to the natural coordinates ξ 1 and ξ 2 , respectively:
∂α1 ∂2 α2 ∂α2 ∂2 α1 ∂α2 ∂2 α1 ∂α1 ∂2 α2
det(J)ξ1 = − + −
∂ξ1 ∂ξ1 ∂ξ2 ∂ξ1 ∂ξ1 ∂ξ2 ∂ξ2 ∂ξ21 ∂ξ2 ∂ξ21
(A.7)
∂α1 ∂2 α2 ∂α2 ∂2 α1 ∂α2 ∂2 α1 ∂α1 ∂2 α2
det(J)ξ2 = − + − +
∂ξ2 ∂ξ1 ∂ξ2 ∂ξ2 ∂ξ1 ∂ξ2 ∂ξ1 ∂ξ22 ∂ξ1 ∂ξ22

Appendix II

An extended version of the generalized mass coefficients of the higher order mass matrix M(τη) is now provided, together with the fundamental
coefficients of L(τη) occurring in Eq. (85), expressed in their discrete form for each τ, η = 0, ..., N + 1.
( )
All the values M(τη) ij , for i, j = 1, 2, 3 of the generalized mass matrix occurring in Eq. (53) are calculated in each point of the CGL discrete grid
(τη)αi αj
already defined in Eq. (72), and they are collected in a IN IM × IN IM matrix denoted with M
̃
ij for i, j = 1, 2, 3:
⎧
⎨ 0 for i ∕
=j
̃ (τη)αi αj = →0(τη)α i α j
M (A.8)
ij
⎩ I ◦ϛα 1 (0)α 2 (0)
for i = j
ij

As can be seen from Eq. (53), in the case of i ∕

= j, the corresponding term of the discrete mass form of the mass matrix is a null matrix of dimension
I N I M × I N I M denoted with 0. The symbol ◦ stands for the well-known Hadamard product [22].
( )
In the same way, the fundamental coefficients Lf(τη) ij for the elastic Winkler matrix introduced in Eq. (27) should be discretized according to Eq.
f(τη)
(28). A matrix of dimension I N I M × I N I M , denoted with L
̃
ij , is thus assessed for each term, as follows:
⎧
⎨ 0 for i ∕
=j
̃ f (τη)α i α j = →f (τη)α i α j
L (A.9)
ij
⎩L ◦ϛα 1 (0)α 2 (0)
for i =j
ij

→f(τη)αi α j
where L ij is the vectorized form of a IN × IM matrix collecting the values assumed at each point of the computational grid by Eq. (28).
( )
On the other hand, each term L(τη) ij , for i, j = 1, 2, 3, of the fundamental matrix L(τη) has been computed in each discrete point of the already
̃ (τη)αi αj :
mentioned CGL grid according to the following expression, leading to the discrete fundamental coefficients L ij

44
F. Tornabene et al. Composite Structures 309 (2023) 116542

̃ (τη)αi αj = → →(τη)αi αj →(τη)αi αj

(τη)αi αj
L L ij(1) ◦ϛα1 (2)α2 (0) + L ij(2) ◦ϛα1 (0)α2 (2) + L ij(3) ◦ϛα1 (1)α2 (1) +
(A.10)
ij
→(τη)αi αj →(τη)αi αj →(τη)αi αj
+ L ij(4) ◦ϛα1 (1)α2 (0) + L ij(5) ◦ϛα1 (0)α2 (1) + L ij(6) ◦ϛα1 (0)α2 (0)

As can be seen, the formulation adopted in the previous equation embeds the previously introduced by-column vectorization of each term Lij(k) i
(τη)α αj

with k = 1, ..., 6 related to the derivatives along the principal coordinates α i , α j = α 1 , α 2 and the employment of the well-known Hadamard’s Product,
identified with the symbol ◦ . Nevertheless, ϛα1 (n)α2 (m) are the weighting coefficient matrices assessed in Eqs. (79)–(80) for the derivatives of n-th and
m-th order calculated along α 1 and α 2 directions. All the terms Lij(k) i j occurring in Eq. (A.9) have been hereafter collected by row.
(τη)α α

First Row of the fundamental operator

→(τη)α1 α1 →o− 2 →(τη)[00]α1 α1
L 11(1) = A 1 ◦ A 11(20)

→(τη)α1 α1 →o− 2 →(τη)[00]α1 α1

L 11(2) = A 2 ◦ A 66(02)

→(τη)α1 α1 →o− 1 →o− 1 →(τη)[00]α1 α1

L 11(3) = 2 A 1 ◦ A 2 ◦ A 16(11)

→(τη)α1 α1 →o− 3 →(τη)[00]α1 α1 α1 (1)α2 (0) → →o− 2 →o− 1 →(τη)[00]α1 α1 α1 (1)α2 (0) →
L 11(4) = − A 1 ◦ A 11(20) ◦ϛ A 1 + A 1 ◦ A 2 ◦ A 11(20) ◦ϛ A2 +
( )
→o− 2 →(τη)[00]α1 α1 →o− 1 →o− 1 α1 (0)α2 (1) →(τη)[00]α1 α1 →o− 1 →(τη)[01]α1 α1 →(τη)[10]α1 α1
+ A 1 ◦ϛα1 (1)α2 (0) A 11(20) + A 1 ◦ A 2 ◦ϛ A 16(11) + A 1 ◦ A 14(10) − A 14(10)

→(τη)α1 α1 →o− 3 →(τη)[00]α1 α1 α1 (0)α2 (1) → →o− 1 →o− 2 →(τη)[00]α1 α1 α1 (0)α2 (1) →
L 11(5) = − A 2 ◦ A 66(02) ◦ϛ A 2 + A 1 ◦ A 2 ◦ A 66(02) ◦ϛ A1 +
( )
→o− 2 →(τη)[00]α1 α1 →o− 1 →o− 1 α1 (1)α2 (0) →(τη)[00]α1 α1 →o− 1 →(τη)[01]α1 α1 →(τη)[10]α1 α1
+ A 2 ◦ϛα1 (0)α2 (1) A 66(02) + A 1 ◦ A 2 ◦ϛ A 16(11) + A 2 ◦ A 46(01) − A 46(01)

→(τη)α1 α1 →o− 2 →o− 1 →(τη)[00]α1 α1 ( α1 (2)α2 (0) → →o− 1 → → ) →o− 2 →o− 1 →(τη)[00]α1 α1 (→o− 1 α1 (1)α2 (0) → α1 (0)α2 (1) →
L 11(6) = A 1 ◦ A 2 ◦ A 12(11) ◦ ϛ A 2 − A 1 ◦ϛα1 (1)α2 (0) A 1 ◦ϛα1 (1)α2 (0) A 2 + A 1 ◦ A 2 ◦ A 16(20) ◦ A 1 ◦ϛ A 1 ◦ϛ A1
)
α1 (1)α2 (1) →
− ϛ A1 +

→o− 1 →o− 2 →(τη)[00]α1 α1 (→o− 1 α1 (0)α2 (1) → α1 (0)α2 (1) → → )

+ A 1 ◦ A 2 ◦ A 66(11) ◦ A 2 ◦ϛ A 1 ◦ϛ A 2 − ϛα1 (0)α2 (2) A 1 +

→o− 1 →o− 2 →(τη)[00]α1 α1 ( α1 (1)α2 (1) → →o− 1 → → )

+ A 1 ◦ A 2 ◦ A 26(02) ◦ ϛ A 2 − A 2 ◦ϛα1 (1)α2 (0) A 2 ◦ϛα1 (0)α2 (1) A 2 +

( ( )
→o− 1 →o− 1 →o− 1 →(τη)[00]α1 α1 →o− 1 α1 (0)α2 (1) →(τη)[00]α1 α1 →o− 1 →(τη)[00]α1 α1 →(τη)[00]α1 α1
+ A 1 ◦ A 2 ◦ A 1 ◦ϛα1 (1)α2 (0) A 12(11) + A 2 ◦ϛ A 26(02) + R 1 ◦ 2 A 24(11) − A 14(20) +

)
→(τη)[01]α1 α1 →(τη)[01]α1 α1 →(τη)[10]α1 α1 →
+ A 14(10) − A 24(01) − A 24(01) ◦ϛα1 (1)α2 (0) A 2 +

( ( )
→o− 1 →o− 1 →o− 1 →(τη)[00]α1 α1 →o− 1 α1 (0)α2 (1) →(τη)[00]α1 α1 →o− 1 →(τη)[00]α1 α1 →(τη)[00]α1 α1
− A 1 ◦ A 2 ◦ A 1 ◦ϛα1 (1)α2 (0) A 16(20) + A 2 ◦ϛ A 66(11) + R 1 ◦ 2 A 46(20) + A 46(11) +

)
→(τη)[01]α1 α1 →(τη)[01]α1 α1 →(τη)[10]α1 α1 → →o− 2 →o− 2 →(τη)[00]α1 α1 α1 (0)α2 (1) → α1 (1)α2 (0) →
− A 46(10) − A 46(01) − A 46(10) ◦ϛα1 (0)α2 (1) A 1 + 2 A 1 ◦ A 2 ◦ A 26(11) ◦ϛ A 1 ◦ϛ A2 +

( ) ( )
→o− 2 →o− 2 →(τη)[00]α1 α1 ( α1 (1)α2 (0) → )o2 →(τη)[00]α1 α1 ( α1 (0)α2 (1) → )o2 →o− 2 →(τη)[00]α1 α1 →o− 1 →(τη)[01]α1 α1 →(τη)[10]α1 α1 →(τη)[11]α1 α1
− A 1 ◦ A 2 ◦ A 22(02) ◦ ϛ A2 + A 66(20) ◦ ϛ A1 − R 1 ◦ A 44(20) + R 1 ◦ A 44(10) + A 44(10) − A 44(00) +

→o− 1 →o− 1 →(τη)[00]α1 α1 →o− 1 α1 (1)α2 (0) →(τη)[01]α1 α1 →o− 1 →o− 1 α1 (0)α2 (1) →(τη)[00]α1 α1
− A 1 ◦ R 1 ◦ϛα1 (1)α2 (0) A 14(20) + A 1 ◦ϛ A 14(10) − A 2 ◦ R 1 ◦ϛ A 46(11) +

→o− 1 →(τη)[01]α1 α1 →o− 1 →o− 2 →(τη)[00]α1 α1 α1 (1)α2 (0) → →o− 1 →o− 2 →(τη)[00]α1 α1 α1 (0)α2 (1) →
+ A 2 ◦ϛα1 (0)α2 (1) A 46(01) + A 1 ◦ R 1 ◦ A 14(20) ◦ϛ R 1 + A 2 ◦ R 1 ◦ A 46(11) ◦ϛ R1 (A.11)

→(τη)α1 α2 →o− 2 →(τη)[00]α1 α2

L 12(1) = A 1 ◦ A 16(20)

→(τη)α1 α2 →o− 2 →(τη)[00]α1 α2

L 12(2) = A 2 ◦ A 26(02)
( )
→(τη)α1 α2 →o− 1 →o− 1 →(τη)[00]α1 α2 →(τη)[00]α1 α2
L 12(3) = A 1 ◦ A 2 ◦ A 12(11) + A 66(11)

( )
→(τη)α1 α2 →o− 3 →(τη)[00]α1 α2 α1 (1)α2 (0) → →o− 2 →o− 1 →(τη)[00]α1 α2 →(τη)[00]α1 α2 →
L 12(4) = − A 1 ◦ A 16(20) ◦ϛ A 1 + A 1 ◦ A 2 ◦ A 11(20) + A 66(20) ◦ϛα1 (0)α2 (1) A 1 +

45
F. Tornabene et al. Composite Structures 309 (2023) 116542

( )
→o− 2 →o− 1 →(τη)[00]α1 α2 →(τη)[00]α1 α2 →(τη)[00]α1 α2 →
+ A 1 ◦ A 2 ◦ A 16(20) − A 16(11) − A 26(11) ◦ϛα1 (1)α2 (0) A 2 +

→o− 2 →(τη)[00]α1 α2 →o− 1 →o− 1 α1 (0)α2 (1) →(τη)[00]α1 α2 →o− 1 →(τη)[01]α1 α2

+ A 1 ◦ϛα1 (1)α2 (0) A 16(20) + A 1 ◦ A 2 ◦ϛ A 66(11) + A 1 ◦ A 15(10) +

→o− 1 →o− 1 →(τη)[00]α1 α2 →o− 1 →(τη)[10]α1 α2 →o− 1 →o− 1 →(τη)[00]α1 α2

− A 1 ◦ R 2 ◦ A 15(11) − A 1 ◦ A 46(10) + A 1 ◦ R 1 ◦ A 46(20)
( )
→(τη)α1 α2 →o− 3 →(τη)[00]α1 α2 α1 (0)α2 (1) → →o− 1 →o− 2 →(τη)[00]α1 α2 →(τη)[00]α1 α2 →
L 12(5) = − A 2 ◦ A 26(02) ◦ϛ A 2 − A 1 ◦ A 2 ◦ A 22(02) + A 66(02) ◦ϛα1 (1)α2 (0) A 2 +

( )
→o− 1 →o− 2 →(τη)[00]α1 α2 →(τη)[00]α1 α2 →(τη)[00]α1 α2 →
+ A 1 ◦ A 2 ◦ A 26(02) + A 16(11) + A 26(11) ◦ϛα1 (0)α2 (1) A 1 +

→o− 2 →(τη)[00]α1 α2 →o− 1 →o− 1 α1 (1)α2 (0) →(τη)[00]α1 α2 →o− 1 →(τη)[10]α1 α2 →o− 1 →o− 1 →(τη)[00]α1 α2
+ A 2 ◦ϛα1 (0)α2 (1) A 26(02) + A 1 ◦ A 2 ◦ϛ A 12(11) − A 2 ◦ A 24(01) + A 2 ◦ R 1 ◦ A 24(11) +

→o− 1 →(τη)[01]α1 α2 →o− 1 →o− 1 →(τη)[00]α1 α2

+ A 2 ◦ A 56(01) − A 2 ◦ R 2 ◦ A 56(02)

→(τη)α1 α2 →o− 2 →o− 1 →(τη)[00]α1 α2 ( α1 (1)α2 (1) → →o− 1 → → )

L 12(6) = A 1 ◦ A 2 ◦ A 11(20) ◦ ϛ A 1 − A 1 ◦ϛα1 (1)α2 (0) A 1 ◦ϛα1 (0)α2 (1) A 1 +

→o− 1 →o− 2 →(τη)[00]α1 α2 (→o− 1 α1 (1)α2 (0) → α1 (0)α2 (1) → → )

+ A 1 ◦ A 2 ◦ A 66(02) ◦ A 2 ◦ϛ A 2 ◦ϛ A 2 − ϛα1 (1)α2 (1) A 2 +

→o− 1 →o− 1 →(τη)[00]α1 α2 (→o− 1 α1 (0)α2 (2) → →o− 2 → → →o− 2 → → →o− 1 → )

+ A 1 ◦ A 2 ◦ A 16(11) ◦ A 2 ◦ϛ A 1 − A 2 ◦ϛα1 (0)α2 (1) A 1 ◦ϛα1 (0)α2 (1) A 2 + A 1 ◦ϛα1 (1)α2 (0) A 1 ◦ϛα1 (1)α2 (0) A 2 − A 1 ◦ϛα1 (2)α2 (0) A 2 +

( ( )
→o− 1 →o− 1 →o− 1 →(τη)[00]α1 α2 →o− 1 α1 (0)α2 (1) →(τη)[00]α1 α2 →o− 1 →(τη)[00]α1 α2 →o− 1 →(τη)[00]α1 α2 →(τη)[00]α1 α2
+ A 1 ◦ A 2 ◦ A 1 ◦ϛα1 (1)α2 (0) A 11(20) + A 2 ◦ϛ A 16(11) + R 1 ◦ A 14(20) − R 2 ◦ A 56(11) + A 56(02) +

) (
→(τη)[01]α1 α2 →(τη)[01]α1 α2 →(τη)[10]α1 α2 → →o− 1 →o− 1 →o− 1 →(τη)[00]α1 α2 →o− 1 →(τη)[00]α1 α2
+ A 56(10) + A 56(01) − A 14(10) ◦ϛα1 (0)α2 (1) A 1 − A 1 ◦ A 2 ◦ A 1 ◦ϛα1 (1)α2 (0) A 16(11) + A 2 ◦ϛα1 (0)α2 (1) A 66(02) +

( ) )
→o− 1 →(τη)[00]α1 α2 →o− 1 →(τη)[00]α1 α2 →(τη)[00]α1 α2 →(τη)[01]α1 α2 →(τη)[01]α1 α2 →(τη)[10]α1 α2 →
+ R 1 ◦ A 46(11) + R 2 ◦ A 15(11) − A 25(02) + A 25(01) − A 15(10) − A 46(01) ◦ϛα1 (1)α2 (0) A 2 +

(
→o− 2 →o− 2 →(τη)[00]α1 α2 ( α1 (1)α2 (0) → )o2 →(τη)[00]α1 α2 ( α1 (0)α2 (1) → )o2
+ A 1 ◦ A 2 ◦ A 26(02) ◦ ϛ A2 + A 16(20) ◦ ϛ A1 +

( ) )
→(τη)[00]α1 α2 →(τη)[00]α1 α2 → → →o− 1 →(τη)[01]α1 α2 →o− 1 →(τη)[10]α1 α2 →o− 1 →o− 1 →(τη)[00]α1 α2
− A 12(11) + A 66(11) ◦ϛα1 (0)α2 (1) A 1 ◦ϛα1 (1)α2 (0) A 2 + R 1 ◦ A 45(10) + R 2 ◦ A 45(01) − R 1 ◦ R 2 ◦ A 45(11) +

( ) ( )
→(τη)[11]α1 α2 →o− 1 →o− 1 α1 (1)α2 (0) →(τη)[00]α1 α2 →(τη)[01]α1 α2 →o− 1 →o− 1 →(τη)[00]α1 α2 →(τη)[01]α1 α2
− A 45(00) − A 1 ◦ R 2 ◦ϛ A 15(11) − ϛα1 (1)α2 (0) A 15(10) − A 2 ◦ R 2 ◦ϛα1 (0)α2 (1) A 56(02) − ϛα1 (0)α2 (1) A 56(01) +

( )
→o− 2 →o− 1 →(τη)[00]α1 α2 α1 (1)α2 (0) → →o− 1 →(τη)[00]α1 α2 α1 (0)α2 (1) →
+ R 2 ◦ A 1 ◦ A 15(11) ◦ϛ R 2 + A 2 ◦ A 56(02) ◦ϛ R2 (A.12)

→(τη)α1 α3 →o− 2 →(τη)[00]α1 α3

L 13(1) = A 1 ◦ A 14(20)

→(τη)α1 α3 →o− 2 →(τη)[00]α1 α3

L 13(2) = A 2 ◦ A 56(02)
( )
→(τη)α1 α3 →o− 1 →o− 1 →(τη)[00]α1 α3 →(τη)[00]α1 α3
L 13(3) = A 1 ◦ A 2 ◦ A 15(11) + A 46(11)

( ( )
→(τη)α1 α3 →o− 1 →o− 1 →(τη)[00]α1 α3 →(τη)[00]α1 α3 →o− 1 →(τη)[00]α1 α3 →(τη)[01]α1 α3 →(τη)[10]α1 α3 →o− 2 →(τη)[00]α1 α3 α1 (1)α2 (0) →
L 13(4) = A 1 ◦ R 1 ◦ A 11(20) + A 44(20) + R 2 ◦ A 12(11) + A 13(10) − A 44(10) − A 1 ◦ A 14(20) ◦ϛ A1
( )
→o− 1 →o− 1 →(τη)[00]α1 α3 α1 (0)α2 (1) → →o− 1 →o− 1 →(τη)[00]α1 α3 →(τη)[00]α1 α3 → →o− 1 →(τη)[00]α1 α3
+ + A 1 ◦ A 2 ◦ A 46(20) ◦ϛ A 1 + A 1 ◦ A 2 ◦ A 14(20) − A 24(11) ◦ϛα1 (1)α2 (0) A 2 + A 1 ◦ϛα1 (1)α2 (0) A 14(20) +
)
→o− 1 →(τη)[00]α1 α3
+ A 2 ◦ϛα1 (0)α2 (1) A 46(11)

46
F. Tornabene et al. Composite Structures 309 (2023) 116542

( ( )
→(τη)α1 α3 →o− 1 →o− 1 →(τη)[00]α1 α3 →(τη)[00]α1 α3
L 13(5) = A 2 ◦ R 1 ◦ A 16(11) + A 45(11)
(
→o− 1 →(τη)[00]α1 α3 →(τη)[01]α1 α3 →(τη)[10]α1 α3 →o− 2 →(τη)[00]α1 α3 α1 (0)α2 (1) → →o− 1 →o− 1 →(τη)[00]α1 α3 α1 (1)α2 (0) → →o− 1 →o− 1 →(τη)[00]α1 α3
+ R 2 ◦ A 26(02) + A 36(01) − A 45(01) − A 2 ◦ A 56(02) ◦ϛ A 2 + − A 1 ◦ A 2 ◦ A 25(02) ◦ϛ A 2 + A 1 ◦ A 2 ◦ A 56(11)
) )
→(τη)[00]α1 α3 → →o− 1 →(τη)[00]α1 α3 →o− 1 α1 (1)α2 (0) →(τη)[00]α1 α3
+ A 56(02) ◦ϛα1 (0)α2 (1) A 1 + + A 2 ◦ϛα1 (0)α2 (1) A 56(02) + A 1 ◦ϛ A 15(11)

( ( ) ( ) )
→(τη)α1 α3 →o− 1 →o− 1 →o− 1 →(τη)[00]α1 α3 →(τη)[00]α1 α3 →o− 1 →(τη)[00]α1 α3 →(τη)[00]α1 α3 →(τη)[01]α1 α3 →(τη)[01]α1 α3 →
L 13(6) = A 1 ◦ A 2 ◦ R 2 ◦ A 12(11) − A 22(02) + R 1 ◦ A 11(20) − A 12(11) + A 13(10) − A 23(01) ◦ϛα1 (1)α2 (0) A 2 +

( ( ) ( )
→o− 1 →o− 1 →o− 1 →(τη)[00]α1 α3 →(τη)[00]α1 α3 →o− 1 →(τη)[00]α1 α3 →(τη)[00]α1 α3 →(τη)[01]α1 α3
+ A 1 ◦ A 2 ◦ R 2 ◦ A 26(02) + A 26(11) + R 1 ◦ A 16(11) + A 16(20) + A 36(01)
) ( ) (
→(τη)[01]α1 α3 → →o− 1 →o− 1 →(τη)[00]α1 α3 →o− 1 α1 (1)α2 (0) →(τη)[00]α1 α3 →o− 1 →o− 1 →(τη)[00]α1 α3
+ A 36(10) ◦ϛα1 (0)α2 (1) A 1 + A 1 ◦ R 1 ◦ϛα1 (1)α2 (0) A 11(20) + R 2 ◦ϛ A 12(11) + A 2 ◦ R 1 ◦ϛα1 (0)α2 (1) A 16(11) +

) (
→o− 1 →(τη)[00]α1 α3 →o− 1 →(τη)[01]α1 α3 →o− 1 α1 (0)α2 (1) →(τη)[01]α1 α3 →o− 2 →(τη)[00]α1 α3 →o− 1 →(τη)[00]α1 α3 α1 (1)α2 (0) →
+ R 2 ◦ϛα1 (0)α2 (1) A 26(02) + A 1 ◦ϛα1 (1)α2 (0) A 13(10) + A 2 ◦ϛ A 36(01) + R 1 ◦ A 14(20) − A 1 ◦ A 11(20) ◦ϛ R1
) ( )
→o− 1 →(τη)[00]α1 α3 α1 (0)α2 (1) → →o− 2 →o− 1 →(τη)[00]α1 α3 α1 (1)α2 (0) → →o− 1 →(τη)[00]α1 α3 α1 (0)α2 (1) →
− A 2 ◦ A 16(11) ◦ϛ R 1 + − R 2 ◦ A 1 ◦ A 12(11) ◦ϛ R 2 + A 2 ◦ A 26(02) ◦ϛ R2 +

( )
→o− 1 →o− 1 →(τη)[00]α1 α3 →o− 1 →(τη)[01]α1 α3 →(τη)[10]α1 α3 →o− 1 →(τη)[10]α1 α3 →(τη)[11]α1 α3
+ R 1 ◦ R 2 ◦ A 24(11) + R 1 ◦ A 34(10) − A 14(10) − R 2 ◦ A 24(01) − A 34(00) (A.13)

Second Row of the fundamental operator

→(τη)α2 α1 →o− 2 →(τη)[00]α2 α1
L 21(1) = A 1 ◦ A 16(20)

→(τη)α2 α1 →o− 2 →(τη)[00]α2 α1

L 21(2) = A 2 ◦ A 26(02)
( )
→(τη)α2 α1 →o− 1 →o− 1 →(τη)[00]α2 α1 →(τη)[00]α2 α1
L 21(3) = A 1 ◦ A 2 ◦ A 12(11) + A 66(11)

( )
→(τη)α2 α1 →o− 3 →(τη)[00]α2 α1 α1 (1)α2 (0) → →o− 2 →o− 1 →(τη)[00]α2 α1 →(τη)[00]α2 α1 →
L 21(4) = − A 1 ◦ A 16(20) ◦ϛ A 1 − A 1 ◦ A 2 ◦ A 11(20) + A 66(20) ◦ϛα1 (0)α2 (1) A 1 +

( )
→o− 2 →o− 1 →(τη)[00]α2 α1 →(τη)[00]α2 α1 →(τη)[00]α2 α1 →
+ A 1 ◦ A 2 ◦ A 16(20) + A 16(11) + A 26(11) ◦ϛα1 (1)α2 (0) A 2 +

( )
→o− 2 →(τη)[00]α2 α1 →o− 1 →o− 1 α1 (0)α2 (1) →(τη)[00]α2 α1 →o− 1 →(τη)[01]α2 α1 →(τη)[10]α2 α1
+ A 1 ◦ϛα1 (1)α2 (0) A 16(20) + A 1 ◦ A 2 ◦ϛ A 12(11) + A 1 ◦ A 46(10) − A 15(10) +

( )
→o− 1 →o− 1 →(τη)[00]α2 α1 →o− 1 →(τη)[00]α2 α1
+ A 1 ◦ R 2 ◦ A 15(11) − R 1 ◦ A 46(20)

( )
→(τη)α2 α1 →o− 3 →(τη)[00]α2 α1 α1 (0)α2 (1) → →o− 1 →o− 2 →(τη)[00]α2 α1 →(τη)[00]α2 α1 →
L 21(5) = − A 2 ◦ A 26(02) ◦ϛ A 2 + A 1 ◦ A 2 ◦ A 22(02) + A 66(02) ◦ϛα1 (1)α2 (0) A 2 +

( )
→o− 1 →o− 2 →(τη)[00]α2 α1 →(τη)[00]α2 α1 →(τη)[00]α2 α1 →
+ A 1 ◦ A 2 ◦ A 26(02) − A 16(11) − A 26(11) ◦ϛα1 (0)α2 (1) A 1 +

( )
→o− 2 →(τη)[00]α2 α1 →o− 1 →o− 1 α1 (1)α2 (0) →(τη)[00]α2 α1 →o− 1 →(τη)[01]α2 α1 →(τη)[10]α2 α1
+ A 2 ◦ϛα1 (0)α2 (1) A 26(02) + A 1 ◦ A 2 ◦ϛ A 66(11) + A 2 ◦ A 24(01) − A 56(01) +

→o− 1 →o− 1 →(τη)[00]α2 α1 →o− 1 →o− 1 →(τη)[00]α2 α1

+ A 2 ◦ R 2 ◦ A 56(02) − A 2 ◦ R 1 ◦ A 24(11)

→o− 1 →o− 2 →(τη)[00]α2 α1 (→o− 1 α1 (1)α2 (0) → α1 (0)α2 (1) → → )

− A 1 ◦ A 2 ◦ A 22(02) ◦ A 2 ◦ϛ A 2 ◦ϛ A 2 − ϛα1 (1)α2 (1) A 2 +

→o− 1 →o− 1 →(τη)[00]α2 α1 (→o− 1 α1 (0)α2 (2) → →o− 2 → → →o− 2 → → →o− 1 → )

− A 1 ◦ A 2 ◦ A 26(11) ◦ A 2 ◦ϛ A 1 − A 2 ◦ϛα1 (0)α2 (1) A 1 ◦ϛα1 (0)α2 (1) A 2 + A 1 ◦ϛα1 (1)α2 (0) A 1 ◦ϛα1 (1)α2 (0) A 2 − A 1 ◦ϛα1 (2)α2 (0) A 2 +

(
→o− 1 →o− 1 →(τη)[01]α2 α1 →(τη)[01]α2 α1 →(τη)[10]α2 α1 →o− 1 α1 (1)α2 (0) →(τη)[00]α2 α1
+ A 1 ◦ A 2 ◦ A 46(10) + A 46(01) − A 25(01) + A 1 ◦ϛ A 26(11) +

( ))
→o− 1 →(τη)[00]α2 α1 →o− 1 →(τη)[00]α2 α1 →o− 1 →(τη)[00]α2 α1 →(τη)[00]α2 α1 →
+ A 2 ◦ϛα1 (0)α2 (1) A 22(02) + R 2 ◦ A 25(02) − R 1 ◦ A 46(20) + A 46(11) ◦ϛα1 (1)α2 (0) A 2 +

47
F. Tornabene et al. Composite Structures 309 (2023) 116542

(
→o− 1 →o− 1 →(τη)[01]α2 α1 →(τη)[01]α2 α1 →(τη)[10]α2 α1 →o− 1 α1 (1)α2 (0) →(τη)[00]α2 α1
− A 1 ◦ A 2 ◦ A 14(10) − A 24(01) − A 56(10) + A 1 ◦ϛ A 66(20) +

( ))
→o− 1 →(τη)[00]α2 α1 →o− 1 →(τη)[00]α2 α1 →o− 1 →(τη)[00]α2 α1 →(τη)[00]α2 α1 →
+ A 2 ◦ϛα1 (0)α2 (1) A 26(11) + R 2 ◦ A 56(11) + R 1 ◦ A 24(11) − A 14(20) ◦ϛα1 (0)α2 (1) A 1 +

( ( ) )
→o− 2 →o− 2 →(τη)[00]α2 α1 ( α1 (1)α2 (0) → )o2 →(τη)[00]α2 α1 ( α1 (0)α2 (1) → )o2 →(τη)[00]α2 α1 →(τη)[00]α2 α1 → →
+ A 1 ◦ A 2 ◦ A 26(02) ◦ ϛ A2 + A 16(20) ◦ ϛ A1 − A 12(11) + A 66(11) ◦ϛα1 (0)α2 (1) A 1 ◦ϛα1 (1)α2 (0) A 2 +

→o− 1 →(τη)[01]α2 α1 →o− 1 α1 (0)α2 (1) →(τη)[01]α2 α1

+ A 1 ◦ϛα1 (1)α2 (0) A 46(10) + A 2 ◦ϛ A 24(01) +
(
→o− 1 →o− 1 →(τη)[00]α2 α1 →o− 1 α1 (0)α2 (1) →(τη)[00]α2 α1
− R 1 ◦ A 1 ◦ϛα1 (1)α2 (0) A 46(20) + A 2 ◦ϛ A 24(11)
)
→(τη)[10]α2 α1 →o− 1 →o− 2 →(τη)[00]α2 α1 α1 (1)α2 (0) → →o− 1 →o− 2 →(τη)[00]α2 α1 α1 (0)α2 (1) → →o− 1 →o− 1 →(τη)[00]α2 α1 →o− 1 →(τη)[01]α2 α1 →(τη)[11]α2 α1
− A 45(10) + A 1 ◦ R 1 ◦ A 46(20) ◦ϛ R 1 + A 2 ◦ R 1 ◦ A 24(11) ◦ϛ R 1 − R 1 ◦ R 2 ◦ A 45(11) + R 2 ◦ A 45(01) − A 45(00)

(A.14)

→(τη)α2 α2 →o− 2 →(τη)[00]α2 α2

L 22(1) = A 1 ◦ A 66(20)

→(τη)α2 α2 →o− 2 →(τη)[00]α2 α2

L 22(2) = A 2 ◦ A 22(02)

→(τη)α2 α2 →o− 1 →o− 1 →(τη)[00]α2 α2

L 22(3) = 2 A 1 ◦ A 2 ◦ A 26(11)

→(τη)α2 α2 →o− 3 →(τη)[00]α2 α2 α1 (1)α2 (0) → →o− 2 →o− 1 →(τη)[00]α2 α2 α1 (1)α2 (0) →
L 22(4) = − A 1 ◦ A 66(20) ◦ϛ A 1 + A 1 ◦ A 2 ◦ A 66(20) ◦ϛ A2 +
( )
→o− 1 →(τη)[01]α2 α2 →(τη)[10]α2 α2 →o− 1 α1 (1)α2 (0) →(τη)[00]α2 α2 →o− 1 α1 (0)α2 (1) →(τη)[00]α2 α2
+ A 1 ◦ A 56(10) − A 56(10) + A 1 ◦ϛ A 66(20) + A 2 ◦ϛ A 26(11)

(
τη α α2 →o− 1 →(τη)[01]α2 α2 →(τη)[10]α2 α2 →o− 2 →(τη)[00]α2 α2 α1 (0)α2 (1) →
¯L(22(5)
) 2
= A 2 ◦ A 25(01) − A 25(01) − A 2 ◦ A 22(02) ◦ϛ A2 +

)
→o− 1 →o− 1 →(τη)[00]α2 α2 α1 (0)α2 (1) → →o− 1 →(τη)[00]α2 α2 →o− 1 α1 (1)α2 (0) →(τη)[00]α2 α2
+ A 1 ◦ A 2 ◦ A 22(02) ◦ϛ A 1 + A 2 ◦ϛα1 (0)α2 (1) A 22(02) + A 1 ◦ϛ A 26(11)

(τη)α α2 →o− 2 →o− 1 →(τη)[00]α2 α2 ( α1 (2)α2 (0) → →o− 1 → → ) →o− 2 →o− 1 →(τη)[00]α2 α2 (→o− 1 α1 (1)α2 (0) → α1 (0)α2 (1) →
¯L22(6)2 = − A 1 ◦ A 2 ◦ A 66(11) ◦ ϛ A 2 − A 1 ◦ϛα1 (1)α2 (0) A 1 ◦ϛα1 (1)α2 (0) A 2 + − A 1 ◦ A 2 ◦ A 16(20) ◦ A 1 ◦ϛ A 1 ◦ϛ A1
)
α1 (1)α2 (1) →
− ϛ A1 +

→o− 1 →o− 2 →(τη)[00]α2 α2 (→o− 1 α1 (0)α2 (1) → α1 (0)α2 (1) → → )

− A 1 ◦ A 2 ◦ A 12(11) ◦ A 2 ◦ϛ A 1 ◦ϛ A 2 − ϛα1 (0)α2 (2) A 1 +

→o− 1 →o− 2 →(τη)[00]α2 α2 ( α1 (1)α2 (1) → →o− 1 → → )

− A 1 ◦ A 2 ◦ A 26(02) ◦ ϛ A 2 − A 2 ◦ϛα1 (1)α2 (0) A 2 ◦ϛα1 (0)α2 (1) A 2 +

( ( )
→o− 1 →o− 1 →o− 1 →(τη)[00]α2 α2 →(τη)[00]α2 α2 →o− 1 →(τη)[00]α2 α2 →o− 1 α1 (0)α2 (1) →(τη)[00]α2 α2
− A 1 ◦ A 2 ◦ R 2 ◦ 2 A 56(02) + A 56(11) + A 1 ◦ϛα1 (1)α2 (0) A 66(11) + A 2 ◦ϛ A 26(02) +

)
→(τη)[01]α2 α2 →(τη)[01]α2 α2 →(τη)[10]α2 α2 →
− A 56(10) − A 56(01) − A 56(01) ◦ϛα1 (1)α2 (0) A 2 +

( ( )
→o− 1 →o− 1 →o− 1 →(τη)[00]α2 α2 →(τη)[00]α2 α2 →o− 1 →(τη)[00]α2 α2 →o− 1 α1 (0)α2 (1) →(τη)[00]α2 α2
+ A 1 ◦ A 2 ◦ R 2 ◦ 2 A 15(11) − A 25(02) + A 1 ◦ϛα1 (1)α2 (0) A 16(20) + A 2 ◦ϛ A 12(11) +

)
→(τη)[01]α2 α2 →(τη)[01]α2 α2 →(τη)[10]α2 α2 →
+ A 25(01) − A 15(10) − A 15(10) ◦ϛα1 (0)α2 (1) A 1 +

(
→o− 2 →o− 2 →(τη)[00]α2 α2 α1 (0)α2 (1) → α1 (1)α2 (0) → →(τη)[00]α2 α2 ( α1 (1)α2 (0) → )o2
+ A 1 ◦ A 2 ◦ 2 A 16(11) ◦ϛ A 1 ◦ϛ A 2 − A 66(02) ◦ ϛ A2
( ) )
→(τη)[00]α2 α2 → o2 →o− 1 →o− 1 →(τη)[00]α2 α2 →o− 1 α1 (0)α2 (1) →(τη)[01]α2 α2 →o− 1 →o− 1 α1 (1)α2 (0) →(τη)[00]α2 α2
− A 11(20) ◦ ϛα1 (0)α2 (1) A 1 + − A 2 ◦ R 2 ◦ϛα1 (0)α2 (1) A 25(02) + A 2 ◦ϛ A 25(01) − A 1 ◦ R 2 ◦ϛ A 56(11) +

→o− 1 →(τη)[01]α2 α2 →o− 1 →o− 2 →(τη)[00]α2 α2 α1 (0)α2 (1) → →o− 1 →o− 2 →(τη)[00]α2 α2 α1 (1)α2 (0) →
+ A 1 ◦ϛα1 (1)α2 (0) A 56(10) + A 2 ◦ R 2 ◦ A 25(02) ◦ϛ R 2 + A 1 ◦ R 2 ◦ A 56(11) ◦ϛ R2 +
( )
→o− 2 →(τη)[00]α2 α2 →o− 1 →(τη)[01]α2 α2 →(τη)[10]α2 α2 →(τη)[11]α2 α2
− R 2 ◦ A 55(02) + R 2 ◦ A 55(01) + A 55(01) − A 55(00) (A.15)

48
F. Tornabene et al. Composite Structures 309 (2023) 116542

→(τη)α2 α3 →o− 2 →(τη)[00]α2 α3

L 23(1) = A 1 ◦ A 46(20)

→(τη)α2 α3 →o− 2 →(τη)[00]α2 α3

L 23(2) = A 2 ◦ A 25(02)
( )
→(τη)α2 α3 →o− 1 →o− 1 →(τη)[00]α2 α3 →(τη)[00]α2 α3
L 23(3) = A 1 ◦ A 2 ◦ A 24(11) + A 56(11)

( ( ) ) (
→(τη)α2 α3 →o− 3 →(τη)[00]α2 α3 α1 (1)α2 (0) → →o− 2 →o− 1 →(τη)[00]α2 α3 α1 (0)α2 (1) → →(τη)[00]α2 α3 →(τη)[00]α2 α3 → →o− 1 →(τη)[01]α2 α3
L 23(4) = − A 1 ◦ A 46(20) ◦ϛ A 1 + A 1 ◦ A 2 ◦ − A 14(20) ◦ϛ A 1 + A 46(20) + A 46(11) ◦ϛα1 (1)α2 (0) A 2 + A 1 ◦ A 36(10)
( ))
→(τη)[10]α2 α3 →o− 1 →(τη)[00]α2 α3 →o− 1 →(τη)[00]α2 α3 →(τη)[00]α2 α3
− A 45(10) + R 1 ◦ A 16(20) + R 2 ◦ A 26(11) + A 45(11) +

→o− 1 →o− 1 →(τη)[00]α2 α3 →o− 2 α1 (1)α2 (0) →(τη)[00]α2 α3

+ A 1 ◦ A 2 ◦ϛα1 (0)α2 (1) A 24(11) + A 1 ◦ϛ A 46(20)
( ( ) ) (
→(τη)α2 α3 →o− 3 →(τη)[00]α2 α3 α1 (0)α2 (1) → →o− 1 →o− 2 →(τη)[00]α2 α3 α1 (1)α2 (0) → →(τη)[00]α2 α3 →(τη)[00]α2 α3 → →o− 1 →(τη)[01]α2 α3
L 23(5) = − A 2 ◦ A 25(02) ◦ϛ A 2 + A 1 ◦ A 2 ◦ A 56(02) ◦ϛ A 2 + + A 25(02) − A 15(11) ◦ϛα1 (0)α2 (1) A 1 + A 2 ◦ A 23(01)
( ))
→(τη)[10]α2 α3 →o− 1 →(τη)[00]α2 α3 →o− 1 →(τη)[00]α2 α3 →(τη)[00]α2 α3
− A 55(01) + R 1 ◦ A 12(11) + R 2 ◦ A 22(02) + A 55(02) +

→o− 1 →o− 1 →(τη)[00]α2 α3 →o− 2 α1 (0)α2 (1) →(τη)[00]α2 α3

+ A 1 ◦ A 2 ◦ϛα1 (1)α2 (0) A 56(11) + A 2 ◦ϛ A 25(02)
( ( ) ( ) )
→(τη)α2 α3 →o− 1 →o− 1 →o− 1 →(τη)[00]α2 α3 →(τη)[00]α2 α3 →o− 1 →(τη)[00]α2 α3 →(τη)[00]α2 α3 →(τη)[01]α2 α3 →(τη)[01]α2 α3 →
L 23(6) = A 1 ◦ A 2 ◦ R 1 ◦ A 16(20) + A 16(11) + R 2 ◦ A 26(02) + A 26(11) + A 36(10) + A 36(01) ◦ϛα1 (1)α2 (0) A 2 +

( ( ) ( ) )
→o− 1 →o− 1 →o− 1 →(τη)[00]α2 α3 →(τη)[00]α2 α3 →o− 1 →(τη)[00]α2 α3 →(τη)[00]α2 α3 →(τη)[01]α2 α3 →(τη)[01]α2 α3 →
+ A 1 ◦ A 2 ◦ R 1 ◦ A 12(11) − A 11(02) + R 2 ◦ A 22(02) − A 12(11) + A 23(01) − A 13(10) ◦ϛα1 (0)α2 (1) A 1 +

( )
→o− 1 →o− 1 →(τη)[00]α2 α3 →o− 1 α1 (1)α2 (0) →(τη)[00]α2 α3 →(τη)[01]α2 α3
+ A 1 ◦ R 1 ◦ϛα1 (1)α2 (0) A 16(20) + R 2 ◦ϛ A 26(11) + ϛα1 (1)α2 (0) A 36(10) +

( )
→o− 1 →o− 1 →(τη)[00]α2 α3 →o− 1 α1 (0)α2 (1) →(τη)[00]α2 α3 →(τη)[01]α2 α3
+ A 2 ◦ R 1 ◦ϛα1 (0)α2 (1) A 12(11) + R 2 ◦ϛ A 22(02) + ϛα1 (0)α2 (1) A 23(01) +

( ) (
→o− 1 →o− 2 →(τη)[00]α2 α3 α1 (1)α2 (0) → →o− 2 →(τη)[00]α2 α3 α1 (1)α2 (0) → →o− 1 →o− 2 →(τη)[00]α2 α3 α1 (0)α2 (1) →
− A 1 ◦ R 1 ◦ A 16(20) ◦ϛ R 1 + R 2 ◦ A 26(11) ◦ϛ R 2 − A 2 ◦ R 1 ◦ A 12(11) ◦ϛ R1 +
)
→o− 2 →(τη)[00]α2 α3 α1 (0)α2 (1) → →o− 1 →o− 1 →(τη)[00]α2 α3 →o− 2 →(τη)[00]α2 α3 →o− 1 →(τη)[01]α2 α3 →o− 1 →(τη)[10]α2 α3 →o− 1 →(τη)[10]α2 α3 →(τη)[11]α2 α3
+ R 2 ◦ A 22(02) ◦ϛ R 2 + R 1 ◦ R 2 ◦ A 15(11) + R 2 ◦ A 25(02) + R 2 ◦ A 35(01) + − R 1 ◦ A 15(10) − R 2 ◦ A 25(01) − A 35(00)

(A.16)
Third Row of the fundamental operator
→(τη)α3 α1 →o− 2 →(τη)[00]α3 α1
L 31(1) = A 1 ◦ A 14(20)

→(τη)α3 α1 →o− 2 →(τη)[00]α3 α1

L 31(2) = A 2 ◦ A 56(02)
( )
→(τη)α3 α1 →o− 1 →o− 1 →(τη)[00]α3 α1 →(τη)[00]α3 α1
L 31(3) = A 1 ◦ A 2 ◦ A 15(11) + A 46(11)

(( ) ) (
→(τη)α3 α1 →o− 3 →(τη)[00]α3 α1 α1 (1)α2 (0) → →o− 2 →o− 1 →(τη)[00]α3 α1 →(τη)[00]α3 α1 → →(τη)[00]α3 α1 α1 (0)α2 (1) → →o− 1 →(τη)[10]α3 α1
L 31(4) = − A 1 ◦ A 14(20) ◦ϛ A1 + A1 ◦A2 ◦ A 14(20) + A 24(11) ◦ϛα1 (1)α2 (0) A 2 + − A 46(20) ◦ϛ A 1 − A 1 ◦ A 13(10)
( ) )
→(τη)[01]α3 α1 →o− 1 →(τη)[00]α3 α1 →(τη)[00]α3 α1 →o− 1 →(τη)[00]α3 α1
− A 44(10) + R 1 ◦ A 11(20) + A 44(20) + R 2 ◦ A 12(11) +

→o− 2 →(τη)[00]α3 α1 →o− 1 →o− 1 α1 (0)α2 (1) →(τη)[00]α3 α1

+ A 1 ◦ϛα1 (1)α2 (0) A 14(20) + A 1 ◦ A 2 ◦ϛ A 15(11)
( )
→(τη)α3 α1 →o− 3 →(τη)[00]α3 α1 α1 (0)α2 (1) → →o− 1 →o− 2 →(τη)[00]α3 α1 →(τη)[00]α3 α1 →
L 31(5) = − A 2 ◦ A 56(02) ◦ϛ A 2 − A 1 ◦ A 2 ◦ A 56(11) − A 56(02) ◦ϛα1 (0)α2 (1) A 1 +

( ) (
→o− 1 →o− 2 →(τη)[00]α3 α1 α1 (1)α2 (0) → →o− 1 →(τη)[10]α3 α1 →(τη)[01]α3 α1 →o− 1 →o− 1 →(τη)[00]α3 α1
+ A 1 ◦ A 2 ◦ A 25(02) ◦ϛ A 2 − A 2 ◦ A 36(01) − A 45(01) − A 2 ◦ R 1 ◦ A 16(11)
)
→(τη)[00]α3 α1 →o− 1 →o− 1 →(τη)[00]α3 α1 →o− 2 α1 (0)α2 (1) →(τη)[00]α3 α1 →o− 1 →o− 1 α1 (1)α2 (0) →(τη)[00]α3 α1
+ A 45(11) + − A 2 ◦ R 2 ◦ A 26(02) + A 2 ◦ϛ A 56(02) + A 1 ◦ A 2 ◦ϛ A 46(11)

49
F. Tornabene et al. Composite Structures 309 (2023) 116542

( ( )
→(τη)α3 α1 →o− 1 →o− 1 →o− 1 →(τη)[00]α3 α1 →(τη)[00]α3 α1 →o− 1 →(τη)[00]α3 α1 →(τη)[01]α3 α1 →(τη)[10]α3 α1 →o− 1 α1 (1)α2 (0) →(τη)[00]α3 α1
L 31(6) = − A 1 ◦ A 2 ◦ R 1 ◦ A 44(20) + A 12(11) + R 2 ◦ A 22(02) + A 44(10) − A 23(01) − A 1 ◦ϛ A 24(11) +
) ( ( )
→o− 1 →(τη)[00]α3 α1 → →o− 1 →o− 1 →o− 1 →(τη)[00]α3 α1 →(τη)[00]α3 α1 →(τη)[00]α3 α1 →(τη)[01]α3 α1 →(τη)[10]α3 α1
− A 2 ◦ϛα1 (0)α2 (1) A 25(02) ◦ϛα1 (1)α2 (0) A 2 + A 1 ◦ A 2 ◦ R 1 ◦ A 16(20) − A 45(11) + Ro−2 1 ◦ A 26(11) + + A 45(01) + A 36(10)
) (
→o− 1 →(τη)[00]α3 α1 →o− 1 α1 (0)α2 (1) →(τη)[00]α3 α1 → →o− 1 →o− 1 →o− 2 →(τη)[00]α3 α1 α1 (1)α2 (0) → α1 (0)α2 (1) →
− A 1 ◦ϛα1 (1)α2 (0) A 46(20) − A 2 ◦ϛ A 56(11) ◦ϛα1 (0)α2 (1) A 1 + A 1 ◦ A 2 ◦ A 1 ◦ A 46(20) ◦ϛ A 1 ◦ϛ A1

→o− 2 →(τη)[00]α3 α1 α1 (1)α2 (0) → α1 (0)α2 (1) →

− A 2 ◦ A 25(02) ◦ϛ A 2 ◦ϛ A2
→o− 2 →(τη)[00]α3 α1 α1 (1)α2 (0) → α1 (1)α2 (0) → →o− 2 →(τη)[00]α3 α1 α1 (0)α2 (1) → α1 (0)α2 (1) → →o− 1 →(τη)[00]α3 α1 α1 (1)α2 (1) →
+ − A 1 ◦ A 24(11) ◦ϛ A 1 ◦ϛ A 2 + A 2 ◦ A 56(11) ◦ϛ A 2 ◦ϛ A 1 − A 1 ◦ A 46(20) ◦ϛ A1 +

)
→o− 1 →(τη)[00]α3 α1 α1 (1)α2 (1) → →o− 1 →(τη)[00]α3 α1 α1 (2)α2 (0) → →o− 1 →(τη)[00]α3 α1 α1 (0)α2 (2) →
+ A 2 ◦ A 25(02) ◦ϛ A 2 + A 1 ◦ A 24(11) ◦ϛ A 2 − A 2 ◦ A 56(11) ◦ϛ A1 +

( ) (
→o− 1 →o− 1 →(τη)[00]α3 α1 →(τη)[01]α3 α1 →o− 1 →o− 1 →(τη)[00]α3 α1
+ A 1 ◦ − R 1 ◦ϛα1 (1)α2 (0) A 44(20) + ϛα1 (1)α2 (0) A 44(10) + A 2 ◦ − R 1 ◦ϛα1 (0)α2 (1) A 45(11)
) ( )
→(τη)[01]α3 α1 →o− 2 →o− 1 →(τη)[00]α3 α1 α1 (1)α2 (0) → →o− 1 →(τη)[00]α3 α1 α1 (0)α2 (1) → →(τη)[00]α3 α1
+ ϛα1 (0)α2 (1) A 45(01) + R 1 ◦ A 1 ◦ A 44(20) ◦ϛ R 1 + A 2 ◦ A 45(11) ◦ϛ R 1 + A 14(20) +

→o− 1 →(τη)[01]α3 α1 →o− 1 →o− 1 →(τη)[00]α3 α1 →o− 1 →(τη)[10]α3 α1 →o− 1 →(τη)[01]α3 α1 →(τη)[11]α3 α1
− R 1 ◦ A 14(10) + R 1 ◦ R 2 ◦ A 24(11) + R 1 ◦ A 34(10) − R 2 ◦ A 24(01) − A 34(00) (A.17)

→(τη)α3 α2 →o− 2 →(τη)[00]α3 α2

L 32(1) = A 1 ◦ A 46(20)

→(τη)α3 α2 →o− 2 →(τη)[00]α3 α2

L 32(2) = A 2 ◦ A 25(02)
( )
→(τη)α3 α2 →o− 1 →o− 1 →(τη)[00]α3 α2 →(τη)[00]α3 α2
L 32(3) = A 1 ◦ A 2 ◦ A 24(11) + A 56(11)

(
→(τη)α3 α2 →o− 1 →o− 2 →(τη)[00]α3 α2 α1 (1)α2 (0) → →o− 1 →o− 1 →(τη)[00]α3 α2 α1 (0)α2 (1) →
L 32(4) = − A 1 ◦ A 1 ◦ A 46(20) ◦ϛ A 1 − A 1 ◦ A 2 ◦ A 14(20) ◦ϛ A1 +

( )
→o− 1 →o− 1 →(τη)[00]α3 α2 →(τη)[00]α3 α2 → →(τη)[10]α3 α2 →(τη)[01]α3 α2 →o− 1 →(τη)[00]α3 α2
+ A 1 ◦ A 2 ◦ A 46(11) − A 46(20) ◦ϛα1 (1)α2 (0) A 2 + A 36(10) − A 45(10) + R 1 ◦ A 16(20) +

( ) )
→o− 1 →(τη)[00]α3 α2 →(τη)[00]α3 α2 →o− 1 →(τη)[00]α3 α2 →o− 1 α1 (0)α2 (1) →(τη)[00]α3 α2
+ R 2 ◦ A 26(11) + A 45(11) − A 1 ◦ϛα1 (1)α2 (0) A 46(20) − A 2 ◦ϛ A 56(11)

(
→(τη)α3 α2 →o− 1 →o− 2 →(τη)[00]α3 α2 α1 (0)α2 (1) → →o− 1 →o− 1 →(τη)[00]α3 α2 α1 (1)α2 (0) →
L 32(5) = − A 2 ◦ A 2 ◦ A 25(02) ◦ϛ A 2 + A 1 ◦ A 2 ◦ A 56(02) ◦ϛ A2 +

( ) ( )
→o− 1 →o− 1 →(τη)[00]α3 α2 →(τη)[00]α3 α2 → →(τη)[10]α3 α2 →(τη)[01]α3 α2 →o− 1 →(τη)[00]α3 α2 →o− 1 →(τη)[00]α3 α2 →(τη)[00]α3 α2
− A 1 ◦ A 2 ◦ A 25(02) + A 15(11) ◦ϛα1 (0)α2 (1) A 1 + A 23(01) − A 55(01) + R 1 ◦ A 12(11) + R 2 ◦ A 22(02) + A 55(02)
)
→o− 1 →(τη)[00]α3 α2 →o− 1 α1 (0)α2 (1) →(τη)[00]α3 α2
− A 1 ◦ϛα1 (1)α2 (0) A 24(11) − A 2 ◦ϛ A 25(02)

( ( )
→(τη)α3 α2 →o− 1 →o− 1 →o− 1 →(τη)[00]α3 α2 →o− 1 →(τη)[00]α3 α2 →(τη)[00]α3 α2 →(τη)[01]α3 α2
L 32(6) = A 1 ◦ A 2 ◦ R 1 ◦ A 16(11) + R 2 ◦ A 26(02) − A 45(11) + A 45(10) +

)
→(τη)[10]α3 α2 →o− 1 α1 (1)α2 (0) →(τη)[00]α3 α2 →o− 1 α1 (0)α2 (1) →(τη)[00]α3 α2 →
+ A 36(01) − A 1 ◦ϛ A 46(11) − A 2 ◦ϛ A 56(02) ◦ϛα1 (1)α2 (0) A 2 +

( ( )
→o− 1 →o− 1 →o− 1 →(τη)[00]α3 α2 →o− 1 →(τη)[00]α3 α2 →(τη)[00]α3 α2 →(τη)[10]α3 α2 →(τη)[01]α3 α2 →o− 1 α1 (1)α2 (0) →(τη)[00]α3 α2
− A 1 ◦ A 2 ◦ R 1 ◦ A 11(20) + R 2 ◦ A 12(11) + A 55(02) − A 13(10) − A 55(01) − A 1 ◦ϛ A 14(20) +
)
→o− 1 →(τη)[00]α3 α2 →
− A 2 ◦ϛα1 (0)α2 (1) A 15(11) ◦ϛα1 (0)α2 (1) A 1 +

(
→o− 1 →o− 1 →o− 2 →(τη)[00]α3 α2 α1 (1)α2 (0) → α1 (0)α2 (1) → →o− 2 →(τη)[00]α3 α2 α1 (1)α2 (0) → α1 (1)α2 (0) →
+ A 1 ◦ A 2 ◦ − A 1 ◦ A 14(20) ◦ϛ A 1 ◦ϛ A 1 + A 1 ◦ A 46(11) ◦ϛ A 1 ◦ϛ A2

→o− 2 →(τη)[00]α3 α2 α1 (0)α2 (1) → α1 (0)α2 (1) →

+ − A 2 ◦ A 15(11) ◦ϛ A 1 ◦ϛ A2 +

→o− 2 →(τη)[00]α3 α2 α1 (0)α2 (1) → α1 (1)α2 (0) → →o− 1 →(τη)[00]α3 α2 α1 (1)α2 (1) → →o− 1 →(τη)[00]α3 α2 α1 (1)α2 (1) → →o− 1 →(τη)[00]α3 α2 α1 (0)α2 (2) →
+ A 2 ◦ A 56(02) ◦ϛ A 2 ◦ϛ A 2 + A 1 ◦ A 14(20) ◦ϛ A 1 − A 2 ◦ A 56(02) ◦ϛ A 2 + + A 2 ◦ A 15(11) ◦ϛ A1
)
→o− 1 →(τη)[00]α3 α2 α1 (2)α2 (0) →
− A 1 ◦ A 46(11) ◦ϛ A2 +

50
F. Tornabene et al. Composite Structures 309 (2023) 116542

( ) (
→o− 1 →o− 1 →(τη)[00]α3 α2 →(τη)[01]α3 α2 →o− 1 →o− 1 →(τη)[00]α3 α2
+ A 1 ◦ − R 2 ◦ϛα1 (1)α2 (0) A 45(11) + ϛα1 (1)α2 (0) A 45(10) + A 2 ◦ − R 2 ◦ϛα1 (0)α2 (1) A 55(02) +
) ( )
→(τη)[01]α3 α2 →o− 2 →o− 1 →(τη)[00]α3 α2 α1 (1)α2 (0) → →o− 1 →(τη)[00]α3 α2 α1 (0)α2 (1) →
+ ϛα1 (0)α2 (1) A 55(01) + R 2 ◦ A 1 ◦ A 45(11) ◦ϛ R 2 + A 2 ◦ A 55(02) ◦ϛ R2 +

→o− 1 →o− 1 →(τη)[00]α3 α2 →o− 1 →(τη)[01]α3 α2 →o− 2 →(τη)[00]α3 α2 →o− 1 →(τη)[01]α3 α2 →o− 1 →(τη)[10]α3 α2 →(τη)[11]α3 α2
+ R 1 ◦ R 2 ◦ A 15(11) − R 1 ◦ A 15(10) + R 2 ◦ A 25(02) − R 2 ◦ A 25(01) + R 2 ◦ A 35(01) − A 35(00) (A.18)

→(τη)α3 α3 →o− 2 →(τη)[00]α3 α3

L 33(1) = A 1 ◦ A 44(20)

→(τη)α3 α3 →o− 2 →(τη)[00]α3 α3

L 33(2) = A 2 ◦ A 55(02)

→(τη)α3 α3 →o− 1 →o− 1 →(τη)[00]α3 α3

L 33(3) = 2 A 1 ◦ A 2 ◦ A 45(11)
(
→(τη)α3 α3 →o− 1 →o− 2 →(τη)[00]α3 α3 α1 (1)α2 (0) → →o− 1 →o− 1 →(τη)[00]α3 α3 α1 (1)α2 (0) → →(τη)[01]α3 α3 →(τη)[10]α3 α3
L 33(4) = A 1 ◦ − A 1 ◦ A 44(20) ◦ϛ A 1 + A 1 ◦ A 2 ◦ A 44(20) ◦ϛ A 2 + A 34(10) − A 34(10) +

)
→o− 1 →(τη)[00]α3 α3 →o− 1 α1 (0)α2 (1) →(τη)[00]α3 α3
+ A 1 ◦ϛα1 (1)α2 (0) A 44(20) + A 2 ◦ϛ A 45(11)

(
→(τη)α3 α3 →o− 1 →o− 2 →(τη)[00]α3 α3 α1 (0)α2 (1) → →o− 1 →o− 1 →(τη)[00]α3 α3 α1 (0)α2 (1) → →(τη)[01]α3 α3 →(τη)[10]α3 α3
L 33(5) = A 2 ◦ − A 2 ◦ A 55(02) ◦ϛ A 2 + A 1 ◦ A 2 ◦ A 55(02) ◦ϛ A 1 + A 35(01) − A 35(01) +

)
→o− 1 →(τη)[00]α3 α3 →o− 1 α1 (1)α2 (0) →(τη)[00]α3 α3
+ A 2 ◦ϛα1 (0)α2 (1) A 55(02) + A 1 ◦ϛ A 45(11)

( )
→(τη)α3 α3 →o− 1 →o− 1 →o− 1 →(τη)[00]α3 α3 →o− 1 →(τη)[00]α3 α3 →(τη)[01]α3 α3 →
L 33(6) = A 1 ◦ A 2 ◦ R 1 ◦ A 14(20) + R 2 ◦ A 24(11) + A 34(10) ◦ϛα1 (1)α2 (0) A 2 +

( )
→o− 1 →o− 1 →o− 1 →(τη)[00]α3 α3 →o− 1 →(τη)[00]α3 α3 →(τη)[01]α3 α3 →
+ A 1 ◦ A 2 ◦ R 1 ◦ A 15(11) + R 2 ◦ A 25(02) + A 35(01) ◦ϛα1 (0)α2 (1) A 1 +

(
→o− 1 →(τη)[01]α3 α3 →o− 1 α1 (0)α2 (1) →(τη)[01]α3 α3 →o− 1 →o− 1 α1 (1)α2 (0) →(τη)[00]α3 α3
+ A 1 ◦ϛα1 (1)α2 (0) A 34(10) + A 2 ◦ϛ A 35(01) + A 1 ◦ R 1 ◦ϛ A 14(20) +

) ( )
→o− 1 →(τη)[00]α3 α3 →o− 1 →o− 1 →(τη)[00]α3 α3 →o− 1 α1 (0)α2 (1) →(τη)[00]α3 α3
+ R 2 ◦ϛα1 (1)α2 (0) A 24(11) + A 2 ◦ R 1 ◦ϛα1 (0)α2 (1) A 15(11) + R 2 ◦ϛ A 25(02) +

( ) (
→o− 2 →o− 1 →(τη)[00]α3 α3 α1 (1)α2 (0) → →o− 1 →(τη)[00]α3 α3 α1 (0)α2 (1) → →o− 2 →o− 1 →(τη)[00]α3 α3 α1 (1)α2 (0) →
− R 1 ◦ A 1 ◦ A 14(20) ◦ϛ R 1 + A 2 ◦ A 15(11) ◦ϛ R 1 − R 2 ◦ A 1 ◦ A 24(11) ◦ϛ R2 +
) (
→o− 1 →(τη)[00]α3 α3 α1 (0)α2 (1) → →o− 2 →(τη)[00]α3 α3 →o− 2 →(τη)[00]α3 α3 →o− 1 →o− 1 →(τη)[00]α3 α3 →o− 1 →(τη)[01]α3 α3
+ A 2 ◦ A 25(02) ◦ϛ R 2 − R 1 ◦ A 11(20) − R 2 ◦ A 22(02) − 2 R 1 ◦ R 2 ◦ A 12(11) − R 1 ◦ A 13(10)
) ( )
→(τη)[10]α3 α3 →o− 1 →(τη)[01]α3 α3 →(τη)[10]α3 α3 →(τη)[11]α3 α3
+ A 13(10) + − R 2 ◦ A 23(01) + A 23(01) − A 33(00) (A.19)

Appendix III

We now present the discrete version of each component of the generalized operator accounted in Eq. (25) for each τ, η-th order of the field variable
kinematic expansion. They are expressed in terms of the generalized GDQ coefficient matrix ϛα1 (n)α2 (m) introduced in Eqs. (79) and (80). Once again, we
recall that ◦ stands for the Hadamard product [22]:

̃ (τη)αi αj = → →(τη)αi αj →(τη)αi αj

(τη)αi αj
O ij O ij(1) ◦ϛα 1 (1)α 2 (0) + O ij(2) ◦ϛα 1 (0)α 2 (1) + O ij(3) ◦ϛα 1 (0)α 2 (0) (A.20)

→(τη)α i α j
In the following, each vectorized term O jk(s) for s = 1, 2, 3 has been reported, setting i, j = 1, 2, 3, k = 1, ..., 9 and τ, η = 0, ..., N + 1.
First column of the generalized operator
→(τη)α i α 1 →o− 1 →(τη)[00]αi α1
O 11(1) = A 1 ◦ A 11(20)

→(τη)α i α 1 →o− 1 →(τη)[00]αi α1

O 11(2) = A 2 ◦ A 16(11)
( )
→(τη)α i α 1 →o− 1 →o− 1 →(τη)[00]αi α1 α1 (1)α2 (0) → →(τη)[00]αi α1 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α1 →(τη)[01]αi α1
O 11(3) = A 1 ◦ A 2 ◦ A 12(11) ◦ϛ A 2 − A 16(20) ◦ϛ A 1 − R 1 ◦ A 14(20) + A 14(10) (A.21)

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→(τη)α i α 1 →o− 1 →(τη)[00]αi α1

O 12(1) = A 1 ◦ A 12(11)

→(τη)α 1 α 1 →o− 1 →(τη)[00]αi α1

O 12(2) = A 2 ◦ A 26(02)
( )
→(τη)α 1 α 1 →o− 1 →o− 1 →(τη)[00]αi α1 α1 (1)α2 (0) → →(τη)[00]αi α1 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α1 →(τη)[01]αi α1
O 12(3) = A 1 ◦ A 2 ◦ A 22(02) ◦ϛ A 2 − A 26(11) ◦ϛ A 1 − R 1 ◦ A 24(11) + A 24(01) (A.22)

→(τη)α i α 1 →o− 1 →(τη)[00]αi α1

O 13(1) = A 1 ◦ A 16(20)

→(τη)α i α 1 →o− 1 →(τη)[00]αi α1

O 13(2) = A 2 ◦ A 66(11)
( )
→(τη)α i α 1 →o− 1 →o− 1 →(τη)[00]αi α1 α1 (1)α2 (0) → →(τη)[00]αi α1 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α1 →(τη)[01]αi α1
O 13(3) = A 1 ◦ A 2 ◦ A 26(11) ◦ϛ A 2 − A 66(20) ◦ϛ A 1 − R 1 ◦ A 46(20) + A 46(10) (A.23)

→(τη)α i α 1 →o− 1 →(τη)[00]αi α1

O 14(1) = A 1 ◦ A 16(11)

→(τη)α i α 1 →o− 1 →(τη)[00]αi α1

O 14(2) = A 2 ◦ A 66(02)
( )
→(τη)α i α 1 →o− 1 →o− 1 →(τη)[00]αi α1 α1 (1)α2 (0) → →(τη)[00]αi α1 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α1 →(τη)[01]αi α1
O 14(3) = A 1 ◦ A 2 ◦ A 26(02) ◦ϛ A 2 − A 66(11) ◦ϛ A 1 − R 1 ◦ A 46(11) + A 46(01) (A.24)

→(τη)α i α 1 →o− 1 →(τη)[00]αi α1

O 15(1) = A 1 ◦ A 14(20)

→(τη)α i α 1 →o− 1 →(τη)[00]αi α1

O 15(2) = A 2 ◦ A 46(11)
( )
→(τη)α i α 1 →o− 1 →o− 1 →(τη)[00]αi α1 α1 (1)α2 (0) → →(τη)[00]αi α1 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α1 →(τη)[01]αi α1
O 15(3) = A 1 ◦ A 2 ◦ A 24(11) ◦ϛ A 2 − A 46(20) ◦ϛ A 1 − R 1 ◦ A 44(20) + A 44(10) (A.25)

→(τη)α i α 1 →o− 1 →(τη)[00]αi α1

O 16(1) = A 1 ◦ A 15(11)

→(τη)α i α 1 →o− 1 →(τη)[00]αi α1

O 16(2) = A 2 ◦ A 56(02)
( )
→(τη)α i α 1 →o− 1 →o− 1 →(τη)[00]αi α1 α1 (1)α2 (0) → →(τη)[00]αi α1 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α1 →(τη)[01]αi α1
O 16(3) = A 1 ◦ A 2 ◦ A 25(02) ◦ϛ A 2 − A 56(11) ◦ϛ A 1 − R 1 ◦ A 45(11) + A 45(01) (A.26)

→(τη)α i α 1 →o− 1 →(τη)[10]αi α1

O 17(1) = A 1 ◦ A 14(10)

→(τη)α i α 1 →o− 1 →(τη)[10]αi α1

O 17(2) = A 2 ◦ A 46(01)
( )
→(τη)α i α 1 →o− 1 →o− 1 →(τη)[10]αi α1 α1 (1)α2 (0) → →(τη)[10]αi α1 α1 (0)α2 (1) → →o− 1 →(τη)[10]αi α1 →(τη)[11]αi α1
O 17(3) = A 1 ◦ A 2 ◦ A 24(01) ◦ϛ A 2 − A 46(10) ◦ϛ A 1 − R 1 ◦ A 44(10) + A 44(00) (A.27)

→(τη)α i α 1 →o− 1 →(τη)[10]αi α1

O 18(1) = A 1 ◦ A 15(10)

→(τη)α i α 1 →o− 1 →(τη)[10]αi α1

O 18(2) = A 2 ◦ A 56(01)
( )
→(τη)α i α 1 →o− 1 →o− 1 →(τη)[10]αi α1 α1 (1)α2 (0) → →(τη)[10]αi α1 α1 (0)α2 (1) → →o− 1 →(τη)[10]αi α1 →(τη)[11]αi α1
O 18(3) = A 1 ◦ A 2 ◦ A 25(01) ◦ϛ A 2 − A 56(10) ◦ϛ A 1 − R 1 ◦ A 45(10) + A 45(00) (A.28)

→(τη)α i α 1 →o− 1 →(τη)[10]αi α1

O 19(1) = A 1 ◦ A 13(10)

→(τη)α i α 1 →o− 1 →(τη)[10]αi α1

O 19(2) = A 2 ◦ A 36(01)
( )
→(τη)α i α 1 →o− 1 →o− 1 →(τη)[10]αi α1 α1 (1)α2 (0) → →(τη)[10]αi α1 α1 (0)α2 (1) → →o− 1 →(τη)[10]αi α1 →(τη)[11]αi α1
O 19(3) = A 1 ◦ A 2 ◦ A 23(01) ◦ϛ A 2 − A 36(10) ◦ϛ A 1 − R 1 ◦ A 34(10) + A 34(00) (A.29)

Second column of the generalized operator

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2
O 21(1) = A 1 ◦ A 16(20)

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2

O 21(2) = A 2 ◦ A 12(11)

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F. Tornabene et al. Composite Structures 309 (2023) 116542

( )
→(τη)α i α 2 →o− 1 →o− 1 →(τη)[00]αi α2 α1 (1)α2 (0) → →(τη)[00]αi α2 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α2 →(τη)[01]αi α2
O 21(3) = A 1 ◦ A 2 ◦ − A 16(11) ◦ϛ A 2 + A 11(20) ◦ϛ A 1 − R 2 ◦ A 15(11) + A 15(10) (A.30)

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2

O 22(1) = A 1 ◦ A 26(11)

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2

O 22(2) = A 2 ◦ A 22(02)
( )
→(τη)α i α 2 →o− 1 →o− 1 →(τη)[00]αi α2 α1 (1)α2 (0) → →(τη)[00]αi α2 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α2 →(τη)[01]αi α2
O 22(3) = A 1 ◦ A 2 ◦ − A 26(02) ◦ϛ A 2 + A 12(11) ◦ϛ A 1 − R 2 ◦ A 25(02) + A 25(01) (A.31)

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2

O 23(1) = A 1 ◦ A 66(20)

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2

O 23(2) = A 2 ◦ A 26(11)
( )
→(τη)α i α 2 →o− 1 →o− 1 →(τη)[00]αi α2 α1 (1)α2 (0) → →(τη)[00]αi α2 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α2 →(τη)[01]αi α2
O 23(3) = A 1 ◦ A 2 ◦ − A 66(11) ◦ϛ A 2 + A 16(20) ◦ϛ A 1 − R 2 ◦ A 56(11) + A 56(10) (A.32)

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2

O 24(1) = A 1 ◦ A 66(11)

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2

O 24(2) = A 2 ◦ A 26(02)
( )
→(τη)α i α 2 →o− 1 →o− 1 →(τη)[00]αi α2 α1 (1)α2 (0) → →(τη)[00]αi α2 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α2 →(τη)[01]αi α2
O 24(3) = A 1 ◦ A 2 ◦ − A 66(02) ◦ϛ A 2 + A 16(11) ◦ϛ A 1 − R 2 ◦ A 56(02) + A 56(01) (A.33)

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2

O 25(1) = A 1 ◦ A 46(20)

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2

O 25(2) = A 2 ◦ A 24(11)
( )
→(τη)α i α 2 →o− 1 →o− 1 →(τη)[00]αi α2 α1 (1)α2 (0) → →(τη)[00]αi α2 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α2 →(τη)[01]αi α2
O 25(3) = A 1 ◦ A 2 ◦ − A 46(11) ◦ϛ A 2 + A 14(20) ◦ϛ A 1 − R 2 ◦ A 45(11) + A 45(10) (A.34)

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2

O 26(1) = A 1 ◦ A 56(11)

→(τη)α i α 2 →o− 1 →(τη)[00]αi α2

O 26(2) = A 2 ◦ A 25(02)
( )
→(τη)α i α 2 →o− 1 →o− 1 →(τη)[00]αi α2 α1 (1)α2 (0) → →(τη)[00]αi α2 α1 (0)α2 (1) → →o− 1 →(τη)[00]αi α2 →(τη)[01]αi α2
O 26(3) = A 1 ◦ A 2 ◦ − A 56(02) ◦ϛ A 2 + A 15(11) ◦ϛ A 1 − R 2 ◦ A 55(02) + A 55(01) (A.35)

→(τη)α i α 2 →o− 1 →(τη)[10]αi α2

O 27(1) = A 1 ◦ A 46(10)

→(τη)α i α 2 →o− 1 →(τη)[10]αi α2

O 27(2) = A 2 ◦ A 24(01)
( )
→(τη)α i α 2 →o− 1 →o− 1 →(τη)[10]αi α2 α1 (1)α2 (0) → →(τη)[10]αi α2 α1 (0)α2 (1) → →o− 1 →(τη)[10]αi α2 →(τη)[11]αi α2
O 27(3) = A 1 ◦ A 2 ◦ − A 46(01) ◦ϛ A 2 + A 14(10) ◦ϛ A 1 − R 2 ◦ A 45(01) + A 45(00) (A.36)

→(τη)α i α 2 →o− 1 →(τη)[10]αi α2

O 28(1) = A 1 ◦ A 56(10)

→(τη)α i α 2 →o− 1 →(τη)[10]αi α2

O 28(2) = A 2 ◦ A 25(01)
( )
→(τη)α i α 2 →o− 1 →o− 1 →(τη)[10]αi α2 α1 (1)α2 (0) → →(τη)[10]αi α2 α1 (0)α2 (1) → →o− 1 →(τη)[10]αi α2 →(τη)[11]αi α2
O 28(3) = A 1 ◦ A 2 ◦ − A 56(01) ◦ϛ A 2 + A 15(10) ◦ϛ A 1 − R 2 ◦ A 55(01) + A 55(00) (A.37)

→(τη)α i α 2 →o− 1 →(τη)[10]αi α2

O 29(1) = A 1 ◦ A 36(10)

→(τη)α i α 2 →o− 1 →(τη)[10]αi α2

O 29(2) = A 2 ◦ A 23(01)
( )
→(τη)α i α 2 →o− 1 →o− 1 →(τη)[10]αi α2 α1 (1)α2 (0) → →(τη)[10]αi α2 α1 (0)α2 (1) → →o− 1 →(τη)[10]αi α2 →(τη)[11]αi α2
O 29(3) = A 1 ◦ A 2 ◦ − A 36(01) ◦ϛ A 2 + A 13(10) ◦ϛ A 1 − R 2 ◦ A 35(01) + A 35(00) (A.38)

Third column of the generalized operator

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3
O 31(1) = A 1 ◦ A 14(20)

53
F. Tornabene et al. Composite Structures 309 (2023) 116542

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3

O 31(2) = A 2 ◦ A 15(11)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3 →o− 1 →(τη)[00]αi α3 →(τη)[01]αi α3

O 31(3) = R 1 ◦ A 11(20) + R 2 ◦ A 12(11) + A 13(10) (A.39)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3

O 32(1) = A 1 ◦ A 24(11)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3

O 32(2) = A 2 ◦ A 25(02)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3 →o− 1 →(τη)[00]αi α3 →(τη)[01]αi α3

O 32(3) = R 1 ◦ A 12(11) + R 2 ◦ A 22(02) + A 23(01) (A.40)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3

O 33(1) = A 1 ◦ A 46(20)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3

O 33(2) = A 2 ◦ A 56(11)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3 →o− 1 →(τη)[00]αi α3 →(τη)[01]αi α3

O 33(3) = R 1 ◦ A 16(20) + R 2 ◦ A 26(11) + A 36(10) (A.41)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3

O 34(1) = A 1 ◦ A 46(11)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3

O 34(2) = A 2 ◦ A 56(02)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3 →o− 1 →(τη)[00]αi α3 →(τη)[01]αi α3

O 34(3) = R 1 ◦ A 16(11) + R 2 ◦ A 26(02) + A 36(01) (A.42)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3

O 35(1) = A 1 ◦ A 44(20)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3

O 35(2) = A 2 ◦ A 45(11)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3 →o− 1 →(τη)[00]αi α3 →(τη)[01]αi α3

O 35(3) = R 1 ◦ A 14(20) + R 2 ◦ A 24(11) + A 34(10) (A.43)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3

O 36(1) = A 1 ◦ A 45(11)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3

O 36(2) = A 2 ◦ A 55(02)

→(τη)α i α 3 →o− 1 →(τη)[00]αi α3 →o− 1 →(τη)[00]αi α3 →(τη)[01]αi α3

O 36(3) = R 1 ◦ A 15(11) + R 2 ◦ A 25(02) + A 35(01) (A.44)

→(τη)α i α 3 →o− 1 →(τη)[10]αi α3

O 37(1) = A 1 ◦ A 44(10)

→(τη)α i α 3 →o− 1 →(τη)[10]αi α3

O 37(2) = A 2 ◦ A 45(01)

→(τη)α i α 3 →o− 1 →(τη)[10]αi α3 →o− 1 →(τη)[10]αi α3 →(τη)[11]αi α3

O 37(3) = R 1 ◦ A 14(10) + R 2 ◦ A 24(01) + A 34(00) (A.45)

→(τη)α i α 3 →o− 1 →(τη)[10]αi α3

O 38(1) = A 1 ◦ A 45(10)

→(τη)α i α 3 →o− 1 →(τη)[10]αi α3

O 38(2) = A 2 ◦ A 55(01)

→(τη)α i α 3 →o− 1 →(τη)[10]αi α3 →o− 1 →(τη)[10]αi α3 →(τη)[11]αi α3

O 38(3) = R 1 ◦ A 15(10) + R 2 ◦ A 25(01) + A 35(00) (A.46)

→(τη)α i α 3 →o− 1 →(τη)[10]αi α3

O 39(1) = A 1 ◦ A 34(10)

→(τη)α i α 3 →o− 1 →(τη)[10]αi α3

O 39(2) = A 2 ◦ A 35(01)

→(τη)α i α 3 →o− 1 →(τη)[10]αi α3 →o− 1 →(τη)[10]αi α3 →(τη)[11]αi α3

O 39(3) = R 1 ◦ A 13(10) + R 2 ◦ A 23(01) + A 33(00) (A.47)

54
F. Tornabene et al. Composite Structures 309 (2023) 116542

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