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A FEM Free Vibration Analysis of Variable Stiffness Composite Plates through Hierarchical Modelling

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17 May 2023

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22 May 2023

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Abstract
Variable Angle Tow (VAT) laminates offer a promising alternative to classical straight fiber composites in terms of design and performance. However, analyzing these structures can be more complex due to the introduction of new design variables. Carrera's Unified Formulation (CUF) has been successful in previous works for buckling, vibrational, and stress analysis of VAT plates. Typically, one-dimensional (1D) and two-dimensional (2D) CUF models are used, with a linear law describing the fiber orientation variation in the main plane of the structure. The objective of this article is to expand the CUF 2D plate finite elements family to perform free vibration analysis of composite laminated plate structures with curvilinear fibers. The primary contribution is the application of Reissner's Mixed Variational Theorem (RMVT) to a CUF finite element model. The Principle of Virtual Displacements (PVD) and RMVT are both used as variational statements for the study of monolayer and multilayer VAT plates dynamic behavior. The proposed approach is compared to Abaqus three-dimensional (3D) reference solutions, classical theories and literature results to investigate the effectiveness of the developed models. Results demonstrate that mixed theories provide the best approximation of the reference solution in all cases.
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Subject: Engineering  -   Aerospace Engineering

1. Introduction

Over the last decades, composite structures have shown interesting properties for aerospace applications, because of their high stiffness-to-weight ratio. Despite this, a common thought is that the potential of fiber reinforced structures could be better exploited by improving the directional properties through the variation of fibers angle along the in-plane directions. The choice to keep the fibre orientation constant in each layer is particularly restrictive for geometries which present geometrical discontinuities like cut-outs. VAT plates are characterized by an in-plane variation of fibres angle, allowing to expand the design space of a specific structure. This is particularly useful for optimization problems, where a wider design space can affect positively the search of an optimal solution. For example, in the context of vibrational analyses, the maximization of fundamental frequencies can be improved by using curvilinear fibres. The complexity of analysis is one of the main disadvantages of VATs, because a greater number of unknowns must be taken into account and unfeasible fibres patterns could be obtained during the optimization process.
Several methods for the study of VATs mechanical response are available in the literature. In the following text, a brief review of these approaches is presented, with a particular focus on free vibration analyses. To the best of authors’ knowledge, the first works that have been presented on the topic were based on the assumption of constant fibers angle within each element in a Finite Element Method (FEM) solution. Therefore, the continuous variation of fibres direction was approximated in a step-wise discrete way. This approach can be used in commercial FEM software tools that, at the moment, cannot handle a continuous fiber variation. Hyer and Charette [1] and Hyer and Lee [2] used this method to improve VATs tensile strength and buckling response, respectively. One of the main disadvantages of this step-wise approach is that, as the true variation is continuous, the discrete representation of fibers angle variation imposes a further approximation. A p-version FEM based on the Third-order Shear Deformation Theory (TSDT) was applied to preform vibrational analyses by Akhavan and Ribeiro [3]. Results showed that fibres variation allows to increase (or decrease) natural frequencies and that thin plates are more affected by this phenomenon if compared with thick ones. Ribeiro and Akhavan [4] used the p-version FEM approach with elements based on the First-Order Shear Deformation Theory (FSDT) to perform non-linear vibration analyses. The advantage of the p-version of the FEM is that the accuracy of the approximation is improved by increasing the order of shape functions over the elements. Vibration analyses were performed on VAT plates with a central circular cut-out considering parabolic fibres by Hachemi et al. [5]. Zhao and Kapania [6] investigated the free vibration of prestressed VAT stiffened plates where plates and stiffeners were modeled separately through Mindlin plate theory and Timoshenko beam theory, respectively. The compatibility conditions at the interface between plate and stiffeners were satisfied by using a transformation matrix. Honda and Narita [7] used the classical plate theory within the Ritz method in order to evaluate the natural frequencies and vibrational modes. An experimental approach was used in Rodrigues et al. [8] for the free vibration analysis of a plate with free boundary conditions subjected to random excitation via an electromagnetic shaker. Subsequently, the results were compared to the ones obtained through FEM, where a four-node isoparametric element based on the Reissner-Mindlin theory was used. Stodieck et al. [9] showed that curvilinear fibres can be useful for improving the aero-elastic response of composite wings. The Rayleigh-Ritz method and classical lamination theory were used to develop a 1D beam model, considering the assumption of null chamber deformation of the wing chord-wise section. The aero-elastic response was computed by introducing quasi-static aerodynamic forces in a model developed for the plate structural analysis. A parametric study showed that by using VATs it is possible to influence wing response both positively and negatively.
Curvilinear fibres can improve the modal response as shown in several works. Abdalla et al. [10] used the classical lamination theory in combination with a successive approximation method in order to solve an optimization problem. Results showed that curvilinear fibres increased the optimal fundamental frequency in comparison with straight ones. A similar approach was presented in Blom et al. [11], where the maximization of the first natural frequency considering manufacturing constraints was obtained for VAT conical shells. In Carvalho, Sohouli et al. [12], a genetic algorithm and shell elements based on FSDT were used for the maximization of the fundamental frequency. The Multi-Scale Two-Level (MS2L) approach allows to split the optimization problem in two parts. The composite is modeled as an equivalent homogeneous anisotropic plate in the first step, which aims to find the ideal distribution of the polar parameters that represent the mechanical design variables. The main goal of a second step is to establish the best stacking sequence in relation to the mechanical properties distribution that has been obtained in the first step. The MS2L method was applied by Montemurro and Catapano [13] to VAT plates in order to optimize the buckling response. In order to evaluate the polar parameters, B-spline surfaces were introduced, while manufacturing constraints were considered during the second step. More details about MS2L approach can be find in Catapano et al. [14], Montemurro and Catapano [15] and Fiordilino et al. [16], where both stiffness and buckling optimization problems were solved.
VAT structures have also been studied by using Carrera’s Unified Formulation, which allows to use an arbitrary expansion order along the thickness of the plate. In this way, both Equivalent Single Layer (ESL) and Layer-Wise (LW) models can be obtained in the context of a specific predefined variational statement, as shown in Carrera [17,18]. Carrera et al. [19] used CUF in order to develop a Navier closed-form solution for the static analysis of isotropic plates under several loading conditions. The same approach was used in Carrera and Giunta [20] in order to perform failure analyses on isotropic plates. A further extension of this method was shown in Giunta et al. [21], where the indentation failure analysis of composite sandwich plates was performed. Giunta et al. [22] performed free vibration analyses of composite beams. In Viglietti et al. [23] and Fallahi et al. [24], free vibration and buckling analyses of VATs were performed through the use of a 1D CUF model. Within this framework, shell models were developed as well for VAT cases in order to perform stress analyses, see Sánchez-Majano et al. [25]. In Pagani and Sánchez-Majano [26,27] and Sánchez-Majano et al. [28], manufacture defects were taken into account by using stochastic techniques. Vescovini and Dozio [29] used the Ritz method within CUF for vibrational and buckling analyses. A generalization of CUF was developed in order to allow the use of different expansions for every component of the displacement vector. Demasi et al. [30] applied this approach to the study of VAT plates with an ESL model. A further advantage of CUF is that it can be used in combination with different variational statements. An alternative to the classic PVD is represented by the RMVT, where both displacements and transverse out-of-plane stresses are considered as primary variables. RMVT has been widely used within CUF for the study of straight fibres composite structures. For example, Carrera and Demasi [31,32] developed RMVT-based CUF models to perform the static analysis of straight fibres plates.
Within the free vibration analysis context, CUF has been applied to the study of VATs considering as variational statement mainly the PVD. For this reason, this work aims to extend this framework with the RMVT formulation in order to develop a family of hierarchical plate finite elements. This will allow to better predict the natural frequencies of composite plates characterized by curvilinear fibres. Section 2 shows the theoretical derivation for free vibration problems. Section 3 presents the numerical results where three cases are investigated. Analyses are performed considering for varying side-to-thickness ratio in order to investigate thin and thick plates and discuss the differences between models based on PVD or RMVT statements. Results are compared towards reference solutions for validation. Concluding observations and remarks are presented in Section 4.

2. Carrera’s Unified Formulation

A plate is flat body whose material points lay in the Cartesian closed point subset
P = Ω × H
of the three-dimensional space R 3 where Ω = x , y : x a , y b ∈ 0 , 1 ⊂ R 2 , H = z : 2 z h ∈ − 1 , 1 , a and b are the dimensions along the two in-plane axes and h measure its thickness along the z-axis and where z ≪ a and b. The global reference system and plate geometry are presented in Figure 1. The displacement field is expressed as:
u = u x u y u z
The strain vector can be divided in two parts, which represent the in-plane and out-of-plane components:
ϵ p = ϵ x x ϵ y y ϵ x y , ϵ n = γ x z γ y z γ z z
The hypothesis of small displacements allows to use a linear strains-displacements relation:
ϵ p = D p u ϵ n = D n Ω + D n z u
where D p , D n Ω and D n z are the following differential operators:
D p = ∂ ∂ x 0 0 0 ∂ ∂ y 0 ∂ ∂ y ∂ ∂ x 0 , D n Ω = 0 0 ∂ ∂ x 0 0 ∂ ∂ y 0 0 0 , D n z = ∂ ∂ z 0 0 0 ∂ ∂ z 0 0 0 ∂ ∂ z
The stress vector is expressed in a similar manner:
σ p = σ x x σ y y σ x y , σ n = σ x z σ y z σ z z
Hooke’s law reads:
σ p = C ˜ p p ϵ p + C ˜ p n ϵ n σ n = C ˜ n p ϵ p + C ˜ n n ϵ n
where the terms C ˜ p p , C ˜ p n , C ˜ n p and C ˜ n n are subcomponents of a material stiffness C ˜ according to the stress and strain ordering in Equations (3) and (6) where the fibres lay in Ω and they are not, in general, aligned with the x-axis. C stands for the stiffness matrix in the global reference system and its components can be written in terms of the young moduli E L and E T , shear moduli G L T and G T T and Poisson’s ratios ν L T and ν T T where subscripts L and T stand for the direction parallel and perpendicular to the fibres, respectively. For further details see Reddy [33].

2.1. Variable Stiffness Composite Plates

Laminated VAT structures are considered in this work. For this reason, the material stiffness coefficients can change layer-wise along the thickness and point-wise along the in-plane directions. The mapping of C into C ˜ reads:
C ˜ = T C T T
Superscript T stands for the transpose operator. The matrix T represents a rotation matrix that depends on an in-plane rotation angle θ . For the sake of brevity, the components of C ˜ and T are not reported here, they can be found in Reddy [33]. in a laminated VAT, the rotation angle θ is a bi-dimensional field in Ω . In this work, two different variation laws are considered for θ , a linear variation law and a parabolic one. The linear law can be expressed according to the following formula:
θ α = Φ + T 0 + T 1 − T 0 d | α |
The angle Φ describes the original direction along which θ varies, α is a generic spatial variable defined as
α = x ′ cos Φ + y ′ sin Φ
where x ′ and y ′ denote a generic in-plane reference system where θ is measured. T 0 and T 1 are the angles between the α -axis and the tangent to a fiber for α equal to zero and d, respectively, see Figure 2. As shown in the figure, the fibres angle is always measured with respect to the x ′ -axis. Further details about the fibres linear variation law can be found in Gürdal et al. [34]. The parabolic law can be expressed according to the following equation:
θ α = Φ + T 0 + tan − 1 γ α d
The parameter γ is used to control the shape of the parabola and it is related to the final fibres angle T 1 as T 1 = tan − 1 ± γ . More details about the parabolic fibres path can be found in Hachemi et al. [5] and Honda et al. [35]. The following notation, based upon the above introduced parameters, is used in order to describe the in-plane linear and parabolic fibres behavior: Φ < T 0 , T 1 > .

2.2. Variational Statements

PVD and RMVT variational statements are considered to derive the governing equations for the free vibration problem for a laminated VAT plate. The fundamental distinction is that the RMVT considers the vector of the out-of-plane stresses σ n as a primary unknown, whereas the PVD considers only displacements as primary variables. For the PVD case, the following variational statement applies:
∫ Ω ∫ H δ ϵ p G T σ p H + δ ϵ n G T σ n H d z d Ω + δ L i n = 0
where the subscript G refers to the components obtained from the geometrical relations in Equation (4), subscript H refers to the components obtained from Hooke’s law in Equation (7). Ω is the in-plane middle surface of the plate and L i n is the inertial work. δ stands for a virtual variation. For the RMVT case, the variational statement is:
∫ Ω ∫ H δ ϵ p G T σ p H + δ ϵ n G T σ n M + δ σ n M T ϵ n G − ϵ n H d z d Ω + δ L i n = 0
The M subscript refers to the transverse stress components considered as primary unknowns in the mixed formulation. For the RMVT formulation, Hooke’s law is rewritten as follows:
σ p H = C ^ p p ϵ p G + C ^ p n σ n M ϵ n H = C ^ n p ϵ p G + C ^ n n σ n M
where C ^ p p , C ^ p n , C ^ n p and C ^ n n are, see Carrera and Demasi [31]:
C ^ p p = C ˜ p p − C ˜ p n C ˜ n n − 1 C ˜ n p C ^ p n = C ˜ p n C ˜ n n − 1 C ^ n p = − C ˜ n n − 1 C ˜ n p C ^ n n = C ˜ n n − 1
The superscript “ − 1 " indicates the inverse of a matrix. The inertial work can be expressed as:
δ L i n = ∫ Ω ∫ H δ u T ρ u ¨ d Ω d z
where ρ is the plate material density and u ¨ represents the acceleration vector.

2.3. Kinematic Assumptions

CUF uses an axiomatic approach along the through-the-thickness direction to represent the primary unknowns, see Carrera [18]. The generic unknown component f = f x , y , z is approximated as:
f x , y , z = F τ z g τ x , y τ = 0 , 1 , ⋯ , N
where f is a displacement component in a formulation derived from the PVD, but it can also be an out-of-plane stress component when a RMVT formulation is considered. F τ is an approximation function in H and g τ is an unknown two-dimensional function in Ω . According to Einstein’s notation, a twice repeated index implies a sum over the index range. Finally, N is the approximation order. Both N and F τ are a-priori defined . This feature of CUF allows to obtain multiple theories in the same formulation. Within CUF, ESL or LW models can be also obtain depending of the support of F τ : in a ESL model F τ : H ↦ R , whereas for a LW model F τ : H k ↦ R where H k = z k : 2 z k h k ∈ − 1 , 1 such that H = ⋃ k = 1 N l H k and H k ∩ H k ′ = ∅ for k ≠ k ′ with k, k ′ = 1 , 2 , ⋯ , N l being N l the total number of laminae and h k the thickness of a generic k lamina such that k = ∑ k = 1 N l h k . The number of unknowns in the ESL case is independent of number of layers in the lamination since the approximation is imposed globally over H . The total stiffness contributes can be seen as a weighted average of each layer stiffness along the thickness. Taylor’s polynomials are considered for the ESL models:
F τ z = z τ τ = 0 , 1 , ⋯ , N
where N is the expansion order. The computational cost of ESL models depends on N only and, for a given N, it is lower than a LW model since this latter depends on the total number of layers in the lamination. ESL are suitable for relatively thick laminates. However, they are unable to accurately predict the behavior of thick plates with a high degree of anisotropy. ESL model have C ∞ -continuity over H because of the used approximation functions, whereas laminated composites presents a C 0 -continuity since the interface between to layers of different materials introduces a change in the slope of the displacements (also known as zig-zag displacements through-the-thickness variation). This behavior can be accommodated within an ESL theory by means of Murakami’s function. This approach is not here considered, for more details refer to Carrera [36]. In a LW model, kinematics of each layer is formulated independently:
f k x , y , z = F b z g b k x , y + F r z g r k x , y + F t z g t k x , y r = 2 , ⋯ , N
where subscripts b and t stand for layer bottom and top, respectively. Congruency at the interface is retrieved via a through-the-thickness assembly procedure similar to that used in the finite element method. For this reason, Lagrange polynomials (which ensure partition of unity) or the following linear combination of Legendre polynomials:
F t z ( ξ k ) = P 0 + P 1 2 , F b z ( ξ k ) = P 0 − P 1 2 , F r z ( ξ k ) = P r − P r − 2 , r = 2 , ⋯ , N
are, typically, used as approximation functions over H k . In Equation (20), ξ k = 2 z k h k ∈ − 1 , 1 and P i = P i ξ k is an i-order Legendre polynomial. Equation (20) create a base where F t and F b are the two linear Lagrange polynomials and F r are a kind of p-version enriching functions since they do not contribute to a base linear combination for ξ k = ± 1 being, by definition, F r ± 1 = 0 . Since LW base functions have local support, inter-layer C 0 -continuity for layers made of different materials is ensured but the computational costs are higher than for ESL models.

2.4. Acronym System

An acronym system is used in order to identify all the derived theories. Figure 3 shows this system. The first letter addresses the approximation level that is applied: ‘E’ denotes ESL models, whereas ‘L’ denotes LW models. The second letter denotes the variational statement: PVD or RMVT are denoted by ‘D’ or ‘M’, respectively. The last number is the order of expansion along the plate thickness. A number at the beginning of the acronym , when present, indicates how many virtual layers have been used to approximate each physical layer in a LW model to improve results for a given approximation order. If this number is not present, only one virtual layer has been used to represent each physical layer.
As an example, in EDN models, the displacement field can be expressed as:
u x = u x 0 + u x 1 z + u x 2 z 2 + ⋯ + u x N z N u y = u y 0 + u y 1 z + u y 2 z 2 + ⋯ + u y N z N u z = u z 0 + u z 1 z + u z 2 z 2 + ⋯ + u z N z N
In a vectorial form:
u = F 0 u 0 + F 1 u 1 + ⋯ + F N u N = F τ u τ τ = 0 , 1 , ⋯ , N
being F τ = z τ and u τ = u τ x , y . Additionally, classical theories can be been taken into account. Classical Lamination Theory (CLT) and First-order Shear Deformation Theory are obtained as a particular case of a first-order ESL theory. FSDT is obtained through the penalization of the u z 1 term, while for CLT also transverse shear stresses are disregarded by using a fictitiously high value of the material shear moduli. The material stiffness matrix is needs to be reduced in a plane-stress sense to overcome thickness locking.
For LDN solutions, only displacements are considered as primary variables:
u k = F 0 u 0 k + F 1 u 1 k + ⋯ + F N u N k = F τ u τ k τ = 0 , 1 , ⋯ , N ; k = 1 , 2 , ⋯ , N l
For LMN solutions, also transverse stresses are treated as primary variables. Indeed, the transverse stresses field can be expressed as:
σ n k = F 0 σ 0 k + F 1 σ 1 k + ⋯ + F N σ N k = F τ σ τ k τ = 0 , 1 , ⋯ , N ; k = 1 , 2 , ⋯ , N l
It can be observed that ESL theories can be considered as a particular case of LW ones. While in the first case the integration along the thickness is performed in order to represent composite’s properties through an unitary layer, for the second case the integration is computed layer by layer. This allows to represent the kinematic of each layer separately for LW models. LDN solutions are obtained with Lagrange polynomials with equally spaced nodes, whereas LMN ones are obtained with Legendre polynomials.

2.5. FE Stiffness Matrices

As far as a FEM solution is concerned, the in-plane domain is discretized into N e subdomains such as Ω = ⋃ e = 1 N e Ω e and and Ω e ∩ Ω e ′ = ∅ for e ≠ e ′ . Shape functions are then introduced for the approximation of the variation over Ω e . In the case of a bi-dimensional model, Equation (17) becomes:
f x , y , z = F τ z N i x , y g τ i τ = 0 , 1 , ⋯ , N , i = 1 , ⋯ , N n
where N i stands for the shape functions and N n is the number of nodes in the used finite element. Classical Lagrangian shape functions are used. They are not here presented for the sake of brevity but they can be found in Bathe [37]. FE stiffness matrices are obtained by the a weak form of the variational principles. In the PVD case, considering Equation (25) the displacements field can be written as:
u = F τ N i q x τ i q y τ i q z τ i = F τ N i q τ i
Through the substitution of Equations (4), (7) and (26) into Equation (12), the weak PVD form can be obtained:
∫ Ω e δ q τ i T [ D p T N i I Z ˜ p p τ s D p N j I + D p T N i I Z ˜ p n τ s D n Ω N j I + D p T N i I Z ˜ p n τ s , z N j I + + D n Ω T N i I Z ˜ n p τ s D p N j I + D n Ω T N i I Z ˜ n n τ s D n Ω N j I + D n Ω T N i I Z ˜ n n τ s , z N j I + + N i I Z ˜ n p τ , z s D p N j I + N i I Z ˜ n n τ , z s D n Ω N j I + N i I Z ˜ n n τ , z s , z N j I + N i I ρ E τ s N j I ] q s j d Ω = − ∫ Ω e δ q τ i T N i I ρ E τ s N j I q ¨ s j d Ω
where:
( Z ˜ w r τ s , Z ˜ w r τ , z s , Z ˜ w r τ s , z , Z ˜ w r τ , z s , z ) = ( C ˜ w r E τ s , C ˜ w r E τ , z s , C ˜ w r E τ s , z , C ˜ w r E τ , z s , z ) : w , r = p , n
( E τ s , E τ , z s , E τ s , z , E τ , z s , z ) = ∫ H ( F τ F s , F τ , z F s , F τ F s , z , F τ , z F s , z ) d z
An axis coordinate as comma preceded subscript stands for a derivative in that coordinate direction. In a compact vectorial form, Equation (27) reads:
δ q τ i T K τ s i j q s j = − δ q τ i T M τ s i j q ¨ s j
where K τ s i j and M τ s i j ∈ R 3 × 3 are Fundamental Nuclei (FN) of the stiffness and mass matrices, respectively. Through the cycles on the indices τ , s , i , j , it is possible to build the stiffness and mass matrices of a finite element. The components of the stiffness FN for the PVD case can be written as:
K x x τ s i j = ∫ Ω e Z ˜ p p 11 τ s N j , x N i , x + Z ˜ p p 16 τ s N j , y N i , x + Z ˜ p p 16 τ s N j , x N i , y + Z ˜ p p 66 τ s N j , y N i , y + Z ˜ n n 44 τ , z s , z N j N i d Ω K x y τ s i j = ∫ Ω e Z ˜ p p 12 τ s N j , y N i , x + Z ˜ p p 16 τ s N j , x N i , x + Z ˜ p p 26 τ s N j , y N i , y + Z ˜ p p 66 τ s N j , x N i , y + Z ˜ n n 45 τ , z s , z N j N i d Ω K x z τ s i j = ∫ Ω e Z ˜ p n 13 τ s , z N j N i , x + Z ˜ p p 36 τ s , z N j N i , y + Z ˜ n n 44 τ , z s N j , x N i + Z ˜ p p 45 τ , z s N j , y N i d Ω K y x τ s i j = ∫ Ω e Z ˜ p p 12 τ s N j , x N i , y + Z ˜ p p 36 τ s N j , y N i , y + Z ˜ p p 16 τ s N j , x N i , x + Z ˜ p p 66 τ s N j , y N i , x + Z ˜ n n 45 τ , z s , z N j N i d Ω K y y τ s i j = ∫ Ω e Z ˜ p p 22 τ s N j , y N i , y + Z ˜ p p 26 τ s N j , x N i , y + Z ˜ p p 26 τ s N j , y N i , x + Z ˜ p p 66 τ s N j , x N i , x + Z ˜ n n 55 τ , z s , z N j N i d Ω K y z τ s i j = ∫ Ω e Z ˜ p n 23 τ s , z N j N i , y + Z ˜ p n 36 τ s , z N j N i , x + Z ˜ n n 45 τ , z s N j , x N i + Z ˜ n n 55 τ , z s N j , y N i d Ω K z x τ s i j = ∫ Ω e Z ˜ n n 44 τ s , z N j N i , x + Z ˜ n n 45 τ s , z N j N i , y + Z ˜ n p 13 τ , z s N j , x N i + Z ˜ n p 36 τ , z s N j , y N i + Z ˜ n n 45 τ , z s , z N j N i d Ω K z y τ s i j = ∫ Ω e Z ˜ n n 45 τ s , z N j N i , x + Z ˜ n n 55 τ s , z N j N i , y + Z ˜ n p 23 τ , z s N j , y N i + Z ˜ p p 36 τ , z s N j , x N i d Ω K z z τ s i j = ∫ Ω e Z ˜ n n 44 τ s N j , x N i , x + Z ˜ n n 45 τ s N j , y N i , x + Z ˜ n n 45 τ s N j , x N i , y + Z ˜ n n 55 τ s N j , y N i , y + Z ˜ n n 33 τ , z s , z N j N i d Ω
The mass FN can be written as:
M τ s i j = ∫ Ω e N i I ρ E τ s N j I d Ω
It is possible to observe that M τ s i j is a diagonal matrix and that, since the plate density is assumed to be constant, the term ρ E τ s can be placed outside the integral.
In the RMVT case, also transverse stresses are a priori approximated:
σ n = F τ N i g x z τ i g y z τ i g z z τ i = F τ N i g τ i
Through the substitution of Equations (4), (14), (26) and (33) into Equation (13), the RMVT weak form can be obtained:
∫ Ω e δ q τ i T [ D p T N i I Z ^ p p τ s D p N j I ] q s j + δ q τ i T [ D p T N i I Z ^ p n τ s N j I + D n Ω T N i I E τ s I N j I + + N i I E τ , z s I N j I ] g s j + δ g τ i T [ N i I E τ s I D n Ω N j I + N i I E τ s , z I N j I + − N i I Z ^ n p τ s D p N j I ] q s j − δ g τ i T N i I Z ^ n n τ s N j I g s j d Ω = − ∫ Ω e δ q τ i T N i I ρ E τ s N j I q ¨ s j d Ω
where:
( Z ^ w r τ s , Z ^ w r τ , z s , Z ^ w r τ s , z , Z ^ w r τ , z s , z ) = ( C ^ w r E τ s , C ^ w r E τ , z s , C ^ w r E τ s , z , C ^ w r E τ , z s , z ) : w , r = p , n
In a compact form:
δ q τ i T K u u τ s i j q s j + δ q τ i T K u σ τ s i j g s j = − δ q τ i T M τ s i j q ¨ s j δ g τ i T K σ u τ s i j q s j + δ g τ i T K σ σ τ s i j g s j = 0
In this case four fundamental nuclei are obtained. The components of the FN for the RMVT case can be written as:
K u u x x τ s i j = ∫ Ω e Z ^ p p 11 τ s N j , x N i , x + Z ^ p p 31 τ s N j , x N i , y + Z ^ p p 13 τ s N j , y N i , x + Z ^ p p 33 τ s N j , y N i , y d Ω K u u x y τ s i j = ∫ Ω e Z ^ p p 12 τ s N j , y N i , x + Z ^ p p 32 τ s N j , y N i , y + Z ^ p p 13 τ s N j , x N i , x + Z ^ p p 33 τ s N j , x N i , y d Ω K u y x τ s i j = ∫ Ω e Z ^ p p 21 τ s N j , x N i , y + Z ^ p p 31 τ s N j , x N i , x + Z ^ p p 23 τ s N j , y N i , y + Z ^ p p 33 τ s N j , y N i , x d Ω K u u y y τ s i j = ∫ Ω e Z ^ p p 22 τ s N j , y N i , y + Z ^ p p 32 τ s N j , y N i , x + Z ^ p p 23 τ s N j , x N i , y + Z ^ p p 33 τ s N j , x N i , x d Ω K u u x z τ s i j = 0 K u u y z τ s i j = 0 K u u z x τ s i j = 0 K u u z y τ s i j = 0 K u u z z τ s i j = 0 K u σ x x τ s i j = ∫ Ω e E τ , z s N j N i d Ω K u σ x z τ s i j = ∫ Ω e Z ^ p n 13 τ s N j N i , x + Z ^ p n 33 τ s N j N i , y d Ω K u σ y y τ s i j = ∫ Ω e E τ , z s N j N i d Ω K u σ y z τ s i j = ∫ Ω e Z ^ p n 23 τ s N j N i , y + Z ^ p n 33 τ s N j N i , x d Ω K u σ z x τ s i j = ∫ Ω e E τ s N j N i , x d Ω K u σ z y τ s i j = ∫ Ω e E τ s N j N i , y d Ω K u σ z z τ s i j = ∫ Ω e E τ , z s N j N i d Ω K u σ x y τ s i j = 0 K u σ y x τ s i j = 0 K σ u x x τ s i j = ∫ Ω e E τ s , z N j N i d Ω K σ u x z τ s i j = ∫ Ω e E τ s N j , x N i d Ω K σ u y y τ s i j = ∫ Ω e E τ s , z N j N i d Ω K σ u y z τ s i j = ∫ Ω e E τ s N j , y N i d Ω K σ u z x τ s i j = − ∫ Ω e Z ^ n p 31 τ s N j , x N i − Z ^ n p 33 τ s N j , y N i d Ω K σ u z y τ s i j = − ∫ Ω e Z ^ n p 32 τ s N j , y N i − Z ^ n p 33 τ s N j , x N i d Ω K σ u z z τ s i j = ∫ Ω e E τ s , z N j N i d Ω K σ u x y τ s i j = 0 K σ u y x τ s i j = 0 K σ σ x x τ s i j = − ∫ Ω e Z ^ n n 11 τ s N j N i d Ω K σ σ x y τ s i j = − ∫ Ω e Z ^ n n 12 τ s N j N i d Ω K σ σ y x τ s i j = − ∫ Ω e Z ^ n n 21 τ s N j N i d Ω K σ σ x x τ s i j = − ∫ Ω e Z ^ n n 22 τ s N j N i d Ω K σ σ x z τ s i j = 0 K σ σ y z τ s i j = 0 K σ σ z x τ s i j = 0 K σ σ z y τ s i j = 0 K σ σ z z τ s i j = 0
The mass FN is the same way of the PVD case, see Equation (32). Since the in-plane integrals are calculated via Gauss quadrature, it is crucial to consider an appropriate number of Gauss points, in accordance with the variational rule of fibers angle.

3. Numerical Results and Discussion

Three cases are analyzed in this work: a cantilever monolayer plate, a clamped multilayer plate and a clamped multilayer plate with a central circular cut out. For each case, a square geometry is considered ( a = b = 1 m). Parametric studies are performed considering different side-to-thickness ratios ( a / h = 100 , 10 , 5 ). Material properties are represented in Table 1 for all the considered analysis case. Reference solutions are represented by an Abaqus 3D model where quadratic elements (C3D20R) have been used. For CUF solutions, nine-node square elements are used. For each case study, a preliminary convergence analysis is carried out to identify the appropriate mesh for both CUF and Abaqus solutions.

3.1. Monolayer plate

The first case corresponds to a cantilever monolayer plate with density ρ = 1540 kg/m 3 . For this problem axes x ′ and y ′ of angle reference system are coincident with axes x and y of the plate. For this reason the length parameter in Equation (9) corresponds to d = b . It is assumed that fibres angle is a linear function of y ′ , see Equation (9). Angle variational law in this case can be expressed as 90 < 0 , 90 > and it is represented in Figure 4. This law has been taken from Viglietti et al. [23]. Reference solution contains 80 elements along each in-plane side and 12 elements along the thickness. The only clamped side of the plate is the one that lays on the x z plane, in correspondence of y ′ = 0 , where fibres form an angle of θ ( 0 ) = 90 ∘ with x axis. For CUF results a 10 × 10 mesh is considered. Table 2 shows the Degrees Of Freedom (DOF) for some considered solutions. It is possible to observe that higher-order CUF models allow a DOF reduction of one order of magnitude in comparison with the Abaqus 3D reference solution. Table 3 shows the first five natural frequencies for a / h = 100 . For this case, classic and higher-order theories show all a very good approximation of the reference solution the maximum difference from the reference solution being 0.4 % for the second natural frequency computed via CLT. Table 4 shows the first five natural frequencies for a / h = 10 . It is possible to observe that classical theories and lower-order ESL ones are now less accurate, specially for the prediction of higher frequencies. For example CLT, FSDT and ED2 models, in correspondence of the third natural frequency, present a percentage error equal to 8.1 % , 1.0 % and 1.1 % , respectively. This can be explained by considering that the side-to-thickness ratio a / h = 10 corresponds to a thick plate. In this case, higher-order theories are needed to obtain an accurate approximation. Since a moderately thick plate is considered, transverse shear stresses affect the solution. This is the reason why CLT, which neglects those stresses, is less close to the reference solution. The best approximations of plate natural frequencies are given by 2LM2 and 3LM4 mixed theories, which show a maximum percentage errors of 0.1 % each for the fourth natural frequency. In particular, it is possible to observe that the 2LM2 solution is globally closer to Abaqus in comparison with the 3LD4 solution, even if the last one is characterized by an higher number of degrees of freedom. Table 5 shows the first five natural frequencies for a / h = 5 . Because of the low side-to-thickness ratio, a very thick plate is considered and lower-order theories do not provide a correct prediction of the natural frequencies. In this case, the inversion of the sixth mode with the fifth one is observed for the CLT model and a corresponding percentage error as high as 27.1 % . On the other hand, a 3LM4 model matches Abaqus reference results.

3.2. Multilayer plate

The second case is taken from Viglietti et al. [23] and corresponds to a multilayer clamped plate with density ρ = 1540 kg/m 3 . The plate is composed by three layers with the same thickness. It is assumed that fibres angle is a function of y ′ . Also in this case, a linear law is considered for fibres path, according to Equation (9). For this problem axes x ′ and y ′ of angle reference system are aligned with axes x and y of the plate, but their origin is placed on the center of the plate ( a / 2 , b / 2 ). In this case b / 2 has been considered as length parameter in Equation (9). The lamination of the plate is 90 < 0 , 45 > for layer 1, 90 < − 45 , − 60 > for layer 2 and 90 < 0 , 45 > for layer 3. The stacking sequence is represented in Figure 5. As for the previous case, the Abaqus reference solution contains 80 elements along each side and 12 elements along the thickness. For CUF results a 10 × 10 mesh is considered. Table 6 shows the first five natural frequencies for a / h = 100 , together with the results presented in Viglietti et al. [23]. In this case the best approximation is given by LM2 and LM4 theories. LM2 and LM4 models have both a maximum percentage error as high as 0.4 % in correspondence of the third frequency. Also classical and low-order theories provide good results since a thin plate is considered. For this reason, transverse stresses do not play an important role. For example the maximum error given by CLT is 2.1 % for the fifth frequency. The case a / h = 10 is presented in Table 7. Here the CLT model shows the inversion of the third and fourth modes. In comparison with the monolayer plate, in this case the modes inversion of the CLT model can be seen for higher side-to-thickness ratios and lower frequencies. For the third-mode, CLT shows a percentage error of 96.0 % , while the best approximation is given by LM4 which has a percentage error of 0.17 % for the same mode. Table 8 shows the first five frequencies for a / h = 5 . In this case lower-order theories have an evident loss of accuracy. The CLT model can predict only the first two modes. Also, FSDT and ED2 models show non-negligible errors, which become bigger with the increase of the frequency. On the other hand, mixed models are able to correctly predict the dynamic behavior of the plate for both low and high frequencies.

3.3. Multilayer plate with central hole

Case 3 is taken from Hachemi et al. [5] and corresponds to a multilayer clamped plate that presents a circular cut-out. The center of the cut-out is placed at plate center ( a / 2 , b / 2 ) and its radius is r = 0.2 m. It is assumed that fibres angle is a parabolic function of x ′ . Like the previous case, x ′ and y ′ axes are parallel respectively to x and y, and their origin is placed at the center of the plate. The angle variational law is defined in Equation (11), considering d = a / 2 . The plate is composed by two layers which have the same thickness and stacking sequence 0 < 0 , ± 30 > , see Figure 6. In this case, the Abaqus reference solution is made of 73728 elements: 4608 elements are defined on the plate plane and 16 elements are defined along the thickness. For CUF results, 128 elements are used on the plate plane. Natural frequencies are expressed through a non-dimensional frequency parameter defined as Ω = ω a 2 ρ h / D 0 , where ω is the natural frequency while D 0 = E 2 h 3 / 12 1 − ν L T ν T L . Table 9 presents the first five non-dimensional frequencies for a / h = 100 . It is possible to observe that the theories show all a good approximation of reference results. Also the percentage errors of FSDT and CLT are less than 2 % . Mixed theories match Abaqus results. Table 10 shows the results for a / h = 10 in order to compare the Abaqus reference solution with the one presented in Hachemi et al. [5] and the solutions obtained with CUF. As already observed in previous cases, classical theories and, in general low-order ones, are not able to provide an accurate approximation of natural frequencies, because of the low side-to-thickness ratio value. It is also possible to notice that this generate the inversion of modes four and five for the CLT case. On the other hand, the best approximation is given by mixed theories, which are closer to Abaqus solution also for high frequencies. The shapes of the modes have been represented in Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11. The first mode shows a simple bending of the plate on the x y plane with a single half-wave along each in-plane direction. The second and the third modes show two half-waves in the y and x directions, respectively. Mode number four shows three half-waves along the plate diagonal between x and y axes. The fifth mode shows three half-waves along the y direction. Finally, Table 11 shows the frequencies for a / h = 5 . Since a thick plate is considered, the effect of transverse stresses is not negligible, which causes classical and lower-order theories to be inaccurate. This can be observed for CLT, which is not able to predict the fourth and fifth modes and has an error as hugh as 69.4 % for the third mode. Considering FSDT, ED4 and LD4 models, this error can be reduced to 4.4 % , 0.5 % and 0.1 % , respectively.

4. Conclusions

In this paper, a new framework for the dynamic analysis of VAT structures is presented. RMVT is developed within CUF in order to obtain a new family of 2D models for the free-vibration analysis of VAT plates. The results are obtained via either RMVT or PVD and are compared in order to show the effective capabilities of the proposed method in the prediction of VAT plates natural frequencies. Abaqus 3D reference solutions and results from Refs. [5,23] are also included in order to present a validation as wide as possible. Linear and parabolic laws are both considered in order to describe the in-plane path of fibres variation. The possibility to use a polynomial order defined a-priori through CUF, and the introduction of the transverse stresses field as primary variable of the problem through RMVT allow to obtain a valid approach for the prediction of VATs dynamic behavior. After the results analysis, the following remarks can be done:
  • Classical theories (FSDT and CLT) provide the best trade off between accuracy and computational costs for thin plates ( a / h = 100 ), whereas they are not able to correctly predict the behavior of thicker plates ( a / h = 10 and 5), specially at high frequencies. The loss of accuracy is more evident for CLT results, since this theory does not consider transverse shear stresses, which become important in thick plates. This error is particularly evident in the second and third, where the inversion of modes can be observed.
  • PVD results show monotonic convergence to the reference solution: the lower the DOF number, the higher the frequency value. For a given mode, frequencies values decrease when higher-order models are employed, and they get closer to reference solution.
  • In all the cases, layer-wise mixed theories yield the best match of the reference 3D solution, independently from the plate geometry or fibres variational law. In every case, a second-order model is more accurate and less computationally expensive than a fourth-order layer-wise PVD model. This is justified by the fact that RMVT considers both displacements and transverse stresses as primary variables, assuring a better approximation of the transverse stresses field into the problem domain improving the overall solution accuracy.
In conclusion, the application of RMVT within CUF has demonstrated significant potential for improving the accuracy and efficiency of modeling VAT plates for free-vibration analyses. The promising results suggest, as future perspectives, the extension to buckling and failure analyses where an accurate and efficient modeling of VAT structures under various loading and operational conditions is required.

Acknowledgments

This research was funded in part by the Luxembourg National Research Fund (FNR), grant reference INTER/ANR/21/16215936 GLAMOUR-VSC. For the purpose of open access, and in fulfillment of the obligations arising from the grant agreement, the author has applied a Creative Commons Attribution 4.0 International (CC BY 4.0) license to any Author Accepted Manuscript version arising from this submission. M. Montemurro is grateful to French National Research Agency for supporting this work through the research project GLAMOUR-VSC (Global–LocAl two-level Multi-scale optimisation strategy accOUnting for pRocess-induced singularities to design Variable Stiffness Composites) ANR-21-CE10-0014.

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Figure 1. Plate geometry and reference system.
Figure 1. Plate geometry and reference system.
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Figure 2. Example of in-plane fiber orientation.
Figure 2. Example of in-plane fiber orientation.
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Figure 3. Acronym system.
Figure 3. Acronym system.
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Figure 4. Stacking sequence, case 1.
Figure 4. Stacking sequence, case 1.
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Figure 5. Stacking sequence, case 2.
Figure 5. Stacking sequence, case 2.
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Figure 6. Stacking sequence, case 3.
Figure 6. Stacking sequence, case 3.
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Figure 7. Mode 1, case 3.
Figure 7. Mode 1, case 3.
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Figure 8. Mode 2, case 3.
Figure 8. Mode 2, case 3.
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Figure 9. Mode 3, case 3.
Figure 9. Mode 3, case 3.
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Figure 10. Mode 4, case 3.
Figure 10. Mode 4, case 3.
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Figure 11. Mode 5, case 3.
Figure 11. Mode 5, case 3.
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Table 1. Material properties for case 1.
Table 1. Material properties for case 1.
Case E L [GPa] E T [GPa] G L T = G T T [GPa] ν L T = ν T T
1 50.0 10.0 5.0 0.25
2 173.0 7.2 3.8 0.29
3 138.0 9.0 7.1 0.30
Table 2. Degrees of freedom, case 1.
Table 2. Degrees of freedom, case 1.
Model DOF
Abaqus 3D 997’515
3LM4 34’398
2LM2 13’230
3LD4 17’199
2LD2 6’615
ED4 6’615
ED2 3’969
FSDT 2’646
CLT 2’646
Table 3. Natural frequencies [Hz], a / h = 100 , case 1.
Table 3. Natural frequencies [Hz], a / h = 100 , case 1.
Mode
1 2 3 4 5
Abaqus 3D 7.397 16.354 37.158 48.025 63.349
3LM4 7.399 16.334 37.164 47.988 63.310
2LM2 7.398 16.333 37.162 47.986 63.309
3LD4 7.400 16.362 37.179 48.053 63.378
2LD2 7.400 16.362 37.179 48.054 63.379
ED4 7.400 16.362 37.179 48.053 63.378
ED2 7.401 16.368 37.186 48.069 63.399
FSDT 7.398 16.363 37.171 48.054 63.388
CLT 7.403 16.414 37.213 48.175 63.537
Table 4. Natural frequencies [Hz], a / h = 10 , case 1.
Table 4. Natural frequencies [Hz], a / h = 10 , case 1.
Mode
1 2 3 4 5
Abaqus 3D 72.229 151.762 338.517 389.336 431.011
3LM4 72.244 151.751 338.577 389.554 431.004
2LM2 72.233 151.705 338.432 389.546 430.824
3LD4 72.250 151.796 338.625 389.587 431.151
2LD2 72.269 151.906 338.939 389.589 431.577
ED4 72.253 151.810 338.669 389.588 431.207
ED2 72.466 153.069 342.179 389.592 435.990
FSDT 72.437 153.021 342.036 389.510 435.853
CLT 73.825 163.064 365.813 389.510 472.565
Table 5. Natural frequencies [Hz], a / h = 5 , case 1.
Table 5. Natural frequencies [Hz], a / h = 5 , case 1.
Mode
1 2 3 4 5
Abaqus 3D 136.723 264.080 389.391 556.394 704.284
3LM4 136.742 264.077 389.557 556.404 704.295
2LM2 136.667 263.747 389.550 555.332 703.121
3LD4 136.755 264.119 389.638 556.511 704.442
2LD2 136.875 264.684 389.643 558.145 706.381
ED4 136.774 264.224 389.640 556.855 704.832
ED2 138.015 269.553 389.651 570.563 721.354
FSDT 137.947 269.463 389.510 570.329 721.159
CLT 146.479 319.929 389.510 696.687 895.089
Table 6. Natural frequencies [Hz], a / h = 100 , case 2.
Table 6. Natural frequencies [Hz], a / h = 100 , case 2.
Mode
1 2 3 4 5
Abaqus 3D 92.18 130.68 194.96 237.56 274.60
Ref. [23] 92.90 132.28 198.97 240.46 278.75
LM4 92.35 131.01 195.77 238.25 275.60
LM2 92.34 130.99 195.74 238.23 275.58
LD4 92.36 131.03 195.81 238.30 275.67
LD2 92.36 131.04 195.84 238.31 275.69
ED4 92.37 131.06 195.88 238.32 275.72
ED2 92.49 131.23 196.16 238.97 276.48
FSDT 92.38 131.01 195.75 238.74 276.20
CLT 93.04 131.85 197.00 242.48 280.40
Table 7. Natural frequencies [Hz], a / h = 10 , case 2.
Table 7. Natural frequencies [Hz], a / h = 10 , case 2.
Mode
1 2 3 4 5
Abaqus 3D 606.67 896.70 1208.24 1313.26 1458.25
Ref. [23] 609.79 903.63 1216.00 1328.41 1469.33
LM4 606.90 897.26 1208.80 1314.85 1459.23
LM2 606.33 896.52 1206.86 1313.56 1457.30
LD4 607.22 897.73 1209.64 1315.80 1460.16
LD2 608.65 901.20 1213.06 1322.93 1465.20
ED4 609.84 905.18 1214.60 1331.82 1469.17
ED2 633.68 941.96 1272.39 1396.16 1540.10
FSDT 632.82 940.46 1271.42 1393.96 1538.74
CLT 921.28 1287.71 2368.22 1885.61 2699.22
Table 8. Natural frequencies [Hz], a / h = 5 , case 2.
Table 8. Natural frequencies [Hz], a / h = 5 , case 2.
Mode
1 2 3 4 5
Abaqus 3D 794.730 1201.916 1439.956 1701.328 1810.250
LM4 794.760 1202.101 1440.092 1701.788 1811.113
LM2 792.734 1199.331 1433.897 1696.266 1805.942
LD4 795.213 1202.777 1441.080 1702.986 1812.317
LD2 799.063 1209.706 1448.714 1713.982 1820.716
ED4 802.019 1216.744 1450.930 1723.900 1825.405
ED2 845.154 1294.481 1523.246 1847.193 1930.364
FSDT 844.048 1292.846 1522.478 1845.945 1928.631
CLT 1790.121 2411.198 - - -
Table 9. Non-dimensional frequencies Ω , a / h = 100 , case 3.
Table 9. Non-dimensional frequencies Ω , a / h = 100 , case 3.
Mode
1 2 3 4 5
Abaqus 3D 87.079 106.407 147.559 184.034 197.096
LM4 87.281 106.622 147.070 184.554 197.522
LM2 87.259 106.593 147.045 184.500 197.489
LD4 87.327 106.704 147.911 184.789 197.969
LD2 87.336 106.719 147.952 184.821 198.022
ED4 87.331 106.708 147.921 184.798 197.984
ED2 87.364 106.768 148.169 184.931 198.228
FSDT 87.184 106.538 148.047 184.525 198.029
CLT 87.387 106.942 150.080 185.420 199.725
Table 10. Non-dimensional frequencies Ω , a / h = 10 , case 3.
Table 10. Non-dimensional frequencies Ω , a / h = 10 , case 3.
Mode
1 2 3 4 5
Abaqus 3LD4 72.645 86.745 104.279 136.366 140.278
Ref. [5] 72.432 86.626 103.910 135.828 139.747
LM4 72.699 86.830 104.307 136.467 140.408
LM2 72.573 86.700 104.051 136.137 140.143
LD4 72.744 86.888 104.376 136.558 140.516
LD2 73.107 87.263 105.144 137.567 141.231
ED4 72.868 86.990 104.630 136.851 140.725
ED2 73.977 88.609 107.143 140.522 143.556
FSDT 74.075 88.782 107.645 141.221 143.885
CLT 84.751 104.166 143.133 190.321 174.656
Table 11. Non-dimensional frequencies Ω , a / h = 5 , case 3.
Table 11. Non-dimensional frequencies Ω , a / h = 5 , case 3.
Mode
1 2 3 4 5
Abaqus 3D 54.333 64.456 70.572 90.875 98.086
LM4 54.326 64.456 70.541 90.866 98.098
LM2 54.038 64.201 70.036 90.292 97.612
LD4 54.388 64.514 70.619 90.956 98.191
LD2 54.875 64.963 71.421 91.868 98.955
ED4 54.554 64.623 70.913 91.224 98.408
ED2 56.062 66.756 73.181 94.442 101.535
FSDT 56.253 66.985 73.702 95.219 102.017
CLT 76.928 95.975 119.513 - -
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