Dynamical Analysis of Long Fiber-Reinforced Laminated Plates with Elastically Restrained Edges

Abstract

This paper presents a variational formulation for the free vibration analysis of unsymmetrically laminated composite plates with elastically restrained edges. The study includes a micromechanics approach that allows starting the study considering each layer as constituted by long unidirectional fibers in a continuous matrix. The Mori-Tanaka method is used to predict the mechanical properties of each lamina as a function of the elastic properties of the components and of the fiber volume fraction. The resulting mechanical properties for each lamina are included in a general Ritz formulation developed to analyze the free vibration response of thick laminated anisotropic plates resting on elastic supports. Comprehensive numerical examples are computed to validate the present method, and the effects of the different mechanical and geometrical parameters on the dynamical behavior of different laminated plates are shown. New results for general unsymmetrical laminates with elastically restrained edges are also presented. The analytical approximate solution obtained in this paper can also be useful as a basis to deal with optimization problems under, for instance, frequency constraints.

1. Introduction

Fiber-reinforced composite laminated plates are extensively used in many engineering applications. The free vibration analysis of these plates plays a very important role in the design of civil, aerospace, mechanical, and marine structures. In addition to the favorable high specific strength and high specific stiffness, fiber-reinforced composite laminates offer the possibility of optimal design through the variation of stacking pattern, angle of fiber orientation, fiber content, and so forth, known as composite tailoring. All these mechanical and geometrical characteristics, as well as the various coupling effects that take place, must be considered in the prediction of the laminates dynamical response to assure that this is reliable, accurate, and adequate to the design requirements.

It is well known that laminated composite plates have relatively low transverse shear stiffness, playing the shear deformation an important role in the global and local behavior of these structures. Among the numerous theories used for laminated plates that include the transverse shear strain, the first-order shear deformation theory (FSDT) [1, 2] is adequate for the computation of global responses (such as natural frequencies) and simultaneously has some advantages due to its simplicity and low computational cost. Many investigations have been reported for free vibration analysis of moderately thick composite laminates using the FSDT kinematics (see for instance [3–13]). However, the results are, in most cases, limited to certain lamination schemes and boundary conditions. As far as the study of thick plates with elastically restrained edges is concerned, most of the previous works are limited to isotropic ones ([14–19] among others). But, limited information is found for the case of thick anisotropic laminated plates resting on elastic supports. For instance, Setoodeh and Karami [20] implemented a layer-wise laminated plate theory linked with three-dimensional elasticity approach for vibration and buckling of symmetric and antisymmetric fiber-reinforced composite plates having elastically restraint edges support and results for cross-ply laminates are presented, whereas Karami et al. [21] applied the differential quadrature method for the free vibration analysis of moderately thick symmetric laminated plates with elastically restrained edges. For the same boundary conditions, semianalytical solutions for the free vibration of angle-ply symmetrically laminated plates were presented by Ashour [22]. Nallim and Grossi [23] also studied the vibration of symmetric laminated plates resting on elastic support employing the Ritz method and beam orthogonal polynomials as approximated functions. These kind of approximate functions (in one or two variables) have been used by many authors to the free vibration analysis of, both homogeneous and nonhomogeneous, plates (Chakraverty et al. [24–26] and Chow et al. [27], among others).

In this paper, a general Ritz formulation for the free vibration analysis of anisotropic laminated plates is developed. All kind of boundary conditions including elastically restrained edges are considered enhancing the study. This feature allows a more realistic analysis of some structural problems. The analysis includes a micromechanical approach (according to the classification of Altenbach et al. [28]), where the average mechanical properties of each anisotropic lamina are estimated from the known characteristics of the fibers and the matrix materials taking into account the fiber volume ratio and the fiber-packing arrangement. At structural level, the dynamic response of the unsymmetrical laminated plate, with elastically restrained edges, is analyzed using the first-order shear deformation theory and the Ritz method with beam orthogonal polynomials as coordinate functions. The approximate analytical solution developed here is very useful to understand, both qualitatively and quantitatively, the behavior of complex laminated plates.

2. Formulation

2.1. Effective Elastic Moduli of Long Fiber-Reinforced Laminae

The micromechanics-based Mori-Tanaka method [29] is used in this section to predict the elastic mechanical properties of the orthotropic unidirectional laminae. This method may be viewed as the simplest mean field approach for inhomogeneous materials that encompass the full physical range of phase volume fraction.

Eshelby’s results [33] show that if an elastic homogeneous ellipsoidal inclusion in an infinite linear elastic matrix is subjected to an eigenstrain 𝜺^𝑇, uniform strain states 𝜺^𝐶 is induced, and it is related to the eigenstrain by the expression 𝜺^𝐶=𝐒^𝐸∶𝜺^𝑇, (1) where 𝐒^𝐸 is the Eshelby tensor, which depends on the reinforcement dimensions and the Poisson ratio of the matrix 𝜈_𝑚. The components of this tensor for a circular, cylindrical inclusion with an infinite length-to-diameter ratio parallel to the 1-axis (parallel to the fiber direction, Figure 1) are 𝑆₁₁₁₁=𝑆₁₁₃₃=𝑆₁₁₂₂=0,𝑆₃₃₃₃=𝑆₂₂₂₂=5−4𝜈_𝑚81−𝜈_𝑚,𝑆₃₃₂₂=𝑆₂₂₃₃=4𝜈_𝑚−181−𝜈_𝑚,𝑆₃₃₁₁=𝑆₂₂₁₁=𝜈_𝑚21−𝜈_𝑚,𝑆₃₂₃₂=3−4𝜈_𝑚81−𝜈_𝑚,𝑆₁₃₁₃=𝑆₁₂₁₂=14. (2) The transformations strains are obtained considering the equivalent homogeneous inclusion for inhomogeneous inclusions developed by Eshelby [33] together with the interaction effects of Mori-Tanaka [29]. These transformations strains are used to equate the total stresses in the inhomogeneities and their equivalent inclusions, as described in the following equation: 𝐂_𝑓∶𝜺^𝑎+𝜺^int+𝜺^𝐶=𝐂_𝑚∶𝜺^𝑎+𝜺^int+𝜺^𝐶−𝜺^𝑇, (3) where 𝐂_𝑓 and 𝐂_𝑚 are the stiffness tensors of fiber and matrix, respectively, 𝜺^𝑎 is the uniform far field strain applied to the domain at infinity, and 𝜺^int is the average elastic strain defined by Mori-Tanaka which is given by 𝜺^int=−𝑘_𝑓𝜺^𝐶−𝜺^T, (4) where 𝑘_𝑓 is the fiber volume fraction.

Figure 1 

General laminated plate resting on elastic supports.

Finally, the stiffness tensor 𝐂 for different unidirectional laminae can be obtained from energy considerations [34] and (1) to (4) as 𝐂=𝐂_𝑚×𝐈−𝑘_𝑓𝐂_𝑓−𝐂_𝑚𝐒^𝐸−𝑘_𝑓𝐒^𝐸−𝐈+𝐂_𝑚⁻¹×𝐂_𝑓−𝐂_𝑚⁻¹, (5) where 𝐈 is the fourth order identity tensor.

Using this method the mechanical properties of unidirectional carbon/epoxy laminae are found considering various fiber volume fractions, and they are depicted in Table 1. These properties, for each unidirectional lamina, are then used in the next section to obtain the reduced constitutive matrix.

Table 1 

Mechanical properties of unidirectional laminae (AS4-3501-6), obtained using Mori-Tanaka method. Fiber and matrix properties 𝐸_𝑓1=225 GPa, 𝐸_𝑓2=15 GPa, 𝐺_𝑓12=15 GPa, 𝐺_𝑓23=7 GPa, 𝜈_𝑓12=0.20; 𝐸_𝑚=4.2 GPa, 𝜈_𝑚=0.34 ([30]).

2.2. General Laminated Plate Resting on Elastic Supports

Let us consider a rectangular fiber-reinforced composite laminated plate, of dimension 𝑎×𝑏 and total thickness ℎ (ℎ=∑ℎ_𝑘, ℎ_𝑘 represents the thickness of a layer). The laminated plate is composed of an arbitrary number of 𝑁_𝑐 orthotropic layers and fibre orientation 𝛽_𝑘(𝑘=1,2,…,𝑁_𝑐) as shown in Figure 1. A rectangular Cartesian coordinate system (𝑥,𝑦,𝑧) is used to represent the plate geometry and the strain–displacement relations. The 𝑥-𝑦 plane coincides with the middle plane of the plate. The displacement field of the first-order shear deformation theory is assumed to be of the form [1, 2] 𝑢𝑥,𝑦,𝑧,𝑡=𝑢₀𝑥,𝑦,𝑡+𝑧𝜙_𝑥,𝑣𝑥,𝑦,𝑡𝑥,𝑦,𝑧,𝑡=𝑣₀𝑥,𝑦,𝑡+𝑧𝜙_𝑦,𝑤𝑥,𝑦,𝑡𝑥,𝑦,𝑧,𝑡=𝑤₀𝑥,𝑦,𝑡, (6) where 𝑡 is the time dimension 𝑢₀, 𝑣₀ and 𝑤₀ denote the mid-surface translational displacements along the 𝑥, 𝑦, and 𝑧 axes, and 𝜙_𝑥, 𝜙_𝑦 are the rotations about 𝑦- and 𝑥-axes, respectively.

The displacement model (6) yields the following kinematic relations: {𝜀}=𝜀₀𝜀^∗₀𝜀+𝑧¹{0}, (7) where {𝜀}^𝑇=𝜀_𝑥,𝜀_𝑦,𝛾_𝑥𝑦,𝛾_𝑦𝑧,𝛾_𝑥𝑧,𝜀₀^𝑇=𝜕𝑢₀,𝜕𝑥𝜕𝑣₀,𝜕𝑦𝜕𝑢₀+𝜕𝑦𝜕𝑣₀,𝜀𝜕𝑥_∗0𝑇=𝜕𝑤₀𝜕𝑦+𝜙_𝑦,𝜕𝑤₀𝜕𝑥+𝜙_𝑥,𝜀₁^𝑇=𝜕𝜙_𝑥,𝜕𝑥𝜕𝜙_𝑦,𝜕𝑦𝜕𝜙_𝑥+𝜕𝑦𝜕𝜙_𝑦.𝜕𝑥 (8) The stress-strain relation of each layer is given by the generalized Hooke’s law as follows: 𝜎_𝑥𝜎_𝑦𝜏_𝑥𝑦𝜏_𝑦𝑧𝜏_𝑥𝑧^(𝑘)=𝑄₁₁𝑄₁₂𝑄₁₆00𝑄₁₂𝑄₂₂𝑄₂₆00𝑄₁₆𝑄₂₆𝑄₆₆00000𝑄₄₄𝑄₄₅000𝑄₄₅𝑄₅₅^(𝑘)𝜀_𝑥𝜀_𝑦𝛾_𝑥𝑦𝛾_𝑦𝑧𝛾_𝑥𝑧, (9) where 𝑄_𝑖𝑗 are the components of the plane-stress reduced constitutive matrix [35] which are function of the elastic constant determined in Section 2.1 and the ply angle 𝛽_𝑘.

2.3. Energy Functional Components

Taking into account (7) and (9), the strain energy due to the laminated plate deflection can be written as 𝑈_𝑝=12_𝑅𝜀₀[𝐴]𝜀₀^𝑇+𝜀₀[𝐵]𝜀₁^𝑇+𝜀₁[𝐵]𝜀₀^𝑇+𝜀₁[𝐷]𝜀₁^𝑇+𝜀^∗₀𝐴^∗𝜀^∗₀^𝑇𝑑𝑥𝑑𝑦, (10) where 𝑅 is the mid-surface area (Figure 1) and the stiffness coefficients [35, 36] are given by (𝐴_𝑖𝑗,𝐵_𝑖𝑗,𝐷_𝑖𝑗)=∫^ℎ/2_−ℎ/2𝑄_𝑖𝑗(1,𝑧,𝑧²)𝑑𝑧(𝑖,𝑗=1,2,6), 𝐴^∗_𝑖𝑗=∫^ℎ/2_−ℎ/2𝑘_𝑖𝑗𝑄_𝑖𝑗𝑑𝑧(𝑖,𝑗=4,5)𝑘_𝑖𝑗being the shear correction factors.

The strain energy corresponding to the elastic edge restraints is given by 𝑈_𝑡=12^𝑎₀𝑡^𝑤₁𝑤₀∣_𝑦=0²+𝑡^𝑤₃𝑤₀∣_𝑦=𝑏²+𝑡^𝑣₁𝑣₀∣_𝑦=0²+𝑡^𝑣₃𝑣₀∣_𝑦=𝑏²+𝑟₁𝜙_𝑦∣_𝑦=0²+𝑟₃𝜙_𝑦∣_𝑦=𝑏²+1𝑑𝑥2^𝑏₀𝑡^𝑤₂𝑤₀∣_𝑥=𝑎²+𝑡^𝑤₄𝑤₀∣_𝑥=0²+𝑡^𝑢₂𝑣₀∣_𝑥=𝑎²+𝑡^𝑢₄𝑣₀∣_𝑥=0²+𝑟₂𝜙_𝑥∣_𝑥=𝑎²+𝑟₄𝜙_𝑥∣_𝑦=0²𝑑𝑦, (11) where 𝑡^•_𝑖 (𝑖=1,…,4 and •=𝑢,𝑣,𝑤) are the elastic translational coefficients and 𝑟_𝑖 (𝑖=1,…,4) are the elastic rotational coefficients.

The kinetic energy is expressed as 1𝑇=2_𝑅𝐼₀𝑢̇²₀+𝑣²₀̇+𝑤̇²₀+2𝐼₁𝑢̇₀𝜙̇_𝑥+𝑣₀̇𝜙̇_𝑦+𝐼₂𝜙̇²_𝑥+𝜙̇²_𝑦𝑑𝑥𝑑𝑦, (12) being 𝐼_𝑖(𝑖=0,1,2) the mass inertias of the plate defined as [35] 𝐼_𝑖=^ℎ/2_−ℎ/2𝜌^(𝑘)𝑧^𝑖𝑑𝑧,(𝑖=0,1,2), (13) where 𝜌^(𝑘) is the material density of the 𝑘-th layer.

3. Application of the Ritz Method for the Free Vibration Analysis

The Ritz method is applied to determine analytical approximate solutions for dynamical behavior of arbitrarily laminated plates resting on elastic supports. During free vibration, the displacements components are assumed split in the spatial and temporal parts, being the last one periodic in time; that is,𝑢₀𝑣𝑥,𝑦,𝑡=𝑈𝑥,𝑦sin𝜔𝑡,₀𝑤𝑥,𝑦,𝑡=𝑉𝑥,𝑦sin𝜔𝑡,₀𝜙𝑥,𝑦,𝑡=𝑊𝑥,𝑦sin𝜔𝑡,_𝑥𝑥,𝑦,𝑡=Φ_𝑥𝜙𝑥,𝑦sin𝜔𝑡,_𝑦𝑥,𝑦,𝑡=Φ_𝑦𝑥,𝑦sin𝜔𝑡, (14) where 𝜔 is the natural frequency in radian.

Putting these displacements into the energy functional components ((10) to (12)) the maximum values of the kinetic energy (𝑇_max) and the strain energies (𝑈_𝑝,max,𝑈_𝑡,max) are derived. Then, the energy functional for free vibration of the laminated plate is given by Π=𝑈_𝑝,max+𝑈_𝑡,max−𝑇_max, (15) which is to be minimized according to the Ritz principle.

3.1. Boundary Conditions and Approximating Functions

There are some options when choosing the unknown functions of displacement components to apply the Ritz method. Particularly, the use of orthogonal polynomials as coordinate functions has important advantages related to numerical stability and fast convergence as has been demonstrated in previous works [23, 37, 38], even for plates with complicated boundary conditions and high degree of anisotropy. For these reasons, in this work, the displacement components are expressed by sets of beam characteristic orthogonal polynomials {𝑝_𝑖^(⋅)(𝑥)}, {𝑞_𝑗^(•)(𝑦)}, (•)=𝑢,𝑣,𝑤,𝜙_𝑥,𝜙_𝑦, resulting in 𝑈𝑥,𝑦≈𝑈_𝑀𝑁𝑥,𝑦=^𝑀𝑁_𝑖=1𝑗=1𝑐^(𝑢)_𝑖𝑗𝑝_𝑖^(𝑢)(𝑥)𝑞_𝑗^(𝑢)𝑦,𝑉𝑥,𝑦≈𝑉_𝑀𝑁𝑥,𝑦=^𝑀𝑁_𝑖=1𝑗=1𝑐^(𝑣)_𝑖𝑗𝑝_𝑖^(𝑣)(𝑥)𝑞_𝑗^(𝑣)𝑦,𝑊𝑥,𝑦≈𝑊_𝑀𝑁𝑥,𝑦=^𝑀𝑁_𝑖=1𝑗=1𝑐^(𝑤)_𝑖𝑗𝑝_𝑖^(𝑤)(𝑥)𝑞_𝑗^(𝑤)𝑦,Φ_𝑥≈Φ_𝑥𝑀𝑁𝑥,𝑦=^𝑁𝑀_𝑖=1𝑗=1𝑐^(𝜙𝑥)_𝑖𝑗𝑝^(𝜙𝑥)_𝑖(𝑥)𝑞^(𝜙𝑥)_𝑗𝑦,Φ_𝑦≈Φ_𝑦𝑀𝑁𝑥,𝑦=^𝑁𝑀_𝑖=1𝑗=1𝑐^(𝜙𝑦)_𝑖𝑗𝑝^(𝜙𝑦)_𝑖(𝑥)𝑞^(𝜙𝑦)_𝑗𝑦, (16) where 𝑐^(𝑢)_𝑖𝑗, 𝑐^(𝑣)_𝑖𝑗, 𝑐^(𝑤)_𝑖𝑗, 𝑐^(𝜙𝑥)_𝑖𝑗, 𝑐^(𝜙𝑦)_𝑖𝑗 are the unknown coefficients, and 𝑀, 𝑁 are the numbers of polynomials in each coordinate.

The procedure for the construction of the orthogonal polynomials has been developed by Bhat [39]. The first members of the sets, 𝑝₁^(•)(𝑥) and 𝑞₁^(•)(𝑦)(•)=𝑢,𝑣,𝑤,𝜙_𝑥,𝜙_𝑦 are obtained as the simplest polynomials that satisfy all the geometrical boundary conditions of the plate in their respective 𝑥 and 𝑦 directions. The higher members of each set are constructed by employing the Gram-Schmidt orthogonalization procedure. The coefficients of the polynomials are chosen in such a way as to make the polynomials orthonormal. However, the functions 𝑝_𝑘^(•)(𝑥) and 𝑞_𝑘^(•)(𝑦) for (•)=𝜙_𝑥,𝜙_𝑦 are obtained from relative rotation conditions starting from polynomials of an order lower than the chosen for the transversal displacements and then applying the sequence of Gram-Schmidt orthogonalization procedure. This particular choice is made to avoid the overestimation of the rate of elastic energy due to the shear respect to the rate due to the bending. This concept has been applied by Auciello and Ercolano [40], to Timoshenko beams, to avoid the shear locking effect and is extended here for laminated plates.

The classical boundary conditions considered in this study are depicted in Table 2. By keeping in mind that in the Ritz method only the geometric boundary conditions need to be satisfied, it is possible to work with any sets of required edge boundary condition and also is very simple the consideration of elastically restrained edges where there are not essential boundary conditions to satisfy.

Table 2 

Notations for various combinations of boundary conditions, in which 𝑛 and 𝑠 indicate the directions normal and tangential to the respective plate edges.

Upon inserting the displacement forms (16) into the energy functional of the system (15), the minimization with respect to the coefficients of the displacement functions is given by 𝜕Π𝜕𝑐^(𝑢)_𝑖𝑗=0,𝜕Π𝜕𝑐^(𝑣)_𝑖𝑗=0,𝜕Π𝜕𝑐^(𝑤)_𝑖𝑗=0,𝜕Π𝜕𝑐^(𝜙𝑥)_𝑖𝑗=0,𝜕Π𝜕𝑐^(𝜙𝑦)_𝑖𝑗=0. (17) From (17) a set of algebraic simultaneous equations is obtained. The number of these equations becomes 5×𝑀×𝑁. The algebraic equations obtained are given as follows, in the form of the generalized eigenvalue problem: 𝐊−𝜔²𝐌{𝐂}={𝟎}, (18) where 𝐊 and 𝐌 are stiffness and inertia matrices, respectively (their expressions are given in the Appendix, {𝐂} contains the unknown coefficients of (16).

For a nontrivial solution, the eigenvalues which make the determinant equal to zero, correspond to the free vibration frequencies.

4. Verification of the Formulation and Numerical Applications

4.1. General Description

The variational algorithm developed in this paper was programmed in Fortran language and used for the free vibration analysis of generally laminated thin and moderately thick laminated plates having different geometric parameters, stacking sequences, material properties, fiber volume fractions, and boundary conditions. The examples considered in this study are confined to laminates with layers of equal thickness, even though the procedure was formulated for plies with arbitrary thickness. In all cases the shear correction factor was taken a 5/6.

Let us introduce the terminology to be used throughout the remainder of the paper for describing the boundary conditions of the considered plates. The designation 𝐶_𝑖𝑆_𝑖𝐹_𝑖𝑆_𝑖, for example, identifies a plate with edges (1) clamped, (2) simply supported, (3) free, and (4) simply supported (see Figure 1) the subscript 𝑖 (𝑖=1,…,4) identifies the in-plane constraints according to Table 2. When the edges are elastically restrained against rotation or translation, the following nondimensional restraint parameters are used𝑇^•_𝑖=𝑎³𝑡^•_𝑖𝐷₀,(),𝑅𝑖=1,…,4and•=𝑢,𝑣,𝑤_𝑖=𝑟_𝑖𝑎𝐷₀,(𝑖=1,…,4), (19)

where 𝐷₀=𝐸₁ℎ³/12(1−𝜈₁₂𝜈₂₁).

The main purposes of the numerical applications presented in this section are twofold. One is to demonstrate the accuracy, the flexibility, and the efficiency of the proposed method and the other is to produce some results which may be regarded as benchmark solutions for other academic research workers and design engineers.

4.2. Validation and Convergence Studies

The accuracy and reliability of the results obtained with the present approach are next demonstrated by comparing them with some selected values published by Shi et al. [5] for moderately thick (𝑎/ℎ=10,20) and thin (𝑎/ℎ=100) arbitrarily clamped laminated plates. The comparison presented in Table 3 authenticates the validity of the present method for arbitrarily laminated plates. Very close agreement for the first sixth nondimensional frequencies 𝜔^∗_𝑖=𝜔_𝑖𝑎²𝜌/(𝐸₂ℎ²) is obtained for all cases and display monotonic convergence tendency to constant values. For thick plates, as shown in Table 3, as number of 𝑁 and 𝑀 is increased from 7 to 10, the frequency parameter decreases merely 0.002% for the first mode and 0.26% for the sixth. For thin plates the relative decreases of the frequency parameters are 0.004% for the first mode and 1.88% for the sixth as the numbers of polynomials 𝑀, 𝑁 are increased from 7 to 10, exhibiting slower convergence rate than that of moderately thick plates. Consequently the number of beam characteristic polynomials used in the following computations for thin and thick plates is chosen as 𝑀=𝑁=7.

Table 3 

Convergence study of frequencies 𝜔^∗_𝑖=𝜔_𝑖𝑎²𝜌/(𝐸₂ℎ²) for a two-layered [0°/45°] square plate. 𝐸₁/𝐸₂=25,  𝐺₁₂=𝐺₁₃=0.5𝐸₂,  𝐺₂₃=0.2𝐸₂,  𝜈₁₂=0.25.

The validation of the proposed methodology for different aspect ratios (𝑎/𝑏) is presented in Table 4, showing a good agreement with Alibeigloo et al. [31] and Reddy [32].

Table 4 

Comparison of fundamental frequency coefficient 𝜔^∗_𝑖=𝜔_𝑖𝑎²𝜌/(𝐸₂ℎ²) for a four layered [45°/−45°/45°/−45°] plate with different aspect ratios 𝑎/𝑏. 𝐸₁/𝐸₂=40, 𝐺₁₂=0.6𝐸₂, 𝐺₁₃=𝐺₂₃=0.5𝐸₂, 𝜈₁₂=0.25.

4.3. Numerical Results and Discussion

Several examples including new results for arbitrarily laminated plates with elastically restrained edges are presented in this section. The elastic properties of the composite materials used here are those shown in Table 1. The influence of different values of fiber volume ratios (𝑘_𝑓) is analyzed in several figures and tables.

Values of the first four frequency parameters 𝜔_𝑖=𝜔_𝑖(𝑎²/ℎ)𝜌/𝐸_𝑓1 for square thick (𝑎/ℎ=10) and thin (𝑎/ℎ=100) unsymmetric laminated plates are shown for increasing values of the translational restraint parameter 𝑇^𝑤_𝑖, in Tables 5(a) and 5(b). Moreover, the influence of rotational restraint parameter 𝑅_𝑖 in the free vibration frequency coefficients is shown in Tables 6(a) and 6(b).

Table 5 

(a) Frequency parameters 𝜔_𝑖 for [0°/45°]and [0°/90°] carbon-epoxy AS4-3501-6 (Table 1), with different translational restraint parameter 𝑇^𝑤_𝑖, 𝑇^𝑢_𝑖=𝑇^𝑣_𝑖=𝑅_𝑖=0 (𝑖=1,…,4), and 𝑎/ℎ=10. (b) Frequency parameters 𝜔_𝑖 for [0°/45°] and [0°/90°] carbon-epoxy AS4-3501-6 (Table 1), with different translational restraint parameter 𝑇^𝑤_𝑖, 𝑇^𝑢_𝑖=𝑇^𝑣_𝑖=𝑅_𝑖=0 (𝑖=1,…,4), and 𝑎/ℎ=100.

Table 6 

(a) Frequency parameters 𝜔_𝑖 for [0°/45°] and [0°/90°] carbon-epoxy AS4-3501-6 (Table 1), with different rotational restraint parameter 𝑅_𝑖, 𝑇^𝑤_𝑖=𝑇^𝑢_𝑖=𝑇^𝑣_𝑖=∞ (𝑖=1,..,4), and 𝑎/ℎ=10. (b) Frequency parameters 𝜔_𝑖 for [0°/45°] and [0°/90°] carbon-epoxy AS4-3501-6 (Table 1), with different rotational restraint parameter 𝑅_𝑖, 𝑇^𝑤_𝑖=𝑇^𝑢_𝑖=𝑇^𝑣_𝑖=∞ (𝑖=1,..,4), and 𝑎/ℎ=100.

In Figures 2–4 the fundamental frequency coefficients 𝜔 corresponding to two laminated square plates are plotted against the restraint parameters 𝑅_𝑖 and 𝑇^𝑤_𝑖. Figure 2 shows the variation of 𝜔 for various values of the rotational restraint 𝑅_𝑖, while Figure 3 shows the variation of 𝜔 for various values of the translational restraint 𝑇^𝑤_𝑖. A major increase of frequency occurs when the elastic restraint values are in the interval 0.1–50. Figure 4 shows the variation of 𝜔 for various values of the rotational and translational restraint parameters: (a) 𝑅_𝑖=0, 𝑇^𝑤_𝑖=𝑆; (b) 𝑅_𝑖=𝑆, 𝑇^𝑤_𝑖=∞, and (c) 𝑅_𝑖=𝑇^𝑤_𝑖=𝑆. The obtained curves illustrate the restraint parameters intervals for which the frequency coefficient 𝜔 is sensitive to 𝑅_𝑖 and 𝑇^𝑤_𝑖.

Figure 2 

Variation of the fundamental frequency coefficient 𝜔 with the rotational restraint parameter 𝑅𝑖, 𝑇𝑤𝑖=𝑇𝑢𝑖=𝑇𝑣𝑖=∞ (𝑖=1,…,4) for square [0°/45°] and [0°/90°] carbon-epoxy (Table 1), with 𝑘𝑓=0.6.

Figure 3 

Variation of the fundamental frequency coefficient 𝜔 with the translational restraint parameter 𝑇𝑤𝑖, 𝑇𝑢𝑖=𝑇𝑣𝑖=𝑅𝑖=0 (𝑖=1,…,4) for square [0°/45°] and [0°/90°] carbon-epoxy (Table 1), with 𝑘𝑓=0.6.

Figure 4 

Variation of the fundamental frequency coefficient 𝜔 with the translational and the rotational restraint parameters 𝑇𝑤𝑖=𝑅𝑖,𝑇𝑢𝑖=𝑇𝑣𝑖=0 (𝑖=1,…,4) for square [0°/45°] and [0°/90°] carbon-epoxy (Table 1), with 𝑘𝑓=0.6 and 𝑎/ℎ=10.

To asses the influence of the aspects ratio 𝑎/𝑏 in the laminated plate response, values of the first four frequency parameters 𝜔_𝑖=𝜔_𝑖(𝑎²/ℎ)𝜌/𝐸_𝑓1 for rectangular thick (𝑎/ℎ=10) unsymmetric laminated plates are shown, for increasing values of the translational restraint parameter 𝑇^𝑤_𝑖 (Table 7(a)) and the rotational restraint parameter 𝑅_𝑖 (Table 7(b)) considering 𝑎/𝑏=1.5 and 𝑎/𝑏=2.

Table 7 

(a) Frequency parameters 𝜔_𝑖 for [0°/45°] carbon-epoxy AS4-3501-6 (Table 1), with different translational restraint parameter 𝑇^𝑤_𝑖, 𝑇^𝑢_𝑖=𝑇^𝑣_𝑖=𝑅_𝑖=0 (𝑖=1,..,4), and 𝑎/ℎ=10. (b) Frequency parameters 𝜔_𝑖 for [0°/45°] carbon-epoxy AS4-3501-6 (Table 1), with different rotational restraint parameter 𝑅_𝑖, 𝑇^𝑤_𝑖=𝑇^𝑢_𝑖=𝑇^𝑣_𝑖=∞ (𝑖=1,…,4), and 𝑎/ℎ=10.

Figure 5 shows the variation of 𝜔 for various values of the rotational restraint 𝑅_𝑖, while Figure 6 shows the variation of 𝜔 for various values of the translational restraint 𝑇^𝑤_𝑖 for rectangular laminated plates.

Figure 5 

Variation of the fundamental frequency coefficient 𝜔 with the rotational restraint parameter 𝑅𝑖, 𝑇𝑤𝑖=𝑇𝑢𝑖=𝑇𝑣𝑖=∞ (𝑖=1,…,4) for different aspect ratios [0°/45°] carbon-epoxy (Table 1), with 𝑘𝑓=0.6.

Figure 6 

Variation of the fundamental frequency coefficient 𝜔 with the translational restraint parameter 𝑇𝑤𝑖, 𝑇𝑢𝑖=𝑇𝑣𝑖=𝑅𝑖=0 (𝑖=1,…,4) for different aspect ratios [0°/45°] carbon-epoxy (Table 1), with 𝑘𝑓=0.6.

To evaluate the effect of different fiber orientation angles (β) and fiber volume fraction on the dynamic properties of the laminates, the variation of the first free vibration coefficient 𝜔₁ is plotted in Figures 7 and 8, considering two lamination stacking sequences, [β/−β] and [0/β]. Two boundary conditions have been included, 𝐶₁𝐶₁𝐶₁𝐶₁ in Figure 7 and 𝑆₁𝑆₁𝑆₁𝑆₁ in Figure 6. It is observed that the [β/−β] laminate is more sensitive to the fiber orientation angle than [0/β] lamination scheme. The adimensional frequency parameter is noticeable higher as the fiber volume fraction 𝑘_𝑓 increases and as the boundary conditions become clamped.

Figure 7 

Effect of the fiber orientation on the first vibration frequency coefficient 𝜔1, for two different fiber volume fraction 𝑘𝑓 with 𝐶1𝐶1𝐶1𝐶1 boundary condition and 𝑎/ℎ=10.

Figure 8 

Effect of the fiber orientation on the first vibration frequency coefficient 𝜔1, for two different fiber volume fraction 𝑘𝑓 with 𝑆1𝑆1𝑆1𝑆1 boundary condition and 𝑎/ℎ=10.

Finally, the first four free vibration coefficients are presented in Table 8 to illustrate the influence of various fiber volume fractions and boundary conditions on the dynamical behavior of an unsymmetric [0°/45°] laminated plate.

Table 8 

Frequency parameters 𝜔_𝑖 for [0°/45°] carbon-epoxy AS4-3501-6 (Table 1), with different fiber volume fraction 𝑘_𝑓 and boundary conditions. 𝑎/ℎ=10.

5. Concluding Remarks

A Ritz approach for free vibration analysis of general laminated plates with edges elastically restrained against translation and rotation is presented in this work. The study includes the effective elastic moduli of each lamina obtained using the Mori-Tanaka mean field theory, which allows taking into account the influence of the fiber volume ratios and the elastic properties of the components (fiber and matrix) into the vibration behavior. The formulation is based on the first-order shear deformation theory, and the generalized displacements are approximate using sets of characteristic orthogonal polynomials generated by the Gram-Schmidt procedure. The consideration of all possible rotational and translational restraints allows generating any classical boundary condition, only approaching the corresponding spring parameter to zero or infinity. The algorithm is computationally efficient, and the solutions are stables and convergent. Close agreement with existing results in the literature is shown and new results are presented in tables and figures which could be useful for design and optimization problems of general long fiber-reinforced laminated plates.

Appendix

The matrices 𝐊 and 𝐌 in (18) are given by[𝐊]=𝐾^𝑢𝑢_{𝑖𝑗𝑘ℎ}𝐾^𝑢𝑣_{𝑖𝑗𝑘ℎ}𝐾^𝑢𝑤_{𝑖𝑗𝑘ℎ}𝐾^𝑢𝜙𝑥_{𝑖𝑗𝑘ℎ}𝐾^𝑢𝜙𝑦_{𝑖𝑗𝑘ℎ}𝐾^𝑣𝑣_{𝑖𝑗𝑘ℎ}𝐾^𝑣𝑤_{𝑖𝑗𝑘ℎ}𝐾^𝑣𝜙𝑥_{𝑖𝑗𝑘ℎ}𝐾^𝑣𝜙𝑦_{𝑖𝑗𝑘ℎ}𝐾^𝑤𝑤_{𝑖𝑗𝑘ℎ}𝐾^𝑤𝜙𝑥_{𝑖𝑗𝑘ℎ}𝐾^𝑤𝜙𝑦_{𝑖𝑗𝑘ℎ}𝐾^{𝜙𝑥𝜙𝑥}_{𝑖𝑗𝑘ℎ}𝐾^{𝜙𝑥𝜙𝑦}_{𝑖𝑗𝑘ℎ}sym𝐾^{𝜙𝑦𝜙𝑦}_{𝑖𝑗𝑘ℎ}, (A.1) where 𝐾^𝑢𝑢_{𝑖𝑗𝑘ℎ}=𝐴_11𝑅𝑝_𝑖^′(𝑢)𝑝_𝑘^′(𝑢)𝑞_𝑗^(𝑢)𝑞_ℎ^(𝑢)𝑑𝑥𝑑𝑦+𝐴_16𝑅𝑝_𝑖^′(𝑢)𝑝_𝑘^(𝑢)𝑞_𝑗^(𝑢)𝑞_ℎ^′(𝑢)+𝑝_𝑖^(𝑢)𝑝_𝑘^′(𝑢)𝑞_𝑗^′(𝑢)𝑞_ℎ^(𝑢)𝑑𝑥𝑑𝑦+𝐴_66𝑅𝑝_𝑖^(𝑢)𝑝_𝑘^(𝑢)𝑞_𝑗^′(𝑢)𝑞_ℎ^′(𝑢)𝑑𝑥𝑑𝑦+𝑡^𝑢₂^𝑏₀𝑝_𝑖^(𝑢)𝑝_𝑘^(𝑢)_𝑥=1𝑞_𝑗^(𝑢)𝑞_ℎ^(𝑢)𝑑𝑦+𝑡^𝑢₄^𝑏₀𝑝_𝑖^(𝑢)𝑝_𝑘^(𝑢)_𝑥=0𝑞_𝑗^(𝑢)𝑞_ℎ^(𝑢)𝐾𝑑𝑦,_{𝑢𝑣𝑖𝑗𝑘ℎ}=𝐴_12𝑅𝑝_𝑖^′(𝑢)𝑝_𝑘^(𝑣)𝑞_𝑗^(𝑢)𝑞_ℎ^′(𝑣)𝑑𝑥𝑑𝑦+𝐴_16𝑅𝑝_𝑖^′(𝑣)𝑝_𝑘^′(𝑢)𝑞_𝑗^(𝑣)𝑞_ℎ^(𝑢)𝑑𝑥𝑑𝑦+𝐴_26𝑅𝑝_𝑖^(𝑣)𝑝_𝑘^(𝑢)𝑞_𝑗^′(𝑣)𝑞_ℎ^′(𝑢)𝑑𝑥𝑑𝑦+𝐴_66𝑅𝑝_𝑖^′(𝑣)𝑝_𝑘^(𝑢)𝑞_𝑗^(𝑣)𝑝_ℎ^′(𝑢)𝐾𝑑𝑥𝑑𝑦,_{𝑢𝑤𝑖𝑗𝑘ℎ}=0,𝐾^𝑣𝑤_{𝑖𝑗𝑘ℎ}𝐾=0,_{𝑢𝜙𝑥𝑖𝑗𝑘ℎ}=𝐵_11𝑅𝑝_𝑖^′(𝑢)𝑝^{′(𝜙𝑥)}_𝑘𝑞_𝑗^(𝑢)𝑞^(𝜙𝑥)_ℎ𝑑𝑥𝑑𝑦+𝐵_16𝑅𝑝_𝑖^′(𝑢)𝑝^(𝜙𝑥)_𝑘𝑞_𝑗^(𝑢)𝑞^{′(𝜙𝑥)}_ℎ𝑑𝑥𝑑𝑦+𝐵_16𝑅𝑝_𝑖^(𝑢)𝑝^{′(𝜙𝑥)}_𝑘𝑞_𝑗^′(𝑢)𝑞^(𝜙𝑥)_ℎ𝑑𝑥𝑑𝑦+𝐵_66𝑅𝑝_𝑖^(𝑢)𝑝^(𝜙𝑥)_𝑘𝑞_𝑗^′(𝑢)𝑝^{′(𝜙𝑥)}_ℎ𝐾𝑑𝑥𝑑𝑦,_{𝑢𝜙𝑦𝑖𝑗𝑘ℎ}=𝐵_12𝑅𝑝_𝑖^′(𝑢)𝑝^(𝜙𝑦)_𝑘𝑞_𝑗^(𝑢)𝑞^{′(𝜙𝑦)}_ℎ𝑑𝑥𝑑𝑦+𝐵_16𝑅𝑝_𝑖^′(𝑢)𝑝^{′(𝜙𝑦)}_𝑘𝑞_𝑗^(𝑢)𝑞^(𝜙𝑦)_ℎ𝑑𝑥𝑑𝑦+𝐵_26𝑅𝑝_𝑖^(𝑢)𝑝^(𝜙𝑦)_𝑘𝑞′_𝑗^(𝑢)𝑞′^(𝜙𝑦)_ℎ𝑑𝑥𝑑𝑦+𝐵_66𝑅𝑝_𝑖^(𝑢)𝑝^{′(𝜙𝑦)}_𝑘𝑞′_𝑗^(𝑢)𝑞^(𝜙𝑦)_ℎ𝐾𝑑𝑥𝑑𝑦,_{𝑣𝑣𝑖𝑗𝑘ℎ}=𝐴_22𝑅𝑝_𝑖^(𝑣)𝑝_𝑘^(𝑣)𝑞_𝑗^′(𝑣)𝑞_ℎ^′(𝑣)𝑑𝑥𝑑𝑦+𝐴_26𝑅𝑝_𝑖^(𝑣)𝑝_𝑘^′(𝑣)𝑞_𝑗^′(𝑣)𝑞_ℎ^(𝑣)+𝑝_𝑖^′(𝑣)𝑝_𝑘^(𝑣)𝑞_𝑗^(𝑣)𝑞_ℎ^′(𝑣)𝑑𝑥𝑑𝑦+𝐴_66𝑅𝑝_𝑖^′(𝑣)𝑝_𝑘^′(𝑣)𝑞_𝑗^(𝑣)𝑞_ℎ^(𝑣)𝑑𝑥𝑑𝑦+𝑡^𝑣₁^𝑎₀𝑝_𝑖^(𝑣)𝑝_𝑘^(𝑣)𝑞_𝑗^(𝑣)𝑞_ℎ^(𝑣)_𝑦=0𝑑𝑥+𝑡^𝑣₃^𝑏₀𝑝_𝑖^(𝑣)𝑝_𝑘^(𝑣)𝑞_𝑗^(𝑣)𝑞_ℎ^(𝑣)_𝑦=1𝐾𝑑𝑥,_{𝑣𝜙𝑥𝑖𝑗𝑘ℎ}=𝐵_12𝑅𝑝_𝑖^(𝑣)𝑝^{′(𝜙𝑥)}_𝑘𝑞_𝑗^′(𝑣)𝑞^(𝜙𝑥)_ℎ𝑑𝑥𝑑𝑦+𝐵_16𝑅𝑝_𝑖^′(𝑣)𝑝^{′(𝜙𝑥)}_𝑘𝑞_𝑗^(𝑣)𝑞^(𝜙𝑥)_ℎ𝑑𝑥𝑑𝑦+𝐵_26𝑅𝑝_𝑖^(𝑣)𝑝^(𝜙𝑥)_𝑘𝑞_𝑗^′(𝑣)𝑞^{′(𝜙𝑥)}_ℎ𝑑𝑥𝑑𝑦+𝐵_66𝑅𝑝_𝑖^′(𝑣)𝑝^(𝜙𝑥)_𝑘𝑞_𝑗^(𝑣)𝑞^{′(𝜙𝑥)}_ℎ𝐾𝑑𝑥𝑑𝑦,_{𝑣𝜙𝑦𝑖𝑗𝑘ℎ}=𝐵_22𝑅𝑝_𝑖^(𝑣)𝑝^(𝜙𝑦)_𝑘𝑞_𝑗^′(𝑣)𝑞^{′(𝜙𝑦)}_ℎ𝑑𝑥𝑑𝑦+𝐵_26𝑅𝑝_𝑖^(𝑣)𝑝^{′(𝜙𝑦)}_𝑘𝑞_𝑗^′(𝑣)𝑞^(𝜙𝑦)_ℎ𝑑𝑥𝑑𝑦+𝐵_26𝑅𝑝_𝑖^′(𝑣)𝑝^(𝜙𝑦)_𝑘𝑞_𝑗^(𝑣)𝑞^{′(𝜙𝑦)}_ℎ𝑑𝑥𝑑𝑦+𝐵_66𝑅𝑝_𝑖^′(𝑣)𝑝^{′(𝜙𝑦)}_𝑘𝑞_𝑗^(𝑣)𝑞^(𝜙𝑦)_ℎ𝐾𝑑𝑥𝑑𝑦,_{𝑤𝑤𝑖𝑗𝑘ℎ}=𝐴_44𝑅𝑝_𝑖^(𝑤)𝑝_𝑘^(𝑤)𝑞_𝑗^′(𝑤)𝑞_ℎ^′(𝑤)𝑑𝑥𝑑𝑦+𝐴_55𝑅𝑝_𝑖^′(𝑤)𝑝_𝑘^′(𝑤)𝑞_𝑗^(𝑤)𝑞_ℎ^(𝑤)𝑑𝑥𝑑𝑦+𝐴_45𝑅𝑝_𝑖^′(𝑤)𝑝_𝑘^(𝑤)𝑞_𝑗^(𝑤)𝑞_ℎ^′(𝑤)+𝑝_𝑖^(𝑤)𝑝_𝑘^′(𝑤)𝑞_𝑗^′(𝑤)𝑞_ℎ^(𝑤)𝑑𝑥𝑑𝑦+𝑡^𝑤₁^𝑎₀𝑝_𝑖^(𝑤)𝑝_𝑘^(𝑤)𝑞_𝑗^(𝑤)𝑞_ℎ^(𝑤)_𝑦=0𝑑𝑥+𝑡^𝑤₂^𝑏₀𝑝_𝑖^(𝑤)𝑝_𝑘^(𝑤)_𝑥=1𝑞_𝑗^(𝑤)𝑞_ℎ^(𝑤)𝑑𝑦+𝑡^𝑤₃^𝑎₀𝑝_𝑖^(𝑤)𝑝_𝑘^(𝑤)𝑞_𝑗^(𝑤)𝑞_ℎ^(𝑤)_𝑦=1𝑑𝑥+𝑡^𝑤₄^𝑏₀𝑝_𝑖^(𝑤)𝑝_𝑘^(𝑤)_𝑥=0𝑞_𝑗^(𝑤)𝑞_ℎ^(𝑤)𝐾𝑑𝑦,_{𝑤𝜙𝑥𝑖𝑗𝑘ℎ}=𝐾𝐴_45𝑅𝑝_𝑖^(𝑤)𝑝^(𝜙𝑥)_𝑘𝑞_𝑗^′(𝑤)𝑞^(𝜙𝑥)_ℎ𝑑𝑥𝑑𝑦+𝐴_55𝑅𝑝_𝑖^′(𝑤)𝑝^(𝜙𝑥)_𝑘𝑞_𝑗^(𝑤)𝑞^(𝜙𝑥)_ℎ,𝐾𝑑𝑥𝑑𝑦_{𝑤𝜙𝑦𝑖𝑗𝑘ℎ}=𝐾𝐴_44𝑅𝑝_𝑖^(𝑤)𝑝^(𝜙𝑦)_𝑘𝑞_𝑗^′(𝑤)𝑞^(𝜙𝑦)_ℎ𝑑𝑥𝑑𝑦+𝐴_45𝑅𝑝_𝑖^′(𝑤)𝑝^(𝜙𝑦)_𝑘𝑞_𝑗^(𝑤)𝑞^(𝜙𝑦)_ℎ,𝐾𝑑𝑥𝑑𝑦_{𝜙𝑥𝜙𝑥𝑖𝑗𝑘ℎ}=𝐾𝐴_55𝑅𝑝^(𝜙𝑥)_𝑖𝑝^(𝜙𝑥)_𝑘𝑞^(𝜙𝑥)_𝑗𝑞^(𝜙𝑥)_ℎ𝑑𝑥𝑑𝑦+𝐷_11𝑅𝑝^{′(𝜙𝑥)}_𝑖𝑝^{′(𝜙𝑥)}_𝑘𝑞^(𝜙𝑥)_𝑗𝑞^(𝜙𝑥)_ℎ𝑑𝑥𝑑𝑦+𝐷_16𝑅𝑝^{′(𝜙𝑥)}_𝑖𝑝^(𝜙𝑥)_𝑘𝑞^(𝜙𝑥)_𝑗𝑞^{′(𝜙𝑥)}_ℎ+𝑝^(𝜙𝑥)_𝑖𝑝^{′(𝜙𝑥)}_𝑘𝑞^{′(𝜙𝑥)}_𝑗𝑞^(𝜙𝑥)_ℎ𝑑𝑥𝑑𝑦+𝐷_66𝑅𝑝^(𝜙𝑥)_𝑖𝑝^(𝜙𝑥)_𝑘𝑞^{′(𝜙𝑥)}_𝑗𝑞^(𝜙𝑥)_ℎ𝑑𝑥𝑑𝑦+𝑟₂^𝑏₀𝑝^(𝜙𝑥)_𝑖𝑝^(𝜙𝑥)_{𝑘𝑥=𝑎}𝑞^(𝜙𝑥)_𝑗𝑞^(𝜙𝑥)_ℎ𝑑𝑦+𝑟₄^𝑏₀𝑝^(𝜙𝑥)_𝑖𝑝^(𝜙𝑥)_𝑘𝑥=0𝑞^(𝜙𝑥)_𝑗𝑞^(𝜙𝑥)_ℎ𝐾𝑑𝑦,_{𝜙𝑥𝜙𝑦𝑖𝑗𝑘ℎ}=𝐾𝐴_45𝑅𝑝^(𝜙𝑥)_𝑖𝑝^(𝜙𝑦)_𝑘𝑞^(𝜙𝑥)_𝑗𝑞^(𝜙𝑦)_ℎ𝑑𝑥𝑑𝑦+𝐷_12𝑅𝑝^{′(𝜙𝑥)}_𝑖𝑝^(𝜙𝑦)_𝑘𝑞^(𝜙𝑥)_𝑗𝑞^{′(𝜙𝑦)}_ℎ𝑑𝑥𝑑𝑦+𝐷_16𝑅𝑝^{′(𝜙𝑥)}_𝑖𝑝^{′(𝜙𝑦)}_𝑘𝑞^(𝜙𝑥)_𝑗𝑞^(𝜙𝑦)_ℎ𝑑𝑥𝑑𝑦+𝐷_26𝑅𝑝^(𝜙𝑥)_𝑖𝑝^(𝜙𝑦)_𝑘𝑞^{′(𝜙𝑥)}_𝑗𝑞^{′(𝜙𝑦)}_ℎ𝑑𝑥𝑑𝑦+𝐷_66𝑅𝑝^(𝜙𝑥)_𝑖𝑝^{′(𝜙𝑦)}_𝑘𝑞^{′(𝜙𝑥)}_𝑗𝑞^(𝜙𝑦)_ℎ𝐾𝑑𝑥𝑑𝑦,_{𝜙𝑦𝜙𝑦𝑖𝑗𝑘ℎ}=𝐾𝐴_44𝑅𝑝^(𝜙𝑦)_𝑖𝑝^(𝜙𝑦)_𝑘𝑞^(𝜙𝑦)_𝑗𝑞^(𝜙𝑦)_ℎ𝑑𝑥𝑑𝑦+𝐷_22𝑅𝑝^(𝜙𝑦)_𝑖𝑝^(𝜙𝑦)_𝑘𝑞^{′(𝜙𝑦)}_𝑗𝑞^{′(𝜙𝑦)}_ℎ𝑑𝑥𝑑𝑦+𝐷_26𝑅𝑝^(𝜙𝑦)_𝑖𝑝^{′(𝜙𝑦)}_𝑘𝑞^{′(𝜙𝑦)}_𝑗𝑞^(𝜙𝑦)_ℎ+𝑝^{′(𝜙𝑦)}_𝑖𝑝^(𝜙𝑦)_𝑘𝑞^(𝜙𝑦)_𝑗𝑞^{′(𝜙𝑦)}_ℎ𝑑𝑥𝑑𝑦+𝐷_66𝑅𝑝^{′(𝜙𝑦)}_𝑖𝑝^{′(𝜙𝑦)}_𝑘𝑞^(𝜙𝑦)_𝑗𝑞^(𝜙𝑦)_ℎ𝑑𝑥𝑑𝑦+𝑟₁^𝑎₀𝑝^(𝜙𝑦)_𝑖𝑝^(𝜙𝑦)_𝑘𝑞^(𝜙𝑦)_𝑗𝑞^(𝜙𝑦)_ℎ𝑦=0𝑑𝑥+𝑟₃^𝑎₀𝑝^(𝜙𝑦)_𝑖𝑝^(𝜙𝑦)_𝑘𝑞^(𝜙𝑦)_𝑗𝑞^(𝜙𝑦)_ℎ𝑦=𝑏𝑑𝑥. (A.2) with 𝐾=5/6[𝐌]=𝑀^𝑢𝑢_{𝑖𝑗𝑘ℎ}𝑀0000_{𝑣𝑣𝑖𝑗𝑘ℎ}𝑀000_{𝑤𝑤𝑖𝑗𝑘ℎ}𝑀00_{𝜙𝑥𝜙𝑥𝑖𝑗𝑘ℎ}0sym𝑀^{𝜙𝑦𝜙𝑦}_{𝑖𝑗𝑘ℎ},𝑀^𝑢𝑢_{𝑖𝑗𝑘ℎ}=𝐼_0𝑅𝑝_𝑖^(𝑢)𝑝_𝑘^(𝑢)𝑞_𝑗^(𝑢)𝑞_ℎ^(𝑢)𝑀𝑑𝑥𝑑𝑦,_{𝑣𝑣𝑖𝑗𝑘ℎ}=𝐼_0𝑅𝑝_𝑖^(𝑣)𝑝_𝑘^(𝑣)𝑞_𝑗^(𝑣)𝑞_ℎ^(𝑣)𝑀𝑑𝑥𝑑𝑦,_{𝑤𝑤𝑖𝑗𝑘ℎ}=𝐼_0𝑅𝑝_𝑖^(𝑤)𝑝_𝑘^(𝑤)𝑞_𝑗^(𝑤)𝑞_ℎ^(𝑤)𝑀𝑑𝑥𝑑𝑦,_{𝜙𝑥𝜙𝑥𝑖𝑗𝑘ℎ}=𝐼_2𝑅𝑝^(𝜙𝑥)_𝑖𝑝^(𝜙𝑥)_𝑘𝑞^(𝜙𝑥)_𝑗𝑞^(𝜙𝑥)_ℎ𝑀𝑑𝑥𝑑𝑦,_{𝜙𝑦𝜙𝑦𝑖𝑗𝑘ℎ}=𝐼_2𝑅𝑝^(𝜙𝑦)_𝑖𝑝^(𝜙𝑦)_𝑘𝑞^(𝜙𝑦)_𝑗𝑞^(𝜙𝑦)_ℎ𝑑𝑥𝑑𝑦. (A.3)

Acknowledgment

The authors wish to thank the economic support of CONICET (PIP no. 0105/2010) and CIUNSa.