Alessandro Ciallella
email: alessandro.ciallella@univaq.it

Francesco D'annibale
email: francesco.dannibale@univaq.it

Dionisio Francesco Dell'isola
email: francesco.dellisola@univaq.it

Del Vescovo

Ivan Giorgio
email: ivan.giorgio@univaq.it

Del Dionisio
email: dionisio.delvescovo@uniroma1.it

Vescovo

Modal Analysis of a Second-Gradient Annular Plate made of an Orthogonal Network of Logarithmic Spiral Fibers

Keywords: Modal analysis, Second gradient surfaces, Homogenized nets, Metamaterials

A dynamical study of small vibrations of an annular system made of an orthogonal network of fibers around the undeformed configuration is performed. Logarithmic spirals characterize the net of fibers connected by deformable cylinders. This particular arrangement is chosen because it produces a tough material with an economy of matter.

Introduction

In this contribution, we analyze the case of a bi-dimensional generalized continuum characterized by a microstructure [START_REF] Eremeyev | Material symmetry group and constitutive equations of micropolar anisotropic elastic solids[END_REF]- [START_REF] De Cicco | A plane strain problem in the theory of elastic materials with voids[END_REF], which bestows on it high performance, namely high strength with a lightweight. An orthogonal net of curved fibers, i.e., logarithmic spirals, constitutes this microstructure optimized to have one of the most efficient patterns for the structural viewpoint, as established in [START_REF] Michell | LVIII The limits of economy of material in framestructures[END_REF].

Among new manufacturing technologies, 3D printing allows us to build these kinds of microstructured mechanical systems, opening unimaginable possibilities until a few decades ago [START_REF] Abali | Additive manufacturing introduced substructure and computational determination of metamaterials parameters by means of the asymptotic homogenization[END_REF]- [START_REF] Yang | Determination of metamaterial parameters by means of a homogenization approach based on asymptotic analysis[END_REF]. For this reason, many researchers initiated the set up of systematic methods for synthesizing new micro-structures and, therefore, conceiving novel materials with a macroscopic behavior that fulfills one or more required functionalities. This way of proceeding is typical of a new branch of generalized continua that focuses on designing the so-called metamaterials [START_REF] Barchiesi | A beam model for duoskelion structures derived by asymptotic homogenization and its application to axial loading problems[END_REF]- [START_REF] Barchiesi | Mechanical metamaterials: a state of the art[END_REF].

In this contribution, we focus our attention on the dynamic response of the considered system of fibers examining its small vibrations around the undeformed configuration. Locally this fibers network has many features in common with the pantographic sheet [START_REF] Isola | Pantographic metamaterials: an example of mathematically driven design and of its technological challenges[END_REF]. Concerning dynamic behavior, we recall the works by [START_REF] Laudato | Linear dynamics of 2D pantographic metamaterials: numerical and experimental study[END_REF]- [START_REF] Turco | A numerical investigation on impulseinduced nonlinear longitudinal waves in pantographic beams[END_REF].

This chapter is organized as follows: in Sect. 8.2, we briefly recall the used model to describe the considered network of fibers. In Sect. 8.3, we provide the modal analysis of the system. Finally, in Sect. 8.4, we draw some conclusions and future developments.

The Model for a Fiber Net Arranged in Logarithmic Spirals

In the continuum mechanics framework, we develop a plate model based on an elastic second gradient theory [START_REF] Fedele | Piola's approach to the equilibrium problem for bodies with second gradient energies. Part I: First gradient theory and differential geometry[END_REF]- [START_REF] Eremeyev | On the well posedness of static boundary value problem within the linear dilatational strain gradient elasticity[END_REF] to study a mechanical system composed of a network of orthogonal fibers. The aim is to take into account the deformation modes of the fibers that, in a classical plate theory, are not considered, namely, geodesic bending and twisting [START_REF] Eremeyev | Linear pantographic sheets: existence and uniqueness of weak solutions[END_REF], [START_REF] Steigmann | Isola F (2015) Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching[END_REF]- [START_REF] Giorgio | Lattice shells composed of two families of curved Kirchhoff rods: an archetypal example, topology optimization of a cycloidal metamaterial[END_REF]. Specifically, the fibers are curved and shaped in the undeformed reference configuration according to logarithmic spirals. To build the net, the fibers are connected with cylindrical pivots, which allow them to have a relative rotation around their pivotal axis [START_REF] Spagnuolo | Contact interactions in complex fibrous metamaterials[END_REF][START_REF] Spagnuolo | Phenomenological aspects of quasiperfect pivots in metallic pantographic structures[END_REF][START_REF] Ciallella | Shear rupture mechanism and dissipation phenomena in bias-extension test of pantographic sheets: Numerical modeling and experiments[END_REF]]. In the examined example, the network is arranged in an annular fashion on a plate (see Fig. 8.1).

According to [START_REF] Giorgio | Lattice shells composed of two families of curved Kirchhoff rods: an archetypal example, topology optimization of a cycloidal metamaterial[END_REF], [START_REF] Ciallella | Deformation patterns in a second-gradient lattice annular plate composed of "spira mirabilis" fibers[END_REF], we define the reference configuration of the elastic surface, S, by the material points 𝑿 = (𝑋 1 , 𝑋 2 ) ∈ Ω ⊂ R 2 that are mapped in the location of the current configuration, 𝒙 = (𝑥 1 , 𝑥 2 , 𝑥 3 ) ∈ R 3 , by the placement map 𝝌 with the standard notation:

𝒙 = 𝝌( 𝑿) = 𝑋 𝛼 𝑒 𝑒 𝑒 𝛼 + 𝑢 𝑖 ( 𝑿) 𝑒 𝑒 𝑒 𝑖 (8.1)
where 𝑢 𝑖 are the components of the displacement field, the Greek indexes range from 1 to 2, the Latin ones from 1 to 3, being Einstein's convention adopted, and the orthonormal vectors {𝑒 𝑒 𝑒 𝑖 } set as a basis that spans the associated vector space of the Euclidian affine space. The logarithmic spirals that represent the middle lines of the fibers are parameterized by 𝜑 and 𝜓 as follows:

𝑋 1 = 𝑅 0 𝑒 ( 𝜑+𝜓) cos(𝜑 -𝜓) 𝑋 2 = 𝑅 0 𝑒 ( 𝜑+𝜓) sin(𝜑 -𝜓) (8.2)
where 𝑅 0 is a constant which specifies the inner radius of the annular plate. In what follows, we adopt the unit-speed parametrization 𝑆 𝛽 (𝛽 = 1, 2) to characterize the two fiber families in the reference configuration given in Eq. (8.2). Therefore, we express the two curvilinear abscissae using the relationships 𝜑 = φ(𝑆 1 ) and 𝜓 = ψ(𝑆 2 ) [START_REF] Ciallella | Deformation patterns in a second-gradient lattice annular plate composed of "spira mirabilis" fibers[END_REF].

The unit tangent vector fields to the fibers in the reference configuration become

𝑫 𝛽 ( 𝑿) = d𝑋 𝛼 d𝑆 𝛽 𝑒 𝑒 𝑒 𝛼 (8.3)
For the two families of fibers in the reference configuration, we have 𝑫 1 • 𝑫 2 = 0; thus, the fibers are orthogonal. The tangent vectors in the current configuration can be evaluated as

𝜆 𝛽 ( 𝑿) 𝒅 𝛽 ( 𝑿) = d 𝝌 d𝑆 𝛽 ( 𝑿) = 𝑭( 𝑿) 𝑫 𝛽 ( 𝑿) (8.4)
where 𝜆 𝛽 = ∥𝑭 𝑫 𝛽 ∥, 𝑭 = ∇ 𝑿 𝝌, and 𝒅 𝛽 represent the unit tangent vector fields in the current configuration. The unit vector normal to the surface S in the current configuration is

𝒏( 𝑿) = 𝒅 1 × 𝒅 2 ∥ 𝒅 1 × 𝒅 2 ∥ (8.5)
To fully describe the kinematics of the fibers, we also introduce two rotation tensor fields, one for each fiber family representative of their cross-section orientation as

𝑹 𝛽 : {𝑒 𝑒 𝑒 𝑖 } ↦ → {𝒅 𝛽 , 𝒏 × 𝒅 𝛽 , 𝒏} (8.6)
and, accordingly, by treating the fibers as an infinite distribution of Kirchhoff's rods [START_REF] Kumar | A novel fourfield mixed FE approximation for Kirchhoff rods using Cartan's moving frames[END_REF]- [START_REF] Greco | An iso-parametric 𝐺 1 -conforming finite element for the nonlinear analysis of Kirchhoff rod. Part I: the 2D case[END_REF], to specify their current geometric features, we define the curvature tensor as follows

𝑾 𝛽 = 𝑹 ⊤ 𝛽 d𝑹 𝛽 d𝑆 𝛽 (8.7)
The skew-symmetric tensor (8.7), indeed, allow us to define the twisting, out-of-plane, and geodesic curvature, i.e.,

𝜅 𝑇 𝛽 = 𝒏 × 𝒅 𝛽 • d𝒏 d𝑆 𝛽 , 𝜅 𝑛𝛽 = 𝒏 • d𝒅 𝛽 d𝑆 𝛽 , 𝜅 𝑔𝛽 = -𝒏 × 𝒅 𝛽 • d𝒅 𝛽 d𝑆 𝛽 (8.8)
respectively. The measures of deformation characterizing the elastic response of the surface S are: i) the fiber stretching,

𝜀 𝛽 = 𝜆 𝛽 -1; ii) the shear distortion angle, 𝛾 = asin ( 𝒅 1 • 𝒅 2 ); iii) the curvature change, Δ𝜅 𝑇 𝛽 = 𝜅 𝑇 𝛽 , Δ𝜅 𝑛𝛽 = 𝜅 𝑛𝛽 , Δ𝜅 𝑔𝛽 = 𝜅 𝑔𝛽 -𝜅 0 𝑔𝛽
in which the superscript '0' refers to the reference configuration and, in particular, the reference geodesic curvature is

𝜅 0 𝑔𝛽 = -𝑒 𝑒 𝑒 3 × 𝑫 𝛽 • d𝑫 𝛽 d𝑆 𝛽 (8.9)
The deformation energy can be assumed as a quadratic form in the measures of deformation [START_REF] Ciallella | Deformation patterns in a second-gradient lattice annular plate composed of "spira mirabilis" fibers[END_REF] as a first attempt to model the elastic behavior of the examined fiber net:

2𝑈 [ 𝝌(•)] = ∫ Ω 2 ∑ 𝛽=1 𝐾 𝑒𝛽 𝜆 𝛽 -1 2 dΩ + ∫ Ω 𝐾 𝑠 𝛾 2 dΩ + ∫ Ω 2 ∑ 𝛽=1 𝐾 𝑔𝛽 (𝜅 𝑔𝛽 -𝜅 0 𝑔𝛽 ) 2 dΩ + ∫ Ω 2 ∑ 𝛽=1 𝐾 𝑛𝛽 (𝜅 𝑛𝛽 ) 2 dΩ + ∫ Ω 2 ∑ 𝛽=1 𝐾 𝑇 𝛽 (𝜅 𝑇 𝛽 ) 2 dΩ (8.10)
in which 𝐾 𝑒𝛽 , 𝐾 𝑠 , 𝐾 𝑔𝛽 , 𝐾 𝑛𝛽 , and 𝐾 𝑇 𝛽 are material stiffnesses related to stretching, shearing, geodesic and normal bending, and twisting, respectively. They are evaluated by the expressions:

𝐾 𝑒 = 𝜂 𝑒 𝑌 𝐴 𝑝 , 𝐾 𝑠 = 𝜂 𝑠 𝐺𝐽 𝑝 ℎ 𝑝 𝑝 2 , 𝐾 𝑇 = 𝜂 𝑇 𝐺𝐽 𝑡 𝑝 , 𝐾 𝑔 = 𝜂 𝑔 𝑌 𝐽 𝑓 𝑔 𝑝 , 𝐾 𝑛 = 𝜂 𝑛 𝑌 𝐽 𝑓 𝑛 𝑝 (8.11)
borrowed by the beam theory. Since the hypotheses behind these expressions are not adequately satisfied, we introduce corrective coefficients, 𝜂 𝑖 , to overtake this issue. Moreover, 𝐴 is the area of the fibers cross-section, 𝐽 𝑡 = 0.196 ℎ 3 𝑏 and 𝐽 𝑓 𝑛 = ℎ 3 𝑏/12, 𝐽 𝑓 𝑔 = 𝑏 3 ℎ/12 are the torsional, the out-of-plane and in-plane bending second moment of area of the fiber cross-sections, respectively; 𝐽 𝑝 = 𝜋 𝑟 4 𝑝 /2 is the torsional second moment of area of the pivot. Because the two families of fibers are equal, so the related stiffnesses are. The variable pitch between two adjacent fibers is denoted by 𝑝.

To define the kinetic energy of the plate, we introduce the skew tensor

𝑽 𝛽 = 𝑹 ⊤ 𝛽 𝑹 𝛽 (8.12)
where the dot denotes differentiation with respect to the time. The significant components of the tensor (8.12) are the angular velocity fields of the cross-section of the fibers, in detail

𝜔 𝛽1 = 𝒏 × 𝒅 𝛽 • 𝒏; 𝜔 𝛽2 = 𝒏 • 𝒅 𝛽 ; 𝜔 𝛽3 = -𝒏 × 𝒅 𝛽 • 𝒅 𝛽 (8.13)
Therefore, the kinetic energy of the equivalent continuum annular plate can be evaluated by

2𝑇 [ 𝝌(•), 𝝌(•)] = ∫ Ω 𝜚 𝑠 ( 𝑢 2 1 + 𝑢 2 2 + 𝑢 2 3 ) dΩ + ∫ Ω 2 ∑ 𝛽=1 𝐽 𝛽1 (𝜔 𝛽1 ) 2 + 𝐽 𝛽2 (𝜔 𝛽2 ) 2 + 𝐽 𝛽3 (𝜔 𝛽3 ) 2 dΩ (8.14)
where 𝜚 𝑠 is the apparent mass density per unit area, 𝐽 𝑖 (the same for the two families of fibers) are related to the inertia of the cross-section of fibers and can be evaluated by the expression 𝜚𝐼 𝑖 /𝑝, being 𝜚 the apparent mass density per unit volume, 𝐼 𝑖 the second moments of area with respect to the directions 𝑫 1 , 𝑫 2 , 𝑒 𝑒 𝑒 3 .

The governing equation of the annular plate, therefore, can be deduced by the stationary-action principle, using the Lagrangian function L = 𝑈 -𝑇.

Modal Analysis

In this section, we evaluate the vibration modes of an annular plate made of a logarithmic spiral network of fibers using the second-gradient bi-dimensional model, linearized around the undeformed configuration.

The modal shapes are evaluated using the finite element software COMSOL Multiphysics, which allows the straightforward implementation of the Lagrangian function, L , based on Eq. (8.10) and (8.14). The finite element discretization employs the Argyris elements since the model is two-dimensional and is characterized by a second-gradient elastic theory. In this context, it is paramount to use interpolating functions in the Sobolev space 𝐻 2 . The same problem might be solved more efficiently with an ad hoc homemade code via the isogeometric formulation [START_REF] Schulte | Isogeometric analysis of fiber reinforced composites using Kirchhoff-Love shell elements[END_REF]- [START_REF] Cazzani | Isogeometric analysis of plane-curved beams[END_REF], discrete Hencky models based on the geometry of the microstructure of the fibers, as done in [START_REF] Turco | Nonlinear waves in pantographic beams induced by transverse impulses[END_REF]- [START_REF] Yildizdag | Modeling and numerical investigation of damage behavior in pantographic layers using a hemivariational formulation adapted for a Hencky-type discrete model[END_REF], or alternatively, as recently proposed, based on swarm robotics [START_REF] Dell'erba | Swarm robotics and complex behaviour of continuum material[END_REF], [START_REF] Dell'erba | Position-based dynamic of a particle system: a configurable algorithm to describe complex behaviour of continuum material starting from swarm robotics[END_REF].

Specifically, a representative sample of an annular plate is investigated. The fibers are assumed to have a rectangular cross-section of size 𝑏 × ℎ, being 𝑏 = 1.6 mm the base and ℎ = 1 mm the height. They are linked together by cylindrical pivots having radius 𝑟 𝑝 = 0.45 mm and height ℎ 𝑝 = 1 mm. The radius of the inner circle is set to be 𝑅 0 = 50 mm and the outer radius is 𝑅 𝑒 = 260 mm. The pitch 𝑝 is geometrically evaluated from a 3D CAD model as shown in Fig. 8.1 by considering the positions of the barycenters of the pivots. In particular, this parameter is characterized by a linear dependence on the annular plate radius according to the relationship:

𝑝 = 𝑚 √ 2 𝑋 1 2 + 𝑋 2 2 + 𝑞 (8.15)
in which 𝑞 = -4 × 10 -2 mm and 𝑚 = 0.1085.

The coefficients 𝜂 𝑖 conceived to correct the material stiffnesses are estimated by a fitting procedure made with the aim of reproducing the same modal shapes and natural frequencies of a more refined three-dimensional elastic model (see, e.g., [START_REF] Rosi | On the validity range of strain-gradient elasticity: a mixed static-dynamic identification procedure[END_REF], [START_REF] Fantuzzi | Free vibration analysis of arbitrarily shaped functionally graded carbon nanotube-reinforced plates[END_REF] for a similar approach). The obtained values are listed in Table 8.1. We assumed the material of the annular plate to be polyamide, homogeneous, and isotropic. It is characterized by Young's modulus 𝑌 = 1600 MPa and Poisson's 𝜈 = 0.4.

The apparent mass density per unit area 𝜚 𝑠 , clearly not uniform, is also estimated using the 3D CAD model. The idea is to divide the annular plate into a large number of parts within cylindrical annuli, evaluate the mass of each piece, and divide it by the equivalent area of the underlying surface. The result of such a procedure is reported in Fig. 8.2 as a function of the radius because of the symmetry of the geometry. The first twelve modal shapes associated with their natural frequencies are determined and shown in Figs. 8.3-8.5. The out-of-plane displacement is displayed by color in the figures. In the same pictures, we also exhibit the same results obtained with a refined three-dimensional model to assess the correctness of the proposed model in capturing the dynamical behavior of the considered system. Table 8.2 reports the eigenfrequencies for both cases analyzed and compares them with the corresponding errors. These results show an excellent agreement for the modal shapes. The natural frequencies are rather close for the two models considered for the first seven modes; the others are still close, but a more considerable discrepancy is detected for the remaining modes. These more significant errors, however, can be easily explained by comparing the length of the unit cell of the network to be homogenized with the wavelength of the modal shape. By increasing the frequency, these two quantities become of a compatible magnitude; therefore, the continuum approach is less suitable for accurately describing the annular plate behavior. Besides, the fact that the qualitative behavior is well captured, after all confirms some results obtained for wide-knit pantographic structures [START_REF] Spagnuolo | Are highergradient models also capable of predicting mechanical behavior in the case of wide-knit pantographic structures[END_REF], even though in a static regime.

Conclusions and Future Perspectives

In this work, the possibility of using a bi-dimensional second gradient model to describe the small oscillations of a network of curved fibers around its undeformed configuration has been examined. The fibers are characterized by a pattern that, in the reference configuration, is given by logarithmic spirals connected by a weak joint.

The results are encouraging since the dynamical response of such a model closely reproduces the behavior obtained by a more refined 3D elastic model, at least for the first modes where the homogenized formulation is significant. However, if we compare the fitting based on the dynamic quantities, i.e., modal shapes and natural frequencies, with the one obtained by static tests, we notice that the coefficients involving the flexural and twisting behavior of the fibers show some differences [START_REF] Ciallella | Deformation patterns in a second-gradient lattice annular plate composed of "spira mirabilis" fibers[END_REF]. This difference in the results of the regression for the stiffnesses is an index of some issues in the proposed model. Indeed, a consistent model must produce the same material parameters in a dynamic or a static regime. Analyzing the nature of the difference, namely related to the flexural and twisting deformation of the fibers and the current shape of the fibers in the considered modal shapes (see Fig. 8.6), it seems reasonable that the coupling between the twisting of a fiber and the bending of the connected fiber plays a considerable role in the behavior of such a system. This observation leads us to believe that the diagonal constitutive behavior given by Eq. (8.10) is too simplistic in this case. Therefore, we plan to examine, both in static and dynamic regimes, coupling constitutive behaviors between the flexural and twisting energy contributions of fibers as future developments to see if employing more comprehensive energy could overcome the detected discrepancy.

Fig. 8 .

 8 Fig. 8.1 A net prototype feasible with 3D printing.

Fig. 8 . 2

 82 Fig. 8.2 Apparent mass density per unit area vs. radius of the annular plate.

Fig. 8 . 3 :Fig. 8 . 4 :Fig. 8 . 5 :

 838485 Fig. 8.3: Modal shapes from the first to the fifth, comparison between a 3D model (a, c, e, g, i) and the reduced 2D model (b, d, f, h, j).

Table 8 . 1 :

 81 The fitted coefficients 𝜂.

	𝜂 𝑒	𝜂 𝑠	𝜂 𝑔	𝜂 𝑛	𝜂 𝑇
	1.0	1.0	0.8	1.9	0.8

Table 8 . 2 :

 82 Natural eigenfrequencies, rad/s.

	Mode #	3D model	2D reduced model	difference %
	1	167.86	162.37	-3.27
	2, 3	180.86	174.13	-3.72
	4, 5	221.52	210.42	-5.01
	6, 7	272.79	271.41	-0.51
	8, 9	322.73	353.95	9.67
	10	382.55	436.73	14.16
	11, 12	385.38	453.67	17.70

Acknowledgements

The authors express their gratitude to Victor Eremeyev for the stimulating discussions and his friendly criticism. Fig. 8.6 Zoom of a modal shape where a flexion-torsion coupling appears.