#### Abstract

Sandwich structures are able to provide enhanced strength, stiffness, and lightweight characteristics, thus contributing to an improved overall structural response. To this sandwich configuration one may associate through-thickness graded core material properties and homogeneous or graded properties nanocomposite skins. These tailor-made possibilities may provide alternative design solutions to specific problem requisites. This work aims to address these possibilities, considering to this purpose a package of three beam layerwise models based on different shear deformation theories, implemented through Kriging-based finite elements. The viscoelastic behaviour of the sandwich core is modelled using the complex method and the dynamic problem is solved in the frequency domain. A set of case studies illustrates the performance of the models.

#### 1. Introduction

Sandwich structural configurations are widely used in situation where an increased strength of the structure and bending stiffness and therefore of its buckling loads and fundamental frequencies is required. When these structures have soft and lightweight cores they confer simultaneously to the global sandwich structure a viscoelastically damped behaviour, which is very useful in several science and engineering applications, to attenuate structural vibration and noise, but also to enhance the fatigue endurance and impact resistance. Studies upon these effects and on the influence of some structural parameters were carried out recently by Wan et al. [1] and by Shen et al. [2]. The latter aspects have been addressed by Idriss et al. [3], which monitored the debonding damage length in the core-skin interface of sandwich specimens through stiffness, hysteresis cycles, and damping ratios changes.

To analyse highly damped structures, it is important to develop computational models able to adequately model their kinematics and therefore to better predict its dynamic response, which as far as possible should be verified and validated when experimental results are available. Unfortunately, validation studies require experimental resources that may not exist and in such cases simulation models are an alternative to this prediction. Researchers have considered different approaches to study these problems. Daya and Potier-Ferry [4] proposed a numerical method for the solution of nonlinear eigenvalue problems. This method associates homotopy and asymptotic numerical techniques and it was applied to the calculation of the natural frequencies and the loss factors of viscoelastically damped sandwich structures. Barkanov et al. [5] developed an inverse technique to characterize the nonlinear mechanical properties of viscoelastic core layers in sandwich panels. This technique was based in vibration tests, allowing preserving the frequency and temperature dependences of the storage and loss moduli of viscoelastic materials. The use of an optimization approach, based on the planning of the experiments and the response surface technique to minimize the error functional, enabled additionally the reduction of the computational cost. Arvin et al. [6] studied the frequency response of sandwich beams with composite faces and viscoelastic cores. The authors considered independent transverse displacements on two faces and linear variations through the depth of the beam core, and the effects of Young modulus, rotational inertia, and core kinetic energy were additionally considered to modify the ‘‘Mead & Markus” theory [7], often used on sandwich beams. The influence of parameters such as the fiber angle, the thicknesses of the faces and core on the loss factors, and natural frequencies were characterized. More recently Ferreira et al. [8] presented a layerwise finite element model for the analysis of sandwich laminated plates with a viscoelastic core and laminated anisotropic face layers. The stiffness and the mass matrices of the element are obtained using Carrera’s Unified Formulation [9], also known as CUF, and the dynamic problem is solved in the frequency domain considering a viscoelastic frequency-dependent core. CUF is based on the formulation of a fundamental nucleus of governing equations from which different theories and finite elements may arise, depending on which is considered the more adequate approach to a specific problem.

In the present work, one studies the dynamic response of soft core sandwich beams, using a set of three Kriging-based layerwise models, developed and implemented for this specific purpose. The Kriging interpolating functions are built by considering a Gaussian function and a polynomial basis as will be latter referred to on this document. These models are based on different shear deformation theories, namely, on the first-order shear deformation theory and on the higher order one due to Pandya and Kant [10]. The reason why these theories were chosen is related to its less computational cost when compared to other higher order theories without loss of performance quality. As the inner layer of these structures is considered to behave in a viscoelastic way, the corresponding elastic properties are modelled using the complex method as some researchers [4, 5, 8] have also considered. The resulting dynamic problem is then solved in the frequency domain.

As mentioned, one considers also in this work the possibility of dispersing graphene in the sandwich face sheets. As one may conclude from more recently published works, a significant research effort is undergoing different scientific areas aiming to achieve a more comprehensive knowledge about graphene, ranging from its single sheet properties characterization to the structural response enhancement potential when considering its applicability. The inclusion of such nanoparticles within a simple epoxy resin leads to a nanocomposites that may reveal higher strength- and stiffness-to-weight ratios, contributing additionally to the improvement of electrical and thermal properties, due to its multifunctional characteristics.

Because of this perception it is considered to be useful, considering the use of these materials associated to sandwich structures. As mentioned, a number of research works have been published on graphene. Atif and Inam [11] discussed the advances in the modelling and simulation of graphene-based polymer nanocomposites. This discussion considered different relevant parameters such as the graphene structure, the topographical features, the interfacial interactions, the dispersion state, the aspect ratio, the weight fraction, and finally the cross-effect among these parameters and the global performance. Another review is due to Papageorgiou et al. [12]. They presented a review dealing with the preparation and physicochemical characterization of graphene-based elastomeric nanocomposites. The processing and characterization of graphene and graphene oxide are described in detail, where the efficient dispersion of graphene in elastomers, the interfacial bonding between the filler and the matrix, or the interactions between the fillers are thoroughly analysed. The use of polymeric matrices is very common, contrarily to the metallic matrices, so the number of works focusing on metallic matrix nanocomposites is much fewer.

Ovid’ko [13] presented an overview of the research efforts both on fabrication and mechanical properties of metal-matrix nanocomposites containing graphene inclusions. A special focus was given to experimental data, with evidence for the enhancement of strength, microhardness, and the modulus of these composites when compared to both unreinforced metals and metal-matrix nanocomposites reinforced with other inclusions. Other works concerning the improvement of composite material properties due to the dispersion of nanoinclusions have been published. Srivastava [14] studied the viscoelastic and nanohardness behaviour of polymer matrix modified by the inclusion of multiwalled carbon nanotubes and graphene nanoplatelets. The experimental results he obtained have shown a significant enhancement in the decomposition temperature and nanohardness of this composite. It was also concluded on the reduction of the elastic modulus of the nanocomposites with the increase of temperature due to the solvent effect.

A study upon the mechanical properties of nanocomposites constituted by an epoxy matrix reinforced with randomly oriented graphite platelets was carried out by Cho et al. [15]. To this purpose they considered the Mori–Tanaka approach in conjunction with molecular mechanics. According to these authors, the moduli of the nanocomposites studied were strongly dependent on the aspect ratio of the reinforcing particles, but not on their size. These results compared favorably with the experimental results of several nanocomposites with different aspect ratios and sizes and graphite particles. More recently, Pedrazzoli et al. [16] carried out a study to determine the properties of hybrid composites made by epoxy reinforced with short glass fibers and exfoliated graphite nanoplatelets, as a function of the glass fibers loading. These authors concluded with the improvement of the tensile modulus, ultimate tensile strength, and impact resistance in the hybrid composites. This improvement is due to the introduction of nanomaterials at the fiber/matrix interface, which enhances the interfacial properties, leading to lighter and stronger composites. The storage modulus and the viscoelastic behaviour of the nanocomposites were also enhanced with the addition of graphene nanoplatelets which indicate a strong interaction between the nanoplatelets and the polymer and the immobilization of the polymer chains. Hadden et al. [17] developed a hierarchical multiscale modelling approach which includes molecular dynamics and micromechanical modelling to characterize the influence of the graphene nanoplatelets volume fraction and its dispersion on the properties of the hybrid composite. The results obtained by the authors allowed them to conclude that these parameters had a significant effect on the transverse mechanical properties of the composite and a not significant influence on the axial properties.

Rashad et al. [18] investigated the effect of adding graphene nanoplatelets on the mechanical properties of a titanium alloy. According to the results of the microstructural characterization achieved, they concluded that the reinforcement graphene particles were uniformly distributed in the matrix, thus acting as an effective reinforcing filler to prevent the deformation. A similar conclusion was obtained in another work of the authors (Rashad et al. [19]), about the influence of the graphene nanoparticle integration on tensile, compressive, and hardness response of aluminium. Rafiee et al. [20] developed a research on the mechanical properties of epoxy nanocomposites with a low weight fraction of graphene platelets, single-walled carbon nanotubes, and multiwalled carbon nanotube additives. The properties assessed were Young’s modulus, the ultimate tensile strength, the fracture toughness, the fracture energy, and the resistance to fatigue crack propagation. According to the authors the graphene platelets outperform carbon nanotube additives in a significant manner. This superiority concerning the mechanical properties enhancement may be related to their higher specific surface area, to an enhanced nanofiller-matrix adhesion/interlocking arising from their wrinkled surface, and to their planar geometry. Also in a recent work, Costa and Loja [21] presented a study on the static behaviour of multiscale nanocomposite plates, where the nanoinclusions can be single-walled and multiwalled carbon nanotubes. A set of parametric studies was carried out to characterize the influence of material and geometrical parameters on the static response of these plates.

In this work, the sandwich beams may have metal or metal-graphene nanocomposite face sheets, and the material properties may also vary through the thickness both for the core and face sheets. As far as the author’s knowledge, no published works were found considering the study of these types of sandwiches with through-thickness graded properties, using layerwise models wherein the interpolation of the primary variables is performed via Kriging shape functions. A set of case studies is carried out to illustrate the performance of these models in the prediction of the dynamic response of soft core sandwich beams and also to characterize the influence of some material and geometric characteristics on it.

#### 2. Materials and Methods

##### 2.1. Constitutive Relations

###### 2.1.1. Core Viscoelastic Material

The sandwich beam structures analysed in the present work are made from isotropic materials, where their skins and core, respectively, are assumed to present an elastic and viscoelastic behaviour. A schematic representation of this sandwich configuration is presented in Figure 1, where the viscoelastic core is identified by the superscript and the outer layers, by , , as they are assumed to behave elastically.

Considering the different models developed and implemented, the most general constitutive relation is written as [23]and the elastic stiffness coefficients are given aswhere , represent the Lamé elasticity constants. For the core viscoelastic material and assuming that Young’s modulus and Poisson’s ratio depend on the frequency [8, 24] one can write them according towhere the loss factor is represented by .

In the present work, one has additionally considered the possibility of a through-thickness nonhomogeneous core shear modulus. Considering two limit values for this modulus, and , respectively, associated to a softer and a stiffer core, these distributions will be described as

This assumed graded distribution is inspired on the volume fraction power law used in functionally graded materials [25, 26]. In the present case the parameter has a tuning effect on the evolution of the shear modulus through the core thickness, and stands for this layer thickness.

These distributions are represented in Figures 2 and 3, for a situation where the viscoelastic inner layer (core) is 1.27 mm thick.

To this illustration, one has also considered the shear modulus of the cores and , respectively, set to and . These shear moduli values were considered also by Arvin et al. [6] in their study although they have assumed homogeneous cores.

###### 2.1.2. Skins Nanocomposite Elastic Material

As mentioned, the outer layers of the sandwich are also considered to be isotropic, having an elastic behaviour. These skins can be made from different metals which may incorporate dispersed graphene nanoplatelets to enhance their mechanical performance. These nanoplatelets are considered to be randomly oriented and perfectly dispersed within the skins matrix material; thus the isotropy assumption continues to be valid. Assuming also the graphene nanoplatelets to act as effective rectangular solid fibers with a width (), length (), and thickness (), the Halpin-Tsai [27] equations can be modified to predict Young’s modulus of this dual-phase nanocomposite (Rafiee et al. [20]) yieldingwhere the volume fraction of the graphene nanoplatelets, , can be expressed as a function of its weight fraction and the density of the phases, and , that constitute the nanocomposite:

In these equations, , denote, respectively, Young’s modulus and the density of the metallic phase. In Figure 4 one can observe the evolution of the nanocomposite Young’s modulus for a graphene weight fraction ranging from 0 to 5%.

To obtain these curves one has considered the material and geometrical properties presented in Table 1, which will also be used in the numerical simulations. The graphene properties used are the same used by Rafiee et al. [20].

If one disperses a 0.1% weight fraction of these graphene nanoplatelets in a polymeric resin (, ) Halpin-Tsai equations give us a prediction for Young’s modulus of the composite achieved, of 3.23 GPa. The same result was obtained by Rafiee et al. [20]. Extending this calculation to the metallic materials in Table 1, if we disperse a 5% weight fraction of graphene nanoplatelets, the results obtained are presented in Table 2.

As mentioned, an effective dispersion of the graphene nanoplatelets is ensured. However it is worth referring that one knows that the improvement of the mechanical response of a composite material is greatly influenced by the quality of the nanoparticles dispersion. This dispersion is difficult to obtain due to Van der Waals forces established among the nanoinclusions and the surrounding matrix, where a number of studies upon this topic exist (Thostenson et al. [28], Hu et al. [29], and Tornabene et al. [30]). The quantitative assessment of the dispersion quality has been addressed by different researchers, namely, by Yazdanbakhsh et al. [31] and Lillehei et al. [32].

In the present work one has also considered that the weight fraction of the graphene nanoplatelets may be distributed in an adjustable, graded way through the thickness of the skins. To this purpose in this work a general formulation for the weight fraction distribution law was considered, with also some similarity to the volume fraction power law used in functionally graded materials [25]:where represents the maximum weight fraction contents and , , and are the thicknesses of the core and outer layers. The exponent allows the adjustment of the distribution to a specified desired pattern evolution. and parameters vary according to the value of the exponent and to the different layers thicknesses, as their goal is to guarantee that the maximum weight fraction never exceeds a specified value. Figure 5 presents the distribution of the weight fraction through skins’ thicknesses, for an illustrative situation where the maximum weight fraction is 2.5% and the layers thicknesses correspond to the ones in the first verification study in Section 3.1.

Although it is not compulsory in the implementation of these distributions, it is assumed in the schematic representation presented in Figure 4 that they have the same exponent in both outer layers. Poisson’s ratio and the density of the nanocomposite are determined by using the Voigt rule of mixtures:

##### 2.2. Layerwise Shear Deformation Models

The layerwise models, developed and implemented to analyse these structures, consider the first-order shear deformation theory [23] and the higher order theory due to Pandya and Kant [10]. The compatibility between the fields that describe the kinematics of each layer is achieved by assuming that the displacement at adjacent layers interfaces is continuous, where no slipping between them occurred. The compatibility equations used to obtain the layerwise description are given in the following:

These equations are used in all cases, regardless of the displacement field selected to model a specific layer.

Considering that all the three layers of the sandwich are modelled with the higher order displacement field proposed by Pandya and Kant [10], one will obtain the higher order layerwise model, here designated as HSDT-HSDT model, which has eleven degrees of freedom per node, , given by

The parameters and denote the thickness coordinates associated to the mid-plane of the elastic layers and , respectively, described in the sandwich coordinate system. The degrees of freedom , , and represent the displacement and rotation around -axis, of a point in the beam mid-plane; and denote the corresponding rotations of the elastic outer layers and the remaining terms are higher order degrees of freedom.

If one considers now the use of the first-order shear deformation theory to model the outer layers of the sandwich, maintaining however the higher order displacement field in the inner layer, the displacement field will become thenwith a total of seven degrees of freedom per node, . To identify this hybrid model, one uses the designation of HSDT-FSDT. Finally, if all the sandwich layers are modelled by the first-order shear deformation displacement field (FSDT-FSDT), the less expensive model, then one will have five degrees of freedom per node and the field is written according to

These three layerwise models were implemented using finite element models whose interpolating functions are derived from the Kriging interpolation technique. These interpolating functions can be obtained aswhere denotes the nodal values vector, associated with a considered domain, in the present case to a finite element domain. The remaining matrices, associated with the polynomial basis () and with the correlation coefficients (), respectively, are defined as

The subscript denotes the order of the polynomial basis, which in the present study was equal to three, and the basis . The other subscript parameter, , represents the number of points used for the establishment of the shape function .

The covariance coefficients of the correlation matrix were determined using the Gaussian function, , where is the Euclidean distance between two given points. The correlation coefficient was set to the value 2 (Loja et al. [33]). It is relevant to note that the use of these interpolants does not require the later use of selective integration for the shear stiffness coefficients matrix. However, an adequate selection of the correlation parameter contributes to an improved performance of the finite element to the problem under study.

##### 2.3. Equilibrium Equations

Considering the beam will be subjected to harmonic excitations, the solution of the forced vibration problem can be obtained by solving the corresponding linear system of equations for each frequency point within the desired frequency range. The equilibrium equations system is the written as follows:where is the Fourier transform of the time domain force history and , are the global stiffness and mass matrices after the boundary conditions imposition. If no excitation forces act the beam, then one will have the free vibration problem:

The vibration mode associated with th frequency is represented by . The complex eigenvalue solutionis constituted by two components, which is the real part of the complex eigenvalue and the square of th frequency and by the modal loss factor associated with that frequency.

#### 3. Results and Discussion

This section is constituted by a first subsection where one considers the use of the present models to analyse cases also studied by other researchers, and a second part where a set of other applications are considered. To these purposes, the beams were discretized in ten elements and a shear correction factor of 5/6 is used in the layers modelled by the first-order shear deformation theory (Reddy [23]). The discretization used derives from previous studies (Loja et al. [34]) where convergence analyses were also carried out.

It is relevant to a facilitated reading of the results to include here some previous notes about the use of higher order displacement approaches. The use of higher order theories allow in general describing the kinematics of the deformation in a closer way when compared to closed-form 3D elasticity theory as demonstrated by Pandya and Kant [10] among others. This is related to the constitution of the displacement field, which in this HSDT case considers a series expansion where higher order terms are not neglected. In this work, when one is considering layers that are described using FSDT, we are only incorporating a linear description of the axial normal strain and a constant shear strain description, for those layers. If one considers the case where the layers are described using the HSDT displacement field in Section 2.2, then the axial strain within those layers will be approximated by a cubic function of the thickness coordinate and the transverse shear by a quadratic one. Moreover, contrarily to the FSDT approach, in this HSDT case we have no transverse inextensibility of the beam, and therefore the transverse normal strain exists and it is approximated by a linear function of the thickness coordinate. According to this, it becomes clearer that the use of FSDT when compared with the HSDT provides a stiffening effect on the structure mechanical response.

Besides, the use of higher order theories does not require the use of shear correction factors, commonly used when the first order shear deformation theory is used.

##### 3.1. Verification Studies

###### 3.1.1. Verification Case 1

A cantilever thin sandwich beam with isotropic face sheets and a viscoelastic core, whose material and geometrical properties are presented in Table 3, is submitted to a unit impulse load at its tip. As exposed in Section 2.2, the developed models have a displacement field defined layer-by-layer and do not require a thickness discretization. The vectors of the generalized degrees of freedom contain all the contributions from the three layers. So the impulse load is applied at the transverse displacement degree of freedom at the beam end node.

The results obtained using the present models can be observed in Table 4, as well as the results obtained by Sainsbury and Zhang [22] which used a finite element model assuming that every layer has the same transverse displacement and that the deformation of the face sheets would follow the thin plate theory. They also considered Poisson’s ratio to be independent of the frequency, contrarily to the present paper, as can be seen in Section 2.1.

A reasonable agreement can be observed among the different models and the reference, taking into account the diverse assumptions in presence. As expected, in a more global appreciation, one finds higher values of frequency associated with the models (HSDT-FSDT and HSDT-HSDT) that involve the higher order displacement field. The differences between the models are not of a great significance, when comparing the fundamental frequencies; however, as expected from the typical performance of the models containing higher order terms, these differences tend to increase when higher modes are considered. Anyway, focusing on the fundamental frequency, although the differences are small (between 0.74% and 2.65%) they exist and obey the known response pattern.

Another reason for this is additionally related to the magnitude of the beam aspect ratio (). The present beam is considered as a thin beam, and, in such cases, we expect that the first-order model (FSDT-FSDT) presents values that are closer to the reference due to the assumptions considered. Concerning the higher order model and more specifically the predicted higher order frequencies, the influence of the higher order degrees of freedom is visible. These latter are not present in the first-order model ((10)–(12)).

###### 3.1.2. Verification Case 2

Another cantilever thin sandwich beam, studied by Daya and Potier-Ferry [4], was now analysed, concerning different values of the core loss factor. The beam was also submitted to a unitary impulse load at its free extremity. The material and geometrical properties of this beam are given in Table 5.

The results obtained with the different beam models, considering different loss factors values for the core, are presented in Table 6.

According to the results obtained, one may observe the same response pattern for the different models. Within each model we may also verify a consistent trend related to the core loss factor. As one observes, the relations between the loss factor associated with each vibration mode and the core loss factor decrease as this latter increases. Also for a specific core loss factor one can see that the previous ratio decreases when one considers successively increasing vibration modes, thus meaning that the energy loss has a decreasing trend with the mode orders identified. Globally we can also conclude that, for the present sandwich beam, the first-order model (FSDT-FSDT) presents a closer agreement with Daya and Potier-Ferry [4] which used an asymptotic numerical method. On their method they assumed plane strain conditions and the structure was discretized in a mesh involving six eight-node quadrilateral elements through the thickness and thirty along the length. As previously shown, in the case of the more flexible layerwise model (HSDT-HSDT) it is also visible that lower values of vibration frequencies arise for higher order modes, which may be attributed to the existence of higher order degrees of freedom in its displacement field.

##### 3.2. Case Studies

###### 3.2.1. Sandwich Beam with Different Skins’ Materials

A soft core sandwich beam with face sheets made of from different materials was analysed in the present case. The thicknesses of the core and the elastic face layers are, respectively, equal to and . The beam has the length and the width is . The materials used on the different layers of the sandwich are given in Table 7, along with the corresponding properties.

In the present study, two boundary conditions were analysed: the clamped-free configuration and the clamped-clamped one. In the first case, the beam was submitted to a unitary impulse load at its tip. In the second beam, this impulse load was applied at the midspan. Table 8 presents the results obtained with the different models, for the cantilever beam.

From Table 8 one may observe that, as expected, the frequencies increase with the relation between the stiffness and the mass of the beam. The beam with magnesium skins, being lighter and possessing also a smaller stiffness, presents the minor values of natural frequencies, though with slight differences when compared to the beam with aluminium face sheets. Contrarily, the beam with titanium outer layers shows higher frequencies. Concerning the loss factors we can conclude that the models that consider in a partial or total way the first-order shear deformation theory present a stiffer behaviour to which correspond also higher values for the frequencies and for the loss factors associated with the modes. As expected, the HSDT-HSDT layerwise model yields lower values when compared to the previous models in this first five sets of frequencies considered.

Considering stiffer boundary conditions, higher values of frequencies are expected, which is also confirmed in Table 9, for all the models. The comments related to Table 8 can also be extended to the results presented in Table 9.

###### 3.2.2. Sandwich Beam with Different Cores’ Materials

One considers now a clamped-clamped beam with the same geometric characteristics as in the previous case study. This beam was submitted to a unitary impulse load at its midspan. In the present case it is intended to analyse the influence of the core softness. To this purpose three different solutions were considered for the core. Their material properties are presented in Table 10. The properties of the skins’ materials are the same as in Table 7.

The results obtained using magnesium and titanium face sheets are, respectively, presented in Tables 11 and 12.

As it is possible to conclude, the softer core () presents a response characterized by lower vibration frequencies and lower loss factors. This is expected due to the global minor stiffness and to a minor energy loss. The sandwich beam that presents greater loss factors is the one where a moderately soft core () is used whereas the higher frequency values correspond to the stiffer core configuration (). As in the previous cases, the higher order model (HSDT-HSDT) presents frequency values that for the higher order modes differ clearly from the ones obtained when the face sheets are modelled using FSDT.

Similar conclusions can be drawn when the face sheets are made from titanium, as presented in Table 12, although the frequencies are globally higher due to the titanium properties.

###### 3.2.3. Sandwich Beam with Graded Metal-Graphene Nanocomposites’ Skins

In the present case study, one has considered sandwich beams with core, according to the previous case study designation, though with smaller aspect ratios (). The thicknesses of the core and of the elastic face layers are, respectively, set to and to . The beam has a width, . The face sheets are made from nanocomposites containing a constant weight fraction of graphene nanoplatelets (GNP) whose properties characterization is given in Table 1. In order to guarantee that a perfect dispersion of the GNP will occur, one considers low weight fractions, within the range [0⋯5]%. In this first analysis a constant weight fraction of 2.5% of GNP is dispersed within an aluminium or titanium matrix phase. The results obtained with the different models are presented in Table 13.

From this table and in addition to the conclusions already drawn respecting the different materials in the face sheets, which in the present case continue to be verified, one observes additionally the stiffening effect of the aspect ratio. In the present case, a different response pattern associated with the hybrid model (HSDT-FSDT) becomes now visible for these lower aspect ratios. In fact, when the beam becomes thicker, the use of higher order degrees of freedom to model the kinematics of the beam core allows obtaining some vibration modes that were only visible until now when using the HSDT-HSDT model, in the thinner beams’ cases. It is also possible to conclude that the loss factors are higher in the case of higher aspect ratios.

A second study within this topic was carried out considering now the graded dispersion of the GNP’s in the titanium phase. This dispersion follows the distribution law stated in (7). A beam aspect ratio of 30 was used and the GNP weight fraction maximum value was set to 5%. The results are presented in Table 14.

From these results one can conclude that concerning the three nanocomposite situations () the frequencies decrease with the increase in the parameter . This is because, for , one has a homogeneous distribution of 5% weight fraction of GNP’s within the titanium phase. In the other cases we have a graded dispersion through the thickness with different dispersion rates, though yielding the maximum value of 5% weight fraction in the outer surfaces of the face sheets. These different rates produce a different distribution of the nanoinclusions thus producing different phases’ mixtures. According to these distributions, which are exemplified in Figure 4, one concludes the dynamic response concordance with the expected. Lower loss factors are found when higher values of vibration frequencies occur. In the present case, the beam with a constant weight fraction of 5% is the stiffest, presenting higher frequencies and lower loss factors.

###### 3.2.4. Sandwich Beam with Graded Core Softness and Nanocomposite Skins

As a final case study one has considered that both the core and the skins of the sandwich beam could present a graded variation of their properties according to Section 2.1. The geometrical properties of the beam are similar to the ones in the previous case study.

This test was first conducted for the situation of the homogeneous graphene-magnesium nanocomposite skins (, ) and next to graded nanocomposite configurations (). The core was considered to go from a softer to a stiffer configuration () and the opposite sense () in a graded form adjusted by the exponent , as seen in Section 2.1. The results obtained are presented in Tables 15 and 16.

From Table 15 one can see that when one considers the transition (softer-stiffer) towards the outer surfaces, that is, when the exponent, , assumes the values 1, 2, and 5, the natural frequencies decrease and the corresponding loss factors increase. When this exponent is set to 0, there exists a homogeneous core, which presents a stiffer behaviour and consequently higher frequency values.

In the opposite way, when one considers the transition (stiffer-softer) the frequencies increase as we can see in Table 16. In this latter situation when an homogeneous core is present thus obtaining lower frequencies. Considering now also a graded profile for the graphene-magnesium nanocomposite skins and a configuration for the core, one obtains the different models and the results are presented in Table 17.

From these results it may be concluded that with the present core and skins graded configurations, one obtains an effective increase on frequency values as the exponent increases. The corresponding loss factors decrease accordingly. However, as one has seen in Table 16, the homogeneous nanocomposite has an enhanced performance concerning the fundamental frequency maximization. However if the aim is the loss factor maximization then the graded configuration () with an adjustment parameter will be preferable.

#### 4. Conclusions

Soft core sandwich structures are able to provide an enhanced strength and stiffness with a simultaneous light weight, allowing for an improved overall structural response, particularly an improved dynamic response concerning the vibration attenuation. The present work addresses this problem, considering a package of three layerwise models which were developed in order to consider the possibility of having through-thickness graded properties at both the core and the skins.

The present study has an innovative character, as the analysis of soft core sandwiches, considering through-thickness graded core material properties, using Kriging-based layerwise models, has not been object of research before. This also includes the use of metal-graphene nanocomposite skins wherein the graphene nanoplatelets are considered to be dispersed in a homogeneous or graded manner.

From the results obtained in the set of case studies considered, it is possible to conclude that the models perform in accordance with the expected according to the theories used. Also it may be concluded that the present layerwise models can constitute an alternative simulation tool.

Higher order theories present an improved quality response when thicker structures are analysed and on general situations where the shear deformation effects may be more severe. The first-order shear deformation theory is commonly used for moderately thick and thin structures, whose behaviour corresponds more closely to the theory assumptions. In accordance with this response pattern and to the verification cases considered, the implemented models present a good performance.

#### Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The author wishes to acknowledge the support given by FCT/MEC through Project LAETA, UID/EMS/50022/2013.