Free vibration and stability analysis of three-dimensional sandwich beams via hierarchical models

Abstract

This paper presents a free vibration and a stability analysis of three-dimensional sandwich beams. Several higher-order displacements-based theories as well as classical models (Euler–Bernoulli’s and Timoshenko’s ones) are derived assuming a unified formulation by a priori approximating the displacement field along the cross-section in a compact form. The governing differential equations and the boundary conditions are derived in a nucleal form that corresponds to a generic term in the displacement field approximation. The resulting fundamental nucleo does not depend upon the approximation order N that is a free parameter of the formulation. A Navier-type, closed form solution is used. Simply supported beams are, therefore, investigated. Slender up to very short beams are considered. As far as free vibrations are concerned, the fundamental natural frequency as well as natural frequencies associated to torsional and higher modes such as sheet face bending and twisting (typical of sandwich structures) are investigated. The stability analysis is carried out in terms of critical buckling stress in the framework of a linearised elastic approach. Results are assessed towards three-dimensional FEM solutions. It is shown that upon an appropriate choice of the approximation order, the proposed models are able to match the three-dimensional reference solutions.

Introduction

In the last decades, sandwich structures have found more and more applications as primary and secondary structural elements in several engineering fields because of their attracting specific properties in stiffness and strength. Besides non-conventional configurations, the faces are, usually, made of composite materials or lightweight metal alloys, whereas the core can be made of carbon or metallic foams, honeycomb made of metallic alloys or Nomex, balsa or plastic materials. The high difference in thickness, stiffness and inertia between the face sheets and the core results in and highly influences “local” face sheets driven modes. An accurate mechanical modelling of these structures is, therefore, very challenging. For these reasons, the formulation of higher-order sandwich models is an interesting and up-to-date research field.

To the best of the authors’ knowledge, one of the first examples of sandwich structures used in civil engineering can be found in Fairbairn [1], whereas Di Taranto [2] was one of the first to present a free vibration analysis of sandwich beams. A first general tractation of sandwich structures can be found in Allen [3]. An assessment of several higher-order and zig–zag displacement-based theories for the free vibration and buckling analysis of laminated composite and sandwich beams was presented by Zhen and Wanji [4]. Several layer-wise models for instability analysis of sandwich beams were assessed by Hu et al. [5]. Kapuria et al. [6] proposed a third-order zig–zag theory for the static, free and forced vibration and buckling analysis of composite and sandwich beams.

The analytical free vibration analysis of sandwich and laminated composite beams was carried out by Kameswara Rao et al. [7] considering a full third-order model. Banerjee [8] developed an analytical solution for the free vibration analysis using the dynamic stiffness method (DSM). The first four modes of a clamped sandwich beam were investigated. The sheet faces were modelled according to Euler–Bernoulli’s kinematic hypotheses, whereas the core was assumed to have shear stiffness only. Within the framework of DSM, Rayleigh’s and Timoshenko’s displacement fields were considered by Banerjee and Sobey [9] for the skins and core mechanics, respectively, whereas Damanpack and Khalili [10] modelled the skins as Euler–Bernoulli’s beams and the axial and transverse displacement components of the core as cubic and quadratic polynomials in the through-the-thickness coordinate. By imposition of the compatibility conditions at layers interface, this latter model was reduced to seven unknown parameters. A family of sinus-refined finite elements for the free vibration analysis of composite beams was presented by Vidal and Polit [11]. Shear correction factors are not required since a global sinusoidal variation of the axial displacement over the cross-section yields a cosine term in the transverse shear strain. A local (layer by layer) polynomial variation is also adopted. The number of models unknowns are reduced by a priori imposing the displacements congruency and the transverse shear stresses equilibrium at layer interfaces.

Volokh and Needleman [12] modelled each layer of a sandwich beam as a Timoshenko’s beam to investigate the buckling when delamination has not yet occurred but the interfaces cannot be considered as perfectly bonded. Ji and Waas [13] used a two-dimensional elasticity approach to investigate the local instabilities (wrinkling and edge buckling) in orthotropic sandwich beams under several boundary conditions. A one-dimensional finite element approach was instead used by Hu et al. [14]. The layer-wise kinematic field by Léotoing et al. [15], [16] was adopted. It consists in Euler–Bernoulli’s kinematics for the skins and a cubic and a quadratic expansion for the core displacement components. The total number of unknown parameters is reduced to five by imposing the compatibility conditions at layers interface (as done in Damanpack and Khalili [10]) and, also, assuming a linear transverse shear stress across the core thickness. The resulting non-linear equations were solved by means of the asymptotic numerical method (see Damil and Potier-Ferry [17]) and results were assessed towards analytical as well as two-dimensional solutions.

The present paper addresses a compact and effective manner to formulate several one-dimensional models for the free vibration and linearised elastic stability analysis of isotropic sandwich beams. These models are derived via a Unified Formulation (UF) that has been previously presented for plates and shells (see Carrera [18], Carrera and Giunta [19] and Giunta et al. [20]) and lately extended to beams, see Carrera et al. [21], [22], Giunta et al. [23], [24], [25] and Catapano et al. [26]. In particular, the free vibration analysis of laminate composite beams was discussed in Giunta et al. [27]. This paper intends to extend that analysis to sandwich beams. This type of structures is made of two materials presenting very different (up to several orders of magnitude) stiffness and inertial properties. The stiff skins are also very thin when compared to the rather thick weak core. This results, as it will be shown by the proposed results, in peculiar vibration modes such as sheet faces bending and symmetric and antisymmetric twisting. An accurate prediction of the frequencies related to these modes calls for higher-order models able to account for the high cross-section deformation occurring within the soft core. It should be noted that classical models consider the cross-section to be rigid on its own plane. A linearised elastic stability analysis is also presented. To the best of authors’ knowledge, a linearised stability analysis within the present UF was first done by D’Ottavio and Carrera [28] for laminated plates and shells. The governing differential equations and the corresponding boundary conditions are obtained via the Principle of Virtual Displacements (PVD). Thanks to a concise notation for the displacement field, they are written in a ‘nucleal’ form that does not depend upon the approximation order. Non-classical deformations, such as warping, can have a significant influence on the response of beams, see Bishop et al. [29]. Via this formulation, classical theories can be easily enhanced in order to account for transverse shear, cross-section in- and out-of-plane warping and rotatory inertia. Classical models, such as Euler–Bernoulli’s (EBT) and Timoshenko’s (TBT), can be obtained as special cases. Governing differential equations are solved via a Navier-type, closed form solution. As far as the free vibrations are concerned, classical as well as higher frequencies are investigated. Analyses are carried out for several length-to-thickness ratios in order to study the natural frequencies and modes and evaluate the accuracy of the proposed theories. The critical buckling stress is also determined. Numerical results show that accurate results can be obtained with reduced computational costs.