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Effect of core plate and panel anisotropy on natural frequency of composite sandwich shell

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https://www.eduzhai.net International Journal of Composite M aterials 2013, 3(6B): 40-52 DOI: 10.5923/s.cmaterials.201310.05 Effect of Core and Face Sheet Anisotropy on the Natural Frequencies of Sandwich Shells with Composite Faces Jörg Hohe Fraunhofer-Institut für Werkstoffmechanik IWM , 79108 Freiburg, Germany Abstract The objective o f the present study is the analysis of the effect of core and face sheet anisotropy on the natural frequencies of plane and doubly curved sandwich structures with laminated composite face sheets and an anisotropic core. For the analysis, a h igher-order sandwich shell theory is adopted. For the special case of a sandwich shell with rectangular projection, an analytical solution is obtained by means of an extended Galerkin procedure. Assuming a harmonic time-dependent response, the problem is transformed into an eigenvalue problem, which can be solved in a nu merically rather efficient manner. The numerical scheme is applied to an analysis of the effect of the face sheet anisotropy induced by fibre angle variations in laminated face sheets consisting of unidirectionally infinite fibre reinforced carbon epoxy plies. Further anisotropy effects derive from the use of honeycomb cores with anisotropy transverse shear moduli. It is observed that anisotropy of core and face sheets may have distinct effects on the lower natural frequencies. Keywords Sandwich St ructures, Co mposite Face Sheets, Anisotropy, Analytical Model, Natural Frequencies 1. Introduction Structural sandwich panels are important elements in modern lightweight structures. The typical sandwich panel is a layered structure according to Figure 1. It consists of three principal layers where two high-density face sheets are adhesively bonded to a low density core. The face sheets carry all in-plane and bending loads whereas the core keeps the face sheets at their desired distance and transmits the transverse norma l and shear loads. The advantage of the sandwich principle is that plates and shells with high bending stiffness may be constructed at an extremely low specific weight (Vinson[13], Zen kert[16]). Typical face sheet materials are thin metal sheets or – especially in high performance applications – co mposite laminates consisting of unidirectionally infin ite carbon or glass fib re rein fo rced p lastic p lies. Depend ing on the stacking sequence of the laminates, the face sheets may feature a d ist inct an isotropy wh ich may b e exp lo ited towards the design of tailored structures with optimized properties complying with any kind of prescribed structural requirement. The core is usually made fro m a weak, low density material such as balsa wood, so lid foam o r a t wo -d imens ion al hon ey co mb ty pe cellu lar st ru ct u re. Es p ecially hon ey co mb co res may feat u re a d is t in ct anisotropy since their transverse shear moduli with respect * Corresponding author: joerg.hohe@iwm.fraunhofer.de (Jörg Hohe) Published online at https://www.eduzhai.net Copyright © 2013 Scientific & Academic Publishing. All Rights Reserved to the two in-plane d irections are usually not identical (Gibson and Ashby[3]). The dynamic response and vibration of sandwich structures is a challenging problem since not only the external geo metry of the plate or shell but also the anisotropy of core and face sheets may affect the response of the structure. Following the pioneering experimental study by Raville and Ueng[11] in 1967, increasing interest has been directed to the vibration of sandwich plates and shells especially during the past two decades. Bardell et al.[1] have analysed the free vibration of plane isotropic sandwich p lates with different external shapes using the fin ite element method. The free v ibration of sandwich plates with laminated anisotropic face sheets has been investigated by Zhou and Li[17] as well as by Kant and Swaminathan[8] using different quasi-analytical models. Yuan and Dawe[14] emp loyed the spline finite strip method to the analysis of the eigenmodes of plane sandwich plates with laminated composite faces. In a later study, this approach has been extended to the problem of plane sandwich plates with stiffeners on one side (Yuan and Dawe[15]). Since the deformat ion behaviour of sandwich structures with soft cores in general is much more co mplex than the response of sandwich structures with stiff, transversely inco mpressible cores, a number of studies use higher-order sandwich models for the analysis of the free vibration problem. Soko linsky et al.[12] have analysed the lower natural frequencies of straight sandwich beams with isotropic core and faces using Frostig’s soft core sandwich beam model. More recently, a study on the effects of the transverse core flexibility on the v ibrational response of International Journal of Composite M aterials 2013, 3(6B): 40-52 41 two-dimensional sandwich plates with quasi-isotropic core and faces using an extended flexib le core model has been provided by Frostig and Thomsen[2]. Meunier and Shenoi[9] have been concerned with a higher-order model accounting for damp ing effects whereas Nayak et al.[10] emp loyed Reddy’s higher-order model. Both studies are concerned with p lane sandwich plates. However, in contrast to the previous studies, anisotropy effects are included for both, the core and the face sheets. In a similar manner, Hause and Librescu[4] analysed the vibration of anisotropic sandwich plates based on an earlier version of the present sandwich shell theory (Hohe and Librescu[5],[6]). Again, the study is restricted to plane sandwich plates. In a preceding study by the present author (Hohe et al.[7]) on the transient response of sandwich structures during and after rapid loading, the effect of curvature has been included. Nevertheless, the study is again restricted to sandwich panels made fro m isotropic materials. Objective of the present contribution is an analysis of the effects of the anisotropy of core and face sheets on the lower natural frequencies of plane sandwich plates as well as cylindrical and doubly curved sandwich shells. Special interest is directed to interaction effects between the external shape and the local core and face sheet anisotropy. The analysis is based on a general model for curved sandwich shells presented earlier by the present author (Hohe and Librescu[5],[6]). The origina l nonlinear model is simp lified and adapted to the requirements of the present linear problem. Based on an extended Galerkin procedure, an analytical d isplacement solution is derived. Assuming harmonic oscillations as the only relevant type of displacement, an eigenvalue problem for the natural frequencies in different eigen modes is obtained. The problem is solved numerically by means of the Newton-Raphson method. In parametric studies the effect of core and face sheet anisotropy on the lower natural frequencies for a sandwich shell with carbon epoxy face sheets and honeycomb core is studied. It is observed that the core and face sheet anisotropy may have different effects on the different eigenfrequencies. Figure 1. Doubly Curved Sandwich Shell 2. Sandwich Shell Model 2.1. Basic Assumptions For the analyses of the present study, the general shell model fo r shallo w sandwich shells presented earlier (Hohe and Librescu[5],[6]) is adopted. The model is re-fo rmulated into a simp lified version comp lying with the requirements of the present problem. Consider a plane or curved sandwich shell according to Figure 1. The face sheets are assumed to consist of composite laminates with constant thickness tf. The core thickness tc is also uniform but much larger than the face sheet thickness tf. The sandwich panel is assumed to be doubly curved with radii of curvature ri which are much larger than the panel thickness so that the conditions of shallow shell theory are satisfied (Figure 1). For the analysis of the sandwich structure, a local Cartesian system xi is introduced, where x1 and x2 are the in-plane direction whereas x3 is the transverse norma l direction. Both, the core and the face sheets are assumed to be linear elastic but anisotropic. For the core, the effective material parameters are assumed to be known directly whereas the face sheet material response is assumed to be given in terms of the laminate stiffness matrix of the classical laminate theory. Since the deflections in the free vibration problem in general are s mall, a geo metrically linear analysis is sufficient. No geometric imperfections are considered, although both features are included in the original model (Hohe and Librescu[5],[6]). 2.2. Kinematic Relati ons For a p rojection of the shell deformat ion behaviour onto the reference surface, the three dimensional displacements are expanded into a power series in terms of x3. Since the material response and the thickness of core and face sheets are rather d ifferent, an effective mu ltilayer model is adopted, treating the three principal layers (core, top and bottom face sheet) separately. Since the face sheets are thin, the Kirchhoff-Love model is adopted for the faces. Thus, the face sheet displacements are given by ( )( ) u1t = u1a + u1d − x3 + t +t f 2 u3a,1 + u3d,1 ( )( ) u2t = u2a + u2d − x3 + t +t f 2 u3a,2 + u3d,2 (1) u3t = u3a + u3d for the top face and ( )( ) u1b = u1a − u1d − x3 − t +t f 2 u3a,1 − u3d,1 ( )( ) u2b = u2a − u2d − x3 − t +t f 2 u3a,2 − u3d,2 (2) u3b = u3a − u3d for the bottom face, where ( ) uia = 1 2 uit + uib ( ) uid = 1 2 uit − uib (3) are the average and the deviation from the average of the 42 Jörg Hohe: Effect of Core and Face Sheet Anisotropy on the Natural Frequencies of Sandwich Shells with Composite Faces face sheet mid -s u rface face sheet d is p lacemen ts u t i and uib. For the core, a higher o rder power series expansion of the displacements is employed in order to account for the transverse compressibility of the central layer in the weak core limit. Considering the compatibility requirements of the core and face sheet displacements at the interfaces at x3 = ±tc/2, the core displacements read u1c = u2c = ( ) u1a − tf 2 u3d,1 − 2 tc x3u1d + tf tc x3u3a,1  +  2 x3 tc 2 −  1 Ω1c  ( ) u2a − tf 2 u3d,2 − 2 tc x3u2d + tf tc x3u3a,2  +  2 x3 tc 2  −1 Ωc2  (4) u= 3c u3a − 2 tc x3u3d where Ω c i are additional d is p lacemen t fu n ctio n s . These additional degrees of freedom describe a quadratic displacement through the layer th ickness in addition to the mid-surface displacements as well as the rotations and thus account for the warp ing of the core. ε 11 = u1,1 − 1 r1 u3 ε 22 = u2,2 − 1 r2 u3 ε 33 = u3,3 (5) ( ) ε 23 = 1 2 u2,3 + u3,2 ( ) ε13 = 1 2 u1,3 + u3,1 ( ) ε12 =1 2 u1,2 + u2,1 are obtained for the three principal layers. 2.3. Equati ons of Motion In the next step, equations of motion have to be determined, wh ich are consistent with the assumptions made in the kinematic considerations. A natural manner to determine an inherently consistent shell theory is the use of Hamilton’s variational principle ∫ (t1 t0 δ U − δW − δT )dt = 0 (6) Fro m Equations (1) to (4), the strains for the three where δU, δW and δT are the variat ions of the strain energy, principal layers are obtained by substituting the expressions the work by the external loads and the kinetic energy into the geometrically linear kinemat ic relation. Under the respectively whereas[t0, t1] is an arbitrary time increment. assumption of the shallow shell limit, the strains In the context of the present multilayer model and the corresponding simplify ing assumptions, the variation of the strain energy is defined by ∫ ∫ ∫ ∫ = δU    − tc 2 σαt βδεαt β dx3 + tc 2 tc 2 +tf σ ic3δε c i3 dx3 + σ b αβ δεαbβ   dx3  dA (7) A  −t f − tc 2 − tc 2 tc 2  where A is the reference surface area of the structure under consideration. As usual, α, β = 1, 2 whereas i, j = 1, 2, 3. The work of the in-plane stresses within the core layer is neglected. Assuming that only trans vers e normal dis tributed loads q3t* and q3b* act on the s urfaces of the top and bottom face sheets , the variation of the work by the external loads read ( ) ∫ = δW q3t*δ u3t + q3b*δ u3b dA A ∫ ∫ ∫ ∫ +   − tc 2  σ t nα δ uαt dx3 + tc 2 tc 2 +t f σ nc3δ u3c dx3 + σ b nα δ uαb   dx3  dxt (8) xt   −tf − tc 2 − tc 2 tc 2   where xn and xt are the normal and tangential direct ion of a local Cartesian coordinate system along the external boundary of the sandwich shell where the prescribed stresses σij* are acting. For the kinetic energy, an additional simp lification is introduced by neglecting all in-plane and rotational inert ia effects since in the free vibration problem, the transverse motion within the x3-direction is the dominant mode of deflect ion. With this assumption and the mass densities ρc and ρf for the core and the face sheets respectively, the variation of the kinetic energy becomes ∫ ∫ ∫ ∫   − tc 2 δT = −ρ f u3tδ u3t dx3 − tc 2 ρ cu3cδ u3c dx3 − tc 2 +tf ρ f u3bδ u3b dx3    dA. (9) A   −tf − tc 2 − tc 2 tc 2   Determining the virtual strains δεij for the three principal layers of the sandwich structure using Equation (5) and substituting the result together with the shell kinematics (1) to (4) into Hamilton’s princip le (6) with the variations of the strain energy, work by the external loads and kinetic energy according to Equations (7) to (9) results in a lengthy International Journal of Composite M aterials 2013, 3(6B): 40-52 43 variational expression. Within this expression, the stresses and the explicit powers of x3 are the only terms wh ich depend on the transverse direction. Hence the stress resultants for the three principal layers { } = Nαt β , Mαt β { ( )} ∫− tc 2 σ t αβ 1, x3 + tc +tf 2 dx3 −tf −t2c { } Nic3, M c i3 − tc 2 = ∫ σ c αβ {1, x3} dx3 (10) −tf −t2c { } = Nαbβ , Mαbβ { ( )} ∫tf + tc 2 σ b αβ 1, x3 − tc +tf 2 dx3 tc 2 with the alternative definit ion { } ({ } { }) = Nαaβ , Mαaβ 1 2 Nαt β , Mαt β + Nαbβ , Mαbβ (11) { } ({ } { }) = Nαdβ , Mαdβ 1 2 Nαt β , Mαt β − Nαbβ , Mαbβ for the face sheets similar to Equation (3) are introduced, by wh ich the dependence of the equation on x3 is eliminated. The equation is integrated by parts wherever possible and the terms with equal dependence on the virtual displace ments δuia and δuid are collected. As a result, a single linear homogeneous equation for the v irtual displacements is obtained. Since the virtual d isplacements are arbit rary and independent, the corresponding coefficients must vanish independently. Fro m the coefficients in the area integral, the equations of motions 0 = N1a1,1 + N1a2,2 0 = N1a2,2 + N a 22,2 0= N1d1,1 + N1d2,2 + 1 tc N1c3 0= N1d2,2 + N d 22,2 + 1 tc N c 23 (12) ( ) ( ) 0 = 1 r1 N1a1 + 1 r2 N 2a2 + M1a1,11 + 2M1a2,12 + M a 22,22 + tc +tf tc N1c3,1 + N c 23,2 + q3a* − mf + 1 2 mc u3a ( ) 0 = 1 r1 N1d1 + 1 r2 N 2d2 + M1d1,11 + 2M1d2,12 + M d 22,22 + N3c3 + q3d* − mf + 1 6 mc u3d are obtained. In a similar manner, the boundary conditions una = una* or Nnan uta = uta* or Nnat und = und* or Nndn utd = utd* or Nndt u3a = u3a* or M a nn,n + 2M a nt ,t + tc +tf tc N c n3 = N a* nn N a* nt N d* nn N d* nt M a* nt ,t + 1 2 N nc3* (13) u3d = u3d* or M d nn,n + 2M d nt ,t = M ndt*,t + t1c M c n3 u3a,n u3d,n = u3a,*n or M nan = u3d,*n or M ndn M a* nn M d* nn follow fro m the boundary integral. The same equations would be obtained by discarding all geo metrically nonlinear terms in the corresponding equations of motion and boundary conditions of the orig inal v. Kármán model (Hohe and Librescu[5],[6]). Notice that so far all equations of the model are independent fro m the material behaviour of core and face sheets. 2.4. Material Model In the present study, sandwich panels with laminated, 44 Jörg Hohe: Effect of Core and Face Sheet Anisotropy on the Natural Frequencies of Sandwich Shells with Composite Faces infinite fibre reinforced face sheets and orthotropic cores are considered. The two face sheets are assumed to be identical and symmetric with respect to their indiv idual central surface. Hence, their materia l response is defined by   N 1a1 N a 22 N1a2  ,     N1d1 N 2d2 N1d2    A1f1 =  (sym) A1f2 A2f2 A1f6 A2f6 A6f6   ε1a1 ε a 22 2ε1a2  ,     ε1d1 ε d 22 2ε1d2     M1a1 M a 22 M1a2   ,    M1d1 M d 22 M1d2   =  D1f1   (sym) D1f2 D2f2 D1f6 D2f6 D6f6   κ1a1 κ a 22 κ1a2   κ1d1 ,    κ d 22 κ1d2   (14) in terms of the average and half difference mid-p lane strains and curvatures εαβa, εαβd, καβa and καβd respectively. The components of the Aijf and Dijf matrices are determined in the usual manner fro m the integration of the components of the reduced stiffness matrices of the indiv idual plies of the face sheets. For the linear elastic, o rthotropic core, the material equations are derived in a similar manner. The material response is defined by  N3c3   A3c3 0 0  ε c 33    N 2c3 N1c3   =  0 0 A4f 4 0 0 A5f5   ε c 23 ε1c3    M c 33   D3c3 0 0 κ3c3  (15)   M c 23 M1c3   =   0 0 D4c4 0 0 D5c5   κ c 23 κ1c3   where εi3c and κi3c are the mid-plane strains and curvatures of the core. The matrix coefficients are determined similar as for the face sheets. 3. Solution Procedure With the kinematic relat ions (1) to (5), the material equations (14) and (15), the equations of motion (12) and the boundary conditions (13), a co mp lete set of equations for the dynamic problem of doubly curved sandwich panels is available. Since the problem in general cannot be solved in closed form, a nu merical solution procedure is required. In order to be able to perform parametric studies in a numerically efficient manner, the present study employs an extended Galerkin p rocedure for the solution. For this purpose, the following considerations are restricted to plane or doubly curved sandwich shells with rectangular projection with edge lengths l1 and l2 with respect to the x1 and x2-directions. Furthermore, the study is restricted to sandwich shells which are simp ly supported along all e xterna l boundaries. In this case, the form ( ) ( ) u3a = wma n sin λamx1 sin µna x2 ( ) ( ) u3d = wpaq sin λdpx1 sin µqd x2 (16) with = λma = ml1π , µna pπ l2 (17) = λpd = pl1π , µqd qπ l2 is an appropriate assumption for the transverse d is p lacemen ts . In Equations (16) and (17), w a mn and wpqd are the modal amplitudes, whereas m, n, p and q are the numbers of sine half waves with respect to the x1 and x2-directions for the respective eigenmode of the free vibration. In the next step, a solution for the in-plane displacements u1a, u1d, u a 2 and u2d is derived, wh ich is co n s isten t with the assumed form (16) and (17) of the transverse displacements. Following the procedure utilized in the previous studies (Hohe and Lib rescu[5],[6]), the assumptions (16) and (17) are substituted into the first four equations of the set (12) of the equations of motion. By this means, a consistent solution for the unknown displacements is obtained which satisfies the first four equilibriu m conditions exactly. The simp ly supported boundary conditions with respect to the transverse deflections u3a and u3d are satisfied identically. All other boundary conditions are satisfied in an integral average sense. The only remain ing unknowns are the modal amp litudes wmna and wpqd of the transverse displacements. Following the concept of the extended Galerkin procedure, the assumption (16) and (17) for the transverse displacements together with a similar assumption for the virtual tran s v ers e displacements δw a mn and δwpqd and together with the consistent solution for the in -plane displacements is substituted into Hamilton’s principle (6) together with the expressions (7) to (8) for the indiv idual virtual energy terms and the stress resultants (10) and (11). The stress resultants are expressed through the material equations (14) and (15) in terms of the strains and curvatures of the three principal layers which are substituted with the displacement exp ressions in terms of the modal amp litudes using the kinematic relations (5) together with Equations (1) to (4). As a result a single homogeneous linear equation for the two v irtual modal amp litudes δwmna and δwp d q is obtained. Since the virtual modal amp litudes are arbitrary and independent from each other, the corresponding coefficients must vanish independently, yielding a set of two coupled second order differential equations for the unknown modal amplitudes wmna and wpqd as a function of time. In contrast to previous studies based on a v. Kármán type nonlinear approach (Hohe and Librescu[5],[6]), a much more simp le linear system is obtained since all geometrical nonlinearit ies were discarded in the present study. The system may be solved as an init ial value prob lem, similar as in preceding studies (e.g. Hohe et al.[7]). In the present study concerning the free vibrations of sandwich structures, a different approach is employed. Assuming harmonic oscillations, the modal amp litudes may be postulated in the form International Journal of Composite M aterials 2013, 3(6B): 40-52 45 wma n = wˆ ma n sin(ω t) wpdq = wˆ pdq sin(ω t) (18) where ŵmna and ŵpqd are the amplitudes and ω is a constant. Substituting Equation (18) into the governing system for the amp litudes wmna and wpqd constitutes a system of the type ( )( )  K11 + ω 2 K21 K12 K22 + ω 2  wˆ ma n wpdq   =  00  (19) where K11, K12, K21 and K22 are coefficients depending on the geometry and material constants of the sandwich panel as well as on the numbers m, n, p and q of the sine half waves in the eigenmode accord ing to Equation (16). Since the system (19) is linear and ho mogeneous, non-trivial solutions for the amplitudes ŵmna and ŵpqd can only exist, if the determinant of the system matrix vanishes. Hence, the natural frequency f = ω 2π (20) for the mode with the numbers m, n, p and q of sine half waves is determined through the smallest positive solution ( )1 ω2 = − 12 ( K11 + K22 ) ± 1 4 ( K11 − K22 )2 + K12 K21 2 (21) of the characteristic equation. 4. Examples 4.1. Vali dati on In a first application, the model is validated against experimental results fro m literature. In their now classical study, Raville and Ueng[11] have reported measurements of the first natural frequencies for simply supported plane rectangular sandwich plates. The panel consists of isotropic alu min iu m face sheets bonded to a honeycomb core. The geometry and the material properties are summarized in Table 1. Table 1. Validation Example (Raville and Ueng[11]) l1[mm] 1828.8 E[GPa] 68.948 G23 [MP a] 51.7 external geometry l2[mm] 1/r1[mm-1] 1219.2 0.0 face sheets (aluminium) ν[-] tf[mm] 0.33 0.46 core (honeycomb) G13 [MP a] tc[mm] 134.5 6.35 1/r2[mm-1] 0.0 ρ f[k g/m3] 3721 ρ c[k g/m3] ≈ 0 The experimental results of Raville and Ueng together with the corresponding numerical results obtained by the procedure described in the previous section are presented in Figure 2. For all considered eigenmodes, an almost perfect agreement is obtained. The largest deviation of experimental and numerical data is obtained for the mode with m = 1 and n = 5 with 5.4%. Hence, the present procedure proves to be accurate. 4.2. Plane Sandwich Plates To study the effect of core and face sheet anisotropy on the natural frequencies and the corresponding eigenmodes, the structural sandwich model is applied in parametric studies concerning different types of sandwich structures. As a first examp le, a p lane square sandwich plate is considered. The plate is assumed to consist of laminated carbon epoxy face sheets with eight plies in a symmetric [0°/±ϑ/90°]s stacking sequence. The materia l data is chosen such that the material is characteristic for a carbon epoxy material with appro ximately 50% fibre vo lu me fraction as it might be processed by a resin transfer mould ing process which becomes increasingly popular for industrial scale applications e.g. in the automotive industry. The core material is chosen in the characteristic range for alu min iu m honeycomb core with anisotropic transverse shear properties. The geometry and material data chosen as a starting point for the parametric studies is summarized in Table 2. Figure 2. Validation 46 Jörg Hohe: Effect of Core and Face Sheet Anisotropy on the Natural Frequencies of Sandwich Shells with Composite Faces Figure 3. Plane Sandwich Panel – Effect of the Transverse Core Stiffness Figure 4. Plane Sandwich Panel – Effect of the Core and Face Sheet Anisotropy International Journal of Composite M aterials 2013, 3(6B): 40-52 47 Table 2. Basic Geometry and Material data for the Parametric Studies l1[mm] external geometry l2[mm] 1/r1[mm-1] 1/r2[mm-1] 1000.0 1000.0 0.0 0.0 El[GPa] face sheets (carbon epoxy,[0°/±ϑ/90°]s) Et[GPa] νlt[-] Glt[GPa] ρf[kg/m3] tply[mm ] 114.0 10.0 0.34 5.4 1550 0.25 core (aluminium honeycomb) G23[MPa] G13[MPa] E3[MPa] ρc[kg/m3] tc[mm] 186.0 98.6 234.0 50.0 30.0 In a first parametric study, the effect of the transverse shear moduli of the core is investigated. For this purpose, the natural frequencies for the eigen modes with (m, n) = (1, 1), (m, n) = (1, 2), (m, n) = (2, 1) as well as (m, n) = (2, 2) which are assumed to be the leading eigen modes are computed for a variety of values fo r the t ransverse shear moduli G23 and G13. The fibre angle ϑ is kept constant at ϑ = 45 so that the face sheets are quasi-isotropic. All other material and geometry parameters are kept constant at their designated values according to Table 2. The results are presented in Figure 3. In all modes, a distinct drop in the first natural frequencies is observed in the weak core limit for G23,G130. In this case, the core loses its stiffness so that the limit case of two uncoupled laminated plates is approached. Due to the decreased stiffness, the eigenfrequencies decrease as well. Fo r large transverse shear moduli G23 and G13, the increasingly stiff core requires an increasing amount of in -plane stretching and compression of the face sheets, since the increasing transverse core stiffness increasingly constrains the relat ive lateral displacements of the face sheets whereas in the weak core limit with vanishing transverse core stiffness, the two face sheets may bend with respect to their indiv idual mid-surfaces rather than with respect to the mid-surface of the entire sandwich structure as in the strong core limit. Depending on the anisotropy of the core, a different order of the four eigen modes with respect to the corresponding eigenfrequencies develops. In this context, e.g. for G23 = 100 M Pa and small G13, f21 is the second natural frequency whereas f12 is the third one. For G13 > 100 M Pa and thus G13 > G23, the two eigenfrequencies exchange their roles and f12 becomes the second eigenfrequency whereas f21 becomes the third one. Hence, for standard honeycomb cores with non-isotropic transverse shear moduli, care has to be taken with respect to its assembly direct ion since a rotated assembly of the core might affect the order of the natural frequencies and thus might result in another eigenmode to become the critica l one. In the next parametric study, the effect of the core and face sheet anisotropy is studied in mo re detail. For this purpose, the fibre angle ϑ is varied over the entire interval[0°, 90°]. In this context, ϑ = 0° constitutes a face sheet layup with six layers orientated towards the x1-direction and only two layers within the x2-direction. Hence, for ϑ = 0°, the x1-direct ion is the strong direction whereas x2 is the weaker direction. For ϑ = 90°, the directions exchange their roles. The case ϑ = 45° constitutes the case of quasi-isotropic face sheets. Five different rat ios G23/G13 for the transverse shear moduli are considered, where the two shear moduli are chosen such that the average transverse shear modulus is (G23 + G13)/2 = 140 MPa. Again, a p lane sandwich plate with all other properties according to Table 2 is considered. The results are presented in Figure 4. For the first eigenmode corresponding to the natural frequency f11, only minor effects of the anisotropy of the core and the face sheets are observed. The eigenfrequency f12 increases with increasing fibre angle ϑ and thus increasing stiffness within the x2-direction forming the direction with two sine half waves (and thus the direction with the shorter modal wave length). The opposite effect is observed for the eigenfrequency f21, since in this case, the number of modal waves within the x1- and x2-directions have been exchanged. Due to the core anisotropy the curves for these two eigenmodes are not obtained as mirror image of each other, except for the case G23/G13 = 1, when the core becomes isotropic. Again, it is observed that the natural frequencies f12 and f21 exchange their order depending on the core and face sheet anisotropy. Hence, care has to be taken in an optimization of the laminate stacking sequences for an improvement of either the stiffness or static strength of the structure, since a variation in the anisotropy of the structure – although possibly advantageous for the static response – might have disadvantageous effects on the dynamic response. Especially, eigen modes, which were in itially non critical might become the leading ones. The effect of the core an isotropy on the natural frequencies depends on the eigenmode considered. As it can be observed in Figure 4, the core anisotropy ratio G23/G13 has a stronger influence on the eigenfrequencies for the t wo modes with m = 2, co mpared to the other two modes. Since in the current parametric study, G23 is larger than G13 (except for G23/G13 = 1), the x2-direction is the direction supplied with an increasing stiffness with increasing deviation of the anisotropy ratio fro m G23/G13 = 1. On the other hand, compared to the modes with m = 1, the eigenmodes with m = 2 feature a shorter modal wave length within the x2-direction. Thus, an increasing core shear stiffness towards this direction results in an increasingly constrained deformation, causing the stronger effects of G23/G13 observed in Figure 4 fo r f21 and f22. 4.3. Curved Sandwich Shells In further parametric studies, the effect of the anisotropy of the honeycomb core and the co mposite face sheets on curved sandwich structures is investigated. Both, cylindrical sandwich shells with either ρ1 = 1/r1 ≠ 0 or ρ2 = 1/r2 ≠ 0 (and the other radius equal to zero) and doubly curved sandwich shells with both ρ1 ≠ 0 and ρ2 ≠ 0 are analysed. 48 Jörg Hohe: Effect of Core and Face Sheet Anisotropy on the Natural Frequencies of Sandwich Shells with Composite Faces Figure 5. Doubly Curved Sandwich Shell – Effect of the Curvature Figure 6. Doubly Curved Geometries In a first study, the effect of the curvature on the natural frequencies is investigated. As an examp le, a sandwich shell with square project ion, quasi-isotropic laminated faces and honeycomb core according to Table 1 is considered. The curvatures ρi are varied over the interval[-1 m-1, 1 m-1]. The results for the first four eigenmodes are presented in Figure 5. Figure 6 gives a sketch on the geometry ranges co ns id ered . In general, a strong effect of the curvature radii on the natural frequencies is observed, since the curvature of the structures strongly influences their (structural) stiffness. The spherical sandwich cap with ρ1 = ρ2 features the strongest structural stiffness. Hence, large eigenfrequencies for all eigenmodes are obtained especially in these cases. Decreasing natural frequencies are obtained, when the geometry approaches the cylindrical case with either ρ1 = 0 or ρ2 = 0. Fu rther decreases are obtained in the limit of the plane sandwich plate with ρ1 = ρ2  0. Nevertheless, for the first natural frequency f11, similar values as for the plane plate are obtained for sandwich saddle shells with ρ1 = -ρ2. Due to the core anisotropy with G23 ≠ G13 (see Table 2), the results for f12 and f21 are dissimilar despite the symmetry of the external geomet ry of the structures under consideration. It is observed that the order of the eigenfrequencies especially for these two eigenmodes depends on the ratio ρ1/ ρ2 of the curvatures, since different curvatures constitute a different structural stiffness with respect to the two spatial directions x1 and x2 and thus different (structural) constraints on the respective deformation. In order to study the effect of the anisotropy of the honeycomb core and the laminated co mposite face sheets in more detail, similar parameter studies as in Figure 4 for the case of a plane sandwich plate are performed for the cases of a cy lindrical sandwich shell with ρ1 = 1 m-1 and ρ2 = 0, a spherical sandwich cap with ρ1 = ρ2 = 1 m-1 as we ll as for a sandwich saddle structure with ρ1=-ρ2= 1 m-1. As before, the fibre angle ϑ is varied over the interval[0°,90°] considering five different core shear stiffness ratios G23/G13 with an average of (G23 + G13)/2 = 140 MPa. All other parameters are kept constant at the values compiled in Table 2. The results are presented in Figures 7, 8 and 9 res p ectiv ely . International Journal of Composite M aterials 2013, 3(6B): 40-52 49 Figure 7. Cylindrical Sandwich Shell – Effect of Core and Face Sheet Anisotropy Figure 8. Spherical Sandwich Cap – Effect of the Curvature

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