Dynamic buckling of flat and curved sandwich panels with transversely compressible core

Composite Structures 74 (2006) 10–24 www.elsevier.com/locate/compstruct

Dynamic buckling of flat and curved sandwich panels with transversely compressible core Jo¨rg Hohe b

a,*

, Liviu Librescu b, Sang Yong Oh

b

a Fraunhofer Institut fu¨r Werkstoffmechanik, Wo¨hlerstr. 11-13, 79108 Freiburg/Brsg., Germany Virginia Polytechnic Institute and State University, Department of Engineering Science and Mechanics, Blacksburg, VA, 24061-0219, USA

Available online 4 May 2005

Abstract In the present study, the effect of the transverse compressibility of the core on the transient dynamic response of structural sandwich panels under rapid loading conditions is investigated. The analysis is based on a higher-order sandwich shell theory in an effective multilayer formulation. The model is based on the standard Kirchhoff–Love hypothesis for the face sheets whereas a first/second order power series expansion is employed for the core. Consistent equations of motion and boundary conditions are derived by means of HamiltonÕs principle. An analytical solution is obtained by an extended Galerkin procedure. The theory is applied to the dynamic buckling and postbuckling analyses of plane and curved sandwich panels subjected to rapidly applied tangential and transverse loads. It is observed that the transverse compressibility of the core can have distinct effects on both, the frequency and the amplitudes of the resulting free oscillations. Due to interactions between the overall oscillation and a local oscillation in the presence of a face wrinkling instability mode, the dynamic response can be chaotic, resulting in oscillating face wrinkling instability modes with unpredictable amplitudes. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Sandwich structure; Compressible core; Higher order theory; Dynamic buckling; Face wrinkling; Transient response

1. Introduction Structural sandwich panels are important elements in modern lightweight construction. The typical sandwich panel is a layered structure according to Fig. 1 consisting of two thin high-density face sheets which are adhesively bonded to a thick core layer made from a lightweight low-density material. The face sheets can consist either of homogeneous metallic materials or of composite laminae whereas cellular structures or solid foams are common core materials. Within the principle of sandwich construction, the face sheets carry the inplane and bending loads whereas the core keeps the face *

Corresponding author. Tel.: +49 761 5142 340; fax: +49 761 5142 110/401. E-mail address: [email protected] (J. Hohe). 0263-8223/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruct.2005.03.003

sheets at their desired distance and transmits the transverse normal and shear loads. Advantage of the sandwich principle is that plates and shells with high bending stiffness and rather low specific weight can be obtained. Due to the presence of the thick core layer made from a weak material, the deformation and buckling behavior of structural sandwich panels is different compared to thin laminated composites and monolayer structures. In addition to the standard overall buckling mode, an additional local face wrinkling instability mode might develop where the face sheets buckle into the core layer whereas the entire structure might remain globally stable. In the early studies on buckling of sandwich panels, both instability modes are treated independently (see e.g. [2,18,19]). Nevertheless, due to the geometrically nonlinear nature of buckling phenomena, the two

J. Hohe et al. / Composite Structures 74 (2006) 10–24

Fig. 1. Structural sandwich panel.

different buckling effects are coupled rather than independent, as it was pointed out, among others, by Frostig et al. [7] as well as by Starlinger and Rammerstorfer [22]. Especially, if the individual buckling loads for both types of structural instability are in the same order of magnitude, strong interaction effects have to be expected. The interaction of overall buckling and the local face wrinkling instability in the static response of structural sandwich panels has been studied theoretically and numerically by da Silva and Santos [3], Wadee and Hunt [24] as well as by the present authors [9,10]. The numerical analysis of coupled instability modes of structural sandwich panels requires a higher order sandwich plate or shell theory which accounts for the transverse compressibility of the core. Different models of this type have been developed during the past decade. Frostig et al. [6,7] provide a multilayer model, which adopts the standard displacement based Kirchhoff–Love model for the face sheets whereas a stress formulation is employed for the core layer. Displacement based models which use a quadratic expansion of the in-plane displacements of the core in conjunction with a linear expansion of the transverse displacements have been provided by Dawe and Yuan [4] as well as by Pai and Palazotto [17]. Both models are geometrically linear. A geometrically nonlinear sandwich shell theory accounting for the transverse compressibility of the core has recently been proposed by the present authors [9,10], based on the geometrically nonlinear, transversely incompressible model presented by Hause et al. [8]. Alternatively to these effective multilayer approaches, Barut et al. [1] have proposed an effective single layer model. With respect to the static buckling of structural sandwich panels, a significant amount of work has been accomplished. An overview of this work can be found e.g. in the review articles by Noor et al. [16], Vinson [23] or Hohe and Librescu [11]. On the other hand, only few studies on the dynamic buckling and time-dependent excitation of laminated and sandwich structures under rapid loading conditions are available. Among the recent studies in this field, the work by Dube et al. [5] on the dynamic buckling of thick laminated spherical caps should be mentioned. Lee [12] has considered the problem of the dynamic buckling of an orthotropic

11

cylindrical shell under rapidly applied axial loads. The transient dynamic response of sandwich structures subject to transverse pressure pulses has been studied e.g. by Ma¨kinen [14] and in a recent paper by Xue and Hutchinson [25]. In another recent paper, Nayak et al. [15] have analyzed the transient dynamic response of sandwich plates at small deflections subjected to a variety of time-dependent excitations. Their study is based on a finite element implementation of ReddyÕs higherorder, effective single layer plate theory. The present paper is concerned with the dynamic buckling of structural sandwich panels under rapidly applied tangential and transverse loads. In this context, special interest is directed to the effect of the transverse compressibility of the core and a possible face wrinkling instability. The study utilizes a higher-order sandwich shell theory which has been employed previously for the analysis of static buckling (Hohe and Librescu [9,10]). The model adopts the standard Kirchhoff–Love hypothesis for the face sheets whereas a higher-order power series expansion is introduced for the core displacements. Consistent equations of motion and boundary conditions are derived by means of HamiltonÕs principle. For the problem of a simply supported sandwich panel with rectangular projection, an analytical solution is obtained by means of an extended Galerkin procedure. The model is applied to the transient dynamic analysis of plane and curved sandwich structures. It is observed that the transverse compressibility of the core has distinct effects on the transient response and especially on the eigenfrequencies. In the presence of both, global buckling and face wrinkling a vibration with chaotic nature might develop. In this case, especially the amplitudes of the local vibration and thus the resulting deformation and stress fields are unpredictable. Furthermore, it is found that the classical sandwich shell models under the assumption of a transversely incompressible core might underestimate the resulting stresses and therefore can be inaccurate.

2. Higher order sandwich shell theory Consider a doubly curved sandwich panel, which is assumed to be symmetric with respect to the global midsurface (see Fig. 1). The thicknesses of the face sheets and the core are denoted by hf and hc, respectively where hf  hc. The shell is doubly curved with radii r1 and r2, respectively. The radii of curvature are assumed to be large compared to the layer thicknesses so that the principles of shallow shell theory apply. For the analysis, a local Cartesian coordinate system xi (i = 1, 2, 3) is defined where xa (a = 1, 2) are the tangential directions whereas x3 denotes the downward normal direction. The three-dimensional displacements vi of the three principal layers of the sandwich structure are individually

12

J. Hohe et al. / Composite Structures 74 (2006) 10–24

expanded into power series with respect to x3. For the face sheets, the classical Kirchhoff–Love model is adopted. Thus, the displacements of the top face sheet are assumed in the form     hc þ hf a hc þ hf d vt1 ¼ ua1 þ ud1  x3 þ u3;1  x3 þ u3;1 2 2 ð1Þ     hc þ hf a hc þ hf d vt2 ¼ ua2 þ ud2  x3 þ u3;2  x3 þ u3;2 2 2 ð2Þ vt3 ¼ ua3 þ ud3

ð3Þ

whereas the displacements of the bottom face sheet are given by     hc þ hf a hc þ hf d b a d v1 ¼ u1  u1  x3  u3;1 þ x3  u3;1 2 2 ð4Þ 

vb2 ¼ ua2  ud2  x3 

hc þ h 2

f

  hc þ hf d a u3;2 þ x3  u3;2 2 ð5Þ

vb3 ¼ ua3  ud3

ð6Þ

where Xca are additional displacement functions describing the warping of the core. It is implicitly understood that the core warping functions Xa as well as the face sheet displacement functions uai and udi depend solely on the in-plane coordinates xa but not on the transverse direction x3. The deformation of the sandwich shell is described in terms of the nonlinear Green–Lagrange strain tensor. In the Ka´rma´n sense, only the transverse displacements are assumed to become large whereas the in-plane displacements remain small. Therefore, only the nonlinear terms related to the in-plane displacements are discarded in the definition of the Green–Lagrange strain tensor. If additionally the shallow shell approach is adopted, the components of the Green–Lagrange strain tensor are given by c11 ¼ v1;1 

1 1 2 v3 þ ðv3;1 Þ þ v3;1 v3;1 r1 2

ð12Þ

c22 ¼ v2;2 

1 1 2 v3 þ ðv3;2 Þ þ v3;2 v3;2 r2 2

ð13Þ

1 c33 ¼ v3;3 þ ðv3;3 Þ2 þ v3;3 v3;3 2 1 1 1 1 c23 ¼ ðv2;3 þ v3;2 Þ þ v3;2 v3;3 þ v3;2 v3;3 þ v3;2 v3;3 2 2 2 2

ð15Þ

where the alternative displacement functions 1 uai ¼ ðuti þ ubi Þ 2

ð7Þ

¼

ua1

hf 2x3 hf  ud3;1  c ud1 þ c x3 ua3;1 þ 2 h h

4ðx3 Þ2 ðhc Þ

2

!  1 Xc1 ð9Þ

vc2

¼

ua2

hf 2x3 hf  ud3;2  c ud2 þ c x3 ua3;2 þ 2 h h

4ðx3 Þ

!

2

ðhc Þ2

1

2x3 d u hc 3

1 1 1 1 c12 ¼ ðv1;2 þ v2;1 Þ þ v3;1 v3;2 þ v3;1 v3;2 þ v3;1 v3;2 2 2 2 2 ð17Þ where v3 denote small initial geometric imperfections with respect to the transverse direction which remain constant during the deformation history. The strain components for the individual layers are obtained by substituting Eqs. (1)–(11) into Eqs. (12)–(17). In this context, the displacement functions Xca are eliminated by the assumption that the antimetric part of the transverse shear strains cca3 of the core layer must vanish. Details are given in a previous paper by the present authors [9]. Consistent equations of motion and boundary conditions are derived from HamiltonÕs principle Z

Xc2 ð10Þ

vc3 ¼ ua3 

1 1 1 1 c13 ¼ ðv1;3 þ v3;1 Þ þ v3;1 v3;3 þ v3;1 v3;3 þ v3;1 v3;3 2 2 2 2 ð16Þ

 1 udi ¼ uti  ubi ð8Þ 2 denoting the average and the half difference of the displacements of the top and bottom face sheets are introduced for convenience. For the core displacements, a second and first order power series expansion is introduced in order to account for the larger thickness of the core layer and to include its transverse compressibility. Considering the displacement compatibility requirements at the core and face sheet interface, the core displacements are given by vc1

ð14Þ

ð11Þ

t1

ðdU  dW  dT Þdt ¼ 0

ð18Þ

t0

where dU, dW and dT are the variations of the strain energy, of the work done by the external loads and of the kinetic energy, respectively, in case of a virtual displacement duai and dudi of the sandwich structure during the

J. Hohe et al. / Composite Structures 74 (2006) 10–24

time interval [t0, t1]. The variations of the energy components dU, dW and dT are expressed in terms of the components sij of the second Piola–Kirchhoff stress tensor and the variations dcij of the Green–Lagrange strain tensor which subsequently are expressed in terms of the virtual displacements duai and dudi by virtue of Eqs. (12)–(17) with Eqs. (1)–(11). In this context, only transverse distributed loads ^ qt3 and ^ qb3 are considered. In a similar manner, only the transverse inertia effects are considered whereas the tangential inertia terms are discarded (see [9] for details). In the resulting expression, the integrals of the stress components with respect to the transverse (x3-) direction are substituted with the in-plane and bending stress resultants    n o Z h2c hc þ hf t t N ab ; M ab ¼ stab 1; x3 þ ð19Þ dx3 c 2 hf h2 n

o Z N bab ; M bab ¼

c

hf þh2

hc 2



N ci3 ; M ci3



¼

Z

hc 2 c

h2

   hc þ h f stab 1; x3  dx3 2

sti3 f1; x3 gdx3 .

ð20Þ

ð21Þ

Similar to the definitions (7) and (8) of the average and difference displacement functions uai and udi , alternative in-plane and bending stress resultants as well as transverse face sheet loads are defined by n o qa3 N aab ; M aab ; ^ 

  o 1 n t N ab þ N bab ; M tab þ M bab ; ^ ¼ qb3 qt3 þ ^ ð22Þ 2 n o qd3 N dab ; M dab ; ^ 

  o 1 n t N ab  N bab ; M tab  M bab ; ^ ¼ qb3 . qt3  ^ ð23Þ 2 The result is integrated by parts wherever feasible and the terms corresponding to the virtual displacements duai and dudi are collected. Since the virtual displacements are arbitrary and independent from each other, the corresponding coefficients must vanish independently. Thus, the following equations of motion are obtained 0 ¼ N a11;1 þ N a12;2

ð24Þ

0 ¼ N a12;1 þ N a22;2

ð25Þ

13

 

 1 ua3;11 þ ua3;12 N a12 N a11 þ 2 ua3;12 þ  ua3;11 þ r1   1 þ ua3;22 þ  ua3;22 þ N a22 þ M a11;11 þ 2M a12;12 r2



 þ M a22;22 þ ud3;11 þ  ud3;11 N d11 þ 2 ud3;12 þ ud3;12 N d12  

 1 hc þ hf  ud3  þ ud3;22 þ  ud3;22 N d22 þ c ud3 h 2

 2

  N c13;1 þ N c23;2  c ud3;1 þ  ud3;1 N c13 h    2 d 1 d c  c u3;2 þ  u3;2 N 23 þ ^qa3  mf þ mc €ua3 ð28Þ h 2  

 1 0 ¼ ua3;11 þ ua3;11 þ ua3;12 N d12 N d11 þ 2 ua3;12 þ  r1   1 þ ua3;22 þ  ua3;22 þ N d22 þ M d11;11 þ 2M d12;12 r2



 þ M d22;22 þ ud3;11 þ  ud3;11 N a11 þ 2 ud3;12 þ ud3;12 N a12  

 2 hc d d d a d  u3   þ u3;22 þ  u3;22 N 22 þ c u3 N c33 h 2   1 c d d f ^ þ q3  m þ m €u3 ð29Þ 6

0¼

where mc and mf are the integrated mass densities of the core and the face sheets respectively. Since the geometric nonlinearities with respect to the in-plane directions have been discarded, the first four equations (24)–(27) are linear whereas two nonlinear equations of motion (28) and (29) are obtained with respect to the transverse normal direction. The corresponding boundary conditions read uan ¼ ^uan

^a or N ann ¼ N nn

ð30Þ

uat ¼ ^uat

^a or N ant ¼ N nt

ð31Þ

udn ¼ ^udn

^ or N dnn ¼ N nn

ð32Þ

udt ¼ ^udt

^d or N dnt ¼ N nt

ð33Þ

d



 or ua3;n þ ua3;n N ann þ ua3;t þ ua3;t N ant



 þ ud3;n þ  ud3;n N dnn þ ud3;t þ  ud3;t N dnt þ M ann;n   1 hc þ hf d a d ^ ant;t þ 1 N ^ cn3  u3  þ 2M nt;t þ c u3 N cn3 ¼ M 2 h 2

ua3 ¼ ^ ua3

ð34Þ



 or ua3;n þ ua3;n N dnn þ ua3;t þ  ua3;t N dnt



 þ ud3;n þ ud3;n N ann þ ud3;t þ  ud3;t N ant þ M dnn;n

ud3 ¼ ^ud3

0 ¼ N d11;1 þ N d12;2 þ

1 c N hc 13

ð26Þ

0 ¼ N d12;1 þ N d22;2 þ

1 c N hc 23

ð27Þ

d

^  þ 2M dnt;t ¼ M nt;t

1 ^c M hc n3

ð35Þ

14

J. Hohe et al. / Composite Structures 74 (2006) 10–24 a

ua3;n ¼ ^ ua3;n

^ or M ann ¼ M nn

ð36Þ

ud3;n ud3;n ¼ ^

^d or M dnn ¼ M nn

ð37Þ

where xn and xt are the normal and tangential directions of the boundary. Prescribed quantities on the external boundaries are denoted by a superimposed hat symbol. The sandwich shell theory presented so far is not restricted to any kind of specific material model. Nevertheless, orthotropic linear elasticity for both, the core and the face sheets is assumed in the remainder of this study. The axes of orthotropy are assumed to coincide with the axes xi of the local coordinate system. Thus, the face sheet in-plane and bending stress resultants N fab and M fab are related to the midsurface and bending f strains cf0 ab and jab of the corresponding face sheet by 0 f 1 0 10 f0 1 c11 N 11 Af11 Af12 0 B f C B CB f0 C f ð38Þ A22 0 A@ c22 A @ N 22 A ¼ @ N f12

0

Af66

ðsym.Þ

1 0 Df11 M f11 B f C B @ M 22 A ¼ @ ðsym.Þ M f12

Df12 Df22

2cf0 12

1 jf11 CB C 0 A@ jf22 A jf12 Df66 0

10

ð39Þ

where Afij and Dfij are the coefficients of the laminae stiffness matrices obtained in the usual manner by integration of the corresponding components of the reduced three-dimensional stiffness matrix. The core stress resultants are related in a similar manner to the core midsurface strains 0 c 1 0 10 c0 1 N 33 Ac33 0 0 c33 B c C B CB c0 C c A44 0 A@ 2c23 A ð40Þ @ N 23 A ¼ @ c0 ðsym.Þ Ac55 N c13 2c13 where the stiffness coefficients Acij are obtained in a similar manner as the face sheet stiffness components Afij from the integration of the corresponding components of the three-dimensional stiffness matrix of the core layer with respect to the layer thickness.

3. Numerical solution procedure The examples considered in Section 4 are related to the dynamic buckling and postbuckling problems of a plane rectangular sandwich plate as well as doubly curved sandwich shells with rectangular projection. The edge lengths with respect to the x1- and x2-axis are denoted by l1 and l2, respectively. The sandwich panels are assumed to be simply supported along all external edges. Thus, all transverse edge displacements are constrained (^ u3 ¼ 0). The prescribed bending moments ^ nn along the external edges are assumed to vanish. M With respect to the tangential directions, either the nor-

^ nn or the corresponding displacemal edge loads N nn ¼ N ments un ¼ ^un are prescribed. Under these conditions, an appropriate representation of the transverse displacements is given by     mp a np ua3 ¼ wamn sin kam x1 sin lan x2 ; kam ¼ ;l ¼ l1 n l2



 pp qp ud3 ¼ wdpq sin kdp x1 sin ldq x2 ; kdp ¼ ; ldq ¼ l1 l2

ð41Þ ð42Þ

where m and n are the number of modal waves with respect to the x1- and x2-directions of the global instability whereas the numbers of modal waves of the face wrinkling instability are denoted by p and q. The only remaining unknown quantities with respect to the transverse displacements are the modal amplitudes wamn and wdpq . The initial geometric imperfections are assumed in the same form as the load-dependent transverse displacements     mp np  ua3 ¼ wamn sin kam x1 sin lan x2 ; kam ¼ ; lan ¼ l1 l2 ð43Þ



 pp qp  ud3 ¼ wdpq sin kdp x1 sin ldq x2 ; kdp ¼ ; ldq ¼ l1 l2 ð44Þ with the same numbers m, n, p and q of modal waves. Nevertheless, the modal amplitudes wamn and wdpq are prescribed quantities which remain constant during the loading history. For the buckling problem, the representation (43) and (44) of the geometric imperfection provides the most critical form (Seide [21]). A solution for the in-plane displacements uaa and uda which is consistent with the assumed form (41) and (42) for the transverse displacements can be obtained by substituting Eqs. (41) and (42) into the constitutive equations (38) and (40), a subsequent substitution of the core and face sheet stress resultants N ca3 ; N aab and M aab into the first four equations of motion (24)–(27) and a repeated solution of the resulting differential equations (see [9] for details). As a result, a consistent solution for the displacement field uai ðxj Þ; udi ðxj Þ is available where the modal amplitudes wamn of the overall buckling mode and wdpq the face wrinkling mode are the only remaining unknowns. The solution satisfies the first four equations of motion (24)–(27) as well as the essential boundary conditions (34)–(37) identically. The nonessential boundary conditions (30)–(33) are satisfied in an integral average sense along the respective external edge. The remaining unknowns wamn and wdpq are determined by means of an extended Galerkin procedure [8] which accounts for the non-fulfilment of the non-essential boundary conditions in a natural manner. Therefore,

J. Hohe et al. / Composite Structures 74 (2006) 10–24

the solution for the displacement field is substituted into the constitutive equations (38)–(40). The obtained expressions for the stress resultants N tab ; N bab ; M tab ; M bab and N ci3 of the top and bottom face sheets, and of the core, respectively, are substituted into the variational equation (18) and the virtual displacements duai and dudi are expressed in terms of the variations dwamn and wdpq of the modal amplitudes using Eqs. (7) and (8) as well as the corresponding consistent solution for the in-plane displacements uaa and uda . Subsequently, the integration in the variational expression (18) is performed and the coefficients for the variations dwamn and wdpq of the modal amplitudes are collected. The result is a single homogeneous linear equation for the variations of the modal amplitudes. Since the variations dwamn and wdpq of the modal amplitudes are arbitrary and independent from each other, the corresponding coefficients must vanish independently. By this procedure, a system of two linear differential equations for the deformation history in terms of the two unknown modal amplitudes wamn and wdpq and the corresponding accelerations is obtained. The system has the structure 3 X 3

i

j X € amn ðtÞ ¼ w C aij ðtÞ wamn ðtÞ wdpq ðtÞ ð45Þ i¼0

€ dpq ðtÞ ¼ w

j¼0

3 X 3 X i¼0

i

j C dij ðtÞ wamn ðtÞ wdpq ðtÞ

ð46Þ

j¼0

where t denotes the time. The coefficients C aij ðtÞ and C dij ðtÞ are lengthy expressions depending on the material properties Afij , Dfij and Acij of the face sheets and the core, respectively, on the geometric parameters hf, hc, la and ra defining the geometry of the sandwich structure as well as on the prescribed external loading history in terms of the distributed transverse normal loads ^ qa ðtÞ and a 3 d ^ ðtÞ or the ^ q3 ðtÞ as well as the tangential edge loads N nn corresponding tangential deflection ^ un ðtÞ. The initial value problem defined by Eqs. (45) and (46) is solved numerically using an explicit fourth-order Runge–Kutta scheme with variable time increments.

and a thickness of hf = 0.41 mm are assumed. The core consists of a transversely incompressible cellular structure with the transverse shear moduli Gc13 ¼ 134.5 MPa and Gc23 ¼ 51.7 MPa, the transverse YoungÕs modulus Ec3 ! 1 and a thickness of hc = 6.4 mm. The density of the core is assumed to be negligible compared to the density of the face sheets. The same problem has been considered experimentally by Raville and Ueng [20]. This study also provides a numerical eigenfrequency analysis based on a simplified model. In order to validate the numerical scheme described in Section 3, the free vibration problem is considered as a transient problem with the initial conditions € amn ðt ¼ 0Þ ¼ 0 as well as wamn ðt ¼ 0Þ ¼ 1 mm and w a d € mn ðt ¼ 0Þ ¼ 0. The transient rewpq ðt ¼ 0Þ ¼ 0 and w sponse for the five oscillation modes with the lowest eigenfrequencies is presented in Fig. 2. As usual in the application of explicit time integration schemes, care has to be exercised to chose a sufficiently small time increment in order to obtain a convergent solution. In a study of convergence, a number of 1000 equidistant time increments is found to result in a convergent solution for all considered oscillation modes. For this time increment, no increase or decrease in the amplitude is observed during the considered period of time, i.e. the modal amplitude wamn after each complete cycle is identical to the corresponding value at the begining of the cycle. Although a fine time discretization is required to obtain a convergent solution, the numerical procedure proves to be rather efficient. Due to the explicit formulation of the algorithm no solution of an implicit system of equations is required. The eigenfrequencies xmn corresponding to the eigenmode with m and n sine half waves with respect to the x1- and x2-directions are determined from the length of the corresponding first oscillation cycle (see Fig. 2). In Table 1, the first 15 eigenfrequencies are compared to the experimental and numerical results by Raville and Ueng [20]. In all cases, an excellent agreement is observed between the numerical results based on the present transient analysis and the experimental results by Raville and Ueng as well as their numerical results

4. Results 4.1. Validation For the validation of the analytical and numerical methods developed in Sections 2 and 3, the free vibration of a plane rectangular sandwich panel is considered. The panel under consideration has the in-plane dimensions l1 = 1828.8 mm, l2 = 1219.2 mm and r1 = r2 = 1, respectively. Isotropic aluminum face sheets with a YoungÕs modulus and a PoissonÕs ratio of Ef = 68.9 GPa and mf = 0.33 respectively, a density of qf = 3721 kg/m3

15

Fig. 2. Eigenfrequencies of a flat sandwich panel.

16

J. Hohe et al. / Composite Structures 74 (2006) 10–24

Table 1 Eigenfrequencies xmn [Hz] for a flat sandwich panel as considered by Raville and Ueng [20] m

n 1

3

4

5

23.0

70.7 69.0 71.0

145.8 152.0 146.0

244.4 246.0 244.0

361.7 381.0 360.0

2

44.8 45.0 45.0

91.5 92.0 91.0

165.9 169.0 165.0

263.5 262.0 263.0

3

80.3 78.0 80.0

126.1 129.0 126.0

199.1 199.0 195.0

4

129.3 133.0 129.0

173.9 177.0 174.0

5

19.3 188.0 191.0

1

2

23.4

Present study Raville and Ueng [20] (exp.) Raville and Ueng [20] (num.) Present study Raville and Ueng [20] (exp.) Raville and Ueng [20] (num.) Present study Raville and Ueng [20] (exp.) Raville and Ueng [20] (num.) Present study Raville and Ueng [20] (exp.) Raville and Ueng [20] (num.) Present study Raville and Ueng [20] (exp.) Raville and Ueng [20] (num.)

obtained in the standard manner from the solution of an eigenvalue problem. Thus, the present scheme for the transient analysis of the structural response of sandwich panels is validated. 4.2. Plane sandwich panel under in-plane compression As a first application, the dynamic buckling of a plane sandwich panel under a rapidly applied in-plane load is investigated. The sandwich panel under consideration has a square geometry with l1 = l2 = 500 mm and r1 = r2 = 1. Isotropic aluminum face sheets with a YoungÕs modulus of Ef = 70 GPa, a PoissonÕs ratio of mf = 0.3, a mass density of qf = 3700 kg/m3 and a thickness of hf = 1 mm are assumed. The core consists of an isotropic low density material with 1% of the stiffness and density as assumed for the face sheets (Ec = 0.7 MPa, mc = 0.3 and qc = 37 kg/m3, respectively). The core thickness is assumed to be hc = 18 mm resulting in a total thickness of 20 mm for the entire structure and a thickness-to-edge-length ratio of 0.04. The structure is loaded by a rapidly applied prescribed deflection of the x2-parallel edge at x1 = l1 which is increased linearly

(a)

from u^1 ¼ 0 to u^1 ¼ 20 mm during a period of Dt = 5 ms. From this time onwards, the prescribed edge load is kept constant. The second x2-parallel edge at x1 = 0 as well as the x2-parallel edge at x2 = 0 are assumed to be fixed. The upper x2-parallel external edge is freely movable within the x1  x2-plane. The initial conditions are wamn ¼ wdpq ¼ w_ amn ¼ w_ dpq ¼ 0. Overall and local dynamic buckling modes with m = 1,n = 1,p = 55 and q = 1 are assumed since these buckling modes prove to be the modes featuring the lowest total strain energy in a static buckling analysis. Initial geometric imperfections with wamn ¼ 0.25 mm and wdpq ¼ 0.001 mm, respectively, are introduced in order to regularize the bifurcation behavior. The overall static response of this structure is examined in Fig. 3a. The problem is analysed by means of the procedure presented by the present authors in an earlier contribution (Hohe and Librescu [9]). For small levels of the overall compressive load N a11 (notice that the standard sign convention that tensile loads are positive whereas compressive loads are negative is adopted throughout the present study), no transverse deflection wamn occurs. If the overall buckling load N a 11 

(b) Fig. 3. Static response of a plane sandwich panel.

J. Hohe et al. / Composite Structures 74 (2006) 10–24

17

(a)

(b)

(c)

(d) Fig. 4. Transient response of a plane sandwich panel.

1000 N/mm is exceeded, a transverse deflection of the plate with a non-zero modal amplitude wamn develops. Due to the geometric imperfection, no bifurcation instability in the rigorous Eulerian sense is observed. In Fig. 3b, the modal amplitude wdpq of the face wrinkling instability is plotted in dependence on the modal amplitude wamn of the overall instability mode, which is related to the applied edge load N a11 through the relation plotted in Fig. 3a. Similar to the overall instability problem, the modal amplitude wdpq of the face wrinkling instability mode vanishes, until the applied edge load reaches a critical level N a 11  1500 N/mm. From this point onward, a face wrinkling mode with non-vanishing modal amplitude develops.

In Fig. 4, the transient response of the sandwich structure is presented in terms of the amplitudes wamn and wdpq of the overall buckling mode and the face wrinkling mode, respectively (Figs. 4a and 4b), the resulting edge load N a11 of the edge with the prescribed edge deflection (Fig. 4c) and the resulting deflection ua2 of the external edge normal to the loaded edge (Fig. 4d). The transient response is considered during the loading process and during the first 2.5 ms of the following free oscillation. In order to investigate the effect of the transverse core compressibility (and thus the possible development of the local face wrinkling instability) results are added which are based on a standard sandwich shell theory with incompressible core (wdpq ¼ 0).

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During the first 2 ms of the loading history, the resulting edge load N a11 increases almost linearly due to the linearly increasing in-plane compression of the sandwich structure by the prescribed edge deflection (see Fig 4c). From t  1.5 ms onwards, the level N a11 of the resulting edge load exceeds the global buckling load, resulting in the development of a non-zero transverse deflection wamn (see Fig. 4a). The rapid increase of the transverse deflection from a non-equilibrium situation exceeds its static counterpart. This effect causes a temporary decrease in the level of the resulting compressive edge load N a11 and a subsequent oscillation in both, the transverse normal deflection ua1 and the resulting edge load N a11 as well as in the resulting deflection ua2 of the x1-parallel edge (see Fig. 4d). At t  3.5 ms, the level N a11 of the resulting compressive edge load for the first time exceeds the buckling load of the face wrinkling instability. From this point onwards, an oscillating face wrinkling instability with a non-zero amplitude wdpq develops (see Fig. 4b). The development of the face wrinkling instability causes an additional weakening of the structure and thus a decrease in the level N a11 of the resulting edge load. Due to the different resulting edge loads in presence and without presence of the face wrinkling instability mode, the development of the face wrinkling instability alters the overall oscillation characteristics of the system. The lower edge load levels N a11 in presence of face wrinkling cause a delay in the development of the overall transverse normal deflection wamn , resulting in an increase in the eigenfrequency related to the overall oscillation (see Fig. 4a). Furthermore, the non-harmonic resulting edge load N a11 results in a non-harmonic oscillation of the system where neither the edge load N a11 nor the transverse deflection wamn are sine functions in time. Another important feature in the dynamic buckling of the considered sandwich panel with transversely compressible core is the fact that the resulting oscillation with the two independent degrees of freedom wamn and wdpq is chaotic. Due to the geometrically nonlinear nature of the deformations in buckling problems with large deflections, the two superimposed oscillations can exchange energy and therefore can affect each other. In this context, the amplitude wdpq of the local oscillation between the second and the third cycle of the face wrinkling mode (starting at t  5.4 ms is less than the amplitude of the local oscillation between the first and second cycle starting at t  4.2 ms. The conditions for the local oscillations during the periods with vanishing average  dpq as well as for the oscillations around amplitude w  dpq (e.g. 3.5 ms 6 t 6 non-zero modal amplitudes w 4.2 ms, 4.6 ms 6 t 6 5.4 ms, etc.) strongly depend on the initial conditions at the beginning of the corresponding periods. Since these conditions are different at the beginning and end of each individual cycle, the oscilla-

tion becomes chaotic, resulting in an unpredictable structural response during a long-term load history. This effect might be crucial, since face wrinkling is a common feature in the dynamic buckling of sandwich structures. A face wrinkling vibration with unpredictable amplitude results in unpredictable local deformation and thus unpredictable local stresses. In this case, even if the resulting stresses might remain well below the static failure load, a fatigue reliability and life-time assessment of the structure becomes impossible. In contrast to static buckling, instability modes cannot only develop within the direction of the corresponding geometric imperfections but also in the opposite direction, depending on the initial conditions at the onset of the development of the instability mode. In this context, the face wrinkling instability around t = 5 ms features a negative modal amplitude wdpq although the d corresponding geometric imperfection wpq is negative.

Furthermore it can be observed in Fig. 4b that the frequency of the local oscillation is not constant but varies with time. This effect is caused by an interaction between the eigenfrequency and the applied load (see e.g. Librescu et al. [13]). In order to investigate the effect of the initial conditions at the beginning and end of the periods with a face wrinkling instability with non-zero average amplitude  dpq in more detail, the structural response of the considw ered sandwich panel during the free oscillation period with constant prescribed edge load ^u1 ¼ 20 mm is studied during a complete cycle wamn ðtÞ of the overall oscillation with different initial conditions for the local oscillation. The results are presented in Figs. 5 and 6. The initial conditions for the overall oscillation are in all cases wamn ðt ¼ 0Þ ¼ 70 mm and w_ amn ðt ¼ 0Þ ¼ 0. In Fig. 5, a local oscillation is considered where the initial conditions are given by wdpq ðt ¼ 0Þ ¼ 0.2 mm and w_ dpq ðt ¼ 0Þ ¼ 0. In addition, two alternative cases of the same oscillation are considered where the oscillation phase angles are shifted by u = 120 and u = 240. For the local oscillations considered in Fig. 6, the phase angle is in all cases u = 120 whereas the initial amplitude is varied. It is observed in Figs. 5 and 6 that the amplitudes of the local oscillations at the end of the considered period of time are rather different from the amplitudes at the beginning of the period. The differences originate from the conditions in the periods of time where the transition from the case of a vanishing average face wrinkling  dpq to a non-zero average face wrinkling amplitude w amplitude (and vice-versa) is made. Since during these periods, the resulting in-plane load N a11 is equal or close to the buckling load of the face wrinkling instability, the natural frequency of the corresponding oscillation is strongly increased. Hence, small variations in the initial conditions during the sub-period with vanishing average

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(a)

(b) Fig. 5. Effect of initial phase angle on the transient response of a plane sandwich panel.

 dpq can result in severe variations of the displacement w structural response in the following sub-period with the development of a face wrinkling instability with a  dpq . These connon-vanishing average modal amplitude w ditions form the initial conditions for the next sub-peri dpq . Thus, od with a vanishing average local deflection w the time-dependent structural response of the considered sandwich structure after one complete cycle wamn ðtÞ of the overall oscillation can be entirely different from the initial conditions at the beginning of the cycle. Thus, the initial conditions for each cycle are different resulting in chaotic oscillations.

4.3. Doubly curved sandwich shell under transverse load In a second example, the dynamic buckling of curved sandwich shells under rapidly applied transverse normal pressure loads is investigated. The sandwich shell under consideration has a square projection with l1 = l2 = 500 mm. The panel is doubly curved with r1 = 500 mm. For the second radius of curvature, two different cases are studied. The first case is the spherical sandwich cap with r2 = r1 = 500 mm whereas a doubly curved sandwich shell with r2 = 1000 mm is considered as the second case. As in the previous example,

(a)

(b) Fig. 6. Effect of initial amplitude on the transient response of a plane sandwich panel.

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(a)

(b)

(c)

(d) Fig. 7. Effect of the curvature on the transient response of a doubly curved sandwich panel.

isotropic aluminum face sheets with a YoungÕs modulus of Ef = 70 GPa, a PoissonÕs ratio of mf = 0.3 a mass density of qf = 3700 kg/m3 and a thickness of hf = 1 mm are assumed. Similar as in the previous example, the core material is isotropic with Ec = 0.7 GPa, mc = 0.3 and qf = 37 kg/m3. Nevertheless, the core thickness is increased to hc = 23 mm resulting in an overall panel thickness of 25 mm and a relative thickness of htotal/ l1 = 0.05. The sandwich structure is simply supported along all four edges. All edges are assumed to be immovable with respect to the tangential directions. The panel is loaded by a distributed transverse normal pressure on the top face sheet which is increased linearly from zero level to ^ qt3 ¼ 20 MPa during the first

2.5 ms of the loading history. Afterwards, the distributed pressure is kept constant. The buckling mode assumed in the dynamic analysis is characterized by the numbers m = 1, n = 1, p = 1 and q = 50 of sine half waves for the overall and the local buckling modes with respect to the two tangential directions respectively. Among all buckling modes, this mode involves the lowest level of overall strain energy in a static analysis and therefore is the first static eigenmode of the system. A geometric imperfection of wdpq ¼ 0.001 mm is assumed for the face wrinkling instability whereas no overall geometric imperfection is considered (wamn ¼ 0). The initial conditions are given by wamn ¼ wdpq ¼ w_ amn ¼ w_ dpq ¼ 0.

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(a)

(b)

(c)

(d) Fig. 8. Effect of the final load level in the transient response of a doubly curved sandwich panel.

In Fig. 7, the transient response is presented for both sandwich shells under consideration during the loading process as well as the during the first 2.5 ms of the following oscillation at constant applied load. The development of the transverse deflections wamn and wdpq is presented in Figs. 7a and 7b, respectively, whereas Figs. 7c and 7d are directed to the resulting average tangential edge loads N a11 and N a22 with respect to the x2- and the x1-parallel external edges, respectively. For comparison, results are added which are based on a standard sandwich model where the transverse core compressibility is not considered so that the effect of the face wrinkling instability is discarded. As it can be observed from Fig. 7a, the spherical sandwich cap with r1 = r2 = 500 mm features a higher

overall stiffness than the doubly curved panel with r1 = 1000 mm resulting in a lower global displacement wamn at the same load level ^qt3 . Due to the steeper increase in the resulting edge load N a22 for the doubly curved panel with r1 = 1000 mm, this panel is the first one where the critical load corresponding to the face wrinkling instability is reached. From t  1.2 ms onwards, an oscillating face wrinkling mode with wdpq 6¼ 0 is present for this sandwich panel (see Fig. 7b). Due to the additional end shortening caused by the development of a face wrinkling deformation with non-zero modal amplitude, the level N a22 of the resulting edge load decreases compared to the analysis under the assumption of a transversely incompressible core. Hence, the overall stiffness of the panel decreases resulting in an overall

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oscillation wamn ðtÞ with an increased amplitude as it can be observed in Fig. 7a. A similar effect occurs in case of the spherical sandwich cap with r1 = r2 = 500 mm. Nevertheless, since in this case the buckling load for the face wrinkling instability is not reached before t  2.1 ms, the effect is delayed for the spherical cap. In both cases it can be observed that the alteration in the resulting edge load due to the consideration of the transverse compressibility of the core and thus the possible development of a face wrinkling instability results in a slight decrease in the frequency of the overall oscillation. With respect to the resulting edge load N a11 along the x2-parallel edge, it can be observed in Fig. 7c that for the doubly curved sandwich panel with r1 = 1000 mm the increased amplitude wamn ðtÞ, obtained within the structural model that includes transverse core compressibility, causes an increase in the amplitudes of N a11 ðtÞ compared to the analysis based on the standard sandwich shell theory that cannot capture the face wrinkling effect. The maximum resulting edge load in consideration of the face wrinkling instability exceeds the corresponding prediction obtained within the analysis that ignores the face wrinkling instability by about 18%. Hence, the discard of transverse core compressibility during the structural analysis would result in a significant underestimation of the resulting edge loads and thus in a significant underestimation of the resulting stresses during dynamic buckling. With respect to the spherical sandwich cap with r1 = r2 = 500 mm, no such effect is observed. Due to the larger curvature of this structure and its increased stiffness compared to the doubly curved panels with different radii of curvature ra, the resulting edge load N a11 for this structure remains compressive throughout the considered loading history with a slight decrease in the level N a11 of the resulting edge load. Hence, inclusion of transverse core compressibility within the shell model can result in both increased or decreased levels of the resulting stresses, depending on the geometry and the loading conditions of the structure. In a final analysis, the effect of the external load level on the dynamic transient buckling response of curved sandwich shells is investigated. Therefore, the doubly curved sandwich shell with r1 = 1000 mm and r2 = 500 mm considered in the previous example is loaded linearly up to two different final load levels ^ qt3 ¼ 10 MPa and ^ qt3 ¼ 20 MPa on the top face sheet. The loading rate in both cases is q_ t3 ¼ 8 MPa/s as before. The results are presented in Fig. 8. It is observed in Fig. 8a that the lower final load level ^ qt3 ¼ 10 MPa results in a lower average transverse deflection wamn of the structure. On the other hand, the amplitudes of the oscillation in the overall deflection are larger for the lower load level. The amplitudes of the oscillation at constant external load level are governed by the conditions at the end of the loading period (t =

1.25 ms and t = 2.5 ms respectively). Since the velocity w_ amn at t = 2.5 ms is much larger than the corresponding velocity at t = 2.5 ms in the case of the larger final load level, the amplitudes for the following oscillation are larger at the lower load level. Another important difference between the oscillations in the global deflection wamn consists in the different natural frequencies obtained at the two final load levels. The differences are caused by load–frequency interaction effects. For the low external load level ^qt3 ¼ 10 MPa, a lower eigenfrequency is obtained since for this case both resulting tangential loads N11 and N22 are in the compressive range (see Figs. 8c and 8d). Hence, the resulting in-plane loading situation for the lower external load level ^qt3 ¼ 10 MPa is closer to any possible buckling load than the situation for the larger external load level ^qt3 ¼ 20 MPa, where the resulting tangential edge load N a11 is well within the tensile range. Strong frequency–load interaction effects are also observed in terms of the modal amplitude wdpq of the face wrinkling instability (see Fig. 8b). For the large final load level ^qt3 ¼ 20 MPa, the buckling load for the face wrinkling instability is exceeded throughout the period t > 1.2 ms. For the low final load level, the corresponding buckling load is only reached during the high-compression periods when the oscillation overall deflection wamn ðtÞ is in the vicinity of its maximum values. In the low-compression periods in between, the buckling load for development of a face wrinkling instability is not exceeded so that the face wrinkling modal amplitude wdpq ðtÞ oscillates around a vanishing mean value (see Fig. 8b). At the transition points between the high- and low-compression periods, the resulting tangential edge load N a22 coincides with the buckling load for the face wrinkling instability. The coincidence of the external load with the buckling loads results in a vanishing eigenfrequency of the corresponding vibration (see e.g. [13]). As in the example of the plane sandwich panel under in-plane compression considered in Section 4.2, this effect results in a chaotic nature of the resulting vibration with unpredictable amplitudes and thus unpredictable local stresses, especially for the face wrinkling oscillation.

5. Conclusions The present study is concerned with the dynamic buckling of flat and curved sandwich panels with transversely compressible core subjected to rapidly applied transient loads. The study utilizes a higher-order sandwich shell theory based on the standard Kirchhoff– Love hypothesis for the face sheets and a higher-order power series expansion for the core. An analytical solution is obtained by means of an extended Galerkin procedure. The transient analysis is performed numerically using an explicit fourth-order Runge–Kutta method. Although the numerical method requires rather small

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time increments for an accurate solution, the scheme proves to be numerically extremely efficient, since— due to the explicit formulation—no solution of any implicit system of equations is required during the individual time steps. The model is validated by a comparison of the eigenfrequencies for the free vibration of a rectangular sandwich panel with experimental data from the literature. An excellent agreement of the eigenfrequencies determined from the transient analysis of the vibration problem by means of the present analytical and numerical scheme with the experimental results is obtained. In the analysis, dynamic buckling of plane and doubly curved structural sandwich panels subject to rapidly applied tangential and transverse loads, a complex transient structural response is obtained. In this context, strong effects of the transverse core compressibility and the possible occurence of the local face wrinkling instability related with this mode of deformation are observed. The development of the face wrinkling instability results in an additional shortening of the structure and thus affects the resulting tangential edge loads in the case of immovable edges. A decrease in the tangential edge loads in general results in a decrease of the eigenfrequency. Therefore, a determination of the eigenfrequencies for structural sandwich panels based on a standard sandwich shell model with transversely incompressible core can yield inaccurate results if a face wrinkling instability mode develops during the loading history. Strong interaction effects between the frequency of the vibration and the applied or resulting edge loads are observed in the vicinity of the corresponding buckling loads. Especially in cases, where face wrinkling occurs only during parts of each cycle of an overall vibration with a lower frequency, the structural response becomes chaotic, since the buckling load for the face wrinkling instability with a vanishing corresponding eigenfrequency is passed twice in each cycle of the overall oscillation. During these transitions between face-wrinkling- and non-face-wrinkling-periods, small alterations and perturbations in the initial conditions can result in large alterations of the structural response in the following period. Due to the geometrically nonlinear nature of the buckling phenomenon, the superposition principle is not valid and an exchange of energy between the superimposed overall and local wrinkling vibration is possible. Hence, the amplitudes of both vibrations—especially the wrinkling vibration—after a large number of oscillation cycles during a long-term load history is unpredictable. This feature is crucial since unpredictable local deformations of the structure can cause unpredictable local stresses. Thus, the hazard of microcracking due to fatigue and therefore a life-time assessment of sandwich structures under dynamic buckling conditions might become impossible.

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Another important feature in the dynamic buckling of sandwich structures under rapidly applied load is the fact that the amplitudes of the free vibration following the transient loading process strongly depend on the conditions during the loading process. In this context, the increased flexibility of the sandwich panel due to the possible development of a face wrinkling deformation can result in an increased amplitude of the free vibration compared to an analysis where the transverse core compressibility and thus the development of a face wrinkling instability are suppressed. In this context, the increased amplitude of the overall vibration might result in an increase of the resulting local stresses. Hence, sandwich shell models without consideration of transverse core compressibility are not necessarily conservative in stress analysis during dynamic buckling, although the presence of a face wrinkling instability in general results in a decrease in the resulting stresses due to a general weakening of the structure. Therefore, several essential features in the dynamic buckling of sandwich structures are not captured by classical model with incompressible core which might yield inaccurate results. Acknowledgements This work has been financially supported by the Office of Naval Research (Composites Program) under Grant No. N00014-02-1-0594. The financial support and the interest and encouragement of the Grant Monitor, Dr. Y.D.S. Rajapakse, are gratefully acknowledged.

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