Dynamic response of anisotropic sandwich flat panels to explosive pressure pulses

ARTICLE IN PRESS

International Journal of Impact Engineering 31 (2005) 607–628

Dynamic response of anisotropic sandwich flat panels to explosive pressure pulses$ Terry Hausea, Liviu Librescub,* a

Mechanical and Manufacturing Engineering, St. Cloud State University, St. Cloud, MN 56301-4498, USA b Engineering Science and Mechanics Department, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0219, USA Received 6 January 2003; received in revised form 12 January 2004; accepted 14 January 2004

Abstract This paper addresses the problem of the dynamic response in bending of flat sandwich panels exposed to time-dependent external blast pulses. The study is carried out in the context of an advanced model of sandwich structures that is characterized by anisotropic laminated face sheets and an orthotropic core layer, and a closed form solution of the dynamic response to a variety of blast pulses is provided. A detailed analysis of the influence of a large number of parameters associated with the particular type of pressure pulses, panel geometry, fiber orientation in the face sheets and, presence of tensile uni/biaxial edge loads on dynamic response is carried out, and pertinent conclusions are outlined. r 2004 Elsevier Ltd. All rights reserved. Keywords: Anisotropic panel; Laminated face sheets; Dynamic response; Explosive blast

1. Introduction The advanced combat aircraft is likely to be exposed, during its operational life, to more severe environmental conditions than in the past. In this sense, blast loads occurring from fuel and nuclear explosions, gust and sonic boom pulses, are likely to act on their structure. The same is valid with military naval vessels whose structure operate under harsh environments and are subjected to severe dynamic loading. Having in view the great advantages related to the use of sandwich structures in the construction of advanced supersonic/hypersonic flight vehicles, reusable space transportation $

Recommended by N. Jones. *Corresponding author. Tel.: +1-540-231-5916; fax: +1-540-231-4574. E-mail address: [email protected] (L. Librescu).

0734-743X/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2004.01.002

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systems, as well as in marine constructions, the study of the dynamic response of sandwich structures to time-dependent external pulses and investigation of conditions resulting in the alleviation of their detrimental effects presents an evident importance toward a rational design of these constructions. In spite of this fact, as the most recent survey papers on sandwich constructions fully attest (see Refs. [1–4]), and with the exception of results displayed in Ref. [5], where the core of the considered sandwich panel is assimilated to a very thick ply in a laminated thick plate, there are not available studies in the specialized literature addressing this topic. The goal of this paper is to fill this gap and provide an approach of dynamic response problem of flat sandwich panels to external time-dependent pressure pulses. In this context, an advanced model of flat sandwich panels is considered in the sense that the face sheets are composed of anisotropic material layers, while the soft core is transversally orthotropic. Moreover, in spite of the complexity of the problem, for a variety of time-dependent blast loads, closed form solutions of the dynamic response are provided.

2. Basic assumptions In this paper, the case of a flat sandwich panel is considered (see Fig. 1). The global mid-plane of the structure s is selected to coincide with that of the core layer. Its points are referred to a curvilinear and orthogonal coordinate system xa ða ¼ 1; 2Þ: The through-the-thickness coordinate x3 is considered positive when measured in the direction of the inward normal. For the sake of convenience, the quantities affiliated with the core layer are identified by a superposed bar, while those associated with the bottom and top face sheets are identified by single and double primes, respectively. Consistent with this convention, the uniform thickness of the core is denoted % while those of the upper and bottom face sheets as h00 and h0 ; respectively. As a result, as 2h; Hð 2h% þ h0 þ h00 Þ is the total thickness of the structure (see Fig. 1). Towards developing the theory of sandwich flat panels, the following assumptions are adopted: (i) the face-sheets are composed of orthotropic material laminae, the axes of orthotropy of the individual plies being rotated with respect to the geometrical axes xa of the structure; (ii) the

Fig. 1. Geometry of the sandwich panel.

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material of the core features orthotropic properties, the axes of orthotropy being parallel to the geometrical axes xi ; (iii) the core layer is capable of carrying transverse shear stresses only, and as a result we deal with a weak core; (iv) a perfect bonding between the face sheets and the core, and between the constituent laminae of the face-sheets is assumed to be valid; (v) the assumption of incompressibility in the transverse normal direction for both the core and face sheets is adopted; (vi) the constituent layers of the faces are assumed to be thin, and as a result, transverse shear effects are neglected in the face sheets; (vii) the structure as a whole, as well as both the top and bottom laminated face-sheets are assumed to exhibit (mechanical and geometrical) symmetry properties with respect to both the mid-plane of the core layer, and about their own mid-planes, and finally; (viii) the blast loads considered in this study are of small to moderate intensities, in the sense of not inducing the damage of the structure.

3. Kinematics In order to be reasonably self-contained, several basic kinematic relationships derived in Refs. [5–7] will be supplied next. 3.1. The 3-D displacement field in the face sheets and core In agreement with the previously stipulated assumptions, the transverse displacement should be uniform through the thickness of the laminate, this implying ð1Þ V30 ðx1 ; x2 ; x3 ; tÞ ¼ V300 ðx1 ; x2 ; x3 ; tÞ ¼ V% 3 ðx1 ; x2 ; x3 ; tÞ  v3 ðx1 ; x2 ; tÞ: Upon discarding transverse shear effects in the face-sheets, the 3-D distribution of the displacement field fulfilling the kinematic continuity conditions at the interfaces between the core and face sheets result as (see Refs. [6–9]), 0 % ð2aÞ V10 ðxa ; x3 ; tÞ ¼ x1 ðxa Þ þ Z1 ðxa ; tÞ  ðx3  aÞqv3 ðxa ; tÞ=qx1 ðhpx 3 ph% þ h Þ; V20 ðxa ; x3 ; tÞ ¼ x2 ðxa ; tÞ þ Z2 ðxa ; tÞ  ðx3  aÞqv3 ðxa ; tÞ=qx2 ;

ð2bÞ

V30 ðxa ; x3 ; tÞ ¼ v3 ðxa;t Þ;

ð2cÞ

% 1 ðxa ; tÞ þ ðh=2Þqv3 ðxa ; tÞ=qx1 g ðhpx % % V% 1 ðxa ; x3 ; tÞ ¼ x1 ðxa;t Þ þ ðx3 =hÞfZ 3 phÞ;

ð3aÞ

% 2 ðxa ; tÞ þ ðh=2Þqv3 ðxa ; tÞ=qx2 g; V% 2 ðxa ; x3 ; tÞ ¼ x2 ðxa ; tÞ þ ðx3 =hÞfZ

ð3bÞ

V% 3 ðxa ; x3 ; tÞ ¼ v3 ðxa;t Þ;

ð3cÞ

and V100 ðxa ; x3 ; tÞ ¼ x1 ðxa ; tÞ þ Z1 ðxa ; tÞ  ðx3 þ aÞqv3 ðxa ; tÞ=qx1

% ðh%  h00 px3 p  hÞ;

ð4aÞ

V200 ðxa ; x3 ; tÞ ¼ x2 ðx2 ; tÞ  Z2 ðxa ; tÞ  ðx3 þ aÞqv3 ðxa ; tÞ=qx2 ;

ð4bÞ

V% 003 ðxa ; x3 ; tÞ ¼ v3 ðxa;t Þ:

ð4cÞ

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In these equations, Vi ðxa ; x3 Þ are the 3-D displacement components in the directions of the coordinates xi : In addition xa ¼ ðV( 0a þ V( 00a Þ=2;

Za ¼ ðV( 0a þ V( 00a Þ=2

ð4d; eÞ

denote the 2-D tangential modified displacement measures, where V( 0a and V( 00a denote the tangential displacements of the points of the mid-planes of the bottom and top face sheets, respectively. For a symmetric sandwich panel, h0 ¼ h00  h define the thickness of the bottom and top face sheets, while a0 ¼ a00 ¼ að h% þ h=2Þ are the distances between the global mid-plane of the structure and the mid-planes of the bottom and top face sheets, respectively. Using the representations of the displacement quantities given by Eqs. (2)–(4) in conjunction with the linearized 3-D strain–displacement relationships, one obtains the distribution of threedimensional strain quantities eij across the panel thickness. In the bottom facings, these are: 9 e011 ¼ e011 þ ðx3  aÞk011 > = % e022 ¼ e022 þ ðx3  aÞk022 for ðhpx 3 ph% þ hÞ; > ; 2e012 ¼ g012 þ ðx3  aÞk012

ð5a–cÞ

in the core layer, e%11 ¼ e%11 þ x3 k% 11 e%22 ¼ e%22 þ x3 k% 22 2e%12 ¼ g% 12 þ x3 k% 22 2e%13 ¼ g% 13 ;

2e%23 ¼ g% 23

9 > > > = > > > ;

% % for ðhpx 3 phÞ;

ð6a–eÞ

and in the upper facings, 9 e0011 ¼ e0011 þ ðx3 þ aÞk0011 = > 00 00 00 % e22 ¼ e22 þ ðx3 þ aÞk22 for ðh%  hpx3 p  h=hÞ: > ; 2e0012 ¼ g0012 þ ðx3 þ aÞk0012

ð7a–cÞ

In Eqs. (5)–(7) the 2-D strain measures are expressed in terms of displacement measures as: for the bottom face sheets: e011 ¼ x1;1 þ Z1;1 g012 ¼ x1;2 þ x2;1 k011 ¼ v3;11 k012 ¼ 2v3;12 ;

ð1$2Þ; þ Z1;2 þ Z2;1 ð1$2Þ;

ð8a–dÞ

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for the core layer: e%11 ¼ x1;1 g% 12 ¼ x1;2 þ x2;1  1 Z1;1 þ 12hv3;11 k% 11 ¼ h%  1 Z1;2 þ Z2;1 þ hv3;12 k% 12 ¼ % h  1 Z1 þ 12hv3;1 þ v3;1 g% 13 ¼ % h

ð1$2Þ;

ð1$2Þ;

ð9a–eÞ

ð1$2Þ;

and for the top face sheets: e0011 ¼ x1;1 þ Z1;1

ð1$2Þ;

g0012 ¼ x1;2 þ x2;1  Z1;2  Z2;1 k0011 ¼ v3;11

ð1$2Þ;

ð10a–dÞ

k0012 ¼ 2v3;12 : It should be remarked that the displacement measure xa and Za as defined by Eqs. (4d,e) belong to stretching and bending problems, respectively. The sign ð1$2Þ accompanying the previous equations that will be used in the remaining of the paper as well, indicates here that the expressions of strain quantities not explicitly written can be obtained from the ones displayed above, upon replacing subscript 1 by 2, and vice-versa. Henceforth, the Greek indices have the range 1, 2 while the Latin indices have the range 1–3, and unless otherwise stated, the Einstein summation convention over the repeated indices is employed. One of the goals of this paper is to explore the problem of determination of the dynamic response of sandwich flat structures that incorporates the directional property of the materials of the face sheets. In particular, this property of fibrous composites can be used to enhance the dynamic response behavior of such structures. In the following sections, the basic equations of sandwich plate theory incorporating the anisotropy of the individual face sheets, and transverse shear effects in the core layer are displayed only to the extent that they are needed in the treatment of the subject considered in this paper.

4. Equations of motion and boundary conditions Before starting to address the problem of determination of the equations of motion and of boundary conditions, one essential remark should be done. This is related to the fact that, in the context of the assumptions of symmetry and linearity of the structural model, the associated stretching and, bending problems result to be decoupled (see e.g. Ref. [9]). Being interested in the dynamic response in bending only, the equations and the quantities related to the stretching motion are discarded.

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The Hamilton’s principle is used to derive the equations of motion and the boundary conditions of the bending theory of sandwich flat panels. This variational principle may be stated as Z t1 dJ ¼ d ðU  W  TÞ dt ¼ 0 ð11aÞ t0

to which Hamilton’s condition stating that dZa ¼ dv3 ¼ 0 at t0 and t1 should be added, where t0 ; t1 are two arbitrary instants of time; U denotes the strain energy; W denotes the work done by external loads; T the kinetic energy of the 3-D body of the sandwich structure, while d denotes the variation operator. The expressions of U; W and T are provided next. Assuming that the constituent material of face sheets exhibit monoclinic symmetry, for weak core sandwich structures we have Z (Z hþ Z h% % h%0 1 0 0 0 # Qabor eab eor dx3 þ Q# 00abor e00ab e00or dx3 dU ¼ d 2 a % 00 h% hh ) Z h% ð11bÞ þ Q% a3o3 e%a3 e%o3 Þ dx3 ds; h%

dW ¼

Z

ð11cÞ

q3 dv3 ds; s

Z

t1

dT dt ¼ m0 t0

Z

Z s

t1

v.3 dv3 ds:

ð11dÞ

t0

In Eq. (11b)

 Qab33 Q33or # Qabor  Qabor  Q3333

ð11eÞ

denote the modified elastic moduli, while m0 is the reduced mass per unit mid-plane area defined as Z hh Z h% Z hþh % 0 % m0 ¼ r% dx3 þ r00ðkÞ dx3 þ r0ðkÞ dx3 ; ð11fÞ h%

h%

h%

r being the mass-density of constituent materials. From Eq. (11a) considered in conjunction with the proper expression of various energies, Eqs. (11b–d), and with the strain–displacement relationships (used as subsidiary conditions) one derive the equations of motion and the boundary conditions. To this end, the following steps should be carried out in Eq. (11a): (i) collection of terms associated to the same virtual displacements; (ii) integration with respect to x3 and integration by parts wherever necessary as to relieve the virtual displacements dZa and dv3 of any differentiation; (iii) consideration of the expressions of global stress resultants and stress couples (to be defined later); and (iv) invoking the arbitrary and independent character of variations dZ1 ; dZ2 ; dv3 : As a result of the previously described procedure, by retaining only the transversal load and transverse inertia terms, the

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equations of motion associated with the bending problem are: dZ1 : L11;1 þ L12;2  N% 13 ¼ 0; dZ2 : L22;2 þ L12;1  N% 23 ¼ 0; 0 0 dv3 : N11 v3;11  N22 v3;22 þ M11;11 þ 2M12;12 þ M22;22 % % þ ð1 þ C1 =hÞðN13;1 þ N% 23;2 Þ þ q3 ðxa;t Þ  c’v3  m0 v.3 ¼ 0:

ð12a–cÞ

0 0 In Eq. (12c), N11 and N22 are the normal edge loads, considered to be positive in compression, while c denotes the transverse viscous damping coefficient. As concerns the associated boundary conditions at the edge xn ¼ const: ðn ¼ 1; 2Þ; these are

Lnn Lnn ¼ B

or

Zn ¼ Z n ;

Lnt ¼ B Lnt

or

Zt ¼ Z t ;

Mnn ¼ M B nn

B

B

or v3;n ¼ B v 3;n ;

% N% n3 ¼ M nt;t þ N% n3 Mnn;n þ 2Mnt;t þ ð1 þ C1 =hÞ B B

or v3 ¼ B v 3:

ð13a–dÞ

Here the subscripts n and t are used to designate the normal and tangential in-plane directions to an edge and hence, n ¼ 1 when t ¼ 2; and vice versa, while the terms underscored by a tilde denote prescribed quantities. It should be noticed that in the special case of flat sandwich panels involving bending only, four boundary conditions have to be prescribed at each edge. This implies that, consistent with the number of boundary conditions, the system of equations governing the bending of sandwich flat panels should be of the eighth order. In Eqs. (12) and (13) the global stress resultants and stress couples are defined as 0 00 þ N11 N11 ¼ N11

ð1$2Þ;

0 N12

00 N12 ¼ þ N12 ; 0 00 % 11  N11 Þ L11 ¼ hðN 0 00 % 12  N12 Þ; L12 ¼ hðN 0 00 M11 ¼ M11 þ M11 0 00 M12 ¼ M12 þ M12 ;

ð1$2Þ; ð1$2Þ; ð14a–fÞ

where C1 ¼ ðh0 þ h00 Þ=4:

ð14gÞ

In addition, ð Þ;a denotes the partial differentiation with respect to surface coordinates xa ; while q3 ðxa ; tÞ denotes the distributed transversal load. In Eqs. (14), the stress resultants and stress couples associated with the bottom facings are n0 Z ðx3 Þk X 0 0 ðs0ab Þk f1; x3  ag dx3 ð15Þ fNab ; Mab g ¼ k¼1

ðx3 Þk1

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and with the weak core is N% a3 ¼

Z

h%

h%

s% a3 dx3 :

ð16Þ

The stress resultants and stress couples for the upper facings can be obtained from Eq. (16) by replacing single primes by double primes, a by a and n0 by n00 : Herein, n0 ð¼ n00 Þ denote the number of constituent layers of the bottom and upper facings, respectively, while ðx3 Þk and ðx3 Þk1 denote the distances from the global reference plane (coinciding with that of the core layer) to the upper and bottom interfaces of the kth layer, respectively. Assuming the upper and lower face sheets to be symmetric with respect to both their mid-planes and with respect to the mid-plane of the entire structure, and considering that the materials of the face sheets exhibit monoclinic symmetry and that the core material is orthotropic, one obtain the constitutive equations. These equations are displayed next. For the bottom face-sheets these are: 0 0 0 0 ¼ A011 e011 þ A012 e022 þ A016 g012 þ E11 k011 þ E12 k022 þ E16 k012 N11

ð$2Þ;

0 0 N12 ¼ A016 e011 þ A026 e022 þ A066 g012 þ E16 k011 0 0 0 0 0 0 0 0 0 M11 ¼ E11 e11 þ E12 e22 þ E16 g12 þ F11 k11 0 0 0 0 0 0 0 0 0 M12 ¼ E16 e11 þ E26 e22 þ E66 g12 þ F16 k11

ð1$2Þ;

0 0 E26 k022 þ E66 k12 ; 0 0 0 0 F12 k22 þ F16 k12 0 0 0 0 F26 k22 þ F66 k12 :

þ þ þ

ð17a–dÞ

The stiffness quantities appearing in Eqs. (17) are defined as fA0or ; B0or ; D0or g

¼

fA00or ; B00or ; D00or g

¼

n0 Z X k¼1

ðx3 Þk

ðx3 Þk1

ðQ# 0or ÞðkÞ ð1; x3 ; x23 Þ dx3

ðo; r ¼ 1; 2; 6Þ; ð18Þ

while 0 Eor ¼ B0or þ aA0or ;

0 00 For ¼ D0or þ 2aB0or þ a2 A0or ¼ For :

ð19a; bÞ

The expression of stress resultants and stress couples for the upper facing can be obtained from their counterparts associated with the bottom faces by replacing the single prime by double primes. In the case when bottom and upper facings feature full symmetry about their own midsurfaces (i.e. of the case considered in this paper), the exact vanishment of terms associated with 0 00 and Eor is occurring. the stiffnesses Eor For the weak core layer considered as an orthotropic body (the axes of orthotropy coinciding with the geometrical axes), the constitutive equations are N% 13 ¼ 2h%K% 2 Q% 55 g% 13 ;

N% 23 ¼ 2h%K% 2 Q% 44 g% 23 ;

where K% 2 is the shear correction factor.

ð20a; bÞ

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5. Governing equations for the flexural motion The equations of flexural motion of sandwich plates encompassing the previously mentioned effects and expressed in terms of displacement quantities are obtained as: dZ1 :A11 Z1;11 þ A66 fZ1;22 þ Z2;12 g þ A12 Z2;12 þ A16 f2Z1;12 þ Z2;11 g þ A26 Z2;22  d1 fZ1 þ av3;1 g ¼ 0; dZ2 :A22 Z2;22 þ A66 fZ2;11 þ Z1;12 g þ A12 Z1;12 þ A26 f2Z2;12 þ Z1;22 g þ A16 Z1;11  d2 fZ2 þ av3;2 g ¼ 0; dv3 :  F11 v3;1111  F22 v3;2222  2F12 v3;1122  4F16 v3;1112  4F26 v3;1222  4F66 v3;1122 0 0 þ d1 afZ1;1 þ av3;11 g þ d2 afZ2;2 þ av3;22 g  N11 v3;11  N22 v3;22  c’v3  m0 v.3 þ q3 ¼ 0:

ð21a–cÞ As is clearly seen, the governing equations associated with bending are expressed in terms of displacement quantities Z1 ; Z2 and v3 : In Eqs. (21) 2K% 2 G% 13 2K% 2 G% 23 d1 ¼ ; d2 ¼ ð22a; bÞ h% h% and Aor ¼ A0or þ A00or : 6. Discretization of the governing equations The problem of determination of dynamic response of sandwich panels to blast pulses reduces to the solution of an associated boundary value problem. Due to the intricacy of the governing equations and of boundary conditions, one should discretize these equations. This process will be carried out via application of the extended Galerkin method (EGM). The method will be illustrated for the case of rectangular sandwich panels simply-supported all-over the contour. For the bending problem, the boundary conditions along the edges xn ¼ 0; Ln ; are Zn ¼ 0; Zt ¼ 0; Mnn ¼ 0; v3 ¼ 0:

ð23a–dÞ

The first step in this process is to fulfil identically both Eqs. (21a,b) and the boundary conditions (23c,d), and fulfil Eq. (21c) and boundary conditions (23a,b) in the Galerkin sense. To this end, the transverse displacement v3 ðx1 ; x2 ; tÞ is represented as to fulfil identically the boundary conditions (23c,d). The pertinent representation is v3 ðx1 ; x3 ; tÞ ¼ qmn ðtÞ sin lm x1 sin mn x2 ;

lm ¼ mp=Li ; mu ¼ np=L3 ;

ð24Þ

where qmn ðtÞ denote the generalized coordinates. As concerns Z1 and Z2 ; we use for these displacements the representations. . X ; ð25aÞ Z1 ðx1 ; x2 ; tÞ ¼ ½H1mn cos lm x1 sin mu x2 þ H2mn sin lm x1 cos mn x2 qmn ðtÞ; m;n

Z2 ðx1 ; x2 ; tÞ ¼ ½I1mn cos lm x1 sin mn x2 þ I2mn sin lm x1 cos mn x2 qmn ðtÞ;

ð25bÞ

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 P where Hmn ; and Imn are yet arbitrary unknown coefficients, whereas the sign m;n indicates that in Eqs. (25) there is no summation over, the indices m and n: Substitution of Eqs. (24) and (25) in Eqs. (21a,b) and identifying the coefficients of the same trigonometric functions yields the expressions of Hmn and Imn that are different of zero, namely af½ðA12 þ A66 Þd2  A22 d1 lm m2n  d1 A66 l3m  d1 d2 lm g ; Dmn af½ðA12 þ A66 Þd1  A11 d2 l2m mn  d2 A66 m3n  d1 d2 mn g ¼ ; Dmn

H1mn ¼ I2mn

ð26a; bÞ

where Dmn ¼ A11 A66 l4m þ ðd2 A11 þ d1 A66 Þl2m þ ðA11 A22  2A12 A66  A212 Þl2m m2n þ ðd1 A22 þ d2 A66 Þm2n þ A22 A66 m4n þ d1 d2 :

ð26cÞ

It should be remarked that Eqs. (26a,b) pass one in another one, that is H1mn $I2mn ; when ð1$2Þ and ðmn $mm Þ; while Eq. (26c) remains invariant under this transformation. Substitution of displacement representations (24) and (25) in Hamilton’s principle, performing the required integrations, keeping in mind that Eqs. (21a,b) and geometric boundary conditions are identically satisfied, and that dqmn are arbitrary, one obtains for the generalized coordinates the equation M q. mn þ C q’ mn þ Kqmn ¼ Fmn :

ð27Þ

In Eq. (27), Mð m0 dmn Þ; Cð cdmn Þ and Fmn denote the mass, damping and the generalized loads, respectively, while Kð Kmn Þ ¼ d1 aH1mn lm þ d2 aI2mn mn þ d1 a2 l2m þ d2 a2 m2n 0 2 0 2 þ F11 l2m þ F22 m2n þ 4F66 l2m m2n þ 2F12 m2n l2m  N11 lm  N22 mn

ð28Þ

and dmn denotes the Kronecker’s delta. The same invariance property as in Eqs. (26a,b) is featured also by Eq. (28). Due to the large front of the explosive blast pulse, one can assume its uniformity over the entire panel area which yields Fmn ðtÞ ¼ where

16dm;2s1 dn;2q1 q3 ðtÞ; ð2s  1Þð2q  1Þp2 (

dm;2s1 ¼

1; m odd ðs ¼ 1; 2yÞ; 0; m even;

ð29aÞ

ð29bÞ

the same definition being valid also for dn;2q1 ’ : From the undamped counterpart of Eq. (27), in the absence of external loads one gets the expression of the natural undamped frequencies o2mn ¼

Kmn : M

ð30Þ

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Having in view Eq. (30) one can express Eq. (27) in a more convenient form as q. mn þ 2Dmn omn q’ mn þ o2mn qmn ¼ Fmn =M;

ð31Þ

where Dmn ¼

c 2Momn

ð32Þ

is the modal viscous damping ratio.

7. Blast loading The dynamic response to sonic-boom and explosive pressure pulses will be investigated. The sonic-boom overpressure can be expressed as follows (see e.g. Ref. [10]): ( Pð1  t=tp Þ for 0otortp ; ð33Þ q3 ðtÞ ¼ 0 for to0 and t > rtp ; where P denotes the peak reflected over pressure, tp denotes the positive phase duration of the pulse measured from the time of impact of the structure, while r denotes the shock pulse length factor. For r ¼ 1 the sonic-boom degenerates into a triangular explosive pulse, while for r ¼ 2 and 3 the pulse corresponds to a symmetric and non-symmetric sonic-boom, respectively. Having in view that Laplace transform will be used to determine the generalized coordinates qmn ðtÞ; it is more appropriate to convert Eq. (33) to its equivalent form as q3 ðtÞ ¼ Pð1  t=tp ÞfHðtÞ  Hðt  rtp Þg;

ð34Þ

where Hð Þ is the Heaviside step function defined as HðtÞ ¼ 1 for tX0 and HðtÞ ¼ 0 for to0: As special cases of Eq. (34), the rectangular and Heaviside pressure pulses can be obtained. For the former case we have q3 ðtÞ ¼ PfHðtÞ  Hðt  tp Þg;

ð35Þ

while for the latter one q3 ðtÞ ¼ P for

8t > 0:

ð36Þ

For the case of sonic-boom pulses, in conjunction with Eq. (34) Eq. (29a) reduces to Fmn ðtÞ ¼ F*mn ð1  t=tp Þ½HðtÞ  Hðt  rtp Þ ;

ð37Þ

where 16Pdm;2s1 dn;2q1 F*mn ¼ 2 p ð2s  1Þð2q  1Þ

ðs; q ¼ 1; 2; yÞ:

ð38Þ

8. Dynamic response. Solution methodology Laplace Transform in time is used to solve Eq. (31) in conjunction with the particular expression of time-dependent external load. To this end, Laplace transform operator L

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defined as Lf g ¼

Z

N

f gest dt

ð39Þ

0

is applied to Eq. (31), where s is Laplace transform variable. – In the case of the sonic-boom pressure pulse we have 1 1 Lfð1  t=tp ÞHðtÞg ¼  2 ; s s tp Lfð1  t=tp ÞHðt  rtp Þg ¼ ð1  rÞ

ð40aÞ ertp s ertp s  2 s s tp

ð40bÞ

and taking zero initial conditions, i.e. qmn ð0Þ ¼ q’ mn ð0Þ ¼ 0 one obtains for qmn in the Laplace transformed space:   F*mn 1 1 ð1  rÞertp s ertp s ;   Qmn ðsÞ ¼ þ m0 ½s2 þ 2Dmn omn s þ o2mn s tp s2 s tp s2

ð40cÞ

ð41aÞ

where Qmn ðsÞ ¼ Lfqmn ðtÞg: Taking the inverse Laplace transform of Eq. (41a) one can readily obtain 

 2Dmn t 2Dmn Dmn omn t F*mn qmn ðtÞ ¼ 1þ   1þ cos Omn t e m0 o2mn tp omn tp tp omn

2  2Dmn þ Dmn omn tp  1 Dmn omn t  e sin Omn t atp    2Dmn t 2Dmn Dmn omn ðtrtp Þ e  1þ   ð1  rÞ þ cos Omn ðt  rtp Þ tp omn tp tp omn   2D2 þ ð1  rÞtp omn Dmn  1  mn sin Omn ðt  rtp Þ Hðt  rtp Þ : Omn tp In Eqs. (41a,b) and next equations qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Omn ¼ omn 1  D2mn

ð41bÞ

ð41cÞ

denotes the damped natural frequency – In the case of a sine pulse: Qmn ðsÞ ¼

p=tp ð1 þ etp s Þ F*mn m0 ½s2 þ ðp=tp Þ2 ½s2 þ 2Dmn omn s þ o2mn

ð42aÞ

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while its inverse counterpart in the time domain becoming tp F*mn p n qmn ðtÞ ¼ a1 cosðpt=tp Þ þ b1 sinðpt=tp Þ þ a2 eDmn omn t cos Omn t m0 tp p

 b2  a2 Dmn omn Dmn omn t þ sin Omn t þ ½a1 cos pðt  tp Þ=tp e Omn b1 tp þ sin pðt  tp Þ=tp þ a2 eDmn omn ðttp Þ cos Omn ðt  tp Þ p  b2  a2 Dmn omn Dmn omn ðttp Þ e sin Omn ðt  tp Þ Hðt  tp Þ : þ Omn

619

ð42bÞ

Herein   o1 2t2p Dmn o2mn n 2 o2mn t2p 2 2 2 2 ½o  p =t 1  o ;  4D 2 mn p mn mn p p2 1  o2mn t2p =p2   b1 ¼ ; o2 t2p 2 2 ðo2mn  p2 =t2p Þ 1  mn o  4D mn mn p2 ( ! )1 o2mn t2p 2 2 2 2 2 2 2 2 2 2 2 2 b2 ¼ ½ð1  omn tp =p Þ  4Dmn omn tp =p ðomn  p =tp Þ 1   4Dmn omn : p2 a1 ¼ a2 ¼

ð42c–eÞ – The solution corresponding to a rectangular pulse: In the Laplace space domain it is F*mn 1 Qmn ðsÞ ¼ ð1  etp s Þ 2 m0 sðs þ 2Dmn omn s þ o2mn Þ

ð43aÞ

while in the time domain it becomes 8 > Dmn F*mn < qmn ðtÞ ¼ 1  eDmn omn t cos Omn t  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eDmn omn t sin Omn t 2 > m0 omn : 1  D2mn 2

39 > = Dmn 6 7 Dmn omn ðttp Þ Dmn omn ðttp Þ cos Omn ðt  tp Þ  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e sin Omn ðt  tp Þ5  41  e > ; 1  D2 mn

 Hðt  tp Þ:

ð43bÞ

– Solution of the response to a Heaviside-pulse: In the Laplace transformed space Qmn ðsÞ ¼

F*mn 1 2 m0 sðs þ 2Dmn omn s þ o2mn Þ

ð44aÞ

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while in the time-domain the solution is 8 9 > > < = Dmn F*mn Dmn omn t Dmn omn t qmn ðtÞ ¼ 1e cos Omn t  qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e sin Omn t : > m0 o2mn > : ; 1  D2mn

ð44bÞ

This expressions of qmn ðtÞ given by Eqs. (41b), (42b), (43b) and (44b) considered in conjunction with Eqs. (24)–(26) provide the dynamic response to the considered external pressure pulses. Needless to say, the undamped counterparts of the dynamic response corresponding to the sonic, sine, rectangular and Heaviside pulses are obtainable by letting Dmn -0 in Eqs. (41b), (42b), (43b) and (44b), respectively.

9. Numerical simulations The numerical simulations are carried out by using for the face sheets and core, the materials whose characteristics are displayed in Tables 1 and 2, respectively. In addition, unless otherwise stated, the characteristics of the square sandwich panel are as follows: L1 ¼ 24 in; h% ¼ 0:5 in; h ¼ 0:075 in D ¼ 0:05: In addition, the considered amplitude of the pressure pulse is P ¼ 200 psi and tp ¼ 0:005 s: In Figs. 2–13 there are highlighted various effects on the dimensionless transverse deflection amplitude time-history v3 ðL1 =2; L2 =2Þ=H  w=H of the central point of sandwich panels impacted by explosive pressure pulses. The type of the pressure pulse is depicted in the inset of each figure. In Fig. 2, the implications of the damping ratios D½ c=ð2m0 o11 Þ on the dimensionless central deflection time-history of the sandwich panel exposed to an explosive blast load are highlighted. The upper and lower face sheets are constituted of five layers each, the stacking sequence being ½45 =  45 =45 =  45 =45 =core=45 =  45 =45 =  45 =45 : The results reveal that in the forced motion regime, but specially in the free motion regime, the increase of D is accompanied by a decrease of the response amplitude.

Table 1 Material properties for the face sheets E1 (Msi)

E2 (Msi)

G12 (Msi)

n12

rf ð1b s2 =in4 Þ

30

0.75

0.37

0.25

14:3  105

Table 2 Material properties for the core G% 13 (Msi)

G% 23 (Msi)

rð1b s2 =in4 Þ

0.0149

0.009

14:97  107

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Fig. 2. Implications of the core damping on deflection time-history of the panel exposed to a triangular explosive pulse.

Fig. 3. Effects of application of tailoring technique in the panel face sheets on deflection time-history of the panel exposed to a triangular explosive pulse.

In Fig. 3 it is highlighted the influence of the ply-angle of face sheets on the dimensionless central deflection time-history of the panel exposed to an explosive pressure pulse. In this case the upper and lower face sheets are constituted of five layers each, the stacking sequence being ½y=  y=y=  y=y=core=y=  y=y=  y=y : From this figure it clearly appears that the tailoring

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Fig. 4. The counterpart of Fig. 2 for the case of a sinusoidal pressure pulse simulating a gust.

Fig. 5. The counterpart of Fig. 3 for the case of the sinusoidal pressure pulse.

technique applied to this panel can play an important role towards alleviating the influence of the blast load in both the forced and free motion regimes. In addition, the results reveal that y ¼ 45 is the best ply-angle, in the sense that for this case one gets the lowest amplitude that damps out as time unfolds. Figs. 4 and 5 display the implications of damping and ply-angle, respectively, in the conditions of a sinusoidal pressure pulse. As in the case of the panel subjected to a triangular pressure pulse,

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Fig. 6. Implications of tp on transversal dimensionless deflection amplitude time-history of sandwich panels under a sinusoidal pressure pulse.

Fig. 7. Effect of ply-angle of face sheets on tranversal dimensionless deflection amplitude time-history of panels under a rectangular pulse.

Fig. 3, also in this case, the ply-angle y ¼ 45 provides a lower deflection of the panel in both the forced and free motion regimes. Related to the implications of the damping, similar conclusions are reached also in this case. In Fig. 6, the implications of the time extension tp of the sinusoidal pulse on the transversal non-dimensional deflection amplitude time-history is highlighted. Having in view that in this case,

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Fig. 8. Counterpart of Fig. 2 for the case of a rectangular pressure pulse.

Fig. 9. Counterpart of Fig. 2 for the case of a step pressure pulse.

tp defines the extension in time of the forced motion, from Fig. 6 it can be seen that its implications are present mainly in that range, while in the free motion regime, these are almost immaterial. In Figs. 7 and 8 the dynamic response to a rectangular pulse is highlighted. While in Fig. 7 the effect of the ply-angle of face sheets is presented, in Fig. 8 that of the damping ratio is supplied. The results reveal that also in the case, y ¼ 45 is the most favorable ply-angle, throughout the

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Fig. 10. Influence of panel aspect ratio on dimensionless central deflection time-history of a square sandwich panel exposed to a triangular pulse. The upper and lower face sheets are in the sequence ½45 =  45 =45 = 45 =45 =core=45 =  45 =45 =  45 =45 ; D ¼ 0:

Fig. 11. Influence of applied/absence of tensile edge loads on dimensionless central deflection time-history of a square sandwich panel whose face sheets feature the stacking-sequence ½45 =  45 =45 =  45 =45 =core=45 = 45 =45 =  45 =45 ; D ¼ 0:

forced and free motion regimes. Related to Fig. 8, the results reveal, as expected, that for D ¼ 0 the oscillation amplitudes remain constant, but of different values in the forced and free motion regimes. However, with the increase of D; a decay of the deflection amplitude in both the forced and free motions as time unfolds is experienced.

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Fig. 12. Counterpart of Fig. 3 for the case of a sonic-boom pressure pulse characterized by r ¼ 3; D ¼ 0:

Fig. 13. Implications of face sheets thickness on the dimensionless deflection amplitude time-history of the panel subjected to a rectangular pulse ðtp ¼ 0:015 sÞ: The stacking sequence of the face sheets is ½ð0 Þ5 =core=ð0 Þ5 ; ðD ¼ 0Þ:

In Fig. 9, the effect of a step pressure pulse in conjunction with that of the damping ratio on the deflection amplitude time-history is highlighted. The results reveal that, in contrast to the previous pulses, in the present case, with the increase of the damping ratio, the deflection amplitude is contained at a fixed amplitude, different of zero.

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In Fig. 10, the influence of the panel aspect ration fð L2 =L1 Þ; when L1 is fixed, coupled with that of a triangular pulse on deflection amplitude time-history is highlighted. The results reveal that f plays a strong role toward the increase/decrease of the response amplitude in both the forced and free motion regimes. In this sense, the results of this plot considered in conjunction with the previous ones enable one to infer that only when f ¼ 1; the minimum response is obtained for y ¼ 45 : In the context of the buckling behavior of sandwich panels, in Ref. [7] it was shown that in the same conditions the panel experiences the maximum buckling load. Although different in their nature, the results appear consistent each other. In Fig. 11 there are revealed implications of the applied tensile edge loads on the dimensionless central deflection time-history amplitude of the panel exposed to a triangular blast load. As is clearly emerging from Fig. 11, the biaxial tensile edge load determine a deflection amplitude lower than that of the panel counterpart acted by uniaxial tensile edge loads, and certainly, much lower than in the case of the absence of tensile edge loads. In Fig. 12 there are highlighted the implications of the ply angle in the conditions of sonic-boom pressure pulse. This case represents the counterpart of Fig. 3 where the response to a triangular blast was considered. As in the case of the panel subjected to a triangular pressure pulse, also in the case, it appears that in contrast to ply-angles y ¼ 0 and 90 ; y ¼ 45 provides a lower deflection of the panel in both the forced and free motion regimes. In Fig. 13 the implications of thickness of face sheets on the dimensionless transverse deflection time-history of the sandwich panel exposed to a rectangular pressure pulse are presented. As emerges from this figure, effects similar to those previously emphasized in the case of other pressure pulses, become evident also in the case of a rectangular pressure pulse. Moreover, the results reveal that the increase of the thickness of face sheets yields a decrease of the deflection amplitude in both the forced and free motions stages.

10. Conclusions An analytical study of the dynamic response in bending of flat sandwich panels characterized by laminated face sheets and a weak core was presented. The adopted solution methodology is based on the extended Galerkin method (EGM) coupled with the Laplace Transform that enables one to get closed-form solutions of the problem. The virtues of this method stem from the fact that, on its basis, the residual terms appearing from the non-fulfillment of both the equation of motion and boundary conditions, are minimized in the Galerkin sense, and as a result, accurate solutions of the response problem to various pressure pulses are reached. As the results obtained in previous works reveal (see e.g. Refs. [11,12]), the EGM provides extremelly accurate solutions, in many situation close to the exact ones. In this paper, the implications of a number of effects, such as of the ply-angle of the material of face sheets, panel aspect ratio, tensile uni/biaxial edge loads, as well as the characteristics of the considered pressure pulses have been investigated. Herein, the dynamic response was approached in the context of a linear theory of sandwich panels. Having in view, however, that this type of structures feature rather high thickness ratios,

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one can expect that the effects of geometric nonlinearities on dynamic response to be quite marginal. Acknowledgements The support of the Office of Naval Research Program on Composites for Marine Structures, Grant N000014-02-1-0594, and the interest, advice and encouragement of the Program Manager, Dr. Y.D.S. Rajapakse, are gratefully acknowledged by L. Librescu. References [1] Noor AK, Burton WS, Bert CW. Computational models for sandwich panels and shells. Appl Mech Rev 1996;4(3):155–99. [2] Altenbach H. Theories for laminated and sandwich plates. A review. Mech Composite Mater 1998;34(3):248–52. [3] Librescu L, Hause T. Recent development in the modeling and behavior of advanced sandwich constructions: a survey. J Composite Struct 2000;48(1–3):1–17. [4] Vinson JR. Sandwich structures. Appl Mech Rev 2001;54(3):201–14. [5] Vinson JR. The behavior of sandwich structures of isotropic and composite materials. Lancester, PA: Technomic Publishing Company; 1999. [6] Librescu L, Hause T, Camarda CJ. Geometrically nonlinear theory of initially imperfect sandwich plates and shells incorporating non-classical effects. AIAA J 1997;35(8):1393–403. [7] Hause T, Librescu L, Johnson TF. Thermomechanical load-carrying capacity of sandwich flat panels. J Thermal Stresses 1998;21(6):627–53. [8] Hause T, Johnson TF, Librescu L. Effect of face-sheets anisotropy on buckling and post-buckling of flat sandwich panels. J Spacecraft Rockets 2000;37(3):331–41. [9] Librescu L. Elastostatics and kinetics of anisotropic and heterogeneous shell-type structures. Leiden, Netherlands: Noordhoff International Publishing; 1975. p. 598. [10] Librescu L, Nosier A. Response of shear deformable elastic laminated composite panels to sonic boom and explosive blast loadings. AIAA J 1990;28(2):345–53. [11] Song O, Ju JS, Librescu L. Dynamic response of anisotropic thin-walled beams to blast and harmonically oscillating loads. Int J Impact Eng 1998;21(8):663–82. [12] Na SS, Librescu L. Optimal dynamic response control of elastically tailored nonuniform thin-walled adaptive beams. J Aircraft 2002;39(3):469–79.