- Email: [email protected]
Linear and non-linear dynamic response of sandwich panels to blast loading
Composites: Part B 35 (2004) 673–683 www.elsevier.com/locate/compositesb
Linear and non-linear dynamic response of sandwich panels to blast loading Liviu Librescua,*, Sang-Yong Oha, Joerg Hoheb a
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Mail Cose (0219) Blacksburg, VA 24061, USA b˙ Fraunhofer-Institute fuer Werkstoffmechanik (IWM), D-79108 Freiburg, Germany Received 17 May 2003; accepted 17 July 2003 Available online 9 April 2004
Abstract The problem of the dynamic response of sandwich flat panels exposed to blast loadings is addressed. The sandwich model includes a number of non-classical effects such as the anisotropy and heterogeneity of face sheets, transverse orthotropy of the core layer, the geometrical non-linearities considered in the von Ka´rma´n sense, as well as the initial geometric imperfections. As concerns the blast pulses considered in this analysis, these are due to either an underwater/in-air explosion, or are due to a pressure wave traveling across the panel span. Implications of the explosive charge, stand-off distance, directionality property of face sheets material, damping ratio, geometric nonlinearity, initial geometric imperfection, and of the characteristics of the blast, on dynamic response and dynamic magnification factor are put into evidence via a parametric study, and pertinent conclusions are outlined. q 2004 Elsevier Ltd. All rights reserved. Keywords: Dynamic response
1. Introduction A growing interest for an extensive integration of sandwich composites in the construction of naval ships and submarines has been manifested in the last decade. This trend was outlined in the extensive review-paper, Ref. [1], as well as in the works of the recent international conference on sandwich constructions, Ref. [2], where the achievements in this area have been presented, and the potential benefits of the incorporation of sandwich composites in the construction of a variety of naval vessels, including the military ones, have been discussed. Some of the underlying reasons and motivation for this interest emerge, among others, from the fact that, compared to metallic hulls, the sandwich ones experience lower structural weight, extended operational life, lower maintenance cost, as well as a range of integrated functions, such as thermal and sound insulation, excellent signature properties, fire safety, good energy absorption, directional properties of the face sheets enabling one an optimized design, and a smooth hydrodynamic surface. * Corresponding author. Tel.: þ 1-540-231-5916; fax: þ1-540-231-4574. E-mail address: [email protected] (L. Librescu). 1359-8368/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2003.07.003
Needless to say, the development of new manufacturing techniques, has further contributed to render the sandwich structures more affordable, and as a result, more attractive for their use. An issue that should be addressed when dealing with sandwich composite structures used in the design of combat ships, is that of their dynamic response to time-dependent external loading of the blast type. In this context one should remark that, to the best of the authors knowledge, with the exception of the works by Moyer et al. [3], Hayman [3], and Ma¨kinen [4] where special issues related to underwater explosion on sandwich panels have been addressed, no other studies on this topic are available in the archival literature. The issue of the dynamic response of sandwich flat panels to time-dependent loads generated by an underwater explosion and by a shock-wave will be considered in the next developments. In the former case, one supposes that we deal with a submerged panel, while in the latter one, with a topside sandwich panel of ship superstructures. In this context, in order to put into evidence the implications of various non-classical effects, such as those of geometrical non-linearities, initial geometric imperfections, anisotropy properties of face sheets and their ply-sequence, transverse shear orthotropy properties of the core layer, etc., an
674
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
advanced model of sandwich constructions will be used. The basic equations of this structural model have been derived in a number of previous works, [5] –[8]. To be reasonably self-contained, 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.
and face sheets, (vi) the layers constituting 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, as well, and finally, (viii) a Lagrangian description of the non-linear model of sandwich structures is adopted, in conjunction with the von-Ka´rma´n concept and consideration of initial geometric imperfections.
2. Basic assumptions. displacement representation The global middle 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 while those of the upper and the core is denoted as 2h; 00 bottom face sheets, as h and h0 ; respectively. As a result, Hð; 2h þ h0 þ h00 Þ is the total thickness of the structure (see Fig. 1). The sandwich model used in this paper is based on the following assumptions: (i) the face-sheets are composed of a number 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 material of the core features orthotropic properties, the axes of orthotropy being parallel to the geometrical axes xa ; (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 is adopted for both the core
3. Kinematics Several basic kinematic relationships derived in Refs. [6,7] will be supplied next. 3.1. The 3D 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 V 03 ðx1 ; x2 ; x3 Þ ¼ V 003 ðx1 ; x2 ; x3 Þ ¼ V 3 ðx1 ; x2 ; x3 Þ ; v3 ðx1 ; x2 Þ ð1Þ Upon discarding transverse shear effects in the face-sheets, the 3D distribution of the displacement field fulfilling the kinematic continuity conditions at the interfaces between the core and face sheet results as (see Refs. [6,7]) V 01 ðxa ; x3 Þ ¼ j1 ðxa Þ þ h1 ðxa Þ 2 ðx3 2 aÞ›v3 ðxa Þ=›x1
ð2aÞ
ðh # x3 # h þ h0 Þ V 02 ðxa ; x3 Þ ¼ j2 ðxa Þ þ h2 ðxa Þ 2 ðx3 2 aÞ›v3 ðxa Þ=›x2
ð2bÞ
V 03 ðxa ; x3 Þ
ð2cÞ
¼ v3 ðxa Þ;
h1 ðxa Þ þ ðh=2Þ›v3 ðxa Þ=›x1 } V 1 ðxa ; x3 Þ ¼ j1 ðxa Þ þ ðx3 =hÞ{ ð2h # x3 # hÞ ð3aÞ h2 ðxa Þ þ ðh=2Þ›v3 ðxa Þ=›x2 } V 2 ðxa ; x3 Þ ¼ j2 ðxa Þ þ ðx3 =hÞ{ ð3bÞ V 3 ðxa ; x3 Þ ¼ v3 ðxa Þ;
ð3cÞ
and V 001 ðxa ; x3 Þ ¼ j1 ðxa Þ 2 h1 ðxa Þ 2 ðx3 þ aÞ›v3 ðxa Þ=›x1
ð4aÞ
ð2h 2 h00 # x3 # 2hÞ
Fig. 1. Geometry of the sandwich flat panels with laminated face sheets.
V 002 ðxa ; x3 Þ ¼ j2 ðxa Þ 2 h2 ðxa Þ 2 ðx3 þ aÞ›v3 ðxa Þ=›x2
ð4bÞ
V 003 ðxa ; x3 Þ
ð4cÞ
¼ v3 ðxa Þ
In these equations, Vi ðxa ; x3 Þ are the 3D displacement components in the directions of the coordinates xi :
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
In addition
ja ¼
ðV^ 0a
þ
V^ 00a Þ=2;
ha ¼
ðV^ 0a
2
V^ 00a Þ=2
ð4dÞ
denote the 2D 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. It should also be mentioned that in the dynamic case, as considered in this paper, the displacement quantities are functions of time, as well. Although this dependence was not explicitly specified, this is automatically implied. In the previous and the following equations, the Greek indices have the range 1, 2, while the Latin indices have the range 1, 2, 3, and unless otherwise stated, Einstein’s summation convention over the repeated indices is employed. In addition, ð·Þ;i denotes partial differentiation with respect to coordinate xi :
4. Distribution of strain quantities across the shell thickness The structure is assumed to feature a stress-free small initial geometric imperfection V 3 ð; V 3 ðxa ÞÞ: Adopting the von-Ka´rma´n concept of small strains and moderately small rotations (see Refs. [6 – 8]), the distribution of 3D strain quantities eij ð; eij ðx1 ; x2 ; x3 Þ across the wall thickness of the sandwich panel results as: In the bottom face-sheets
675
tangential and the transverse shear strain measures, respectively. Their expressions in terms of the 2D displacement measures in the face-sheets and core are supplied next. 4.1. 2D strain –displacement relationships Bottom face sheets 1011 ¼ j1;1 þ h1;1 þ
1 2 v þ v3;1 v 3;1 2 3;1
ð1 , 2Þ
ð8aÞ
g012 ¼ j1;2 þ j2;1 þ h1;2 þ h2;1 þ v3;1 v3;2 þ v 3;1 v3;2 þ v3;1 v 3;2 ð8bÞ k011 ¼ 2v3;11
ð1 , 2Þ
ð8cÞ
k012 ¼ 22v3;12
ð8dÞ
Core Layer 111 ¼ j1;1 þ
1 2 v þ v 3;1 v3;1 2 3;1
ð1 , 2Þ
g12 ¼ j1;2 þ j2;1 þ v3;1 v3;2 þ v 3;1 v3;2 þ v3;1 v 3;2 1 1 k11 ¼ h1;1 þ hv3;11 ð1 , 2Þ 2 h 1 k12 ¼ {h1;2 þ h2;1 hv3;12 } h 1 1 g13 ¼ h1 þ hv3;1 þ v3;1 2 h
ð9aÞ ð9bÞ ð9cÞ ð9dÞ
ð1 , 2Þ
ð9eÞ
Top face sheets 10011 ¼ j1;1 2 h1;1 þ
1 2 v þ v 3;1 v3;1 2 3;1
ð1 , 2Þ
ð10aÞ
e011 ¼ 1011 þ ðx3 2 a0 Þk011
ð5aÞ
g0012 ¼ j1;2 þ j2;1 2 h1;2 2 h2;1 þ v3;1 v3;2 þ v 3;1 v3;2 þ v3;1 v 3;2 ð10bÞ
e022 ¼ 1022 þ ðx3 2 a0 Þk022
ð5bÞ
k0011 ¼ 2v3;11
2e012 ¼ g012 þ ðx3 2 a0 Þk012
ð5cÞ
k0012 ¼ 22v3;12
In the core layer e 11 ¼ 111 þ x3 k11
ð6aÞ
e 22 ¼ 122 þ x3 k22
ð6bÞ
2e12 ¼ g12 þ x3 k12
ð6cÞ
2e13 ¼ g13
ð6dÞ
2e23 ¼ g23
ð6eÞ
In the top face-sheets e0011 ¼ 10011 þ ðx3 þ a00 Þk0011
ð7aÞ
e0022
ð7bÞ
¼
10022
þ ðx3 þ a
00
Þk0022
2e0012 ¼ g0012 þ ðx3 þ a00 Þk0012
ð7cÞ
In these equations 111 ; 122 ; 112 ð; g12 =2Þ and 113 ð; g13 =2Þ; 123 ¼ ðg23 =2Þ denote the 2D (in the sense of 1ij ð; 1ij ðx1 ; x2 ÞÞ
ð1 , 2Þ
ð10cÞ ð10dÞ
Herein and in the forthcoming equations, the sign ð1 , 2Þ indicates that, from the equations accompanied by this sign, companion equations, not explicitly displayed, can be obtained by replacing subscript 1 by 2, and vice versa. As a mere remark, due to inclusion of geometrical nonlinearities, the inherent stretching-bending coupling emerges in the kinematic equations, Eqs. (8) –(10). Eqs. (9a) – (9e) are applicable in general to strong core sandwich structures. For the weak core sandwich structures, only g13 and g23 are relevant.
5. Equations of motion and boundary conditions Hamilton’s principle is used to derive the equations of motion and the boundary conditions of geometrically non-linear theory of sandwich flat panels. This may be
676
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
stated as ðt1 dJ ¼ d ðU 2 W 2 TÞdt ¼ 0
ð11Þ
t0
where t0 ; t1 are two arbitrary instants of time, U; W and T denote the strain energy, the work done by surface tractions, edge loads and body forces, and the kinetic energy of the 3D body of the sandwich structure, respectively, while d denotes the variation operator. The expressions of U; W and T are not displayed here. These can be found in Refs. [6 –8]. From Eq. (11) considered in conjunction with the proper expression of various energy quantities and with the strain– displacement relationships (used as subsidiary conditions), carrying out the integration with respect to x3 and integrating by parts wherever necessary as to relieve the virtual displacements of any differentiation and invoking the arbitrary and independent character of the variations dh1 ; h2 ; dj1 dj2 and dv3 throughout the entire domain of the plate and within the time interval ½t0 ; t1 ; using the expression of global stress resultants and stress couples (to be defined later), one derive the equations of motion and the boundary conditions relevant to weak core sandwich panels. By retaining only the transversal load, and the transverse inertia and transverse viscous damping terms, one obtain. The equations of motion:
dj1 : N11;1 þ N12;2 ¼ 0
ð12aÞ
dj2 : N22;2 þ N12;1 ¼ 0
ð12bÞ
dh1 : L11;1 þ L12;2 2 N 13 ¼ 0
ð12cÞ
dh2 : L22;2 þ L12;1 2 N 23 ¼ 0
ð12dÞ
dh3 : N11 ðv3;11 þ v 3;11 Þ þ 2N12 ðv3;12 þ v 3;12 Þ þ N22 ðv3;22 þ v 3;22 Þ þ ðM11;11 þ 2M12;12 þ M22;22 Þ N 13;1 þ N 23;2 Þ þ q3 ¼ m0 v€ 3 þ C_v3 þ a=hð
ð12eÞ
where C is the transverse viscous damping coefficient, while the overdots denote time-derivatives. In addition, the boundary conditions that are consistent to the previously displayed equations of motion are as follows: Nnn ¼ Nnn ~
or jn ¼ jn ~
Nnt ¼ Nnt ~
or jt ¼ jt ~ or hn ¼ hn ~
Lnn ¼ Lnn ~
ð13aÞ ð13bÞ . X
ð13cÞ
In these equations subscripts n and t are used to designate the normal and tangential in-plane directions to an edge and, hence, nP ¼ 1 when t ¼ 2; and vice versa. In addition, the symbol n;t indicates that no summation over the indices n and t is implied, while the terms underscored by a tilde denote prescribed quantities. The displayed equations of motion and static boundary conditions are expressed in terms of the global stress resultants and stress couples as N11 ¼ N 011 þ N 0011 N12 ¼ N 012 þ N 0012 L11 ¼
011 hðN
L12 ¼
012 hðN
2
N 0011 Þ
2
N 0012 Þ
M12 ¼
M 012
þ
Lnt ¼ Lnt or ht ¼ ht ~ ~ Mnn ¼ Mnn or v3;n ¼ v3;n ~ ~ Nnt ðv3;t þ v 3;t Þ þ Nnn ðv3;n þ v 3;n Þ þ Mnn;n þ 2Mnt;t
ð13dÞ
N n3 ¼ Mnt;t þ N n3 or v3 ¼ v3 þ ða=hÞ ~ ~ ~
ð13fÞ
ð13eÞ
ð14aÞ ð14bÞ
ð1 , 2Þ
ð14cÞ ð14dÞ
M11 ¼ M 011 þ M 0011
ð1 , 2Þ
M 0012
ð14eÞ ð14fÞ
In addition, in Eqs. (12) and (13), ðN 0ab ; M 0ab ÞðN 00ab ; M 00ab Þ; and N ab are the 2D stress resultants and stress couples measures associated with the lower and upper face sheets and with the core layer, respectively. These equations are defined as {N 0ab ; M 0ab } ¼
N ððx3 Þk X k¼1
N a3 ¼
ðh 2h
ðx3 Þk21
ðS0ab Þk {1; x3 2 a}dx3
S a3 dx3 ða; b ¼ 1; 2Þ
ð15aÞ
ð15bÞ
where Sij are the components of the Piola –Kirchhoff stress tensor. In addition, N is the number of constituent layers in the bottom face sheets, (equal to that in the top faces), whereas ðx3 Þk and ðx3 Þk21 are the distances from the global mid-plane of the structure to the upper and lower interfaces of the kth layer, respectively. As concerns the 2D constitutive equations associated to the sandwich structures, these have been supplied, e.g. in Refs. [6,7], and will not be repeated here.
6. Governing system For the problem at hand, the mixed representation of the governing equations will be considered. This formulation is done in terms of the Airy’s potential function fðx1 ; x2 Þ; and the 2D displacement measures v3 ; h1 and h2 : To this end, by expressing the stress resultants in terms of the Airy’s potential function fð; fðxv ÞÞ as: Nab ¼ cav cbr f;vr
n;t
ð1 , 2Þ
ð16Þ
the equilibrium Eqs. (12a) and (12b) are identically fulfilled, where cab denotes the 2D permutation symbol. Having in view that by virtue of representation Eq. (16), the two equilibrium Eqs. (12a) and (12b) are eliminated, in order to ensure single valued displacements, the compatibility equation for the tangential strain measures has to be fulfilled.
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
677
For flat sandwich panels featuring a weak core this equation is:
moduli. For more details about the derivation of these equations the reader is referred to [6,7].
111;22 þ 122;11 2 g12;12 2 2v23;12 þ 2v3;11 v3;22 þ 2v3;11 v3;22 þ 2v3;11 v 3;22 2 4v3;12 v 3;12 ¼ 0
7. Blast loads
ð17Þ
where
7.1. Underwater explosive shock loading
111 ¼ 1011 þ 10011
ð18aÞ
122 ¼ 1022 þ 10022
ð18bÞ
g12 ¼ g012 þ g0012
ð18cÞ
Making use of the partially inverted form of constitutive equations considered in conjunction with Eq. (16), Eq. (17) can be expressed in terms of the basic unknown function as
When a charge is exploded underwater, (see Fig. 2), an instantaneous pressure pulse, or shock wave is generated and transmitted in all directions (see Refs. [9,10]). The pressure time-history at a fixed location starts with an instantaneous pressure increase to a peak followed by a decrease. The expression of the pressure that corresponds to the incident wave pulse is given by
Ap22 f;1111 þ Ap11 f;2222 2 2Ap16 f;1222 2 2Ap26 f;2111
Pi ðtÞ ¼ qm e2ðt2t1 Þ=a t $ t1
ðAp66
2Ap12 Þf;1122
2v23;12
þ þ 2 þ 2v3;11 v3;22 þ 2v3;11 v3;22 þ 2v3;11 v 3;22 2 4v3;12 v 3;12 ¼ 0
ð19Þ
On the other hand, the equilibrium Eqs. (12c) – (12e) represented in terms of displacement quantities and of Airy’s function are as follows: A11 h1;11 þ A16 h2;11 þ A66 h1;22 þ ðA12 þ A66 Þh2;12 h1 þ av3;1 Þ ¼ 0 13 =hÞð þ 2A16 h1;12 þ A26 h2;22 2 ð2K 2 G ð20aÞ A22 h2;22 þ A26 h1;22 þ A66 h2;11 þ ðA12 þ A66 Þh1;12 h2 þ av3;2 Þ ¼ 0 23 =hÞð þ 2A26 h2;12 þ A16 h1;11 2 ð2K 2 G ð20bÞ
f;22 ðv3;11 þ v 3;11 Þ22f;12 ðv3;12 þ v 3;12 Þ2F11 v3;1111 2F22 v3;2222 24F16 v3;1112 24F26 v3;1222 G 13 ðh1;1 þav3;11 Þ 22ðF12 þ2F66 Þv3;1122 þð2K 2 a=hÞ{ þ G23 ðh2;2 þav3;22 Þ}2m0 v€ 3 2C_v3 þq3 ¼0 ð20cÞ In these equations by virtue of the structural symmetry of the sandwich panel. Aab ;ðA0ab ¼A00ab Þ; Fb; ;ðF 0ab ¼F 00ab Þ; h;ðh0 ¼h00 Þ ð21aÞ whereas the stiffness quantities Apab represent the inverted counterparts of Aab : Herein {A0vr ; B0vr ; D0vr } ¼
N ððx3 Þk X k¼1
ðx3 Þk21
ðv; r ¼ 1; 2; 6Þ
where t 2 t1 (ms) is the time elapsed after the arrival of the shock pulse, where t1 ¼ ðR þ zÞ=c is the time to reach a point at which the pressure is measured; c is the speed of sound in seawater ( ¼ 1476 m/s); qm denotes the peak magnitude of the pressure (MPa) in the shock front, while a denotes the constant that describes the exponential decay (ms). Both qm and a depend on the charge weight Q (kg) and the stand-off distance RðmÞ from the charge to the target. As concerns the pressure due to the reflected shock wave, this is given by Pr ðtÞ ¼
qm ½2 e2ðtþt1 Þ=mr a 2 ð1 þ mr Þe2ðtþt1 Þ=a 1 2 mr
PðtÞ ¼ Pi ðtÞ þ Pr ðtÞ
ð24Þ
For qm and a that are involved in Eqs. (22) and (23), approximate expressions will be used in the present numerical simulations. These are (see Refs. [10,11]): !1:15 Q1=3 qm ¼ 56:6 ðMPaÞ ð25aÞ R
^ 0vr ÞðkÞ ð1; x3 ; x23 Þdx3 ; ðQ ð21bÞ
ð21cÞ
while K 2 ; is the transverse shear correction factor associated to the core layer. By virtue of the symmetry with respect to ^ 0vr ¼ Q ^ 00vr ; where the global mid-surface, N 0 ¼ N 00 and Q ^ vr ¼ Qvr 2 Qv3 Qr3 =Q33 ; Q ^ vr denoting the reduced elastic Q
ð23Þ
In this equation mr ð; m0 =rcaÞ is the mass ratio, where m0 is the panel mass per unit area and r is the fluid mass density that in sea water is r ¼ 1000 kg/m3. It results that the total pressure at a certain distance is
and F 0vr ¼ D0vr 2 2aB0vr 2 a2 A0vr
ð22Þ
Fig. 2. Sandwich panel subjected to underwater explosion.
678
a ¼ 0:08 þ Q
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
1=3
Q1=3 R
!20:23 ðmsÞ
ð25bÞ
where the coefficients in Eq. (25) correspond to the HB £ TNT explosive. The good accuracy of these expressions has been validated by comparing their predictions with the experimental data. As is was reported (see Refs. [5,10]), the agreement is good up to one decay length, which is equivalent to pressures larger than about one third of the peak value. It should also be mentioned that: (i) the supplied pressure expression is not applicable in the case of contact explosion, i.e. when the target is located in the immediate proximity of the explosive charge, and that (ii) the effect of cavitation was not accounted for. Moreover, as Eq. (24) reveals, the pressure generated by the passage of wave through the core was not accounted for. 7.2. Air-blast loading In addition to blasts produced by underwater explosions, in combat operations, the structure of warships can be exposed to air-blasts generated by explosions and by shockwave disturbances produced by an aircraft flying at supersonic speeds, or by any supersonic projectile, rocket or missile. In the latter case, the blast pulse is referred to as sonic-boom. Its time-history is described as an N-shaped pulse, whose negative phase direction is included as a variable in the analysis (see Refs. [14 –16]). In connection with the in-air explosion, one should remark that Eqs. (22) –(25) remain valid also for this case. What differentiates these expressions for the two events there are the values of the speed of sound, c; and the mass density r, in water and air. While for the sea water their values have been supplied, in air we have c ¼ 320 m/s and r ¼ 1:20 kg/m. As a result, essential differences in the pressure time-histories featured in the two types of explosions should occur. For the same charge weight and stand-off distance, in Fig. 3 there are depicted the time-histories of the normalized total pressure exerted on the panel front face and generated by the two explosion events. The results show that in the case of the in-air explosion, Fig. 3a, the decay, of the pressure, as time unfolds, is much slower than in the case of the underwater explosion, Fig. 3b, this rendering the effects of the in-air explosion much more severe than in the latter case. In the case of underwater explosion, Fig. 3b shows that at time t . 0:02 ms after the plate impact, the pressure becomes negative. In this case, cavitation is likely to occur at the plate surface. Having in view the large blast wave front generated by the explosion as compared to the relatively small dimensions of the panel, one assumes with sufficient accuracy that the pressure is uniform over the entire panel that is impacted at normal incidence. This assumption is adopted in the cases
Fig. 3. Normalized total pressure time-histories on the panel front face by an in-air (a), and underwater (b), explosions (Q ¼ 9 kg, R ¼ 3 m).
of the underwater/in-air explosions, as well as in that of the panel impacted at normal incidence by a sonic-boom pulse. A unified expression of the normal shock load applicable, when is properly specialized, to the sonic-boom, triangular blast and Heaviside pulse is given by Eq. (26) ! t PðtÞ ¼ qm 1 2 ½HðtÞ 2 db Hðt 2 rtp Þ ð26Þ tp This equation provided in Ref. [12] includes as special cases the pressure pulses above mentioned, and considered separately in Refs. [13 – 16]. In this equation and the next ones, HðtÞ denotes the Heaviside step function, db is a tracer that can take the values 1 or 0 depending on whether the sonic-boom or triangular blast load is considered, respectively, qm denotes the peak reflected pressure in excess to the ambient one, tp denotes the positive phase duration of the pulse measured from the time of impact of the structure, r denotes the shock pulse length factor. For r ¼ 1; the sonic-boom blast degenerates into a triangular explosive pulse, while for r ¼ 2; a symmetric
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
sonic-boom pulse is obtained. When r ¼ 1 and tp ! 1; Eq. (26) defines a Heaviside pulse. The case of an air-blast traveling in tangential direction to the panel span will also be considered. The pressure time-history for this latter case is represented as (see Ref. [17]) PðtÞ ¼ qm e2hðct2x1 Þ Hðct 2 x1 Þ
ð27Þ
where c is the wave speed in the medium surrounding the structure, while h is an exponent determining the character of the blast decay. Notice that when h ¼ 0; from Eq. (27), a Heaviside pulse is reached.
8. Solution methodology The governing equation systems, Eqs. (19) and (20) in conjunction with the explicit form of the pressure pulse and the boundary conditions has to be solved as to determine the time-history of displacements h1 ; h2 ; v3 ; and of the Airy’s function F: This constitutes an essential step toward determination of the full response time history, that is of strains and stresses, as well. From the mathematical point of view, the problem at hand reduces to the solution of a dynamic non-linear boundary value problem. Herein, the sandwich rectangular panel is assumed to be simply supported all over the contour. Consistent with the order twelve of the governing system, six boundary conditions have to be prescribed at each edge. Assuming the case of edges unloaded and immovable in the tangential direction, normal to the panel edge, the boundary conditions (see Refs. [6 –8]), are:
j1 ¼ 0 N12 ¼ 0; h1 ¼ 0; h2 ¼ 0; M11 ¼ 0; v3 ¼ 0 on x1 ¼ 0; L1
679
Table 1 Elastic coefficients for the materials of face sheets and core layers Face sheets Core Layer
E1 (GPa) 207 13 (GPa) G 0.1027
E2 (GPa) 5.17 23 (GPa) G 0.0621
G12 (GPa) 2.55
n12 0.25 rc 16
rf (kg/m3) 1588.22
9. Results The numerical results presented here have considered the case of face sheets laminated from the same orthotropic material, whose axes of orthotropy are rotated with respect to the geometrical axes of the panel by an angle u: For the face sheets, a material characterized by a rather large orthotropicity ratio was considered. Its properties, as well as those of the core layer are given in Table 1. In all numerical simulations, unless otherwise stated, the stacking sequence of face sheets is as follows: ½0=908=0=908=0=core=0=908=0=908=0; and the geometrically non-linear structural model was used. In Fig. 4 it is depicted the dimensionless deflection w=H time-history of the central point of a square panel ðL1 =L2 ¼ 1Þ; for selected values of the explosive weight. Needless to say, here and in the following plots, time is measured from the instant of the plate impact by the incident waves. Herein w ; v3 ðL1 =2; L2 =2Þ: The simulations have been carried out for the geometrically non-linear panel featuring a prescribed small initial geometric imperfection at the center of the panel d0 ð; v 3 ðL1 =2; L2 =2Þ=HÞ ¼ 0:2: It is assumed also that the stand-off distance R ¼ 3:1 m. It is seen that the structural damping ratio zð; C=2m0 v1 Þ where v1 is the undamped fundamental frequency (that for this case is z ¼ 0:05), has a strong effect of attenuating the oscillation induced by the explosion. In Fig. 5, it is highlighted the effect of the stand-off distance on the dimensionless panel deflection. The results
and
j2 ¼ 0; N12 ¼ 0; h1 ¼ 0; h2 ¼ 0; M22 ¼ 0; v3 ¼ 0 on x2 ¼ 0;L2 Due to the intricacy of the present boundary value problem, an approximate solution methodology based on Extended Galerkin Method (EGM) coupled with the satisfaction of the tangential boundary conditions on an average sense was used. This methodology was presented in details in Refs. [7] and [8], and in a more general context in Ref. [18], and will not be repeated here. It should be stressed that the predictions provided by the application of the EGM are of an extreme accuracy, these being identical, in many instances to the exact ones. In this sense, the reader is referred to Ref. [19] where an exact superpostion of predictions on dynamic response obtained via EGM and Laplace Transform Method was reported.
Fig. 4. Effect of the explosive weight on dimensionless deflection timehistory of the center panel subject to underwater explosion (z ¼ 0:05; d0 ¼ 0:2; R ¼ 3:1 m, L1 =H ¼ 20).
680
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
Fig. 5. The effect of the stand-off distance on dimensionless deflection timehistory of the panel subjected to underwater explosion (z ¼ 0:05; d0 ¼ 0:2; Q ¼ 18 kg; L1 =H ¼ 20).
Fig. 7. Effects of the structural damping on dimensionless velocity timehistory of the center panel subjected to underwater explosion (d0 ¼ 0:2; Q ¼ 9 kg; R ¼ 3:1 m; L1 =H ¼ 20).
reveal that the increase of the stand-off distance yields a decrease of the transverse deflection. However, results not displayed here reveal that for a fixed explosive charge, beyond a certain value of the stand-off distance (in this case for R . 6 m), the decay of deflection is almost immaterial. In Fig. 6, there are highlighted the effects of the underwater and in-air explosions. In this sense, for the same explosive charge and stand-off distance, there are displayed the deflection time-histories for the two cases. Notice that due to much higher deflection induced by the in-air explosion, in order of being able to represent the responses on the same graph, the normalization factors for the deflection corresponding to the explosion in-air and underwater have been selected as 2H and H; respectively. The results in this plot reveal that, as concerns the panel deflection, the effects of the explosion in-air are more dramatic than the ones in sea water.
In Fig. 7, there are presented the effects of the structural damping z on the dimensionless speed time-history of the center panel. Notice that the normalizing factor for the panel speed is the speed of sound in water. The beneficial effects of the structural damping become evident form this figure. In Fig. 8 the obtained results are based on the following architecture of the sandwich panel ½u= 2 u=u= 2 u=u= core=u= 2 u=u= 2 u=u=: The plots in this figure reveal that the ply-angles in the face sheets can play a significant role toward the attenuation, without weight penalties, of the intensity of the oscillatory motion resulting from the underwater explosion. However, as is revealed, the optimal ply-angle rendering the minimum dynamic deflection depends strongly on the 23 : 13 and G values of transverse shear moduli of the core, G For relatively high values of G13 and G23 ; as emerging from Fig. 8a, the optimal ply-angle is u ¼ 458; whereas for a material of the core more flexible in transverse shear, (such as the foam), the results of Fig. 8b reveal that u ¼ 908 is a more favorable ply-angle. As a result, for sandwich panels, implementation of the tailoring technique in the face sheets should always be considered in conjunction with the parameters defining the flexibilities in transverse shear of the core. Fig. 9 displays the normalized deflection time-histories of the central panel that correspond to an in-air explosion, for selected values of the stand-off distance. Although the explosive charge is rather small, Q ¼ 0:89 N the effects are very strong, specially for reduced stand-off distances. In Fig. 10 the effects of the explosive weight for an in-air explosion are depicted. Although the stand-off distance in rather large, (R ¼ 15:2 m), the effects of the considered weights that are rather small, are still quite strong. In Fig. 11 the effects of a traveling blast as expressed by Eq. (27), on the dimensionless central deflection of the panel are depicted. For two thickness ratios of the sandwich
Fig. 6. Effects of the underwater and in-air explosions on sandwich panel deflection time-histories (z ¼ 0:05; d0 ¼ 0:2; Q ¼ 9 kg, L1 =H ¼ 15).
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
681
Fig. 10. Effects of the in-air explosive weight on dimensionless deflection time-history of the center panel. (z ¼ 0:05; d0 ¼ 0:2; R ¼ 15:2 m; L1 =H ¼ 20).
Fig. 8. Effect of the ply-angle of face sheets on dimensionless deflection time-history of the sandwich panel under underwater explosion (8a) 13 ¼ 103 MPa; G 23 ¼ 5G 13 ; R ¼ 3:1 m; (z ¼ 0:05; d0 ¼ 0:2; Q ¼ 18 kg; G 13 ¼ 10:3 MPa; G23 ¼ 5G13 ). L1 =H ¼ 20Þ; (8b) ðG
Fig. 9. Effects of stand off distance on dimensionless deflection time-history of the panel subjected to an in-air explosion. (z ¼ 0:05; d0 ¼ 0:2; Q ¼ 0:9 kg; L1 =H ¼ 20).
panels, L1 =H ¼ 10 and 15, both the linear and nonlinear dynamic responses are depicted. The results reveal that for the thinner panel ðL1 =H ¼ 15Þ; the effects of the non-linearities are significant in the sense of contributing to the diminution of the vibration amplitude. However, when the panel is moderately thick ðL1 =H ¼ 10Þ; their effect becomes quite immaterial. In Fig. 12 there are highlighted effects of the wave propagation in the medium surrounding the panel subjected to a traveling blast. For this case it appears that c ¼ 50 m/s is the most critical propagation blast speed. However, (see Ref. [17]), depending on the characteristics of the panel, other values of c can become more critical. Finally, in Fig. 13 there is displayed the variation of the dynamic magnification function, (DMF) to a symmetric sonic-boom blast ðr ¼ 2Þ; of the center of the sandwich
Fig. 11. Effects of linear and nonlinear modeling coupled with that of the relative panel thickness on dynamic response of the sandwich panel subjected to a traveling blast. (qm ¼ 1:38 MPa; d0 ¼ 0; h ¼ 1; c ¼ 50 m/s).
682
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
the damped and undamped structure are supplied. It is seen that for this moderately thick sandwich panel the geometrically non-linearities play still a significant beneficial role toward the decay of the DMF. The same beneficial role, in the sense of the diminution of the DMF, is played by the structural damping. In the same context, as it clearly emerges from Fig. 13, the spikes that occur when z ¼ 0; completely disappear in the presence of moderate damping. In a simplified context this trend was also reported in Ref. [20]. It should also be observed that for the undamped and the linearized model the DMF is around 2, a value also indicated in Ref. [21]. However, a rather small additional damping yields a drastic drop of the DMF. At the same time, in the latter case, the beneficial effect of geometric nonlinearities decays considerably. Fig. 12. Effects of the velocity of the blast propagation on nonlinear dynamic response of the sandwich panel center exposed to a traveling blast (qm ¼ 1:38 MPa; d0 ¼ 0:2; z ¼ 0:05; h ¼ 1; L1 =H ¼ 15).
10. Conclusions panel vs. vtp ; where v is the fundamental undamped frequency. The sonic-boom pulse is represented in the inset of the figure. The DMF constituting a measure of the severity of the dynamic response, is defined as the ratio of the largest (in the absolute sense) dynamic deflection to the static deflection due to a load equal to the peak dynamic load. In this figure, comparisons of the variation of the DMF for the linear and non-linear structural models, and for
The problem of the dynamic response of sandwich flat panels subjected to explosive type loadings has been addressed. Loadings produced by both underwater/in-air explosions and by in-air blasts due to a sonic-boom or a traveling pressure wave have been considered. The implications of a number of structural and geometrical characteristics of the sandwich panel, as well as of the ones related to the blast have been highlighted and related conclusions have been drawn.
Fig. 13. Effects of geometrical non-linearities and structural damping on the DMF of the sandwich panel center exposed to a symmetric ðr ¼ 2Þ sonic-boom blast. (d0 ¼ 0; L1 =H ¼ 15; qm ¼ 1:38 MPa).
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
In this context, both the geometrical non-linearities and the effects of the unavoidable geometric imperfections have been implemented in the structural model. Needless to say, the obtained results can be extended without any difficulty as to determine the time-histories of strain and stress components of the various points of the structure. These items are essential toward determining the failure of the structure. Other issues that have still to be addressed are the ones involving the incorporation of the cavitation effect on the dynamic response of submerged sandwich panels exposed to an explosion, and also the effects played by transverse normal compressibility of the core layer (see Ref. [22]). These issues will be addressed in the forthcoming papers.
Acknowledgements The financial 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.
References [1] Mouritz AP, Gellert E, Burchhill P, Challis K. Review of advanced composite structures for naval ships and submarines. Compos Struct 2001;53:21–41. [2] Vinson JR, Rajapakse YDS, Carlsson LE, editors. Sixth International Conference on Sandwich Structures, Boca Raton, London: CRC Press; 2003. [3] Hayman B. Underwater explosion loading on foam-cored sandwich panels. Sandwich Construction 3, EMAS. Proceedings from the Third International Conference on Sandwich Construction, May 11–15, Southampton, England; 1995. [4] Ma¨kinen K. Underwater shock loaded sandwich structures. Department of Aeronautics, Royal Institute of Technology, Report 99-01 Stockholm, Sweden; 1999. [5] Librescu L. Elastostatics and Kinetics of anistropic and heterogeneous shell-type structures. Nordhoff International Publishing, Leydon, Netherlands. Ch. VI, 1975; 515 –32.
683
[6] Librescu L, Hause T, Camarda CJ. Geometrically nonlinear theory of initially imperfect sandwich plates and shells incorporating nonclassical effects. AIAA J 1997;35(8):1392 –403. [7] Hause T, Librescu L, Camarda CJ. Postbuckling of anisotropic flat and doubly-curved sandwich panels under complex loading conditions. Int J Solids Struct 1998;35:3007–28. [8] Librescu L, Hause T, Johnson TF. Buckling and nonlinear response of sandwich curved panels to combined mechanical loads—implications of face-sheets elastic tailoring. J Sandwich Struct Mater 2000;2: 246–69. [9] Cole RH. Underwater explosions. New York: Dover Publications; 1965. [10] Shin YS, Geers TL. Response of marine structures to underwater explosions. International short course notebook, shock and vibration research, Monterey, CA; 1994. [11] Jiang J, Olsen MD. Modeling of underwater shock-induced response of thin plate structures. Report No. 39, Department of Civil Engineering, University of British Columbia, Vancouver, BC; 1994. [12] Marzocca P, Librescu L, Chiocchia G. Aeroelastic response of 2D lifting surfaces to gust and arbitrary explosive loading signature. Int J Impact Engng 2001;25(1):67–85. [13] Birman V, Bert CW. Behavior of laminated plates subjected to conventional blast. Int J Impact Engng 1987;6(3):145–55. [14] Gottlieb JJ, Ritzel OV. Analytical study of sonic boom from supersonic projectiles. Progr Aerospace Sci 1988;25:131 –88. [15] Librescu L, Nosier A. Response of shear deformable elastic laminated composite panels to sonic boom and explosive blast loadings. AIAA J 1990;28(2):345–52. [16] Nosier A, Librescu L, Frederick D. The effects of time-dependent excitation on the oscillatory motion of viscously damped laminated composite flat panels. In: Voyiadjis GZ, Karamanlidis D, editors. Advances in the theory of plates and shells. Amsterdam: Elsevier; 1990. p. 249–68. [17] Volmir AS. Nonlinear dynamics of plates and shells. Moskow: Nauka; 1972. in Russian. [18] Librescu L, Meirovitch L, Na SS. Control of cantilevers vibration via structural tailoring and adaptive materials. AIAA J 1997;35(8):1309–15. [19] Song O, Ju JS, Librescu L. Dynamic response of anisotropic thinwalled beams to blast and harmonically oscillating loads. Int J Impact Engng 1998;21(8):663–82. [20] Cheng DH, Benveniste JE. Sonic boom effects on structures—a simplified approach. Trans NY Acad Sci, Ser II 1968;30:457 –78. [21] Vinson JR. The behavior of sandwich structures of isotropic and composite materials. Lancaster, Basel: Technomic Publication Co. Inc; 1999. [22] Hohe J, Librescu L. A nonlinear theory for doubly curved anisotropy sandwich shells with transversely compressible core. Int J Solids Struct 2003;40:1059 –88.
Linear and non-linear dynamic response of sandwich panels to blast loading Liviu Librescua,*, Sang-Yong Oha, Joerg Hoheb a
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Mail Cose (0219) Blacksburg, VA 24061, USA b˙ Fraunhofer-Institute fuer Werkstoffmechanik (IWM), D-79108 Freiburg, Germany Received 17 May 2003; accepted 17 July 2003 Available online 9 April 2004
Abstract The problem of the dynamic response of sandwich flat panels exposed to blast loadings is addressed. The sandwich model includes a number of non-classical effects such as the anisotropy and heterogeneity of face sheets, transverse orthotropy of the core layer, the geometrical non-linearities considered in the von Ka´rma´n sense, as well as the initial geometric imperfections. As concerns the blast pulses considered in this analysis, these are due to either an underwater/in-air explosion, or are due to a pressure wave traveling across the panel span. Implications of the explosive charge, stand-off distance, directionality property of face sheets material, damping ratio, geometric nonlinearity, initial geometric imperfection, and of the characteristics of the blast, on dynamic response and dynamic magnification factor are put into evidence via a parametric study, and pertinent conclusions are outlined. q 2004 Elsevier Ltd. All rights reserved. Keywords: Dynamic response
1. Introduction A growing interest for an extensive integration of sandwich composites in the construction of naval ships and submarines has been manifested in the last decade. This trend was outlined in the extensive review-paper, Ref. [1], as well as in the works of the recent international conference on sandwich constructions, Ref. [2], where the achievements in this area have been presented, and the potential benefits of the incorporation of sandwich composites in the construction of a variety of naval vessels, including the military ones, have been discussed. Some of the underlying reasons and motivation for this interest emerge, among others, from the fact that, compared to metallic hulls, the sandwich ones experience lower structural weight, extended operational life, lower maintenance cost, as well as a range of integrated functions, such as thermal and sound insulation, excellent signature properties, fire safety, good energy absorption, directional properties of the face sheets enabling one an optimized design, and a smooth hydrodynamic surface. * Corresponding author. Tel.: þ 1-540-231-5916; fax: þ1-540-231-4574. E-mail address: [email protected] (L. Librescu). 1359-8368/$ - see front matter q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2003.07.003
Needless to say, the development of new manufacturing techniques, has further contributed to render the sandwich structures more affordable, and as a result, more attractive for their use. An issue that should be addressed when dealing with sandwich composite structures used in the design of combat ships, is that of their dynamic response to time-dependent external loading of the blast type. In this context one should remark that, to the best of the authors knowledge, with the exception of the works by Moyer et al. [3], Hayman [3], and Ma¨kinen [4] where special issues related to underwater explosion on sandwich panels have been addressed, no other studies on this topic are available in the archival literature. The issue of the dynamic response of sandwich flat panels to time-dependent loads generated by an underwater explosion and by a shock-wave will be considered in the next developments. In the former case, one supposes that we deal with a submerged panel, while in the latter one, with a topside sandwich panel of ship superstructures. In this context, in order to put into evidence the implications of various non-classical effects, such as those of geometrical non-linearities, initial geometric imperfections, anisotropy properties of face sheets and their ply-sequence, transverse shear orthotropy properties of the core layer, etc., an
674
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
advanced model of sandwich constructions will be used. The basic equations of this structural model have been derived in a number of previous works, [5] –[8]. To be reasonably self-contained, 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.
and face sheets, (vi) the layers constituting 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, as well, and finally, (viii) a Lagrangian description of the non-linear model of sandwich structures is adopted, in conjunction with the von-Ka´rma´n concept and consideration of initial geometric imperfections.
2. Basic assumptions. displacement representation The global middle 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 while those of the upper and the core is denoted as 2h; 00 bottom face sheets, as h and h0 ; respectively. As a result, Hð; 2h þ h0 þ h00 Þ is the total thickness of the structure (see Fig. 1). The sandwich model used in this paper is based on the following assumptions: (i) the face-sheets are composed of a number 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 material of the core features orthotropic properties, the axes of orthotropy being parallel to the geometrical axes xa ; (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 is adopted for both the core
3. Kinematics Several basic kinematic relationships derived in Refs. [6,7] will be supplied next. 3.1. The 3D 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 V 03 ðx1 ; x2 ; x3 Þ ¼ V 003 ðx1 ; x2 ; x3 Þ ¼ V 3 ðx1 ; x2 ; x3 Þ ; v3 ðx1 ; x2 Þ ð1Þ Upon discarding transverse shear effects in the face-sheets, the 3D distribution of the displacement field fulfilling the kinematic continuity conditions at the interfaces between the core and face sheet results as (see Refs. [6,7]) V 01 ðxa ; x3 Þ ¼ j1 ðxa Þ þ h1 ðxa Þ 2 ðx3 2 aÞ›v3 ðxa Þ=›x1
ð2aÞ
ðh # x3 # h þ h0 Þ V 02 ðxa ; x3 Þ ¼ j2 ðxa Þ þ h2 ðxa Þ 2 ðx3 2 aÞ›v3 ðxa Þ=›x2
ð2bÞ
V 03 ðxa ; x3 Þ
ð2cÞ
¼ v3 ðxa Þ;
h1 ðxa Þ þ ðh=2Þ›v3 ðxa Þ=›x1 } V 1 ðxa ; x3 Þ ¼ j1 ðxa Þ þ ðx3 =hÞ{ ð2h # x3 # hÞ ð3aÞ h2 ðxa Þ þ ðh=2Þ›v3 ðxa Þ=›x2 } V 2 ðxa ; x3 Þ ¼ j2 ðxa Þ þ ðx3 =hÞ{ ð3bÞ V 3 ðxa ; x3 Þ ¼ v3 ðxa Þ;
ð3cÞ
and V 001 ðxa ; x3 Þ ¼ j1 ðxa Þ 2 h1 ðxa Þ 2 ðx3 þ aÞ›v3 ðxa Þ=›x1
ð4aÞ
ð2h 2 h00 # x3 # 2hÞ
Fig. 1. Geometry of the sandwich flat panels with laminated face sheets.
V 002 ðxa ; x3 Þ ¼ j2 ðxa Þ 2 h2 ðxa Þ 2 ðx3 þ aÞ›v3 ðxa Þ=›x2
ð4bÞ
V 003 ðxa ; x3 Þ
ð4cÞ
¼ v3 ðxa Þ
In these equations, Vi ðxa ; x3 Þ are the 3D displacement components in the directions of the coordinates xi :
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
In addition
ja ¼
ðV^ 0a
þ
V^ 00a Þ=2;
ha ¼
ðV^ 0a
2
V^ 00a Þ=2
ð4dÞ
denote the 2D 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. It should also be mentioned that in the dynamic case, as considered in this paper, the displacement quantities are functions of time, as well. Although this dependence was not explicitly specified, this is automatically implied. In the previous and the following equations, the Greek indices have the range 1, 2, while the Latin indices have the range 1, 2, 3, and unless otherwise stated, Einstein’s summation convention over the repeated indices is employed. In addition, ð·Þ;i denotes partial differentiation with respect to coordinate xi :
4. Distribution of strain quantities across the shell thickness The structure is assumed to feature a stress-free small initial geometric imperfection V 3 ð; V 3 ðxa ÞÞ: Adopting the von-Ka´rma´n concept of small strains and moderately small rotations (see Refs. [6 – 8]), the distribution of 3D strain quantities eij ð; eij ðx1 ; x2 ; x3 Þ across the wall thickness of the sandwich panel results as: In the bottom face-sheets
675
tangential and the transverse shear strain measures, respectively. Their expressions in terms of the 2D displacement measures in the face-sheets and core are supplied next. 4.1. 2D strain –displacement relationships Bottom face sheets 1011 ¼ j1;1 þ h1;1 þ
1 2 v þ v3;1 v 3;1 2 3;1
ð1 , 2Þ
ð8aÞ
g012 ¼ j1;2 þ j2;1 þ h1;2 þ h2;1 þ v3;1 v3;2 þ v 3;1 v3;2 þ v3;1 v 3;2 ð8bÞ k011 ¼ 2v3;11
ð1 , 2Þ
ð8cÞ
k012 ¼ 22v3;12
ð8dÞ
Core Layer 111 ¼ j1;1 þ
1 2 v þ v 3;1 v3;1 2 3;1
ð1 , 2Þ
g12 ¼ j1;2 þ j2;1 þ v3;1 v3;2 þ v 3;1 v3;2 þ v3;1 v 3;2 1 1 k11 ¼ h1;1 þ hv3;11 ð1 , 2Þ 2 h 1 k12 ¼ {h1;2 þ h2;1 hv3;12 } h 1 1 g13 ¼ h1 þ hv3;1 þ v3;1 2 h
ð9aÞ ð9bÞ ð9cÞ ð9dÞ
ð1 , 2Þ
ð9eÞ
Top face sheets 10011 ¼ j1;1 2 h1;1 þ
1 2 v þ v 3;1 v3;1 2 3;1
ð1 , 2Þ
ð10aÞ
e011 ¼ 1011 þ ðx3 2 a0 Þk011
ð5aÞ
g0012 ¼ j1;2 þ j2;1 2 h1;2 2 h2;1 þ v3;1 v3;2 þ v 3;1 v3;2 þ v3;1 v 3;2 ð10bÞ
e022 ¼ 1022 þ ðx3 2 a0 Þk022
ð5bÞ
k0011 ¼ 2v3;11
2e012 ¼ g012 þ ðx3 2 a0 Þk012
ð5cÞ
k0012 ¼ 22v3;12
In the core layer e 11 ¼ 111 þ x3 k11
ð6aÞ
e 22 ¼ 122 þ x3 k22
ð6bÞ
2e12 ¼ g12 þ x3 k12
ð6cÞ
2e13 ¼ g13
ð6dÞ
2e23 ¼ g23
ð6eÞ
In the top face-sheets e0011 ¼ 10011 þ ðx3 þ a00 Þk0011
ð7aÞ
e0022
ð7bÞ
¼
10022
þ ðx3 þ a
00
Þk0022
2e0012 ¼ g0012 þ ðx3 þ a00 Þk0012
ð7cÞ
In these equations 111 ; 122 ; 112 ð; g12 =2Þ and 113 ð; g13 =2Þ; 123 ¼ ðg23 =2Þ denote the 2D (in the sense of 1ij ð; 1ij ðx1 ; x2 ÞÞ
ð1 , 2Þ
ð10cÞ ð10dÞ
Herein and in the forthcoming equations, the sign ð1 , 2Þ indicates that, from the equations accompanied by this sign, companion equations, not explicitly displayed, can be obtained by replacing subscript 1 by 2, and vice versa. As a mere remark, due to inclusion of geometrical nonlinearities, the inherent stretching-bending coupling emerges in the kinematic equations, Eqs. (8) –(10). Eqs. (9a) – (9e) are applicable in general to strong core sandwich structures. For the weak core sandwich structures, only g13 and g23 are relevant.
5. Equations of motion and boundary conditions Hamilton’s principle is used to derive the equations of motion and the boundary conditions of geometrically non-linear theory of sandwich flat panels. This may be
676
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
stated as ðt1 dJ ¼ d ðU 2 W 2 TÞdt ¼ 0
ð11Þ
t0
where t0 ; t1 are two arbitrary instants of time, U; W and T denote the strain energy, the work done by surface tractions, edge loads and body forces, and the kinetic energy of the 3D body of the sandwich structure, respectively, while d denotes the variation operator. The expressions of U; W and T are not displayed here. These can be found in Refs. [6 –8]. From Eq. (11) considered in conjunction with the proper expression of various energy quantities and with the strain– displacement relationships (used as subsidiary conditions), carrying out the integration with respect to x3 and integrating by parts wherever necessary as to relieve the virtual displacements of any differentiation and invoking the arbitrary and independent character of the variations dh1 ; h2 ; dj1 dj2 and dv3 throughout the entire domain of the plate and within the time interval ½t0 ; t1 ; using the expression of global stress resultants and stress couples (to be defined later), one derive the equations of motion and the boundary conditions relevant to weak core sandwich panels. By retaining only the transversal load, and the transverse inertia and transverse viscous damping terms, one obtain. The equations of motion:
dj1 : N11;1 þ N12;2 ¼ 0
ð12aÞ
dj2 : N22;2 þ N12;1 ¼ 0
ð12bÞ
dh1 : L11;1 þ L12;2 2 N 13 ¼ 0
ð12cÞ
dh2 : L22;2 þ L12;1 2 N 23 ¼ 0
ð12dÞ
dh3 : N11 ðv3;11 þ v 3;11 Þ þ 2N12 ðv3;12 þ v 3;12 Þ þ N22 ðv3;22 þ v 3;22 Þ þ ðM11;11 þ 2M12;12 þ M22;22 Þ N 13;1 þ N 23;2 Þ þ q3 ¼ m0 v€ 3 þ C_v3 þ a=hð
ð12eÞ
where C is the transverse viscous damping coefficient, while the overdots denote time-derivatives. In addition, the boundary conditions that are consistent to the previously displayed equations of motion are as follows: Nnn ¼ Nnn ~
or jn ¼ jn ~
Nnt ¼ Nnt ~
or jt ¼ jt ~ or hn ¼ hn ~
Lnn ¼ Lnn ~
ð13aÞ ð13bÞ . X
ð13cÞ
In these equations subscripts n and t are used to designate the normal and tangential in-plane directions to an edge and, hence, nP ¼ 1 when t ¼ 2; and vice versa. In addition, the symbol n;t indicates that no summation over the indices n and t is implied, while the terms underscored by a tilde denote prescribed quantities. The displayed equations of motion and static boundary conditions are expressed in terms of the global stress resultants and stress couples as N11 ¼ N 011 þ N 0011 N12 ¼ N 012 þ N 0012 L11 ¼
011 hðN
L12 ¼
012 hðN
2
N 0011 Þ
2
N 0012 Þ
M12 ¼
M 012
þ
Lnt ¼ Lnt or ht ¼ ht ~ ~ Mnn ¼ Mnn or v3;n ¼ v3;n ~ ~ Nnt ðv3;t þ v 3;t Þ þ Nnn ðv3;n þ v 3;n Þ þ Mnn;n þ 2Mnt;t
ð13dÞ
N n3 ¼ Mnt;t þ N n3 or v3 ¼ v3 þ ða=hÞ ~ ~ ~
ð13fÞ
ð13eÞ
ð14aÞ ð14bÞ
ð1 , 2Þ
ð14cÞ ð14dÞ
M11 ¼ M 011 þ M 0011
ð1 , 2Þ
M 0012
ð14eÞ ð14fÞ
In addition, in Eqs. (12) and (13), ðN 0ab ; M 0ab ÞðN 00ab ; M 00ab Þ; and N ab are the 2D stress resultants and stress couples measures associated with the lower and upper face sheets and with the core layer, respectively. These equations are defined as {N 0ab ; M 0ab } ¼
N ððx3 Þk X k¼1
N a3 ¼
ðh 2h
ðx3 Þk21
ðS0ab Þk {1; x3 2 a}dx3
S a3 dx3 ða; b ¼ 1; 2Þ
ð15aÞ
ð15bÞ
where Sij are the components of the Piola –Kirchhoff stress tensor. In addition, N is the number of constituent layers in the bottom face sheets, (equal to that in the top faces), whereas ðx3 Þk and ðx3 Þk21 are the distances from the global mid-plane of the structure to the upper and lower interfaces of the kth layer, respectively. As concerns the 2D constitutive equations associated to the sandwich structures, these have been supplied, e.g. in Refs. [6,7], and will not be repeated here.
6. Governing system For the problem at hand, the mixed representation of the governing equations will be considered. This formulation is done in terms of the Airy’s potential function fðx1 ; x2 Þ; and the 2D displacement measures v3 ; h1 and h2 : To this end, by expressing the stress resultants in terms of the Airy’s potential function fð; fðxv ÞÞ as: Nab ¼ cav cbr f;vr
n;t
ð1 , 2Þ
ð16Þ
the equilibrium Eqs. (12a) and (12b) are identically fulfilled, where cab denotes the 2D permutation symbol. Having in view that by virtue of representation Eq. (16), the two equilibrium Eqs. (12a) and (12b) are eliminated, in order to ensure single valued displacements, the compatibility equation for the tangential strain measures has to be fulfilled.
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
677
For flat sandwich panels featuring a weak core this equation is:
moduli. For more details about the derivation of these equations the reader is referred to [6,7].
111;22 þ 122;11 2 g12;12 2 2v23;12 þ 2v3;11 v3;22 þ 2v3;11 v3;22 þ 2v3;11 v 3;22 2 4v3;12 v 3;12 ¼ 0
7. Blast loads
ð17Þ
where
7.1. Underwater explosive shock loading
111 ¼ 1011 þ 10011
ð18aÞ
122 ¼ 1022 þ 10022
ð18bÞ
g12 ¼ g012 þ g0012
ð18cÞ
Making use of the partially inverted form of constitutive equations considered in conjunction with Eq. (16), Eq. (17) can be expressed in terms of the basic unknown function as
When a charge is exploded underwater, (see Fig. 2), an instantaneous pressure pulse, or shock wave is generated and transmitted in all directions (see Refs. [9,10]). The pressure time-history at a fixed location starts with an instantaneous pressure increase to a peak followed by a decrease. The expression of the pressure that corresponds to the incident wave pulse is given by
Ap22 f;1111 þ Ap11 f;2222 2 2Ap16 f;1222 2 2Ap26 f;2111
Pi ðtÞ ¼ qm e2ðt2t1 Þ=a t $ t1
ðAp66
2Ap12 Þf;1122
2v23;12
þ þ 2 þ 2v3;11 v3;22 þ 2v3;11 v3;22 þ 2v3;11 v 3;22 2 4v3;12 v 3;12 ¼ 0
ð19Þ
On the other hand, the equilibrium Eqs. (12c) – (12e) represented in terms of displacement quantities and of Airy’s function are as follows: A11 h1;11 þ A16 h2;11 þ A66 h1;22 þ ðA12 þ A66 Þh2;12 h1 þ av3;1 Þ ¼ 0 13 =hÞð þ 2A16 h1;12 þ A26 h2;22 2 ð2K 2 G ð20aÞ A22 h2;22 þ A26 h1;22 þ A66 h2;11 þ ðA12 þ A66 Þh1;12 h2 þ av3;2 Þ ¼ 0 23 =hÞð þ 2A26 h2;12 þ A16 h1;11 2 ð2K 2 G ð20bÞ
f;22 ðv3;11 þ v 3;11 Þ22f;12 ðv3;12 þ v 3;12 Þ2F11 v3;1111 2F22 v3;2222 24F16 v3;1112 24F26 v3;1222 G 13 ðh1;1 þav3;11 Þ 22ðF12 þ2F66 Þv3;1122 þð2K 2 a=hÞ{ þ G23 ðh2;2 þav3;22 Þ}2m0 v€ 3 2C_v3 þq3 ¼0 ð20cÞ In these equations by virtue of the structural symmetry of the sandwich panel. Aab ;ðA0ab ¼A00ab Þ; Fb; ;ðF 0ab ¼F 00ab Þ; h;ðh0 ¼h00 Þ ð21aÞ whereas the stiffness quantities Apab represent the inverted counterparts of Aab : Herein {A0vr ; B0vr ; D0vr } ¼
N ððx3 Þk X k¼1
ðx3 Þk21
ðv; r ¼ 1; 2; 6Þ
where t 2 t1 (ms) is the time elapsed after the arrival of the shock pulse, where t1 ¼ ðR þ zÞ=c is the time to reach a point at which the pressure is measured; c is the speed of sound in seawater ( ¼ 1476 m/s); qm denotes the peak magnitude of the pressure (MPa) in the shock front, while a denotes the constant that describes the exponential decay (ms). Both qm and a depend on the charge weight Q (kg) and the stand-off distance RðmÞ from the charge to the target. As concerns the pressure due to the reflected shock wave, this is given by Pr ðtÞ ¼
qm ½2 e2ðtþt1 Þ=mr a 2 ð1 þ mr Þe2ðtþt1 Þ=a 1 2 mr
PðtÞ ¼ Pi ðtÞ þ Pr ðtÞ
ð24Þ
For qm and a that are involved in Eqs. (22) and (23), approximate expressions will be used in the present numerical simulations. These are (see Refs. [10,11]): !1:15 Q1=3 qm ¼ 56:6 ðMPaÞ ð25aÞ R
^ 0vr ÞðkÞ ð1; x3 ; x23 Þdx3 ; ðQ ð21bÞ
ð21cÞ
while K 2 ; is the transverse shear correction factor associated to the core layer. By virtue of the symmetry with respect to ^ 0vr ¼ Q ^ 00vr ; where the global mid-surface, N 0 ¼ N 00 and Q ^ vr ¼ Qvr 2 Qv3 Qr3 =Q33 ; Q ^ vr denoting the reduced elastic Q
ð23Þ
In this equation mr ð; m0 =rcaÞ is the mass ratio, where m0 is the panel mass per unit area and r is the fluid mass density that in sea water is r ¼ 1000 kg/m3. It results that the total pressure at a certain distance is
and F 0vr ¼ D0vr 2 2aB0vr 2 a2 A0vr
ð22Þ
Fig. 2. Sandwich panel subjected to underwater explosion.
678
a ¼ 0:08 þ Q
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
1=3
Q1=3 R
!20:23 ðmsÞ
ð25bÞ
where the coefficients in Eq. (25) correspond to the HB £ TNT explosive. The good accuracy of these expressions has been validated by comparing their predictions with the experimental data. As is was reported (see Refs. [5,10]), the agreement is good up to one decay length, which is equivalent to pressures larger than about one third of the peak value. It should also be mentioned that: (i) the supplied pressure expression is not applicable in the case of contact explosion, i.e. when the target is located in the immediate proximity of the explosive charge, and that (ii) the effect of cavitation was not accounted for. Moreover, as Eq. (24) reveals, the pressure generated by the passage of wave through the core was not accounted for. 7.2. Air-blast loading In addition to blasts produced by underwater explosions, in combat operations, the structure of warships can be exposed to air-blasts generated by explosions and by shockwave disturbances produced by an aircraft flying at supersonic speeds, or by any supersonic projectile, rocket or missile. In the latter case, the blast pulse is referred to as sonic-boom. Its time-history is described as an N-shaped pulse, whose negative phase direction is included as a variable in the analysis (see Refs. [14 –16]). In connection with the in-air explosion, one should remark that Eqs. (22) –(25) remain valid also for this case. What differentiates these expressions for the two events there are the values of the speed of sound, c; and the mass density r, in water and air. While for the sea water their values have been supplied, in air we have c ¼ 320 m/s and r ¼ 1:20 kg/m. As a result, essential differences in the pressure time-histories featured in the two types of explosions should occur. For the same charge weight and stand-off distance, in Fig. 3 there are depicted the time-histories of the normalized total pressure exerted on the panel front face and generated by the two explosion events. The results show that in the case of the in-air explosion, Fig. 3a, the decay, of the pressure, as time unfolds, is much slower than in the case of the underwater explosion, Fig. 3b, this rendering the effects of the in-air explosion much more severe than in the latter case. In the case of underwater explosion, Fig. 3b shows that at time t . 0:02 ms after the plate impact, the pressure becomes negative. In this case, cavitation is likely to occur at the plate surface. Having in view the large blast wave front generated by the explosion as compared to the relatively small dimensions of the panel, one assumes with sufficient accuracy that the pressure is uniform over the entire panel that is impacted at normal incidence. This assumption is adopted in the cases
Fig. 3. Normalized total pressure time-histories on the panel front face by an in-air (a), and underwater (b), explosions (Q ¼ 9 kg, R ¼ 3 m).
of the underwater/in-air explosions, as well as in that of the panel impacted at normal incidence by a sonic-boom pulse. A unified expression of the normal shock load applicable, when is properly specialized, to the sonic-boom, triangular blast and Heaviside pulse is given by Eq. (26) ! t PðtÞ ¼ qm 1 2 ½HðtÞ 2 db Hðt 2 rtp Þ ð26Þ tp This equation provided in Ref. [12] includes as special cases the pressure pulses above mentioned, and considered separately in Refs. [13 – 16]. In this equation and the next ones, HðtÞ denotes the Heaviside step function, db is a tracer that can take the values 1 or 0 depending on whether the sonic-boom or triangular blast load is considered, respectively, qm denotes the peak reflected pressure in excess to the ambient one, tp denotes the positive phase duration of the pulse measured from the time of impact of the structure, r denotes the shock pulse length factor. For r ¼ 1; the sonic-boom blast degenerates into a triangular explosive pulse, while for r ¼ 2; a symmetric
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
sonic-boom pulse is obtained. When r ¼ 1 and tp ! 1; Eq. (26) defines a Heaviside pulse. The case of an air-blast traveling in tangential direction to the panel span will also be considered. The pressure time-history for this latter case is represented as (see Ref. [17]) PðtÞ ¼ qm e2hðct2x1 Þ Hðct 2 x1 Þ
ð27Þ
where c is the wave speed in the medium surrounding the structure, while h is an exponent determining the character of the blast decay. Notice that when h ¼ 0; from Eq. (27), a Heaviside pulse is reached.
8. Solution methodology The governing equation systems, Eqs. (19) and (20) in conjunction with the explicit form of the pressure pulse and the boundary conditions has to be solved as to determine the time-history of displacements h1 ; h2 ; v3 ; and of the Airy’s function F: This constitutes an essential step toward determination of the full response time history, that is of strains and stresses, as well. From the mathematical point of view, the problem at hand reduces to the solution of a dynamic non-linear boundary value problem. Herein, the sandwich rectangular panel is assumed to be simply supported all over the contour. Consistent with the order twelve of the governing system, six boundary conditions have to be prescribed at each edge. Assuming the case of edges unloaded and immovable in the tangential direction, normal to the panel edge, the boundary conditions (see Refs. [6 –8]), are:
j1 ¼ 0 N12 ¼ 0; h1 ¼ 0; h2 ¼ 0; M11 ¼ 0; v3 ¼ 0 on x1 ¼ 0; L1
679
Table 1 Elastic coefficients for the materials of face sheets and core layers Face sheets Core Layer
E1 (GPa) 207 13 (GPa) G 0.1027
E2 (GPa) 5.17 23 (GPa) G 0.0621
G12 (GPa) 2.55
n12 0.25 rc 16
rf (kg/m3) 1588.22
9. Results The numerical results presented here have considered the case of face sheets laminated from the same orthotropic material, whose axes of orthotropy are rotated with respect to the geometrical axes of the panel by an angle u: For the face sheets, a material characterized by a rather large orthotropicity ratio was considered. Its properties, as well as those of the core layer are given in Table 1. In all numerical simulations, unless otherwise stated, the stacking sequence of face sheets is as follows: ½0=908=0=908=0=core=0=908=0=908=0; and the geometrically non-linear structural model was used. In Fig. 4 it is depicted the dimensionless deflection w=H time-history of the central point of a square panel ðL1 =L2 ¼ 1Þ; for selected values of the explosive weight. Needless to say, here and in the following plots, time is measured from the instant of the plate impact by the incident waves. Herein w ; v3 ðL1 =2; L2 =2Þ: The simulations have been carried out for the geometrically non-linear panel featuring a prescribed small initial geometric imperfection at the center of the panel d0 ð; v 3 ðL1 =2; L2 =2Þ=HÞ ¼ 0:2: It is assumed also that the stand-off distance R ¼ 3:1 m. It is seen that the structural damping ratio zð; C=2m0 v1 Þ where v1 is the undamped fundamental frequency (that for this case is z ¼ 0:05), has a strong effect of attenuating the oscillation induced by the explosion. In Fig. 5, it is highlighted the effect of the stand-off distance on the dimensionless panel deflection. The results
and
j2 ¼ 0; N12 ¼ 0; h1 ¼ 0; h2 ¼ 0; M22 ¼ 0; v3 ¼ 0 on x2 ¼ 0;L2 Due to the intricacy of the present boundary value problem, an approximate solution methodology based on Extended Galerkin Method (EGM) coupled with the satisfaction of the tangential boundary conditions on an average sense was used. This methodology was presented in details in Refs. [7] and [8], and in a more general context in Ref. [18], and will not be repeated here. It should be stressed that the predictions provided by the application of the EGM are of an extreme accuracy, these being identical, in many instances to the exact ones. In this sense, the reader is referred to Ref. [19] where an exact superpostion of predictions on dynamic response obtained via EGM and Laplace Transform Method was reported.
Fig. 4. Effect of the explosive weight on dimensionless deflection timehistory of the center panel subject to underwater explosion (z ¼ 0:05; d0 ¼ 0:2; R ¼ 3:1 m, L1 =H ¼ 20).
680
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
Fig. 5. The effect of the stand-off distance on dimensionless deflection timehistory of the panel subjected to underwater explosion (z ¼ 0:05; d0 ¼ 0:2; Q ¼ 18 kg; L1 =H ¼ 20).
Fig. 7. Effects of the structural damping on dimensionless velocity timehistory of the center panel subjected to underwater explosion (d0 ¼ 0:2; Q ¼ 9 kg; R ¼ 3:1 m; L1 =H ¼ 20).
reveal that the increase of the stand-off distance yields a decrease of the transverse deflection. However, results not displayed here reveal that for a fixed explosive charge, beyond a certain value of the stand-off distance (in this case for R . 6 m), the decay of deflection is almost immaterial. In Fig. 6, there are highlighted the effects of the underwater and in-air explosions. In this sense, for the same explosive charge and stand-off distance, there are displayed the deflection time-histories for the two cases. Notice that due to much higher deflection induced by the in-air explosion, in order of being able to represent the responses on the same graph, the normalization factors for the deflection corresponding to the explosion in-air and underwater have been selected as 2H and H; respectively. The results in this plot reveal that, as concerns the panel deflection, the effects of the explosion in-air are more dramatic than the ones in sea water.
In Fig. 7, there are presented the effects of the structural damping z on the dimensionless speed time-history of the center panel. Notice that the normalizing factor for the panel speed is the speed of sound in water. The beneficial effects of the structural damping become evident form this figure. In Fig. 8 the obtained results are based on the following architecture of the sandwich panel ½u= 2 u=u= 2 u=u= core=u= 2 u=u= 2 u=u=: The plots in this figure reveal that the ply-angles in the face sheets can play a significant role toward the attenuation, without weight penalties, of the intensity of the oscillatory motion resulting from the underwater explosion. However, as is revealed, the optimal ply-angle rendering the minimum dynamic deflection depends strongly on the 23 : 13 and G values of transverse shear moduli of the core, G For relatively high values of G13 and G23 ; as emerging from Fig. 8a, the optimal ply-angle is u ¼ 458; whereas for a material of the core more flexible in transverse shear, (such as the foam), the results of Fig. 8b reveal that u ¼ 908 is a more favorable ply-angle. As a result, for sandwich panels, implementation of the tailoring technique in the face sheets should always be considered in conjunction with the parameters defining the flexibilities in transverse shear of the core. Fig. 9 displays the normalized deflection time-histories of the central panel that correspond to an in-air explosion, for selected values of the stand-off distance. Although the explosive charge is rather small, Q ¼ 0:89 N the effects are very strong, specially for reduced stand-off distances. In Fig. 10 the effects of the explosive weight for an in-air explosion are depicted. Although the stand-off distance in rather large, (R ¼ 15:2 m), the effects of the considered weights that are rather small, are still quite strong. In Fig. 11 the effects of a traveling blast as expressed by Eq. (27), on the dimensionless central deflection of the panel are depicted. For two thickness ratios of the sandwich
Fig. 6. Effects of the underwater and in-air explosions on sandwich panel deflection time-histories (z ¼ 0:05; d0 ¼ 0:2; Q ¼ 9 kg, L1 =H ¼ 15).
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
681
Fig. 10. Effects of the in-air explosive weight on dimensionless deflection time-history of the center panel. (z ¼ 0:05; d0 ¼ 0:2; R ¼ 15:2 m; L1 =H ¼ 20).
Fig. 8. Effect of the ply-angle of face sheets on dimensionless deflection time-history of the sandwich panel under underwater explosion (8a) 13 ¼ 103 MPa; G 23 ¼ 5G 13 ; R ¼ 3:1 m; (z ¼ 0:05; d0 ¼ 0:2; Q ¼ 18 kg; G 13 ¼ 10:3 MPa; G23 ¼ 5G13 ). L1 =H ¼ 20Þ; (8b) ðG
Fig. 9. Effects of stand off distance on dimensionless deflection time-history of the panel subjected to an in-air explosion. (z ¼ 0:05; d0 ¼ 0:2; Q ¼ 0:9 kg; L1 =H ¼ 20).
panels, L1 =H ¼ 10 and 15, both the linear and nonlinear dynamic responses are depicted. The results reveal that for the thinner panel ðL1 =H ¼ 15Þ; the effects of the non-linearities are significant in the sense of contributing to the diminution of the vibration amplitude. However, when the panel is moderately thick ðL1 =H ¼ 10Þ; their effect becomes quite immaterial. In Fig. 12 there are highlighted effects of the wave propagation in the medium surrounding the panel subjected to a traveling blast. For this case it appears that c ¼ 50 m/s is the most critical propagation blast speed. However, (see Ref. [17]), depending on the characteristics of the panel, other values of c can become more critical. Finally, in Fig. 13 there is displayed the variation of the dynamic magnification function, (DMF) to a symmetric sonic-boom blast ðr ¼ 2Þ; of the center of the sandwich
Fig. 11. Effects of linear and nonlinear modeling coupled with that of the relative panel thickness on dynamic response of the sandwich panel subjected to a traveling blast. (qm ¼ 1:38 MPa; d0 ¼ 0; h ¼ 1; c ¼ 50 m/s).
682
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
the damped and undamped structure are supplied. It is seen that for this moderately thick sandwich panel the geometrically non-linearities play still a significant beneficial role toward the decay of the DMF. The same beneficial role, in the sense of the diminution of the DMF, is played by the structural damping. In the same context, as it clearly emerges from Fig. 13, the spikes that occur when z ¼ 0; completely disappear in the presence of moderate damping. In a simplified context this trend was also reported in Ref. [20]. It should also be observed that for the undamped and the linearized model the DMF is around 2, a value also indicated in Ref. [21]. However, a rather small additional damping yields a drastic drop of the DMF. At the same time, in the latter case, the beneficial effect of geometric nonlinearities decays considerably. Fig. 12. Effects of the velocity of the blast propagation on nonlinear dynamic response of the sandwich panel center exposed to a traveling blast (qm ¼ 1:38 MPa; d0 ¼ 0:2; z ¼ 0:05; h ¼ 1; L1 =H ¼ 15).
10. Conclusions panel vs. vtp ; where v is the fundamental undamped frequency. The sonic-boom pulse is represented in the inset of the figure. The DMF constituting a measure of the severity of the dynamic response, is defined as the ratio of the largest (in the absolute sense) dynamic deflection to the static deflection due to a load equal to the peak dynamic load. In this figure, comparisons of the variation of the DMF for the linear and non-linear structural models, and for
The problem of the dynamic response of sandwich flat panels subjected to explosive type loadings has been addressed. Loadings produced by both underwater/in-air explosions and by in-air blasts due to a sonic-boom or a traveling pressure wave have been considered. The implications of a number of structural and geometrical characteristics of the sandwich panel, as well as of the ones related to the blast have been highlighted and related conclusions have been drawn.
Fig. 13. Effects of geometrical non-linearities and structural damping on the DMF of the sandwich panel center exposed to a symmetric ðr ¼ 2Þ sonic-boom blast. (d0 ¼ 0; L1 =H ¼ 15; qm ¼ 1:38 MPa).
L. Librescu et al. / Composites: Part B 35 (2004) 673–683
In this context, both the geometrical non-linearities and the effects of the unavoidable geometric imperfections have been implemented in the structural model. Needless to say, the obtained results can be extended without any difficulty as to determine the time-histories of strain and stress components of the various points of the structure. These items are essential toward determining the failure of the structure. Other issues that have still to be addressed are the ones involving the incorporation of the cavitation effect on the dynamic response of submerged sandwich panels exposed to an explosion, and also the effects played by transverse normal compressibility of the core layer (see Ref. [22]). These issues will be addressed in the forthcoming papers.
Acknowledgements The financial 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.
References [1] Mouritz AP, Gellert E, Burchhill P, Challis K. Review of advanced composite structures for naval ships and submarines. Compos Struct 2001;53:21–41. [2] Vinson JR, Rajapakse YDS, Carlsson LE, editors. Sixth International Conference on Sandwich Structures, Boca Raton, London: CRC Press; 2003. [3] Hayman B. Underwater explosion loading on foam-cored sandwich panels. Sandwich Construction 3, EMAS. Proceedings from the Third International Conference on Sandwich Construction, May 11–15, Southampton, England; 1995. [4] Ma¨kinen K. Underwater shock loaded sandwich structures. Department of Aeronautics, Royal Institute of Technology, Report 99-01 Stockholm, Sweden; 1999. [5] Librescu L. Elastostatics and Kinetics of anistropic and heterogeneous shell-type structures. Nordhoff International Publishing, Leydon, Netherlands. Ch. VI, 1975; 515 –32.
683
[6] Librescu L, Hause T, Camarda CJ. Geometrically nonlinear theory of initially imperfect sandwich plates and shells incorporating nonclassical effects. AIAA J 1997;35(8):1392 –403. [7] Hause T, Librescu L, Camarda CJ. Postbuckling of anisotropic flat and doubly-curved sandwich panels under complex loading conditions. Int J Solids Struct 1998;35:3007–28. [8] Librescu L, Hause T, Johnson TF. Buckling and nonlinear response of sandwich curved panels to combined mechanical loads—implications of face-sheets elastic tailoring. J Sandwich Struct Mater 2000;2: 246–69. [9] Cole RH. Underwater explosions. New York: Dover Publications; 1965. [10] Shin YS, Geers TL. Response of marine structures to underwater explosions. International short course notebook, shock and vibration research, Monterey, CA; 1994. [11] Jiang J, Olsen MD. Modeling of underwater shock-induced response of thin plate structures. Report No. 39, Department of Civil Engineering, University of British Columbia, Vancouver, BC; 1994. [12] Marzocca P, Librescu L, Chiocchia G. Aeroelastic response of 2D lifting surfaces to gust and arbitrary explosive loading signature. Int J Impact Engng 2001;25(1):67–85. [13] Birman V, Bert CW. Behavior of laminated plates subjected to conventional blast. Int J Impact Engng 1987;6(3):145–55. [14] Gottlieb JJ, Ritzel OV. Analytical study of sonic boom from supersonic projectiles. Progr Aerospace Sci 1988;25:131 –88. [15] Librescu L, Nosier A. Response of shear deformable elastic laminated composite panels to sonic boom and explosive blast loadings. AIAA J 1990;28(2):345–52. [16] Nosier A, Librescu L, Frederick D. The effects of time-dependent excitation on the oscillatory motion of viscously damped laminated composite flat panels. In: Voyiadjis GZ, Karamanlidis D, editors. Advances in the theory of plates and shells. Amsterdam: Elsevier; 1990. p. 249–68. [17] Volmir AS. Nonlinear dynamics of plates and shells. Moskow: Nauka; 1972. in Russian. [18] Librescu L, Meirovitch L, Na SS. Control of cantilevers vibration via structural tailoring and adaptive materials. AIAA J 1997;35(8):1309–15. [19] Song O, Ju JS, Librescu L. Dynamic response of anisotropic thinwalled beams to blast and harmonically oscillating loads. Int J Impact Engng 1998;21(8):663–82. [20] Cheng DH, Benveniste JE. Sonic boom effects on structures—a simplified approach. Trans NY Acad Sci, Ser II 1968;30:457 –78. [21] Vinson JR. The behavior of sandwich structures of isotropic and composite materials. Lancaster, Basel: Technomic Publication Co. Inc; 1999. [22] Hohe J, Librescu L. A nonlinear theory for doubly curved anisotropy sandwich shells with transversely compressible core. Int J Solids Struct 2003;40:1059 –88.