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Blast resistance and energy absorption of foam-core cylindrical sandwich shells under external blast
Composite Structures 94 (2012) 3174–3185
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Blast resistance and energy absorption of foam-core cylindrical sandwich shells under external blast Michelle S. Hoo Fatt ⇑, Harika Surabhi Department of Mechanical Engineering, The University of Akron, Akron, OH 44325-3903, United States
a r t i c l e
i n f o
Article history: Available online 22 May 2012 Keywords: Blast Cylindrical sandwich shell Foam core
a b s t r a c t The early time, through-thickness stress wave response of a foam-core, composite sandwich cylindrical shell under external blast is examined in this paper. Solutions for the transient response of the facesheets were derived as stress waves propagated through an elastic–plastic, crushable foam core. These solutions were found to be in good agreement with results from finite element analysis. The blast response of the composite sandwich cylindrical shell was shown to be affected by the magnitude and duration of the pressure pulse. High amplitude, low duration (impulsive) pressure pulses induced the greatest energy absorption. Low amplitude, long duration pressure pulses caused minimal energy absorption. The amount of energy absorbed increased and the failure load decreased with increasing core thickness. Sandwich shells with foams of varying density, compressive modulus and crushing resistance were also examined. The sandwich shells with the foam of the highest density, compressive modulus and crushing resistance (Divinycell HCP100) were found to be the most blast resistant to failure even though no energy was absorbed by them. Per unit weight, however, the shells with a lighter, less stiff and strong, Divinycell H200 foam core were more blast resistant to failure than shells with a Divinycell HCP100 foam core. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Foam-core composite sandwich shells are being increasingly used in naval and other military applications because of excellent properties such as high specific stiffness and strength, corrosion resistance, and reduced magnetic signatures. Sandwich shells with crushable foam cores can also offer better resistance to external blast when compared to monolithic shells because the foam cores are able to absorb some of the blast energy through shock attenuation and plastic work dissipation. The objective of this study is to develop analytical solutions for the early time response of a composite sandwich cylindrical shell subjected to external blast. The term ‘‘early time response’’ refers to a through-thickness wave propagation phase involving the formation of elastic–plastic stress waves in the foam core as they reflect from one facesheet to another. The crushable foam core is considered as an elastic–plastic, energy dissipating medium, which may attenuate shock waves and absorb some of the blast energy during this time period. Most of the recent work on the blast response of composite sandwich cylindrical shells has been done where the core was considered as a purely elastic material [1–4], and very few studies have dealt with the elastic–plastic behavior of the core. This study covers the elastic–plastic wave response of the foam and the ⇑ Corresponding author. Tel.: +1 330 972 6308; fax: +1 330 972 6027. E-mail address: [email protected] (M.S. Hoo Fatt). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.05.013
interaction of the foam stress waves with the facesheets during blast. The analytical solution for the transient response is restricted to the early time response of the sandwich shell, i.e., within the several reflections of the wave transit time through the core, because it is in this regime that the foam core undergoes plastic work dissipation. The previous solution developed by Hoo Fatt and Pothula [5] for dynamic pulse buckling of a thin composite cylindrical shell is extended to find the equations of motion for the outer and inner facesheets. Shell fracture is also investigated using laminate shell theory and failure criteria. The analytical solution obtained is also compared to finite element analysis (FEA) using ABAQUS standard analysis software to gage its accuracy. Finally, the analytical solution derived is used to understand mechanisms and parameters that are responsible for increasing blast resistance and energy absorption of the composite sandwich cylindrical shell.
2. Problem formulation Consider the composite sandwich cylinder with geometry shown in Fig. 1a. The outer and inner facesheets are thin, fiberreinforced laminated shells with density qf, thickness h and radii r1 and r2, respectively. The core is made of crushable foam with thickness H and density qc. The sandwich shell is subjected to external blast, i.e., a uniformly-distributed pressure pulse loading described as
M.S. Hoo Fatt, H. Surabhi / Composite Structures 94 (2012) 3174–3185
3175
Fig. 1. Foam core, composite sandwich cylinder under external blast: (a) geometry and (b) stresses and deformations during loading.
t pðtÞ ¼ po 1 DT
ð1Þ
where po is the peak pressure, DT is the pulse duration and t is time. The analysis is limited to long shells, where the length of the shell is 20 times greater than its mean radius. This combined with the fact that the pressure load does not vary along the shell longitudinal axis allows one to consider the composite sandwich cylinder in plane strain. Each facesheet deforms as a ring subjected to radial pressure from the external blast and/or radial stresses propagating back and forth in the core. As indicated in Fig. 1b, the radial deformation of the outer and inner facesheets are denoted by w1 and w2, respectively, while the tangential deformation of the outer and inner facesheets are denoted by v1 and v2, respectively. The facesheets oscillate while stress waves propagate through the foam. The equations of motion of each facesheet are derived by considering the pressure pulse, interactions with the foam (incident and reflected stress propagating in the foam), shell kinetic energy and the shell strain energy. The propagation of radial stresses r and deformations w in the foam is governed by the wave equation. The boundary conditions of this wave equation are determined by the equations of motion of the outer and inner facesheets.
of the outer and inner facesheets. At these strain levels, the core density does not change substantially and the compressive stress–strain response of the crushable foam is described as elastic-perfectly plastic, as shown in Fig. 2a. In the sandwich cylindrical shell, the circumferential and axial dimensions are larger than the radial dimension during through-thickness wave propagation. This allows one to approximate the foam stress–strain response to be in a state of uniaxial strain because of the deformation constraint both in the circumferential and axial directions. The counterpart stress–strain response under uniaxial strain conditions for an elastic-perfectly plastic material is shown in Fig. 2b. The constrained elastic modulus E0 , yield strength q0 and elastic–plastic tangent modulus E0p are derived in Appendix A and are as follows:
4G þK 3 ð3K þ 4GÞq q0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð24G2 þ 27K 2 Þ
E0 ¼
ð3Þ ð4Þ
and
4G 1 þK E0p ¼ 3 3
hpffiffiffi i2 pffiffiffiffi 3KI1 = J 2 þ 4G h i KI21 =J 2 þ 4G
ð5Þ
3. Propagation of plane radial waves in core The equation governing the propagation of radial deformation w in the core is given by
r2 w
1 @2w ¼0 c20 @t 2
ð2Þ
qffiffiffiffiffiffiffiffiffiffiffiffi @ 1 @2 @ where r2 ¼ 1r @r r @r þ r2 @h2 and the wave speed is c0 ¼ E0t =qc and E0t is the tangent modulus of the foam under uniaxial strain constraint. The tangent modulus of the foam is determined from the compressive stress–strain of the foam, which is elastic–plastic. 3.1. Uniaxial strain It is necessary to limit our analysis to the small strains in the foam (less than 20%) because larger strains would induce failure
Fig. 2. Compressive stress–strain behavior of crushable foam: (a) uniaxial stress and (b) uniaxial strain.
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where G = E/(2(1 + m)) is the shearp modulus, K = E/(3(1 pffiffiffi 2m)) is the ffiffiffiffi bulk modulus, I1 = rr + 2rh and J 2 ¼ ðrr rh Þ= 3 are the first invariant of the stress tensor and the second invariant of the deviatoric stress tensor, respectively, and rr and rh are the radial and tangential stresses in the foam, respectively. The radial and tangential stresses in the foam are related to each other by the foam yield criteria. A relation between these stresses is given in Appendix A for a crushable foam yield criterion with isotropic hardening. Hence, the elastic–plastic tangent modulus can be represented in terms of radial stress only. The elastic wave speed in the foam is
ce ¼
sffiffiffiffiffi E0
qc
cp ¼
qc
ð7Þ
ð9Þ
_ 1i and w _ 1r are the incident and reflected wave velocities. where w 3.2.1. Elastic wave propagation The radial incident stress is related to the incident velocity by
ð10Þ
The reflected stress is given by
r1r ¼ qc ce w_ 1r
ð17Þ
ð18Þ
Eliminating r1max from Eqs. (17) and (18) gives
cp 0 _ 1 max w _ 1Þ _ 1 max þ qc ce ðw q þ qc cp w ce
Since the plastic wave speed is slower than the elastic wave speed, the plastic loading wave is overtaken by the elastic unloading wave at time t3 where
t 3 ¼ t 2 þ cp
ðt 2 t1 Þ ðce cp Þ
ð19Þ
where t1 and t2 are the times at which plastic waves and elastic unloading waves begin to propagate, respectively. The incident stress on the inner facesheet is not only delayed by the wave transit time te but it is unaffected by the induced plastic stresses at the outer foam. Jump conditions across the elastic loading–unloading boundary give the incident velocity and stress on the inner facesheet as
_ 1 max þ w _ 1 ht t e i _ 2 ¼ Ve w w
ð20Þ
and
r2 ¼ qc ce ðV e w_ 1 max þ w_ 1 ht te iÞ
ð21Þ
where the symbol hi is used to denote shifted time, i.e.,
_ 1 ht t e i ¼ w
0; t < te _ w1 ðt t e Þ; t P te
ð22Þ
ð12Þ
Hence the total stress in the foam at the outer facesheet is
r1 ¼ r1i þ qc ce ðw_ 1i w_ 1 Þ
_ 1 max V e ½r1 max q0 ¼ qc cp ½w
ð11Þ
Substituting Eq. (9) into Eq. (11) gives
r1r ¼ qc ce ðw_ 1i w_ 1 Þ
ð16Þ
_ 1 max are the maximum stress and particle velocwhere r1max and w ity at the outer facesheet, respectively. The momentum jump condition across the plastic wave front gives
ð8Þ
Here the incident stress results from stress wave propagation in the foam. The pressure pulse load is not considered as an incident stress; it acts on the opposite side of the outer facesheet. Compatibility requires that the particle velocity in the foam nearest to the outer facesheet is
r1i ¼ qc ce w_ 1i
cp 0 _1 q þ qc cp w ce
Elastic unloading then takes place when the plastic stress or particle velocity is at a maximum value. Figure 3c shows the wave fronts in the foam during elastic unloading. The momentum jump condition across the elastic unloading wave front gives
r1 ¼ 1
The total radial foam stress acting on the outer facesheet r1 is the sum of an incident stress r1i and a reflected stress r1r:
_ 1i þ w _ 1r _1¼w w
_1w _ 1 max ½r1 r1 max ¼ qc ce ½w
3.2. Foam stress acting on outer facesheet
ð15Þ
Thus the plastic stress at the outer facesheet is
ð6Þ
Unlike the elastic modulus, the elastic–plastic tangent modulus varies with radial stress. The constrained plastic modulus shown in Fig. 2b indicates that the plastic wave speed is less than the elastic waves speed. An average constrained plastic modulus for plastic strains less than 20% may be used to approximate a constant plastic wave speed (see Appendix A). The radial stress in the foam during wave propagation is discussed below.
r1 ¼ r1i þ r1r
_ 1 V e brp1 q0 c ¼ qc cp ½w
rp1 ¼ 1
while the plastic wave speed is
sffiffiffiffiffi E0p
When the outer foam stress at the outer facesheet reaches q0 , plastic waves start to propagate from r1 to r2 with the plastic wave speed cp. Since E0p < E0 , the plastic wave speed is slower than the elastic wave speed. As shown in Fig. 3b, an elastic region with stress q0 and particle velocity Ve = q0 /(qcce) thus precedes the plastic wave front. Momentum conservation across the plastic wave front (momentum jump condition) gives
3.3. Foam stress acting on inner facesheet
ð13Þ
The total foam stress acting on the inner facesheet is
The above equation is the resistive foam stress acting on the outer facesheet, as indicated in Fig. 1b.
r2 ¼ r2i þ r2r
3.2.2. Elastic–plastic wave propagation The propagation of elastic–plastic waves in the foam from the outer facesheet to the inner facesheet is described in Fig. 3a–c. Elastic waves first propagate from r1 to r2 with the elastic wave speed ce, as shown in Fig. 3a. The elastic wave transit time through core thickness is
where r2i and r2r are the incident and reflected stresses respectively. The velocity of the inner facesheet is
te ¼
H ce
ð14Þ
_ 2i þ w _ 2r _2 ¼w w
ð23Þ
ð24Þ
_ 2i and w _ 2r are the incident and reflected wave velocities. where w 3.3.1. Elastic wave propagation Figure 4a–c show the interaction of incident and reflected stresses in the foam near the inner facesheet. The incident stress
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Fig. 3. Propagation of elastic–plastic waves from outer facesheet: (a) elastic loading, (b) elastic–plastic loading, and (c) elastic unloading.
propagates from the outer facesheet and was shown in Fig. 4a. The reflected stress is
r2r ¼ qc ce w_ 2r
ð25Þ
_ 20 w _ 2i ½q0 r2i ¼ qc ce ½w
Substituting Eq. (24) into Eq. (25) gives
r2r ¼ qc ce ðw_ 2i w_ 2 Þ
ð26Þ
ð27Þ
The above expression is the foam stress acting on the inner facesheet, as shown in Fig. 1b.
ð29Þ
_ 20 from Eqs. (28) and (29) give Eliminating w
Hence the total stress acting on the inner facesheet is
r2 ¼ r2i þ qc ce ðw_ 2i w_ 2 Þ
_ 20 is the particle velocity in elastic zone ahead of the plastic where w wave front. The jump condition across the incident/reflected elastic wave front gives
rp2 ¼ 1
cp 0 cp _ 2i w _ 2Þ q þ r2i þ qc cp ðw ce ce
ð30Þ
An elastic unloading wave will propagate from the inner facesheet when the plastic stress is at a maximum value. The elastic unloading wave is shown in Fig. 4c. The momentum jump condition across the elastic unloading wave front gives
3.3.2. Elastic–plastic wave propagation Plasticity begins when the total stress at the inner facesheet reaches q0 . This causes a plastic wave front in addition to the elastic wave front, as shown in Fig. 4b. Both the incident and reflected stresses propagate with cp in the plastic zone. Momentum conservation across the plastic wave front gives
_ 21 are the peak plastic stress and particle velocity where r2max and w ahead of the wave front. Furthermore, the momentum jump condition across the plastic wave front gives
_2w _ 20 brp2 q0 c ¼ qc cp ½w
_ 21 w _ 20 ½r2 max q0 ¼ qc cp ½w
ð28Þ
_2w _ 21 ½r2 r2 max ¼ qc ce ½w
Fig. 4. Propagation of elastic–plastic waves from inner facesheet: (a) elastic loading, (b) elastic–plastic loading, and (c) elastic unloading.
ð31Þ
ð32Þ
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Finally, the momentum jump condition across the incident/reflected elastic wave front gives
_ 20 w _ 2i ½q0 r2i ¼ qc ce ½w
ð33Þ
_ 21 and w _ 20 from Eqs. (31)–(33) gives Eliminating w
r2
cp 0 cp _ 2i max q þ r2i max ðr2i max r2i Þ þ qc cp w ¼ 1 ce ce _ 2i max w _ 2i Þ qc cp w _ 2 max w _ 2Þ _ 2 max þ qc ce ðw qc ce ðw
uniform radial impulse. Following the inextensionality condition, w01 ¼ f1 a0 . This condition implies that cn = bn/n and dn = an/n. Hence,
w1 ¼
1 X 1 n¼2
n
½an ðs1 Þ sin nh bn ðs1 Þ cos nh
ð41Þ
Similarly, for imperfections
ð34Þ
_ 2i max and w _ 2 max are the maximum stress, incident where r2imax, w particle velocity and particle velocity at the start of unloading.
f1i ¼
ð42Þ
n¼2
and
w1i ¼
4. Transient response of facesheets
1 X ½dn ðs1 Þ cos nh þ cn ðs1 Þ sin nh
1 X 1 n¼2
n
½dn ðs1 Þ sin nh cn ðs1 Þ cos nh
ð43Þ
The solution methodology for the outer and inner facesheet response is an extension of a previous solution on impulsivelyloaded laminated shells by Hoo Fatt and Pothula [5]. In this study, however, both outer and inner facesheets are considered with initial imperfections and the load duration is non-impulsive. To incorporate imperfections, the total radial and tangential deformations for the outer facesheet are expressed as
The inner facesheet deformations are similarly expressed in terms of a normalized radial deflection f2 = w2/r2, tangential deflection w2 = v2/r2 and time s2 = ct/r2. The Fourier series representation for deflections of the inner facesheet are
w1 ðh; tÞ ¼ w1 ðh; tÞ þ w1i ðhÞ
and
ð35Þ
and
v 1 ðh; tÞ ¼ v 1 ðh; tÞ þ v 1i ðhÞ
ð36Þ
where w1(h, t) is the radial deformation, measured positive inward from the initial imperfection w1i(h), and v1(h, t) is the tangential deformation, measured anti-clockwise from the initial imperfection v1i(h). Similarly the total radial and tangential deformations for the inner facesheet are expressed as
w2 ðh; tÞ ¼ w2 ðh; tÞ þ w2i ðhÞ
f2 ¼ e0 ðs2 Þ þ
w2 ¼
1 X en ðs2 Þ n¼2
n
sin nh
f n ð s2 Þ cos nh n
ð45Þ
The imperfections associated with the inner facesheet are
f2i ¼
1 X ½en ðs2 Þ cos nh þ kn ðs2 Þ sin nh
ð46Þ
n¼2
and
w02i ¼
1 X en ðs2 Þ n¼2
v 2 ðh; tÞ ¼ v 2 ðh; tÞ þ v 2i ðhÞ
ð44Þ
n¼2
ð37Þ
and
1 X ½en ðs2 Þ cos nh þ fn ðs2 Þ sin nh
n
sin nh
kn ðs2 Þ cos nh n
ð47Þ
ð38Þ 4.2. Lagrange’s equations of motion
4.1. Fourier series representation tangential Define a normalized radial deflection f1 = w1/rq 1,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi deflection w1 = v1/r1 and time s1 = ct/r1, where c ¼ A22 =ðqf hÞ is the wave speed in the circumferential direction of the outer facesheet. The Fourier series representation of normalized radial and tangential displacement of the outer facesheet are 1 X f1 ¼ a0 ðs1 Þ þ ½an ðs1 Þ cos nh þ bn ðs1 Þ sin nh
ð39Þ
n¼1
The radial and tangential facesheet deformations are derived from Lagrange’s equations of motion:
d @T @T @U þ ¼ Qn dt @ q_ n @qn @qn
where T is the kinetic energy, U is the strain energy, qn is a generalized coordinate, q_ n is the generalized velocity and Qn is a generalized force. The generalized forces are obtained from virtual work dW as follows:
and
w1 ¼
1 X
Qn ¼ ½cn ðs1 Þ cos nh þ dn ðs1 Þ sin nh
ð40Þ
n¼1
where n is the mode number. The term at n = 0 or a0(s) is the breathing mode whereby the shell goes in and out of hoop compression. The term at n = 1 denotes rigid body motion, while the terms for n P 2 are bending modes. Under uniform pressure load, there can be no rigid body motion of the shell. Hence, the terms involving n = 1 are omitted in this analysis. Inextensional deformations of thin rings and shells originated from Rayleigh [6], where he showed that displacement due to the extension of mid-surface are negligibly small in comparison with displacements due to bending. Goodier and McIvor [7] later demonstrated that the amplitude of the extensional modes were indeed negligible compared to the amplitude of the bending modes in studying the stability of an isotropic elastic shell subjected to
ð48Þ
@ðdWÞ @ðdqn Þ
ð49Þ
The kinetic energy for the outer and inner facesheets (T1 and T2, respectively) are
T1 ¼
1 A22 r1 2
2 Z 2p " @f 1
0
@ s1
þ
@w1 @ s1
2 # dh
ð50Þ
2 # dh
ð51Þ
and
1 T 2 ¼ A22 r2 2
2 Z 2p " @f 2
0
@ s2
@w2 þ @ s2
where A22 is the membrane stiffness of the facesheet in the hoop direction. Following Lindberg and Florence [8], the elastic strain energy for the outer and inner facesheets
M.S. Hoo Fatt, H. Surabhi / Composite Structures 94 (2012) 3174–3185
(U1 and U2, respectively) are
Z 2p h 0 2 1 0 0 0 U 1 ¼ A22 r 1 w1 f1 þ w01 f1 f02 1 2f1 w1 þ 2f1 f1i 2 r0 i 2 2f01 w01 2f1 w01i þ a21 f001 þ f1 dh
Substituting Eqs. (56) and (57) into Eq. (48) and evaluating generalized forces from Eqs. (59) and (49) give the equations of motion for Fourier coefficients. For the breathing mode (n = 0),
ð52Þ
and
Z 2p h 0 2 0 0 0 w2 f2 þ w02 f2 f02 2 2f2 w2 þ 2f2 f2i r0 2 i 2f02 w02 2f2 w02i þ a22 f002 þ f2 dh
2 1
1X d a0 2 r 10 þ a0 1 þ a21 ðn2 2Þ a2n þ bn ¼ p 4 n¼2 ds21
i d an n2 h 2 þ ðn 1Þ2 a21 ðn2 2Þa0 an ds21 n2 þ 1 2
ð53Þ
where a ¼ D22 = r 21 A22 , a22 ¼ a21 r21 =r 22 , ½ 0 ¼ @½ =@h and D22 is the bending stiffness of the facesheet in the hoop direction. The virtual work associated with the outer facesheet is
¼
2 1
dW 1 ¼ A22 r 1
r 1 df1 dh ½p
ð54Þ
n2 ðn2 2Þ 1n dn a0 r n2 þ 1
i d bn n2 h 2 n2 ðn2 2Þ ^ 1n þ 2 ðn 1Þ2 a21 ðn2 2Þa0 bn ¼ dn a0 r 2 n2 þ 1 ds1 n þ 1 2
ð62Þ
¼ pr1 =A22 is a normalized pressure load and r 1 ¼ r1 r1 =A22 where p is the normalized stress in the foam at the outer facesheet. If the foam is purely elastic, r1 is given by Eq. (13), and if the foam is elastic–plastic, r1 is given by Eq. (16). In contrast, the virtual work associated with the inner facesheet is
Z 2p
ð55Þ
2 ¼ r2 r 2 =A22 is the normalized stress in the foam acting on where r the inner facesheet. If the foam is purely elastic, r2 is given by Eq. (27), and if the foam is elastic–plastic, r1 is given by Eqs. (30) and (34). 4.3. Outer facesheet equations of motion
da0 ds1
2
1 2 X
1 n þ1 þ 2 n¼2 n2
d a0 r 10 þ a0 ¼ p ds21
ð63Þ
i d an n2 h 2 þ 2 ðn 1Þ2 a21 ðn2 2Þa0 an 2 ds1 n þ 1 2
¼
n2 ðn2 2Þ 1n ; n P 2 dn a0 r n2 þ 1
ð64Þ
i d bn n2 h 2 þ 2 ðn 1Þ2 a21 ðn2 2Þa0 bn 2 ds1 n þ 1 2
Substituting Eqs. (39), (41), (42) and (43) into Eqs. (50) and (52) gives the kinetic energy as
T 1 ¼ pA22 r 1
The breathing mode is in general coupled with the bending modes. 2 When deflections are small, a2n and bn are negligible compared to an and bn. In addition, a21 1 because it is on the order of (h/r1)2. Eqs. (60)–(62) then reduce to 2
r 2 df2 dh
o
"
ð61Þ
and
o
dW 2 ¼ A22 r 2
ð60Þ
For modes of n P 2
1 U 2 ¼ A22 r 2 2
Z 2p
3179
dan ds1
2
dbn þ ds1
2 !#
¼
ð56Þ
n2 ðn2 2Þ cn a0 r^ 1n ; n P 2 n2 þ 1
ð65Þ
The above equations of motion are solved with the following initial conditions: dan/ds1(0) = 0, dbn/ds1(0) = 0, an(0) = 0, and bn(0) = 0 for n = 0 and n P 2.
and the strain energy as
(
1 h i 1X U 1 ¼ pA22 r 1 a20 1 þ a21 þ a21 ðn2 1Þ2 ðn2 2Þa0 2 n¼2 ) 1
X 2 a2n þ bn ðn2 2Þðdn an þ cn bn Þa0
4.4. Inner facesheet equations of motion The equations of motion for the inner facesheet are derived similarly to those of the outer facesheet. For the breathing mode
ð57Þ
n¼2
Since deflections are small, terms higher than second order have been neglected. 1 is represented by FouTo calculate the virtual work, assume r rier series:
r 1 ¼ r 10 þ
1 X
1 X
n¼2
n¼2
r 1n cosðnhÞ þ
r^ 1n sinðnhÞ
1 X
pA22 r1 r 1n dan
n¼2
1 X
pA22 r1 r^ 1n dbn
n¼2
where da0, dan, and dbn are virtual displacement amplitudes.
ð59Þ
ð66Þ
and for modes of n P 2:
i d en n2 h 2 n2 ðn2 2Þ þ 2 en e0 þ r 2n ðn 1Þ2 a22 ðn2 2Þe0 en ¼ 2 n2 þ 1 ds2 n þ 1 2
ð58Þ
1i0 is the zero component, r 1in is the nth cosine component, where r ^ 1in is the nth sine component of r 1 . The components follow and r from the Fourier series representation of the foam velocities in Eqs. (13) and (16). The virtual work is then
r 10 da0 dW 1 ¼ 2pA22 r 1 ½p
2
d e0 20 þ e0 ¼ r ds22
ð67Þ and
i d fn n2 h 2 n2 ðn2 2Þ ^ 2n þ 2 ðn 1Þ2 a22 ðn2 2Þe0 fn ¼ kn e0 þ r 2 n2 þ 1 n þ 1 ds2 2
ð68Þ 20 is the zero component, r 2n is the nth cosine component, where r ^ 2n is the nth sine component of r 2 . The components follow and r from the Fourier series representation of the foam velocities into Eqs. (27), (30), and (34). The equations of motion for the inner facesheet are solved using the following initial conditions: den/ ds2(0) = 0, dfn/ds2(0) = 0, en(0) = 0, and fn(0) = 0, for n = 0 and n P 2.
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5. Shell failure
Thus the value of the hoop strain at which failure occurs is given by
To examine failure of the outer facesheet, one calculates hoop strains from the facesheet deformation and use laminate shell theory and failure criteria to determine failure. Shell fracture of the woven E-Glass/Vinyl Ester facesheet is examined in this section. The total hoop strain in the shell is given by
ef ¼
eh ¼ ehm þ zjh
ð69Þ
For the orthotropic facesheets the maximum tangential stress would occur at the outer plies where the bending strains are maximum, i.e., at z = ±h/2:
h 2
eh ¼ ehm jh
ð70Þ
Here ehm = a0 and the curvature is given by
jh ¼
" # 1 X 1 ao an ðn2 1Þ cos nh r1 n¼2
ð71Þ
The above curvature varies with time t and angular position h. It has a maximum value at h = 0,p. Denote the curvature at h = 0 as
j0
" # 1 X 1 2 ¼ ao an ðn 1Þ r1 n¼2
ð72Þ
and the curvature at h = p as
jp ¼
" # 1 X 1 ao þ an ð1Þn ðn2 1Þ r1 n¼2
ð73Þ
A modified Hashin–Rotem is used to examine lamina failure of the woven roving E-Glass/Vinyl Ester [9]. According to the modified Hashin–Rotem failure criteria, the failure of the composite occurs when
jrx j ¼ 1 if XT jrh j ¼ 1 if XT jsxh j ¼1 SL
jrx j ¼ 1 if XC jrh j > 0 or ¼ 1 if XC
rx > 0 or
rx < 0
ð74Þ
rh
rh < 0
ð75Þ ð76Þ
minðX T ; X c Þ Q 22
The above expression describes the maximum allowable strain based on a Hashin–Rotem composite failure criterion. 6. An example Consider a sandwich shell with E-Glass Vinyl Ester facesheets of thickness h = 2 mm, outer radius r1 = 93.5 mm and inner radius r2 = 66.5 mm and 25 mm-thick Divinycell H100 core. The facesheets are given initial imperfections f1i = d2 cos 2h and f2i = e2 cos 2h, where d2 = 0.1h/r1 and e2 = 0.1h/r2. A MATLAB program was written to solve for the outer and inner facesheet response when po = 2 MPa and DT = 1 ms. Finite element analysis (FEA) using ABAQUS Dynamic Implicit analysis was done in addition to gage the accuracy of the analytical solution. Dynamic, Explicit analysis, although less computationally expensive, could not be used for this type of problem because of numerical damping used to stabilize the explicit algorithm in ABAQUS Explicit. The direct-integration method provided for Dynamic, Implicit analysis in ABAQUS Standard is the Hilber–Hughes Taylor operator, which is an extension of the trapezoidal rule. Automatic time increment with specified half-step residual was used. A parametric study was done to determine the appropriate size of the half-step residual that would yield accurate results. No numerical damping was specified in the problem. The FEA model of the sandwich cylinder in plane strain is shown in Fig. 5. Continuum plane strain elements with full integration (CPE4) were chosen. The composite material properties were specified in a local cylindrical coordinate (RTZ) according to the ABAQUS User Manual, Version 6.7 [10]. The facesheet was modeled as orthotropic elastic material. The foam was modeled as crushable foam with isotropic hardening, using only the through-thickness compressive stress–strain curve of the foam. Plasticity properties for the PVC H100 foam were taken from Ref. [11]. It was found that under this pressure pulse, plastic dissipation in the foam occurs only near the inner facesheet. Figure 6 shows the radial stress–strain curves in the foam near the outer facesheet (Point C in Fig. 5) and near the inner facesheet (Point D in Fig. 5). The analytical solution also confirms that foam plasticity was localized near the inner facesheet. Finally, there is good agreement in
For an orthotropic shell, the relationship between the principal stress and strains are given by
8 9 > > < rx = > :
2
Q 11
Q 12
rh ¼ 6 4 Q 12 Q 22 > sxh ; 0 0
9 38 > < ex > = 7 0 5 eh > > : cxh ; Q 66 0
ð77Þ
where Q 11 ¼ E11 =ð1 m12 m21 Þ, Q 22 ¼ E22 =ð1 m12 m21 Þ, Q 21 ¼ m12 E22 =ð1 m12 m21 Þ, and Q 66 ¼ G12 . In the plane strain problem, the stresses in the shell are reduced to
rx Q 12 eh rh ¼ Q 22 eh
ð78Þ ð79Þ
Since Q 22 > Q 12 , rh > rx and failure of the shell will occur in tangential direction. Substituting Eq. (79) into Eq. (75) one gets
Q 22 eh ¼ 1 if XT
rh > 0
ð80Þ
rh < 0
ð81Þ
and
Q 22 eh ¼ 1 if XC
ð82Þ
Fig. 5. Finite element model of sandwich cylindrical shell.
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near the inner facesheet and near the outer facesheet. For all load durations, plasticity begins near the inner facesheet. When the load duration is 1 ms or longer, there is no plasticity in the foam near the outer facesheet up to the point of failure. When the load duration is below 1 ms, plastic work is associated with multiple wave reflections and the foam near the inner and outer facesheets undergo hysteresis before failure. The energy absorption before shell failure increases substantially with shorter load durations, as shown in Fig. 8b. The energy absorption shown in Fig. 8b were taken from FEA. 7.2. Core parameters
Fig. 6. Radial stress–strain response in foam near outer and inner facesheets: (po = 2 MPa, DT = 1 ms).
the outer and inner facesheet radial deformations and velocities between the analytical solution and FEA, as shown in Fig. 7a and b. The outer and inner facesheet motions are defined at Points A and B in Fig. 5, respectively. 7. Parametric studies The proposed analytical model is used to understand mechanisms and parameters that are responsible for increasing blast resistance and energy absorption of the composite sandwich shell. This is an important step in the design of blast resistant structures. In all the cases below, the facesheets are given initial imperfections: f1i = d2cos 2h and f2i = e2cos 2h, where d2 = 0.1h/r1 and e2 = 0.1h/r2. The influence of load and core parameters is discussed below. 7.1. Load parameters The peak pressure and load duration of the blast loads described in Eq. (1) varies with many factors such as weight of charge and stand-off distance. Figure 8a shows the combination of peak pressures and load duration that would just cause sandwich shell failure. For all load cases the sandwich shell fails when the outer (pulse-loaded) facesheet ruptures. In general, higher peak pressures can be sustained with lower load durations. However, it was also found that maximum load to failure did not vary significantly for load duration of 1 ms and longer. Also shown in Fig. 8a is the peak pressure that would just induce plastic work in the foam
(a)
The effect of core thickness on the strength and energy absorption of the sandwich shell was examined by keeping the woven E-Glass/Vinyl Ester facesheet thickness at 2 mm and the mean sandwich shell radius at 80 mm but varying the Divinycell H100 core thickness from 6.25 mm to 50 mm (1/4–2 in.). Figures 9a and b show the peak pressure to failure with a load duration of 1 ms and the corresponding energy absorbed by the Divinycell H100 core as they vary with the core thickness. The peak pressure to failure generally decreases with increasing core thickness, whereas the energy absorbed increases with increasing core thickness. The fact that the peak pressure to failure is lower for a thicker sandwich shell is counter-intuitive since a thicker sandwich shell would offer higher bending stiffness. However, the failure of the sandwich cylindrical shell involves local rupture of the outer facesheet during the through-thickness wave propagation phase. During this phase, the outer facesheet deformation is influenced by reflected stress waves in the core. Since the reflected stress wave from the inner facesheet travels over a shorter distance in the thinner cores, it reaches the outer facesheet earlier thereby preventing outer facesheet rupture. The effect of core material properties was also investigated by replacing the Divinycell H100 foam with four other foams, which are listed in Table 1. The facesheets were 2 mm thick with r1 = 93.5 mm and r2 = 66.5 mm. Two core thicknesses were examined, a 25 mm-thick core and a 50 mm-thick core. As indicated in Fig. 10a, the sandwich shells with foam cores of higher stiffness and crush strength sustain higher peak pressures when the load duration is 1 ms. In all cases the outer facesheet ruptured, but as Fig. 10b indicates there was no core plasticity or energy absorption in the sandwich shells with 25 mm-thick Klegecell R300 and Divinycell HCP100 cores and the sandwich shell with the 50 mm-thick Divinycell HCP100 core. The crushing resistance of the Klegecell R300 and Divinycell HCP100 were high enough to keep the core in the elastic regime in these scenarios. However,
(b)
Fig. 7. Deflection and velocity during elastic–plastic response (po = 2 MPa DT = 1 ms): (a) deflection and (b) velocity.
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Fig. 8. Effect of load duration on (a) peak pressure and (b) energy absorption to failure.
Fig. 9. Effect of core thickness on panel strength: (a) peak pressure to failure and (b) energy absorption.
Table 1 Material properties of 0/90 woven roving E-glass/vinyl ester and several foams.
Density (kg/m3) E11 (+) (GPa) E22 (+) (GPa) E33 (+) (GPa) E11 () (GPa) E22 () (GPa) E33 () (GPa) m12 = m21 m13 = m23 m31 = m32 G12 = G21 (GPa) G23 = G32 (GPa) G13 = G31 (GPa) q (MPa)
eD XT (MPa) XC (MPa) YT (MPa) YC (MPa) SL (MPa) ST (MPa)
E-Glass/vinyl ester
Divinycell H30
Divinycell H100
Divinycell H200
Klegecell R300
Divinycell HCP100
1391.3 17 17 7.48 19 19 – 0.13 0.28 0.12 4.0 1.73 1.73 – – 270 200 270 200 40 31.6
36 0.044 0.044 0.044 0.027 0.027 0.027 0.25 0.25 0.25 0.013 0.013 0.013 0.3 0.85 0.57 0.29 0.57 0.29 0.35 0.35
100 0.149 0.149 0.149 0.105 0.105 0.105 0.31 0.31 0.31 0.0438 0.0438 0.0438 1.66 0.8 3.2 1.53 3.5 1.53 1.47 1.47
200 0.277 0.277 0.277 0.293 0.293 0.293 0.3 0.3 0.3 0.110 0.110 0.110 4.35 0.7 6.4 4.36 6.4 4.36 3.86 3.86
300 – – – 0.338 0.338 0.338 0.23 0.23 0.23 0.123 0.123 0.123 7.8 0.285 – – – – – –
400 – – – 0.340 0.340 0.340 0.3 0.3 0.3 0.131 0.131 0.131 10.3 0.15 – – – – 7.4 7.4
the crushing resistance of the foam is not the only parameter governing energy absorption of the sandwich cylindrical shell. The sandwich shell with the 50 mm-thick Klegecell R300 core went exhibited plasticity near the inner facesheet of the sandwich shell before it failed, while the sandwich shell with the 25 mm-thick
Klegecell R300 core did not. Thus there is a minimum core thickness for energy absorption of the sandwich shell with the Klegecell R300 core. The stiffness and crush strength of the foam generally increase at the expense of increasing foam density or weight (see Table 1).
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Fig. 10. Effect of core type: (a) peak pressure and (b) energy absorption to failure.
Fig. 11. Failure pressure per unit weight: (a) 25 mm-thick core and (b) 50 mm-thick core.
In Fig. 11a and b, the peak pressure to failure is normalized by the weight per unit length of the shell cL, which is defined as follows:
"
2 2 2 2 # h h h h cL ¼ qf g p r1 þ r1 þ r2 þ r2 2 2 2 2 " # 2 2 h h ð83Þ r2 þ þ qc g p r1 2 2 where g is the acceleration due to gravity. The core thickness in Fig. 11a and b is 25 mm and 50 mm, respectively. Per unit weight, the sandwich shell with the Divinycell H200 foam core is the most blast resistant shell of all considered, surpassing sandwich shells made with Klegecell R300 and Divinycell HCP100 foams. Furthermore, even the shell with the 50 mm-thick Divinycell H30 core is more blast resistant per unit weight than shells of comparable thickness and made with Klegecell R300 and Divinycell HCP100 foam cores. This parametric study elucidates the importance of considering wave propagation effects in designing blast resistant sandwich structures. 8. Concluding remarks Stress wave analysis has been done on a composite sandwich cylindrical shell subjected to external blast. The facesheets and core of the sandwich shell consisted of orthotropic elastic and crushable (elastic–plastic) foam materials, respectively. Failure of the sandwich shell under intense dynamic loading was due to rupture of the outer facesheet. The sandwich shell response was
shown to be affected by the magnitude and duration of the pressure pulse. High amplitude, low duration (impulsive) pressure pulses induced high energy absorption associated with multiple wave reflections. Low amplitude, long duration pressure pulses caused minimal energy absorption in the foam nearest to the inner facesheet. It was found that the amount of energy absorbed increases and the failure load decreases with increasing core thickness of Divinycell PVC H100. Five different foams were also considered in this study. In order of increasing density, these were Divinycell H30, H100, H200, Klegecell R300 and Divinycell HCP100. The compressive modulus and crush resistance of these foams generally increase with foam density so that the Divinycell H30 foam had the lowest compressive modulus and crush resistance, while the Divinycell HCP100 foam had the highest compressive modulus and crush resistance. The sandwich shells with Divinycell HCP100 foam core were found to be the most blast resistant to failure even though no energy was absorbed by them. However, the Divinycell HCP100 foam was also the heaviest and per unit weight, the Divinycell H200 foam was more blast resistant to failure than it. A reason for this was that the lower crush resistance of the Divinycell H200 foam allowed plastic work dissipation in the core before shell failure. Acknowledgements The authors gratefully acknowledge financial support from Dr. Yapa Rajapakse at the Office of Naval Research under Grant No. N00014-11-1-0485.
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Appendix A. Crushable elastic–plastic foam in uniaxial strain The constrained elastic modulus, yield strength and elastic– plastic tangent modulus for the foam under uniaxial strain compression are derived for a crushable foam in this section. The foam is assumed to be an isotropic material with the uniaxial stress compressive elastic–plastic properties, and isotropic hardening is also assumed during plastic deformation. These assumptions are valid if the stress state in the foam is primarily compressive, but is only approximately true if the foam experiences a multi-axial stress state because most foams are anisotropic. It is also assumed that the compressive strains are limited to 20%, which is low enough to avoid foam densification and appreciable stiffening. A.1. Elastic behavior
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 1 f ¼ 2J 2 þ I21 q ¼ 0 3
Under uniaxial strain, I1 = rr + 2rh and Eq. (A5), one gets
rh ¼ K
2G 3
ðA10Þ pffiffiffi pffiffiffiffi J2 ¼ 1= 3ðrr rh Þ. From
er
ðA11Þ
Combining Eqs. (A6) and (A11) to eliminate er gives
rh ¼
3K 2G rr 3K þ 4G
ðA12Þ
Hence,
Under uniaxial strain, the strain eij and the stress rij are given by
eij ¼ ½er ; 0; 0
9K rr 3K þ 4G
ðA13Þ
pffiffiffiffi 1 6G rr J 2 ¼ pffiffiffi 3 3K þ 4G
ðA14Þ
I1 ¼
ðA1Þ and
and
rij ¼ ½rr ; rh ; rh
ðA2Þ
where rr and rh are the radial and tangential stresses in the foam, respectively. The strain deviator eij and the stress deviator sij are thus
eij ¼
pffiffiffi most foams compressed topaffiffiffiffiuniaxial strain of about 20%, a ¼ 3= 2. Hence, in terms of I1 and J 2 , the yield criterion for the foam is
1 er ½2; 1; 1 3
ðA3Þ
Substituting Eqs. (A13) and (A14) into Eq. (A10) gives a value of constrained flow strength q0 as
qð3K þ 4GÞ
r1 ¼ q0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð24G2 þ 27K 2 Þ
ðA15Þ
and
sij ¼ ½sr ; sh ; sh
ðA4Þ
During purely elastic behavior, the constitutive relation between the stress and strain is
rij ¼ 2Geij þ K ekk dij
ðA5Þ
where dij is the Kronecker delta function, G = E/(2(1 + m)) is the shear modulus, K = E/(3(1 2m)) is the bulk modulus, and E and m are the Young’s modulus and Poisson’s ratio in the uniaxial stress state. Substituting the stresses and strains defined in Eqs. (A1)–(A3) into Eq. (A5) gives
4 3
rr ¼ Ger þ K er ¼
To find the elastic–plastic tangent modulus, the total strain increment during plastic deformation of the foam is decomposed into elastic and plastic components:
deij ¼ deeij þ depij
ðA6Þ
4 GþK 3
Under uniaxial strain, the strain increment deij and the stress increment drij are given by
deij ¼ ½der ; 0; 0
ðA7Þ
drij ¼ ½drr ; drh ; drh
ðA18Þ
deij ¼
1 der ½2; 1; 1 3
ðA19Þ
and
dsij ¼ ½dsr ; dsh ; dsh To find the constrained yield strength, one must consider a yield criterion for crushable foam. Deshpande and Fleck [12] proposed the following crushable foam yield criterion:
^ q¼0 f ¼r
ðA8Þ
^ is an equivalent stress given by where r
2 1 2 2 a2 i re þ a rm
1þ
ðA17Þ
The strain deviator increment deij and the stress deviator increment dsij are also given as
A.2. Plastic yielding
r^ ¼ h
ðA16Þ
and
4 G þ K er 3
Thus the constrained modulus E0 is given by
E0 ¼
A.3. Elastic–plastic behavior
ðA9Þ
3
pffiffiffiffiffiffiffi
r^ ¼ I1 =3 is the mean stress, re ¼ 3J2 is the effective or Mises stress, a is the pressure sensitivity pffiffiffifficoefficient, I1 = rii is the first
invariant of the stress tensor and J2 ¼ ð1=2sij : sij Þ1=2 is the deviatoric stress. Equation (A8) describes a yield surface that is elliptical in rm re space, while a defines the aspect ratio of the ellipse. For
ðA20Þ
The constitutive relation for the elastic strain increment follows from Eq. (A5), while the plastic strain increment is obtained from the flow rule. The complete strain–stress relation is
deij ¼
dsij dI1 @f þ dij þ dk @ rij 2G 9K
ðA21Þ
where dk is an undetermined factor such that
dk ¼
0;
when f < 0 or f ¼ 0 when df < 0
> 0; when f ¼ 0 and df ¼ 0
ðA22Þ
The form of dk is determined from the consistency condition
df ¼
@f drij ¼ 0 @ rij
ðA23Þ
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Now consider a yield pffiffiffiffi function expressed in general terms of stress invariant I1 and J2 :
pffiffiffiffi f ¼ F I1 ; J2 q ¼ 0
ðA24Þ
FEA
Chen and Han [13] gives
drij
@f G @f ¼ 2Gdeij þ Kdekk dij dk 3K dij þ pffiffiffiffi pffiffiffiffi sij @I1 J2 @ J2
!
ANA
ðA25Þ
and
dk ¼
@f ffiffiffi smn demn 3Kdekk @I@f1 þ pGffiffiffi p J2 @ J 2 2
2 @f ffiffiffi 9K @I@f þ G p 1
@
ðA26Þ
J2
Hence for the foam yield function specified in Eq. (A10), one gets that under uniaxial strain constraint
drr
Fig. A1. Compressive stress–strain behavior of Divinycell H100 crushable foam.
8 > <4G
hpffiffiffi i2 9 pffiffiffiffi > = þ 4G = J 3 KI 1 2 1 h i þK ¼ der 2 > > 3 3 : ; KI1 =J 2 þ 4G
ical and FEA solutions. The plastic modulus varies with strain or stress. An average plastic modulus E0p is defined by
E0p ¼ ðA27Þ
1 ðr20 q0 Þ
Z r20
E0p E0 drr
ðA31Þ
q0
where r20 is the stress at 20% strain found from integration of Eq. (A29).
The constrained elastic–plastic tangent modulus is thus
4G 1 þK E0p ¼ 3 3
hpffiffiffi i2 pffiffiffiffi 3KI1 = J 2 þ 4G h i KI21 =J 2 þ 4G
References
ðA28Þ
pffiffiffi pffiffiffiffi where I1 = rr + 2rh and J2 ¼ ðrr rh Þ= 3. The radial and tangential stresses in the foam are related to each other during plastic foam by the foam yield criteria defined in Eq. (A10). As an example, the plastic portion of the stress–strain curve for Divinycell PVC H100 foam can be obtained by integrating the plastic modulus:
Z rr q0
drr
¼ 0 Ep E0
Z
er
der
ðA29Þ
ee
0
where ee ¼ qE0 is the elastic limit strain. Here E0p is given in terms of rr only because during yielding
rh ¼
1 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y 2 r2r
ðA30Þ
For Divinycell H100 foam, Y = 1.66 MPa and the resulting elastic– plastic stress–strain curve after numerical integration is shown in Fig. A1. Note that there is very good agreement between the analyt-
[1] Han JH, Kardomateas GA, Simitses GJ. Elasticity, shell theory and finite element results for the buckling of long sandwich cylindrical shells under external pressure. Composites Part B 2004;35(6–8):591–8. [2] Li R, Kardomateas GA, Simitses GJ. Nonlinear response of a shallow sandwich shell with compressible core to blast loading. J Appl Mech 2008;75(6):1–10. [3] Li R, Kardomateas GA. A high-order theory for cylindrical sandwich shells with flexible cores. J Mech Mat Struct 2009;4(7–8):1453–67. [4] Rahmani O, Khalili SMR, Malekzadeh K. Free vibration response of composite sandwich cylindrical shell with flexible core. Compos Struct 2010;92(5):1269–81. [5] Hoo Fatt MS, Pothula SG. Dynamic pulse buckling of composite shells subjected to external blast. Compos Struct 2010;92(7):1716–27. [6] Lord Rayleigh. Theory of sound. New York: Dover; 1945. [7] Goodier JN, McIvor IK. The elastic cylindrical shell under nearly uniform radial impulse. J Appl Mech 1964;31:259–66. [8] Lindberg HE, Florence AL. Dynamic pulse buckling. Dordrecht: Martinus Nijhoff Publishers; 1987. [9] Daniel IM. Failure of composite materials. Strain 2007;43(1):4–12. [10] ABAQUS, Inc. ABAQUS User Manual, Version 6.7. Dassault Systèmes; 2007. [11] Mines RAW, Alias A. Numerical simulation of the progressive collapse of polymer composite sandwich beams under static loading. Composites Part A 2002;33(1):11–26. [12] Deshpande VS, Fleck NA. Isotropic constitutive model for metallic foams. J Mech Phys Solids 2000;48(6-7):1253–76. [13] Chen W-F, Han D-J. Plasticity for structural engineers. Ft. Lauderdale: J. Ross Publishing; 2007.
Contents lists available at SciVerse ScienceDirect
Composite Structures journal homepage: www.elsevier.com/locate/compstruct
Blast resistance and energy absorption of foam-core cylindrical sandwich shells under external blast Michelle S. Hoo Fatt ⇑, Harika Surabhi Department of Mechanical Engineering, The University of Akron, Akron, OH 44325-3903, United States
a r t i c l e
i n f o
Article history: Available online 22 May 2012 Keywords: Blast Cylindrical sandwich shell Foam core
a b s t r a c t The early time, through-thickness stress wave response of a foam-core, composite sandwich cylindrical shell under external blast is examined in this paper. Solutions for the transient response of the facesheets were derived as stress waves propagated through an elastic–plastic, crushable foam core. These solutions were found to be in good agreement with results from finite element analysis. The blast response of the composite sandwich cylindrical shell was shown to be affected by the magnitude and duration of the pressure pulse. High amplitude, low duration (impulsive) pressure pulses induced the greatest energy absorption. Low amplitude, long duration pressure pulses caused minimal energy absorption. The amount of energy absorbed increased and the failure load decreased with increasing core thickness. Sandwich shells with foams of varying density, compressive modulus and crushing resistance were also examined. The sandwich shells with the foam of the highest density, compressive modulus and crushing resistance (Divinycell HCP100) were found to be the most blast resistant to failure even though no energy was absorbed by them. Per unit weight, however, the shells with a lighter, less stiff and strong, Divinycell H200 foam core were more blast resistant to failure than shells with a Divinycell HCP100 foam core. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Foam-core composite sandwich shells are being increasingly used in naval and other military applications because of excellent properties such as high specific stiffness and strength, corrosion resistance, and reduced magnetic signatures. Sandwich shells with crushable foam cores can also offer better resistance to external blast when compared to monolithic shells because the foam cores are able to absorb some of the blast energy through shock attenuation and plastic work dissipation. The objective of this study is to develop analytical solutions for the early time response of a composite sandwich cylindrical shell subjected to external blast. The term ‘‘early time response’’ refers to a through-thickness wave propagation phase involving the formation of elastic–plastic stress waves in the foam core as they reflect from one facesheet to another. The crushable foam core is considered as an elastic–plastic, energy dissipating medium, which may attenuate shock waves and absorb some of the blast energy during this time period. Most of the recent work on the blast response of composite sandwich cylindrical shells has been done where the core was considered as a purely elastic material [1–4], and very few studies have dealt with the elastic–plastic behavior of the core. This study covers the elastic–plastic wave response of the foam and the ⇑ Corresponding author. Tel.: +1 330 972 6308; fax: +1 330 972 6027. E-mail address: [email protected] (M.S. Hoo Fatt). 0263-8223/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compstruct.2012.05.013
interaction of the foam stress waves with the facesheets during blast. The analytical solution for the transient response is restricted to the early time response of the sandwich shell, i.e., within the several reflections of the wave transit time through the core, because it is in this regime that the foam core undergoes plastic work dissipation. The previous solution developed by Hoo Fatt and Pothula [5] for dynamic pulse buckling of a thin composite cylindrical shell is extended to find the equations of motion for the outer and inner facesheets. Shell fracture is also investigated using laminate shell theory and failure criteria. The analytical solution obtained is also compared to finite element analysis (FEA) using ABAQUS standard analysis software to gage its accuracy. Finally, the analytical solution derived is used to understand mechanisms and parameters that are responsible for increasing blast resistance and energy absorption of the composite sandwich cylindrical shell.
2. Problem formulation Consider the composite sandwich cylinder with geometry shown in Fig. 1a. The outer and inner facesheets are thin, fiberreinforced laminated shells with density qf, thickness h and radii r1 and r2, respectively. The core is made of crushable foam with thickness H and density qc. The sandwich shell is subjected to external blast, i.e., a uniformly-distributed pressure pulse loading described as
M.S. Hoo Fatt, H. Surabhi / Composite Structures 94 (2012) 3174–3185
3175
Fig. 1. Foam core, composite sandwich cylinder under external blast: (a) geometry and (b) stresses and deformations during loading.
t pðtÞ ¼ po 1 DT
ð1Þ
where po is the peak pressure, DT is the pulse duration and t is time. The analysis is limited to long shells, where the length of the shell is 20 times greater than its mean radius. This combined with the fact that the pressure load does not vary along the shell longitudinal axis allows one to consider the composite sandwich cylinder in plane strain. Each facesheet deforms as a ring subjected to radial pressure from the external blast and/or radial stresses propagating back and forth in the core. As indicated in Fig. 1b, the radial deformation of the outer and inner facesheets are denoted by w1 and w2, respectively, while the tangential deformation of the outer and inner facesheets are denoted by v1 and v2, respectively. The facesheets oscillate while stress waves propagate through the foam. The equations of motion of each facesheet are derived by considering the pressure pulse, interactions with the foam (incident and reflected stress propagating in the foam), shell kinetic energy and the shell strain energy. The propagation of radial stresses r and deformations w in the foam is governed by the wave equation. The boundary conditions of this wave equation are determined by the equations of motion of the outer and inner facesheets.
of the outer and inner facesheets. At these strain levels, the core density does not change substantially and the compressive stress–strain response of the crushable foam is described as elastic-perfectly plastic, as shown in Fig. 2a. In the sandwich cylindrical shell, the circumferential and axial dimensions are larger than the radial dimension during through-thickness wave propagation. This allows one to approximate the foam stress–strain response to be in a state of uniaxial strain because of the deformation constraint both in the circumferential and axial directions. The counterpart stress–strain response under uniaxial strain conditions for an elastic-perfectly plastic material is shown in Fig. 2b. The constrained elastic modulus E0 , yield strength q0 and elastic–plastic tangent modulus E0p are derived in Appendix A and are as follows:
4G þK 3 ð3K þ 4GÞq q0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð24G2 þ 27K 2 Þ
E0 ¼
ð3Þ ð4Þ
and
4G 1 þK E0p ¼ 3 3
hpffiffiffi i2 pffiffiffiffi 3KI1 = J 2 þ 4G h i KI21 =J 2 þ 4G
ð5Þ
3. Propagation of plane radial waves in core The equation governing the propagation of radial deformation w in the core is given by
r2 w
1 @2w ¼0 c20 @t 2
ð2Þ
qffiffiffiffiffiffiffiffiffiffiffiffi @ 1 @2 @ where r2 ¼ 1r @r r @r þ r2 @h2 and the wave speed is c0 ¼ E0t =qc and E0t is the tangent modulus of the foam under uniaxial strain constraint. The tangent modulus of the foam is determined from the compressive stress–strain of the foam, which is elastic–plastic. 3.1. Uniaxial strain It is necessary to limit our analysis to the small strains in the foam (less than 20%) because larger strains would induce failure
Fig. 2. Compressive stress–strain behavior of crushable foam: (a) uniaxial stress and (b) uniaxial strain.
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where G = E/(2(1 + m)) is the shearp modulus, K = E/(3(1 pffiffiffi 2m)) is the ffiffiffiffi bulk modulus, I1 = rr + 2rh and J 2 ¼ ðrr rh Þ= 3 are the first invariant of the stress tensor and the second invariant of the deviatoric stress tensor, respectively, and rr and rh are the radial and tangential stresses in the foam, respectively. The radial and tangential stresses in the foam are related to each other by the foam yield criteria. A relation between these stresses is given in Appendix A for a crushable foam yield criterion with isotropic hardening. Hence, the elastic–plastic tangent modulus can be represented in terms of radial stress only. The elastic wave speed in the foam is
ce ¼
sffiffiffiffiffi E0
qc
cp ¼
qc
ð7Þ
ð9Þ
_ 1i and w _ 1r are the incident and reflected wave velocities. where w 3.2.1. Elastic wave propagation The radial incident stress is related to the incident velocity by
ð10Þ
The reflected stress is given by
r1r ¼ qc ce w_ 1r
ð17Þ
ð18Þ
Eliminating r1max from Eqs. (17) and (18) gives
cp 0 _ 1 max w _ 1Þ _ 1 max þ qc ce ðw q þ qc cp w ce
Since the plastic wave speed is slower than the elastic wave speed, the plastic loading wave is overtaken by the elastic unloading wave at time t3 where
t 3 ¼ t 2 þ cp
ðt 2 t1 Þ ðce cp Þ
ð19Þ
where t1 and t2 are the times at which plastic waves and elastic unloading waves begin to propagate, respectively. The incident stress on the inner facesheet is not only delayed by the wave transit time te but it is unaffected by the induced plastic stresses at the outer foam. Jump conditions across the elastic loading–unloading boundary give the incident velocity and stress on the inner facesheet as
_ 1 max þ w _ 1 ht t e i _ 2 ¼ Ve w w
ð20Þ
and
r2 ¼ qc ce ðV e w_ 1 max þ w_ 1 ht te iÞ
ð21Þ
where the symbol hi is used to denote shifted time, i.e.,
_ 1 ht t e i ¼ w
0; t < te _ w1 ðt t e Þ; t P te
ð22Þ
ð12Þ
Hence the total stress in the foam at the outer facesheet is
r1 ¼ r1i þ qc ce ðw_ 1i w_ 1 Þ
_ 1 max V e ½r1 max q0 ¼ qc cp ½w
ð11Þ
Substituting Eq. (9) into Eq. (11) gives
r1r ¼ qc ce ðw_ 1i w_ 1 Þ
ð16Þ
_ 1 max are the maximum stress and particle velocwhere r1max and w ity at the outer facesheet, respectively. The momentum jump condition across the plastic wave front gives
ð8Þ
Here the incident stress results from stress wave propagation in the foam. The pressure pulse load is not considered as an incident stress; it acts on the opposite side of the outer facesheet. Compatibility requires that the particle velocity in the foam nearest to the outer facesheet is
r1i ¼ qc ce w_ 1i
cp 0 _1 q þ qc cp w ce
Elastic unloading then takes place when the plastic stress or particle velocity is at a maximum value. Figure 3c shows the wave fronts in the foam during elastic unloading. The momentum jump condition across the elastic unloading wave front gives
r1 ¼ 1
The total radial foam stress acting on the outer facesheet r1 is the sum of an incident stress r1i and a reflected stress r1r:
_ 1i þ w _ 1r _1¼w w
_1w _ 1 max ½r1 r1 max ¼ qc ce ½w
3.2. Foam stress acting on outer facesheet
ð15Þ
Thus the plastic stress at the outer facesheet is
ð6Þ
Unlike the elastic modulus, the elastic–plastic tangent modulus varies with radial stress. The constrained plastic modulus shown in Fig. 2b indicates that the plastic wave speed is less than the elastic waves speed. An average constrained plastic modulus for plastic strains less than 20% may be used to approximate a constant plastic wave speed (see Appendix A). The radial stress in the foam during wave propagation is discussed below.
r1 ¼ r1i þ r1r
_ 1 V e brp1 q0 c ¼ qc cp ½w
rp1 ¼ 1
while the plastic wave speed is
sffiffiffiffiffi E0p
When the outer foam stress at the outer facesheet reaches q0 , plastic waves start to propagate from r1 to r2 with the plastic wave speed cp. Since E0p < E0 , the plastic wave speed is slower than the elastic wave speed. As shown in Fig. 3b, an elastic region with stress q0 and particle velocity Ve = q0 /(qcce) thus precedes the plastic wave front. Momentum conservation across the plastic wave front (momentum jump condition) gives
3.3. Foam stress acting on inner facesheet
ð13Þ
The total foam stress acting on the inner facesheet is
The above equation is the resistive foam stress acting on the outer facesheet, as indicated in Fig. 1b.
r2 ¼ r2i þ r2r
3.2.2. Elastic–plastic wave propagation The propagation of elastic–plastic waves in the foam from the outer facesheet to the inner facesheet is described in Fig. 3a–c. Elastic waves first propagate from r1 to r2 with the elastic wave speed ce, as shown in Fig. 3a. The elastic wave transit time through core thickness is
where r2i and r2r are the incident and reflected stresses respectively. The velocity of the inner facesheet is
te ¼
H ce
ð14Þ
_ 2i þ w _ 2r _2 ¼w w
ð23Þ
ð24Þ
_ 2i and w _ 2r are the incident and reflected wave velocities. where w 3.3.1. Elastic wave propagation Figure 4a–c show the interaction of incident and reflected stresses in the foam near the inner facesheet. The incident stress
M.S. Hoo Fatt, H. Surabhi / Composite Structures 94 (2012) 3174–3185
3177
Fig. 3. Propagation of elastic–plastic waves from outer facesheet: (a) elastic loading, (b) elastic–plastic loading, and (c) elastic unloading.
propagates from the outer facesheet and was shown in Fig. 4a. The reflected stress is
r2r ¼ qc ce w_ 2r
ð25Þ
_ 20 w _ 2i ½q0 r2i ¼ qc ce ½w
Substituting Eq. (24) into Eq. (25) gives
r2r ¼ qc ce ðw_ 2i w_ 2 Þ
ð26Þ
ð27Þ
The above expression is the foam stress acting on the inner facesheet, as shown in Fig. 1b.
ð29Þ
_ 20 from Eqs. (28) and (29) give Eliminating w
Hence the total stress acting on the inner facesheet is
r2 ¼ r2i þ qc ce ðw_ 2i w_ 2 Þ
_ 20 is the particle velocity in elastic zone ahead of the plastic where w wave front. The jump condition across the incident/reflected elastic wave front gives
rp2 ¼ 1
cp 0 cp _ 2i w _ 2Þ q þ r2i þ qc cp ðw ce ce
ð30Þ
An elastic unloading wave will propagate from the inner facesheet when the plastic stress is at a maximum value. The elastic unloading wave is shown in Fig. 4c. The momentum jump condition across the elastic unloading wave front gives
3.3.2. Elastic–plastic wave propagation Plasticity begins when the total stress at the inner facesheet reaches q0 . This causes a plastic wave front in addition to the elastic wave front, as shown in Fig. 4b. Both the incident and reflected stresses propagate with cp in the plastic zone. Momentum conservation across the plastic wave front gives
_ 21 are the peak plastic stress and particle velocity where r2max and w ahead of the wave front. Furthermore, the momentum jump condition across the plastic wave front gives
_2w _ 20 brp2 q0 c ¼ qc cp ½w
_ 21 w _ 20 ½r2 max q0 ¼ qc cp ½w
ð28Þ
_2w _ 21 ½r2 r2 max ¼ qc ce ½w
Fig. 4. Propagation of elastic–plastic waves from inner facesheet: (a) elastic loading, (b) elastic–plastic loading, and (c) elastic unloading.
ð31Þ
ð32Þ
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Finally, the momentum jump condition across the incident/reflected elastic wave front gives
_ 20 w _ 2i ½q0 r2i ¼ qc ce ½w
ð33Þ
_ 21 and w _ 20 from Eqs. (31)–(33) gives Eliminating w
r2
cp 0 cp _ 2i max q þ r2i max ðr2i max r2i Þ þ qc cp w ¼ 1 ce ce _ 2i max w _ 2i Þ qc cp w _ 2 max w _ 2Þ _ 2 max þ qc ce ðw qc ce ðw
uniform radial impulse. Following the inextensionality condition, w01 ¼ f1 a0 . This condition implies that cn = bn/n and dn = an/n. Hence,
w1 ¼
1 X 1 n¼2
n
½an ðs1 Þ sin nh bn ðs1 Þ cos nh
ð41Þ
Similarly, for imperfections
ð34Þ
_ 2i max and w _ 2 max are the maximum stress, incident where r2imax, w particle velocity and particle velocity at the start of unloading.
f1i ¼
ð42Þ
n¼2
and
w1i ¼
4. Transient response of facesheets
1 X ½dn ðs1 Þ cos nh þ cn ðs1 Þ sin nh
1 X 1 n¼2
n
½dn ðs1 Þ sin nh cn ðs1 Þ cos nh
ð43Þ
The solution methodology for the outer and inner facesheet response is an extension of a previous solution on impulsivelyloaded laminated shells by Hoo Fatt and Pothula [5]. In this study, however, both outer and inner facesheets are considered with initial imperfections and the load duration is non-impulsive. To incorporate imperfections, the total radial and tangential deformations for the outer facesheet are expressed as
The inner facesheet deformations are similarly expressed in terms of a normalized radial deflection f2 = w2/r2, tangential deflection w2 = v2/r2 and time s2 = ct/r2. The Fourier series representation for deflections of the inner facesheet are
w1 ðh; tÞ ¼ w1 ðh; tÞ þ w1i ðhÞ
and
ð35Þ
and
v 1 ðh; tÞ ¼ v 1 ðh; tÞ þ v 1i ðhÞ
ð36Þ
where w1(h, t) is the radial deformation, measured positive inward from the initial imperfection w1i(h), and v1(h, t) is the tangential deformation, measured anti-clockwise from the initial imperfection v1i(h). Similarly the total radial and tangential deformations for the inner facesheet are expressed as
w2 ðh; tÞ ¼ w2 ðh; tÞ þ w2i ðhÞ
f2 ¼ e0 ðs2 Þ þ
w2 ¼
1 X en ðs2 Þ n¼2
n
sin nh
f n ð s2 Þ cos nh n
ð45Þ
The imperfections associated with the inner facesheet are
f2i ¼
1 X ½en ðs2 Þ cos nh þ kn ðs2 Þ sin nh
ð46Þ
n¼2
and
w02i ¼
1 X en ðs2 Þ n¼2
v 2 ðh; tÞ ¼ v 2 ðh; tÞ þ v 2i ðhÞ
ð44Þ
n¼2
ð37Þ
and
1 X ½en ðs2 Þ cos nh þ fn ðs2 Þ sin nh
n
sin nh
kn ðs2 Þ cos nh n
ð47Þ
ð38Þ 4.2. Lagrange’s equations of motion
4.1. Fourier series representation tangential Define a normalized radial deflection f1 = w1/rq 1,ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi deflection w1 = v1/r1 and time s1 = ct/r1, where c ¼ A22 =ðqf hÞ is the wave speed in the circumferential direction of the outer facesheet. The Fourier series representation of normalized radial and tangential displacement of the outer facesheet are 1 X f1 ¼ a0 ðs1 Þ þ ½an ðs1 Þ cos nh þ bn ðs1 Þ sin nh
ð39Þ
n¼1
The radial and tangential facesheet deformations are derived from Lagrange’s equations of motion:
d @T @T @U þ ¼ Qn dt @ q_ n @qn @qn
where T is the kinetic energy, U is the strain energy, qn is a generalized coordinate, q_ n is the generalized velocity and Qn is a generalized force. The generalized forces are obtained from virtual work dW as follows:
and
w1 ¼
1 X
Qn ¼ ½cn ðs1 Þ cos nh þ dn ðs1 Þ sin nh
ð40Þ
n¼1
where n is the mode number. The term at n = 0 or a0(s) is the breathing mode whereby the shell goes in and out of hoop compression. The term at n = 1 denotes rigid body motion, while the terms for n P 2 are bending modes. Under uniform pressure load, there can be no rigid body motion of the shell. Hence, the terms involving n = 1 are omitted in this analysis. Inextensional deformations of thin rings and shells originated from Rayleigh [6], where he showed that displacement due to the extension of mid-surface are negligibly small in comparison with displacements due to bending. Goodier and McIvor [7] later demonstrated that the amplitude of the extensional modes were indeed negligible compared to the amplitude of the bending modes in studying the stability of an isotropic elastic shell subjected to
ð48Þ
@ðdWÞ @ðdqn Þ
ð49Þ
The kinetic energy for the outer and inner facesheets (T1 and T2, respectively) are
T1 ¼
1 A22 r1 2
2 Z 2p " @f 1
0
@ s1
þ
@w1 @ s1
2 # dh
ð50Þ
2 # dh
ð51Þ
and
1 T 2 ¼ A22 r2 2
2 Z 2p " @f 2
0
@ s2
@w2 þ @ s2
where A22 is the membrane stiffness of the facesheet in the hoop direction. Following Lindberg and Florence [8], the elastic strain energy for the outer and inner facesheets
M.S. Hoo Fatt, H. Surabhi / Composite Structures 94 (2012) 3174–3185
(U1 and U2, respectively) are
Z 2p h 0 2 1 0 0 0 U 1 ¼ A22 r 1 w1 f1 þ w01 f1 f02 1 2f1 w1 þ 2f1 f1i 2 r0 i 2 2f01 w01 2f1 w01i þ a21 f001 þ f1 dh
Substituting Eqs. (56) and (57) into Eq. (48) and evaluating generalized forces from Eqs. (59) and (49) give the equations of motion for Fourier coefficients. For the breathing mode (n = 0),
ð52Þ
and
Z 2p h 0 2 0 0 0 w2 f2 þ w02 f2 f02 2 2f2 w2 þ 2f2 f2i r0 2 i 2f02 w02 2f2 w02i þ a22 f002 þ f2 dh
2 1
1X d a0 2 r 10 þ a0 1 þ a21 ðn2 2Þ a2n þ bn ¼ p 4 n¼2 ds21
i d an n2 h 2 þ ðn 1Þ2 a21 ðn2 2Þa0 an ds21 n2 þ 1 2
ð53Þ
where a ¼ D22 = r 21 A22 , a22 ¼ a21 r21 =r 22 , ½ 0 ¼ @½ =@h and D22 is the bending stiffness of the facesheet in the hoop direction. The virtual work associated with the outer facesheet is
¼
2 1
dW 1 ¼ A22 r 1
r 1 df1 dh ½p
ð54Þ
n2 ðn2 2Þ 1n dn a0 r n2 þ 1
i d bn n2 h 2 n2 ðn2 2Þ ^ 1n þ 2 ðn 1Þ2 a21 ðn2 2Þa0 bn ¼ dn a0 r 2 n2 þ 1 ds1 n þ 1 2
ð62Þ
¼ pr1 =A22 is a normalized pressure load and r 1 ¼ r1 r1 =A22 where p is the normalized stress in the foam at the outer facesheet. If the foam is purely elastic, r1 is given by Eq. (13), and if the foam is elastic–plastic, r1 is given by Eq. (16). In contrast, the virtual work associated with the inner facesheet is
Z 2p
ð55Þ
2 ¼ r2 r 2 =A22 is the normalized stress in the foam acting on where r the inner facesheet. If the foam is purely elastic, r2 is given by Eq. (27), and if the foam is elastic–plastic, r1 is given by Eqs. (30) and (34). 4.3. Outer facesheet equations of motion
da0 ds1
2
1 2 X
1 n þ1 þ 2 n¼2 n2
d a0 r 10 þ a0 ¼ p ds21
ð63Þ
i d an n2 h 2 þ 2 ðn 1Þ2 a21 ðn2 2Þa0 an 2 ds1 n þ 1 2
¼
n2 ðn2 2Þ 1n ; n P 2 dn a0 r n2 þ 1
ð64Þ
i d bn n2 h 2 þ 2 ðn 1Þ2 a21 ðn2 2Þa0 bn 2 ds1 n þ 1 2
Substituting Eqs. (39), (41), (42) and (43) into Eqs. (50) and (52) gives the kinetic energy as
T 1 ¼ pA22 r 1
The breathing mode is in general coupled with the bending modes. 2 When deflections are small, a2n and bn are negligible compared to an and bn. In addition, a21 1 because it is on the order of (h/r1)2. Eqs. (60)–(62) then reduce to 2
r 2 df2 dh
o
"
ð61Þ
and
o
dW 2 ¼ A22 r 2
ð60Þ
For modes of n P 2
1 U 2 ¼ A22 r 2 2
Z 2p
3179
dan ds1
2
dbn þ ds1
2 !#
¼
ð56Þ
n2 ðn2 2Þ cn a0 r^ 1n ; n P 2 n2 þ 1
ð65Þ
The above equations of motion are solved with the following initial conditions: dan/ds1(0) = 0, dbn/ds1(0) = 0, an(0) = 0, and bn(0) = 0 for n = 0 and n P 2.
and the strain energy as
(
1 h i 1X U 1 ¼ pA22 r 1 a20 1 þ a21 þ a21 ðn2 1Þ2 ðn2 2Þa0 2 n¼2 ) 1
X 2 a2n þ bn ðn2 2Þðdn an þ cn bn Þa0
4.4. Inner facesheet equations of motion The equations of motion for the inner facesheet are derived similarly to those of the outer facesheet. For the breathing mode
ð57Þ
n¼2
Since deflections are small, terms higher than second order have been neglected. 1 is represented by FouTo calculate the virtual work, assume r rier series:
r 1 ¼ r 10 þ
1 X
1 X
n¼2
n¼2
r 1n cosðnhÞ þ
r^ 1n sinðnhÞ
1 X
pA22 r1 r 1n dan
n¼2
1 X
pA22 r1 r^ 1n dbn
n¼2
where da0, dan, and dbn are virtual displacement amplitudes.
ð59Þ
ð66Þ
and for modes of n P 2:
i d en n2 h 2 n2 ðn2 2Þ þ 2 en e0 þ r 2n ðn 1Þ2 a22 ðn2 2Þe0 en ¼ 2 n2 þ 1 ds2 n þ 1 2
ð58Þ
1i0 is the zero component, r 1in is the nth cosine component, where r ^ 1in is the nth sine component of r 1 . The components follow and r from the Fourier series representation of the foam velocities in Eqs. (13) and (16). The virtual work is then
r 10 da0 dW 1 ¼ 2pA22 r 1 ½p
2
d e0 20 þ e0 ¼ r ds22
ð67Þ and
i d fn n2 h 2 n2 ðn2 2Þ ^ 2n þ 2 ðn 1Þ2 a22 ðn2 2Þe0 fn ¼ kn e0 þ r 2 n2 þ 1 n þ 1 ds2 2
ð68Þ 20 is the zero component, r 2n is the nth cosine component, where r ^ 2n is the nth sine component of r 2 . The components follow and r from the Fourier series representation of the foam velocities into Eqs. (27), (30), and (34). The equations of motion for the inner facesheet are solved using the following initial conditions: den/ ds2(0) = 0, dfn/ds2(0) = 0, en(0) = 0, and fn(0) = 0, for n = 0 and n P 2.
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5. Shell failure
Thus the value of the hoop strain at which failure occurs is given by
To examine failure of the outer facesheet, one calculates hoop strains from the facesheet deformation and use laminate shell theory and failure criteria to determine failure. Shell fracture of the woven E-Glass/Vinyl Ester facesheet is examined in this section. The total hoop strain in the shell is given by
ef ¼
eh ¼ ehm þ zjh
ð69Þ
For the orthotropic facesheets the maximum tangential stress would occur at the outer plies where the bending strains are maximum, i.e., at z = ±h/2:
h 2
eh ¼ ehm jh
ð70Þ
Here ehm = a0 and the curvature is given by
jh ¼
" # 1 X 1 ao an ðn2 1Þ cos nh r1 n¼2
ð71Þ
The above curvature varies with time t and angular position h. It has a maximum value at h = 0,p. Denote the curvature at h = 0 as
j0
" # 1 X 1 2 ¼ ao an ðn 1Þ r1 n¼2
ð72Þ
and the curvature at h = p as
jp ¼
" # 1 X 1 ao þ an ð1Þn ðn2 1Þ r1 n¼2
ð73Þ
A modified Hashin–Rotem is used to examine lamina failure of the woven roving E-Glass/Vinyl Ester [9]. According to the modified Hashin–Rotem failure criteria, the failure of the composite occurs when
jrx j ¼ 1 if XT jrh j ¼ 1 if XT jsxh j ¼1 SL
jrx j ¼ 1 if XC jrh j > 0 or ¼ 1 if XC
rx > 0 or
rx < 0
ð74Þ
rh
rh < 0
ð75Þ ð76Þ
minðX T ; X c Þ Q 22
The above expression describes the maximum allowable strain based on a Hashin–Rotem composite failure criterion. 6. An example Consider a sandwich shell with E-Glass Vinyl Ester facesheets of thickness h = 2 mm, outer radius r1 = 93.5 mm and inner radius r2 = 66.5 mm and 25 mm-thick Divinycell H100 core. The facesheets are given initial imperfections f1i = d2 cos 2h and f2i = e2 cos 2h, where d2 = 0.1h/r1 and e2 = 0.1h/r2. A MATLAB program was written to solve for the outer and inner facesheet response when po = 2 MPa and DT = 1 ms. Finite element analysis (FEA) using ABAQUS Dynamic Implicit analysis was done in addition to gage the accuracy of the analytical solution. Dynamic, Explicit analysis, although less computationally expensive, could not be used for this type of problem because of numerical damping used to stabilize the explicit algorithm in ABAQUS Explicit. The direct-integration method provided for Dynamic, Implicit analysis in ABAQUS Standard is the Hilber–Hughes Taylor operator, which is an extension of the trapezoidal rule. Automatic time increment with specified half-step residual was used. A parametric study was done to determine the appropriate size of the half-step residual that would yield accurate results. No numerical damping was specified in the problem. The FEA model of the sandwich cylinder in plane strain is shown in Fig. 5. Continuum plane strain elements with full integration (CPE4) were chosen. The composite material properties were specified in a local cylindrical coordinate (RTZ) according to the ABAQUS User Manual, Version 6.7 [10]. The facesheet was modeled as orthotropic elastic material. The foam was modeled as crushable foam with isotropic hardening, using only the through-thickness compressive stress–strain curve of the foam. Plasticity properties for the PVC H100 foam were taken from Ref. [11]. It was found that under this pressure pulse, plastic dissipation in the foam occurs only near the inner facesheet. Figure 6 shows the radial stress–strain curves in the foam near the outer facesheet (Point C in Fig. 5) and near the inner facesheet (Point D in Fig. 5). The analytical solution also confirms that foam plasticity was localized near the inner facesheet. Finally, there is good agreement in
For an orthotropic shell, the relationship between the principal stress and strains are given by
8 9 > > < rx = > :
2
Q 11
Q 12
rh ¼ 6 4 Q 12 Q 22 > sxh ; 0 0
9 38 > < ex > = 7 0 5 eh > > : cxh ; Q 66 0
ð77Þ
where Q 11 ¼ E11 =ð1 m12 m21 Þ, Q 22 ¼ E22 =ð1 m12 m21 Þ, Q 21 ¼ m12 E22 =ð1 m12 m21 Þ, and Q 66 ¼ G12 . In the plane strain problem, the stresses in the shell are reduced to
rx Q 12 eh rh ¼ Q 22 eh
ð78Þ ð79Þ
Since Q 22 > Q 12 , rh > rx and failure of the shell will occur in tangential direction. Substituting Eq. (79) into Eq. (75) one gets
Q 22 eh ¼ 1 if XT
rh > 0
ð80Þ
rh < 0
ð81Þ
and
Q 22 eh ¼ 1 if XC
ð82Þ
Fig. 5. Finite element model of sandwich cylindrical shell.
M.S. Hoo Fatt, H. Surabhi / Composite Structures 94 (2012) 3174–3185
3181
near the inner facesheet and near the outer facesheet. For all load durations, plasticity begins near the inner facesheet. When the load duration is 1 ms or longer, there is no plasticity in the foam near the outer facesheet up to the point of failure. When the load duration is below 1 ms, plastic work is associated with multiple wave reflections and the foam near the inner and outer facesheets undergo hysteresis before failure. The energy absorption before shell failure increases substantially with shorter load durations, as shown in Fig. 8b. The energy absorption shown in Fig. 8b were taken from FEA. 7.2. Core parameters
Fig. 6. Radial stress–strain response in foam near outer and inner facesheets: (po = 2 MPa, DT = 1 ms).
the outer and inner facesheet radial deformations and velocities between the analytical solution and FEA, as shown in Fig. 7a and b. The outer and inner facesheet motions are defined at Points A and B in Fig. 5, respectively. 7. Parametric studies The proposed analytical model is used to understand mechanisms and parameters that are responsible for increasing blast resistance and energy absorption of the composite sandwich shell. This is an important step in the design of blast resistant structures. In all the cases below, the facesheets are given initial imperfections: f1i = d2cos 2h and f2i = e2cos 2h, where d2 = 0.1h/r1 and e2 = 0.1h/r2. The influence of load and core parameters is discussed below. 7.1. Load parameters The peak pressure and load duration of the blast loads described in Eq. (1) varies with many factors such as weight of charge and stand-off distance. Figure 8a shows the combination of peak pressures and load duration that would just cause sandwich shell failure. For all load cases the sandwich shell fails when the outer (pulse-loaded) facesheet ruptures. In general, higher peak pressures can be sustained with lower load durations. However, it was also found that maximum load to failure did not vary significantly for load duration of 1 ms and longer. Also shown in Fig. 8a is the peak pressure that would just induce plastic work in the foam
(a)
The effect of core thickness on the strength and energy absorption of the sandwich shell was examined by keeping the woven E-Glass/Vinyl Ester facesheet thickness at 2 mm and the mean sandwich shell radius at 80 mm but varying the Divinycell H100 core thickness from 6.25 mm to 50 mm (1/4–2 in.). Figures 9a and b show the peak pressure to failure with a load duration of 1 ms and the corresponding energy absorbed by the Divinycell H100 core as they vary with the core thickness. The peak pressure to failure generally decreases with increasing core thickness, whereas the energy absorbed increases with increasing core thickness. The fact that the peak pressure to failure is lower for a thicker sandwich shell is counter-intuitive since a thicker sandwich shell would offer higher bending stiffness. However, the failure of the sandwich cylindrical shell involves local rupture of the outer facesheet during the through-thickness wave propagation phase. During this phase, the outer facesheet deformation is influenced by reflected stress waves in the core. Since the reflected stress wave from the inner facesheet travels over a shorter distance in the thinner cores, it reaches the outer facesheet earlier thereby preventing outer facesheet rupture. The effect of core material properties was also investigated by replacing the Divinycell H100 foam with four other foams, which are listed in Table 1. The facesheets were 2 mm thick with r1 = 93.5 mm and r2 = 66.5 mm. Two core thicknesses were examined, a 25 mm-thick core and a 50 mm-thick core. As indicated in Fig. 10a, the sandwich shells with foam cores of higher stiffness and crush strength sustain higher peak pressures when the load duration is 1 ms. In all cases the outer facesheet ruptured, but as Fig. 10b indicates there was no core plasticity or energy absorption in the sandwich shells with 25 mm-thick Klegecell R300 and Divinycell HCP100 cores and the sandwich shell with the 50 mm-thick Divinycell HCP100 core. The crushing resistance of the Klegecell R300 and Divinycell HCP100 were high enough to keep the core in the elastic regime in these scenarios. However,
(b)
Fig. 7. Deflection and velocity during elastic–plastic response (po = 2 MPa DT = 1 ms): (a) deflection and (b) velocity.
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Fig. 8. Effect of load duration on (a) peak pressure and (b) energy absorption to failure.
Fig. 9. Effect of core thickness on panel strength: (a) peak pressure to failure and (b) energy absorption.
Table 1 Material properties of 0/90 woven roving E-glass/vinyl ester and several foams.
Density (kg/m3) E11 (+) (GPa) E22 (+) (GPa) E33 (+) (GPa) E11 () (GPa) E22 () (GPa) E33 () (GPa) m12 = m21 m13 = m23 m31 = m32 G12 = G21 (GPa) G23 = G32 (GPa) G13 = G31 (GPa) q (MPa)
eD XT (MPa) XC (MPa) YT (MPa) YC (MPa) SL (MPa) ST (MPa)
E-Glass/vinyl ester
Divinycell H30
Divinycell H100
Divinycell H200
Klegecell R300
Divinycell HCP100
1391.3 17 17 7.48 19 19 – 0.13 0.28 0.12 4.0 1.73 1.73 – – 270 200 270 200 40 31.6
36 0.044 0.044 0.044 0.027 0.027 0.027 0.25 0.25 0.25 0.013 0.013 0.013 0.3 0.85 0.57 0.29 0.57 0.29 0.35 0.35
100 0.149 0.149 0.149 0.105 0.105 0.105 0.31 0.31 0.31 0.0438 0.0438 0.0438 1.66 0.8 3.2 1.53 3.5 1.53 1.47 1.47
200 0.277 0.277 0.277 0.293 0.293 0.293 0.3 0.3 0.3 0.110 0.110 0.110 4.35 0.7 6.4 4.36 6.4 4.36 3.86 3.86
300 – – – 0.338 0.338 0.338 0.23 0.23 0.23 0.123 0.123 0.123 7.8 0.285 – – – – – –
400 – – – 0.340 0.340 0.340 0.3 0.3 0.3 0.131 0.131 0.131 10.3 0.15 – – – – 7.4 7.4
the crushing resistance of the foam is not the only parameter governing energy absorption of the sandwich cylindrical shell. The sandwich shell with the 50 mm-thick Klegecell R300 core went exhibited plasticity near the inner facesheet of the sandwich shell before it failed, while the sandwich shell with the 25 mm-thick
Klegecell R300 core did not. Thus there is a minimum core thickness for energy absorption of the sandwich shell with the Klegecell R300 core. The stiffness and crush strength of the foam generally increase at the expense of increasing foam density or weight (see Table 1).
M.S. Hoo Fatt, H. Surabhi / Composite Structures 94 (2012) 3174–3185
3183
Fig. 10. Effect of core type: (a) peak pressure and (b) energy absorption to failure.
Fig. 11. Failure pressure per unit weight: (a) 25 mm-thick core and (b) 50 mm-thick core.
In Fig. 11a and b, the peak pressure to failure is normalized by the weight per unit length of the shell cL, which is defined as follows:
"
2 2 2 2 # h h h h cL ¼ qf g p r1 þ r1 þ r2 þ r2 2 2 2 2 " # 2 2 h h ð83Þ r2 þ þ qc g p r1 2 2 where g is the acceleration due to gravity. The core thickness in Fig. 11a and b is 25 mm and 50 mm, respectively. Per unit weight, the sandwich shell with the Divinycell H200 foam core is the most blast resistant shell of all considered, surpassing sandwich shells made with Klegecell R300 and Divinycell HCP100 foams. Furthermore, even the shell with the 50 mm-thick Divinycell H30 core is more blast resistant per unit weight than shells of comparable thickness and made with Klegecell R300 and Divinycell HCP100 foam cores. This parametric study elucidates the importance of considering wave propagation effects in designing blast resistant sandwich structures. 8. Concluding remarks Stress wave analysis has been done on a composite sandwich cylindrical shell subjected to external blast. The facesheets and core of the sandwich shell consisted of orthotropic elastic and crushable (elastic–plastic) foam materials, respectively. Failure of the sandwich shell under intense dynamic loading was due to rupture of the outer facesheet. The sandwich shell response was
shown to be affected by the magnitude and duration of the pressure pulse. High amplitude, low duration (impulsive) pressure pulses induced high energy absorption associated with multiple wave reflections. Low amplitude, long duration pressure pulses caused minimal energy absorption in the foam nearest to the inner facesheet. It was found that the amount of energy absorbed increases and the failure load decreases with increasing core thickness of Divinycell PVC H100. Five different foams were also considered in this study. In order of increasing density, these were Divinycell H30, H100, H200, Klegecell R300 and Divinycell HCP100. The compressive modulus and crush resistance of these foams generally increase with foam density so that the Divinycell H30 foam had the lowest compressive modulus and crush resistance, while the Divinycell HCP100 foam had the highest compressive modulus and crush resistance. The sandwich shells with Divinycell HCP100 foam core were found to be the most blast resistant to failure even though no energy was absorbed by them. However, the Divinycell HCP100 foam was also the heaviest and per unit weight, the Divinycell H200 foam was more blast resistant to failure than it. A reason for this was that the lower crush resistance of the Divinycell H200 foam allowed plastic work dissipation in the core before shell failure. Acknowledgements The authors gratefully acknowledge financial support from Dr. Yapa Rajapakse at the Office of Naval Research under Grant No. N00014-11-1-0485.
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Appendix A. Crushable elastic–plastic foam in uniaxial strain The constrained elastic modulus, yield strength and elastic– plastic tangent modulus for the foam under uniaxial strain compression are derived for a crushable foam in this section. The foam is assumed to be an isotropic material with the uniaxial stress compressive elastic–plastic properties, and isotropic hardening is also assumed during plastic deformation. These assumptions are valid if the stress state in the foam is primarily compressive, but is only approximately true if the foam experiences a multi-axial stress state because most foams are anisotropic. It is also assumed that the compressive strains are limited to 20%, which is low enough to avoid foam densification and appreciable stiffening. A.1. Elastic behavior
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 1 f ¼ 2J 2 þ I21 q ¼ 0 3
Under uniaxial strain, I1 = rr + 2rh and Eq. (A5), one gets
rh ¼ K
2G 3
ðA10Þ pffiffiffi pffiffiffiffi J2 ¼ 1= 3ðrr rh Þ. From
er
ðA11Þ
Combining Eqs. (A6) and (A11) to eliminate er gives
rh ¼
3K 2G rr 3K þ 4G
ðA12Þ
Hence,
Under uniaxial strain, the strain eij and the stress rij are given by
eij ¼ ½er ; 0; 0
9K rr 3K þ 4G
ðA13Þ
pffiffiffiffi 1 6G rr J 2 ¼ pffiffiffi 3 3K þ 4G
ðA14Þ
I1 ¼
ðA1Þ and
and
rij ¼ ½rr ; rh ; rh
ðA2Þ
where rr and rh are the radial and tangential stresses in the foam, respectively. The strain deviator eij and the stress deviator sij are thus
eij ¼
pffiffiffi most foams compressed topaffiffiffiffiuniaxial strain of about 20%, a ¼ 3= 2. Hence, in terms of I1 and J 2 , the yield criterion for the foam is
1 er ½2; 1; 1 3
ðA3Þ
Substituting Eqs. (A13) and (A14) into Eq. (A10) gives a value of constrained flow strength q0 as
qð3K þ 4GÞ
r1 ¼ q0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð24G2 þ 27K 2 Þ
ðA15Þ
and
sij ¼ ½sr ; sh ; sh
ðA4Þ
During purely elastic behavior, the constitutive relation between the stress and strain is
rij ¼ 2Geij þ K ekk dij
ðA5Þ
where dij is the Kronecker delta function, G = E/(2(1 + m)) is the shear modulus, K = E/(3(1 2m)) is the bulk modulus, and E and m are the Young’s modulus and Poisson’s ratio in the uniaxial stress state. Substituting the stresses and strains defined in Eqs. (A1)–(A3) into Eq. (A5) gives
4 3
rr ¼ Ger þ K er ¼
To find the elastic–plastic tangent modulus, the total strain increment during plastic deformation of the foam is decomposed into elastic and plastic components:
deij ¼ deeij þ depij
ðA6Þ
4 GþK 3
Under uniaxial strain, the strain increment deij and the stress increment drij are given by
deij ¼ ½der ; 0; 0
ðA7Þ
drij ¼ ½drr ; drh ; drh
ðA18Þ
deij ¼
1 der ½2; 1; 1 3
ðA19Þ
and
dsij ¼ ½dsr ; dsh ; dsh To find the constrained yield strength, one must consider a yield criterion for crushable foam. Deshpande and Fleck [12] proposed the following crushable foam yield criterion:
^ q¼0 f ¼r
ðA8Þ
^ is an equivalent stress given by where r
2 1 2 2 a2 i re þ a rm
1þ
ðA17Þ
The strain deviator increment deij and the stress deviator increment dsij are also given as
A.2. Plastic yielding
r^ ¼ h
ðA16Þ
and
4 G þ K er 3
Thus the constrained modulus E0 is given by
E0 ¼
A.3. Elastic–plastic behavior
ðA9Þ
3
pffiffiffiffiffiffiffi
r^ ¼ I1 =3 is the mean stress, re ¼ 3J2 is the effective or Mises stress, a is the pressure sensitivity pffiffiffifficoefficient, I1 = rii is the first
invariant of the stress tensor and J2 ¼ ð1=2sij : sij Þ1=2 is the deviatoric stress. Equation (A8) describes a yield surface that is elliptical in rm re space, while a defines the aspect ratio of the ellipse. For
ðA20Þ
The constitutive relation for the elastic strain increment follows from Eq. (A5), while the plastic strain increment is obtained from the flow rule. The complete strain–stress relation is
deij ¼
dsij dI1 @f þ dij þ dk @ rij 2G 9K
ðA21Þ
where dk is an undetermined factor such that
dk ¼
0;
when f < 0 or f ¼ 0 when df < 0
> 0; when f ¼ 0 and df ¼ 0
ðA22Þ
The form of dk is determined from the consistency condition
df ¼
@f drij ¼ 0 @ rij
ðA23Þ
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M.S. Hoo Fatt, H. Surabhi / Composite Structures 94 (2012) 3174–3185
Now consider a yield pffiffiffiffi function expressed in general terms of stress invariant I1 and J2 :
pffiffiffiffi f ¼ F I1 ; J2 q ¼ 0
ðA24Þ
FEA
Chen and Han [13] gives
drij
@f G @f ¼ 2Gdeij þ Kdekk dij dk 3K dij þ pffiffiffiffi pffiffiffiffi sij @I1 J2 @ J2
!
ANA
ðA25Þ
and
dk ¼
@f ffiffiffi smn demn 3Kdekk @I@f1 þ pGffiffiffi p J2 @ J 2 2
2 @f ffiffiffi 9K @I@f þ G p 1
@
ðA26Þ
J2
Hence for the foam yield function specified in Eq. (A10), one gets that under uniaxial strain constraint
drr
Fig. A1. Compressive stress–strain behavior of Divinycell H100 crushable foam.
8 > <4G
hpffiffiffi i2 9 pffiffiffiffi > = þ 4G = J 3 KI 1 2 1 h i þK ¼ der 2 > > 3 3 : ; KI1 =J 2 þ 4G
ical and FEA solutions. The plastic modulus varies with strain or stress. An average plastic modulus E0p is defined by
E0p ¼ ðA27Þ
1 ðr20 q0 Þ
Z r20
E0p E0 drr
ðA31Þ
q0
where r20 is the stress at 20% strain found from integration of Eq. (A29).
The constrained elastic–plastic tangent modulus is thus
4G 1 þK E0p ¼ 3 3
hpffiffiffi i2 pffiffiffiffi 3KI1 = J 2 þ 4G h i KI21 =J 2 þ 4G
References
ðA28Þ
pffiffiffi pffiffiffiffi where I1 = rr + 2rh and J2 ¼ ðrr rh Þ= 3. The radial and tangential stresses in the foam are related to each other during plastic foam by the foam yield criteria defined in Eq. (A10). As an example, the plastic portion of the stress–strain curve for Divinycell PVC H100 foam can be obtained by integrating the plastic modulus:
Z rr q0
drr
¼ 0 Ep E0
Z
er
der
ðA29Þ
ee
0
where ee ¼ qE0 is the elastic limit strain. Here E0p is given in terms of rr only because during yielding
rh ¼
1 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Y 2 r2r
ðA30Þ
For Divinycell H100 foam, Y = 1.66 MPa and the resulting elastic– plastic stress–strain curve after numerical integration is shown in Fig. A1. Note that there is very good agreement between the analyt-
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