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Dynamic response of double-layer rectangular sandwich plates with metal foam cores subjected to blast loading
Accepted Manuscript
Dynamic response of double-layer rectangular sandwich plates with metal foam cores subjected to blast loading Jianxun Zhang , Renfang Zhou , Mingshi Wang , Qinghua Qin , Yang Ye , T.J. Wang PII: DOI: Reference:
S0734-743X(17)31074-6 https://doi.org/10.1016/j.ijimpeng.2018.08.016 IE 3161
To appear in:
International Journal of Impact Engineering
Received date: Revised date: Accepted date:
8 December 2017 12 July 2018 29 August 2018
Please cite this article as: Jianxun Zhang , Renfang Zhou , Mingshi Wang , Qinghua Qin , Yang Ye , T.J. Wang , Dynamic response of double-layer rectangular sandwich plates with metal foam cores subjected to blast loading, International Journal of Impact Engineering (2018), doi: https://doi.org/10.1016/j.ijimpeng.2018.08.016
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Highlights
The analytical model for dynamic response of double-layer sandwich plates under blast loading is developed. The so-called ‘bounds’ are obtained by using approximate yield criteria.
Membrane mode solutions are obtained for dynamic response.
The analytical model captures numerical results reasonably.
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Dynamic response of double-layer rectangular sandwich plates with metal foam cores subjected to blast loading Jianxun Zhang, Renfang Zhou, Mingshi Wang, Qinghua Qin*, Yang Ye, T. J. Wang State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace,
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Xi’an Jiaotong University, Xi’an 710049, China
Abstract. In this paper, the analytical and numerical analyses are conducted to predict the dynamic response of fully clamped double-layer rectangular sandwich plates with metal foam cores
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subjected to blast loading. The so-called ‘bounds’ of analytical solutions for dynamic response of double-layer sandwich plates are obtained by using the inscribing and circumscribing squares of the exact yield locus. Also, the membrane mode solution for large deflection of the double-layer sandwich plates is presented. The effect of the double-layer factor on the dynamic response of double-layer sandwich plates is also discussed. The finite element calculation is carried out to study
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the dynamic response of the double-layer sandwich plates subjected to blast loading. Good
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agreement is achieved between the analytical predictions and numerical results. It is demonstrated that the impact resistance of the double-layer sandwich plate is better than that of the monolayer one
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with the same mass subjected to the higher impulse.
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Keywords: Double-layer sandwich plate; Yield criterion; Core layer-by-layer compression;
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Dynamic response; Large deflection.
1. Introduction
Sandwich structures are widely used in plenty of engineer fields due to the various advantages,
such as aircraft, aerospace, high speed train, and ship. Sandwich structures are consisted of stiff and strong face sheets and lightweight cores. To satisfy different requirements, several kinds of lightweight cores are developed, for example, honeycomb [1, 2], corrugated core [3], pyramidal *
Corresponding author. [email protected] (Q.H. Qin), Tel/Fax: +86-29-82664382. 2
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truss [4-6], metal foam [7-10] and woven material [11]. Compared with monolayer sandwich structures, investigations have proven that multilayer sandwich structures provide much more choices for design of structures [12,13]. In the past decades, the dynamic response of monolayer sandwich structures has been investigated extensively. Fleck and Deshpande [14] developed an analytical model for the dynamic response of the fully clamped sandwich beam subjected to uniform blast loading. Qiu et al. [15]
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obtained the so-called ‘bounds’ of analytical solutions for dynamic response of the clamped circular sandwich plates subjected to blast loading. Qin and Wang [16] obtained the analytical solutions for the impulsive response of fully clamped metal sandwich beams by using the membrane factor method. Zhang et al. [17] developed a plastic-string model to predict the impulsive response of
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corrugated metal sandwich plates with unfilled and foam-filled sinusoidal plate cores. Zhu et al. [18] and Cui et al. [19] obtained the so-called ‘bound’ solutions for the dynamic response of square sandwich plates subjected to blast loading. Employing the yield criterion including the effect of core strength [20], Qin et al. [21] obtained the analytical solutions for the dynamic response of fully
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clamped rectangular sandwich plates subjected to blast loading. Zhu et al. [22] and Cui et al. [23] experimentally studied the deformation and failure of metal sandwich plates with aluminum foam
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core and pyramidal lattice cores subjected to air blast loading, respectively. Comparing to monolayer sandwich structures, there were few investigations on impact
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response of multilayer sandwich structures. Dharmasena et al. [13] experimentally and numerically studied the dynamic response of a multilayer sandwich plates with corrugated cores subjected to
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impulsive loads incidenting from water. The results show that the multilayer sandwich plates significantly reduced the transmitted pressures of an impulsive load. Kılıçaslan et al. [24]
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experimentally and numerically investigated the crushing of single- and double-layer aluminum trapezoidal corrugated core sandwich panels at quasi-static and dynamic impact loading. Süsler et al. [25] analytically and numerically investigated the nonlinear dynamic behavior of simply supported tapered sandwich plates with multilayer face sheets subjected to air blast loading, and analytical results are in good agreement with numerical results. Wang et al. [26] numerically studied the dynamic response of a multilayer honeycomb sandwich structure under blast loading, and obtained the multi-objective optimization design that can apparently enhance the shielding performance of the protective structure without increasing the mass. Xiong et al. [27] carried out the low-velocity 3
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impact tests to study failure mechanisms and energy absorption of two-layer carbon fiber composite sandwich plates with pyramidal truss cores. Al-Shamary et al. [28] carried out drop weight impact experiment to investigate the low-velocity impact response of one-layer, two-layer and three-layer composite sandwich plates. The three-layer sandwich plate exhibits high energy absorption compared with the one-layer sandwich plate. Chen et al. [29] experimentally studied the low-velocity compression behavior, failure mechanism and energy absorption of multilayer woven
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lattice sandwich panels. Relative to quasi-static compressed panel, the impacted woven textile sandwich panel has much greater energy absorption induced by greater dynamic strength. Chen et al. [30] performed the projectile impact resistance of multilayer sandwich plates with corrugation, hexagonal honeycomb and pyramidal truss cores. At low impact velocity, multiplying layer can
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reduce the maximum central deflection of bottom face sheet of three sandwich plates except pyramidal truss ones in high relative density. Cao et al. [31] experimentally and numerically investigated the dynamic response of multilayer corrugated core sandwich plates using impact Hopkinson bar under out-of-plane compressive impact loading. It was found the bending of the
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interlayer plate contributes to reduce the force oscillation during successive folding. There is little work on the analytical solutions for impact response of double-layer sandwich
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plates under blast loading. The work of this paper is to investigate the dynamic response of fully clamped double-layer rectangular sandwich plates subjected to blast loading. This paper is
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organized as follows. In section 2, analytical analyses for the impulsive response of a fully clamped double-layer rectangular sandwich plate are presented. In section 3, employing the yield criteria for
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double-layer sandwich structures, analytical solutions are derived to predict the dynamic response of fully clamped double-layer sandwich plates subjected to blast loading. In section 4, finite element
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analysis is carried out. In section 5, analytical predictions are compared with numerical results, and the effect of the double-layer factor on the dynamic response of the double-layer sandwich plates is discussed. Conclusions are presented in section 6. 2. Statement of the problem Consider a fully clamped double-layer rectangular metal foam core sandwich plate with length 2L and width 2B subjected to blast loading of uniformly distributed impulse I per unit area, as shown in Fig. 1. Two identical top and bottom face sheets with thickness h and interlayer sheet with 4
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thickness hm are assumed to perfectly bond to the identical top and bottom metal foam cores with thickness c. It is assumed that the face sheets and interlayer sheets are made from rigid-perfectly plastic
material
with
yield
strength
f .
The
foam
cores
are
idealized
as
a
rigid-perfectly-plastic-locking ( r p p l ) material with plateau stress c and densification strain
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D . To describe the relation between top and bottom face sheets and interlayer sheet, a double-layer factor is defined as hm 2h .
The analytical model for dynamic response of fully clamped metallic sandwich beams subjected to blast loading over the entire span was developed by Fleck and Deshpande [14]. They
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estimated the time of the monolayer sandwich structure by analyzing a plastic shock wave propagating through the metal foam core. It should be noted that the assumption of three-stage deformation is largely depended on the ratio of time between the core compression and overall deformation of the sandwich plate. For the double-layer sandwich structure, the response time for core compression may depend on several parameters, such as core mass, core strength, core height,
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thickness of face sheet and pulse of load. In general, the time period of the core compression is at
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least 6 times smaller and for most of cases 10 times smaller than the overall structural response time of the sandwich structure. The detailed discussion of it can be found in Appendix A. Then, the
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coupling between the core compression and beam bending and stretching can be decoupled. In the core compression phase it is assumed that there is momentum transferred from the top face sheet to
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the core and bottom face sheet, and no momentum is transferred to any other portion of the beam. Then, any bending effects are neglected. This phase can be reduced to the one-dimensional core
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compression problem. See more details in Refs. [14, 32]. Here, this assumption is adopted to investigate the impulsive response of the double-layer
sandwich plates.
2.1. Core compression stage It is assumed that the impulse I per unit area imparts to the top face sheet with a velocity V0 = I f h . According to the momentum conservation law, the final common velocity V f of the
sandwich plate including two cores and two face sheets and interlayer sheet at the end of core 5
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compression stage is given by
I
Vf =
(1)
2 f h f hm +2c c
Then, neglecting the rate effect, it is assumed that the absorb energy U a for unit area is dissipated in the process of core compression, and the kinetic energy loss Ua can be expressed as
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h 1 2 1 Ua 2h I 2 2 f h where
hm 2h , c . 4c f
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h
(2)
The core layer-by-layer compression is assumed in the core compression stage. First, the impact energy is dissipated by the top core till the top foam core is densified. If the impact energy is not dissipated by the top core, the residual energy would be dissipated by the bottom core till the
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4c 2 h 2h D , the bottom core bottom core attains densification. When I I 0 I 0 h 1 2 1 2
I2
(3)
I
L f f
, =
c c , c . L f
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I
4c h 2h 2
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where
h 1 2 1 2
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c1
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keeps the initial situation while the top core is compressed to the strain c1 ,
When I I 0 , the top core attains the densification strain c1 D , and the bottom core is compressed to the strain c 2
c2 I 2
h 1 2 1 2 4c 2 h 2h
D
(4)
2.2. Bending and stretching phase At the end of the core compression phase, the top and bottom face sheets, the interlayer sheet, 6
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and the top and bottom core have a common velocity V2, and then the sandwich plate is brought to rest by plastic bending and stretching. Jones [33] developed an approximate theoretical approach to analyze the dynamic plastic behavior of monolithic rectangular plates considering influence of finite deflection. It is assumed that the initially rectangular flat plate is divided into four rigid regions which are separated by r straight hinge lines with length l j , as sketched in Fig. 2. Based on the
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principle of energy dissipation rate balance, the governing equation is given by r
p w wdA Nw M j dl j j 1 l j
A
(5)
where the external pressure p is applied on the whole area A of the plate uniformly, is mass
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per unit area of the sandwich plate, w is transverse deflection, w is transverse velocity, w is transverse accelerate, j is the relative angular velocity across the hinge line l j , and N and M are the membrane force and bending moment per unit length, respectively. Herein, this procedure is employed to study the dynamic response of the fully clamped double-layer sandwich plate
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subjected to the initial velocity.
The double-layer rectangular sandwich plate is assumed to be divided into four regions by nine
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straight hinge lines, and the transverse displacement profile is assumed to be linear. The front and side views of the transverse displacement fields are as follows,
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y 1 B W0 , w 1 x W , 0 B tan
Region
(6)
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Region
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as shown in Fig. 2, where W0 is the transverse displacement of the center of the plate, is determined by the upper bound theory and = arctan
3 B L B L [34], x is the 2
horizontal coordinate and y is the vertical coordinate. 3. Analytical solutions 3.1. The so-called ‘bounds’ of the analytical solutions The yield criterion for compressed double-layer sandwich cross-section is presented in Appendix B. Circumscribing yield locus can be expressed as 7
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M M p , N Np
(7)
Inscribing yield locus can be expressed as
M M p , N N p
(8)
where the inscribing coefficient factor is given in Appendix C, as shown in Fig. 3. Taking the similar analysis in Jones [34], combining Eqs. (5), (7) and (8), the non-dimensional
double-layer rectangular sandwich plate are given by
Wm
Wm a1I L 2 c 3a3a4b1b2
1 2
2 Aa c 1 2 1 b1a3 1 2
T
T f 1 a4b2 arctan L f 3a3b1
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and
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maximum central deflection of the bottom face sheet and structural response time of the
(9)
(10)
for the choice of circumscribing yield locus, and
Wm a1I L 2 c 3a3a4b1b2 2
2a Ac 2 1 a3b1 2
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Wm and
(11)
T f 1 a4b2 arctan L f 3a3b1
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T
(12)
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for the choice of inscribing yield locus, where L B , a1 3 tan , a2 cot ,
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a3 2 cot tan , a4 2 tan , b1 2h , b2 2h , =
a1 I 4 a2 Ac 2
a3b1 . 3a4b2
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3.2. Membrane mode solution The effect of bending moment decreases with the increasing the deflection, while the structural
response is mainly determined by membrane force only in theoretical analysis for the dynamic response of circular monolithic plate and circular sandwich plate [34, 35]. Here, the analytical method is extended to study the large deflections of rectangular double-layer sandwich plates subjected to blast loading. Herein, neglecting the bending moment M and letting N = N p in Eq. (5), the membrane 8
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mode solutions for the non-dimensional maximum central deflection Wm of the bottom face sheet and structural response time T of double-layer sandwich plates are given by Wm
Wm a1 I L 2 3a3a4b1b2 c
(13)
and
T f L f 2 3
a4b2 a3b1
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T
4. Finite element analysis
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Using the ABAQUS / Explicit software, the dynamic response of the double-layer sandwich
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plates is numerically analyzed subjected to blast loading. The double-layer sandwich plate is modelled using 8-node brick element with reduced integration (C3D8R). All the displacements are fully clamped at the ends of the sandwich beams. A uniformly distributed velocity field applied on the top face sheet is predefined. Mesh sensitivity study reveals that further refinements do not
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improve the accuracy of the calculations appreciably.
The top and bottom face sheets and interlayer sheets obey J 2 plastic flow theory and
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Deshpande-Fleck model [36] is used to describe the plastic crushable behavior of the metal foam core implemented in ABAQUS, which allows the shape change of yield surface due to differential
where
1
1 3
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ˆ 2
(15)
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ˆ c 0
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hardening along the hydrostatic and deviatoric axes. The yield function for the foam core is
2
2 e
2 m2
(16)
with e 3sij sij 2 being von Mises effective stress, sij the deviatoric stress, m kk 3 the mean stress and the shape factor of yield surface. The associated plastic flow rule is adopted and the plastic Poisson’s v p is given by
p 1 2 3 p p 2 1 3
2
(17)
22
11
9
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Face sheets and interlayer sheets are made from aluminum alloy with Young’s modulus E ft 70 GPa , yield strength f 200 MPa , Poisson’s ratio f 0.3 , density f 2700 kg/m3
and linear strain hardening modulus E f t 330MPa . Following the data of Ashby et al. [37], the aluminum foam core has Young’s modulus Ec 1GPa , yield strength c 2MPa , density
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c 270kg/m3 , relative density c f 0.1 , elastic Poisson’s ratio vce 0.3 , plastic Poisson’s ratio vcp 0 . The foam core has a long plateau-stress platform c that continues up to the densification strain D 0.7 , and the stress rises steeply with a large tangent modulus
Ect 0.5GPa beyond the densification strain. Geometrical parameters of the double-layer
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sandwich plates are as follows: length L 1m , core thickness c 0.02m , face sheet thickness
h 0.002m , interlayer sheet thickness hm 0.004m , i.e. =0.01 , c =0.02 , h =0.1 , width B 0.5m for rectangular sandwich plates and width B 1m for square sandwich plates are
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considered, i.e. =2 and =1 .
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5. Results and discussion
Analytical solutions and numerical results for the dynamic response of the rectangular ( 2 )
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and square ( 1 ) metal double-layer sandwich plates subjected to blast loading are shown in Figs.
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4 and 5, respectively. The so-called ‘bounds’ for the maximum central deflection Wm of the bottom face sheet and structural response time T of metal double-layer sandwich plates versus impulse
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I are shown in Figs. 4 and 5, respectively. Also, the membrane mode solutions for structural
response of double-layer sandwich plates are given. It is seen that the analytical predictions derived by employing the circumscribing and inscribing yield locus bound the numerical results well, and membrane mode solutions are in good agreement with numerical results. The analytical model neglects the effects of elasticity, strain hardening and strain rates of materials, wrinkling of face sheets, shear force, and reduction of the momentum provided by the supports in the core compression phase. For higher impulsive loading, the discrepancy may be due to the assumption of full densification of the core in analytical solutions while there is no distinct core densification in 10
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the numerical results. Analytical solutions for the structural response time T versus impulse I agree well with numerical results, as shown in Figs. 4(b) and 5(b), respectively. The effect of the double-layer factor on the maximum central deflection Wm of bottom face sheet versus impulse I of rectangular double-layer sandwich plates is shown in Fig. 6. Analytical prediction derived by employing the inscribing yield locus is employed in Figs. 6 and 7.
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It is seen that in the lower impulse, the bigger the double-layer factor is, the bigger the maximum central deflection of the double-layer sandwich plates is. In the higher impulse I 0.0017 , the maximum central deflection for the case of 0 is bigger than the other cases.
It means that the impact resistance of double-layer sandwich plates is better than that of monolayer
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sandwich plate in the higher impulse I 0.0017 .
Comparisons of the maximum central deflection Wm of bottom face sheet versus impulse I of the rectangular double-layer sandwich plate and solid monolithic plate with the same mass are
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shown in Fig. 7. The sandwich plate has 0.5 , 0.01, 2 , h =0.1 , c =0.02 , =0.1, and the density, length and width of solid monolithic plate are same as those of the sandwich plate.
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It is seen that the maximum central deflection of double-layer sandwich plate is smaller than that of solid monolithic plate when I 0.00125 , while the maximum central deflection of double-layer
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sandwich plate is bigger than that of solid monolithic plate when I 0.00125 . It means that the
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double-layer sandwich plate performs better impact resistance than the solid monolithic plate under high impulsive loading I 0.00125 .
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Numerical results of the maximum central deflection Wm of bottom face sheet versus impulse
I of rectangular double-layer sandwich plates, monolayer sandwich plates and solid monolithic
plates with the same mass in the low impulse are shown in Fig. 8. The double-layer sandwich plates have face sheet thickness h 0.0032m and interlayer sheet thickness hm 0.0016m . Three kinds of plates have width B 0.5m and length L 1m . The core height and top/bottom core height are identical for the monolayer and double-layer sandwich plates. It is seen that the maximum central deflection of double-layer sandwich plates is smaller than those of monolayer sandwich 11
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plates and solid monolithic plates with the same mass in the low impulse. Therefore, the double-layer sandwich plate performs better impact resistance than the monolayer sandwich plate and solid monolithic plate with same mass in the low impulse range. 6. Conclusions A theoretical analysis was conducted to predict the dynamic response of fully clamped
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rectangular double-layer sandwich plates with metal foam cores subjected to blast loading. The compressed yield criterion of double-layer sandwich cross-sections was given based on core layer-by-layer compression assumption. Employing the inscribing and circumscribing yield loci, the so-called ‘bounds’ of the analytical solutions for the dynamic response of double-layer sandwich
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plates were obtained. Also, membrane mode solutions were obtained for large deflections of the double-layer sandwich plates. The present analytical predictions agree well with the numerical results. The results show that the impact resistance of the double-layer sandwich plate is better than that of the monolayer sandwich plate with the same mass subjected to the higher impulse
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I 0.0017 , while the double-layer sandwich plate performs better impact resistance than the solid
monolithic plate with same mass under high impulsive loading I 0.00125 . It can be concluded
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that the membrane force plays an important role in large deflection of double-layer rectangular sandwich plates.
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It should be noted that the assumption that "the coupling between the core compression, and beam bending and stretching are decoupled" may be mandatory and simplified for double-layer
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sandwich plates. The good agreement between the present analytical predictions and FE results may indicate that the assumption used in derivation of the analytical model are appropriate to some
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extent. A detailed experimental study of the clamped rectangular double-layer sandwich plates with metal foam cores subjected to blast loading is necessary to fully validate the model developed and it will be one of our future work.
Acknowledgments. The authors gratefully acknowledge the financial supports of NSFC (11502189, 11572234, 11321062, and 11372235), Natural Science Basic Research Plan in Shaanxi Province of China (2017JM1020), China Postdoctoral Science Foundation funded project (2015M572546), and 12
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the Fundamental Research Funds for the Central Universities.
Appendix A. Discussion on response time for core compression Fig. A1 shows the numerical results for time period of the core compression and overall structural response time of the sandwich structure under the low impulse loading I =0.001 and
t f L f
VL f . The double-layer c f
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high impulse loading I =0.0035 , in which t
and V =
sandwich plates have face sheet thickness h 0.0032m , interlayer sheet thickness hm 0.0016m ,
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width B 0.5m and length L 1m . In the case of low impulse loading I =0.001, the velocity of top face sheet decreases, and the velocities of bottom face sheet and interlayer sheet increase till velocities of top, bottom face sheet and interlayer sheet keep almost same. In the case of high impulse loading I =0.0035 , first the velocity of top face sheet decreases and the velocities of
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bottom face sheet and interlayer sheet increase, and then the velocities of top face sheet and interlayer sheet keep nearly same. At last, the velocities of top, bottom face sheet and interlayer
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sheet keep almost same. Fig. A2 shows the ratio k of time period of the core compression to overall structural response time of the double-layer sandwich plates under the impulse loading using the
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Finite element analysis. It is seen that the ratio k is smaller than 16%. In other words, the time period of the core compression is at least 6 times smaller and for most of cases 10 times smaller
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than overall structural response time of the sandwich structure. The assumption in the analytical solutions is verified.
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The response time for core compression strongly depends on core strength, core height,
thickness of face sheet and pulse of load. The discussion details how these parameters affect the results using the Finite element analysis are presented as follows. Fig. A3(a) shows the effect of core strength on the response time Tc
Tc f L f
for core compression of double-layer
sandwich plates under impulsive loading I 0.001 . The double-layer sandwich plates have face sheet thickness h 0.0032m , interlayer sheet thickness hm 0.0016m , width B 0.5m and 13
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length L 1m . It is seen that the response time for core compression almost decreases with the increase of core strength for the cases of I 0.001 and I 0.0035 , except for the case of the small core strength under impulse loading I 0.0035 . The effects of core height c and thickness of face sheet h on the response time Tc for
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core compression of double-layer sandwich plates under impulsive loading I 0.001 are shown in Figs. A3(b) and (c), respectively. The double-layer sandwich plates have interlayer sheet thickness, width B 0.5m and length L 1m . In Fig. A3(b), the face sheet thickness h 0.002m , and the core height c 0.02m in Fig. A3(c). It is seen that the response time for core compression almost increases with the increase of core height for two cases, while almost keeps the constant when
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c 0.02 under impulse loading I 0.001 . Also, the response time for core compression almost increases with the increase of thickness of face sheet h under impulse loading I 0.0035 , and it almost keeps the constant under impulse loading I 0.001 .
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Fig. A4 shows the effect of pulse of load I on the response time Tc for core compression of
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double-layer sandwich plates under impulsive loading, in which 0.01 . The double-layer sandwich plates have face sheet thickness h 0.0032m , interlayer sheet thickness hm 0.0016m ,
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width B 0.5m and length L 1m . The response time for core compression increases with increase of pulse of load, and then decreases with increase of pulse of load. Therefore, the optimal
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design of the double-layer sandwich structure may result in better resistance to the blast loading.
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Appendix B. Yield criterion for the compressed double-layer sandwich cross-sections The appendix contains the expressions of the yield criterion for the compressed double-layer
sandwich cross-sections based on core layer-by-layer compression assumption. Similar to Zhang et al. [38], the compressed yield criterion of the double-layer sandwich cross-section in the generalized stress space (N, M) can be derived based on core layer-by-layer compression assumption. For the sake of brevity, the derivation is omitted and the expression of the compressed yield criterion of the double-layer sandwich cross-section is 14
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(A1)
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2 2 2h 1 1 m C 1 c1 2h n 1 c1 c1 c 2 0, A 1 2 A 2 h 1 1 n 2h 1 2 1 4 h 2 2 2h 1 c1 2h 2hn n 2 m A C 1 1 c1 1 1 1 c 1 2 h 1 2 h + 1 4h 1 0, n c 1 c 2 A 2 2h 1 2h 1 1 2 2h 2 2 h 2 h n 1 0, n m A 2h 1 2h 1 2 2h 1 c 2 1 1 2h 1 c 2 n m C c 2 c1 A 2 1 1 c 2 2 2 h 1 2h 2 h n A 1 1 1 c 2 0, c2 2h 1 2h 1 2 2 1 C 1 2h 2h 1 c 2 1 n 2h m 1 c 2 A 2 2 1 2 h 1 + 1 h c 2 c1 1 1 2h 1 2 0, n 1 c2 c1 c2 A 1 2 2 2 h 1
where
4h 1 1 2h c1 c 2 M N , n , A 4h 2 , c1 c 2 1 2 1 Mp Np
C
2h 2h 1 2 2 c1 c 2 , 1 1
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m
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in which the fully plastic axial force N p and bending moment M p are
N p f 2h hm 2 c c
(A2)
and
2
h 1 M p f h m 2 f hc c hm c c c 2 c1 c 2 f hc c1 c 2 . 2 2
(A3)
The predicted yield surface ( N N p , M M p ) for the metal double-layer sandwich
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cross-section with different core compressive strain c1 and c 2 based on core layer-by-layer compression assumption are shown in Figs. B1(a) and B1(b), respectively. It is seen that the present yield criterion can be applied to double-layer sandwich structures with different core compressive strain, core strength, face sheet and interlayer sheet materials and geometries.
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Appendix C. Inscribing coefficient for compressed double-layer sandwich cross-sections Based on the core layer-by-layer assumption, the inscribing coefficient is given by 1 4k0 1 , 2k0 2 k 4k2 k1 1 , 2 k 2 4k k 4 3 3 , 2
J1 0
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J1 0 and J 2 0 , J2 0
where
k1
M
A
2
,
4h 1 c 2 1 c 2 2h A 1 c 2 1
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k0
2h
2h 1 1 2
2
,
c2
2 2h 1 c 2 +A
AC
k3
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2 2h c 2 c1 2 2 1 c2 C 1 c 2 1 4h 2 2 1 c 2 2 1 , k2 2 2 2 2 h 1 1 c2
k4
2h
2
,
2h 2 2 c 2 2h c 2 c1 1 c 2 c1 2 , 2 2h 2h 2
2 h 4h 2 J1 1, 2h A 1 J2
2 h 1
2h
4h 2 1 2 2h 1 2 c1 c 2 c 2 c1 1 2 . A 1 2
16
(B1)
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[5] Yin S, Wu LZ, Yang JS, Ma L, Nutt S. Damping and low-velocity impact behavior of filled composite pyramidal lattice structures. J Sandw Struct Mater 2017; 19(5): 525-543.
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[6] Lou J, Wu LZ, Ma L, Xiong J, Wang B. Effects of local damage on vibration characteristics of
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composite pyramidal truss core sandwich structure. Compos B 2014; 62: 73-87. [7] Zhou HY, Ma GW, Li JD, Zhao ZY. Design of metal foam cladding subjected to close-range
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blast. J Perform Constr Facil 2014; 29: 04014110. [8] Sun YL, Li QM. Dynamic compressive behaviour of cellular materials: A review of
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phenomenon, mechanism and modelling. Int J Impact Eng 2018; 112: 74-115. [9] Zhang JX, Qin QH, Yang Y, Yu XH, Chen SJ, Wang TJ. Large-deflection bending of clamped
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metal foam-filled rectangular tubes. Int J Appl Mech 2017; 9(3): 1750043. [10] Li ZB, Zheng ZJ, Yu JL, Qian CQ, Lu FY. Deformation and failure mechanisms of sandwich beams under three-point bending at elevated temperatures. Compos Struct 2014; 111: 285-290. [11] Fan HL, Zhou Q, Yang W, Zheng JJ. An experiment study on the failure mechanisms of woven textile sandwich panels under quasi-static loading. Compos B 2010; 41: 686-692. [12] Fan HL, Yang W, Zhou Q. Experimental research of compressive responses of multi-layered woven textile sandwich panels under quasi-static loading. Compos B 2011; 42: 1151-1156. [13] Dharmasena K, Queheillalt D, Wadley H, Chen Y, Dudt P, Knight D, Wei Z, Evans A. Dynamic 17
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response of a multilayer prismatic structure to impulsive loads incident from water. Int J Impact Eng 2009; 36: 632-643. [14] Fleck NA, Deshpande VS. The resistance of clamped sandwich beams to shock loading. ASME J Appl Mech 2004; 71: 386-401. [15] Qiu X, Deshpande VS, Fleck NA. Dynamic response of a clamped circular sandwich plate subject to shock loading. ASME J Appl Mech 2004; 71: 637-645.
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[16] Qin QH, Wang TJ. A theoretical analysis of the dynamic response of metallic sandwich beam under impulsive loading. Eur J Mech A/Solids 2009; 28: 1014-1025.
[17] Zhang JX, Qin QH, Wang TJ. Compressive strength and dynamic response of corrugated metal sandwich plates with unfilled and foam-filled sinusoidal plate cores. Acta Mech 2013; 224:
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759-775.
[18] Zhu F, Wang ZH, Lu GX, Nurick G. Some theoretical considerations on the dynamic response of sandwich structures under impulsive loading. Int J Impact Eng 2010; 37: 625-637. [19] Cui XD, Zhao LM, Wang ZH, Zhao H, Fang DN. A lattice deformation based model of
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metallic lattice sandwich plates subjected to impulsive loading. Int J Solids Struct 2012; 49: 2854-2862.
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[20] Qin QH, Wang TJ. An analytical solution for the large deflections of a slender sandwich beam with a metallic foam core under transverse loading by a flat punch. Compos Struct 2009; 88:
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509-518.
[21] Qin QH, Yuan C, Zhang JX, Wang TJ. Large deflection response of rectangular metal sandwich
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plates subjected to blast loading. Eur J Mech A/Solids 2014, 47: 14-22. [22] Zhu F, Zhao LM, Lu GX, Wang ZH. Structural response and energy absorption of sandwich
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panels with an aluminium foam core under blast loading. Adv Struct Eng 2008; 11: 525-536. [23] Cui XD, Zhao LM, Wang ZH, Zhao H, Fang DN. Dynamic response of metallic lattice sandwich structures to impulsive loading. Int J Impact Eng 2012; 43: 1-5. [24] Kılıçaslan C, Odacı İK, Güden M. Single- and double-layer aluminum corrugated core sandwiches under quasi-static and dynamic loadings. J Sandw Struct Mater 2016; 18(6): 667-692. [25] Süsler S, Türkmen HS. Kazancı Z. Nonlinear dynamic analysis of tapered sandwich plates with multi-layered faces subjected to air blast loading. Int J Mech Mater Des 2017; 13: 429-451. [26] Wang ZQ, Zhou YB, Wang XH, Zhang XL. Multi-objective optimization design of a 18
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multi-layer honeycomb sandwich structure under blast loading. Proc Inst Mech Eng D-J Automob Eng 2017; 231(10): 1449-1458. [27] Xiong J, Vaziri A, Ma L, Papadopoulos J, Wu LZ. Compression and impact testing of two-layer composite pyramidal-core sandwich panels. Compos Struct 2012; 94: 793-801. [28] Al-Shamary AKJ, Karakuzu R, Özdemir O. Low-velocity impact response of sandwich composites with different foam core configurations. J Sandw Struct Mater 2016; 18(6): 754-768.
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[29] Chen HL, Zheng Q, Wang P, Fan HL, Zheng JJ, Zhao L, Jin FN. Dynamic anti-crushing behaviors of woven textile sandwich composites: Multilayer and gradient effects. J Compos Mater 2015; 49(25): 3169-3179.
[30] Chen LM, Du B, Zhang J, Zhou H, Li DS, Fang DN. Numerical study on the projectile impact
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resistance of multi-layer sandwich panels with cellular cores. Lat Am J Solids Struct 2016; 13: 2876-2895.
[31] Cao BT, Hou B, Li YL, Zhao H. An experimental study on the impact behavior of multilayer sandwich with corrugated cores. Int J Solids Struct 2017; 109: 33-45.
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[32] Qiu X, Deshpande VS, Fleck NA. Impulsive loading of clamped monolithic and sandwich beams over a central patch. J Mech Phys Solids 2005; 53: 1015-1046.
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[33] Jones N. A theoretical study of the dynamic plastic behavior of beams and plates with finite-deflections. Int J Solids Struct 1971; 7: 1007-1029.
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[34] Jones N. Structural impact. Cambridge: Cambridge University; 1989. [35] Qin QH, Wang TJ. Impulsive loading of a fully clamped circular metallic foam core sandwich
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plate. Proceedings of the 7th International Conference on Shock & Impact Loads on Structures, Beijing, China; 2007, p. 481-488.
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[36] Deshpande VS, Fleck NA. Isotropic constitutive models for metallic foams. J Mech Phys Solids 2000; 48: 1253-1283. [37] Ashby MF, Evans AG, Fleck NA, Gibson LJ, Hutchinsion JW, Wadley HNG. Metal foams: a design guide. Oxford: Butterworth Heinemann; 2000. [38] Zhang JX, Qin QH, Xiang CP, Wang TJ. Plastic analysis of multilayer sandwich beams with metal foam cores. Acta Mech 2016; 227(9): 1-15.
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Figure Captains Fig. 1. Sketch of a fully clamped double-layer rectangular sandwich plate subjected to blast loading. Fig. 2. Plastic hinge line pattern and the transverse displacement fields in a fully clamped rectangular plate. Fig. 3. Sketch of the approximate yield locus of the compressed sandwich cross-section ( 1 ,
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0.01 , h =0.1 , c =0.02 , =0.1 ) based on core layer-by-layer compression ( c1 0.5 , c 2 0 ).
Fig. 4. Analytical solutions and numerical results for (a) the maximum central deflection Wm of
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the bottom face sheet and (b) structural response time T as a function of the impulse I for the double-layer rectangular sandwich plates ( 1 , 2 , 0.01, h =0.1 , c =0.02 , =0.1). Fig. 5. Analytical solutions and numerical results for (a) the maximum central deflection Wm of the bottom face sheet and (b) structural response time T as a function of the impulse I for the
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double-layer square sandwich plates ( 1 , 1 , 0.01 , h =0.1 , c =0.02 , =0.1).
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Fig. 6. The effect of the double-layer factor on maximum central deflection Wm of the
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double-layer sandwich plates versus impulse I ( 2 , 0.01 , h =0.1 , c =0.02 , =0.1). Fig. 7. Comparisons of the maximum central deflection Wm of bottom face sheet versus impulse
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I of rectangular double-layer sandwich plate and solid monolithic plate with the same mass.
Fig. 8. Numerical results of the maximum central deflection Wm of bottom face sheet versus
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impulse I of rectangular double-layer sandwich plates, monolayer sandwich plates and solid monolithic plates with the same mass in the low impulse. Fig. A1. The time period of the core compression and overall structural response time of the double-layer sandwich plates under the impulse loading (a) I =0.001, and (b) I =0.0035 . Fig. A2. The ratio of time period of the core compression to overall structural response time of the double-layer sandwich plates under the impulse loading.
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Fig. A3. The relations between the response time of the double-layer sandwich plates for core compression and the parameters. (a) Core strength, (b) core height, and (c) thickness of face sheet. Fig. A4. The response time of the double-layer sandwich plates for core compression under different pulse of load. Fig. B1. The yield surface based on core layer-by-layer compression assumption with (a) core strain
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( 1 , 0.01 , h =0.1 , c =0.02 ) and (b) double-layer factor ( 0.01 , h =0.1 , c =0.02 ,
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c1 D , c 2 0.2 ).
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Fig. 1. Sketch of a fully clamped double-layer rectangular sandwich plate subjected to blast loading.
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Fig. 2. Plastic hinge line pattern and the transverse displacement fields in a fully clamped
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rectangular plate.
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Fig. 3. Sketch of the approximate yield locus of the compressed sandwich cross-section ( 1 ,
0.01 , h =0.1 , c =0.02 , =0.1 ) based on core layer-by-layer compression ( c1 0.5 ,
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c 2 0 ).
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Fig. 4. Analytical solutions and numerical results for (a) the maximum central deflection Wm of the bottom face sheet and (b) structural response time T as a function of the impulse I for the double-layer rectangular sandwich plates ( 1 , 2 , 0.01, h =0.1 , c =0.02 , =0.1).
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(a)
(b)
Fig. 5. Analytical solutions and numerical results for (a) the maximum central deflection Wm of the bottom face sheet and (b) structural response time T as a function of the impulse I for the double-layer square sandwich plates ( 1 , 1 , 0.01 , h =0.1 , c =0.02 , =0.1).
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Fig. 6. The effect of the double-layer factor on maximum central deflection Wm of the
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double-layer sandwich plates versus impulse I ( 2 , 0.01 , h =0.1 , c =0.02 , =0.1).
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Fig. 7. Comparisons of the maximum central deflection Wm of bottom face sheet versus impulse
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I of rectangular double-layer sandwich plate and solid monolithic plate with the same mass.
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Fig. 8. Numerical results of the maximum central deflection Wm of bottom face sheet versus impulse I of rectangular double-layer sandwich plates, monolayer sandwich plates and solid
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monolithic plates with the same mass in the low impulse.
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(a)
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Fig. A1. The time period of the core compression and overall structural response time of the double-layer sandwich plates under the impulse loading (a) I =0.001, and (b) I =0.0035 .
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Fig. A2. The ratio of time period of the core compression to overall structural response time of the
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double-layer sandwich plates under the impulse loading.
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(b)
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Fig. A3. The relations between the response time of the double-layer sandwich plates for core
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compression and the parameters. (a) Core strength, (b) core height, and (c) thickness of face sheet.
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Fig. A4. The response time of the double-layer sandwich plates for core compression under
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different pulse of load.
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(a)
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Fig. B1. The yield surface based on core layer-by-layer compression assumption with (a) core strain ( 1 , 0.01 , h =0.1 , c =0.02 ) and (b) double-layer factor ( 0.01 , h =0.1 , c =0.02 ,
c1 D , c 2 0.2 ). 34
Dynamic response of double-layer rectangular sandwich plates with metal foam cores subjected to blast loading Jianxun Zhang , Renfang Zhou , Mingshi Wang , Qinghua Qin , Yang Ye , T.J. Wang PII: DOI: Reference:
S0734-743X(17)31074-6 https://doi.org/10.1016/j.ijimpeng.2018.08.016 IE 3161
To appear in:
International Journal of Impact Engineering
Received date: Revised date: Accepted date:
8 December 2017 12 July 2018 29 August 2018
Please cite this article as: Jianxun Zhang , Renfang Zhou , Mingshi Wang , Qinghua Qin , Yang Ye , T.J. Wang , Dynamic response of double-layer rectangular sandwich plates with metal foam cores subjected to blast loading, International Journal of Impact Engineering (2018), doi: https://doi.org/10.1016/j.ijimpeng.2018.08.016
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Highlights
The analytical model for dynamic response of double-layer sandwich plates under blast loading is developed. The so-called ‘bounds’ are obtained by using approximate yield criteria.
Membrane mode solutions are obtained for dynamic response.
The analytical model captures numerical results reasonably.
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Dynamic response of double-layer rectangular sandwich plates with metal foam cores subjected to blast loading Jianxun Zhang, Renfang Zhou, Mingshi Wang, Qinghua Qin*, Yang Ye, T. J. Wang State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace,
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Xi’an Jiaotong University, Xi’an 710049, China
Abstract. In this paper, the analytical and numerical analyses are conducted to predict the dynamic response of fully clamped double-layer rectangular sandwich plates with metal foam cores
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subjected to blast loading. The so-called ‘bounds’ of analytical solutions for dynamic response of double-layer sandwich plates are obtained by using the inscribing and circumscribing squares of the exact yield locus. Also, the membrane mode solution for large deflection of the double-layer sandwich plates is presented. The effect of the double-layer factor on the dynamic response of double-layer sandwich plates is also discussed. The finite element calculation is carried out to study
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the dynamic response of the double-layer sandwich plates subjected to blast loading. Good
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agreement is achieved between the analytical predictions and numerical results. It is demonstrated that the impact resistance of the double-layer sandwich plate is better than that of the monolayer one
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with the same mass subjected to the higher impulse.
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Keywords: Double-layer sandwich plate; Yield criterion; Core layer-by-layer compression;
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Dynamic response; Large deflection.
1. Introduction
Sandwich structures are widely used in plenty of engineer fields due to the various advantages,
such as aircraft, aerospace, high speed train, and ship. Sandwich structures are consisted of stiff and strong face sheets and lightweight cores. To satisfy different requirements, several kinds of lightweight cores are developed, for example, honeycomb [1, 2], corrugated core [3], pyramidal *
Corresponding author. [email protected] (Q.H. Qin), Tel/Fax: +86-29-82664382. 2
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truss [4-6], metal foam [7-10] and woven material [11]. Compared with monolayer sandwich structures, investigations have proven that multilayer sandwich structures provide much more choices for design of structures [12,13]. In the past decades, the dynamic response of monolayer sandwich structures has been investigated extensively. Fleck and Deshpande [14] developed an analytical model for the dynamic response of the fully clamped sandwich beam subjected to uniform blast loading. Qiu et al. [15]
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obtained the so-called ‘bounds’ of analytical solutions for dynamic response of the clamped circular sandwich plates subjected to blast loading. Qin and Wang [16] obtained the analytical solutions for the impulsive response of fully clamped metal sandwich beams by using the membrane factor method. Zhang et al. [17] developed a plastic-string model to predict the impulsive response of
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corrugated metal sandwich plates with unfilled and foam-filled sinusoidal plate cores. Zhu et al. [18] and Cui et al. [19] obtained the so-called ‘bound’ solutions for the dynamic response of square sandwich plates subjected to blast loading. Employing the yield criterion including the effect of core strength [20], Qin et al. [21] obtained the analytical solutions for the dynamic response of fully
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clamped rectangular sandwich plates subjected to blast loading. Zhu et al. [22] and Cui et al. [23] experimentally studied the deformation and failure of metal sandwich plates with aluminum foam
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core and pyramidal lattice cores subjected to air blast loading, respectively. Comparing to monolayer sandwich structures, there were few investigations on impact
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response of multilayer sandwich structures. Dharmasena et al. [13] experimentally and numerically studied the dynamic response of a multilayer sandwich plates with corrugated cores subjected to
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impulsive loads incidenting from water. The results show that the multilayer sandwich plates significantly reduced the transmitted pressures of an impulsive load. Kılıçaslan et al. [24]
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experimentally and numerically investigated the crushing of single- and double-layer aluminum trapezoidal corrugated core sandwich panels at quasi-static and dynamic impact loading. Süsler et al. [25] analytically and numerically investigated the nonlinear dynamic behavior of simply supported tapered sandwich plates with multilayer face sheets subjected to air blast loading, and analytical results are in good agreement with numerical results. Wang et al. [26] numerically studied the dynamic response of a multilayer honeycomb sandwich structure under blast loading, and obtained the multi-objective optimization design that can apparently enhance the shielding performance of the protective structure without increasing the mass. Xiong et al. [27] carried out the low-velocity 3
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impact tests to study failure mechanisms and energy absorption of two-layer carbon fiber composite sandwich plates with pyramidal truss cores. Al-Shamary et al. [28] carried out drop weight impact experiment to investigate the low-velocity impact response of one-layer, two-layer and three-layer composite sandwich plates. The three-layer sandwich plate exhibits high energy absorption compared with the one-layer sandwich plate. Chen et al. [29] experimentally studied the low-velocity compression behavior, failure mechanism and energy absorption of multilayer woven
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lattice sandwich panels. Relative to quasi-static compressed panel, the impacted woven textile sandwich panel has much greater energy absorption induced by greater dynamic strength. Chen et al. [30] performed the projectile impact resistance of multilayer sandwich plates with corrugation, hexagonal honeycomb and pyramidal truss cores. At low impact velocity, multiplying layer can
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reduce the maximum central deflection of bottom face sheet of three sandwich plates except pyramidal truss ones in high relative density. Cao et al. [31] experimentally and numerically investigated the dynamic response of multilayer corrugated core sandwich plates using impact Hopkinson bar under out-of-plane compressive impact loading. It was found the bending of the
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interlayer plate contributes to reduce the force oscillation during successive folding. There is little work on the analytical solutions for impact response of double-layer sandwich
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plates under blast loading. The work of this paper is to investigate the dynamic response of fully clamped double-layer rectangular sandwich plates subjected to blast loading. This paper is
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organized as follows. In section 2, analytical analyses for the impulsive response of a fully clamped double-layer rectangular sandwich plate are presented. In section 3, employing the yield criteria for
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double-layer sandwich structures, analytical solutions are derived to predict the dynamic response of fully clamped double-layer sandwich plates subjected to blast loading. In section 4, finite element
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analysis is carried out. In section 5, analytical predictions are compared with numerical results, and the effect of the double-layer factor on the dynamic response of the double-layer sandwich plates is discussed. Conclusions are presented in section 6. 2. Statement of the problem Consider a fully clamped double-layer rectangular metal foam core sandwich plate with length 2L and width 2B subjected to blast loading of uniformly distributed impulse I per unit area, as shown in Fig. 1. Two identical top and bottom face sheets with thickness h and interlayer sheet with 4
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thickness hm are assumed to perfectly bond to the identical top and bottom metal foam cores with thickness c. It is assumed that the face sheets and interlayer sheets are made from rigid-perfectly plastic
material
with
yield
strength
f .
The
foam
cores
are
idealized
as
a
rigid-perfectly-plastic-locking ( r p p l ) material with plateau stress c and densification strain
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D . To describe the relation between top and bottom face sheets and interlayer sheet, a double-layer factor is defined as hm 2h .
The analytical model for dynamic response of fully clamped metallic sandwich beams subjected to blast loading over the entire span was developed by Fleck and Deshpande [14]. They
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estimated the time of the monolayer sandwich structure by analyzing a plastic shock wave propagating through the metal foam core. It should be noted that the assumption of three-stage deformation is largely depended on the ratio of time between the core compression and overall deformation of the sandwich plate. For the double-layer sandwich structure, the response time for core compression may depend on several parameters, such as core mass, core strength, core height,
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thickness of face sheet and pulse of load. In general, the time period of the core compression is at
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least 6 times smaller and for most of cases 10 times smaller than the overall structural response time of the sandwich structure. The detailed discussion of it can be found in Appendix A. Then, the
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coupling between the core compression and beam bending and stretching can be decoupled. In the core compression phase it is assumed that there is momentum transferred from the top face sheet to
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the core and bottom face sheet, and no momentum is transferred to any other portion of the beam. Then, any bending effects are neglected. This phase can be reduced to the one-dimensional core
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compression problem. See more details in Refs. [14, 32]. Here, this assumption is adopted to investigate the impulsive response of the double-layer
sandwich plates.
2.1. Core compression stage It is assumed that the impulse I per unit area imparts to the top face sheet with a velocity V0 = I f h . According to the momentum conservation law, the final common velocity V f of the
sandwich plate including two cores and two face sheets and interlayer sheet at the end of core 5
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compression stage is given by
I
Vf =
(1)
2 f h f hm +2c c
Then, neglecting the rate effect, it is assumed that the absorb energy U a for unit area is dissipated in the process of core compression, and the kinetic energy loss Ua can be expressed as
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h 1 2 1 Ua 2h I 2 2 f h where
hm 2h , c . 4c f
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h
(2)
The core layer-by-layer compression is assumed in the core compression stage. First, the impact energy is dissipated by the top core till the top foam core is densified. If the impact energy is not dissipated by the top core, the residual energy would be dissipated by the bottom core till the
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4c 2 h 2h D , the bottom core bottom core attains densification. When I I 0 I 0 h 1 2 1 2
I2
(3)
I
L f f
, =
c c , c . L f
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I
4c h 2h 2
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where
h 1 2 1 2
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c1
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keeps the initial situation while the top core is compressed to the strain c1 ,
When I I 0 , the top core attains the densification strain c1 D , and the bottom core is compressed to the strain c 2
c2 I 2
h 1 2 1 2 4c 2 h 2h
D
(4)
2.2. Bending and stretching phase At the end of the core compression phase, the top and bottom face sheets, the interlayer sheet, 6
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and the top and bottom core have a common velocity V2, and then the sandwich plate is brought to rest by plastic bending and stretching. Jones [33] developed an approximate theoretical approach to analyze the dynamic plastic behavior of monolithic rectangular plates considering influence of finite deflection. It is assumed that the initially rectangular flat plate is divided into four rigid regions which are separated by r straight hinge lines with length l j , as sketched in Fig. 2. Based on the
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principle of energy dissipation rate balance, the governing equation is given by r
p w wdA Nw M j dl j j 1 l j
A
(5)
where the external pressure p is applied on the whole area A of the plate uniformly, is mass
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per unit area of the sandwich plate, w is transverse deflection, w is transverse velocity, w is transverse accelerate, j is the relative angular velocity across the hinge line l j , and N and M are the membrane force and bending moment per unit length, respectively. Herein, this procedure is employed to study the dynamic response of the fully clamped double-layer sandwich plate
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subjected to the initial velocity.
The double-layer rectangular sandwich plate is assumed to be divided into four regions by nine
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straight hinge lines, and the transverse displacement profile is assumed to be linear. The front and side views of the transverse displacement fields are as follows,
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y 1 B W0 , w 1 x W , 0 B tan
Region
(6)
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Region
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as shown in Fig. 2, where W0 is the transverse displacement of the center of the plate, is determined by the upper bound theory and = arctan
3 B L B L [34], x is the 2
horizontal coordinate and y is the vertical coordinate. 3. Analytical solutions 3.1. The so-called ‘bounds’ of the analytical solutions The yield criterion for compressed double-layer sandwich cross-section is presented in Appendix B. Circumscribing yield locus can be expressed as 7
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M M p , N Np
(7)
Inscribing yield locus can be expressed as
M M p , N N p
(8)
where the inscribing coefficient factor is given in Appendix C, as shown in Fig. 3. Taking the similar analysis in Jones [34], combining Eqs. (5), (7) and (8), the non-dimensional
double-layer rectangular sandwich plate are given by
Wm
Wm a1I L 2 c 3a3a4b1b2
1 2
2 Aa c 1 2 1 b1a3 1 2
T
T f 1 a4b2 arctan L f 3a3b1
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and
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maximum central deflection of the bottom face sheet and structural response time of the
(9)
(10)
for the choice of circumscribing yield locus, and
Wm a1I L 2 c 3a3a4b1b2 2
2a Ac 2 1 a3b1 2
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Wm and
(11)
T f 1 a4b2 arctan L f 3a3b1
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T
(12)
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for the choice of inscribing yield locus, where L B , a1 3 tan , a2 cot ,
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a3 2 cot tan , a4 2 tan , b1 2h , b2 2h , =
a1 I 4 a2 Ac 2
a3b1 . 3a4b2
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3.2. Membrane mode solution The effect of bending moment decreases with the increasing the deflection, while the structural
response is mainly determined by membrane force only in theoretical analysis for the dynamic response of circular monolithic plate and circular sandwich plate [34, 35]. Here, the analytical method is extended to study the large deflections of rectangular double-layer sandwich plates subjected to blast loading. Herein, neglecting the bending moment M and letting N = N p in Eq. (5), the membrane 8
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mode solutions for the non-dimensional maximum central deflection Wm of the bottom face sheet and structural response time T of double-layer sandwich plates are given by Wm
Wm a1 I L 2 3a3a4b1b2 c
(13)
and
T f L f 2 3
a4b2 a3b1
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T
4. Finite element analysis
(14)
Using the ABAQUS / Explicit software, the dynamic response of the double-layer sandwich
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plates is numerically analyzed subjected to blast loading. The double-layer sandwich plate is modelled using 8-node brick element with reduced integration (C3D8R). All the displacements are fully clamped at the ends of the sandwich beams. A uniformly distributed velocity field applied on the top face sheet is predefined. Mesh sensitivity study reveals that further refinements do not
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improve the accuracy of the calculations appreciably.
The top and bottom face sheets and interlayer sheets obey J 2 plastic flow theory and
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Deshpande-Fleck model [36] is used to describe the plastic crushable behavior of the metal foam core implemented in ABAQUS, which allows the shape change of yield surface due to differential
where
1
1 3
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ˆ 2
(15)
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ˆ c 0
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hardening along the hydrostatic and deviatoric axes. The yield function for the foam core is
2
2 e
2 m2
(16)
with e 3sij sij 2 being von Mises effective stress, sij the deviatoric stress, m kk 3 the mean stress and the shape factor of yield surface. The associated plastic flow rule is adopted and the plastic Poisson’s v p is given by
p 1 2 3 p p 2 1 3
2
(17)
22
11
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Face sheets and interlayer sheets are made from aluminum alloy with Young’s modulus E ft 70 GPa , yield strength f 200 MPa , Poisson’s ratio f 0.3 , density f 2700 kg/m3
and linear strain hardening modulus E f t 330MPa . Following the data of Ashby et al. [37], the aluminum foam core has Young’s modulus Ec 1GPa , yield strength c 2MPa , density
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c 270kg/m3 , relative density c f 0.1 , elastic Poisson’s ratio vce 0.3 , plastic Poisson’s ratio vcp 0 . The foam core has a long plateau-stress platform c that continues up to the densification strain D 0.7 , and the stress rises steeply with a large tangent modulus
Ect 0.5GPa beyond the densification strain. Geometrical parameters of the double-layer
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sandwich plates are as follows: length L 1m , core thickness c 0.02m , face sheet thickness
h 0.002m , interlayer sheet thickness hm 0.004m , i.e. =0.01 , c =0.02 , h =0.1 , width B 0.5m for rectangular sandwich plates and width B 1m for square sandwich plates are
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considered, i.e. =2 and =1 .
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5. Results and discussion
Analytical solutions and numerical results for the dynamic response of the rectangular ( 2 )
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and square ( 1 ) metal double-layer sandwich plates subjected to blast loading are shown in Figs.
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4 and 5, respectively. The so-called ‘bounds’ for the maximum central deflection Wm of the bottom face sheet and structural response time T of metal double-layer sandwich plates versus impulse
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I are shown in Figs. 4 and 5, respectively. Also, the membrane mode solutions for structural
response of double-layer sandwich plates are given. It is seen that the analytical predictions derived by employing the circumscribing and inscribing yield locus bound the numerical results well, and membrane mode solutions are in good agreement with numerical results. The analytical model neglects the effects of elasticity, strain hardening and strain rates of materials, wrinkling of face sheets, shear force, and reduction of the momentum provided by the supports in the core compression phase. For higher impulsive loading, the discrepancy may be due to the assumption of full densification of the core in analytical solutions while there is no distinct core densification in 10
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the numerical results. Analytical solutions for the structural response time T versus impulse I agree well with numerical results, as shown in Figs. 4(b) and 5(b), respectively. The effect of the double-layer factor on the maximum central deflection Wm of bottom face sheet versus impulse I of rectangular double-layer sandwich plates is shown in Fig. 6. Analytical prediction derived by employing the inscribing yield locus is employed in Figs. 6 and 7.
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It is seen that in the lower impulse, the bigger the double-layer factor is, the bigger the maximum central deflection of the double-layer sandwich plates is. In the higher impulse I 0.0017 , the maximum central deflection for the case of 0 is bigger than the other cases.
It means that the impact resistance of double-layer sandwich plates is better than that of monolayer
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sandwich plate in the higher impulse I 0.0017 .
Comparisons of the maximum central deflection Wm of bottom face sheet versus impulse I of the rectangular double-layer sandwich plate and solid monolithic plate with the same mass are
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shown in Fig. 7. The sandwich plate has 0.5 , 0.01, 2 , h =0.1 , c =0.02 , =0.1, and the density, length and width of solid monolithic plate are same as those of the sandwich plate.
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It is seen that the maximum central deflection of double-layer sandwich plate is smaller than that of solid monolithic plate when I 0.00125 , while the maximum central deflection of double-layer
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sandwich plate is bigger than that of solid monolithic plate when I 0.00125 . It means that the
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double-layer sandwich plate performs better impact resistance than the solid monolithic plate under high impulsive loading I 0.00125 .
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Numerical results of the maximum central deflection Wm of bottom face sheet versus impulse
I of rectangular double-layer sandwich plates, monolayer sandwich plates and solid monolithic
plates with the same mass in the low impulse are shown in Fig. 8. The double-layer sandwich plates have face sheet thickness h 0.0032m and interlayer sheet thickness hm 0.0016m . Three kinds of plates have width B 0.5m and length L 1m . The core height and top/bottom core height are identical for the monolayer and double-layer sandwich plates. It is seen that the maximum central deflection of double-layer sandwich plates is smaller than those of monolayer sandwich 11
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plates and solid monolithic plates with the same mass in the low impulse. Therefore, the double-layer sandwich plate performs better impact resistance than the monolayer sandwich plate and solid monolithic plate with same mass in the low impulse range. 6. Conclusions A theoretical analysis was conducted to predict the dynamic response of fully clamped
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rectangular double-layer sandwich plates with metal foam cores subjected to blast loading. The compressed yield criterion of double-layer sandwich cross-sections was given based on core layer-by-layer compression assumption. Employing the inscribing and circumscribing yield loci, the so-called ‘bounds’ of the analytical solutions for the dynamic response of double-layer sandwich
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plates were obtained. Also, membrane mode solutions were obtained for large deflections of the double-layer sandwich plates. The present analytical predictions agree well with the numerical results. The results show that the impact resistance of the double-layer sandwich plate is better than that of the monolayer sandwich plate with the same mass subjected to the higher impulse
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I 0.0017 , while the double-layer sandwich plate performs better impact resistance than the solid
monolithic plate with same mass under high impulsive loading I 0.00125 . It can be concluded
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that the membrane force plays an important role in large deflection of double-layer rectangular sandwich plates.
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It should be noted that the assumption that "the coupling between the core compression, and beam bending and stretching are decoupled" may be mandatory and simplified for double-layer
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sandwich plates. The good agreement between the present analytical predictions and FE results may indicate that the assumption used in derivation of the analytical model are appropriate to some
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extent. A detailed experimental study of the clamped rectangular double-layer sandwich plates with metal foam cores subjected to blast loading is necessary to fully validate the model developed and it will be one of our future work.
Acknowledgments. The authors gratefully acknowledge the financial supports of NSFC (11502189, 11572234, 11321062, and 11372235), Natural Science Basic Research Plan in Shaanxi Province of China (2017JM1020), China Postdoctoral Science Foundation funded project (2015M572546), and 12
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the Fundamental Research Funds for the Central Universities.
Appendix A. Discussion on response time for core compression Fig. A1 shows the numerical results for time period of the core compression and overall structural response time of the sandwich structure under the low impulse loading I =0.001 and
t f L f
VL f . The double-layer c f
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high impulse loading I =0.0035 , in which t
and V =
sandwich plates have face sheet thickness h 0.0032m , interlayer sheet thickness hm 0.0016m ,
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width B 0.5m and length L 1m . In the case of low impulse loading I =0.001, the velocity of top face sheet decreases, and the velocities of bottom face sheet and interlayer sheet increase till velocities of top, bottom face sheet and interlayer sheet keep almost same. In the case of high impulse loading I =0.0035 , first the velocity of top face sheet decreases and the velocities of
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bottom face sheet and interlayer sheet increase, and then the velocities of top face sheet and interlayer sheet keep nearly same. At last, the velocities of top, bottom face sheet and interlayer
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sheet keep almost same. Fig. A2 shows the ratio k of time period of the core compression to overall structural response time of the double-layer sandwich plates under the impulse loading using the
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Finite element analysis. It is seen that the ratio k is smaller than 16%. In other words, the time period of the core compression is at least 6 times smaller and for most of cases 10 times smaller
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than overall structural response time of the sandwich structure. The assumption in the analytical solutions is verified.
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The response time for core compression strongly depends on core strength, core height,
thickness of face sheet and pulse of load. The discussion details how these parameters affect the results using the Finite element analysis are presented as follows. Fig. A3(a) shows the effect of core strength on the response time Tc
Tc f L f
for core compression of double-layer
sandwich plates under impulsive loading I 0.001 . The double-layer sandwich plates have face sheet thickness h 0.0032m , interlayer sheet thickness hm 0.0016m , width B 0.5m and 13
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length L 1m . It is seen that the response time for core compression almost decreases with the increase of core strength for the cases of I 0.001 and I 0.0035 , except for the case of the small core strength under impulse loading I 0.0035 . The effects of core height c and thickness of face sheet h on the response time Tc for
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core compression of double-layer sandwich plates under impulsive loading I 0.001 are shown in Figs. A3(b) and (c), respectively. The double-layer sandwich plates have interlayer sheet thickness, width B 0.5m and length L 1m . In Fig. A3(b), the face sheet thickness h 0.002m , and the core height c 0.02m in Fig. A3(c). It is seen that the response time for core compression almost increases with the increase of core height for two cases, while almost keeps the constant when
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c 0.02 under impulse loading I 0.001 . Also, the response time for core compression almost increases with the increase of thickness of face sheet h under impulse loading I 0.0035 , and it almost keeps the constant under impulse loading I 0.001 .
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Fig. A4 shows the effect of pulse of load I on the response time Tc for core compression of
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double-layer sandwich plates under impulsive loading, in which 0.01 . The double-layer sandwich plates have face sheet thickness h 0.0032m , interlayer sheet thickness hm 0.0016m ,
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width B 0.5m and length L 1m . The response time for core compression increases with increase of pulse of load, and then decreases with increase of pulse of load. Therefore, the optimal
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design of the double-layer sandwich structure may result in better resistance to the blast loading.
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Appendix B. Yield criterion for the compressed double-layer sandwich cross-sections The appendix contains the expressions of the yield criterion for the compressed double-layer
sandwich cross-sections based on core layer-by-layer compression assumption. Similar to Zhang et al. [38], the compressed yield criterion of the double-layer sandwich cross-section in the generalized stress space (N, M) can be derived based on core layer-by-layer compression assumption. For the sake of brevity, the derivation is omitted and the expression of the compressed yield criterion of the double-layer sandwich cross-section is 14
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(A1)
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2 2 2h 1 1 m C 1 c1 2h n 1 c1 c1 c 2 0, A 1 2 A 2 h 1 1 n 2h 1 2 1 4 h 2 2 2h 1 c1 2h 2hn n 2 m A C 1 1 c1 1 1 1 c 1 2 h 1 2 h + 1 4h 1 0, n c 1 c 2 A 2 2h 1 2h 1 1 2 2h 2 2 h 2 h n 1 0, n m A 2h 1 2h 1 2 2h 1 c 2 1 1 2h 1 c 2 n m C c 2 c1 A 2 1 1 c 2 2 2 h 1 2h 2 h n A 1 1 1 c 2 0, c2 2h 1 2h 1 2 2 1 C 1 2h 2h 1 c 2 1 n 2h m 1 c 2 A 2 2 1 2 h 1 + 1 h c 2 c1 1 1 2h 1 2 0, n 1 c2 c1 c2 A 1 2 2 2 h 1
where
4h 1 1 2h c1 c 2 M N , n , A 4h 2 , c1 c 2 1 2 1 Mp Np
C
2h 2h 1 2 2 c1 c 2 , 1 1
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m
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in which the fully plastic axial force N p and bending moment M p are
N p f 2h hm 2 c c
(A2)
and
2
h 1 M p f h m 2 f hc c hm c c c 2 c1 c 2 f hc c1 c 2 . 2 2
(A3)
The predicted yield surface ( N N p , M M p ) for the metal double-layer sandwich
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cross-section with different core compressive strain c1 and c 2 based on core layer-by-layer compression assumption are shown in Figs. B1(a) and B1(b), respectively. It is seen that the present yield criterion can be applied to double-layer sandwich structures with different core compressive strain, core strength, face sheet and interlayer sheet materials and geometries.
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Appendix C. Inscribing coefficient for compressed double-layer sandwich cross-sections Based on the core layer-by-layer assumption, the inscribing coefficient is given by 1 4k0 1 , 2k0 2 k 4k2 k1 1 , 2 k 2 4k k 4 3 3 , 2
J1 0
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J1 0 and J 2 0 , J2 0
where
k1
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A
2
,
4h 1 c 2 1 c 2 2h A 1 c 2 1
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k0
2h
2h 1 1 2
2
,
c2
2 2h 1 c 2 +A
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k3
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2 2h c 2 c1 2 2 1 c2 C 1 c 2 1 4h 2 2 1 c 2 2 1 , k2 2 2 2 2 h 1 1 c2
k4
2h
2
,
2h 2 2 c 2 2h c 2 c1 1 c 2 c1 2 , 2 2h 2h 2
2 h 4h 2 J1 1, 2h A 1 J2
2 h 1
2h
4h 2 1 2 2h 1 2 c1 c 2 c 2 c1 1 2 . A 1 2
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(B1)
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response of a multilayer prismatic structure to impulsive loads incident from water. Int J Impact Eng 2009; 36: 632-643. [14] Fleck NA, Deshpande VS. The resistance of clamped sandwich beams to shock loading. ASME J Appl Mech 2004; 71: 386-401. [15] Qiu X, Deshpande VS, Fleck NA. Dynamic response of a clamped circular sandwich plate subject to shock loading. ASME J Appl Mech 2004; 71: 637-645.
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[16] Qin QH, Wang TJ. A theoretical analysis of the dynamic response of metallic sandwich beam under impulsive loading. Eur J Mech A/Solids 2009; 28: 1014-1025.
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[34] Jones N. Structural impact. Cambridge: Cambridge University; 1989. [35] Qin QH, Wang TJ. Impulsive loading of a fully clamped circular metallic foam core sandwich
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[36] Deshpande VS, Fleck NA. Isotropic constitutive models for metallic foams. J Mech Phys Solids 2000; 48: 1253-1283. [37] Ashby MF, Evans AG, Fleck NA, Gibson LJ, Hutchinsion JW, Wadley HNG. Metal foams: a design guide. Oxford: Butterworth Heinemann; 2000. [38] Zhang JX, Qin QH, Xiang CP, Wang TJ. Plastic analysis of multilayer sandwich beams with metal foam cores. Acta Mech 2016; 227(9): 1-15.
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Figure Captains Fig. 1. Sketch of a fully clamped double-layer rectangular sandwich plate subjected to blast loading. Fig. 2. Plastic hinge line pattern and the transverse displacement fields in a fully clamped rectangular plate. Fig. 3. Sketch of the approximate yield locus of the compressed sandwich cross-section ( 1 ,
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0.01 , h =0.1 , c =0.02 , =0.1 ) based on core layer-by-layer compression ( c1 0.5 , c 2 0 ).
Fig. 4. Analytical solutions and numerical results for (a) the maximum central deflection Wm of
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the bottom face sheet and (b) structural response time T as a function of the impulse I for the double-layer rectangular sandwich plates ( 1 , 2 , 0.01, h =0.1 , c =0.02 , =0.1). Fig. 5. Analytical solutions and numerical results for (a) the maximum central deflection Wm of the bottom face sheet and (b) structural response time T as a function of the impulse I for the
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double-layer square sandwich plates ( 1 , 1 , 0.01 , h =0.1 , c =0.02 , =0.1).
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Fig. 6. The effect of the double-layer factor on maximum central deflection Wm of the
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double-layer sandwich plates versus impulse I ( 2 , 0.01 , h =0.1 , c =0.02 , =0.1). Fig. 7. Comparisons of the maximum central deflection Wm of bottom face sheet versus impulse
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I of rectangular double-layer sandwich plate and solid monolithic plate with the same mass.
Fig. 8. Numerical results of the maximum central deflection Wm of bottom face sheet versus
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impulse I of rectangular double-layer sandwich plates, monolayer sandwich plates and solid monolithic plates with the same mass in the low impulse. Fig. A1. The time period of the core compression and overall structural response time of the double-layer sandwich plates under the impulse loading (a) I =0.001, and (b) I =0.0035 . Fig. A2. The ratio of time period of the core compression to overall structural response time of the double-layer sandwich plates under the impulse loading.
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Fig. A3. The relations between the response time of the double-layer sandwich plates for core compression and the parameters. (a) Core strength, (b) core height, and (c) thickness of face sheet. Fig. A4. The response time of the double-layer sandwich plates for core compression under different pulse of load. Fig. B1. The yield surface based on core layer-by-layer compression assumption with (a) core strain
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( 1 , 0.01 , h =0.1 , c =0.02 ) and (b) double-layer factor ( 0.01 , h =0.1 , c =0.02 ,
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c1 D , c 2 0.2 ).
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Fig. 1. Sketch of a fully clamped double-layer rectangular sandwich plate subjected to blast loading.
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Fig. 2. Plastic hinge line pattern and the transverse displacement fields in a fully clamped
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rectangular plate.
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Fig. 3. Sketch of the approximate yield locus of the compressed sandwich cross-section ( 1 ,
0.01 , h =0.1 , c =0.02 , =0.1 ) based on core layer-by-layer compression ( c1 0.5 ,
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c 2 0 ).
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Fig. 4. Analytical solutions and numerical results for (a) the maximum central deflection Wm of the bottom face sheet and (b) structural response time T as a function of the impulse I for the double-layer rectangular sandwich plates ( 1 , 2 , 0.01, h =0.1 , c =0.02 , =0.1).
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(a)
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Fig. 5. Analytical solutions and numerical results for (a) the maximum central deflection Wm of the bottom face sheet and (b) structural response time T as a function of the impulse I for the double-layer square sandwich plates ( 1 , 1 , 0.01 , h =0.1 , c =0.02 , =0.1).
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Fig. 6. The effect of the double-layer factor on maximum central deflection Wm of the
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double-layer sandwich plates versus impulse I ( 2 , 0.01 , h =0.1 , c =0.02 , =0.1).
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Fig. 7. Comparisons of the maximum central deflection Wm of bottom face sheet versus impulse
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I of rectangular double-layer sandwich plate and solid monolithic plate with the same mass.
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Fig. 8. Numerical results of the maximum central deflection Wm of bottom face sheet versus impulse I of rectangular double-layer sandwich plates, monolayer sandwich plates and solid
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monolithic plates with the same mass in the low impulse.
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Fig. A1. The time period of the core compression and overall structural response time of the double-layer sandwich plates under the impulse loading (a) I =0.001, and (b) I =0.0035 .
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Fig. A2. The ratio of time period of the core compression to overall structural response time of the
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(b)
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Fig. A3. The relations between the response time of the double-layer sandwich plates for core
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Fig. A4. The response time of the double-layer sandwich plates for core compression under
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Fig. B1. The yield surface based on core layer-by-layer compression assumption with (a) core strain ( 1 , 0.01 , h =0.1 , c =0.02 ) and (b) double-layer factor ( 0.01 , h =0.1 , c =0.02 ,
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