Accepted Manuscript
The dynamic response of sandwich panels with cellular metal cores to localized impulsive loading
Lin Jing, Zhihua Wang, Longmao Zhao
PII: S1359-8368(16)30059-2
DOI: 10.1016/j.compositesb.2016.03.035 Reference: JCOMB 4136
To appear in: Composites Part B Received Date: 5 August 2015 Revised Date: 7 January 2016 Accepted Date: 13 March 2016
Please cite this article as: Jing L, Wang Z, Zhao L, The dynamic response of sandwich panels with cellular metal cores to localized impulsive loading, Composites Part B (2016), doi: 10.1016/
j.compositesb.2016.03.035.
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
M AN US CR IP T
AC CE PT ED
The dynamic response of sandwich panels with cellular metal cores to localized impulsive loading
Lin Jing1,* Zhihua Wang2 Longmao Zhao2
1State Key Laboratory of Traction Power, Southwest Jiaotong University, Chengdu 610031, China
2Institute of Applied Mechanics and Biomedical Engineering, Taiyuan University of Technology, Taiyuan 030024,
China
Abstract: The deformation/failure modes and dynamic response of peripherally clamped square monolithic and sandwich panels of localized impulsive loading were investigated experimentally by metallic foam projectile impact. The sandwich panels comprise three different types of cellular metallic cores, i.e., closed-cell aluminum foam core, open-cell aluminum foam core and aluminum honeycomb core. Experimental results show that all the sandwich panels present mainly large global inelastic deformation with obvious local compressive failure in the central area, except for those open-cell foam core sandwich panels. The dynamic response of sandwich panels is sensitive to the applied impulse and their geometrical configurations. Based on the experimental investigation, a theoretical analysis was developed to predict the dynamic response of sandwich panels by employing a comprehensive yield locus and a modified classic monolithic panel theory.
A comparison of experimental results and theoretic predictions was made, and a good agreement was then found. These findings are very useful to guide the engineering applications of metallic sandwich structures for the protection purpose.
Keywords: A. Foams; B. Impact behavior; C. Mechanical testing; Sandwich panel
1. Introduction
The employment of sandwich structure, which is a special form comprising a combination of two thin stiff metallic/composite skins and a softer low-density cellular metallic core, continues to be of much academic and industrial interests [1-3]. These sandwich structures have the favorable ability to undergo large plastic deformation at a relatively long low plateau stress, attributing to the devisable microstructure of cellular metallic cores, resulting in a wide use in many protective engineering as shock-resistance components and energy absorbers to resist in blast, shock or impact loads [2-4]. With the development of using a metallic foam projectile to simulate shock loading on a structure [5], which is safe and simple to conduct in a laboratory setting, corresponding studies on the dynamic response of such sandwich structures under metallic foam projectile impact has thus become increasingly attractive to guide the engineering applications.
Over the past decade, a large number of studies of cellular metal core sandwich structures have
M AN US CR IP T
AC CE PT ED
been widely reported focusing on the deformation/failure modes, dynamic structural response and energy absorption, and so on. Using a drop weight machine, the low-velocity impact behavior of sandwich structures with different types of face-sheets and cellular metal cores has been investigated [6-11]. Subsequently, based on a one-dimension plastic shock wave analysis, the dynamic response of sandwich beams [12-14] and panels [15-17] has been widely studied using a metallic foam projectile impact technique, as mentioned above, by numerous researchers. All these researches show that sandwich structures have a higher shock resistance than the corresponding solid monolithic counterparts of equal mass. Some typical dynamic failure modes such as face-sheet yielding or wrinkling, core compression or shear, and interfacial failure have also been demonstrated experimentally. For blast-resistance cases, a four-cable ballistic pendulum was employed to investigate the dynamic response of blast-loaded flat and curved sandwich panels [18-20], respectively. By detonating explosive discs in very close proximity range, Nurick et al [21] studied the inelastic response of aluminum alloy honeycomb core sandwich panels under approximately uniformly distributed loading. Meanwhile, the corresponding finite element analyses were conducted to further study the dynamic response, failure mechanism, energy absorption capability and regimes of behavior of such sandwich structures [22-26].
Theoretically, Fleck and Deshpande [27] developed an analytical model for the finite deflection response of clamped sandwich beams subjected to shock loading, which has become a theoretical frame of studying the shock resistance of sandwich structures. In their model, the whole response of sandwich structures was split into three sequential stages, that is, fluid-structure interaction phase, core compression, and overall bending and stretching phase. Qiu et al [4, 28] extended this analytical model for clamped sandwich beams subjected to impulsive loading over a central loading patch and axisymmetric sandwich plates to a spatially uniform air or underwater shock, respectively. More recently, by incorporating a unified yield criterion considering the effect of core strength into the Fleck-Deshpande model, some theoretical analyses for the response of sandwich beams [29], panels [30] and shells [31] of impulsive loading have been reported.
However, due to the coupling influences of sandwich topologies, loading method and manufacturing process, a comparative study on the shock resistance of different core topology sandwich structures still remains to need to be fully understood so as to quantify the structural advantages of sandwich design.
In this study, a comparative experimental study of sandwich panels with three different types of cellular metallic cores under localized impulsive loading was conducted, mainly focusing on the deformation/failure modes, dynamic response and failure mechanism. Based on experimental
M AN US CR IP T
AC CE PT ED
investigations, a theoretical analysis was developed to predict the dynamic response of sandwich panels. The experimental results were finally compared with theoretical predictions.
2. Experimental process 2.1 Specimens
Square sandwich panel specimens with the length of side 300 mm were fabricated by two thin Al-2024 aluminum alloy face-sheets and a cellular metallic core using epoxy adhesive, as shown in Fig. 1. Three thickness face-sheets (h = 0.5 mm, h = 0.8 mm and h = 1.0 mm) and three different cellular cores (closed-cell or open-cell metallic foam core, and aluminum honeycomb core) were used. The mechanical properties of face-sheet material, which were measured by the standard quasi-static tests, are as follows: Young's modulus E = 72.4 GPa, Shear modulus G = 28 GPa, Poisson's ratio ν= 0.33, density ρ= 2700 kg/m3, and yield stress σfY = 75.8 MPa.
(Fig. 1)
The closed-cell and open-cell aluminum foam core materials were supplied by Hongbo Metallic Material Company (China). Three different core thicknesses c (10 mm, 20 mm and 30 mm) were used for closed-cell cores while three different average cell sizes dc (0.75 mm, 1.5 mm and 2.5 mm; they are determined by the corresponding SEM photographs as shown in Ref. [14]) were chose for open-cell cores, respectively. The closed-cell foam is with density of 308 kg/m3 (i.e., relative densityρ ≈0.11), which the open-cell foams are with an approximate relative density of 0.40. The aluminum honeycomb core (supplied by HexWebR.com), which is made of aluminum 5052, comprises a square array of hexagonal cells, with cell length lc = 3.18 mm and three values of cell-wall thickness tc (i.e., 0.018 mm, 0.025 mm and 0.038 mm). The face-sheet thickness of 0.8 mm and core thickness of 12.5 mm were set for all aluminum honeycomb core sandwich panels.
Typical quasi-static uniaxial compressive stress versus strain curves for these three types of cellular metal core materials are shown in Figs. 2 (a) - (c). An energy-efficiency based approach [32] with the following equations was employed to calculate the plateau stresses and densification strains of these cellular core materials, and the corresponding results are included in Table 1.
( ) ( ) ( )
aa cr
c c a
d
ε ε ε ε
ε σ
ε ε ε σ
η
=
=
∫
(1)
( )
=0=D
d d
ε
ε ε
ε
η (2)
( )
cr a
c pl
a cr
d ε ε
ε ε σ σ
ε ε
=
∫
−(3) where η(εa) is the energy absorption efficiency; εa is a given nominal strain and σc(ε) is the
M AN US CR IP T
AC CE PT ED
corresponding stress value; εcr and εD are the strain at the yield point and densification strain, respectively; σpl is the plateau stress.
(Fig. 2) (Table 1)
For comparison, square monolithic solid plates with side length of 300 mm and thickness of 2.0 mm were also tested, aiming to quantify the structural advantages of sandwich panels. Meanwhile, the quasi-static punch tests were conducted by loading centrally the monolithic and sandwich plates with a flat cylindrical steel punch, whose diameter equals to that of foam projectiles.
2.2 Experimental set-up
Impact tests were conducted by loading the panels over a central area with closed-cell Alporas foam (supplied by Shinko Wire Company, Germany) projectiles using a gas gun apparatus.
Cylindrical foam projectiles with the diameter dp = 36.5 mm and different lengths lp of 45 - 80 mm, were fired from a 37 mm diameter bore and 4.5 m long gas gun at different initial velocities vp of 50.4 - 272.7 m/s, generating a projectile momentum I =ρpAplpvp, where ρp and Ap are the density and unit area of foam projectiles, respectively. Fig. 3 shows a photograph of the overall experimental setup, where a specially designed fixture was used to clamp specimens while a laser displacement transducer was employed to measure the deflection history response of specimens.
For complete, a comparison of the typical deformed profile of the foam projectile with length of 80 mm before and after impacting at the tested specimen was also included in Fig. 3.
(Fig. 3)
In order to better grasp the dynamic deformation mechanism of sandwich panels, eight strain gauges (BE-120-3AA, zemic.com, China) with an operating range of 2.0%, a nominal resistance of 119.9 Ω ± 0.1 and a gauge factor of 2.21, were mounted on the front and back face-sheets to measure the strain-time history response at the key points of specimens, as shown in Fig. 4.
(Fig. 4)
3. Results and discussion
Different levels of initial momentum were imparted onto the monolithic and sandwich configuration specimens by varying the length and impact velocity of metallic foam projectiles. A summary of the details of specimen configuration, projectile momentum and the measured central point deflections of specimens is listed in Table 2.
(Table 2)
M AN US CR IP T
AC CE PT ED
3.1 Deformation/failure modes
Compared to monolithic plates, sandwich specimens can fail in various modes mainly depending on their structural configuration and the levels of initial momentum. The dynamic residual deformation and failure modes of specimens can be observed by deflection profiles of post-tested sandwich panels with three different cores, as shown in Fig. 5. Local indentation deformation of front face-sheets was the primary mode for all the core topology sandwich panels, while the penetration failure in the central area of front face-sheets also occurs for those closed-cell foam core sandwich panels with thinner face-sheets or subjected to larger initial momentum. It is seen from Figs. 5 (a), (b) and (d) that the obvious core compression deformation can be observed for closed-cell foam core and aluminum honeycomb core sandwich panels, but little core compression (as shown in Fig. 5 (c)) is observed for those open-cell foam core specimens. For some closed-cell foam core specimens, the cores also can fail via shearing, as shown in Fig. 5 (b). Delamination between the back face-sheet and crushed closed-cell foam core is evident near the compression or shear region of core, as shown in Figs. 5 (a) and (b).
(Fig. 5)
For the purpose of comparison, a monolithic plate and a closed-cell foam core sandwich panel was loaded quasi-statically to have an equal central point deflection with the corresponding nominally identical dynamically-loaded specimen, respectively. A comparison of the quasi-static and dynamic deformed profile of specimens is shown in Fig. 6. It could be found that the quasi-static response of both monolithic plate and sandwich panel is dominated by membrane action with stationary plastic hinges at the mid-span and supports. However, the deformed profile of the dynamically loaded specimen is continuously curved due to the existence of traveling plastic hinges. A larger degree of core compression can be observed for the dynamically-loaded sandwich panels than that quasi-statically-loaded one.
(Fig. 6)
3.2 Shock resistance performance
The shock resistance performance of sandwich panels is evaluated by the permanent central point deflection of the back face-sheet. A comparison of the shock resistance performance of monolithic plates and sandwich panels is shown in Fig. 7, where the specimen deflections are plotted as a function of initial projectile momentum in the following dimensionless form. It is clear that the normalized deflection increases with the increasing normalized impulse for all the specimens; and the open-cell foam core sandwich panels have the largest deflection values, and then followed by
M AN US CR IP T
AC CE PT ED
the monolithic and closed-cell foam specimens; the lowest is aluminum honeycomb core sandwich panels. This indicates that, for the identical specimen mass condition, all the sandwich panels have a better shock resistance performance than monolithic plates except the open-cell foam core sandwich panels; and aluminum honeycomb core sandwich panels are the best, and then the closed-cell core sandwich panels.
W =W L (4)
0 fY f
I I
µA σ ρ
= (5)
where, W is the permanent back face-sheet central point deflection of specimens; L is the half side length of plates; I, µ, A0, ρf and σfY are initial impulse, the mass of plates per unit area, loading area, density and yield strength of face-sheets, respectively.
(Fig. 7)
Similar to sandwich beams [5, 12-14] and shells [17, 20], the shock resistance performance of sandwich panels is also sensitive to their geometrical configuration. This influence is discussed in term of face-sheet thickness (closed-cell and open-cell foam core specimens), core thickness (closed-cell foam core specimens), cell size of the foam (open-cell foam core specimens) and cell-wall thickness (aluminum honeycomb core specimens), respectively; as shown in Figs. 8 (a) - (d). It is shown that the normalized deflection of specimens decreases with the increase of normalized face-sheet thickness, core thickness and cell-wall thickness, and increases with the increase of the normalized cell size. In the other words, increasing of face-sheet or cell-wall thickness and decreasing of cell size of cellular metallic cores can enhance the structural stiffness of specimens, as a result of the enhancement of the shock resistance performance of sandwich panels. An increase of core thickness of sandwich specimens also can enhance their resistance to shock loading by dissipating a large amount of energy via core compression.
(Fig. 8)
3.3 Strain-time history curves
Two typical sets of strain-time history curves corresponding to an open-cell foam core sandwich panel (OF4) and a closed-cell core sandwich specimen (CF10) are shown in Figs. 9 and 10, respectively. For the open-cell foam core sandwich panel, it can be found from Figs. 9 (a) and (b) that the deformation of face-sheets was dominated by plastic tension along the diagonal lines. The maximum strain value occurs at the central area subjected to impact loading, and the deformation magnitude reduces with the increase of the distance apart from the center. From the time lag of
M AN US CR IP T
AC CE PT ED
peak strain, it is obvious that the plastic hinges move along the diagonals of face-sheets. For comparison, the strain signals (strain curves 4 and 8) along the horizontal axis but have identical distance apart from the center with strain gauge 3 (front face-sheet) and 7 (back face-sheet) were measured. It is seen from strain signals 3 and 4 that the horizontal axis of the front face-sheet was subjected to compressive deformation but with nearly same magnitude as that of the diagonal.
Moreover, it should be noted that, due to the relatively small deformation of the back face-sheet, the deformation of those gauges near the clamped end (strain curves 7 and 8) is almost equal to zero.
(Fig. 9)
As shown in Figs. 10 (a) and (b), the deformation mechanism of closed-cell foam core sandwich panels is not consistent with that of open-cell core sandwich specimen. The front face-sheet was subjected to compressive deformation along the horizontal axis, and was subjected to plastic tension deformation along the diagonal line except at the central area (strain gauge 1). The tensile strain signal of gauge 1 is caused by the local indentation deformation of the specimen in the loading area. It must be noted that the strain gauge 1 fails in the test process, resulting in a very smooth section in the strain-time curve during the late stage of response, as shown in Fig. 10 (a).
Since the most part of energy was dissipated by core compression, the deformation of the back face-sheet is very small, as a result of a set of very weak strain signals shown in Fig. 10 (b).
(Fig. 10)
4. Theoretical analysis model
Based on experimental results, a theoretical analysis was developed here to predict the dynamic response of sandwich panels. Considering a clamped square sandwich panel with face-sheet thickness of h, core thickness of c, and side length of 2L subjected to an impulse I, as shown in Fig. 11. The face-sheet is considered as a rigid-ideally plastic material shown in Fig. 12 (a) with yield strength of σfY and density of ρf. The core with density of ρc is assumed to be with a constant compressive stress σc (only in the transverse direction of the panel with no lateral expansion) up to a densification strain εD; and it is taken to rigid beyond densification, as shown in Fig. 12 (b). Due to the significant difference of the time-periods, the whole response of the sandwich panel under impact loading is decoupled temporally and is split to three sequential stages [27, 28], i.e., (i) transfer of impulse to the front face-sheet; (ii) core compression; (iii) overall bending and stretching.
(Fig. 11) (Fig. 12)
M AN US CR IP T
AC CE PT ED
4.1 Transfer of impulse to the front face-sheet
The impulse is transmitted to the front face-sheet of the sandwich panel, and the front face-sheet is assumed to obtain instantly an initial velocity v0 over the area of A0 while the rest of the panel is stationary.
0 0 f
v I
Aρ h
= (6)
The kinetic energy of the front face-sheet can be written as
2 0
2 0 f
K I
A ρ h
= (7)
4.2 Core compression
In this stage, it is assumed that the momentum is transferred immediately to the core and back face-sheet below the loading patch, and no momentum is transferred to any other portion of the panel. At the end of this stage, the front face-sheet, core and back face-sheet obtain a final common velocity v1 of the sandwich panel cross-section over the loading area, which is calculated by the momentum conservation as
1
0(2 f c )
v I
A ρ h ρc
= + (8) The kinetic energy of the sandwich panel obtained at the end of this stage is given by
( )
2 1
2 0 2 f c
K I
A ρ h ρc
= + (9)
The energy dissipation by face-sheets is neglected compared to the contribution of the core. Hence, the energy absorption of the sandwich panel during first two stages can be calculated as
0 1
Ep =K −K , (10) or
0 1
0
1
p 2
K K q
E K q
− +
= =
+ (11) where q is the mass ratio of the core to face-sheet, and q = (ρc c) / (ρf h).
Meanwhile, the energy absorption also can be written by
0 d
p c
E =A
σ
∆c (12) whereσ
cd =ψσ
cis the dynamic compressive plateau stress of the core; ψ is an empirical constant,M AN US CR IP T
AC CE PT ED
and ∆c is the core compression value.
Then the average compressive strain εc of the core is obtained by
2
2
0 0
1
2 2
p
c d
c c f
E q I
A c q A hc
ε σ σ ρ
= = +
+ (13) It should be noted that when the resulting average compressive strain εc exceeds the densification strain εD, which indicates that the core is fully compressed and would be impossible practically, so then it is set equal to εD. Although the core compression time of sandwich panels can be obtained by neglecting the core mass [27], considering it is relatively very small compared to the whole structural response time, the central point deflection response of the back face-sheet of the sandwich panel is neglected in this stage.
4.3 Overall bending and stretching
After the first two stages, the compression deformation of the cellular core completed, and the corresponding problem can be simplified to predict the dynamic response of a sandwich panel with a core thickness c′=
(
1−εc)
cand a velocity v1 over the central loading area of A0. Since the central patch of the cellular metallic core has been fully densified after subjected to impact loading through the first two stages, the mechanical behavior of the sandwich panel in the third stage may be considered to reduce to the monolithic plate case. Here, a classic monolithic plate theoretical model is modified and employed for the sandwich panel by incorporating a comprehensive yield criterion.4.3.1 Yield criterion
A modified yield criterion has been widely used for sandwich structures with thin but strong face-sheets and a thick but weak cellular metallic core [28], which is governed by
0 0 1
M M + N N = (14) where M and N are bending moment and membrane force, respectively. The longitudinal plastic membrane force N0 is assumed to be insensitive to the degree of core compression and is given by
0 c 2 fY
N =σc+ σ h (15) The plastic bending moment includes the bending moment over the uncompressed section M0u and over the compressed section M0c, i.e., M0= M0u+ M0c.
2
0 (c )
u fY c 4
M =σ h + +h σ c (16a) and
M AN US CR IP T
AC CE PT ED
2
0 (c )
c fY c 4
M =σ h ′+ +h σ c′ (16b)
In this study, a more comprehensive yield criterion, which has been used in Refs. [29-31], is adopted for sandwich panels. In this yield criterion, the cellular metallic core is assumed to be with the identical properties in compression and tension, and the influences of core strength and the plastic bending and stretching are considered. The corresponding yield locus is expressed as
( )
( )
1 14
2 2
0 2 2
0
=
+ + + +
N N h
h h M
M
σ σ
σ for
h N
N 0 2
0 ≤ +
≤ σ σ (17a)
( ) ( ) ( )
( )
2
2
0
0
2 1 1 2
0 4 1
N h h
M N
M h h
σ σ
σ
+ + − − +
+ =
+ +
for +2 ≤ N0 ≤1N σ h
σ (17b)
whereσ σ σ= c fY andh=hc. Whenσ <<1andh<<1, Eq. (17) reduces to the yield criterion for a sandwich section with thin but strong face-sheets and a thick and weak core, i.e., Eq. (14).
Here, the circumscribing square locus is written as M0
M = (18a)
N0
N = (18b) Likewise, the inscribing locus can be described as
M = ζ M
0 (19a) N0N =ζ (19b) where
1 2 2
1 2
2 3 2 2 2
1 4 1
8 (1 ) 0
2
4 8 (1 ) 0
2
s h h
s
s s s
h h
σ ζ
σ
+ −
+ − ≤
=
+ − + − >
(20)
with
2
1 2
( 2 )
4 (1 )
s h
h h
σ
σ σ
= +
+ + ,
2 2
(1 )(3 8 ) ( 2 ) 1 s h
h σ σ
σ
− +
= +
+ , and
3
2(1 2 )
1 ( 2 )
s h
σ + h
= − + .
4.3.2 Governing equations and solution
Jones [33] developed a theoretical analysis model to predict the maximum permanent central deflection and response duration of monolithic plates struck at the centre by a rigid mass impact,
M AN US CR IP T
AC CE PT ED
which retains the effects of the finite transverse deflection and geometry change. In this study, as stated earlier, since the central cellular metallic core of the sandwich panel was fully densified through the first two stages, the monolithic plate theory of Jones [33] could be employed here to predict the mechanical response of the sandwich panel in stage III. The assumed kinematically admissible transverse velocity field is shown Fig. 13, which can be written for the quadrant with positive values of x and y by
( 1 ) 0 , 0
w W & = & − x L ≤ ≤ x L ≤ ≤ y x
(21a)and
( 1 ) 0 , 0
w W & = & − y L ≤ ≤ x y ≤ ≤ y L
(21b)where w and W are the transverse deflection of the panel at the arbitrary point and the plate centre, respectively; w&and W& are the associated velocity. And the behavior of the other three quadrants is the same as that of this quadrant due to the structural symmetry.
(Fig. 13)
The transverse velocity profile in Fig. 13 requires plastic hinges at the clamped ends and along the diagonals of the sandwich panel. The governing equation is given by
( )
0 1
r
GWW A wwdA m lm M Nw m mdl
µ θ
− −∫ = ∑ ∫ +
=
&
&& & && & (22)
where G is the mass of the striker; µ is the mass per unit area of the sandwich panel; lm and r are the length and number of plastic hinges, respectively; θ&mis the relative angular velocity across a plastic hinge.
Since the sandwich panel obtains a uniform velocity field in the central area after the impact of the foam projectile, G vanishes in Eq. (22) in this stage, that is,
( )
0 1
r
wwdA M Nw dl
A m lm m m
µ θ
−∫ = ∑ ∫ +
=
&
&& & (23)
Substituting Eq. (18) for the circumscribing square yield locus and Eq. (21) for transverse velocity field into Eq. (23), then the governing equation is written as
( )
2 2
2
W&&+α W= − +c h α (24)
where
( )
2 0
2
24 2
M h c L
α = µ + (25)
M AN US CR IP T
AC CE PT ED
and with the corresponding initial conditions (0) 0
W = (26)
(0) 1
W& =v (27)
( ) 0
W T& = (28)
Here, we proceed to introduce the following non-dimensional variables c
h
h = (29a)
c = c L
(29b)f
c ρ
ρ
ρ = (29c)
c fY
σ σ σ =
(29d)f fY
L T T
ρ
= σ (29e)
0 fY f
I I
µA σ ρ
=
(29f)
Combining Eqs. (24), (26), (27) and (28), the normalized maximum central point deflection of the back face-sheet and the corresponding normalized response time can be obtained as
(
1 2)
cos sin(
1 2)
W I
W c h t t c h
L
α α
= = + +
δ
− + (30)( )
1
11
tan
1 2 T I
c h
δ
−δ
= +
(31) where
( ) ( )
{ }
( )( )
2 2
6 8 4 2 1 1
2 1 2
c c
h h
h h h
ε σ ε
δ ρ
+ − + − −
= + + (32)
Similarly, using the same analysis and applying the inscribing yield locus gives
(
1 2)
cos I sin(
1 2)
W c h
ζ α
tζ α
t c h= + +
ζ δ
− + (33)( )
1
11
tan 1 2
T I
c h
ζ δ ζ δ
−
= +
(34)
M AN US CR IP T
AC CE PT ED
5. Model validation with experimental results
The measured maximum back face-sheet central point deflections of sandwich panels, as listed in Table 2, were compared with theoretical predication to validate the analytical model. The loading segment, specimen geometry, and material parameters of specimens used in the analytical model are kept the same as those of experiments. The normalized theoretical prediction deflections of two typical sets of sandwich panels with identical geometrical configuration, employing the circumscribing and inscribing yield loci, are plotted in Fig. 14 compared with the experimental data, as a function of the normalized impulse. It is clear that the prediction deflections are agreed well with those experimental data for sandwich panels with closed-cell and open-cell cores. It should be noted that all the aluminum honeycomb core sandwich panels were tested under an approximate equivalent impulse value, aimed to investigate the influence of cell-wall thickness;
the experimental and prediction deflections is hard to be plotted as a function of the impulse. Thus, the corresponding comparison of the aluminum honeycomb core sandwich panels was omitted.
(Fig. 14)
6. Conclusions
The dynamic response of clamped square sandwich panels, with three different types of cellular metal cores, to localized impulsive loading were investigated experimentally and theoretically.
The deformation modes and mechanism, shock resistance performance of dynamically loaded specimens were discussed. Based on the experimental results, a theoretical analysis was developed by incorporating a modified classic monolithic plate theory into the three-stage framework model of sandwich structures. The whole response of the impulse-loaded sandwich panel is split to three sequential stages, i.e., transfer of impulse to the front face-sheet, core compression, and overall bending and stretching. A comprehensive yield criterion, which covers the influences of core strength and the plastic bending and stretching, is employed to predict the "upper" and "lower"
bounds of the back face-sheet central point deflections and response time. The following main conclusions are drawn out:
(1) All the sandwich panels mainly present the large global inelastic deformation with obvious local failure in the central loading area, except for those open-cell foam core specimens.
The quasi-static response of sandwich panels is dominated by membrane action with stationary plastic hinges at the mid-span and supports, while the dynamic response is with the traveling plastic hinges.
(2) For the identical specimen mass condition, all the sandwich panels have a better shock resistance performance than monolithic plates except the open-cell foam core sandwich panels;
and aluminum honeycomb core sandwich panels are the best, and then the closed-cell core
M AN US CR IP T
AC CE PT ED
sandwich panels. Increasing of face-sheet thickness, core thickness or cell-wall thickness, and decreasing of cell size of cellular metallic cores, can enhance the shock resistance performance of sandwich panels.
(3) A good agreement between experimental data and theoretical predictions was obtained, and therefore demonstrates the validity of the proposed analytical model.
Acknowledgements
The authors greatly appreciate the financial support by the National Natural Science Foundation of China (Grant Nos. 11402216 and 51475392) and by the State Key Laboratory of Traction Power (2014TPL_T09).
References
[1]. Lu G, Yu TX. Energy absorption of structures and materials. Cambridge: Woodhead Publishing Ltd., 2003.
[2]. Ashby MF, Evans AG, Fleck NA, Gibson LJ, Hutchinson JW and Wadley HNG. Metal foams:
A design guide. Oxford: Butterworth-Heinemann, 2000.
[3]. Gibson LJ, Ashby MF. Cellular solids: structure and properties. 2nd ed. Cambridge:
Cambridge University Press, 1997.
[4]. Jing L, Wang ZH, Zhao LM. Dynamic response of cylindrical sandwich shells with metallic foam cores under blast loading—Numerical simulations. Composite Structures, 2013, 99:
213-223.
[5]. Radford DD, Deshpande VS, Fleck NA. The use of metal foam projectiles to simulate shock loading on a structure. International Journal of Impact Engineering, 2005, 31: 1152-1171.
[6]. Bhuiyan MA, Hosur MV, Jeelani S. Low-velocity impact response of sandwich composites with nanophased foam core and biaxial braided face sheets. Composites Part B, 2009, 40:
561-571.
[7]. Yu JL, Wang EH, Li JR, Zheng ZJ. Static and low-velocity impact behavior of sandwich beams with closed-cell aluminum-foam core in three-point bending. International Journal of Impact Engineering, 2008, 35: 885-894.
[8]. Kursun A, Senel M, Enginsoy HM, Bayraktar E. Effect of impactor shapes on the low velocity impact damage of sandwich composite plate: Experimental study and Modelling.
Composites Part B, 2015. (in press)
[9]. Baral N, Cartie DDR, Partridge IK, Baley C, Davies P. Improved impact performance of marine sandwich panels using through-thickness reinforcement: Experimental results.
Composites Part B, 2010, 41: 117-123.
[10]. Imielinska K, Guillaumat L, Wojtyra R, Castaings M. Effects of manufacturing and face/core bonding on impact damage in glass/polyester-PVC foam core sandwich panels. Composites
M AN US CR IP T
AC CE PT ED
Part B, 2008, 39: 1034-1041.
[11]. Schubel PM, Luo JJ, Daniel IM. Low velocity impact behavior of composite sandwich panels.
Composites: Part A, 2005, 36: 1389-1396.
[12]. Xie QH, Jing L, Wang ZH, Zhao LM. Deformation and failure of clamped shallow sandwich arches with foam core subjected to projectile impact. Composites Part B, 2013, 44: 330-338.
[13]. Rubino V, Deshpande VS, Fleck NA. The dynamic response of end-clamped sandwich beams with a Y-frame or corrugated core. International Journal of Impact Engineering, 2008, 35:
829-844.
[14]. Jing L, Wang ZH, Ning JG, Zhao LM. The dynamic response of sandwich beams with open-cell metal foam cores. Composites: Part B, 2011, 42: 1-10.
[15]. Radford DD, McShane GJ, Deshpande VS, Fleck NA. The response of clamped sandwich plates with metallic foam cores to simulated blast loading. International Journal of Solids and Structures, 2006, 43: 2243-2259.
[16]. Yahaya MA, Ruan D, Lu G, Dargusch MS. Response of aluminum honeycomb sandwich panels subjected to foam projectile impact-an experimental study. International Journal of Impact Engineering, 2015, 75: 100-109.
[17]. Jing L, Wang ZH, Zhao LM. Response of metallic sandwich shells subjected to projectile impact-Experimental investigations. Composite Structures, 2014, 107: 36-47.
[18]. Zhu F, Zhao LM, Lu GX, Wang ZH. Deformation and failure of blast-loaded metallic sandwich panels-Experimental investigations. International Journal of Impact Engineering, 2008, 35: 937-951.
[19]. Shen JH, Lu G, Wang ZH, Zhao LM. Experiments on curved sandwich panels under blast loading. International Journal of Impact Engineering, 2010, 37: 960-970.
[20]. Jing L, Wang ZH, Shim VPW, Zhao LM. An experimental study of the dynamic response of cylindrical sandwich shells with metallic foam cores subjected to blast loading. International Journal of Impact Engineering, 2014, 71: 60-72.
[21]. Nurick GN, Langdon GS, Jacob YCN. Behavior of sandwich panels subjected to intense air blast-Part 1: Experiments. Composite Structures, 2009, 91: 433-441.
[22]. Qiu X, Deshpande VS, Fleck NA. Finite element analysis of the dynamic response of clamped sandwich beams subjected to shock loading. European Journal of Mechanics A/Solids, 2003, 22: 801-814.
[23]. Tilbrook MT, Deshpande VS, Fleck NA. The impulsive response of sandwich beams:
analytical and numerical investigation of regimes of behavior. Journal of the Mechanics and Physics of Solids, 2006, 54: 2242-2280.
[24]. Jing L, Xi CQ, Wang ZH, Zhao LM. Energy absorption and failure mechanism of metallic
M AN US CR IP T
AC CE PT ED
cylindrical sandwich shells under impact loading. Materials and Design, 2013, 52: 470-480.
[25]. Tilbrook MT, Deshpande VS, Fleck NA. Underwater blast loading of sandwich beams:
Regimes of behavior. International Journal of Solids and Structures, 2009, 46: 3209-3221.
[26]. Zhu F, Zhao LM, Lu G, Gad E. A numerical simulation of the blast impact of square metallic sandwich panels. International Journal of Impact Engineering, 2009, 36: 687-699.
[27]. Fleck NA, Deshpande VS. The resistance of clamped sandwich beams to shock loading. J.
Appl. Mech. Trans. ASME, 2004, 71: 386-401.
[28]. Qiu X, Deshpande VS, Fleck NA. Dynamic response of a clamped circular sandwich plate subject to shock loading. Journal of Applied Mechanics, 2004, 71: 637-645.
[29]. Qin QH, Wang TJ. A theoretical analysis of the dynamic response of metallic sandwich beams under impulsive loading. European Journal of Mechanics A/Solids, 2009, 28:
1014-1025.
[30]. Zhu F, Wang ZH, Lu G, Nurick G. Some theoretical considerations on the dynamic response of sandwich structures under impulsive loading. International Journal of Impact Engineering, 2010, 37: 625-637.
[31]. Jing L, Wang ZH, Zhao LM. An approximate theoretical analysis for clamped cylindrical sandwich shells with metallic foam cores subjected to impulsive loading. Composites: Part B, 2014, 60: 150-157.
[32]. Li QM, Magkiriadies I, Harrigan JJ. Compressive strain at the onset of densification of cellular solids. J Cell Plast 2006, 42: 371-392.
[33]. Jones N. Impact loading of ductile rectangular plates. Thin-Walled Structures, 2012, 50:
68-75.
M AN US CR IP T
AC CE PT ED
Table 1 Calculation results of plateau stress and densification strain of cellular metals Variables Closed-cell
foam
Open-cell foam Aluminum honeycomb 0.75
mm
1.50 mm
2.50 mm
0.018 mm
0.025 mm
0.038 mm Plateau stress
(MPa) 2.15 12.0 14.2 13.9 0.95 1.79 3.13
Densification
strain 0.63 0.46 0.45 0.42 0.83 0.82 0.80
M AN US CR IP T
AC CE PT ED
Table 2 Summary of the details of dynamic tests on monolithic and sandwich plates
Specimen Core type Face-sheet
thickness (mm)
Core thickness (mm)
Cell size (mm)
Cell-wall thickness (mm)
Length of projectiles (mm)
Velocity of projectiles (m/s)
Impulse (Ns)
Deflection (mm)
M1 Monolithic 2.0 - - - 50 96.2 1.31 6.8
M2 Monolithic 2.0 - - - 50 121.4 1.70 9.6
M3 Monolithic 2.0 - - - 50 133.0 1.80 12.7
M4 Monolithic 2.0 - - - 50 168.9 2.49 14.3
M5 Monolithic 2.0 - - - 50 198.4 2.77 16.3
M6 Monolithic 2.0 - - - 50 208.3 2.98 18.8
CF1 Closed-cell foam 0.5 10 - - 80 50.4 0.74 0.3
CF2 Closed-cell foam 0.5 10 - - 80 82.2 1.37 4.2
CF3 Closed-cell foam 0.5 10 - - 80 142.1 2.14 10.7
CF4 Closed-cell foam 0.5 10 - - 80 183.8 2.92 14.1
CF5 Closed-cell foam 0.5 10 - - 80 158.5 2.26 11.0
CF6 Closed-cell foam 0.8 10 - - 80 158.7 2.27 6.3
CF7 Closed-cell foam 1.0 10 - - 80 151.5 2.22 5.2
CF8 Closed-cell foam 1.0 10 - - 80 186.6 2.96 15.2
CF9 Closed-cell foam 1.0 20 - - 80 181.8 2.89 10.2
CF10 Closed-cell foam 1.0 30 - - 80 190.5 3.03 3.1
OF1 Open-cell foam 1.0 10 2.5 - 45 176.5 2.90 3.7
OF2 Open-cell foam 1.0 10 2.5 - 45 214.3 3.51 7.7
OF3 Open-cell foam 1.0 10 2.5 - 45 215.2 3.53 8.0
OF4 Open-cell foam 1.0 10 2.5 - 45 238.2 3.81 8.1
OF5 Open-cell foam 1.0 10 2.5 - 45 272.7 4.58 12.4
OF6 Open-cell foam 1.0 10 1.5 - 45 271.0 4.54 5.2
OF7 Open-cell foam 1.0 10 0.75 - 45 269.1 4.53 4.4
AH1 Aluminum honeycomb 0.8 12.5 3.18 0.018 80 138.9 1.88 4.4
AH2 Aluminum honeycomb 0.8 12.5 3.18 0.018 80 142.7 1.93 4.6
AH3 Aluminum honeycomb 0.8 12.5 3.18 0.025 80 153.8 2.08 4.3
AH4 Aluminum honeycomb 0.8 12.5 3.18 0.025 80 137.9 1.87 4.1
AH5 Aluminum honeycomb 0.8 12.5 3.18 0.038 80 144.4 1.95 2.7
AH6 Aluminum honeycomb 0.8 12.5 3.18 0.038 80 145.6 1.97 3.0
M AN US CR IP T
AC CE PT ED
(a) Closed-cell metallic foam core
(b) Open-cell metallic foam core
(c) Aluminum honeycomb core
Fig. 1 Photographs of sandwich panels with three different cellular metal cores
M AN US CR IP T
AC CE PT ED
(a) Closed-cell foam
(b) Open-cell foam
(c) Aluminum honeycomb
Fig. 2 Quasi-static compressive stress-strain responses of three types of cellular metals
M AN US CR IP T
AC CE PT ED
Fig. 3 A photograph of the overall experimental setup
M AN US CR IP T
AC CE PT ED
Fig. 4 Schematic diagram of locations of strain gauges mounted on specimens (front view)
M AN US CR IP T
AC CE PT ED
Fig. 5 Photographs of the dynamic deformed profiles of post-tested sandwich panels with three different cores: (a) the specimen CF10, (b) the specimen CF6, (c) the specimen OF5, and (d) the
specimen AH5
M AN US CR IP T
AC CE PT ED
Fig. 6 Photographs of the deformed profiles of monolithic and sandwich specimens: (a) monolithic plate M6 under dynamic loading; (b) monolithic plate under quasi-static loading; (c) sandwich
panel CF4 under dynamic loading; and (d) sandwich panel under quasi-static loading.
M AN US CR IP T
AC CE PT ED
Fig. 7 Relationships between the normalized deflection of monolithic and sandwich specimens and the normalized impulse
M AN US CR IP T
AC CE PT ED
(a) Effect of face-sheet thickness (b) Effect of core thickness
(c) Effect of cell size (d) Effect of cell-wall thickness Fig. 8 Relationships between the normalized deflection of various core sandwich panels and key
normalized geometrical parameters
M AN US CR IP T
AC CE PT ED
(a) Front face-sheet
(b) Back face-sheet
Fig. 9 Strain-time history curves of the open-cell foam core sandwich panel (the specimen OF4)
M AN US CR IP T
AC CE PT ED
(a) Front face-sheet
(b) Back face-sheet
Fig. 10 Strain-time history curves of the closed-cell foam core sandwich panel (the specimen CF10)
M AN US CR IP T
AC CE PT ED
Fig. 11 Sketch of the loading geometry of the clamped sandwich panel
M AN US CR IP T
AC CE PT ED
Fig. 12 Material characteristics of a sandwich panel: (a) face-sheets and (b) the core
M AN US CR IP T
AC CE PT ED
Fig. 13 The assumed kinematically admissible transverse velocity field for the sandwich panel
M AN US CR IP T
AC CE PT ED
(a) Closed-cell foam core
(b) Open-cell foam core
Fig. 14 The normalized deflections of various core type sandwich panels as a function of the normalized impulse
