Dynamic response of square sandwich plates with a metal foam core subjected to low-velocity impact

Dynamic response of square sandwich plates with a metal foam core subjected to low-velocity impact

Accepted Manuscript Dynamic response of square sandwich plates with a metal foam core subjected to low-velocity impact Qinghua Qin , Xiaoyu Zheng , J...

944KB Sizes 6 Downloads 289 Views

Accepted Manuscript

Dynamic response of square sandwich plates with a metal foam core subjected to low-velocity impact Qinghua Qin , Xiaoyu Zheng , Jianxun Zhang , Chao Yuan , T.J. Wang PII: DOI: Reference:

S0734-743X(17)30360-3 10.1016/j.ijimpeng.2017.09.011 IE 2988

To appear in:

International Journal of Impact Engineering

Received date: Revised date: Accepted date:

20 April 2017 26 August 2017 10 September 2017

Please cite this article as: Qinghua Qin , Xiaoyu Zheng , Jianxun Zhang , Chao Yuan , T.J. Wang , Dynamic response of square sandwich plates with a metal foam core subjected to low-velocity impact, International Journal of Impact Engineering (2017), doi: 10.1016/j.ijimpeng.2017.09.011

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.

ACCEPTED MANUSCRIPT

Highlights A dynamic analytical model is developed to predict the dynamic response of metal foam core sandwich plates struck by heavy mass with low-velocity.



Large deflection effect is included in analysis by considering the interaction between the plastic bending and stretching.



The dynamic response and energy absorption of the sandwich plates depend on the impact location of the striker.

AC

CE

PT

ED

M

AN US

CR IP T



ACCEPTED MANUSCRIPT Submitted to IJIER2

Dynamic response of square sandwich plates with a metal foam core subjected to low-velocity impact Qinghua Qin, Xiaoyu Zheng, Jianxun Zhang, Chao Yuan, T. J. Wang

CR IP T

State Key Laboratory for Strength and Vibration of Mechanical Structures, Shaanxi Engineering Laboratory for Vibration Control of Aerospace Structures, School of Aerospace, Xi‟an Jiaotong University, Xi‟an, 710049, China

Abstract: An analytical model is developed to predict the dynamic response of fully clamped

AN US

square sandwich plate with metal foam core struck transversely by a heavy mass with low-velocity. Large deflection effect is incorporated in analysis by considering the interaction between the plastic bending and stretching. The analytical expressions are obtained for the

M

structural deflection, the structural response time and the impact force. The finite element

ED

results validate the accuracy of the analytical model and good agreement is achieved.

PT

Keywords: Sandwich plate; Metal foam core; Low-velocity impact; Large deflection;

AC

CE

Dynamic plasticity.



Corresponding author: [email protected]

ACCEPTED MANUSCRIPT

1. Introduction Sandwich structure has been paid more attention and widely used in a number of critical engineering due to its excellent advantages over monolithic solid structure of the same mass. The sandwich structure has been considered to be a promising alternative to the monolithic

CR IP T

solid structure. The structural behavior of sandwich structure depends on the material properties, the geometries of the face-sheet, the core and boundary condition, etc.[1-3]. Metal foam is a kind of lightweight material with high stiffness-to-weight ratio, high

AN US

strength-to-weight ratio, novel physical and mechanical properties, such as nearly isotropic, high energy absorption, good sound damping, non-combustibility, easy fabrication to form curved configurations and integrated face-sheets [4-9]. The metal foam is then selected as the

M

core of sandwich structure.

Much work has been carried out to investigate the dynamic response of solid monolithic

ED

structure previously. Symonds [10] examined the dynamic response of fully clamped

PT

monolithic solid beam subjected to impulsive loading over a central patch in small deflection. Then the interaction between of bending moment and axial force was firstly considered by

CE

Onat and Prager [11] for the large deflection problems. Haythornwaite [12] obtained

AC

analytical solution for fully clamped beam with solid rectangular cross-section under transversely concentrated loading. Parkes [13] experimentally investigated the fully clamped monolithic solid beam struck by a mass at any location across its span and conducted theoretical analysis with a rigid-perfectly plastic assumption. Nonaka [14-16], D Oliveira [17] and Liu and Jones [18] theoretically studied the effects of finite displacement and transverse shear on the dynamic response and found that the shear force may play a significant role as

ACCEPTED MANUSCRIPT

the deflection is small. Shen and Jones [19] obtained the dynamic and quasi-static solutions for the fully clamped monolithic solid beam struck at midspan. Moreover, the quasi-static method can be used to estimate the dynamic plastic response of monolithic solid beam struck by a heavy mass with low-velocity. Comparisons between the quasi-static theoretical

CR IP T

predictions and experimental results were presented for the fully clamped monolithic solid beam struck by a mass at midspan and good agreement was achieved [20].

As it to monolithic solid plates, a theoretical rigid-plastic method retaining the influence

AN US

of large transverse displacements was developed to predict the maximum permanent transverse displacements or damage, of ductile circular and rectangular plates subjected to a pressure pulse causing plastic strains [21]. Also, This method was extended to examine the

M

dynamic response of monolithic solid plates with various aspect ratios subjected to impulsive loading and good agreement between the theoretical predictions and experimental results was

ED

found [22, 23]. Wen et al. [24] experimentally investigated the dynamic response for the fully

PT

clamped circular mild steel plates struck by blunt-faced projectiles. Zaera et al. [25] studied the dynamic axisymmetric response of circular plates subjected to impulsive pressure

CE

loadings, comparing favorably with the experimental results reported by Bodner and Symonds

AC

[26]. Moreover, they observed that the rotatory inertia can be neglected, which is consistent with the observation [27]. This method [21] was then extended to study the impact

response

of circular plates [28, 29] and square plates [29] struck by a solid mass at the center and good agreement was found between theoretical predictions and experimental results of the maximum permanent transverse displacements. The perforation of ductile metal circular and rectangular plates with a range of aspect ratios struck normally by cylindrical projectiles at the

ACCEPTED MANUSCRIPT

center was experimentally explored by Jones et al. [30, 31]. Subsequently, Jones [32] extended above theoretical method to predict the maximum permanent transverse displacements or ductile damage for a range of aspect ratios rectangular plates struck by a rigid mass at the center. The strain rate sensitivity to the dynamic inelastic response of the

CR IP T

plates was further identified [33]. Hazizan and Cantwell [34, 35] employed a simple energy balance model to predict the maximum impact force and energy absorption of the composite sandwich beams with

AN US

aluminum honeycomb and polymer foam cores subjected to low-velocity impact. Crupi and Montanini [36] experimentally investigated the low-velocity impact behavior of the metal foam core sandwich beams and gave some complex collapse modes. Yu et al. [37, 38]

M

experimentally observed that deformation and failure mechanism of the low-velocity impact of metal foam core sandwich beam are similar to those under the quasi-static loading and

ED

three competing collapse modes are summarized as face yield, core shear and indentation,

PT

while the force-displacement histories for two loading conditions are different. Using a yield criterion for metallic sandwich structure considering the effect of core strength, an analytical

CE

solution for the large deflections of slender sandwich beam with metallic foam core under

AC

transverse loading by a flat punch was derived by Qin and Wang [39]. They gave the analytical solutions for the low-velocity impact response of fully clamped slender metal foam core sandwich beam struck by a heavy mass while the effect of local denting was neglected in the analysis [40]. The results demonstrate that the quasi-static solutions can be approximately applied to predict the impact response of sandwich beam, when the striker mass is much greater than that of the sandwich beam. Qin and Wang [41] extended the

ACCEPTED MANUSCRIPT

quasi-static method of analysis for predicting low-velocity impact response of the metal foam core sandwich beam including the local denting and found that the energy absorption of the sandwich beam may be overestimated if the effect of local denting is neglected in the analysis. Zhang et al. [42] employed this quasi-static method to study the low-velocity impact response

CR IP T

of fully clamped geometrically asymmetric sandwich beam with metal foam core. Further, they obtained the analytical solution of the dynamic response of slender multilayer sandwich beams with metal foam cores subjected to low-velocity impact [43]. However, to the authors‟

AN US

knowledge, no attention is paid to the dynamic response of the metal foam core sandwich plate subjected to low-velocity impact considering the axial stretch induced by large deflections.

M

The purpose of the present work is to study the dynamic response of the fully clamped sandwich plates struck by a heavy mass with low-velocity. The analytical model is developed

ED

to predict the large deflection dynamic response of sandwich plate. Finite element calculations

PT

are conducted and effect of different parameters on the dynamic response is discussed. Comparisons between the analytical predictions and the calculations are used to assess the

CE

fidelity of the analytical model.

AC

2. Analytical model

Consider a fully clamped metal sandwich square plate of length 2L with two identical face

sheets of thickness h and core thickness c struck by a heavy mass G with low-velocity V0 , as shown in Fig. 1. The densities of face-sheets and core are  f and  c , respectively. The two face-sheets are assumed to be perfectly bonded to the core and obey the rigid perfectly plastic law with yield strength  f . The metal foam core is modeled as a rigid-perfectly-plastic

ACCEPTED MANUSCRIPT

locking (r-p-p-l) material with a plateau-stress  c and a critical densification strain  D , as shown in Fig. 2. Neglecting the effect of elasticity of the face sheets and core may fail to accurately describe the behavior of the sandwich structure. The impact point is determined by a vector (2a, 2b) from the top left corner of the plate, where 0  a  L and 0  b  L , as

CR IP T

shown in Fig. 3. We assume that the heavy mass G remains in contact with the upper face sheet of sandwich plate after impact. Then, the deformation of sandwich plate can be divided into four

AN US

rigid regions (i.e. RegionⅠ, Ⅱ, Ⅲ, Ⅳ) separated by four straight plastic hinges, as sketched in Fig. 3. Based on the principle of energy dissipation rate balance, the governing equation is given by [29]

j

GW0W0   WWdA     M  NW  r dlr r 1

lr

(1)

M

A

The terms on the left hand side of Eq. (1) are the work rate due to the inertia force, where A is

ED

the total area of the plate and    c c  2 f h  is the mass per unit area; W is the transverse

PT

deflection, W is the transverse velocity and W is the transverse acceleration; W0 and W0

CE

denote the transverse velocity and acceleration of sandwich plate at the impact point, respectively. The terms on the right hand side are the energy dissipated in the plastic hinges,

AC

where M and N are the bending moment and membrane force, respectively; lr is the length of the plastic hinge;  r is the relative angular rotation rate across the hinge line; r is the number of the plastic hinge lines. Some comments on the assumptions underlying the above dynamic response model of sandwich plate are necessary. The assumption implies that the portion of energy absorption of the local denting under the striker nose is much smaller than that of global deformation of

ACCEPTED MANUSCRIPT

sandwich plate. Then the energy absorption of local denting can be neglected in analysis. Similar to the monolithic solid plate [21], the velocity fields for the sandwich plate are assumed to be linear. The front and side views of transverse velocity fields are shown in Fig.4, such that

CR IP T

Region I

Region II

(2)

Region III

Region IV

AN US

W0   2a  x   2a W0  2b  2b  y   W  W0   2 L  2a  x  2 L  a   W0  2 L  2b  y   2  L  b 

where x and y are the horizontal and vertical coordinates, respectively. The relative angular rotation rate across the hinge line can be given by

CE

PT

ED

M

W0   2a W0  2b    W0  2 L  a   W0  2  L  b 

AC

Combination of Eqs.(2) and (3), we have

Region I Region II (3)

Region III Region IV

ACCEPTED MANUSCRIPT

2

 W0   W0       a   b 

2

Region I&II

2

 W0   W0       b   La

2

2

 W0   W0       L a   L b  2

 W0   W0       a   Lb 

Region II&III (4)

2

Region III&IV

2

CR IP T

 1 2  1   2   1  2  1   2

Region IV&I

We can obtain the maximum deflection and response time by employing the linear

as



A

AN US

velocity profiles in Fig. 4. Then the second term on the left hand side of Eq. (1) is calculated

2 3

WWdA   L2W0W0

(5)

M

As can be seen from Eq.(5) ,the work rate due to the inertia force is independent of the impact

ED

location.

Similarly, the term on the right hand side of Eq. (1) based on the internal hinges is

j1

PT

calculated as

   M  NW  dl

CE

r 1

lr

r

r

L L   N  L L        M  W0 W0 2  a b L a L b   

(6)

AC

The term on the right hand side of Eq. (1) based on the boundary hinges is given by j2

L L  L L ( M  NW ) r dlr        MW0 lr  a b L a L b 

 r 1

(7)

Substituting Eqs. (5)-(7) into Eq. (1), we have

2 N     G   L2 W0  W0  2 M 3 2   where  =

(8)

L L L L is a geometrical factor determined by the position of impact    a b La Lb

ACCEPTED MANUSCRIPT

point. For the case of impact point located at the center of the sandwich plate, then

a  b  L 2 and   8 . According to the assumed transverse velocity profiles and momentum conservation, the initial conditions are taken as

W0  t  0  0

respectively. 2.1 The so-called ‘bounds’ of the solutions

3GV0 3G  4 L2

(9b)

AN US

W0  t  0  

CR IP T

and

(9a)

Employing the approximate yield criteria Eqs. (A3a) and (A3b) for the sandwich

M

cross-section in Appendix A, we can give the analytical solutions for structural deflections

ED

and response time. Then Eq. (8) is re-written as

PT

and

W0   2W0    2

(10a)

W0   2W0   2

(10b)

CE

2 M p  Np , 2  and  is defined by Eq.(3b) in Appendix 2 G  2  L2 3 2  G  2  L 3

AC

where  2 

A.

Combining Eqs. (9) and (10), we can obtain the deflections of the sandwich plate versus

response time at the impact point,

W0  t   and

GV0 sin  t 

G  4 L 3 2



2 2 cos  t    2 2

(11a)

ACCEPTED MANUSCRIPT



 t



2 W0  t    cos G  4 L2 3   2 GV0 sin

2  t  2 





(11b)

respectively. When the velocity W0  t  T   0 , the deformation ends and the central deflection attains maximum value W0 f  W0  t  T  ,

W0 f 

 G  4  L 3  2

for the circumscribing locus at T 

for the inscribing locus at T 

2

2



2 2

(12a)

  GV0  , and arctan  2 2    G  4 L 3   1

2 GV0   2   G  4 L2

  G  4  L2 3  2

3  4 2

2  2 

AN US

W0 f 

3  4

CR IP T

2  GV0   2  G  4 L2

(12b)

  GV0 . arctan  2 2     G  4 L 3   1

M

The non-dimensional parameters are introduced as follows,

ED

c GV02 t W0 c G W0  ,c  , G  ,  ,t  , Ek  f  f c3 c  2h L 4  L2  c  2h   f  f

CE

AC

and

PT

The non-dimensional initial conditions of Eqs. (9a) and (9b) are re-written as

W0  t  0  0

dW0  t  0 3c  dt 2  3G  1   2h

(13a)

Ek G

(13b)

Thus, the non-dimensional maximum deflection is

  4h 1  h     3Ek G  6G  1   2h   W0 f   1  1 2 2   2h 1  2h     3G  1 4h 1  h     

for the circumscribing locus, and

(14a)

ACCEPTED MANUSCRIPT

  4h 1  h     3Ek G  6G  1   2h   W0 f   1  1 2 2   2h 1  2h     3G  1 4h 1  h     

(14b)

for the inscribing locus. 2.2 The solutions based on the yield locus

CR IP T

According to the energy equilibrium of Eq. (8), the governing equations for the dynamic response of the fully clamped sandwich plate subjected to low-velocity impact are obtained by using the yield locus Eq. (A1) and the associated flow rule of the sandwich cross-section. The non-dimensional governing equations for dynamic response can be expressed as

AN US

3 c 2 4h 1  h     1  2h  d 2W0  dt 2 4  6G  1    2h  and

0  W0 

2 3 c 2 1  2h  d 2W0 3 c 1  2h    1  W0   dt 2 4  6G  1    2h  4  6G  1    2h 

ED

and

PT

2 d 2W0 3 c   2h 1  2h   W0  0 dt 2 4  6G  1    2h 

CE

When the motion ceases, i.e.

(15a)

3

M

2

1 1  2h

1  W0  1 1  2h

(15b)

2

W0  1

(15c)

dW0  t   0 , we can obtain the non-dimensional maximum dt

AC

central deflection,

W0 f  1 

3G  6G  1 Ek

  3G  1   2h 1  2h  2

2



2  2h 2  4h  1  8 8h 4  16h 3  11h 3  2h 

  2h 1  2h 

3

(16) The reaction force between the striker and the sandwich plate is

F

4    2h  G F dW02  N pL   2h  c 1  2h  dt 2

(17)

ACCEPTED MANUSCRIPT

Substituting Eq. (17) into Eqs. (15a-c), we can give the relations between the reaction force and the deflection,

F

3 Gc 4h 1  h    

 6G  1   2h 

0  W0 

1 1  2h

(18a)

and

 6G  1   2h 

W0 

3 Gc 1  2h 

 6G  1   2h 

and

3 Gc 1  2h 

 6G  1

W0

1  W0  1 1  2h

W0  1

(18b)

(18c)

AN US

F

2

CR IP T

F

3 Gc 1  2h    1

3. Finite element analysis

In order to validate the aforementioned analytical model, the explicit time integration

M

technique in the commercial finite element (FE) code ABAQUS (version 6.11) is used to

ED

model the dynamic response of the sandwich plate transversely struck by a heavy mass with low-velocity. Eight node brick element with reduced integration (type C3D8R in ABAQUS

PT

notation) is selected. Damping associated with the bulk viscosity in ABAQUS/Explicit is

CE

switched off by setting the bulk viscosity to be zero. There are 120000 elements for the case of the full sandwich plate with two and six

AC

elements in the thickness direction of each face-sheet and core, respectively. The striker is modeled as a rigid-roller that is an analytical rigid-body with a concentrated point mass. The contact between the sandwich plate and the striker is modeled by using the contact pair surface algorithm with a frictionless contact option in ABAQUS/Explicit. The vertical, horizontal and rotational displacements of nodes at the edges of the sandwich plate are zero. All calculations are presented for sandwich plates with the length 2L  200mm , the

ACCEPTED MANUSCRIPT

face-sheet thickness h  1mm and the core thickness c  4mm . We have h  0.25 and

c  0.04 . Diameter of the loading roller is d  10mm , then d  d 2L  0.05 . A mesh sensitivity study demonstrates that further mesh refinements did not improve the calculation results apparently.

CR IP T

The face-sheet obeys J2 flow theory of plasticity. Deshpande Fleck model [44] is used to model the plastic crushable behavior of metal foam core, which allows a change of the shape of yield surface due to differential hardening along the hydrostatic and deviatoric axes. The

AN US

yield function  for the foam core is

  ˆ   c  0 where ˆ 2 

1 1    3

2



2 e

(19)

  2 m 2  with  e  3 sij s ji 2 being von Mises effective stress,

M

sij the deviatoric stress,  m   kk 3 the mean stress and  the shape factor of yield surface.

The plastic Poisson‟s ratio  cp for uniaxial compression is 2

(20)

PT

ED

 p 1 2   3  cp   22p  11 1   32

The face-sheets are assumed to be made of stainless steel with yield strength

CE

 f  200 MPa, elastic modulus E f  200 GPa, elastic Poisson‟s ratio  ef  0.3 , density

AC

 f  8000 kg m3 and linear hardening modulus Etf  0.01E f .The isotropic foam core is made from the same material as the face-sheets and has the relative density   0.1 (core density c  800 kg m3 ), yield strength  c  0.5 f [6], elastic modulus Ecf   E f , elastic Poisson‟s ratio  ec  0.3 and plastic Poisson‟s ratio  cp  0 .The foam core has a plateau stress  c that continues up to densification strain  D  0.7 . The stress rises steeply with a very large tangent modulus Ect  0.1E f beyond the densification strain.

ACCEPTED MANUSCRIPT

Five impact locations (i.e. P1, P2, P3, P4 and P5) are considered, as shown in Fig.5. The dimensionless energies with the mass and velocities of the strikers are listed in Table 1. Five typical FE deflection modes (at the selected impact points) of the sandwich plate struck by a mass G  83 with velocity V0  4 m s are shown in Fig. 6. It can be seen that the

CR IP T

deflection modes are global deformation at impact points P1, P2 and P3, while the local deformations are apparent at impact points P4 and P5. The corresponding FE results for the face-sheet deflections versus time of the sandwich plate are shown in Fig. 7. It can be seen

AN US

that both the deflections increase non-linearly before they reach the maximum values. When the striker separates from the upper face-sheet, the sandwich plate unloads and elastically rebounds. The maximum deflection of the sandwich plate is defined as the peak value of the

4. Results and discussion

M

lower face-sheet in the deflection-time history.

ED

Comparisons of the analytical predictions and FE results for the non-dimensional

PT

maximum deflection of the sandwich plate against the non-dimensional initial kinetic energy of the striker are shown in Fig. 8. The analytical predictions by using the yield locus lie in the

CE

so-called „bounds‟ of the analytical predictions by using the inscribing and circumscribing

AC

squares of the yield criterion. The analytical predictions by using the yield criterion are in good agreement with the FE results for the cases of P1, P2 and P3. For the cases of P4 and P5, the analytical model using the yield locus underestimates the maximum transverse deflections of sandwich plates at low initial kinetic energy while it overestimates the maximum transverse deflections at high initial kinetic energy. This may be the reason that the effects of both shear force and local denting at the impact region of the sandwich plates increase when the impact

ACCEPTED MANUSCRIPT

location is close to the boundaries. However, these are neglected in the analytical model. It also can be seen that the analytical model by using inscribing locus can capture the FE results at low initial kinetic energy of the striker while the analytical model by using circumscribing locus can capture the FE results at high initial kinetic energy of the striker.

CR IP T

Fig. 9 shows comparisons of the FE results for the non-dimensional maximum deflections of the sandwich square plate as the non-dimensional impact energy equals with different striker mass. As can be seen, the maximum deflections of the sandwich plate

AN US

subjected to a heavy mass impact with low velocity strongly depend on the initial kinetic energy of the striker while are independent of the striker mass or velocity with the same initial impact energy. Moreover, the FE results agree well with the theoretical predictions. A comparison between the analytical predictions and FE results of impact force versus

M

deflection of the sandwich plate struck with a mass G  83 and velocity V0  4 m s is

ED

plotted in Fig. 10. The FE results of the impact force increase with the deflection of lower

PT

face sheet monotonously, while there is a platform following with a fall before the deflection attains the depth of the sandwich cross-section for the analytical predictions. The analytical

CE

model overestimates the impact force as the deflection is less than the depth of the

AC

cross-section. It is argued that this is due to the fact that the local deformation of the upper face sheet at impact location, combined with the complexity of contact and elasticity of the materials are neglected in the analytical model. However, good agreement is achieved between the analytical predictions and the FE results as the deflections exceed the depth of the cross-section. The FE results and analytical predictions for the non-dimensional maximum deflections of

ACCEPTED MANUSCRIPT

the five specific locations against the non-dimensional initial kinetic energy of the striker with

G  83 and 188 are shown in Fig.11. It is noted that the analytical predictions overestimate the maximum deflections at high initial kinetic energy of the striker. The discrepancies become more and more apparent as the impact location moves from impact point P1 to P5. It is

CR IP T

possible that the effect of transverse shear increases when the impact location is close to the boundaries.

Fig.12 shows the FE results for the energy absorption of each part of the sandwich plate

AN US

versus time. The energy absorption proportion of the core rises sharply until attains the peak value. Subsequently, it decreases until the stable value. On the contrary, the energy absorption proportion of the front face falls sharply until reaches a minimum value. Subsequently, it

M

increases to be a stable value. The energy absorption proportion of the back face rises monotonously to a stable value. It can be seen that the energy absorption of the face-sheets

ED

takes a great proportion for a given initial kinetic energy, while the energy absorption

PT

proportion of the core decreases with the increase of the initial kinetic energy. The energy absorption proportions of each part at impact point P2 against the

CE

non-dimensional initial kinetic energy of the striker are plotted in Fig.13. The energy

AC

absorption proportion of the core decreases with the increase of the initial kinetic energy, while those of both the face-sheets increase. The energy absorption proportion of the core is large at the low initial kinetic energy while it is small at the initial kinetic energy. Fig.14 plots the energy absorption proportions of each part at five locations for the cases of Ek  180 and Ek  320 , respectively. The energy absorption proportion of the core increases when the impact location moves from the impact points P1 to P5, while those of

ACCEPTED MANUSCRIPT

both the face-sheets decrease. Moreover, the more the location is close to the boundaries, the more the energy absorption proportion of the core is large. The energy absorption proportion of the face-sheets is large for the case of the high initial kinetic energy. 5. Concluding remarks

CR IP T

The low-velocity impact response of the sandwich square plates with metal foam core struck by heavy mass is studied theoretically and numerically. The analytical solutions are obtained by using the yield criterion considering the interaction between the bending moment

AN US

and axial force and lie in the so-called “bounds” of dynamic response given by using the inscribing and circumscribing yield loci. The present analytical predictions show good agreement with FE results. The results demonstrate that the dynamic response of the sandwich

M

plates depends on the impact location of the striker strongly. The energy absorption of core

close to the boundaries.

PT

Acknowledgments.

ED

decreases with the initial kinetic energy increasing while it increases with the impact location

CE

The authors gratefully acknowledge the financial supports of NSFC (11372235, 11321062, 11572234 and 11502189), China Postdoctoral Science Foundation funded project

AC

(2015M572546).

Appendix A: Yield criteria for sandwich cross-section[39] The yield criterion for the sandwich cross-section considering the interaction between of the bending moment and axial force is

ACCEPTED MANUSCRIPT 2    2h    | m |  n2  1 0 | n | 2   2h 4 h 1  h         2h    2h  | n | 2h    2 | n | 1  0  | m |  | n | 1    2h 4h 1  h    

(A1)

where    c  f , h  h c , m  M M p and n  N N p with M p and N p denoting the fully

axis force can be defined by

M p   f h c  h   c

AN US

N p   c c  2 f h

c2 4

CR IP T

plastic bending moment and axial force, respectively. The fully plastic bending moment and

(A2)

The yield locus for a circumscribing square is

| M | M p , | N | N p

(A3a)

and the yield locus for an inscribing square is

M

M  M p , N  N p

ED

where

CE

PT

 1  4k0  1 , 8h 2 1  h    2  0  2 k0   ,  k12  4k2  k1 ,8h 2 1  h    2  0   2

  2 h   ,   4h 1  h     1     3  8h   1 ,   2 h   2

AC

k0

k1

k2  1 

2 1    .   2h

2

(A3b)

ACCEPTED MANUSCRIPT

References [1]

Allen H. Analysis and design of structural sandwich panels, 1969. Robert Maxwell, MC, MP. 1969.

[2] Noor AK, Burton WS, Bert CW. Computational models for sandwich panels and shells.

CR IP T

Applied Mechanics Reviews. 1996;49:155-99. [3] Abrate S. Impact on composite structures, 1998. Cambridge University Press, Cambridge; 2000.

AN US

[4] Gibson LJ, Ashby MF. Cellular solids: structure and properties: Cambridge university press; 1997.

[5] Evans AG, Hutchinson J, Ashby M. Multifunctionality of cellular metal systems.

M

Progress in Materials Science. 1998;43:171-221.

[6] Ashby MF, Evans T, Fleck NA, Hutchinson J, Wadley H, Gibson L. Metal Foams: A

ED

Design Guide: A Design Guide: Elsevier; 2000.

PT

[7] Gibson L. Mechanical behavior of metallic foams. Annual review of materials science. 2000;30:191-227.

CE

[8] Banhart J. Manufacture, characterisation and application of cellular metals and metal

AC

foams. Progress in materials science. 2001;46:559-632. [9] Evans AG, Hutchinson JW, Fleck NA, Ashby M, Wadley H. The topological design of multifunctional cellular metals. Progress in Materials Science. 2001;46:309-27.

[10] Martin J, Symonds PS. Mode approximations for impulsively-loaded rigid-plastic structures. Journal of the Engineering Mechanics Division. 1966;92:43-66. [11] Onat E, Prager W. Limit analysis of arches. Journal of the Mechanics and Physics of

ACCEPTED MANUSCRIPT

Solids. 1953;1:77-89. [12] Haythornthwaite R. Beams with full end fixity. Engineering. 1957;183:110. [13] Parkes EW.The permanent deformation of an encastre beam struck transversely at any point in its span. ICE Proceedings: Thomas Telford; 1958. p. 277-304.

CR IP T

[14] Nonaka T. Some Interaction Effects in a Problem of Plastic Beam Dynamics—Part 1: Interaction Analysis of a Rigid, Perfectly Plastic Beam. Journal of Applied Mechanics. 1967;34:623-30.

AN US

[15] Nonaka T. Some Interaction Effects in a Problem of Plastic Beam Dynamics—Part 2: Analysis of a Structure as a System of One Degree of Freedom. Journal of Applied Mechanics. 1967;34:631-7.

M

[16] Nonaka T. Some Interaction Effects in a Problem of Plastic Beam Dynamics—Part 3: Experimental Study. Journal of Applied Mechanics. 1967;34:638-43.

ED

[17] D Oliveria JG. Beams under lateral projectile impact. Journal of the Engineering

[18]

PT

Mechanics Division. 1982;108:51-71. Liu J, Jones N. Dynamic response of a rigid plastic clamped beam struck by a mass

CE

at any point on the span. International journal of solids and structures. 1988;24:251-70.

AC

[19] Shen WQ, Jones N. A comment on the low speed impact of a clamped beam by a heavy striker. Journal of Structural Mechanics. 1991;19:527-49.

[20] Jones N. Quasi-static analysis of structural impact damage. Journal of Constructional Steel Research. 1995;33:151-77. [21] Jones N. A theoretical study of the dynamic plastic behavior of beams and plates with finite-deflections. International Journal of Solids and Structures. 1971;7:1007-29.

ACCEPTED MANUSCRIPT

[22] Jones N, Baeder RA. An experimental study of the dynamic plastic behavior of rectangular plates. 1972. [23] Jones N. A literature review of the dynamic plastic response of structures: MIT Department of Ocean Engineering; 1974.

CR IP T

[24] Wen HM, Jones N. Experimental Investigation into the Dynamic Plastic Response and Perforation of a Clamped Circular Plate Struck Transversely by a Mass. ARCHIVE Proceedings of the Institution of Mechanical Engineers Part C Journal of Mechanical

AN US

Engineering Science 1989-1996 (vols 203-210). 1994;208:113-37.

[25] Zaera R, Arias A, Navarro C. Analytical modelling of metallic circular plates subjected to impulsive loads. International Journal of Solids & Structures. 2002;39:659-72.

M

[26] Bodner SR, Symonds PS. Experiments on viscoplastic response of circular plates to impulsive loading. Journal of the Mechanics & Physics of Solids. 1977;27:91-113.

ED

[27] Jones N, Gomes dO, J. Dynamic Plastic Response of Circular Plates With Transverse

PT

Shear and Rotatory Inertia. Journal of Applied Mechanics. 1978;47:27-34. [28] Jones N, Kim S-B, Li Q. Response and failure of ductile circular plates struck by a mass.

CE

Journal of pressure vessel technology. 1997;119:332-42.

AC

[29] Jones N. On the mass impact loading of ductile plates. Defence Science Journal. 2003;53:15-24.

[30] Jones N, Birch R, Duan R. Low-velocity perforation of mild steel rectangular plates with projectiles having different shaped impact faces. Journal of Pressure Vessel Technology. 2008;130:031206. [31] Jones N, Birch RS. Low Velocity Perforation of Mild Steel Circular Plates With

ACCEPTED MANUSCRIPT

Projectiles Having Different Shaped Impact Faces. Journal of Pressure Vessel Technology. 2008;130:1047-57. [32] Jones N. Impact loading of ductile rectangular plates. Thin-walled structures. 2012;50:68-75.

CR IP T

[33] Jones N. Dynamic inelastic response of strain rate sensitive ductile plates due to large impact, dynamic pressure and explosive loadings. International Journal of Impact Engineering. 2014;74:3-15.

AN US

[34] Hazizan MA, Cantwell W. The low velocity impact response of foam-based sandwich structures. Composites Part B: Engineering. 2002;33:193-204.

[35] Hazizan MA, Cantwell W. The low velocity impact response of an aluminium

M

honeycomb sandwich structure. Composites Part B: Engineering. 2003;34:679-87. [36] Crupi V, Montanini R. Aluminium foam sandwiches collapse modes under static and three-point

International

Journal

of

Impact

Engineering.

PT

2007;34:509-21.

bending.

ED

dynamic

[37] Yu J, Wang X, Wei Z, Wang E. Deformation and failure mechanism of dynamically

CE

loaded sandwich beams with aluminum-foam core. International Journal of Impact

AC

Engineering. 2003;28:331-47. [38] Yu J, Wang E, Li J, Zheng Z. 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;35:885-94. [39] Qin QH, Wang T. An analytical solution for the large deflections of a slender sandwich beam with a metallic foam core under transverse loading by a flat punch. Composite

ACCEPTED MANUSCRIPT

Structures. 2009;88:509-18. [40] Qin QH, Wang TJ. Low-velocity heavy-mass impact response of slender metal foam core sandwich beam. Composite Structures. 2011;93:1526-37. [41] Qin Q, Wang TJ. Low-velocity impact response of fully clamped metal foam core beam

incorporating

local

denting

effect.

Composite

Structures.

CR IP T

sandwich

2013;96:346-56.

[42] Zhang J, Qin Q, Xiang C, Wang Z, Wang TJ. A theoretical study of low-velocity impact

AN US

of geometrically asymmetric sandwich beams. International Journal of Impact Engineering. 2016;96:35-49.

[43] Zhang J, Qin Q, Xiang C, Wang TJ. Dynamic response of slender multilayer sandwich

M

beams with metal foam cores subjected to low-velocity impact. Composite Structures. 2016;153:614-23.

ED

[44] Deshpande V, Fleck N. Isotropic constitutive models for metallic foams. Journal of the

AC

CE

PT

Mechanics and Physics of Solids. 2000;48:1253-83.

ACCEPTED MANUSCRIPT

Figure captions V0 V0

c+2h

AN US

c

(a)

(b)

(c)

AC

CE

PT

2L

ED

M

2a

2b

CR IP T

G

2L

(d)

Fig. 1 Sketch of a fully clamped sandwich square plate struck by a heavy mass G with an initial low velocity V0 . (a) Isometric view, (b) front view, (c) side view and (d) top view.

CR IP T

ACCEPTED MANUSCRIPT

(a)

(b)

Fig. 2 Material characteristics of metal sandwich plates. (a) Face sheets and (b) metal foam

AC

CE

PT

ED

M

AN US

core.

ACCEPTED MANUSCRIPT

2a 2b

Ⅱ Ⅲ

y

2L



CR IP T

x

AN US



ED

M

2L

AC

CE

PT

Fig. 3 Plastic hinge lines in the fully clamped sandwich square plate struck by a heavy mass.

ACCEPTED MANUSCRIPT

2a

y

O

2 L  2a

W

(a)

x

2b

2 L  2b

AN US

O

CR IP T

W0

W

W0

M

(b)

AC

CE

PT

ED

Fig. 4 The transverse velocity fields of sandwich plate. (a) Front view and (b) side view.

ACCEPTED MANUSCRIPT

0.5L

P5

0.25L

2L

P2

P4

AN US

2L

P1

CR IP T

P3

M

0.5L

0.25L

AC

CE

PT

ED

Fig. 5 The typical impact locations at the sandwich plate for FE analysis.

ACCEPTED MANUSCRIPT

(b) P2

CR IP T

(a) P1

(d) P4

AN US

(b) P3

(e) P5

Fig. 6 Deformation modes for the impact response of a sandwich plate with h  0.25 ,

c  0.04 ,   0.1 ,   0.05 and  D  0.7 struck by a heavy mass with the mass G  83 and

AC

CE

PT

ED

M

velocity V0  4 m s at (a) P1, (b) P2, (c) P3, (d) P4, (e) P5.

CR IP T

ACCEPTED MANUSCRIPT

(b) P2

(d) P4

AC

CE

PT

ED

(c) P3

M

AN US

(a) P1

(e) P5 Fig. 7 The FE results of face-sheet deflections versus time for the impact response of a sandwich plate with h  0.25 , c  0.04 ,   0.1 ,   0.05 and  D  0.7 struck by a heavy mass with the mass G  83 and velocity V0  4 m s at (a) P1, (b) P2, (c) P3, (d) P4, (e) P5.

CR IP T

ACCEPTED MANUSCRIPT

(b) P2

AN US

(a) P1

(d) P4

(e) P5

AC

CE

PT

ED

M

(c) P3

Fig. 8

Comparisons of the analytical predictions and FE results for the non-dimensional

maximum deflections of the sandwich square plate with h  0.25 , c  0.04 ,   0.1 ,

  0.05 and  D  0.7 as a function of the non-dimensional impact energy of the striker at (a) P1, (b) P2, (c) P3, (d) P4, and (e) P5.

CR IP T

ACCEPTED MANUSCRIPT

(b) P2

(d) P4

AC

CE

PT

ED

(c) P3

M

AN US

(a) P1

(e) P5

Fig. 9 Comparisons of FE results for the non-dimensional maximum deflections of the sandwich square plate with h  0.25 , c  0.04 ,   0.1 ,   0.05 and  D  0.7 as the non-dimensional impact energy equals with different striker mass at (a) P1, (b) P2, (c) P3, (d) P4, and (e) P5.

CR IP T

ACCEPTED MANUSCRIPT

(b) P2

(d) P4

AC

CE

PT

ED

(c) P3

M

AN US

(a) P1

(e) P5 Fig. 10 Comparisons of the analytical predictions and FE results for the non-dimensional impact force versus deflections of the sandwich square plate with h  0.25 , c  0.04 ,

  0.1 ,   0.05 and  D  0.7 at (a) P1, (b) P2, (c) P3, (d) P4, and (e) P5.

(b)

AC

CE

PT

ED

M

AN US

(a)

CR IP T

ACCEPTED MANUSCRIPT

Fig. 11 Comparisons of the analytical predictions and FE results for the non-dimensional maximum deflections at impact locations of the sandwich square plate with h  0.25 ,

c  0.04 ,   0.1 ,   0.05 and  D  0.7 as a function of the non-dimensional impact energy of the striker. (a) G  83 and (b) G  188 .

CR IP T

ACCEPTED MANUSCRIPT

(b)

AN US

(a)

(d)

M

(c)

Fig. 12 The energy absorption of each part of the sandwich square plate with h  0.25 ,

ED

c  0.04 ,   0.1 ,   0.05 and  D  0.7 . (a) Ek  45 , (b) Ek  180 , (c) Ek  320 and (d)

AC

CE

PT

Ek  405 .

AN US

CR IP T

ACCEPTED MANUSCRIPT

Fig. 13 The energy absorption proportions of the sandwich square plate at impact point P2

h  0.25

,

c  0.04

,

  0.1 ,   0.05 and  D  0.7 as a function of the

M

with

AC

CE

PT

ED

non-dimensional impact energy of the striker.

AC

CE

PT

ED

M

AN US

(a)

CR IP T

ACCEPTED MANUSCRIPT

(b)

Fig. 14 The energy absorption proportions of the sandwich square plate with h  0.25 ,

c  0.04 ,   0.1 ,   0.05 and  D  0.7 .(a) Ek  180 and (b) Ek  320 .

ACCEPTED MANUSCRIPT

Table captions Table 1 Parameters for the strikers in FE analysis. Ek

0.5

1.25

1

5

1.5

11.25

2

20

2.5

31.25

3

45

83

144

4

80

5

125 180

ED

6

245

8

320

PT

7

CE

9

AC

G (kg)

405

V0 (m/s)

G

Ek

CR IP T

V0 (m/s)

AN US

64

G

M

G (kg)

0.5

2.8125

1

11.25

1.5

25.3125

2

45

3

101.25

4

180

5

281.25

6

405

188