Theoretical analysis of aluminum honeycomb sandwich panel supported by reinforced concrete wall under low-speed impact load

Nomenclature

D _a

the total energy absorbed by the energy absorbing structure

E _S

the energy absorption per unit area

h

the thickness of face panel

H

the thickness of honeycomb core

K _l

the kinetic energy loss

l _c

length of the honeycomb

L _L

the length of sandwich panel

L _W

the width of sandwich panel

M ₁

mass of impactor

N ₀

the full plastic film force of the upper panel material of per unit area

P _m

average axial crushing force

R

half of the width of the square hammer head

S and L

the stroke and total length of the energy absorbing device, respectively

S _e

energy amount absorbed by unit mass material

t _c

the single honeycomb wall’s thickness

U _f

the full plastic large deflection deformation energy consumption of upper panel

V ₁ and V ₂

the impact velocity and rebound velocity

W

displacement of upper panel

α

the mass error coefficient during production

δ

impact indentation depth

ε _r

the radial tensile strain of upper panel

ξ

half of the width of depression area

σ _f

the yield stress of face panel material

σ _p

the out-of-plane impact platform stress of honeycomb core

П

the potential energy of whole system

1 Introduction

As excellent low-weight energy absorbents, porous materials play vital roles in energy consumption, cushioning and protection in shipping, aviation, and other fields, such as a thin-walled aviation buffer filled with porous materials. Honeycomb structures belong to the most mature and common periodic porous materials in current use. Its impact resistance and dynamic response under impact have been widely studied. However, in civil engineering, there have been few studies on the impact protection of honeycomb materials as energy absorbing materials for reinforced concrete (RC) components, especially RC walls.

Hu and Lu [1] used experiments and numerical simulation to study hexagonal honeycombs under impact and obtained an analytical formula for the dynamic crushing strength. Hu et al. [2,3,4] established a theoretical model of hexagonal honeycombs under y-directional and x-directional crushing. Khan et al. [5] carried out an experimental study on the mechanical behavior of aluminum honeycomb under in-plane and out-of-plane impact. Xu et al. [6] studied the platform stress of aluminum honeycomb with double wall thickness under out-of-plane dynamic impact, gave the corresponding semi-empirical formula for predicting the platform stress, and studied the energy absorption characteristics of the aluminum honeycomb. Ashab et al. [7,8] conducted out-of-plane impact tests on honeycombs with different specifications. An impact indenter smaller than the size of the honeycomb specimen was used to impact the honeycomb. The proportion of honeycomb tear energy in energy absorption during the impact process was studied, and the proportion of tear energy in energy absorption ranged from 11 to 22%.

Most theoretical analysis of honeycomb materials involves the study of the mechanical properties of honeycomb materials. Yazdanparast and Rafiee [9] developed a computational technique for determining the effective in-plane properties of hexagonal core honeycombs. Harkati et al. described the out-of-plane properties of a curved-wall honeycomb structure that was evaluated using analytical models and finite element techniques to calculate the out-of-plane properties of a curved-wall honeycomb structure [10].

Research on the impact resistance of honeycomb sandwich structures mostly focuses on the honeycomb sandwich structure. Qi et al. [11] studied the dynamic response of an aluminum honeycomb sandwich plate under low-speed impact by finite element simulation. Gao et al. [12] studied the impact resistance of carbon fibre reinforced plastics/aluminum honeycomb sandwich panels by numerical simulation and experimental results.

At present, one of the main applications of honeycomb materials in civil engineering are the impact protection of piers. Pan et al. [13,14] used “U” shaped thin-walled steel plates with glass fiber reinforced plastic-filled honeycomb as an energy absorbing structure that provides protection to overpass bridge piers, and found that the protection structure provides good crashworthiness and functioned well for protecting the bridge pier. Chang et al. [15] summarized the important advances in the basic structural design of honeycombs to obtain new honeycomb with various improved mechanical properties; this study is based on the design of basic cellular structure and not on civil application scenarios.

Thus, theoretical research and energy absorption estimation of honeycomb aluminum as a protective layer for impact protection have not been studied extensively, which prevents the further application of honeycomb energy absorption in civil engineering. Owing to the lack of systematic theoretical support, innovation in practical application is often compromised, and the theoretical modeling and energy dissipation theory of honeycomb sandwich panel protective RC wall under low speeds and flat-head impact loads need to be further studied.

2 Impact deformation

2.1 Impact process

Aluminum honeycomb material is a typical material composed of hexagonal aluminum foil cavity cells arranged regularly. The classic honeycomb structure anisotropy direction of the honeycomb sandwich panel is shown in Figure 1.

Figure 1

Structure and anisotropy direction of honeycomb.

When the honeycomb sandwich panel is used as the impact protective layer to protect the RC wall, considering the local characteristics of the impact, the overall in-plane dimension of the honeycomb sandwich panel is 1 m, i.e., L _L = L _W = 1 m, H = 40 mm, and the thickness of the panel placed on either sides of the honeycomb is 3 mm.

A pendulum test is used to simulate the impact with large mass and low speed. The measured impact speed of the hammer is 2.3 m/s, the impact direction of the hammer is the out-of-plane direction of the honeycomb, the mass of the entire pendulum is 2 t, and a 340 mm × 340 mm × 50 mm cast iron backing plate was set between the hammerhead and the honeycomb sandwich panel, which is the impact indenter. The entire test restraint and bonding position of the honeycomb sandwich plate are shown in Figure 2.

Figure 2

Pendulum impact test. (a) Pendulum test and (b) numerical model.

The upper beam of the RC wall can be considered simply supported, while the lower beam is assumed to be fixed. RC wall can be regarded as the immovable back support of the honeycomb aluminum sandwich panel. The impact load parameters in the finite element model (FEM) are consistent with the test. The upper beam is simply supported, and the lower beam is fixed in the FEM.

2.2 Impact deformation

The indentation of the protective layer on the honeycomb sandwich panel after the pendulum impact is shown in Figure 3(a), the deformation of the honeycomb cladding panel obtained by the built FEM is shown in Figure 3(b), and the ideal indentation model established according to the actual impact deformation and panel deformation of numerical simulation is shown in Figure 3(c).

Figure 3

Deformation in concave area by impact pressing. (a) Actual deformation. (b) FEM deformation. (c) Ideal indentation model.

Compared to the local breaking phenomenon under high energy and strain rates when the back is empty or when filled with water, and the interlayer cracking and failure phenomenon between the panel and the core layer, the sandwich panel supported by the RC wall has good integrity under the local accident impact. In this case, only local depression occurs, there is subsidence within 10 cm around the pit, and there is no cracking and failure between the panel and the core layer.

2.3 Description of impact process

It can be inferred that under ideal conditions, when the impact load with the flat indenter acts on the protective layer, the impact process on the protective anti-impact layer is shown in Figure 4.

Figure 4

Impact process.

Under the impact of the rigid hammer, the cast iron plate acts as a square rigid impact head. Under the impact of the high-energy rigid flat head, the area around the flat head on the impact surface of the sandwich panel will yield, and the face panel which is in contact with the impact indenter (upper panel) and honeycomb core on the impact contact surface will sink, and the recessed honeycomb core will undergo progressive folding and buckling to form a square flat pit (as shown in Figure 4) surrounded by a transitional area (as shown in Figure 5). The upper panel and honeycomb core in the depression area bear all the energy consumption on the sandwich panel. It is found that the panel in contact with the RC wall (lower panel) hardly undergoes deformation. After the impact process is completed, the lower panel remains unchanged without local deformation.

Figure 5

Depression area of protective layer.

3 Impact theory model under square intender

3.1 Impact contact force

According to Chai and Zhu [16], the large mass impact pressing process of a sandwich plate is closely related to its constraints, and the entire process can be considered as quasi-static pressing. The impact depression area in the ideal state is established as shown in Figure 6, and considering the area in the circle as an example, 2 R is the width of the square hammer head.

Figure 6

Indentation model.

The entire impact depression area is symmetrical about the x and y axes. In addition to the displacement field expression, the following analysis is based on the half model of the impact area. The structural models of regions A and C are established along the x-axis, as shown in Figure 7(a), and the structural models of corner regions B and C are established along the diagonal of region C, as shown in Figure 7(b). The shape of region B is regarded as a quarter circle whose radius is ξ − R.

Figure 7

Schematic profile of the deforming zone under a square flat indenter. (a) Schematic diagram of B and A. (b) Schematic diagram of B and C.

Based on the research in ref. [17], the expression of displacement field in one quarter of impact area is given as equation (1).

(1) W ( x , y ) = δ , 0 < x < R , 0 < y < R , δ 1 − x − R ξ − R 2 , R < x < ξ , 0 < y < R , δ 1 − y − R ξ − R 2 , 0 < x < R , R < y < ξ , δ 1 − x 2 + y 2 − 2 R ξ − R 2 , R < x < ξ , R < y < ξ .

The potential energy of the whole system is [18] as follows:

(2) Π = U f + U h − W .

In equation (2), U _f is the full plastic large deflection deformation energy consumption of impact surface aluminum plate, which can be written as:

(3) U f = 4 ∫ A N 0 ε ( x , y ) d A + 2 ∫ C N 0 ε r d C .

In equation (3), N ₀ is the full plastic film force of the upper panel material per unit area:

(4) N 0 = σ f h .

In equation (3), ε _r and ε(x,y) is the radial tensile strain of upper panel, and the tensile strain of the panel in the polar coordinate system can be approximately as follows [18]:

(5) ε r = 1 2 ∂ w ∂ r 2 .

Equation (5) can be transformed into rectangular coordinate system:

(6) ε ( x , y ) = 1 2 ∂ w ( x , y ) ∂ x ⋅ ∂ x ∂ r 2 = 1 2 ∂ w ( x , y ) ∂ x ⋅ x 2 + y 2 x 2 .

Substitute equations (4)–(6) in equation (3) to obtain:

(7) U f = 4 3 σ f h δ ( ξ − R ) + π 3 σ f h δ 2 .

In equation (2), U _h is the energy consumption of honeycomb compression of honeycomb core:

(8) U h = ∫ V σ p d V = ∫ R ξ σ p W ( x , y ) ⋅ 2 R x d x + 2 R 2 σ p δ + ∫ 2 R ξ − R + 2 R σ p W ( r ) ⋅ π r d r .

Obtained as equation (9):

(9) U h = 1 6 σ p δ ( π ξ 2 + 4 R ξ + 8 R 2 ) .

In equations (8) and (9), σ _p is the out-of-plane impact platform stress of the honeycomb core under low-speed impact, which can be generally obtained from equation (10):

(10) σ p = 5.6 t c l c 5 / 3 σ c .

In equation (10), t _c and l _c are the honeycomb wall’s thickness and length of the honeycomb, respectively, and σ _c is the yield stress of honeycomb matrix material.

In equation (2), W is the external work done by the impact force:

(11) W = ∫ 0 δ F d δ .

Substitute equations (7), (9), and (11) in equation (2) to obtain the total system potential energy of the impact structure:

(12) Π = 4 3 σ f h δ ( ξ − R ) + π 3 σ f h δ 2 + 1 6 σ p δ ( π ξ 2 + 4 R ξ + 8 R 2 ) − ∫ 0 δ F d δ .

∂ Π ∂ δ = 0 , the impact force is expressed as equation (13):

(13) F = 4 3 σ f h ( ξ − R ) + 2 π 3 σ f h δ + 1 6 σ p ( π ξ 2 + 4 R ξ + 8 R 2 ) .

Note: in equation (13), ref. [18] defines a series of dimensionless parameters to analyze the dimensionless impact force of the system, and a more concise expression of dimensionless impact force can be obtained. The initial impact force for the square plane indenter is as follows:

(14) F 0 = 4 R 2 σ p .

It can also be seen from ref. [14] that:

(15) ξ = ( 2 N 0 δ / σ p ) 1 / 2 .

Then, the impact force F can be finally reduced to:

(16) F = 4 3 N 0 [ ( 2 N 0 δ / σ p ) 1 / 2 − R ] + 2 π 3 N 0 δ + 1 6 σ p [ π ( 2 N 0 δ / σ p ) + 4 R ( 2 N 0 δ / σ p ) 1 / 2 + 8 R 2 ] .

Resulting in:

(17) F = f ( δ ) .

3.3 Energy dissipation of protective panel

3.3.1 Energy dissipation theory of honeycomb core

To protect key structures and components that may be subjected to various impact and explosion loads, a sacrificial layer can be created on the outer wall as an energy absorption device to ensure the safety of the structural system. This form of protection can be realized by refitting or upgrading the existing structural system and absorbing the instantaneous impact energy through the inelastic behavior of the covering material that will resist the impact and explosion load. For the impact protection of RC members, the overall structure of the aluminum honeycomb impact cover is shown in Figure 8.

Figure 8

Mechanical response of protective panel. (a) Typical protection unit. (b) Idealized crushing process.

The covering layer can contain countless units to work together to protect the object. For the covering layer of aluminum honeycomb material as the web, the honeycomb hole and its upper and lower panels act as a protective unit. The overburden system reduces the impact force by acting on the base plate and propagating the pulse for a long time. Assuming that the energy absorption is mainly due to the plastic deformation of the web, the energy of the aluminum honeycomb sandwich panel is mainly consumed by the plastic deformation or fracture of the aluminum honeycomb core.

The efficiency of the energy absorption device can be characterized by the specific energy (S _e ), which is defined as the amount of energy absorbed by the material per unit mass [19]:

(20) S e = D a m ,

where D _a is the total energy absorbed by the energy absorbing structure, and m is the total mass of the energy absorbing device.

The energy absorption efficiency can also be measured in terms of stroke utilization [19]:

(21) S te = S L .

In equation (21), S _te is the stroke utilization rate; S and L are the stroke and total length of the energy absorbing device, respectively.

For aluminum honeycomb as an energy absorbing structure, the values and evaluation of energy absorption efficiency collected by Ezra [20] are shown in Table 1.

Table 1

Energy absorption of aluminum honeycomb structure

Parameter	S e (kJ/kg)	S te	Force stroke characteristic	Comment
Aluminum honeycomb (crushing)	15–30	0.7	Constant when S te = 0.7	Full bearing capacity, reliable

Energy absorbed per unit area E S of honeycomb sandwich with thickness H is as follows:

(21) E S = ρ s SHS e S = ρ s HS e = 4156.922 t α HS e l ,

where the specific energy S _e is taken as the minimum value of 15 and the maximum value of 30, α is the mass error coefficient during production, E _S is the energy absorption per unit area, the maximum and minimum values of E _S (kJ) can be estimated by the formula:

(22) E S min = 62353.83 t α H l ,

(23) E S max = 124707.66 t α H l ,

where H is the thickness of honeycomb sandwich layer; t is the thickness of aluminum foil on the honeycomb wall; l is the side length of honeycomb hole; α is the quality error coefficient of aluminum honeycomb, which shall be determined according to the retained samples of this batch of products of the manufacturer.

3.3.2 Energy dissipation theory of honeycomb crushing

When such inelastic collision energy consumption occurs in the honeycomb cover, considering that the velocity before impact of the impactor is V ₁ and the mass is M ₁, the total mass of the impacted RC wall and honeycomb cover is M ₂, and the recovery coefficient is regarded as 0 (completely inelastic collision), according to the equation of momentum conservation [19]:

(24) M 1 V 1 = ( M 1 + M 2 ) V 3 .

where V ₃ is the common velocity of two colliding objects at the moment after collision. Then, the kinetic energy loss is [19] as follows:

(25) K l = 1 2 M 1 V 1 2 − 1 2 ( M 1 + M 2 ) V 3 2 .

Use equation (24) to eliminate V ₃, and equation (25) can be written as [19]:

(26) K l = ( M 1 V 1 2 / 2 ) / ( 1 + M 1 / M 2 ) ,

where 1 2 M 1 V 1 2 is the initial kinetic energy of the impactor M ₁.

Considering the constraints of the honeycomb covering layer and RC wall of the impacted object, the actual collision process remains static, that is, equation (26) can be written as:

(27) K l = 1 2 M 1 V 1 2 .

The energy absorbed by a single thin-walled honeycomb structure undergoing dynamic progressive buckling [21] with reference to the average axial crushing force of P _m is as follows:

(28) D a = P m Δ ,

where ∆ is the total axial crushing amount.

For the typical failure mode of honeycomb sandwich plate in ref. [22], the axial crushing distance ∆ can be regarded as the displacement of impact compression region δ. Then, the total number of honeycombs n in the impact compression area of the flat top indenter is as follows:

(29) n = S S l ,

where S is the area of the flat pit; S _l is the area of a single honeycomb. Under special circumstances, for example, when the flat top square indenter is used:

(30) n = 4 R 2 S l = 8 R 2 l 2 3 3 = 1.5396 R 2 l 2 .

When the impact energy is completely absorbed by the honeycomb hole [19]:

(31) D a = n P m Δ = K l .

Then, when one end of the energy absorption device remains stationary:

(32) 1.5396 R 2 l 2 P m Δ = M 1 V 1 2 2 .

The predicted total crushing distance is as follows:

(33) δ = M 1 V 1 2 2 n P m = M 1 V 1 2 2 P m ⋅ l 2 1.5396 R 2 .

4 Conclusion

Based on the results of a simulated pendulum impact test and impact with large mass and low velocity, a deformation model of the porous sandwich panel supported by the RC wall with large mass flat square indenter is established, and a series of theories related to the model are deduced using an axisymmetric model of the impact depression area.

1. The ideal theoretical model of indentation deformation of an aluminum honeycomb sandwich panel impacted by a flat square indenter can be divided into a flat pit area and transitional area. The shape of the flat pit area is consistent with that of the flat top indenter. The transition area during square head indentation can be divided into quarter circle transition area on the four corners of the square indenter and rectangle area on the four sides.

2. On the basis of conclusion (1), the relationship between the impact force and the pressing depth and the size of the pressing area is obtained.

3. On the basis of conclusion (2), the energy balance model of the sandwich plate subjected to square flat impact indenter is extended.

4. According to the theory of energy dissipation of aluminum honeycomb sandwich layer, the general law of energy dissipation of the sandwich plate for RC wall protection is obtained, and the formula for estimating the crushing distance of the covering layer based on the performance the of honeycomb material is given.