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Composites Part A journal homepage: www.elsevier.com/locate/compositesa

On dynamic response of corrugated sandwich beams with metal foam-ﬁlled folded plate core subjected to low-velocity impact ⁎

Qinghua Qina,c, Wei Zhanga, Shiyu Liua, Jianfeng Lia, Jianxun Zhanga, , L.H. Pohb,

T

⁎

a

State Key Laboratory for Strength and Vibration of Mechanical Structures, School of Aerospace, Xi’an Jiaotong University, Xi’an 710049, China Department of Civil and Environmental Engineering, National University of Singapore, Singapore 117576, Singapore c State Key Laboratory of Explosion Science and Technology, Beijing Institute of Technology, Beijing 100081, China b

A R T I C LE I N FO

A B S T R A C T

Keywords: Metal sandwich beam Foam-ﬁlled core Low-velocity impact Yield criterion Large deﬂection

The paper focuses on fully clamped corrugated sandwich beams with metal foam-ﬁlled folded plate core to investigate its response subjected to low-velocity impact. The yield criteria for the metal foam-ﬁlled corrugated sandwich beam cross-section are obtained by considering the strength eﬀects of metal foam and folded plate. Based on the yield criteria, dynamic and quasi-static models are developed to analytically predict the large deﬂections of corrugated sandwich beams, respectively, which agree well with ﬁnite element results. Furthermore, it is shown that the strain hardening of face sheets and folded plate do not signiﬁcantly inﬂuence the low-velocity impact response.

1. Introduction Due to the excellent comprehensive properties over monolithic structures, sandwich structures are widely adopted in critical engineering ﬁelds, such as aerospace, aircraft, vehicle, high speed train and marine industries. A conventional sandwich structure comprises of two stiﬀ face sheets separated by a lightweight core, for example, metal foam [1,2], honeycomb [3], corrugated plate [4,5], pyramidal truss [6,7], and Kagome lattice [8]. Sandwich structures with metal foam core are widely used as energy absorption members due to a long region of plateau stress after initial failure, however their load-carrying capability is limited by the relatively low peak strength [9]. On the contrary, sandwich structures with lattice cores are widely used as primary load-carrying structures because of their high peak loads. However their energy absorption performances are somewhat limited since the residual load-carrying capacities usually drop rapidly upon reaching the peak values [6,10]. In order to combine the advantageous properties of the two systems, the hybrid lattice/foam-cored sandwich structures have been designed. In engineering applications, sandwich structures are often utilized for protection against low-velocity impact, e.g. dropped tools, hailstones, and runway debris. For design purposes, an eﬃcient analysis that adequately captures the impact response is required. This motivates the focus of the paper. Over the past decades, low-velocity impact behaviors of

⁎

conventional foam-cored or lattice-cored sandwich structures have been investigated extensively. Crupi et al. [11], Yu et al. [12,13] and Tan et al. [14] experimentally studied the dynamic response of metal foam core sandwich structures, and diﬀerent dynamic failure modes were observed with diﬀerent geometry and material properties of the face sheet and core. Crupi et al. [15], St-Pierre et al. [16] and Zhang et al. [17] experimentally investigated the low-velocity impact response of sandwich structures with diﬀerent lattice cores such as honeycomb, corrugated plate, Y-frame and pyramidal truss. Experimental results revealed that diﬀerent topological cores induced diﬀerent collapse modes, which inﬂuence the capability of impact resistance. Based on these experimental observations, much theoretical work to predict the impact response of sandwich structures has also been carried out. Hazizan and Cantwell [18,19] theoretically predicted the low-velocity impact response of foam-based and aluminum honeycomb sandwich structures with an energy-balance model. Foo et al. [20] extended the validity of this model beyond the elastic regime. In addition, an analytical spring-mass model [21,22] has also been proposed for the same purpose. Li et al. [23] developed an elastic–plastic model to predict the dynamic response of a simply supported composite sandwich beam, in which the idealized bending hinge was adopted. Furthermore, Wang and co-authors [24–26] theoretically predicted the large deﬂection lowvelocity impact response of slender metal foam core sandwich beam with symmetric and asymmetric face sheets, where a yield criterion incorporating the eﬀect of core strength was adopted. Also, some

Corresponding authors. E-mail addresses: [email protected] (J. Zhang), [email protected] (L.H. Poh).

https://doi.org/10.1016/j.compositesa.2018.08.015 Received 28 May 2018; Received in revised form 11 August 2018; Accepted 13 August 2018 Available online 15 August 2018 1359-835X/ © 2018 Elsevier Ltd. All rights reserved.

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interesting conclusions were reported based on the numerical simulations of sandwich structures under low-velocity impact [20,27,28]. Similar work has also been done on the impact behavior of hybrid lattice/foam-cored sandwich structures. Yazici et al. [29] experimentally and numerically investigated the blast resistance of foam-ﬁlled corrugated core steel sandwich structures. It was shown that the addition of foam inﬁll strengthens the performance of sandwich panels against blast. Zhang et al. [30] experimentally studied the energy absorption and low velocity impact response of polyurethane foam ﬁlled pyramidal lattice core sandwich panels. Compared to sandwich panels with only lattice or foam ﬁlled cores, the results suggest that the loadcarrying capacity and energy absorption eﬃciency of foam ﬁlled sandwich panels can exceed the values given by the sum of the two component cores. Zhang et al. [31] theoretically investigated the compressive strengths and blast responses of corrugated sandwich plates with unﬁlled and foam-ﬁlled sinusoidal plate core. Therewith, the compressive strengths of unﬁlled and foam-ﬁlled sinusoidal plate cores were derived and a simpliﬁed plastic-string model was developed to predict the large deﬂection blast response. However, to the authors’ knowledge, there is little work on the theoretical investigations of lowvelocity impact response of hybrid lattice/foam-cored sandwich structures. The objective of this work is to investigate low-velocity impact response of fully clamped corrugated sandwich beams with metal foamﬁlled folded plate core. The paper is organized as follows. In Section 2, the problem formulation is presented. In Section 3, the yield criteria of metal foam-ﬁlled corrugated sandwich beam cross-section are derived, considering both the eﬀects of metal foam and folded plate. In Section 4, analytical models for the large deﬂection responses of the fully clamped corrugated sandwich beams with metal foam-ﬁlled folded plate core subjected to low-velocity impact are developed. In Sections 5 and 6, comparisons between numerical simulations and analytical predictions are performed. Finally, concluding remarks are presented in Section 7.

Fig. 2. Sketches of corrugated sandwich beams. (a) Metal foam-ﬁlled sandwich beam, (b) half unit cell of the unﬁlled sandwich beam, and (c) half unit cell of the metal foam-ﬁlled sandwich beam.

plate and metal foam are ρf , ρfc and ρc , respectively. It is assumed that the face sheets and folded plate obey the rigid-perfectly plastic (r-p-p) constitutive relation with yield strengths σf and σfc , while the metal foam core follows the rigid-perfectly plastic-locking(r-p-p-l) material with yield strength σc and densiﬁcation strain εD , as shown in Fig. 3.

3. Yield criterion Consider a sandwich cross-section shown in Fig. 4. It is assumed that the sandwich cross-section has a fully plastic stress distribution resulting from a combination of a bending moment M and an axial force N. The distance between the plastic neutral surface and the external surface of the bottom face sheet is denoted by H = ξ × (c + 2h) , where ξ ∈ [0, 1]. Then the axial force N and the bending moment M can be given by

2. Problem formulation Consider a corrugated sandwich beam of span 2L with metal foamﬁlled folded plate core fully clamped at its two vertical surfaces. A heavy mass Gs of initial velocity VI struck the beam at a distance L1 from left support as depicted in Fig. 1. Two identical face sheets, each of thickness h , are assumed to be perfectly bonded to the hybrid core of thickness c comprising of folded plates and ﬁlled metal foam. Sketches of corrugated sandwich beam, half unit cell of unﬁlled and metal foamﬁlled sandwich beams are provided in Fig. 2, respectively. The width of half unit cell is denoted as b , the thickness and angle of inclination of the folded plate are bc and θ , and the densities of face sheets, folded

b

N=

⎧ 2σf b [h−ξ (c + 2h)] + (σfc−σc ) sincθ c + σc bc, ⎨ [(σfc−σc ) bc + σc b ](c + 2h)(1−2ξ ), sin θ ⎩

0⩽ξ⩽ h c + 2h

⩽ξ

h c + 2h 1 ⩽ 2

(1) and

Fig. 1. Sketch of a fully clamped corrugated sandwich beam with metal foam-ﬁlled folded plate core struck by a heavy mass with low-velocity. 108

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M

=

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ σf b (c + 2h)2ξ (1−ξ ), 0⩽ξ⩽

h c + 2h

1 {σ b [(c + 2h)2−c 2] 4 f 1 h ⩽ξ⩽ 2 c + 2h

Fig. 3. Material properties of (a) the face sheets, folded plate and (b) the metal foam.

b

+ [(σfc−σc ) sincθ + σc b ][c 2−(c + 2h)2 (1−2ξ )2]},

(2) respectively. The fully plastic axial force NP and bending moment MP correspond to ξ = 0 and ξ = 1/2 respectively, to give

NP = 2σf bh + (σfc−σc )

bc c + σc bc sin θ

(3)

and

MP = σf bh (c + h) +

1 b [(σfc−σc ) c + σc b ] c 2. 4 sin θ

(4)

Combination of Eqs. (1)–(4) leads to

Fig. 4. Distributions of the strain and stress on cross-section of half unit cell of corrugated sandwich beam under bending moment and axial force. (a) 0 ⩽ ξ ⩽ and (b)

h c + 2h

⩽ξ⩽

1 . 2

109

h c + 2h

Composites Part A 114 (2018) 107–116

Q. Qin et al. 2ξ (1 + 2h¯ )

⎧1− , b¯ 2h¯ + (σ¯ fc − σ¯c ) c + σ¯c ⎪ sin θ ⎪ n= b¯ ⎨ [(σ¯fc − σ¯c ) sincθ + σ¯c ](1 + 2h¯)(1 − 2ξ ) , ⎪ b¯ 2h¯ + (σ¯ fc − σ¯c ) c + σ¯c ⎪ sin θ ⎩

0⩽ξ⩽ h¯ 1 + 2h¯

h¯ 1 + 2h¯

⩽ξ⩽

1 2

(5)

and ¯ 2

4ξ (1 − ξ )(1 + 2h) ⎧ , b¯ 4h¯ (1 + h¯ ) + (σ¯ fc − σ¯c ) c + σ¯c ⎪ sin θ ⎪ m= b¯ [(σ¯ fc − σ¯c ) c + σ¯c ](1 + 2h¯ )2 (1 − 2ξ )2 ⎨ sin θ , ⎪ 1− b¯ 4h¯ (1 + h¯ ) + (σ¯ fc − σ¯c ) c + σ¯c ⎪ sin θ ⎩

h¯ 1 + 2h¯

0⩽ξ⩽ h¯ 1 + 2h¯

⩽ξ⩽

1 2

(6)

where n = N / NP , m = M / MP , σ¯fc = σfc / σf , σ¯c = σc / σf , h¯ = h/ c and b¯c = bc / b . Eliminating the parameter ξ , the yield criterion for metal foam-ﬁlled corrugated sandwich beam cross-section can be written as

⎧|m| +

(A + 2B )2 n2 A [A + 4B (1 + B )]

⎨ |m| + ⎩

[(A + 2B ) | n | +1 − A]2 − (1 + 2B )2 A + 4B (1 + B )

= 1,

0 ⩽ |n| ⩽ = 0,

A A + 2B

A A + 2B

,

⩽ |n| ⩽ 1

Fig. 6. Yield loci of metal foam-ﬁlled corrugated sandwich cross-sections. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

(7a)

where A = (σ¯fc−σ¯c ) b¯c /sin θ + σ¯c , B = h¯ . For an unﬁlled corrugated sandwich beam cross-section, i.e. σ¯c = 0 , the above yield criterion can be reduced to

⎧ ⎪|m| + ⎨ |m| + ⎪ ⎩

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(σ¯c + 2h¯ )2 n2 4σ¯c h¯ (1 + h¯ ) + σ¯c2

= 1,

{(σ¯c + 2h¯ ) |n| + 1 − σ¯c }2 − (1 + 2h¯ )2 4h¯ (1 + h¯ ) + σ¯c

0 ⩽ |n| ⩽ = 0,

σ¯c σ¯c + 2h¯

σ¯c σ¯c + 2h¯

⩽ |n| ⩽ 1

, (7c)

where A = σ¯c , B = h¯ . Particularly, as σ¯c = σ¯fc = 1, which corresponds to a monolithic solid structure, Eq. (7a) is reduced to the well-known yield criterion of the monolithic rectangular solid cross-section [32], b¯c 2 ) sin θ n2 = 1, b¯ b¯ [4h¯ (1 + h¯ ) + σ¯ fc c ] σ¯ fc c sin θ sin θ b¯ b¯ {(2h¯ + σ¯ fc c ) |n| + 1 − σ¯ fc c }2 − (1 + 2h )2 sin θ sin θ + b¯ 4h¯ (1 + h¯ ) + σ¯ fc c sin θ

|m| +

|m|

(2h¯ + σ¯ fc

0 ⩽ |n| ⩽

= 0,

b¯c sin θ b¯ 2h¯ + σ¯ fc c sin θ

σ¯ fc

b¯c sin θ b¯ 2h¯ + σ¯ fc c sin θ

σ¯ fc

|m| + n2 = 1.

(8)

Fig. 5 shows the yield loci for the metal rectangular sandwich crosssections with various core strengths. It is clear that the yield loci expands outwards with increasing core strength. Adopting an associated plastic ﬂow rule for the sandwich beam, we have

,

⩽ |n| ⩽ 1 (7b)

b¯ σ¯fc sincθ ,

B = h¯ . where A = When the core strength is equal to that of the web plate or the thickness of web plate is zero, i.e. σ¯fc−σ¯c = 0 or b¯c = 0 , Eq. (7a) can be reduced to

Np ε ̇ ∂Φp ∂Φp dm · = / =− , Mp κ ̇ dn ∂ (N / Np) ∂ (M / Mp)

(9)

where Φp is the yield function, ε ̇ and κ̇ are the rates of extension and

Fig. 5. Yield loci for the metal rectangular sandwich cross-sections with various core strengths (θ = 45°). (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.) 110

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Fig. 7. Overall deformation pattern of a fully clamped corrugated sandwich beam with metal foam-ﬁlled folded plate cores. (a) Transverse velocity proﬁle and (b) free body diagram.

the mass-beam system after impact is given by

curvature of the neutral surface of the sandwich beam, respectively. Substituting Eq. (7a) into Eq. (9), we obtain 1

B

V0 =

A

0 ⩽ |n| ⩽ A + 2B ⎧ ( 2 + A ) c|n|, ε̇ | κ ̇ |= c . A ⎨ [(A + 2B )|n| + 1−A], ⩽ |n|⩽1 A + B 2 2 ⎩

b

Similar to monolithic solid beam [33], we can obtain the circumscribing and inscribing loci of yield criterion Eq. (7a), (11)

|n| = k and |m| = k ,

(12)

−[L1 (1−

where k is the inscribing coeﬃcient. The sketch of circumscribing and inscribing loci are illustrated in Fig. 6. From Eq. (7a), the inscribing coeﬃcient k can be given by

⎧ ⎪ k= ⎨ ⎪ ⎩

1 + 4k 0 − 1 , 2k 0 k12 + 4k2 − k1 2

where k 0 =

8B2 (B + 1)−A2 ⩾ 0

k1 =

4B (B + 1) + A + 2(A + 2B )(1 − A) , (A + 2B )2

k2 =

2 − A + 2B . A + 2B

eL ≈

4.1. Dynamic response

and

If the depth to span aspect ratio of the sandwich beam is large enough, the metal foam-ﬁlled corrugated sandwich beam deforms in a global manner neglecting the local denting eﬀect of impact point, as shown in Fig. 7. We assume that there are three stationary plastic hinges developed at the impact location and the end supports, respectively. The global deformation pattern of the neutral axis of the sandwich beam remains straight. Then, the velocity proﬁle is assumed to be linear,

ψ1 ≈

W02 2L1

(18)

W0 L1

(19)

respectively. For simplicity, we introduce the following dimensionless parameters:

G∗ =

x

⎧W0̇ (1− 2L − L1 ), 0 ⩽ x ⩽ (2L−L1) , ⎨W0̇ (1 + x ), −L1 ⩽ x < 0 L1 ⎩

(17)

where eL1 and eL2 are the axial extensions concentrated at the left support and the impact point, respectively, as shown in Fig. 7(b). The total elongation and the rotation angular of left segment are

(13)

4. Analytical model

Ẇ =

(16)

eL = eL1 + eL2, ,

(A + 2B )2 , A [4B (B + 1) + A]

L1 G L (2L−L1) ¨ ) Gs + b 1 ] W0 = FW0 + 2M , 2L 3

where F ≈ N for moderate deﬂection. It is assumed that the left segment L1 of the sandwich beam have a total extension eL ,

8B2 (B + 1)−A2 < 0 ,

(15)

where Gb = ρc c + (ρfc −ρc ) sincθ + 2ρf h is the mass per unit length of the metal foam-ﬁlled corrugated sandwich beam. Considering the moment of momentum of the beam with respect to point A, we can obtain

(10)

|n| = 1 and |m| = 1,

Gs VI , Gs + Gb L

Gs , 2Gb L

L1∗ =

L1 , L

c¯ = =

(14)

where W0̇ is the velocity at the impact point. According to the law of momentum conservation, the velocity V0 of

c , L

b¯

ρ∗ =

W0 , c + 2h

(ρfc −ρc ) sincθ + ρc ρf V0∗ =

,

W0∗

V0 . σf / ρf

From Eqs. (10), (18) and (19), the relationship between deﬂection W0∗ and dimensionless axial force n takes the form 111

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2

⎧ A (1 + 2B) (W0∗)3 + W0∗, ⎪ 3[A + 4B (1 + B)] ⎪ (1 + 2B)3 (W0∗)3 + 3(1 + 2B)2 (A − 1)(W0∗)2 ⎪ + 3(1 + 2B)3W ∗ + A − 1 ⎪ 0 , α 2βUK∗ = 3[A + 4B (1 + B )](1 + 2B ) ⎨ ⎪ (A + 2B)(1 + 2B) (W0∗)2 + ⎪ A + 4B (1 + B) ⎪ 4(1 + 2B)3 + 3(2A + 2B − 1)(1 + 2B)2 + A − 1 , ⎪ 3[A + 4B (1 + B )](1 + 2B ) ⎩

0 ⩽ W0∗ < 1 1 + 2B

1 1 + 2B

⩽ W0∗ < 1

W0∗ ⩾ 1

(23) where α = (1 + 1/2G∗)−1, β =

L1∗ (2−L1∗)(1

+ 1/3G∗) ,

UK∗

=

Gs VI2 2P0 (c + 2h)

,

4Mp

P0 = L . Considering the equilibrium of the striker in the vertical direction, the reaction force between the striker and metal foam-ﬁlled corrugated sandwich beam can be deﬁned as

¨ 0. P = −Gs W

(24)

Substituting Eq. (21) into Eq. (24), we obtain the relationship of the normalized reaction force Pr∗ versus the normalized maximum impact point deﬂection W0∗, i.e. 2

1 ⎧ 1 [ A (1 + 2B) (W0∗)2 + 1], 0 ⩽ W0∗ ⩽ 1 + 2B ⎪ β A + 4B (1 + B) ⎪ (1 + 2B){(1 + 2B)[(W ∗)2 + 1] + 2(A − 1) W ∗} 1 0 0 Pr∗ = , 1 + 2B ⩽ W0∗ ⩽ 1 β [A + 4B (1 + B )] ⎨ ⎪ 2(A + 2B)(1 + 2B) ∗ W0∗ ⩾ 1 ⎪ β [A + 4B (1 + B)] W0 , ⎩

(25)

Pr∗

= Pr / P0 . where Similarly, using the circumscribing and inscribing yield criteria, the so-called ‘bounds’ of the dynamic solutions are derived. Substituting the circumscribing yield criterion Eq. (11) into Eq. (16), the governing equation can be written as −[L1 (1−

L1 G L (2L−L1) ¨ ) Gs + b 1 ] W0 = NP W0 + 2MP . 2L 3

The maximum deﬂection

Fig. 8. Comparisons of the dynamic and quasi-static solutions of the normalized maximum deﬂection of fully clamped sandwich beams. (a) G∗ = 10 and (b) G∗ = 200 . (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

W0∗c =

W0∗c

and reaction force

2B ) α 2βUK∗

4(A + 2B )(1 + A + 4B (1 + B ) ( (A + 2B )(1 + 2B ) A + 4B (1 + B )

(26)

Prc∗

are given by

+ 1 −1)

(27)

and

W0∗ =

⎧

A + 2B |n|, A (1 + 2B )

⎨ (A + 2B) | n | +1 − A , 1 + 2B ⎩

0 ⩽ |n| ⩽ A A + 2B

A A + 2B

Prc∗ =

.

⩽ |n| ⩽ 1

(20)

2

2L1∗ (2 − L1∗)(3G∗ + 1)(ρ∗ + 2B )

(28)

respectively. Similarly, substituting the inscribing yield criterion Eq. (12) into Eq. (16), we have

Substituting Eqs. (3), (4), (7) and (20) into Eq. (16) yields 1 ⎧ A (1 + 2B) W0∗2 + 1, 0 ⩽ W0∗ < 1 + 2B ⎪ 4B (1 + B) + A ⎪ ∗2 ∗ ∗ 1 ¯¨ 0 = (1 + 2B)[(1 + 2B)(W0 + 1) + 2(A − 1) W0 ] , ηW ⩽ W0∗ < 1 , 4B (1 + B ) + A 1 + 2B ⎨ ⎪ 2(A + 2B)(1 + 2B) ∗ W0∗ ⩾ 1 ⎪ 4B (1 + B) + A W0 , ⎩

1 2(A + 2B )(1 + 2B ) ∗ [ W0 + 1] β A + 4B (1 + B )

W0∗i =

4(A + 2B )(1 + 2B ) α 2βUK∗ A + 4B (1 + B ) ( + 1 −1) (A + 2B )(1 + 2B ) k [A + 4B (1 + B )]

(29)

and (21)

Pri∗ =

W0∗

. Note that when where η = − 2 ⩾ 1, the sand3¯c [4B (1 + B ) + A](1 + 2B ) wich beam is dominated by axial force and behaves like a plastic string. The initial conditions are

k 2(A + 2B )(1 + 2B ) ∗ [ W0 + 1] β A + 4B (1 + B )

(30)

respectively. 4.2. Quasi-static response

∗ Ẇ 0 |t = 0 = V0∗ and W0∗ |t = 0 = 0.

(22) The quasi-static response may be eﬀective to predict the dynamic response of sandwich beam with metal foam core struck by a heavy mass when the mass of the striker is much larger than that of the sandwich beam [24–26]. Herein, this method is extended to predict the low-velocity impact response of the corrugated sandwich beams with

Combining Eqs. (21) and (22) yields the relationship of the normalized initial impact kinetic energy UK∗ versus the normalized maximum impact point deﬂection W0∗,

112

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Fig. 9. The proﬁles of maximum principal plastic strain of metal foam-ﬁlled corrugated sandwich beam ( ρ¯ = 0.1 , Etf = 0.001 Ef , G∗ = 80 , VI = 2 m/s ). (a) t = 0 ms ; (b) t = 6 ms ; (c) t = 27 ms . (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

metal foam-ﬁlled folded plate core. Consider the quasi-static analysis of a fully clamped corrugated sandwich beam with metal foam-ﬁlled folded plate cores subjected to a concentrated loading P at the impact point O, as shown in Fig. 1. The transverse deﬂection proﬁle and free body diagram of the sandwich beam are similar to those in Fig. 7. Thus, the moment equilibrium equation is

4M + 2FW0−PL1 (2L−L1)/ L = 0.

UK∗ 2

=

(31)

Substituting Eqs. (3), (4), (7a) and (20) into Eq. (31), we obtain the load–deﬂection relation as

1 ⎧ ∗ 1 ∗ { A (1 + 2B) (W0∗)3 + W0∗ }, 0 ⩽ W0∗ < 1 + 2B ⎪ L1 (2 − L1 ) 3[A + 4B (1 + B)] ⎪ (1 + 2B)3 (W0∗)3 + 3(1 + 2B)2 (A − 1)(W0∗)2 + 3(1 + 2B)3W0∗ + A − 1 1 , 1 + 2B ⩽ W0∗ < 1 ⎪ 3[A + 4B (1 + B )](1 + 2B ) L1∗ (2 − L1∗)

⎨ 1 (A + 2B )(1 + 2B ) ∗ 2 ⎪ L1∗ (2 − L1∗) { A + 4B (1 + B) (W0 ) + ⎪ 3 2 ⎪ 4(1 + 2B) + 3(2A + 2B − 1)(1 + 2B) + A − 1 }, 3[A + 4B (1 + B )](1 + 2B ) ⎩

(35)

2

⎧ ∗ 1 ∗ [ A (1 + 2B) (W0∗)2 + 1] 0 ⩽ W0∗ ⩽ 1 1 + 2B ⎪ L1 (2 − L1 ) A + 4B (1 + B) ⎪ (1 + 2B){(1 + 2B)[(W0∗)2 + 1] + 2(A − 1) W0∗} 1 ∗ Pr∗ = ⩽ W 0 ⩽ 1 . 1 + 2B [A + 4B (1 + B )] L1∗ (2 − L1∗) ⎨ ⎪ 2(A + 2B)(1 + 2B) ∗ W0∗ ⩾ 1 ⎪ [A + 4B (1 + B)] L1∗ (2 − L1∗) W0 ⎩

It can be seen that the dynamic solutions of Eqs. (23) and (25) are reduced to the quasi-static solutions Eqs. (32) and (35) when G∗ ≫ 1. Similarly, the so-called ‘bounds’ of the dynamic solutions Eqs. (27)–(30) can also be reduced to the following so-called ‘bounds’ of the quasistatic solutions,

(32)

It is assumed that the initial kinetic energy UK of the striker is dissipated by the plastic deformation of the metal foam-ﬁlled corrugated sandwich beam,

UP = UK =

1 Gs VI2. 2

W0∗c =

∫0

W0

P (W0) dW0.

4(A + 2B )(1 + 2B ) L1∗ (2−L1∗) UK∗ A + 4B (1 + B ) ( + 1 −1) (A + 2B )(1 + 2B ) A + 4B (1 + B ) (36)

(33) and

The absorbed plastic deformation energy UP of the metal foam-ﬁlled corrugated sandwich beam can be calculated as

UP =

.

W0∗ ⩾ 1

Prc∗ =

(34)

and

Combination of Eqs. (32) to (34) leads to 113

1 2(A + 2B )(1 + 2B ) ∗ [ W0 + 1] β A + 4B (1 + B )

(37)

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Fig. 11. The analytical predictions and FE results for the normalized impact force versus deﬂection curves of the corrugated sandwich beams with (a) Etf = 0.001Ef and (b) Etf = 0.01Ef . (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

Fig. 10. Comparisons between the analytical predictions and FE results for the normalized maximum deﬂection of the corrugated sandwich beam versus normalized kinetic energy of the striker. (a) G∗ = 80 and (b) VI = 2 m/s . (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

W0∗i =

5. Finite element analysis

4(A + 2B )(1 + 2B ) L1∗ (2−L1∗) UK∗ A + 4B (1 + B ) ( + 1 −1) (A + 2B )(1 + 2B ) k (A + 4B (1 + B ))

Using the commercial software ABAQUS/Explicit, ﬁnite element (FE) simulations are carried out to investigate the low-velocity impact response of sandwich beams with metal foam-ﬁlled folded plate core transversely struck by a heavy mass. The striker is modeled as a rigid roller. The face sheets, the metal foam and the folded plate are modeled by using eight node bilinear brick elements with reduced integration (C3D8R). Appropriate mesh reﬁnements near the clamped end and the loading location are conducted. A mesh sensitivity checked in calculations revealed that additional mesh reﬁnement did not change the results appreciably. The typical case of the metal foam-ﬁlled corrugated sandwich beam struck at mid-span is considered and symmetric boundary condition is applied on sandwich cross-section at mid-span of the beam. All displacements of nodes at the ends of the sandwich beam are zero. Damping associated with the bulk viscosity in ABAQUS/Explicit is switched oﬀ by setting the bulk viscosity to be zero. Frictionless contact is adopted between the sandwich beam and the striker. The sandwich beam has a half span of L = 200mm. The width of the half unit cell is b = 9mm. The thickness of the top and bottom face sheets, the hybrid core and the web plate are h = 1mm, c = 8 mm and bc = 0.707 mm, respectively. The inclination angle of web plate is θ = 45°. Radius of the

(38) and

Pri∗ =

k 2(A + 2B )(1 + 2B ) ∗ [ W0 + 1] β A + 4B (1 + B )

(39)

respectively, where subscripts c and i denote the circumscribing and inscribing yield criterion. Fig. 8(a) and (b) compare the predictions of dynamic, quasi-static and the ‘bounds’ models for the normalized maximum deﬂections W0∗ versus the normalized maximum initial impact kinetic energy UK∗ , for G∗=10 and 200, respectively. A slight discrepancy between the predictions of the dynamic and quasi-static solutions for G∗ = 10 is observed in Fig. 8a. When G∗ = 200 , however, the dynamic solutions agree well with the quasi-static ones in Fig. 8b. Furthermore, the predictions of both dynamic and quasi-static models using the yield criteria lie within the predictions ‘bounds’ using the circumscribing and inscribing yield criteria. We thus conclude that the quasi-static model can rationally capture the low-velocity impact response of corrugated sandwich beams with metal foam-ﬁlled folded plate core when G∗ ≫ 1. 114

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Comparisons between the analytical predictions and FE results of the normalized maximum deﬂection W0∗of the corrugated sandwich beams with the strain hardening Etf = 0.001Ef and Etf = 0.01Ef versus normalized initial kinetic energy Uk∗ of the striker are shown in Fig. 10. In Fig. 10(a), the mass ratio G∗ = 80 is ﬁxed and the striker velocity VI is varied, and the striker velocity VI = 2 m/s is ﬁxed and the mass ratio G∗ is varied in Fig. 10(b). It can be seen that the quasi-static model agrees well with FE results for the lower kinetic energy Uk∗ of the striker, while the quasi-static model may somewhat overestimate the maximum deﬂections for the higher kinetic energy Uk∗ of the striker. Moreover, both FE results and quasi-static predictions of the deﬂections using the yield criterion lie in the ‘bounds’ using the circumscribing and inscribing yield criteria. The strain hardening of the face sheets and folded plate has slight eﬀect on the dynamic response of metal foam-ﬁlled corrugated sandwich beam. Figs. 11 and 12 show comparisons of the analytical model and FE results of the normalized impact force Pr∗ versus the normalized deﬂection W0∗ of the corrugated sandwich beam with Etf = 0.001Ef and Etf = 0.01Ef . It is clear that FE results have strong oscillation at the initial stage of the normalized impact force–deﬂection curves and the forces increase steadily to the maximum values due to the axial constrains. The analytical model is in good agreement with FE results of impact force of the corrugated sandwich beam with the lower strain hardening Etf = 0.001Ef , while slightly underestimates FE results of impact force for higher strain hardening Etf = 0.01Ef . It is argued that the present analytical model neglects the eﬀect of the strain hardening of materials of the corrugated sandwich beams. Similarly, both FE results and quasi-static predictions of the impact force using the yield criterion lie in the ‘bounds’ using the circumscribing and inscribing yield criteria. 7. Concluding remarks Low-velocity impact response of fully clamped corrugated sandwich beams with metal foam-ﬁlled folded plate core was investigated analytically and numerically. The yield criteria for the metal foam-ﬁlled corrugated sandwich beam cross-section were obtained. Employing the yield criteria, dynamic and quasi-static solutions were determined for the large deﬂections of fully clamped metal foam-ﬁlled corrugated sandwich beams. FE simulations were conducted and good agreement was achieved between the analytical predictions and FE results. Comparisons of analytical predictions and FE results reveal that strain hardening of the face sheets and folded plate has slight eﬀects on the dynamic response of fully clamped metal foam-ﬁlled corrugated sandwich beams. Both FE results and quasi-static predictions using the yield criterion lie in the ‘bounds’ using the circumscribing and inscribing yield criteria.

Fig. 12. The analytical model and FE results for the normalized impact force versus deﬂection curves of the corrugated sandwich beams with (a) G∗ = 80 and (b) G* = 160. (For interpretation of the references to colour in this ﬁgure legend, the reader is referred to the web version of this article.)

loading roller is R = 0.5 mm. Both face sheets and folded plate obey J2 ﬂow theory of plasticity. We assume that they are made of stainless steel with yield strength σf = 200 MPa , elastic modulus Ef = 200 GPa , elastic Poisson’s ratio νef = 0.3, density ρ = 7900 kg/m3 , and linear hardening modulus Etf = 0.001Ef and Etf = 0.01Ef . The metal foam is modeled as a plastic crushable continuum [34]. It is assumed that the crushable foam is made from the same material as the face sheet and has density ρ = 790 kg/m3 , yield strength σc = 0.5ρσ ¯ f = 10 MPa , elastic modulus Ec = ρ¯2 Ef = 2 GPa [9], elastic Poisson’s ratio νec = 0.3, plastic Poisson’s ratio νpc = 0 , a plateau stress σc and a densiﬁcation strain εD = 0.5. Beyond densiﬁcation, we assume that the metal foam obeys a linear hardening law with tangent modulus Etc = 20 GPa .

Acknowledgments The authors are grateful for ﬁnancial support from NSFC (11572234 and 11502189), opening project of State Key Laboratory of Explosion Science and Technology (Beijing Institute of Technology, KFJJ18-07M), Natural Science Basic Research Plan in Shaanxi Province of China (2017JM1020), China Postdoctoral Science Foundation funded project (2015M572546), and the Fundamental Research Funds for the Central Universities.

6. Results and discussion References The proﬁles of maximum principal plastic strain in the metal foamﬁlled corrugated sandwich beam before, during and after the impact are shown in Fig. 9. It can be seen that the sandwich beam deforms in a global manner without obvious local denting at the impact location and there are large plastic strain at the impact location and the end supports. This may provide a justiﬁcation that the assumptions made in the analytical model is reasonable.

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