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Abstract

A new simplified analysis is carried out to predict the dynamic response of clamped metal sandwich plates with various lattice cores subjected to impulsive loading. Based on the yield condition for sandwich plate cross-section, the interaction of bending moment and axial force for the section is decoupled. The simplified analytical solutions of the maximum midspan deflection and structural response time are obtained. Moreover, neglecting the effect of bending moment, a simplified plastic-string model is developed. Comparisons of the present theoretical predictions with previous analytical and numerical results are conducted and good agreement is achieved for a wide range of sandwich core topologies.

Keywords

Metal sandwich plate lattice core impulsive loading large deflection dynamic response

Introduction

Sandwich structures have been widely used because of their excellent mechanical performance compared with monolithic structures, such as lightweight, high specific strength, high specific stiffness.^1–3 Usually, sandwich structures consist of two thin face sheets and a lightweight core and the excellent properties of such structures mainly depend on the innovative core topologies design. Thus, several kinds of core topologies for sandwich structures have emerged, such as metal foams, honeycombs, pyramidal lattice truss core.^2,4–6

Over the last decade, the shock resistances of sandwich structures with various lightweight cores subjected to blast loading have been investigated extensively. Xue and Hutchinson^7,8 have made a comparison between the monolithic solid plates and the metal sandwich plates with pyramidal truss core, square honeycomb core, and folded plate core under the same conditions and results indicated that the blast resistance of metal sandwich plates is superior. Also, numerous numerical simulations were carried out to study the dynamic response of metal sandwich structures with a variety of core topologies subjected to air blast loading and underwater blast loading.^9–12 Meanwhile, many researchers^13–16 experimentally studied the shock resistance of metal sandwich structures using the metal foam projectiles which can be adopted to simulate the shock loading. These experiments also demonstrated that the sandwich structures have better shock resistances than monolithic solid structures under the same conditions.

Much effort of establishing theoretical models to predict the blast response of metal sandwich structures has also been devoted. Fleck and Deshpande¹⁷ proposed a theoretical model to analyze the structural response of clamped sandwich beams under uniform impulsive loading, in which the dynamic response was sequentially divided into three steps. Moreover, based on the classical yield locus of sandwich beam section, the approximate yield criterions were employed. Subsequently, Qiu et al.^9,18 adopted this analytical method to research the structural response of a clamped sandwich beam under blast loading acting in the central region and conducted finite element (FE) simulation to validate this theoretical model. However, the yield criterion in the aforementioned analysis does not take into account the effect of core strength. In order to capture the structural response more accurately, a new yield criteria considering the effect of core strength for the symmetric and asymmetric sandwich cross-sections was proposed by Wang and co-authors,^19–22 and a membrane factor method²³ was used to predict the large deflection impulsive response of clamped metal circular sandwich plates and sandwich beams based on this new yield criterion.^24–26 Results showed that the solutions of impulsive response lie in the upper and lower bounds using this method and are accurate enough to capture the structural response. The plastic limit loci of interaction of bending moment and axial force result in several complications for analyses of dynamic response of the sandwich structure in above models. Therefore, it is necessary to establish a simplified prediction model for decoupling the bending moment and the axial force to capture the large deflection response for the sandwich structure with a wide range of core topologies.

The objective of the study is to establish a new simplified analysis method to capture the structural response of clamped metal sandwich plates with various lattice cores under impulsive loading. The organization of this article is as follows. First, the basic problem is described in section “Problem formulation.” Then, section “Analytical solutions for decoupling model” deals with the decoupling model of the structural response and section “Analytical solutions for plastic–string model” proposes a further simplified plastic-string model. Moreover, comparisons of the present theoretical predictions with the previous analytical and numerical results of the sandwich plates with a wide range of lattice core topologies are carried out in section “Results and discussion.” Finally, section “Concluding remarks” outlines concluding remarks.

Problem formulation

Herein, a metal sandwich plate with lattice core subjected to impulsive loading $I$ per unit length is considered, as shown in Figure 1. The clamped constrains are applied at two ends of the sandwich plate with the span $2 L$ . Two sides of lattice core with thickness $c$ are assumed to be bonded two identical face sheets with thickness $h$ perfectly. As a necessary simplification, the two metal face sheets are assumed to be rigid–perfectly plastic material with yield strength $σ_{f}$ and density $ρ_{f}$ , as shown in Figure 2(a). The core is assumed to be a rigid-perfectly plastic locking (r-p-p-l) material with normal compressive strength $σ_{n}$ , longitudinal strength $σ_{l}$ , densification strain $ε_{D}$ and density $ρ_{c}$ , as shown in Figure 2(b). Effects of the strain rates of the face sheets and core material are neglected in the present analytical models.

Figure 1.

Sketch of a fully clamped sandwich plate subjected to impulsive loading.

Figure 2.

Material properties of sandwich structure: (a) face sheets and (b) core.

For this problem, Fleck and Deshpande¹⁷ assumed that the structural response can be sequentially divided into three steps: fluid–structure interaction, core compression, and beam bending/stretching. As the complexity of the fluid–structure interaction, this step is neglected and an impulse $I$ per unit length is assumed to be applied at the front face sheet uniformly in analysis. Moreover, it is assumed that the reduction of momentum caused by the supports is neglected. Therefore, based on the conservation of momentum, the front face sheet will have an initial velocity $v_{0} = I / I (ρ_{f} (ρ_{f} h)$ before the core compression step and the total structure will have a common velocity at the end of core compression step, that is

v_{f} = \frac{I}{2 ρ_{f} h + ρ_{c} c}

(1)

For the core compression step, the plastic energy is dissipated by the core deformation and the average compressive strain $ε_{c}$ along the direction of the entire thickness of the core is given by, that is¹⁷

ε_{c} = \frac{{\bar{I}}^{2} (2 \bar{h} + \bar{ρ}) (\bar{h} + \bar{ρ})}{2 {\bar{σ}}_{n} \bar{h}}

(2)

where $\bar{I} = I / I [(2 ρ_{f} h + ρ_{c} c) \sqrt{σ_{f} / ρ_{f}}]$ , $ρ = ρ_{c} / ρ_{c} ρ_{f}$ , ${\bar{σ}}_{n} = σ_{n} / σ_{f}$ , $\bar{h} = h / c$ , and $\bar{c} = c / L .$ If the average compressive strain $ε_{c}$ attains the densification strain $ε_{D}$ , $ε_{c}$ is replaced by the value $ε_{D}$ .

When the core compression step ends, the total sandwich plate attains the uniform common velocity $v_{f}$ and the structure will be subjected to a combination of plastic bending and longitudinal stretching. According to the magnitude of the deflection, the main effect of plastic bending and longitudinal stretching is different. For small deflections, the sandwich plate is dominated by the bending, while for large deflections, the sandwich plate is dominated by the axial force. For the monolithic solid plate, Schubak et al.²⁷ uncoupled the dynamic response into an initial bending-only step and plastic string step using the approximate yield condition and gave the critical deflection for the transition from bending-only to plastic string. Similarly, Yuan et al.²⁸ gave a critical deflection for sandwich beam, that is

w_{0 r} = \frac{2 M_{p}}{N_{p}}

(3)

where $M_{p}$ and $N_{p}$ are the fully plastic bending moment and the fully plastic axial force, respectively. For the compressed sandwich plate cross-section, the fully plastic bending moment can be expressed as

M_{p} = σ_{f} bh [c (1 - ε_{c}) + h] + σ_{l} \frac{b c^{2} (1 - ε_{c})}{4}

(4)

while the fully plastic axial force is assumed to be insensitive to the degree of the compression of the core¹⁷ and it can be expressed as

N_{p} = σ_{l} bc + 2 σ_{f} bh

(5)

where $b$ is the width of the sandwich plate.

Therefore, the structural response of sandwich plate under impulsive loading can be redivided into three stages in the following order: stage I is the stage of core compression; stage II is the stage of pure bending; stage III is the stage of plastic string. In addition, there exits two phases of the structural motion: in phase I, there are two stationary plastic hinges at the fixed ends and other two traveling plastic hinges toward the midspan; in phase II, two traveling plastic hinges merge and keep stationary at the midspan. Moreover, according to the magnitude of the impulse loading, the transition from stage II to stage III may occur in either phase of the structural motion. For the high impulse loading, the transition will occur in phase I of the motion, while for the low impulse loading, the transition will occur in phase II of the motion. Then, there exits two critical impulse loading values $I_{cr 1}$ and $I_{cr 2}$ ( $I_{cr 2} > I_{cr 1}$ ),²⁸ that is

I_{cr 1} = \frac{M_{p}}{L} \sqrt{\frac{12 G}{N_{p}}}

(6)

and

I_{cr 2} = \frac{2 M_{p}}{L} \sqrt{\frac{6 G}{N_{p}}}

(7)

where $G = 2 ρ_{f} h + ρ_{c} c$ is the total mass per unit length of sandwich plate.

According to the above analysis, we can get the following three cases. For the case of $I \leq I_{cr 1}$ , the impulse loading is too low and the critical deflection value $w_{0 r}$ will not be attained, so the whole structural response will only be dominated by the bending moment. When the impulse loading is adequate, that is, $I > I_{cr 1}$ , the critical deflection value $w_{0 r}$ can be attained and the whole structural response will be dominated not only by the bending moment but also by the axial force. For this case, there are still two kinds of cases, of which, when $I_{cr 1} < I \leq I_{cr 2}$ , sandwich plate will attain the critical deflection value $w_{0 r}$ in phase II of the motion, and when $I > I_{cr 2}$ , sandwich plate will attain the critical value $w_{0 r}$ in phase I of the motion. We will analyze them separately as follows.

Analytical solutions for decoupling model

The midspan deflection did not attain the critical value $w_{0 r}$ , $I \leq I_{cr 1}$

Similar to Fleck’s analysis,¹⁷ sketches of the velocity profile and the free-body diagram for the sandwich plate are given in Figures 3 –5. When the motion keeps in phase I, there will be two stationary plastic hinges at the fixed ends and other two traveling plastic hinges toward the midspan, as sketched in Figure 3(a). The central portion between two traveling plastic hinges keeps the constant velocity $v_{f}$ , while the adjacent segment of length $ξ$ at each end rotates about the fixed ends. Based on the conversation of the moment of momentum for the half sandwich plate about a fixed end in Figure 3(b), we have the governing equation

\frac{d}{d t} [G (L - ξ) v_{f} (ξ + \frac{L - ξ}{2}) + \int_{0}^{ξ} \frac{G v_{f} x^{2}}{ξ} d x] = - 2 M_{p}

(8)

Figure 3.

Sketch of the velocity profile and the free-body diagram for the case of $I \leq I_{cr 1}$ for small deflection response: (a) velocity profile in phase I, (b) free-body diagram of the half plate in phase I, (c) velocity profile in phase II, and (d) free-body diagram of the half plate in phase II.

Figure 4.

Sketch of the velocity profile and the free-body diagram for the case of $I_{cr 1} < I \leq I_{cr 2}$ for large deflection response of sandwich plate: (a) velocity profile in phase II and (b) free-body diagram of the half plate in phase II.

Figure 5.

Sketch of the velocity profile and the free-body diagram for the case of $I \geq I_{cr 2}$ for large deflection response of sandwich plate: (a) velocity profile in phase I, (b) free-body diagram of the half plate in phase I, (c) velocity profile in phase II, and (d) free-body diagram of the half plate in phase II.

Thus, we obtain the relation of $ξ$ and $t$

ξ = \sqrt{\frac{12 M_{p}}{I} t}

(9)

Then, the two traveling plastic hinges will merge into one plastic hinge at midspan with the continuation of the motion, that is, $ξ (t_{I}) = L$ . Thus, the end time of phase I is

t_{I} = \frac{I L^{2}}{12 M_{p}}

(10)

Correspondingly, the midspan deflection $w_{I}$ at the end time of phase I is

w_{I} = v_{f} t_{I} = \frac{I^{2} L^{2}}{12 G M_{p}}

(11)

Next, the motion will keep in phase II. For this case, two plastic hinges at the fixed ends and one plastic hinge at the midspan will all keep stationary, as sketched in Figure 3(c). Based on the conversation of moment of momentum for the half sandwich plate about a fixed end in Figure 3(d), we have the governing equation

\frac{d}{dt} [\int_{0}^{L} \frac{G {\overset{\cdot}{w}}_{0} x^{2}}{L} dx] = - 2 M_{p}

(12)

Solving equation (12), we have

{\overset{\cdot\cdot}{w}}_{0} = - \frac{6 M_{p}}{G L^{2}}

(13)

Using the initial condition ${\overset{\cdot}{w}}_{0} (t_{I}) = v_{f}$ , $w_{0} (t_{I}) = w_{I}$ , we further obtain

{\overset{\cdot}{w}}_{o} = - \frac{6 M_{p}}{G L^{2}} t + \frac{3 I}{2 G}

(14)

and

w_{0} (t) = w_{I} + \int_{t_{I}}^{t} {\overset{\cdot}{w}}_{0} dt = - \frac{3 M_{p}}{G L^{2}} t^{2} + \frac{3 I}{2 G} t - \frac{I^{2} L^{2}}{48 G M_{p}}

(15)

When ${\overset{\cdot}{w}}_{0} (t_{f}) = 0$ , the final response time and the final midspan deflection can be obtained

t_{f} = \frac{I L^{2}}{4 M_{p}}

(16)

w_{f} = \frac{I^{2} L^{2}}{6 G M_{p}}

(17)

The midspan deflection attained the critical value $w_{0 r}$ in the phase Π of the structural motion, $I_{cr 1} < I \leq I_{cr 2}$

When the impulsive loading is moderate, the critical midspan deflection will attained in phase II of the motion. Before the midspan deflection attained the critical value $w_{0 r}$ , the velocity profile, the free-body diagram, and the governing equations are the same as the case of $I \leq I_{cr 1}$ , as shown in Figure 3(a)–(d). When the critical midspan deflection attained in phase II of the motion, the velocity profile and the free-body diagram of the sandwich plate are sketched in Figure 4. Then, the sandwich plate will respond like a plastic string and be governed by the following differential equation²⁷

G \frac{\partial^{2} w}{\partial t^{2}} - N_{p} \frac{\partial^{2} w}{\partial x^{2}} = 0

(18)

A good approximation solution of equation (18) can be expressed as^29,30

w (x, t) = w_{0} (t) \cdot \cos (\frac{π}{2 L} x)

(19)

Then, the final midspan deflection $w_{0} (t)$ is governed by

{\overset{\cdot\cdot}{w}}_{0} + \frac{π^{2} N_{p}}{4 G L^{2}} w_{0} = 0

(20)

with initial conditions at time $t_{r}$ , which corresponding to the end of the bending stage. Solving equation (20), we can get

t_{f} = t_{r} + \frac{1}{a} \arctan (\frac{{\overset{\cdot}{w}}_{0 r}}{a w_{0 r}})

(21)

and

w_{0} (t_{f}) = w_{f} = \sqrt{w_{0 r}^{2} + {(\frac{{\overset{\cdot}{w}}_{0 r}}{a})}^{2}}

(22)

where $a = π \sqrt{N_{p} / G} / (2 L)$ , and details of solution process are presented in Appendix 1.

As the midspan deflection and the kinetic energy of the sandwich plate in the two stages should be equal, we can get the midspan velocity initial condition by equating the kinetic energies at $t_{r}$ ²⁷

{\overset{\cdot}{w}}_{0 r} = \sqrt{\frac{2}{L} \int_{0}^{L} {[\overset{\cdot}{w} (x, {t_{r}}^{-})]}^{2} d x}

(23)

Combining equations (19) and (23) yields

{\overset{\cdot}{w}}_{0 r} = \sqrt{\frac{2}{L} \int_{0}^{L} {[\overset{\cdot}{w} (x, {t_{r}}^{-})]}^{2} d x} = \sqrt{\frac{2}{L} \int_{0}^{L} {(\frac{x}{L} v_{0 r})}^{2} d x} = \frac{\sqrt{6}}{3} v_{0 r}

(24)

Using the critical deflection $w_{0 r}$ , equations (18) and (19), we can obtain

v_{0 r} = \sqrt{\frac{2 I^{2}}{G^{2}} - \frac{24 M_{p}^{2}}{G L^{2} N_{p}}}

(25)

Then, the midspan velocity at the critical deflection is

{\overset{\cdot}{w}}_{0 r} = \frac{\sqrt{6}}{3} v_{0 r} = \sqrt{\frac{4 I^{2}}{3 G^{2}} - \frac{16 M_{p}^{2}}{G L^{2} N_{p}}}

(26)

Thus, the final response time is

t_{f} = \frac{I L^{2}}{4 M_{p}} - L \sqrt{\frac{I^{2} L^{2}}{18 M_{p}^{2}} - \frac{2 G}{3 N_{p}}} + \frac{2 L}{π} \sqrt{\frac{G}{N_{p}}} \arctan (\frac{L}{π M_{p}} \sqrt{\frac{4 N_{p} I^{2}}{3 G} - \frac{16 M_{p}^{2}}{L^{2}}})

(27)

and the final midspan deflection is

w_{f} = \sqrt{\frac{4 M_{p}^{2}}{N_{p}^{2}} + \frac{16 I^{2} L^{2}}{3 π^{2} G N_{p}} - \frac{64 M_{p}^{2}}{π^{2} N_{p}^{2}}}

(28)

The midspan deflection attained the critical value $w_{0 r}$ in phase I of the structural motion, $I > I_{cr 2}$

As the increase of the impulsive loading, the critical midspan deflection value $w_{0 r}$ may be attained in phase I of the structural motion. For this case, the velocity profile and the free-body diagram of sandwich plate are sketched in Figure 5. It is noted that the central portion between two traveling plastic hinges will keep the velocity $v_{f}$ constant in phase I. Thus, critical time $t_{r}$ is obtained as

t_{r} = \frac{w_{0 r}}{v_{f}} = \frac{2 G M_{p}}{I N_{p}}

(29)

According to the critical time $t_{r}$ and equation (8), we can get

ξ = \frac{2 M_{p}}{I} \sqrt{\frac{6 G}{N_{p}}}

(30)

Similarly, solving equation (23) yields

{\tilde{w}}_{0 r} = \sqrt{\frac{2}{L} \int_{0}^{L} {[\tilde{w} (x, t_{r}^{-})]}^{2} d x} = \sqrt{\frac{2}{L} (\int_{0}^{ξ} {(\frac{x}{ξ} v_{f})}^{2} d x + \int_{ξ}^{L} v_{f}^{2} d x)} = \sqrt{v_{f}^{2} (2 - \frac{4 ξ}{3 L})}

(31)

The midspan velocity at the critical deflection is

{\overset{\cdot}{w}}_{0 r} = \sqrt{\frac{2 I^{2}}{G^{2}} - \frac{8 I M_{p}}{3 L} \sqrt{\frac{6}{N_{p} G^{3}}}}

(32)

Thus, the final response time is

t_{f} = \frac{2 G M_{p}}{I N_{p}} + \frac{2 L}{π} \sqrt{\frac{G}{N_{p}}} \arctan (\frac{I L}{π M_{p}} \sqrt{\frac{2 N_{p}}{G} - \frac{8 M_{p}}{3 I L} \sqrt{\frac{6 N_{p}}{G}}})

(33)

and the final midspan deflection is

w_{f} = \sqrt{\frac{4 M_{p}^{2}}{N_{p}^{2}} + \frac{8 I^{2} L^{2}}{π^{2} G N_{p}} - \frac{32 IL M_{p}}{3 π^{2}} \sqrt{\frac{6}{{GN}_{p}^{3}}}}

(34)

For simplicity, we introduce the following dimensionless parameters

α_{1} = 4 \bar{h} (1 - ε_{c} + \bar{h}) + {\bar{σ}}_{l} (1 - ε_{c})

(35a)

α_{2} = \bar{ρ} + 2 \bar{h}

(35b)

α_{3} = {\bar{σ}}_{l} + 2 \bar{h}

(35c)

Then, the aforementioned formulations of the final response time and midspan deflection have the following dimensionless forms

{\bar{t}}_{f} = {\begin{matrix} \frac{\bar{I} α_{2}}{\bar{c} α_{1}}, 0 < \bar{I} \leq \frac{\bar{c} α_{1}}{2} \sqrt{\frac{3}{α_{2} α_{3}}} \\ \frac{\bar{I} α_{2}}{\bar{c} α_{1}} - \sqrt{\frac{8 α_{2}^{2} {\bar{I}}^{2}}{9 {\bar{c}}^{2} α_{1}^{2}} - \frac{2 α_{2}}{3 α_{3}}} + \frac{2}{π} \sqrt{\frac{α_{2}}{α_{3}}} \arctan (\frac{4}{π {\bar{c}}^{2} α_{1}} \sqrt{\frac{4 {\bar{c}}^{2} {\bar{I}}^{2} α_{2} α_{3}}{3} - {\bar{c}}^{4} α_{1}^{2}}), \frac{\bar{c} α_{1}}{2} \sqrt{\frac{3}{α_{2} α_{3}}} < \bar{I} \leq \frac{\bar{c} α_{1}}{2} \sqrt{\frac{6}{α_{2} α_{3}}} \\ \frac{\bar{c} α_{1}}{2 \bar{I} α_{3}} + \frac{2}{π} \sqrt{\frac{α_{2}}{α_{3}}} \arctan (\frac{4 \bar{I}}{π \bar{c} α_{1}} \sqrt{2 α_{2} α_{3} - \frac{2 \bar{c} α_{1}}{3 \bar{I}} \sqrt{6 α_{2} α_{3}}}), \bar{I} > \frac{\bar{c} α_{1}}{2} \sqrt{\frac{6}{α_{2} α_{3}}} \end{matrix}

(36)

and

{\bar{w}}_{f} = {\begin{matrix} \frac{2 {\bar{I}}^{2}}{3 {\bar{c}}^{3} α_{1} α_{2}}, 0 < \bar{I} \leq \frac{{\bar{c}}^{2} α_{1}}{2} \sqrt{\frac{3 α_{2}}{α_{3}}} \\ \sqrt{\frac{{\bar{c}}^{2} α_{1}^{2}}{4 α_{3}^{2}} + \frac{16 {\bar{I}}^{2}}{3 π^{2} {\bar{c}}^{2} α_{2} α_{3}} - \frac{4 {\bar{c}}^{2} α_{1}^{2}}{π^{2} α_{3}^{2}}}, \frac{{\bar{c}}^{2} α_{1}}{2} \sqrt{\frac{3 α_{2}}{α_{3}}} < \bar{I} \leq \frac{{\bar{c}}^{2} α_{1}}{2} \sqrt{\frac{6 α_{2}}{α_{3}}} \\ \sqrt{\frac{{\bar{c}}^{2} α_{1}^{2}}{4 α_{3}^{2}} + \frac{8 {\bar{I}}^{2}}{π^{2} {\bar{c}}^{2} α_{2} α_{3}} - \frac{8 \bar{I} α_{1}}{3 π^{2}} \sqrt{\frac{6}{α_{2} α_{3}^{3}}}}, \bar{I} > \frac{{\bar{c}}^{2} α_{1}}{2} \sqrt{\frac{6 α_{2}}{α_{3}}} \end{matrix}

(37)

where ${\bar{t}}_{f} = t_{f} / (L \sqrt{ρ_{f} / σ_{f}})$ , ${\bar{w}}_{f} = w_{f} / L$ .

Analytical solutions for plastic–string model

For the dynamic response of the clamped metal sandwich plates under impulsive loading, as the deflection increases, the effect of the bending moment diminishes and the main effect of structural response is by the axial (membrane) force. Correspondingly, the effect of the bending moment can be ignored in analysis when the deflection of the sandwich plate is larger. Therefore, the impulsive response prediction of a clamped metal sandwich plate in Figure 1 can be obtained by a plastic-string model. The governing equation of plastic-string response can be given by equation (18). The form of the solution is

w_{0} (t) = \frac{{\overset{\cdot}{w}}_{0 I}}{a} \sin (at) + w_{0 I} \cos (at)

(38)

Using the initial condition $w_{0 I} (t = 0) = 0$ , ${\overset{\cdot}{w}}_{0 I} (t = 0) = v_{0} = I / G$ , we obtain

\cos (at) = 0

(39)

Therefore, the final response time is

t_{f} = \frac{π}{2 a} = L \sqrt{\frac{G}{N_{p}}}

(40)

and the final midspan deflection is

w_{f} = \frac{v_{0}}{a} = \frac{2 IL}{π} \sqrt{\frac{1}{G N_{p}}}

(41)

Correspondingly, the dimensionless forms are

{\bar{t}}_{f} = \sqrt{\frac{α_{2}}{α_{3}}}

(42)

and

{\bar{w}}_{f} = \frac{2 \bar{I}}{π} \sqrt{\frac{α_{2}}{α_{3}}}

(43)

Results and discussion

For the verification of the present theoretical models, comparisons between the present theoretical prediction results and the previous analytical and numerical results of the metal sandwich plates with various lattice cores are conducted.

Numerical studies on the impulsive response of sandwich plates with unfilled and metal foam-filled sinusoidal cores are carried out by Zhang et al.³¹ and be adopted here. For the metal foam-filled sinusoidal core sandwich plates, the core web was filled by the metal foam. In the FE analysis, the commercial FE code ABAQUS/Explicit was employed. The whole model was established with eight-node linear brick element with reduced integration (C3D8R). Since the boundary of plate was constrained during blasting, all the displacements and rotations of the nodes at ends of the plate were set to zero. The friction was not considered and a general contact algorithm with a frictionless contact option was used in the numerical calculation. The dimensionless geometrical dimensions and material parameters are listed as $\bar{c} = 0.1$ , $\bar{h} = 0.08$ , $f = 0.04$ , ${\bar{σ}}_{n} = 0.00365$ , ${\bar{σ}}_{l} = 0.04$ , ${\bar{ρ}}_{c} = 0.04$ for unfilled sinusoidal core sandwich plate and $\bar{c} = 0.1$ , $\bar{h} = 0.0608$ , $f = 0.04$ , ${\bar{σ}}_{n} = 0.02285$ , ${\bar{σ}}_{l} = 0.0592$ , ${\bar{ρ}}_{c} = 0.04$ for foam-filled sinusoidal core sandwich plate, where $f$ is the volume fraction of sinusoidal core.

Figure 6 shows the comparisons for the relation of the normalized maximum midspan deflection ${\bar{w}}_{f}$ and the normalized impulsive loading $\bar{I}$ of the present theoretical predictions and the previous analytical predictions²⁵ and FE results of sandwich plates with unfilled and foam-filled sinusoidal core. It can be found that the present decoupling model has the same accuracy with the previous membrane factor method comparing to FE results at low impulsive loadings, while has better accuracy than that of the previous membrane factor method at high impulsive loading. It demonstrates that the present decoupling model is effective and the interaction of plastic bending and longitudinal stretching can be neglected in analytical analysis to avoid the complexity in mathematics. Moreover, the present plastic-string model is in excellent agreement with FE results on the whole impulsive loadings. It may be the reasons that the dynamic behavior of sandwich plates with various lattice cores is dominated by membrane force in large deflections, while the bending moment has small effect in large deflection. Therefore, the effect of bending moment can be ignored further in the simplified analysis for the sandwich plates with lattice cores. Correspondingly, the governing equation is much simpler and more available for engineering applications. The similar results can be obtained for the relation of the non-dimensional structural response time ${\bar{t}}_{f}$ and the normalized impulsive loading $\bar{I}$ of the sandwich plate, as shown in Figure 7.

Figure 6.

Analytical and numerical results for the normalized maximum midspan deflection of the sandwich plate versus the normalized impulsive loading: (a) the unfilled sinusoidal plate core and (b) foam-filled sinusoidal plate core.

Figure 7.

Analytical and numerical results for the non-dimensional structural response time of the sandwich plate versus the normalized impulsive loading: (a) the unfilled sinusoidal plate core and (b) foam-filled sinusoidal plate core.

Numerical results of sandwich plates with the pyramidal truss core, square honeycomb core, and corrugated core under impulsive loading that carried out by Xue and Hutchinson⁸ are adopted too. Comparisons of the present theoretical predictions and the previous theoretical predictions²⁵ as well as the FE results for the normalized maximum midspan deflection of sandwich plates with the pyramidal truss core, square honeycomb core, and corrugated core are shown in Figure 8. It is also observed that the present analytical model is in excellent agreement with FE results and has better accuracy than that of the previous membrane factor method at high impulsive loading. The good agreement between the analytical model and FE results may justify that the high initial peak, non-uniform collapsing, and fluctuating crushing behavior of the lattice cores using r-p-p-l model have small effect on the global deformation of the sandwich plates with various lattice cores subjected to impulsive loading and these can further prove the applicability of the present model.

Figure 8.

Analytical and numerical results for the normalized maximum midspan deflection versus the normalized impulsive loading of sandwich plates: (a) pyramidal truss core, (b) square honeycomb core, and (c) corrugated plate core.

Concluding remarks

A new simplified analysis for the dynamic response of clamped metal sandwich plates with various lattice cores under impulsive loading is conducted. The simplified theoretical solutions for the maximum midspan deflection and structural response time are derived by decoupling the moment–axial force interaction. Comparisons of the present theoretical predictions with the previous analytical and numerical results are carried out and good agreement is achieved for a wide range of sandwich core topologies. It is shown that the present theoretical models are simple and efficient to predict the impulsive response of the metal sandwich plates with various lattice cores in engineering applications.

Footnotes

Appendix 1

Handling Editor: Wensu Chen

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors gratefully acknowledge the financial supports of NSFC (11372235, 11572234, and 11502189), China Postdoctoral Science Foundation funded project (2015M572546).

ORCID iD

Wei Zhang

References

Gibson

Ashby

. Cellular solids: structure and properties. Cambridge: Cambridge University Press, 1997.

Ashby

Evans

Fleck

et al . Metal foams: a design guide. Boston, MA: Butterworth-Heinemann, 2000.

Qiu

. Some topics in recent advances and applications of structural impact dynamics. Appl Mech Rev 2011; 38: 4001.

Wadley

HNG

Dharmasena

Queheillalt

et al . Dynamic compression of square honeycomb structures during underwater impulsive loading. J Mech Mater Struct 2007; 2: 2025–2048.

. Dynamic crushing strength of hexagonal honeycombs. Int J Impact Eng 2010; 37: 467–474.

Wang

et al . Mechanical behavior of the sandwich structures with carbon fiber-reinforced pyramidal lattice truss core. Mater Design 2010; 31: 2659–2663.

Xue

Hutchinson

. Preliminary assessment of sandwich plates subject to blast loads. Int J Mech Sci 2003; 45: 687–705.

Xue

Hutchinson

. A comparative study of impulse-resistant metal sandwich plates. Int J Impact Eng 2004; 30: 1283–1305.

Qiu

Deshpande

Fleck

. Finite element analysis of the dynamic response of clamped sandwich beams subject to shock loading. Eur J Mech A: Solid 2003; 22: 801–814.

10.

Vaziri

Hutchinson

. Metal sandwich plates subject to intense air shocks. Int J Solid Struct 2007; 44: 2021–2035.

11.

Liu

Tian

et al . Sandwich plates with functionally graded metallic foam cores subjected to air blast loading. Int J Mech Sci 2014; 84: 61–72.

12.

Huang

Zhang

Huang

et al . Dynamic response of aluminum corrugated sandwich subjected to underwater impulsive loading: experiment and numerical modelling. Int J Impact Eng 2017; 109: 78–91.

13.

Radford

Deshpande

Fleck

. The use of metal foam projectiles to simulate shock loading on a structure. Int J Impact Eng 2005; 31: 1152–1171.

14.

Rubino

Deshpande

Fleck

. The dynamic response of clamped rectangular Y-frame and corrugated core sandwich plates. Eur J Mech A: Solid 2009; 28: 14–24.

15.

Wang

Jing

Ning

et al . The structural response of clamped sandwich beams subjected to impact loading. Compos Struct 2011; 93: 1300–1308.

16.

Jing

Wang

et al . Energy absorption and failure mechanism of metallic cylindrical sandwich shells under impact loading. Mater Design 2013; 52: 470–480.

17.

Fleck

Deshpande

. The resistance of clamped sandwich beams to shock loading. J Appl Mech 2004; 71: 386–401.

18.

Qiu

Deshpande

Fleck

. Impulsive loading of clamped monolithic and sandwich beams over a central patch. J Mech Phys Solids 2005; 53: 1015–1046.

19.

Qin

Wang

. An analytical solution for the large deflections of a slender sandwich beam with a metallic foam core under transverse loading by a flat punch. Compos Struct 2009; 88: 509–518.

20.

Qin

Zhang

Wang

et al . Large deflection of geometrically asymmetric metal foam core sandwich beam transversely loaded by a flat punch. Int J Aerospace Lightweight Struct 2011; 1: 23–46.

21.

Qin

Wang

et al . A yield criterion and plastic analysis for physically asymmetric sandwich beam with metal foam core. Int J Appl Mech 2013; 5: 1350037.

22.

Wang

Qin

Zhang

et al . Low-velocity impact response of geometrically asymmetric slender sandwich beams with metal foam core. Compos Struct 2013; 98: 1–14.

23.

Stronge

. Large deflections of a rigid-plastic beam-on-foundation from impact. Int J Impact Eng 1990; 9: 115–126.

24.

Qin

Wang

. Impulsive loading of a fully clamped circular metallic foam core sandwich plate. In: Proceedings of the seventh international conference on shock and impact loads on structures, Beijing, China, 17–19 October 2007, pp.481–488. CI-Premier.

25.

Qin

Wang

. A theoretical analysis of the dynamic response of metallic sandwich beam under impulsive loading. Eur J Mech A: Solids 2009; 28: 1014–1025.

26.

Qin

Wang

Zhao

. Large deflections of metallic sandwich and monolithic beams under locally impulsive loading. Int J Mech Sci 2009; 51: 752–773.

27.

Schubak

Anderson

Olson

. Simplified dynamic analysis of rigid plastic beams. Int J Impact Eng 1989; 8: 27–42.

28.

Yuan

Qin

Wang

. Simplified analysis of large deflection response of a metal sandwich beam subjected to impulsive loading. Acta Mech 2015; 226: 3639–3651.

29.

Vaziri

. Finite deflection dynamic analysis of rigid-plastic beams. Vancouver, BC, Canada: University of British Columbia, 1985.

30.

Vaziri

Olson

Anderson

. Dynamic response of axially constrained plastic beams to blast loads. Int J Solid Struct 1987; 23: 153–174.

31.

Zhang

Qin

Wang

. Compressive strengths and dynamic response of corrugated metal sandwich plates with unfilled and foam-filled sinusoidal plate cores. Acta Mech 2013; 224: 759–775.

Simplified analysis of large deflections for metal sandwich plates with various lattice cores subjected to impulsive loading

Abstract

Keywords

Introduction

Problem formulation

Analytical solutions for decoupling model

The midspan deflection did not attain the critical value w 0 r , I ≤ I cr 1

The midspan deflection attained the critical value w 0 r in the phase Π of the structural motion, I cr 1 < I ≤ I cr 2

The midspan deflection attained the critical value w 0 r in phase I of the structural motion, I > I cr 2

Analytical solutions for plastic–string model

Results and discussion

Concluding remarks

Footnotes

Appendix 1

Declaration of conflicting interests

Funding

ORCID iD

References

The midspan deflection did not attain the critical value $w_{0 r}$ , $I \leq I_{cr 1}$

The midspan deflection attained the critical value $w_{0 r}$ in the phase Π of the structural motion, $I_{cr 1} < I \leq I_{cr 2}$

The midspan deflection attained the critical value $w_{0 r}$ in phase I of the structural motion, $I > I_{cr 2}$