Keywords

1 Introduction

Many modern architectures, such as factories, airports, train stations, theaters, etc., are large-span shell structures lacking internal supports. In the event of an accidental explosion, the impact of the explosion shock wave can easily lead to building collapse. Therefore, implementing effective blast engineering measures to reduce the explosive dynamic response of shelter structures and enhance the survivability of buildings is of paramount importance. Currently, buildings often employ methods such as reinforced concrete structures, steel structures, and the construction of blast walls as blast protection measures. However, these protection measures are time-consuming to deploy, construction-intensive, and limited in terms of size. There is a need for a structurally simple, easily deployable, cost-effective, and highly blast-resistant architectural structural protection scheme.

One of the protective measures is to limit the deformation of the building structure during impact to prevent further accidents. Therefore, a protective structure capable of absorbing impact energy has been developed to reduce the risk of accidents. The protective structure must be able to effectively absorb dynamic impact energy and should also be lightweight to reduce the load on the building and effectively lower costs, as shown in Fig. 1.

The Poisson’s ratio of a material refers to the ratio of transverse strain to axial strain. Structures or materials with a negative Poisson’s ratio are called auxetic structures or materials. Negative Poisson’s ratio means that when an axial force is applied to the material, it tends to increase in the direction perpendicular to the applied force [1].

Under the impact load, materials with a positive Poisson’s ratio will move outward from the impacted area in the direction perpendicular to the impact load, while materials with a negative Poisson’s ratio will contract towards the impacted area. At this time, the local density of the material with a negative Poisson’s ratio increases and the modulus increases, thus exhibiting better impact resistance and blast resistance compared to other honeycomb structures as shown in Fig. 2.

Fig. 1.
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Diagram of blast-resistant honeycomb sandwich protective structure.

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Fig. 2.
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The deformation of the material under impact compression.

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Due to the unique properties of materials with a negative Poisson’s ratio, which originate from their internal structure, there is a growing interest in optimizing methods to further enhance their performance [2,3,4]. Researchers are actively exploring various approaches to improve the properties of materials with a negative Poisson’s ratio, such as optimizing the internal structure of the material to enhance its auxetic properties. This can involve designing specific patterns or arrangements of the structure’s constituents to achieve desired mechanical behaviour.

By focusing on these research areas and utilizing optimization methods, it is possible to further enhance the performance of materials with a negative Poisson’s ratio and unlock their potential for various applications, including in protective materials [5,6,7].

In order to enhance the blast-resistance capabilities of buildings, this paper proposes a honeycomb sandwich protective structure based on a negative Poisson’s ratio structure. The stress-strain relationship of the structure under impact loads was analysed using the finite element method. Furthermore, a combination of response surface methodology and genetic algorithm was employed to optimize the structural parameters of the protective structure [8].

2 Numerical Modelling and Validation of Auxetic Structure

The structural parameters of the auxetic structure depicted in Fig. 3 is presented. Preliminary modelling and simulation need to be conducted to establish precise parameters and validate the structure.

Fig. 3.
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Diagram of blast-resistant honeycomb sandwich protective structure.

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The initial model chosen for this purpose is based on the work by references [9] and [10]. The geometric dimensions of the model can be found in Table 1.

Table 1. The geometric dimensions.
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In this paper, a finite element simulation was conducted to investigate the deformation process of the proposed impact-resistant protective shell, as shown in Fig. 4. The numerical analysis of the in-plane impact characteristics of honeycomb materials was conducted using the ANSYS/Explicit Dynamic Finite Element software. In the calculations, the matrix material utilized was aluminium alloy, assumed to exhibit ideal elastoplastic behaviour. The material parameters for the aluminium alloy were set as follows: shear modulus Es = 27.6 GPa, yield stress σy = 680 MPa, density ρ = 2.7 × 103 kg/m3, and Poisson’s ratio ν = 0.3.

Fig. 4.
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Finite element mode of the blast-resistant honeycomb structure.

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The cell walls were modelled using the SHELL163 shell element. To ensure convergence, five integration points were defined along the thickness direction. Additionally, a single-sided automatic contact algorithm was applied in the calculations. The surfaces of the rigid plate and the external surface of the honeycomb specimen were considered to be smooth, with no friction between them.

The proposed impact-resistant protective shell subjected to an initial velocity of 50 m/s and a duration of 3 ms on its upper surface (Fig. 5).

Fig. 5.
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Finite element mode of the blast-resistant honeycomb structure.

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The process of deformation occurring in auxetic structure under low-speed impact can be divided into four stages (Fig. 6): elastic zone, plateau zone, enhancement zone, and densified zone. The graph below illustrates this, with the x-axis representing strain and the y-axis representing stress.

Fig. 6.
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Stress strain curve of auxetic structure.

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The force-deformation behaviour of an auxetic structure is initially unstable when subjected to impact. The elastic zone is also very brief. In the plateau zone, the stress of the structure fluctuates within a small range around a constant value. However, as the strain increases, the stress no longer fluctuates around a constant value but gradually increases. The structure enters the plateau stress-enhanced zone, and the strain at this point is known as the plateau-enhanced strain. As the strain of the auxetic structure further increases, the honeycomb walls start to compress, and the slope of the stress-strain curve rapidly increases and approaches a constant value. The structure enters the densified zone, and the strain at this point is known as the densified strain. Densification occurs because as the deformation increases, the voids between the honeycomb cells gradually fill up. Ideally, densified strain should be reached when the voids are filled. However, in practical situations, the densification strain is generally smaller than the ideal case. This is because in real-world scenarios, some of the rods in the auxetic structure experience compression and friction between each other, leading to densified strain even when some voids still exist.

Due to the instability and transient nature of the elastic region of an auxetic structure under impact loads, the energy absorbed during the entire deformation process is relatively small. Therefore, when studying the theoretical model of how much energy a negative Poisson’s ratio structure can absorb under impact loads, the focus is primarily on two stages: the platform region and the platform stress-enhanced region. The theoretical results are used to represent the energy absorbed by the negative Poisson’s ratio structure from the onset of the impact load until densification. From the onset of the impact load until densification, the absorbed energy E can be expressed as the sum of the energy absorbed in the platform region E1 and the energy absorbed in the platform stress-enhanced region E2, i.e., E = E1 + E2. This can be represented on the equivalent stress-strain curve as the area enclosed by the equivalent stress-strain curve and the horizontal axis.

Clearly, in the case of constant corrugated plate thickness and tube wall thickness, the microstructure of the negative Poisson’s ratio influences the shape of the equivalent stress-strain curve, thereby affecting the energy absorption capacity of the structure.

3 Response Surface Methodology and Genetic Algorithm Optimization

Based on this research foundation, it is necessary to further optimize the structure with the energy absorption-to-mass ratio as the optimization goal. In order to expedite the optimization process, this paper adopts an optimization method based on response surface models. The Latin hypercube experimental design method was used to randomly select 15 sample points, and a polynomial response surface model was used to fit the simulation results of these 15 sample points, in order to obtain the model parameters that result in the maximum energy absorption-to-mass ratio.

3.1 Style and Spacing

The response surface methodology involves fitting a polynomial regression equation to model the complex nonlinear relationship between the optimization objective and the design variables [8]. The regression equation for a multivariate quadratic response surface approximation model is given by:

$$ y\left( x \right) = \beta_{0} + \sum\limits_{i = 1}^{n} {\beta_{i} x_{i} } + \sum\limits_{i = 1}^{n} {\beta_{ii} x_{i}^{2} } + \sum\limits_{i < j}^{n} {\beta_{ij} x_{i} x_{j} } + \varepsilon $$
(1)

where y(x) represents the predicted response, x1, x2, …, xn are the design variables, β0, β1, β2, …, βn are the regression coefficients, βii are the quadratic coefficients, βij are the cross-product coefficients, and ε is the error term.

The negative Poisson’s ratio structure introduced in this paper is determined by three independent variables: the lateral spacing W1, the longitudinal spacing W2, and the pipe diameter D. Any change in one of these variables will have a certain impact on the structural strength. However, in order to expedite the search for optimal results, this paper selects two of these variables, lateral spacing W1 and longitudinal spacing W2, as the influencing factors, while keeping the pipe diameter D, which has a relatively small impact on the structure, constant.

This paper uses the Latin hypercube sampling method for experimental design. The range of values for the lateral spacing W1 is set to [6.5, 8], and the range of values for the longitudinal spacing W2 is set to [16, 20]. A total of 15 random samples were taken.

Based on the previous analysis, the energy absorbed by the impact-resistant structure from the onset of impact to densification, denoted as E, can be calculated using the finite element method. After matrix calculation, the energy-mass ratio response surface model is obtained as follows:

$$ y = - 5422.13 + 920.08x_{1} + 246.85x_{2} - 32.03x_{1}^{2} - 1.72x_{2}^{2} - 25.41x_{1} x_{2} $$
(2)

The response surface fitting model is shown in the Fig. 7, from which it can be seen that the optimization problem has a global optimal advantage. The two-dimensional variable optimization of the response surface model is carried out, and the optimal advantage is found to be x = 6.821 mm, y = 18.65 mm, and the maximum energy-mass ratio is 158.3J/kg. The optimized impact energy mass ratio is greater than the original structure’s impact energy mass ratio of 137.2 J/kg.

Fig. 7.
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Response surface fitting model.

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4 Discussion and Conclusions

In this paper, theoretical analysis and optimization simulation of the energy absorption process of the new negative Poisson’s ratio structural model under impact load are carried out, and the following conclusions are obtained:

  1. (1)

    The impact calculation model of the new negative Poisson ratio structure is established, and the stress-strain curve of the impact-resistant structure is obtained from the finite element calculation results, so as to obtain the energy-absorption energy-mass ratio.

  2. (2)

    By using the Latin hypercube experimental design method, the variables related to the structure parameters of negative Poisson ratio were randomly sampled, and 15 sample points were obtained. Through the finite element calculation results of 15 sample points, the response surface model of the new negative Poisson ratio structure was obtained by response surface fitting technology.

  3. (3)

    According to the response surface model, the optimal impact model parameters of the negative Poisson ratio impact resistant structure were found with the energy-absorption-mass ratio as the optimization objective, and the impact resistance finite element calculation of the optimal model was re-performed. The calculation results verified the effectiveness of the optimization results.