Meta-Material Layout for the Protection of Buried Steel Pipes against Surface Explosion

Abstract

This paper reports on preliminary yet consistent studies and results around the concept of meta-material applied to the protection of buried gas transmission pipelines. The capacity of the proposed meta-material layout in attenuating and dissipating the energy induced by a surface explosion is described in general terms, and then it is examined for a set of nine realistic cases. The formulation of the band gaps, which are considered zones of mitigation for the incident waves of certain frequencies, composes the core of the analysis. For the calculation of the band gaps that target a specific range of frequencies, the 1D periodic structures’ theory is adopted, and the results have been verified numerically via COMSOL. The layout is tested for nine cases of surface explosions via finite element analyses in ABAQUS, using the CONWEP model for simulating the surface explosions. Extremely satisfying results are demonstrated regarding the reduction in the vertical and horizontal displacements of the buried steel pipe. The outer goal of the present study is to spotlight the implementation of meta-material concepts for the efficient blast protection of underground structures, addressing a major hazard for this type of structure and a gap in the current literature.

1. Introduction

Extreme incidents (e.g., bombing and ballistic attacks) along with conflicts between states and territories have shown a significant increase worldwide. This reality is escalating the demands for more robust and effective protection for structures and the built environment against these hazards. Critical Infrastructures (CI) are of high importance for the vitality of current cities and states and, for this reason, are frequently considered targets for attacks. Their interconnection and interdependency expose them even more to these hazards and intensify their vulnerabilities. Due to this emerging problem, more states are adopting resiliency frameworks for the protection of CI, and they are enhancing their infrastructure systems’ capacity to withstand and recover from these events. To this goal, the contribution of the meta-materials could be important and of high value for the blast protection of the CI.

The use of meta-materials in civil engineering applications was conceptualized relatively recently (within the previous decade) and was mainly focused on the anti-seismic protection of buildings and structures. Layered periodic foundations consisting of concrete and rubber components were developed and tested widely against the mitigation of the seismic waves and energy [1,2,3,4], while experimental results validated the enhanced behavior towards seismic waves [1,5]. The specific methodology also extended to two- [6] and three-dimension seismic isolation [7]. The concept of meta-foundation was developed for the protection of fuel and liquid storage tanks, using locally resonant materials [8,9], and also tested experimentally [10]. Recently, the investigation of a non-linear periodic foundation was presented, placing roller bearings between the concrete layers of the foundation [11].

Meta-materials and their design philosophy also extend to other civil engineering applications aimed at effectively protecting structures against blast phenomena. A pioneering concept in this field was the creation of metaconcrete, where lead spheres are coated with soft materials and embedded into the concrete [12], followed by experimental validation [13,14,15] and studies on locally resonant concrete structures under blast load [16,17]. Another type of relevant structure is meta-sandwich panels. The use of sandwich structures for energy absorption applications has had widespread adoption in the past [18], but now it is examined and extended under the prism of the meta-materials theory. Meta-sandwich panels with meta-cores as a sacrificial cladding, mitigating the blast energy, were proposed in [19,20], along with studies on composite meta-panel of resonators embedded in the core under blast load, including viscous damping [21]. Τhe explosion vibration attenuation effect of plate-like metastructures, based on the dynamic meta-plate theory, has also been investigated [22]. Various other concepts were developed within the broader field of meta-materials for the same scope. Some of them are the locally resonant woodpile metamaterials for impact and blast mitigation [23], elastic metamaterials with negative effective mass prohibiting the impact stress wave propagation [24], or resonant structures in resin matrixes for the mitigation of the blast wave [25]. The major research activity focused on the protection of buildings or structures against blast-wave phenomena occurred above the ground surface. Neither the examination of the performance for the proposed solutions in the underground space nor the development of meta-material concepts oriented to the blast protection of the structures below the ground surface has been considered so far.

The current landscape in the blast protection of buried structures is targeted more at the evaluation and the assessment of the performance under blast loading for various geometric and material characteristics of the structures. Comparative studies have been conducted on box shape and semi ellipse and circular and horseshoe shape tunnels under a surface explosion [26]. The circular and the square-shaped tunnels were the first ones to suffer more blast damage, respectively [27], or D-shaped and circular tunnels under dynamic loading [28]. The increase in tunnel-lining thickness provided beneficial effects in reducing the blast-induced strains and displacements [29,30,31]. Furthermore, the increase in the structure’s burial depth constantly remains a factor in minimizing the deformation due to blast loading, both for tunnels [27,32] and pipelines [33,34]. Finally, mitigation measures in the philosophy of protective wave barriers have also been adopted for existing tunnels for explosion risk [35], but on a small scale and without examining the feasibility of this solution for the protection of other structures, even though the experimental validation of the specific solution for vibration isolation [36]. It has also been observed that the studies of tunnels’ performance under blast loading and mitigation strategies of their blast damage exceed those for pipelines.

This study proposes a highly efficient solution for the blast protection of the pipelines against surface explosions, based on the periodicity of the layout’s design and the enhanced dynamic properties that are shown in the attenuation of the blast wave energy. Τhe novelty of this study lies in the feasibility of the proposed solution and the exploitation of meta-materials for the protection of underground structures against surface explosion. Thus, covering a major gap in the present literature and spotlighting the potential extension of the use of meta-materials for blast protection in various underground structures (e.g., tunnels and underground storage tanks). The band gaps were obtained from the analytical model and verified with numerical results via the finite element program of COMSOL. The convergence was almost precise. Various scenarios of surface explosions were tested, and the analyses showed great results. The response of the buried steel pipe was close to isolation against the effects of the surface explosion. The analyses were conducted via the finite element program ABAQUS, and the CONWEP model was adopted for the simulation of the surface explosions.

2. Background

2.1. Definition of the Frequency Domain

The special characteristic of the meta-materials is their ability to demonstrate “blind” zones for specific frequency ranges of transmitted waves. These zones are referred to as band gaps. This results in the mitigation or even denial of transmission to the waves of those frequencies. So, the awareness of the waves’ main frequency domain is crucial, both for the selection of component materials and the design of the meta-material.

Therefore, the calculation of the main frequency ranges for the surface blast-induced waves travelling within the soil was conducted. The calculation was based on the expression of the magnitude of the subsurface free-field shock stress, P(t), at a given time t, as proposed in [37] and described below, and a Fast Fourier Transformation (FFT) of this pressure–time history, the results of which are shown in Figure 1.

$$\mathrm{P}\left(\mathrm{t}\right)={ \mathrm{P}}_{\mathrm{o}}{\mathrm{e}}^{-\mathrm{a}\frac{\mathrm{t}}{{\mathrm{t}}_{\mathrm{a}}}}$$

(1)

where the ${\mathrm{P}}_{\mathrm{o}}$ is the peak free-field shock stress, α a constant typically depending on the specific site (geologic media), and ${\mathrm{t}}_{\mathrm{a}}$ the arrival time depending on the distance between the blast source and the depth of the structure (the depths selected for the Fast Fourier Transformations were usually used for steel gas transmission pipes).

The qualitative expression of the Fourier curve remains the same for both the different quantities of explosives and different distances between the soil depth and the blast source. Only the magnitude of single-sided amplitude $\left|\mathrm{P}1 \left(\mathrm{f}\right)\right|$ changes, according to the distance between the soil depth and the blast source. The frequency range continues until ~8–9 kHz, but it is assumed that the waves of these frequency values do not contribute to the respective level of pressure loading in the underground structure.

The frequency range of main interest for the meta-material design and the desirable band gaps is defined between 0–200 Hz as the biggest portion of blast energy and pressure is transferred via waves within this range of frequencies. Theoretically, the band gaps reproduce themselves infinitely, so practically, band gaps will continue to appear also for frequencies beyond the selected range.

2.2. Simplification on the Analytical Simulation Model of Surface Explosion

The adopted analytical model for the form of the pressure loading on the pipe is shown in Figure 2, and it is close to similar analytical models [38]. The blast waves generated due to surface explosion were considered as 1D shear (S) waves, transmitting towards the radial direction and applied vertically to the upper part of the pipe. It was assumed that the vertical component of the shock wave would be the largest contributor to the total damage. This simplification was necessary for the adaptation of the specific meta-material methodology, which is analyzed in the next chapter. No analytical calculation of pressures or stresses based on this model was taken into consideration.

3. Methodology and Materials

3.1. Periodic Materials’ Theory

The 1D periodic materials theory is based on the elastic wave propagation towards a theoretically infinite periodic structure and the boundary conditions between the composite materials. For this reason, we assumed a periodic structure and a so-called unit cell for the two different material layers, as shown in Figure 3.

Assuming the u (z,t) as the component of displacement in the y direction, and an elastic wave propagating along direction z, then the equation of motion (Equation (2)) in each layer is considered [1]:

$$\frac{{\partial }^2{\mathrm{u}}_{\mathrm{i}}}{\partial {\mathrm{t}}^2}={\mathrm{C}}_{\mathrm{i}}^2\frac{{\partial }^2{\mathrm{u}}_{\mathrm{i}}}{\partial {{\mathrm{z}}_{\mathrm{i}}}^2}$$

(2)

where ${\mathrm{u}}_{\mathrm{i}}$ represents the vertical displacement of each layer, and ${\mathrm{C}}_{\mathrm{i}}$ is the shear (${\mathrm{C}}_{\mathrm{t}}$) or the longitudinal (${\mathrm{C}}_{\mathrm{p}}$) wave velocity of each layer, respectively.

The general solution of Equation (2) is considered to be Equation (3), which also leads to the definition of the shear stress (τ) equation (Equation (4)).

$${\mathrm{U}}_{\mathrm{i}}({\mathrm{z}}_{\mathrm{i}})={ \mathrm{A}}_{\mathrm{i}}\sin \left(\frac{{\mathsf{\omega }\mathrm{z}}_{\mathrm{i}}}{{\mathrm{C}}_{\mathrm{ti}}}\right)+{\mathrm{B}}_{\mathrm{i}}\cos \left(\frac{{\mathsf{\omega }\mathrm{z}}_{\mathrm{i}}}{{\mathrm{C}}_{\mathrm{ti}}}\right)$$

(3)

$${\tau }_{\mathrm{i}}({\mathrm{z}}_{\mathrm{i}})={ \mathsf{\mu }}_{\mathrm{i}}\frac{\partial {\mathrm{u}}_{\mathrm{i}}}{\partial {\mathrm{z}}_{\mathrm{I}}}={ \mathsf{\mu }}_{\mathrm{i}}\mathsf{\omega }\left[{\mathrm{A}}_{\mathrm{i}}\cos \left(\frac{{\mathsf{\omega }\mathrm{z}}_{\mathrm{i}}}{{\mathrm{C}}_{\mathrm{ti}}}\right)-{\mathrm{B}}_{\mathrm{i}}\sin \left(\frac{{\mathsf{\omega }\mathrm{z}}_{\mathrm{i}}}{{\mathrm{C}}_{\mathrm{ti}}}\right)\right]/{\mathrm{C}}_{\mathrm{ti}}$$

(4)

where ω is considered the angular frequency, $\mathrm{I}, {\mathrm{I}}_{\mathrm{i}}$ are unknown constants, related to the amplitude of each layer’s oscillation

For the determination of the four unknown constants (A₁, A₂, B₁, B₂), two for each layer, respectively, the boundary and continuity conditions were taken into consideration. Equations (5) and (6) refer to the continuity conditions in the interface between the layers, while Equations (7) and (8) refer to the periodic conditions, exploiting the Bloch–Floquet theorem [1].

$${\mathrm{u}}_1({\mathrm{h}}_1)={ \mathrm{u}}_2\left(0\right)$$

(5)

$${\tau }_1({\mathrm{h}}_1)={ \tau }_2\left(0\right)$$

(6)

$${\mathrm{u}}_1\left(0\right){\mathrm{e}}^{\mathrm{ikh}}={ \mathrm{u}}_2\left({\mathrm{h}}_2\right)$$

(7)

$${\tau }_1\left(0\right){\mathrm{e}}^{\mathrm{ikh}}={ \tau }_2\left({\mathrm{h}}_2\right)$$

(8)

where k is considered the wavenumber.

By substituting Equations (5) and (7) into Equations (3) and (4), the four equations can be set as:

$$\left[\begin{array}{cccc}\sin \left(\frac{{\mathsf{\omega }\mathrm{h}}_1}{{\mathrm{C}}_{\mathrm{t}1}}\right)& \cos \left(\frac{{\mathsf{\omega }\mathrm{h}}_1}{{\mathrm{C}}_{\mathrm{t}1}}\right)& 0& -1\\ {\mathsf{\mu }}_1{\mathrm{C}}_{\mathrm{t}2}& -{\mathsf{\mu }}_1{\mathrm{C}}_{\mathrm{t}2}\sin \left(\frac{{\mathsf{\omega }\mathrm{h}}_1}{{\mathrm{C}}_{\mathrm{t}1}}\right)& -{\mathsf{\mu }}_2{\mathrm{C}}_{\mathrm{t}1}& 0\\ 0& {\mathrm{e}}^{\mathrm{ikh}}& -\sin \left(\frac{{\mathsf{\omega }\mathrm{h}}_2}{{\mathrm{C}}_{\mathrm{t}2}}\right)& -\cos \left(\frac{{\mathsf{\omega }\mathrm{h}}_2}{{\mathrm{C}}_{\mathrm{t}2}}\right)\\ {\mathsf{\mu }}_1{\mathrm{C}}_{\mathrm{t}2}{\mathrm{e}}^{\mathrm{ikh}}& 0& -{\mathsf{\mu }}_2{\mathrm{C}}_{\mathrm{t}1}\cos \left(\frac{{\mathsf{\omega }\mathrm{h}}_2}{{\mathrm{C}}_{\mathrm{t}2}}\right)& {\mathsf{\mu }}_2{\mathrm{C}}_{\mathrm{t}1}\sin \left(\frac{{\mathsf{\omega }\mathrm{h}}_2}{{\mathrm{C}}_{\mathrm{t}2}}\right)\end{array}\right]\left[\begin{array}{c}{\mathrm{A}}_1\\ {\mathrm{B}}_1\\ {\mathrm{A}}_2\\ {\mathrm{B}}_2\end{array}\right]=0$$

(9)

In order for a non-trivial solution to exist, the determinant of the co-efficient matrix must be equal to zero. Thus, a dispersion relation for ω as a function of k (Equation (10)) is given [1].

$$\cos \left(\mathrm{k}\times \mathrm{h}\right)=\cos \left(\frac{{\mathsf{\omega }\mathrm{h}}_1}{{\mathrm{C}}_{\mathrm{t}1}}\right)\cos \left(\frac{{\mathsf{\omega }\mathrm{h}}_2}{{\mathrm{C}}_{\mathrm{t}2}}\right)-\frac{1}{2}\left(\frac{{\mathsf{\rho }}_1{\mathrm{C}}_{\mathrm{t}1}}{{\mathsf{\rho }}_2{\mathrm{C}}_{\mathrm{t}2}}+\frac{{\mathsf{\rho }}_2{\mathrm{C}}_{\mathrm{t}2}}{{\mathsf{\rho }}_1{\mathrm{C}}_{\mathrm{t}1}}\right)\sin \left(\frac{{\mathsf{\omega }\mathrm{h}}_1}{{\mathrm{C}}_{\mathrm{t}1}}\right)\sin \left(\frac{{\mathsf{\omega }\mathrm{h}}_2}{{\mathrm{C}}_{\mathrm{t}2}}\right)$$

(10)

Based on this equation, and the scattering in the first Brillouin zone (k $\in$ [−π/h, π/h]), the dispersion curve and the analytical band gaps could be calculated for the interested frequency range. The method and the dispersion relationship refer to rectangular coordinates, but it is assumed that the results in the band gaps’ formation of the respective problem in cylindrical coordinates (as with the problem of this paper) will have minor deviations. So, the design of the meta-material will follow the presented theory.

3.2. Materials and Layout Structure

The proposed layout consisted of eight layers and two component materials, periodically set. The thickness of each layer was 5.00 mm. A solution of more and thinner layers was preferred, instead of one with fewer and thicker layers, in order for the layout’s dynamic features to be considered closer to those of an infinite structure. The materials chosen for the meta-material layout and their characteristics are presented in

Table 1. Properties of the chosen materials for the meta-material layout.

Properties	Material 1 Polyurethane Foam	Material 2 Rubber
Density ρ (kg/m3)	900	1300
Young’s Modulus E (Pa)	1.47 × 108	58,000
Poisson’s ratio ν	0.42	0.463

. Polyurethane foam has been used in the past for the protection of pipelines against surface explosions [39], but not as part of a meta-material concept. The characteristics of the polyurethane foam were derived from [19] and rubber from the commercial market. In Figure 4, the proposed eight-layer layout solution bonded around the steel pipe is presented visually.

3.3. Analytical and Numerical Results

The analytical band gaps of the meta-material layout were calculated based on the dispersion relationship of Equation (2) and then compared to the numerical results via COMSOL (Figure 5). The specific structure led to the creation of five attenuation zones (i.e., the band gaps) within the desired frequency range and are presented in

Table 2. Presentation of the first five (analytically calculated) band gaps of the meta-material layout, within the desirable frequency range.

Attenuation Zone No.	Frequency Range (Hz)
1	24–39
2	54–78
3	88–117
4	124–156
5	162–195

. The band gaps also expanded beyond the targeted frequency range, creating wider attenuation zones than the initial ones. The convergence between analytical and numerical results is almost identical. The only exception was for the second band gap, which presented divergences for a small range of k, and was calibrated.

4. Finite Element Modelling

4.1. Soil Modelling

The soil block was modelled using the Drucker–Prager Cap Model, which is available in ABAQUS/Explicit. The model was adopted due to its account for stress path dependency and provision of soil hardening and softening behavior. The specific soil model has been considered appropriate for modelling soil behavior under blast loading and suggested for use also in other similar studies [37,40,41]. The exact parameters of the Drucker–Prager Cap Model are presented in

Table 3. Drucker-Prager Cap model parameters for silty clay.

Properties	Values
Density ρ (kg/m3)	1920
Young’s Modulus E (MPa)	51.7
Poisson’s ratio ν	0.45
Material cohesion d (MPa)	0.036
Material angle of friction β	24
Cap eccentricity R	0.02
Initial cap yield surface position εv	0.02
Transition surface radius a	0.0
Cap hardening behavior (stress vs. plastic strain) (MPa vs. %)	2.75, 0.00 4.83, 0.02 5.15, 0.04 6.20, 0.08

.

The soil block in the finite element model represents a 100 × 100 × 50-m domain, as Figure 6 shows. The dimensions were chosen accordingly so that the reflection of the artificial pressure from the domain boundaries could be prevented [37]. The mesh was refined in the area near the pipe and the meta-material layout, and a detailed picture of the mesh in this area is also provided in Figure 6. The elements became relatively orthogonally smooth near the pipe area. The meta-material layout was tied to the steel pipe in order for the joint response to be secured.

4.2. Blast Pressure Loading and Case Studies

The modelling of the dynamic blast loading due to surface explosion was conducted using the CONWEP algorithm, which is available in ABAQUS/Explicit. This method is empirical on the condition that the distance between the explosive charge and the point of consideration should be greater than the radius of that charge [37,41]. The step time of the blast loading is defined as 0.03 s, as this is the duration required for the air-blast generated pressure to reach various soil depths and attenuate under the 0.50 MPa, considering a typical charge scenario of 100 kg TNT [37]. For the same TNT quantity, the peak pressure is almost 6.00 MPa for soil depth at 1.115 m, so it is considered acceptable that the value of 0.50 MPa is set as a lower pressure limit for defining the step time.

This research studied the behavior of the pipe, with and without the use of the meta-material layout for blast protection. The categorisation of the cases was based on the distance between the point of explosion charge and the pipe and the height of the explosion charge above the ground surface. Subdivisions refer to the amount of the TNT explosive charges (

Table 4. The case studies considered in the finite element analyses, with and without the meta-material layout.

Case No.	Subcase	TNT Charge (kg)	Distance from Pipe (m)	Height above Ground Surface (m)
1	a	100	5.00	0.50
	b	200	5.00	0.50
	c	400	5.00	0.50
2	a	100	2.50	0.50
	b	200	2.50	0.50
	c	400	2.50	0.50
3	a	100	5.00	0.60
	b	200	5.00	0.60
	c	400	5.00	0.60

). The distances were chosen to be as in [41], while the explosives charges are close to the ones in [37,41]. Three different categories of TNT explosive charges were used in the analyses: 100 kg, 200 kg, and 400 kg. All of the charges were assumed to have a spherical shape and the explosions occurred near the center of the ground surface. The characteristics of the steel gas transmission pipe are those for a common one and refer to radius r = 1.00 m, thickness t = 1.50 cm, ρ = 7890 kg/m³, E = 209 × 10⁹ Pa and v = 0.275.

5. Results

5.1. Vertical Displacements in the Pipe Crown

The vertical displacements on the pipe crown, calculated for each study case, are presented in Figure 7, Figure 8 and Figure 9. Each displacement is distracted from the outer surface of the pipe and for the reference point below the center of the ground surface. As was expected, the level of vertical displacements in the pipe crown varies according to the distance of the explosion source from the pipe and its height above the ground surface. The most damaging case is in Case 2, in which the explosion occurs closer to the pipe and takes place 0.5 m above the ground surface. The impressive results of the meta-material layout are presented, as it maintains the pipe practically isolated against the surface explosion blast effects. The maximum displacement did not exceed 0.1 mm for all the cases where the pipe is protected via the layout.

5.2. Horizontal Displacements in the Pipe Spring Line

The horizontal displacements on the pipe spring line, calculated for each study case, are presented in Figure 10, Figure 11 and Figure 12. Each displacement is distracted from the outer surface of the pipe and for the reference point below and close to the center of the ground surface. As was expected, the qualitative behavior towards displacement mitigation proved to be almost the same for the vertical displacements. The most damaging case is again Case 2, in which the explosion takes place closer to the pipe and 0.5 m above the ground surface. The impressive results of the meta-material layout are also demonstrated here, as the maximum displacement did not again exceed 0.1 mm. The horizontal displacements are bigger than the vertical ones for the respective cases, as was expected.

6. Conclusions

The notion of meta-materials is a very recent and promising design methodology in the field of civil engineering. It is capable of providing shielding for the underlying structures against various types of dynamic nature hazards. The authors tried to implement this methodology against blast and shock hazards for the protection of underground structures such as pipelines. The main assumptions that have been taken into consideration for the analytical resolution and design of the meta-material layout proved reasonable. The design methodology becomes simpler without affecting the final results, and the response of the meta-material layout proved efficient. The vertical displacements in the pipe crown, which ranged between 3.0 and 10.5 mm for the studied cases, were reduced to under 0.1 mm. The same qualitative response was also demonstrated for the horizontal displacements in the pipe spring line. The range of values was between 8.0 and 16.0 mm, but after the use of the proposed layout, the expected mitigation led to displacements under 0.1 mm. The suggested solution offered almost isolation conditions to the examined structure against surface explosion and various levels of blast loadings.

The goal of the authors was both to expand the range of civil engineering applications that this innovative technology can be implemented but also to offer feasible solutions for the designers and the operators of these infrastructure systems. The novelty of the present study lies in the covering of a double gap in the literature, as it proposes a blast-resistant meta-material concept for buried pipelines against surface explosion. The authors hope to the further development of meta-material concepts for the blast protection of other underground structures, such as tunnels or underground storage tanks. The research can also proceed with the validation of these solutions against various blast phenomena, such as underground or underwater explosions. In many cases, the meta-material design philosophy exhibits conditions of isolation against these hazards for the underlying structures. The goal of its exploitation for more civil and geotechnical projects is both challenging and worthy of further research.