Performance assessment of rigid polyurethane foam core sandwich panels under blast loading

Abstract

The performance of rigid polyurethane foams, as an energy absorbent core of sandwich panels covered with two exterior steel sheets, was investigated numerically through finite element methods. After verifying the finite element model, numerical studies were conducted to investigate the role of thickness and density of the foam layer in the response behavior of sandwich panels under blast loads. A set of cylindrical polyurethane foam specimens were manufactured at five different nominal densities, 90, 140, 175, 220, and 250 kg/m³, and their stress–strain curves were evaluated using uniaxial compression tests. The test data were then employed to define characteristics of the polyurethane foams in the finite element model. Based on the results of finite element analysis runs, the optimum density of the foam layer was determined by assessing two response parameters including the peak pressure transmitted to the back face of the foam layer and the maximum deflection of sandwich panel. These response parameters were found to be affected differently by variations in the density of the foam layer within the panel. An increase in the thickness of the foam layer, to a certain extent, was found to be beneficial to the mitigation capability of sandwich panel.

Keywords

Blast resistant sandwich panels polyurethane foam sandwich panel blast loading blast pressure mitigation core density

Introduction

Given the increasing terrorist attacks over the past two decades around the world, a significant number of research studies have focused on blast protection of urban infrastructures and public buildings. Among various solutions proposed by different researchers, the concept of using a blast resistant sandwich panel (Guruprasad and Mukherjee, 2000a, 2000b; Kotzialis et al., 2005) or a spaced-apart structure (in which a gap between layers might be also filled with air) (Balandin et al., 2017) has drawn a lot of attention due to their ease of application and predictable response behavior. Two main parts of every blast resistant sandwich panels include the exterior face sheet and an inner soft core. These sandwich structures have various energy dissipation mechanisms, such as bending and stretching of the face sheet, as well as compression and shear of the core, which enable the cladding to decrease the blast pressure transmitted to the protected structure (Palanivelu et al., 2011; Xia et al., 2017).

Foams, either metallic or polymeric, regarding to their porous structure and significant deformability, have proved themselves quiet desirable for energy dissipation purposes (Gibson and Ashby, 1999; Roy, 2009; Boey, 2009). Boey (2009) examined the pressure mitigation ability of polyurethane (PU) foam under the impact of a projectile both experimentally and numerically, and concluded PU foam may be able to reduce the impact pressure even up to 90%. Ong et al. (2011) used an energy absorbent PU foam layer in a multi-layer armor configuration, and showed the great ability of the foam in absorbing the momentum and mitigating the impact pressure. Kitagawa et al. (2009) examined the performance of porous layers including sand and PU foams in energy dissipation and response mitigation under blast loading. Their investigation showed that the ability of mitigating the blast overpressure in porous materials increases with a decrease in the density (i.e. an increase in the porosity ratio) of the material. Furthermore, in the case of using PU foam, the maximum transmitted pressure due to the blast can drop by approximately 90%. The energy dissipation ability stems from the porous structure of the PU foam.

Lee et al. (1986) studied shock wave attenuation in various polymeric foams and concluded that PU foams have the greatest ability of shock wave energy dissipation among other types of foams. Hartman et al. (2006) examined foams with different porosity ratios and compared their ability in blast response mitigation. They concluded that the mid-range density/porosity foams have the most desirable performance. In addition, the pressure attenuation ability of foam may not necessarily increase by decreasing its density.

This case study aims to set up a numerical method by which an optimum characteristic can be found for PU foam cores in sandwich panels under blast loading condition. The goal of the experimental part of the study is to define material properties of the foams with different densities in the Ansys-Autodyn FE-Software (AUTODYN, 2006). The influence of density and thickness of the PU foam on the blast response of a foam core sandwich panel is investigated numerically using finite element (FE) method. The results of a previous experimental study (Yazici et al., 2014) on blast response of PU foam core sandwich panels are used to verify the FE model.

PU foam specimens

This section provides a brief background on the PU polymers, and includes a report on the manufacturing and testing of the rigid PU foams investigated in this study.

Heat producing reaction of diisocyanate and polyol commonly forms PU polymers. Depending on their physical properties, PU foams can be classified into two major groups: flexible and rigid foams. The main ingredients of both foams include diisocyanate and polyols. However, the variations in the quantity of these ingredients may result in a flexible or a rigid foam. In addition, the type of isocyanates and polyols used in the mix syrup greatly influences the properties of PU (Sarier and Onder, 2007). “Rigid PU foams” which are used in this study generally show significantly greater values of yield stress (σ_y) and modulus of elasticity (E).

After purchasing the compounds, a cup test was employed to determine the essential properties such as cream time, string gel time, and free rise density of the mix syrup. For the mix formula used in this study, values of these parameters were evaluated in the lab using the cup test procedure outlined in ASTM D7487-08 (2008). Table 1 includes the results of PU cup tests. The room temperature was 31°C.

Table 1.

Results of polyurethane cup test.

Cup test no.	Mixed sample weight (gr)	Cream time (s)	String gel time (s)	Free rise density (kg/m³)
1	40	21	57	140
2	50	20	55	140.5
3	60	21	57	141.2
Average		20.6	56.3	140.56

According to ASTM D7487-08 (2008), the free rise density is the density of a PU foam that is prepared in an open cup. The free rise density of the PU foam of this study was approximately 140 kg/m³ (see Table 1). Therefore, foam density values equal to or less than 140 kg/m³ may be achieved in the air pressure without the application of any outer pressure. For the density values beyond the free rise density, a strong retaining setup is needed to prevent the compound from expanding.

Figure 1 shows a special mold setup constructed and used for casting PU foams of this study. The main components of the mold setup as seen in Figure 1(a) include the following: (a) a steel hollow cylinder of 110 mm inner diameter and 200 mm length split into two half-cylinders to ease the safe removal of the foam, (b) an aluminum cap of 120 mm inner diameter fixed at the bottom of the mold setup as a bed to hold the mold cylinder in place, (c) a removable aluminum cap of 120 mm inner diameter having a 5 mm diameter hole to exhaust the trapped air, and (d) two fixing bolts and nuts. Figure 1(b) shows the assembled mold. Five cylindrical PU foam specimens of 105 mm diameter and 200 mm height with nominal mass densities of 90, 140, 175, 220, and 250 kg/m³ were produced. To cast the PU foams with nominal densities of 90 and 140 kg/m³, the upper cap of the mold was not installed, as these densities were either below or approximately equal to the free rise density of the foam (see Table 1).

Figure 1.

PU foam casting mold: (a) parts of the mold setup and (b) assembled mold.

In the case of casting the 90 kg/m³ PU foam, 4 g of chlorofluorocarbon (CFC) 141b liquid gas, as a leaven agent, was added to the syrup in order to keep the foam density below its free rise density. To achieve densities beyond the free rise density of the foam, the upper cap of the mold was tightened on its place by means of the fixing nuts shown in Figure 1(a) right after pouring and mixing the syrup in the mold. Figure 2(a) shows a typical PU foam cylinder casted using the mold setup shown in Figure 1. After the installation of the steel half-cylinders, their inner surface was lubricated. Next, a PVC tube was pushed inside the cylinder to prevent the compound from sticking to the inner surface of the steel cylinder. The mixed syrup was then poured inside the PVC tube within the steel cylinder, and the upper cap was secured on its place.

Figure 2.

Preparation of PU foam specimens: (a) a typical PU foam cylinder removed from the mold setup shown in Figure 1(b), (b) PU foam disks cut from the original foam cylinder, and (c) PU foam disk specimens with PVC exterior rings removed and ready for test.

As seen in Figure 2(b), each PU foam cylinder of 200 mm height was cut into smaller disks (i.e. disk specimens) of 20 mm thickness. The disk specimens were extracted from the bottom, the middle, and the top of each PU foam cylinder. Next, the PVC rings around the disk specimens were removed prior to implementing uniaxial compression tests.

The three disk specimens of each nominal mass density were subjected to unconstrained uniaxial compression test under a quasi-static strain rate according to ISO 844 (2014). Figure 3 shows the uniaxial compression test setup.

Figure 3.

Uniaxial compression test setup.

Figure 4(a) and (b) shows typical deformed foam specimens of 175 and 250 kg/m³ nominal mass densities, respectively, after completion of the compression test. As seen in these figures, the plastic deformation of the foam specimen with lower density was found to be larger. However, both foam specimens experienced a significant height shortening as compared to their original height (Figure 4(c)).

Figure 4.

An overview of typical foam specimens after and before testing: (a) nominal density 175 kg/m³ (after testing), (b) nominal density 250 kg/m³ (after testing), and (c) nominal density 250 kg/m³ (before testing).

Figure 5 shows the uniaxial compression stress–strain response of the PU foams with different nominal densities. Each diagram specified by a marker represents the average values of the compression tests conducted on three foam specimens of the same density. The stress and strain values reported in Figure 5 represent the nominal values. As seen in Figure 5, the stress–strain curve of the rigid PU foam under uniaxial compression exhibits three definite regions, namely, a linear elastic region, a stress plateau, and a densification region. The densification region appears when the stress values are increased rapidly without any appreciable increase in the strain values. This phenomenon is related to the densification of the collapsed cell walls under compressive loads that are applied on the foam. The densification region does not play any significant role in the energy absorption of the foam as the substance in this region starts to behave like a solid rather than a porous material. In this study, all of the test specimens were loaded until they showed, to some extent, a densification behavior.

Figure 5.

Quasi-static compressive stress–strain relationships for PU foams of various densities.

Table 2 includes the yield stress and modulus of elasticity evaluated for foams of different nominal densities. The true densities of the foams are also listed in Table 2. The nominal densities in fact represent the rounded values of the true densities. As seen in Table 2, both the modulus of elasticity and yield stress increase as the foam density increases.

Table 2.

Mechanical properties of the polyurethane foams of different densities.

True density ρ₀ (kg/m³)	Nominal density (kg/m³)	Yield stress σ_y (MPa)	Modulus of elasticity (MPa)
88.1	90	1.02	12.2
139.7	140	1.90	27.9
177.2	175	2.40	36.3
222.0	220	3.40	50.7
247.6	250	4.50	61.0

FE simulation

A 3D FE model was developed in Ansys-Autodyn Explicit FE-Software (AUTODYN, 2006) based on a previous experimental work (Yazici et al., 2014) to examine the effects of the foam core density and thickness on the dynamic response of a sandwich panel under blast loading. Details and verification of the FE model are provided in the following subsections.

Geometry, loading, and boundary conditions

The experimental data and FE analysis results of a previous study conducted by Yazici et al. (2014) were used to develop and verify the FE model of this article. A foam core sandwich panel with plan dimensions of 203 × 50.8 mm consisted of two steel face sheets, namely, front face (FF) and back face (BF), and an inner PU foam core that was bonded to the two steel face sheets was subjected to blast loading by means of a shock tube apparatus (Figure 6). To completely bond the PU foam core to the face sheets, the “join” capability of AUTODYN software was used. If two parts are joined, AUTODYN automatically finds and joins all coincident nodes in the two parts. The software default value of 0.05 was selected for the join tolerance in this study (AUTODYN, 2006).

Figure 6.

Shock tube experimental setup.

The typical profile of pressure (as a 10-point piecewise-linear curve) that was generated by the shock tube (Yazici et al., 2014) is shown in Figure 7. The thickness of the low-carbon steel face sheets (FF and BF) and the PU foam core was 3.2 and 25 mm, respectively. The panel was symmetrically aligned with the center of the shock tube, and two simple knife-edge supports, spanning in the Y direction, were located at the BF of the panel at a distance of 152.4 mm from each other. The translational degrees of freedom in the Z and Y directions, and the rotational degrees of freedom about X and Z axes, were constrained in the mentioned supports. The exit muzzle inner diameter of the shock tube was 38.1 mm, and the FF of the sandwich panel was set normal to the axis of the shock tube. It is worth noting that the incident peak pressure of the shock wave was chosen to be 1.1 MPa and the reflected peak pressure of approximately 5.5 MPa was obtained. A high-speed side view camera recorded the deflections of the panel.

Figure 7.

A typical profile of shock tube pressure (Yazici et al., 2014).

The steel sheets and the foam core were modeled using hexagonal Lagrangian elements with 6 degrees of freedom in each node. A Lagrangian solver was selected for the steel sheets, and an Arbitrary Lagrange Euler (ALE) solver was used for the foam core due to its large deformations (AUTODYN, 2006). The ALE method of space discretization is a hybrid of the Euler and Lagrange methods. It allows redefining the grid continuously as the calculation proceeds. The significant advantage of ALE is its ability to reduce the difficulties caused by severe mesh distortions encountered by the Lagrange method. It allows a FE analysis run to continue without casual interruptions.

A mesh sensitivity analysis was performed (see Table 5) to verify the accuracy and processing speed. According to the mesh sensitivity analysis, a mesh size of 2 × 2 mm in X–Y plane was selected for all three layers of the sandwich panel. In addition, the size of elements in the cross-sectional plane of steel and foam layers was selected to be approximately 1 and 2 mm, respectively. The experimentally evaluated pressure profile shown in Figure 7 was applied on the FF of the panel as a 10-points stress boundary condition to define the blast loading in FE model shown in Figure 8.

Figure 8.

FE model in Ansys-Autodyn.

Material properties

Steel

The properties of steel material include modulus of elasticity (E), 205 GPa; Poisson’s ratio (ν), 0.29; and density (ρ), 7.85 g/cm³. These were selected in conformance with Yazici et al. (2014). For steel material, the Johnson–Cook Strength Model (Johnson, 1983; Schwer, 2007) was used as the equation of state (EOS). In this model, yield stress is expressed as

\bar{σ} = [A + B {({\bar{ε}}^{p l})}^{n}] [1 + C . l n (\frac{{\dot{\bar{ε}}}^{p l}}{ε})] (1 - {\hat{θ}}^{m})

(1)

where $\bar{σ}$ is yield stress at nonzero strain rate; ${\bar{ε}}^{p l}$ is the equivalent plastic strain; and ${\dot{\bar{ε}}}^{p l}$ is the equivalent plastic strain rate. The term $(1 - {\hat{θ}}^{m})$ in equation (1) represents the effect of temperature and $\hat{θ} = (θ - θ_{r o o m}) / (θ_{m e l t} - θ_{r o o m})$ , where $θ$ is the absolute temperature; $θ_{r o o m}$ is the room temperature, and $θ_{m e l t}$ is the melting temperature of steel. The values of material constants A, B, C, n, and m are presented in Table 3 (Yazici et al., 2014).

Table 3.

The nominal values of Johnson–Cook strength model parameters for low carbon steel.

Shear modulus (GPa)	Yield stress (MPa)A	Hardening constant (MPa)B	Hardening exponentn	Strain rate constantC	Thermal softening exponentm	Ref. strain rate(1/s)
78.84	220	499.87	0.228	0.017	0.917	1

PU foam

The following components must be addressed in the FE modeling of the compaction of porous materials under dynamic loading (AUTODYN, 2006). The values of parameters used in the model verification is given in Table 4.

Table 4.

The values of parameters used for the FE model verification of the 44 kg/m³ PU foam.

Modulus of elasticity (MPa)	Bulk modulus (MPa)	Shear modulus (MPa)	Compaction curve	Hydro tensile limit (GPa)	Poisson’s ratio	DIF
0.24	0.08	0.12	According to Figure 9	–2	0	2

FE: finite element; PU: polyurethane; DIF: dynamic increase factor.

EOS

An EOS describes the hydrodynamic response of a material. This is the primary response for solids at high deformation rates when the hydrodynamic pressure is far greater than the yield stress of the material (AUTODYN, 2006).

Based on the nature of the phenomena and according to the literature (Roy, 2009; Boey, 2009; Ong et al., 2011), a linear shock EOS is generally used to model the hydrodynamic response of the PU foam. For most solids and many liquids over a wide range of pressure, there is an empirically verified linear relationship between Up (particle velocity) and U (shock velocity): U = C₀ + SUp. Data for this EOS can be found in various references and for many of the materials in the explicit material library. For PU materials, the values are C₀ = 4569 m/s and S = 1.49 (Roy, 2009; Boey, 2009; Ong et al., 2011).

However, in this article, due to the simplicity of the strength model, a linear EOS was used. The bulk modulus value required for linear EOS is given in Table 4. A bulk modulus can be used to define a linear, energy independent EOS. Combined with a shear modulus property, this material definition is equivalent to using linear elasticity.

Strength model

Solid materials may initially respond elastically, but under highly dynamic loadings, they may reach stress states that exceed their yield stress and deform plastically. In this study, a porous-crushable foam strength model (Goel et al., 2012; Bryson, 2009; Matsagar, 2016) was used for the foam material. This model provides a simple approach to simulate the crushing characteristics of foam materials under impact loading conditions (non-cyclic loading). The strain rate dependency of the foam materials was included in the model by defining a dynamic increase factor (DIF) of 2 as suggested by Chen et al. (2002). Chen et al. (2002) showed that in compressive stress–strain curves of PU foams with densities of 78–455 kg/m³, the stress values under quasi-static condition are approximately 50% of the corresponding dynamic values. The rate dependency of the PU foam stems from two facts: (a) the strain-rate dependency of the solid PU of the cell walls and (b) the compressibility of the air trapped in the closed cells (Chen et al., 2002). In the porous-crushable foam strength model employed in this article, the pressure and stress deviators are updated in any increment n + 1 as below (AUTODYN, 2006; Livermore Software Technology Corporation, n.d.)

P^{n + 1} = P^{n} + K {\dot{ε}}_{v}^{\frac{n + 1}{2}} Δ t^{\frac{n + 1}{2}}

(2)

S_{i j}^{n + 1} = S^{n} + 2 G (ε_{i j}^{\frac{n + 1}{2}} - δ_{i j} {\dot{ε}}_{v}^{\frac{n + 1}{2}}) Δ t^{\frac{n + 1}{2}}

(3)

where P represents the pressure, S is the principal normal stress, and K and G are the bulk modulus and shear modulus, respectively. Parameter $ε_{i j}$ refers to principal strain and ${\dot{ε}}_{v}$ represents the volumetric strain rate. $δ_{i j}$ and Dt $Δ t$ are Kronecker delta and the time increment in each step, respectively. A zero Poisson’s ratio is considered for the foam material (AUTODYN, 2006).

The “principal stress” versus “volumetric strain” piecewise linear curves of PU foam specimens were calculated using the compaction curves obtained from the experiment in a quasi-static rate. The volumetric strain, $ε_{v}$ , is defined as the natural log of the volume ratio: it is the ratio of the original volume, V₀, to the current volume, V, as follows (AUTODYN, 2006)

ε_{v} = \ln (\frac{V_{0}}{V})

(4)

The foam model is strain rate dependent and crushes one-dimensionally with a Poisson’s ratio that is essentially zero. The elastic modulus is considered constant and the stress is updated assuming elastic behavior.

Figure 9 shows the principle stress versus volumetric strain curve of the PU foam with density of 44 kg/m³. This curve was used to define the properties of the verification panel’s PU foam core. Figure 10 also shows the principal stress versus volumetric strain of the PU foams manufactured in the lab and used in the parametric study.

Figure 9.

Uniaxial compression stress versus volumetric strain for the PU foam core of a mass density of 44 kg/m³ (Yazici et al., 2014).

Figure 10.

Principal stress versus volumetric strain for the PU foams of various densities.

Material failure model

Solid materials usually fail under extreme loading conditions which result in crushing or cracking of the material. In this article, a hydrodynamic tensile limit of –2 GPa was used in the failure model (Roy, 2009; Boey, 2009).

FE model verification

Figure 11(a) and (b) shows the FF and BF deflections of the PU-foam core panel, respectively. As seen in these figures, results of the FE simulation carried out in this article are generally in a good agreement with both the experimental observations and the results of FE simulation of Yazici et al. (2014). The deflections evaluated by the FE model of current article are in average 14.2% off from the experimental observations. The size of FEs used to discretize the foam material of the sandwich panel plays a crucial role in the accuracy of the FE analysis results. Increasing the size of elements in the FE model would in general lead to more rigid responses. Based on the trial analysis runs conducted in this study, when the size of elements in the cross-sectional plane of the foam layer was reduced from 2 to 1 mm, the accuracy of results was improved. However, the improved accuracy was achieved at the expense of a significantly increased (i.e. approximately three times longer) analysis time. As the main objective of this research was to perform a parametric study, a large number of FE analysis runs on sandwich panels of different foam density and thickness was needed to be conducted. Therefore, with regard to the FE model sufficient comparative accuracy, the size of FEs at the cross-sectional plane of the foam layer, with a compromise between analysis time and accuracy, was selected to be 2 mm in all analysis runs. Table 5 provides the mesh sensitivity analysis results.

Figure 11.

Time history of deflections of the sandwich panel: (a) front face (FF) and (b) back face (BF).

Table 5.

Results of mesh sensitivity analysis.

	X-Y plane (element size along Z axis for PU and steel is 2 and 1 mm, respectively)		Z-axis (element size within X–Y plane is 2 × 2 mm² for all three layers)
	For all three layers		PU foam (element size along Z axis is 1 mm for steel)		Steel plates (element size along Z axis is 2 mm for PU)
Approximate element size (mm)	1 × 1	2 × 2	1	2	1	2
Error (%) in panel deflection with respect to the experimental results	12.7	14.2	11.5	14.2	14.2	Shear lock
Ratio of analysis time	4	1	3	1	1	–

PU: polyurethane.

Figure 12 shows the deformation and failure modes of the sandwich panel at 0.5, 1, 1.5, 2, 2.5, and 3 ms after the initiation of the shock load. A comparison between the experimental observation (Figure 12(a)) and the FE analysis results (Figure 12(b)) indicates the effectiveness of the FE model in the response simulation of the sandwich panel. At the time instant of 1 ms, the slapping effect is well modeled by the FE model.

Figure 12.

Deformation/failure modes of the sandwich panel in time instances of 0.5, 1, 1.5, 2, 2.5, and 3 ms after the initiation of shock loading: (a) experimental observations (Yazici et al., 2014) and (b) FE simulation (current article).

A DIF of 2 has been recommended by Chen et al. (2002) for the PU foams of density values of 78–455 kg/m³ that are subjected to a strain rate of 1000–5000 s^–1. The peak value of the strain rate in the PU foams of this study is approximately 1250 s^–1. As such, a DIF of 2 has been employed in the FE analysis runs of this article.

Parametric study

The foam core of the sandwich panel described in the FE model of “FE simulation” section was replaced with a set of PU-foam layers of different mass density and thickness values. Three different face sheet thicknesses were used for every analysis to ensure that the effect of face sheet thickness on the panel’s behavior was also considered. Except for the three parameters mentioned above, other conditions and dimensions were exactly the same as those described in “Geometry, loading, and boundary conditions” section of this article.

Foam core density—pressure attenuation

In order to study the effect of foam density on its pressure attenuation ability, five foam core layers of 25 mm thickness having different densities of 90, 140, 175, 220, and 250 kg/m³ were placed into the foam-core sandwich panel described in “Geometry, loading, and boundary conditions” section. The pressure transmitted to the back of the foam layer (i.e. between foam layer and the BF steel sheet) was determined at the center of panel in the X–Y plane. In order to consider the effect of face sheet thickness on the results, each panel was modeled with three different face sheet thicknesses of 2.6, 3.2, and 4.8 mm.

Figure 13 shows the peak pressure transmitted to the BF of the foam layer for each foam density and face sheet thickness investigated in this research. As seen in this figure, regardless of the face sheet thickness value, the peak pressure drops significantly, as the foam density decreases from 250 to 90 kg/m³. The foam with a density of nearly 90 kg/m³ shows an enhanced ability in blast energy dissipation. However, it should be noted that at densities lower than approximately 90 kg/m³, the energy dissipation ability of the foam is decreased.

Figure 13.

Variations of the peak pressure, transmitted to the back face of the foam layer, with the foam density for sandwich panels of different metal face sheet thicknesses.

As seen in Figure 13 at the foam density of 44 kg/m³, the peak pressure induced in the back of the foam core is increased significantly. It is postulated that in such foams, the excess number of closed cells diminishes the mechanical strength of the cell walls. As such, the foam exhibits a different response behavior and its energy dissipation ability is decreased. Given the weak mechanical properties of the foam of 44 kg/m³ density value, the FF could deflect easily, without influencing the BF of the panel. According to the experimental observations reported in Yazici et al. (2014), a sort of slapping effect occurred between the FF and BF, which resulted in an increased pressure in this particular case. The slapping effect was verified in the finite element method of this study, as the strain values within the entire foam material, along the application of the explosive load, exceeded the extreme value obtained from the material test (Figure 9). As such, the foam elements were totally eliminated at the central region of the panel, and the FF slapped the BF. In order to measure the magnitude of the peak pressure that was developed in the foam material bonded to the BF of the panel, a large erosion criteria value was assigned to the two rows of the foam elements at the vicinity of the BF. Accordingly, since these elements were not eliminated, the peak pressure was evaluated by the FE analysis (see Figure 13).

Figure 13 clearly shows that regardless of the thickness of the metal face sheets, there is a certain lower limit for the PU-foam density to remain effective in the response mitigation of sandwich panel. Given the weak energy absorption capability of the foam of 44 kg/m³ mass density, it was excluded from the parametric studies of this article.

Foam core density—BF deflection of panel

The BF deflection of the panel was measured for different foam core densities and face sheet thicknesses. As seen in Figure 14, regardless of the face sheet thickness, the maximum BF deflection is decreased as the foam core density increases from 90 to 250 kg/m³. These results that are in agreement with the previous observations (Hassan et al., 2012) are relevant to the increased foam flexural rigidity and fracture toughness with increasing foam density (Uday et al., 2014; Marsavina et al., 2013). There is an approximately linear relationship between the BF deflection and the foam density (see Figure 14). In addition, a decrease in the face sheet thickness increases the effect of foam density on the deflection of the panel.

Figure 14.

Variations of the maximum deflection at the back face (BF) of the panel with the foam density for the sandwich panels of different metal face sheet thicknesses.

Figure 15 shows the deformed status of the sandwich panels with an initial foam core thickness of 25 mm and face sheet thickness of 3.2 mm. The mass density of the foam material in the deformed sandwich panels shown in Figure 15 is 44, 90, and 220 kg/m³, respectively. It can be seen that under the same loading condition, the sandwich panels of lower foam densities experience more deformations. No tension strength has been considered for the foam material in the FE model. Thus, for the areas in which the foam core is subjected to tension effects (e.g. at the two ends of the panel, where a differential deformation occurs between the FF and BF), the foam material is detached from the steel sheet. Therefore, the propagation of the transverse pressure in the foam core forces the detached foam to move outward.

Figure 15.

Deformation of the sandwich panels with different foam core densities at the instant of maximum back face deflection (the initial thickness of the foam layer was 25 mm, and the thickness of the metal face sheets is 3.2 mm in all cases).

Optimum foam density

In “Foam core density—pressure attenuation” and “Foam core density—BF deflection of panel” sections, the influence of the foam mass density on the maximum transmitted pressure and the BF deflection of a sandwich panel were investigated. It was concluded that an increase in the foam density decreases the BF deflection of the panel due to the enhanced flexural rigidity of the foam core. However, the magnitude of pressure transmitted to the BF of the panel is increased with increasing mass density. This is due to the decreased porosity of the foam. Therefore, an increase in the foam density may adversely affect the pressure attenuation efficiency of the foam. As such, it seems necessary to find an optimum value for the mass density of the foam that compromises between the transmitted pressure and the BF deflection of the sandwich panel.

Figure 16(a) to (c) shows the variations of peak pressure and BF deflection with foam density at sandwich panels having metal face sheet thicknesses of 2.6, 3.2, and 4.8 mm, respectively. As seen in Figure 16(a), for the thinnest metal sheet with a thickness of 2.6 mm, a sort of balance exists between the maximum pressure and the maximum BF deflection of the panel for a foam density of approximately 140 kg/m³. For the other foam mass density values, one of the response parameters is boosted, while the other one is diminished. Therefore, a mass density of approximately 140 kg/m³ is deemed as the optimum value for the sandwich panel having metal face sheets of 2.6 mm thickness. For the sandwich panels of relatively thicker metal sheets (i.e. the thickness values of 3.2 and 4.8 mm), the influence of foam density on the BF deflection becomes insignificant. However, the peak pressure transmitted to the BF of the foam layer remains sensitive to the foam density. According to Figure 16(b), for metal sheets of 3.2 mm thickness, the two curves intersect at a foam density of approximately 90 kg/m³. As such, this value may be viewed as the optimum value for this sandwich panel. For panels of 4.8 mm thick metal sheets, a foam material of even a lower mass density may be employed to achieve further reduction in the blast pressure that is transmitted to the BF of the foam layer.

Figure 16.

Variations of the maximum pressure and the maximum back face (BF) deflection with foam core density at the sandwich panels with metal face sheet thicknesses of (a) 2.6 mm, (b) 3.2 mm, and (c) 4.8 mm.

According to Figure 16(a) to (c), the optimum foam mass density in a sandwich panel is dependent on the thickness of its metal face sheets. The optimum density is decreased with increasing thickness of the metal sheets.

Foam core thickness—pressure mitigation and BF deflection of panel

To examine the influence of the foam layer thickness on the response of the sandwich panel, three different thicknesses, namely, 25, 45, and 75 mm, were used for the foam layer of sandwich panels in the FE model. The nominal mass density of the foam material was chosen to be 140 kg/m³, as the optimum mass density evaluated in the previous section for the sandwich panel of the thinnest metal face sheets (see Figure 16(a)). The analysis outputs include the maximum pressure transferred to the BF of the foam layer and the maximum deflection at the BF of the sandwich panel.

As seen in Figure 17, the thickness of the foam layer significantly affects the maximum BF deflection of the panel. The maximum values of the BF deflection decrease approximately 50% in the foam layer with 75 mm thickness as compared to that of 25 mm.

Figure 17.

Foam core thickness versus the BF deflection of the panel for three different face sheet thicknesses.

Figure 18 shows the variations of peak pressure developed in the BF of the foam layer with respect to the foam layer thickness. As seen, the peak pressure is decreased with increasing thickness of the foam layer. The most pressure attenuation that is achieved belongs to the sandwich panel having face sheets of 2.6 mm thickness, where the peak pressure in a foam layer of 75 mm thickness is only 7.5% lower than that of the foam of 25 mm thickness. As seen in Figures 17 and 18, both the panel deflections and pressure attenuation capabilities are affected by the foam density.

Figure 18.

The pressure transmitted to the back of the foam core versus foam core thickness for three different face sheet thicknesses.

Figure 19 shows the deformation of the sandwich panels with different foam core thicknesses, namely, 25, 45, and 75 mm. The deformed shapes correspond to the instant at which the BF deflection is a maximum. A close examination of Figure 19 shows that in the case of the foam thickness of 75 mm, a kind of wrinkle in the middle height at the center of the foam core is created. The impact loading that is applied on the FF of the panel develops a shock wave that propagates within the panel. As the shock wave reaches the BF of the panel, a reflected wave forms and propagates backward within the panel until it reaches the FF and re-reflected to the foam substance. It is postulated that the continuous re-reflection of the shock wave between the FF and BF compresses the foam core elements toward each other and causes a kind of wrinkle in the foam material.

Figure 19.

Maximum deformation in sandwich panels with different foam core thicknesses, namely, 25, 45, and 75 mm (foam mass density 140 kg/m³ and face sheet thickness 3.2 mm).

Total energy absorption (TEA)

In general, the area under the “load-displacement” plot indicates the TEA. This area may be calculated from $\int_{0}^{δ_{f}} Pd δ$ , where P represents the axial crushing load and $δ$ is the corresponding axial crushing displacement with a final value of $δ_{f}$ . The specific energy absorption (SEA) is the energy absorbed per unit mass of the material (i.e. TEA/mass) and is crucial to the design of parts that require weight reduction (Zhu et al., 2008). As an important parameter, the maximum internal energy that is absorbed by the panel and its different components are cited in Table 6. (The information given by this table belongs to the foam core mass densities of 90, 175, and 250 kg/m³.) Two different foam core thickness values of 25 and 75 mm that are embedded in the panels of 2.6 and 4.8 mm steel face sheets have been taken into account in Table 6.

Table 6.

Maximum internal energy absorption of the panel and its different components (J).

Steel face sheet thickness (mm)		2.6				4.8
Core mass density (kg/m3)	Core thickness (mm)	Panel	BF	PU foam	FF	Panel	BF	PU foam	FF
90	25	51.2	10.1	24	17.1	29.5	6.4	13.2	9.9
	75	63.4	2.3	33.8	27.3	36.6	0.9	23.7	11.5
175	25	35.8	10.2	14.7	10.9	20.9	6.5	7.7	6.7
	75	46	2.5	32.1	11.4	27.1	1.5	18.9	6.7
250	25	31.9	10.1	11.6	10.2	18.9	6	6.7	6.2
	75	39.7	2.9	27.8	9	23.7	1.7	16.7	5.3

PU: polyurethane; BF: back face; FF: front face.

When the explosive energy is transferred to the panel, it will be dissipated via the panel plastic deformations. The initial energy that is transferred to the structure (ET) is essentially the sum of the kinetic (EK) and internal energy (EI, also known as deformation energy, ED). The kinetic energy will be reduced over time, while the internal energy of the system will be increased (Zhu et al., 2008). Based on the literature (Fleck and Deshpande, 2004) for the sandwich panels subjected to blast load, the whole deformation can be split into three phases:

Phase I: In this phase, the blast impulse is delivered onto the FF of sandwich structure, and the FF attains an initial velocity while the rest of the structure is stationary.

Phase II: The core is compressed while the BF is stationary.

Phase III: The BF starts to deform, the whole structure is deformed at the same velocity, and finally the structure is brought to rest by plastic bending and stretching.

Given the impulse delivered on the FF (I), with the impulse transmission, the FF obtains an initial velocity, v₀, given below

v_{0} = \frac{I}{A ρ_{f} h_{f}}

(5)

where A is the exposed area, and ρ_f and h_f are the material density and thickness of face sheets, respectively. Based on the conservation of momentum, the kinetic energy of the FF is calculated by equation (6), which is the total energy of the structure obtained from the blast load

W_{I} = \frac{I^{2}}{2 A ρ_{f} h_{f}}

(6)

At the end of Phase II, the whole structure would have the identical velocity, and the kinetic energy can be calculated by

W_{I I} = \frac{I^{2}}{2 A (2 ρ_{f} h_{f} + H_{c} ρ_{c})}

(7)

where ρ_c and H_c are the mass density and thickness of the core, respectively. This part of energy would be dissipated by plastic bending and stretching of the panel in Phase III.

An examination of Table 6 indicates that in most cases, the foam core absorbs a significantly larger amount of energy as compared to the steel face sheets. For a given thickness, the foam energy absorption is decreased with increasing mass density. For a given mass density, the foam energy dissipation capability is boosted with an increase in the foam thickness. An increase in the thickness of the steel face sheet results in an increased stiffness, decreased deformation, and eventually a decreased energy absorption capability.

Conclusion

In this article, the blast response of sandwich panels made of inner PU foam layer with two outer metal sheets was investigated through FE methods. The objective was to examine the influence of mass density and thickness of the foam layer on the dynamic blast response of the sandwich panel for a given shock profile. The following includes the major conclusions of this study:

The peak pressure transmitted to the BF of the foam layer (as a measure for the energy dissipation ability of the sandwich panel) is not always decreased with decreasing mass density of the foam material. At density value of approximately 90 kg/m³ for the sandwich panels studied in this article, any further decrease in the foam density undermines the pressure attenuation ability of the foam.

The maximum deflection at the BF of a sandwich panel is decreased with increasing mass density of the foam layer.

Since the mass density of the foam material has different effects on the response parameters of a sandwich panel, an intermediate value that compromises between the response parameters must be selected as the optimum density value. The value of optimum mass density is affected by the thickness of metal sheets of the panel. The value of nearly 140 kg/m³ was found to be an optimum value for the sandwich panels having metal sheets of 2.6 mm thickness. At this optimum density value, a compromise is made between the magnitude of peak pressure transformed to the BF of the foam layer and the BF deflection of the panel. For the panels of thicker metal sheets (3.2 and 4.8 mm), the optimum foam density value is decreased to 90 kg/m³ and lower.

Using a foam core of optimum mass density, an increase in the thickness of the foam layer is beneficial to the blast mitigation ability of sandwich panel. Both the peak pressure transferred to the BF of the foam layer and the BF deflection of the sandwich panel were found to be sensitive to the variations in the thickness of the foam layer.

For a given foam thickness, variations in the thickness of the metal face sheets significantly affect the BF deflection of the panel.

For the foam nominal mass densities of 90 kg/m³ and greater, the TEA capability of the foam core is decreased with increasing foam mass density.

An increase in the foam core thickness results in an increased energy absorption capability. In addition, the foam core absorbs a significantly larger amount of energy as compared to the other components of the panel.

An increase in the thickness of the steel face sheet results in a decreased maximum internal energy that is absorbed by the panel.

Footnotes

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Hamid Toopchi-Nezhad

References

ASTM D7487-08 (2008) Practice for Polyurethane Raw Materials Polyurethane Foam Cup Test (Vol. D7487–08). West Conshohocken, PA: ASTM International.

AUTODYN (2006) Theory Manual, Century Dynamics. Canonsburg, PA: ANSYS, Inc.

Balandin

Bragov

Zefirov

, et al. (2017) Experimental and numerical study of the impact interaction of a rigid impactor with a combined target. Combustion, Explosion, and Shock Waves 53(1): 116–121.

Boey

(2009) Investigation of shock wave attenuation in porous materials. MSc dissertation, Naval Postgraduate School, Monterey, CA.

Bryson

(2009). Impact Response of Polyurethane. MSc dissertation, Washington State University, Washington, DC.

Chen

Winfree

(2002) High-strain-rate compressive behavior of a rigid polyurethane foam with various densities. Experimental Mechanics 42(1): 65–73.

Fleck

Deshpande

(2004) The resistance of clamped sandwich beams to shock loading. Journal of Applied Mechanics 71: 1–16.

Gibson

Ashby

(1999) Cellular Solids: Structure and Properties. Cambridge: Cambridge University Press.

Goel

Matsagar

Marburg

, et al. (2012) Comparative performance of stiffened sandwich foam panels under impulsive loading. Journal of Performance of Constructed Facilities 27(5): 540–549.

10.

Guruprasad

Mukherjee

(2000a) Layered sacrificial claddings under blast loading Part I—analytical studies. International Journal of Impact Engineering 24(9): 957–973.

11.

Guruprasad

Mukherjee

(2000b) Layered sacrificial claddings under blast loading Part II—experimental studies. International Journal of Impact Engineering 24(9): 975–984.

12.

Hartman

Larsen

Boughton

(2006) Blast Mitigation Capabilities of Aqueous Foam. Albuquerque, NM: Sandia National Laboratories Press.

13.

Hassan

Guan

Cantwell

, et al. (2012) The influence of core density on the blast resistance of foam-based sandwich structures. International Journal of Impact Engineering 50: 9–16.

14.

ISO 844 (2014) Rigid cellular plastics—Determination of compression properties.

15.

Johnson

(1983) A constitutive model and data for materials subjected to large strains, high strain rates, and high temperatures. In: Proceedings of the 7th international symposium on ballistics, The Hague, 19–21 April.

16.

Kitagawa

Yamashita

Takayama

, et al. (2009) Attenuation properties of blast wave through porous layer Shock Waves. New York: Springer, pp. 73–78.

17.

Kotzialis

Derdas

Kostopoulos

(2005) Blast Behaviour of Plates with Sacrificial Cladding.on Computational Mechanics. In: Paper presented at the 5th GRACM international congress, Limassol, 29 June–1 July.

18.

Lee

Wang

Sung

, et al. (1986) The Attenuation of Shock Waves in PU Foam and Its Application Shock Waves in Condensed Matter. New York: Springer, pp. 687–692.

19.

20.

Marsavina

Linul

Voiconi

, et al. (2013) A comparison between dynamic and static fracture toughness of polyurethane foams. Polymer Testing 32(4): 673–680.

21.

Matsagar

(2016) Comparative performance of composite sandwich panels and non-composite panels under blast loading. Materials and Structures 49(1–2): 611–629.

22.

Ong

Boey

Hixson

, et al. (2011) Advanced layered personnel armor. International Journal of Impact Engineering 38(5): 369–383.

23.

Palanivelu

Van Paepegem

Degrieck

, et al. (2011) Close-range blast loading on empty recyclable metal beverage cans for use in sacrificial cladding structure. Engineering Structures 33(6): 1966–1987.

24.

Roy

(2009) Investigation of advanced personnel armor using layered construction. MSc dissertation, Naval Postgraduate School, Monterey, CA.

25.

Sarier

Onder

(2007) Thermal characteristics of polyurethane foams incorporated with phase change materials. Thermochimica Acta 454(2): 90–98.

26.

Schwer

(2007) Optional strain-rate forms for the Johnson cook constitutive model and the role of the parameter Epsilon_0. In: Paper presented at the 6th European LS_DYNA users’ conference, Gothenburg, 29–30 May.

27.

Uday

Varma

CST

Varma

BNK

, et al. (2014) The influence of rigid foam density on the flexural properties of glass fabric/epoxy-polyurethane foam sandwich composites. International Journal of ChemTech Research 6(6): 3314–3317.

28.

Xia

Bennett

(2017) An Analytical Model of Linear Density Foam–protected Structure Under Blast Loading. International Journal of Protective Structures 8(3): 454–472.

29.

Yazici

Wright

Bertin

, et al. (2014) Experimental and numerical study of foam filled corrugated core steel sandwich structures subjected to blast loading. Composite Structures 110: 98–109.

30.

Zhu

Zhao

, et al. (2008) Structural response and energy absorption of sandwich panels with an aluminum foam core under blast loading. Advances in Structural Engineering 11: 525–536.