Design of multifunctional polymethacrylimide foam sandwich with excellent radar absorption capability and compressive performance

Abstract

A composite multifunctional radar absorption polymethacrylimide (RAPMI) foam sandwich (MRAPS) was constructed. Analytical models for out-of-plane and in-plane compression of the MRAPS were established. Three-dimensional failure mechanism maps and specific strength cubic cloud maps were drawn. Finally, a multiobjective particle swarm optimization (MOPSO) algorithm was used to achieve an integrated design, and a specific calculation example was tested and verified through experiments. The resulting sandwich with a 43.0 mm thickness core and 0.13 g/cm³ density achieves 90% effective absorption above broadband (RL ≤ −10 dB) in the 2–18 GHz range. As a radar absorption absorber, the MRAPS increased the engineering value and strategic importance of PMI foam for structural components in the military and aerospace sectors.

Keywords

Sandwich panel polymethacrylimide foam compression radar absorption structure optimization

Introduction

Foam sandwich structures have been widely used in recent years owing to their excellent mechanical performance and simple preparation process. Polymethacrylimide (PMI) foam is a closed-cell rigid foam material that exhibits excellent mechanical performance; it has been widely used as the sandwich core in construction, civil, marine, aerospace, military, and other engineering fields.¹ In addition, PMI foam exhibits some excellent functional characteristics.² However, PMI foam itself is an electromagnetic wave (EMW)–transmitting material. Under the ubiquitous complex electromagnetic environment, improving the EMW absorption performance of PMI foam will considerably enhance its engineering value. Currently, numerous studies are being conducted on microwave absorption materials serving as coatings. Microwave absorption structures can often easily achieve wider bandwidth absorption and multifunctional integrated design. Li et al.,³ Hui et al.,⁴ and Chen et al.⁵ prepared various structural absorbers with satisfactory thermal conductivity, sound absorption, and microwave absorption performance based on different substrate and absorbing materials. Fan et al.⁶ and Wang et al.⁷ fabricated different composite absorbing lattices or sandwich structures. Through simulation, theoretical design, and experiments, these structures were proven to have strong mechanical properties and excellent broadband absorption abilities. Only few studies have been conducted on microwave absorption PMI foam. Song et al.⁸ prepared radar absorption PMI (RAPMI) foam; experimental results indicated that RAPMI foam has excellent radar absorption ability and mechanical properties. In this study, a composite multifunctional RAPMI foam sandwich (MRAPS) structure was constructed based on previous research.⁸ The reflection loss (RL) of the multilayer MRAPS was analyzed. An analytical model for the MRAPS under quasi-static compressive load was established. The size effects of independent conditions were analyzed using three-dimensional (3D) failure maps and cubic cloud maps. Finally, optimization was performed using a multiobjective particle swarm optimization (MOPSO) algorithm. Six MRAPS specimens were designed, manufactured, and tested based on the optimized results. The results aligned well with the predictions and demonstrated excellent broadband radar absorption and compressive load–bearing capabilities of the MRAPS. Furthermore, the sandwich structure offers the advantages of cost-effectiveness and easy processing, rendering it suitable for mass production. These advantages strengthen practical value of this load-bearing and microwave absorption–integrated multifunctional composite sandwich structure in military and aerospace engineering.

Electromagnetic model

When EMW is vertically incident on the surface of a homogeneous material, the input impedance Z _in is expressed as follows:

Z_{i n} = Z_{0} {(\frac{μ_{r}}{ε_{r}})}^{\frac{1}{2}} \tanh {j (\frac{2 π f t}{c}) {(ε_{r} μ_{r})}^{\frac{1}{2}}},

(1)

Z_{0} = {(\frac{μ_{0}}{ε_{0}})}^{\frac{1}{2}},

(2)

where Z ₀ represents the impedance of free space; ɛ _r represents the relative permittivity; μ _r represents the relative magnetic permeability; c represents the speed of light in vacuum; f represents the frequency; t represents the thickness; and ɛ ₀ and μ ₀ represent the relative permittivity and magnetic permeability of free space, respectively. The MRAPS can be regarded as a multilayer absorption material. The relative permittivity and magnetic permeability of the 1_st, 2_nd…, and N_th layers can be expressed as ɛ _r1, u _r1 ɛ _r2, u _r2… ɛ _rN, u _rN. The impedances between the 1_st and 2_nd, 2_nd and 3_rd… N_th and N + 1_th layers can be expressed as Z _in1, Z _{in
2}… Z _inN, where Z _inN can be delivered based on the following iterative equations:^9,10

Z_{i n N} = Z_{c N} \frac{Z_{i n (N - 1)} + Z_{c N} \tanh {j (\frac{2 π {f t}_{N}}{c}) {(ε_{r N} u_{r N})}^{\frac{1}{2}}}}{Z_{c N} + Z_{i n (N - 1)} \tanh {j (\frac{2 π {f t}_{N}}{c}) {(ε_{r N} u_{r N})}^{\frac{1}{2}}}},

(3)

where t _N represents the thickness of the N_th layer and Z _cN represents the wave impedance of the N_th layer. Moreover, Z _cN can be expressed as follows:

Z_{c N} = Z_{0} \sqrt{\frac{u_{r N}}{ε_{r N}} .}

(4)

Combining equations (1)–(4), the RL of the MRAPS can be obtained as follows:

R L = 20 \lg | \frac{Z_{i n N} - Z 0}{Z_{i n N} + Z 0} | .

(5)

when RL is less than −10 dB, approximately 90% of the corresponding band EMWs are absorbed.

Mechanical models

Foam model

The compressive strength σ _c, shear strength τ _c, compressive modulus E _c, and shear modulus G _c of RAPMI foam can be expressed as follows:¹¹

\begin{array}{l} \frac{σ_{c}}{σ_{y s}} = A_{1} {(\frac{ρ_{c}}{ρ s})}^{1.5} + B_{1} (\frac{ρ_{c}}{ρ_{s}}) \\ \frac{E_{c}}{E_{s}} = A_{2} {(\frac{ρ_{c}}{ρ_{s}})}^{2} + B_{2} (\frac{ρ_{c}}{ρ_{s}}) \\ \frac{τ_{c}}{σ_{y s}} = A_{3} {(\frac{ρ_{c}}{ρ_{s}})}^{\frac{2}{3}} \\ \frac{G_{c}}{E_{s}} = A_{4} {(\frac{ρ_{c}}{ρ_{s}})}^{2} + B_{4} \frac{ρ_{c}}{ρ_{s}} \end{array},

(6)

where σ _ys represents the compressive strength of PMI particles, ρ _c represents the foam density, ρ _s represents the particle density, and E _s represents the particle modulus. Moreover, A ₁– A ₄, B ₁– B ₂, and B ₄ are constants, with the following values:

A_{1} = 1.55, B_{1} = - 0.1, A_{2} = 1.41, B_{2} = 0.14, A_{3} = 0.097, A_{4} = 0.56, {and B}_{4} = 0.05

of RAPMI foam are suggested in Ref. [8].

Out-of-plane compression

The schematics of out-of-plane and in-plane compression for the MRAPS are shown in Figure 1. The FR-4 GFRC panel was selected as the top skin, while the CFRC panel was selected as the bottom skin. The performance parameters of CFRC and GFRC panels for forming the MRAPS are shown in Table 1. The schematics of out-of-plane compression are shown in Figure 1(a). The performance was determined based on the compressive strength of the core. Ref. [8] indicates that the mechanical properties of the RAPMI foam with different absorbers wt% are similar (maximum difference of ≤2.5%). Therefore, the strength between layers can be regarded as equivalent. The out-of-plane compressive failure strength σ _c1 of the MRAPS can be expressed as follows:

σ_{c 1} = σ_{c} = [A_{1} {(\frac{ρ_{c}}{ρ_{s}})}^{1.5} + B_{1} (\frac{ρ_{c}}{ρ_{s}})] σ_{y s} .

(7)

Table 1.

Mechanical parameters of T700 CFRC and FR-4 GFRC woven laminates.

Material	Property	Value	Description
GFRC woven laminates	E _f1t	27 GPa	Longitudinal tensile modulus
	E _f1c,	25 GPa	Longitudinal compressive modulus
	σ _f1t	470 MPa	Longitudinal tensile strength
	σ _f1c	446 MPa	Longitudinal compressive strength
	G ₁	6 GPa	In-plane shear modulus
	τ ₁	53 MPa	In-plane shear strength
	ν ₁	0.12	Poisson’s ratio
	ρ ₁	2.68 g/cm³	Density
CFRC woven laminates	E _f2t	57 GPa	Longitudinal tensile modulus
	E _f2c,	56 GPa	Longitudinal compressive modulus
	σ _f2t	767 MPa	Longitudinal tensile strength
	σ _f2c	670 MPa	Longitudinal compressive strength
	G ₂	4.5 GPa	In-plane shear modulus
	τ ₂	106 MPa	In-plane shear strength
	ν ₂	0.09	Poisson’s ratio
	ρ ₂	1.81 g/cm³	Density

The conversion equation between σ _c1 and critical load P _c1 can be expressed as follows:

σ_{c 1} = \frac{P_{c 1}}{w L_{0}},

(8)

where L ₀ represents the sandwich length and w represents the width.

In-plane compression

In-plane compressive loads typically occur on structural sidewalls. The schematics and potential failure modes of the MRAPS under in-plane compressive conditions are shown in Figure 1(b). When the sandwich is under an in-plane compressive load, the conversion equation between compressive strength σ _c2 and critical load P _c2 can be expressed as follows:¹²

σ_{c 2} = \frac{P_{c 2}}{2 {w h}_{f}},

(9)

where h _f represents the skin thickness. Four potential failure modes have been reported for foam sandwiches¹³: Euler buckling (EB), skin fracture (SF), skin wrinkling (SW), and shearing buckling (SB). EB failure usually occurs for a large length-to-thickness ratio. The failure strength of EB can be expressed as follows:

σ_{e u l e r} = \frac{2 π^{2}}{L_{1^{2}} {w h}_{f}} {(E I)}_{e q},

(10)

where L ₁ represents the length of the sandwich column and (EI) _eq represents the equivalent bending stiffness. The neutral axis is a surface in the sandwich where the bending stress is zero during flexural deformation. Because of the difference between the top and bottom skins of the MRAPS, the position of the neutral axis does not coincide with that of the geometric midplane. An x–y coordinate system with a geometric center as the origin was established. The coordinates for the neutral axis of the MRAPS can be expressed as follows:^14,15

y_{n} = \frac{\int_{- \frac{h}{2} - h f}^{- \frac{h}{2}} E_{f 2 t} y d y + \int_{- \frac{h}{2}}^{\frac{h}{2}} E_{c} y d y + \int_{\frac{h}{2}}^{\frac{h}{2} + h f} E_{f 1 c} y d y}{\int_{- \frac{h}{2} - h f}^{- \frac{h}{2}} E_{f 2 t} d y + \int_{- \frac{h}{2}}^{\frac{h}{2}} E_{c} d y + \int_{\frac{h}{2}}^{\frac{h}{2} + h f} E_{f 1 c} d y},

(11)

where h represents the core height. By solving and scaling the results of equation (11), the coordinates can be expressed as follows:

y_{n} = \frac{(h_{f} + h) (E_{f 1 c} - E_{f 2 t})}{2 (E_{f 1 c} + E_{f 2 t})} .

(12)

In this study, the coordinate can be approximately calculated as −0.16 h. Therefore, the bending stiffness of the MRAPS can be expressed as follows:

{(E I)}_{e q} = \frac{{w h}_{f^{3}} (E_{f 1 c} + E_{f 2 t})}{12} + \frac{w h^{3} E c}{12} + \frac{{w h}_{f} E_{f 1 c} {(0.5 h_{f} + 0.66 h)}^{2}}{4} + \frac{{w h}_{f} E_{f 2 t} {(0.5 h_{f} + 0.34 h)}^{2}}{4} .

(13)

Based on $E_{f 1 c} > > E_{c}$ , $E_{f 2 t} {≫ E}_{c}$ , and $h ≫ h_{f}$ , (EI) _eq can be simplified as follows:

{(E I)}_{e q} = \frac{{w h}_{f} [E_{f 1 c} {(0.5 h_{f} + 0.66 h)}^{2} + E_{f 2 t} {(0.5 h_{f} + 0.34 h)}^{2}]}{4} .

(14)

By combining Equations (10) and (14), the EB failure strength can be expressed as follows:

σ_{e u l e r} = π^{2} \frac{[E_{f 1 c} {(0.5 h_{f} + 0.66 h)}^{2} + E_{f 2 t} {(0.5 h_{f} + 0.34 h)}^{2}]}{2 L_{1}^{2}} .

(15)

The SF failure strength under in-plane compressive conditions depends on the compressive strength of the skins. Because of the difference in the upper and lower skins of the MRAPS, the failure strength can be expressed as follows:

σ_{s f} = {(σ_{f 1 c}, σ_{f 2 c})}_{\min} .

(16)

SW failure usually occurs in a sandwich with softer skin. The SW failure strength can be expressed as follows:^12,16,17

σ_{sw} = 0.5 {[{(E_{f 1 c}, E_{f 2 c})}_{\min} E c G c]}^{\frac{1}{3}} .

(17)

By combining equations (6) and (17), the final equation for the SW failure strength can be expressed as follows:

σ_{sw} = 0.5 {E_{s}^{2} {(E_{f 1 c}, E_{f 2 c})}_{\min} [A_{2} {(\frac{ρ_{c}}{ρ_{s}})}^{2} + B_{2} (\frac{ρ_{c}}{ρ_{s}})] [A_{4} {(\frac{ρ_{c}}{ρ_{s}})}^{2} + B_{4} \frac{ρ_{c}}{ρ_{s}}]}^{\frac{1}{3}} .

(18)

The SB failure strength of the sandwich under in-plane compressive strength conditions can be approximated as the shear stiffness of the core. The SB failure strength of the MRAPS can be expressed as¹³ follows:

σ_{s b} = \frac{h G C}{2 h_{f}},

(19)

By combining equations (6) and (19), the SB failure strength can be expressed as follows:

σ_{s b} = \frac{h E_{s}}{2 h_{f}} [A 4 {(\frac{ρ_{c}}{ρ_{s}})}^{2} + B 4 \frac{ρ_{c}}{ρ_{s}}] .

(20)

The in-plane compressive strength of the MRAPS should be a minimum of the four aforementioned failure modes, which can be expressed as follows:

σ_{c 2} = {(σ_{e u l e r}, σ_{s w}, σ_{s f}, σ_{s b})}_{\min} .

(21)

Size effects

Out-of-plane compression

The size effect analysis under the out-of-plane compressive load of the MRAPS is shown in Figure 2. Because σ _c1 is only a function of ρ _c /ρ _s, the change mode is consistent with that of σ _c. The density of sandwich ρ ₀ can be expressed as follows:

ρ_{0} = \frac{(ρ_{1} + ρ_{2}) h_{f} + h ρ_{c}}{(2 h_{f} + h)} .

(22)

Figure 2(a) shows the dimensionless surface of the out-of-plane compressive specific strength, while Figure 2(b) shows the projection cloud map. σ_c1/ρ₀ increases with increasing ρ_c/ρ_s and decreases with increasing h_f/h. When h_f/h is small, the increase rate of σ_c1/ρ₀ with ρ_c/ρ_s is slow. The growth rate gradually increases with increasing h_f/h. When ρ_c/ρ_s is small, σ_c1/ρ₀ remains almost unchanged. The rate of decrease in σ_c1/ρ₀ with h_f/h accelerates rapidly with increasing ρ_c/ρ_s.4.2 In-plane compression

A failure map is an essential tool for analyzing the size effects of a sandwich under different external loads. Initially, a two-dimensional (2D) failure map was proposed by Triantafillou et al.,^13,18 but later, Wei et al.¹⁹ introduced the concept of a 3D failure map and analyzed the failure mechanism of composite honeycomb sandwiches. The in-plane compression strength of the MRAPS can be expressed as a function: $σ_{c 2} = f (ρ_{c} / ρ_{s}, h_{f} / h, h / L_{1})$ . Therefore, the 3D failure map can effectively reflect the entire failure modes under different size configurations for all situations. From the application viewpoint, the range of variables is limited from 0.01 to 0.3. The 3D failure and cubic cloud maps are shown in Figure 3. The 3D failure map obtained under the in-plane compressive load is shown in Figure 3(a). The intercepted 2D failure maps are shown in Figure 3(b). Notably, when h/L ₁ is small, EB is the primary failure mode. However, when h/L ₁ > 0.162, EB failure mode does not occur and SB, SW, and SF become the three competing failure modes. When ρ _c /ρ _s is < 0.153, SB is the primary failure mode and SW only occurs when h _f /h is small. When ρ _c /ρ _s exceeds 0.153, SF becomes the dominant failure mode. When ρ _c /ρ _s is > 0.261, the sandwich completely fails through SF. The in-plane compressive cubic cloud map is shown in Figure 3(c). Color and transparency simultaneously reflect the value of the in-plane compressive specific strength σ _c2 /ρ ₀. Σ _c2 /ρ ₀ decreases monotonically with increasing h _f /h, because an increase in skin thickness contributes more to the increase in sandwich density than that in in-plane compressive strength. Therefore, when $h_{f} / h = 0.01$ , σ _c2 /ρ ₀ reaches maximum. The cloud map of $h_{f} / h = 0.01$ is shown in Figure 3(d). The three competing failure modes are SW, SF, and EB. σ _c2 /ρ ₀ in the EB area is significantly lower than in the SW and SF regions. Furthermore, σ _c2 /ρ ₀ at the junction area is higher than that around it. The maximum point is calculated to be on the line between SW and SF. The coordinates are calculated to be (0.155, 0.155). The maximal σ _c2 /ρ ₀ is calculated to be 19.7 × 10⁵ m²/s².

Design and manufacture

Multicondition optimization

Optimization of the sandwich is important as it guides practical engineering. The preparation before optimization should be as follows:

(1) Select a density of the RAPMI foam and test its EM parameters for different absorbent wt% (density is usually used as the standard to distinguish types of PMI foam in engineering).

(2) Calculate the maximum specific strength of the MRAPS under independent mechanical conditions.

The importance of radar absorption performance and mechanical performance is considered equal (weight 1:1). Furthermore, W ₁ and W ₂ represent the scaling factors of the absorption bandwidth and specific strength, respectively, and can be expressed as follows:

W_{1} = \frac{| A B |}{| f_{2} - f_{1} |},

(23)

W_{2} = Min (\frac{f_{1} (x, y)}{M_{1}}, \frac{f_{2} (x)}{M_{2}}),

(24)

where ( f ₁, f ₂) represents the target absorption frequency band of RL ≤ −10 dB and AB represents the actual absorption frequency band. In this study, f ₁ = 2 GHz and f ₂ = 18 GHz. Moreover,

| A B |

represents the actual absorption bandwidth. The variables

x = h_{f} / h

and

y = h / L_{1}

are introduced. Furthermore, M ₁ and M ₂ represent the maximum specific strengths of the corresponding density under three independent conditions; f ₁ (x,y) and f ₂(x) are the specific strength functions and can be expressed as

σ_{c 2} / ρ_{0} {= f}_{1} (x, y)

and

σ_{c 1} / ρ_{0} {= f}_{2} (x)

, respectively. The variable u _i is introduced and satisfied as

u_{i} = t_{i} (i = 1, 2 \dots N, N + 1) .

According to the previous analysis, the absorption performance of the MRAPS can be expressed only as a function of the thickness of each layer and can be expressed as follows:

R L = F_{N} = F (u_{1}, u_{2}, \dots u_{N}, u_{N + 1})

. The minimum thickness is sought to reduce the weight and increase the designability. The MOPSO algorithm is applied, and the equations can be expressed as follows:

{\begin{cases} F i n d x, y, u_{1}, u_{2} . . ., u_{N}, u_{N + 1} \\ M a x i m i z e W_{1} + W_{2} \\ M i n i m i z e \sum_{i = 1}^{N + 1} u_{i} \\ s . t . 0.01 \leq x, y \leq 0.3 \\ u_{i} \geq 0 \\ 0.5 \leq u_{N + 1} \leq 2 \\ F_{N} \leq - 10 \\ f \subseteq A B \subseteq (2, 18) \\ W_{1} \geq δ_{1}, δ_{1} \in (0, 1) \\ W_{2} \geq δ_{2}, δ_{2} \in (0, 1) \end{cases},

(25)

where δ ₁ and δ ₂ represent the minimal scaling factors.

Calculated examples

The optimization example for an input density of 75 kg/m³ was calculated. The selectable wt% gradients were 1, 1.2, 1.5, 1.8, and 2 wt%, where N = 2→5 and δ ₁ = δ ₂ = 0.9. The EM parameters of these types of RAPMI foam are obtained from Ref. [8]. The effect of CFRC laminates is similar to that of metal panels, where $Z_{i n 0} = 0$ . The relative permittivity and magnetic permeabilities of the GFRC panel are 4.98 + 0.159i and 1, respectively.²⁰ Figure 4 shows the optimization results. The calculated relationship between Min (å u _i) and Max ( W ₁ + W ₂) is shown in Figure 4(a). When W ₁ + W ₂ ≤ 1.93, the main change is from W ₁, indicating that the increase in å u _i exerts a higher impact on EM properties than on mechanical performance. The skin thickness of GFRC ( u _N+1) was selected as 0.5 mm. This is because the compression ratio strength and absorption bandwidth decrease with increasing x . An increase in x considerably influences absorption performance. Min (å u _i) cannot be lower than 40.6 mm, or W ₂ will be < 0.9. Under the same sandwich thickness, Max ( W ₁ + W ₂) increases with increasing N, indicating that an increase in N increases the optimized absorption bandwidth. Furthermore, as N increases, the initial solution of Max ( W ₁ + W ₂) gradually increases from 1.81 to 1.85, indicating that under the condition of fixed 90% maximal specific strength ( W ₂ = 0.9), the absorption bandwidth at 2–18 GHz can directly achieve 90% maximum above ( W ₁ ≥ 0.91). When W ₁ reaches 1, the change in Max ( W ₁ + W ₂) is independent of N and the increase becomes considerably slower as thickness increases. The minimum value of the turning point is not less than 1.63. Figure 4(b) shows the EM optimization results for N = 2→5. The optimal Max ( W ₁+ W ₂) with h = 42 mm is selected. The calculated results are shown in Table 2. $| A B |$ increases with larger N, which is consistent with the previous result.⁸ The stronger absorption is mainly concentrated in the S and C bands. When N = 5, the absorption peak can reach −38.7 dB at 3.8 GHz. Figure 4(c) presents the out-of-plane compressive specific strength curve. When f ₂(x)/M₂ ≥ 0.9, h _f /h is lower than 0.013. When the core thickness is greater than 40.6 mm, h _f /h ≤ 0.012 and f ₂(x)/M₂ > 0.92 can be ensured. Figure 4(d) shows the 2D failure cloud map. The area that satisfies f ₁ (x, y)/M₁ > 0.9 is on the left. The main failure mode is SW. Furthermore, to avoid EB failure, h/L ₁ must be higher than 0.045 and h _f /h lower than 0.012.

Table 2.

Electromagnetic optimization results of MRAPS.

N	Stacking(%)	u_i (mm)	$\| AB \|$ (GHz)	Peak absorption (dB)
2	2%/1	25/17	14.6	−34.3
3	2%/1.2%/1	24.5/5.5/12	15.3	−28.3
4	2%/1.8%/1.5%/1	12/16.2/4.8/10	15.7	−33.1
5	2%/1.8%/1.5%/1.2%/1	8.4/1/25/2.6/5	16	−38.7

Sandwich manufacture

Figure 5 shows the fabrication process sketches for the MRAPS. First, the multilayer RAPMI foam with N = 2 and 3 was chosen and prepared. The single-layer RAPMI prepolymer with ρ _c = 75 kg/m³ in the mold was prepared according to Ref. [8] and placed from top to bottom according to the absorbent wt% of 1%, 1.2%, and 2% with thicknesses of 12, 5.5, and 24.5 mm (N = 3) and from 1% to 2% with thicknesses of 17 and 25 mm (N = 2). Further, they are placed near a radiation source. The crosslinking agent is added to initiate a crosslinking reaction between the interfaces while being protected by nitrogen gas with N₂ and Ar (Figure 5(a) and (b)). The multilayer RAPMI foam is formed after post-treatment in the foaming furnace. Subsequently, the T700 CFRC and FR-4 GFRC panels (Shandong Carbon Fiber Materials Technology Co., Ltd) were used as the bottom and top skins, respectively. The plies are symmetrical and orthotropic. The thickness is 0.5 mm, with four layers for the CFRC panel and three layers for the GFRC panel. HY-JM-05-40 epoxy adhesive film with a shear strength of 18.7 MPa (Hubei Hangyu New Materials Co., Ltd) was used to connect the skin and core. The MRAPS was obtained by heating at 120°C for 3 h under 0.1 MPa pressure in an autoclave (Figure 5(c)). Figure 5(d) shows the actual multilayer RAPMI foam. Figure 5(e)) shows the actual MRAPS structure. Six types of specimens were formed, where 1#–2# were used for EM tests and 3#–4# were used for mechanical tests. The structural parameters are shown in Table 3.

Table 3.

Structural parameters of formed MRAPS specimens.

Specimen	Layers	L₀ (mm)	w (mm)	h (mm)	h_f (mm)
1#	2	180	180	43	0.5
2#	3	180	180	43	0.5
3#	2	50	50	43	0.5
4#	3	50	50	43	0.5
5#	2	300	80	43	0.5
6#	3	300	80	43	0.5

Experimental test and discussion

Reflectivity tests

The reflectivity test was performed using an arcuate method at the Material Preparation Technology Innovation Center (Hebei, China) according to the standard GJB2038A–2011 procedure (Figure 6). Figure 6(a) shows the experimental setup. Figure 6(b) shows a schematic of the arcuate method. A vector network analyzer (Keysight E5063A, Keysight Technologies Limited Company, China) was used for data collection. Figure 6(c) and (d) show the test results. Overall, the experiments follow the established theory. The best-fit of the waveband is at 2–4 GHz. The overall absorption effect is stronger than predicted. For 1#, the minimum peak absorption at 3.23 GHz in the S band is −44.9 dB. In the C- and X-bands, two wave troughs with similar peak absorption are observed. In the Ku-band, a wave trough of −23.2 dB was observed. Subsequently, the absorption capacity rapidly decreased and exceeded −10 dB at 17.3 GHz. The overall absorption bandwidth reached 15.3 GHz, exceeding the predicted value by 4.84%. For 2#, the peak absorption was −28.5 dB at 3.05 GHz in the S band; moreover, the curve trends were similar to those of 1#. Troughs at 6.20, 8.12, 11.9, and 13.0 GHz with reflectivities of −23.8, −23.4, −22.2, and −16.1 dB, respectively were observed. The results exceeded −10 dB at 17.7 GHz. The overall absorption bandwidth reached 15.7 GHz, exceeding the prediction by 2.62%. The main reasons for this error are as follows: (1) there is a deviation in the tested EM parameter of the RAPMI foam, (2) errors caused by operation in experiments are inevitable, and (3) the imperfect homogeneous distribution of absorbents causes interference cancellation between EM waves, improving the absorption capacity.

Mechanical test

The in-plane and out-of-plane compression performances of the MRAPS were tested using an Instron 5982 universal material testing machine (Instron Corporation, USA) based on the standards ASTM C365-05 and ASTM C364-16. A high-definition camera (Canon EOS R5, Canon Corporation, Japan) was used to capture the experimental process. LED lights were used to supplement light sources. The drop rate of the indenter or platen was set to 5 mm/min. Figure 7 shows the test results. Figure 7(a) shows the out-of-plane compression performance of samples 3# and 4#, which have similar maximum loads and compressive moduli (Figure 7(b)). The compression process can be divided into three stages. The load increases linearly during the elastic deformation stage. When the load reaches its maximum, it slightly decreases and subsequently enters the platform stage. The change in the load during this stage is small; moreover, the platform stage of sample 3# is longer than that of sample 4#. Finally, as compression enters the densification stage, the load rapidly increases. Notably, whether for sample 3# or 4#, densification always occurs first from a single layer, followed by overall densification when the platen drops. The maximum loads of samples 3# and 4# were slightly lower than predicted. The in-plane compression test results for samples 5# and 6# are shown in Figure 7(c) and (d), respectively. After the linear stage, samples 5# and 6# simultaneously undergo SW failures. However, sample 5# subsequently undergoes EB failure. As the degree of buckling increases, the core gradually undergoes shear failure, resulting in a serrated decrease in load. The SW part of sample 6# undergoes debonding failure, leading to an instant decrease in load. Because of the eccentric force, sample 6# gradually exhibits a buckling state. As the indenter decreases, the debonding area of the right skin gradually expands. The load changes smoothly and gradually decreases. The maximum load of sample 5# is higher than the predicted force, while that of sample 6# is slightly lower, which demonstrates the accuracy of the prediction.

Microwave absorption mechanism

The EMW absorption of the MRAPS is primarily dependent on the RAPMI foam in the core. On the one hand, the multilayer foam causes EMW scattering between different layers. On the other hand, the single-layer foam itself achieves considerable EM energy loss inside. The microwave absorption mechanism of the RAPMI foam is shown in Figure 8. When EMWs are incident on the absorber surface, the good impedance matching performance⁸ of RAPMI foam enables most EMWs to enter its interior. Because RAPMI foam mainly includes dielectric loss absorbers with carbon fibers and SiC fibers as the absorbents, this study primarily analyzes the dielectric loss mechanism. The SEM image shows that absorbents are mainly distributed on the cell walls. Numerous pores of varied sizes that are prone to multiple reflections and scattering are observed. Moreover, the interface polarization and dipole polarization effects caused by the mixed absorbents⁸ at the microscale inside the foam considerably enhance the dielectric loss ability.

Comparison

Figure 9 shows a comparison column graph of the tested specific strength, absorption bandwidth, predicted values, and maximum. Figure 9(a) shows the comparison of specific strengths under in-plane compression conditions. The specific strength of sample 5# is 2.1% higher than the predicted value, while that of sample 6# is lower than the predicted value and is 87% of the maximum, which is slightly lower than the design requirement (90%). The out-of-plane compressive specific strength of samples 3# and 4# is slightly lower than predicted but can meet the 90% maximum above (Figure 9(b)). The absorption bandwidths of samples 1# and 2# are higher than the predicted values and can achieve 95% and 98% of full-band at 2–18 GHz for 90% above absorption, respectively (Figure 9(c)). Because of the inevitable errors in processing and experimentation, these comparative results effectively indicate the successful design. The molded MRAPS likewise demonstrate excellent compression performance and radar absorption capability.

Conclusion

A composite radar absorption PMI foam sandwich was manufactured. Its electromagnetic and mechanical models were established and analyzed. The out-of-plane compression specific strength of the MRAPS increases with increasing ρ _c /ρ _s. When h _f /h is small, the rate of increase with ρ _c /ρ _s is slow. As h _f /h increases, the growth rate gradually increases. The in-plane compression 3D failure maps and specific strength cubic cloud maps of the MRAPS were drawn. The failure corresponds to the maximum specific strength within the range is SW or SF. The theoretical maximum in-plane compression specific strength is calculated to be 19.7 × 10⁵ m²/s². The sandwich was designed and optimized using an MOPSO algorithm. Both out-of-plane and in-plane compressive specific strength and absorption bandwidth of the MRAPS decrease as h _f /h increases. Therefore, the skin thickness must be as small as possible. SW becomes the main compressive failure mode. Meanwhile, skin thickness considerably influenced the absorption performance. The results were verified through experiments and demonstrated excellent mechanical properties and radar absorption ability. For a weight ratio of 1:1 and a core density of 75 kg/m³, the optimized MRAPS experimentally demonstrated effective bandwidth absorption above 15.1 GHz. Moreover, the maximum out-of-plane and in-plane compressive specific strengths reached 1.25 × 10⁴ and 1.44 × 10⁶ m²/s², respectively.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: We are grateful for the support provided by National Natural Science Foundation of China under Grant No. 52205137.

CRediT authorship contribution statement

Writing – original draft, Methodology, Data curation, Conceptualization, Software; Writing – review & editing, Validation, Supervision, Project administration, Methodology, Investigation; Investigation, Methodology; Resources; Investigation, Resources, Visualization, Funding acquisition.

ORCID iD

Shijun Song

Data availability statement

Data will be made available on request.*

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Specimen	Layers	L₀ (mm)	w (mm)	h (mm)	h_f (mm)
1#	2	180	180	43	0.5
2#	3	180	180	43	0.5
3#	2	50	50	43	0.5
4#	3	50	50	43	0.5
5#	2	300	80	43	0.5
6#	3	300	80	43	0.5

Specimen	Layers	L₀ (mm)	w (mm)	h (mm)	h_f (mm)
1#	2	180	180	43	0.5
2#	3	180	180	43	0.5
3#	2	50	50	43	0.5
4#	3	50	50	43	0.5
5#	2	300	80	43	0.5
6#	3	300	80	43	0.5

Specimen	Layers	L₀ (mm)	w (mm)	h (mm)	h_f (mm)
1#	2	180	180	43	0.5
2#	3	180	180	43	0.5
3#	2	50	50	43	0.5
4#	3	50	50	43	0.5
5#	2	300	80	43	0.5
6#	3	300	80	43	0.5