Introduction

Aircraft electromagnetic environment is becoming more complex and deterioration owing to the generation of electromagnetic radiation sources has developed from natural lightning flash to artificial electromagnetic radiation. What's worse is artificial electromagnetic radiation tends to ultrahigh frequency, high power and strong electric field with the development of electromagnetic wave technology such as radar system, high altitude electromagnetic pulse (HEMP) and high power microwave (HPM) weapons [1]. Electromagnetic interference (EMI) will harm human health, affect communication quality and cause irreparable damage to some sensitive electronic components in aircraft. Physical damage can be aggravated in high frequency and high power microwave conditions due to microwave heating effect [2]. Electromagnetic radiation protection is an indispensable part in aircraft design.

An effective electromagnetic protection method is making protective objects enclosed by electromagnetic shielding or absorbing material to ensure electromagnetic information safety and mitigating human health hazards. Main effective mechanisms in absorbing material include reflection, absorption and multiple reflections to shield against electromagnetic radiation [3]. Metallic material with high conductivity exhibits strong electromagnetic reflection and provides adequate shielding. But this does not mean that high electrical conductivity is a decisive criterion for electromagnetic shielding. In reality, the volume resistivity in the order of 1 Ω·cm is sufficient [4]. The electromagnetic reflection mechanism of material depends critically on mobile charge carriers that interact with the electromagnetic field. Therefore, electromagnetic shielding capability of material can be improved by enhancing the connectivity of the electrical conduction path [5].

Polymer composites have been largely applied in EMI protection due to their lightweight, corrosion resistance and wide bandwidth properties compared to traditional metal [6]. Conductivity fillers such as metal powder [3] or carbon-based fillers [7,8,9,10,11] are mixed into polymer matrix to obtain better EMI shielding performance by enhancing the connectivity of electrical conductive networks in the matrix. Besides, the filler content has to balance the electromagnetic shielding effectiveness (SE) and mechanical properties [12]. It can be inferred that the effective microwave properties of composite are affected by the size and microstructural of particle filler. The relationship between the electromagnetic properties of composites and particle size, resistivity and saturation magnetization is described by Bruggeman effective medium theory [13]. Wu L Z et al. [14] study the influence of particle size on microwave properties and conclude that the effective permeability of composite is strongly dependent on magnetic particle size. It recommends that the filler particle size should be in the micrometer or nanometer scale due to the mechanical or electromagnetic properties of composite is significantly enhanced [15, 16].

The protective effect of EMI shielding material mainly depends on the electromagnetic wave reflection ability. However, the absorbing material can avoid secondary electromagnetic radiation due to due to electromagnetic wave attenuation in material. Absorbing mechanisms are classified as dielectric loss originated from polarization [17] and magnetic loss caused by relaxation effect. There are typically hysteresis, resonance and eddy current in absorbing material during the magnetization process [18]. Therefore, both magnetic metal and dielectric fillers mixed into polymers can result in synergistic loss effects to promote microwave absorbing ability [19]. To date, magnetic materials such as Fe3O4, Ni–Zn ferrite and carbonyl iron powders (CIP) have relatively high permeability and appropriate permittivity, which are candidates for fabricating broadband microwave absorbing composite in the frequency of 2–18 GHz [20]. What calls for attention is that carbonyl iron powders filled into silicon rubber (SR) have been widely applied in microwave absorbing owing to their low price, low coercivity and high saturation magnetization [21,22,23]. Besides, silicone rubber (SR) has excellent thermal stability, weathering resistance and better adhesion with particles. CIP/SR absorbing composite is widely applied in the aviation field as electromagnetic functional material.

CIP/SR composites can be classified into spherical, flake and core–shell according to the type of filler particles. To data, there are many works have been done to investigate the electromagnetic performance of different types of CIP/SR composite. It reveals that flake CIP milled from spherical particles can reduce eddy current loss at high frequency as well as improve permeability [24]. Min D et al. [25] prepare the flake carbonyl iron/epoxy resin composite in a magnetic field and the results indicate that higher permeability and modest permittivity are obtained, which is a promising approach for fabrication of magnetic absorbing materials with wider working frequencies range. CIP/SR absorbing material performs well from low to high frequency and absorption peak shift to lower frequency band with the increase in CIP content, but absorbing performance inclines to stability when the weight fraction is close to 70% [26]. In addition, core–shell type of CIP/SR composite with magnetic cores and conductivity shells can also enhance dielectric loss and obtain proper impedance matching to some extent [27, 28]. Some works have been done to modify spherical carbonyl iron powder coated with silver [23], silica [29], graphene oxide [30] and epoxy silicon resin [31] to improve absorbing efficiency and broaden absorbing bandwidth. However, there is electromagnetic thermal effect in CIP/SR composite no matter what type of carbonyl iron powder due to their high magnetic and dielectric loss.

Recently, the interaction between electromagnetic composite and microwave begins to be concerned as well as the electromagnetic thermal effect. Microwave absorption is a process of electromagnetic energy converted into heat energy. Some works have been done to study the thermal effect of dielectric composites exposed to microwave radiation. Xu X Y et al. [32] study the dielectric heating effect of aramid/epoxy composite exposed to microwave radiation in a numerical method and investigate thermal damage characteristics through the experimental approach. The results indicate that thermal damage shape is related to fiber orientation and quasi-isotropic laminate shows better resistance to microwave radiation compared to unidirectional composites for the same thickness. Li Y et al. [33] study the heat transfer of carbon fiber composite during microwave curing based on a numerical model, which considers the microwave attenuation and exothermal reaction in the anisotropic composite. The result evidence the rapidity and uniformity of microwave heating. Li J L et al. [34] investigate the crack propagation in pegmatite caused by microwave thermal effect through a thermo-mechanical fully coupled model, which reveals the thermal stress damage mechanism in composite exposed to microwave radiation from the microscopic perspective. However, there are few studies on the microwave thermal effect of CIP/SR composite or absorbing materials, let alone the thermal damage of absorbing composite under UHF or high power microwave radiation.

Overall, many work have been done to make an effort to improve the electromagnetic property of absorbing composites by all means. But the interaction mechanism and physical effects of absorbing material exposed to high-frequency microwave radiation are deserved to be investigated. In this paper, the electric field response, heat transfer and thermal stress damage in CIP/SR composite exposed to microwave will be studied through establishing a numerical model coupled with heat transfer, electromagnetic wave and mechanics theory. The relationship between the electric field and temperature distribution of absorbing composite will be revealed. Thermal stress and expansive deformation of composite will be analyzed quantitatively. The effect of microwave frequency on electromagnetic shielding, heat transfer and deformation damage of CIP/SR composite materials will be studied.

Property characterization of CIP/SR composite

CIP/SR composite material and absorbing mechanism

The carbonyl iron powder is mixed well after ultrasonic dispersion by adding an appropriate amount of hydroxyl silicone oil, coupling agent KH-570 and dilution solvent. Carbonyl iron powder is blended proportionally in silicone rubber on the open roll mill for 30 min at ambient temperature. The mixture is vulcanized into molding on the plate vulcanization machine at 175 °C and 15 MPa. Post-vulcanization is carried out in a vacuum drying chamber at 100 °C for 4 h. CIP/SR composite sample can be prepared by slowly curing at ambient temperature for more than 24 h in a drying oven (Fig. 1).

Figure 1
figure 1

Schematic of microwave absorbing mechanism in CIP/SR composite

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The morphology of carbonyl iron particles is characterized by scanning electron microscopy (SEM) as shown in Fig. 2. Carbonyl iron powder has a regular spherical morphology which is uniformly dispersed in silicone rubber. Carbonyl iron powders have smooth surface due to silicon rubber covered. Most of the CIP particles are in the range of 2–4 μm. The mass fraction of carbonyl iron powders is 78% and the density of CIP/SR composite is 3.407 g/cm3. CIP/SR sample with 2 mm is prepared by curing process to further measure the electromagnetic parameter. Detailed information about this absorbing material is given in Table 1.

Figure 2
figure 2

SEM micrographs of carbonyl iron powders

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Table 1 Characteristics of CIP/SR composite
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The performance parameters of CIP / SR composites are obtained in preparation for the subsequent numerical calculation. CIP/SR composite can be regarded as isotropic material. Tensile test is performed on the universal tensile testing machine (WDW-20E) to obtain the elastic modulus. The loading rate is 1 mm / min and the sampling frequency is 2 Hz. The elastic modulus of CIP/SR is 1.06 MPa, which indicates this composite is easy to deform and belongs to soft material. Considering the high resistance of CIP / SR composites, volume resistance is obtained by high resistance weak current tester (ST2255) according to the test standard of GB/T 1410–2006. The radius of the composite sample is 50 mm and the thickness is 2 mm. Electrical conductivity of CIP/SR is 9.03 × 10–11 S/m. It indicates that CIP/SR composite can be treated as an insulating material. There is not forming a continuous conductive network in silicon rubber with 78 wt% carbonyl iron powder, which could not effectively improve the electrical conductivity of insulating silicone rubber. Thermal conductivity of this composite is measured by the HotDisk thermal constant analyzer system at room temperature. The sample is made into a disc with 2 mm thickness and a radius of 20 mm. The sensor is placed between two samples and temperature rises after current applied. The heating power is set at 0.4 W for 5 s. Temperature over time is recorded for calculating thermal conductivity of CIP/SR, which value is 0.987 W/(m K). Thermal conductivity of CIP/SR is higher than that of silicon rubber with 0.27 W/(m K). Besides, thermal expansion coefficient is measured on the thermal dilatometer (DIL402C). Test temperature ranges from 20 to 200 °C and the heating rate is about 5 °C/min. Thermal expansion coefficient is calculated by the ratio of deformation to temperature increment. Thermal expansion coefficient of CIP/SR is about 130 × 10–6 K−1. The properties of CIP/SR composite are presented in Table 2.

Table 2 Material properties of CIP/SR composite
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In order to understand the interaction between particle reinforced composites and electromagnetic waves, the absorbing mechanism of CIP/SR composite is described by a schematic illustrated in Fig. 1. As the electromagnetic wave propagates to CIP/SR composite, a part of incident wave is reflected on the surface of absorbing material. On the other hand, the remaining electromagnetic waves enter into composite and then attenuated. Considering silicone rubber belongs to nonmagnetic material and dielectric loss is lower [35], electromagnetic wave absorption mainly depends on multiple reflections between particles and energy dissipation in carbonyl iron powder.

Electromagnetic performances

The equivalent complex permittivity εr and permeability μr reflect the polarization and hysteresis characteristics of the absorbing material to some extent, which are described in complex form and expressed as Eq. (1) and (2). The real parts (ε´, μ´) and imaginary parts (ε", μ") represent energy storage and electromagnetic loss, respectively.

$$\varepsilon_{r} = \varepsilon^{^{\prime}} - j\varepsilon^{^{\prime\prime}} = \varepsilon^{^{\prime}} \left( {1 - j\;\tan \delta_{\varepsilon } } \right)$$
(1)
$$\mu_{r} = \mu^{^{\prime}} - j\mu^{^{\prime\prime}} = \mu^{^{\prime}} \left( {1 - j\;\tan \delta_{\mu } } \right)$$
(2)

Here, tan δε denotes dielectric loss tangent and tan δμ represent magnetic loss tangent.

The complex permittivity ε and permeability μ of absorbing material is related to the frequency and measured by vector network analyzer (HP8722) based on coaxial transmission and reflection method. The vector network analysis and measurement system are schematically illustrated in Fig. 3a. Powder filled composite is pressed into coaxial sample rings with inner diameter of 3 mm and outer diameter of 7 mm installed in a coaxial air-filled line as shown in Fig. 3b. According to the coaxial wire analysis theory, the relationship between electromagnetic parameter (ε, μ) and scattering coefficients (S11, S22, S12, S22) has been described by standard Nicolson Ross Weir (NRW) method in detail [36]. Once the scattering coefficients are measured, permittivity and permeability of CIP/SR composite can be calculated.

Figure 3
figure 3

Sketch of electromagnetic parameters measurement

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In the frequency range 2–18 GHz, electromagnetic parameters ε´, ε", μ´and μ" of CIP/SR composite with 2 mm thickness are shown in Fig. 4a and b. Real parts permittivity ε´ increases slowly as the frequency increased, while the growth rate of imaginary part permittivity ε" accelerates after 14 GHz. On the whole, both real and imaginary parts of permittivity are changed in a small range with the increase in frequency. Although the weight fraction is different, the permittivity and permeability result of CIP/SR composite has a similar trend with that of reference [24]. Besides, the fluctuation range of complex permittivity and permeability are relatively close. The dielectric loss tangent ε"/ε´of CIP/SR composite is increasing in the frequency range of 2–18 GHz according to Fig. 4c. It indicates that dielectric loss of CIP/SR composite in high frequency band is higher than that in low frequency band. In addition, real part permeability μ´ is larger than the imaginary part permeability μ". But both of them are shown a declining trend as the frequency increased. The magnetic loss tangent μ"/μ´of CIP/SR composite increases in the frequency range of 2–13 GHz and then decreases as shown in Fig. 4c. The results indicate that magnetic loss of CIP/SR composite is higher than that of the dielectric loss because this composite belongs to magnetic material.

Figure 4
figure 4

Electromagnetic properties of CIP/SR composite, a complex permittivity, b complex permeability, c dielectric and magnetic loss tangent

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On the other hand, impedance matching affects the electromagnetic wave entering into absorbing composite. Reflection loss (RL) is an important quantity to characterize the reflectivity of absorbing material. The reflection loss of CIP/SR can be calculated by complex permittivity and permeability according to the transmission line theory, which is expressed by Eq. (3) [37].

$$RL\left( {dB} \right) = 20\log_{10} \left| {\frac{{Z_{in} + Z_{0} }}{{Z_{in} - Z_{0} }}} \right|$$
(3)
$$Z_{in} = Z_{0} \sqrt {\frac{\mu }{\varepsilon }} \tanh \left( {jd\left( {\frac{2\pi f}{{c_{0} }}} \right)\sqrt {\mu \varepsilon } } \right)$$
(4)
$$Z_{0} = \sqrt {\frac{{\mu_{0} }}{{\varepsilon_{0} }}}$$
(5)

Here, Zin and Z0 are the input impedances of absorbing composite and characteristic impedances of free space, respectively. ε0 and μ0 are the permittivity and permeability of free space. Besides, the f, d and c0 denote the frequency, the thickness of absorbing composite and the velocity of electromagnetic in free space, respectively.

Consequently, reflection loss of CIP/SR composite is calculated by Eq. (3), (4) and (5) using MATLAB software. Reflection spectrum is shown in Fig. 5 within the frequency bands of 2 ~ 18 GHz. Reflectivity value is negative and the smaller value represents stronger absorption performance of CIP/SR composite. The absorbing peak of CIP/SR composite appears at 8.5 GHz and the peak value is −31.8 dB. Besides, the results show that effective absorption bandwidth below −10 dB is about 4.4 GHz, which ranges from 6.4 GHz to 10.8 GHz. This CIP/SR composite exhibits better absorption performance in the frequency band below 10 GHz.

Figure 5
figure 5

Reflection spectrum of CIP/SR composite

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A numerical method for multi-physics coupling

Metallic materials can directly reflect electromagnetic waves, and electromagnetic energy storage or conversion inside of these materials is operating in low efficiency. In contrast, dielectric and magnetic loss will induce temperature rising and results in thermal expansion in lossy media or absorbing material under electromagnetic wave radiation, especially high power microwave. Therefore, it is a complex process involving multi-physics coupling when ICP/RS composite is exposed to high frequency microwave. To further investigate the mechanism and interior laws, a numerical model is established based on the theory of electromagnetics, heat transfer and mechanics. In this section, the coupling relationship between physical fields will be discussed. The simplification and establishment of this coupled numerical model will be described in detail. Simulation of microwave radiation sources in different frequency bands will be solved.

Governing equations

Electromagnetic wave equation

Electromagnetic wave propagation in space or medium should be treated as an attenuation process. In the frequency domain, instantaneous Maxwell's equations can be transformed into time-harmonic forms by inverse Fourier transform, which is expressed as Eq. (6) and (7) [38].

$$\nabla \times {\varvec{H}} = \sigma {\varvec{E}} + j\omega \varepsilon_{0} \varepsilon_{r} {\varvec{H}}$$
(6)
$$\nabla \times {\varvec{E}} = - j\omega \mu_{0} \mu_{r} {\varvec{H}}$$
(7)

Therefore, the governing equation of electromagnetic wave is rewritten as Eq. (8) to solve electric field distribution in the whole calculation domain when refers to electromagnetic wave propagation in the medium [39].

$$\nabla \times \mu_{r}^{ - 1} \left( {\nabla \times {\varvec{E}}} \right) - k_{0}^{2} \left( {\varepsilon_{r} - \frac{j\sigma }{{\omega \varepsilon_{0} }}} \right){\varvec{E}} = 0$$
(8)
$$k_{0} = \omega \sqrt {\varepsilon_{0} \mu_{0} } = \frac{\omega }{{c_{0} }}$$
(9)

Here, E, H, ω, k0 and σ represent electric field intensity, magnetic field intensity angular frequency, electric conductivity and vacuum wavenumber, respectively.

Heat transfer equation

Part of microwave electromagnetic energy is converted into heat energy due to electromagnetic heating in absorbing material. Absorbed energy Qe in loss material can be calculated by Eq. (10) according to the Poynting theorem [40]. In addition, thermal radiation from the surface to ambient will result in energy dissipation, which can be calculated by Stefan-Boltzmann's law [41] and defined as Eq. (11). The governing equation of heat transfer in solid is described by the energy equation and expressed as a partial differential Eq. (12), in which electromagnetic heating Qe and thermal radiation Qr are added as heat energy source items [42].

$$Q_{e} = Q_{rh} + Q_{ml} = \frac{1}{2}{\text{Re}} \left( {{\varvec{J}} \cdot {\varvec{E}}^{*} + i\omega {\varvec{B}} \cdot {\varvec{H}}^{*} } \right)$$
(10)
$$Q_{r} = \varepsilon \sigma_{0} \left( {T_{amb}^{4} - T^{4} } \right)$$
(11)
$$\rho C_{p} \frac{\partial T}{{\partial t}} = \nabla \cdot \left( {k\nabla T} \right) + Q_{e} + Q_{r}$$
(12)

Here, ρ, Cp k, ε, Qrh and Qml denote the density, specific heat capacity, thermal conductivity, surface emissivity, resistive loss and magnetic loss of absorbing material, respectively. The value of Stefan-Boltzmann constant σ0 is 5.67 × 10–8 W·m−2·K−4. Tamb represents environmental temperature.

Thermal expansion equation

Considering thermal-stress coupling, there is inevitably thermal deformation in absorbing material under the constraint condition. CIP/SR composite is treated as isotropic linear elastic and thermal strain εth can be calculated by Eq. (13). Thermal strain relates to thermal expansion coefficient α and temperature increment.

$$\varepsilon_{th} = \alpha \cdot \left( {T - T_{0} } \right)$$
(13)

In this case, the relationship between stress and strain is rewritten as a thermoelastic constitutive Eq. (14) according to Duhamel-Neumann law [43].

$$\sigma_{s} = D \cdot \left( {\varepsilon_{el} - \varepsilon_{th} } \right)$$
(14)

Here, σs represents stress, εel denotes the stress resulted from elastic and D is elasticity matrix defined as Eq. (15).

$$D = \frac{E}{{\left( {1 + \upsilon } \right)\left( {1 - 2\upsilon } \right)}}\left[ {\begin{array}{*{20}c} {1 - \upsilon } & \upsilon & \upsilon & 0 & 0 & 0 \\ \upsilon & {1 - \upsilon } & \upsilon & 0 & 0 & 0 \\ \upsilon & \upsilon & {1 - \upsilon } & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1 - 2\upsilon }{2}} & 0 & 0 \\ 0 & 0 & 0 & 0 & {\frac{1 - 2\upsilon }{2}} & 0 \\ 0 & 0 & 0 & 0 & 0 & {\frac{1 - 2\upsilon }{2}} \\ \end{array} } \right]$$
(15)

Here, E and υ denote Young's modulus and Poisson's ratio of CIP/SR composite.

The coupled numerical model

Studying the characteristics of absorbing materials under microwave radiation needs to be carried out in a shielding chamber. Microwave source is generated from power modulator and achieves directional radiation through an antenna. The microwave radiation process is illustrated in Fig. 6a, and the simplified numerical model is shown in Fig. 6b. This numerical model is comprised of air domain, perfectly matched layer (PML), pyramidal horn antenna and CIP/SR composite. The distance Ld between pyramidal horn antenna and CIP/SR composite plate is 150 mm and the geometrical size of absorbing material plate is 200 mm × 200 mm × 2 mm.

Figure 6
figure 6

a Schematic of absorbing material irradiated by microwave, b numerical model for calculation

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Calculation domain in this numerical model needs to be defined as a spherical region that is large enough to contain a horn antenna and composite plate. Besides, the function of the truncated domain is simulating the far domain and absorbing electromagnetic wave without reflection. In numerical method, PML absorbing boundary is more accurate than scattering boundary conditions when dealing with electromagnetic radiation in free space or anechoic chamber. PLM layer is established on the outermost layer of spherical air domain with a thin thickness. The thickness of PML domain is set to wavelength λ. High power microwave source is generated by defining a port that ensures electromagnetic energy entering into horn antenna. Microwave feeding pattern is defined as TE10 and the port type is rectangular. In addition, antenna walls are regarded as perfect electric conductors.

To reflect dielectric and magnetic loss mechanism in CIP/SR composite, the value of complex permittivity and permeability are originated from experiment data according to Fig. 4. Initial temperature T0 in the whole domain is set to 293.15 K and thermal radiation coefficient on composite plate surfaces is about 0.95. Considering this composite plate is usually fixed on the windows in shielding enclosure. A fixed constraint boundary is defined on all side faces of CIP/SR composite to evaluate thermal stress and deformation caused by thermal expansion. Finite element mesh generation is controlled by all of these physical fields. The mesh size depends on frequency in the electromagnetic domain. Mesh in PML is generated by sweeping scheme and the other region is disassembled by hexahedron element. Besides, the maximum mesh size should be no larger than λ/5.

Electromagnetic wave equation is calculated in the whole domain due to electromagnetic wave is excited to form antenna and then propagates from air to CIP/SR composite. Electromagnetic loss and thermal effect mainly occur in composite materials. Electromagnetic, thermal and mechanical coupling equations are solved in CIP/SR composite domain. Numerical calculation is performed on COMSOL Multiphysics software by coupling with electromagnetic wave equation, heat transfer equation and thermal expansion equation. Electromagnetic heating source can be calculated by electromagnetic wave equation and added into heat transfer equation based on secondary development technology in COMSOL software. Subsequently, temperature distribution of CIP/SR composite can be obtained from heat transfer calculation. This temperature profile is applied to composite as a thermal load. Thermal stress and deformation response analysis can be achieved by thermo-mechanical coupling calculation. This multi-physical coupled model is solved by a fully coupled method based on frequency domain and transient study. The detailed coupling analysis algorithm is shown in Fig. 7.

Figure 7
figure 7

Flow chart of coupled numerical method

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Microwave radiation method

The purpose of antenna design is to generate microwave radiation source in a certain frequency range. Pyramidal horn antenna is comprised of a feeding rectangular waveguide and a horn aperture, which is shown in Fig. 8. The geometric parameters AH, BE, LH and LE of horn aperture can be calculated based on antenna design theory [44], when the inner dimensions a, b and l in feeding rectangular waveguide are given. Optimum horn antenna is designed by adjusting aperture parameters to obtain maximum gain for given slant lengths in the E and H planes.

Figure 8
figure 8

Pyramidal horn antenna design

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Aperture parameters AH and BE are calculated by Eq. (16) to ensure the best performance of pyramidal horn in E and H plane.

$$A_{H} = \sqrt {3\lambda R_{1} } \;,\;\;B_{E} = \sqrt {2\lambda R_{2} }$$
(16)

Here, R1 and R2 denote the slant length of the aperture in E and H field direction, respectively.

The gain G of a pyramidal horn antenna can be derived from Eq. (17).

$$G = \frac{4\pi }{{\lambda^{2} }}\varepsilon_{ap} A_{H} \cdot B_{E}$$
(17)

Here, aperture efficiency εap is a dimensionless factor that depends on horn length and its value is 0.511 in the optimum pyramidal horn. The λ represents wavelength.

In addition, the width of aperture AH in E field direction can be calculated by a fourth-order Eq. (18), the approximate solution of which is expressed as Eq. (19). In this sense, the width of aperture BE in H field direction is also determined for a given gain according to Eq. (17). The relationship between aperture horn length in the E and H field direction needs to satisfy Eq. (20) to ensure this pyramidal horn can be realized [45].

$$A_{H}^{4} - aA_{H}^{3} + \frac{{3bG\lambda^{2} }}{{8\pi \varepsilon_{ap} }}A_{H} = \frac{{3G^{2} \lambda^{4} }}{{32\pi^{2} \varepsilon_{ap}^{2} }}$$
(18)
$$A_{H} = 0.45\lambda \sqrt G$$
(19)
$$L_{E} = L_{H} = \frac{{A_{H} - a}}{3\lambda }A_{H}$$
(20)

Thermal effect evaluation of CIP/SR under microwave radiation

Response analysis of microwave radiation source

Response analysis of microwave radiation source aims to study the characteristics of electromagnetic microwave radiated from the antenna to free space. The electromagnetic field distribution results of free radiation are beneficial to investigate the influence of absorbing materials on electromagnetic radiation. Based on the above antenna design theory, pyramidal horn antennas in corresponding frequency bands are designed to generate microwave with a frequency of 2–18 GHz. Geometry size of horn antenna can be calculated in different frequency bands such as G, C, X, and Ku when the standard rectangular waveguide is selected. The detail size of pyramidal horn antenna is calculated to achieve a gain of 17 dB, which are given in Table 3. Geometry size of pyramidal horn antenna is replaced by self-definited variable. Pyramidal horn antenna model is established by parametric modeling method and performs in COMSOL Multiphysics with MATLAB.

Table 3 Size of pyramidal horn antenna in different frequency bands
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The direction of electromagnetic wave propagation in space depends on the antenna directivity. It needs to generate directional radiation of electromagnetic wave to absorbing material. Generally, smaller main lobe width means more concentrated radiation as well as stronger directionality. In the case of X band, antenna radiation patterns are shown in Fig. 9 and the horn antenna gain is calculated through numerical method. The main lobe width is about 25°in E-plane and the main lobe width is about 20°in H-plane. In addition, the maximum gain appears when the angle φ is 180°, which denotes this antenna horn in X band has the most concentrated radiation and directivity in the + y direction.

Figure 9
figure 9

Gain of pyramidal horn antenna

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On the other hand, research of electromagnetic field from antenna to air domain is devoted to better understanding the radiation and propagation characteristics of microwave. Electric and magnetic field distribution is shown in Fig. 10 when feeding pattern is defined as TE10 rectangular waveguide mode with center frequency 9.375 GHz. The results show that microwave radiated from antenna horn in the form of spherical waves. Electromagnetic wave intensity decreases with the increase in propagation distance. Both in E-plane and H-plane, the peaks and troughs of electric or magnetic field distribution appear at the same position along y direction, which indicates electric field and magnetic field are perpendicular to each other. The maximum electric field is 9.77 × 104 V/m and the maximum magnetic field is 175 A/m in horn interior under the condition of input power is 1 kW.

Figure 10
figure 10

a Electrical field distribution in E-plane and b magnetic field distribution in H-plane

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Quantitative evaluation of thermal and deformation

In this numerical model, CIP/SR composite is defined as a dielectric and magnetic loss material. Permittivity and permeability of CIP/SR composite are 9.94–0.35j and 1.5–0.875j, respectively, when frequency is 9.37 GHz. Electric field distribution is shown in Fig. 11, which demonstrates the high-frequency radiation process and electromagnetic response characteristics of CIP/SR composite. Electric field profile exhibits wave morphology between horn and composite. Compared with the radiation from antenna to free space, the electric field intensity is reduced. It indicates that part of electromagnetic wave is reflected on the CIP/SR composite surface as shown in Fig. 11a and then generating interference between reflecting and incoming wave. There is still electromagnetic wave reflection for CIP/SR absorbing composite. The incident wave can be attenuated by reflected electromagnetic wave.

Figure 11
figure 11

a Radiation characteristics in the whole calculation domain and b electrical field distribution on CIP/SR composite

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The reason for reflection on the surface is that CIP/SR composite has poor impedance matching at 9.37 GHz. Electric field is attenuated in the region behind CIP/SR composite plate owing to loss mechanism. On the other hand, electric field profile of CIP/SR composite exhibits wave morphology and central symmetrical distribution as shown in Fig. 11b. Electric field intensity decreases gradually from the center to the edge of this composite plate. The maximum value of electric field is 4.24 kV/m and the minimum value is 362 V/m. Electric field is mainly concentrated in the central region and extends along the z and y axis, which relates to antenna gain characteristic in E-plane and H-plane as well as the CIP/SR composite belongs to isotropic material.

Part of electromagnetic wave absorbed by CIP/SR composite is converted into heat energy. Thermal effect of CIP/SR composite exposed to microwave can be quantitatively evaluated through electromagnetic-thermal coupling calculation. Temperature distribution of CIP/SR composite is shown in Fig. 12 when the microwave radiation last 2 s at 9.37 GHz. The temperature of composite rises by 36  °C within 2 s and high temperature region is located in the center. In comparison, temperature on the surface close to antenna is higher than that on the reverse side. Besides, temperature profile is similar to that of electric field. To some extent, strong electric field intensity contributes to dielectric polarization, hysteresis loss, and energy conversion efficiency in absorbing material. The result indicates that there is a linear relationship between electric field and temperature.

Figure 12
figure 12

Temperature distribution on the surfaces of CIP/SR at 2 s

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Thermal effect will inevitably lead to expansive deformation in CIP/SR composite, which is a soft material. Thermal stress and deformation generate mechanical damage to some degree. CIP/SR composite has high flexibility and low elastic modulus, which makes thermal expansion more visible. Thermal stress distribution and deformation degree are evaluated by thermal–mechanical coupling calculation. The stress, strain and displacement distribution of CIP/SR composite is shown in Fig. 13 when the input power of microwave is 1 kW.

Figure 13
figure 13

Thermal stress and expansion of CIP/SR at 2 s

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Thermal stress is distributed symmetrically and concentrates in high temperature region. The maximum thermal stress is about 3.9 kPa that occurs in the center area of composite plate. Thermal stress on the surface close to antenna is higher which means intense thermal expansion. CIP/SR composite is in tension state along Y direction. Thermal strain distribution is related to temperature and thermal expansion coefficient. Thermal expansion causes bulge in the center of CIP/SR composite and the maximum displacement is about 0.66 mm. The displacement contour of CIP/SR can be characterized as Gauss distribution.

Overall, this numerical coupling model effectively predicates the distribution of electric, magnetic, temperature and thermal stress in absorbing composite exposing to directional microwave radiation. Numerical results reveal that part of the incident microwave is reflected on CIP/SR composite surface. The rest microwave is attenuated in CIP/SR and transferred into heating energy. Thermal effect of absorbing originates from electromagnetic heat accumulation with time. However, the primary reason is loss mechanism of carbonyl iron powder in silicon rubber. In fact, CIP/SR composite is normally designed as an absorbing patch and stuck on the surface of object with adhesive. It will generate local stress concentration and deformation in absorbing composite subjected to microwave directional radiation. Moreover, CIP/SR composite are soft material which is prone to thermal deformation. Thermal damage to CIP/SR composite should not be ignored.

Thermal response characteristics of CIP/SR composite in broadband

Intense electromagnetic pulse or complex electromagnetic environment typically covers a wide range of frequencies. In addition, permeability and permittivity of CIP/SR composite are frequency dependent. Therefore, electromagnetic loss properties and thermal effect of CIP/SR are different at different frequencies. In this section, some typical frequency points are selected within 2–18 GHz and the corresponding complex permeability and permittivity of CIP/SR are redefined. Besides, horn antenna size also needs to be redesigned to ensure the TE10 mode wave generated from rectangular waveguide with the same gain. Temperature distribution of CIP/SR composite in different frequency points is calculated through parametric sweep analysis in COMSOL software, and the results are shown in Fig. 14.

Figure 14
figure 14

Temperature distribution of CIP/SR at different frequency points

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Temperature contours reveal that thermal effect is obvious in the central region of composite due to directional radiation for the whole selected frequency points. However, temperature distribution shows discontinuous and exhibits wrinkle shape along the X-axis direction in the case of lower frequency, which is typical at 6.5 GHz., Temperature distribution trends more concentrated and smoother as frequency increases. The reason is that microwave has stronger penetration in low frequency than that in high frequency and microwave propagation in the medium is in the form of fluctuation. Electromagnetic waves passing through loss media are absorbed more fully and converted into heat energy. When the input power remains 1 kW, temperature of CIP/SR increases with frequency according to the curve of peak temperature. It indicates that the thermal effect of absorbing materials is more serious under high frequency microwave radiation than that of low frequency microwave.

Subsequently, corresponding thermal stress of CIP/SR with fixed boundary is calculated and the results at selected frequency points are shown in Fig. 15. Thermal stress distribution is linearly correlated with temperature profile. Similarly, thermal stress distribution of composite under low frequency microwave radiation shows more dispersed than that in high frequency. There is a point probe defined to record thermal stress at the center point on composite surface close to the radiation source. Thermal stress concentration exhibited in the central part and the maximum value is 8.12 kPa at 17.5 GHz. Considering the permittivity and permeability of CIP/SR varying within a narrow range, thermal stress increases with frequency at a near-linear trend according to the curve in Fig. 15.

Figure 15
figure 15

Thermal stress profile of CIP/SR at different frequency points

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Both of thermal stress and expansion analysis is devoted to evaluating the mechanical damage of absorbing materials caused by electromagnetic thermal effect. For quantitative comparison of deformation degree, a straight line through the center point and paralleling to X axis is defined on composite surface. Displacement field on this intercept line is presented in Fig. 16. All of these curves are shown as bell-shaped distribution with symmetry. Besides, displacement curves tend to relatively flat at low frequency but display sharply at high frequency. The maximum deformation of CIP/SR composite is 1.42 mm when the frequency point is 17.5 GHz. The results indicate that there is obvious thermal bulge deformation on the CIP/SR composite. Overall, the results indicate that local thermal stress and expansion tends to be more serious as frequency increases.

Figure 16
figure 16

Thermal deformation of CIP/SR at different frequency points

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Conclusions

It is studied the micro distribution characteristics of carbonyl iron powder particles in silicone rubber. The electromagnetic properties of CIP/SR composites are investigated in microwave band based on experiment method. The shielding and absorbing mechanism of particle reinforced electromagnetic protection material is discussed. Besides, a coupled numerical model has been presented for estimating thermal and mechanical effects in CIP/SR absorbing composite radiated by microwave. It mainly studied the generation mechanism of electromagnetic heating and quantitative characterized thermal stress and expansion deformation based on coupling method. The following main conclusion can be obtained.

  1. (1)

    The absorbing peak of CIP/SR composite in the reflection loss curve appears at 8.5 GHz and effective absorption bandwidth below -10 dB is about 4.4 GHz. It reveals that microwave absorption band of CIP/SR with 78 wt% particle filling rate is mainly concentrated in C and X band.

  2. (2)

    Although CIP/SR composite is an absorbing material, electromagnetic wave reflection still occurs on the surface. Incident microwave is attenuated by the reflected electromagnetic wave. Electromagnetic wave interference appears in the region between antenna and composite plate. Electromagnetic wave intensity is reduced.

  3. (3)

    The distribution of temperature and electric field in CIP/SR composite is similar. Microwave heating of absorbing composite is originated from electric polarization and hysteresis loss. The temperature of CIP/SR composite increases by 36 °C within 2 s when the frequency is 9.37 GHz and input power is 1 kW.

  4. (4)

    Temperature rise and thermal deformation show a linear correlation with frequency range from 2 to 18 GHz. Thermal stress concentration and thermal expansion in CIP/SR composite trends to be more serious as frequency increases. The maximum stress and deformation are 8 kPa and 1.6 mm, respectively, at 17.5 GHz.