# Numerical and Experimental Verification of Spherically Layered Ice Model for Analysis on Hail Impacting Aviation Structures

Authors: Gang Luo; Fengqi Zhang; Wanting Cao; Haiyang Zhang; Lulu Liu; Wei Chen
Journal: International Journal of Aerospace Engineering
 (2022)

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## Abstract

Aviation structures are required to be resistant to impact damage from hail. Therefore, aiming to gain a better understanding of such impacts, the formation mechanisms, spherically layered structures, and layer-density distributions of real hail were investigated. Based on analysis of the influence of different constitutive model parameters on the impact load on a metal target plate, a layered finite-element hail model was established, and this and several equivalent hail models were simulated and compared in terms of the dynamic responses during impact with a Ti6Al4V alloy plate target. The results showed that the characteristic values of the contact forces and impact displacements of spherically layered hail were lower than those for equivalent hailstones with the same mass or diameter. The spherically layered modeling of hail could provide a method for fine analysis of the phenomenology of hailstone impacts.


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## Body

## 1. Introduction

Aircraft and aero engines may encounter hail impacts during operation, and this can very easily cause impact damage to various aviation structures and endanger safety [1, 2]. Many countries stipulate minimum hail-related safety requirements for aircraft and aero engines in the form of airworthiness regulations, and these also specify basic analysis and test methods to achieve these safety requirements for the purposes of verification [3–5]. When the hail impacts the aircraft and engine structure, damage such as depression, tear, and penetration of the structure surface usually occurs [6–11]. The reason is that the relative kinetic energy of the hail is transmitted to the structure, and the concentrated effect of the impact energy causes the shear damage of the structure. The degree of the above damage depends on the following factors. First is the relative kinetic energy of the hail projectile, that is, the mass of the hail and the relative velocity at the time of impact; researchers generally regard it as an engineering problem and can grasp the corresponding laws through variable parameter analysis. Second is the material mechanical properties and mechanical strength of the structure. People have carried out a lot of research on metals and composites and achieved fruitful results; this part is not the focus of this paper, so it is not reviewed. Thirdly, the dynamic mechanical properties and impact contact mechanism of ice are the basic starting point of this paper. The research on the dynamic mechanical properties and impact contact mechanism of ice mainly focuses on the following aspects:
(1)
Test method for mechanical properties of ice under high strain rateResearchers have developed two classical test methods. One is to use SHPB (Split Hopkinson Pressure Bar) equipment to carry out dynamic compression and shear tests of ice specimens, so as to obtain the mechanical properties of ice under compression. The limitation of this method is that it is unable to systematically test the mechanical performance parameters of ice in tension and collapse states. The second is to use a rigid or flexible target impact test to test the dynamic load of the ice specimen impacting the target, so as to deduce and obtain the compression/tensile mechanical properties of ice by phenomenological analysis. The above two methods are the mainstream test methods for the dynamic mechanical properties of ice. In addition, a series of methods for preparing standard ice test pieces without cracks or bubbles are derived, such as freezing, casting, and die casting. Shazly et al. used SHPB equipment to carry out high strain-rate compression tests on single-crystal ice specimens in the strain-rate range 60–1400 s−1, and they found that the strength of ice increases with the strain rate, while the microstructure of the ice has little effect on its strength [12]. Kim and Keune obtained the structure in the higher strain-rate range (up to 2600 s−1) using SHPB tests of cylindrical ice specimens and further verified the strain-rate sensitivity of the ice strength [13]. Wu and Prakash used SHPB measurements to study the dynamic response of lake-water and distilled-water ice under dynamic uniaxial compression and combined compression and shear loads (oblique impact); they found that the strength of lake-water ice containing impurities was higher than that of distilled-water ice [14]. Regarding the form of and preparation methods for ice test pieces, most researchers have used molds made of metal, plastic, resin, or other materials to freeze spherical or cylindrical test specimens, obtaining good quality via temperature control and other processes. Juntikka and Olsson used plastic molds to freeze 34- and 48 mm diameter ice ball test specimens with no bubbles or cracks [15]. Fasanella et al. used polyurethane molds with plastic–elastic nesting to freeze cylindrical compression-test specimens. The deformation of the plastic–elastic nesting offsets the internal forces generated during freezing [16]. Combescure et al. prepared single-crystal ice specimens using a slow freezing speed and formed cylindrical ice specimens by turning [17]. Some researchers have used different methods to obtain ice test specimens. Kim and Kedward used a set of nylon molds with movable top covers and an inner diameter of 12.7 mm to eliminate the prestress in the freezing process and prepare impact compression-test specimens with good quality [18]. They also prepared flat ice balls with 10–11 layered structures by multiple casting in spherical molds [18]. Lavoie et al. used two hemispherical molds with pressure blocks to compress snow into ice balls to prepare equivalent ice ball test specimens with specific densities [19].
(2)
Mechanical behavior and constitutive model of ice under high strain rateResearchers have found that ice is a strain-rate-sensitive material, and its strength increases with the increase of the strain rate. The strain-rate-sensitive effect must be considered in the constitutive model of ice. At present, the phenomenological constitutive model that comprehensively considers the yield strength, tensile/compressive failure stress, and strain-rate-sensitive factors of stress can better describe the dynamic mechanical behavior of ice. The ice material model developed by Kim and Kedward using DYNA3D adopted failure strain and tensile failure criteria, and they only considered hydrostatic pressure without considering the influence of strain-rate sensitivity [18]. Park and Kim developed an ice material model using the ABAQUS software package [20]. This model requires manual setting of the yield strength, failure tensile stress, and other calculation control parameters, such as the volume viscosity. Pernas-Sánchez et al. established a constitutive model of ice under high strain rates based on the Drucker–Prager plasticity criterion [21]. Carney et al. developed a phenomenological constitutive model of ice, taking into account the strain-rate independence of the tensile/compressive failure stress of ice, the strain-rate sensitivity of the flow stress, and the ability of ice to continue to bear load after crushing; this model has been verified by many tests and has now been officially included as the Mat155 material model in LS-DYNA [22]. Tippmann developed a strain-rate-dependent material model to simulate the high-speed impact of spherical ice, and this is characterized by three-boundary compressive-strength–strain-rate curves based on measured rate-dependent strength data [23]. Song et al. established a mesoscopic numerical simulation method for ice containing bubbles using a meshless arbitrary Lagrangian–Eulerian multimaterial method; by simulating SHPB tests, they also studied the influence of the number and aggregation of bubbles in ice on its dynamic mechanical properties. They found that the dynamic strength of ice decreases nonlinearly with increasing bubble volume ratio [24].Considering the application background of hail impact on aviation structures, new problems worthy of research appear. At present, no matter the experimental research, mechanical property research, or simulation research of ice, the research object is almost frozen transparent ice, while hail in nature has a more complex layered structure. Each layer of hail has different porosities and densities due to its formation reasons and environmental differences. Obviously, its mechanical properties are different from those of artificially frozen uniform material ice balls, resulting in the following problems:(1)
Due to the stratification of natural hail with different densities, is the action process and failure behavior of high-density ice and low-density ice under high-speed impact consistent? At present, there are few studies on the similarities and differences of its mechanical properties(2)
In the current general constitutive model, there are few studies on which parameters have the key influence mechanism on the impact load of ice with different densities(3)
In the hail impact test and analysis of aviation structures, artificial frozen ice balls are often used as alternative projectiles for the impact test. Because the specific gravity and strength of artificial frozen ice are different from those of natural hail, the uniform material finite element model developed on this basis cannot fully reproduce the natural hail structure, physical properties, and mechanical properties under the impact state, and the analysis results may be distorted or too safe and the strength redundancy design of the structure maybe too largeBased on the above considerations, this paper analyzed the causes and symbolic layered structure parameters of natural hail and explored the key factors affecting the impact load in the current ice constitutive model by the impact mechanical properties of ice with different densities. Two kinds of ice ball test pieces with different densities were prepared by casting and die casting, and the impact load test was carried out. The impact failure process and load transfer mechanism of ice with different densities were pointed out, and the numerical simulation method of ice impact with different densities was verified. According to the historical data, a layered hail finite element model equivalent to natural hail is “assembled,” and the response law of hail impacting a metal target with different structures and parameters is explored. It lays a foundation for further trial production, test, and impact mechanics analysis of artificial stratified hail.


## 2. Origin and Structure Analysis of Real Hail

The formation mechanisms of hail have been studied for a long time. Nelson analyzed the growth mechanism of hail in two super-cell storms in the Great Plains of the United States using Multiple Doppler Radar data and a three-dimensional hail-growth model. He pointed out that hail growth was mainly concentrated in a layer at temperatures warmer than −25°C, and the formation and growth of hail embryos might involve multiple up-and-down circulation transportation processes, with the hail embryos coming from many different regions [25]. Later, Rasmussen and Heymsfield developed a detailed model of the melting, shedding, and wet growth of spherical hail; in the study, a parametric modeling method to quantitatively describe the alternating growth of dry and wet hail was proposed [26]. Based on the analysis of a large number of documents, Allen et al. systematically described the microphysical structure and physical characteristics of hail development. With regard to the processes leading to hail growth and the environmental parameters of hail occurrence, size, and intensity, they obtained an equivalent density relationship curve based on hail mass and diameter parameters [27] (Figure 1). Giammanco et al. conducted uniaxial compression tests on more than 900 pieces of natural hail. In comparisons with frozen ice balls with initial cracks, they found that the average compressive strength of natural hail was higher than that of frozen ice balls. However, the data dispersion was very high, and the so-called “hardness” of hail was found to greatly affect its compressive strength in tests [28]. So far, researchers only have a general understanding that the density of hard hail is greater than that of soft hail, and there has been no systematic study to explain the quantitative relationship between the density and strength of hail [29].Figure 1
Equivalent density curve based on hail mass and diameter parameters [27].To examine the specific structure of hail, Prodi applied specific-gravity measurement technology to X-ray micrographs. He proposed that the formation of hail was due to the alternating growth of “dry” and “wet” ice; water will penetrate into the porous structure formed by dry growth, ultimately forming layered structures with different specific gravities after freezing. X-ray micrographs of local sections of collected hail are shown in Figure2 [30].Figure 2
Stratification structure of hailstones with different specific gravities (New Raymer, CO, USA, 10 June 1969) [30].It can be seen that real hailstones can be regarded as spheres formed from approximately spherical layered shells composed of ice with different specific gravities and different crystallization modes. Due to the above structure and formation processes, natural hail tends to have a specific gravity in the range 0.8–0.9. In the analysis of hail impacts, the artificial hail used in testing can be made equivalent to a concentric spherical-shell model with different layers, different thicknesses, and different mechanical properties.


## 3. Parameter Analysis of Dynamic Constitutive Model of Ice under High-Speed Impact

### 3.1. Modeling of an Ice Ball Impacting a Metal Target

It is complex to systematically carry out dynamic performance tests of ice samples with different physical properties. Therefore, in this research, a group of relatively simple impact-contact finite-element models was firstly used to analyze the effects on the impact contact load of changing the parameters of the constitutive model. Then, the parameters and change law that had the greatest influence on the impact contact load among the model parameters were explored for subsequent analysis. A finite-element model of a 25 mm diameter ice ball positively impacting the face of a fixed 60 mm diameter circular target plate at a speed of 150 ms−1 was established in LS-PrePost (pre- and postprocessing software provided by Livermore Software Technology), and the calculation time was set as 0.5 ms. The ice ball was established using an SPH (Smoothed Particle Hydrodynamics) method, and the target plate was constructed from 1045 steel using the Lagrange method with the constitutive model of Plastic-Kinematic, and the constitutive parameters are shown in Table 1. The finite-element model is shown in Figure 3.Table 1
Constitutive model parameters of 1045 steel.
CategoryDensity(kg/m3)Young’s modulus(GPa)Poisson’s ratioYield stress(GPa)Tangent modulus(GPa)Value78502110.30.3556Figure 3
Model of 25 mm diameter ice ball impacting a metal target.In this study, the dynamic constitutive model of ice proposed by Carney et al. [22] and the Mat155 constitutive model (plasticity_compression_tension) were adopted, and this was described by the keyword “tabulated_compaction” to define its equation of state. In the study, the specific gravity, Young’s modulus, Poisson’s ratio, compression cut-off pressure, and tensile cut-off pressure in the constitutive model were varied for the calculations. The ranges of these modeling parameters are shown in Table 2.Table 2
Variation of constitutive model parameters of ice.
ParameterValueSpecific gravity0.6–1.0Young’s modulus6×109 to 10×109PaPoisson’s ratio0.30–0.34Parameter of strain rateDefault value (0)Compression cut-off pressure2×106 to 6×106PaTension cut-off pressure−0.5×106 to 0.1×106Pa


### 3.2. Preliminary Analysis of an Ice Ball Impacting a Target

At an impact speed of 150 ms−1, taking an ice ball with specific gravity of 0.9 as an example, the impact process of the central slice is shown in Figure 4. As can be seen, when the ice ball is just touching the target plate, it produces an initial impact, and the internal contact area of the ice ball quickly enters plasticity and fails. Then, a shock wave propagates inside the ice ball in the opposite direction to its velocity, producing longitudinal cracks, and the ice ball as a whole breaks and fails quickly. It then continues to act on the target plate in the form of a dynamic powder to complete the energy exchange, and an impact load is generated on the target plate.Figure 4
Typical impact process of the central slice of an ice ball.During the numerical simulations, five parameters—the specific gravity, Young’s modulus, tension/compression cut-off pressure, and Poisson’s ratio—of the ice balls were varied within the ranges noted in Table1. The results of changing each of these model parameters individually are shown in Figures 5–9.Figure 5
Impact loads of ice balls with different specific gravities.Figure 6
Impact loads of ice balls with different elastic moduli.Figure 7
Impact loads of ice balls with different compression cut-off pressure values.Figure 8
Impact load of ice balls with different tension cut-off pressure values.Figure 9
Impact loads of ice balls with different Poisson’s ratios.As can be seen in Figure5, the load curves of ice balls impacting a rigid target plate include an initial impact and a constant-flow state. When the specific gravity of the ice ball is low, the impact load curve has a relatively obvious initial impact stage. After the initial impact peak and secondary peak, it enters a relatively obvious constant-flow stage, in which about 60% of the impulse is transmitted to the target plate through the initial impact. With the gradual increase of specific gravity, the peak value of the impact load also increases obviously, but the constant-flow stage becomes less obvious. The waveform of the impact load becomes steeper and tends to form a clear triangle. About 80% of the impulse is transmitted to the target plate through the initial impact, which shows that changing the specific gravity has a great impact on the peak value and the waveform of the impact load in the simulations. As shown in Figures 6–9, the impact load peaks and waveforms of ice balls with the same specific gravity were not significantly affected by changing Young’s modulus, cut-off pressures, or Poisson’s ratio within the noted ranges. Therefore, in the subsequent modeling and numerical simulations, it can be preliminarily determined that changing the specific gravity in the constitutive model will have the most significant effect on the dynamic mechanical properties of the ice ball during an impact.


## 3.1. Modeling of an Ice Ball Impacting a Metal Target

It is complex to systematically carry out dynamic performance tests of ice samples with different physical properties. Therefore, in this research, a group of relatively simple impact-contact finite-element models was firstly used to analyze the effects on the impact contact load of changing the parameters of the constitutive model. Then, the parameters and change law that had the greatest influence on the impact contact load among the model parameters were explored for subsequent analysis. A finite-element model of a 25 mm diameter ice ball positively impacting the face of a fixed 60 mm diameter circular target plate at a speed of 150 ms−1 was established in LS-PrePost (pre- and postprocessing software provided by Livermore Software Technology), and the calculation time was set as 0.5 ms. The ice ball was established using an SPH (Smoothed Particle Hydrodynamics) method, and the target plate was constructed from 1045 steel using the Lagrange method with the constitutive model of Plastic-Kinematic, and the constitutive parameters are shown in Table 1. The finite-element model is shown in Figure 3.Table 1
Constitutive model parameters of 1045 steel.
CategoryDensity(kg/m3)Young’s modulus(GPa)Poisson’s ratioYield stress(GPa)Tangent modulus(GPa)Value78502110.30.3556Figure 3
Model of 25 mm diameter ice ball impacting a metal target.In this study, the dynamic constitutive model of ice proposed by Carney et al. [22] and the Mat155 constitutive model (plasticity_compression_tension) were adopted, and this was described by the keyword “tabulated_compaction” to define its equation of state. In the study, the specific gravity, Young’s modulus, Poisson’s ratio, compression cut-off pressure, and tensile cut-off pressure in the constitutive model were varied for the calculations. The ranges of these modeling parameters are shown in Table 2.Table 2
Variation of constitutive model parameters of ice.
ParameterValueSpecific gravity0.6–1.0Young’s modulus6×109 to 10×109PaPoisson’s ratio0.30–0.34Parameter of strain rateDefault value (0)Compression cut-off pressure2×106 to 6×106PaTension cut-off pressure−0.5×106 to 0.1×106Pa


## 3.2. Preliminary Analysis of an Ice Ball Impacting a Target

At an impact speed of 150 ms−1, taking an ice ball with specific gravity of 0.9 as an example, the impact process of the central slice is shown in Figure 4. As can be seen, when the ice ball is just touching the target plate, it produces an initial impact, and the internal contact area of the ice ball quickly enters plasticity and fails. Then, a shock wave propagates inside the ice ball in the opposite direction to its velocity, producing longitudinal cracks, and the ice ball as a whole breaks and fails quickly. It then continues to act on the target plate in the form of a dynamic powder to complete the energy exchange, and an impact load is generated on the target plate.Figure 4
Typical impact process of the central slice of an ice ball.During the numerical simulations, five parameters—the specific gravity, Young’s modulus, tension/compression cut-off pressure, and Poisson’s ratio—of the ice balls were varied within the ranges noted in Table1. The results of changing each of these model parameters individually are shown in Figures 5–9.Figure 5
Impact loads of ice balls with different specific gravities.Figure 6
Impact loads of ice balls with different elastic moduli.Figure 7
Impact loads of ice balls with different compression cut-off pressure values.Figure 8
Impact load of ice balls with different tension cut-off pressure values.Figure 9
Impact loads of ice balls with different Poisson’s ratios.As can be seen in Figure5, the load curves of ice balls impacting a rigid target plate include an initial impact and a constant-flow state. When the specific gravity of the ice ball is low, the impact load curve has a relatively obvious initial impact stage. After the initial impact peak and secondary peak, it enters a relatively obvious constant-flow stage, in which about 60% of the impulse is transmitted to the target plate through the initial impact. With the gradual increase of specific gravity, the peak value of the impact load also increases obviously, but the constant-flow stage becomes less obvious. The waveform of the impact load becomes steeper and tends to form a clear triangle. About 80% of the impulse is transmitted to the target plate through the initial impact, which shows that changing the specific gravity has a great impact on the peak value and the waveform of the impact load in the simulations. As shown in Figures 6–9, the impact load peaks and waveforms of ice balls with the same specific gravity were not significantly affected by changing Young’s modulus, cut-off pressures, or Poisson’s ratio within the noted ranges. Therefore, in the subsequent modeling and numerical simulations, it can be preliminarily determined that changing the specific gravity in the constitutive model will have the most significant effect on the dynamic mechanical properties of the ice ball during an impact.


## 4. Verification of Impact Load on Ice Balls with Different Specific Gravities

To verify the influence of changing the specific gravity on the results of the numerical simulations of hail impacts, two groups of dynamic load impact tests of ice balls with different specific gravities were designed, and these were used to test the impact loads and verify the accuracy and effectiveness of the numerical simulation method.


### 4.1. Preparation of Ice Ball Specimens with Different Specific Gravities

In this study, 25 mm diameter ice balls with different specific gravities were used, and their impact loads were tested using a target plate with a dynamic impact-force sensor. The specific gravities of the two kinds of ice ball test pieces were determined using different freezing and pressing preparation methods. Figure10 shows a photograph of the two halves of the aluminum mold in which the test specimens were frozen.Figure 10
Aluminum ice ball mold.When using the ice ball freezing process described in a previous report [31], the specific gravity of the ice balls was found to be 0.99. The low-specific-gravity ice ball test specimens were prepared using a controllable-specific-gravity mold, and a schematic diagram and photographs of this mold are shown in Figure 11.Figure 11
Pressing preparation mold, showing (a) schematic diagram, (b) assembled mold, and (c) separated mold.
(a)(b)(c)It is difficult to prepare ice with low specific gravity using ordinary low-temperature freezing. Therefore, in these experiments, a device for quickly preparing ice balls with different specific gravities by compressing ice chips was designed. Using this device, control of the specific gravity can be achieved by using a particular weight of crushed ice calculated according to a preset specific gravity. As shown in Figure11, this device was composed of a cavity made from stainless steel, an upper compression rod, and a lower compression rod with vent holes. The end faces of the upper and lower compression rods were numerically processed into hemispherically concave surfaces with a diameter of 25 mm such that they would just meet to create a spherical cavity.When preparing ice balls using this device, an ice crusher was used to break prefrozen ice into a powder, and ice chips of the appropriately calculated weight were placed into the cavity. These ice chips were then pressed to a specified position using the compression rod, compacting them between the upper and lower compression rods into an artificial ice ball with a set specific gravity. This was then left to stand for 1 min to allow the small initial cracks in the ice ball to integrate as much as possible. This was then placed into a freezer at −18°C for storage. In this test, the specific gravity of the pressed low-specific-gravity ice balls was controlled to 0.80. Figures12(a) and 12(b) show examples of simply frozen (specific gravity 0.99) and pressed test specimens (specific gravity 0.80). In this way, the formation process of natural hail was prepared through the three steps of pressing, standing, and freezing of broken ice, so as to effectively press the air into the tiny gap inside the ice ball. The ice structure inside the ice ball was fused through standing, and the ice structure inside the ice ball was formed at low temperature through freezing to form a low-specific-gravity ice ball with a large number of fine air gaps, so as to simulate a fine structure of low-specific-gravity layer of natural hail.Figure 12
Ice ball specimens with specific gravities of (a) 0.99 and (b) 0.80.
(a)(b)


### 4.2. Experimental Device

The tests were carried out using a 50 mm air-gun test system. The 5-meter-long barrel enabled the projectile to be pushed at a lower acceleration, and this helped to maintain its integrity. The projectile launching speed was controlled by the injection pressure of the air-gun chamber, and the ice ball test specimens were placed in an aluminum sabot with a foam lining. The sabot was separated from the separator at the muzzle when it launched, allowing the ice ball test piece to flow to the target. The speed and attitude of the ice ball before hitting the target were calibrated and recorded by high-speed photography of no less than 30,000 FPS (Frames per Second). A dynamic load sensor was fastened onto the target plate seat through front and rear mounting bolt tooling and positioning studs, and this was installed on a rectangular steel base, which was fixed onto a triangular rigid support platform. The dynamic load sensor measured the contact forces between the front face of the sensor and the tooling of the front mounting bolt. The output voltage signal thus described the curve of the end-face contact load over time, and this was stored on a computer using a high-speed data-acquisition system. A schematic and photograph of the whole test system are shown in Figure13, and its basic performance parameters are shown in Table 3.Figure 13
Impact test and measurement system: (1) compressed air chamber, (2) launch valve, (3) quick-load device, (4) barrier, (5) sabot separator, (6) automatic injection control system, (7) computer, (8) high-speed data-acquisition system, (9) light, and (10) high-speed camera.Table 3
Basic performance parameters of the test system.
DeviceTypeBasic performance parameterAir cannonAC-05050 mm caliber, launching ice balls with a maximum speed of 320 ms−1Dynamic force sensorKD3005CMaximum range 40 kNSampling frequency not less than 100 kHzHigh-speed cameraV1212>30,000 FPS at640×480 pixelsData-acquisition systemDongHuaSampling frequency 1 MHz


### 4.3. Results and Analysis

In the tests, two 25 mm diameter ice ball projectiles with different specific gravities were launched at a predetermined speed of 150 ms−1 to impact the target. A total of four groups of tests were carried out. The test working-condition parameters obtained from the actual tests are shown in Table 4, and representative images of the impact processes of the two types of ice ball impacting the target plate are shown in Figure 14. It can be seen that the sabot was successfully separated in the separator, and both kinds of the ice ball projectile accurately and squarely hit the target plate.Table 4
Working-condition parameters of the impact tests.
NumberSpecific gravity of projectileImpact speed (ms−1)Peak load (N)1-10.99151.018,185.621-20.99149.617,755.912-10.80148.913,505.802-20.80151.514,179.34Figure 14
Impact processes of ice balls with specific gravities of (a) 0.99 and (b) 0.80.
(a)(b)The ice ball with a specific gravity of 0.99 was transparent before hitting the target plate. At the moment of impact, the load was rapidly transmitted through the ice ball; the internal structure was unable to bear this load, and cracks rapidly appeared, making the inside of the ice ball opaque. With the progress of the impact process, cracks were initiated at the surface of the ice ball and propagated from the contact point to the opposite side. Under the joint action of surface and internal cracks, the ice ball structure collapsed and continued to act on the target surface in the form of a particle flow to create a continuous impact load.The impact process of the ice ball with a specific gravity of 0.80 was similar to that of the ice ball with a specific gravity of 0.99. However, the appearance of the ice ball was initially opaque due to it containing a large number of interconnected cavities and bubbles. As such, crack initiation and propagation in the ice ball at the moment of impact could not be observed. The initiation time of the crack on the surface of the ice ball with a specific gravity of 0.80 was close to that of the ice ball with a specific gravity of 0.99, but the dissipation rate was faster. The reason for this is that the strength of the low-specific-gravity ice ball was lower than that of the high-specific-gravity ice ball.According to relevant documents [13, 18, 21, 23], the impact failure process of the frozen transparent ice ball was generally divided into four stages, as shown in Figure 15.Figure 15
The impact failure process of frozen transparent ice ball.The stage I of the ice ball impacting the target plate can be regarded as the crack initiation stage. Once the ice ball specimen impacts the target, cracks and local failures will occur immediately. When the specimen continues to impact the target plate, the number of cracks will increase. Two deformation waves with equal size but opposite direction will form at the contact point between the specimen and the target plate. When the deformation wave reaches the critical value, it will lead to local failure. Stage II can be regarded as the crack propagation stage. Under unconstrained conditions, the crack will grow in the direction of load and in the opposite direction of impact until the end of the ice ball. Stage III can be regarded as the collapse stage. At this stage, the main crack has fully developed. As the ice ball specimen continues to impact the target plate, the ice ball part that does not contact the target plate can no longer maintain its original geometric shape. At this time, it begins to collapse, and the failed broken ice will flow from the impact center to the edge of the target plate. Stage IV can be considered as the fragmentation stage. At this stage, the whole ice ball specimen is broken. As the impact energy is absorbed, the part that does not contact the target plate will not flow along the radial direction of the target plate. If the impact energy is small enough, part of the broken ice will remain on the target plate. However, according to the high-speed photography results of the impact test of low-specific-gravity ice ball, the failure process is faster and more complete than that of the transparent ice ball, as shown in Figure16.Figure 16
The four impact failure processes of low-specific-gravity ice ball.The four impact failure processes of the low-specific-gravity ice ball can be described as follows: rapid crack propagation in stage I, overall failure in stage II, particle flow in stage III, and flow termination in stage IV. Compared with the failure process of the frozen high-specific-gravity ice ball, the characteristic of low-specific-gravity ice ball impact failure is that crack initiation and propagation and overall failure occur in a very short time, so that under the condition of maintaining almost complete geometric shape, the inside of the ice ball has been completely broken and has lost its bearing capacity, and then, the whole ice ball impacts the surface of the target plate in a granular or powder state. Correspondingly, although the crack of frozen ice ball extends to all spheres, the broken particle state is only in the part close to the target plate, and the nonimpacted sphere still has a certain bearing capacity and interacts with the target plate until the end of impact. Therefore, theoretically, the frozen high-specific-gravity ice ball can maintain a high impact load in a certain period of time, while the low-specific-gravity ice ball can only maintain a low level of impact load in a certain period of time due to its rapid failure and collapse. The impact load curves of two groups of ice balls are shown in Figure17.Figure 17
Impact load–time histories of the two types of ice ball.It can be seen that the duration of each of the impact tests was about 0.16 ms. According to calculations of the ice ball diameter and speed, the standard impact time was 0.167 ms. The load-time curves obtained from the tests thus verify their effectiveness. The load peak of the 0.99-specific-gravity ice ball was about 18,000 N, and that of the 0.80-specific-gravity ice ball was about 14,000 N. The impact load peak of the low-specific-gravity ice ball was significantly lower than that of the high-specific-gravity ice ball. Through analysis, it was found that(1)
the total mass of the low-specific-gravity ice ball was lower than that of the high-specific-gravity ice ball, so at the same impact speed, the momentum transmitted to the target plate and the load peak was lower(2)
the internal structure of the low-specific-gravity ice ball is fragile and loose, so the transmission speed of the stress wave caused by the impact load is low and the internal response is slight; these effects are mapped to the sensor surface, and this is reflected in the low speed of the load rising from zero to maximum, and the impact consumption time of the low-specific-gravity ice ball will be slightly longer than that of the high-density ice ball; the load curve presents a “platform” effectThe tests showed that the impact load level of the low-specific-gravity ice ball was lower than that of the high-specific-gravity ice ball.


### 4.4. Comparison with Numerical Simulation Results

For further analysis, a full three-dimensional finite-element model of the two kinds of ice ball impacting the target plate under the test conditions was established, and this is shown in Figure18. The finite-element models of the ice balls were created using the SPH method, and these were set to impact the end face of the test system axially. The middle of the hollow-ring load sensor module was pierced with a positioning stud. Through the positioning stud, the sensor was clamped by the rear and front nut tooling. The rear end of the rear nut tooling was constrained in the direction and translation directions. The sensor body and nut tooling were made from Ti6Al4V alloy, the diameter of the end face cylindrical tooling of the test system was 60 mm, and the material was 1045 steel. The Ti6Al4V and 1045 steel materials used the follow-up plastic constitutive model, and the parameters were selected from Ref. [32]; the 1045 steel’s constitutive model parameters are indicated in Table 1, and the Ti6Al4V alloy’s constitutive model parameters are shown in Table 5. The sensor and tooling were modeled using the Lagrange method. The fixed-contact mode was applied between the nut, stud, and end face, and the surface-contact mode was applied between the sensor and the nut and stud. In the modeling, the other parameters in the dynamic constitutive model and the equation of state were set to those used in a previous report [22].Figure 18
(a) Finite-element model of load sensor for ice ball impact target and (b) detail of the sensor model.
(a)(b)Table 5
Ti6Al4V alloy’s constitutive model parameters.
CategoryDensity(kg/m3)Young’s modulus(GPa)Poisson’s ratioYield stress(GPa)Tangent modulus(GPa)Strain-rate parameter(C)Strain-rate parameter(P)Hardening parameter(β)Value44001130.341.00994.712476403.82060.7661In the tests and simulations, we found that the change trend of the impact load curve was highly related to the parameters of the constitutive model and the equation of state, and the duration and peak value of the load were highly related to the support stiffness of the sensor test system. In many reports, the simulation of such dynamic load sensor test systems was realized by establishing a mass–spring system [22, 33–35]. In these models, the spring and mass were connected in either parallel or series according to the fixed support mode of the sensor, and the stiffness of the spring was obtained through modal analysis [22]. The simulated value of the impact load obtained by this method was in good agreement with the experimental value.In this study, considering that the stiffness of the spring system obtained through modal analysis may have a strong nonlinear response and deviate from the modal analysis value under high-speed impact deformation, a rapid but more approximate support stiffness adjustment method was adopted to verify the effectiveness of the constitutive model while improving the calculation efficiency. Specifically, several manual iterations were adopted during the analysis; Young’s modulus value of the 1045 steel material of the rear tooling nut was reduced by a factor of ten; that is to say, the value of Young’s modulus was set as 21.1 GPa, and the load simulation curve of the numerical simulation was obtained. After the load curve was examined by a fast Fourier transform, a filtering mode with a threshold of 13,000 Hz was determined. In this way, the high-frequency vibration clutter was filtered, and a comparison diagram showing the variations in the test and simulation values over time could be plotted. The results of this are shown in Figure19.Figure 19
Comparison between numerical simulation and experimental measurement curves of impact load for specific gravities of (a) 0.99 and (b) 0.80.
(a)(b)It can be seen that the numerical simulation results have been effectively verified by the test results. The peak error of the impact load was very small. In the example impact of an ice ball with a specific gravity of 0.99, the simulation curve of the impact load was basically consistent with the test curve. In the example impact of an ice ball with a specific gravity of 0.80, the simulation value of the impact load peak was in good agreement with the test value. However, comparing the impact load test curves, the action time of the simulation curve was too short to accurately simulate the platform effect of the impact load of the low-density ice ball. The reason for this phenomenon is that the waveform details of the impact load curve of the ice are controlled by its state equation. The state equation parameters used in conjunction with the dynamic constitutive model of ice developed in a previous report [22] were calibrated using simple frozen ice ball impact tests. Therefore, a numerical simulation method using the corresponding parameters can accurately reflect the impact process and effect of a frozen ice ball (the high-specific-gravity ice ball in this study). However, when these parameters were used in the simulation of a low-specific-gravity ice ball, due to its low bearing capacity and rapid collapse, the state equation parameters describing its flow state will be different. Accurate calibration is therefore needed. From the point of view of the modeling method in this paper, the accuracy of the original model parameters was acceptable. Table 6 shows a comparative analysis of the errors between the numerical simulation and test results. According to these results, it can be considered that variable-density-parameter modeling can better reflect the impact loads of ice balls with different specific gravities.Table 6
Error comparison between ice ball impact tests and numerical simulation results.
Specific gravityPeak loadTest value (N)Simulation value (N)Error (%)1-10.9918 185.6216 989.73−6.581-20.9917 755.91−4.322-10.8013 505.8013 701.16+1.452-20.8014 179.34−3.37


### 4.5. Finite-Element Modeling of Hail with Layered Structure

With reference to the layered structure of real hail in nature, LS-PrePost was used to establish a layered ice ball model. It should be noted that natural hail has different shapes. In fact, their shapes are not only approximate spherical but also irregular block, sheet, ellipsoid, bowl with wave edge, or even crown. However, their common feature is stratification. Therefore, the most common layered spherical hail is still selected as the modeling and analysis object in this paper. The modeling process is shown in Figure20. According to this process, a spherically layered ice ball model was established based on data about the structure and layered specific gravity of natural hail with a diameter of 29 mm (as shown in Figure 2 [30]). The constitutive models of the different ice layers used the parameters given in Section 3.1; the layers were given the specific gravities listed in Table 7, and the parameters of other types of equivalent hail models are shown in Table 8. In view of the large number of applications of titanium alloy structures in aviation structures, a model of layered hail impacting a 2 mm thick TI6AL4V target plate fixed on four sides at a speed of 150 ms−1 was established, and this is shown in Figure 21. The parameters used for the constitutive model of the TI6AL4V target plate were those given in a previous report [32].Figure 20
Layered ice ball modeling process.Table 7
Modeling parameters.
Outward expansion29 mm diameter natural hailLayer thickness (mm)Specific gravityFirst layer (core)70.71Second layer30.85Third layer60.82Fourth layer20.87Table 8
Modeling types and parameters.
TypeConfigurationDiameter (mm)Mass (g)Equivalent specific gravityALayered2910.80.847BEquivalent2911.60.9CEquivalent2910.80.847DEquivalent28.410.80.9Figure 21
Model of layered ice ball impacting Ti6Al4V target.


## 4.1. Preparation of Ice Ball Specimens with Different Specific Gravities

In this study, 25 mm diameter ice balls with different specific gravities were used, and their impact loads were tested using a target plate with a dynamic impact-force sensor. The specific gravities of the two kinds of ice ball test pieces were determined using different freezing and pressing preparation methods. Figure10 shows a photograph of the two halves of the aluminum mold in which the test specimens were frozen.Figure 10
Aluminum ice ball mold.When using the ice ball freezing process described in a previous report [31], the specific gravity of the ice balls was found to be 0.99. The low-specific-gravity ice ball test specimens were prepared using a controllable-specific-gravity mold, and a schematic diagram and photographs of this mold are shown in Figure 11.Figure 11
Pressing preparation mold, showing (a) schematic diagram, (b) assembled mold, and (c) separated mold.
(a)(b)(c)It is difficult to prepare ice with low specific gravity using ordinary low-temperature freezing. Therefore, in these experiments, a device for quickly preparing ice balls with different specific gravities by compressing ice chips was designed. Using this device, control of the specific gravity can be achieved by using a particular weight of crushed ice calculated according to a preset specific gravity. As shown in Figure11, this device was composed of a cavity made from stainless steel, an upper compression rod, and a lower compression rod with vent holes. The end faces of the upper and lower compression rods were numerically processed into hemispherically concave surfaces with a diameter of 25 mm such that they would just meet to create a spherical cavity.When preparing ice balls using this device, an ice crusher was used to break prefrozen ice into a powder, and ice chips of the appropriately calculated weight were placed into the cavity. These ice chips were then pressed to a specified position using the compression rod, compacting them between the upper and lower compression rods into an artificial ice ball with a set specific gravity. This was then left to stand for 1 min to allow the small initial cracks in the ice ball to integrate as much as possible. This was then placed into a freezer at −18°C for storage. In this test, the specific gravity of the pressed low-specific-gravity ice balls was controlled to 0.80. Figures12(a) and 12(b) show examples of simply frozen (specific gravity 0.99) and pressed test specimens (specific gravity 0.80). In this way, the formation process of natural hail was prepared through the three steps of pressing, standing, and freezing of broken ice, so as to effectively press the air into the tiny gap inside the ice ball. The ice structure inside the ice ball was fused through standing, and the ice structure inside the ice ball was formed at low temperature through freezing to form a low-specific-gravity ice ball with a large number of fine air gaps, so as to simulate a fine structure of low-specific-gravity layer of natural hail.Figure 12
Ice ball specimens with specific gravities of (a) 0.99 and (b) 0.80.
(a)(b)


## 4.2. Experimental Device

The tests were carried out using a 50 mm air-gun test system. The 5-meter-long barrel enabled the projectile to be pushed at a lower acceleration, and this helped to maintain its integrity. The projectile launching speed was controlled by the injection pressure of the air-gun chamber, and the ice ball test specimens were placed in an aluminum sabot with a foam lining. The sabot was separated from the separator at the muzzle when it launched, allowing the ice ball test piece to flow to the target. The speed and attitude of the ice ball before hitting the target were calibrated and recorded by high-speed photography of no less than 30,000 FPS (Frames per Second). A dynamic load sensor was fastened onto the target plate seat through front and rear mounting bolt tooling and positioning studs, and this was installed on a rectangular steel base, which was fixed onto a triangular rigid support platform. The dynamic load sensor measured the contact forces between the front face of the sensor and the tooling of the front mounting bolt. The output voltage signal thus described the curve of the end-face contact load over time, and this was stored on a computer using a high-speed data-acquisition system. A schematic and photograph of the whole test system are shown in Figure13, and its basic performance parameters are shown in Table 3.Figure 13
Impact test and measurement system: (1) compressed air chamber, (2) launch valve, (3) quick-load device, (4) barrier, (5) sabot separator, (6) automatic injection control system, (7) computer, (8) high-speed data-acquisition system, (9) light, and (10) high-speed camera.Table 3
Basic performance parameters of the test system.
DeviceTypeBasic performance parameterAir cannonAC-05050 mm caliber, launching ice balls with a maximum speed of 320 ms−1Dynamic force sensorKD3005CMaximum range 40 kNSampling frequency not less than 100 kHzHigh-speed cameraV1212>30,000 FPS at640×480 pixelsData-acquisition systemDongHuaSampling frequency 1 MHz


## 4.3. Results and Analysis

In the tests, two 25 mm diameter ice ball projectiles with different specific gravities were launched at a predetermined speed of 150 ms−1 to impact the target. A total of four groups of tests were carried out. The test working-condition parameters obtained from the actual tests are shown in Table 4, and representative images of the impact processes of the two types of ice ball impacting the target plate are shown in Figure 14. It can be seen that the sabot was successfully separated in the separator, and both kinds of the ice ball projectile accurately and squarely hit the target plate.Table 4
Working-condition parameters of the impact tests.
NumberSpecific gravity of projectileImpact speed (ms−1)Peak load (N)1-10.99151.018,185.621-20.99149.617,755.912-10.80148.913,505.802-20.80151.514,179.34Figure 14
Impact processes of ice balls with specific gravities of (a) 0.99 and (b) 0.80.
(a)(b)The ice ball with a specific gravity of 0.99 was transparent before hitting the target plate. At the moment of impact, the load was rapidly transmitted through the ice ball; the internal structure was unable to bear this load, and cracks rapidly appeared, making the inside of the ice ball opaque. With the progress of the impact process, cracks were initiated at the surface of the ice ball and propagated from the contact point to the opposite side. Under the joint action of surface and internal cracks, the ice ball structure collapsed and continued to act on the target surface in the form of a particle flow to create a continuous impact load.The impact process of the ice ball with a specific gravity of 0.80 was similar to that of the ice ball with a specific gravity of 0.99. However, the appearance of the ice ball was initially opaque due to it containing a large number of interconnected cavities and bubbles. As such, crack initiation and propagation in the ice ball at the moment of impact could not be observed. The initiation time of the crack on the surface of the ice ball with a specific gravity of 0.80 was close to that of the ice ball with a specific gravity of 0.99, but the dissipation rate was faster. The reason for this is that the strength of the low-specific-gravity ice ball was lower than that of the high-specific-gravity ice ball.According to relevant documents [13, 18, 21, 23], the impact failure process of the frozen transparent ice ball was generally divided into four stages, as shown in Figure 15.Figure 15
The impact failure process of frozen transparent ice ball.The stage I of the ice ball impacting the target plate can be regarded as the crack initiation stage. Once the ice ball specimen impacts the target, cracks and local failures will occur immediately. When the specimen continues to impact the target plate, the number of cracks will increase. Two deformation waves with equal size but opposite direction will form at the contact point between the specimen and the target plate. When the deformation wave reaches the critical value, it will lead to local failure. Stage II can be regarded as the crack propagation stage. Under unconstrained conditions, the crack will grow in the direction of load and in the opposite direction of impact until the end of the ice ball. Stage III can be regarded as the collapse stage. At this stage, the main crack has fully developed. As the ice ball specimen continues to impact the target plate, the ice ball part that does not contact the target plate can no longer maintain its original geometric shape. At this time, it begins to collapse, and the failed broken ice will flow from the impact center to the edge of the target plate. Stage IV can be considered as the fragmentation stage. At this stage, the whole ice ball specimen is broken. As the impact energy is absorbed, the part that does not contact the target plate will not flow along the radial direction of the target plate. If the impact energy is small enough, part of the broken ice will remain on the target plate. However, according to the high-speed photography results of the impact test of low-specific-gravity ice ball, the failure process is faster and more complete than that of the transparent ice ball, as shown in Figure16.Figure 16
The four impact failure processes of low-specific-gravity ice ball.The four impact failure processes of the low-specific-gravity ice ball can be described as follows: rapid crack propagation in stage I, overall failure in stage II, particle flow in stage III, and flow termination in stage IV. Compared with the failure process of the frozen high-specific-gravity ice ball, the characteristic of low-specific-gravity ice ball impact failure is that crack initiation and propagation and overall failure occur in a very short time, so that under the condition of maintaining almost complete geometric shape, the inside of the ice ball has been completely broken and has lost its bearing capacity, and then, the whole ice ball impacts the surface of the target plate in a granular or powder state. Correspondingly, although the crack of frozen ice ball extends to all spheres, the broken particle state is only in the part close to the target plate, and the nonimpacted sphere still has a certain bearing capacity and interacts with the target plate until the end of impact. Therefore, theoretically, the frozen high-specific-gravity ice ball can maintain a high impact load in a certain period of time, while the low-specific-gravity ice ball can only maintain a low level of impact load in a certain period of time due to its rapid failure and collapse. The impact load curves of two groups of ice balls are shown in Figure17.Figure 17
Impact load–time histories of the two types of ice ball.It can be seen that the duration of each of the impact tests was about 0.16 ms. According to calculations of the ice ball diameter and speed, the standard impact time was 0.167 ms. The load-time curves obtained from the tests thus verify their effectiveness. The load peak of the 0.99-specific-gravity ice ball was about 18,000 N, and that of the 0.80-specific-gravity ice ball was about 14,000 N. The impact load peak of the low-specific-gravity ice ball was significantly lower than that of the high-specific-gravity ice ball. Through analysis, it was found that(1)
the total mass of the low-specific-gravity ice ball was lower than that of the high-specific-gravity ice ball, so at the same impact speed, the momentum transmitted to the target plate and the load peak was lower(2)
the internal structure of the low-specific-gravity ice ball is fragile and loose, so the transmission speed of the stress wave caused by the impact load is low and the internal response is slight; these effects are mapped to the sensor surface, and this is reflected in the low speed of the load rising from zero to maximum, and the impact consumption time of the low-specific-gravity ice ball will be slightly longer than that of the high-density ice ball; the load curve presents a “platform” effectThe tests showed that the impact load level of the low-specific-gravity ice ball was lower than that of the high-specific-gravity ice ball.


## 4.4. Comparison with Numerical Simulation Results

For further analysis, a full three-dimensional finite-element model of the two kinds of ice ball impacting the target plate under the test conditions was established, and this is shown in Figure18. The finite-element models of the ice balls were created using the SPH method, and these were set to impact the end face of the test system axially. The middle of the hollow-ring load sensor module was pierced with a positioning stud. Through the positioning stud, the sensor was clamped by the rear and front nut tooling. The rear end of the rear nut tooling was constrained in the direction and translation directions. The sensor body and nut tooling were made from Ti6Al4V alloy, the diameter of the end face cylindrical tooling of the test system was 60 mm, and the material was 1045 steel. The Ti6Al4V and 1045 steel materials used the follow-up plastic constitutive model, and the parameters were selected from Ref. [32]; the 1045 steel’s constitutive model parameters are indicated in Table 1, and the Ti6Al4V alloy’s constitutive model parameters are shown in Table 5. The sensor and tooling were modeled using the Lagrange method. The fixed-contact mode was applied between the nut, stud, and end face, and the surface-contact mode was applied between the sensor and the nut and stud. In the modeling, the other parameters in the dynamic constitutive model and the equation of state were set to those used in a previous report [22].Figure 18
(a) Finite-element model of load sensor for ice ball impact target and (b) detail of the sensor model.
(a)(b)Table 5
Ti6Al4V alloy’s constitutive model parameters.
CategoryDensity(kg/m3)Young’s modulus(GPa)Poisson’s ratioYield stress(GPa)Tangent modulus(GPa)Strain-rate parameter(C)Strain-rate parameter(P)Hardening parameter(β)Value44001130.341.00994.712476403.82060.7661In the tests and simulations, we found that the change trend of the impact load curve was highly related to the parameters of the constitutive model and the equation of state, and the duration and peak value of the load were highly related to the support stiffness of the sensor test system. In many reports, the simulation of such dynamic load sensor test systems was realized by establishing a mass–spring system [22, 33–35]. In these models, the spring and mass were connected in either parallel or series according to the fixed support mode of the sensor, and the stiffness of the spring was obtained through modal analysis [22]. The simulated value of the impact load obtained by this method was in good agreement with the experimental value.In this study, considering that the stiffness of the spring system obtained through modal analysis may have a strong nonlinear response and deviate from the modal analysis value under high-speed impact deformation, a rapid but more approximate support stiffness adjustment method was adopted to verify the effectiveness of the constitutive model while improving the calculation efficiency. Specifically, several manual iterations were adopted during the analysis; Young’s modulus value of the 1045 steel material of the rear tooling nut was reduced by a factor of ten; that is to say, the value of Young’s modulus was set as 21.1 GPa, and the load simulation curve of the numerical simulation was obtained. After the load curve was examined by a fast Fourier transform, a filtering mode with a threshold of 13,000 Hz was determined. In this way, the high-frequency vibration clutter was filtered, and a comparison diagram showing the variations in the test and simulation values over time could be plotted. The results of this are shown in Figure19.Figure 19
Comparison between numerical simulation and experimental measurement curves of impact load for specific gravities of (a) 0.99 and (b) 0.80.
(a)(b)It can be seen that the numerical simulation results have been effectively verified by the test results. The peak error of the impact load was very small. In the example impact of an ice ball with a specific gravity of 0.99, the simulation curve of the impact load was basically consistent with the test curve. In the example impact of an ice ball with a specific gravity of 0.80, the simulation value of the impact load peak was in good agreement with the test value. However, comparing the impact load test curves, the action time of the simulation curve was too short to accurately simulate the platform effect of the impact load of the low-density ice ball. The reason for this phenomenon is that the waveform details of the impact load curve of the ice are controlled by its state equation. The state equation parameters used in conjunction with the dynamic constitutive model of ice developed in a previous report [22] were calibrated using simple frozen ice ball impact tests. Therefore, a numerical simulation method using the corresponding parameters can accurately reflect the impact process and effect of a frozen ice ball (the high-specific-gravity ice ball in this study). However, when these parameters were used in the simulation of a low-specific-gravity ice ball, due to its low bearing capacity and rapid collapse, the state equation parameters describing its flow state will be different. Accurate calibration is therefore needed. From the point of view of the modeling method in this paper, the accuracy of the original model parameters was acceptable. Table 6 shows a comparative analysis of the errors between the numerical simulation and test results. According to these results, it can be considered that variable-density-parameter modeling can better reflect the impact loads of ice balls with different specific gravities.Table 6
Error comparison between ice ball impact tests and numerical simulation results.
Specific gravityPeak loadTest value (N)Simulation value (N)Error (%)1-10.9918 185.6216 989.73−6.581-20.9917 755.91−4.322-10.8013 505.8013 701.16+1.452-20.8014 179.34−3.37


## 4.5. Finite-Element Modeling of Hail with Layered Structure

With reference to the layered structure of real hail in nature, LS-PrePost was used to establish a layered ice ball model. It should be noted that natural hail has different shapes. In fact, their shapes are not only approximate spherical but also irregular block, sheet, ellipsoid, bowl with wave edge, or even crown. However, their common feature is stratification. Therefore, the most common layered spherical hail is still selected as the modeling and analysis object in this paper. The modeling process is shown in Figure20. According to this process, a spherically layered ice ball model was established based on data about the structure and layered specific gravity of natural hail with a diameter of 29 mm (as shown in Figure 2 [30]). The constitutive models of the different ice layers used the parameters given in Section 3.1; the layers were given the specific gravities listed in Table 7, and the parameters of other types of equivalent hail models are shown in Table 8. In view of the large number of applications of titanium alloy structures in aviation structures, a model of layered hail impacting a 2 mm thick TI6AL4V target plate fixed on four sides at a speed of 150 ms−1 was established, and this is shown in Figure 21. The parameters used for the constitutive model of the TI6AL4V target plate were those given in a previous report [32].Figure 20
Layered ice ball modeling process.Table 7
Modeling parameters.
Outward expansion29 mm diameter natural hailLayer thickness (mm)Specific gravityFirst layer (core)70.71Second layer30.85Third layer60.82Fourth layer20.87Table 8
Modeling types and parameters.
TypeConfigurationDiameter (mm)Mass (g)Equivalent specific gravityALayered2910.80.847BEquivalent2911.60.9CEquivalent2910.80.847DEquivalent28.410.80.9Figure 21
Model of layered ice ball impacting Ti6Al4V target.


## 5. Dynamic Response Analysis of Layered Ice Ball Impacting Ti6Al4V Target

### 5.1. Impact Process

The processes of the central slices of the four types of ice ball models impacting the Ti6Al4V target are shown in Figure22. The overall impact processes of the four kinds of ice ball produced basically the same deformation trend in the target plate. After the ice ball contacted the target plate, the failure occurred first in the front impact center, and the failure area expanded rapidly to the periphery of the ice ball, resulting in the crushing of the whole ice ball. In this process, the conversion of impact energy was completed, causing bending deformation of the impact center of the target plate.Figure 22
Processes of the four types of hail impacting the target plate.In the comparison of the impact processes of hailstones of equal diameters, it can be seen that the failure mode of the layered ice ball was the same as that of its equivalent ice balls, and the parts of the different layers were compressed and failed layer by layer due to the impact. In the comparison of the impact processes of hailstones of equal mass, it can be seen that the expansion mode of the failure area of the ice ball with the small diameter was different from that of the ice ball with the large diameter. The failure area not only rapidly expanded to the circumferential positions on the ice ball but also rapidly expanded to the center of the ice ball. This was similar to the situation of a ball impacting a rigid target, indicating that the geometric size and equivalent specific gravity of the ice ball have a certain influence on its failure mode.


### 5.2. Contact Forces

A comparison of the contact forces of the four types of ice ball models impacting the target plate is shown in Figure23. The type-B ice ball, which had the largest mass, had the largest kinetic energy. The peak value of the contact force hitting the target plate was therefore the largest. In the comparison of the impact contact forces of the other three types of ice ball, it can be seen that the peak value of the impact contact force of the layered ice ball was the smallest, and there was slight fluctuation. The reason for this was that the properties of each layer of the layered ice ball were different, and the propagation speed of the stress wave in each was different, causing slight fluctuation of the load as the different layers were crossed. The impact load peaks of the type-C and type-D ice ball models were basically the same. The reason for this was that their mass and kinetic energy were the same. Because the equivalent specific gravity of the type-D ice ball was higher than that of the type-C ice ball, the initial impact load of the type-C ice ball was less than that of the type-D ice ball. At the same time, because the diameter of the type-C ice ball was large and contacted the target plate earlier, the time sequence of the contact force was faster than that of the type-D ice ball.Figure 23
Contact forces of the four types of ice ball.


### 5.3. Maximum Displacement of Target Plate Center

The maximum displacements of the center of the impact target plate with the four types of ice ball models are shown in Figure24. As shown in the figure, the center displacement caused by the impact of the equivalent-structure ice ball with the largest mass (type B) on the target plate was the largest, the displacement caused by the impact of the layered-structure ice ball (type A) was the smallest, and the displacements caused by the impact of the other types of ice ball models were basically the same. However, they were all greater than the displacements caused by the ice ball with a layered structure; according to the comparison of status between the ice ball of layered structure and those of equivalent structure, it was indicated that there may be energy loss in the process of the stress wave passing through the layered structure and interlayer collapse.Figure 24
Maximum displacements of the center of the impact target plate.


### 5.4. Maximum Effective Stress in the Center of the Target Plate

The calculated maximum effective stress levels produced by the four types of ice balls impacting the target plate are shown in Figure25. As shown, the impact stress value from the layered structure was the smallest, and the values from the other three equivalent ice ball models were basically the same. The reason for this was that the overall deformation of the target plate caused by the impact of the other three equivalent ice ball models was relatively large, while the local deformation of the target plate caused by the impact of the layered structure was relatively large. Therefore, the equivalent stress on the back of the target plate was the largest, and the stress fluctuation range was large due to the rapid rebound of the target plate.Figure 25
Maximum effective stress values produced by the impacts of the four types of ice ball.


### 5.5. Maximum Effective Plastic Strain in the Center of the Target Plate

The maximum effective plastic strain values produced by the four types of ice ball model impacting the target plate are shown in Figure26. Each impact led to a plastic state in the impacted area of the target plate. Among the models, the plastic strain in the target plate caused by the impact of the layered ice ball was the smallest, indicating that the impact energy loss of the layered ice ball was large and was not completely passed to the target plate. The plastic strain values caused by the impact of the two equivalent ice ball models with equal diameter were basically the same. The plastic strain caused by the impact of the type-D ice ball, which had the smallest diameter, was relatively small. At the same time, considering the simulation results shown in Figure 19, it can be seen that the center and overall failure of the ice ball with the small diameter and high density were the fastest under the impact. As such, the plastic strain value of the impact center was reduced overall. This shows that the geometric size and density of the ice ball have a certain influence on the local deformation and plastic strain state of the target plate.Figure 26
Maximum effective plastic strain produced by the impacts of the four types of ice ball.


### 5.6. Energy Absorbed by Target Plate

The energy absorbed by the target plate from the impacts of the four types of ice ball models is shown in Figure27. As shown, each impact causes the target plate to absorb energy. Among them, the layered-structure ice ball caused the least energy to be absorbed when impacting the target plate, while the type-B ice ball had the largest mass, so the energy transmitted to the target plate was also the highest. The type-C and type-D ice balls had the same initial kinetic energy, but their impact energies were relatively concentrated because the diameter of the type-D ice ball was slightly smaller. Therefore, the overall energy conversion degree caused by the impact was slightly higher than that of the type-C ice ball.Figure 27
Energy absorbed by the target plate from the four types of ice ball during impact.


### 5.7. Summary

To sum up, during the generation of hail, different freezing methods lead to different densities of hail layers. Between these different layers, due to the different wave impedances, which are determined by the ice density and stress-wave velocity of each layer, the stress wave generated by instantaneous high-speed impact reflects and transmits when passing through the layers. This results in additional energy loss compared with the impact of a homogeneous ice ball. The energy loss is shown in the reduction of the dynamic response level of the Ti6Al4V target in the numerical simulations. The next research focus will be divided into two parts: first is preparing real layered hail with concentric spherical shells for impact-test verifications, and second, in the concentric spherical-shell finite-element hail model, analysis work is planned that will use cohesive interlayer transition elements to reveal a more accurate interlayer impact energy transmission mode and mechanism.


## 5.1. Impact Process

The processes of the central slices of the four types of ice ball models impacting the Ti6Al4V target are shown in Figure22. The overall impact processes of the four kinds of ice ball produced basically the same deformation trend in the target plate. After the ice ball contacted the target plate, the failure occurred first in the front impact center, and the failure area expanded rapidly to the periphery of the ice ball, resulting in the crushing of the whole ice ball. In this process, the conversion of impact energy was completed, causing bending deformation of the impact center of the target plate.Figure 22
Processes of the four types of hail impacting the target plate.In the comparison of the impact processes of hailstones of equal diameters, it can be seen that the failure mode of the layered ice ball was the same as that of its equivalent ice balls, and the parts of the different layers were compressed and failed layer by layer due to the impact. In the comparison of the impact processes of hailstones of equal mass, it can be seen that the expansion mode of the failure area of the ice ball with the small diameter was different from that of the ice ball with the large diameter. The failure area not only rapidly expanded to the circumferential positions on the ice ball but also rapidly expanded to the center of the ice ball. This was similar to the situation of a ball impacting a rigid target, indicating that the geometric size and equivalent specific gravity of the ice ball have a certain influence on its failure mode.


## 5.2. Contact Forces

A comparison of the contact forces of the four types of ice ball models impacting the target plate is shown in Figure23. The type-B ice ball, which had the largest mass, had the largest kinetic energy. The peak value of the contact force hitting the target plate was therefore the largest. In the comparison of the impact contact forces of the other three types of ice ball, it can be seen that the peak value of the impact contact force of the layered ice ball was the smallest, and there was slight fluctuation. The reason for this was that the properties of each layer of the layered ice ball were different, and the propagation speed of the stress wave in each was different, causing slight fluctuation of the load as the different layers were crossed. The impact load peaks of the type-C and type-D ice ball models were basically the same. The reason for this was that their mass and kinetic energy were the same. Because the equivalent specific gravity of the type-D ice ball was higher than that of the type-C ice ball, the initial impact load of the type-C ice ball was less than that of the type-D ice ball. At the same time, because the diameter of the type-C ice ball was large and contacted the target plate earlier, the time sequence of the contact force was faster than that of the type-D ice ball.Figure 23
Contact forces of the four types of ice ball.


## 5.3. Maximum Displacement of Target Plate Center

The maximum displacements of the center of the impact target plate with the four types of ice ball models are shown in Figure24. As shown in the figure, the center displacement caused by the impact of the equivalent-structure ice ball with the largest mass (type B) on the target plate was the largest, the displacement caused by the impact of the layered-structure ice ball (type A) was the smallest, and the displacements caused by the impact of the other types of ice ball models were basically the same. However, they were all greater than the displacements caused by the ice ball with a layered structure; according to the comparison of status between the ice ball of layered structure and those of equivalent structure, it was indicated that there may be energy loss in the process of the stress wave passing through the layered structure and interlayer collapse.Figure 24
Maximum displacements of the center of the impact target plate.


## 5.4. Maximum Effective Stress in the Center of the Target Plate

The calculated maximum effective stress levels produced by the four types of ice balls impacting the target plate are shown in Figure25. As shown, the impact stress value from the layered structure was the smallest, and the values from the other three equivalent ice ball models were basically the same. The reason for this was that the overall deformation of the target plate caused by the impact of the other three equivalent ice ball models was relatively large, while the local deformation of the target plate caused by the impact of the layered structure was relatively large. Therefore, the equivalent stress on the back of the target plate was the largest, and the stress fluctuation range was large due to the rapid rebound of the target plate.Figure 25
Maximum effective stress values produced by the impacts of the four types of ice ball.


## 5.5. Maximum Effective Plastic Strain in the Center of the Target Plate

The maximum effective plastic strain values produced by the four types of ice ball model impacting the target plate are shown in Figure26. Each impact led to a plastic state in the impacted area of the target plate. Among the models, the plastic strain in the target plate caused by the impact of the layered ice ball was the smallest, indicating that the impact energy loss of the layered ice ball was large and was not completely passed to the target plate. The plastic strain values caused by the impact of the two equivalent ice ball models with equal diameter were basically the same. The plastic strain caused by the impact of the type-D ice ball, which had the smallest diameter, was relatively small. At the same time, considering the simulation results shown in Figure 19, it can be seen that the center and overall failure of the ice ball with the small diameter and high density were the fastest under the impact. As such, the plastic strain value of the impact center was reduced overall. This shows that the geometric size and density of the ice ball have a certain influence on the local deformation and plastic strain state of the target plate.Figure 26
Maximum effective plastic strain produced by the impacts of the four types of ice ball.


## 5.6. Energy Absorbed by Target Plate

The energy absorbed by the target plate from the impacts of the four types of ice ball models is shown in Figure27. As shown, each impact causes the target plate to absorb energy. Among them, the layered-structure ice ball caused the least energy to be absorbed when impacting the target plate, while the type-B ice ball had the largest mass, so the energy transmitted to the target plate was also the highest. The type-C and type-D ice balls had the same initial kinetic energy, but their impact energies were relatively concentrated because the diameter of the type-D ice ball was slightly smaller. Therefore, the overall energy conversion degree caused by the impact was slightly higher than that of the type-C ice ball.Figure 27
Energy absorbed by the target plate from the four types of ice ball during impact.


## 5.7. Summary

To sum up, during the generation of hail, different freezing methods lead to different densities of hail layers. Between these different layers, due to the different wave impedances, which are determined by the ice density and stress-wave velocity of each layer, the stress wave generated by instantaneous high-speed impact reflects and transmits when passing through the layers. This results in additional energy loss compared with the impact of a homogeneous ice ball. The energy loss is shown in the reduction of the dynamic response level of the Ti6Al4V target in the numerical simulations. The next research focus will be divided into two parts: first is preparing real layered hail with concentric spherical shells for impact-test verifications, and second, in the concentric spherical-shell finite-element hail model, analysis work is planned that will use cohesive interlayer transition elements to reveal a more accurate interlayer impact energy transmission mode and mechanism.


## 6. Conclusion

In this paper, a modeling method for spherically layered hail was examined, and numerical simulations of the impact with a Ti6Al4V target were carried out with this and equivalent hail models with equal diameter and equal mass. Before getting the conclusion, it is still necessary to explain some simplifications and approximations in the research. These problems include the following: using a sphere to characterize the shape of natural hail and using tension-compression equation of state with strain rate to approximately characterize the flow mechanical properties of low-specific-gravity ice, although it can be seen from the numerical simulation results that frozen ice balls are more suitable to use this kind of equation of state. There is still room to further improve the accuracy of the simulation results of specific-gravity ice by using other equations of state. These works will be paid attention to and deeply studied in the future.The following conclusions were obtained.(1)
The test results indicated that the frozen high-specific-gravity ice ball could maintain a high impact load in a certain period of time, while the low-specific-gravity ice ball could only maintain a low level of impact load in a certain period of time due to its rapid failure and collapse; that is to say, the impact load of the ice ball is inversely proportional to its specific gravity(2)
A layered hail model can be produced for numerical simulations using general finite-element software, and corresponding results were obtained. The layered hail model and its equivalent hail models had similar impact processes and failure modes when impacting a target plate(3)
In the lower-order response eigenvalues, such as the impact load and impact displacement of the target plate, the response value of the equivalent hail model with uniform specific gravity was greater than that of the layered hail model, and the basic change trend was directly proportional to the equivalent specific gravity and geometric size of the model(4)
In the higher-order response eigenvalues of target stress and strain, the responses of the four hail models to impacting the target were coupled with the hail size and the macroscopic and local deformation state of the target, resulting in a more complex dynamic response(5)
The reason for the lower response level of the Ti6Al4V target when impacted by the layered hail model is that the wave impedance of each layer, which is determined by its density and stress-wave velocity, is different. The stress wave generated by the instantaneous high-speed impact reflects and transmits when passing through the hail layers, resulting in the loss of additional energy when compared with the impact of a homogeneous ice ball(6)
By replacing the approximate parameters in this model with the verified dynamic mechanical performance parameters of ice with different specific gravities, a more accurate layered hail model can be obtained, and a quasireal hail modeling and verification method reflecting the interactions between different layers needs to be developed



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*Source: 1806623-2022-06-17.xml*