## Abstract

This is a theoretical study about ice particle impact onto a rigid wall. It is motivated by the need to model the process of ice crystal accretion or damage caused by an ice particle impacts. A quasi-one-dimensional model of ice particle impact and deformation is developed. Spherical, cylindrical and conical shapes of the ice crystals are analysed. The model is able to predict particle residual height, the force produced by impact and the collision duration. The theoretical predictions agree well with the available experimental data.

## 1. Introduction

The reason for interest to better understand ice particle impact is mainly in the attempt to model and predict potential damage which such impact can cause on solid structures. It is important for ship building and the design of the aircraft, arctic and space research. Moreover, ice crystal impact in hot environments, e.g. in aircraft engines or on heated measurement instruments can lead to ice accretion. Melting and subsequent shedding of the accumulated ice layer can result in even greater damage, e.g. impact onto aircraft compressor stages.

Experimental investigation of impact in laboratories using artificial, simulated ice particles is the main source of the knowledge about mechanisms of particle deformation and break-up [1]. The impact is observed using high-speed video systems. Many studies are focused on characterization of the kinematics of the post-impact fragments [2–5]. In particular, the restitution coefficient of the ice particles is measured [6,7]. Images of a high-speed hail ice impact showing details of its deformation and fragmentation, captured using a high-speed video system, can be found in [8]. One such time sequence is shown in figure 1. Similar images of cylindrical ice crystals or crystals with a conical tip shape can be found in [9]. Moreover, the time evolution of the force produced by impact has been measured by Combescure *et al.* [9]. The knowledge about the magnitude and duration of this force can help evaluate the damage produced by ice hail collision on various materials, in particular, composite plates widely used in the aircraft industry.

A full three-dimensional computation of drop deformation and break-up [8,10–12] after impact onto composite and rigid substrates provides more details about the velocity and stress fields in the impacting particle and to evaluate different fracture models. In [13], the crater formation in ice by a solid projectile is computed. Impact of long ice cylinders onto a rigid wall is investigated numerically in [14]. In a numerical study [15], an impact of a brittle sphere is simulated. The mechanisms of the sphere fragmentation is described, which is very similar to the processes of ice hail break-up.

Accurate numerical simulations of an impact of a single ice particle requires extensive computational resources. This is why modelling of ice crystals accretion [16–18] or material damage by collision with a polydisperse ice crystal cloud requires simplified analytical models or semi-empirical correlations. Not many such analytical models can be found for ice impact description. It is not surprising that many ideas for such modelling appear first in the fields of penetration mechanics or transport of granular materials, where penetration, deformation, erosion or fragmentation of impactors and targets are often described using hydrodynamic theories [19–22].

Ice crystal impact is a rather complicated process which includes several typical stages, characterized by different time and length scales.

Typical stages of ice particle impact are

— initial moment of particle collision, which is characterized by the development of a shock wave in both the particle and target; the duration of this stage is much shorter than the total duration of the collision. The dynamics of the shock wave is expressed by the Rankine–Hugoniot jump conditions;

— elastic deformation of both particle and target, which is described using the Herztian theory [23,24]. It is relevant only for relatively small impact velocities and will not be considered here in detail because the elastic effects are assumed to be small in comparison to the plastic stresses; and

— plastic deformation of the particle when the stress at the contact with the target reaches the yield pressure. During this stage, as an ice crystal is a semi-brittle material [25], the particle consists of a crushed zone near the target surface and a cracked zone, where relatively large fragments can be created. The energy dissipated in the plastic region goes on the heating of the material and on its fragmentation, namely on creation of the new free surfaces of the fragments.

The main objective of the present work is a theoretical investigation of particle collision with a dry substrate, leading to the particle deformation and crushing. The model is based on the assumption that the thickness of the crushed region near the wall is much smaller than the particle initial size. The equations of motion of the rear part of the particle are formulated, based on the assumption that the pressure at the wall is approximately equal to the nearly constant compressive failure stress of ice. The model predicts the evolution of the particle height, duration of the deformation process and the force generated by the particle on a rigid wall. The agreement between the theoretical predictions and experimental data for ice particles of characteristic sizes larger than 20 mm is rather good. As the model is based on the first principles, it can be potentially used also for the description of impacts of submillimeter ice particles, relevant to the case of ice accretion.

## 2. Deformation and crushing of a rigid/semi-brittle particle impacting onto a perfectly rigid wall

### (a) Spherical impactor

Consider the impact of a particle of radius *R*_{0} onto a perfectly rigid wall. The impact velocity is denoted by *U*_{0}. The impact of a particle onto a rigid wall is analogous to an impact of a long plastic rod of Taylor [19], who described the rod deformation by a propagating plastic wave, where the pressure is equal to the compressive failure stress *Y* . The rod consists of the rear part, which moves as a rigid body, and a stationary plastic region near the wall. The main rod deformation occurs at the plastic wavefront.

We apply such a model to the description of a spherical particle. The theory in [19] can be significantly simplified for impact of a semi-brittle ice crystal. In this case the plastic region, characterized by intensive particle deformation, is rather thin. This thin plastic layer separates the rear part of the particle, where the deformation is small, from the thin region near the rigid wall of a compressed crushed crystal material. The pressure in this region leads to a fast radial ejection of fine ice particles along the target. The rear part of the sphere continue to move as a rigid body. Its shape is approximated by a truncated sphere. The solution is thus reduced to the theory of Andrews [26], which is presented below with some modifications, allowing to describe larger particle deformations. We are interested in describing high impact velocity of crystals leading to the deformation significantly exceeding the elastic strains, and eventually to crystal break-up. In these cases, the elastic effects are negligibly small and are not considered in the present analysis. Moreover, the momentum in the axial direction of the crystal flow in the thin plastic and crushed regions is also neglected.

Main geometrical parameters of the crushing particle are defined in figure 2. Denote *b*(*t*)=*R*_{0}[1−*δ*(*t*)] as the distance of the particle centre from the undeformed surface of the wall and *u*(*t*) as its instantaneous velocity. Here *δ*(*t*) is a dimensionless displacement of the particle centre after first instant of impact, scaled by the initial particle radius. The expressions for the radius of the impression *a*(*δ*) and the volume of the rear solid part of the particle *V* (*δ*) can be derived from geometrical considerations, assuming that the thickness of the plastic and the crushed zones (shown in figure 3) is much smaller than the sphere radius

The force applied to the particle from the wall can be estimated as *πa*^{2}*Y* , where *Y* is the yield strength of the material. The equation of motion of the particle is thus determined by the axial particle momentum

The instantaneous velocity of the particle is defined as *u*≡−*db*/*dt*=*R*_{0}*dδ*/*dt*. Differential equation (2.3) can be thus rewritten for *u* with the help of (2.1) and (2.2)

The solution of this equation in dimensionless form
*Y* is constant.

The maximum centre displacement, *s*=0, is the solution of the equation
*δ*<2 for any positive *s*_{0}. The solution for *a*, obtained with the help of (2.1), are shown in figure 4*a*.

Linearizing (2.7) yields an approximate solution
*s*_{0}<2 (figure 4*a*).

The evolution of the centre displacement in time is obtained by integrating (2.6) in the following form
*ξ* is a dummy variable.

The expressions for the residual maximum impression radius, *a*_{res}, after collision, and the total duration of impact,

The impression radius *a*_{res} (shown in figure 4*b* as a function of *s*_{0}) and the material properties of the particle determine the fracture of the ice particle, in particular, the compressive failure stress *Y* and the fracture toughness *K*_{c}, determine the fracture of the ice caused by the cracks development. Finally, the relative volume of the crushed part of the particle is

### (b) Cylinder impact onto a rigid wall

The case of cylinder impact onto a rigid wall is analogous to a sphere. As only the geometrical relations are different but the physical idea is the same, we dispense with the derivation of equations and present only the resulting expressions
*L* and *L*_{0} are the instantaneous and initial cylinder lengths. The residual impactor length and the typical deformation duration are

### (c) Impact of a conical particle

Consider the case of a normal, axisymmetric impact of a conical particle of the initial volume *V* _{0}. The dimensionless particle velocity
*α* is the half angle. The maximum dimensionless particle displacement and the approximate total time of collision, obtained from (2.15), are

## 3. Results and discussion

It is known that the value of the yield stress of ice depends on the strain rate. For low values of the strain rate, the value *Y* _{stat}=5.2 MPa was obtained in [8]. We define a characteristic velocity ^{−1}, noting that the ice density is *ρ*=920 kg m^{−3}.

In order to estimate the average compressive failure stress at higher impact velocities we roughly estimate the value of the force applied on the wall by the impacting sphere, which is the ratio of the initial momentum of the particle *MU*_{0} to the total duration of the plastic deformation *M* is the initial mass of the particle. The expression for the force is obtained with the help of (2.11)

The expression for the effective compressive failure stress, obtained by fitting the data from [8] is

The comparison of the estimated values for *Y* to that from experiments are shown in figure 5*a*. The range of the estimated compressive failure stress is the same as calculated in [8]. From these data it is not clear how to determine the average compressive failure stress for particles with different diameters. The value of *Y* is usually determined as a function of a strain rate *h*_{def} of the deformation layer can be estimated from the balance of inertia of the particle material and stresses in the deformation layer, which leads to the relation *ρu*^{2}/*t*∼*Y*/*h*_{def}. Taken the typical deformation time from (2.11) yields

The dependence of the estimated values for *Y* on the characteristic strain rates, determined in (3.3) is shown in figure 5*b*. The scatter of the data is not surprising. Similar scatter was reported also in [8]. It can be explained by the not perfectly spherical shape of the particle and by the possible dependence of the failure stress on time due to the development of the cracks in the deforming material. It should be noted that the data for *Y* is estimated for the ice temperature −10°C. These values can significantly depend on the temperature.

In figure 6, the theoretically predicted and measured [8] values for the peak force are compared. Perfect agreement corresponds to the dashed line. The agreement indicates that the influence of the particle size on the value of the effective compressive failure stress is rather small.

The value of the peak force can significantly depend on the shape of the particle [28]. The expressions for the maximum force generated by a cylindrical particle of a length *L*_{0} and by a cone of the initial volume *V* _{0} are obtained using the equations for the collision time (2.14) and (2.16), respectively

In figure 7, the experimentally measured values of the peak force generated by the ice particles of various nose shapes [28] are compared with the theoretical predictions by (3.4). The velocity of the impacting particle is approximately *U*_{0}=62 m s^{−1}. The agreement between the predictions and the experimental data is rather good.

In figure 8, the experimental data for the instant corresponding to the measured peak force [8,29] are compared with the predicted duration of impact,

In figure 9, the theoretically predicted evolution of the ice particle height is compared with the experimental data [8,9] for cylindrical and spherical ice particles. The agreement is rather good for the value of the compressive failure stress *Y* =10 MPa, corresponding to *U*_{0}=60 m s^{−1}.

## 4. Conclusion

In this study, a quasi-one-dimensional model of ice particle impact onto a perfectly rigid wall is developed. The model is based on the assumption that the material of the ice crystal is semi-brittle and the value of the compressive failure stress is assumed constant during impact.

The value of the effective compressive failure stress is estimated from the experiments as a function of the scaled impact velocity. The obtained ice-hardening behaviour agrees with the existing knowledge about the material properties of ice.

The model is able to predict the total duration of the collision, the magnitude of the force generated by particle impact onto the wall, evolution of the impacting particle height in time and its residual height. The theoretical predictions agree well with the existing experimental data.

## Author' contributions

I.V.R. developed the theoretical model. C.T. has critically contributed to the interpretation of the results. Both authors have written and revised the manuscript.

## Competing interests

We have no competing interests.

## Funding

The research leading to these results has received funding from the European Union Seventh Framework Programme FP7/2007-2013 under grant agreement n°ACP2-GA-2012-314314 (HAIC - High Altitude Ice Crystals).

## Acknowledgements

The authors thank Tobias Hauk from Airbus Group Innovations (Munich) for motivating discussions.

- Received July 30, 2015.
- Accepted October 22, 2015.

- © 2015 The Author(s)