Introduction
Erosion impacts are a widely studied phenomena, the understanding of which has yielded valuable insights into the physical mechanisms at play, which has in turn allowed for the improvement of technologies which either rely on preventing erosion based damage or utilizing the resulting energies. The study of erosion induced fracture, particularly for small indenter sizes, has however often been seen as at best a secondary concern, despite certain technologies being notably susceptible (e.g. parametric resonance screens [1] ). Additionally, conventional models often only examine prescribed indenter shapes, potentially meaning that important effects may be overlooked.
Introduction
The need for differing models for small sized indenters became apparent during experimental research examining erosion fracture, through studies into the threshold velocities of impacting particles [2, 3] . These investigations revealed an interesting phenomenon, namely that values of the threshold velocity for particles with a relatively large radius are approximately the same, whereas reducing the radius leads to a significant increase in the threshold velocities. It was concluded that, as the impact duration primarily depends on the particle size, the set of threshold velocity values have to be divided into two branches: one static, for large-size particles, and the other dynamic, for small-sized indenters. However the conventional approach when studying the strength problem only permits one to calculate values of the threshold velocities by applying the critical stress criterion, and as such can only explain the static branch [4] .
Introduction
A more effective analytical model, capable of explaining the nature of origin for both the static and dynamic branches, was developed in [5] . The initial model provided the solution of the contact problem for a spherical particle impacting an elastic half-space, with the approach later being generalized for particles of different shapes [6, 7] . These approaches looked at the problem of erosion induced fracture from an energy costs point of view. In particular they examined the threshold fracture energy, which is the kinetic energy required to induce erosion based fracture. This was made possible by utilizing an incubation time based fracture criterion to estimate the parameters of the tensile stress pulses which characterize erosion fracture.
Introduction
As these models developed however, it soon became clear that there was an unexplained ramoswilliam@example.com. In [6] , which utilized the Hertz solutions for erosion impact, it was observed that there existed a minimum threshold fracture energy in the case of a spherical, but not cylindrical, indenter. The subsequent investigation [7] , which utilized the Shtaerman-Kilchevsky theory of quasi-static blunt impact for an indenter whose arbitrary shape function: z = B 1−λ r λ describes a set of smooth paraboloids, observed that there existed a critical shape parameter λ * = 5.5, with indenters characterized by λ < λ * having a unique minimum threshold fracture energy, while no such minimum existed when λ > λ * . However neither paper was able to provide a physical explanation for this critical value. A later paper attempted to unearth clues by improving on the accuracy of the previous approach, achieved by examining fixed spacial points for which the stress function was maximized [8] , however this was similarly unsuccessful in explaining the value of the critical shape parameter.
Introduction
While these previous models successfully incorporated various dynamic aspects into the modeling, notable dynamic effects where ignored. For example, the existence of a supersonic stage, where the speed of the contact zone boundary exceeds the elastic wave propagation velocity in the condensed matter, was shown in [9, 10] . Borodich adduced the proof of this statement for the convex contact surfaces based on geometrical principles in [11] . It is noteworthy that only the quasi-static approaches of Hertz and Shtaerman for the contact problem were used during the previous analysis.
Introduction
In this paper we attempt to show that the mystery surrounding this parameter was due to a failure of previous approaches to take into account the dynamic properties of the elastic medium. Towards this end, we will examine the threshold fracture energy, as well as the critical shape parameter λ * , in more detail by incorporating supersonic modeling of the initial stages of the impact. We consider the supersonic stage in a similar way to that applied by Argatov for a spherical indenter in [12] . Utilizing the general solution proposed by Borodich [11] we create a model for which the maximal impact stress and threshold fracture energy are more sensitive to the physical characteristics of the impact. This will notably ensure that the impact characteristics take account of the wave velocity within the impacted medium. The choice of Borodich's solution is stipulated by the fact that it describes only the supersonic stage and is based on the common principles which have historically been used by other researchers [9, 10, [13] [14] [15] . It should be also noted that both theories neglect the effect of friction at the contact interface, which does play a role, especially for oblique impacts [16] . This simplifying assumption is often made in the literature [17] in the case of normal impact incidence, and as this paper is primarily aiming to examine the viability and effects of such an approach only the simplest form of each regime will be examined.
Introduction
The The general solution of the contact problem during the supersonic stage was presented by Borodich [11] , and allows one to obtain the temporal dependencies of both the contact force and the penetration depth. General formulas for an arbitrary convex shaped indenter were presented, with specific shapes in the form of a cone, elliptic paraboloid and a pyramid being evaluated. We use the results from [11] to find the mechanical parameters of the process in the supersonic regime for a axisymmetric blunt indenter, defined by the shape function:
Introduction
where λ > 1 and B is a constant having dimension [length] . Then the value S(w) of contact area is related to the penetration depth w as:
Introduction
In an accordance with Borodich solution, the time dependence for the penetration depth w can be implicitly calculated as follows:
Introduction
where V 0 and m are initial velocity and mass of the projectile respectively, ρ is elastic media density and c is the velocity of the longitudinal waves. Some simple algebraic transformations of (3) yield:
Introduction
Thus we know the function F (y, µ), which implicitly describes the relationship between the penetration depth w and the time t
Introduction
It should be noted that F (y, µ) is a monotonic increasing function over 0 < y < 1 for every 1 < µ < 3, and its value increases from zero to infinity. It is however possible to determine the inverse function, F −1 , which allows us to obtain the particular solution for a considered projectile shape.
Introduction
As a result, the supersonic indentation of a blunt indenter into the elastic halfplane is defined by the following relationship (5):
Introduction
while the other problem parameters: the indenter velocity, V d (t), and total force, P d (t), are given by the following relations:
Introduction
The end of the supersonic stage, denoted t * , is that moment in time when the outward velocity of the contact area becomes equal to velocity of the longitudinal waves c. The value of t * can be calculated as the solution to the equatioṅ
Introduction
where the longitudinal wave speed is that of a one-dimensional solid, namely:
Introduction
equation (8) can be written in the form:
Introduction
Thus, one can now evaluate the time t * and calculate boundary values of the penetration depth w * = w d (t * ), contact radius, a * = a d (t * ), and the projectile velocity
Introduction
Moreover, one can observe that t * → 0 as ρ → 0. Unfortunately there are no exact solutions for the stress distribution during the supersonic stage, only the normal stresses for the central point r = 0 of projectile
Subsonic stage. Shtaerman-Kilchevsky theory
During the subsonic stage (t > t * ) we utilize the Shtaerman-Kilchevsky theory, which has previously been applied to the case of quasi-static blunt impact by Argatov, Mishuris, Petrov [7] . The relationship between the contact force P qs (t) and penetration depth w qs (t) can be written as:
Subsonic stage. Shtaerman-Kilchevsky theory
where
Subsonic stage. Shtaerman-Kilchevsky theory
here E and ν are the Young modulus and Poisson ratio for the elastic media, Γ(x) is a Gamma function. The equation of motion for a projectile with mass m can be written as follows:
Subsonic stage. Shtaerman-Kilchevsky theory
The solution obtained for the supersonic stage gives following initial conditions [7] :
Subsonic stage. Shtaerman-Kilchevsky theory
Multiplying both sides of equation (12) by dw dt before integrating, we obtain:
Subsonic stage. Shtaerman-Kilchevsky theory
The maximum penetration depth w 0 can be calculated by substituting V qs = 0 into equation (14) w 0 =
Subsonic stage. Shtaerman-Kilchevsky theory
A second integration of (14) gives us the the temporal dependence of the penetration depth w qs (t) during the subsonic stage
Subsonic stage. Shtaerman-Kilchevsky theory
where
Subsonic stage. Shtaerman-Kilchevsky theory
Thus, the estimated time, t qs , when the penetration reaches its maximum value w 0 is as follows
Subsonic stage. Shtaerman-Kilchevsky theory
where t 0 is time duration of the load stage. Using (15) this can transformed into:
Subsonic stage. Shtaerman-Kilchevsky theory
Since damping occurs, due to the quasi static nature of the formulation, the duration of the unload stage t * * − t 0 exaggerates the value t 0 . For this reason we have to solve the equation (12) one more time with the new initial conditions:
Subsonic stage. Shtaerman-Kilchevsky theory
The first integration give us the projectile velocity in the reverse direction:
Subsonic stage. Shtaerman-Kilchevsky theory
Thus, the value of t * * can be determine from the following expression:
Subsonic stage. Shtaerman-Kilchevsky theory
Note that t * * − t 0 > t 0 . Indeed, the latter is equivalent to
Coupling of the solutions for supersonic and subsonic stages
Because of the peculiarities of the dynamic solution by Borodrich and quasi-static solution by Shtaerman, the function which describes the temporal dependence of the contact area radius will be discontinuous at the time t * . This is due to the fact that the supersonic and subsonic stages have different physical principles determining the relationship between penetration depth and contact area, and the penetration depth w(t) is continuous in this model (by design). In order to avoid this irregularity, let us introduce the smooth function χ(x), which continuously matches the two solutions within the interval [0.95t
Coupling of the solutions for supersonic and subsonic stages
Also, we employ this function χ(x) to sew the contact force P (t), the equation of which also has a discontinuity at the point t * .
Coupling of the solutions for supersonic and subsonic stages
Additionally the mean contact area, which allows us to calculate the stress field under the projectile, is defined as p 0 = P/(πa 2 ) and can be approximated in a similar way:
Coupling of the solutions for supersonic and subsonic stages
Since Borodich's solution does not contain any information about the surface stresses, we enrich the quasi-static solution by extrapolating the corresponding results into the supersonic stage. Thus, the distribution of the contact pressure beneath an indenter with the considered shape is given by
Coupling of the solutions for supersonic and subsonic stages
where r 0 = r 0 (r, t) = r/a(t) is the dimensionless radial coordinate. Similarly the radial stress will be approximated by
Coupling of the solutions for supersonic and subsonic stages
The maximum radial stresses are achieved at the edge of the contact area, where r 0 = 1
Coupling of the solutions for supersonic and subsonic stages
As a result, at any time during the penetration process, the approximated solution to the problem will be given by the formulas (25),(26),(28). Note that all of these representations are exact near the ends of the time interval. Indeed, for very small times, Borodich's formulas give the exact solution for the supersonic regime, while the Shtaerman solution precisely describes the projectile behavior during the quasistatic stage.
Estimation of the energy costs for the fracture initiation

Incubation time fracture criterion
The previous analysis [7] allows us to produce a lower bound on the fracture pulse of the tensile stresses. Additionally, in [8] , it was shown that instead considering the stress pulses for fixed points on the material surface doesn't qualitatively alter the final behavior of the threshold energy in relation to the impact duration. Hence, in the present paper, we employ the lower estimate of the fracture pulses with a view to avoiding additional computational difficulties. The threshold amplitudes of pulses caused by particle impacts are evaluated using an incubation time approach, which can be effectively applied to explain a number of general dynamic fracture effects caused both by high strain rate and short pulse loading [5, 18, 19] . The corresponding fracture criterion, employed to determine the threshold pulse parameters, is referred to as the incubation time based fracture criterion. In its simplest form, for a brittle fracture, this criterion can be written as follows:
Incubation time fracture criterion
where σ c is the static strength of the material and τ is the incubation time of the fracture. This criterion has been successfully used to model a multitude of fracture problems, and has proved itself to be a quite simple and effective method of fracture prediction. It permits us to determine the necessary conditions required for fracture over a wide range of impacts.
Estimation of the energy costs for the fracture initiation

Description of the algorithm
The algorithm used to calculate the threshold fracture energy for a fixed value of λ utilizes the fact that the stress function at the fracture front is a positively valued function, which monotonically increases with increasing v 0 . As such we define the following function:
Description of the algorithm
which has the property that:
Description of the algorithm
For any arbitrary δ > 0. As such a simple iterative procedure can be utilized in order to obtain v * 0 to a desired level of accuracy.
Description of the algorithm
A second concern is to ensue consistency of results between this paper and the previous investigation [7] . In order to achieve this the constants B and m will have to be chosen in such a way as to coincide as ρ → 0. This is completed in the first step of the algorithm, and as such the first step could be discarded when carrying out a more general investigation.
Description of the algorithm
With this in place, the algorithm is defined as follows:
Description of the algorithm
(1) Choose an arbitrary t p , which will approximate t 0 . From this calculate the constants B and m using equations (50) and (57) in [7] . Additionally compute an initial expected value of the initial velocity required for fracture initiation, v p , using equation (56) in the aforementioned paper.
Description of the algorithm
Note: While the value t p → t 0 as ρ → 0, the two will not be related in this formulation. It is simply an arbitrary starting point chosen to ensure the results are consistent with previous investigations, and is discarded after this step.
Description of the algorithm
(2) Using the value v p create an initial interval in which v * 0 , the initial velocity required for fracture initiation, is expected to occur (e.g. 
Results for fixed λ
The material constants used in calculations are those for gabbro-diabase [20] , which is the same as used in previous investigations of this phenomena [7, 8] . The values are as follows: E = 6.2 × 10 9 N/m 2 , ν = 0.26, σ c = 44.04 × 10 6 N/m 2 , τ = 44 µs, while the density of the indenter is given by: ρ 1 = 2400 kg/m 3 . Results are displayed in Fig. 1 , for λ = {2, 3, 4, 5}. It is clear from these results that the threshold fracture energy decreases as the material density ρ increases, however the effect is only apparent for impacts with short durations. This makes intuitive sense, given that v 0 decreases monotonically as the impact duration is increased, and as such the supersonic stage will play a far greater role in rapid impacts.
Results for fixed λ
Additionally, it is immediately apparent from the results in Fig. 1 that the critical shape parameter λ * will decrease when eroding a more dense material. Furthermore, given that the threshold fracture energy seems to tend to zero for all λ in these graphs, it is safe to assume that the critical shape parameter will have λ * → 1 rapidly as the material density is increased.
3.3.
Critical shape parameter λ * for various material densities
Obtaining the critical shape parameter
In previous investigations [7, 8] it was shown that there existed a critical shape parameter, λ * , with values λ < λ * exhibiting a unique non-zero threshold fracture energy (the initial energy required for fracture initiation), while no such minimum existed for λ > λ * . These studies however assumed a zero-valued density of the impacted medium, and could give no physical explanation for the obtained value λ * = 5.5. Here we will attempt to expand on this work, by examining whether the critical shape parameter varies with the density of the impacted material, and as such a physical explanation of the phenomena can be postulated.
Obtaining the critical shape parameter
The method of obtaining the critical shape parameter λ * relies on noting the behaviour of the threshold fracture energy for very small impact durations t * * . As demonstrated in Fig. 2 , when λ < λ * , the derivative of the threshold fracture energy ϵ 0 will be negative, while this derivative will be positive for λ > λ * and zero for the critical shape parameter. As such the algorithm is designed to obtain the sign of this derivative near the origin for a range of λ, before using the results to iteratively work towards the desired critical shape parameter. A full explanation of the algorithm is given below. • Arbitrary initial points 0 < t 1 p < t 2 p are taken 1 . These approximate t 0 as in the previous algorithm, and must be sufficiently small to predict the behaviour of the derivative of ϵ 0 as t * * → 0.
Obtaining the critical shape parameter
• The threshold fracture energy, ϵ 0 , is obtained, for each value of
Obtaining the critical shape parameter
, using the algorithm outlined in Sect. 3.2.1, to a required level of accuracy.
Obtaining the critical shape parameter
• The derivative of ϵ 0 , with respect to t * * , is approximated between t 1 p and t 2 p . (3) With the value of the derivative obtained for each λ ∈ λ p , a new interval in which λ * exists for the given ρ is obtained. (4) A new λ p is defined using the obtained interval for λ * , and steps 2-3 are repeated until a desired level of accuracy is reached.
Obtaining the critical shape parameter
This algorithm allows one to obtain the value of λ * for a given material density ρ, and as such by repeating the we can begin to study the relationship between the impacted material density and the threshold fracture energy. In practice we have found that it is often simpler to instead begin with a value λ, and then iterate to discover the density ρ * for which it is the critical shape parameter.
Results for various impacted material densities
Numerical simulations were conducted using the same values for the impact parameters as in Sect. 3.2.2, with the addition of taking t p = {2, 2.5} × 10 −6 seconds. The results obtained are shown in Fig. 3 .
Results for various impacted material densities
It is clear from this that the critical shape parameter λ * decreases when impacting materials with higher densities. There is however a clear divergence from previous results which must be expanded upon, which isn't made obvious in the above figure. In previous investigations, it has been the case that taking λ = λ * will yield a non-unique minimum energy required for fracture. However this will not always be the case under the revised scheme. In order to explain this further, we define the following:
Results for various impacted material densities
An example of the threshold fracture energy, evaluated taking ρ = ρ * when λ = 3 is given in Fig. 4 . Figure 4 . Value of the threshold fracture energy ϵ 0 , evaluated when λ = 3 for ρ = 0 (blue line), ρ = ρ * (black crosses) and ρ = ρ * × 1 ± 0.05 (green crosses, red circles). It is clear that a unique minimum occurs for both ρ ≤ ρ * , however we will have ϵ 0 → 0 for ρ > ρ * .
Results for various impacted material densities
It is clear from this that the minimum energy required for fracture is not unique, however as any increase in the impacted material density results in having ϵ 0 → 0 as May 20, 2016 Philosophical Magazine Erosion˙dip9˙FinalwTemp t * * → 0 it is the highest value of λ for which a minimum threshold fracture energy exists. As a result, while this value of λ * still maintains its most important quality, that of defining the point when a change in case between having a unique threshold fracture energy and none existing occurs, it is of a weaker form than described in previous papers.
Discussion and Conclusions
An approach for calculating the threshold fracture energy during erosion impacts with supersonic stages has been derived, and algorithms for obtaining numerical results have been implemented. The results indicate that the critical shape parameter, λ * , above which no minimum energy for fracture initiation exists, decreases rapidly as the density of the impacted material increases. This provides evidence to support the proposal that the critical value of the shape parameter λ * = 5.5 only emerged in previous modeling due to the fact that the models were only based on the quasi-static approach to contact problem. These results resolve the unanswered questions from previous investigation into this phenomena, showing the dynamic nature of this parameter, and that the value of λ * depends on the supersonic stage duration determined by the geometry of the contact surfaces and the density of media.
Discussion and Conclusions
It should be noted that there are possible variations to the approach suggested in this paper, namely as a result of the hypotheses used when gluing together the dynamic and static stages of the contact problem. However there is every reason to believe that they are not meaningful and that the main result would be the same. Also the provided reasoning shows that considering the case of a spherical shaped indenter is more convenient and is naturally supported when constructing such models. The smooth shape of the contact surfaces allows one to avoid irregularities in the solution and unnatural behavior of the process characteristics. the some tending to infinity while others tend to zero, inevitably resulting in a loss of accuracy, particularly during the supersonic stage. This provides not just an additional reason for accepting a lower bound on the value of t 1 p , but also an additional constraint which should be applied to t 2 p . Namely, in order to ensure that inaccuracies in the numerical results don't adversely effect our estimation of the gradient, we must have that the interval t p is sufficiently large to ensure a significant difference in the values of the threshold fracture energy at either end of the interval. This will decrease the tightness of the lower bound obtained on the value of λ * , but must be done to ensure reliable results.
Discussion and Conclusions
Combining the above with the results of numerical experiments it was found that, when maintaining t 1 p as a constant rather than as a function of λ, the best balance between accuracy and ensuring physically meaningful results was obtained when taking t 1 p ≈ 2 × 10 −6 , t 2 p ≈ 2.5 × 10 −6 seconds.

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