## 1 Introduction
[24], this method seems to be adapted to a high-velocity impact such as hail or bird strikes, widely used in the wind or ballistic industries despite the discovery of issues related to a smoothing function and material interfaces [25]. Additionally, [PERSON] et al. [24] explain that the Lagrangian-Eulerian method uses the Lagrangian method to mesh the body while using the Eulerian method to mesh the projectile. The Eulerian mesh allows the projectile to continuously adopt the mesh while deforming, allowing it to remain uniform when undergoing large deformations. There exists a further version of the Lagrangian-Eulerian method called the Arbitrary Lagrangian-Eulerian approach (ALE), where an arbitrary motion of the body optimises the shape of the meshing elements. Also, [PERSON] and [PERSON] [26] argue that the use of FEA implementation of peridynamics may help avoid many of the well-known difficulties in describing the interface between a penetrator and a target when using conventional finite elements. In particular, the use of truss elements to describe the target is expected to reduce or eliminate problems with mesh tangling and distortion. In the current study, initial evaluations were undertaken with a representative (typical) 2.0 mm diameter droplet, assuming a frictionless surface. This work then investigated the effect of a friction coefficient varying from frictionless (zero friction coefficient) to a maximum possible friction coefficient of 0.5. In addition, this work also investigates the effect of contact and droplet size, which has not been investigated previously in detail. Theoretical equations provide a solid framework for validating the numerical results. One of the most useful equations is the water hammer pressure, as described by several authors, including [PERSON] et al. [5]. The original waterhammer equation estimated rain droplet impact pressure, assuming a one-dimensional impact on a rigid body with constant speed of sound and density [27]. Later, the modified waterhammer equation improved estimates for wind-turbine blade erosion by considering both water and solid properties. Figure 2: **(a)** Stress wave behaviour under impingement on the multilayer material and (b) corresponding numerical simulation [22; 23]. However, droplet impacts are more complex, involving compressive, shear, and Rayleigh waves, as well as wave reflections--all influenced by the impact conditions and material properties [27]. Additionally, after the compressive wave passes the contact edge, water jetting (lateral outflow) occurs, causing decompression and reducing initial pressures by up to an order of magnitude [28]. Among early works, a simplified 1-D WH equation was the most widely used to approximate the peak pressure during impacts when only pure elastic material characterisation is assumed. It is used in this work to derive an appropriate analytical compressive stress history development. This equation can be used to estimate the pressure peak caused by a liquid droplet impacting a flat surface. The modified water hammer pressure considers more parameters than the original version, incorporating various properties for liquid and solid components. (a) Water hammer pressure:  
\[P=\rho_{0}c_{0}V_{0} \tag{1}\]  
With:  
\(\rho_{0}\): Density of water (kg/m\({}^{3}\))  
\(c_{0}\) : Speed of sound in water (m/s)  
\(V_{0}\) : Velocity of the fluid (m/s)  
(b) Modified water hammer pressure:  
\[P=\frac{V\rho_{l}c_{l}\rho_{s}c_{s}}{\rho_{l}c_{l}+\rho_{s}c_{s}} \tag{2}\]  
With:  
\(\rho_{l}\) : Density of liquid (kg/m\({}^{3}\))  
\(c_{l}\) : Speed of sound in liquid (m/s)  
\(\rho_{s}\) : Density of solid (kg/m\({}^{3}\))  
\(c_{s}\) : Speed of sound in solid (m/s)  
\(V\): Velocity of the fluid (m/s)  
Moreover, the water hammer pressure equations given by Equations (1) and (2) do not account for the droplet impact size, which will have a significant impact on the peak pressure, and there are no analytical equations available in the earlier work to the authors' knowledge.