**3. Single Droplet Impact Modelling Considering Viscoelastic Material Characterization**

### *3.1. Thin Coatings mechAnical Properties at High-Strain-Rates*

The waterdrop impact introduces a very high-rate pressure transient build-up. The viscoelastic material experiences a very rapidly increasing stress field that leads to a distortion and a subsequent strain relaxation. The large deformation stress-strain behavior of elastomeric materials is strongly dependent on strain rate. ISO 18872 standard [23] is defined for high strain rate testing of polymeric materials. In [24–26] is analyzed for particular materials the deformation behavior over a wide range in strain rates. The problem is widely studied in the literature in regards of different mechanical properties, chemistry systems (molecular transitions and relaxations) and loading cases (considering tensile or compression). For our droplet impact analysis and modelling, it is important to note that the Ultimate Strength σ*<sup>u</sup>* characterization represents an important input parameter because is directly related with the erosion strength and is exponentially related to lifetime estimation (see Equations (12) and (14) respectively). Its rate dependent value [26] is an important source of deviation on the modelling accuracy. Representative engineering stress-strain plots of a polyurethane-based polymer material under dynamic tension loading with three curves per selected strain rate level can be obtained from [26]. Characterizing LEP materials at high strain rates is difficult, even at small amplitudes because the regime of interest at a very high frequency is limited.

The highly transient material behavior during waterdrop collisions require to define the range of frequency of its data set. Stiffness of a polymer is measured as a modulus, a ratio of stress to strain at a certain stage of deformation. LEP polymers are viscoelastic materials and as a result their mechanical and acoustical property will depend very much upon measurement frequency and temperature, [27–31]. Viscoelastic variation in application of solid particle erosion analysis under high speed impact conditions is reported in [18].

This material behavior can be obtained from the frequency response data from Dynamic Mechanical Thermal Analysis (DMTA) where a sinusoidal strain is imposed on a rectangular sample as a function of temperature, see Figure 9.

**Figure 9.** DMTA developed Testing for prototype LEP used in this work, only valid for low frequencies 1Hz up to 100 Hz but considering Temperature variation.

For the dynamic experiments the modulus is complex *E*\* and is given by *E*\* = *E*- + *iE*" whereby *E*- describes the elastic or energy storage component of the modulus and *E*" the loss of energy as heat in a cycle deformation. The modulus of a viscoelastic material is a function of time as well as temperature which is the basis for time-temperature superposition principles which may be used to predict the temperature-frequency behavior of a polymer.

The Dynamic Mechanical Thermal Analysis (DMTA) is the appropriate test to determine the viscoelastic properties, however, the information provided by this technique is only valid up to a frequency of 100 Hz, and therefore, it is not useful in the present context.

The Time-Temperature superposition principle may be required to determine temperature-dependent mechanical properties in a broad range of frequencies. It also may consider transforming the data from the frequency to the time domain for the computational analysis (by performing convolution calculations and inverse Fourier transform on *Eand E*" data set).

On the other hand, Dielectric Thermal Analysis (DETA) supplies information on the molecular motion up to a frequency of 10<sup>7</sup> Hz by means of measuring the complex dielectric permittivity (ε\*) over the entire frequency range, so the regime of interest, see Figure 10. It gives complete information for shifts in Tg transitions depending on frequency and temperature variations, but it has a lack on the mechanical values of *E* and *E*" since it only gives us dielectric data. Thus, it is necessary to obtain with additional time-temperature superposition the relevant mechanical data and converting it to the complex Young modulus (*E*\*). To that end, a series of mathematical models capable of performing such interconversion may be applied [32–34].

**Figure 10.** DETA dielectric testing was developed with 3 different temperatures for prototype LEP used in this work, valid for high frequencies up to 10 MHz but not contemplating mechanical properties definition.

A direct measurement of the required mechanical elastic properties in high frequency ranges can be obtained from Ultrasonic measurements as detailed in the literature [28–30]. There is a good correlation between the ultrasonic properties' attenuation α and sound velocity *C* and the elastic modulus properties:

$$E' = \rho \mathbb{C}^2 \quad ; \quad E'' = \frac{\rho \mathbb{C}^3}{\pi f} a \tag{18}$$

The speed of sound is temperature and frequency dependent so are the acoustic impedance. In next section, further analysis is developed for better understanding the effect of most important parameters that may affect the frequency development of the stress-strain-time evolution and hence the consideration of appropriate speed of sound measurement as input modelling data.

#### *3.2. Stress-Strain Frequency Range Analysis during Droplet Impingement*

The working frequency range definition of a given single droplet impact is a complex phenomenon that needs appropriate 3D Stress-Strain analysis out of the scope of this work and depends on many operational variables. In Springer model it is simplified analytically considering the impact as a

step-like function defined by means the droplet size, the water and coating speed of sound and the impact speed (see first section of this document). In this work, the impact velocity of the droplets is defined for the conditions of the collision without differentiation of the rotor speed or the gravity effects of the rain droplets. The impact pressure is then considered through Water-hammer pressure that depends on the speed of impact and the liquid and surface impedances. The total impact duration *tL* depends on the droplet diameter and the speed of sound in the liquid,

$$t\_L = \frac{2d}{\mathcal{C}\_L} \tag{19}$$

It can be observed in Figure 11 this step wise force pulse definition yields a Fourier Transform decomposition with not valid frequencies due to the abrupt change on the time required to build up and down the contact forces.

**Figure 11.** Impact force step wise definition and its corresponding Fourier Reconstruction (where impact duration depends on droplet diameter and the peak value depends on the velocity of droplet impact and liquid-coating relative impedances).

A first approximation of the problem would be to consider droplet with shape completely round with diameters in the range of 1–4 mm, so the corresponding duration of impacts *tL* are 1.35–5.4 μs and if We assume that the time to build up and down is at least half of the impact duration time then, <sup>1</sup> <sup>2</sup> *tL,* give as a relation for the frequencies of that force pulse with values of 0.18–0.74 MHz, respectively.

In order to improve understanding the stress-strain development in the LEP system, we will introduce different modelling cases of analysis with alternative to Springer model assumptions:


is neglected in this work to avoid complex liquid-coating contact modelling following Springer assumptions simplification.

• 1D formulation examines the impact of a liquid droplet treating the problem only as tensile-compression event. This simplification is applicable since shear stresses and shear material characterization are out of the scope of the fatigue analysis case involved.

The simplified model proposed that considers these assumptions, see Figure 12, has been numerically developed in Open Modellica [19]. The algorithm that includes the material models is outlined in Figure 13. This LEP configuration is defined for rain erosion testing performed at PolyTech Test & Validation A/S according to DNV-GL-RP-0171 [22], see Figure 14.

**Figure 12.** LEP configuration considering viscoelastic material models, multilayer and 1D droplet impact event.


**Table 1.** Initial Reference Input data used for the impact modelling of RET testing.


**Figure 13.** (**a**) Reference multilayer configuration for RET coupons. Liquid droplet and each material layer are defined by the input mechanical parameters of Table 1; (**b**) corresponding numerical configuration outline implemented in OpenModelica.

**Figure 14.** Rain erosion test facility and three specimens used at PolyTech Test & Validation A/S according to DNV-GL-RP-0171 [22] for the analysis and experimental validation.

The numerical procedure was implemented in a general LEP configuration according to Rain Erosion Testing coupons. Simulations of the stress-strain behavior caused in the multilayer system are computed for a given 1D discretization through the thickness position solving for a set of material nodes that belong to a particular homogenized layer. The Equilibrium equation to be accomplished for any two consecutive nodes in the multilayer system is given by

$$m\_i \frac{d^2 \mathbf{x}\_i}{dt^2} = F\_{i-1,i} - F\_{i,i+1} \tag{20}$$

where *layer k* defines *nk* nodes, *node i*−1 defines position *xi*−<sup>1</sup> and *node i* position *xi*. The material models implemented to state a given layer k give us distinctive stress-strain behavior that can be modelled as:

• Pure elastic model, where *A* is the impact area defined by the droplet size and *E* is the elastic modulus

$$F\_{i-1,i} = -\frac{A \cdot E}{\left(\mathbf{x}\_i^0 - \mathbf{x}\_{i-1}^0\right)} \left(\mathbf{x}\_i - \mathbf{x}\_{i-1} - \mathbf{x}\_i^0 + \mathbf{x}\_{i-1}^0\right) \tag{21}$$

• Kelvin-Voight (KV) viscoelastic model, where η is the viscosity,

$$F\_{i-1,i} = -\frac{A \cdot E}{\left(\mathbf{x}\_i^0 - \mathbf{x}\_{i-1}^0\right)} \left(\mathbf{x}\_i - \mathbf{x}\_{i-1} - \mathbf{x}\_i^0 + \mathbf{x}\_{i-1}^0\right) - \frac{A \cdot \eta}{\left(\mathbf{x}\_i^0 - \mathbf{x}\_{i-1}^0\right)} \left(\frac{d\mathbf{x}\_i}{dt} - \frac{d\mathbf{x}\_{i-1}}{dt}\right) \tag{22}$$

and considering appropriate estimation of the viscosity attenuation observed in as:

$$\begin{array}{rcl}\sigma\_{\text{total}} = \sigma\_{\text{s}} + \sigma\_{d} & \rightarrow & \sigma = E\varepsilon + \eta \frac{d\varepsilon}{dt} \\ \varepsilon\_{\text{total}} = \varepsilon\_{\text{s}} = \varepsilon\_{d} & & \\ E' = E' + iE'' = E' + i2\pi f\eta & \rightarrow & \eta = \frac{E''}{2\pi f} \end{array} \tag{23}$$

• Havriliak-Negami H-N viscoelastic model [32–34], where *E*<sup>∞</sup> define the unrelaxed or glassy modulus, and *Eo* is the relaxed rubbery modulus and τ is the relaxation time, see Figures 15 and 16.

$$F\_{i-1,i} + \tau \frac{dF}{dt} = -\frac{A \cdot Eo}{\left(\mathbf{x}\_i^0 - \mathbf{x}\_{i-1}^0\right)} \left(\mathbf{x}\_i - \mathbf{x}\_{i-1} - \mathbf{x}\_i^0 + \mathbf{x}\_{i-1}^0\right) - \frac{A \cdot Eo}{\left(\mathbf{x}\_i^0 - \mathbf{x}\_{i-1}^0\right)} \left(\frac{d\mathbf{x}\_i}{dt} - \frac{d\mathbf{x}\_{i-1}}{dt}\right) \tag{24}$$

**Figure 15.** H-N model. Storage Modulus variation with Frequency. E<sup>∞</sup> unrelaxed modulus.

**Figure 16.** H-N model. Relaxation Time dependence on Temperature.

This simplified computational tool allow us to treat as parameters the material models (as pure elastic, Kelvin-Voight, Havriliak-Negami, etc.) and their related properties (density, storage modulus, loss modulus, tan delta, speed of sound, thickness, etc.), the operational variables (impact velocity, droplet diameter size, droplet density, droplet speed of sound, etc.). In order to quantify the strain rate analysis of the single droplet impact simulation, we will evaluate different cases considering the effect on variations in coating-substrate thickness, viscoelastic material properties, droplet size and droplet impact velocity from a reference configuration used in RET testing (Figure 13). The input data values for these prototype materials of Table 1 where defined initially from previous testing results and here are used for the exposed modelling procedure in order to discuss Stress-Strain frequency range analysis during droplet impingement.

The strain-stress evolution with time is evaluated at different locations of the LEP coating for appropriate comparison. The specific location of the analysis through the layer thickness is defined as variable e\_x for the strain and variable s\_x for the stress, where x is defined at surface x = 0, interface x = 100, or any intermediate positions with x = 25, 50 or 75 referring all to the given % of the layer thickness, see Figure 13.

A first result for the analysis of the reference testing LEP configuration is plotted in Figure 17. It is observed the strain evolution with time at the surface of the coating layer e\_0 comparing two

cases: Coat\_1 using a pure elastic modelling of the coating material compared with Coat\_2 using a Kelvin-Voight modelling.

**Figure 17.** Strain evolution at the coating surface for the reference LEP configuration for RET coupons with input data defined in Table 1. Material models comparison.

In addition, the spectrogram of the strain evolution with time for a given location e\_x is calculated with the Fourier transform applied in short-time periods though the duration of the impact analysis (0–50 μs). The strain frequency decomposition during the impact event, provide us a plot of the dominant frequency spectrum with a range of values (measured as power (dB) over Frequency (Hz)), for each time analysis period. This procedure allows us to estimate indirectly the highly transient strain-rate variations for the single droplet impact event.

Figures 18 and 19 show as a first simplified approximation, the effect of the inclusion of the attenuation consequence due to the material modelling definition. The reason for such comparison is to clarify that the material properties are frequency dependent so the input data for the modelling. This assumption is important to consider when we define the speed of sound as a constant variable in our analysis. A first conclusion for the developed case is that the most dominant frequencies occur during the first stage of the impact and that Kelvin-Voight material modelling is appropriate to avoid additional frequency noise due to the lack of attenuation when using pure elastic material models.

**Figure 18.** Spectrogram for strain evolution at the coating surface for the Reference multilayer configuration. Material models analysis for Pure elastic consideration.

**Figure 19.** Spectrogram for strain evolution at the coating surface for the Reference multilayer configuration. Material models analysis for Kelvin-Voight consideration.

The computational analysis limits the frequency in a range of 0.5–2 MHz for this initial set-up conditions.
