**2. Wear Erosion Lifetime Prediction Modelling from Fundamental Material Properties. A Review of Springer Model**

In this section a review of the Springer model is exposed in order to be used in the next sections for wear erosion lifetime analysis depending on the material impedance measurements as input modelling data.

The progression of erosion can be experimentally measured with applicable Rain Erosion Testing. One method is in terms of the average erosion depth versus time or mass loss versus time (directly related to the number of impacts, see Figure 5). There is initially an incubation period in which damage progresses without perceptible change in the material weight loss. After a sufficient amount of fatigue degradation has accumulated, the material tends to lose mass with a constant erosion rate. This marks the end of the incubation period and a steady mass loss period begins, where the weight loss varies nearly linearly with time.

**Figure 5.** (**a**) Evolution of weight loss on experimental rain erosion testing coupons and lifetime prediction model defining the incubation period and mass removal rate (**b**) Springer Model based Fatigue life N approximation related with the material ultimate strength σu, the parameter "erosion strength", σ<sup>e</sup> and the parameter b that includes the fatigue knee at the endurance limit σI. Adapted from: [13].

Springer analytical model [13] quantitatively predicts the erosion of coated materials under the previously untested conditions. The erosion evolution can be approximated by two straight lines as depicted in Figure 5 with

$$\begin{aligned} m &= 0 & 0 &< n\_{\rm ic} \\ m &= \alpha\_{\rm c} (n - n\_{\rm ic}) & n\_{\rm ic} &< n &< n\_{\rm fc} \end{aligned} \tag{1}$$

where the mass loss *m* produced by a given number of droplets impacts *n*, can be estimated once the incubation time of the coating *nic* and the slope of the erosion rate on the coating α*<sup>c</sup>* are identified.

In order to establish these parameters, the stress history of the coating and the substrate is assessed analytically. It is affected by the shockwave progression due to the vibro-acoustic properties of each layer, and by the frequency of the repeated water droplet impacts (see Figure 6).

**Figure 6.** Stress wave pattern in the coating and in the substrate for the time intervals related with coating thickness hc and its wave speed Cc. (**a**) Stress wave contact at interface; (**b**) Stress wave consecutive interactions.

Upon impingement on the coating, two different wave fronts travel into the liquid and coating respectively. The wave front in the coating further advances towards the coating-substrate interface, where a portion of the stress wave is reflected back into the coating and the remaining part is transmitted

to the substrate. Due to this reflection a new wave is now advancing in the coating with a different amplitude depending on the relative acoustic impedances of the coating and substrate,

$$q\eta\_{\rm L\varepsilon} = \frac{Z\_{\rm L} - Z\_{\rm c}}{Z\_{\rm L} + Z\_{\rm c}} \quad ; \quad q\rho\_{\rm 5\varepsilon} = \frac{Z\_{\rm s} - Z\_{\rm c}}{Z\_{\rm s} + Z\_{\rm c}} \tag{2}$$

where *Z* = ρ*C* is the impedance of the material, ρ is the density and *C* the elastic wave speed (the speed of sound of the medium). *ZL*, *ZC*, and *ZS* are the elastic impedances of the consecutive materials (i.e., in our problem they are the liquid (L), coating (C), and substrate (S) layers). ϕ*Lc* defines the relative impedance parameter defined on the liquid-coating interface and ϕ*sc* on the substrate-coating one.

This 1D formulation, see Figure 6, examines only the normal impact of a liquid droplet with diameter *d*, onto a two layered structure with the first layer formed by the coating and the second layer by the substrate (assumed semi-infinite) with thickness hs > 2d Cs CL , which means in fact that the reflections from a subsequent substrate additional layers are not considered in the fatigue analysis.

The magnitude of the traveling waves propagating upwards the coating-liquid interface, and traveling waves propagating downwards the coating-substrate interface, are expressed with the *k* number of reflections as depicted in Figure 6:

$$\begin{array}{l} \frac{\sigma\_{2k}}{\sigma\_{1}} = \frac{1 + q\varsigma\_{\rm sc}}{1 - q\varsigma\_{\rm sc}\varrho\_{1\varepsilon}} \left[1 - \left(\varvarrho\_{\rm sc}\varrho\_{1\varepsilon}\right)^{k}\right] \\\frac{\sigma\_{2k-1}}{\sigma\_{1}} = \frac{\sigma\_{2k}}{\sigma\_{1}} - \varvarrho\_{\rm sc}\left(\varvarrho\_{\rm sc}\varrho\_{1\varepsilon}\right)^{k-1} \end{array} \tag{3}$$

where the Water-hammer Pressure defines the initial impact pressure σ<sup>1</sup> = *P*

$$P = \frac{VZ\_L\cos(\theta)}{\left(\frac{Z\_L}{Z\_c} + 1\right)}\tag{4}$$

That depends on the droplet impact speed *V* and its impact angle with cos(θ). The stabilized stress at the interface coating-substrate can be approximated as

$$
\sigma\_{\infty} = \sigma\_1 \lim\_{k \to \infty} \sigma\_{2k} = \sigma\_1 \frac{1 + \varphi\_{\infty}}{1 - \varphi\_{\infty} \varphi\_{L\varepsilon}} = \sigma\_1 \frac{1 + \frac{Z\_l}{Z\_{\varepsilon}}}{1 + \frac{Z\_l}{Z\_s}} \tag{5}
$$

After a long enough period of time, the stresses at both the coating surface and the substrate interface approaches to the constant value σ∞, which is also the stress that would occur in the substrate after impingement in the absence of the coating layer. An example on its use within the project is depicted on Figure 7. One can obtain an analytical value of the stress evolution on the coating during the droplet impact. It is an alternative simplified computation to the algorithm presented in the previous section based on a 3D numerical modelling.

**Figure 7.** Variation of stress at coating-substrate interface for three different top coating LEP material candidates for the same substrate. It can be observed the capability to avoid peak values with the appropriate selection of the material impedances.

To evaluate the average stress values at the coating-liquid and coating-substrate interfaces during the duration of the impact, it is introduced a parameter *k* that depends on the average number of reflections in the coating layer

$$k = \frac{1 - e^{-\gamma}}{1 - \psi\_{Lc}\psi\_{sc}}\tag{6}$$

where the coating thickness *hc* enters its computation through the parameter γ that depends on it and also on the droplet diameter *d*. It may be calculated as

$$\gamma = \frac{2\mathcal{C}\_{\varepsilon} \left(\frac{Z\_{\perp}}{Z\_{\ast}} + 1\right)d}{\mathcal{C}\_{\varepsilon} \left(\frac{Z\_{\perp}}{Z\_{\ast}} + 1\right) \left(\frac{Z\_{\ast}}{Z\_{\ast}} + 1\right)\hbar\_{\mathbb{C}}} \tag{7}$$

Finally, the average stress on the coating surface at *x* = *0* is defined with σ*<sup>o</sup>* as

$$\begin{split} \sigma\_{\theta} &= \frac{P(\psi\_{\kappa} + 1)}{(1 - \psi\_{Lc}\psi\_{\kappa})} \Big( 1 - \frac{(1 - c^{\gamma})(\psi\_{Lc} + 1)\psi\_{\kappa}}{\gamma(\psi\_{\kappa} + 1)} \Big) \\ \sigma\_{\theta} &= \frac{VZ\_{l}\cos(\theta)(\psi\_{\kappa} + 1)}{\left(\frac{Z\_{l}}{Z\_{\kappa}} + 1\right)(1 - \psi\_{Lc}\psi\_{\kappa})} \Big( 1 - \frac{(1 - c^{\gamma})(\psi\_{Lc} + 1)\psi\_{\kappa}}{\gamma(\psi\_{\kappa} + 1)} \Big) \end{split} \tag{8}$$

If the value of the relative impedance parameter of the substrate-coating interface equals zero, ψ*sc* = 0, so the coating material is considered the same as the substrate, and this expression reduces to σ*<sup>o</sup>* = σ<sup>1</sup> = *P* that may be used to compute the average stress for homogeneous materials.

The average stress on the coating-substrate interface at *x* = *h* is defined then with σ*<sup>h</sup>* as

$$\begin{array}{l} \sigma\_{\hbar} = \frac{P(\psi\_{\kappa\epsilon} + 1)}{(1 - \psi\_L \psi\_{\kappa\epsilon})} \Big( 1 - \frac{(1 - \epsilon^{\gamma})\psi\_{L\epsilon}\psi\_{\kappa\epsilon}}{\mathcal{I}} \Big) \\ \sigma\_{\hbar} = \frac{VZ\_{\mathbb{L}}\cos(\theta)(\psi\_{\kappa\epsilon} + 1)}{\left(\frac{Z\_{\mathbb{L}}}{Z\_{\mathbb{L}}} + 1\right)(1 - \psi\_{L\epsilon}\psi\_{\kappa\epsilon})} \Big( 1 - \frac{(1 - \epsilon^{\gamma})\psi\_{L\epsilon}\psi\_{\kappa\epsilon}}{\mathcal{I}} \Big) \end{array} \tag{9}$$

And, as above, if the value of the relative impedance parameter of the substrate-coating interface equals zero, ψ*sc* = 0, so the coating material is considered the same as the substrate, this expression reduces to σ*<sup>o</sup>* = σ*<sup>h</sup>* = *P*.

The incubation period of time neglecting mass loss prior the erosion develops at a given rate, as depicted in Figure 5, is analyzed with fatigue concepts. It may be estimated applying Miner's rule to the impingement force cycles and considering the averaged stress values with the equivalent dynamic stress σ*<sup>e</sup>* per unit area on the impact locations, see Figure 5c.

An approximation value for the fatigue life *N* is then given by a function of the equivalent dynamic stress σ*<sup>e</sup>*

$$\begin{array}{l} N = \left(\frac{\sigma\_{\rm nc}}{\sigma\_{\rm c}}\right)^{b\_{\rm c}}\\ b\_{\rm c} = \frac{b\_{2c}}{\log\_{10}\left(\frac{\sigma\_{\rm nc}}{\sigma\_{\rm lc}}\right)} \end{array} \tag{10}$$

where the subscript *c* indicates to the coating material, *bc* defines the fatigue slope, *b2c* matches to the "knee" in the fatigue curve (that may be estimated with its endurance limit σ*I*) and the coating ultimate tensile strength σ*uc* is defined for N = 1.

A parameter of the material "strength" *Sc* is introduced with a semi-empirical approach and depends on the poison coefficient ν*<sup>c</sup>* (included to consider the location of the impact force on the radial averaged stress), and other relevant properties of both the coating material and substrate treated previously,

$$\begin{array}{l} S\_{\varepsilon} = \frac{4(b\_{\varepsilon} - 1)\sigma\_{\text{uc}}}{(1 - 2\nu\_{\varepsilon})\left[1 - \left(\frac{\sigma\_{L}}{\sigma\_{\text{uc}}}\right)^{b\_{\varepsilon} - 1}\right]} \,\,\,\,\stackrel{\text{\tiny\prime}}{\quad}\\ S\_{\varepsilon} = \frac{4(b\_{\varepsilon} - 1)\sigma\_{\text{uc}}}{(1 - 2\nu\_{\varepsilon})} \end{array} \tag{11}$$

An important issue for fatigue analysis is how to consider the effect of the fatigue slope parameter for the coating *bc* since it is difficult to obtain experimentally for typical LEP elastomeric materials. Equation (11) may be simplified assuming that σ*Ic* < σ*uc* and *bc* 1.

It may be stated as an equivalent erosion resistance parameter for the coating *Sec* including the damping effect of the coating described previously by means of the average number of reflections *k* and the relative impedance parameter ψ*sc* that acts on the interface wave reflections,

$$S\_{\rm ac} = \frac{4(b\_c - 1)\sigma\_{\rm ac}}{(1 - 2\nu\_c)\left(2k\left|\psi\_{\rm sc}\right| + 1\right)}\tag{12}$$

Fatigue life of the material is then estimated with the number of impacts during the incubation time period as

$$m\_{\dot{\&}}^{\*} = a\_1 \left(\frac{S\_{cc}}{\sigma\_o}\right)^{a\_2} \tag{13}$$

where *a*<sup>1</sup> and *a*<sup>2</sup> can be considered determined constants that may be fitted experimentally, *Sec* represents the erosion strength of the material and depends on its fundamental properties defined in Equation (12) and the averaged stress of the coating surface during the impact event defined in Equation(8). In [1], the parameter values where defined as

$$m\_{\rm ic}^{\*} = 7 \times 10^{-6} \left(\frac{S\_{\rm ac}}{\sigma\_{\rm o}}\right)^{5.7} \tag{14}$$

That may also be expressed in terms of the number impacts per site when considering the circular projected area of the droplet with a given diameter *d*

$$m\_{\rm ic} = \frac{8.9}{d^2} \left(\frac{S\_{\rm cc}}{\sigma\_o}\right)^{5.7} \tag{15}$$

where using appropriates units allow one to predict the number of impacts per m<sup>2</sup> at which the coating material starts to develop erosion with a given erosion rate that may also be computed from the previous estimated parameters as

$$a\_{\mathfrak{c}} = \frac{7.3310^{-5} d^3 \rho\_{\mathfrak{c}} \sigma\_{\mathfrak{o}} \mathfrak{a}^4}{S\_{\mathfrak{N}} \mathfrak{a}^4} \tag{16}$$

The equivalent analysis may be used to determine the erosion strength at interface coat-substrate instead of surface. Accordingly, Equations (11) and (15) are written introducing the fundamental properties of the substrate as

$$\mathcal{S}\_{\rm cs} = \frac{\mathfrak{a}\_{\rm ill} - \frac{8.9}{d^2} \left( \frac{\mathcal{S}\_{\rm ff}}{\mathcal{O}\_h} \right)^{5/7}}{4(b\_s - 1)a\_{\rm ns}} \stackrel{\rm S/T}{\to} \frac{4(b\_s - 1)a\_{\rm ns}}{\left( (2k|\psi\_{\rm ref}| + 1) \right)} \tag{17}$$

As we have previously stated from [13] in order to predict the incubation time and the mass removal rate, the stress history in the coating and in the substrate has to be identified analytically or numerically. It is affected by the shockwave progression due to the vibro-acoustic properties of each layer, and by the time interval of the repeated water droplet impacts. Fatigue life of the material is then calculated, and the model can be applied to estimate the stress at different locations through the thickness, i.e., the coating surface or at the coating–substrate interface. Nevertheless, it is assumed that the bond and adhesion of the boundary interface is ideally perfect, so the modelling does not account for the microstructural imperfections and lack of adhesion of such interfaces and does not account either for the shear stresses developed on the 3D impact event.

Considering for previous assumptions, the method has been applied successfully for wear erosion damage in [15]. In that case, the erosion strength of the coating material defined in Equation (12) was empirically obtained by means of the RET (Rain Erosion Testing) testing as a unique value instead of obtaining the fundamental properties values separately.

Figure 8 shows a complete map of the liquid droplet, coating LEP and substrate (primer or filler) material impedances as input parameters of the modelling with the related equations previously stated. The impedance of the LEP thin coating and substrate materials need to be characterized and used as input data in the modelling. The appropriate variable working frequency range depending on the impact and material settings is analysed in next section and defined so the corresponding impedance characterization with Ultrasonic testing for such measurements.

**Figure 8.** Diagram of liquid, coating and substrate material impedances and operational parameters affecting rain erosion performance.
