Upon impact between a droplet and the blade surface, transient stresses occur and propagate in microseconds. By numerical modeling, local strains up to 80 % and strain rates as high as 103 to 106 s⁻¹ are predicted . The actual magnitude and orientation of stresses imposed by a droplet impact depend on the impact velocity and size of the droplet, and the hyper visco-elastic and acoustic properties of the surface material. Research by showed experimentally how a single impact from a high speed water jet can cause fracture. Distinct fracture patterns depended on the material properties. For a hard elastic polymer, they observed ring-shaped cracks at the surface, some distance from the impact center. For rubber and other soft polymers, cracks were observed inside the bulk of the material, below the impact center. For concentric repeated impacts, as in the single point impact fatigue test (SPIFT), similar crack patterns are observed . The fatigue cracks initiate after a number of impacts, the number depending on the impact velocity. The location and orientation of the cracks are also directly linked to the stress patterns regarding the impact on the specific materials. For hard elastic coatings, the ring-shaped cracks initiate at the surface, then propagate at each subsequent impact, in a cone shape into the material. For soft hyper visco-elastic materials, the fatigue cracks initiate inside the material, beneath the impact center, and progress radially in a star-shaped pattern.

For the randomly distributed droplet impacts in rain erosion, an infinitesimal material element will be exposed to transient loads at different magnitudes and orientations . The position and orientation of the maximum local peak stress will change from one impact to another. In this case, cracks may occur at randomly distributed inhomogeneities and defects, and the distinct crack patterns, as observed in single point impact fatigue, may not be present. In both cases, it is a reasonable hypothesis that crack growth depends on three factors: (1) the rate and magnitude of the transient stresses and strains that are generated by the impacting droplets and controlled by the hyper visco-elastic properties of the impacted material, (2) the position, orientation, and length of existing cracks, and (3) the material's resistance to crack initiation and growth. presented a fracture-mechanics-based model for droplet impact on an elastic substrate with a crack. They established a criterion to predict whether a crack would propagate upon impact of a spherical liquid projectile, depending on the radius, density, and velocity of the projectile, the crack length, and the elastic modulus and critical crack intensity factor of the impacted material. They defined the “damage threshold velocity” as a function of the abovementioned parameters.

For soft visco-elastic materials, the stresses and strains at impact are time and rate dependent . This may also be the case for the parameters governing crack growth in rate-dependent materials, as demonstrated by .

explore the Springer model for a range of materials. The Springer model does not account for rate-dependent properties. However, Hoksbergen notes that, for visco-elastic materials, the material data entering the model must be obtained at the relevant high strain rates. make the Springer analysis based on actual fatigue data. They fit the model with RET data by assuming a correlation between the strain rate and the ultimate strength of the material. Later research by concludes that the visco-elastic properties, obtained by a DMA test, govern the rain erosion performance. perform RET at different droplet impact frequencies while maintaining the impact speed. They describe a threshold impact frequency below which the LEP shows an elastic high-cycle fatigue damage mechanism, and above which it changes to a brittle low-cycle fatigue behavior.

Using a single point impact fatigue test, demonstrates that the number of impacts to crack initiation depends not only on the impact velocity but also decreases with decreasing time interval between impacts. This effect is partly due to hysteresis-induced heating and partly due to the time-dependent recovery between subsequent impacts. examine similar correlations, applying a pulsating jet type of rain erosion test. They conclude that increased impact frequency results in fewer impacts before the end of incubation occurs. They attribute this to the material recovery effect between successive impacts. They also apply dry intervals, where the test is paused for a period and then resumed. Dry intervals lead to increased lifetime of the coatings, as it is also described by for the whirling arm RET. Time-dependent recovery can be observed directly in the cyclic tensile loading of visco-elastic materials. When cyclic tension load is applied with a constant amplitude of elongation, and the material is not allowed enough time to recover fully between each load cycle, the material gradually loses stiffness, and the load-displacement loop converges after a number of cycles. The work of deformation, or strain energy, is then reduced from the initial cycle to the state where the loops stabilize . This, known as the Mullins effect, or stress softening, can be either reversible or irreversible. examined the effect of pauses for rubber in a plane strain fatigue crack growth test. They applied sequences of loading pulses, with fixed strain amplitudes, separated by pauses, dwell times, and observed that the crack growth rates increased for increased dwell times. It may seem contradictory that increased dwell time leads to longer life in RET while it causes increased crack growth rates in fracture mechanics testing . However, this is likely because the droplet impact test is load or energy driven, because the impact energy in each successive droplet is constant, whereas the fatigue crack growth test is displacement controlled, and, consequently, stress softening causes the stiffness, maximum load, and strain energy to decrease with increasing cycle number. Dwell times allow the material to recover and the next load sequence will resume at a higher stiffness compared to the later cycles of the previous sequence. In fatigue crack growth testing, the time interval between individual load pulses, the dwell period, can also affect the crack growth rate .

The crack growth behavior of materials is studied in the field of fracture mechanics, which correlates crack growth with material properties and loading conditions. The critical energy release rate, Gc, is a material property that determines the energy per area required to propagate a crack. The energy release rate, G, for an elastically loaded specimen with a crack can be equated with the potential strain energy that is released when the crack propagates. The corresponding property for crack growth on thin sheets made of materials such as rubber is often denoted tearing energy, T . The pure shear tensile test, also called the plane strain test with one or two slits, is often used for crack growth testing. The specimen is wide and has a short distance between the grips. Hence the material is constrained from contraction in the width direction. The cracks can grow in steady state because the test specimen, when correctly proportioned, has a stress-free section behind the crack tips and a section of uniform plane strain ahead of the crack tips. From an energy balance analysis shows that the tearing energy is given by 1T=wh0. By J integral analysis, also arrives at J=wh0, where w is the strain energy density in the section of plane strain, and h0 is the initial length of the specimen (distance between grips) in the non-strained condition. T is also used as the load parameter in the plane strain fatigue crack growth test , where for rubber there is a fatigue limit of T, below which the crack growth is negligible . demonstrated that the strain energy function depends on the loading rate and correlates with the visco-elastic properties.

Thus, two essential parameters must be determined throughout a fatigue crack growth (FCG) test using the plane strain test setup. One is the lengths of cracks versus the number of cycles. The other is the actual tearing energy imposed during growth of the fatigue cracks. For the latter, the strain energy density in the plane strain zone must be determined. Different approaches are used to measure and calculate the actual strain energy density and tearing energy imposed on the crack during FCG testing. , using a single-edge notch specimen tension test setup, used the positive part of the area under the force-deflection (FD) curve (the strain energy), the length of the un-cracked ligament, and the initial height to determine the tearing energy. determined the crack growth rate vs. tearing energy curve using a strain energy density function constructed from the loading response of an uncracked specimen. For both methods, edge effects affect the accuracy of the strain energy density determined for the plane strain zone.

For the plane strain tensile test, the grips need to be wide to ensure the plane strain constraint. Rubber-like materials are difficult to grip because of their softness and the large strains often applied. When using traditional tensile grips with plane jaw faces, the material tends to slip when a tensile strain is applied, because the material contracts in the thickness direction in and near the gauge section. Molded test specimens with beads and specially designed grips are used in some laboratories . However, it is often desirable to test specimens cut from sheet material. For this purpose, a varying grip pressure can be a solution. presented a grip design, clamping a sheet between a knurled cylinder and a semicircular part, allowing for a varying compression ratio along the grip section.

A material's resistance against rain erosion can be characterized by its ability to resist damage initiation and progression and by the rate at which damage propagates once it has initiated . Several authors suggest that the rain erosion performance of LEP materials may be correlated with fracture toughness and crack sensitivity . In design, testing, and analysis of wind turbine blade structures, fracture mechanics has become a well established field of research . Fracture mechanics testing and analysis is also applied to cyclically loaded elastomer materials, such as compounds for tires and drive belts . However, fracture-mechanics-based analysis and mechanical characterization of leading-edge protection for wind turbine blades have not been found in the literature by the present authors. The present paper proposes to introduce fracture mechanics as a novel direction in the analysis and mechanical characterization of leading-edge protection materials and systems for wind turbine blades.

Recent literature shows that fatigue crack growth is an important damage mode in leading-edge erosion on wind turbine blades. Several testimonies from partners in the IEA Wind Task 46 and at the annual International Symposium on Leading Edge Erosion indicate that localized cracking, compared to progressive erosion, is becoming an increasingly dominant mode of damage for modern LEP systems. This is also the case for the LEP system examined in the present paper. Rain-erosion testing and subsequent X-ray computed tomography analysis of the eroded sample revealed locally occurring cracks surrounded by virtually undamaged material; see Fig. .

In rain erosion, droplets impacting on the surface generate stress waves, which propagate through the material, potentially causing transient opening and closing of the cracks. Under such cyclic loading, cracks may propagate due to the increase in strain energy in the material near the crack tip. The correlations between crack tip loading, and fatigue crack growth rate can be characterized by fracture mechanics testing.

Although fracture mechanics concepts, such as the critical tearing energy threshold for negligible crack growth, are well established for a range of elastomer materials and applications, the corresponding knowledge for leading-edge protection (LEP) systems remains limited. In particular, the influence of loading conditions representative of liquid droplet impact (such as loading rate, dwell period, and load amplitude) on the tearing energy threshold governing fatigue crack growth in modern LEP materials has not been investigated.

The paper therefore introduces a fracture mechanics test to characterize an LEP material. The study investigates the effect of dwell time, load amplitude, and tearing energy on the fatigue crack growth rate in a polyurethane-based LEP material containing pre-existing cracks.

The paper is structured with a methods section (Sect. 2), introducing a novel method for determining the tearing energy T in fatigue crack growth testing, and describing the experimental test setup, design considerations, and data analysis. Section 3 contains the results and related observations and discussion. Section 4 presents the conclusions, perspectives, and suggestions for future work. Appendix  provides a list of abbreviations and general nomenclature. It also includes two tables that will be referenced throughout the paper.

The plane strain test with two slits, also known as the pure shear test, was chosen for fatigue crack growth (FCG) testing because it emulates steady-state crack growth and simplifies the analysis due to a plane strain zone between the crack tips . A two-slit configuration was chosen because it gives symmetric loading on the fixture and load cell, as well as twice the quantity of crack growth data, da/dN, per test. Please consider the setup in Fig. . A thin sheet is clamped at the upper and lower edges, where it is constrained from deformation in the horizontal direction or 1-direction. The specimen is shown in the unloaded condition where the cracks are closed, and the dimensions are width l0, length h0, and thickness b0. At the free edges it has slits or pre-cracks of length a1 and a2.

In this work, the tearing energy will be determined by comparing two stages of the FCG test, at cycle N and at cycle N+ΔN. In each load cycle, the test specimen is exposed to a uniform displacement u2(t) along the x1, x3 plane at h0/2; see Fig. . Three zones, A, B, and C, of different deformation states can be identified. The following assumptions apply to the analysis. (1) The cracks are sufficiently long to leave zone A stress free. (2) The distance between the two crack tips is long enough to have zone C, which is not affected by the strain field near the crack tips. (3) The aspect ratio Lc/h0 of zone C is high enough to ensure a uniform plane strain condition in zone C with zero strain in the x1 direction. (4) Each load cycle applies the same elongation u2 to the specimen, and the hysteresis behavior (cyclic stress–strain behavior) is independent of the number of load cycles throughout the analysis.

The strain energy applied to stretch the specimen is divided into two contributions, corresponding to zone B and zone C. So W=WB+WC. The strain energy, WC, of the plane strain zone will be determined by following the FCG test from cycle N to cycle N+ΔN. In this interval the two cracks grow by Δa1 and Δa2, and consequently, the width of zone C of plane strain is reduced from LC to LC-(Δa1+Δa2), as illustrated in Fig. .

As the cracks propagate with increasing cycle count, the remaining ligament between the crack tips decreases; consequently, the load and the work required to stretch the specimen, the strain energy W, decrease proportionally. Zone A is stress-free, and zone B, which includes the complicated stress fields at the crack tip, is identical at both stages. Thus, the change in strain energy from cycle N to cycle N+ΔN corresponds to the contribution of the part of zone C, of width (Δa1+Δa2), which is present in cycle N and has been removed in cycle N+ΔN.

The strain energy contribution of the removed part of zone C is given by ΔW=WN-WN+ΔN, as depicted in Fig. .

In cycle N, the strain energy is 2WN=WC,N+WB. The contribution from zone C is 3WC,N=wLCb0h0, where w is the strain energy density in zone C. While WB remains constant, in cycle N+ΔN, the contribution to the strain energy of zone C is reduced to WC,N+ΔN=w(LC-(Δa1+Δa2))b0h0. The difference in strain energy ΔW=WN-WC,N+ΔN from cycle N to cycle N+ΔN can now be written 4ΔW=w(Δa1+Δa2)bh0. Isolating w and inserting it in Eq. () yields 5T=1b0ΔWΔ(a1+a2). Equation () will be used to determine T in the FCG test.

The test fixture designed for the present study adapts to the concept of partially circular grip faces as illustrated in Fig. . The inner grip face is 90° of a cylinder with radius ri=11.5 mm. Its concave counterpart has a slightly larger radius, ro, leaving space for the test material between the two. The grip pressure at 90° is determined by the tension of the bolts, whereas at 0° the specimen is restricted by the geometry of the fixture and compressed to fit in the gap ro-ri, which is approximately two-thirds of the specimen thickness. The tests reported in this paper were conducted using a prototype fixture manufactured by 3D printing in PLA plastic.

The fixture was mounted in an Instron E3000 electro-pulse test machine. The test specimens were cut into rectangular shapes of 80×46 mm. Precuts of 15 mm in length were cut with a doctor's blade. The specimens were clamped in the test fixture shown in Fig. . The fixture design enables a gripping zone of 80 mm width, and the gripping mechanism maintains grip even at high strains and substantial contraction in the thickness direction. The specimens were clamped in the full width. The initial free length between the grips was 10 mm.

In rain erosion, the specific impact frequency (ASTM G73), fi=a/bΨV, is a factor indicating the frequency of impacts per projected area, a, of an impacting liquid body of volume, b. In a rain field with evenly distributed spherical droplets of diameter, d, the specific impact frequency, fi, is calculated by Eq. (). 6fi=32dΨVimpact where Vimpact is the impact velocity of the droplet relative to the exposed body. The volume concentration of liquid in a rain field, Ψ, is obtained from the rainfall rate, I˙, and the terminal falling velocity of the droplet, Vdroplet, as 7Ψ=I˙Vdroplet.

In the crack growth test, the period between successive droplet impacts is simulated by applying a dwell time, td, between load pulses. The dwell time was chosen based on impact frequencies computed from data, using the findings from (Please see Tables  and ). Although, the characteristic impact period is within the range from 1 to 100 s, the dwell times employed in this study are 0.1 and 1 s. The parameters were chosen due to time constraints and observable differences in stable hysteresis loops.

The displacement control module utilizes a time-dependent sinusoidal displacement profile to simulate the transient impact response between a wind turbine blade and a single water droplet. The waveform is characterized by a pulse time, tp, and a dwell time, td. The minimum-to-maximum displacement ratio is set to zero to replicate a scenario in which the transient stress response from a single droplet impact has fully decayed before the next impact occurs, controlled through ϕ, representing the load-active values of time during the fatigue process.

The displacement profile is given in Eq. () and depicted in Fig. : 8u2(t)=u2max⁡21+sin⁡2πϕ-π2,0≤ϕ≤10,otherwiseϕ=mod(t,td+tp)/tp.

When the pulse displacement is applied, after a running-in period, the two cracks typically grow at approximately constant and similar rates. During the test, data of load and position peak values, as well as full hysteresis loops, are acquired at regular intervals. In parallel, a digital camera is configured to capture an image of the specimen at the peak displacement and crack opening at set intervals. The test continues until a minimum peak load criterion is reached, typically when the remaining section between the crack tips is reduced to approximately 10–20 mm.

The strain energy, W, applied to the specimen during a single cycle, is computed as the integral of the load over the displacement curve from minimum to maximum elongation, at set intervals, during the FCG experiment. The crack lengths, a1 and a2, are determined by measuring the change in pixels from cycle N to N+ΔN, followed by conversion using the pixel-to-millimeter ratio. The work and crack distances are plotted against the number of cycles, as illustrated in Fig. a and b. Domain II constitutes the analysis region of interest. This interval is reached when the cracks grow at approximately similar rates, da1/dN≈da2/dN, and the material has reached stable values in terms of dW/d(a1+a2). The crack growth rates and tearing energy are determined from the slope in domain II, as seen in Fig. b and c.

Figures  and show, respectively, the load–displacement curves for an un-cracked specimen, illustrating the hysteresis loops of the initial cycle and cycle number 2000, and the corresponding maximum loads as a function of cycle number for different dwell times.

For a dwell time of 0.1 s, the maximum load and strain energy decrease from approximately 221 N and 0.28 J in cycle 1 to 126 N and 0.15 J in cycle 2000; see Fig. a. After the test, the material is left to recover for 10 min. Then, a new test sequence is done with a dwell time of 1 s. In the first cycle, the material has almost recovered to its initial state with a maximum load and strain energy of 215 N and 0.27 J, followed by a reduction at cycle 2000 to 171 N and 0.22 J, Fig. b. The material is left to recover for an additional 10 min, and the test is repeated with a dwell time of 10 s. Again, the material almost recovers to its original state with a measured maximum load and strain energy of 218 N and 0.28 J, followed by a reduction at cycle 2000 to 211 N and 0.27 J, Fig. c. Here, the material almost fully recovers between the successive cycles.

The number of cycles required to stabilize the maximum load decreases with increasing dwell time, indicating that the material's load-carrying capacity (energy uptake) is significantly reduced when insufficient time is allowed for relaxation to happen between successive load cycles Fig. a and b. Additionally, the gradient of the load–unload curve from the initial to stabilization phase undergoes a notable change for td=0.1 but not for td=10 s. Moreover, the ability to restore load-carrying capacity and curve shape suggests reversible molecular changes within the rubbery phase of the elastomeric system, attributed to the Mullins effect , commonly referred to as stress softening. Most notably, the Mullins effect depends on the dwell time (i.e., the time between two cycles). When the dwell time is reduced to 1 s or 0.1 s, both stiffness and energy density decrease. This dependence shows that the stress-softening response, stiffness, and strain energy density require a run-in period before reaching stable values/conditions, as seen in Fig. a and b. The electro-pulse test machine was unable to maintain a constant load amplitude as the specimen stiffness decreased. This is the reason why load cycles with low stiffness result in higher maximum displacement, as seen in Fig. a and b.

Each test is conducted and analyzed as described in the methods section. Here, we describe in detail the observations, features, and analysis of three selected tests, namely tests no. 1, 2, and 3, for which differing crack growth behaviors are observed.

Figure  shows plots of the numerical data extracted from these three tests, with crack lengths versus cycle number in the first column, loading strain energy versus cycle number in the second column, and loading strain energy versus the sum of crack lengths in the third column. The absolute value of the linear regression dW/d(a1+a2) is the basis for calculating the tearing energy using Eq. (). For each individual test, da1/dN, da2/dN, and dW/d(a1+a2) are evaluated in cycle interval II, where the rates are approximately constant. The data are summarized in Table .

In the typical scenario, the two cracks grow at approximately constant, and similar rates. However, for some tests, discrepancies are observed, which will be addressed in the following.

For test no. 1, Fig. , the two cracks grow at a rate of around 5⋅10-6 mm per cycle for almost 570 000 cycles. Subsequently, the right crack instantly jumps to a rate of 4⋅10-4 mm per cycle, also observed within domain III in Fig. a. Such an abrupt change in crack growth rate was also seen in another specimen. It may indicate that this test was loaded just below a threshold tearing energy and that the threshold was exceeded for one crack after 570 000 cycles. This may be caused by a slight misalignment of the grips or uneven gripping along the width of the test specimen. It may also be a consequence of the slight increase in peak displacement as the specimen stiffness decreases due to the growing cracks.

Another feature is observed for test no. 2, Fig. , where the two cracks initially grow at identical rates until around 5000 cycles, when the right crack stops while the left crack continues, highlighted as the crack arrest region in Fig. d. At around 10000 cycles, the right crack finds its way around what appears to be a tough inhomogeneity, and the crack resumes propagation. It then accelerates until around 20000 cycles, where it reaches its initial rate and grows at the same rate as the left crack. Looking carefully at the photos, it can also be seen that the right crack has followed a path that has been affected by the tough inhomogeneity.

Figure  shows images from test no. 3, where both cracks grow at a relatively constant rate of approximately 1.3⋅10-3 mm per cycle throughout the experiment. A kink angle appears after 5397 cycles in the left crack, changing the propagation direction of both cracks. This behavior may arise from local material inhomogeneity or a slight misalignment or variation in the gripping pressure along the width of the specimen. The change in propagation direction will modify the local strain and stress fields due to a change in principal orientation relative to the global loading direction. However, the calculated work of loading does not indicate any significant change associated with the kink formation, nor is a deviation observed in the crack growth rate, as seen in Fig. a and b. This indicates that the tearing energy, which governs crack advance, is not significantly affected by changes in the direction of crack growth.

These observations indicate that a threshold tearing energy exists below which the fatigue crack growth rate is of the order of nanometers per cycle and above which the rates are in the micrometer per cycle range. This threshold may be somewhere in the interval T=2000–2400 J m⁻². The da/dN versus tearing energy values for all test in the present campaign are presented in the next subsection.

Figure a shows the crack growth rate da/dN versus the maximum strain for dwell times of td∈0.1,1 s.

The strain threshold between slow and fast crack growth is higher for the experiments conducted with a dwell time of td=0.1 s compared to the data points obtained using a dwell time equivalent to td=1 s. The relative percentage difference is approximately εdifference=27 %, as depicted in Fig. a. Following the trend lines in the growth regimes, it can be stated that for a given input strain, the crack grows faster for td=1 s than for td=0.1 s. Accordingly, for a given crack growth rate, the quantity of strain required to reach an equivalent growth rate is greater for td=0.1 s than for td=1 s. This implies that longer dwell times allow for a greater quantity of strain energy to be imparted into the system for a given peak strain, thus increasing the tearing energy that drives both crack initiation and accelerated crack growth.

However, it can be observed from Fig. b that, when inspecting the tearing energy, T, versus the crack growth rate, da/dN, the data tend to collapse for the two dwell times, td∈[0.1,1] s, although with a slight difference in tearing energy threshold of Tdifference=10 %. The crack growth rate over tearing energy is largely independent of dwell time for the values tested. Fitting a power law to both td=0.1 and td=1 data for da/dN values above 10-4 mm per cycle gives da/dN=3.98⋅10-11⋅T2.19. In contrast, finds that dwell time does affect the same correlation in rubber compounds for tires. The preliminary data suggest a tearing energy threshold, below which crack growth is negligible. With the reservation of limited data points, the threshold is estimated by computing Tthress=Tmean≈2100 J m⁻² of the data below da/dN<10-4 mm per cycle with a Tstd≈200 J m⁻².

The test data used to calculate the crack growth rate, da/dN, and the tearing energy, T=1/b0⋅dW/(da1+da2), are the load, the position of the actuator, the positions of the crack tips, and the initial thickness of the sheet. The relative uncertainties in load, position of the actuator, and thickness of the specimen are roughly 1 % each. The largest source of uncertainty in the crack growth rate is related to the positions of the crack tips, which are determined by digital image analysis. The pixel size corresponds to 0.02 mm, and the uncertainty in the crack tip position is estimated to be ±3 pixels. Below the threshold, the crack growth and tearing energy are typically evaluated in 1 mm of crack growth intervals, leading to combined uncertainties greater than 10 %. Above the threshold, the evaluation length is roughly 10 mm and the uncertainty is roughly 2 %.