# Damage in Rubble Mound Breakwaters. Part I: Historical Review of Damage Models

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## Abstract

**:**

## 1. Introduction

## 2. Historical Review of Damage Models for Rubble Mound Breakwaters

_{S}) is lower than a certain function (f). This function depends on the n parameters (p

_{1}, p

_{2}, …, p

_{n}) influencing stability and an empirical coefficient (K) determined by the parameters not directly accounted for in the stability equation (see Appendix B for the rest of symbols).

#### 2.1. First Approaches to Hydraulic Stability Characterization. Models from Iribarren and Hudson

_{D}is tabulated for each armor unit type as a function of the H/H

_{D=0}ratio. H

_{D}

_{=0}is the limit wave height that produces no damage, considered as less than 1% volume of units eroded relative to the total volume of stones in the active armor layer (see Appendix B for the rest of symbols). At the same time, the H/H

_{D=0}ratio was related to a certain percentage of damage, which means that Hudson’s model was probably one of the firsts providing quantitative information about the level of damage. The latter was possible owing to a standardized method for damage profiling, developed by the U.S. Army Engineer Waterways Experiment Station (WES). Despite not considering water depth or wave period, Hudson recommended his model as an initial approximation of the major forces from both breaking and non-breaking waves. However, he pointed out that “there is some doubt as to which of the various wave heights in natural wave trains should be selected as the design wave”. In fact, the Shore Protection Manual (SPM) suggested first H

_{1/3}[27] and, afterwards, H

_{1/10}[28] as an alternative wave height parameter in Equation (2) regarding irregular waves, without offering a clear justification for this modification.

#### 2.2. 70s to 80s: Intensive Research on the Stability of Rubble Mound Breakwaters. Van der Meer’s Formulae

- (1)
- (2)
- Thompson and Shuttler [41] presented a detailed study on riprap stability in 1975. They concluded that “the erosion damage caused by irregular waves on a riprap slope is itself a random variable” and that the method for positioning the stones highly affects damage evolution. Long-term and short-term experiments were tested with impermeable core, finding no relation with depth or wave period, but with mean number of zero crossing waves, among others. In addition, it was suggested that damage tends to reach equilibrium or, in other words, that damage curves are meant to be asymptotic.
- (3)
- Bruun and Günbak [32] assured in 1976 that, regarding breakwater stability, “the significance of wave period is clearly demonstrated.” They started to work on a risk-based approach in the design of sloping structures considering Iribarren’s number.
- (4)
- Whillock and Price [42] presumably coined the concept of “fragility” in 1976 when indicating that elements with high void ratio and designed to interlock, such as dolosse, were associated with a reduction in the safety margin as failure is approached. They also supported the idea of quarry stones being more stable to oblique wave attack than to normal attack. However, this assumption was denied for armor units that are susceptible to drag forces. Indeed, they demonstrated that the overall stability for a particular test with dolosse decreased up to an angle of incidence of 60°. In line with the concept of “fragility”, Magoon and Baird [43] underlined in 1977 the importance of rocking movements in the breakage of armor units designed to interlock.
- (5)
- Losada and Giménez-Curto [44] proposed in 1979 a stability model for the initiation of damage by means of design curves depending on armor unit type and Iribarren’s number. They used Iribarren’s data to fit their model, and compared it satisfactorily with the results from Hudson [26] and Ahrens and McCartney [40]. However, they stated the difficulty to establish a comparison between experimental results undertaken in different laboratories because of divergences in both experimental process and damage criteria. For this reason, it was remarked that “To obtain general criteria on the behaviour of rubble mound breakwaters under wave action, laboratories should establish uniform experimental procedures and criteria.”
- (6)
- Losada and Giménez-Curto [45] studied the influence of oblique incidence in 1982. They concluded that, for gravity armor units, the stability of steep slopes under oblique wave attack is not worse than for normal incidence. On the contrary, they found that for high interlocking armor units (such as dolosse or tetrapods) oblique wave attack is hazardously worse than normal incidence, in agreement with Whillock and Price [42].
- (7)
- Broderick and Ahrens [9] presented in 1982 a technical paper on scale effects using the large-scale tests from Ahrens [39]. These tests, with wave heights up to 1.83 m and periods up to 11.3 s were compared with a 1:10 Froude scaled model. They found a 20% reduction in the zero-damage stability numbers from the large-scale tests whereas run-up increased about 20%. This was identified to be due to the improper modeling of the flow regime within the filter layer for the small-scale experiments and to the lack of penetration of the wave run-up into the filter layer. Scale effects were less severe at high levels of damage and the shapes of the damaged profiles for tests with the same relative depth were very similar. Furthermore, at the zero-damage level, wave period had less influence on the small scale. In their study, the widely used dimensionless erosion area (S) was firstly proposed.
- (8)
- Jensen [46] published in 1984 a thorough monograph on rubble-mound breakwaters. He suggested the parameter H
_{13.6%}instead of H_{S}as a descriptor for wave height in both deep waters (where wave height is usually characterized by a Rayleigh distribution) and shallow waters (where a certain percentage of waves break). Also, in order to reach a stable damage level, scaled storms representing at least 8 to 10 hours in prototype were recommended. Furthermore, possibly the first formula on rear slope stability was proposed. - (9)
- Hedar [47] developed in 1986 a complete stability model taking into account the water depth, wave height at breaking, rubble mound slope, the internal friction angle, a permeability function, and two empirical coefficients.

_{0}), number of waves (N

_{w}), equivalent cube length (D

_{n50}), relative excess specific weight (Δ), breakwater slope (cotα), and permeability of the core (P). Furthermore, the aforementioned dimensionless erosion area (S) was introduced in the formulae as follows:

_{n50}was considered in the formulation. Moreover, after trying two different Pierson-Moskowitz spectra (a wideband and a narrowband), no relation was found between damage and either spectral shape or wave groupness. Strong differences between monochromatic and irregular tests were pointed out, together with a damage dependence on the number of waves. Indeed, as it was reported by different authors, these formulae are only valid for 1000 to 7000 waves and it tends to overestimate the damage for more than 8000 waves. In 1998, VdM [51] related the Hudson’s “no-damage” criteria and filter exposure (failure criteria) to different values of S depending on breakwater slope. Also, an adaptation of his single storm formulae to calculate the damage caused by more than one storm event was proposed.

_{o}, the number of units displaced out of the armor layer within a strip width of one equivalent cube length. This equivalent cube length was assumed to be equal to D

_{n50}for cubes, 0.65 h for tetrapods and 0.7 h for accropode.

#### 2.3. Studies Based on VdM’s. Van Gent’s Formula

- (1)
- Using exactly the same rocks tested by VdM, Latham et al. [54] demonstrated in 1988 the dependency of damage regarding armor shape. An additional coefficient to consider the armor shape effect was suggested: as rock units become more rounded they also become more unstable, especially under surging waves.
- (2)
- Medina and McDougal [55] highlighted in 1990 some shortcomings of VdM formulae, such as the complexity of the equation or the overestimation of damage for more than 7000 waves. They were especially critical with the independence of VdM’s model on wave groupness. In fact, a small but systematic higher stability for random waves from narrowband spectra was detected in VdM tests. A simpler stability model was alternatively proposed.
- (3)
- Kaku et al. [56] found in 1991 that VdM’s damage levels were not accurate enough for forecasting models. Therefore, they proposed a new empirical model assuming that the damage level approaches to an asymptotic equilibrium for the same wave energy. They also included the initiation of armor movement based on the similarity between the stability number and the Shields parameter used for sediment transport. However, in 1992, Smith et al. [57] indicated the difficulties of both empirical formulae in accounting for the complex wave and structural interactions affecting breakwater reshaping, mainly because they were built up under static stability conditions.
- (4)
- Melby and Hughes [3] utilized in 2003 part of the small-scale laboratory data of VdM to fit a stability equation derived from basic principles for uplift, sliding, and rolling incipient motion. It was based on the assumption that the maximum wave momentum flux at the toe of the structure is proportional to the maximum wave forces on armor units.
- (5)
- Vidal et al. [58,59] proposed in 2004 a modification of VdM’s formulae after accomplishing a comparison between the results from Thompson and Shuttler [41], Losada and Giménez-Curto [44], and VdM [50]. This modification consists of using H
_{50}instead of H_{S}. H_{50}is defined as the average wave height of the 50 highest waves reaching the rubble-mound breakwater. This new way of describing wave height parameter is further discussed in Section 2.4. - (6)
- Mertens [60] attempted in 2007 to transform the datasets of Thompson and Shuttler [41] and VdM [50] into comparable information with the one generated by Van Gent et al. [61] in 2003. He also reported some deviations in VdM data because of the influence of stone roundness. Following this line, Verhagen and Mertens [62] proposed in 2009 a methodology for unifying the formulation for both deep and shallow waters. To accomplish this, a correction factor based on the Iribarren’s number was added in order to incorporate the effect of the foreshore. For the correct application of this method they claimed for an accurate calculation of the wave height and wave period at the toe of the breakwater, including a precise determination of H
_{2%}and T_{m−1,0}.

_{2%}/H

_{S}was added. However, after noticing that the influence of this parameter was small, as so was the influence of wave period regarding the data scatter due to other reasons, a single and simpler formula was proposed. In the new formula the permeability was incorporated in a direct way by the nominal diameter of the core material:

#### 2.4. Wave Groupness and Wave Height Parameter Discussion. H_{50} Parameter by Vidal et al.

_{1/3}(in the third edition [27]) and H

_{1/10}(in the fourth edition [28]) for irregular waves. Indeed, for some authors H

_{1/10}is extremely conservative, and for others, such as Jensen et al. [65], H

_{1/20}is reported to be a more suitable parameter. Despite H

_{S}is, even nowadays, the most extended parameter for the characterization of irregular waves, it does not give enough information about the highest waves of the series which, in fact, are the most likely to be responsible for armor damage development. Other studies discussing the adequacy of alternative wave height descriptors are highlighted below:

- (1)
- Teisson [66] presented in 1990 a statistical approach for characterizing the duration of extreme storms and its consequences on breakwater damage. In the study it was stated that “to select H
_{S}as design wave height in Hudson formulae assumes that the associated storm will last for only 10 minutes: this choice could lead to an under estimation of breakwater design.” Alternatively, the expression H_{D}= 1.18 H_{S}t^{0.095}was proposed for the calculation of the design wave height. Teisson tried to relate regular and irregular wave effects on stability. Furthermore, by an integrated theoretical approach, he developed a step-by-step methodology for calculating cumulated damage during storms, assuming that a storm can be described by a sequence of significant wave heights steps with a certain duration each. - (2)
- Vidal et al. [67] suggested in 1995 that the wave height parameter should contain information about the distribution of the highest waves, the length of the time series and the number of times it is recycled to achieve a given degree of damage. The longer the test is, the higher waves are likely to attack the structure, i.e., damage evolution after testing two time series with a certain H
_{S}and a duration t will be different compared to testing a unique time series with the same H_{S}and duration of 2t. Based on numerical simulation, they reported variations in the damage parameter exceeding 50% when considering H_{S}after testing JONSWAP spectra with the same H_{S}and T_{p}but different random seeds. They also confirmed that the highest waves were related to wave groupness. An H_{n}parameter, directly related to test duration and suitable also for breaking conditions (i.e., where wave height distribution during storms can depart from Rayleigh’s due to non-linearity), was proposed for a better characterization of wave-damaging energy. This new parameter was expected to facilitate the comparison of stability results obtained in different investigations, including those carried out with regular waves. Initially, H_{n}=H_{100}was proposed, but in Vidal et al. [58,59] this parameter was adjusted to H_{50}after comparing the datasets from Thompson and Shuttler [41], Losada and Giménez-Curto [44] and VdM [50]. In addition, a new formula was developed based on VdM equations. The lack of consistency of the Rayleigh distribution for breaking conditions was also accounted for by other authors, such as Battjes and Groenendijk [68], which suggested a Weibull distribution for damage models in shallow foreshores after a spectral analysis, or Méndez and Castanedo [69], which provided a model for the depth-limited distribution of the highest waves in a sea state. - (3)
- Jensen et al. [65] tested both regular and irregular waves for identifying a wave height parameter within the irregular waves corresponding to the wave height of a regular series with a similar damage level on the structure. In line with Vidal et al. [67], they ended up in 1996 with an H
_{n}parameter, but in this case a n-value of approximately 250 was found to be more suitable. - (4)
- Medina [70] claimed in 1996 for non-stationary stochastic models as more adequate for modeling real waves. He defined five conditions for any rational armor damage model to properly take into account the storm duration, such as damage must necessarily increase with the duration of the storm under random wave attack in deep water conditions. A wave-to-wave exponential model accomplishing these conditions was proposed. The model depends on the number of waves and introduces an asymptotic maximum damage to the armor layer under a constant regular wave attack. It also depends on a mean damage parameter consisting of the number of regular waves causing 63% of the maximum asymptotic damage, which is linked to the concept of mean lifetime of the structure. Medina compared the results with the models from Teisson and Vidal (based on different assumptions but accomplishing most of the five aforementioned conditions) and applied the new method to a real case: the partial failure of Zierbana breakwater (Port of Bilbao, Spain) under construction in February 1996. In Gómez-Martín and Medina [71], the wave-to-wave exponential model was slightly modified and the mean damage parameter was found to be dependent on Iribarren’s number. They also designed a neural network (NN) applicable to random waves in non-stationary conditions, finding that the estimation of accumulated armor damage using both wave-to-wave exponential method and NN model showed a good agreement to damage observations.

#### 2.5. The 90s: Probabilistic Approaches and Damage Progression Models. Melby and Kobayashi’s Model

- (1)
- (2)
- In 1992, the final report from PIANC [73] provided a review on random wave’s armor stability models. In the same year, Koev [74] studied statistically a homogeneous set of 21 armor layer stability formulae developed under regular waves, and proposed a regression model valid for 0.04 ≤ H/L ≤ 0.1 and 1.1 ≤ cotα ≤ 20.
- (3)
- In 1994, Vidal et al. [75] revised the available methodologies for the calculation of the armoring hydraulic stability, both for breakwater heads and trunks. Formulations and design recommendations for berm breakwaters, low-crested breakwaters and conventional breakwaters were also included.

- (1)
- Hughes [76] presented in 1993 a publication of reference in physical modeling. It discussed the principles of dimensional analysis, scale effects, and similitude criteria including specific similitude requirements for different coastal structures. Also, considerations about movable-bed models, generation of gravity waves in laboratory, and a discussion on laboratory measurements and data analysis were included. Scale effects were further addressed in 2004 by Tirindelli et al. [77], who focused on wave impacts, run-up, overtopping, structure deformation, porous flow, and flow forces on plants and organisms.
- (2)
- Davies et al. [78] summarized in 1994 the different methods for damage measurement, describing the different techniques available for this purpose. Using experimental data, a comparison between damage measured with a profiler and damage defined by stone counting was carried out, with good agreement for low levels of damage. In addition, the sliding failure of the armor layer was investigated.
- (3)
- Burcharth et al. [79] suggested in 1999 a methodology for scaling core material in small scale rubble mound breakwater models. As it was first identified by Broderick and Ahrens [9], this kind of experiments can be subjected to significant scale effects when the flow type through the model core is different than in prototype: in a Froude small-scale model the flow through the core is usually laminar whereas in a full-scale core the flow is mainly turbulent. Indeed, Hegde and Srinivas [80] demonstrated experimentally that as core porosity is increased the stability is also increased. De Jong [81] ratified the obtention of lower values of damage after scaling the core according to Burcharth’s methodology. Additional information on core permeability and damage, together with prototype data, can be found in Reedijk et al. [82].

- (1)
- Wang and Peene [85] possibly headed in 1990 the first attempt on the development of a fully probabilistic model of rubble mound armor stability, based on the stochastic nature of both wave forces and resistance forces. After pointing out that “the random nature of the resistant force offered by the armor blocks has not been seriously addressed at all,” they examined the behavior of interlocking resistance and the random nature of breakwater stability through laboratory experiments. A theoretical probabilistic model containing six random variables was proposed, which behaviors were not all known at that time, not even hitherto. Instead of computing wave loading in a conventional way, they calculated the resistance of the armor layer by pulling out the units with a motorized lift line (static stability test), recording the force history with a load cell. They applied a modified version of the Kolmogorov-Smirnov D-test on five data sets (with about 120 samples each) with tetrapods and dolosse, after testing different bed slopes, pull-out directions and locations, placement methods and unit sized. They concluded that the resistance of tetrapods and dolosse could be treated as a random variable with a log-normal distribution. Furthermore, comparing these results with the ones carried out with stones, they cast a doubt on the stabilizing effects of interlocking properties of artificial units.
- (2)
- Carver and Wright [86] pointed out in 1991 the random variations in the stability response of stone-armored rubble mound breakwaters after carrying out stability tests with depth limited irregular waves. They concluded that “repeat testing is a must,” indeed repeating each spectrum six or seven times. Also, they registered how the lower stabilities occurred at the lowest values of h/L in shallow water, i.e., regarding the longer wave periods.
- (3)
- Medina [70], as detailed in Section 2.4, recommended in 1996 non-stationary stochastic models for being more adequate for modeling real waves.
- (4)
- Burcharth [87] published in 1997 the Chapter Reliability-Based Design of Coastal Structures as part of a book which summed up the advances in coastal and ocean engineering. Different sources of uncertainty were identified, together with a probabilistic methodology for single failure mode probability analysis, including formulations and examples with Level III methods, Level II methods and FORM (first-order reliability method). Additionally, further examples of probabilistic methodologies applied to breakwater design can be found in Castillo et al. [88,89], Mínguez et al. [90], Tørum et al. [91], or Gouldby et al. [92]. Burcharth [87] also set a discussion on the probability analysis of failure mode systems, typically faced using fault trees. Some practical examples on this topic can be found in van Gent and Pozueta [93], who studied the rear-side stability of rubble mound structures after being overtopped, or Campos et al. [94], who addressed the effects of a fuse parapet failure in other failure modes of a caisson breakwater using Monte Carlo simulations.
- (5)
- Similar to Wang and Peene [85], Hald [8] and Hald and Burcharth [95] also chose an alternative approach away from correlating wave parameters directly to armor layer stability. For this purpose, they studied the wave-induced loading by means of a force transducer connected to an 8mm steel rod attached to a single stone of average size made in coated plastic foam. The rest of the armor layer was made of conventional natural stones. Largest forces were found to take place in a normal direction and upslope, and a dimensionless force model of the normal and the peak force was proposed as a function of wave parameters. A log-normal distribution was found to be suitable for describing the limit pullout forces. From the force model, they derived a stability model in 1998 based on a lifting criterion, obtaining comparable scatter with respect to the equations from Hudson [25] and VdM [50].

_{D=0}. These approaches are helpful for design purposes, but they are not time dependent and they assume starting from a non-damaged structure. In a context where, not only the design was meant to be addressed, but also the evolution of the structure regarding maintenance strategies and useful-life total costs, hydraulic "static" stability concept moved on toward damage progression models. The latter are aimed to predict the evolution of rubble mound’s geometry by means of a quantitative damage parameter, usually the dimensionless erosion area (S).

_{n}, N

_{S}is the stability number based on the highest one-third wave heights from a zero-upcrossing analysis, T

_{m}is the mean period, b is an empirical coefficient introduced for long-duration tests, and a

_{s}is related to breakwater slope, permeability and an empirical coefficient derived from the tests. Therefore, it is an iterative damage progression model that allows the calculation of the damage at the instant t

_{n+1}based on the damage level at the instant t

_{n}and the incident wave conditions between t

_{n}and t

_{n+1}represented by a constant value of H

_{S}and T

_{m}.

#### 2.6. Researches on Breakwater’s Damage in the XXI Century. Castillo et al.’s Model

^{®}by means of pressure clamps. Similar experiments were discussed by Pardo et al. [111] in 2013 after accounting for the three armor randomness indexes (ARIs) introduced by Medina et al. [112]. These experiments were measured with a laser scanner in order to quantify the randomness in the placement of cube and Cubipod

^{®}. Further information on the influence of initial placement can be found in Yagci and Kapdasli [113,114], Gürer et al. [115], Van Buchem [116], or Medina et al. [117]. Recently, Marzeddu et al. [118] analyzed different approaches for taking into account wave storm representation on damage measurements. Also, Clavero et al. [119] proposed a methodology to improve breakwater design and to assimilate the data from different wave flumes after analyzing the bulk wave dissipation in the armor layer of rock and cube armored small-scale models.

_{0}and σ

_{0}depend on the initial conditions. Note that damage is not normal according to the model, but transformed damage, i.e., (D*-γ)1/b, is instead.

## 3. Conclusions

- (1)
- Hydraulic instability of armor layers is a complex process because of the stochastic nature of both wave loading, initiation of movement, and damage progression. The highly non-linear flow over the slope, involving wave breaking, together with the variable shape of armor units and their random placement deals necessarily with a probabilistic concept of damage initiation and progression, as it was pointed out by many authors.
- (2)
- The concept of “damage” in a rubble mound breakwater, understood as the partial or total loss of its functionality, is subjected to different interpretations. Not every structure is designed under the same functional requirements and there is a wide variety of typologies. In addition, not every structure presents the same fragility. For instance, what is considered as damage in single-layered structures might not be equivalent in multi-layered structures. Also, each type of armor unit presents a singular behavior against wave action. These facts complicate the correlation between quantitative damage descriptors and qualitative damage levels such as the ones defined in Vidal et al. [140].
- (3)
- The parameterization and measurement of damage has been just slightly addressed in this paper due to the extensive information available. Indeed, Part II of this review [6] is aimed entirely on this topic. The dimensionless erosion area (S) from Broderick and Ahrens [9] is the most widely damage index nowadays. However, it seems that there is not a worldwide standard on how to measure it, which is crucial for reproducibility and for a consistent comparison between the results from different laboratories. Not only that, regarding the random nature of damage and the frequent accomplishment of damage initiation/progression tests in coastal laboratories, this kind of experiments needs to be reproducible. Thus, a concise methodology ideally agreed by the scientific community and shared worldwide may be helpful.
- (4)
- As exposed in Section 2.4, an adequate selection of wave action parameters is crucial for correlating damage just with the waves of the irregular train directly responsible of the hydraulic instability of the armor layer.
- (5)
- Damage has a spatial component that cannot be completely addressed with the solely characterization of the well-known dimensionless erosion area (S). Most studies aimed to develop empirical or semi-empirical damage formulations are conducted on wave flumes, using physical profilers or visual counting for characterizing damage. However, the recent advances and affordability of scanning and photogrammetric techniques [141] allow nowadays a more complete analysis of the geometrical evolution of the armoring both in wave tank’s models and in full-scale prototypes. Also, the increase in the resolution of global navigation satellite systems and other technological advances such as drones, can help to foster damage monitoring in real structures.
- (6)
- Indeed, taking into account that the structural response is a random variable, its characterization is necessarily linked, not only to repeating experiments in laboratory, but also to monitoring prototypes. Only in this way the models can be properly calibrated.

**Table 1.**Review of hydraulic stability formulae (extended from Hald [8]). See Appendix B for the list of symbols.

RESEARCHER | FORMULA | RESEARCHER | FORMULA |
---|---|---|---|

de Castro (1933) [4] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={\left\{\frac{{\rho}_{w}}{K}{\left(\mathrm{cot}\alpha +1\right)}^{2}\sqrt{\mathrm{cot}\alpha -\frac{2}{{S}_{r}}}\right\}}^{1/3}$ | Rybtchevsky (1964) [23] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={\left(K\frac{H}{L}\right)}^{1/3}\mathrm{cos}\alpha \mathrm{sin}\alpha $ |

Iribarren (1938) [5] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}=\left(\mathrm{cos}\alpha -\mathrm{sin}\alpha \right){\left(\frac{{\rho}_{w}}{K}\right)}^{1/3}$ | Iribarren(1965) [11] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}=\left(\mathrm{tan}\varphi \mathrm{cos}\alpha -\mathrm{sin}\alpha \right){\left(\frac{1}{K}\right)}^{1/3}$ |

Mathews(1948) | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={\left\{K\frac{H}{T}{\left(\mathrm{cos}\alpha -0.75\mathrm{sin}\alpha \right)}^{2}\right\}}^{1/3}$ | Metelicyna (1967) [24] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={K}_{}^{1/3}\mathrm{cos}\left(23\xba+\alpha \right)$ |

Epstein, Tyrrel (1949) [16] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={K}_{}^{1/3}\left(\mathrm{tan}\varphi -\mathrm{tan}\alpha \right)$ | SPM (1973) [27] | $\frac{{H}_{S}}{\mathsf{\Delta}{D}_{n50}}={\left({K}_{D}\mathrm{cot}\alpha \right)}^{1/3}$ |

Hickson, Rodolf (1951) [17] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={\left(K\frac{H}{T}\right)}^{1/3}\mathrm{tan}\left(45\xb0-\frac{\alpha}{2}\right)$ | Losada, Gim.-Curto(1979) [44] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={\gamma}_{w}^{1/3}{\left\{{K}_{1}\left(\xi -{\xi}_{0}\right)\mathrm{exp}\left({K}_{2}\left(\xi -{\xi}_{0}\right)\right)\right\}}^{-1/3}$ |

Larras (1952) [13] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={K}_{}^{1/3}\left(\mathrm{cos}\alpha -\mathrm{sin}\alpha \right)\frac{\mathrm{sinh}\left(4\pi h/L\right)}{\left(2\pi H/L\right)}$ | SPM (1984) [28] | $\frac{{H}_{1/10}}{\mathsf{\Delta}{D}_{n50}}={\left({K}_{D}\mathrm{cot}\alpha \right)}^{1/3}$ |

Hudson, Jackson (1953) [18] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={K}_{}^{1/3}\frac{\left(\mathrm{tan}\varphi \mathrm{cos}\alpha -\mathrm{sin}\alpha \right)}{\mathrm{tan}\varphi}$ | Hedar (1986) [47] | $\frac{{H}_{b}}{\mathsf{\Delta}{D}_{n50}}={\left(\frac{6}{\pi}\right)}^{1/3}\frac{{K}_{2}{f}_{1}\left(\gamma \right)\mathrm{cos}\alpha}{{K}_{1}\left(\frac{{h}_{b}}{{H}_{b}}+0.7\right)\left(\mathrm{tan}\varphi +2\right)}$ ${f}_{1}\left(\gamma \right)=permeabilityfunction$ |

Beaudevin (1955) [14] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={\left\{K\left(\frac{\mathrm{cot}\alpha -0.8}{1.12-0.15\mathrm{cot}\alpha}\right)\right\}}^{1/3}$ | Medina and McDougal (1990) [55] | $\frac{{H}_{S}}{\mathsf{\Delta}{D}_{n50}}=\frac{1.86}{1.27}\sqrt{\frac{2}{\mathrm{ln}{N}_{w}}}{\left({K}_{D}\mathrm{cot}\alpha \right)}^{1/3}$ |

Hudson(1958) [22] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={\left({K}_{D}\mathrm{cot}\alpha \right)}^{1/3}$ | Wang and Peene (1990) [85] | $\frac{\tilde{w}}{{W}_{D}}=\frac{{\left({f}_{c}\right)}_{n}^{3}{\left[{C}_{S}/{C}_{V}\right]}_{n}^{3}{H}_{n}^{3}}{{R}_{n}^{3}{D}_{n50}^{3}{\left[f\left(\theta \right)\right]}_{n}}$ $\begin{array}{l}seeWangandPeene(1990)\\ forthelistofsymbols\end{array}$ |

Goldschtein, Kononenko (1959) [21] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={\left({K}_{}{\mathrm{tan}}^{1.83}\alpha \right)}^{1/3}$ | Koev (1992) [74] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={\left(\left(\frac{\mathrm{cot}{\left(\alpha \right)}^{{K}_{2}}}{{K}_{1}}\right){\left(\frac{H}{L}\right)}^{{K}_{3}}\right)}^{1/3}$ |

SN-92-60(1960) [22] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={\left(K\frac{H}{L}\sqrt{1+{\mathrm{cot}}^{3}\alpha}\right)}^{1/3}$ | Hald and Burcharth (2000) [95] | $\frac{{H}_{S}}{\mathsf{\Delta}{D}_{n50}}={\left(\frac{1}{{K}_{1}}\frac{1+{\mathrm{sin}}^{2}\alpha}{{\mathrm{tan}}^{{K}_{2}}\alpha}\right)}^{\frac{1}{{K}_{3}}}$ |

Svee(1962) [20] | $\frac{H}{\mathsf{\Delta}{D}_{n50}}={K}_{}^{1/3}\mathrm{cos}\alpha $ | - |

**Table 2.**Review of damage progression’s models. See Appendix B for the list of symbols.

RESEARCHER | FORMULA |
---|---|

Van der Meer (1988) [51] | $\frac{{H}_{S}}{\mathsf{\Delta}{D}_{n50}}=\{\begin{array}{l}6.2{\xi}_{0}^{-0.5}{P}^{0.18}{\left(\frac{S}{\sqrt{{N}_{w}}}\right)}^{0.2}\hspace{1em}\to forplungingwaves\\ {\xi}_{0}^{P}\sqrt{\mathrm{cot}\alpha}{P}^{-0.13}{\left(\frac{S}{\sqrt{{N}_{w}}}\right)}^{0.2}\to forsurgingwaves\end{array}$ |

Teisson (1990) [66] | ${N}_{d}\left(t\right)=A{\left({\displaystyle \sum {H}_{S,i}^{B/C}\mathsf{\Delta}{t}_{i}}\right)}^{C}$ |

Kaku et al. (1991) [56] | $S={S}_{e}\left(1-{e}^{-K\cdot {N}_{w}}\right)$ |

Medina (1996) [70] | $D\left(H,T,{N}_{w}\right)={D}_{0}\left(H,T\right)\left[1-{e}^{\frac{-{N}_{w}}{n63\%}}\right]$ |

Melby and Kobayashi (1998) [96] | $\overline{S}\left(t\right)=\overline{S}\left({t}_{n}\right)+{a}_{s}{N}_{s}^{5}\left(\frac{{t}^{b}-{t}_{n}^{b}}{{T}_{m}^{b}}\right);{t}_{n}\le t\le {t}_{n+1}$ |

Melby and Kobayashi (1999) [98] | ${\left[\overline{S}\left(t\right)\right]}^{1/b}={\left[\overline{S}\left({t}_{n}\right)\right]}^{1/b}+{\left({a}_{s}{N}_{s}^{5}\right)}^{1/b}\frac{t-{t}_{n}^{}}{{T}_{m}^{}};{t}_{n}\le t\le {t}_{n+1}$ |

Melby and Hughes (2003) [3] | ${N}_{m}=\{\begin{array}{l}0.5{\left(\frac{S}{\sqrt{{N}_{w}}}\right)}^{0.2}{P}^{0.18}{\left(\mathrm{cot}\alpha \right)}^{0.5}\hspace{1em}\to forplungingwaves\\ 0.5{\left(\frac{S}{\sqrt{{N}_{w}}}\right)}^{0.2}{P}^{0.18}{\left(\mathrm{cot}\alpha \right)}^{0.5-P}{s}_{m}{}^{-P/3}\to forsurgingwaves\end{array}$ |

Van Gent et al. (2003) [61] | $\frac{{H}_{S}}{\mathsf{\Delta}{D}_{n50}}=1.75{\left(\frac{S}{\sqrt{{N}_{w}}}\right)}^{0.2}\sqrt{\mathrm{cot}\alpha}\left(1+\frac{{D}_{n50,core}}{{D}_{n50}}\right)$ |

Gómez-Martín and Medina (2004) [71] | $D\left(H,Ir,{N}_{w}\right)={D}_{0}\left(H,Ir\right)\left[1-{2}^{\frac{-{N}_{w}}{n50\%}}\right]$ |

Vidal et al. (2006) [59] | $\begin{array}{l}\frac{{H}_{50}}{\mathsf{\Delta}{D}_{n50}}=\{\begin{array}{l}4.44{P}^{0.18}{S}^{0.2}{\xi}_{0}^{-0.5}\hspace{1em}\to for{\xi}_{0}{\xi}_{mc}and\mathrm{cot}\alpha \le 4\\ 0.716\sqrt{\mathrm{cot}\alpha}{P}^{-0.13}{S}^{0.2}{\xi}_{0}^{P}\to for{\xi}_{0}\ge {\xi}_{mc}or\mathrm{cot}\alpha \ge 4\end{array}\\ {\xi}_{mc}={\left(6.2{P}^{0.31}\sqrt{\mathrm{tan}\alpha}\right)}^{\frac{1}{P+0.5}}\end{array}$ |

Castillo et al. (2012) [130] | ${F}_{{D}^{\ast}\left({t}^{\ast}\right)}\left(D\right)=\mathsf{\Phi}\left(\frac{{\left(D-\gamma \right)}^{1/b}-{\mu}_{0}-k{t}^{\ast}}{\sqrt{{\sigma}_{0}^{2}+r{t}^{\ast}}}\right)$ |

**Table 3.**Timeline of some of the most relevant models on armor layer’s stability, together with the parameters/properties accounted for and the innovations introduced by each of them. See Appendix B for the list of symbols.

RESEARCHER | Wave Action | Breakwater | Others | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

R | I | T | Ir | N_{w} | h | a | AT | P | D_{0} | DPM | CDF | |

Iribarren (1938) [5] | ||||||||||||

Mathews (1948) | ||||||||||||

Larras (1952) [13] | ||||||||||||

Hudson (1958) [22] | ||||||||||||

SPM (1973) [27] | H_{1/3} | |||||||||||

Losada & G.C. (1979) [44] | ||||||||||||

SPM (1984) [28] | H_{1/10} | |||||||||||

Hedar (1986) [47] | H_{b} | |||||||||||

VdM (1988) [51] | H_{1/3} | |||||||||||

Teisson (1990) [66] | H_{1/3} | |||||||||||

Kaku et al. (1991) [56] | H_{1/3} | |||||||||||

Medina (1996) [70] | H_{i} | |||||||||||

M & K (1998, 1999) [96,98] | H_{1/3} | |||||||||||

G.M.& Medina (2004) [71] | H_{i} | |||||||||||

Van Gent et al. (2003) [61] | H_{1/3} | |||||||||||

Vidal et al. (2006) [59] | H_{50} | |||||||||||

Castillo et al. (2012) [130] | H | |||||||||||

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Special Shaped Armor Units

**Table A1.**Main special shaped armor units. Adapted from The Rock Manual [2].

Armor Unit | Country | Year | Armor Unit | Country | Year |
---|---|---|---|---|---|

Tetrapod | France | 1950 | Seabee | Australia | 1978 |

Tribar | USA | 1958 | Accropode | France | 1980 |

Stabit | UK | 1961 | Shed | UK | 1982 |

Akmon | The Netherlands | 1962 | Haro | Belgium | 1984 |

Tripod | The Netherlands | 1962 | Hollow Cube | Germany | 1991 |

Dolos | South Africa | 1963 | Core-Loc | USA | 1996 |

Cob | UK | 1969 | A-Jack | USA | 1998 |

Antifer Cube | France | 1973 | Cubipod^{®} | Spain | 2006 |

^{®}was recommended. Van Buchem [116] also addressed stability of cubes by means of a packing density parameter.

## Appendix B. List of Symbols

A, B, C | Empirical coefficients (in Table 2) |

ANN | Artificial neural network |

AT | Armor type (in Table 3) |

A_{e} | cross-sectional eroded area |

a_{s} | coefficient of the damage progression model of M&K [98] |

b | coefficient of the damage progression model of M&K [98] and, independently, of Castillo et al. [130] |

CDF | Cumulative distribution function |

D | damage defined in a generic way |

D* | dimensionless damage defined in a generic way in the model of Castillo et al. [130] |

D_{n50} | median nominal diameter or equivalent cube size: D_{n50} = (W_{50}/ρ_{a})^{1/3} |

D_{0} | initial damage |

D*_{0} | initial dimensionless damage in the model of Castillo et al. [130] |

D_{0} (H, T) | Asymptotic maximum damage defined in Medina [70] |

D_{0} (H, Ir) | Asymptotic maximum damage defined in Gómez-Martín and Medina [71] |

DPM | Damage progression model |

ELM | Extreme learning machine |

g | acceleration of gravity |

H | wave height |

H_{b} | wave height at breaking in Hedar [47] |

H_{D = 0} | maximum wave height producing no damage |

H_{n} | average wave height of the n highest waves in a sea state |

H_{1/n} | average wave height of the N/n highest waves in a sea state composed of N waves |

H_{S} = H_{1/3} | significant wave height |

H_{n%} | wave height exceeded by the n% highest waves in a sea state |

h | water depth |

h_{b} | water depth at breaking in Hedar [47] |

I | Irregular waves (in Table 3) |

Ir or ξ = tanα/(H/L)^{0.5} | Iribarren’s number, also referred as surf similarity parameter |

k | wave action parameter in the model of Castillo et al. [130] |

K | empirical coefficient in hydraulic stability models |

K_{D} | empirical coefficient in the hydraulic stability model of Hudson [25] |

K, K_{1}, K_{2}, K_{3} | empirical coefficient in hydraulic stability models. Note that despite using same notation in Table 1, the values of these coefficients are different for each model. |

L | wavelength |

L_{0} = gT^{2}/2π | deep water wavelength |

M&K | Melby and Kobayashi |

n63% | number of regular waves causing 63% of the maximum asymptotic damage (Medina [70]) |

n50% | number of regular waves causing 50% of the maximum asymptotic damage (Gómez-Martín and Medina [71]) |

N_{d} | number of displaced stones |

N_{m} | stability number using wave momentum flux (Melby and Hughes [3]) |

N_{S} = H_{s}/(Δ∙D_{n50}) | stability number |

N_{w} | number of waves |

Probability density function | |

p_{1}, p_{2}, …p_{n} | parameters influencing armor layer stability in Equation 1. |

P | permeability |

r | wave action parameter in the model of Castillo et al. [130] |

R | regular waves (in Table 3) |

S = A_{e}/D^{2}_{n50} | dimensionless erosion area |

S_{e} | equilibrium damage level, introduced by Kaku et al. [56] |

S_{r} = γ_{s} / γ_{w} | submerged-related density |

SWL | Still water level |

t | time |

t* = t/T_{m} | relative duration, also referred as mean number of waves |

T | wave period |

T_{m} | mean wave period |

T_{p} | peak wave period |

VdM | Van der Meer |

W_{50} | armor unit weight exceeded by 50% of the armor units. |

WES | Waterways Experiment Etation |

α | sea-side armor slope |

Δ = (γ_{s} - γ_{w})/ γ_{w} | relative excess specific weight |

γ | breakwater parameter in the model of Castillo et al. [130] |

γ_{w} =ρ_{w∙g} | specific weight of water |

γ_{s} = ρ_{s∙g} | specific weight of armor units |

φ | internal friction angle |

μ_{0} | Initial mean damage parameter in the model of Castillo et al. [130] |

σ_{0} | Initial standard deviation damage parameter in the model of Castillo et al. [130] |

ξ_{0} =tanα/(H/L_{0})^{0.5} | deep water Iribarren’s number |

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**MDPI and ACS Style**

Campos, Á.; Castillo, C.; Molina-Sanchez, R.
Damage in Rubble Mound Breakwaters. Part I: Historical Review of Damage Models. *J. Mar. Sci. Eng.* **2020**, *8*, 317.
https://doi.org/10.3390/jmse8050317

**AMA Style**

Campos Á, Castillo C, Molina-Sanchez R.
Damage in Rubble Mound Breakwaters. Part I: Historical Review of Damage Models. *Journal of Marine Science and Engineering*. 2020; 8(5):317.
https://doi.org/10.3390/jmse8050317

**Chicago/Turabian Style**

Campos, Álvaro, Carmen Castillo, and Rafael Molina-Sanchez.
2020. "Damage in Rubble Mound Breakwaters. Part I: Historical Review of Damage Models" *Journal of Marine Science and Engineering* 8, no. 5: 317.
https://doi.org/10.3390/jmse8050317