#
Numerical Model Study of Prototype Drop Tests on Cube and Cubipod^{®} Concrete Armor Units Using the Combined Finite–Discrete Element Method

^{1}

^{2}

^{*}

## Abstract

**:**

^{®}, used on rubble-mound breakwaters and coastal structures, through a numerical methodology using the combined finite–discrete element method (FDEM). A numerical modeling methodology is developed to reproduce the results of an experimental examination published by Medina et al. (2011) of a free-fall drop test performed on a 15 t conventional Cubic block and a 16 t Cubipod

^{®}unit. The field results of the Cube drop tests were used to calibrate the model. The numerically simulated response to the Cubipod

^{®}test is then discussed in the context of a validation study. The calibration process and validation study provide insights into the sensitivity of breakage to tensile strength and collision angle, as well as a better understanding of the crushing and cracking damage of this unit under drop test impact conditions.

## 1. Introduction

_{D}or similar stability factor, but also the structural integrity that should be taken into consideration for CAU selection [8], because no matter how high the packing density or the placing tolerances achieved, the possibility of a significant number of rocking units under storm conditions cannot be discounted. The scientific and engineering community is, therefore, looking for a way to select the type of CAU that better balances hydraulic stability and structural integrity. This is because the maximum stress within a concrete unit that may cause it to crack is not necessarily linked directly to the ‘averaged’ hydrodynamic loads assumed from knowledge of the incoming wave parameters or the static contact forces from the weight of overlying or neighbor units. Rather, it is likely to be a consequence of stochastic time and space variations in unit stability and the impact loads caused by the unavoidable collisions between CAUs while rocking or rolling under storm conditions. Where and when hydraulic loads from drag, inertia, and buoyancy exceed the combined resistance forces of its weight and the frictional and interlocking contact forces mobilized with neighbor units, locally varying unit displacements and, hence, rocking and rolling impacts may occur. The proportion of units exhibiting rocking, therefore, depends on the storm event severity, the inherent conservatism in the design, the type of units, and the construction quality achieved in meeting the unit designer’s construction specification. In one extreme example in the early days of deploying Core-Locs and before tighter packing densities were adopted for Core-Loc structures, the small Port St Francis breakwater suffered 35 broken units from 800. The high breakage was mostly attributed to the storm arriving during construction and the low dimensionless packing density of 0.58 adopted [9].

## 2. Experimental Prototype Drop Tests Provided by Medina et al. (2011)

## 3. The Finite–Discrete Element Method

**Solidity**, which is a leading combined finite-discrete element method (FDEM) code that has discrete fracturing modelling capabilities. Solidity uses a GUI preprocessor,

**GiD**(CIMNE)2 to apply boundary conditions and material properties, as well as to generate tetrahedral meshes for the FDEM solver. In this study, Solidity is used to simulate the behavior of CAUs during free-fall tests, using a three-dimensional fracture model, while

**Paraview**is the software package used in post-processing.

## 4. Prototype Drop Test: Numerical Simulation

- Velocity-constrained boundary conditions were applied to the base of the plate. The bottom surface of the base was restrained to zero velocity in all directions. These conditions guaranteed that there was no rigid body motion of the base. Of course, it is just a simplification of reality. In the field tests, after the fall of the units, the plate gets damaged and moves.
- The cohesive zone fracture model was applied to the brittle breakable CAU blocks, whereas a viscoelastic constitutive model was applied to represent the behavior of the unbreakable platform.
- The acceleration of gravity g was set to be 9.8 m/s
^{2}. - To set up the drop height of the units, the unit was positioned very close (~1 mm) above the platform, and the impact speed in m/s was then assigned using the following formula coming from an energy balance:

_{n}and the shear displacement δ

_{s}between triangular surfaces N

_{1}N

_{2}N

_{3}and N

_{4}N

_{5}N

_{6}, respectively, were calculated according to the constitutive law presented in Figure 5.

_{t}, and, for shear stress τ, it means shear strength f

_{s}. A Mohr–Coulomb criterion with a tension cutoff was used to determine the shear strength on the basis of the normal stress acting perpendicular to the shear direction [14,30]. Fracture energy G

_{f}is a material property, which defines the energy needed for fracture surface to propagate per unit area, and it is the area under the graph (Figure 5).

^{−7}m

^{3}. Due to its greater volume, the Cubipod had 607,059 elements, with a minimum element edge of 0.0141 m and a minimum volume of 5.87 × 10

^{−7}m

^{3}.

^{−8}s.

## 5. Calibration

- Angle of impact,
- Tensile strength of the CAU,
- Platform stiffness.

#### 5.1. The Influence of the Angle of Drop on the Free-Fall Test Results

- 1.5°,
- 2°,
- 2.5°.

#### 5.2. The Influence of the Tensile Strength on the Free-Fall Test Results

- 2.5 MPa,
- 3 MPa,
- 3.5 MPa,
- 4.5 MPa.

## 6. Free-Fall Test Results

^{2}) generated from different heights of drop (cm) at the end of the test for both the Cube and the Cubipod unit.

#### 6.1. Cube Free-Fall Test Results

_{1}(Pa) in the cut plane perpendicular to the y-direction, where tensile stress is positive and compressive stress is negative. The right-hand column shows the three-dimensional fracture development in the Cube unit, where the red color represents fractured surfaces.

_{1}, it was possible to observe how the edge of the Cube experienced compression and the stress wave was at the point of spreading inside the unit (Figure 15). In frame 84, it is worth noting that the fractures along the edge continued to develop but there were new vertical cracks along the y-direction on the bottom side of the Cube (Figure 16). In frame 110, new fractures were generated parallel to the x-direction (Figure 17). Thus, three different main directions of fracture were revealed. The velocity and stress trends were the same until frame 170, in which the right-hand edge of the Cube increased its velocity (Figure 18). Note that when the Cube dropped, at the beginning, it reduced its velocity; however, later, it bounced off the base and the upper part could again move with increasing velocity. In frame 200, all different types of fractures were linked with each other. The main fracture damage was in the area subjected to tensile stress where pieces of concrete split from the unit (Figure 19 and Figure 20). Finally, in the last frame (249), in the 2 m drop with tensile strength of 4 MPa, the 45° fracture planes were clearly seen in embryonic form but not fully traversing (Figure 21). By increasing the drop height above 2.25 m, it is considered highly probable that the fully traversing fracture surfaced at 45° would have become evident.

#### 6.2. Cubipod Free-Fall Test Results

## 7. Validation Discussion

## 8. Concluding Remarks

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CAU | Concrete armour unit |

FEA | Finite element analysis |

FDEM | Finite–discrete element method |

AD | Anvil drop |

RLM | Relative loss of mass |

FEM | Finite element method |

DEM | Discrete element method |

h | Height of fall |

g | Acceleration of gravity |

v | Impact speed |

δ_{n} | Normal displacement |

δ_{s} | Shear displacement |

σ | Normal stress |

τ | Shear stress |

f | Peak stress |

f_{t} | Tensile strength |

f_{s} | Shear strength |

Δt | Time step |

G | Fracture energy |

μ | Friction coefficient between the units and the base |

ρ | Density |

E | Young’s modulus |

ν | Poisson’s ratio |

η | Mass damping coefficient |

c | Cohesion |

φ | Internal friction angle |

l | Minimum edge length |

σ_{1} | Maximum principal stress in the cut plane perpendicular to the z-direction |

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**Figure 1.**(

**a**) Dimensions (mm) of the 16 t Cubipod®; (

**b**) relative loss of mass (RLM) (%) in Cube and Cubipod® prototype drop tests.

**Figure 2.**(

**a**) Cube and Cubipod prototype position just before being released during the AD test; (

**b**) setup of the 15 t Cube and 16 t Cubipod of the AD numerical test as the unit strikes the steel anvil base.

**Figure 9.**Last frame of a Cube AD test with a tensile strength of 2.5 MPa, axes convention: z-vertical (yellow), y-horizontal (green), x-horizontal (red).

**Figure 13.**Trend of the fractured area (m

^{2}) with the height of drop (cm) of the Cube and the Cubipod.

**Figure 23.**Frame 1, t = 3.3 × 10

^{−8}s. Axes convention: z-vertical (green), y-horizontal (yellow), x-horizontal (red).

**Figure 31.**Frame 249, t = 8.217 × 10

^{−6}s alternative views. Top view of a section perpendicular to z-direction (

**left**); front view perpendicular to y-direction showing surface visible cracks (

**right**).

**Figure 33.**Comparison of fractures generated at the end of the tests of the 15 t Cube from a height of 2 m. (

**a**) Frames of the sample C-03 obtained from videos (Supplementary Materials); (

**b**) last frame (259) of the numerical simulation.

Material Types | Concrete | Steel |
---|---|---|

Density ρ (kg·m^{−3}) | 2340 | 7850 |

Young’s modulus E (GPa) | 35 | 200 |

Poisson’s ratio υ | 0.3 | 0.28 |

Mass damping coefficient η | 1.10 × 10^{5} | 2.40 × 10^{4} |

Penalty number (Pa) | 1.75 × 10^{10} | 1.00 × 10^{11} |

Tensile Fracture Energy (J·m^{−2}) | 190 | - |

Shear Fracture Energy (J·m^{−2}) | 350 | - |

Tensile strength (MPa) | 4 | - |

Cohesion c (MPa) | 16.5 | - |

Internal friction angle φ (°) | 30 | - |

1 | Anvil drop (AD); the prototype is dropped with its bottom face as close to parallel to the platform as is practically possible. |

2 | The licensor is CIMNE and AMCG group (Applied Modeling and Computation Group) at Imperial College London is the licensee. |

3 | Catastrophic failure means that the unit splits into two parts along the main fracture path. |

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**MDPI and ACS Style**

Scaravaglione, G.; Latham, J.-P.; Xiang, J. Numerical Model Study of Prototype Drop Tests on Cube and Cubipod^{®} Concrete Armor Units Using the Combined Finite–Discrete Element Method. *J. Mar. Sci. Eng.* **2021**, *9*, 460.
https://doi.org/10.3390/jmse9050460

**AMA Style**

Scaravaglione G, Latham J-P, Xiang J. Numerical Model Study of Prototype Drop Tests on Cube and Cubipod^{®} Concrete Armor Units Using the Combined Finite–Discrete Element Method. *Journal of Marine Science and Engineering*. 2021; 9(5):460.
https://doi.org/10.3390/jmse9050460

**Chicago/Turabian Style**

Scaravaglione, Giulio, John-Paul Latham, and Jiansheng Xiang. 2021. "Numerical Model Study of Prototype Drop Tests on Cube and Cubipod^{®} Concrete Armor Units Using the Combined Finite–Discrete Element Method" *Journal of Marine Science and Engineering* 9, no. 5: 460.
https://doi.org/10.3390/jmse9050460