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The common failure mechanism for brittle rocks is known to be axial splitting which happens parallel to the direction of maximum compression. One of the mechanisms proposed for modelling of axial splitting is the sliding crack or so called, “wing crack” model. Fairhurst-Cook model explains this specific type of failure which starts by a pre-crack and finally breaks the rock by propagating 2-D cracks under uniaxial compression. In this paper, optimization of this model has been considered and the process has been done by a complete sensitivity analysis on the main parameters of the model and excluding the trends of their changes and also their limits and “peak points”. Later on this paper, three artificial intelligence algorithms including Particle Swarm Intelligence (PSO), Ant Colony Optimization (ACO) and genetic algorithm (GA) has been used and compared in order to achieve optimized sets of parameters resulting in near-maximum or near-minimum amounts of wedging forces creating a wing crack.

The study of rock failure under compression is a matter of importance in rock mechanics. Two main mechanisms, including axial splitting and shear failure, have been identified for this type of rock failure [

Splitting parallel to the direction of maximum compression is a common type of macroscopic fracture of a brittle rock in the vicinity of its surface. Modelling of brittle failure is one of the greatest challenges of material failure analysis which results from the irreversible and very rapid propagation and connection of cracks, in a process called fracturing.

Materials like natural ice and rock are heterogeneous and crystalline with different behaviors under variation of applying forces. Generally, their behavior can be categorized in two main types, including ductile behavior under high confining pressures, and brittle behavior under low confining pressures [

One of the main mechanisms proposed for modelling of axial splitting is the sliding crack or “wing crack” model which was originally proposed for a 2D crack in a plate [

The ultimate rock strength and the orientation of the macroscopic failure plane depend on the confining pressure applying on the rock. Some models for micro-cracking under compression (e.g. [

As discussed before, under uniaxial compression, the tension cracks grow at a sharp angle relative to the orientation of overall failure plane with confining pressure. Variation of this “crack angle” changes the amount of wedging force with specific but different trends, resulting different amounts of opening in wing cracks and “ultimate strengths” for rocks to fail under compression. Excluding these trends is another purpose of this paper.

As it can be seen in

The amount of this wedging force (f) which creates the wing crack can be determined from the following equation:

f = 2.a.σ.β(α) (1)

where “σ” represents the maximum principal stress, “a” represents the crack amplitude, “f” represents the wedging force and “β(α)” is a function of crack angle which can be determined from the following equation:

β(α) = sin^{2}(α).cos(α).(1-tan(α).tan(μ)) (2)

In the above equation, “μ” represents the internal frictional angle of the rock.

In this study, a complete sensitivity analysis performed to find the optimized amounts of parameters in Fairhurst-Cook Model. Three main parameters

including crack amplitude (a), crack angle (α) and maximum principal stress (σ) and a fixed amount of 89 degrees for internal friction angle (μ) have been considered in this sensitivity analysis. According to this model, these parameters have the main roles in determining the amount of wedging force which creates a two-dimensional wing crack, while the internal friction angle (μ) should be determined according to the rock sample which is being cracked.

Three different stages have been considered in the sensitivity analysis process. In the first stage of this process, crack amplitude analyzed and this process repeated for the second and third stages by analyzing crack angle (α) and maximum principal stress (σ). Performing this sensitivity analysis in Matlab environment, results in

As it can be seen in

As it is shown in

Another fact which can be excluded from

In the next stage of analyzing, the change of wedging force creating wing crack by variation of crack angle has been examined. For this purpose, one of the curves in

angle (α = 37.7˚) acts like a border line which limits the increase of wedging force by increasing the amount of the relevant parameter.

For the third stage of this sensitivity analysis which includes analyzing the wedging force variation by changing the maximum principal stress (σ), three different amounts of maximum principal stress considered in a fixed amount of crack amplitude (a). In this stage, the variation of wedging force by changing the crack angle in different amounts of maximum principal stress has been investigated and by its excluded trends, plotted in

As it can been seen in

According to this sensitivity analysis, optimized amounts of each parameter by fixing other relevant parameters have been excluded, but finding the optimum amounts of parameters without fixing others is much more important. For this reason, using some of the efficient algorithms of artificial intelligence for optimization is suggested. In this research three algorithms including Ant Colony Optimization (ACO), Particle Swarm Intelligence (PSO) and Genetic Algorithm (GA) were used and their results have been compared in order to find the most efficient method in general or separate parts (before and after the peak

point angle) and also to achieve the more reliable results.

Ant colony optimization (ACO) is a population-based metaheuristic algorithm which can be used to find approximate solutions to hard and discrete optimization problems. ACO is an algorithm for finding optimal paths that is based on foraging behavior of some ant species searching for food [

Particle Swarm Intelligence originally formed based on the movement of organisms in animal swarms such as bird flocks or fish schools to simulate their social behavior [

This algorithm can be summarized in four main steps, which are repeated until the stopping condition is satisfied:

・ Assigning initially random positions and velocities for all of the particles in the search-space

・ Evaluation of the fitness of each individual particle

・ Updating the individual and global best positions

・ Updating the velocity and position of each particle [

In past several years, PSO Algorithm has been successfully applied in many researches and different application areas. It is demonstrated that PSO algorithm gives better results in a faster, cheaper way compared with other methods [

Genetic Algorithm (GA) is a heuristic optimizer algorithm based on the evolutionary ideas of natural selection and principles of “survival of the fittest” from “Charles Darwin”. Genetic Algorithms are commonly used to generate proper solutions to optimization problems by relying on bio-inspired operators such as “crossover” and “mutation”.

GA simulates the natural selection process among individuals (also called “phenotypes”), each of which representing a possible solution. Each possible solution has a set of properties and is known as a “chromosome” (also called “genotype”). Representing the solutions as binary strings of 0 s and 1 s are more common, but other types such as “real strings” are also applicable. The initially created individuals then lead through the process of evolution [

Starting with a randomly generated population of chromosomes, the evolution occurs as a process of fitness based selection and recombination to produce a better population in each iteration called a “generation” [

In each generation, the fitness of every individual in the population which is usually the value of the objective function in the problem, is being evaluated and the GA creates a new population by a new group of chromosomes with resulted fitness values. In this process, firstly, parents are selected to mate, based on their fitness, producing “offspring”, so better solutions with more fitness are given better chance to reproduce by crossover operation. The offspring inherit characteristics from both parents, but not equally. As parents mate and produce offspring, some new rooms must be freed for the newly generated chromosomes.

Since the population contains more information than each individual fitness, GA combines the good information hidden in a solution with good information from another one in the mating pool, in order to produce new solutions with good information inherited from both parents [

Each generation will contain, on average, more good genes than the previous one. Once the population is not producing much better solutions than previous generations, the algorithm is said to have converged to a specific set of solutions for the problem. Eventually, the algorithm terminates when either it converged to a proper solution with satisfactory fitness level or a specific number of generations has been produced [

As it has been discussed earlier, three different optimization algorithms including Ant Colony Optimization (ACO), Particle Swarm Intelligence (PSO) and Genetic Algorithm (GA) have been used in order to find the optimum or near optimum parameters influencing the wedging force which creates a wing crack. This optimization has been performed in several cases including general maximizing and minimizing of the wedging force and also separate maximizing and minimizing of this parameter, isolated in before and after the peak point angle (37.7˚).

The First algorithm was ACO and it has been performed by a colony of 46 ants trying to find the optimum amounts of crack angle (α), crack amplitude (a), maximum principal stress (σ) and frictional angle (μ) in 100 iterations. Two specific upper and lower limits have been considered for each parameter in order to specify their valid range of variation. This range was from 0 to 120 degrees for

crack angle, from 0 to 89.999 degrees for frictional angle, from 0.001 to 0.046 meters for crack amplitude and finally 10 to 100 mega-Pascal for maximum principal stress. The results of this optimization are illustrated in

The second optimization algorithm was Particle Swarm Intelligence (PSO) and it has been performed in 50,000 iterations with 10,000 swarms and 4 particles for each swarm, representing crack angle (α), crack amplitude (a), maximum principal stress (σ) and frictional angle (μ). Just like the previous algorithm, two specific upper and lower limits with the same amounts have been considered for each parameter in order to specify their valid range of variation. The results of this optimization are also illustrated in

The third and final optimization algorithm was Genetic Algorithm (GA). In this study, real genetic algorithm used and it has been performed with 200 chromosomes and 4 genes for each chromosome. These genes represented the related parameters including crack angle (α), crack amplitude (a), maximum principal stress (σ_{1}) and frictional angle (μ) and each chromosome represented a resulting amount of wedging force (f). The optimization process repeated for 4000 iterations with the same upper and lower limits for different parameters and like the previous stages, its results is illustrated in

These steps repeated several times, in order to achieve better results of wedging force without exiting the valid range for each parameter and as it can be seen in this table, Particle Swarm Intelligence (PSO) algorithm achieved better results with more amount of wedging force at the end. So these amounts of parameters will be more proper to use as the optimum amounts and this method is more efficient in general maximizing of wedging force in wing crack model comparing to the ACO and GA algorithms. These steps have been repeated for the general minimization stage with the same ranges for the parameters. The results are shown in

Crack Angle (α) [deg] | Crack Amplitude (a) [m] | Maximum Principal Stress (σ) [Mpa] | Frictional Angle (μ) [deg] | Wedging Force (f) [KN] | |
---|---|---|---|---|---|

Ant Colony Optimization (ACO) | 34.658 | 0.045 | 100.000 | 26.000 | 1586.823 |

Particle Swarm Intelligence (PSO) | 55.393 | 0.045 | 96.812 | 0.0886 | 3266.574 |

Genetic Algorithm (GA) | 52.874 | 0.043 | 97.009 | 1.977 | 3055.086 |

Crack Angle (α) [deg] | Crack Amplitude (a) [m] | Maximum Principal Stress (σ) [Mpa] | Frictional Angle (μ) [deg] | Wedging Force (f) [MN] | |
---|---|---|---|---|---|

Ant Colony Optimization (ACO) | 118.637 | 0.045 | 100.000 | 89.000 | −0.352 |

Particle Swarm Intelligence (PSO) | 102.752 | 0.014 | 86.183 | 89.996 | −34.347 |

Genetic Algorithm (GA) | 93.887 | 0.044 | 79.444 | 89.999 | −397.802 |

As it can be understood from

For better understanding and easier access to the best algorithms and optimum amounts of parameters, a summary of these methods and results are shown in

In the first part of this study, a complete sensitivity analysis performed on the main parameters of Fairhurst-Cook Model to determine their relations and

State relative to the peak point angle (37.7˚) | Crack Angle (α) [deg] | Crack Amplitude (a) [m] | Maximum Principal Stress (σ) [Mpa] | Frictional Angle (μ) [deg] | Wedging Force (f) [KN] | |
---|---|---|---|---|---|---|

Particle Swarm Intelligence (PSO) | Before | 36.684 | 0.046 | 95.710 | 12.878 | 2082.467 |

Genetic Algorithm (GA) | Before | 36.568 | 0.043 | 97.994 | 1.412 | 2358.560 |

Particle Swarm Intelligence (PSO) | After | 55.393 | 0.045 | 96.812 | 0.0886 | 3266.574 |

Genetic Algorithm (GA) | After | 52.874 | 0.043 | 97.009 | 1.977 | 3055.086 |

State relative to the peak point angle (37.7˚) | Crack Angle (α) [deg] | Crack Amplitude (a) [m] | Maximum Principal Stress (σ) [Mpa] | Frictional Angle (μ) [deg] | Wedging Force (f) [MN] | |
---|---|---|---|---|---|---|

Particle Swarm Intelligence (PSO) | Before | 28.640 | 0.044 | 46.570 | 89.948 | −0.490 |

Genetic Algorithm (GA) | Before | 37.515 | 0.045 | 98.897 | 89.998 | −72.087 |

Particle Swarm Intelligence (PSO) | After | 102.752 | 0.014 | 86.183 | 89.996 | −34.347 |

Genetic Algorithm (GA) | After | 93.887 | 0.044 | 79.444 | 89.999 | −397.802 |

Case | Best Algorithm | Crack Angle (α) [deg] | Crack Amplitude (a) [m] | Maximum Principal Stress (σ) [Mpa] | Frictional Angle (μ) [deg] | Wedging Force (f) [KN] |
---|---|---|---|---|---|---|

General Maximizing | PSO | 55.393 | 0.045 | 96.812 | 0.0886 | 3266.574 |

General Minimizing | GA | 93.887 | 0.044 | 79.444 | 89.999 | −397,802 |

Maximizing Before The Peak Point Angle | GA | 36.568 | 0.043 | 97.994 | 1.412 | 2358.560 |

Maximizing After The Peak Point Angle | PSO | 55.393 | 0.045 | 96.812 | 0.0886 | 3266.574 |

Minimizing Before The Peak Point Angle | GA | 37.515 | 0.045 | 98.897 | 89.998 | −72,087 |

Minimizing After The Peak Point Angle | GA | 93.887 | 0.044 | 79.444 | 89.999 | −397,802 |

trends of change. Concluded results from this sensitivity analysis are summarized as below:

・ The trend of changes for each of these parameters reversed in a specific amount of crack angle (α) as a “peak point”.

・ This peak point is about 60.7 degrees in the analysis of crack amplitude and also maximum principal stress and it is about 37.7 degrees in crack angle analysis.

・ In all of above cases, straight relations between that specific parameter and the amount of wedging force observed before the peak point, but it has changed to reverse relations after that point.

・ The peak point virtually acts like a border line and limits the increase of wedging force by increasing the crack angle.

・ The reason for the reversion of wedging force sign is thought to be the reversion of force from compression to tensile with the same magnitude order; which means the biggest force still remains the biggest one, but in a reverse direction and it happens because of the changes in crack angle, making an angle of more than 90 ̊ between the direction of wedging force and its applying area.

Knowing the amount of peak points can be helpful for understanding the ultimate rock strength under uniaxial compression and also to apply optimum wedging force on a rock sample and create a wing crack. In order to achieve this optimum wedging force, a set of optimum amounts of related parameters should be used. For this purpose, three different optimizer algorithms (e.g. ACO, PSO and GA) were used and compared to identify the best algorithms and optimum results. Although it is not guaranteed that these resulted amounts of parameters are complete optimum, but they are expected to have proper applications as the local and near-global optimums with satisfactory resulting wedging force to create a wing crack.

The authors declare no conflicts of interest regarding the publication of this paper.

Najjarpour, M. and Jalalifar, H. (2018) Optimization of Fairhurst-Cook Model for 2-D Wing Cracks Using Ant Colony Optimization (ACO), Particle Swarm Intelligence (PSO), and Genetic Algorithm (GA). Journal of Applied Mathematics and Physics, 6, 1581-1595. https://doi.org/10.4236/jamp.2018.68134

a = crack amplitude

f = wedging force

l = crack semi-length

α = crack angle

β(α) = model function depending on crack angle

σ = maximum principal stress

μ = internal friction angle