# Peridynamic Analysis of Rail Squats

^{1}

^{2}

^{*}

## Abstract

**:**

## Featured Application

## Abstract

## 1. Introduction

## 2. Methods

#### 2.1. State-Based Peridynamic Theory

#### 2.2. Computational Model

#### 2.3. Model of a Rail

^{3}, Poisson’s ratio 0.3, and Young’s modulus 189.9 GPa, obtained from [40]. The LPS model is the peridynamic equivalent to the elastic material model in continuum mechanics. It has been selected because the applied loads do not cause the material to exceed its yield strength.

^{−7}m. It should have no effect on the PD model’s accuracy.

#### 2.4. Fatigue Damage Model Parameters

^{10}N/m

^{3}(equivalent to 50 MPa) has been applied to nodes within one $\delta $ of both the top and bottom, and damage is disabled for nodes within $3\delta $ from the top and bottom, to avoid unphysical behavior near the boundary conditions. Crack growth speed data only from phase II are required, so switching to phase II at low damage reduces the simulation time. The damage required for transition from phase I to phase II has been, therefore, set to 0.017. For the trial simulation, ${A}_{II}^{\prime}=1e6$ and ${m}_{II}=\mathrm{4.00.}$ An LPS material model with the same parameters as for the rail head simulation is used. The first simulation (with ${A}_{II}^{\prime}$) ran for 163,100 cycles, after which the crack turned upward, so Equation (23) is no longer accurate; the second simulation runs for 13,275,999 cycles until the crack splits in two. Figure 7 shows the simulation with ${A}_{II}$ at cycle 309,999 (top) and step 13,275,999 (bottom). The number of cycles is large because a low applied stress causes fatigue damage to increase slowly.

#### 2.5. Boundary Conditions

^{3}, from the elastic pressure (data from Figure 5f in [48]) can be computed from a modified ellipsoid’s formula:

^{3}), $x,z$ are node coordinates (m), and $h$ is the node size (m). Since loads are applied to a $1\delta $ (three node spacings) thick layer, the computed value at a position $\left(x,z\right)$ has been divided by 3 and applied to each of three nodes under this position.

## 3. Results

## 4. Discussion and Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Rail surface defects: (

**a**) white etching layer (WEL)-related rail studs (multiple studs); (

**b**) a rolling contact fatigue (RCF)-related rail squat (single squat).

**Figure 2.**The most conservative two-dimensional (2D) case when it could be thought that a crack has appeared. Some bonds are drawn curved to avoid overlapping.

**Figure 3.**A model of a rail: (

**a**) an Ansys model with solid elements; (

**b**) peridynamic (PD) mesh-free discretization with the load area highlighted.

**Figure 4.**The displacement in the cross-section of an undamaged model: (

**a**) the Y displacement finite element (FE) model; (

**b**) the Y displacement PD model; (

**c**) the X displacement FE model; (

**d**) the X displacement PD model. Deformations are increased 50 times.

**Figure 7.**A single edge notch (SEN) specimen at: (

**a**) 3,099,999 cycles; (

**b**) 13,275,999 cycles. Displacements are increased 10 times.

**Figure 8.**Half of the load area divided into four parts with a tri-linear function for each part. Functions describe the shear traction stress values in the load area. The other half of the load area is a mirror image. The axis directions and node size are the same as in the rail head’s model.

**Figure 9.**Surface shear traction data from [48] (shown with symbols) and the tri-linear functions used to describe the shear traction values in the load area.

**Figure 10.**The cross-section (x-y plane) along the middle of the rail head in the longitudinal direction. Damage is shown in the top part of the model after: (

**a**) 37,000; (

**b**) 42,500; (

**c**) 42,850; and (

**d**) 42,884 cycles.

**Figure 11.**The cross-section (x-z plane) along the middle of the rail head in the transversal direction. Damage is shown in the top half of the model after: (

**a**) 37,000; (

**b**) 42,500; (

**c**) 42,850; and (

**d**) 42,884 cycles.

**Figure 12.**An ultrasonic rail squat measurement: (

**a**) crack depths at each grid point; (

**b**) top view of the rail surface.

Parameter | Volume, m^{3} | % Difference |
---|---|---|

Cubic | 1.25000 × 10^{−10} | 0.00% |

Min | 1.18960 × 10^{−10} | −4.83% |

Max | 1.29750 × 10^{−10} | 3.80% |

Average | 1.26464 × 10^{−10} | 1.17% |

**Table 2.**Maximum and minimum displacement values in the X and Y directions from the finite-element (FE) and Peridynamic (PD) simulations.

Value | X | Y | ||||
---|---|---|---|---|---|---|

FE, m | PD, m | Difference | FE, m | PD, m | Difference | |

Max | 2.03 × 10^{−5} | 1.90 × 10^{−5} | −6.95% | 2.35 × 10^{−6} | 2.55 × 10^{−6} | 7.92% |

Min | −6.59 × 10^{−7} | −8.30 × 10^{−7} | 20.61% | −4.69 × 10^{−5} | −5.07 × 10^{−5} | 7.48% |

Phase I | Phase II | |
---|---|---|

A | 426.00 | 25,237.48 |

m | 2.77 | 4.00 |

ε_{∞} | 0.00186 | -- |

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**MDPI and ACS Style**

Freimanis, A.; Kaewunruen, S. Peridynamic Analysis of Rail Squats. *Appl. Sci.* **2018**, *8*, 2299.
https://doi.org/10.3390/app8112299

**AMA Style**

Freimanis A, Kaewunruen S. Peridynamic Analysis of Rail Squats. *Applied Sciences*. 2018; 8(11):2299.
https://doi.org/10.3390/app8112299

**Chicago/Turabian Style**

Freimanis, Andris, and Sakdirat Kaewunruen. 2018. "Peridynamic Analysis of Rail Squats" *Applied Sciences* 8, no. 11: 2299.
https://doi.org/10.3390/app8112299