# Updated Smoothed Particle Hydrodynamics for Simulating Bending and Compression Failure Progress of Ice

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

#### 2.1. Ice Elasto-Plastic Constitutive Model

## 3. Failure Model in the SPH Framework

#### 3.1. Drucker–Prager Model

#### 3.2. Numerical Errors in Computational Plasticity

#### 3.3. Cohesion Softening

## 4. SPH Formulations and Corrective SPH Method

#### 4.1. The Particle Approximation and Spatial Derivatives of SPH

#### 4.2. Artificial Stress Method

#### 4.3. Boundary Conditions

#### 4.4. Corrective SPH Method

- (1)
- Calculate the values of ${\dot{\epsilon}}^{\alpha \beta}$ and ${\dot{\sigma}}^{\alpha \beta}$ from Equations (47), (17) or (19).
- (2)
- Calculate the stress components ${\sigma}^{\alpha \beta}$ based on the obtained stress rate ${\dot{\sigma}}^{\alpha \beta}$.
- (3)
- Check the stress state and judge whether the corresponding stress need to be corrected: if $-{\alpha}_{\varphi}{I}_{1}^{n}+{k}_{c}<\sqrt{{J}_{2}^{n}}$, the stress need to be modified by Equation (25).
- (4)
- Implement Cohesion softening model based on Equation (27).

## 5. Numerical Simulations

#### 5.1. Elastic Vibration of a Cantilever Beam

^{7}N/m

^{2}, the Poisson’s ratio is $\upsilon =0.3$, and the mass density is P = 1 kg/m

^{3}. External excitation force P = 1000$g(t)$ and $g(t)$ is a function related to time.

^{−1}. Figure 7 shows the comparison of the displacement in y direction of the free end of the cantilever beam ($y$) between the SPH and SPH_SFDI results with 10,000 particles and the finite element method (FEM) solution from Long [45]. This shows that the displacement time histories computed by the SPH_SFDI method shares a better agreement with the FEM data than the SPH result.

#### 5.2. Four-Point Pending of Ice Beam

#### 5.3. Uniaxial Compressive Test of Ice Specimen

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 3.**The cohesion softening results by the different softening coefficient $k$ in the uniaxial compression test in section 5.2: (

**a**) the relationship between the cohesion and accumulated plastic strain; and (

**b**) the corresponding stress–strain curves.

**Figure 5.**The stress comparison of theoretical value and numerical results: the traditional formula (Equation (46)) and the SFDI scheme (Equation (47)).

**Figure 8.**The time histories of displacement $y$ obtained by: (

**a**) SPH and (

**b**) SPH_SFDI with different particle numbers.

**Figure 12.**Enlarged partial views of accumulated plastic strain of the failure path in the ice beam: (

**a**) standard SPH; and (

**b**) SPH_SFDI.

**Figure 14.**Comparison of force versus time plot of the ice beams by numerical results and experimental data with different velocities of two moving upward supports: (

**a**) ${V}_{1}$ = 0.001842 m/s; and (

**b**) ${V}_{2}$ = 0.003225 m/s.

**Figure 15.**Comparisons of the force time histories among experimental data and SPH_SFDI results with different flow rules.

**Figure 16.**Snapshot of the failure paths of accumulated plastic strain in SPH_SFDI results with different flow rules: (

**a**) associative flow rule; and (

**b**) non-associative flow rule.

**Figure 18.**Comparisons of the stress–strain curve between experimental data and numerical results: (

**a**) standard SPH and SPH_SFDI with non-associative flow rule; and (

**b**) only SPH_SFDI with different flow rules.

**Figure 19.**Comparisons of (

**a**) the typical fracture pattern among experimental results of Zhang [48] (experimental fracture crack: red lines); and the standard SPH and SPH_SFDI results at different time: (

**b**) $t$ = 0.69 s; and (

**c**) $t$ = 1.38 s (contours of accumulated plastic strain).

**Figure 20.**Comparison of the SPH_SFDI results with different flow rules (contours of accumulated plastic strain).

**Figure 21.**Comparisons of the bulge fracture pattern among (

**a**) the bulge fracture patterns among the experiment results by Zhang [48], (

**b**) the standard SPH and SPH_SFDI results (The color legend is the accumulated plastic strain).

Approach | ${\mathit{F}}^{\prime}$(${\mathit{V}}_{1}$) | ${\mathit{t}}^{\prime}$(${\mathit{V}}_{1}$) | $\mathit{\delta}$(${\mathit{V}}_{1}$) | ${\mathit{F}}^{\prime}$(${\mathit{V}}_{2}$) | ${\mathit{t}}^{\prime}$(${\mathit{V}}_{2}$) | $\mathit{\delta}$(${\mathit{V}}_{2}$) |
---|---|---|---|---|---|---|

EXP | 6.87 kN | 0.40 s | 1.29 mm | 6.87 kN | 0.57 s | 1.05 mm |

SPH_SFDI | 6.95 kN | 0.39 s | 1.24 mm | 6.98 kN | 0.59 s | 1.15 mm |

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**MDPI and ACS Style**

Zhang, N.; Zheng, X.; Ma, Q.
Updated Smoothed Particle Hydrodynamics for Simulating Bending and Compression Failure Progress of Ice. *Water* **2017**, *9*, 882.
https://doi.org/10.3390/w9110882

**AMA Style**

Zhang N, Zheng X, Ma Q.
Updated Smoothed Particle Hydrodynamics for Simulating Bending and Compression Failure Progress of Ice. *Water*. 2017; 9(11):882.
https://doi.org/10.3390/w9110882

**Chicago/Turabian Style**

Zhang, Ningbo, Xing Zheng, and Qingwei Ma.
2017. "Updated Smoothed Particle Hydrodynamics for Simulating Bending and Compression Failure Progress of Ice" *Water* 9, no. 11: 882.
https://doi.org/10.3390/w9110882