# Influence of Partially Debonded Interface on Elasticity of Syntactic Foam: A Numerical Study

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## Abstract

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## 1. Introduction

## 2. Geometric and Physical Model Concepts

- Mixed model (Figure 1a): There is a mixture of microspheres having both bonded and debonded interfaces. For each microsphere, the interface is either fully bonded or fully debonded.
- Partially debonded model (Figure 1b): Each microsphere has an interface which consists of two distinctly divided regions of bonded and debonded states. When the debonded region is smaller than the bonded region, there is a cap-type gap in one side of the interface. The cap-type connection exists in the opposite situation.
- Discontinuously bonded model (Figure 1c): The interface is discontinuously bonded meaning the microsphere and matrix are sparsely interconnected at the interface.

## 3. Numerical Results

#### 3.1. Mixed Model

#### 3.2. Partially Debonded Model

#### 3.3. Discontinuously Bonded Model

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Schematics of debonding geometries for (

**a**) the mixed model, (

**b**) the partially debonded model and (

**c**) the discontinuously bonded model.

**Figure 2.**(

**a**) Model geometry for the mixed model, with 30 randomly distributed microspheres, showing the debonded geometry for a debonding fraction of 13.3% (four microspheres out or a total of 30 microspheres are debonded); (

**b**) Representative model geometries of the partially debonded model for $\xi $ = 12.5%, 50% and 75%; (

**c**) Model geometries for the discontinuously bonded models.

**Figure 3.**Finite element meshes for (

**a**) the mixed model; (

**b**) matrix for the partially debonded and discontinuously bonded models; (

**c**) microsphere of the partially debonded model; (

**d**) microsphere of the discontinuously bonded model; and (

**e**) cross-section image of meshes used for the microsphere wall in the partially debonded model.

**Figure 4.**Five different representative elementary volumes (RVEs) for the mixed model, having the same $\xi $ = 50% with differently selected debonded microspheres. In the figure, bonded microspheres are shown in dark grey and debonded microspheres are transparent.

**Figure 5.**(

**a**) Elastic modulus and (

**b**) Poisson’s ratio obtained by the mixed model for a microsphere volume fraction of 30%, as a function of debonding fraction in tension and compression.

**Figure 6.**Equivalent stress fields at 1% tensile and compressive loads for the mixed model, with $\xi $ = 100% which is complete debonding for all microspheres. Strains are magnified by a factor of 10 for better visualization.

**Figure 7.**(

**a**) Elastic moduli and (

**b**) Poisson’s ratios obtained by the partially debonded model for a microsphere volume fraction of 30%, as a function of the debonding fraction. Black and blue lines represent directions in the isotropic plane and out-of-plane, respectively. Lines with and without circular marks indicate compressive and tensile elastic components, respectively.

**Figure 8.**(

**a**) Cross-section view of the equivalent stress field of the partially debonded model subjected to 1% shear loading ($\xi $ = 75 %). Strain is 10 times larger than actual for easier visualization. (

**b**) Evolution of shear modulus ${G}_{tl}$ with increasing $\xi $.

**Figure 9.**(

**a**) Schematic of rotation of coordinates and (

**b**) elastic moduli of the partially debonded model with random orientation of debonding direction.

**Figure 10.**(

**a**) Elastic moduli as a function of $\xi $ for the mixed and partially debonded models. (

**b**) Compressive and tensile moduli for $\xi $ = 50%, obtained by different models. The abbreviations used are: P.D. (partially debonded) and D.B. (discontinuously bonded).

**Figure 11.**Equivalent stress distributions in microspheres for models with $\xi $ = 50%. (

**a**) Mixed model; (

**b**) partially debonded model; and (

**c**) discontinuously bonded model.

**Table 1.**Compressive and tensile elastic moduli obtained by the mixed model for $\xi $ = 50% with five different RVEs consisting of randomly selected debonded microspheres.

Model No.1 | Model No.2 | Model No.3 | Model No.4 | Model No.5 | Average | |
---|---|---|---|---|---|---|

Compressive modulus (MPa) | 3035.8 | 3027.8 | 3033.5 | 3033.5 | 3039.2 | 3033.9 ± 3.7 |

Tensile modulus (MPa) | 2496.1 | 2504.8 | 2510.3 | 2500.4 | 2515.7 | 2505.5 ± 7.0 |

Model Type | 8 Splits | 24 Splits | 48 Splits |
---|---|---|---|

Compressive modulus (${E}^{c}$, MPa) | 2929.4 | 2919.4 | 2922.5 |

Tensile modulus (${E}^{t}$, MPa) | 2767.4 | 2881.2 | 2934.9 |

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**MDPI and ACS Style**

Cho, Y.J.; Kang, Y.; Lee, Y.C.; Park, Y.; Lee, W.
Influence of Partially Debonded Interface on Elasticity of Syntactic Foam: A Numerical Study. *Materials* **2017**, *10*, 911.
https://doi.org/10.3390/ma10080911

**AMA Style**

Cho YJ, Kang Y, Lee YC, Park Y, Lee W.
Influence of Partially Debonded Interface on Elasticity of Syntactic Foam: A Numerical Study. *Materials*. 2017; 10(8):911.
https://doi.org/10.3390/ma10080911

**Chicago/Turabian Style**

Cho, Yi Je, Youngjeong Kang, Young Cheol Lee, Yongho Park, and Wookjin Lee.
2017. "Influence of Partially Debonded Interface on Elasticity of Syntactic Foam: A Numerical Study" *Materials* 10, no. 8: 911.
https://doi.org/10.3390/ma10080911