# Assessment of Compressive Mechanical Behavior of Bis-GMA Polymer Using Hyperelastic Models

^{1}

^{2}

^{3}

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^{7}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Hyperelastic Constitutive Model

_{m}are the material parameters, J

_{el}is the elastic volume ratio, ${\overline{I}}_{1}$ and ${\overline{I}}_{2}$ are the first and second deviatoric strain invariants, which with the assumption of full incompressibility, are defined as [6,53,54]:

_{1}, λ

_{2}and λ

_{3}are the principal stretches and the subscript el refers to the elastic limit. The detailed information about the equations and models could be found elsewhere [47,48,49,50,51,52].

_{u}), which is derived from the strain energy density by applying the principle of virtual work as follows:

_{u}is the stretch in the loading direction.

## 3. Material and Experiment Method

#### 3.1. Sample Preparation

^{2}LED light for 60 s from each side (i.e., top, bottom, and surrounding). In the next step, all of the cured Bis-GMA specimens were ejected from the mold. The top and bottom surfaces of the specimens were smoothened with 400–2000 grit abrasive papers. The final height of all samples was 10 ± 0.05 mm.

#### 3.2. Compression Test Method

#### 3.3. Nano-Indentation Test Method

## 4. Finite Element Simulation

## 5. Hybrid Experimental-Computational Approach

^{.}First, by selecting a group of constitutive models that approximate the material point behavior similar to the nominal stress–strain obtained from the experiment, and second, through the internal analysis of the FE model based on the selected models (first round), to examine the accuracy of the predicted results in terms of structural response and deformation of the polymer sample. Once the best hyperelastic constitutive model is identified, the mechanical characterization of the Bis-GMA polymer is completed. In the final step, the hybrid approach leads to attaining a validated FE model and simulation process. The hybrid approach is recommended to be used for the mechanical characterization and the prediction of the mechanical behavior of other hyperelastic polymers under different quasi-static monotonic loads.

## 6. Results and Discussion

#### 6.1. Compression Test Results

#### 6.2. Examination of Bis-GMA Polymer Hyperelastic Behavior Through Nano-Indentation Test

#### 6.3. FE Simulation Results

#### 6.3.1. Initial Selection of the Proper Hyperelastic Models

#### 6.3.2. Prediction of the Bis-GMA Polymer Structural Response

#### 6.3.3. Prediction of the Bis-GMA Polymer Structural Deformation

#### 6.3.4. Identification of the Best Hyperelastic Model

_{ij}s are hyperelastic material constants which were calculated by a curve fitting process, as described in Section 2. These material constants were determined in this study for the Bis-GMA polymer as:

#### 6.4. Determination of the Bis-GMA Polymer True Stress–Strain Curve

## 7. Conclusion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Geometry (

**left**) and mesh configuration (

**right**) of FE model representing the Bis-GMA polymer under compression loading condition.

**Figure 2.**Flowchart of the hybrid experimental-computational approach to determining the mechanical behavior of hyperelastic polymers.

**Figure 4.**The mean values of nominal stress versus nominal strain obtained from the experimental results.

**Figure 5.**The loading-unloading responses of Bis-GMA polymer in the nano-indentation test (indentation loads of 3 μN (

**a**) and 6 μN (

**b**)).

**Figure 6.**The resultant stress–strain curves that fitted using the hyperelastic models provided in Table 2.

**Figure 7.**The predicted load-displacement curves and stress–strain responses of the system in comparison with the experimental data.

**Figure 8.**Structural deformation of the Bis-GMA polymer specimen under different compressive deformation (CD).

**Figure 9.**Contour plots of the radial deformation of the FE model incorporating polynomial N = 2 (

**a**), Van der Waals (

**b**) and Yeoh (

**c**) hyperelastic models, and plot of transverse deformation (

**d**) of a selected path (

**e**) on the FE models.

**Figure 10.**The true stress–strain curves obtained from the structural response of the model using different hyperelastic equations.

**Table 1.**The strain energy potential models used in the simulation of Bis-GMA polymer under monotonic compressive load.

Model Name | Equation | Detail |
---|---|---|

Arruda-Boyce form [47] | $\begin{array}{l}U=\mu \{\frac{1}{2}({\overline{I}}_{1}-3)+\frac{1}{20{\lambda}_{m}^{2}}({{\overline{I}}_{1}}^{2}-9)+\frac{11}{1050{\lambda}_{m}^{4}}({{\overline{I}}_{1}}^{3}-27)\\ +\frac{19}{7000{\lambda}_{m}^{6}}({{\overline{I}}_{1}}^{4}-81)+\frac{519}{673750{\lambda}_{m}^{8}}({{\overline{I}}_{1}}^{5}-243\}+\frac{1}{D}\left(\frac{{J}_{el}^{2}-1}{2}-\mathrm{ln}{J}_{el}\right)\end{array}$ | - |

Polynomial form [48] | $U={\displaystyle \sum _{i+j=1}^{N}{C}_{ij}{({\overline{I}}_{1}-3)}^{i}{({\overline{I}}_{2}-3)}^{j}+{\displaystyle \sum _{i=1}^{N}\frac{1}{{D}_{i}}({J}_{el}}-1{)}^{2i}}$ | N = 1, 2 |

Reduced polynomial form [49] | $U={\displaystyle \sum _{i=1}^{N}{C}_{i0}{({\overline{I}}_{1}-3)}^{i}+{\displaystyle \sum _{i=1}^{N}\frac{1}{{D}_{i}}({J}_{el}}-1{)}^{2i}}$ | N = 1, 2, …, 6 |

Ogden form [50] | $U={\displaystyle \sum _{i=1}^{N}\frac{2{\mu}_{i}}{{\alpha}_{i}^{2}}({\overline{\lambda}}_{1}^{{\alpha}_{i}}+{\overline{\lambda}}_{2}^{{\alpha}_{i}}+{\overline{\lambda}}_{3}^{{\alpha}_{i}}-3)+{\displaystyle \sum _{i=1}^{N}\frac{1}{{D}_{i}}({J}_{el}}-1{)}^{2i}}$ | N = 1, 2, …, 6 |

Yeoh form [51] | $\begin{array}{l}U={C}_{10}({\overline{I}}_{1}-3)+{C}_{20}{({\overline{I}}_{1}-3)}^{2}+{C}_{30}{({\overline{I}}_{1}-3)}^{3}\\ +\frac{1}{{D}_{1}}{({J}_{el}-1)}^{2}+\frac{1}{{D}_{2}}{({J}_{el}-1)}^{4}+\frac{1}{{D}_{3}}{({J}_{el}-1)}^{6}\end{array}$ | Reduced polynomial N = 3 |

Van der Waals form [52] | $U=\mu \left\{-({\lambda}_{m}^{2}-3)\left[\mathrm{ln}(1-\eta )+\eta \right]-\frac{2}{3}a{\left(\frac{\overline{I}-3}{2}\right)}^{\frac{3}{2}}\right\}+\frac{1}{D}\left(\frac{{J}_{el}^{2}-1}{2}-\mathrm{ln}{J}_{el}\right)$ | - |

Commercial Name | Chemical Name | Molecular Formula | Molecular Weight (g/mol) | Manufacturer |
---|---|---|---|---|

Bis-GMA | 2,2-bis[4-(2-hydroxy-3-methacryloxypropoxy)phenyl propane | C_{29}H_{36}O_{8} | 512.59 | Sigma-Aldrich Inc., St. Louis, MO, USA |

Camphorquinone | 2,3-bornadenione | C_{10}H_{14}O_{2} | 166 | Sigma-Aldrich Inc., St. Louis, MO, USA |

DMAEMA | 2-(dimethylamino) ethyl methacrylate | C_{7}H_{14}NO_{2} | 157 | Sigma-Aldrich Inc., St. Louis, MO, USA |

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**MDPI and ACS Style**

Karimzadeh, A.; Ayatollahi, M.R.; Rahimian Koloor, S.S.; Bushroa, A.R.; Yahya, M.Y.; Tamin, M.N. Assessment of Compressive Mechanical Behavior of Bis-GMA Polymer Using Hyperelastic Models. *Polymers* **2019**, *11*, 1571.
https://doi.org/10.3390/polym11101571

**AMA Style**

Karimzadeh A, Ayatollahi MR, Rahimian Koloor SS, Bushroa AR, Yahya MY, Tamin MN. Assessment of Compressive Mechanical Behavior of Bis-GMA Polymer Using Hyperelastic Models. *Polymers*. 2019; 11(10):1571.
https://doi.org/10.3390/polym11101571

**Chicago/Turabian Style**

Karimzadeh, Atefeh, Majid Reza Ayatollahi, Seyed Saeid Rahimian Koloor, Abd Razak Bushroa, Mohd Yazid Yahya, and Mohd Nasir Tamin. 2019. "Assessment of Compressive Mechanical Behavior of Bis-GMA Polymer Using Hyperelastic Models" *Polymers* 11, no. 10: 1571.
https://doi.org/10.3390/polym11101571