# Modelling the Nanomechanical Responses of Biofilms Grown on the Indenter Probe

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Finite Element Model (FEM)

#### 2.2. Linear Viscoelastic Model

#### 2.3. Nonlinear Viscoelastic Model

_{j}(j = 1, 2, 3), shear modulus (μ

_{i}), and strain stiffening (α

_{i}), which is given by,

#### 2.4. Analytical Viscoelastic Model

_{tip}is the radius of indenter and R

_{nominal}is the nominal effective radius.

## 3. Results

#### 3.1. New Predictive Model Based on the Analytical Viscoelastic Model

#### 3.2. An Alternative Method

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Von Mises stress contours in thinner biofilm with varied Poisson’s ratio of 0.1 and 0.47 at indentation penetration of 0.1 µm.

**Figure 3.**The comparison of the simulated force-displacement curves and the analytical model (Equation (4)), when the thickness of biofilm is 0.5 μm and the Poisson’s ratio varies between 0.1 and 0.47.

**Figure 4.**Comparison of force-displacement curves determined by the finite element model (FEM) and refined linear viscoelastic model Equation (10) with geometry term Equation (11), with the biofilm thickness of 0.5 µm and the Poisson’s ratio varies between 0.1 and 0.47.

**Figure 5.**Comparison of force-displacement curves of FEM and refined linear viscoelastic model Equation (10) with geometry term Equation (12), with the biofilm thickness of 3 µm and the Poisson’s ratio varies between 0.1 and 0.47.

**Figure 6.**Von Mises stress contours in thicker biofilm with varied strain stiffening component of 0.5 and 4.

**Figure 7.**Force versus indentation depth curves comparing the analytical viscoelastic model Equation (4) to the finite element (FE) simulations, when the thickness of biofilm is 0.5 μm, Poisson’s ratio is 0.4999, strain stiffening component ranges from 0.5 to 4.

**Figure 8.**Comparison of force-displacement curves determined by FEM and refined nonlinear viscoelastic model Equation (10) with geometry term Equation (13), with the biofilm thickness of 0.5 µm and the strain stiffening component varying between 0.5 and 4.

**Figure 9.**Comparison of force-displacement curves determined by FEM and refined nonlinear viscoelastic model Equation (10) with geometry term Equation (14), with the biofilm thickness of 3 µm and the strain stiffening component varying between 0.5 and 4.

**Figure 10.**Deviation between the FEM results and Equation (10) or Equation (17), when biofilm thickness is 0.5 µm.

**Figure 11.**Deviation between the FEM results and Equation (10) or Equation (17), when biofilm thickness is 3 µm.

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**MDPI and ACS Style**

Xia, Y.; Duan, P.; Chen, J.
Modelling the Nanomechanical Responses of Biofilms Grown on the Indenter Probe. *Processes* **2018**, *6*, 84.
https://doi.org/10.3390/pr6070084

**AMA Style**

Xia Y, Duan P, Chen J.
Modelling the Nanomechanical Responses of Biofilms Grown on the Indenter Probe. *Processes*. 2018; 6(7):84.
https://doi.org/10.3390/pr6070084

**Chicago/Turabian Style**

Xia, Yuqing, Pengfei Duan, and Jinju Chen.
2018. "Modelling the Nanomechanical Responses of Biofilms Grown on the Indenter Probe" *Processes* 6, no. 7: 84.
https://doi.org/10.3390/pr6070084