# Numerical Approach to Simulate the Mechanical Behavior of Biodegradable Polymers during Erosion

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## Abstract

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## 1. Introduction

#### 1.1. Kinetic Models for Erosion and Degradation

#### 1.2. Constitutive Models for Polymers

#### 1.3. Approaches to Simulate the Mechanical Behaviour during Hydrolytic Degradation

## 2. Materials and Methods

#### 2.1. Kinetic Models for Erosion and Degradation

#### 2.2. Erosion Damage

#### 2.3. Constitutive Model

#### 2.3.1. Hyperelastic Model

#### 2.3.2. Elastoplastic Model

#### 2.4. Finite Element Implementation

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Zhang, X.; Espiritu, M.; Bilyk, A.; Kurniawan, L. Morphological behaviour of poly(lactic acid) during hydrolytic degradation. Polym. Degrad. Stab.
**2008**, 93, 1964–1970. [Google Scholar] [CrossRef] - Deng, M.; Zhou, J.; Chen, G.; Burkley, D.; Xu, Y.; Jamiolkowski, D.; Barbolt, T. Effect of load and temperature on in vitro degradation of poly(glycolide-co-L-lactide) multifilament braids. Biomaterials
**2005**, 26, 4327–4336. [Google Scholar] [CrossRef] [PubMed] - Weir, N.A.; Buchanan, F.J.; Orr, J.F.; Farrar, D.F.; Dickson, G.R. Degradation of poly-L-lactide. Part 2: Increased temperature accelerated degradation. Proc. Inst. Mech. Eng. Part H, J. Eng. Med.
**2004**, 218, 321–330. [Google Scholar] [CrossRef] - Wise, J.; Gillen, K.; Clough, R. An ultrasensitive technique for testing the Arrhenius extrapolation assumption for thermally aged elastomers. Polym. Degrad. Stab.
**1995**, 49, 403–418. [Google Scholar] [CrossRef] - Guo, M.; Chu, Z.; Yao, J.; Feng, W.; Wang, Y.; Wang, L.; Fan, Y. The effects of tensile stress on degradation of biodegradable PLGA membranes: A quantitative study. Polym. Degrad. Stab.
**2016**, 124, 95–100. [Google Scholar] [CrossRef] - Li, D. Extended layerwise method of laminated composite shells. Compos. Struct.
**2016**, 136, 313–344. [Google Scholar] [CrossRef] - Miller, N.; Williams, D. The in vivo and in vitro degradation of poly(glycolic acid) suture material as a function of applied strain. Biomaterials
**1984**, 5, 365–368. [Google Scholar] [CrossRef] - Dreher, M.L.; Nagaraja, S.; Batchelor, B. Effects of fatigue on the chemical and mechanical degradation of model stent sub-units. J. Mech. Behav. Biomed. Mater.
**2016**, 59, 139–145. [Google Scholar] [CrossRef] - Fan, Y.B.; Li, P.; Zeng, L.; Huang, X.J. Effects of mechanical load on the degradation of poly(d,l-lactic acid) foam. Polym. Degrad. Stab.
**2008**, 93, 677–683. [Google Scholar] [CrossRef] - Li, P.; Feng, X.; Jia, X.; Fan, Y. Influences of tensile load on in vitro degradation of an electrospun poly(l-lactide-co-glycolide) scaffold. Acta Biomater.
**2010**, 6, 2991–2996. [Google Scholar] [CrossRef] - Gan, Z.; Yu, D.; Zhong, Z.; Liang, Q.; Jing, X. Enzymatic degradation of poly(ε-caprolactone)/poly(dl-lactide) blends in phosphate buffer solution. Polymer
**1999**, 40, 2859–2862. [Google Scholar] [CrossRef] - Williams, J. Stress Analysis of Polymers; Ellis Horwood Series in Engineering Science; E. Horwood: New York, NY, USA, 1980. [Google Scholar]
- Tsuji, H.; Ikada, Y. Properties and morphology of poly(L-lactide). II. Hydrolysis in alkaline solution. J. Polym. Sci. Part Polym. Chem.
**1998**, 36, 59–66. [Google Scholar] [CrossRef] - Tsuji, H.; Ikada, Y. Properties and morphology of poly(L-lactide). 4. Effects of structural parameters on long-term hydrolysis of poly(L-lactide) in phosphate-buffered solution. Polym. Degrad. Stab.
**2000**, 67, 179–189. [Google Scholar] [CrossRef] - Tsuji, H.; Nakahara, K. Poly(L-lactide). IX. Hydrolysis in acid media. J. Appl. Polym. Sci.
**2002**, 86, 186–194. [Google Scholar] [CrossRef] - Grizzi, I.; Garreau, H.; Li, S.; Vert, M. Hydrolytic degradation of devices based on poly(dl-lactic acid) size-dependence. Biomaterials
**1995**, 16, 305–311. [Google Scholar] [CrossRef] - Therin, M.; Christel, P.; Li, S.; Garreau, H.; Vert, M. In vivo degradation of massive poly(α-hydroxy acids): Validation of In vitro findings. Biomaterials
**1992**, 13, 594–600. [Google Scholar] [CrossRef] [PubMed] - Li, S.; Garreau, H.; Vert, M. Structure-property relationships in the case of the degradation of massive poly(α-hydroxy acids) in aqueous media. J. Mater. Sci. Mater. Med.
**1990**, 1, 198–206. [Google Scholar] [CrossRef] - Li, S.M.; Garreau, H.; Vert, M. Structure-property relationships in the case of the degradation of massive aliphatic poly-(α-hydroxy acids) in aqueous media. J. Mater. Sci. Mater. Med.
**1990**, 1, 123–130. [Google Scholar] [CrossRef] - Vert, M.; Li, S.; Garreau, H. More about the degradation of LA/GA-derived matrices in aqueous media. J. Control. Release
**1991**, 16, 15–26. [Google Scholar] [CrossRef] - Sevim, K.; Pan, J. A model for hydrolytic degradation and erosion of biodegradable polymers. Acta Biomater.
**2018**, 66, 192–199. [Google Scholar] [CrossRef] [PubMed] - Zhang, T.; Zhou, S.; Gao, X.; Yang, Z.; Sun, L.; Zhang, D. A multi-scale method for modeling degradation of bioresorbable polyesters. Acta Biomater.
**2017**, 50, 462–475. [Google Scholar] [CrossRef] [PubMed] - Moore, J.E.; Soares, J.S.; Rajagopal, K.R. Biodegradable Stents: Biomechanical Modeling Challenges and Opportunities. Cardiovasc. Eng. Technol.
**2010**, 1, 52–65. [Google Scholar] [CrossRef] - Bardenhagen, S.; Stout, M.; Gray, G. Three-dimensional, finite deformation, viscoplastic constitutive models for polymeric materials. Mech. Mater.
**1997**, 25, 235–253. [Google Scholar] [CrossRef] - Grabow, N.; Bünger, C.M.; Sternberg, K.; Mews, S.; Schmohl, K.; Schmitz, K.P. Mechanical Properties of a Biodegradable Balloon-expandable Stent From Poly(L-lactide) for Peripheral Vascular Applications. J. Med. Devices
**2006**, 1, 84–88. [Google Scholar] [CrossRef] - Bergström, J.; Boyce, M. Constitutive modeling of the large strain time-dependent behavior of elastomers. J. Mech. Phys. Solids
**1998**, 46, 931–954. [Google Scholar] [CrossRef] - Bergström, J.; Kurtz, S.; Rimnac, C.; Edidin, A. Constitutive modeling of ultra-high molecular weight polyethylene under large-deformation and cyclic loading conditions. Biomaterials
**2002**, 23, 2329–2343. [Google Scholar] [CrossRef] - Boyce, M.C.; Parks, D.M.; Argon, A.S. Large inelastic deformation of glassy polymers. part I: Rate dependent constitutive model. Mech. Mater.
**1988**, 7, 15–33. [Google Scholar] [CrossRef] - Drozdov, A.; Gupta, R. Constitutive equations in finite viscoplasticity of semicrystalline polymers. Int. J. Solids Struct.
**2003**, 40, 6217–6243. [Google Scholar] [CrossRef] - Breche, Q.; Chagnon, G.; Machado, G.; Nottelet, B.; Garric, X.; Girard, E.; Favier, D. A non-linear viscoelastic model to describe the mechanical behavior’s evolution of biodegradable polymers during hydrolytic degradation. Polym. Degrad. Stab.
**2016**, 131, 145–156. [Google Scholar] [CrossRef] - Gleadall, A.; Pan, J.; Kruft, M.A. An atomic finite element model for biodegradable polymers. Part 2. A model for change in Young’s modulus due to polymer chain scission. J. Mech. Behav. Biomed. Mater.
**2015**, 51, 237–247. [Google Scholar] [CrossRef] - Gleadall, A.; Pan, J.; Ding, L.; Kruft, M.A.; Curcó, D. An atomic finite element model for biodegradable polymers. Part 1. Formulation of the finite elements. J. Mech. Behav. Biomed. Mater.
**2015**, 51, 409–420. [Google Scholar] [CrossRef] - Khan, K.A.; El-Sayed, T. A phenomenological constitutive model for the nonlinear viscoelastic responses of biodegradable polymers. Acta Mech.
**2013**, 224, 287–305. [Google Scholar] [CrossRef] - Muliana, A.; Rajagopal, K. Modeling the response of nonlinear viscoelastic biodegradable polymeric stents. Int. J. Solids Struct.
**2012**, 49, 989–1000. [Google Scholar] [CrossRef] - Soares, J.S.; Rajagopal, K.R.; Moore, J.E. Deformation-induced hydrolysis of a degradable polymeric cylindrical annulus. Biomech. Model. Mechanobiol.
**2010**, 9, 177–196. [Google Scholar] [CrossRef] - Taguti, M.V.H.; Françoso, A.; Ribeiro, M.L.; Vieira, A.F.C. Numerical approach to simulate the mechanical behavior of biodegradable structures considering degradation time and heterogeneous stress field. Polym. Eng. Sci.
**2020**, 60, 1566–1578. [Google Scholar] [CrossRef] - Vieira, A.C.; Vieira, J.C.; Ferra, J.M.; Magalhaes, F.D.; Guedes, R.M.; Marques, A.T. Mechanical study of PLA-PCL fibers during in vitro degradation. J. Mech. Behav. Biomed. Mater.
**2011**, 4, 451–460. [Google Scholar] [CrossRef] [PubMed] - Vieira, A.C.; Guedes, R.M.; Tita, V. Constitutive modeling of biodegradable polymers: Hydrolytic degradation and time-dependent behavior. Int. J. Solids Struct.
**2014**, 51, 1164–1174. [Google Scholar] [CrossRef] - Wang, Y.; Han, X.; Pan, J.; Sinka, C. An entropy spring model for the Young’s modulus change of biodegradable polymers during biodegradation. J. Mech. Behav. Biomed. Mater.
**2010**, 3, 14–21. [Google Scholar] [CrossRef] - Chen, Y.; Zhou, S.; Li, Q. Mathematical modeling of degradation for bulk-erosive polymers: Applications in tissue engineering scaffolds and drug delivery systems. Acta Biomater.
**2011**, 7, 1140–1149. [Google Scholar] [CrossRef] - Farrar, D.; Gillson, R. Hydrolytic degradation of polyglyconate B: The relationship between degradation time, strength and molecular weight. Biomaterials
**2002**, 23, 3905–3912. [Google Scholar] [CrossRef] - Laycock, B.; Nikolić, M.; Colwell, J.M.; Gauthier, E.; Halley, P.; Bottle, S.; George, G. Lifetime prediction of biodegradable polymers. Prog. Polym. Sci.
**2017**, 71, 144–189. [Google Scholar] [CrossRef] - Gopferich, A.; Langer, R. Modeling of polymer erosion. Macromolecules
**1993**, 26, 4105–4112. [Google Scholar] [CrossRef] - Göpferich, A. Polymer bulk erosion. Macromolecules
**1997**, 30, 2598–2604. [Google Scholar] [CrossRef] - Thombre, A.G.; Himmelstein, K.J. Modelling of drug release kinetics from a laminated device having an erodible drug reservoir. Biomaterials
**1984**, 5, 250–254. [Google Scholar] [CrossRef] - Boland, E.L.; Shirazi, R.N.; Grogan, J.A.; McHugh, P.E. Mechanical and Corrosion Testing of Magnesium WE43 Specimens for Pitting Corrosion Model Calibration. Adv. Eng. Mater.
**2018**, 20, 1800656. [Google Scholar] [CrossRef] - Saconi, F.; Diaz, G.H.; Vieira, A.C.; Ribeiro, M.L. Experimental Characterization and Numerical Modeling of the Corrosion Effect on the Mechanical Properties of the Biodegradable Magnesium Alloy WE43 for Orthopedic Applications. Materials
**2022**, 15, 7164. [Google Scholar] [CrossRef] [PubMed] - Taguti, M.V.H. Metodologia Multidimensional para Previsão do Comportamento mecâNico de PolíMeros Biodegradáveis Durante DegradaçãO Hidrolítica. Master’s Thesis, Escola de Engenharia de São Carlos, Universidade de São Paulo, São Paulo, Brazil, 2022. [Google Scholar]

**Figure 7.**Degradation experiment of biogradable polymers regarding different stress fields [36].

**Figure 13.**(

**a**) Initial hyperelastic von Mises stress field, (

**b**) initial elastoplastic model Stress field, and (

**c**) initial (0 weeks) molecular weight field for stress-free.

**Figure 14.**Mass loss vs time for the stress-free condition, the elastoplastic model and the hyperelastic models.

**Figure 15.**10 weeks molecular weight field for (

**a**) stress-free, (

**b**) elastoplastic, and (

**c**) hyperelastic model.

**Figure 16.**20 weeks molecular weight field for (

**a**) stress-free, (

**b**) elastoplastic, and (

**c**) hyperelastic model.

**Figure 17.**30 weeks molecular weight field for (

**a**) stress-free, (

**b**) elastoplastic, and (

**c**) hyperelastic model.

**Figure 18.**40 weeks molecular weight field for (

**a**) stress-free, (

**b**) elastoplastic, and (

**c**) hyperelastic model.

**Figure 19.**50 weeks molecular weight field for (

**a**) stress-free, (

**b**) elastoplastic, and (

**c**) hyperelastic model.

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**MDPI and ACS Style**

Vieira, A.F.C.; Da Silva, E.H.P.; Ribeiro, M.L.
Numerical Approach to Simulate the Mechanical Behavior of Biodegradable Polymers during Erosion. *Polymers* **2023**, *15*, 1979.
https://doi.org/10.3390/polym15091979

**AMA Style**

Vieira AFC, Da Silva EHP, Ribeiro ML.
Numerical Approach to Simulate the Mechanical Behavior of Biodegradable Polymers during Erosion. *Polymers*. 2023; 15(9):1979.
https://doi.org/10.3390/polym15091979

**Chicago/Turabian Style**

Vieira, André F. C., Enio H. P. Da Silva, and Marcelo L. Ribeiro.
2023. "Numerical Approach to Simulate the Mechanical Behavior of Biodegradable Polymers during Erosion" *Polymers* 15, no. 9: 1979.
https://doi.org/10.3390/polym15091979