# Modelling of Environmental Ageing of Polymers and Polymer Composites—Modular and Multiscale Methods

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## Abstract

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## 1. Introduction

#### Terminology

## 2. Ageing and Degradation Modelling of Composite Microconstituents

#### 2.1. Polymer Degradation Models

#### 2.1.1. Predicting Polymer Properties from Chemical Structure

#### 2.1.2. Degradation Mechanisms

#### 2.1.3. Diffusion and Leaching

_{ij}, and D

_{11}, D

_{22}and D

_{33}are the diffusion constants in directions 1, 2 and 3, respectively.

#### 2.1.4. Swelling and Plasticization

_{f}the fibre volume fraction, ${\epsilon}_{f}$ is the fibre swelling strain, ${\epsilon}_{m}$ is the matrix swelling strain, $W$ is the moisture content in the matrix, ${W}_{c}$ the moisture content in the composite, ${\beta}_{y}$ is the transverse swelling coefficient.

#### 2.1.5. Hydrolysis

_{n}. Modelling hydrolysis as a first-order kinetic mechanism, the hydrolytic damage ${d}_{h}$ is defined as follows [121]:

#### 2.2. Fibre Degradation Models

#### 2.2.1. An Introduction to Glass Fibre Degradation

#### 2.2.2. Chemical Reactions during Hydrolytic Degradation of Glass Fibres

#### 2.2.3. Molecular Mechanism and Kinetics of Degradation

#### 2.2.4. Modelling of Mass Loss and Radius Reduction of Glass Fibres

_{2}is the major component in virtually all types of glass [153], and SiO

_{2}dominates the dissolution process, or at least does so in Phase II. The DCZOK model can be used to describe dissolution for each element separately and for the total mass loss [141]. The analytical DCZOK model equations for radius reduction during Phase I and Phase II are depicted in Equation (20) [141]:

#### 2.2.5. Modelling Crack Growth and Strength Loss of Glass Fibres

#### 2.2.6. Carbon Fibre Degradation

#### 2.2.7. Aramid Fibre Degradation

_{2}) into nitro groups (-NO

_{2}) [164]. The hydrolysis pathway undergoes splitting of the molecule into two fragments, causing an increase in amine (-NH

_{2}) and acid end groups (-COOH) [164]. The thermal decomposition mechanism produces the breaking of the polymeric chain into two fragments, increasing the number of amine end groups (-NH

_{2}). According to Yang, this reaction simultaneously produces random and specific thermal degradation lower molecular weight fragments such as MPD-I monomers, cyclic dimers, and cyclic trimers [165]. The degradation rate depends on the fiber polydispersity. The kinetic constants following all three possible mechanisms increase linearly as polydispersity increases [164].

#### 2.2.8. Basalt Fibre Degradation

#### 2.2.9. Natural Fibre Degradation

#### 2.3. Interphase Degradation Models

## 3. Modular and Multiscale Approaches

#### 3.1. Modular Approach

#### 3.2. Multiscale Simulation Frameworks

#### 3.3. Direct Numerical Simulation (DNS)

#### 3.4. Analytical Homogenization (AH)

#### 3.5. Numerical Homogenization (NH)

#### 3.6. Computational Homogenization (CH)

^{2}—the idea is to concurrently model two or more scales without making any constitutive assumptions during upscaling [218,219]. This is achieved through a continuous scale link enforced by embedding an independent RVE model at each integration point of the macroscopic domain and solving the associated microscopic boundary-value problem every time the macroscopic constitutive response needs to be computed.

#### 3.7. Multiple Time Scales and Other Approaches

#### 3.8. Accelerating Multiscale Simulations

- When performing long-term predictions of material degradation under cyclic environmental exposure, with transient simulations requiring a large number of time steps;
- As part of many-query applications such as design optimization, in which evaluating each trial design requires a complete set of high-fidelity simulations to be run;
- When employing models as part of Structural Health Monitoring (SHM) frameworks requiring inverse problems to be solved on the fly as new sensor measurements are obtained.

#### 3.9. Model Order Reduction (MOR)

#### 3.10. Machine Learning (ML) Approaches

## 4. Emerging Trends in Degradation Modelling of Biodegradable Polymers

#### 4.1. Biodegradation—An Introduction to Terms and Definitions

#### 4.2. Data-Driven Approach to Elucidate Degradation Trends

_{g}values are <T

_{Ocean}. PA (polyamide/Nylon) is the most used fishing gear material belonging to the yellow group (Nylon 66 and Nylon 6). Nylon is very persistent in the oceans despite the possible fragmentation, having a very long lifetime and contributing to a major extent to e.g. ghost fishing and plastic debris [262].

^{−1}values for these very hydrophobic plastics show lower densities, which allows them to float near the sea surface. The ranking is shown in Figure 4, and tends to correlate with the propensity for polyester degradation. However, plastics with T

_{g}values > T

_{Ocean}, including polymers such as PLA, PLLA and PET, degrade more slowly than expected. The degradation of PLA in seawater is very slow, in contrast to degradation under composting conditions [264]; see also Table 3 in [18]. This shows that multiple metrics are required to understand the degradation of polymers in the ocean. Therefore, crystallinity, enthalpy of fusion, Tg, molecular weight, and LogP(SA)

^{−1}values were studied in pairs to find patterns of degradation. Figure 13 compares crystallinity and enthalpy of melting with values of LogP(SA)

^{−1}for both biotic and abiotic conditions. Surface erosion was calculated using the surface area of each plastic (SAbulk), mass loss and the number of days in ocean water. To obtain the systematic diversity of hydrophobicity values, the number of -CH

_{2}- units in the monomer structures was calculated between 5 for PPS and 11 for PPSeb. As a result, Min et al. recently demonstrated that the enzymatic degradation of polyesters with T

_{g}values below the ocean temperature is faster than for abiotic hydrolysis. In the case of abiotic hydrolysis, biodegradation and photoinitiated processes co-occur. It is likely that the decrease in molecular weight due to abiotic hydrolysis or photoinitiated reactions could facilitate biotically induced processes, whereas enzymatic hydrolysis could promote abiotic hydrolysis. It appears that abiotic hydrolysis (see Figure 13a,c) is probably more sensitive to increases in hydrophobicity, enthalpy of melting, and crystallinity compared to biotic processes. Biotic processes exhibit faster rates in more hydrophobic polyesters, such as PPPim) and PPSub. Moreover, the comparison of polyesters and PA (e.g., Nylon 6 and Nylon 6,6) demonstrates that biotic and abiotic processes also occur in semi-crystalline plastics, but crystallinity slows down these processes. More details are provided in [62].

#### 4.3. Laboratory vs. Field Experiments: Outlook

## 5. Economic Role of Degradation Modelling

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Time scale and critical points for materials’ life show the “two-edged sword” nature of the degradation processes (clipart is attributed to “Scale Clipart Transparent—Creative Commons Clipart”).

**Figure 2.**Schematic representation of the environmental degradation of polymers: chemical and physicochemical degradation factors [16].

**Figure 4.**The wide range of hydrophobicity of plastics [62].

**Figure 5.**The seven developed modules of the Reinforcement Modular Group within the Modular Paradigm reproduced from [139].

**Figure 6.**The interphase flaw is formed and gets filled with water, reproduced from [13].

**Figure 7.**Modelling toolbox modules for predicting environmental ageing of composites. Red coloured modules indicate that such models are not yet available, whereas blue ones are complete (and described in the previous chapter).

**Figure 9.**The FE2 framework for hygrothermal ageing was proposed by Rocha et al. [209].

**Figure 10.**The different steps are involved in the biodegradation process [18].

**Figure 12.**Flow chart for calculating hydrophobicity [62].

**Figure 13.**The influence of crystallinity and hydrophobicity on degradation [62]. The computational LogP(SA)

^{−1}values are plotted versus the enthalpy of melting for abiotic hydrolysis (

**a**), the enthalpy of melting for biotic processes (

**b**), the % crystallinity for abiotic hydrolysis (

**c**), the % crystallinity for biotic processes (

**d**).The size of circles and colour is equivalent to surface erosion (in mg cm

^{−2}day

^{−1}) in artificial seawater.

**Figure 14.**Real and virtual degradable experiments according to the study of Yamawaki et al. [265].

**Table 1.**Reinforcement comparison by fibre type, i.e., glass, carbon, basalt, aramid, and their market share, cost range and mechanical properties (in the unaged state), reproduced from [124].

Fibre Type ^{1} | Market Share [%] | Cost Range [$/kg] | Tensile Strength [GPa] | Young’s Modulus [GPa] |
---|---|---|---|---|

E-Glass | ~70% | 1.3–2.6 | 3.45–3.5 | 72.5–73.5 |

E-CR-Glass | 1.2–3 | 2–3.625 | 72.5–83 | |

AR-Glass | 2.5–3 | 1.7–3.5 | 72–175 | |

C-Glass | 1–2.5 | 3.3 | 69 | |

A-Glass | 2–3 | 3.3 | 72 | |

S/S-2-Glass | 16–26 | 4.6–4.9 | 86–89 | |

R-Glass | 16–26 | 4.4 | 86 | |

PAN Type Carbon | ~12% | 15–120 | 1.8–7.0 | 230–540 |

HS Carbon | 20–120 | 3.31–5 | 228–248 | |

IM Carbon | 25–120 | 4.1–6 | 265–320 | |

HM Carbon | 25–120 | 1.52–2.41 | 393–483 | |

UHM Carbon | 30–120 | 2.24 | 724 | |

Basalt | ~11% | 5 | 4.84 | 89 |

Aramid/Kevlar | ~7% | 15–30 | 2.6–3.4 | 55–127 |

^{1}Fibre type abbreviations expanded: E-Glass [Electric], E-CR-Glass [Electric/Corrosion Resistant], AR-Glass [Alkali Resistant], C-Glass [Chemical], A-Glass [Alkali], S/S-2-Glass [Strength], R-Glass [Reinforcement], HS Carbon [High Strength], IM [Intermediate Modulus], HM Carbon [High Modulus], UHM Carbon [Ultra High Modulus].

**Table 2.**A condensed list of recent works modelling ageing in heterogeneous materials at multiple scales. Model abbreviations: DNS (Direct Numerical Simulation), AH (Analytical Homogenization), NH (Numerical Homogenization), CH (Computational Homogenization).

Ref(s) | Material(s) | Model | Process(es) |
---|---|---|---|

[194,195,196] | GFRP | DNS | Diffusion, swelling |

[104] | CFRP | DNS | Diffusion, swelling |

[197] | Flax/Epoxy | DNS | Diffusion, swelling |

[198] | Woven composites | AH | Diffusion |

[112] | GFRP | AH/NH | Swelling |

[199] | GFRP | AH | Degradation |

[200] | High-Vf polymers | AH/NH | Diffusion |

[201] | GFRP | AH | Diffusion, swelling |

[202] | GFRP | NH | Diffusion |

[203] | GFRP | NH | Diffusion, degradation, fracture |

[204] | Woven composites | NH | Diffusion |

[205,206] | GFRP | NH | Diffusion, degradation |

[207] | Concrete | CH | Diffusion, swelling, degradation, fracture |

[208] | Polyamide composites | CH | Diffusion, swelling, degradation, damage |

[202,209,210] | GFRP | CH | Diffusion, swelling, degradation, fracture |

[211] | Titanium composites | Other | Diffusion, degradation |

[212] | Braided composites | Other | Degradation, fracture |

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**MDPI and ACS Style**

Krauklis, A.E.; Karl, C.W.; Rocha, I.B.C.M.; Burlakovs, J.; Ozola-Davidane, R.; Gagani, A.I.; Starkova, O. Modelling of Environmental Ageing of Polymers and Polymer Composites—Modular and Multiscale Methods. *Polymers* **2022**, *14*, 216.
https://doi.org/10.3390/polym14010216

**AMA Style**

Krauklis AE, Karl CW, Rocha IBCM, Burlakovs J, Ozola-Davidane R, Gagani AI, Starkova O. Modelling of Environmental Ageing of Polymers and Polymer Composites—Modular and Multiscale Methods. *Polymers*. 2022; 14(1):216.
https://doi.org/10.3390/polym14010216

**Chicago/Turabian Style**

Krauklis, Andrey E., Christian W. Karl, Iuri B. C. M. Rocha, Juris Burlakovs, Ruta Ozola-Davidane, Abedin I. Gagani, and Olesja Starkova. 2022. "Modelling of Environmental Ageing of Polymers and Polymer Composites—Modular and Multiscale Methods" *Polymers* 14, no. 1: 216.
https://doi.org/10.3390/polym14010216