# Modelling of Environmental Ageing of Polymers and Polymer Composites—Durability Prediction Methods

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## Abstract

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## 1. Introduction

_{2}emissions, which can be partially achieved by reducing the mass of vehicles; and (2) new manufacturing methods, which can increase the production rate and reduce the unit cost for components, such as forming. Among the major benefits these materials offer are the high stiffness to weight ratio, which makes a strong case for the transportation industry, and the good durability, which has supported the use of composites in aggressive environments. Polymers and composite materials are often exposed to environmental influences such as water, humidity, elevated temperatures, pH, mechanical stress, and their combinations. Thus, environmental factors negatively impact the performance, affecting their durability [1,2,3,4]. Several reviews have been recently presented on the physical and chemical phenomena [2,5,6], including one from the authors [7].

## 2. Models for Predicting Material Durability and Service Lifetime

#### 2.1. Rate Models

#### 2.1.1. Arrhenius Model

_{a}is the process activation energy, R is the universal gas constant, and T is the absolute temperature. The degradation rate is proportional to the inverse time for degradation of a mechanical property for a given value set by the lifetime criterion, and log(t) vs. 1/T is a linear function with the slope E

_{a}/R (Figure 2). The Arrhenius relationship is widely used for lifetime predictions of polymers and composites through monitoring ultimate mechanical properties and their retention, e.g., tensile strength, interfacial shear strength, creep strength, and fatigue strength [6,21].

_{a}can be determined by dynamic thermal mechanical analysis (DMTA) assessing T

_{g}dependence on the test frequency [28,29].

#### 2.1.2. Eyring’s Model

_{1}(s

^{−}

^{1}) is a material constant and $\gamma $ is the coefficient linked to the activation volume; σ is the applied stress. Eyring’s activated flow theory is widely used to assess plasticity-controlled failure in thermoplastic polymers and composites [31,69,70,71,72,73,74,75,76,77,78,79,80]. A detailed discussion is given in Section 2.3.

#### 2.1.3. Zhurkov’s Model

#### 2.2. Superposition Principles

#### 2.2.1. Time–Temperature Superposition Principle

_{0}and T

_{1}, differ only by a time scale defined by the temperature shift functions ${a}_{T0}$ and ${a}_{T1}$. Creep compliances J can reach the same values at different time moments t

_{0}and t

_{1}, i.e., $J\left({t}_{0},{T}_{0}\right)=J\left({t}_{1},{T}_{1}\right)$ or according to Equation (5) ${t}_{0}{a}_{T0}$=${t}_{1}{a}_{T1}$ and $\mathrm{log}{t}_{0}-\mathrm{log}{t}_{1}=\mathrm{log}{a}_{T1}-\mathrm{log}{a}_{T0}$. For simplicity, the reference shift factor is typically taken as unity ${a}_{T0}=1$. Thus, the creep compliance curves in logarithmic time axes are parallel for different T and shifted to each other for values $\mathrm{log}{a}_{T}$. The long-term prediction is represented by the generalised master curve (Figure 3). The concept of time-shift factors allows one to estimate the ratio between times required for a certain decrease in a mechanical property at two different operating temperatures [8,22]. Similar considerations are valid for other accelerated factors and related superposition principles. Applicability of TTSP has also been a validated methodology to determine the energy limit and stress threshold of linear viscoelastic behaviour of various polymers [37,38,84].

_{g}and T

_{g}+100 °C [89]:

_{g}[89]:

_{g}, although with a different ${E}_{a}$ value, i.e., slope in $\mathrm{log}{a}_{T}\mathrm{vs}.1/T$ line (Figure 3). A generalised relationship for the time–temperature shift factor valid in a full operating temperature range is defined as follows [85,90]:

_{g}, respectively.

#### 2.2.2. Time–Moisture Superposition Principle

_{g}drop. The Fox model [52,53], also known as Simha–Boyer model [96], is among the most known models used for the prediction of T

_{g}variations with $w$:

_{g}as an indicator of polymer chain mobility related to its free volume, Krauklis et al. developed the time–temperature-plasticization superposition principle [28]. The moisture (called plasticization) shift factors were determined by the Arrhenius-type equation (Equation (9)), changing operating temperatures to T

_{g}of the dry and plasticized (${T}_{gw})$ polymer [28]:

_{g}changes with the test frequency, was found to be the same for dry and moisture-plasticized polymer. The moisture shift factors $\mathrm{log}{a}_{w}$ calculated by Equation (15) correlated well with those determined by a common shifting of the creep compliance curves [28].

_{g}changes.

#### 2.2.3. Time–Stress Superposition Principle

_{g}), while Equation (17) is valid for glassy polymers (T < T

_{g}). TSSP could be applied to both creep and creep-recovery data. As demonstrated by the example of PA6,6 fibres [59], $\mathrm{log}{a}_{\sigma}$ is identical in both cases, pointing to linear viscoelastic behaviour.

#### 2.2.4. Time-Ageing Time Superposition

_{g}is affected by physical ageing, i.e., a phenomenon related to the evolution of thermodynamic state manifesting as a reduction in free volume and changes in molecular configuration [105,106]. Structural rearrangements increase T

_{g}and material stiffening manifested via the increased strength and lower creep [44,96,107]. The time-ageing time superposition principles (TASP) are formulated considering the ageing time as a factor altering the relaxation spectra of a polymer (Equation (5)) [108]. Similarly to other superposition-based methods, time-dependence of material properties, which invalidates the use of Boltzmann superposition principles, is taken into account by applying the effective time-domain approach [108,109]. The real-time is normalised by the time-dependent relaxation time such that the relaxation dynamics remains invariant with respect to the effective time.

_{g}to a temperature below T

_{g}. The time the material spends below its T

_{g}is referred to as the ageing time ${t}_{ag}$. Meanwhile, the temperature of ageing and the cooling rate are crucial parameters that determine the extent of physical ageing [107]. Short-term creep tests are conducted on samples with different ${t}_{ag}$ and the long-term master curve is constructed by horizontal shifting momentary data to a reference time ${t}_{ag0}$. The duration of these tests should be much shorter (at least by a factor of ten [88]) than the ageing time to exclude ageing effects during the test. The ageing shift factor $\mathrm{log}{a}_{ag}$ is expressed as follows [44,88,108,110]:

_{g}.

#### 2.3. Plasticity-Controlled Failure

#### 2.4. Parametric Methods for Creep

#### 2.5. Fatigue Prediction Methods

#### 2.5.1. Factors Affecting Fatigue Damage

- (i)
- Material related factors: fibre type and dimensions, matrix type, fibre volume content, reinforcement structure (unidirectional, multidirectional, woven, braided, spatially reinforced, etc.), laminate stacking sequence, etc.
- (ii)
- Testing related factors: loading conditions (stress ratio, cyclic frequency, monotonic/variable frequency, axial/multiaxial loading, force/displacement-controlled loading), and environmental conditions (temperature, humidity, water/salt water, UV).
- (iii)
- Manufacturing and storage-related factors: manufacturing process, inherent defects and voids, thermal or ageing pre-history, etc.

#### 2.5.2. Classification of Fatigue Models

_{min}/σ

_{max}. Stress S may have different definitions: stress amplitude S

_{a}= (σ

_{max}− σ

_{min})/2, mean stress S

_{m}= (σ

_{max}+ σ

_{min})/2, or normalised stress divided by a reference such as ultimate strength S

_{u}. Constant life diagrams (CLD), also called Goodman-type diagrams, are obtained by plotting S

_{a}vs. S

_{m}and presenting isolife lines with $N=const$, i.e., endurance limits (Figure 12b). Many examples of fatigue life models can be found in the literature [121,134]. Fatigue life predictions are commonly obtained by fitting a set of experimental data, in most cases by the Basquin-type power law equation [26,78,124]:

_{0}) or stiffness (E/E

_{0}) with respect to the number of cycles (parameters with subscripts 0 are related to the initial undamaged materials property). The rate of strength degradation is typically defined as a function of several factors:

#### 2.5.3. Fatigue Prediction under the Environmental Impact

_{g}. Under the coupled influence of temperature and absorbed water, the water effect, related to both accelerated viscoelastic response of the polymer matrix and triggering additional damage mechanisms, is taken into account via the modified temperature shift functions. This methodology has been used by Gagani et al. [26], Zhou and Wu [133], and Fatemi et al. [146,161].

## 3. Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 4.**Creep curves of vinylester with different equilibrium moisture contents (w

_{0}, w

_{1}, w

_{2}) and the master curve constructed by applying TMSP. The Boltzmann–Volterra equation calculates the line for the linear viscoelastic solid and time–moisture shift function given by Equation (13). Data are taken from [50].

**Figure 5.**Creep curves for PA6,6 fibres at various creep stresses (

**a**) and master curve obtained by TSSP (

**b**). Adopted with permission from Ref. [59]. Copyright 2017 Willey.

**Figure 6.**Strength vs. ageing time for amine-based epoxy conditioned in seawater up to saturation (wet) and in an inert atmosphere (dry). Adopted with permission from Ref. [96]. Copyright 2019 Elsevier.

**Figure 9.**Strain rate dependencies of the yield stress in tensile tests and applied stress in creep tests (

**a**) and a correlation between the plastic flow rate and time-to-failure according to Equation (24) (

**b**) for glass fibre reinforced isostatic polypropylene composites [72].

**Figure 10.**Residual recovery strain vs. total creep strain for polypropylene filled with different contents of MWCNT. Data obtained in creep-recovery tests under various loads and creep times; one point corresponds to one creep-recovery test. Data reproduced from [80].

**Figure 11.**LMP master curve (

**a**) and Monkman–Grant correlation ${t}_{r}$ vs. ${\dot{\epsilon}}_{min}$ (

**b**) for HDPE under various temperatures [57].

**Figure 12.**Classification of fatigue models and the principal ways for predicting the environmental impact (e.g., temperature T and water content w). Representative methods for fatigue analysis: (

**a**) S–N curve; (

**b**) constant life diagram; (

**c**) residual strength/stiffness dependence on the number of cycles; (

**d**) damage function; (

**e**) flowchart of progressive damage analysis.

**Figure 14.**S–N master curves for dry and conditioned GFRP (

**a**) and superimposed environmental master curve with the definition of equivalent temperature (

**b**) [26].

**Figure 15.**Formulation of accelerated testing methodology by Nakada and Miyano. Adapted with permission from Ref. [53]. Copyright 2009 Elsevier.

**Figure 16.**Larson–Miller master curves for polypropylene (PP), neat and reinforced with talc (PP-T), and glass fibres (PP-G) at R = 0.1 and 0.3. Adapted with permission from Ref. [78]. Copyright 2016 Elsevier.

**Figure 17.**Constant life diagrams for plain-woven CFRP aged in seawater for different times. Adapted with permission from Ref. [167]. Copyright 2019 Elsevier.

**Table 1.**A condensed list of recent works on methods for predicting long-term mechanical properties of polymers and polymer composites.

Prediction Method | Material | Property | Ref. |
---|---|---|---|

Rate models | |||

Arrhenius model | GFRP | Tensile strength | [22,30] |

GFRP | ILSS | [27] | |

GFRP | Fatigue ILSS | [26] | |

GFRP bars | Tensile strength | [8] | |

CFRP/GFRP rods | ILSS | [23] | |

BFRP bars | Residual tensile strength | [24] | |

GFRP rods | Bond strength | [25] | |

Eyring’s model | PA6,6, PC, CFRP | Creep failure time | [31] |

Zhurkov’ model | PP | Fatigue strength | [32] |

Superposition principles | |||

Time–temperature (TTSP) | Epoxy | Creep compliance | [28,33] |

Epoxy | Stress relaxation | [34] | |

Filled epoxy | Stiffness/Relaxation modulus | [35] | |

PMMA | Creep compliance | [36] | |

Polyvinyl chloride, epoxy | Stress threshold of LVE | [37,38] | |

Flax/vinylester | Creep compliance | [39] | |

CFRP | Creep compliance | [40,41] | |

CFRP, GFRP | Static/creep/fatigue strength | [42,43] | |

Time–moisture (TMSP) | Epoxy | Creep compliance | [28,33] |

Epoxy | Relaxation/storage modulus | [44,45,46,47,48] | |

Epoxy-based compounds | Relaxation modulus | [49] | |

Vinylester | Creep strain | [50] | |

Polyester | Creep strain | [51] | |

PA6, PA6,6 | Storage modulus | [47,52] | |

CFRP, GFRP | Fatigue strength | [53] | |

Time–stress (TSSP) | PA6 | Creep strain | [54] |

PMMA | Creep compliance | [36,55,56] | |

HDPE | Creep strain/lifetime | [57] | |

Polycarbonate | Creep compliance | [58] | |

PA6,6 fibres | Creep strain | [59] | |

Glass/PA, PP, HDPE | Creep compliance | [60] | |

HDPE/wood flour | Creep strain | [61] | |

Graphite/epoxy FRP | Creep strain | [62] | |

Kevlar yarns, PA6, epoxy | Creep strain (stepped isostress test) | [63,64,65] | |

Coupled | |||

TTSP + TMSP | Epoxy | Creep compliance | [28] |

TTSP + TMSP | PA6,6 | Storage modulus | [52] |

TTSP + TMSP | Acrylate-based polymers | Storage modulus | [66] |

TTSP + TMSP | CFRP, GFRP | Static/creep/fatigue strength | [53,67] |

TTSP + TSSP | HDPE/wood flour | Creep strain | [61] |

TASP+TMSP | Epoxy, polyester | Creep compliance, stress relaxation | [44,45] |

TTSP+TASP | Epoxy | Relaxation modulus | [34,68] |

TTSP+TASP+TSSP | PMMA | Creep strain | [55] |

Plasticity-controlled failure | PP, PP/CNT, glass/PP, carbon/PEEK, PC/GF, PA6 | Lifetime (tensile, creep, fatigue) | [69,70,71,72,73,74] |

PA6,6, PC, CFRP | Creep lifetime | [31] | |

Parametric methods | HDPE | Creep lifetime (Larson–Miller, Monkman–Grant) | [57] |

GFRP | Creep lifetime (Monkman–Grant) | [75] | |

Rubber-bonded composite | Creep lifetime (Larson–Miller) | [76] | |

Adhesive anchor in concrete | Creep lifetime (Monkman–Grant) | [77] | |

Short fibre thermoplastics | Fatigue lifetime (Larson–Miller) | [78,79] |

**Table 2.**A condensed list of recent works modelling fatigue under environmental impacts (T and w are associated with temperature and water effects, respectively).

Factor | Material/Testing Details | Prediction Method (s) | Author, Ref. |
---|---|---|---|

T T + w | Short fibre-reinforced thermoplastic composites, R = −1, 0.1, 3, 0.25–10 Hz | TTSP for S–N curves (dry and wet). S–N curve “normalisation”; strength degradation model with temperature-dependent parameters | Fatemi et al. [146,161,166] |

T T + w | GFRP, four-point bending, R = 0.1, 4 Hz | TTSP for S–N curves (dry and wet) | Gagani et al. [26] |

T | UD, braided, GFRP, CFRP; R = 0.1, 10, −0.8, −1; f = 3.3, 5, 10 Hz. | TTSP for S–N curves | Zhou et al. [133] |

T | PP, PP/talc, PP/glass R = −1, 0.1, 0.3 | Larson–Miller parametrisation for S–N curves; strength degradation model with temperature-dependent parameters. | Eftekhari et al. [78,79] |

T T + w | CFRP, GFRP; tension, bending | S–N master curves by TTSP held for viscoelastic properties and static strength of the polymer matrix | Miyano et al. [42,53,67,90,162] |

T | CFRP (AS4/PEEK) cross-ply, quasi-isotropic, R = 0.1; 5 Hz. | S–N curve “normalisation” | Jen et al. [164] |

T | 2.5D woven CFRP; R = 0.1; 10 Hz | S–N curve “normalisation”, residual stiffness model accounting temperature effect | Song et al. [165] |

T | Weave GFRP R = 0.1; 5 Hz | Strength degradation model with two temperature-dependent parameters | Cormier et al. [136] |

w | GFRP UD, biaxial, vinylester, R = 0.1, 5 Hz | Strength degradation model with parameters related to hydrothermal ageing time | Acosta et al. [169] |

w | Plain-woven GFRP, R = 0.1, −0.52, 10; 5 Hz; seawater | Strength degradation model; S–N and CLT diagrams—model with ageing time-dependent parameters | Koshima et al. [167] |

T | Cross-ply, quasi-isotropic, woven FRP composites | Cumulative fatigue damage model with temperature-dependent parameters determined in constant strain rate tests | Mivehchi et al. [149] |

T + w | GFRP UD, R = 0.1, 2; 10 Hz; fresh, sea water | Stiffness degradation model with three material parameters dependent on environmental conditions; S–N curves | Tang et al. [151] |

T | Weave woven CFRP/epoxy laminates; R = 0.1; 20 Hz | Stiffness degradation model with damage function dependent on temperature | Khan et al. [152] |

w | CFRP woven; 3-point bending, R = 0.1, 1 Hz; seawater | Strain-life curves with two parameters depending on samples ageing | Prabhakar et al. [145] |

w | Epoxy resin, R = 0.1 | Viscoelastic/viscoplastic model with continuum damage accelerated by water plasticization | Rocha et al. [102] |

T, UV | Triaxial CFRP laminates; R = −1; thermal cycles; 750 h UV | Stochastic analysis: Monte Carlo simulation for S–N curves of different guarantee areas depending on the ageing state | Mossalam et al. [137] |

w | CFRP UD, cross-ply, bending, R = 0.1, 10 Hz, seawater | FEA modelling: virtual crack closure technique, water-induced accelerated crack propagation | Meng et al. [156] |

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**MDPI and ACS Style**

Starkova, O.; Gagani, A.I.; Karl, C.W.; Rocha, I.B.C.M.; Burlakovs, J.; Krauklis, A.E.
Modelling of Environmental Ageing of Polymers and Polymer Composites—Durability Prediction Methods. *Polymers* **2022**, *14*, 907.
https://doi.org/10.3390/polym14050907

**AMA Style**

Starkova O, Gagani AI, Karl CW, Rocha IBCM, Burlakovs J, Krauklis AE.
Modelling of Environmental Ageing of Polymers and Polymer Composites—Durability Prediction Methods. *Polymers*. 2022; 14(5):907.
https://doi.org/10.3390/polym14050907

**Chicago/Turabian Style**

Starkova, Olesja, Abedin I. Gagani, Christian W. Karl, Iuri B. C. M. Rocha, Juris Burlakovs, and Andrey E. Krauklis.
2022. "Modelling of Environmental Ageing of Polymers and Polymer Composites—Durability Prediction Methods" *Polymers* 14, no. 5: 907.
https://doi.org/10.3390/polym14050907