Substantial gains and savings of resources of time and money can be gained through the use of modelling and simulation to understand material system performance. Since for development of the new materials validation is expensive and timeconsuming, the bottleneck is time and funding—modelling might be the way to replace testing programs, which would be beneficial for providing new innovative materials faster to the market.
1. Arrhenius Model
The Arrhenius model is widely used when temperature is the dominant accelerating factor in ageing. It is assumed that a single dominant degradation mechanism does not change during the exposure period, while the degradation rate is accelerated with an increase of exposure temperature
^{[1]}. The Arrhenius relation is given by
$$K\left(T\right)=A\mathrm{exp}\left(\frac{{E}_{a}}{RT}\right)or\mathrm{ln}K\left(T\right)=\frac{{E}_{a}}{RT}+\mathrm{ln}A$$
where
K is a reaction rate or degradation rate,
A is a constant related to material and degradation process,
E_{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 1). 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
^{[2]}^{[3]}.
Figure 1. Lifetime prediction according to the Arrhenius model.
The durability prediction methodology in most studies is based on the time shift concept
^{[4]}^{[3]}^{[5]}^{[6]}^{[7]}^{[8]}^{[9]}^{[10]}. According to Equation (1), the time shift factor (TSF) for two different exposure temperatures
T1 and (
T1<
T2) can be calculated as
$$TSF=\frac{{t}_{1}}{{t}_{2}}=\frac{A\mathrm{exp}\left({E}_{a}/R{T}_{2}\right)}{A\mathrm{exp}\left({E}_{a}/R{T}_{1}\right)}=\mathrm{exp}\left[\frac{{E}_{a}}{R}\left(\frac{1}{{T}_{1}}\frac{1}{{T}_{2}}\right)\right]$$
Equation (2) has been further used in the predictive methods based on superposition principles and assessment of the temperature shift factors. The activation energy in Equations (1) and (2) is commonly evaluated by thermal analysis methods, e.g., differential scanning calorimetry or thermogravimetric analysis by measuring heat capacity changes or mass losses under different heating rates. Alternatively, E_{a} can be determined by dynamic thermal mechanical analysis (DMTA) assessing T_{g}_{ }dependence on the test frequenc ^{[11]}^{[12]}.
Some recent studies considering the Arrhenius model and TSF approach for predicting the longterm strength of various FRP are listed in Table 1. Note that this methodology can be applied for assessment of tensile strength ^{[5]}^{[13]}, interlaminar shear strength ^{[6]}^{[9]}^{[10]} or bond strength ^{[8]}, under both static and fatigue loadings ^{[9]}. The accelerating temperature effect can also be coupled with other factors, e.g., absorbed water. For instance, Gagani et al. ^{[9]} applied the time shift concept to assess interlaminar shear fatigue lifetime of GFRP considering the effects of temperature and water immersion. The glass transition temperature of the material, and its decrease due to absorbed water, was used in Equation (2), enabling representation of both dry and water saturated samples in the same Arrheniusbased master curve.
Table 1. A condensed list of recent works on methods for predicting longterm mechanical properties of polymers and polymer composites.
Prediction Method 
Material 
Property 
Ref. 
Rate models 



Arrhenius model 
GFRP 
Tensile strength 
^{[5]}^{[13]} 

GFRP 
ILSS 
^{[10]} 

GFRP 
Fatigue ILSS 
^{[9]} 

GFRP bars 
Tensile strength 
^{[4]} 

CFRP/GFRP rods 
ILSS 
^{[6]} 

BFRP bars 
Residual tensile strength 
^{[7]} 

GFRP rods 
Bond strength 
^{[8]} 
Eyring’s model 
PA6,6, PC, CFRP 
Creep failure time 
^{[14]} 
Zhurkov’ model 
PP 
Fatigue strength 
^{[15]} 
Superposition principles 



Time–temperature (TTSP) 
Epoxy 
Creep compliance 
^{[11]}^{[16]} 

Epoxy 
Stress relaxation 
^{[17]} 

Filled epoxy 
Stiffness/Relaxation modulus 
^{[18]} 

PMMA 
Creep compliance 
^{[19]} 

Polyvinyl chloride, epoxy 
Stress threshold of LVE 
^{[20]}^{[21]} 

Flax/vinylester 
Creep compliance 
^{[22]} 

CFRP 
Creep compliance 
^{[23]}^{[24]} 

CFRP, GFRP 
Static/creep/fatigue strength 
^{[25]}^{[26]} 
Time–moisture (TMSP) 
Epoxy 
Creep compliance 
^{[11]}^{[16]} 

Epoxy 
Relaxation/storage modulus 
^{[27]}^{[28]}^{[29]}^{[30]}^{[31]} 

Epoxybased compounds 
Relaxation modulus 
^{[32]} 

Vinylester 
Creep strain 
^{[33]} 

Polyester 
Creep strain 
^{[34]} 

PA6, PA6,6 
Storage modulus 
^{[30]}^{[35]} 

CFRP, GFRP 
Fatigue strength 
^{[36]} 
Time–stress (TSSP) 
PA6 
Creep strain 
^{[37]} 

PMMA 
Creep compliance 
^{[19]}^{[38]}^{[39]} 

HDPE 
Creep strain/lifetime 
^{[40]} 

Polycarbonate 
Creep compliance 
^{[41]} 

PA6,6 fibres 
Creep strain 
^{[42]} 

Glass/PA, PP, HDPE 
Creep compliance 
^{[43]} 

HDPE/wood flour 
Creep strain 
^{[44]} 

Graphite/epoxy FRP 
Creep strain 
^{[45]} 

Kevlar yarns, PA6, epoxy 
Creep strain (stepped isostress test) 
^{[46]}^{[47]}^{[48]} 
Coupled 



TTSP + TMSP 
Epoxy 
Creep compliance 
^{[11]} 
TTSP + TMSP 
PA6,6 
Storage modulus 
^{[35]} 
TTSP + TMSP 
Acrylatebased polymers 
Storage modulus 
^{[49]} 
TTSP + TMSP 
CFRP, GFRP 
Static/creep/fatigue strength 
^{[36]}^{[50]} 
TTSP + TSSP 
HDPE/wood flour 
Creep strain 
^{[44]} 
TASP+TMSP 
Epoxy, polyester 
Creep compliance, stress relaxation 
^{[27]}^{[28]} 
TTSP+TASP 
Epoxy 
Relaxation modulus 
^{[17]}^{[51]} 
TTSP+TASP+TSSP 
PMMA 
Creep strain 
^{[38]} 
Plasticitycontrolled failure 
PP, PP/CNT, glass/PP, carbon/PEEK, PC/GF, PA6 
Lifetime (tensile, creep, fatigue) 
^{[52]}^{[53]}^{[54]}^{[55]}^{[56]}^{[57]} 

PA6,6, PC, CFRP 
Creep lifetime 
^{[14]} 
Parametric methods 
HDPE 
Creep lifetime (Larson–Miller, Monkman–Grant) 
^{[40]} 

GFRP 
Creep lifetime (Monkman–Grant) 
^{[58]} 

Rubberbonded composite 
Creep lifetime (Larson–Miller) 
^{[59]} 

Adhesive anchor in concrete 
Creep lifetime (Monkman–Grant) 
^{[60]} 

Short fibre thermoplastics 
Fatigue lifetime (Larson–Miller) 
^{[61]}^{[62]} 
2. Eyring’s Model
The reaction rate of a process can rely on several stressors. For example, nonthermal stresses such as humidity, voltage and mechanical stress may also play a significant role in accelerating degradation ^{[1]}. The Eyring model is based on chemical reactionrate theory and describes how the rate of degradation of a material varies with stress. It is assumed that the contribution of each stressor to the reaction rate is independent; thus, one could multiply the respective stress contributions to the rate of reaction. The model is closely related to the Arrhenius model and is based on the fact that the logarithm of the reaction rate is inversely proportional to absolute temperature (Equation (1)).
According to Eyring’s thermal activation flow theory, the strain rate (or the characteristic time) is given by the relationship
$$\dot{\epsilon}=\frac{1}{t}={A}_{1}\mathrm{exp}\left(\frac{{E}_{a}\gamma \sigma}{RT}\right)$$
where
A1 (s
^{−}^{1}) is a material constant and
γ is the coefficient linked to the activation volume;
σ is the applied stress. Eyring’s activated flow theory is widely used to assess plasticitycontrolled failure in thermoplastic polymers and composites
^{[14]}^{[52]}^{[53]}^{[54]}^{[55]}^{[56]}^{[57]}^{[58]}^{[59]}^{[60]}^{[61]}^{[62]}^{[63]}.
3. Zhurkov’s Model
Zhurkov developed the kinetic theory of strength of solids using temperature and tensile stress
^{[15]}^{[64]}. The relationship for calculation of the fracture lifetime
t_{f} is similar to Equation (3), while the coefficient
γ referred to as the lethargy coefficient is linked to lattice structure and defects in it. In the general case of nonisothermal tests and
σ varying in time (e.g., creep and fatigue tests), the fracture probability with account of the linear damage accumulation concept is given by the following equation
^{[15]}^{[58]}^{[65]}:
$$\underset{0}{\overset{{t}_{f}}{{\displaystyle \int}}}\frac{dt}{{t}_{0}\mathrm{exp}\left(\frac{{E}_{a}\gamma \left(\sigma \left(t\right)\right)}{RT\left(t\right)}\right)}=1$$
where
t_{0} is the time constant.
Zhurkov’s model initially developed for metals often could not give accurate lifetime predictions for polymers owing to the sensitiveness of their mechanical properties to strain rate and uncertain distinctions between the elastic and plastic ranges. As a result, the parameters involved in Equation (4) are interrelated stress, temperature, and strainrate dependent functions. The kinetic concept of strength is applied to model fatigue damage evolution ^{[15]}^{[66]}. Hur et al. developed a modified Zhurkov’s fatigue life model introducing strainratedependent lethargy coefficient and applied it to polypropylene reinforced with glass fibres ^{[15]}. The stressbased failure cycles in the ranges of low and highcycle fatigue were predicted and successfully validated by the proposed modified strainrate model. The kinetic strength model applied to lifetime predictions of biodegradable polymers is discussed in the review paper by Laycock et al. ^{[67]}. For this type of material, Zhurkov’s equation is coupled with broader biodegradation models enabling assessment of stress effects on the lifetime via lowering the activation energy for chain scission.