# A Three-Scale Model of Basic Mechanical Properties of Nafion

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The mechanical properties of Nafion are explained and modeled on the basis of Kafka’s general mesomechanical model and confronted with experimental results. In this approach, Nafion is looked upon as a composite consisting of three constituents: a crystalline Nafion, amorphous Nafion, and water. Taking into account the degree of hydration, its elastic, elastic-plastic, and hysteretic properties are discussed and modeled. It is shown how the interaction between the three constituents manifests itself on the macroscale.

## Keywords

Nafion mechanical properties mesomechanics material structure hydration## Nomenclature

## General mechanics

*σ*_{ij}stress tensor

*δ*_{ij}*σ*isotropic part of

*σ*_{ ij }(*σ*= σ_{ ii }*/*3)*s*_{ij}deviatoric part of

*σ*_{ ij }(*σ*_{ ij }− δ_{ ij }*σ*)*ε*_{ij}strain tensor

- δ
_{ij}*ε* isotropic part of

*ε*_{ ij }(*ε*=*ε*_{ ii }*/*3)*e*_{ij}deviatoric part of

*ε*_{ ij }(*ε*_{ ij }−*δ*_{ ij }*ε*)*δ*_{ij}Kronecker’s delta

*E*Young’s modulus

- ν
Poisson’s ratio

*μ*= (1+ν)/*E*deviatoric elastic compliance

*ρ*= (1 − 2ν) /*E*isotropic elastic compliance.

## Specific symbols

- \( \overline{\Big|} \)
overbar that relates the symbol | to its macroscopic value — the average in the representative volume element RVE

- |
^{ω}or |_{ω} index that relates the symbol | to the

*ω*-constituent — the average in the subvolume of RVE that is filled in by the*ω*-constituent*ω*=*e*elastic constituent in the general two-phase model

*ω*=*n*inelastic constituent in the general two-phase model

*ω*=*a*amorphous constituent in Nafion

*ω*=*c*crystalline constituent in Nafion

*ω*=*w*water comprised in Nafion

*ω*=*wa*aggregate of two constituents in Nafion: of the amorphous Nafion with water

- |
_{ij} Einstein’s notation

\( {\varepsilon}_{ij}^{\prime }={\varepsilon}_{ij}-{\overline{\varepsilon}}_{ij}; \)

*δ*_{ij}*ε*′isotropic part of

*ε*_{ ij }^{′}*e*_{ij}^{′}deviatoric part of

*ε*_{ ij }^{′}*σ*_{ij}^{′}stress related to

*ε*_{ ij }^{′}similarly as*σ*_{ ij }is related to*ε*_{ ij }*δ*_{ij}*σ*′isotropic part of

*σ*_{ ij }^{′}*s*_{ij}^{′}deviatoric part of

*σ*_{ ij }^{′}*ν*_{e}{*ν*_{n}}volume fraction of the elastic {inelastic} constituent in the general two-phase model

*ν*_{a}{*ν*_{w}}volume fraction of the amorphous Nafion {of water} in the aggregate of amorphous Nafion with water

*V*_{ω}volume fraction of the

*ω*- constituent in the total Nafion (*ω*=*a*,*c*,*w*,*wa*);\( {R}_a^c=\frac{V_c}{V_a}; \)

*c*_{a}elastic limit of

*s*_{11}^{ a }in the*a*-constituent*C*_{wa}elastic limit of

*s*_{11}^{ wa }in the*wa*-constituent*n*_{ω}structural parameter (

*ω*=*a*,*c*,*w*,*wa*)- |
^{Ω} superscript that relates the symbol | to the respective Ω -specimen of Nafion

- Ω =
*H* hydrated specimen

- Ω =
*D* dry specimen

*p*=*V*_{ c }^{ H }*η*_{ c }*μ*_{ c }+*V*_{ wa }^{ H }*η*_{ wa }*μ*_{ wa }^{ H }\( q=\frac{\mu_c}{V_c^H}\left(p\left|+{\eta}_c{\eta}_{wa}{\mu}_{wa}^H\right.\right) \).

- \( \overset{\cdot }{h} \)
formal variable equal to 0 in elasticity and to \( \overset{\cdot }{\lambda } \) in plasticity

- |
_{L} value of | at the elastic limit.

## Notes

### Acknowledgements

This work was supported by the Czech Science Foundation within projects P108/10/1296 and 103/09/2101. Acknowledged is also the support through the Institutional Project RVO: 68378297. D. Vokoun would like to thank Dr. M. Paidar for fruitful discussions.

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