Paper Acar

Mech Time-Depend MaterDOI 10.1007/s11043-012-9187-8

Modeling of hydro-thermo-mechanical behaviorof Nafion NRE212 for Polymer Electrolyte MembraneFuel Cells using the Finite Viscoplasticity Theory Basedon Overstress for Polymers (FVBOP)

Ozgen U. Colak · Alperen Acar

Received: 23 January 2012 / Accepted: 9 August 2012© Springer Science+Business Media, B. V. 2012

Abstract The primary aim of this work is to present the modifications made to the FiniteViscoplasticity Theory Based on Overstress for Polymers (FVBOP). This is a unified statevariable theory and the proposed changes are designed to account for humidity and temper-ature effects relevant to the modeling of the hydrothermal deformation behavior of ionomermembranes used in Polymer Electrolyte Membrane Fuel Cells (PEMFC). Towards that end,the flow function, which is responsible for conferring rate dependency in FVBOP, is modi-fied. A secondary objective of this work was to investigate the feasibility of using the storagemodulus obtained by Dynamic Mechanical Analysis (DMA) in place of the elasticity mod-ulus obtained from conventional tensile/compressive tests, and find the correlation betweenthe storage modulus and the elasticity modulus. The numerical simulations were juxtaposedagainst data from tensile monotonic loading and unloading experiments on perfluorosulfonicacid (PFSA) membrane Nafion NRE212 samples which are used extensively as a membranematerial in PEMFC. The deformation behavior was modeled at four different temperatures(298, 323, 338, and 353 K—all values below the glass transition temperature of Nafion) andat three water content levels (3, 7 and 8 % swelling). The effects of strain rate, temperature,and hydration were captured well with the modified FVBOP model.

Keywords Nafion · VBO · Temperature · Humidity · Strain rate dependency · PEMFC

1 Introduction

Increasing demands for energy and various factors such as economic stability, burgeoningcosts for identifying and extracting newer reserves of fossil fuels, and environmental factorsare driving the search for alternative fuels. One of these alternative fuels is hydrogen, which,in conjunction with fuel cell technologies, presents a clean and efficient energy option. Eventhough there are many types of fuel cells, Proton Exchange Membrane Fuel Cells (PEMFC),

O.U. Colak (�) · A. AcarDepartment of Mechanical Engineering, Yildiz Technical University, Istanbul, Turkeye-mail: [email protected]

mailto:[email protected]

Mech Time-Depend Mater

Fig. 1 Schematic of PEMFC(Tang et al. 2006)

which convert hydrogen energy to electrical energy, have been at the forefront of extensiveindustrial and academic research for some time.

The main component of a PEMFC is the membrane electrode assembly (MEA), whichconsists of a polymer electrolyte membrane with a catalyst layer and porous carbon electrodesupport on either side. The role of MEAs in PEMFCs is to conduct protons from anodeelectrode to the cathode electrode, and at the same time to act as an electronic insulator andgas barrier to prevent mixing of oxygen and hydrogen. Schematic of PEMFC is given inFig. 1.

To enable proton conduction, the polymer electrolyte membrane needs to be humidified.In response to changes in temperature and moisture, polymer electrolyte membrane experi-ences expansion and contraction. The different thermal expansion and swelling coefficientsof these various materials in MEA introduce hygrothermal stresses during the operation cy-cles (start/shut down) of the fuel cell. Hygrothermal stresses play an important role in thefailure of the MEA by creating pinholes in the membrane or delamination of the polymermembrane and gas diffusion layers (Kundu et al. 2005). Strength and stability problems ofpolymer electrolyte membranes are the main impediment for their unsuitability for use in theautomotive industry. Therefore, the characterization of the mechanical behavior of PEMsexperimentally and the ability to perform reliable lifetime predictions through numericalmodeling which incorporates the effects of temperature and hydration are of interest for thedesign of robust PEMFCs (Tang et al. 2006).

Membranes commonly used in PEM fuel cells are composed of perfluorosulfonic acid(PFSA) membranes, and the most common commercially available membrane is Nafionproduced by DuPont. Even though, a large volume of the experimental data on polymerelectrolyte membranes used in PEMFCs is available in the scientific literature, the numberof works devoted to modeling is quite limited. Also, most of the modeling efforts assumethat the material behavior is rate independent for simplification, whereas the polymericmaterials used in polymer electrolyte membranes exhibit rate dependence. The mechani-cal properties of ionomer membranes Nafion 112, 117, NR111, N1110, 115 and NRE212in tensile test conditions and using a Dynamic Mechanical Analysis (DMA) are reportedin Silberstein and Boyce (2010), Tang et al. (2006), Kundu et al. (2005), Solasi et al.(2007), Satterfield and Benziger (2009), Majsztrik et al. (2007, 2008). The effect of tem-perature and hydration has been investigated in these works. The modeling of the mechan-


ical behavior of ionomer membranes in polymer electrolyte membrane fuel cells (PEMFC)has been done mostly using a linear elasticity (Tang et al. 2006), and linear viscoelastic-viscoplastic (Solasi et al. 2007), and elasto-plastic (Kusoglu et al. 2006) constitutive mod-els. Recently, some researchers have applied nonlinear viscoelastic-viscoplastic modelsfor modeling uniaxial behavior of polymer membranes (see Silberstein and Boyce 2010;Yoon and Huang 2011).

The mechanical response of the membranes subjected to a hygrothermal cycle has beeninvestigated by Kusoglu et al. (2006). Linear-elastic, perfectly plastic laws are used as thematerial model for Nafion with the added thermal and swelling expansion coefficients. In thework of Solasi et al. (2007) mentioned above, in addition to the experimental work, modelingwork similar to that by Kusoglu et al. (2006) was performed. While a linear viscoelastic-plastic model was used for the membranes, the authors do comment that a more completenonlinear viscoplastic model is desirable in order to predict the stress–strain values moreprecisely.

A nonlinear viscoelastic-viscoplastic model by Bergström–Boyce was used to simulatemonotonic tensile behavior of Nafion NR 111 by Yoon and Huang (2011). In this work, hy-dration and temperature dependent empirical equations for elastic modulus are introducedand only the monotonic loading behaviors of Nafion NR 111 are simulated at different tem-perature and humidity values. Silberstein and Boyce (2010) conducted uniaxial tension testsat the strain rate of 0.01 s−1 at temperatures from 298 to 373 K and swelling percentagesfrom 3 to 9 %. The elasto-viscoplastic models, named Model I and II, are used. Model Iwas used to simulate only monotonic loading behavior at different temperatures, hydrationlevels, and strain rates. By including a back stress feature to the model (Model II), loading-unloading and reloading behavior of Nafion NRE212 is modeled at two different strain rates(1×10−1 s−1 and 1×10−3 s−1) and temperatures (353 and 298 K) even though experimentswere performed at three different strain rates and five various temperature and humidity val-ues. Using Model II somewhat improved the unloading behavior; however, neither modelcould collectively capture the strain rate, temperature and humidity dependence of Nafionvery well. Some inconsistencies are also observed in the experimental results.

Dynamic mechanical analysis (DMA) is a widely used characterization technique forpolymeric materials. Here, the samples are typically subjected to a periodic mechanicalstrain or stress, and temperature ramps can also be implemented simultaneously. The mate-rial parameters measured by means of DMA are the storage modulus E′, which is relatedto stiffness of the material (elastic response), and the loss modulus E′′, which is the mea-sure of energy dissipation (viscous response) of polymers. The E′/E′′ ratio labeled tan δ

provides a measure of the mechanical damping. These different moduli allow better char-acterization of the material, because a material’s propensity to return or store energy (E′)and lose energy (E′′) can be quantified individually leading to a calculation of the damping(Menard 1999). These three parameters vary significantly with temperature and frequency,especially around polymer relaxation phenomena, such as glass transition or sub-glass tran-sition (Menard 1999). Therefore, DMA is used to determine the glass transition temperature(Tg) of materials as well.

One of the aims in this work was to investigate applicability of the storage modulusobtained by DMA as the elasticity modulus obtained from the conventional tests and tofind the correlation between the storage modulus and the elasticity modulus. DMA hasbeen used extensively to characterize parameters mentioned above (Tg , loss and storagemodulus, etc.) (see Kundu et al. 2005; Osborn et al. 2007; Sgreccia et al. 2010). However,a few works relate the modulus obtained by DMA to the conventional mechanical proper-ties (Deng et al. 2007; Richeton et al. 2005). Stiffness variations (storage modulus) for var-


ious polymers (thermoset, thermoplastics, amorphous, semi-crystalline, linear, crosslinked,etc.) over a wide range of temperature ranging from the glassy to rubbery state havebeen reported by Mahieux and Reifsnider, Richeton and others (see Richeton et al. 2005;Mahieux and Reifsnider 2001; Mahieux et al. 2001, 2002). A modulus (storage modu-lus) versus temperatures curve is typically obtained by DMA. All transitions (beta, glassy,and flow) can be seen in these curves. Therefore, Mahieux and Reifsnider (2001, 2002),Mahieux et al. (2001) introduced the temperature-dependent storage modulus in terms ofthree major property transitions observed in polymers and polymer matrix composites us-ing DMA. However, in their work, this stiffness parameter has not been used for simulatingstress–strain behavior of polymers or polymer matrix composites. Considering the workby Mahieux and Reifsnider (2001, 2002), Mahieux et al. (2001), Richeton et al. (2005)proposed a model for the prediction of the stiffness modulus for a wide range of tempera-ture and frequencies/strain rates. This new formulation has been validated for two amor-phous polymers (polymethyl-methacrylate, PMMA and polycarbonate, PC) using DMAand uniaxial compression testing. The elasticity modulus obtained from uniaxial compres-sion tests at different temperature and strain rates are simulated by a proposed rate andtemperature-dependent formulation. However, similarly to Mahieux and Reifsnider (2002),this formulation has not been used in a constitutive model for simulating stress–strain be-havior. Temperature-dependent moduli of two cured epoxy systems and two silica-epoxynanocomposites were experimentally measured by DMA and mechanical tests at varioustemperatures and correlation between DMA and mechanical testing data is established inDeng at al. (2007).

In this work, a constitutive model for modeling of hydro-thermo-mechanical behaviorof Nafion NRE212, used as membrane material in PEMFCs, over a wide range of temper-ature, humidity, and strain rate is proposed. The modified constitutive model is the FiniteViscoplasticity Theory Based on Overstress for Polymers (FVBOP), see Krempl and Glea-son (1996), Krempl (1996), Krempl and Tachiban (1998), Colak and Krempl (2005), Colak(2005), Hassan et al. (2011). The modifications have been performed such that tempera-ture and humidity effects can be incorporated. Recognizing that the working conditionsof PEMFCs include cyclic variations in temperature and hydration, the monotonic uniax-ial loading and unloading behavior of Nafion NRE212 is modeled at different strain rates(0.1, 0.01 and 0.001 s−1), temperatures (25, 50, 65, 80 ◦C) and hydrations (3, 7 and 8 %swelling) using the modified unified state variable theory (VBOP). The storage moduluscurve of Nafion NRE212 is modeled using the equation given by Mahieux and Reifsnider(2002) and used as the elastic parameter which is dependent on temperature. This approachof modeling using shifting modulus value is inspired by the work of Deng et al. (2007).Experimental data obtained by Silberstein and Boyce (2010) are compared to the simulationresults.

2 Nafion NRE212

The most commonly used membrane material in PEM fuel cells is perfluorosulfonic acid(PFSA) membrane. Among the PFSA polymers used in fuel cells, the most favored one hasbeen the sulfonated tetrafluoroethylene-based fluoropolymer, a copolymer with the tradename Nafion®, developed and manufactured by Walther Grot of DuPont in the 1960s. Thematerial is generated by the copolymerization of a perfluorinated vinyl ether comonomerwith tetrafluoroethylene (Mauritz and Moore 2004).


Fig. 2 Chemical structure ofNafion

The chemical structure of Nafion is shown Fig. 2. The fluorinated backbone of the poly-mer is essentially polytetrafluoroethylene (PTFE) (Teflon®), which gives Nafion good me-chanical strength and resistance to harsh chemical environments. Fluorinated ether linkagesterminate in sulfonate groups.

Despite much research, Nafion membrane morphology is still a subject of debate, seeMauritz and Moore (2004) and Galperin et al. (2009). There are two distinct hydropho-bic and hydrophilic regions in Nafion. The hydrophobic region is a semi-crystalline regionwhich is Teflon®-like. The hydrophilic regions consist of sulfonate groups, which swell andchange size/shape with water absorption and allow water and proton transport (Majsztrik etal. 2007, 2008).

The equivalent weight (EW), which is the ratio of the number of grams of polymer permole of sulfonic acid groups of the material in the acid form and when completely dry, canbe varied and strongly affects mechanical properties. Increasing EW, which means decreas-ing sulfonic acid group concentration, improves mechanical properties but decreases protonconductivity. Nafion with 1100 EW is typically used in PEM fuel cell applications sinceit has a reasonable balance of proton conductivity and mechanical integrity (Mauritz andMoore 2004).

Nafion’s specifications are designated by a system of numbers. The first two numbersdenote EW while the third and possible fourth digits denote dry membrane thickness inthousandths of an inch. (For example, Nafion NRE212 is non-reinforced dispersion castfilm based on Nafion with a 2100 EW and 0.002 in thickness.)

3 Finite Viscoplasticity Theory Based on Overstress for Polymers (FVBOP)

There are two classes of continuum theories. The first includes classical plasticity theoriesin which all time effects, such as rate sensitivity, creep, relaxation and strain recovery, areexcluded. The second contains viscoplasticity theories which assume that inelastic deforma-tion is rate-dependent even at low homologous temperatures.

Viscoplasticity models represented by unified state variable theories do not permit theseparation of creep and plasticity. One of the unified state variable theories is ViscoplasticityTheory based on Overstress (VBO) developed by Krempl and his co-workers for metallicmaterials (see Krempl 1996; Krempl and Tachiban 1998; Colak and Krempl 2003, 2005;Colak 2004). State variables in the model are defined as macroscopic integrators of eventsassociated with microstructure changes and cannot be directly measured or controlled.

VBO model is based on standard linear solid (SLS) model with extensive modificationsto the properties of each element. Rheological representation of SLS is given in Fig. 3.

Governing equation for SLS model is given in Eq. (1):

dε

dt+ E2

ηε = 1

E1

dσ

dt+ σ

η

(1 + E2

E1

)(1)


Fig. 3 Standard linear solidmodel (SLS)

When Eq. (1) is written in overstressed form, Eq. (2) is obtained:

ε = σ

E1+ σ − Bε

C(2)

where B = E1E2E1+E2

and C = ηE1E1+E2

, E1, and E2 are stiffnesses of springs and η is viscosityfunction.

In Eq. (2), (σ − Bε) is called overstress. This overstress concept is used to develop VBOmodel. It is initially developed for modeling the mechanical behavior of metallic materials.However, due to the similarities observed in the polymeric and metallic materials’ behavior,VBO is modified to capture mechanical behavior of polymers as well, see Colak (2005),Dusunceli and Colak (2008, 2006).

With changes in temperature and frequency (or rate), significant variations are observedin the viscoelastic parameters which are strongly dependent on molecular motions and seg-mental mobility. Therefore, any factor which affects macromolecular mobility, such as age-ing and crystallinity, leads to significant changes in viscoelastic properties of polymers.Increasing strain rate increases the strength of the material, since the polymer chains cannotfind enough time for recovery. When temperature is increased, the thermal energy of poly-mer molecules exceeds that of van der Waals interactions between the molecular chains andweak crosslinking interactions between sulfonic acid groups. Chains are able to move easilywhen the force is applied. As a result, the yield stress, ultimate stress and elasticity modulusdecrease. The effect of hydration on mechanical behavior of polymers is due to the ionicregions which absorb water and change the interactions between chains.

For modeling material behavior of polymeric materials, two main mechanisms areneeded: molecular interaction (stretching and orientation of molecular network) and inter-molecular interaction. In VBOP, resistance to deformation due to secondary bonds, whichare present between molecular chains (intermolecular interactions), is captured by includingthe characteristics of a nonlinear dashpot. Intermolecular resistance depends on rate, tem-perature and hydration. Characteristics of a nonlinear spring are used in the model to capturethe resistance to deformation due to primary bonds. Stretching and changing in the orien-tation of molecular network are defined by nonlinear spring element. The nonlinear springand dashpot are in parallel. For elastic behavior, linear spring is added to the parallel systemas well. A rheological representation of VBOP model is depicted in Fig. 4.

Finite VBO is obtained by replacing the ordinary time derivative by an objective one inthe small deformation viscoplasticity theory based on overstress (Colak 2004). The additivedecomposition of the rate of deformation tensor D is used for modeling finite deformation.The rate of deformation tensor is decomposed into elastic and viscoplastic parts. The flowlaw for finite deformation theory of VBO is

d = de + dvp = 1 + ν

CEs + 3

2F

[Γ

D

]s − gΓ

(3)


Fig. 4 Rheologicalrepresentation of VBOP model

where s and g are respectively the deviators of the Cauchy stress tensor, σ , and the equilib-rium stress, G, which is the stress that the material can sustain at rest. E is Young’s modulusand ν is the elastic Poisson’s ratio; de and dvp are deviators of elastic and viscoplastic rateof deformation tensor D, respectively; ‘◦’ denotes the objective rate. The main differencebetween VBO and VBOP is the parameter C, given by C = 1 − λ(|G − K|/A)α , where λ

and α are model parameters, K kinematic stress and A isotropic stress. For metals, C isusually set to one for representing linear unloading behavior. Γ is the overstress invariantwith the dimension of stress defined by

Γ 2 = 3

2(s − g) : (s − g), (4)

F [ ] is the positive, increasing flow function with the dimension of 1/time and F [0] = 0.The symbol [ ] stands for “the function of”. The flow function F [ ] is set as a power law. Itis responsible for modeling nonlinear rate sensitivity and is given by

F [ ] = B

(Γ

D

)m

(5)

with B as a universal constant having the dimension of 1/time. D is the drag stress, whichcan be considered as another state variable with a growth law. However, in this study it is aconstant.

One of the state variables is the equilibrium stress. The equilibrium stress is similar butnot quite the same as the back stress in rate-independent plasticity models. In plasticitymodels the back stress is considered as the repository for kinematic hardening, whereasin VBOP the repository for kinematic hardening is the kinematic stress. The equilibriumstress is the stress that must be overcome to generate inelastic deformation. The growth lawfor the deviatoric objective equilibrium stress, which is the rate-independent contribution tohardening, is

g = ψsE

+ ψF

[Γ

D

](s − gΓ

− g − kA

)+

(1 − ψ

E

)k (6)

where ψ is shape function bounded by Et < ψ < E. It affects the transition from the quasi-elastic to the inelastic region. The isotropic stress A is a scalar state variable for modeling


rate-independent cyclic hardening (or softening) behavior. Its effect is similar to the isotropichardening in rate-independent plasticity (Krempl 1996).

The evolution of the shape function is given in Eq. (7):

ψ = ψ1 +(

C2 − ψ1

exp(C3|εvp|))

, ψ1 = C1

(1 + C4

( |G|A + |K| + Γ ζ

))(7)

where C1, C2, C3, C4 and ζ are material parameters determined using transition regionsfrom elastic to viscoplastic responses of the stress–strain curves.

The difference between Cauchy stress and equilibrium stress is called overstress. In theelastic region of stress–strain curve, bond stretching occurs. When the stress is increasedbeyond a certain level, some chains in amorphous and crystalline phase overcome the sec-ondary interactions (Van der Walls) and irreversible slippage develops and yielding occurs(Yoon and Huang 2011). The equilibrium stress is overcome to generate inelastic deforma-tion. Whenever yielding occurs, overstress becomes nonzero. Then entangled chains startlocking up and strain hardening results.

During deformation, hardening (dynamic recovery) occurs due to entanglement (disen-tanglement) of polymer chains. To simulate dynamic recovery and hardening, the hardeningterms, which are the first two terms in the evolution equation of the equilibrium stress, andthe dynamic recovery term, the third term in Eq. (6), are included into the evolution equationof the equilibrium stress tensor.

A tensor-valued kinematic stress k is introduced to model the Bauschinger effect, andtension/compression asymmetry through initial conditions. It also sets the tangent modulusEt at the maximum inelastic strain of interest. The tangent modulus can be positive, zero ornegative. The growth law for the objective kinematic stress is given as follows:

k = EtF

[Γ

D

]s − gΓ

(8)

One of the fundamental principles that all constitutive equations have to satisfy is the prin-ciple of objectivity or frame indifference. According to this principle, constitutive equationsmust be invariant under a change of reference frame. Tensor rates used in constitutive equa-tions need to be objective. A co-rotational objective rate of a tensor A is denoted by

A = A + AΩ − ΩA (9)

where A is the material rate with respect to the basis of A. A is objective rate of A and Ω isa skew-symmetric spin tensor. Various forms of Ω can be evaluated, such Jaumann, Green–Naghdi, Logarithmic rates. Since skew-symmetric spin tensor is zero for uniaxial loading,objective rate of any tensor will be equal to material rate.

4 Modification of FVBOP model to account for temperature and humidity effect

The present form of VBOP is capable of modeling uniaxial or multiaxial, monotonic orcyclic behavior of wide classes of polymeric materials (such as thermoset, thermoplastics,semi-crystalline or amorphous, etc.) (see Colak 2005; Hassan et al. 2011; Dusunceli andColak 2006, 2008) and metallic materials at room temperature (see Colak and Krempl 2003,2005; Colak 2004). The modifications in FVBOP have been performed such that temper-


Table 1 Transition temperaturesand modulus values on the onsetof transitions (see Almeida andKawano 1999;Osborn et al. 2007)

T1 (Tβ ) T2 (Tg) T3 (Tf )

253 ◦K 373 ◦K 503 ◦K

E1 = 1550 MPa E2 = 700 MPa E3 = 14.2 MPa

ature and humidity effects can be modeled. From the phenomenological point of view, itis known that elasticity and tangent modulus, yield stress and ultimate stress depend ontemperature. In the modeling of polymeric material behavior at different temperatures, twomethods have been used to account for temperature. In the first, some of the material param-eters are defined as varying with respect to temperature (as a function such as parabolic). Inthe second, some of the state variables or material functions are described through functionsof temperature that are physically related. The second method is especially useful for vis-cous effects (Chaboche 2008). A similar approach can be followed for humidity effects aswell. In this work, both methods are used to model hydro-thermo-mechanical behavior ofpolymeric materials.

Departing from currently reported research, this work uses the result of DMA based de-termination of the storage modulus as a function of temperature to establish the temperaturedependence of the modulus of elasticity. First, the applicability of the storage modulus byDMA as the elasticity modulus obtained from conventional tests is investigated. Then, thecorrelation between storage modulus and elasticity modulus is derived. The storage mod-ulus versus temperature curve of Nafion NRE212 is modeled using the equation given byMahieux and Reifsnider (2002) and used as the temperature-dependent elastic parameterwith the idea of shifting modulus inspired by the work of Deng et al. (2007).

4.1 Correlation between storage modulus and elasticity modulus

Temperature-dependent elasticity modulus is determined using the storage modulus dataobtained from DMA test results. In order to determine storage modulus versus temperaturecurve mathematically, Eq. (10) proposed by Mahieux and Reifsnider (2002) is used:

E(T ) = (E1 − E2) exp

(−

(T

T1

)m1)+ (E2 − E3) exp

(−

(T

T2

)m2)

+ E3 exp

(−

(T

T3

)m3)(10)

Temperature-dependent storage modulus equation uses the three transition temperature val-ues T1, T2, T3 (sub-glass (β) transition, glass (α) transition, flow) and the modulus valuesat the onset of transitions E1, E2, E3 as parameters. Exponential parameters, m1, m2, m3,are Weibull constants which are statistical parameters. Those parameters are almost con-stant for all amorphous thermoplastic polymers (m1 = 1.5, m2 = 20, m3 = 20) (Mahieuxand Reifsnider 2002).

Transitions temperatures of Nafion determined by Almeida and Kawano (1999) and Os-born et al. (2007) are used in this work. Experimental work of Silberstein and Boyce (2010)is used to determine the modulus values and compared to the storage modulus curve ob-tained by Eq. (10). Transition temperatures and modulus values on the onset of transitionsof Nafion are given in Table 1. Comparison of storage modulus obtained by Eq. (10) andDMA result of Silberstein and Boyce (2010) is given in Fig. 5.


Fig. 5 Comparison of storagemodulus obtained by Eq. (10)and the DMA experimental resultof Silberstein and Boyce (2010)

Fig. 6 Comparison of elasticitymodulus varying withtemperature and storage moduluscurve obtained from Eq. (10)

The storage modulus is related to the elastic contribution of the total viscoelastic behavior(Menard 1999). The elasticity modulus is defined as resistance to elastic deformation andobtained from the slope of the stress–strain curve in the elastic region. Therefore, these twomoduli definitions relate to the same physical phenomena—elastic deformation. Figure 6shows the comparison of elasticity modulus varying with temperature determined from theuniaxial tension experiments of Silberstein and Boyce (2010) and storage modulus curveobtained from Eq. (10). As seen from the figure, the magnitudes of elasticity modulus andstorage modulus are not the same. But it could be noticed that the variation of modulusvalues with varying temperature follows the same trend.

Storage and elasticity moduli relate to the same physical phenomena, as explained above,but the technique and instruments used to determine these moduli are different. Indeed,a DMA device can be considered as a miniature tensile testing machine. But the size ofthe specimen and the loading frame is much smaller than a conventional tensile testingmachine. Therefore, the stiffness of a tensile testing machine is much greater than that of aDMA device. This can lead to a DMA overestimating the value of the material modulus. Thework of Deng et al. (2007), mentioned in previous sections, proposes an easy and effective


Fig. 7 Shifted and unshiftedstorage modulus curves andelasticity modulus varying withtemperature. Experimental data isobtained by uniaxial tension testsat different temperatures bySilberstein and Boyce (2010)

way to neglect those effects originated by experimental technique, therefore the value ofthe modules becomes slightly equal. If a straightforward vertical shifting is applied on thestorage modulus curve, the values of the two moduli overlap. Figure 7 shows a comparisonof −600 MPa shifted storage modulus curve and elasticity modulus values varying withtemperature.

One fact should be clarified as a constraint of this shifting phenomenon. The shiftingphenomenon is not valid for temperatures above Tg . For temperatures above Tg , the shiftedmodulus values will be negative which is physically meaningless. Therefore the shifting isvalid until the start of glass transition.

Following the procedures explained above, temperature-dependent elasticity modulus isobtained as given in Eq. (11):

E(T ) = 850 exp

(−

(T

253

)1.5)+ 658.8 exp

(−

(T

372

)20)

+ 14.2 exp

(−

(T

503

)20)− 600 (11)

This technique of determining the temperature-dependent elasticity modulus by a singleDMA test derives a more efficient, easy and thrifty experimental work. Moreover, this tech-nique neglects the necessity to assure a large number of homogenous specimens.

4.2 Variable tangent modulus

The viscoplastic region of Nafion NRE212 is highly nonlinear. Moreover, temperature de-pendency of the viscoplastic region of stress–strain curve cannot be ignored as is seen inFig. 10. Therefore a nonlinear and temperature-dependent tangent modulus is defined as inEqs. (12a) and (12b):

Et0(T ) = −0.03∗T + 16 (12a)

Et = Et0(1 + ea|εvp|)β

2(12b)


Fig. 8 Stress–strain curves atdifferent strain rates for ahypothetical material. The flowfunction is given in Eq. (5)

where T is temperature in Celsius. Equation (12a) is determined from the uniaxial ten-sion tests performed at different temperatures at the same loading rate by Silberstein andBoyce (2010), considering of the slopes of the viscoplastic part of the stress–strain curvesdetermined at different locations. Equation (12a) provides the initial temperature-dependenttangent modulus, while Eq. (12b) is determined as a function of inelastic strain in order tomodel the nonlinear viscoplastic behavior of Nafion NRE212 (a = 3 and β = 0.96).

4.3 Temperature and humidity dependent flow function

Defining elasticity modulus and tangent modulus as function of temperature is not enoughto model hydro-thermo-mechanical behavior of polymers. Therefore, the flow function F [ ],which is responsible for rate dependency in VBOP model, is modified to account for tem-perature and humidity effects as well. The power law form of flow function in VBOP isgiven in Eq. (5).

When the effects of temperature, humidity and rate on the mechanical behavior of mate-rials are compared, it is observed that increasing temperature or humidity leads to a decreasein the material properties such as elasticity modulus, yield stress, etc. On the other hand, in-creasing strain rate increases the mechanical properties. Therefore, the effect of strain rateor temperature–hydration is inversely correlated. In the present form of VBOP, strain ratedependency is captured with flow function which is in the form of power law. With modi-fications, in addition to strain rate dependency, temperature and hydration dependency arealso captured by flow function. This analogy of temperature–hydration and strain rate effectsleads to the idea of modifying flow function in FVBOP as given in Eq. (13):

F [] = B

[δD

m

(Γ

δD

)m

+ ξD

n

(Γ

ξD

)n](13)

where δ and ξ are the parameters for temperature and humidity, respectively. Equation (13)is introduced considering the viscoplastic potential assumed as power function in Chaboche(Almeida and Kawano 1999).

Since flow function is responsible for rate dependency also, it is needed to investigatehow the rate dependency change with the made modifications in flow function, F [ ]. Whenthe present form of flow function is used, the stress–strain behavior for an arbitrary materialparameters is obtained as in Fig. 8.


Fig. 9 Stress–strain curves atdifferent strain rates for ahypothetical material. Modifiedflow function used is given inEq. (13)

Table 2 δ temperature-dependent parameter used in thesimulations, ξ = 0.5

T (◦K) 298 323 338 353

δ 1 0.32 0.13 0.001

The difference between Cauchy (σ ) and equilibrium stress (G) is the overstress whichis a rate-dependent component. If the modified form of F [ ] is used, stress–strain responsesare obtained as shown in Fig. 9.

As is seen from figures, the rate dependency is still captured. Depending upon materialparameters chosen in flow function, the rate dependency can be simulated.

5 Modeling temperature and humidity dependent deformation responseof Perfluorosulfonic Acid (PFSA) membrane Nafion NRE212

For the conduction of protons in PEMFCs, the polymer electrolyte membrane needs to behumidified. As a function of temperature and moisture, polymer electrolyte membrane ex-periences expansion and contraction. Determination of the mechanical behavior of PEMsexperimentally and prediction of the mechanical behavior and its dependence on temper-ature and hydration and lifetime of the membrane precisely is of interest for the designof robust PEMFCs. Therefore, uniaxial tension behavior of perfluorosulfonic acid (PFSA)membrane Nafion NRE212 is modeled at four different temperatures (25, 50, 65, 80 ◦C) andat three water contents (3, 7 and 8 % swelling). Large-strain monotonic loading and unload-ing tensile behavior of Nafion NRE212 is highly rate, temperature and hydration dependent.Uniaxial tensile tests are conducted at the strain rates of 0.001 to 0.1 1/s and temperaturesfrom 25 to 100 ◦C (298 to 353 K) and 3, 7 and 8 % swelling by Silberstein and Boyce(2010). The experimental data are compared to simulation results.

A humidity value of 50 % is used in the uniaxial tension experiments at different tem-peratures. Since humidity is kept constant, parameter related to humidity is kept constant(ξ = 0.5 (for %50 RH)) and simulations are performed for different δ’s and variable elas-ticity modulus obtained from DMA and tangent modulus given in Eqs. (12a) and (12b).Temperature-dependent parameters (δ) used in the simulations are given in Table 2.

Simulation results of uniaxial stress–strain behaviors of Nafion NRE212 at differenttemperatures are presented in Fig. 10. As seen in the figure, the modified VBO model is


Fig. 10 Nafion NRE212experimental and simulationresponses under uniaxial straincontrolled loading and unloading(strain rate = 0.01 1/s) atdifferent temperatures and50 % RH. Experimental data areobtained from Silberstein andBoyce (2010)

Table 3 ξ hydration-dependentparameters used in thesimulations

Swelling ratio (%) Dry 3 7 8

ξ 0.5 0.17 0.1 0.07

now capable of modeling the mechanical response of Nafion which is highly temperature-dependent even in the elastic region. Using the modified flow function, the temperature-dependent elasticity modulus and variable tangent modulus are used to account for temper-ature dependency, temperature-dependent elastic region, yield stress and the nonlinear vis-coplastic region. In addition to temperature-dependent behavior, the modeling of unloadingbehavior of polymeric materials is still an issue. Most of the material models in the literatureare not able to simulate nonlinear unloading behavior. On the other hand, VBOP is capableof capturing nonlinear unloading behavior as seen in Fig. 10. The C function is responsiblefor this nonlinear behavior in addition to shape function which affects the transition fromelastic or viscoelastic to viscoplastic region.

For investigating hydration effect on the mechanical response of Nafion NRE212, spec-imens are left in the water for a variable amount of time in accordance with desired watercontent. Then uniaxial tension experiments are performed by the method of Silberstein andBoyce (2010). The swelling percentage was calculated from the change in the distance be-tween dots marked on the specimen from the dry state to the hydrated state at the start of thetest.

A hydration-dependent coefficient ξ (given in Table 3) was used in the modified formof the flow function to model the hydration effects on the mechanical behavior of NafionNRE212. The change in the flow function due to the hydration-dependent coefficient ξ

changes the overstress, and this reduction of overstress captures the onset of yield phe-nomenon. Hydration dependence of on elasticity and the tangent moduli could be neglectedcompared to temperature dependency. As seen from Fig. 11, the stress–strain behaviors ofNafion NRE212 under various swelling percentages (humidity) are captured well. Swellingdecreases the network density. Therefore the mechanical stiffness decreases. The strain ratein simulations given in Fig. 11 is also 0.01 s−1 as used in the experimental work.

Strain rate dependency is also simulated and depicted in Fig. 12. Simulation results arenot compared to experimental data obtained by Silberstein and Boyce (2010) since therewere some inconsistencies in their experiments about rate dependency. In the aforemen-tioned work by Silberstein and Boyce (2010), in their Fig. 3, which is labeled “True stress–


Fig. 11 Nafion NRE212experimental and simulationresponses under uniaxial straincontrolled loading and unloading(strain rate = 0.01 1/s) atdifferent swelling percentages atroom temperature. Experimentaldata are obtained fromSilberstein and Boyce (2010)

Fig. 12 Rate dependency ofNafion NRE212 under uniaxialtension

true strain behavior at 25 ◦C at multiple strain rates (inset: logarithmic rate dependence ofyield stress)”, the square-dotted line (rate 0.01 s−1) and in Fig. 5(a), which is labeled “Truestress–true strain curve at 0.01 s−1 and (a) as a function of temperature”, the circle-dottedline (25 ◦C) do not match. The results are different even though the loading conditions arethe same. Therefore, only simulation results of FVBOP are depicted in Fig. 12.

Trial and error analysis is used for determination of material parameters. The other ma-terial parameters used in the simulations are given in Table 4.

6 Conclusions

The Finite Viscoplasticity Theory Based on Overstress for Polymers (FVBOP) is modified tosimulate hydro-thermo-mechanical behavior of Nafion NRE212 which is used as membranematerial in PEMFCs. The storage modulus versus temperature curve of Nafion NRE212 ismodeled using the equation given by Mahieux and Reifsnider (2002) and used as the elas-tic parameter which is dependent on temperature in the modeling with the idea of shiftingmodulus values inspired by the work of Deng et al. (2007). Uniaxial tension behavior of


Table 4 VBO parameters usedin the simulations Young’s Modulus (E) Varies with temperature

Tangent Modulus (Et ) Varies with temperature

Shape Function c1 135 MPa

c2 55 MPa

c3 and c4 0.31, 0.2

ζ 2

Flow Function B 1 1/s

D 38 MPa

m 6

n 10

Izotropic Stress, A 5 MPa

λ,α 0.25 and 0.15

perfluorosulfonic acid (PFSA) membrane Nafion NRE212 is modeled at four different tem-peratures (25, 50, 65, 80 ◦C) and three swelling percentages (3, 7 and 8 %). Experimentaldata obtained by Silberstein and Boyce (2010) are compared with simulation results. Strainrate, temperature and hydration dependent monotonic loading and unloading large deforma-tion behavior of Nafion NRE212 membrane is captured well with the modified FVBOP.

Acknowledgement The support of the Scientific and Technological Research Council of Turkey (TUBITAK)is gratefully acknowledged. The Project No. 108M521.

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