Z. angew. Math. Phys. 53 (2002) 160–1660010-2571/02/010160-7 $ 1.50+0.20/0c© 2002 Birkhauser Verlag, Basel

Zeitschrift fur angewandteMathematik und Physik ZAMP

The Bailey criterion: Statistical derivation and applicationsto interpretations of durability tests and chemical kinetics

Alan D. Freed and Arkady I. Leonov ∗

Abstract. The Bailey durability criterion, well known in mechanics of materials, has also beenused in other fields of study such as the kinetics of chemical reactions. This paper rationalizesthe wide applicability of this criterion in terms of Markovian statistical properties of systems.Two particular cases are discussed as examples of the general approach: durability of a wideclass of solid materials and cure (cross-linking) reactions employed for fabrications of thermosetsand rubbers.

Keywords. Durability, Markovian stochastic process, damage, cure, cross-linking.

1. Introduction

For many materials, whose durability (or lifetime) is tested under time dependentsimple loading conditions, it is possible to use the Bailey criterion of durability [1]:

∫ t?

0

dt

τ(σ(t)

) = 1. (1)

Here τ(σ) is the durability function determined experimentally under constantstress σ , and t? is the durability under an arbitrarily given active loading processσ(t) .

The following restrictions seem to play an important role for applicability ofEq. (1) to the durability of materials with various structures:1. Irreversibility of the damage process,2. Independence of the various damage accumulations, and3. Absence of memory effects in every damage accumulation.

Materials that obey these restrictions are typical hard materials such as metals,glasses, minerals, hard plastics, thermoset resins, and highly extended polymericfibers. These hard materials have such common features as incremental deforma-tions and an absence of memory effects.

∗ Corresponding author.

Vol. 53 (2002) The Bailey criterion 161

The above three conditions, along with the discrete character of the damageformation in materials, are in a sense quite general. They indicate that relation(1) might be applied to various different kinetic processes, whose behaviors arepossible to describe using discrete stochastic processes of the Poisson type [2].Indeed, Eq. (1) has been used in [3] to describe the non-isothermal cure kineticsfor rubbers and thermosets.

2. Statistical derivation

Consider a physical system that under external actions undergoes random orderedtransitions within a discrete set of states 0, 1, 2, . . . , n, . . . , exposed in increasingorder. Let ξ(t) be the Markovian, purely discontinuous stochastic process, whichat some discrete time instants accepts values 0, 1, 2, . . . , n, . . . in increasing order.The function ξ(t) characterizes the changes in state of this system over the timeinterval (0, t) . Let q(x; t, t+dt) be the transition probability characterizing thechange of state x of the system in the time interval (t, t+dt) . It is assumed that

q(x; t, t+∆t) = K(θ)∆t + o(∆t). (2)

Here K(θ) > 0 , with θ being a positive external parameter that reflects theinternal actions imposed on the system. This parameter might in general betime dependent (i.e., θ = θ(t) ). The Markovian assumption for the process ξ(t)results in a statistical independence of changes in the state of the system in non-intersecting time intervals. It can also be assumed that the probability of morethan one change of ξ(t) in small time intervals ∆t is negligible (i.e., equal too(∆t) ). Then the probability of a n -fold change in the state of the system overthe time interval (0, t) is described by the well-known Poisson distribution [2]:

Pn(t) =λn(t) e−λ(t)

n!, (n = 0, 1, 2, . . .), (3)

wherein

λ(t) =∫ t

0

K(θ(t′)

)dt′. (4)

Due to Eq. (3), the value λ(t) is equal to the average number of changes in thestate of the system:

〈nt〉 =∞∑

n=0

nPn(t) =∞∑

n=0

nλn(t)

n!e−λ(t) = λ(t). (5)

It is now assumed that the state of the system becomes, in a sense, criticalwhenever the average number of changes of state in the system reaches a certainlimiting value N . Given a time dependence for the internal parameter θ(t) ,values for N are reached at different time instants tN , depending on the choice

162 A. D. Freed and A. I. Leonov ZAMP

of θ(t) . Equations (4) and (5) therefore combine to produce the relation for tN :

〈ntN〉 ≡ N = λ(tN ) =

∫ tN

0

K(θ(t)

)dt. (6)

Let us now introduce two variables, τ and t? , defined as:

τ = tN |θ=const , t? = tN |θ=var . (7)

When θ = const , the variable τ is determined from Eq. (6) as follows:

τ = τ(θ) =N

K(θ)

∣∣∣∣θ=const

. (8)

Because the function K(θ) does not explicitly depend on time t , Eq. (8) alsodefines the dependence τ

(θ(t)

). Then expressing the function K(θ) via Eq. (8)

and substituting it into Eq. (6) yields:∫ t?

0

dt

τ(θ(t)

) =1N

∫ t?

0

K(θ(t)

)dt = 1. (9)

Equation (9) is Bailey’s criterion (1) applied to an external time-dependent pa-rameter θ(t) . Differentiating Eq. (3) with respect to time yields:

dPN (t)dt

= K(θ(t)

)(N − λ(t)

)λN−1(t) e−λ(t)

N !. (10)

Equation (10) shows that the time instant t? ( = tN ) when the system reachesits critical condition is characterized by a maximum of probability PN (t) .

The parameter of relative change in the system, νn , can also be introduced asthe ratio of current change in the state of the system, n , to the critical one, N ,viz.:

νn =n

N. (11)

Whenever the external parameter characterizing the system is kept constant, thendue to Eqs. (6) and (11),

νn =tntN

, or ν(t) =t

ν(t)=

t

τ(θ = const). (12)

It is remarkable that, in this case, the parameter νn does not explicitly dependon the external parameter θ . This is a direct result of the above Markovianassumptions. Whenever the parameter θ is time dependent (i.e., θ = θ(t) ), thenthe parameter νn becomes explicitly represented as a functional of the functionθ(t) :

νn =

∫ tn

0K

(θ(t)

)dt∫ tN

0K

(θ(t)

)dt

, or ν(t) =

∫ t

0K

(θ(t′)

)dt′∫ t?

0K

(θ(t′)

)dt′

. (13)

Equations (12) and (13) are convenient for interpretations of experimental obser-vations.

Vol. 53 (2002) The Bailey criterion 163

3. Applications

3.1. Durability

The Bailey criterion has been successfully applied many times to durability testsdone on those classes of materials discussed in the introduction. In this case theparameter θ may be interpreted as the stress σ applied to a specimen of a testedmaterial in simple experiments like uniaxial stretching, with the stochastic pro-cess ξ(t) describing the number of ‘damages’ accumulated in the specimen underapplied stress. Here the Markovian assumption means independence between thevarious existing damages during their accumulation. At the last stage of materiallifetime, when the material has accumulated a lot of damages, the assumption ofthe damage independence is, of course, not valid anymore. At this stage voidsand micro-cracks start to collide to form the magisterial crack, which finally re-sults in rupture of the material specimen. Nevertheless, it is observed that thislast period of interaction is very short when compared to the whole lifetime du-ration. Therefore, for an overwhelming duration time of the lifetime interval, theaccumulation of material damages can be considered as statistically independent.Consequently, neglecting the duration of the last period, the function τ(σ) inEq. (8) can be viewed as approximately describing the whole lifetime period. Anindirect confirmation of this viewpoint is that the relative damage accumulationunder constant stress in durability (creep) tests of the aforementioned materialsis commonly proportional to the ratio of duration of the current time interval tothe lifetime of material [4], as it is described by Eqs. (11) and (12).

Additionally, if the function τ(σ) in Eq. (8) and the amount of damages,N , are known, then one can use Eq. (8) for establishing a relation between thedependence of the critical state of damage, characterized by the function τ(σ)under a constant stress σ , and such a fundamental characteristic of the materialas the main factor K(σ) in the transition probability (2).

For the above materials where Markovian conditions seem to be valid, theEuring-Zhurkov formula [4, 5],

τ(σ, T ) = τ0 exp(

U − γσ

RT

), (14)

has been established and widely verified for describing material lifetime underisothermal and constant-stress loading conditions. In Eq. (14), T is the ‘absolute’(Kelvin) constant temperature, σ is the constant stress, U is the activationenergy per mole, R is the universal gas constant, τ0 is the typical sub-molecularperiod of oscillation, reciprocal to the Debay frequency, when T →∞ (which forcrystals is about 10−12 sec.), and γ is a typical scale of periodicity in the solidstructure (e.g., crystal periodicity for crystalline materials). Equation (14) meansthat durability is in fact the result of thermo-fluctuations, which can eventuallydestroy a material even without applying a stress. The stress action just decreasesthe potential barrier U in the stress direction [4, 5], and therefore facilitates the

164 A. D. Freed and A. I. Leonov ZAMP

thermo-fluctuations.It should be mentioned that the Bailey relation (1) is applicable to formula

(14) not only when σ = σ(t) , but also when temperature variations T = T (t) areapplied with or without the presence of a time-dependent stress action.

3.2. Chemical reactions

The Bailey criterion can also be applied for describing the kinetics of some chemicalreactions. Here it is assumed that the kinetics can be approximately described bya one-stage, gross (or ‘macroscopic’) kinetic equation that lumps together manymicroscopic stages of reaction. A typical example of this type is a thermoset cross-linking (cure) reaction (e.g., see [3]) where an initial low-molecular weight liquidresin is finally cross-linked into a solid-like 3D polymer network. Another exampleof such a cure reaction is the vulcanization of ‘green’ rubber that behaves as aviscoelastic liquid. The macromolecules of the rubber are cross-linked during thecure reaction by a curing agent (say, sulfur) in a solid-like 3D polymer network [6].In actuality, the whole curing process is very complicated and cannot be describedas a one-stage reaction. However, its very important initial stage can be treated inthis way up to a point of gelation (or ‘scorch’ in the case of rubber vulcanization),when the material loses its fluidity. It seems that up to this point all Markovianassumptions are satisfied. Here the stochastic function ξ(t) describes the amountof cross-links emerged during the reaction up to current time t , with the externalparameter θ denoting temperature. For such reactions, an interpretation of thegeneral approach in §2 is straightforward. Therefore the gelation time τ(T ) , foundexperimentally under isothermal conditions, can be easily described by the Baileyequation (1) under non-isothermal conditions given via the function T (t) . In thiscase, the ‘non-steady cure durability’ τ

(T (t)

)should be substituted in Eq. (1)

instead of the ‘non-steady stress durability’ τ(σ(t)

)of the prior section.

For the above class of chemical reactions, the Bailey criterion can also be di-rectly derived from the initial cure kinetics. In this case, the reaction convergence,the ratio of current cross-link numbers to the ‘final’ cross-link number, is definedas a function α(t) ( 0 ≤ α ≤ 1 ) that monotonically increases in time and satisfiesthe initial condition: α(0) = 0 . It is convenient to introduce the monotonicallydecreasing function defined as: β(t) = 1 − α(t) . Evidently, β(0) = 1 . Thefunction β(t) is generally described by the kinetic equation,

dβ

dt= −k(T ) f(β). (15)

The function f(β) is assumed to be a regular and monotonically increasing pos-itive function in the interval 0 ≤ β ≤ 1 , satisfying the condition f(β) = o(β)as β → 0 . Simple specifications of function f(β) have been considered in theliterature [3, 6]. Also, the temperature dependence of reactivity, k(T ) , is assumed

Vol. 53 (2002) The Bailey criterion 165

to be described by the Arrhenius relation:

k(T ) = k0 exp(−Ur

RT

), (16)

where Ur is the activation energy of chemical reaction and k0 is a constant pre-exponential factor. It is easy to see that under the above conditions a uniqueregular solution β(t) of Eq. (16) with the stated initial condition does exist andtends toward zero as t →∞ .

Consider a non-isothermal reaction governed by a given function T (t) . Thefirst integral of Eq. (15), with an account of its initial condition, can be presentedby the form:

Φ(β) ≡∫ 1

β

dx

f(x)=

∫ t

0

k(T (t′)

)dt′. (17)

It is now assumed that the gel transition (scorch) occurs when the function β(t)reaches a certain value β? . This assumption is in accordance with many observa-tions and measurements. Then due to Eq. (17), the time t? for reaching gelationunder a non-isothermal condition is found for a given function T (t) as follows:

1Φ(β?)

∫ t?

0

k(T (t)

)dt = 1. (18)

A comparison of Eqs. (18) and (6) demonstrates that parameter Φ(β?) plays therole of a critical number of cross-links when gelation happens. The analog of Eq.(8) becomes:

τ(T ) =Φ(β?)k(T )

∣∣∣∣T=const

, (19)

and the Bailey criterion (1) is now presented as:∫ t?

0

dt

k(T (t)

) = 1. (20)

The main conclusion of this paper is that the Bailey criterion is independent ofthe physical mechanism of a governing kinetic process, and can be used in thosecases where the general Markovian assumptions are valid.

Acknowledgment

Author (AIL) is grateful for NASA support via Research Grant #NCC3–752.

References

[1] J. Bailey. An attempt to correlate some tensile strength measurements on glass: III. GlassInd. 20 (1939), 95–99.

166 A. D. Freed and A. I. Leonov ZAMP

[2] M. S. Bartlett. An Introduction to Stochastic Processes. The University Press, Cambridge,1955.

[3] A. I. Leonov, N. I. Basov and Yu. V. Kazankov. Fundamentals of Injection Molding ofThermosets and Rubbers. (In Russian), Chimiya, Moscow, 1977, pp. 56–59.

[4] G. M. Bartenev and Yu. S. Zuyev. Strength and Failure of Visco-Elastic Materials. PergamonPress, New York, 1968.

[5] S. Glasston, K. G. Laidler, H. Eyring. The Theory of Rate Processes. McGraw Hill, NewYork, 1941.

[6] R. Ding and A. I. Leonov. A kinetic model for sulfur accelerated vulcanization of a naturalrubber compound. J. Appl. Polym. Sci., 61 (1996), 455–463.

Alan D. FreedPolymers BranchMS 49-3NASA Glenn Research Center21000 Brookpark RoadCleveland, OH 44135USAe-mail: Alan.D.Freed@grc.nasa.gov

Arkady I. LeonovDepartment of Polymer EngineeringThe University of AkronAkron, OH 44325-0309USAe-mail: leonov@uakron.edu

(Received: June 14, 2001)

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