# A model for the thermoviscoelastic behavior of physically aged polymers.

1 INTRODUCTIONThe paper is concerned with the non-isothermal viscoelastic behavior of physically aged polymers. Our objective is twofold: (i) to derive a constitutive model for the long-term viscoelastic response (when the characteristic time of observation exceeds the characteristic time of material aging) in amorphous polymers, and (ii) to demonstrate the effect of physical aging on residual stresses built up in polymeric articles cooled after curing. We confine ourselves to small strains when the mechanical response in polymers may be treated as linear. However, the obtained results can be easily extended to nonlinear viscoelastic media as well.

Physical aging of amorphous and semicrystalline polymers has been the focus of attention in the past two decades, see, e.g., Struik (1), Kovacs et al. (2), Chai and McCrum (3), Tant and Wilkes (4), Rendell et al. (5), McKenna (6), Hodge (7), to mention a few. Aging is treated as a strong dependence of the mechanical response in a viscoelastic solid on the elapsed time [t.sub.e], when the medium is quenched from above the glass transition temperature [[Theta].sub.g] to some temperature [Theta] below [[Theta].sub.g] and annealed for the time [t.sub.e] at the temperature [Theta] before loading.

Short-term static and dynamic tests for amorphous polymers demonstrate that the elapsed time [t.sub.e] affects mainly the retardation (relaxation) spectra, while elastic moduli (initial and equilibrium) remain unchanged. This permits creep (relaxation) master-curves to be constructed by using short-term creep (relaxation) curves measured at various elapsed times (the time-aging time superposition principle (1)).

For a number of polymers, the long-term response can be successfully predicted with the use of the master curve based on the concept of reduced time (1). However, for several polymeric materials, this approach entails significant deviations between results of numerical simulation and observations. Struik (1) demonstrated that short-term master-curves do not describe creep in poly(vinyl chloride) when the duration of measurements is comparable with the characteristic time of aging. Similar results were obtained by Matsumoto (8) for poly(ether imide) and polycarbonate, by Read et al. (9, 10) and Dean et al. (11) for polypropylene and poly(vinyl chloride), and by Woo and Kuo (12) for poly(ether imide). Discrepancies between observations and their theoretical prediction are usually ascribed to changes in limiting compliances caused by physical aging (13).

To predict observations in long-term creep tests, Read et al. (9, 10) proposed a model which employs some parameters measured in high-frequency dynamic tests. That model is an extension of a model developed previously by Struik (1). The first objective of the present paper is to develop a constitutive model which would describe the long-term isothermal viscoelastic behavior of amorphous polymers based on experimental data obtained in short-term static tests only. Our aim is to derive relatively simple constitutive equations that can be used to assess stresses built up in aged viscoelastic solids with complicated geometry, and that do not require complicated dynamic tests to find adjustable parameters. For this purpose, we do not model explicitly suppression of the [Beta]-relaxation peaks caused by physical aging (9-11), but account for this phenomenon implicitly treating the rates of breakage and reformation of adaptive links as functions of the elapsed time.

Another question of interest is the effect of physical aging on the non-isothermal response in viscoelastic media. This issue is of essential importance for polymer engineering, since basic stages of polymer and polymer-composite processing (heating a soft polymer for curing and cooling a hard article) are essentially non-isothermal (14-16). Physical aging of amorphous polymers at different (but fixed) temperatures was analyzed by Kovacs (17) and Kovacs et al. (18). The effect of temperature on isothermal physical aging has been recently studied experimentally in a number of works, see, e.g., Lee and McKenna (19), Espinoza and Aklonis (20), Mariani et al. (21). The non-isothermal behavior of aged media in quench-up-quench (T-jump) tests was analyzed by Kovacs et at (2), Chai and McCrum (3, 13), Lagasse et al. (22), Struik (23), McKenna (24), McKenna et al. (25), to mention a few. Uniform cooling (heating) of aged polymers was considered by Hutchinson and Kovacs (26) and Kovacs and Hutchinson (27).

The other objective of this paper is to apply the constitutive model to the analysis of stresses in a viscoelastic spherical pressure vessel cooled from the glass transition temperature to the room temperature, and to demonstrate that the material aging significantly affects both stresses and displacements in the polymeric article. For this purpose, we determine adjustable parameters of the model for poly(vinyl chloride) and analyze the cooling process numerically. The effect of physical aging on residual stresses in inorganic glasses has been considered by a number of authors, e.g., Lee et al. (28), Narayanaswamy and Gardon (29), Gardon and Narayanaswamy (30), Narayanaswamy (31), Tackels and Crochet (32), Crochet and Denayer (33), Soules et al. (34), Mauch and Jackle (35). For application of the Narayanaswamy theory to calculating residual stresses in plastics, see Struik (36). Unlike previous studies, confined to stresses built up in an annealed plate, we (i) analyze deformation of a spherical pressure vessel, where physical aging incorporates with the stress inhomogeneity, and (ii) employ new constitutive equations.

Our constitutive relations are based on the theory of transient networks, see, e.g., Green and Tobolsky (37), Yamamoto (38), Lodge (39), Tanaka and Edwards (40). According to conventional models of temporal networks, a polymeric chain is treated as a beam that does not permit axial deformations and which does not resist bending (an incompressible strip). Mechanical energy of a chain is neglected, and the effect of macrostrains on its free energy is taken into account through the configuration entropy which is treated as a function of the end-to-end vector. That model is quite acceptable at high temperatures (when the average distance between junctions is rather large, and the contribution of entropy into the free energy exceeds that of the mechanical energy), and it adequately predicts the response in rubber polymers and in polymeric gels.

Drozdov (41) proposed a model of adaptive links that can describe the isothermal behavior of aged polymers below the glass transition temperature. At low temperatures, the average distance between junctions becomes relatively small (due to the essential growth in the number of entanglements), which implies an abrupt decrease in the configurational entropy. On the contrary, the specific mechanical energy increases significantly (because of the growth in elastic moduli), and its contribution into the specific free energy becomes predominant. For a glassy polymer, the configurational entropy is neglected, and polymeric chains are thought of as purely elastic springs (links), which break and emerge because of thermal motion.

In the standard models of transient networks, rates of reformation and breakage of active chains are treated as independent functions to be prescribed. Unlike those studies, we demonstrate that the breakage function can be expressed in terms of the reformation rate with the use of a balance equation similar to that derived by Yamamoto (38) and Tanaka and Edwards (40). The reformation rates for physically aged polymers are assumed to depend on the current temperature and the time elapsed after cooling below the glass transition temperature. Bearing in mind the objectives of this study, we accept an explicit dependence of the shift factor on time. However, the effect of elapsed time on the reformation rates may be interpreted as a dependence of the relaxation spectrum on the free-volume fraction (volume relaxation), on the specific enthalpy (enthalpy recovery), or on the fictive temperature (the Tool concept of fictive temperature). According to simplified theories of physical aging, changes in fictive temperature, enthalpy, and free volume occur with the same rate of approaching thermodynamic equilibria (27). Experimental data for several polymers demonstrate decoupling of structural and stress relaxation (24, 25) and different rates of the free volume and enthalpy recovery (42). However, we do not dwell on this question in the present study.

The exposition is organized as follows. In section 2 we begin with a version of the model of transient networks and establish some connections between the rates of breakage and reformation for elastically active chains. As a result, we arrive at differential equations for the numbers of active chains of various kinds similar to equations in population dynamics. Coefficients in these equations are assumed to depend on the current temperature and time. We postulate that (in the simplest cases) the effects of these parameters can be separated and confirm our assertion by experimental data. In section 3 constitutive equations are introduced for uniaxial loading. Adjustable parameters in these equations are determined in section 4 using data obtained in short-term creep tests. To validate the model, results of numerical simulation are compared with experimental data obtained in long-term tests, and fair agreement is demonstrated between observations and their prediction. The constitutive relations are extended to three-dimensional loading in section 5. In section 6 the model is applied to predict residual stresses and strains built up in a polymeric pressure vessel under cooling. The effects of material and structural parameters on stresses and displacements in the vessel are analyzed numerically.

2 A MODEL OF TRANSIENT NETWORKS

According to the Tanaka-Edwards theory (40, 43), a viscoelastic medium is treated as a network consisting of polymeric chains connected to junctions by sticky functional groups. An active chain (whose both ends are connected to separate junctions) is modeled as an elastic link. Snapping of one end of an active chain from a junction is tantamount to breakage of the link. When a dangling chain captures one of the junctions in its neighborhood, a new active chain is created.

We assume that M different kinds of active chains exist that correspond to M different relaxation times. In the polymer dynamics theory (44), micro-Brownian motion of chain molecules is determined by potentials which are characterized by several length scales (bond length, persistence length, coil diameter, etc.). From this standpoint, any kind of active chains corresponds to interactions with some characteristic length (time). On the other hand, different kinds of active chains correspond to different network elements (long chains, slide and entanglement chains, short chains, etc. (45)). Finally, several kinds of active chains allow us to distinguish entanglements and temporal (physical) crosslinks with different lifetimes.

Let [Mathematical Expression Omitted] initial chains not be involved in the reformation process (permanent chemical crosslinks and reversible physical crosslinks whose lifetimes exceed the test duration). Denote by [X.sub.m](L [Tau]) the number of chains of the mth kind which have arisen before instant r and exist at instant t. The quantities [Mathematical Expression Omitted] and [X.sub.m](t, [Tau]) refer to chains located in a unit volume in the reference configuration.

We introduce the relative rates of reformation

[Mathematical Expression Omitted], (1)

where [Mathematical Expression Omitted] = [X.sub.m](0, 0), and the breakage functions [g.sub.m](t, [Tau]) which equal the relative number of active chains of the ruth kind existing at instant [Tau] and broken before instant t. Evidently,

[Mathematical Expression Omitted],

[Mathematical Expression Omitted]. (2)

Substitution of expressions (2) into the formula

[X.sub.m](t, t) = [X.sub.m](t, 0) + [integral of] [Delta][X.sub.m] / [Delta][Tau] (t, [Tau]) d[Tau] between limit t and 0.

yields

[Mathematical Expression Omitted]. (3)

Equation 3 is similar to the integral equations derived by Yamamoto (88) and Tanaka and Edwards (40). Our approach, however, differs from the conventional theory of transient networks, since the reformation process is described entirely by the functions [X.sub.m] of two instants, t and [Tau], while the standard balance equation is written for the chain-distribution function that depends on the current instant t and the end-to-end vector. The conservation Eq 3 extends a similar equation derived in (41), where the effect of permanent crosslinks was neglected.

First, we consider viscoelastic media with time-independent mechanical properties. The latter means that the total numbers of active chains [X.sub.m] (t, t) remain constant, the rates of reformation [[Gamma].sub.m] are independent of time, and the breakage functions [g.sub.m] depend only on the difference between the instant of creation r and the current instant t:

[Mathematical Expression Omitted]. (4)

It follows from Eqs 3 and 4 that

[g.sub.m0](t) = [[Gamma].sub.m0] [integral of] [1 - [g.sub.m0]([Tau])] d[Tau] between limits t and 0.

Differentiation of this equality implies that

[dg.sub.m0] / dt (t) = [[Gamma].sub.m0] [1 - [g.sub.m0](t)], [g.sub.m0](0) = 0. (5)

Solving Eq 5, we find that

[g.sub.m0](t) = 1 - exp (-[[Gamma].sub.m0]t). (6)

We substitute expression (6) into Eq 2, use Eq 4, and obtain

[Mathematical Expression Omitted],

[Mathematical Expression Omitted]. (7)

It follows from Eqs 4 and 7 and the formula

[X.sub.m](t, [Tau]) = [X.sub.m](t, t) - [integral of] [[Delta].sub.m] / [Delta]s (t, s)ds, between limits t and [Tau],

that the total namber (per unit volume) of active chains arisen before instant [Tau] and existing at instant t

[Mathematical Expression Omitted] (8)

is calculated as

[Mathematical Expression Omitted], (9)

where

[Mathematical Expression Omitted]. (10)

Let us consider a subsystem of a network of chains of the mth kind. We assume that the subsystem contains [v.sub.m]([Tau]) active chains at instant [Tau] and calculate the number of active chains [v.sub.m](t) at an arbitrary instant t. By analogy with Eq 2, we write

[v.sub.m](t) = [v.sub.m]([Tau]) [1 - [g.sub.m](t, [Tau])]. (11)

Differentiation of Eq 11 with the use of Eqs 4 and 5 implies that

1 / [v.sub.m](t) [dv.sub.m] / dt (t) = -[[Gamma].sub.m0]. (12)

According to Eq 12, formula (5) means that the relative rate of breakage equals [[Gamma].sub.m0] for any subsystem of chains of the mth kind in the transient network.

For viscoelastic media with time-dependent properties, Eqs 4 are not valid, and the balance equation (3) contains three unknown functions [X.sub.m](t, t), [[Gamma].sub.m](t), and gin(t, [Tau]). According to the standard approach, to calculate the numbers of active chains [X.sub.m](t, t), the chain generation rates [[Gamma].sub.m] and the chain breakage rates [Delta][g.sub.m]/[Delta]t are prescribed as functions of the current state of the transient network, e.g., (40, 43, 46). Unlike those works, we assume that Eq 5 for the breakage functions [g.sub.m](t, [Tau]) remains true both for aged and non-aged media:

[Delta][g.sub.m] / [Delta]t (t, [Tau]) = [[Gamma].sub.m](t)[1 - [g.sub.m] (t, [Tau])], [g.sub.m]([Tau], [Tau]) = 0. (13)

Given reformation rates [[Gamma].sub.m](t), Eq 13 is treated as a governing equation for the breakage functions [g.sub.m](t, [Tau]). After solving this equation, Eq 3 is employed to calculate the total number of adaptive links [X.sub.m](t, t) the mth kind.

It follows from Eqs 11 and 13 that the functions

[Mathematical Expression Omitted] (14)

satisfy the differential equations

1 / [n.sub.m](t) [dn.sub.m] / dt (t) = -[[Gamma].sub.m](t), [n.sub.m](0) = 1,

1 / [N.sub.m](t, [Tau]) [Delta][N.sub.m] / [Delta]t(t, [Tau]) = - [[Gamma].sub.m](t), [N.sub.m]([Tau], [Tau]) = [[Gamma].sub.m]([Tau]]). (15)

Equations 15 imply that breakage and reformation of adaptive links in an aged viscoelastic medium are determined by the functions [[Gamma].sub.m](t) reciprocal to the relaxation times [T.sub.m](t). Confining ourselves to thermorheologically simple media, we assume that the relaxation times [T.sub.m] and the reformation rates [[Gamma].sub.m] with different indices m change in time similarly to each other

[T.sub.m](t) / [T.sub.m0] = [[Gamma].sub.m0] / [[Gamma].sub.m](t) = a(t). (16)

The function a(t) characterizes the effects of temperature [Theta] and the elapsed time [t.sub.e] on the relaxation spectrum. We accept the separability hypothesis, which states that the influences of temperature and aging are independent of each other

a = [a.sub.[Theta]] ([Theta]) [a.sub.[t.sub.e]]([t.sub.e]).(17)

The separability principle for the absolute temperature [Theta] and some measure of deviations from the thermodynamic equilibrium of a glass goes back to Tool (47), who proposed to treat the shift factor [a.sub.[t.sub.e]] as a function of the fictive temperature [[Theta].sub.f]. The Tool model was extended by Narayanaswamy (31), who suggested a phenomenological equation for the function [a.sub.[t.sub.e]] similar to the Arrhenius formula. Using the molecular kinetics approach, Adam and Gibbs (48) derived a theory, where the coefficient [a.sub.[t.sub.e]] was expressed in terms of the configurational entropy. According to the KAHR concept (2), the shift factor [a.sub.[t.sub.e]] depends on the free volume fractionf. The above approaches were combined by Scherer (49), who incorporated the Narayarmswamy and the Adam-Gibbs equations.

As common practice, the shift factor [a.sub.[Theta]] obeys the Arrbenius equation

log [a.sub.[Theta]] = [Delta]H / Rln10 (1 / [Theta] - 1 / [[Theta].sup.0]), (18)

where [Delta]H is the activation enthalpy, [[Theta].sup.0] is the reference temperature, and R is the Boltzmann constant. To determine the adjustable parameter [Delta]H, a mastercurve is constructed by shifts along the time axis of short-term creep (relaxation) curves measured at different temperatures [Theta] for the same elapsed time [t.sub.e] and plotted in the bi-logarithmic coordinates.

To account for the effect of elapsed time on the relaxation spectra, a simple phenomenological equation is introduced for the shift function ate, see, e.g., (1, 23, 50),

[Mathematical Expression Omitted], (19)

where [Mathematical Expression Omitted] is the reference elapsed time, and C is the aging rate, which is close to unity. To determine the adjustable parameter C, a master-curve is constructed by shifts along the time axis of short-term creep (relaxation) curves measured at the same temperature [Theta] for different elapsed times [t.sub.e] and plotted in the bi-logarithmic coordinates.

In the general case, the activation enthalpy [Delta]H depends on the elapsed time [t.sub.e], while the aging rate C is affected by the temperature [Theta]. The separability principle (17) states that these coupling effects may be neglected. To validate this assertion, we present experimental data for an epoxy glass and poly(vinyl chloride). Figures 1 and 2 demonstrate that the shift factors [a.sub.[Theta]] and [a.sub.[t.sub.e]] are practically independent of [t.sub.e] and [Theta], respectively.

The standard KAHR model (2) captures (i) nonlinearity with respect to the magnitude of deviations from the thermodynamic equilibrium, (ii) asymmetry of relaxation curves caused by positive and negative temperature jumps, and (iii) nonexponentiality of memory effects. That model contains at least two adjustable parameters (an analog of the activation enthalpy for the fictive temperature in the Narayanaswamy formula and the fractional exponent in the KWW equation for deviations of the fictive temperature), which are found by fitting experimental data. Determination of material parameters is not an easy problem, see a discussion of this issue in (51, 52), and significant deviations can occur between experimental data and their prediction, in particular, for relatively rapid cooling, see, e.g., (53, 54). This entails that Eqs 17 through 19 with C [approximately equal to] 1 may be even preferable compared to conventional models for physical aging, provided these relations have some physical basis. Our objective now is to demonstrate that at the initial stage of the aging process, the simplified equation (19) follows from the KAHR model.

According to the one-parameter version of the KAHR model (27), after quenching of a polymer from the reference temperature [[Theta].sup.0] to some temperature [Theta], the fictive temperature [[Theta].sub.f] obey the linear kinetic equation

[d[Theta].sub.f] / [dt.sub.e] = [Theta] - [[Theta].sub.f] / T, [[Theta].sub.f](0) = [[Theta].sup.0], (20)

where the characteristic time of annealing T is calculated by the formula

T = [T.sub.0][a.sub.[Theta]]([Theta])[a.sub.[[Theta].sub.f]]([[Theta].sub.f]). (21)

Here [T.sub.0] is the characteristic time at the reference temperature [[Theta].sup.0], [a.sub.[Theta]] is the shift factor caused by changes in the absolute temperature, and [a.sub.[[Theta].sub.f]] is the shift factor caused by structural relaxation. By accepting a linear dependence of the logarithm of the shift factor [a.sub.[[Theta].sub.f]] on the deviation of the fictive temperature [[Theta].sub.f] from its equilibrium value [Theta], we arrive at the formula

ln [a.sub.[[Theta].sub.f]] = B(([Theta] - [[Theta].sub.f]), (22)

where B is an adjustable parameter. Combining Eqs 20 to 22, we find that

d[Delta][[Theta].sub.f] / [dt.sub.e] = -[Delta][[Theta].sub.f] exp (B[Delta][[Theta].sub.f]) / [T.sub.0][a.sub.[Theta]]([Theta]), [Delta][[Theta].sub.f](0) = [[Theta].sup.0] - [Theta], (23)

where

[Delta][[Theta].sub.f] = [[Theta].sub.f] - [Theta].

Integration of Eq 23 implies that

[Mathematical Expression Omitted]. (24)

Introducing the exponential integral function

Ei (-x) = -[integral of] exp(-t) / t dt between limit [infinity] and x

we present Eq 24 as

Ei (-B[Delta][[Theta].sub.f]([t.sub.e]) - Ei (-B[Delta][[Theta].sub.f](0)) = -[t.sub.e] / [T.sub.0][a.sub.[Theta]([Theta]). (25)

For sufficiently large x;

Ei (-x) = exp(-x) / -x.

Applying this formula to Eq 25 and assuming that the difference of temperatures [[Theta].sup.o] - [Theta] is rather large (which entails that the second term in the left-hand side of Eq 25 may be neglected), we obtain

exp(-B[Delta][[Theta].sub.f])([t.sub.e])) / B[Delta][[Theta].sub.f}([t.sub.e]) = [t.sub.e] / [T.sub.0][a.sub.[Theta]]([Theta]). (26)

Since for sufficiently large

exp(-x) / x [less than or equal to] exp (-x),

Eq 26 yields

exp(-b[Delta][[Theta].sub.f]([t.sub.e])) [greater than or equal to] [t.sub.e] / [T.sub.0][a.sub.[Theta]]([Theta]).

Combining this inequality with Eq 22, we arrive at the formula

[a.sub.[[Theta].sub.f]] [greater than or equal to] [t.sub.e]/[T.sub.0][a.sub.[Theta]]([Theta]). (27)

On the other hand, for sufficiently large x,

exp(-x) / x [greater than or equal to] exp[-(1 + [Kappa])x],

where [Kappa] is an arbitrarily small positive quantity. It follows from this estimate and Eq 26 that

exp(-B[Delta][[Theta].sub.f]([t.sub.e])) [less than or equal to] ([t.sub.e] / [T.sub.0][a.sub.[Theta]]([Theta])).sup.1/1+[Kappa]],

which, together with Eq 22, implies that

[a.sub.[[Theta].sub.f]] [less than or equal to] ([t.sub.e] / [T.sub.0][a.sub.[Theta]]([Theta]). (28)

Combining Eqs 27 and 28, we arrive at the formula

ln [a.sub.[[Theta].sub.f]] [approximately equal to] ln [t.sub.e] / [T.sub.0][a.sub.[Theta]]([Theta]). (29)

The function [a.sub.[[Theta].sub.f]]([t.sub.e]) in Eq 29 determines shift of a creep (relaxation) curve observed after an elapsed time with respect to the curve obtained immediately after quenching. According to Eq 29, shift of a creep (relaxation) curve measured after the elapsed time [t.sub.e] with respect to the creep (relaxation) curve obtained after the reference time of aging [Mathematical Expression Omitted] is calculated as

[Mathematical Expression Omitted], (30)

which coincides with Eq 19 with C [approximately equal to] 1.

According to Eq 30, the model (17) through (19) is equivalent to the one-parameter version of the KAHR model provided that the difference between the reference temperature [[Theta].sup.0] and the current temperature [Theta] is sufficiently large. This implies that formula (19) is not valid in the immediate proximity of the glass transition temperature [[Theta].sub.g] (which is confirmed by experimental data for an epoxy glass (19) and for polycarbonate and polystyrene (23)). However, hypotheses (17) through (19) are quite acceptable to predict the viscoelastic behavior of polymers under cooling provided that the initial stage of the process (when [[Theta].sub.g] - [Theta] [less than] 10 [degrees] C, say) is rather short and it does not affect strongly the mechanical response.

3 CONSTITUTIVE EQUATIONS FOR A VISCOELASTIC MEDIUM

In this section, we employ the model of transient reversible networks to derive constitutive relations for a viscoelastic medium at uniaxial deformation. The obtained equations will be extended to three-dimensional loading in section 5.

An active chain of the mth kind is modeled as a linear elastic spring with a rigidity [Mathematical Expression Omitted] (for simplicity, we assume that rigidities of chains of different kinds coincide and they are temperature-independent). The natural (stress-free) configuration of a dangling chain that captures a junction at instant [Tau] coincides with the actual configuration of the network at that instant. This assertion is equivalent to the hypothesis that stress in a dangling chain is totally relaxed before the chain catches a new sticky junction (40).

The potential energy of an active chain is calculated as

[Mathematical Expression Omitted], (31)

where [[Epsilon].sub.*] (t, [Tau]) is the strain at the current instant t in an active chain created at instant [Tau] Let [Mathematical Expression Omitted] be the unit vector directed along a chain at the instant of its formation, and [Mathematical Expression Omitted] the (macroscopic) strain tensor in a viscoelastic medium at instant t. Evidently,

[Mathematical Expression Omitted], (32)

where the dot stands for inner product. For uniaxial deformation, we set

[Mathematical Expression Omitted], (33)

where [Mathematical Expression Omitted] is the unit eigenvector of the strain tensor. Substitution of expressions (32) and (33) into Eq 31 implies that

[Mathematical Expression Omitted], (34)

where the angles [Phi] and [Theta] determine the position of the vector [Mathematical Expression Omitted] in spherical coordinates connected with the vector [Mathematical Expression Omitted].

We suppose that strain energy density of the entire transient network equals the sum of strain energy densities of active chains (40). Summing up potential energies (34) and assuming the medium to be isotropic, we find that the swain energy density [Mathematical Expression Omitted] (per unit volume in the reference configuration) of active chains of the mth kind created within the interval [[Tau], [Tau] + d[Tau]] is determined as follows:

[Mathematical Expression Omitted].

By analogy with this equality, the strain energy density [Mathematical Expression Omitted] (per unit volume in the reference configuration) of the initial active chains of the ruth kind equals

[Mathematical Expression Omitted].

Calculation of the integrals implies that

[Mathematical Expression Omitted], (35)

where [Mathematical Expression Omitted]. Summing potential energies 135) for active chains existing at instant t, we obtain the total potential energy of the network

[Mathematical Expression Omitted]. (36)

Combining Eqs 8 and 36, we arrive at the formula

[Mathematical Expression Omitted]. (37)

On the other hand. substitution of expressions (10) and (14) into Eq 36 implies that

[Mathematical Expression Omitted], (38)

where [Mathematical Expression Omitted] The stress [Sigma](t) is determined by the formula (e.g. (55))

[Sigma](t) = [Delta]W(t) / [Delta][Epsilon](t). (39)

Equations 37 and 39 result in the constitutive equation

[Mathematical Expression Omitted]. (40)

Assuming the network to be strain-free at the initial instant t = 0 and integrating Eq 40 by parts, we arrive at the constitutive relation

[Sigma](t) = c [integral of] X(t, [Tau]) d[Epsilon]/dt ([Tau]) d[Tau] between limit t and 0, (41)

which expresses the Boltzmann superposition principle in linear viscoelasticity (the Risz theorem). The latter means that the model of transient networks describes an arbitrary linear dependence of the current stress [Sigma](t) on the entire history of strains provided the function X (which determines reformation of active chains) is chosen appropriately.

It follows from Eqs 38 and 39 that

[Sigma](t) = [Mu] {[[Eta].sub.0] + [summatio of] [[Eta].sub.m][n.sub.m](t)][Epsilon](t) where m = 1 to M + [summation of] [[Eta].sub.m] where m = 1 to M [integral of] [N.sub.m](t, [Tau])[[Epsilon]([Tau])d[Tau]} between limits t and 0.

Bearing in mind that

[[Eta].sub.0] = 1 -[summation of] [[Eta].sub.m] where m = 1 to M,

we obtain

[Sigma](t) = [Mu] {[1 - [Zeta](t)][Epsilon](t) - [summation of] [[Eta].sub.m] where m = 1 to M [integral of] [N.sub.m](t, [Tau])d[Tau]} between limits t and 0, (42)

where

[Zeta](t) = [summation of] [[Eta].sub.m][[Zeta].sub.m](t), [[Zeta].sub.m](t) = 1 -[n.sub.m](t) where m = 1 to M [integral of] [N.sub.m](t, [Tau])d[Tau] between limit t and 0. (43)

It follows from Eqs 15 and 43 that the functions [[Zeta].sub.m](t) obey the ordinary differential equations

[d[Zeta].sub.m] / dt (t) = -[[Gamma].sub.m](t)[[Zeta].sub.m](t), [[Zeta].sub.m](0) = 0. (44)

Equations 43 and 44 imply that [[Zeta].sub.m](t) equals zero identically, the function [Zeta](t) vanishes, and Eq 42 is presented as

[Sigma](t) = [Mu][[Epsilon](t) - [summation of] [[Eta].sub.m] where m = 1 to M [integral of] [N.sub.m](t, [Tau])[Epsilon]([Tau])d[Tau] between limits t and 0]. (45)

Formula (45) provides a constitutive equation for a thermorheologically simple, aged, linear, viscoelastic medium. The functions [N.sub.m](t, [Tau]) in the right-hand side of Eq 45 satisfy Eqs 15, where the reformation rates [[Gamma].sub.m] are determined by Eq 16. The parameters [[Gamma].sub.m0] in Eq 16 equal reformation rates in a medium quenched from the glass transition temperature [[Theta].sub.g] to the reference temperature [[Theta].sup.0] and annealed for the reference time [Mathematical Expression Omitted]. The shift factor a is Eq 16 is expressed in terms of the temperature [Theta] and time t with the use of Eqs 17 through 19.

4 PREDICTION OF LONG-TERM CREEP RESPONSE

To demonstrate advantages of the model, we consider uniaxial tension of a viscoelastic specimen quenched from the glass transition temperature [[Theta].sub.g] to some reference temperature [[Theta].sup.0] and loaded at the temperature [[Theta].sup.0] (after annealing for a time [t.sub.e]) according to the program

[Mathematical Expression Omitted]. (46)

Our objective is to predict the viscoelastic response of the specimen in long-term tests by using data obtained in short-term experiments.

It follows from Eqs 45 and 46 that the strain [[Epsilon].sub.[t.sub.e]](t) = [Epsilon]([t.sub.e] + t) is governed by the Volterra integral equation

[[Epsilon].sub.[t.sub.e]](t) - [summation of] [[Eta].sub.m] where m = 1 to M [integral of] [N.sub.m](t, [Tau])[[Epsilon].sub.[t.sub.e]]([Tau])d[Tau] between limits t and 0 = [Sigma] / [Mu]. (47)

For isothermal tests, we set

[a.sub.[Theta]] = 1.

In short-term tests, the shift factor [a.sub.[t.sub.e]] may be treated as independent of t,

[a.sub.[t.sub.e]] = [a.sub.[t.sub.e]]([t.sub.e]).

Combining these assertions with Eqs 16 and 17, we find that

[[Gamma].sub.m](t) = [[Gamma].sub.m0] / [a.sub.[t.sub.e]]([t.sub.e]. (48)

It follows from Eqs 15 and 48 that

[N.sub.m](t, [Tau]) = [[Gamma].sub.m0]/[a.sub.[t.sub.e]] ([t.sub.e] exp [-[[Gamma].sub.m0](t - [Tau])/[a.sub.[t.sub.e]]([t.sub.e])].

Substituting this expression into Eq 47, we obtain after simple algebra

[Mathematical Expression Omitted]. (49)

where the relaxation measure Q(t) equals

Q(t) = -[summation of] [[Eta].sub.m] where m = 1 to M [1 - exp(-[[Gamma].sub.m0]t / [a.sub.[t.sub.e]]([t.sub.e])], (50)

and the superposed dot denotes the derivative with respect to time. The solution of Eq 49 reads

[[Epsilon].sub.[t.sub.e]](t) = [1 + C(t)] [Sigma] / [Mu]], (51)

where the creep measure C(t) is connected with the relaxation measure Q(t) by the linear Volterra equation

[Mathematical Expression Omitted]. (52)

We begin with a set of short-term creep curves [[Epsilon].sub.[t.sub.e]] = [[Epsilon].sub.[t.sub.e]](t) measured at the reference temperature [[Theta].sup.0] for various elapsed times [t.sub.e] and find the shift factor [a.sub.[t.sub.e]] as a function of the elapsed time [t.sub.e], see Fig. 3.

The short-term creep master-curve (obtained by horizontal shifts of creep curves and reduced to the reference elapsed time [Mathematical Expression Omitted]) is approximated by expression (51). where the creep measure C(t) is presented by a truncated Prony series

C(t) = [summation of] [X.sub.m][1 - exp(-[[Gamma].sub.m0]t)] where m = 1 to M]. (53)

Given M and the reformation rates [[Gamma].sub.m0], the coefficients [X.sub.m] and the elastic modulus [Mu] are found to ensure the best fitting of experimental data, see Fig. 4.

To determine the relaxation measure Q(t) corresponding to the creep measure (53), we solve the Volterra integral equation

[Mathematical Expression Omitted] (54)

Combining Eqs 53 and 54, we find that

C(t) = -[summation of] [X.sub.m][1 - exp(-[[Gamma].sub.m0]t) where m = 1 to M - [Pm](t)], (55)

where the functions

[P.sub.m](t) = [[Gamma].sub.m0] [integral of] exp[-[[Gamma].sub.m0](t - [Tau])]Q([Tau])d[Tau] between limit t and 0.

satisfy the ordinary differential equations

[dP.sub.m] / dt (t) = [[Gamma].sub.m0][Q(t) - [P.sub.m](t)], [P.sub.m](0) = 0. (56)

Equations 55 and 56 are integrated numerically to calculate the dependence Q = Q(t), which, in turn, is approximated by the truncated Prony series, see Eq 50,

[Mathematical Expression Omitted].

The adjustable parameters [[Eta].sub.m] are found to ensure the best fit of the relaxation measure Q(t), see Fig. 5.

Given M, C, [Eta].sub.m], and [[Gamma].sub.m0], the material response in long-term creep tests is determined by the integral equation (47), where the functions [N.sub.m](t, [Tau]) obey Eqs 15. Introducing the notation

[[Psi].sub.m](t) = [integral of] [N.sub.m](t, [Tau])[J.sub.[t.sub.e]]([Tau])d[Tau] between limit t and 0, [J.sub.[t.sub.e]](t) = [[Epsilon].sub.[t.sub.e]](t)/[Sigma], (57)

we present Eq 47 in the form

[J.sub.[t.sub.e]](t) = [[Mu].sup.-1] + [summation of] [[Eta].sub.m][[Psi].sub.m(t) where m = 1 to m. (58)

According to Eqs 15 and 57, the functions [[Psi].sub.m](t) are governed by the ordinary differential equations [d[Psi].sub.m]/dt(t) = [[Gamma].sub.m0]/[a.sub.[t.sub.e]](t) [[J.sub.[t.sub.e]](t) - [[Psi].sub.m](t)], [[Psi].sub.m](0) = 0.

We determine the long-term creep compliance [J.sub.[t.sub.e]](t) by solving numerically Eqs 58 and 59 and compare results of calculations with experimental data. Figure 6 demonstrates fair agreement between experimental data and their numerical prediction, which means that the model may be used to describe the long-term isothermal response in aged polymers.

5 THREE-DIMENSIONAL LOADING

To extend the constitutive equation (45) to threedimensional deformations, we assume that (i) a viscoelastic medium is incompressible, and (ii) the deviatoric components of the stress tensor, [Mathematical Expression Omitted], and the strain tensor, [Mathematical Expression Omitted], are connected by the same relationship as stresses and strains under uniaxial loading. This implies that the first invariant [Epsilon] of the strain tensor obeys the equation

[Epsilon] = 3[Alpha]([Theta] - [[Theta].sup.0]), (60)

where [Alpha] is a (temperature-independent) coefficient of thermal expansion, while the deviatoric part of the stress tensor is calculated as

[Mathematical Expression Omitted]. (61)

As usually, we write [Mu] as a coefficient in the constitutire equation for uniaxial loading (where [Mu] stands for the Young modulus E) and 2[Mu] in the constitutive relation for three-dimensional loading (where [Mu] means the shear modulus G).

The incompressibility condition is a consequence of the hypothesis that the potential energy of the entire network equals the sum of potential energies of individual chains. To account for compressibility of the material under consideration, a correlation should be established between volume changes and changes in the energy of interaction between active chains, and an appropriate expression for the energy of interaction should be added to the right-hand sides of Eqs 37 and 38. We do not dwell on this question, since a constitutive model for compressible reversible networks will be studied elsewhere.

To demonstrate that Eq 61 implies the constitutive Eq 45 for uniaxial loading of an incompressible specimen, we present the deformation tensor [Mathematical Expression Omitted] in the form

[Mathematical Expression Omitted], (62)

where [Mathematical Expression Omitted] are unit vectors of a Cartesian coordinate frame. The incompressibility condition implies that the strain tensor [Mathematical Expression Omitted] coincides with its deviatoric part [Mathematical Expression Omitted]. Substitution of expression (62) into Eq 61 results in

[Mathematical Expression Omitted], (63)

where

[Mathematical Expression Omitted],

[Mathematical Expression Omitted]. (64)

The Cauchy stress tensor [Mathematical Expression Omitted] is determined by the formula

[Mathematical Expression Omitted], (65)

where p is pressure and [Mathematical Expression Omitted] is the unit tensor. Combining Eqs 63 and 65, we find that

[Mathematical Expression Omitted],

where

[[Sigma].sub.1](t) = -p(t) + [s.sub.1](t), [[Sigma].sub.2](t) = -p(t) - [s.sub.2](t). (66)

It follows from Eq 66 and the boundary condition on the lateral surface of the specimen

[[Sigma].sub.2](t) = 0

that p(t) = -[s.sub.2](t). Substitution of this expression and Eq 64 into Eq 66 implies that the only non-zero component of the stress tensor [Mathematical Expression Omitted] equals

[[Sigma].sub.1](t) = 3G[[Epsilon](t) - [summation of] [[Eta].sub.m] where m = 1 to M [integral of] [N.sub.m](t, [Tau])[Epsilon]([Tau])d[Tau] between limit t and 0]. (67)

The shear modulus G is connected with the Young modulus E by the formula

G = E / 2(1 + v), (68)

where v is Poisson's ratio. Setting [Upsilon] = 0.5 in formula (68) and substituting this expression into Eq 67, we arrive at the constitutive equation (45) for uniaxial loading of an incompressible specimen.

Equations 60 and 61 with temperature-independent coefficient of thermal expansion [Alpha], shear modulus [Mu], and relaxation intensities [[Eta].sub.m] provide an analog of a thermorheologically simple viscoelastic solid with a temperature-dependent relaxation spectrum. The novelty of our approach is that (i) the model accounts for physical aging of polymers, and (ii) it treats non-conventionally the effect of time-varying temperature on the relaxation times.

It is worth noting, that the concept of transient networks permits the effect of temperature on the shear modulus [Mu] and the coefficient of thermal expansion to be taken into account. A model of a thermorheologically complex viscoelastic medium (where both initial and equilibrium moduli are assumed to depend on temperature, but the effect of physical aging is neglected) is discussed in (56).

By analogy with Eq 57, we introduce the tensors

[Mathematical Expression Omitted] (69)

and present the constitutive Eq 61 as

[Mathematical Expression Omitted]. (70)

It follows from Eqs 15 and 69 that the functions [Mathematical Expression Omitted] satisfy the ordinary differential equations

[Mathematical Expression Omitted]. (71)

According to Eqs 70 and 71, to calculate stresses for a given deformation program [Mathematical Expression Omitted], it is not necessary to solve partial differential equations (15), but it suffices to integrate ordinary differential equations (71).

6 COOLING OF A VISCOELASTIC PRESSURE VESSEL

Fair agreement between numerical prediction of long-term viscoelastic behavior with experimental data demonstrated in section 4 for poly(ether imide) [similar results for polystyrene and poly(vinyl chloride) are presented elsewhere] shows that the model adequately describes physical aging of polymers. However, the question remains whether the effect of aging is important for applied problems in polymer engineering, i.e., whether this effect should be taken into account when residual stresses are calculated in polymeric articles cooled after curing. Since cooling of polymers is a quite complicated thermochemical process (curing at different temperatures ensures different conversion levels, which change the glass transition temperature, which, in turn, affects the aging rate, etc.), no experimental works are known, where the effect of physical aging on residual stresses is analyzed experimentally. In this paper, we study numerically cooling of a spherical pressure vessel on a metal mandrel and demonstrate that physical aging significantly affects stresses and displacements.

To describe the response in the polymeric thickwalled shell, we employ the constitutive model (60), (70), and (71) for an incompressible viscoelastic solid. As common practice, polymers in the glassy state are compressible media. However, since the effect of compressibility on stresses and displacements in thickwalled spheres and cylinders under cooling is rather weak (it does not exceed 10 to 15% when Poisson's ratio v changes from 0.0 to 0.5, see (57)), we neglect it in the present study.

We consider a thick-walled polymeric spherical shell with inner radius [R.sub.1] and outer radius [R.sub.2]. The viscoelastic shell is located on a metal mandrel: thick-walled elastic shell with inner radius [R.sub.0] and outer radius [R.sub.1]. At the initial instant t = 0, the structure is in the stress-free state at the glass transition temperature of the polymer [[Theta].sub.g]. To neglect residual stresses bufit up at curing, we assume that the vessel was annealed above [[Theta].sub.g] for a time necessary for the stresses to relax and was aged at the glass transition temperature [[Theta].sub.g] for a time to before cooling.

At cooling with a constant rate q, temperature of the oven [[Theta].sub.0] changes in time linearly

[Mathematical Expression Omitted], (72)

where [[Theta].sub.r] is room temperature, [T.sub.1] = ([[Theta].sub.g] - [[Theta].sub.r])/q, [T.sub.2] is the total time of cooling, and time t equals zero when the cooling process starts,

Temperature in the polymeric shell [Theta](t, r) at instant t at a point with radius r obeys the heat conduction equation

[Delta][Theta] / [Delta]t = [Kappa] ([[Delta].sup.2][Theta] / [Delta][r.sup.2] + 2 / r [Delta][Theta] / [Delta]r), (73)

where the temperature conductivity [Kappa] is assumed to be temperature-independent. Experimental data show that [Kappa] decreases in temperature below [[Theta].sub.g]. but rather slow: about 20% per 100 [degrees] C, see (58).

Temperature of the mandrel satisfies Eq 73 as well. However, since the temperature conductivity of metals exceeds that of polymers by about three orders of magnitude. we may suppose that temperature in the mandrel reaches its equilibrium distribution at any instant t and neglect the derivative with respect to time on the left-hand side of Eq 73. Neglecting heat exchange on the inner surface of the mandrel,

[Delta][Theta] / [Delta]r [where] r = [R.sub.0] = 0,

we find that temperature of the mandrel should be uniform,

[Theta](t, r) = [[Theta].sub.*](t), (74)

which implies that

[Delta][Theta]/[Delta]r [where] r = [R.sub.1] = 0.

Bearing in mind this equality, we write boundary conditions for Eq 73 as

[Delta][Theta] / [Delta]r (t, [R.sub.1] = 0, [Theta](t, [R.sub.2]) = [[Theta].sub.0](t). (75)

The initial condition for Eq 73 reads

[Theta](t, r) - [[Theta].sub.g]. (76)

Denote by u(t, r) radial displacement at instant t at a point with radius r. The first invariant [Epsilon] of the strain tensor is calculated as

[Epsilon] [Delta]u / [Delta]r + 2u / r, (77)

while the non-zero components of the deviatoric part of the strain tensor in the spherical coordinates {r, [Theta]. [Phi]] equal

[e.sub.r] = 2 / 3 F, [e.sub.[Theta]] = [e.sub.[Phi]] = -1 / 3 F, = [Delta]u / [Delta]r = u / r. (78)

We begin with deformation of the mandrel, which obeys the constitutive equations of an isotropic elastic medium with temperature-independent bulk modulus K, shear modulus G and coefficient of thermal expansion [[Alpha].sub.*]

[Mathematical Expression Omitted]. (79)

We combine Eqs 77 to 79, find the non-zero components of the stress tensor [Mathematical Expression Omitted], substitute them into the equilibrium equation

[Delta][[Sigma].sub.r] / [Delta]r + 2 / r ([[Sigma].sub.r] - [[Sigma].sub.[Theta]]) = 0, (80)

and obtain

[[Delta].sup.2]u / [Delta][r.sup.2] + 2 / r [Delta]u / [Delta]r - 2u / [r.sup.2] = 0.

The solution of this equation reads

u = [[Beta].sub.1](t)r [[Beta].sub.2](t)[r.sup.-2], (81)

where [[Beta].sub.1] and [[Beta].sub.2] are functions to be determined. Substitution of expression (81) into Eqs 77 to 79 implies that

[[Sigma].sub.r] = 3K[[[Beta].sub.1](t) - [[Alpha].sub.*] ([[Theta].sub.*] - [[Theta].sub.g])] - 4G[[Beta].sub.2](t)[r.sup.-3]. (82)

Combining Eqs 81 and 82 and using the boundary condition [[Sigma].sub.r](t [R.sub.0]) = 0, we express the stress [[Sigma].sub.[r.sup.*]],(t) = [[Sigma].sub.r](t, [R.sub.1]) on the interface r = [R.sub.1] in terms of the radial displacement [u.sub.*](t) = u(t, [R.sub.1]),

[Mathematical Expression Omitted]. (83)

We now analyze stresses and displacements in the polymeric shell, which is governed by the constitutive equations of a linear, aged, incompressible, viscoelastie medium (60) and (70). We substitute expression (77) into Eq 60 and integrate the obtained equation. Assuming that no delamination occurs on the interface [this hypothesis is quite acceptable, since the coefficient of thermal expansion for polymers a exceeds that for metals [[Alpha].sub.*] by an order), we find that

[Mathematical Expression Omitted]. (84)

It follows from Eqs 78 and 84 that

[Mathematical Expression Omitted]. (85)

Equations 71 and 78 imply that the non-zero components [[Psi].sub.r, m] and [[Psi].sub.[Theta], m] = [[Psi].sub.[Phi], of the tensors [Mathematical Expression Omitted] obey the differential equations

[Mathematical Expression Omitted],

[Mathematical Expression Omitted].

The non-zero components of the deviatoric part [Mathematical Expression Omitted] of the stress tensor are calculated as

[Mathematical Expression Omitted]. (86)

It follows from Eqs 78 and 86 that

[s.sub.r] - [s.sub.[Theta]] = 2[Mu] (F - [summation of] [[Eta].sub.m][[Psi].sub.m] where m = 1 to M, (87)

where the functions [[Psi].sub.m] = [[Psi].sub.r,m] = [[Psi].sub.[Theta], m] satisfy the differential equations

[Delta][[Psi].sub.m]/[Delta]t (t, r) = [[Gamma].sub.m](t, r}[F(t, r) - [[Psi].sub.m](t, r)], [[Psi].sub.m](0, r) = 0. (88)

We integrate Eq 80 from [R.sub.1] to [R.sub.2] and employ the boundary conditions

[[Sigma].sub.r](t, [R.sub.1]) = [[Sigma].sub.[r.sup.*]](t), [[Sigma].sub.r](t, [R.sub.2] = 0.

Bearing in mind that [[Sigma].sub.r] - [[Sigma].sub.[Theta]] = [s.sub.r] - [s.sub.[Theta]] and using Eq 87, we obtain

[Mathematical Expression Omitted]. (89)

Substitution of expressions (83) and (85) in Eq 89 yields

[Mathematical Expression Omitted]. (90)

Let [T.sup.*] be the characteristic time of cooling. We introduce the notation

[Mathematical Expression Omitted],

and rewrite the governing equations (73), (85), (88), and (90) in the dimensionless form

[Mathematical Expression Omitted],

[Mathematical Expression Omitted]. (91)

where

[Mathematical Expression Omitted].

Equations 91 are solved numerically for a poly(vinyl chloride) pressure vessel cooled from the glass transition temperature [[Theta]g] = 65.7 [degrees] C, see (59), to room temperature [[Theta].sub.r] = 23 [degrees] C. We set T(*) = 0.1 (s), which allows us to account for the relaxation times of the order of 1 (s). The dimensionless cooling rate [q.sup.*] = [qT.sup.*]/[[Theta].sub.g] equals 3.0 [multiplied by] [10.sup.-6], which corresponds to the cooling rate of the oven of about 0.01 ([degrees] C/s). The dimensionless temperature conductivity [[Kappa].sup.*] is taken to be 8.0 [multiplied by] [10.sup.-5], which corresponds to [Kappa] = 0.2 ([mm.sup.2]/s), see (58), and the inner radius [R.sub.1] = 0.05 (m). Numerical analysis shows that for these parameter and [R.sub.2]/[R.sub.1] = 2.0, outer surface of the vessel is cooled down to room temperature during about 1 (hrs) and the entire vessel is cooled down within 5 (hrs), see Fig. 7.

To determine the Young modulus [Mu] in Eq 45 and the parameters [[Eta].sub.m] we construct a master-curve using short-term creep curves measured at various temperatures below the glass transition temperature [[Theta].sub.g], and approximate it by Eqs 51 and 53, see Fig. 8. Afterwards, we calculate the relaxation measure Q(t) with the use of Eqs 55 and 56 and approximate it by Eq 50. The adjustable parameters are collected in Table 2. By using the obtained Young modulus, we calculate the shear modulus [Mu] by Eq 68 with v = 0.5, which implies that [Mu] = 0.3035 (GPa).

Table 1. The Adjustable Parameters [[Gamma].sub.m0], [[Chi].sub.m] and [[Eta].sub.m] for Poly(Ether Imide) Ultem-1000 (GE Plastics) at [[Theta].sup.0] = 156.4 [degrees] C. [[Gamma].sub.m0] ([s.sup.-1]) [[Chi].sub.m] [[Eta].sub.m] 0.000001 5.5410 0.3046 0.000010 0.3638 0.4206 0.000100 0.1911 0.1512 0.001000 0.0565 0.0570 0.010000 0.0292 0.0251 0.100000 0.0114 0.0110 1.000000 0.0291 0.0285 Table 2. The Adjustable Parameters [[Gamma].sub.m0], [[Chi].sub.m] and [[Eta].sub.m] for Poly(Vinyl Chloride) at [[Theta].sub.g] = 65.7 [degrees] C. [[Gamma].sub.m0] ([s.sup.-1]) [[Chi].sub.m] [[Eta].sub.m] 0.0005 3.6187 0.1764 0.0050 0.3733 0.2787 0.0500 0.3369 0.1311 0.5000 0.1545 0.1259 5.0000 0.1409 0.1331

To determine the shift factors [a.sub.[Theta]] and [a.sub.[t.sub.e]] for poly (vinyl chloride), we employ experimental data obtained by Struik (1) and Schwarzl and Zahradnik (59), see Figs. 9 and 10.

The dimensionless radial displacements [u.sup.*] on the inner and outer surfaces of the pressure vessel are plotted versus time t in Fig. 11, while the dimensionless intensity of stresses [S.sup.*] = ([s.sub.r] - [s.sub.[Theta]])/(3[mu][Alpha][[Theta].sub.g]) at the final instant [T.sub.2] = 5 (hrs) is plotted versus the dimensionless radius [r.sup.*] in Fig. 12. Calculations are carried out for

[R.sub.2]/[R.sub.1] = 2.0, [R.sub.0]/[R.sub.1] = 0.9, G/[Mu] = 10.0, [[Alpha].sub.*]/[Alpha] = 0.2, v = 0.3,

where v is Poisson's ratio of the mandrel. The ratios of material parameters correspond to a poly(vinyl chloride) shell on an aluminum mandrel. However, these parameters were changed in numerical simulation to analyze their effect on stresses and displacements in the vessel.

The following conclusions are drawn:

1. The radial displacements increase in time and tend to some ultimate values. The radial displacement on the outer boundary of the polymeric shell is practically independent of the aging time [t.sub.0], whereas the radial displacement on the inner boundary increases with the growth of elapsed time.

2. The limiting displacements are practically independent of the rate of cooling (when [q.sup.*] is changed in the range from 2.0 [multiplied by] [10.sup.-6] to 4.0 [multiplied by] [10.sup.-6]) and of the ratio of shear moduli G/[Mu] (when it changes from 5 to 20). An increase in temperature conductivity [k.sup.*] leads to the growth of the radial displacements (both current and limiting).

3. The only material parameter which affects significantly the radial displacements is the ratio of coefficients of thermal expansion [Mathematical Expression Omitted]. With the growth of [Mathematical Expression Omitted], the radial displacements increase, while the effect of physical aging becomes weaker, see Fig. 13.

4. The influence of the ratio of inner and outer radii [R.sub.2]/[R.sub.1] on the radial displacements is rather weak (when this ratio changes from 1.5 to 3.0), while thickness of the mandrel affects essentially the radial displacements. When the ratio [R.sub.0]/[R.sub.1] grows from 0.9 to 0.98, the radial displacement on the inner boundary increases by twice, and the effect of aging becomes more pronounced.

5. The dimensionless quantity [S.sup.*] is negative in the vicinity of the inner boundary and increases monotonically in [r.sup.*]. Physical aging affects dramatically the stress distribution: in an aged vessel the difference of stresses [[Sigma].sub.r] - [[Sigma].sub.[Theta]] is negative, while in a non-aged one it becomes positive in the vicinity of the outer boundary, see Fig. 12.

6. The stress intensity depends extremely weakly on the ratio of shear moduli G/[Mu], as well as on the rate of cooling and on the thermal conductivity. Some changes may be noted only in a neighborhood of the inner boundary of the pressure vessel.

7. The ratio of coefficients of thermal expansion affects essentially the stress intensity: with the growth of [Mathematical Expression Omitted] the difference of stresses increases and becomes positive across the pressure vessel, see Fig. 14.

8. An increase in the outer radius of the cylinder affects the stresses rather weakly, but it enhances the effect of aging. On the contrary, an increase in the inner radius of the mandrel makes the influence of material aging less pronounced (except for some vicinity of the outer boundary, where the effect of the mandrel is negligible).

The above conclusions are in qualitative agreement with available experimental data for cooling of polymer plates (60) and polymer cylinders (61).

7 CONCLUDING REMARKS

A new model is derived for the long-term thermoviscoelastic response in physically aged polymers. An aged viscoelastic medium is treated as a transient network, where links between junctions break (snapping of ends of elastically active chains from temporal junctions) and emerge (reformation of dangling chains). It is assumed that M different kinds of active chains exist, which correspond to different characteristic lengths of interaction

between chains.

A balance equation (3) is developed for the numbers of elastically active chains of various kinds. based on the presentation (12), kinetic equations (15) are proposed. Coefficients in these equations (the reformation rates) are assumed to satisfy the superposition principle (16), where the shift factor a is a function of temperature and elapsed time, see Eq 17.

Based on the concept of temporal networks, the constitutive equations (45) and (61) are derived for a linear viscoelastic medium subjected to aging. Adjustable parameters are determined by fitting experimental data obtained in short-term creep tests for poly(ether imide) and poly(vinyl chloride). To validate the model, its prediction for long-term creep tests is compared with experimental data. Results of numerical analysis demonstrate fair agreement with observations, see Fig. 6.

The model is applied to analyze residual stresses and strains built up in a polymeric pressure vessel cooled on a metal mandrel from the glass transition temperature to room temperature. Using adjustable parameters for poly(vinyl chloride) found in short-term creep tests, we study the effect of material and structural parameters of the vessel on stresses and displacements. It is shown that physical aging of polymers affects significantly residual stresses, and its influence should be taken into account in predicting stresses in polymeric articles cooled after curing.

ACKNOWLEDGMENT

Financial support by the Israel Ministry of Science (grant 9641-1-96) is gratefully acknowledged.

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Author: | Drozdov, Aleksey D. |
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Publication: | Polymer Engineering and Science |

Date: | May 1, 1998 |

Words: | 10074 |

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