modeling the viscoelastoplastic behavior of amorphous glassy polymers

Modeling the Viscoelastoplastic Behavior of Amorphous Glassy Polymers

ALEKSEY D. DROZDOV

Institute for Industrial Mathematics 4 Hmachtom Street

Beersheba, 8431 1 Israel

Constitutive equations are derived for the viscoelastic and viscoplastic responses of amorphous glassy polymers at isothermal loading with small strains. A polymer is modeled as an ensemble of cooperatively relaxing regions (0 that are rearranged at random times as they are thermally agitated. At relatively low stresses, CRRS are bridged by links that ensure that the macro-strain in a specimen coincides with micro-strains in relaxing regions. When the average stress exceeds a threshold strength for the links, some of them break, and relaxing domains begin to slide with respect to one another. Rearrangement of CRRS reflects the viscoelastic behavior, whereas sliding of micro-domains is associated with the viscoplastic response of glassy polymers. Kinetic equations are proposed for the evolution of viscoplastic strains and the strength of an ensemble of links. These equations are vedied by comparison with experimental data in tensile relaxation tests and in tensile and compressive tests with constant strain rates. Fair agreement is demonstrated between results of numerical simulation and observations for a polyurethane resin and poly(methy1 methacrylate).

INTRODUCTION strain rate on the apparent yield stress (the maximum

model is derived for the viscoelastic and visco- A plastic responses of amorphous glassy polymers. The time-dependent behavior of glassy polymers has been a focus of attention during the past decade (1-15). This may be explained by the use of new experimental techniques that allow complicated programs of loading to be carried out at various temperatures.

This study is concerned with the response of glassy polymers in uniaxial relaxation tests and in uniaxial tests with constant strain rates at small strains. The latter means that the nonlinear terms are disregarded in the expression for the longitudinal strain in terms of the axial elongation. We confine ourselves to relatively slow loadings and to temperatures in the sub-Tg region, where Tg is the glass transition temperature. This implies that crazing and other phenomena associated with brittle fracture are excluded from the consideration. The objective is to develop constitutive equations that describe viscoelastoplasticity of amorphous polymers at the micro-level, on the one hand, and that are simple enough to be used in the numerical analysis of three-dimensional deformations of solid polymers with complicated geometry, on the other. We aim to design stress-strain relations that reflect (i) thermally induced changes in elastic moduli and the characteristic times of relaxation and (ii) the effects of temperature and

stress on the stress-strain curve). The viscoelastic response of glassy polymers is de-

scribed within the Adam-Gibbs theory of cooperative relaxation (16). An amorphous polymer is treated as an ensemble of cooperatively rearranging regions (CRR). A CRR is thought of as a globule consisting of scores of neighboring strands of long chains that change their position simultaneously because of large-angle reorien- tation of strands [see, e.g., Dyre (17) and Sollich (18)). The characteristic length of a relaxing region amounts to several nanometers [see Rizos and Ngai (19) and the references therein]. In the phase space, a CRR is treated as a point trapped in its potential well on the energy landscape. A CRR spends most time at the bottom level of its trap. At random instants, it hops to higher energy levels as it is thermally agitated. We adopt the transition-state theory [see Goldstein (2O)J. according to which some liquid-like (reference) level exists on the energy landscape. When a CRR reaches the reference state in a hop, it totally relaxes. Visco- elasticity of a glassy polymer is treated as sequential rearrangements of CRRs trapped in potential wells with various depths.

CRRS are connected by links (which are associated with temporary crosslinks, entanglements and van der Waals forces). At relatively low stresses, the links

1762 POLYMER ENGINEERING AND SCIENCE, OCTOBER 2001, Vol. 41, No. 10

Modeling the Viscoehtoplastic Behavior of Amorphous Glassy Polymers

prevent mutual displacements of CRRS, which implies that the micro-strains in relaxing regions coincide with the macro-strain in a specimen. When the average stress in an ensemble of CRRs exceeds a threshold strength for links, some of them break, and relaxing regions change their positions with respect to each other. The mutual displacement of CRRs reflects the viscoplastic response at the micro-level.

Cooperatively rearranged regions are associated with "more cohesive domains" observed in amorphous glassy polymers by means of low frequency Raman scattering, whereas the ensemble of links between CRRs is associated with "less cohesive spaces percolating through the material" [Mennet et aL (2111.

Different links between CRRS are assumed to have different strengths, which implies that two stages can be distinguished in the process of their annihlla . tion. At the first stage, individual links with relatively low strengths break, which is tantamount to the nucleation of quasi-point defects in the theory proposed by GSell et aL (22). At the other stage, the defects in the ensemble of links aggregate into clusters (shear micro-domains), the creation of which designates the beginning of viscoplastic flow. This paper focuses on modeling the coalescence process for quasi-point defects and its in- fluence on the time-dependent response of solid polymers.

The exposition is organized as follows. The study begins with a description of thermally induced rearrangement of CRRs. Afterwards, we develop constitutive relations for the viscoelastic response of glassy polymers and kinetic equations for the evolution of viscoplastic strain and average strength of an ensemble of links between CRRS. Results of numerical simulation are compared with experimental data for polyurethane and poly(methy1 methacrylate). The work is concluded with a discussion and final remarks.

KINETICS OF REARRANGEMENT

The position of the liquid-like state at the glass transition temperature Tg is accepted as the zero energy level on the energy landscape. The depth of a potential well where a CRR is trapped with respect to the zero energy level is denoted as w 2 0. We suppose that the position of the reference state on the energy landscape is affected by temperature and denote by a ( T ) the ascent of the liquid-like state at a temperature T < Tg with respect to its position at Tg. The function a [ T ) satisfies the natural condition a(T,) = 0.

Denote by X , the concentration of traps per unit mass of the bulk medium. We assume that X, is a constant (independent of temperature and loading history). Rearrangement of relaxing regions is character- ized by the function X ( t , T , w), which equals the concentration of CRRs (per unit mass at time t) trapped in cages with potential energy w, which had last been rearranged before instant 7 I t. The distribution of traps is determined by the probability density, p(t , w).

of traps with energy w at time t. The functions p( t , w) is expressed in terms of the function X ( t , 7, w) by the formula

where X ( t , t, w) is the concentration of CRRS trapped in cages with potential energy w at instant t The study concentrates on equilibrated polymers with a steady- state distribution of traps (the function p is independent of time). Bearing in mind that in the sub-Te region the distribution of traps is substantially affected by temperature, we set p = p f T , w). With reference to the random energy model [see. e.g., Dyre (17)1, we assume the probability density p to be quasi-Gaussian,

P(T, w) = 0 (w < O), (2)

where W ( T ) and ZIT) are parameters of the Gaussian distribution and the constant C is found from the condition

[ P ( T . w)dw = 1.

Let q(z)dz be the probability for a CRR to reach (in a hop) the energy level that exceeds the bottom level of its cage by some value z' E [z, z + dz]. In agreement with Bouchaud and Mezard (231, we postulate that

q(z) = a exp ( - az),

where a is a material constant. The probability to reach the reference state in an arbitrary hop reads

Q(T, w) = lrn q ( z ) d ~ = exp [-a(w + ~ ( T ) ) I . w+O(T)

Denote by To = r,(T) the attempt rate (the average number of hops in a trap per unit time). The rate of rearrangement R equals the product of the attempt rate, r,, by the probability of reaching the reference state in a hop, Q,

T(T) = r,(T) exp [ -dk(T)] ,

where r(T) is the relaxation rate. Equating the relative rates of rearrangement to the function R, we arrive at the differential equations

1 ax - (t, 0, W) = -R(T, w).

X(t, 0, w ) at

[ E ( t , T , w)]-' ( t , T , w) = -R(T, w).

The solutions of these equations with the initial condition X(0, 0, w) = X,p(T, w) (see Eq 1) read

POLYMER ENGINEERING AND SCIENCE, OCTOBER 2001, Vol. 41, No. 10 1 763

Aleksey D. Drozdov

where 1 ax

F(7. w ) = - - ( t . 7 , w) . xo a7 I t=.r

The number N(t , w) of relaxing regions (per unit mass) in traps with potential energy w rearranged within the interval [t , t + dt] equals the sum of the numbers of CRRs rearranged for the first time at instant t and those that have last been rearranged at some instant T E [0, t ] and reach the reference state at time t,

ax N(t , W ) = - - (t, 0, w) - at

(t. T , w)dT. (51

Neglecting the duration of a hop [a few picoseconds, see Dyre (17)] compared to the characteristic time of relaxation in the sub-Tg region, we find that N(t , LO) coincides with the number of CRRs returning to traps with potential energy w within the interval [t, t + dt],

N(t , W ) = XoF(t, w).

Substitution of Eqs 4 and 5 into this equality results in the Volterra equation

F(t, w ) = R(T, w) p(T, w ) exp [-R(T, w)t] { t F ( T . W) eXp [-R(T, w)(t - T ) ] d T } +I,

The solution of this equation is given by

F(t, w) = R(T, w)p(T, w), which, together with Eq 4, implies that

CONSTITUTIVE EQUATIONS

We assume that the macro-strain in a specimen, E,

equals the sum of the viscoelastic strain, ce, and the viscoplastic strain, ~ p ,

(7)

The viscoelastic strain, E,, is the average strain in a relaxing region (neglecting the distribution of micro- strains, we suppose that the micro-strain in any CRR coincides with E ~ ) , and the viscoplastic strain, ep. is the average strain induced by mutual displacements of CRRs.

A CRR is assumed to totally relax when it reaches the liquid-like state, which means that the natural (stress- free) state of a CFW after rearrangement coincides with the deformed state of the bulk medium at the instant of rearrangement. For uniaxial deformation the

.(t) = €&) + EP(t).

strain from the natural configuration of a relaxing region to its actual confguration at time t is given by

E(t, 7) = Ee(t) - Ee('), (8)

where 7 cr t is the last instant when the region was rearranged. A CRR is modeled as a linear elastic medium with the mechanical energy

1 +(t, T ) = - ce2 (t, T ) ,

2

where c = c(T) is the rigidity of a relaxing region (which is assumed to be independent of the energy of trap wl. Summing the mechanical energies of CRRs and neglecting the energy of their interaction, we find the strain energy density of a polymer (per unit mass)

Substitution of Eqs 7 and 8 into this equality results in

@(t) = c[(@ - Ep(t))2 l m X ( t , 0, w)dw 0 2

Because relaxing regions are modeled as elastic media, the average stress in a CRR, o(t), is expressed in terms of the macro-strain, &), by the conventional formula

where p is mass density in the stress-free state. Com- bining this equality with Eq 9, we arrive at the stress- strain relation

u(t) = Eo(T)[ ( E ( t ) - cp( t ) ) 6 X(t, 0, w)dw

where E,(TI = pc(T)X, is an analog of initial Young's modulus.

KINETICS OF PLASTIC FLOW The evolution of viscoplastic strain, Ep(t), is described

by constitutive equations similar (to some extent) to those in the viscoplasticity theories based on overstress [see Krempl et aL (24)l or back stress [see Boyce et aL (25)l. For definiteness, we consider active loading processes with

1 764 POLYMER ENGINEERING AND SCIENCE, OCTOBER 2001, Val. 41, No. 10

de dt

k(t ) = - (t) 2 0,

and postulate that the material time associated with straining, A, is proportional to the macro-strain, E,

where A" is a material constant. The following hypotheses are introduced:

1. The rate of viscoplastic strain is entirely determined by the average strength of an ensemble of links, g , which depends on temperature and loading history. Viscoplastic deformations occur when the current stress a(t) exceeds the strength g ( t ) . The rate of change in the viscoplastic strain % (with respect to the internal time A) is proportional to the difference u - g .

where K ; is a material parameter and H(x) is the Heaviside function,

1, x 2 0, 0, x < 0. H ( x ) =

2. For a stress-free specimen, the threshold strength has a fured value g. (the strength of links that prevent mutual displacements of CRRs). The quantity g. decreases with temperature, T, in agreement with the theory of thermally activated processes: the growth of temperature results in an increase in the amplitude of thermal fluctuations which, in turn, reduces the average strength of links.

3. The strength of the ensemble of links decreases under loading. Two stages in the evolution of strength are distinguished at the micro-level. The first is associated with breakage of individual links (mechanically induced nucleation of quasi-point defects), which takes place at relatively small strains (in the region of linear visco- elasticity and in the transition domain between linear and nonlinear viscoelastic responses). At the other stage, when the concentration of broken links (quasi-point defects) becomes sufficiently large, the nucleation process becomes energeti- cally unsustainable and is replaced by coalescence of micro-flaws produced by broken links (aggregation of quasi-point defects into shear micro-domains). It is assumed that coalescence of micro-defects is responsible at the micro-level for the characteristic features of the yield-like response in glassy polymers.

4. To avoid speculative hypotheses about the kinetics of nucleation of quasi-point defects (which is rather poorly studied experimentally), we disre- gard the first stage of microstructural evolution. To account for its effect on the mechanical behavior at the macro-level, the average strength of

(12)

links in a stress-free polymer, g., is replaced by the average strength of links at the beginning of the stage of coalescence of micro-defects, go. The latter parameter is a function of temperature (because g. depends on T) and the strain rate, P , during the nucleation period. Assuming the rate of breakage of links between CRRs (or, in other terms, the rate of nucleation of quasi-point defects) to be finite, we expect that go increases with I :

at slow loading, the duration of the stage of breakage of individual links is rather large, which implies that the average strength of the ensemble of non-broken links (at the end of this stage) is low: at relatively rapid straining, the number of links annihilated during the first stage of microstructural evolution is small, which results in a relatively high strength go at the beginning of the state of aggregation of micro-defects.

Modeling the Viscoelastoplastic Behavior o f A m r p h u s Glassy Polymers

POLYMER ENGtNEERiNG AND SCENCE, OCTOBER 2007, VoI, 41, No. 10 1765

5. At the stage of aggregation of micro-defects, when the current stress u exceeds the threshold stress, the average strength of links, g , is altered. Changes in the average strength of the ensemble of links are determined by two mechanisms. The first is associated with a decrease in g driven by coalescence of flaws induced by destruction of individual links. The rate of decrease in the strength (with respect to the internal time A) is assumed to be proportional to the difference u - g . Bearing in mind that g is a nonnegative function, we set

ds = - K ; g ( u - g)H(u - 9) . dA (13)

where K l is a constant. Eguation 13 ensures that g ( t ) = 0 for any t h to. provided that g( to) = 0.

6. The other mechanism for changes in the average strength, g, reflects thermally induced healing of micro-defects. This process is assumed to be similar to craze healing in glassy polymers [see Plum- mer and Donald (26) and the references therein]. Because the healing process is driven by thermal fluctuations, its kinetics is analogous to that for the reformation process in a temporary network [see Tanaka and Edwards (27)]. This means that the rate of reformation (per unit time t ) for bridges between CRRS is proportional to the number of broken links,

(14)

where K , is a constant. In the general case, K i , Kz and K , are functions

of temperature T and loading history. To reduce the number of adjustable parameters, we suppose the effect of these factors to be rather weak and treat them as constants. Introducing the notation K , = KYA" and K , = KzX", we find from Eqs 1 1 to 14 that the functions % and g obey the nonlinear differential equations

Aleksey D. Drozdov

I 1 c\

Bearing in mind that the stage of annihilation of individual links is disregarded, the initial conditions for Eqs 15 are given by

EJO) = 0, g ( 0 ) = go. (16)

Equations 2, 3, 4, 6, 10, 15 and 16 uniquely determine the stress u(t) in a specimen for a given straining pro- gram, c( t ) . These equations contain four functions of temperature, E,(T), W(T), Z(T) and T(T), one function of temperature and strain rate, go(T, h) , and three constants K,, K, and K3 (without loss of generality, the parameter a in Eq 3 is set to unity). The total number of adjustable parameters in the model is less than that in conventional stress-strain relations in viscoelastoplasticity based on the overstress concept [Krempl et aL (241.

COMPARISON WITH EXPERIMENTAL DATA

Adjustable parameters are determined by fitting experimental data in tensile relaxation tests and in tensile and compressive tests with constant strain rates. First, we consider obsenrations in an isothermal relaxation test with

0, t < 0, { €0. t > 0, E ( t ) =

where the strain E, does not exceed the yield strain. It follows from Eqs 15 and 16 that

.p(t) = 0, g( t ) = g.(T). Substitution of these expressions and Eqs 2, 3, 4, 6, 1 7 into Eq 10 results in

E(T, t ) = C(T)E,(T)

+ T(T) exp (- w)t (w - W(T))2 exp [-( 2Z2(T)

where E(T, t ) = a(t)/eo is the current Young's modulus. We begin with matching experimental data for a

polyurethane resin (PU) at various temperatures T. The specimens and the experimental procedure are described in Letemer and G'Sell(28). Given a parameter E,, the quantities Z, Wand T are found by using the steepest-descent procedure. Afterwards, the initial Young's modulus E, is determined by the least-squares technique. mure 1 demonstrates excellent agreement between

experimental data and results of numerical simulation. Adjustable parameters of the model are plotted versus the degree of supercooling AT = Tg - T in Figs. 2 to 4, which show that the functions E,(T), Z(T), W(T) and r (T) are correctly approximated by the phe- nomenological relations

1

2

E

I .5

3

1

2

E

I .5

(1.0

-2.0 1Og t 4.0

Fig. 1 . Young's modulus E GPa versus time t s for PU at temperature T "C. Circles: experimental data (281. Solid lines: results of numerical simulation. Curue I: T = 25.0: curve 2: T = 35.0: c m e 3: T = 50.0: curve 4: T = 75.0.

Eo = a, + a,AT,

log W = co + c,AT,

Z = bo + blAT,

log r = do + dlAT (18)

with adjustable parameters ai, bi, ci and di . The value of E, at T = 75°C deviates from Eq 18, which may be explained by noticeable changes in the elastic moduli in the close vicinity of the glass transition temperature Tg

We proceed with matching experimental data in a tensile test with a constant strain rate,

0, t < 0, .(t) = { i,t, t 5 0.

2 a 1

I 1 I I

(19)

2b

U.(J 40.0 AT 80.0 Fig. 2. Initial Young's modulus E, GPa -us the degree of supercooling ATK. Symbols: treatment of observations (28, 31). Solid lines: appmximdion of the experimental data by Eq 18. Curoe I: PU, a, = 1.8409, al = 0.0097: m s 2: PMMA, 2a.a0= 0.1419, a, = 0.0604,2b:a0= 1.1940,a1 = 0.0097.


Modeling the Viscoelastoplastic Behavior of Amorphous Glassy Polymers

2

1

0.0 40.0 AT 80.0 Fig. 3. The dimensionless parameter Z versus the degree of supercooling ATK. Symbols: treatment of observations (28, 31). Solid lines: approximation of the experimental data by Eq 18. Curue 1 : PU, bo = -0.6291. b, = 0.2231; curve 2: PMMA, bo = -0.4021, bl = 0.9945.

We integrate Eqs 2, 3, 4, 6, 15, 16 and 19 with the material parameters Z, W and r found in relaxation tests and determine the quantities go and Ki that min- imize deviations between results of numerical simulation and observations. Because the initial Young's modulus Eo found in relaxation tests underestimates stresses in the initial region of the stress-strain curve, the value of Eo is found by fitting experimental data in the interval E E [0, 0.02).

First, we approximate the stress-strain curve at the strain rate io = 0.01 min-l measured at ambient temperature (T = 25°C). The quantities go and Ki are found

by the steepest-descent procedure. Afterwards, we fx the constants Ki and match experimental data at other temperatures and other strain rates with the only adjustable parameter, go. Figures 5 and 6 demonstrate good agreement between experimental data and results of numerical analysis. The initial threshold strength go is plotted versus i,, in Fig. 7, which reveals that in the range of strain rates under consideration the function go(io) is fairly well appromated by the rela- tionship (conventionally used to describe the effect of strain rates on the yield stress)

with adjustable parameters Pr.

IT

50 .0

0.0

0.U E 0.05 Fig. 5. Stress cr Mpa versus strain E for PU loaded at temperature T "C with the strain mte k o = 0.01 min-I. Circles: experimental data 128). Solid lines: results of numerical simulation.Cwvel:T= 25.0:curue2:T= 50.0.

1 i

0.0 1 I I I I I I I J 0.0 o . i) 40.0 AT 80.0

Rg. 4. The dimensionless pammeter W ( u n t d circles) and the relawation rate r s-, ltilled circles) uersus the degree of supercoding A T K for PU. Symbols: treatment of observations (281. Solid lines: apprawimation of the experimental data by Eq 18. Curve 1 : co = 0.5807, c, = 0.0152; curve 2: d, = 0.4780, d, = 0.0146.

n

60 0

0.0 0.0 E 0.05

Fig. 6. Stress cr MPa wrsus strain c for PU loaded at ambient temperature with the strain rate Lo min-'. Circles: experimental data 128). Solid lines: results of numerical simulation. CW 1 : €0 = 0.1; c u ~ e 2: €0 = 0.01; cur~e 3: Go = 0.005.

POLYMER ENGINEERING AND SCIENCE, OCT06ER 2007, Yo/. 47, No. 70 1767

Aleksey D. Drozdov

11111.11

!/,I

511.0

0.0 -5.U log& 0.u

Fig. 7. The threshold strength go MPa versus the stmin rate i, min-' at ambient temperature. Circles: treatment of obser- cations (28, 30). Solid lines: approximation of the experimen-

curue 2: PMMA, Po = 97.922, PI = 20.176. tal data by 20. CW 1: PU, Po = 86.301, P1 = 12.621:

We now match experimental data for poly(methy1 methacrylate) (PMMA) at ambient temperature. Be- cause of the lack of observations in relaxation tests and in tests with constant strain rates for the same specimens, we use experimental data in relaxation tests presented in Cizmecioglu et al. (29) (the relaxation master curve corresponding to various aging times after quench from 165°C) and data in tests with constant strain rates exposed by Berthoud et al. (30). We begin with fitting observations in tensile relaxation test, Eq 17, with the strain E, = 0.005. Figure 8 demonstrates fair agreement between experimental data and results of numerical simulation. Using material

parameters found in the relaxation test, we approximate observations in compressive tests with constant strain rates. By analogy with the analysis of experimental data for polyurethane, we determine the coefi- cients Ki by fitting data in a test with the strain rate i, = 0.01 s-l, fur their values, and match observations for other strain rates by using the only adjustable parameter go. Figure 9 shows excellent agreement between experimental data and results of numerical simulation. The function go(;) is depicted in Fig. 7, which demonstrates that Eq 20 correctly approximates observations.

To assess the effect of temperature on the viscoelastoplastic response of polflmethyl methacrylate), we fit observations in relaxation tests [data are adopted from Mariani et al. (31)l and in tests with constant strain rates [observations are taken from Chen et aL (13)] at various temperatures in the sub-2'' region. Because experimental data in relaxation tests are provided for a relatively small interval of time (about 3 decades), we reduce the number of adjustable parameters in constitutive equations by setting W = 0 and r = 1.06 s-l

(the latter value is found from the condition of the best fit of observations at T = 50°C). Relaxation curves for other temperatures are matched by using only two adjustable constants, Eo and C . Figure 10 reveals good agreement between results of numerical simulation and experimental data. The function E,(T) is depicted in Fig. 2 and the function C(T) is plotted in Fig. 3, which show that observations are fairly well approximated by Eq 18.

Experimental data in tensile tests with constant strain rates are presented in Rg. 11 together with the results of numerical analysis (the initial Young's modulus, E,, is taken directly from the data in relaxation tests). This Figure also exhibits good agreement between observations and results of numerical simulation. Material

1 (1

1 . 0 log t 6.0

Q. 8. Young's modulus E GPa versus time t s for PMMA at room temperature. Circles: experimental data (29). Solid line: results of numerical simulation with E, = 2.04 GPa, Z = 14.8, W = 12.4andF= 0.05s-'.

0.0 O.OG t 0.12 Fig. 9. Stress D MPa wrsus strain E for PMMA loaded with the strain rate 6 , s-I at ambient temperature. circles: experimental data (30). Solid lines: results of numerical simulation. Curve I : do =. curve 3: curve 2: i , = 5 x io = cur~e 4: €, = 5 x cur~e 5: i, = 10-4 .


Modeling the Viscwlastoplastic Behavior of Amorphous Glassy Polymers

2.0

E

1.0

!I 0

Fig. 10.

" 5 L I I I 1 I I 1.0 log t 4.0

Young's modulus E GPa versus time t sfor PMMA at temperature T "C. Circles: experimental data (31). Solid lines: results of nwnerical sim~&tion. Curue 1 : T = 50; curve 2: T = 7 5 ; m 3 : T = 85;curue4:T= 90;curue5:T= 100.

parameters go and Ki for polyurethane and poly(methy1 methacrylate) are listed in Table 1, which demonstrates that these quantities have the same order of magni- tude for the two polymers.

DISCUSSION Fgures 1 , 8 and 10 demonstrate that the stress-

strain relations correctly determine the response of polyurethane and poly(methy1 methacrylate) in relaxation tests, whereas Figs. 5, 6, 9 and 11 show that the constitutive equations quantitatively describe the stress-strain curves in tensile and compressive tests at various temperatures and strain rates.

The initial Young's modulus Eo decreases with temperature T (Fg. 2) in accord with conventional observations for the effect of temperature on elastic moduli. The function Eo(T) is fairly well approximated by 4 18, where adjustable parameters ai take Merent values in the close vicinity of the glass transition temperature (from Tg - 20 K to Tg) and below Tg - 20 K Fol- lowing Letenier and GSell (28), we use Tg = 80°C for polyurethane, which corresponds to the glass transition temperature for hard segments of the block co- polymer. Dynamic mechanical tests demonstrate that the effect of soft (poly-01) segments (which are in the rubbeIy state at ambient temperature) on the mechanical response of polyurethane is negligible (28). In agreement with Cizmecioglu et aL (29), we adopt the value Tg = 105°C for poly(methy1 methacrylate).

I ^ n

0.0 0.05 E 0.1 9. 11. Stress IS Mpa versus strain E for PMMA loaded with the strain rate ko = 0.1 min-' at ternpemture T "C. Circles: experimental data (13). Solid lines: results of numerical simulation. Curue 1: T = 50; curve 2: T = 75.

The parameters W and Z that characterize the distribution of potential wells with various energies (see Eq 2) decrease with temperature (F'lgs. 3 and 4), which is in good agreement with the conventional picture for the growth in the ruggedness of the energy landscape with a n increase in the degree of supercooling [see Appignanesi et aL (3211.

The relaxation rate r is determined by Eq 3 as a product of the attempts rate and the factor exp [-n(T)], where N T ) is the ascent of the reference energy level at temperature Twith respect to that at the glass transition temperature. According to the theory of thermally activated processes, the attempt rate r, increases with T. Because the position of the liquid-like energy level on the energy landscape de- scends with temperature, it is rather dimcult to pre- dict thermally induced changes in the relaxation rate, r, a priori. Flgure 4 demonstrates that the relaxation rate grows with the degree of supercooling AT, whereas r is set to be a constant in matching data plotted in

The only parameter in the constitutive equations that depends on the rate of straining is the strength of links between CRRs at the beginning of the stage of coalescence of micro-defects, go. Figure 7 reveals that go increases with the strain rate go, in agreement with our scenario for the kinetics of plastic flow. The experimental data are fairly well described by Eq 20, where the constants pi are rather close for the two polymers.

Fig. 10.

Table 1. Adjustable Parameters go and K, for Polyurethane and Poly(methy1 methacrylate).

Polymer T"C 90 MPa K, MPa-l K2 MPa-l K3 min-'

PU PU PMMA PMMA

25.0 50.0 50.0 75.0

64.0 36.5 36.0 32.5

0.022 0.022 0.07 0.07

0.8 0.8 1 .o 1 .o

35.0 35.0 20.0 20.0

~~~~~~ ~~ ~~ ~ ~ ~

POLYMER ENGINEERING AND SCIENCE, OCTOBER 2001, Vol. 41, No. I 0 1769

Aleksey D. Drozdov

The latter implies that soft segments in a block co- polymer (PU) act in a way similar to that for links between CRRS in a homopolymer (PNIMA).

The model correctly approximates observations in tests with constant strain rates even when several material parameters are determined in relaxation tests for samples with dif€erent thermal histories (for exam- ple, specimens described in Cizmecioglu et cd (29) were molded at 165°C. aged at 60°C and quenched to room temperature, whereas those of Berthoud et aL (30) were annealed at 140°C and slowly cooled to ambient temperature). This may be explained by the facts that (i) elastic moduli are weakly affected by thermal pretreatment, and (ii) although thermally induced changes in relaxation times are substantial, their effect on the stress-strain curves at various strain rates is relatively weak (because of a limited range of strain rates under consideration). The effect of thermal pretreatment on adjustable parameters of the model will be the subject of a subsequent study.

CONCLUDING REMARKS A constitutive model has been derived for the time-

dependent response of glassy polymers at small shins. An amorphous polymer is treated as an ensemble of CRRs trapped in their cages. In a stress-free medium, relaxing regions are bridged by links which ensure that the macro-strain in a specimen coincides with the micro-strains in CRRS. When the stress exceeds an average threshold strength for links, some of them break providing an opportunity for relaxing regions to change their positions with respect to each other. Kinetic equations are developed for the evolution of the threshold strength and the viscoplastic strain. The stress-strain relations are verified using experimental data for polyurethane and poly(methy1 methacrylate) at various temperatures and strain rates. Fair agreement is demonstrated between experimental data and results of numerical simulation.

ACKNOWLEDGIUENT

I would like to express my gratitude to Prof. E. Krempl, who read a preliminary version of the paper and made some valuable suggestions. This work was supported by the Israeli Ilrlinistry of Science through grant 1202- 1-00.

REFERENCES 1. C. M. Bordonaro and E. Krempl, Polyrn Eng. Sci, 32,

2. C. Xiao, J. Y. Jho, and A. F. Yee, Macromkcules, 27,

3. 0. A. Hasan and M. C. Boyce, Polym Eng. Sci, 35, 331

4. E. Krempl and C. M. Bordonaro, Polyrn Eng. Sci, 35.

5. R. Quinson, J. Perez, Y. Germain, and J. M. Murraciole,

6. E. Kontou, J. AppL PoZyrn Sci. 61, 2191 (1996). 7. R. Quinson. J. Perez, M. Rink, and A. Pavan, J. Mater.

8. L. David, R. Quinson, C. Gauthier, and J. Perez, Pofgn.

9. R. Quinson, J. Perez, M. Rink, and A. Pavan, J. Mater.

1066 (1992).

2761 (1994).

(1995).

310 (1995).

Polymer, 36, 743 (1995).

Sci. 31, 4387 (1996).

Eng. Sci, 37, 1633 (1997).

Sci, 32, 1371 (1997). 10. G. Spathis and C. Maggana, Polymer, 38,2371 (1997). 11. P. Tordjeman, L. Teze, J. L. Halary, and L. MOMerie,

12. C. Zhang and I. D. Moore, Polyrn Eng. Sci. 37,404, 414

13. X. Chen, P. Tong, and R. Wang. J . Mech Phys. So&,

14. E. Krempl and C. M. Bordonaro, Ink J. Plasticity, 14,

15. A. Pegoretti, A. Guardini, C. Migliaresi, and T. Ricco,

16. G. Adam and J. H. Gibbs, J. Chem. Phys. , 43, 139

17. J. C. Dyre, Phys. Reu. B, 81, 12276 (1995). 18. P. Sollich, Phys. Reu. E, 58, 738 (1998). 19. A. K Rizos and K. L. Ngai, Phys. Reu. E, 59.612 (1999). 20. M. Goldstein, J. Chem Phys., 51, 3728 (1969). 21. A. Mermet, E. Duval, S. Etienne, and C. G'Sell. Po@mer,

22. C. G'Sell, H. El Bari, J. Perez, J. Y. CavaiUe, and G. P.

23. J.-P. Bouchaud and M. Mezard, J. Phys. A. 30, 7997

P d w Eng. Sci, 37, 1621 (1997).

(1997).

46, 139 (1998).

245 (1998).

Polymer, 41, 1857 (2000).

(1 965).

37,615 (1996).

Johari, Mater. Sci Eng. A, 110,223 (1989).

I1 9971. .- - ~ ~ ,- 24. E. Krempl, J. J. McMahon, and D. Yao, Meek Mater., 5,

35 (1986). 25. M. C. Boyce, D. M. Parks, and A. S. Argon, M e c h Mater.,

26. C . J. G. Plummer and A. M. Donald, J. Mater. Sci, 24,

27. F. Tanaka and S. F. Edwards, Macromolecules. 21, 1516

28. Y. Leterrier and C. G'Sell, J. Mater. Sci, 29,4209 (1988). 29. M. Cizmecioglu, R F. Fedors, S. D. Hong, and J. Moa-

30. P. Berthoud, T. Baumberger, C. G'Sell, and J.-M. Hiver,

31. P. Mariani, R. Frassine, M. Rink, and A. Pavan, Polym

32. G. A. Appignanesi, R. Montani, and A. Femandez, Phys-

7, 15 (1988).

1399 (1989).

( 1992).

canin, Pdyrn Erg. Sci, 21.940 (1981).

Phys. Rev. B, 59, 14313 (1999).

Eng. Sci. 36,2750 (1996).

ica A, 262:. 349 ( 1999).


modeling the viscoelastoplastic behavior of amorphous glassy polymers

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