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Nonequilibrium statistical thermodynamic theory for viscoelasticity of polymers
Pergamon
J. Mech. Phys. Solids, Vol. 46, No. I, pp. 1399152, 1998 0 1997 Elsetier Science Ltd Printed in Great Britain. All rights reserved 002225096/97 %17.00 + 0.00 PII: soo22-5096(97)ooo10-0
NONEQUILIBRIUM STATISTICAL THEORY FOR VISCOELASTICITY XIAOHONG
THERMODYNAMIC OF POLYMERS
CHEN1_, PIN TONG
Center for Advanced Engineering Materials, Department of Mechanical Engineering, The Hong Kong University of Science & Technology. Clear Water Bay, Kowloon, Hong Kong
and REN WANG Department
of Mechanics
(Received
and Engineering
18 November
Science, Peking University,
1995
Beijing
100871, P.R. China
; in revised,form 29 Januarx 1997)
ABSTRACT In this paper, we propose a new molecular relaxation mechanism for polymers by considering the change in the actual microscopic structure under macroscopic stress fields. The effects of both intramolecular and intermolecular forces on the inner rotation and the relative slippage of links are taken into account, A constraint potential function, along with a constraint tensor, is introduced to describe the constraint exerted by the surrounding medium. A unified three-dimensional constitutive framework for the viscoelasticity of polymers including thermal effects is established by making use of nonequilibrium statistical thermodynamics, which can be reduced to James and Guth’s (1943) non-Gaussian polymer network theory for rubber elasticity. The model compares well with the experimental data for PMMA and plasticized PVC over a wide range of temperatures. 0 1997 Elsevier Science Ltd. All rights reserved Keywords
: A. thermomechanical
processes,
B. constitutive
behavior,
B. viscoelastic
material.
INTRODUCTION Since macromolecules are densely packed together in polymeric materials, both the intramolecular and intermolecular forces have significant influence on the relaxation process. When a thermodynamic force is exerted on a segment through the surrounding medium, the Gibbs free energy of links changes and the equilibrium condition breaks down. The segment responds by reorienting the links to a new lower free energy state. Nevertheless, there exists an energy barrier for local structural rearrangement due to the constraint of the surrounding medium. Thus it takes time for the links to jump over the energy barrier. The response rate depends on the ratio between the height of the energy barrier and the average kinetic energy of links. Most of the current molecular theories for polymers (Eyring, 1936; Robertson, 1966; Haward and Thackray, 1968; Argon, 1973; Boyce et al., 1989, 1993) are based on t Currently
on leave from the Department
of Mechanics 139
and Engineering
Science, Peking
University.
X. CHEN
140
et al.
shear-induced thermally activated mechanisms. In these theories, however, the shift in the height of the energy barrier for the microscopic relaxation process was assumed to depend only on the macroscopic stress, and the back stress due to rubber elasticity was introduced in the constitutive relation as the internal variable. O’Dowd and Knauss (1995) generalized the theory for rubber elasticity to include the effect of broad-spectral rate dependence by taking into account the molecular orientation and the volumetric strain, so that the nonlinear large deformation viscoelasticity characteristics can be reproduced. In this paper, we describe the constraint exerted on links by the surrounding medium by introducing a constraint potential function along with a constraint tensor. We propose a new molecular mechanism for the relaxation of polymers by considering the effects of the combined inner rotation and local relative slippage of links on the reorganization of microscopic structure. Based on this molecular mechanism, the viscoelastic constitutive relation of polymers can be expressed directly in terms of microscopic state variables and their time derivatives. This is due without introducing the back stress. Energetic elasticity and rubber elasticity are two limiting cases. Both the softening effect within a moderate range of deformation and the hardening effect at large deformation are natural consequences of the competition between the thermallymechanically activated mechanism for the inner rotation and the relative slippage of links and the stiffening mechanism due to finite chain extensibility. Universal ratetemperature equivalence is derived from the dependence of the constraint tensor on temperature, with the Arrhenius equation and the WLF equation as special cases (Ward, 1983 ; Ferry, 1980). Thus, this model is a new attempt to unify the various polymer constitutive theories from the viewpoint of nonequilibrium statistical thermodynamics. This approach is quite different from those by Haward and Thackray, Boyce and Arruda, and O’Dowd and Knauss.
MOLECULAR
RELAXATION
MECHANISM
Since polymer chains are densely packed together in the condensed state, the motion of a macromolecular chain is restricted due to the existence of both intramolecular and intermolecular forces. The portion of a chain between two neighboring junctions is called a segment. When an external field is applied, segments are subjected to thermodynamic forces transferred by the surrounding medium. The structural rigidity results from both the intramolecular and intermolecular forces. For T < Tb (where Tb is the brittle-to-ductile transition temperature), polymers display energetic elasticity owing to the change of bond length and bond angle since the inner rotation and the relative slippage of links are fundamentally frozen. Fracture takes place before relaxation is thermally-mechanically stimulated. For T,, d T < T,,(where T, is the glass transition temperature), although the average degree of constraint exerted on the links by the surrounding medium is still very large, the energy barrier for both the inner rotation and the relative slippage of links can be lowered under external load. The cooperative motion of links can be induced by the thermodynamic force, which leads to the conformational change of segments. Thus polymers display anelasticity predominantly.
Statistical
thermodynamic
theory
for viscoelasticity
of polymers
Fig. I. A labeled segment. The dark circles stand for the atoms in the labeled segment, and the grey circles stand for the atoms in other segments. The thin line locates the actual position of the labeled segment which is designated by R(s,; t), the thick line stands for the equilibrium position of the labeled segment which is designated by R(s, ; t), where so is the coordinate of arc length.
The definition (Reichl, 1980)
of the Gibbs
free energy G for a thermodynamic
G=
U-TS-XY,
system is given by
(1)
where U is the internal energy, T is the absolute temperature, S is the entropy, Y is a set of generalized forces, and X is the corresponding generalized displacements. As shown in Fig. 1, the actual position of the labeled segment is R(s, ; t) and its equilibrium position is ii&; t), where s0 denotes the position of a link, the smallest unit of motion in a segment. If a thermodynamic force f is exerted on the labeled segment along the direction of its end-to-end vector, the links in the labeled segment can be divided into two subsystems. One subsystem denoted as A comprises the links whose Gibbs free energy increases, and the other denoted as B comprises the links whose Gibbs free energy decreases. The averaged Gibbs free energy per link for each subsystem undergoes a shift under the thermodynamic force. It can be written as :
142
X. CHEN
et al.
exp{-g+}=~Zexp~~)sinBdB=
1-ex;(-8),
(2)
(3)
where kB is the Boltzmann constant, 1 is the link length, f is the magnitude of the thermodynamic force f, p =J/kBT, & = GA/n,, tje = GJn,, GA and Gs denote, respectively, the Gibbs free energy for each subsystem, nA and nB denote, respectively, the number of links in each subsystem, and 8’ is the Gibbs free energy per link without the thermodynamic force. The Langevin equation for micro-Brownian motion of a segment can be written as
cxso ;0 - WQo ; 0at-&so ; 0) =
_ MR(so ; 0 -&o
;
0)
+r(so
.
7
aR
t>
>
(4)
where &o ; t) = R(so ; t) -& ; t) stands for the fluctuation around the equilibrium position. The term on the left-hand side is the drag force owing to the difference between the local velocity and the mean field velocity. The drag coefficient is a secondorder tensor, which can be taken as [(so; t) = {[(l + Y$-Y&, ; t) 0 u(so; t)] for the axisymmetric case, where c is a scalar with the dimension of a drag coefficient, I is the identity tensor, u(so ; t) is the unit vector along the direction of a&, ;t)/ds,, and yi is a parameter describing the anisotropic property of the drag coefficient tensor. The first term on the right-hand side is the force corresponding to the constraint potential g(R(so ; t) -ii(s, ; t)) exerted by the surrounding medium, and the second term is the random force whose distribution is assumed to be Gaussian. The constraint potential g(R(so ; t) - ii(s, ; t)) can be approximated by g(R(s,;t)--(Soit))
=s,$
E2~~{l+COS[~+~~~‘2(R~(~~;t)-~i(~O;t))]},
(3
I I
which can be expanded in the form of Taylor’s series near the equilibrium position g(R(so ; t)-ik(s,
; t)) = T i
12~f(R,(~o ;t)-&so
; t))*,
where u. is a parameter with the dimension of energy, and o, is a principal value of o(so ; t), a second-order tensor describing the degree of constraint exerted by the surrounding medium. For the axisymmetric case, the constraint tensor can be taken as w(s,, ; t) = o[( 1+ yo)I- y&s,, ; t) @ u(s, ; t)], where o is the scalar constraint coefficient, and yWis a parameter describing the anisotropic property of the constraint tensor. One of the principal values of o(so ; 1) is o, = o along the principal direction parallel to a8(so ; t)/ as, and the other two are w2 = o3 = (1 + yW)walong the principal directions perpendicular to a&s,; t)/as,. Using (4) and (5), we can write the Langevin equation in the principal coordinate system as
Statistical
thermodynamic
theory
for viscoelasticity
of polymers
143
(7)
By making use of the Fokker-Planck equation equivalent to the Langevin equation, Kramers (Risken, 1989) calculated the escape rate of a particle sitting in a deep potential well near x = X,in as
(8) where p is the friction coefficient, g(x) denotes the energy barrier function, g”(x) is the second derivative of g(x), and Ag = gmax-gmin is the height of the energy barrier. Therefore, the intrinsic rate for a link jumping over the energy barrier by microBrownian motion along the principal direction in the absence of the thermodynamic force f is given by (9) where gymin”= Ig;,,,"I
=
$,h$,
Ag; = &,l’lu,.
The intrinsic frequency for a link jumping over the energy barrier by the combined inner rotation and local relative slippage can be taken as the maximum of vy, VPand v’:: 0:) =
max(vy, vi, v!) = !!CY$!?exp(-sT)=$exp(-z),
(10)
where R is the gas constant, AG, = 4A~.4+01~/31r~ is the energy barrier for the inner rotation and local relative slippage of a mole of such links, which depends on the intermolecular cohesive force and steric hindrance, and A,, is the Avagadro constant. From (7) we see that l/L o d escribes the extent of the constrained micro-Brownian motion of a link and plays an important role similar to the free volume in the glassy transition of polymers. Under the thermodynamic force f, the Gibbs free energy for the links in the two subsystems undergoes a different change, leading to the height shift for the energy barrier. The mean frequency for a link in subsystem A jumping over the energy barrier is B, =
P
iJoexp l-
and for a link in subsystem B is
w-P)
(11)
144
X. CHEN
ifB =
V’exp
et al.
= Be-Q0 kBT
( )
P
.-0
exp(B) - 1’
(12)
The change rate for the number of links in each subsystem can be written as
(13)
where nA+ nB = n, and n denotes the total number of links in the segment. The solution for (13) is
(14) where ni and ni denote, respectively, the initial number of links in each subsystem. The governing equation for the change in the projection of link length along the direction of the thermodynamic force for subsystems A and B can be written as :
dl, - = vA(leA-fIA), dt
-dl, = dt
(15) B&-Is),
where Z> and I> are, respectively, the equilibrium values for the average projection length of links in subsystems A and B ; G and & can be expressed in terms of the thermodynamic force :
(16)
The solution for (15) is
Statistical
thermodynamic
theory
for viscoelasticity
of polymers
145
where 12 and 1: denote, respectively, the initial value of the average projection length of links in subsystems A and B. The relation between the end-to-end vector of the segment and the thermodynamic force exerted can be written as r = rA +r,
= nAIA fn,l,,
(18)
where nA and n, are given by (14), and IA and Is are given by ( 17). As ~‘)t -+ K!, the system reaches the equilibrium state. We get
I- ev-P) 2 sinh(fl)
” (19)
ev(P>
-
2 sinh(B)
1
”
(20)
As the system reaches equilibrium, the relation between the projection of the endto-end vector of the segment and the thermodynamic force exerted becomes
(21) which is the same as the James-Guth network theory for rubber elasticity based upon the non-Gaussian distribution function (Ward, 1983). The Deborah number is defined as the ratio of the frequency of the external force or the deformation rate to the intrinsic relaxation frequency, i.e. De = w,/B” or De = k/V”. For the Deborah number De >> 1, the microscopic structural rearrangement cannot be observed during the experimental period because it takes time for links to get enough energy to jump the deep energy barrier of inner rotation and relative slippage. Polymers display essentially energetic elasticity, which is provided by the change of bond length and bond angle. For the Deborah number De << 1, the microscopic structural rearrangement can be completed through the combined inner rotation and relative slippage of links and a new dynamic equilibrium state can be reached rapidly. Energetic elasticity turns into rubber elasticity as a result of the conformation change of segments. For the Deborah number De - 1, the rate of the microscopic structural reorganization is of the same order as the frequency of the external force or the deformation rate. The system is in a nonequilibrium state. The thermally-mechanically activated barrier jumping process consumes energy resulting in energy dissipation. Anelasticity dominates at this rate. We have just identified the physical mechanism of the rate effects of polymers at the molecular level.
146
X. CHEN
RATE-TEMPERATURE
et al.
EQUIVALENCE
For T < Tg, the average kinetic energy for the thermal motion of links is small compared with the constraint potential barrier for inner rotation and local relative slippage of links. Under the thermodynamic force, the Gibbs free energy of links will be shifted. The links that have relatively high Gibbs energy are mechanically-thermally activated first. Then the confirmation of segments containing activated links is changed through the cooperative motion of links. By (lo), we get the shift factor defined by the intrinsic frequency ivy:
(22) where the subscript s denotes the value at the reference temperature. At low temperature, we assume that w12 has little dependence on temperature, i.e. o12 z o,12, and we have (23) Equation (23) has the form of Arrhenius equation, similar to the result of the barrier model (Ward, 1983). As temperature increases, the average kinetic energy of the thermal motion of molecules increases while the intermolecular constraint force decreases, so larger units of motion can be activated. As the relaxation time for the motion of segments decreases to the same order as the testing time scale, glass transition occurs. For a segment consisting of n links, we can obtain the expression for its intrinsic relaxation frequency following almost the same procedure as for a link : (24) where co, is the constraint parameter describing the degree of the restriction on the motion of a segment consisting of n links by the surrounding medium, and AG, = 4A,uoni20,/3n2 is the constraint potential barrier for the motion of a mole of such segments, depending on the intermolecular cohesive force and steric hindrance. By (24), we get the shift factor defined by the intrinsic frequency ijz : (25) Near the glass transition temperature T,, we assume that n120, satisfies the following relation : 1 p=CIT’ 121’0,
T- T,
Statistical
thermodynamic
where c( is the proportional temperature. We have
theory
factor,
for viscoelasticity
and
147
of polymers
T2 is the true thermodynamic
as T+
n12co, -+ co
transition
T,
nl%o, --+1/cc at high temperature. Substituting (26) into (25), we have lna,
= 1,;
= 21n ns
We expand T, and get :
T(T, - T,) + 4nl’o,,u,, 3r.r2k,T, T,(T- Tz)
the first term on the right-hand
side of (27) at the reference
In ar C: (TlogaT=mlOC;+~-T,’ which has the form of the WLF equation CJZ satisfy :
(I - $$)’
(Ferry,
(27) temperature
TJ
1980). The two parameters
(28) C; and
At high temperature, the rate-temperature equivalence is again reduced to the form of Arrhenius equation on account that n12co, + l/cc, which is suitable to describe the high temperature creep behavior of polymers.
CONSTITUTIVE
RELATION
Polymeric materials can display energetic elasticity, anelasticity, and entropic elasticity accompanying viscous flow under different conditions owing to the existence of multiple units of motion. Next we will present a unified three-dimensional constitutive relation suitable for describing the relaxation and transition of polymeric materials under small as well as large deformation. The multiplicative decomposition of the deformation gradient is given by F = F”F”‘, where F” is the elastic deformation gradient and F”’ is the anelastic deformation gradient. We choose F” = Fe’, det(F”“) = 1. The representation of the kinematics of inelastic deformation has been discussed by Boyce et al. (1989) in detail. The velocity gradient L can be written as L = D+W
= I$--’
L” = D’+W” 1
L”’ = D”‘+W””
= L’+F’L”‘F”~‘.
= pFe-‘,
(30)
= B”‘F”“~ ‘,
Due to the nonuniformity of the polymer network, the links with higher kinetic energy and subjected to weaker constraint may be activated first with the assistance of thermodynamic force, whereas the links with lower kinetic energy and subjected
148
X. CHEN
et al.
to stronger constraint are still being frozen. The segments which contain activated links can deform through the conformation change caused by the synergetic motion of links, while the segments which contain no activated links may only deform through the change of bond length and bond angle. Therefore the macroscopic Cauchy stress tensor T consists of two parts. The elastic part Tb comes from the frozen segments whose deformation is totally supplied by the change of bond length and bond angle, and the anelastic part T” comes from the activated segments whose deformation is mainly supplied by the conformation change. The macroscopic Cauchy stress tensor T is determined by the total deformation gradient F and the elastic deformation gradient F” (Anand, 1979 ; Boyce et al., 1989) : T = Tb+T” =f”(t)J-‘T”[lnF]+f”(t)J-‘_S?[lnF”],
(31)
where J = det(F”), Y is the fourth-order elastic modulus tensor, andfb(t) andf”(t) are, respectively, the fraction of frozen segments and activated segments at time t. Let PJtlt’) be the probability of the segments with n links that are generated at time t’ and still deform affinely together with the macroscopic object at time t. Considering the relaxation of segments, we obtain the governing equation :
dPn(tlt’) ~ = -2(~(t))(Pr,(tlf)) dt
(32)
By the assumption of affine deformation, the evolution equation for the end-to-end vector r(r, t’) of an activated segment which is generated at time t’ and still deforms affinely at time t under the macroscopic velocity gradient field is given as follows : % r(t, t') = L""(t) *r(t, t’).
(33)
We take the end-to-end vector r of a segment as the state variable and the force f exerted on the segment as the corresponding thermodynamic force. By making use of the approach proposed by Jongschaap (1990) who considered the parity of variables with respect to macroscopic time reversal, we can express the microscopic stress tensor u as I
u = -pI+
(S
h(t’)P(tlt’)r(t, cc
t’) @ f(t, t’) dt’ , >
(34)
where p is the pressure, I is the second-order unit tensor, h(t’) is the generating rate for the activated segments at time t’, and (. . .) stands for the ensemble average. The relation between the end-to-end vector r(t, t’) and the corresponding thermodynamic force f(t, t’) is given by (18). The anelastic part of the macroscopic Cauchy stress T” and the microscopic stress d satisfy T”
=
J-‘FeaFeT.
(35)
Statistical
thermodynamic
theory
for viscoelasticity
of polymers
149
At the initial equilibrium state, there are N,, segments per unit volume. The fraction of frozen segments and activated segments is, respectively, fi and ,ft. Using the Boltzmann distribution law, we have
(36a)
(36b) Since the generation of activated segments is originated from the activated process of the frozen links, the generating rate for the activated segments can be taken as
J’: (r(t”))
Thus the fraction
of frozen segments
and activated
dt”
(t’ > 0).
segments
is
.f”(t> = 1-f”(t),
fb(t) =f’i exp {-[I
(37)
(38a) $(r(t’))
dl}.
(38b)
Our constitutive framework can be simplified by the eight-chain network model in which the network is made up of groups of eight chains connecting the center to the corners of a unit cube (Arruda and Boyce, 1993).
ANALYSIS
OF EXPERIMENTAL
DATA
We carried out the tensile experimental study on PMMA and plasticized PVC at the constant strain rate & = 1.67 x 10e3 SK’ for temperatures ranging from room temperature to the glass transition temperature. We applied the unified constitutive equation to the experimental data. The unknown parameters are fitted by the method of nonlinear regression. The glass transition temperature T, is 105” for PMMA and 50 for plasticized PVC. Young’s modulus at the glassy state Eg is 3.3 GPa for PMMA and 2.2 GPa for plasticized PVC. We assume n = 20 as the number of links contained in a segment. We chose Cf = 17.44, C’“, = 51.6 in (29) as the universal constants with the glass transition temperature Tg being the reference temperature (Ferry, 1980). We obtain for PMMA c.w12w,, = T&p, AG, = A,,k,Tg(2.303Cf
= 7.33,
-2Cg?/Tg+2)
= 131.6 kJjmo1,
(39a) (39b)
150
X. CHEN
et al.
and for plasticized PVC cm12w,, = T&p,
AC, = A,k,Tg(2.303Ct
= 6.26,
-2Cg2/Tg+2)
(4Oa)
= 112.4 kJ/mol.
(4Ob)
For the PMMA experimental data at 28°C 75°C and 108”C, the initial number of segments per unit volume is fitted as N, = 1.02 x 102’ rnp3, the constraint potential barrier for inner rotation and local relative slippage of a mole of links as AG, = 10.0 kJ/mol, the reference relaxation frequency of links at the glass transition temperature Tg as vyp = 5.8 x lop2 Hz, and the reference relaxation frequency of segments at the glass transition temperature Tg as ij& = 1.6 x 1O-3 Hz. The experimental curve at 50°C for PMMA is compared with the theoretical curve predicted by the above parameters. For the plasticized PVC experimental data at 30°C 50°C and 65°C the initial number of segments per unit volume is fitted as No = 6.51 x 1O27rnp3, the constraint potential barrier for inner rotation and local relative slippage of a mole of links as AG, = 16.8 kJ/mol, the reference relaxation frequency of links at the glass transition temperature Tg as ijyg = 7.0 x lop2 Hz, and the reference relaxation frequency of segments at the glass transition temperature Tg as !j& = 1.9 x lo-’ Hz. As shown in Figs 2 and 3, the present constitutive model describes successfully the viscoelastic properties of PMMA and plasticized PVC within the wide range of temperatures. Both the small-strain yield behavior and the large-strain hardening effect are the intrinsic features of the present model. They are predicted without having to introduce an additional criterion. Below the glass transition temperature, the conformation change rate of segments lags behind the macroscopic deformation rate at the initial stage of loading because it takes time for links to overcome the 70
60
0.00
.05
.I0
.I5
.20
.25
.30
lnh Fig. 2. Comparison of theoretical model with experimental data for PMMA. Thin lines denote experimental true stress+logarithmic strain curve; thick lines denote theoretical true stress-logarithmic strain curve.
Statistical
thermodynamic
theory
for viscoelasticity
of polymers
IS1
40 T=30°C \
30
T=50°C Z E b
20
10
/
0
0.00
T=65’C
___---.I 8
--.05
Fig. 3. Comparison of theoretical experimental true stress-logarithmic
.lO
_.----I
I
I
.15
.20
.25
InX model with experimental data for plasticized strain curve, thick lines denote theoretical strain curve.
.30
PVC. Thin lines denote true stress-logarithmic
potential barrier for inner rotation and local relative slippage in the thermally~ mechanically activated process. In this case, the macroscopic deformation is mostly provided by the change of bond length and bond angle. Thereafter the anelastic deformation mechanism starts due to the conformation change of segments. As the macroscopic deformation rate totally results from the anelastic deformation, stress reaches a peak and yielding occurs. Under large deformation, the polymer chains will be reoriented and straightened, leading to the hardening effect. Within the moderate deformation range, whether softening or hardening behavior occurs is dependent on the competition between the thermally-mechanically activated mechanism and the stiffening mechanism. Over the glass transition zone, the shape of the stress-strain curve changes sharply and the magnitude of the tensile modulus drops two to three orders.
CONCLUSION In this paper, a new molecular relaxation mechanism was presented by considering the competition between two sets of links that are divided according to the change in their Gibbs free energy under a thermodynamic force. Compared to the molecular theories proposed by Robertson (1966), Haward and Thackray (1968), Boyce and Arruda (1989, 1993), and O’Dowd and Knauss (1995), the present nonequilibrium statistical thermodynamical theory identifies the evolution law of microscopic structure through the thermally-mechanically activated mechanism of the conformation change of segments. Comparing with the experimental data from PMMA and plas-
152
X. CHEN et al.
ticized PVC, we showed that the present unified constitutive model can successfully predict the viscoelastic behaviors of PMMA and plasticized PVC within a wide range of temperatures. Both the softening effect within the moderate deformation range and the hardening effect at large deformation are the inherent features of the present constitutive model. We have built a bridge between the energetic elasticity and the rubber elasticity at the molecular level using the nonequilibrium statistical thermodynamics.
REFERENCES Anand, L. (1979) On H. Hencky’s approximate strain energy function for moderate deformation. J. Appl. Mech. 46, 78-82. Argon, A. S. (1973) A theory for the low-temperature plastic deformation of glassy polymers. Phil. Mug. 28, 839-864.
Arruda, E. M. and Boyce, M. C. (1993) Three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41, 389412. Boyce, M. C., Weber, G. G. and Parks, D. M. (1989) On the kinematics of finite strain plasticity. J. Mech. Phys. Solids 37, 647-665. Eyring, H. (1936) Viscosity, plasticity and diffusion as examples of absolute reaction rates. J. Chem. Phys. 4,283. Ferry, J. D. (1980) Viscoelastic Properties of Polymers, 3rd edn. John Wiley & Sons, Chichester. Haward, R. N. and Thackray G. (1968) The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics. Proc. Roy. Sot. A 302,453472. James, H. M. and Guth, E. (1943) Theory of the elastic properties of rubber. J. Chem. Phys. l&455481. Jongschaap, R. J. J. (1990) Towards a unified formulation of microrheological models. In Rheological Modelling : Thermodynamical and Statistical Approaches, Proceedings, Saint Feliu de Guixols, Spain, 1990, ed. J. Casas-Vazquez and D. Jou. Springer-Verlag, Berlin. O’Dowd, N. P. and Knauss, W. G. (1995) Time dependent large principal deformation of polymers. J. Mech. Phys. Solids 43, 771-792. Reichl, L. E. (1980) A Modern Course in Statistical Physics. University of Texas Press, Austin, TX. Risken, H. (1989) The Fokker-Plunck Equations, 2nd edn. Springer-Verlag, Berlin. Robertson, R. E. (1966) Theory for the plasticity of glassy polymers. J. Chem. Phys. 44, 3950. Ward, I. M. (1983) Mechanical Properties of Solid Polymers, 2nd edn. John Wiley & Sons, Chichester.
J. Mech. Phys. Solids, Vol. 46, No. I, pp. 1399152, 1998 0 1997 Elsetier Science Ltd Printed in Great Britain. All rights reserved 002225096/97 %17.00 + 0.00 PII: soo22-5096(97)ooo10-0
NONEQUILIBRIUM STATISTICAL THEORY FOR VISCOELASTICITY XIAOHONG
THERMODYNAMIC OF POLYMERS
CHEN1_, PIN TONG
Center for Advanced Engineering Materials, Department of Mechanical Engineering, The Hong Kong University of Science & Technology. Clear Water Bay, Kowloon, Hong Kong
and REN WANG Department
of Mechanics
(Received
and Engineering
18 November
Science, Peking University,
1995
Beijing
100871, P.R. China
; in revised,form 29 Januarx 1997)
ABSTRACT In this paper, we propose a new molecular relaxation mechanism for polymers by considering the change in the actual microscopic structure under macroscopic stress fields. The effects of both intramolecular and intermolecular forces on the inner rotation and the relative slippage of links are taken into account, A constraint potential function, along with a constraint tensor, is introduced to describe the constraint exerted by the surrounding medium. A unified three-dimensional constitutive framework for the viscoelasticity of polymers including thermal effects is established by making use of nonequilibrium statistical thermodynamics, which can be reduced to James and Guth’s (1943) non-Gaussian polymer network theory for rubber elasticity. The model compares well with the experimental data for PMMA and plasticized PVC over a wide range of temperatures. 0 1997 Elsevier Science Ltd. All rights reserved Keywords
: A. thermomechanical
processes,
B. constitutive
behavior,
B. viscoelastic
material.
INTRODUCTION Since macromolecules are densely packed together in polymeric materials, both the intramolecular and intermolecular forces have significant influence on the relaxation process. When a thermodynamic force is exerted on a segment through the surrounding medium, the Gibbs free energy of links changes and the equilibrium condition breaks down. The segment responds by reorienting the links to a new lower free energy state. Nevertheless, there exists an energy barrier for local structural rearrangement due to the constraint of the surrounding medium. Thus it takes time for the links to jump over the energy barrier. The response rate depends on the ratio between the height of the energy barrier and the average kinetic energy of links. Most of the current molecular theories for polymers (Eyring, 1936; Robertson, 1966; Haward and Thackray, 1968; Argon, 1973; Boyce et al., 1989, 1993) are based on t Currently
on leave from the Department
of Mechanics 139
and Engineering
Science, Peking
University.
X. CHEN
140
et al.
shear-induced thermally activated mechanisms. In these theories, however, the shift in the height of the energy barrier for the microscopic relaxation process was assumed to depend only on the macroscopic stress, and the back stress due to rubber elasticity was introduced in the constitutive relation as the internal variable. O’Dowd and Knauss (1995) generalized the theory for rubber elasticity to include the effect of broad-spectral rate dependence by taking into account the molecular orientation and the volumetric strain, so that the nonlinear large deformation viscoelasticity characteristics can be reproduced. In this paper, we describe the constraint exerted on links by the surrounding medium by introducing a constraint potential function along with a constraint tensor. We propose a new molecular mechanism for the relaxation of polymers by considering the effects of the combined inner rotation and local relative slippage of links on the reorganization of microscopic structure. Based on this molecular mechanism, the viscoelastic constitutive relation of polymers can be expressed directly in terms of microscopic state variables and their time derivatives. This is due without introducing the back stress. Energetic elasticity and rubber elasticity are two limiting cases. Both the softening effect within a moderate range of deformation and the hardening effect at large deformation are natural consequences of the competition between the thermallymechanically activated mechanism for the inner rotation and the relative slippage of links and the stiffening mechanism due to finite chain extensibility. Universal ratetemperature equivalence is derived from the dependence of the constraint tensor on temperature, with the Arrhenius equation and the WLF equation as special cases (Ward, 1983 ; Ferry, 1980). Thus, this model is a new attempt to unify the various polymer constitutive theories from the viewpoint of nonequilibrium statistical thermodynamics. This approach is quite different from those by Haward and Thackray, Boyce and Arruda, and O’Dowd and Knauss.
MOLECULAR
RELAXATION
MECHANISM
Since polymer chains are densely packed together in the condensed state, the motion of a macromolecular chain is restricted due to the existence of both intramolecular and intermolecular forces. The portion of a chain between two neighboring junctions is called a segment. When an external field is applied, segments are subjected to thermodynamic forces transferred by the surrounding medium. The structural rigidity results from both the intramolecular and intermolecular forces. For T < Tb (where Tb is the brittle-to-ductile transition temperature), polymers display energetic elasticity owing to the change of bond length and bond angle since the inner rotation and the relative slippage of links are fundamentally frozen. Fracture takes place before relaxation is thermally-mechanically stimulated. For T,, d T < T,,(where T, is the glass transition temperature), although the average degree of constraint exerted on the links by the surrounding medium is still very large, the energy barrier for both the inner rotation and the relative slippage of links can be lowered under external load. The cooperative motion of links can be induced by the thermodynamic force, which leads to the conformational change of segments. Thus polymers display anelasticity predominantly.
Statistical
thermodynamic
theory
for viscoelasticity
of polymers
Fig. I. A labeled segment. The dark circles stand for the atoms in the labeled segment, and the grey circles stand for the atoms in other segments. The thin line locates the actual position of the labeled segment which is designated by R(s,; t), the thick line stands for the equilibrium position of the labeled segment which is designated by R(s, ; t), where so is the coordinate of arc length.
The definition (Reichl, 1980)
of the Gibbs
free energy G for a thermodynamic
G=
U-TS-XY,
system is given by
(1)
where U is the internal energy, T is the absolute temperature, S is the entropy, Y is a set of generalized forces, and X is the corresponding generalized displacements. As shown in Fig. 1, the actual position of the labeled segment is R(s, ; t) and its equilibrium position is ii&; t), where s0 denotes the position of a link, the smallest unit of motion in a segment. If a thermodynamic force f is exerted on the labeled segment along the direction of its end-to-end vector, the links in the labeled segment can be divided into two subsystems. One subsystem denoted as A comprises the links whose Gibbs free energy increases, and the other denoted as B comprises the links whose Gibbs free energy decreases. The averaged Gibbs free energy per link for each subsystem undergoes a shift under the thermodynamic force. It can be written as :
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et al.
exp{-g+}=~Zexp~~)sinBdB=
1-ex;(-8),
(2)
(3)
where kB is the Boltzmann constant, 1 is the link length, f is the magnitude of the thermodynamic force f, p =J/kBT, & = GA/n,, tje = GJn,, GA and Gs denote, respectively, the Gibbs free energy for each subsystem, nA and nB denote, respectively, the number of links in each subsystem, and 8’ is the Gibbs free energy per link without the thermodynamic force. The Langevin equation for micro-Brownian motion of a segment can be written as
cxso ;0 - WQo ; 0at-&so ; 0) =
_ MR(so ; 0 -&o
;
0)
+r(so
.
7
aR
t>
>
(4)
where &o ; t) = R(so ; t) -& ; t) stands for the fluctuation around the equilibrium position. The term on the left-hand side is the drag force owing to the difference between the local velocity and the mean field velocity. The drag coefficient is a secondorder tensor, which can be taken as [(so; t) = {[(l + Y$-Y&, ; t) 0 u(so; t)] for the axisymmetric case, where c is a scalar with the dimension of a drag coefficient, I is the identity tensor, u(so ; t) is the unit vector along the direction of a&, ;t)/ds,, and yi is a parameter describing the anisotropic property of the drag coefficient tensor. The first term on the right-hand side is the force corresponding to the constraint potential g(R(so ; t) -ii(s, ; t)) exerted by the surrounding medium, and the second term is the random force whose distribution is assumed to be Gaussian. The constraint potential g(R(so ; t) - ii(s, ; t)) can be approximated by g(R(s,;t)--(Soit))
=s,$
E2~~{l+COS[~+~~~‘2(R~(~~;t)-~i(~O;t))]},
(3
I I
which can be expanded in the form of Taylor’s series near the equilibrium position g(R(so ; t)-ik(s,
; t)) = T i
12~f(R,(~o ;t)-&so
; t))*,
where u. is a parameter with the dimension of energy, and o, is a principal value of o(so ; t), a second-order tensor describing the degree of constraint exerted by the surrounding medium. For the axisymmetric case, the constraint tensor can be taken as w(s,, ; t) = o[( 1+ yo)I- y&s,, ; t) @ u(s, ; t)], where o is the scalar constraint coefficient, and yWis a parameter describing the anisotropic property of the constraint tensor. One of the principal values of o(so ; 1) is o, = o along the principal direction parallel to a8(so ; t)/ as, and the other two are w2 = o3 = (1 + yW)walong the principal directions perpendicular to a&s,; t)/as,. Using (4) and (5), we can write the Langevin equation in the principal coordinate system as
Statistical
thermodynamic
theory
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of polymers
143
(7)
By making use of the Fokker-Planck equation equivalent to the Langevin equation, Kramers (Risken, 1989) calculated the escape rate of a particle sitting in a deep potential well near x = X,in as
(8) where p is the friction coefficient, g(x) denotes the energy barrier function, g”(x) is the second derivative of g(x), and Ag = gmax-gmin is the height of the energy barrier. Therefore, the intrinsic rate for a link jumping over the energy barrier by microBrownian motion along the principal direction in the absence of the thermodynamic force f is given by (9) where gymin”= Ig;,,,"I
=
$,h$,
Ag; = &,l’lu,.
The intrinsic frequency for a link jumping over the energy barrier by the combined inner rotation and local relative slippage can be taken as the maximum of vy, VPand v’:: 0:) =
max(vy, vi, v!) = !!CY$!?exp(-sT)=$exp(-z),
(10)
where R is the gas constant, AG, = 4A~.4+01~/31r~ is the energy barrier for the inner rotation and local relative slippage of a mole of such links, which depends on the intermolecular cohesive force and steric hindrance, and A,, is the Avagadro constant. From (7) we see that l/L o d escribes the extent of the constrained micro-Brownian motion of a link and plays an important role similar to the free volume in the glassy transition of polymers. Under the thermodynamic force f, the Gibbs free energy for the links in the two subsystems undergoes a different change, leading to the height shift for the energy barrier. The mean frequency for a link in subsystem A jumping over the energy barrier is B, =
P
iJoexp l-
and for a link in subsystem B is
w-P)
(11)
144
X. CHEN
ifB =
V’exp
et al.
= Be-Q0 kBT
( )
P
.-0
exp(B) - 1’
(12)
The change rate for the number of links in each subsystem can be written as
(13)
where nA+ nB = n, and n denotes the total number of links in the segment. The solution for (13) is
(14) where ni and ni denote, respectively, the initial number of links in each subsystem. The governing equation for the change in the projection of link length along the direction of the thermodynamic force for subsystems A and B can be written as :
dl, - = vA(leA-fIA), dt
-dl, = dt
(15) B&-Is),
where Z> and I> are, respectively, the equilibrium values for the average projection length of links in subsystems A and B ; G and & can be expressed in terms of the thermodynamic force :
(16)
The solution for (15) is
Statistical
thermodynamic
theory
for viscoelasticity
of polymers
145
where 12 and 1: denote, respectively, the initial value of the average projection length of links in subsystems A and B. The relation between the end-to-end vector of the segment and the thermodynamic force exerted can be written as r = rA +r,
= nAIA fn,l,,
(18)
where nA and n, are given by (14), and IA and Is are given by ( 17). As ~‘)t -+ K!, the system reaches the equilibrium state. We get
I- ev-P) 2 sinh(fl)
” (19)
ev(P>
-
2 sinh(B)
1
”
(20)
As the system reaches equilibrium, the relation between the projection of the endto-end vector of the segment and the thermodynamic force exerted becomes
(21) which is the same as the James-Guth network theory for rubber elasticity based upon the non-Gaussian distribution function (Ward, 1983). The Deborah number is defined as the ratio of the frequency of the external force or the deformation rate to the intrinsic relaxation frequency, i.e. De = w,/B” or De = k/V”. For the Deborah number De >> 1, the microscopic structural rearrangement cannot be observed during the experimental period because it takes time for links to get enough energy to jump the deep energy barrier of inner rotation and relative slippage. Polymers display essentially energetic elasticity, which is provided by the change of bond length and bond angle. For the Deborah number De << 1, the microscopic structural rearrangement can be completed through the combined inner rotation and relative slippage of links and a new dynamic equilibrium state can be reached rapidly. Energetic elasticity turns into rubber elasticity as a result of the conformation change of segments. For the Deborah number De - 1, the rate of the microscopic structural reorganization is of the same order as the frequency of the external force or the deformation rate. The system is in a nonequilibrium state. The thermally-mechanically activated barrier jumping process consumes energy resulting in energy dissipation. Anelasticity dominates at this rate. We have just identified the physical mechanism of the rate effects of polymers at the molecular level.
146
X. CHEN
RATE-TEMPERATURE
et al.
EQUIVALENCE
For T < Tg, the average kinetic energy for the thermal motion of links is small compared with the constraint potential barrier for inner rotation and local relative slippage of links. Under the thermodynamic force, the Gibbs free energy of links will be shifted. The links that have relatively high Gibbs energy are mechanically-thermally activated first. Then the confirmation of segments containing activated links is changed through the cooperative motion of links. By (lo), we get the shift factor defined by the intrinsic frequency ivy:
(22) where the subscript s denotes the value at the reference temperature. At low temperature, we assume that w12 has little dependence on temperature, i.e. o12 z o,12, and we have (23) Equation (23) has the form of Arrhenius equation, similar to the result of the barrier model (Ward, 1983). As temperature increases, the average kinetic energy of the thermal motion of molecules increases while the intermolecular constraint force decreases, so larger units of motion can be activated. As the relaxation time for the motion of segments decreases to the same order as the testing time scale, glass transition occurs. For a segment consisting of n links, we can obtain the expression for its intrinsic relaxation frequency following almost the same procedure as for a link : (24) where co, is the constraint parameter describing the degree of the restriction on the motion of a segment consisting of n links by the surrounding medium, and AG, = 4A,uoni20,/3n2 is the constraint potential barrier for the motion of a mole of such segments, depending on the intermolecular cohesive force and steric hindrance. By (24), we get the shift factor defined by the intrinsic frequency ijz : (25) Near the glass transition temperature T,, we assume that n120, satisfies the following relation : 1 p=CIT’ 121’0,
T- T,
Statistical
thermodynamic
where c( is the proportional temperature. We have
theory
factor,
for viscoelasticity
and
147
of polymers
T2 is the true thermodynamic
as T+
n12co, -+ co
transition
T,
nl%o, --+1/cc at high temperature. Substituting (26) into (25), we have lna,
= 1,;
= 21n ns
We expand T, and get :
T(T, - T,) + 4nl’o,,u,, 3r.r2k,T, T,(T- Tz)
the first term on the right-hand
side of (27) at the reference
In ar C: (TlogaT=mlOC;+~-T,’ which has the form of the WLF equation CJZ satisfy :
(I - $$)’
(Ferry,
(27) temperature
TJ
1980). The two parameters
(28) C; and
At high temperature, the rate-temperature equivalence is again reduced to the form of Arrhenius equation on account that n12co, + l/cc, which is suitable to describe the high temperature creep behavior of polymers.
CONSTITUTIVE
RELATION
Polymeric materials can display energetic elasticity, anelasticity, and entropic elasticity accompanying viscous flow under different conditions owing to the existence of multiple units of motion. Next we will present a unified three-dimensional constitutive relation suitable for describing the relaxation and transition of polymeric materials under small as well as large deformation. The multiplicative decomposition of the deformation gradient is given by F = F”F”‘, where F” is the elastic deformation gradient and F”’ is the anelastic deformation gradient. We choose F” = Fe’, det(F”“) = 1. The representation of the kinematics of inelastic deformation has been discussed by Boyce et al. (1989) in detail. The velocity gradient L can be written as L = D+W
= I$--’
L” = D’+W” 1
L”’ = D”‘+W””
= L’+F’L”‘F”~‘.
= pFe-‘,
(30)
= B”‘F”“~ ‘,
Due to the nonuniformity of the polymer network, the links with higher kinetic energy and subjected to weaker constraint may be activated first with the assistance of thermodynamic force, whereas the links with lower kinetic energy and subjected
148
X. CHEN
et al.
to stronger constraint are still being frozen. The segments which contain activated links can deform through the conformation change caused by the synergetic motion of links, while the segments which contain no activated links may only deform through the change of bond length and bond angle. Therefore the macroscopic Cauchy stress tensor T consists of two parts. The elastic part Tb comes from the frozen segments whose deformation is totally supplied by the change of bond length and bond angle, and the anelastic part T” comes from the activated segments whose deformation is mainly supplied by the conformation change. The macroscopic Cauchy stress tensor T is determined by the total deformation gradient F and the elastic deformation gradient F” (Anand, 1979 ; Boyce et al., 1989) : T = Tb+T” =f”(t)J-‘T”[lnF]+f”(t)J-‘_S?[lnF”],
(31)
where J = det(F”), Y is the fourth-order elastic modulus tensor, andfb(t) andf”(t) are, respectively, the fraction of frozen segments and activated segments at time t. Let PJtlt’) be the probability of the segments with n links that are generated at time t’ and still deform affinely together with the macroscopic object at time t. Considering the relaxation of segments, we obtain the governing equation :
dPn(tlt’) ~ = -2(~(t))(Pr,(tlf)) dt
(32)
By the assumption of affine deformation, the evolution equation for the end-to-end vector r(r, t’) of an activated segment which is generated at time t’ and still deforms affinely at time t under the macroscopic velocity gradient field is given as follows : % r(t, t') = L""(t) *r(t, t’).
(33)
We take the end-to-end vector r of a segment as the state variable and the force f exerted on the segment as the corresponding thermodynamic force. By making use of the approach proposed by Jongschaap (1990) who considered the parity of variables with respect to macroscopic time reversal, we can express the microscopic stress tensor u as I
u = -pI+
(S
h(t’)P(tlt’)r(t, cc
t’) @ f(t, t’) dt’ , >
(34)
where p is the pressure, I is the second-order unit tensor, h(t’) is the generating rate for the activated segments at time t’, and (. . .) stands for the ensemble average. The relation between the end-to-end vector r(t, t’) and the corresponding thermodynamic force f(t, t’) is given by (18). The anelastic part of the macroscopic Cauchy stress T” and the microscopic stress d satisfy T”
=
J-‘FeaFeT.
(35)
Statistical
thermodynamic
theory
for viscoelasticity
of polymers
149
At the initial equilibrium state, there are N,, segments per unit volume. The fraction of frozen segments and activated segments is, respectively, fi and ,ft. Using the Boltzmann distribution law, we have
(36a)
(36b) Since the generation of activated segments is originated from the activated process of the frozen links, the generating rate for the activated segments can be taken as
J’: (r(t”))
Thus the fraction
of frozen segments
and activated
dt”
(t’ > 0).
segments
is
.f”(t> = 1-f”(t),
fb(t) =f’i exp {-[I
(37)
(38a) $(r(t’))
dl}.
(38b)
Our constitutive framework can be simplified by the eight-chain network model in which the network is made up of groups of eight chains connecting the center to the corners of a unit cube (Arruda and Boyce, 1993).
ANALYSIS
OF EXPERIMENTAL
DATA
We carried out the tensile experimental study on PMMA and plasticized PVC at the constant strain rate & = 1.67 x 10e3 SK’ for temperatures ranging from room temperature to the glass transition temperature. We applied the unified constitutive equation to the experimental data. The unknown parameters are fitted by the method of nonlinear regression. The glass transition temperature T, is 105” for PMMA and 50 for plasticized PVC. Young’s modulus at the glassy state Eg is 3.3 GPa for PMMA and 2.2 GPa for plasticized PVC. We assume n = 20 as the number of links contained in a segment. We chose Cf = 17.44, C’“, = 51.6 in (29) as the universal constants with the glass transition temperature Tg being the reference temperature (Ferry, 1980). We obtain for PMMA c.w12w,, = T&p, AG, = A,,k,Tg(2.303Cf
= 7.33,
-2Cg?/Tg+2)
= 131.6 kJjmo1,
(39a) (39b)
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X. CHEN
et al.
and for plasticized PVC cm12w,, = T&p,
AC, = A,k,Tg(2.303Ct
= 6.26,
-2Cg2/Tg+2)
(4Oa)
= 112.4 kJ/mol.
(4Ob)
For the PMMA experimental data at 28°C 75°C and 108”C, the initial number of segments per unit volume is fitted as N, = 1.02 x 102’ rnp3, the constraint potential barrier for inner rotation and local relative slippage of a mole of links as AG, = 10.0 kJ/mol, the reference relaxation frequency of links at the glass transition temperature Tg as vyp = 5.8 x lop2 Hz, and the reference relaxation frequency of segments at the glass transition temperature Tg as ij& = 1.6 x 1O-3 Hz. The experimental curve at 50°C for PMMA is compared with the theoretical curve predicted by the above parameters. For the plasticized PVC experimental data at 30°C 50°C and 65°C the initial number of segments per unit volume is fitted as No = 6.51 x 1O27rnp3, the constraint potential barrier for inner rotation and local relative slippage of a mole of links as AG, = 16.8 kJ/mol, the reference relaxation frequency of links at the glass transition temperature Tg as ijyg = 7.0 x lop2 Hz, and the reference relaxation frequency of segments at the glass transition temperature Tg as !j& = 1.9 x lo-’ Hz. As shown in Figs 2 and 3, the present constitutive model describes successfully the viscoelastic properties of PMMA and plasticized PVC within the wide range of temperatures. Both the small-strain yield behavior and the large-strain hardening effect are the intrinsic features of the present model. They are predicted without having to introduce an additional criterion. Below the glass transition temperature, the conformation change rate of segments lags behind the macroscopic deformation rate at the initial stage of loading because it takes time for links to overcome the 70
60
0.00
.05
.I0
.I5
.20
.25
.30
lnh Fig. 2. Comparison of theoretical model with experimental data for PMMA. Thin lines denote experimental true stress+logarithmic strain curve; thick lines denote theoretical true stress-logarithmic strain curve.
Statistical
thermodynamic
theory
for viscoelasticity
of polymers
IS1
40 T=30°C \
30
T=50°C Z E b
20
10
/
0
0.00
T=65’C
___---.I 8
--.05
Fig. 3. Comparison of theoretical experimental true stress-logarithmic
.lO
_.----I
I
I
.15
.20
.25
InX model with experimental data for plasticized strain curve, thick lines denote theoretical strain curve.
.30
PVC. Thin lines denote true stress-logarithmic
potential barrier for inner rotation and local relative slippage in the thermally~ mechanically activated process. In this case, the macroscopic deformation is mostly provided by the change of bond length and bond angle. Thereafter the anelastic deformation mechanism starts due to the conformation change of segments. As the macroscopic deformation rate totally results from the anelastic deformation, stress reaches a peak and yielding occurs. Under large deformation, the polymer chains will be reoriented and straightened, leading to the hardening effect. Within the moderate deformation range, whether softening or hardening behavior occurs is dependent on the competition between the thermally-mechanically activated mechanism and the stiffening mechanism. Over the glass transition zone, the shape of the stress-strain curve changes sharply and the magnitude of the tensile modulus drops two to three orders.
CONCLUSION In this paper, a new molecular relaxation mechanism was presented by considering the competition between two sets of links that are divided according to the change in their Gibbs free energy under a thermodynamic force. Compared to the molecular theories proposed by Robertson (1966), Haward and Thackray (1968), Boyce and Arruda (1989, 1993), and O’Dowd and Knauss (1995), the present nonequilibrium statistical thermodynamical theory identifies the evolution law of microscopic structure through the thermally-mechanically activated mechanism of the conformation change of segments. Comparing with the experimental data from PMMA and plas-
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X. CHEN et al.
ticized PVC, we showed that the present unified constitutive model can successfully predict the viscoelastic behaviors of PMMA and plasticized PVC within a wide range of temperatures. Both the softening effect within the moderate deformation range and the hardening effect at large deformation are the inherent features of the present constitutive model. We have built a bridge between the energetic elasticity and the rubber elasticity at the molecular level using the nonequilibrium statistical thermodynamics.
REFERENCES Anand, L. (1979) On H. Hencky’s approximate strain energy function for moderate deformation. J. Appl. Mech. 46, 78-82. Argon, A. S. (1973) A theory for the low-temperature plastic deformation of glassy polymers. Phil. Mug. 28, 839-864.
Arruda, E. M. and Boyce, M. C. (1993) Three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. J. Mech. Phys. Solids 41, 389412. Boyce, M. C., Weber, G. G. and Parks, D. M. (1989) On the kinematics of finite strain plasticity. J. Mech. Phys. Solids 37, 647-665. Eyring, H. (1936) Viscosity, plasticity and diffusion as examples of absolute reaction rates. J. Chem. Phys. 4,283. Ferry, J. D. (1980) Viscoelastic Properties of Polymers, 3rd edn. John Wiley & Sons, Chichester. Haward, R. N. and Thackray G. (1968) The use of a mathematical model to describe isothermal stress-strain curves in glassy thermoplastics. Proc. Roy. Sot. A 302,453472. James, H. M. and Guth, E. (1943) Theory of the elastic properties of rubber. J. Chem. Phys. l&455481. Jongschaap, R. J. J. (1990) Towards a unified formulation of microrheological models. In Rheological Modelling : Thermodynamical and Statistical Approaches, Proceedings, Saint Feliu de Guixols, Spain, 1990, ed. J. Casas-Vazquez and D. Jou. Springer-Verlag, Berlin. O’Dowd, N. P. and Knauss, W. G. (1995) Time dependent large principal deformation of polymers. J. Mech. Phys. Solids 43, 771-792. Reichl, L. E. (1980) A Modern Course in Statistical Physics. University of Texas Press, Austin, TX. Risken, H. (1989) The Fokker-Plunck Equations, 2nd edn. Springer-Verlag, Berlin. Robertson, R. E. (1966) Theory for the plasticity of glassy polymers. J. Chem. Phys. 44, 3950. Ward, I. M. (1983) Mechanical Properties of Solid Polymers, 2nd edn. John Wiley & Sons, Chichester.