A model of adaptive links in finite viscoelasticity

Mechmfics Reseau~ Communicatiom, Vol. 24, No. 2, pp. 161-166, 1997 Copyrisht © 1997 Elmvi~ Scieme Ltd Plinted in the USA~ An riglm reunved 0093-6413/97 $17.00 + .00

Pergamon

PIIS0093..6413(97)00008-6

A MODEL OF ADAPTIVE LINKS IN FINITE VISCOELASTICITY

A.D. DROZDOV Institute for Industrial Mathematics Ben-Gurion University of the Negev 22 Ha-Histadrut Street, Be'ersheba 84213, Israel. (Received 30 October 1996. accepted for print 18 December 1996)

1. I N T R O D U C T I O N The paper is concerned with a new model for the nonlinear viscoelastic behavior of polymers at finite strains. The model is based on the Tobolsky concept [7] of adaptive links (entanglements) between polymeric molecules. According to it, a polymeric medium is treated as a system of parallel links which arise and collapse. By assuming the rates of breakage and reformation for adaptive links to depend on strain energy density, we arrive at the constitutive model which does not permit the effects of time and strains to be separated. The objective of the present paper is to derive a new constitutive model in viscoelasticity, which does not obey the separability principle [4], and to determine adjustable parameters by fitting experimental data for a number of polymers. 2. A M O D E L O F A D A P T I V E L I N K S A viscoelastic medium is modeled as a system of parallel elastic springs (links between polymeric molecules) which arise and collapse. It is assumed that )(0 initial links are not involved in the reformation process, and M different kinds of replacing links exist which correspond to M kinds of entanglements [12]. The number of links of the ruth kind arisen before instant r and existing at instant t is denoted as X,,(t, r). We introduce the relative rate of reformation 1 ox,,,

~(~) = Y;~ o-~ (~'~)' the breakage function g,~(t, r)

(1)

where )(m = X,,(0, 0), and which equals the relative number of adaptive links of the ruth kind existed at instant r and broken to instant t. Evidently that OX,,, 8x,,, (~, ~)[1 - z~(t, ~)]. x,~(t, 0) = R~[I - g~,(t,0)l, 0r (t, r) = --g~r (2) We substitute expressions (1) and (2) into the formula rt 0A~ x ~ ( t , 0 = x = ( t , 0 ) + J0 -gTJ ( t ' r ) d r ' and suppose that the viscoelastic material is non-aging, which means that

xm(t,t) =

~,

~(t) = ~o,

161

g,,(t,r) = g~0(t - ~).

(3)

162

A.D. DROZDOV

As a result, we find that

/o'I

gmo(O =

1-

Differentiation of this equality yields

d~t°(t)

= 7,~0[1 - gmo(t)l,

gin0(0) = 0. J

(4)

Resolving Eq. (4) with respect to g,~o(t) and using Eq. (2), we find that the relaxation kernel of a viscoelastic medium is presented in the form of a truncated Prony series [4]. To account for the effect of strains on the rate of reformation -y,~ (which implies that Eqs. (3) are not longer fulfilled), we assume that Eq. (4) remains true for aging as well as for non-aging media. Returning to the initial notation, we present this equation as

Ogm(t,T) Ot

= 7m(t)[1 -gin(t, T)],

gm(v,r) = 0.

(5)

10Xm.( t ,

r)

(6)

7re(r).

(7)

It follows from Eqs. (2) and (5) that the functions

rim(t) = Xm(t,O) '

Nm(t,T)-

Xm

satisfy the differential equations

! dt (t) nm(t) 1

= -Tin(t),

ON, n (t,

Nm(t, r) Ot

nm(0) = 1,

r) = -%~(t),

Nm(r, r) =

By analogy with the time-temperature superposition principle, the rates of reformation 7,,, are assumed to change similarly to each other with the growth of strains 7m

= a.

(s)

"Tin0 For a discussion of the mechanical meaning of Eq. (8), see [1]. A similar hypothesis (in terms of a reduced time) was employed in the Schapery model [11] at infinitesimal strains. Unlike the free-volume theory of viscoelasticity [8, 10], we express the shift function a in terms of the potential energy W0 for non-replaced links In a = A0 ~ 0 ,

(9)

where A0 is an adjustable parameter. Phenomenological equations similar to Eq. (9) were suggested in [13, 14] for polymeric melts. Let the response in an adaptive link of the ruth kind obey the constitutive equation of an incompressible, isotropic, hyperelastic solid, the potential energy Wm of which depends on the first invariants I1 and I3 of a Finger tensor F. For the initial links (existed at t = 0), we set fi' = ~'°(t), where FO(t) is the Finger tensor for transition from the initial to the actual configuration. For links arising at instant r, we set F = ~b°(t, r), where/~°(t, r) is the relative Finger tensor for transition from the actual configuration at instant r to the actual

ADAPTIVE LINKS IN FINITE VISCOELASTICITY

7O

15

or

a

I

i

I

I

I

I

1.0

I

)~

1.3

I

I

I

I

163

I

.0

I

I

)X

A

1

1.2

B

Figure 1: The longitudinal stress a (MPa) versus the extension ratio ,X in tension with a constant rate of strain t. Circles: experimental data obtained in [6]; solid lines: prediction of the model. (A) cellulose nitrate - curve 1: t = 0.02 (rain-l), /~ = 716 (MPa), ~ = 0.90, 70 = 0.62 (rain-1) and A0 = 0.6448 (MPa-~); curve 2: t = 0.002 (rain-l), /J = 680 (MPa), 7/ = 0.92, 70 = 0.075 (min -1) and A0 = 0.5098 (MPa-{). (B) cellulose acetate - t = 0.01 (rain-l), ~u = 190 (MPa), 0 = 0.90, 70 = 0.48 (min -1) and A0 = 1.4569 (MPa-~). configuration at instant t. Summing up the potential energies for a d a p t i v e links, we o b t a i n the t o t a l pot~ential energy of the system at instant t M IV(t) = XoWo(Ik(F°(t)) + ~_. [Xm(t, O)W,,,(Ik(F°(t)) + fot ~ T m

^ (t, r)Wm(Ik(F*(t,

r))dr]

rn= l M

= r]°ff'°(I~(P°(t)) + ~

t

rl'n[n'~(t)~V"(Ik(F°(t)) + fo N,.(t, r)I?Vm(h(_k*(t, r))dr],(lO)

rn=l

where

M = )(o + ~

)?m Xm,

%, = -~--,

l)dm = W,,X'.

(11)

m----1

Using the Lagrange variational principle and Eq. (10), and applying an a p p r o a c h derived in [3], we o b t a i n t h e following constitutive equation for a viscoelastic m e d i u m which does not obeys the s e p a r a b i l i t y principle M

5(t) = -p(t)~l + 2 { % O e ( t )

t

rl,,[nm(t)Om(t) + fo N,,(t, r)~}:(t, r)dr]}.

+ ~

(12)

m---- ~,

Here & is t h e Cauchy stress tensor, p is pressure, ] is the unit tensor, 2

2

6..(t) = ~ ~..~(t)IPo(t)] k, k=l

~

6*..(t, r) = ~ ~:dt, r)[~*(t, ~

~)1~,

k=l

- 0---~-(ff(0),

ow., . ~,l(t,r) = ~aw..(g(t,~))+ff(t ' r)--~-2(I~(t,r)),

01~2(t,r ) = ow... -~= (I~(t,r)),

164

A.D. DROZDOV

where I~(t) = Ik(P°(t)), I~(t, r) = I~(P*(t, r)). 3. U N I A X I A L

DEFORMATION

OF A S P E C I M E N

To analyze the model (7) to (9), and (12), we consider uniaxial tension of a specimen z: = A(t)Xl,

x2 = Ao(t)x2,

(13)

x~ = ~o(t)x~,

where Xi and xi are Cartesian coordinates in the initial and actual configuration, respectively. It follows from the incompressibility condition I3(fi'°(t)) = 1 that A0(t) = A-~(t).

(:4)

Substitution of expressions (13) and (14) into Eq. (12) implies that ~(t) = ~(t)~,~: + o o ( t ) ( : 2 ~ +

~),

where el are unit vectors of the Cartesian coordinate flame in the initial configuration, and M

or(t) = --p(t) + 2r/o[~ol(t) + kOo~(t)A2(t)]A2(t) + 2 ~ rlm{nm(t)[~ml(t ) + kOm2(t)A2(t)] m-----1

×A~(t) + fotNm(t, r)[~o,(t, r) + ~=2(t, r ) ( ~ ) 2 1 ( A ( t )

2

M

ao(t) = --p(t) + 2r/o[kOo,(t) + k~o2(t)A-l(t)]A-'(t) + 2 ~ ,m{nn~(t)[qlml(t) + @m2(t)),-'(t)] m-~ l

XA-:(t) + f0 Nm(t'r)[@Ol(t'r) + kOOm~(t'r)-~]'~ ar~"

(15)

To satisfy boundary conditions on the lateral surface of the specimen, we set ao = 0. Excluding pressure p from this equality and Eqs. (15), we find that M

M

a(t) = 2[r/O~Ol(t) + E ~mnm(t)kOml(t)][A2(t) -- A-:(t)] + 2[r/0tl/02(t) + ~ rlmnm(t)~Pm2(t)] rnml

rn=l M

t

×W(t) - ~-~(t)]+ 2 r~n m l ,~ f0 N,~(t,r){k~m,(t,r)_(zkr])° , ~(t) , (~(~))~]}&. +~,~2(t,,)[(:~) -

A(t)

A(r) 1

[,-'7-7, 2 _ A(t) ~ (16)

Let us suppose that any adaptive link obeys the constitutive equation of a neo-Hookean elastic medium with the strain energy density :(I,

-

3).

(17)

Combining Eqs. (16) and (17), we obtain M

a(t)

=

#{[1 - y~ rim(1 - r~m(t))l[A2(t)

- A-l(t)]

m=l

,

,,, A(t),~

Nm(t'rJtt-7?: )) -

A(T)

(:s)

ADAPTIVE LINKS IN FINITE VISCOELASTICITY

165

100

100 a

i

I

[

I

i

I

1.0

I

I

A

I

I

2.0

I

I

I

I

I

1.0

A

I

I

A

I

2.0

B

Figure 2: The longitudinal stress a (MPa) v e r s u s the extension ratio ~, in tension with a constant rate of strain t = 0.0984 (min-1). Circles: experimental data obtained in [5], solid lines: prediction of the model. (A) poly(vinyl chloride) - / ~ = 260 (MPa), t / = 0.92, 7o = 1.12 (rain-t), A0 = 0.0860 ( M P a - ] ) ; (B) Lexan polycarbonate - g = 210 (MPa), r / = 0.90, 70 = 0.84 (min-1), A0 = 0.0664 ( M P a - ~ ) .

It follows from Eqs. (8), (9), and (17) that the functions nm(t) and Nm(t, r) satisfy Eqs. (7) with 7re(t) = 7m0 exp{A[A(t) + 2A-~(t) - 3]½}, (19) where A is an adjustable parameter. To fit experimental data for cellulose nitrate, cellulose acetate, poly(viny~ chloride), and polycarbonate under tension with a constant rate of strain A(t) = 1 + it, we set M = 1. The latter means that only two types of links are taken into account which are responsible for bond stretching and conformational change, see [2]. Figs. 1 and 2 demonstrate fair agreement between results of numerical simulation and experimental data, which implies that the model (7), (12), and (17) may be employed to predict the nonlinear viscoelastic behavior of polymers. 4. C O N C L U D I N G

REMARKS

A new constitutive model is derived for the nonlinear behavior of viscoelastic media which do not obey the separability principle. According to this model, a polymeric material is treated as a system of M different kinds of adaptive links, and the rates of reformation for the adaptive links irrcrease with the growth of the strain energy density for non-replaced bonds. Adjustable parameters of the constitutive equations are found by fitting experimental data for a number of polymers under tension with a constant rate of strain. Results of numerical analysis demonstrate fair prediction of experimental data for deformations up to 200 per cent. Small deviations observed in Fig. 2 may be explained by the unsufficient number (M = 1) of different kinds of adaptive links (which is equivalent to the unsufficient number of relaxation times taken into account). As common practice, quasi-static tensile tests with a constant rate of strain are employed to reveal the characteristic features of the viscoplastic behavior of polymers, see, e.g., [5, 6, 9]. The point A = ~,, where the stress reaches its local maximum is identified as the yield point (which means that plastic strains arise at )~ > )~,), the decreasing branch of the a(~)

166

A.D. DROZDOV

curve immediately after the yield point is explained by the material softening caused by non-homogeneous deformation, while the increasing branch of this curve (at large ~ values) is explained by the material hardening due to configurational changes, see [2, 9]. The dependencies plotted in Figs. 1 and 2 and typical of the viscoplastic behavior of polymers are obtained in the framework on a nonlinear constitutive model for a viscoelastic medium without softening and hardening. This does not mean that the main postulates of viscoplasticity should be revised. Our purpose is to demonstrate that a more complicated program of loading (compared with tension with a constant rate of strain) is necessary to characterize the viscoelasto-plastic behavior of polymers. REFERENCES

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