Rubber Elasticity - Continuum Theories

7/28/2019 Rubber Elasticity - Continuum Theories

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where QI and QII are the amounts of heat released to the low-temperaturereservoir (T I) and absorbed from the high-temperature reservoir (T II),respectively.

While the entropy change is zero for either system during the reversibleadiabatic steps (see Figures 9.12c and 9.13c), it must be emphasized that theentropy change is greater than zero for an irreversible adiabatic process. Anexample for an elastomer is “letting go” of a stretched rubber band.

9.8.4 An Elastomer Thermal Cycle Demonstration

The elastomer thermal cycle is demonstrated in Figure 9.14 (51). A bicycle

wheel is mounted on a stand, with a source of heat on one side only. Stretchedrubber bands replace the spokes. On heating, the stress that the stretchedrubber bands exert is increased so that the center of gravity of the wheel isdisplaced toward 9 o’clock in the drawing. The wheel then rotates counter-clockwise (52).

Each of the steps in Figure 9.13 may be traced in Figure 9.14, although noneof the steps in Figure 9.13 are purely isothermal or adiabatic, and then of course they are not strictly reversible. Steps 1 to 2 in Figure 9.13 occur at 6o’clock in Figure 9.14, where there is a (near) adiabatic length increase due togravity. At 3 o’clock, at T II, heat is absorbed (nearly) isothermally, and thelength decreases, doing work.At 12 o’clock,corresponding to steps 3 to 4,thereis an adiabatic length decrease due to gravity. Last, at 9 o’clock, steps 4 to 1,there is a (nearly) isothermal length increase, and heat is given off to the sur-roundings at T I, and work is done on the elastomer.

9.9 CONTINUUM THEORIES OF RUBBER ELASTICITY

9.9.1 The Mooney–Rivlin Equation

The statistical theory of rubber elasticity is based on the concepts of randomchain motion and the restraining power of cross-links; that is, it is a molecular

9.9 CONTINUUM THEORIES OF RUBBER ELASTICITY 453

Figure 9.14 A thermally rotated wheel, employing an elastomer as the working substance (51).



theory. Amazingly, similar equations can be derived strictly from phenome-nological approaches, considering the elastomer as a continuum. The bestknown such equation is the Mooney–Rivlin equation (53–56),

(9.48)

which is sometimes written in the algebraically identical form,

(9.49)

Equations (9.48) and (9.49) appear to have a correction term for equation(9.34), with an additional term being added. However, they are derived fromquite different principles.

According to equation (9.34), the quantity s /(a - 1/a 2) should be a con-stant. Equation (9.49), on the other hand, predicts that this quantity dependson a :

(9.50)

Plots of s /(a - 1/a 2) versus 1/a are found to be linear, especially at low elon-

gation (see Figure 9.15) (57). The intercept on the a -1 = 0 axis yields 2C 1, andthe slope yields 2C 2. The value of 2C 1 varies from 2 to 6 kg/cm2, but the valueof 2C 2, interestingly, remains constant near 2 kg/cm2. Appendix 9.2 describesa demonstration experiment that illustrates both rubber elasticity [see equa-tion (9.34)] and the nonideality expressed by equation (9.50).

On swelling, the value of 2C 2 drops rapidly (see Figure 9.16) (57), reachinga value of zero near v2 (volume fraction of polymer) equal to 0.2. This samedependence is observed for the same polymer in different solvents, differentlevels of cross-linking for the same polymer, or (as shown) different polymersentirely (57).

The interpretation of the constants 2C 1 and 2C 2 has absorbed much time;the results are inconclusive (42). It is tempting but generally considered

incorrect to equate 2C 1 and nRT (r 2i

––

/r 20

––

). The original derivation of Mooney(46) shows that 2C 2 has to be finite, but it does not indicate its value relativeto 2C 1. According to Flory (41), the ratio 2C 2/2C 1 is related to the loosenesswith which the cross-links are embedded within the structure. Trifunctionalcross-links have larger values of 2C 2/2C 1 than tetrafunctional cross-links, forexample (58).

As indicated above,2C 2 decreases with the degree of swelling. FurthermoreGee (59) showed that during stress relaxation, swelling increased the rate of approach to equilibrium. Ciferri and Flory (60) showed that 2C 2 is markedly

s

a a a -= +

12

22 1

2C

C

s a

a a

= +Ê Ë

ˆ ¯ -Ê Ë

ˆ ¯ 2

2 11

2

2C

C

s a a a

= -Ê Ë

ˆ ¯ + -Ê

Ë ˆ ¯ 2

12 1

11 2 2 3

C C

454 CROSS-LINKED POLYMERS AND RUBBER ELASTICITY



reduced by swelling and deswelling the sample at each elongation. Thesamples, actually measured dry, had 2C 2 values about half as large after theswelling–deswelling operation as those measured before.These results suggestthat the magnitude of 2C 2 is caused by nonequilibrium phenomena. Gumbrellet al. (57) stated it in terms of the reduced numbers of conformations avail-able in the dry state versus the swollen state.

Other possible explanations include non-Gaussian chain or network statis-tics (see Section 9.10.6) and internal energy effects (42). The latter, bearing on

the front factor, will be treated in Section 9.10.

9.9.2 Generalized Strain–Energy Functions

Following the work of Mooney, more generalized theories of the stress–strainrelationships in elastomers were sought. The central problem was how tocalculate the work, W , stored in the body as strain energy.

Rivlin (56) considered the most general form that such strain–energy func-tions could assume. As basic assumptions, he took the elastomer to be incom-


Figure 9.15 Plot of s /(a - 1/ a

2

) versus a

-1

for a range of natural rubber vulcanizates. Sulfurcontent increases from 3 to 4%, with time of vulcanization and other quantities as variables

(57).



pressible and isotropic in the unstrained state. Symmetry conditions requiredthat the three principal extension ratios, a 1, a 2, and a 3, depend only on evenpowers of the a ’s. In three dimensions (see Figure 9.17), the simplest functionsthat satisfy these requirements are

(9.51)

(9.52)

(9.53)

where I 1, I 2, and I 3 are termed strain invariants.The third strain invariant is equal to the square of the volume change,

(9.54) I V

V 3

0

2

1= Ê Ë

ˆ ¯ =

I 3 12

22

32= a a a

I 2 12 22 22 32 32 12= + +a a a a a a

I 1 12

22

32= + +a a a


Figure 9.16 Dependence of C 2 on v 2 for synthetic rubber vulcanizates (57). Polymers:

᭺, butadiene–styrene, (95/5); , butadiene–styrene, (90/10); ᭻, butadiene–styrene, (85/15);

᭹, butadiene–styrene, (75/25); ᭪, butadiene–styrene, (70/30); X, butadiene–acrylonitrile,

(75/25).



which under the assumption of incompressibility equals unity. Alternate for-mulations have been proposed by Valanis and Landel (61) and by Ogden (62),which have been reviewed by Treloar (42).

Consider the deformation of a cube (Figure 9.17). The work that is storedin the body as strain energy can be written (63).

(9.55)

where the s ’s are the stresses.The work, in a more general form, can be expressed as a power series (56):

(9.56)

Equation (9.56) is written so that the strain energy term in question vanishesat zero strain.

For the lowest member of the series, i = 1, j = 0, and k = 0,

(9.57)

which is functionally identical to the free energy of deformation expressed inequation (9.31). For the case of uniaxial extension,

(9.58)

and noting equation (9.32) and equation (9.54),

a a a

2 3

1 21

= = Ê Ë

ˆ ¯

a a 1 =

W C I = -( )100 1 3

W C I I I ijk

i j k

i j k

= -( ) -( ) -( )=

•

Â 1 2 3

0

3 3 1, ,

W d d da s a s a s a ( ) = + +Ú Ú Ú 1 1 2 2 3 3


Figure 9.17 An elastomeric cube. (a ) Undeformed and (b ) deformed states, showing princi-

pal stresses and strains.



Equation (9.57) can now be written

(9.59)

and the stress can be written [see equation (9.54)]

(9.60)

This equation is readily identified with equation (9.34), suggesting (for this

case only) that

(9.61)

On retention of an additional term in equation (9.55), with i = 0, j = 1, andk = 0,

(9.62)

which leads directly to the Mooney–Rivlin equation, equation (9.49).Interestingly, if we retain one more term, C 200, an equation of the form

(63)

(9.63)

can be written, where

(9.64)

(9.65)

and

(9.66)

Equation (9.63),with two additional terms over the statistical theory of rubberelasticity, fits the data quite well (see Figure 9.18) (64).

Because no particular molecular model was assumed, theoretical valuescannot be assigned to C , C ¢, and C ≤ , nor can any molecular mechanisms beassigned. These phenomenological equations of state, however, accuratelyexpress the form of the experimental stress–strain data.

¢¢ =C C 4 200

¢ = +( )C C C 2 4 200 010

C C C = -( )2 6100 200

s a

a a a

= +¢

+ ¢¢Ê Ë

ˆ ¯ -Ê Ë

ˆ ¯ C

C C 2

2

1

W C I C I = -( ) + -( )100 1 010 23 3

2 100

2

02

C nRT r

r

i=

s ∂

∂a a

a = = -Ê

Ë ˆ ¯

W C 2

1100 2

W C = +Ê Ë

ˆ ¯ 100

2 2a

a




9.10 SOME REFINEMENTS TO RUBBER ELASTICITY

The statistical theory of rubber elasticity has undergone significant and con-tinuous refinement, resulting in a series of correction terms. These are some-times omitted and sometimes included in scientific and engineering research,as the need for them arises. In this section we briefly consider some of these.

9.10.1 The Inverse Langevin Function

The Gaussian statistics leading to equation (9.34) are valid only for relativelysmall strains—that is, under conditions where the contour length of the chainis much more than its end-to-end distance. In the region of high strains, wherethe ratio of the two parameters approaches 1

–3 to 1–2 , this limit is exceeded.

Kuhn and Grün (65) derived a distribution function based on the inverseLangevin function. The Langevin function itself can be written

(9.67)

and was first applied to magnetic problems. In this case

(9.68)

where n¢ is the number of links of length l . (It must be pointed out that thequantity n¢ in this case need not be identical with the number of mers in thechains.) Thus the quantity n¢l represents a measure of the contour length of the chain, and r /n¢l is the fractional chain extension. Of course, for the inverseLangevin function of interest here,

Lr

n l ¢( ) =

¢b

L x x x

( ) = -coth1

9.10 SOME REFINEMENTS TO RUBBER ELASTICITY 459

Figure 9.18 Mooney–Rivlin plot for sulfur-vulcanized natural rubber. Solid line, equation

(9.48); dotted line, equation (9.63).

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Rubber Elasticity - Continuum Theories