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Nonlinear E lasticity

8.1

Molecular Approach to Rubber Elasticity

Many of today’s challenging design problems involve materials such as

. Rubber materials

might be most easily characterized by the stretching and relaxing of a rubber

band. The

butadiene rubber

(BR),

natural rubber

(NR), or

elastomers

of rubber, the ability to recover intial dimensions after

large strain, was not possible with natural latex until Charles Goodyear

discovered vulcanization in 1939. Vulcanization is a chemical reaction known

as cross-linking which turned liquid latex into a non-meltable solid

(

resilience

). Cross-linked rubber would also allow considerable stretching

with low damping; strong and stiff at full extension, it would then retract

rapidly (rebound). One of the ﬁrst applications was rubber-impregnated

cloth, which was used to make the sailor’s “mackintosh.” Tires continue to

be the largest single product of rubber although there are many, many other

applications. These applications exhibit some or all of rubber’s four charac-

teristics, viz. damping in motor mounts, rebound/resilience in golf ball cores,

or simple stretching in a glove or bladder. While thermoset rubber remains

dominant in rubber production, processing difﬁculties have led to the devel-

opment and application of

thermoset

(TPEs). These materials

are easier to process and are directly recyclable. While TPEs are not as rubber-

like as the thermosets, they have found wide application in automotive fascia

and as energy-absorbing materials.

There are several reasons why designing with plastic and rubber materials

is more difﬁcult than with metals. For starters, the stress-strain response,

that is, the constitutive response, is quite different. Figure 8.1A shows the

stress-strain curves for a mild steel specimen along with the response of a

natural rubber used in an engine mount. Note that the rubber specimen

strain achieves a much higher stretch value than the steel. The dashed ver-

tical line in Figure 8.1B represents the strain value of the mild steel at failure.

This value is much less than the 200% strain the rubber underwent without

failing. In fact, many rubber and elastomer materials can obtain 300 to 500%

thermoplastic elastomers

FIGURE 8.1A

Nominal stress-stretch curve for mild steel.

FIGURE 8.1B

Nominal stress-stretch curve for natural rubber. Note the large stretch compared to the mild

steel curve.

strain. Highly cross-linked and ﬁlled rubbers can result in materials not

intended for such large strains. Two-piece golf ball cores are much stiffer

than a rubber band, for instance. Each of these products is designed for

different strain regimes. A golf ball’s maximum strain would be on the order

of 40 to 50%. Its highly crosslined constitution is made for resilience, not for

large strain. The rubber stress-strain curve exhibits nonlinear behavior from

the very beginning of its deformation, whereas steel has a linear regime

below the yield stress.

The reason rubber materials exhibit drastically different behavior than

metals results from their sub-microscopic characteristics. Metals are crystal-

line lattices of atoms all being, more or less, well ordered: in contrast, rubber

material molecules are made up of carbon atoms bonded into a long chain

resembling a tangled collection of yarn scraps. Since the carbon-carbon (C-

C) bond can rotate, it is possible for these entangled long chain polymers to

rearrange themselves into an inﬁnite number of different conformations.

While the random coil can be treated as a spring, true resilience requires a

cross-link to stop viscous ﬂow. In a thermoset rubber, a chemical bond, often

with sulfur, affords the tie while physical entanglements effect the same

function in a TPE material. The degree of cross-linking is used to control the

rubber’s stiffness.

In the context of this book’s coverage of continuum mechanics, material

make-up on the micro-scale is inconsistent with the continuum assumption

discussed in Chapter One. However, a rubber elasticity model can be derived

from the molecular level which somewhat represents material behavior at

the macroscopic level. In this chapter, rubber elasticity will be developed

from a ﬁrst-principle basis. Following that, the traditional continuum

approach is developed by assuming a form of the strain energy density and

using restrictions on the constitutive response imposed by the second law

of thermodynamics to obtain stress-stretch response.

One of the major differences between a crystalline metal and an amorphous

polymer is that the polymer chains have the freedom to rearrange them-

selves. The term conformation is used to describe the different spatial ori-

entations of the chain. Physically, the ease at which different conformations

are achieved results from the bonding between carbon atoms. As the carbon

atoms join to form the polymer chain, the bonding angle is 109.5°, but there

is also a rotational degree of freedom around the bond axis. For most macro-

molecules, the number of carbon atoms can range from 1,000 to 100,000.

With each bond having a certain degree of freedom to orient itself, the

number of conformations becomes quite large. Because of this substantial

amount, the use of statistical thermodynamics may be used to arrive at

rubber elasticity equations from ﬁrst principles. In addition to the large

number of conformations for a single chain, there is another reason the

statistical approach is appropriate: the actual polymer has a large number

of different individual chains making up the bulk of the material. For

FIGURE 8.2

A schematic comparison of molecular conformations as the distance between molecule’s ends

varies. Dashed lines indicated other possible conformations.

instance, a cubic meter of an amorphous polymer having 10,000 carbon

atoms per molecule would have on the order of 10

molecules (McCrum

et al., 1997). Clearly, the sample is large enough to justify a statistical

approach.

At this point, consider one particular molecule, or polymer chain, and its

conformations. The number of different conformations the chain can obtain

depends on the distance separating the chain’s ends. If a molecule is formed

Separating the molecule’s ends, the length L would mean there is only one

possible conformation keeping the chain intact. As the molecule’s ends get

closer together, there are more possible conformations that can be obtained.

Thus, a Gaussian distribution of conformations as a function of the distance

between chain ends is appropriate. Figure 8.2 demonstrates how more con-

formations are possible as the distance between the molecule’s ends is

reduced.

Molecule end-to-end distance,

segments each having length

, the total length would be

, is found by adding up all the segment

lengths, as is shown in Figure 8.3 . Adding the segment lengths algebra-

ically gives distance

from end-to-end, but it does not give an indication of

the length of the chain. If the ends are relatively close and the molecule is

long there will be the possibility of many conformations. Forming the mag-

nitude squared of the end-to-end vector,

, in terms of the vector addition of

individual segments

=⋅= + + +

llLl

⋅

llLl

+++

(8.1-1)

where

segment of the molecule chain. Mul-

tiplying out the right-hand side of Eq 8.1-1 leads to

is the vector deﬁning the i

~ i

= + ⋅ +⋅ + + ⋅

l l

(8.1-2)

−

This would be the square of the end-to-end vectors for one molecule of the

polymer. In a representative volume of the material there would be many

chains from which we may form the mean square end-to-end distance

FIGURE 8.3

A freely connected chain with end-to-end vector

∑ l

+ ⋅ +⋅ + + ⋅

l l

(8.1-3)

−

The bracketed term in Eq 8.1-3 is argued to be zero from the following logic.

Since a large number of molecules is taken in the sample, it is reasonable

that for every individual product there will be another segment pair

product which will equal its negative. The canceling segment pair does not

necessarily have to come from the same molecule. Thus, the bracketed term

in Eq 8.1-3 is deemed to sum to zero, leaving a simple expression for the

mean end-to-end distance

l ~ 12

⋅

= l

(8.1-4)

The mean end-to-end distance indicates how many segments, or carbon

atoms, are in a speciﬁc chain. To address the issue of how the end-to-end

distances are distributed throughout the polymer, a Gaussian distribution is

assumed. Pick the coordinate’s origin to be at one end of a representative

chain. Figure 8.3 shows this for a single chain, with the other end of the chain

in an inﬁnitesimal volume

located by the vector

. The probability of the

chain’s end lying in the volume

is given by

−

() =

Prdr

(8.1-5)

( )

πρ

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