Holpzaphel_-_Nonlinear-Solid-Mechanics-a-Contin

For a more comprehensive survey of the general concepts of statistical mechanics, a terminology which was introduced by Gibbs, the reader may be referred to the books by FLORY '[1969], TRELOAR [l975J, WEINER [.1983], CALLEN .[1985, Part II] and.

:MARK and ERMAN [1988]; see also the review paper by GUTH [1966].

.Rubber ve.rsus 'hard' solids. One of the remarkable differences between rubber and 'hard' solids, such as metals, glasses, ceramics, crystals, etc. lies in the effect of temperature. The foilowing crucial physical properties, ·explored and quantified in a set of experiments by Joule, exhibit the distinctive behavior of rubber (see JOULE [.1859.t

p.1.05]):

(i)a piece of vulcanized rubber subjected to a weight produces a slight .cooling e./fect in the very low strain range and changes to a heating effect by increasing the weight, and

(ii)rubber will contract its .length under tension when .its temperature 'israised (it is not very known that healthy human and animal arteries a]so shrink upon heating,

a phenomenon that was pointed out in the early work of ROY [ .1880-1882] for the first time). <:~;~~i;- _'.21 .;

These results are based on a (previous) simple qualitative observation by GOUGH

[1805] that a rapidly ...stretched rubber band (adiabatic straining) brought into slight contact with the lips as a sensitive detector feels warm. On the other hand a stretched rubber band in thermodynamic equilibrium feels cold after releasing the tension. This

(thermoelastic) coupling phenomenon entered the literature as the .so-called Gough-

.Joule etfect. Note that the behavior of a metallic spring .is in striking contrast to a rubber band. A metaIIic spring coo"ls continuously ·On elastic stretching. Th.is is the opposite behavior of a rubber "band which warms on stretching, a remarkable experimental observation (see Box l).

The properties of rubber are well-known aboye the glass transition temperature

{see, for example, CYR [1988], WARD and HADLEY :[1993]), and are characterized by

(i)extremely long~range extensibility, typica.lly 300-500% extension for vulcanized natural rubber (i.e., without carbon black or other reinforcing :fillers) and even more for synthetic rubber (generating low mechanical stresses), accompanied by

(ii)full recovery to the initial dimensions without mechanical and thermal hysteresis within the Io\ver temperature domain of the 'rubbery' region.

Below the glass transition temperature, "flexibility .and mobility of the chains are so reduced that rubber behaves like a brittle glass (the material consists of rigid crystals).. The glass transition te·mperature, for ·example, for natural rubber and butyl rubber is

- 73°C (see MARK and ERMAN [1.988, Chapter 2])..

For an ide.al rubber, for ·which by definition one property is incompressibility (the volume remains constant (locally and ·globally) during a mechanical process), the re-

-tractive force is, however, .purely determined by changes in entropy and the internal energy does not change withclefomi'ati·oii_iic'a'n, ·re..a-signi'fiC'arit"'characteristic of rub-

ber elasticity. We term this type of rubber-like material 'entropic elastic'.

However, elasticity of metals, glasses, ·ceramics or crystals arises bas.ically through removing atoms from their equilibrium positions, accompanied by rapid internal en- ergy changes, while the entropy does not change at all (see, for example, HILL [1975] and ERICKSEN (1977]). These materials with a regular atomic structure and usually with high strength are typically called 'energetic elastic'. They exhibit, in general, substantial volume changes on deformations and stand "in sha~p contrast to rubber~like solids. For a general overview, the corresponding concepts in polymers {ideal rubber) and 'hard' solids (metals) are summarized in Box l.

Natural rubber (cis-polyisoprene) does not recover completely. In order to achieve di·me.nsional stability and deformations which are completely reversible in the 'rub- bery' state, vulcanization of the rubber (typically done for commercial products) is required. Within a vulcanization. process polymer chains are che·mica.lly connected to ·other cha.ins at different :Jocation.s to produce a cross-linked monolithic three-dimensio- nal network (see SPERLING [1992]). On the other hand crystallization occurring in the highly stretched rubber is influenced by vulcanization. Crystallization is lower for a higher concentration of sulfur used in the vulcanization process (see TRELOAR [1975., pp. 1.6-23]).

It is important to note that the material properties of highly stretched crystalline rubber become anisotropic and that .the heat of crystallization is much larger than produced by the mentioned thermoelastic Gough-Joule effect. Moreover, due)to frictional losses during the deformation process an additional heat is generated. In addition, it is mentioned that real networks contain defects (see, for example, MULLINS and

THOMAS [1960] and SCANLAN [ 1960]). Crystallinity effects and network imperfectioris lie outside the .scope of this text..

7.2 Thermoelasticity of Macroscopic Networks

A typical vulcanized rubber may ·be considered as the assembly of lon.g-chain molecules. Each chain is attached at both ends and thus produces one giant molecule, which we call the (molecular) network. · From the irregular three-dimensional network we draw condusions regarding its material properties. In the following, motivated by statistical thermodynamics, we describe the material properties of rubber through the Helmholtz free-energy function '11.

network contains 1V chains per unit volume which is often referred to as the -network density.

We consider first a representative single polymer ·chain in space detached from the network, which .means that the chain is taken out of the network. Our aim is to compute the entropy of this chain and to study .its thermodynamic behavior. The chain with contour length .L is cross-linked at the tie points 0 and P of the network (see

Figure 7.2). One end of the chain is attached to the fixed ·orig.in 0 of the xh :c2, :r:3- coordinate syste·m. The other end is given :by the vector ·r

pointing to P and characterizing a certain numb.er of different shapes .

Figure 7.2 A representative single polymer chain OP detached from the network.

Further, we assume a so-called Gaussian chain which is defined so that the distance

Hence, we follow the context of the Gaussian statistical theory of elasticity which is valid for problems where only small strains are involved.

The contour length L of a single chain is commonly considered to be an assembly of n (statisti~al) segments joined together, ·each of length l so that L = nl,. We suppose t~at .there is no correlation between the directions of the successive segments. Based on this simple mechanical model we may determine the so-called mean valuer of the end-to-end distance r for this freely jointed chain .observed at one instant of time. This

explanation the interested reader is referred to the textbooks by, for example, :McCRUM et al. :[I 997, Section 2.8] or WARD and HAD.LEY .[.1993, Section 3.3] and left to study Exerc.ise 1 on p. 320.

We wish to calculate the probability that the tie point P lies within the infinitesi- mal volume element of size dv = d:z:1 dx2 d:c3 at P (see Figure 7.2). By analogy with eq. (7.1) we may introduce the probability densities p(:.r:.2) and p(x3 ) which are associ .. ated with the J:raxis and the x3 -axis, respectively. .In addition, it is possible to show that p(.1:1 ) depends only on :i:1 (p(:z:2 ) only on :i:2 and p(x:i) only on :c3 ) provided that n, is large and x1 , x 2 , x3 are much smaller than the contour length L of the chain.

Generalizing the relation (7. l) to three dimensions we may find the probability density p(x 1, 1;2 , ~'t:1), now per unit volume, that the tie point P of the freely jointed chain occurs in the infinitesimal volume element dv (see Figure 7.2). Under the restrictions

considered it is the product of the independent probability densities according to .

.P(:x1, X2, X:i)&z:-1 d:1:2cfa:3 = p(:1:1 )p(X2)JJ(X3 )&i:1 d:1:2dX3

where r 2 = :i:i + :z:~ + ~1:5 is the square of the distance between the tie points 0 and P for this detached Gaussian chain and parameter b denotes a measure of a representative length. As for the one-dimensional case the maximum of the Gaussian function (7.4) occurs at -r = 0. The Gaussian distribution function, as given in (7.4), represents a sufficiently accurate solution to the stochastic problem in question.

EXAMPLE 7.1 In this example consider a freely jointed Gaussian chain with tie and P and calculate all possible conformations of the chain at any given value

end-to-end distance r irrespective of direction. Furtherm-ore, show that the :measure of the representative length b controls the mean-square value -r2 according to

to be j~00 r2p(r)dv/ J;~ p(r)dv, with the probability density p given by eq. (7.4)2.

Spherical shell

du = 47rr2 dr

Figure 7.3 Sphcrica:I shell which defi ncs all possible conformations of a representative Gaus- s·ian chain OP irrespective of direction.

.Solution. The restriction to a particular direction in space .as cons·idered in the

previous analysis is not appropriate .anymore, so we take into account al1 directions of the vector r equally. Doing so, the tie point P does not move within the infinitesimal rectangular block, .rather within an infinitesimal volume ·dv of a spherical shell which

IS

(7.6)

The infinitesimal volume is defined between the inner radius rand the outer radius r + dr from the ·Other tie point, which is fixed at the origin 0 of the coordinate system (see Figure 73). The required probability p(r)dr that the chain length lies .in the interval between r and r + dr, is, by means of (7.4h,

The function (7.7) represents the Gaussian distribution of the distance r for a set of free chains. In the r-distribution function, no restriction on the direction of the vector r is involved.. The maximum of function (7.7), i.e. the ·most probable value of r, is obtained by differentiation of p(r) with respect to r. This maximum occurs at r = 1/ b.

.In order to compute the important mean-square value of r, i.e. r2, we ·find after

which proves (7.5). We know froin eq. (7.3) that the mean-square value r2 depends on the (statistical) segments n .and their lengths l (recall that nl is the contour length L of

the chains). Hence, we conclude that bis a measure of a representative length. •

The entropy of a single chain. On the basis of the statistical concept of thermo-

dynamics we now .determine the entropy r}i of a representative single Gaussian chain i whose -ends .are located at specified points in space. The chain "is assumed to be taken out (detached) from the network. We apply Boltzmann's equation (or what Einstein called the Boltz·mann :principle) relating thermodynamic entropy· and the probability of a thermodynamic state (molecular conformations). Hence, from the statistica1 point of view, the entropy r/i of a single chain is defined to be proportional to the logarithm of the probability density v(r) and varies with the end-to-end distance .r according to

Boltzmann's ·equation, whic.h is given in the form

where a denotes a constant entropy with respect .to a reference levet which need not be specified here in more detail. The universal constant of proportionality k = 1.38 ·

10-2~JNm/K denotes Boltzmann's constant.

The famous relation between entropy and probabi"Jity was published by the Aus-

trian physicist Boltzma1111 in 1877 (at this time he worked as .a professor for physics

in Graz). The term 'Boltzmann's constant' fork and the mathematical formulation of the principle in the form of S = klogll' due to to Planck. This form was carved on Boltzmann's gravestone at the ~zentra.lfriedhof' in Vienna in 1933.

Substituting (7.7)2 into (7.9) we obtain finally

.incorporates the constant entropy o.. The measure of the representative length b for this Gaussian chain detached from the network

is given by

316 ·7 Thermodynamics of Materials ·

(see eq. (7 .5)), where r 2out denotes the mean-square value of the end-to-end distance of this un-cross-linked free chain out of the network. It is an intrinsic property of the -chain molecule and is independent of volume changes. As can be seen from expression (7.10) the entropy tends to its largest value for r --+ 0, as expected (see the considerations in the previous section).

The elasticity of a molecular network.. In order to determine the elasticity of a molecular network, we choose, without loss of generality~ the case of .a homogeneous de.formation state of a rubber block given by the _principal stretches ,\a, a = l, 2, .3.

Further, we introduce two crucial assumptions:

(i)there is no change in vo1ume on deformation., the material is idealized as totally incompressible (incompressibility assumption), i.e. /\ 1,\2 ,,.\J = 1,

(ii)changes in the length and orientation of lines marked on -chains in a network are identical to changes in lines marked on the corresponding dimensions of the macroscopic rubber sample (affine motion assumption)~

Thus, we refer to Fig':lre 7.4, which shows the .affine motion of a representative ·Gaussian chain with one end at the origin.

Figure 7.4 Undeformed and deformed confi_gurations or a representative Gaussian chain.

In the undeformed configuration of a network the end-to-end distance of the chain

318 7 Thermodynamics of Materials

holds, where rfiin is the mean-square value of the end-to-end distance roin of the assembled chains i1l the specimen in the undeformed state.

Combining (7.16) and (7.'11) with (7.14) we find the entropy change of the network independent of the parameter b., i.e.

(7.17)

The term rfiin/r 2olit accounts for the different end-to-end distances of chains in the network and detached fro.m the network.

We have learnt in the previous section that .for an ideal rubber the internal energy e does not change with deformation at all. Hence, from the Legendre transformation (4.152) it follows .that for an .isothermal process the change in the Helmholtz freeenergy function '11 is ~'11 = -8~r/. As a consequence of the Gaussian statistical theory of a molecular network, using the fundamental expression eq. (7.17), we find

finally that

According to (6A) we have assumed that the free energy .is zero in the undeformed con- figuration. This important result .shows that, within the scope of the Gaussian statistical

theory, the only quantities pertaining to the molecular network are the tota1 number of chains N contained .in the network (per unit volume) and rfiin/r2out·

On comparison with relation (6.128) it emerges that (7.18) represents the s.impJe neo-Hookean model with the physical parameter /J,, known as the shear modulus, which

:is proportional to the.concentration of network chains .N, given as

,,.2

(see, in addition, FLORY [1956] or TRELOAR [1975, p. U4]). A simple .method for determining the number of chains N per unit volume is to measure the shear modulus

µfor rubber.

For high chain extensions the end-to-end distance r .is close to or .equal to the contour :length L. Therefore, condition (7.2) cannot be satisfied anymore and the Gaus-